Digital-communication.pdf

Department of Communications Engineering
Digital Communications
CME 624 May 2016
Lecture Guide
Prof. Okechukwu C. Ugweje
Complexity High
APK
M-ary PSK
QPR
CPFSK - optimal detection
MSK
OQPSK
QAM, QPSK
BPSK
Low
OOK - envelope detection
DQPSK
DPSK
CPFSK -discriminator detection
FSK - noncoherent detection
Sampler
f B
s  2
Quantizer
L k
 2
x n
( )
xk 
xk
( )
x n
x t
( )
© Prof. Okey Ugweje 1
Federal University of Technology, Minna
Lecture Guide Contents
Module 1: Introduction and Overview
 Course Introduction
 Review of linear systems
 Review of Random Variables
 Review of Random Processes:
Autocorrelation, Cross-correlation, Power
spectral density, Energy Spectral Density
 Overview of digital communication systems
 Why digital communication?, Goals in
communication system design, Digital
signal nomenclature
Module 2: Source Encoding & Decoding
 Elements of Digital Communication System
 Formatting of Analog Information
 Sampling, Quantization and Coding
 Compounding and Encoding
 Speech & Image Coding Techniques
 Line Coding Techniques & Pulse Shaping
 Inter Symbol Interference (ISI)
 Controling ISI
 Equalization
Module 3: Baseband Communication
Digital Baseband Communication Systems
 Digital Transmission & Reception
Techniques
 Noise in Communication Systems
 Detection of Binary Signal in Gaussian
Noise
 Optimum Receivers: Maximum Likelihood
Receiver, Matched Filtering, Correlation
Receiver
 Correlator
 Matched Filter
 Coherent & Noncoherent Detection
 Probability of Error for Binary Antipodal
Systems
Lecture Guide Contents
Module 4: Bandpass Communication
 Modulation and Demodulation
 Why Modulate?, Modulation categories
 Basic Binary Modulation Schemes: BPSK,
BFSK, BPSK
 Others Modulation Schemes: DPSK,
QPSK, OQPSK, M_ary Signaling
 Comparisons of Digital Modulation
Schemes
 Detection of Binary Signals
 Error Performance (Bit and Symbol Error)
Module 5: Multiplexing and Multiple Access
 Multiplexing techniques
 Frequency-Division Multiplexing
 Time-Division Multiplexing
 Code-Division Multiplexing
 Multiple Access
 Frequency Division Multiple Access
 Time Division Multiple Access
 Code Division Multiple Access
Module 6: Spread Spectrum
 What is Spread Spectrum?/Significance of
Spreading
 Basic Characteristics of SS System
 Classifications of Spread Spectrum
 Direct Sequence Spread Spectrum
 Summary of Direct Sequence Techniques
 Frequency Hopped Spread Spectrum
 Direct Sequence vs. Frequency Hopping
Digital Communication System
Module 1
Introduction and Overview
 Review of Linear Systems (Signals and Systems)
 Review of Probability and Random Signals

 Introductions
 Course Outline/Syllabus
 Course Calendar
 Course Overview
Introduction and Handout
 Note:
 Some of the material contained in Module 1 is a review
of prerequisite materials covered in undergraduate
classes such as:
 Signals and Systems
 Communications and Signal Processing
 Random Signals and Processes
 Some of the materials are included in this section for
your benefit
 It is your responsibility to review most of the material in
this Module
 Most materials in this section can be found in Chapter
1 and the Appendix of the recommended textbook
 Signals and Systems
 Continuous Convolution
 Parseval’s’ theorem
 Linear Transform
 Fourier Transform Techniques
 Concept of Bandwidth/ Filtering
Signals and Systems
Signals - 1
Signals are used to convey information
Signals and waveforms (voltage, current and intensity)
are central to communication and signal processing
Signals can be viewed either in time or frequency
domain
A signal is any physical quantity that varies with time,
space, or any other independent variables
Often, the independent variables for most signals is
“time”
Theoretical signals can be described mathematically,
graphically or in tabular form
Real signals are however difficult to describe, and more
often can be described approximately

Signals - 2
Mathematically, a signal is defined as a function of one
or more independent variables, e.g.,
x(t) = 10t
x(t) = 5t2
s(x,y) = 3x + 2xy + 10y2
Sometimes the functional dependence on the
independent variable is not precisely known, e.g.,
speech signal
Sometimes a signal is a combination of other signals
e.g., sum of sinusoid of different amplitudes,
frequency & phase
 
1
( ) ( )sin 2 ( ) ( )
n
i i i
i
s t A t F t t
 

 

Signals - 3
Mathematically, a signal is defined as a function of one or
more independent variables, e.g.,
 x(t) = 10t
 x(t) = 5t2
 s(x,y) = 3x + 2xy + 10y2
Sometimes the functional dependence on the independent
variable is not precisely known, e.g., speech signal
Sometimes a signal is a combination of other signals
 e.g., sum of sinusoid of different amplitudes, frequency & phase
Signals are the inputs outputs, and internal functions that
the systems process or produce, such as voltage,
current, pressure, displacements, intensity, etc.
 
1
( ) ( )sin 2 ( ) ( )
n
i i i
i
s t A t F t t
 

 

Signals - 4
The variable time may be continuous or discrete and the
value of the signal may be represented as
 Continuous-valued x(t)
 Discrete-valued x(nts)
 Quantized xQ(t), and
 Digital x[n]
These types of signals occur at different stages of the
process
Other variables (distance, angle, etc.) can also be the
independent variable, especially for 2-D signals like
images and video
Physical realizable signals must
 Have time duration
 Occupy finite frequency spectrum
 Are continuous (as in analog signal)
 Have finite peak value, and
 Are real-valued
All real-world signals will have these properties
Sometimes we use mathematical signal models which violate
these conditions
 e.g., Dirac delta function (or impulse function)
The most commonly used analog signals are the sinusoidal
signals (sine, cosine, etc.)
In communication systems, we are concerned with info
bearing signals that evolve as a function of the independent
variable, t
Signals - 5

Systems - 1
When signals are corrupted by noise, they no longer convey
the required information directly, hence they often require
processing
 Radio receivers are especially sensitive to noise
Signals are processed by systems, which may modify them
or extract additional information from them
Thus, a system is an entity that processes a set of signals
(inputs) to yield another set of signals (outputs)
A system can also be associated to the signal as in the
source or sink of the signal
A system may be made up of physical components
(hardware realization), as in electrical, mechanical, or
hydraulic systems, or it may be an algorithm (software
realization) that computes an output from an input signal
Systems - 2
 Many systems have signals that are not wanted (commonly
known as noise or interference)
 A system is a device, process, or algorithm that, given an
input x(t), produces an output y(t)
 A system is characterized by its input (excitation or forcing
function), its output (response), and the rules of operation
(internal functions)
 From a communication engineers’ viewpoint, a system is a law
that assigns output signals to various input signals
 Systems may be realized as an integration of sub-systems or
as a single entity
 In practice, systems with feedback is of great importance
Systems - 3
Systems may be classified functionally as in
Analyzers, Synthesizers, Transducers, Channels,
Filters, and Equalizers, etc.
or descriptively as in
linear, nonlinear, causal, discrete, continues, time
invariant, etc.
Examples of Systems
Electronic systems: resistors, inductors, Radio/TV,
phone networks, sonar and radar, guidance &
navigation, satellite, lab instrumentation, biomedical
instrumentation, etc.
Mechanical systems: loudspeakers, microphones,
vibration analyzers, springs, dampers
Systems - 4
To understand the behavior of systems
(electronic/mechanical), the response to inputs
(usually signals) must be understood
Terminology of Systems
State:
Variables that allow us to determine the energy level
of the system
All physical systems are referenced to zero-energy
state, e.g., ground state, rest state, relaxed state
Initial Conditions
The initial conditions or initial state is the state of the
system before an input is applied

Broad Classification of Systems
 We are
interested only
on the systems
that intersect the
dotted path.
Distributed
Parameters
SYSTEMS
Lumped Parameters
Stochastic Deterministic
Continuous Time Discrete Time
Nonlinear Linear
Nonlinear Linear
Time
Varying
Time
Invariant
Time
Varying
Time
Invariant
Systems - 5 Department of Communications Engineering
Operation on Linear Systems
 An operator, T, is a rule to transform one function to another
 Additive
 Homogeneous
 Principle of Superposition
 Superposition implies both additive & homogeneous rules
 If a system fails either rule, the function is nonlinear
 Addition or homogeneity is sufficient condition to test for
linearity
T x t y t
( ) ( )
  
T x t x t T x t T x t
1 2 1 2
( ) ( ) ( ) ( )
  
k p k p k p
T Kx t KT x t
( ) ( )
    
T Ax t Bx t AT x t BT x t
1 2 1 2
( ) ( ) ( ) ( )
  
k p k p k p
Systems - 6
Linear Time-Invariant (LTI) Systems
Linear systems are characterized by the ability to accept
input and produce output in response to the input
Most communication systems can be modeled as linear
systems with signals forming the input and output functions
h(t)
h[n]
H(ejw)
H(f)
H(z)
LTI
y(t)
y[n]
Y(ejw)
Y(f)
Y(z)
x(t)
x[n]
x(ejw)
X(f)
X(z)
Time Function
Pole-Zero Plot
Difference Equation
H - Function
Frequency Function
Why study signals and systems?
In signals and systems theory we study the definition
and description of signals, and the behavior of systems
under different conditions
Signals form the inputs, outputs and internal
functions of systems
In electrical & computer engineering, the understanding
of signals and the behavior of systems is of immense
importance
Communication engineers are concerned with systems
which transmit, receive, and process signals carrying
information
Hence before one can characterize a system, one must
be able to characterize the system

Size of a Signal - 1
 The size of a signal is the value of the strength of the
signal
 The signal strength may be measures in its entirety
or in a given interval
 Such a measure must consider not only the signal
amplitude, but also its duration
 There are two major ways of determining the signal
strength
1. Signal Energy
 A signal is classified as energy-type if its energy Eg is
finite (0<Eg<)
 Energy may be computed in either time or frequency
domain, whichever is easier using the following
formula
 where G(f) is the Fourier transform of g(t)
 All time-limited signals of finite amplitude are energy
signals
 Energy signals have zero power
 Since signal energy also depends on the “load” the actual
signal energy should be normalized by the load R
2 2 2
/2
lim
/2
( ) ( ) ( )
T
g T
T
E g t dt g t dt G f df
 
  

  
   (unit)2s
2. Signal Power
 A signal is power-type if its power Pg is finite (0<Pg<)
 The power Pg of a signal can be computed using the
formula
 Notice that the signal power is the time-average
(mean) of the signal amplitude squared
 Most periodic signals are power-type signals
 For periodic signals Eg & Pg can be computed by
integrating over one period
/ 2
lim lim
/ 2
2 2
1 1
2 ( ) ( )
T T
T T
T T
g T T
P g t dt g t dt
 
 

 
 (unit)2
Important Signal Classifications
Deterministic and Random Signals
 Value of the signal is known or not known at all
times
Periodic and Non-periodic Signals
Analog (Continuous-Time) and Discrete Signals
 Exists for all times t vs. exists at discrete time
only
Signals and Spectra - 1
0
( ) ( ),
x t x t T t
      

 Energy- and Power-Type Signals
with waveform
 Unit Impulse Function
.5 2 2
.5
lim ( ) ( )
T
X T
T
E x t dt x t dt

 

 
 
.5 2 2
.5
1 1
lim ( ) ( )
T
x T
T T T
P x t dt x t dt

 

 
 
( ) 1, ( ) 0 0
t dt t for t
 


  

0
( ) ( ) ( )
o
x t t d x t
  


 

.5 2
.5
( )
T
T
x T
E x t dt

 
.5 2
.5
1 1
( )
T
T T
x x T
T T
P E x t dt

  
 Others
 Even and Odd Signals
 Real and Complex Signals
 Causal and Noncausal
Spectral Density
 Energy Spectral Density
 Power Spectral Density
 For periodic signals, the PSD is given by
2
2
( )
2
0
( ) '
( )
( ) ( ) is defined as energy spectral density
( )
X
X
f
X
X
E x t dt df Parseval s Theorem
x f
f df f
f df

 

 
 



  
 
 

 



2
2
2
2
1
( )
T
T
X n
n
P x t dt power
C
T




  

 
2
0
( )
X n
n
G f C f nf




 
Examples
1. Example 1
 Signal Power
2. Example 2
 Signal Energy
3. Example 3
 Signal Energy

 Some Important or Common Signals & Functions
 Sinusoidal Signal
 Complex Exponential (harmonics)
 Unit Step Function [denoted by u(t)]
 Ramp Function [denoted by r(t)]
 Rectangular Pulse Function [denoted by rect(t) or
(t)]
 Triangular Pulse Function[denoted by (t)]
 Sign (Signum) Function [denoted by sgn(t)]
 Sinc Function [denoted by sinc(t)]
 Impulse (Delta, Dirac) Function [denoted by (t)]
 Operations on Signals
 Amplitude Scaling
 Amplitude Shifting
 Time Shifting
 Displaces a signal in time without changing its
shape
( ) ( )
"+"shifts the signal left by
"-" shifts the signal right by (delayed)
y t x t 


 
 Time Scaling
 Slows down or speeds up time which results in signal
compression or stretching
 The expression
 Reflection or Folding
 A scaling operation with  = -1  x(t) = x(-t)
 The mirror image of x(t) about the y-axis through t = 0
 Operations in Combinations
 x(t)  delay (shift right) by   x(t-)
 compress by   x(t-)
 x(t)  compress by   x(t)
 delay (shift right) by /  x(t-)
( )
t
y t x

 
  
 
 Some useful signal operations and models
 Continuous/Discrete Convolution
 Parseval’s’ theorem
 Hilbert Transform
Concept of Bandwidth and Filtering
 Some Important Properties of Signals
 DC Value
 Is the time average of a signal or the time average
over a finite interval [t1, t2]
 Average Power
 The ensemble average
 RMS Value

 Fourier Series and Transform
 Definition and Properties
 Important Fourier transform cases
 Energy and power spectral density
 Different Types of Sampling Techniques
 Idea Sampling
 Natural Sampling
 Sample-and-Hold
Examples
4. Example 4
 Periodicity of Signal
5. Example 5
 Even and Odd Signals
 Even  x(t) = x(-t)
 Odd  x(t) = -x(-t)
6. Example 6
 Even and Odd Signals
 
0
( )
g t g t T
 
Examples
7. Example 7 : Convolution
 Convolution is a technique of finding the zero state
response of LTI system
8. Example 8: Convolution
h(t) y(t)
x(t)
( ) ( ) ( ) ( ) ( ) ( ) ( )
y t x t h t x h t d x t h d
     
 
 
     
 
Fourier Transform Table

Fourier Transform Pair
Examples
9. Example 9: Fourier Transform
10.Example 10: Fourier Transform
13.Example 13: Inverse Fourier Transform
X f F x t x t e j ftdt
( ) ( ) ( )
  
z


2
x t F X f X f e j ftdf
( ) ( ) ( )
  z



1 2
 Probability Theory
 Distribution Functions
 Density Functions
 Expectations
 Random Processes, etc
Review of Probability and
Random Signals
Please review the course
CME621:Stochastic
Processes
Examples – Random Signals
14. Example 14
 Random Signals
15. Example 15
 Random Processes

Module 2
Source Encoding & Decoding
 Elements of Digital Communication
 Formatting of Analog Signal
 Sampling and Quantization
 Compounding
 Encoding and Line Coding Techniques
 Intersymbol interference
Elements of Digital
Communication System
Elements of Digital Communication - 1
 Each of these blocks represents one or more transformations
 Each block identifies a major signal processing function which changes or
transforms the signal from one signal space to another
 Some of the transformation block overlap in functions
Elements of Digital Communication - 2
Format Multiplex
Channel
Encoder
Source
Encoder
Spread
Modulate
Format Demultiplex
Channel
Decoder
Source
Decoder
Despread
Demodulate
&
Detect
Performance
Measure
Bits or
Symbol
To other
destinations
From other
sources
Digital
input
Digital
output
Source
bits
Source
bits
Channel
bits
Carrier & symbol
synchronization
Channel
bits
$
mi
n s
mi
l q
Pe
Multiple
Access
Waveforms
Multiple
Access
Tx
Rx

Why Digital Communications? - 1
1. Advantages
 Two-state signal representation
 Hardware is more flexible
 Hardware implementation is flexible and permits the use of
microprocessors, mini-processors, LSI or VLSI, etc.
 Low cost
 With LSI/VLSI, implementation cost is reduced
 Easy to regenerate the distorted signal
 Repeaters can detect a digital signal and retransmit a new,
clean (noise free) signal
 Hence, prevent accumulation of noise along the path
Less subject to distortion and interference
 Digital system is more immune to channel noise/ distortion
Easier and more efficient to multiplex several digital
signals
 Digital multiplexing techniques – TDMA and CDMA - are
easier to implement than analog techniques such as FDMA
Can combine different signal types – data, voice,
TV, text, etc.
 It is possible to combine both format for transmission
through a common medium
Can use packet switching
Encryption and privacy techniques are easier to
implement
Better overall performance
 Inherently more efficient than analog techniques in
realizing the exchange of SNR for bandwidth
2. Disadvantages
 Requires reliable “synchronization”
 Requires A/D conversions at high data rate
 Requires larger bandwidth (require BW efficient
MODEM)
 Banalog = W Hz
 Bdigital = nW Hz
– where n is the # of bits used to quantize the amplitude
of the signal
 Generally an increase in complexity over analog
system
 To maximize transmission rate, R, e.g., symbols per sec
 To minimize bit error rate, Pe, or Pb
 To minimize required power, Eb/No (or ~ly required signal
power)
 To minimize required systems bandwidth, W
 To maximize system utilization, U
 To minimize system complexity, Cx
Goals in Communication System Design
R U Pe W Cx Eb/No
• In most practical
applications trade-
offs are necessary

 Information Source
Discrete output values, e.g. Keyboard (1~26 (A~Z) symbols)
Analog signal source information is continuous valued
 Textual Message
A meaningful sequence of character or symbols, e.g.,
 How are you? I am ok, thank you; I feel like a million dollars!
 Character
 Member of an alphanumeric/symbol (A ~ Z, 0 ~ 9)
 Characters can be mapped into a sequence of binary digits
using one of the standardized codes such as
 ASCII: American Standard Code for Information
Interchange
 Others: EBCDIC, Hollerith, Baudot, Murray, Morse, etc.
Digital Signal Nomenclature - 1
Symbol
 A digital message made up of groups of k-bits considered as a unit
 A member of source alphabet. May or may not be binary, e.g. 2
symbol binary, 4 symbol PSK, 128 symbol ASCII
Digital Message
Messages constructed from a finite # of symbols (26 letters, 10
numbers, “space” and punctuation marks).
 Hence a text is a digital message with about 50 symbols
Morse-coded telegraph message is a digital message
constructed from 2 symbols “Mark” and “Space”
M_ary
A digital message constructed with M symbols
 Digital Waveform
 Current or voltage waveform that represents a digital symbol
 Binary Digit (Bit)
Fundamental unit of info made up of 2 symbols (0 and 1)
Quantity of info carried by a symbol with probability P = ½
 Bit: number with value 0 or 1
 n bits: digital representation for 0, 1, … , 2n
 Byte or Octet, n = 8
 Computer word, n = 16, 32, or 64
 n bits allows enumeration of 2n possibilities
 n-bit field in a header
 n-bit representation of a voice sample
 Message consisting of n bits
 The number of bits required to represent a message is a measure
of its information content
 More bits → More content
Binary Stream (or bit stream or baseband signal)
 A sequence of binary digits, e.g., 10011100101010
Block
 Information that occurs in
a single block
 Text message
 Data file
 JPEG image
 MPEG file
 Size = Bits / block
or bytes/block
 1 kbyte = 210 bytes
 1 Mbyte = 220 bytes
 1 Gbyte = 230 bytes
Stream
• Information that is
produced & transmitted
continuously
– Real-time voice
– Streaming video
• Bit rate = bits / second
– 1 kbps = 103 bps
– 1 Mbps = 106 bps
– 1 Gbps =109 bps

Examples of Block Information
Type Method Format Original Compressed
(Ratio)
Text Zip,
compress
ASCII Kbytes-
Mbytes
(2-6)
Fax CCITT
Group 3
A4 page
200x100
pixels/in2
256
kbytes
5-54 kbytes
(5-50)
Color
Image
JPEG 8x10 in2 photo
4002 pixels/in2
38.4
Mbytes
1-8 Mbytes
(5-30)
© Prof. Okey Ugweje Federal University of Technology, Minna 53
 L number of bits in message
 R bps speed of digital transmission system
 L/R time to transmit the information
 tprop time for signal to propagate across medium
 d distance in meters
 c speed of light (3x108 m/s in vacuum)
Use data compression to reduce L
Use higher speed modem to increase R
Place server closer to reduce d
Delay = tprop + L/R = d/c + L/R seconds
Transmission Delay
Bit Rate
 Actual rate at which info is transmitted per second
Baud Rate
 The rate at which bits are transmitted, i.e. # of signaling elements per
second
Bit Error Rate
 The probability that one bit is in error, Pb, or simply the probability of
error, Pe
Data Rate
 The rate at which info is transferred in bits per second
 If binary symbols are independent & equiprobable, the bit rate = baud
rate
Character Rate
 Characters transmitted per second
Bit Rate of Digitized Signal
Bandwidth Ws Hertz: how fast the signal changes
 Higher bandwidth → more frequent samples
 Minimum sampling rate = 2 x Ws
Representation accuracy: range of approximation error
 Higher accuracy
→ smaller spacing between approximation values
→ more bits per sample

Th e s p ee ch s i g n al l e v el v a r ie s w i th t i m(e)
Stream Information
A real-time voice signal must be digitized &
transmitted as it is produced
Analog signal level varies continuously in time
Sampling Rate and Bandwidth
A signal that varies faster needs to be sampled more
frequently
Bandwidth measures how fast a signal varies
 What is the bandwidth of a signal?
 How is bandwidth related to sampling rate?
1 ms
1 1 1 1 0 0 0 0
. . . . . .
t
x2(t)
1 0 1 0 1 0 1 0
. . . . . .
t
1 ms
x1(t)
Bandwidth of General Signals
 Not all signals are periodic
 E.g. voice signals varies according
to sound
 Vowels are periodic, “s” is noiselike
 Spectrum of long-term signal
 Averages over many sounds, many
speakers
 Involves Fourier transform
 Telephone speech: 4 kHz
 CD Audio: 22 kHz
s (noisy ) | p (air stopped) | ee (periodic) | t (stopped) | sh (noisy)
X(f)
f
0 Ws
“speech”
Analog vs. Digital Communications
Analog Digital
Older technology Newer technology
Used to design mainly for voice Used to design for data and voice
Inefficient for data Efficient for data
Noisy and error prone Noise can be easily filtered out
Lower speeds Higher speeds
High overhead Low overhead
Info is precise since recorded,
transmitted or displayed
continuously in time
Digital is accurate since info is displayed in
terms of values; but we don't know if the
precise value is displayed
Interpretation of display is harder Interpretation of display is easier
More test options
Discrete-level information
Performance measured with SNR Performance measured with BER

Analog vs. Digital Transmission
Analog transmission: all details must be reproduced accurately
Sent
Sent
Received
Received
Distortion
Attenuation
Digital transmission: only discrete levels need to be reproduced
Distortion
Attenuation
Simple Receiver:
Was original pulse
positive or
negative?
Bandwidth Dilemma
All bandwidth criteria have in common the attempt to
specify a measure of the width, W, of a nonnegative
real-valued spectral density defined for all frequencies
f < ∞
The single-sided power spectral density for a single
heterodyned pulse xc(t) takes the analytical form:
(1.73)
2
sin ( )
( )
( )
c
x
c
f f T
G f T
f f T


 

  

 
Different Bandwidth Criteria
(a) Half-power
bandwidth.
(b) Equivalent
rectangular or noise
equivalent bandwidth.
(c) Null-to-null bandwidth.
(d) Fractional power
containment
bandwidth.
(e) Bounded power
spectral density.
(f) Absolute bandwidth.
Digital Communication Transformations

Formatting of Analog
Signal
Baseband Systems
Formatting Textual Data (messages, character, symbols)
Formatting Analog Information
Sampling (see prerequisite section)
Quantization
Line Coding
Encoding and Decoding of Messages
(Baseband Systems)
Multiplex
Channel
Encoder
Spread
Modulate
Demultiplex
Channel
Decoder
Despread
Demodulate &
Detect
Bits or
Symbol
To other
destinations
From other
sources
Source bits
Source bits Channel bits
Carrier and symbol
synchronization
Channel bits

mi
l q
mi
l q

Pe
Multiple
Access
Waveforms
Multiple
Access
Format
Source
Decoder
Digital
output
Digital
input
Source
Encoder
Format
Performance
Measure
Pulse
Modulation
Digital Communication Transformations - 1
67
© Prof. Okey Ugweje Federal University of Technology, Minna
Transmit and Receive Formatting
 Transition from info source  digital symbols  info sink
Sampler Quantizer Coder
Waveform
Encoder
(Modulator)
Transmitter
Channel
Receiver
Waveform
Detector
LPF Decoder
Digital Information
Textual
Information
Analog
Information
Format
Analog
Information
Textual
Information
Digital Information
Source
Sink

Character Coding (Textual Info)
A textual info is a sequence of alphanumeric characters
Characters are encoded into bits
Groups of k bits can be combined to form new digits or
symbols of size M
A symbol set of size M is referred to as M-ary system
Textual
Message
Encoder
Group of k bits
M=2k
Waveform
Encoder
(Modulator)
... 01101 ... M_ary
2k
M 
Character coding, messages and symbols
Alphanumeric and symbolic characters are encoded
into digital bits using one of several standard formats
 ASCII
 EBCDIC
 Others Baudot, Hollerith, Morse
Example 16:
In ASCII alphabets, numbers, and symbols are encoded
using a 7-bit code
A total of 27 = 128 different characters can be
represented using a 7-bit unique ASCII code
1 0
1
0
1
1
0
1
0
1
0 0 1 1 1 0
0
0
0
0 1
7-bit ASCII
16_ary digits
(symbols)
A
U S
1 5 C
9
6 1
b7 b1
b2
b3
b4
b5
b6
b8
7-bit ASCII
Least significant
Most significant
Parity

Digital Representation of Analog Signals
Most practical signal of interest are analog in nature
e.g., speech
biological signals
seismic signals
radar signals
sonar, and
various communication signals (audio, video, text, etc)
Conversion to digital form is necessary
Interface
(A/D)
Analog
Signal
Digital
Signal
Sampling
Digitization of Analog Signals
1. Sampling: obtain samples of x(t) at uniformly spaced
time intervals
2. Quantization: map each sample into an approximation
value of finite precision
 Pulse Code Modulation: telephone speech
 CD audio
3. Compression: to lower bit rate further, apply additional
compression method
 Differential coding: cellular telephone speech
 Subband coding: MP3 audio
 Compression discussed in Chapter 12
Transmitter Side Encoding
(Formatting Analog Information)
Structure of Digital Communication Transmitter
Analog-to-Digital (A/D) Conversion
Sampling Quantization
Digital
Modulation
Input
Signal
Transmitted
Signal
Transmitter
Sampler Quantizer
xa(t)
Analog signal
A/D Converter
Discrete-time
signal
Quantized
signal
x[n] xq
(n)
Quantized
Output Signal
Analog Input
Signal

Sampling - 1
A/D conversion involves a 2 step process:
Sampling (Review 341 course notes)
 Converts CT analog signal x(t) to DT continuous value signal
xs(t)
 Obtained by taking the “samples” of x(t) at DT intervals, Ts
 xs(t) is discrete time signal (but still continuous valued)
 Proper sampling must satisfy Nyquist theorem
 Sampling does not introduce error or distortion
Quantization
 Converts DT continuous valued signal to DT discrete valued
signal
Sampling
Continuous
Time Analog
Signal
Discrete-time
continuous-valued
signal
Illustration of sampling:
Sampling - 2
78
© Prof. Okey Ugweje
Sampling Theorem (section 2.4.1)
Let the signal x(t) be bandlimited @ B (or fm), with
Fourier Transform (or spectrum) X(f)
x(t) can be perfectly reconstructed provided Rs 
2B (fs  2fm)
2B is called the Nyquist Rate
If Rs < 2B, aliasing (overlapping of spectra) results
If signal is not strictly bandlimited, then it must be
passed through LPF before sampling
Sampling - 3
The first step in PCM is sampling.
The analog signal is sampled every Ts sec, where Ts is the
sample interval or period.
The inverse of the sampling interval is the sampling rate or
sampling frequency and denoted by fs, where fs = 1/Ts.
Sampling - 4

 There are 3 sampling methods.
 Ideal (or Impulse) Sampling
 Natural Sampling
 Sample-and-Hold
 Practical Sampling
 Flat-Top Sampling
Covered in 4400:341
Communications and
Signal Processing
Sampling - 5
 In ideal sampling, pulses from the analog signal are sampled.
This method is ideal and cannot be easily implemented.
 In natural sampling, a high-speed switch is turned on for only
the small period of time when the sampling occurs. The result is
a sequence of samples that retains the shape of the analog
signal.
 The most common sampling method, called sample and hold,
however, creates flat-top samples by using a circuit.
Sampling - 6
Ideal Sampling (or Impulse Sampling)
Natural Sampling (or Gating)
Sample-and-Hold
( ) ( ) ( )
( ) ( ) ( ) ( )
x t x t x t
s
x t t nTs x nTs t nTs
n n

 

 
   
 
 
Sampling - 7
x t x t x t x t c j nf t
e
s p n s
n
( ) ( ) ( ) ( )
  


2
( ) '( ) ( )
( ) ( ) ( )
x t x t p t
s
x t t n p t
T s
n

 

 
  

 
 

 
For all sampling techniques
If fs > 2B then we recover x(t) exactly
If fs < 2B) spectral overlapping known as aliasing
will occur
Sampling - 8
According to the Nyquist theorem, the
sampling rate must be
at least 2 times the highest frequency
contained in the signal.
Note

 First, we can sample a signal only if the signal is band-limited. A
signal with an infinite bandwidth cannot be sampled.
 Second, the sampling rate must be at least 2 times the highest
frequency, not the bandwidth.
 If the analog signal is low-pass, the bandwidth and the highest
frequency are the same value.
 If the analog signal is bandpass, the bandwidth value is lower than
the value of the maximum frequency
Please Note
17.Example 17
Consider the analog signal x(t) given by
What is the Nyquist rate for this signal?
Can this signal be reconstructed at the receiver at
the Nyquist rate?
18.Examples 18
Sampling
19.Examples 19
Sampling
     
( ) 100sin
50 300 100
x t t t t
  
  
3cos cos
Examples
Speech:
 Telephone quality speech has a bandwidth of 4 kHz
 Most digital telephone systems are sampled at 8000
samples/sec
Audio:
 The highest frequency the human ear can hear is
approximately 15 kHz
 CD quality audio are sampled at rate of 44,000
samples/sec
Video:
 The human eye requires samples at a rate of at least
20 frames/sec to achieve smooth motion
Practical Sampling Rates
Quantization & Pulse
Code Modulation

Quantization - 1
Sample values require infinite # of bits for perfect
representation since sampler output still continuous in
amplitude
 each sample can take on any value, e.g. 4.752, 0.001, etc
 the number of possible values is infinite
To transmit as a digital signal we must restrict the # of
possible values to finite bits
Sampler Quantizer
x(t)
Analog signal
A/D Converter
Discrete-time signal Quantized signal
x[n] xq
(n)
Analog
Input
signal
Quantized
output signal
Quantization - 2
Definition:
 Quantization is the process of approximating
continuous-valued samples with a finite number of
bits
Quantizer
 device that operates on a discrete-time signal to
produce finite # of amplitudes by approximating the
sampled values
 maps each sampled value to one of pre-assigned
output levels
 the process of “rounding off” a sample according to
some rule
 e.g., suppose we must round to the nearest tenth,
then:
4.752  4.8
0.001  0
 rounds off the sample values to the nearest
discrete value in a set of L quantum levels
 quantized samples xq(n) are discrete in time (by
virtues of sampling) and discrete in amplitude (by
virtue of quantization)
 Because we are approximating the analog sample
values by using finite # of levels, L, error is
introduced during quantization
Quantization - 3
Definition
number, size, location of its quantizing cell
boundaries, and step size of the quantization process
Quantization Resolution
# of bits, n, used to represent each sample
where L = number of levels
more bits results in better fidelity
 However, the bit rate is higher and more bandwidth is required
Xq
(nT)
X[nT] Quantizer
random process
Quantizer Model and Definitions - 1
n L
 log2

Telephone systems typically use 8 bits of resolution
 64 kbps
CD players use 16 bits of resolution/channel
 705.6 kbps (mono)
Quantization error = difference of xs(t) and xq(nT)
Unlike sampling quantization is an irreversible
process
It results in signal distortion
Quantizer Model and Definitions - 2
Illustration and Description of Quantization - 1
Operational Description
Process of approximating DT continuous valued samples
with a finite # of bits
the process of “rounding off” a sample according to some
rule maps each sampled value to one of pre-assigned
output levels, L
quantized samples xq(n) are discrete in time and discrete
in amplitude
the approximation introduces errors
LPF Sampler Quantizer Encoder
input
signal
Binary
codes
Range over which a quantizer will operate
Vmax, Vmin (Vp, -Vp)
Peak-to-peak voltage range
Vpp = Vp – (-Vp) = 2Vp
 
max
min
max
2
/
max
V
Dynamic Range
V
V k
L
V L

  
 Dynamic Range depends on the
resolution of the converter
 min detectable signal variation is
Vmax/L volts =
 ~ quantization step size, q

Mathematically
 Sampled values are converted to one of L allowable
levels, m1, m2, …, mL, according to some desired rule
 Output is a sequence of levels, Xq(t)
 Improvement can be achieved by careful selection of xi's
and mi's
 Let X be a random variable representing a sample of data
X kT m if x x kT x
q s i k s k
( ) , ( )
  
1
X t X kT if kT t k T
q q s s s
( ) ( ), ( )
   1
Quantizer
+
x
e t x x
( ) 
 
 ( ) ( )
x f x x e t
  
( )
e t x x
 

Then, the quantized value of X is given by
If a quantizer has L quantization levels
Then, with the endpoints, we have L+1 values
This implies that
 ( )
X f X

  ,  ,  , , 
X x x x xL
 1 2 3 
k p
 ,  ,  , ,  ,  , 
x x x x where x x
L L
0 1 2 0

k p    
x x x X f X X
k k k
     
1
 ( ) 
In Tabular Form
k xk xk xk

  
  
  
 

1
1 3 35
2 3 2 2 5
3 2 1 15
4 1 0 0 5
5 0 1 0 5
6 1 2 15
7 2 3 2 5
8 3 35

.
.
.
.
.
.
.
.
In Concise Form
 {-3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5}
 Why?
We assume that all points are
quantized to the nearest
quantization level
This determines the position of the
borders of the quantization regions

Transfer Functions
 Graphical representation
of the input and output
characteristics of the
quantizer
 Quantizer’s input/output characteristics ~ simple staircase
graphs
x1 x2 x6
x5
x4
y6
y7
y3
y2
y1
y5
x3
x nTs
a f
x nT
q s
a f
output
input
(odd # of levels)
x1 x2
x5
x4
y6
y3
y2
y1
y5
x3
x nTs
a f
x nT
q s
a f
output
input
(even # of levels)
MIDTREAD MIDRISER
Nonuniform Biased
Biased
(Truncation)
Zero assigned
to a quantization
level
Zero assigned
to a decision level
Uniform (linear) vs. Nonuniform
Uniform => equally spaced quantization levels
Nonuniform => levels not equally spaced
Scalar vs. Vector
Scalar => operates on each output separately
Vector => works on several samples at a time
Many signals exhibit strong correlation between samples
This implies that RX(t)  RX(t + TS)
– e,.g., in speech correlation b/w adjacent samples =0.9
quantizing 2 or more samples at a time exploits this
correlation
Classification of Quantizers - 1
Differential Pulse-Code Modulation (DPCM)
quantizes the prediction error rather than the actual
signal samples
uses a linear prediction filter

Adaptive DPCM (ADPCM)
allows the spacing between quantization levels to be
changed on the fly
used to avoid “slope overload”
Delta modulation
1-bit DPCM
Vocoding (Voice Coding)
Transmits a mathematical model of a set of samples
rather than actual samples
Uniform Quantizer (UQ) - 1
A uniform quantizer is a quantizer for which
Has equal quantization levels
Each sample is approximated within a quantile interval
Optimal when the input pdf is uniform
i.e. all values within the range are equally likely
Most ADC’s are implemented using UQ
Error of a UQ is bounded by
 
1
ˆ ˆ , 0,1, ..., 1
k k
x x q k L
     
  
q
e
q
2 2
x
q
2
1
q
0

q
2
Uniform Quantizer (UQ) - 1
Uniform Quantization Transfer function
Output
signal
Input signal
2 4 6 8
-8 -6 -4
-2
2
4
6
-6
-4
-2
Uniform 3 bit Quantizer
X(t)
Xq
(t)
2 p
V
q
L

Nonuniform Quantizer (NQ) - 1
NQ have unequally spaced levels
 spacing chosen to optimize the SNR
Characterized by:
 Variable step size
 Quantizer step size depend on signal pdf
Basic principle ~ use variable level sizes at regions
with variable pdf
 concentrate q-levels in areas of largest pdf
 use small (large) step size for weak (strong) signals

Practically, NQ is realized by
sample compression followed
by UQ
Compression transforms the
input variable X to another
variable Y using a nonlinear
transformation
Output signal
Xq(t)
Input signal
X(t)
X X
X
X X
X
X
X
X X
X
X
X
Advantages:
NQ yields a higher average SNR than UQ when the
pdf is nonuniform which is usually the case in
practice
The rms value of
the noise power is
proportional to the
sampled values
hence distortion is
minimized
Mathematical Description of Quantizer - 1
Quantization adds random “noise” to the true value of
the sample
Process can be interpreted as an additive noise process
Let the quantizer error variance be
where fX(x) is the probability density function
2 2 2
ˆ ˆ
( ) ( ) ( ) ( )
X X
x x f x dx x x f x dx
  
 
   
 
Quantizer
+
 
x t
   
ˆ
( )
e t x t x t
 
   
ˆ ( ) ( )
x t f x x t e t
  
The variance corresponds to the average quantization
noise power, i.e.,
In NQ, we wish to make small when fX(x) is large
We can accept larger when fX(x) is small
Want to minimize average noise variance
MSE penalizes large errors more than small errors
 
2 2
2
ˆ
( ) ( )
ˆ X
E x x f x dx
x x
 

  
  

  See eqn. 13.13
 
2
ˆ
x x

 
2
ˆ
x x


Signal-to-quantization noise ratio (SQNR) (or
simply SNR)
From above equation, average SNR can be written as
 
 
 
 
2
2
2
2 2
2
2
{ }
( )
( )
{ } { }
ˆ
( ) ( )
ˆ
avg
X
X
Signal Power
S
NoisePower
N
E x
E e t
x f x dx
E x E x
D x x f x dx
E x x




 

 
 


  



We have assumed
1. e(t) is uniformly distributed
2. {e(t)} is a stationary white noise process, i.e. e(j)
and e(k) are uncorrelated for j = k
3. e(t) is uncorrelated with the input signal x(t), and
4. signal sample xs(t) is zero mean and stationary
As a rule of thumb, each bit of quantization increases
the SNR by 6 dB provided that
a) xs(t) has a uniform distribution, and
b) the quantizer is a uniform quantizer
If the input signal is a sequence, then
1
2
0
1
[ ]
N
S s
n
P x n
N


 
1
2
0
1
[ ]
N
N
n
P e n
N


 
1
2
0
1
2
0
[ ]
[ ]
N
s
S n
N
N
n
x n
P
SNR
P e n





 

Signal power
Noise power
Signal-to-noise ratio
Given
q = step size, max quantization error is
where L = 2n is the # of quantization levels
The noise variance of the quantization error is given by
L/2 –1 positive levels
L/2 –1 negative levels
1 zero level
1
pp pp
V V
q
L L
 

SNR for Uniform Quantizer - 1
2 2 2 2
1 1
2 2 2
2 2 2
2
3 2
2
( ) ( ) ( ) ( )
1
3 12
q q q
q q q
q q
q
q
error p e de e de e de
q
e
q

  
  
  
  

Equation 13.12
L –1 level
L –2 intervals
This is the MSE
(noise variance)

Given
q = step size
max quantization error is
where L = 2n is the # of quantization levels
Peak signal power
Average quantization noise power
1
pp pp
V V
q
L L
 

2
2
pp
peak signal
V
P 
 
  
 
Assuming Vpp is peak power
centered around zero (±Vpp/2)
 
2
2
2
12 12
pp
average
V
q
P
L
 
For UQ with nonuniform inputs use the formula
Therefore, if a quantizer is (a) uniform with L levels,
(b) input is uniform pdf, then SNR is
This is the peak signal power to the average
quantization error power
S
N avg
E x
E x x
FH IK 

 
{ }

2
2
l q
2
2
2
3
2
12
4
peak signal pp
L
avg average q pp
P V
S
SNR L
P V
N



  
 
    
   
 
   
 
See eqn. 2.20
SNR for Uniform Quantizer n- 3
D = 2 = MSE
We can also find the peak signal power to the peak
quantization error power
Peak signal power
Peak quantization noise power
The quantization error is at worst half the
distance between quantization levels
The power of this error is therefore
2
2
pp
peak signal
V
P 
 
  
 
2
2
2 2
pp
peak q
V
q
P
L

 
 
 
   
   
 Therefore the SNR is
Hence, there are two SNRs: Peak-to-Average and
Peak-to-Peak
For the peak, since L = 2n, SNR = 22n or in decibels
gain, each additional bit (doubling L) increases SNR
by 6 dB
Same technique is used to compute the SNR of a NQ
S
N
n dB
dB
n
FH IK  
10 2 6
10
2
log c h
S
N
n dB
averageSNR
peak SNR
dB
e j a f
   
R
S
T
6
0
4 77
 
,
. ,
2
2 2
2
4
4
peak signal pp
peak peak q pp
P V
S
SNR L L
P V
N


  
 
   
 
   
 
   
 

Non-uniform Quantization - 1
For many classes of signals, UQ is not efficient
E.g., in speech signal smaller amplitudes predominate
and larger amplitudes are relatively rare
UQ will be wasteful for speech signals since many of
the quantizing levels are rarely used
An efficient scheme is to employ a non-uniform
quantizing method
Variable step sizes
smaller steps for small amplitudes
Let x = input
q(x) = quantized version
e(x) = x - q(x) = error
p(x) = pdf of x
122
NQ operates in 2 regions (linear and saturation)
Let Emax = saturation amplitude of the quantizer
The noise variance is given by
 
max
max
2
2
2
2
2
0
2 2
2 2
0
2 2
( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
q
E
E
Lin sat
E x q x
e x p x dx
e x p x dx
e x p x dx e x p x dx

 




 
 
 
 
 
 
 
 
 see eqn. 13.14
123
For NQ, error is amplitude dependent
 can be formulated into discrete outputs as in UQ
where xn is a quantizer level
Note: In Chapter 13, your textbook uses N instead of L
2
1
1
2 2
0
2 ( ) ( )
L
n
x
Lin xn
n
e x p x dx
 


  
2
Lin

2
2 2 2
2
3 2
1 1 1
3
2
0 0 0
2 ( ) 2 ( ) 2 ( )
12 12
3
qn
L L L
qn
x
n n
Lin n n n n
n n n
x
q q
x p x p x p x q


  
  

  
  
If we consider a quantile interval qn = (xn+1 – xn) and
assume e(x)  x
124

Error is the weighted sum of error powers in each
quantile
weighted by p(xn)qn
If the quantizer has uniform quantiles (i.e., UQ), then
If the Q does not operate in the saturation region, then
 
 
2
2
1
2 2
0
1
2
0
2
2
2 ( )
12
1
2
12 2
1
2 1
12 12
2 2
L
L
Lin n n n
n
n n
n n
q p x q
q q
q L
q
L
q q
q L





 
 
   

 
 
 
 
  
 
  
 
2 2
q Lin
 

125
##Uniform vs. Nonuniform Quantization
Let
Numerical integration will indicate that
However, NQ will yield a better result
The “best” possible quantizer has
NQ can give better performance for most signals than UQ
f x e
X
x
( ) 

1
2
2
2

 . ,  . ,  . ,  .
x x x x
1 1494 2 0498 3 0498 4 1494
   
l q
D E x
 
01188 1
2
. , [ ]
S
N
dB
avg
F
H
I
K  F
H
I
K 
10
1
01188
9 25
10
log
.
.
S
N avg
dB
FH IK  12 0
.
126
Types of Noise in Quantizer
Overload Noise (Saturation Noise)
when input signal > Lmax resulting in clipping of signal
Granularity Noise (Quantization Noise)
when L are not finely spaced apart enough to accurately
approximate input signal
 Truncation or Rounding error
This type of noise is signal dependent
Timing Jitter
Error caused by a shift in the sampler position
Easily isolated with stable clock reference and power
supply isolation
127
Reading Assignment:
Differential Quantization
Is used to reduce the dynamic range
Interpolation from previous value if samples are
correlated
Correlation can be increased by oversampling
Important/Practical Systems Using Quantization - 1
x
Differeence
Value
(k+2)T
(k+3)T
kT
Actual data
predited (linear interpolation)
Oversampling Predictor Differential
more samples/sec fewer samples/sec
128

Differential PCM (DPCM)
Delta Modulation
Linear Predictive Coding
Adaptive Predictive Coding
Important/Practical Systems Using Quantization - 2
129
20.Example 20
 Quantization
21.Example 21
 Uniform Qantrizer
Example 22: (uniform quantization)
Sampler
f B
s  2
Quantizer
2n
L 
x n
( )
xk 
xk
( )
x n
x t
( )
 n = # of binary bits used to
represent each sample
 fs = sampling frequency or
sampling rate
 = quantized
value of x(t)
2q
1
2 q
q k
x
ˆk
x
3q
2q
 q

3q

3
2 q
5
2 q
7
2 q
1
2 q

3
2 q

5
2 q

7
2 q

111
110
101
100
011
010
001
000
ˆ ˆ[ ] [ ]
k q
x x n x n
 
Uniform Quantizer
130
 Let the quantization level be {1,3,5,7}. Assume that
the input signal to a quantizer have the pdf shown
a) Compute the signal mean power
b) Compute the mean square error at the quantizer
output
c) Compute the output SNR
d) How would you change the distribution of the
quantization level in order to decrease the
distortion?
Example - Quantization
f x
x
else
x
( )
,
,

 
R
S
T
32 0 8
0
1
4
x t
( )
8
f x
( )
131
Federal University of Technology, Minna 132
Companding

Companding - 1
Quantization along with sampling is used to generate
a Pulse Code Modulated (PCM) signal.
Using quantization, the instantaneous voltage value of
an analog signal is quantized into 28 (256) discrete
signal levels
With each sample, the signal is instantaneously
measured and adjusted to match one of the 256
discrete voltage levels
The adjustments of the voltage levels (256 discrete
levels), introduces some signal distortion
133
Companding - 2
This distortion (quantizing noise) is greater for low-
amplitude signals than for high-amplitude signals.
A technique called companding is used to correct this
problem
a method that compresses and divides the lower-
amplitude signals into more voltage levels and
provides more signal detail at the lower-voltage
amplitudes
134
Companding - 3
Definition: Companding is a process of COMpressing the
signal at the Tx and exPANDING the signal at the Rx
Compressor
S/H +
ADC
Transmitter
Expander DAC Receiver
Regenerative
Repeater
Signal
Input
Signal
Output
Signal
In
Signal
Out
Transmitter Side
Receiver Side
LPF
LPF
ADC
DAC
law
law
amplitude of one of the
signals is compressed
135
Companding - 4
Why Compand?
improve resolution (enhance SQNR) of weak
signals by
enlarging the signal, or
decreasing quantization step size
improves resolution of strong signals by
reducing the signal or
increasing the required quantization step size
reducing the # of bits required in the ADC & DAC
while reducing the dynamic range or improving the
SQNR
136

Companding - 5
Since NQ are expensive and difficult to make, we
compand the signal and then use UQ
after compression, input of quantizer will have ly
uniform pdf
Companding introduces nonlinearity into the signal
maps nonuniform pdf into something resembling
uniform pdf
137
Companding - 6
Companding is important for speech signals and has
been standardized for telephone interconnect around
the world
Two standards of companding techniques
US standard called -law algorithm
European standard called A-law algorithm
 conversion is required when calls are made between
countries using different algorithms.
138
Input/Output Relationship
 Y = log X is the most commonly used compander
 Taking the log of Y = log X reduces the dynamic range since
0
0
x t
x
( )
max
  0

y t
y
( )
max
1
1
0
1.0
-1.0
0
1.0
x t
x
( )
max
1.0
y t
y
( )
max
 
log if 0
1
e x x
x  

139
Types of Companding - 1
-Law Companding (North & South America,
Japan)
where
x and y represent the input and output voltages
 is a constant number determined by
experiment
y x y
x
x
x
y
x
x x
x
y
x
x x
x
e
e
e
e
e
( )
log
log
sgn( )
log
,
log
log
,
max
max
max
max
max
max
max
max

 FH IK
L
NM O
QP


FH IK FH IK 
FH IK
L
NM O
QP FH IK 
R
S
|
|
|
T
|
|
|
1
1
1
1








a f
140

In U.S., telephone lines uses  = 255
Samples 4 kHz speech waveform at 8,000
sample/sec
Encodes each sample with 8 bits, L = 256 quantizer
levels
Hence data rate R = 64 kbit/sec
 = 0 corresponds to uniform quantization
141
A-Law Companding (Europe, China, Russia, Asia,
Africa)
where
 x and y represent the input and output voltages
 A is a constant number determined by experiment, A = 87.6
You can find the companding gain by differentiating
the output
y x
y
A x
x
A
x
x
x A
y
A x
x
A
x
A
x
x
e
e
( )
sgn( ),
log
log
sgn( ),
max
max
max
max
max
max


 
 
 FH IK
L
NM O
QP

 
R
S
|
|
|
T
|
|
|
1
0 1
1
1
1 1
G
d
dx
y x
x
 ( )
 0
See eqn. 2.23
142
Encoding
Quantizer output  is one of L possible signal levels
For binary transmission, each quantized sample is
mapped into an n-bit binary word
Encoding is the process of representing each of
the L outputs of the quantizer by an n-bit code
word
one-to-one mapping - no distortion introduced
xa(t)
Analog
signal
A/D Converter
Discrete-Time
signal
Quantized
signal
x[n] xq[n]
Sampler Quantizer
Line
Coder
an
Encoding - 1
144

Pulse Code Modulation (PCM) is commonly used
PCM refers to a digital baseband signal that is
generated directly from the quantizer output
Sometimes PCM is used interchangeably with
quantization
Encoding - 2
145
Pulse Modulation Techniques - 1
Recall that analog signals can be represented by a
sequence of discrete samples (output of sampler)
APM results when some characteristic of the pulse
(amplitude, width or position) is varied in
correspondence with the data signal
Can be obtained either by Natural or Flat top Sampling
146
 Two Types:
Pulse Amplitude Modulation (PAM)
 The amplitude of the periodic pulse train is varied in
proportion to the sample values of the analog signal
Pulse Time Modulation
 Encodes the sample values into the time axis of the digital
signal
 Pulse Width Modulation (PWM)
– Constant amplitude, width varied in proportion to the
signal
 Pulse Duration Modulation (PDM)
– sample values of the analog waveform are used in
determining the width of the pulse signal
147
148

Pulse Code Modulation (PCM) - 1
Sample
Quantize
Assign Code #
Convert to Binary #s
Analog PCM
149
See Figure 2.16
150
Quantization and encoding of a sampled signal
152

153
154
 Advantages of PCM
 Relatively inexpensive
 Easily multiplexed
 PCM waveforms from different sources can be
transmitted over a common digital channel (TDM)
 Easily regenerated:
 useful for long-distance communication  e.g., telephone
 Better noise performance than analog system
 Modem is all digital, thus affording reliability, stability and is
readily adaptable to integrated circuits
 Signals may be stored and time-scaled efficiently (e.g.,
satellite communication)
 Efficient codes are readily available
 Disadvantage
 Requires wider bandwidth than analog signals
155
Implementation of A/D Converters
Serial Input Output (SIO) circuit converts quantization
level to a sequence of bits n = log2 L
ADC SIO
 ( )
x f x

x n bits
Quantizer
Sampler Quantizer Coder
xa(t)
Analog signal
A/D Converter
Discrete-Time signal Quantized signal Digital signal
x[n] xq[n]
n
156

Comparison of Practical ADCs
 Counting or Ramp ADC
 Test value is incremented in equal steps until
it is equal to input sample
 Serial or Successive Approximation ADC
 Uses binary search to narrow range of input
sample until desired accuracy is reached
 Parallel or Flash ADC
 Input samples compared with all possible
quantization levels at once
157
Speech Coding
Speech Coding - 1
Introduction To Speech Coding
To date, most source encoding techniques is based on
the -law or the A-law companding of A/D and D/A
converters
They are often referred to as CODECS
A CODEC is a device designed to convert analog
signals, such as voice, into PCM-compressed samples
to be sent into digital carries
The process is reversed at the receiver
The term CODEC is an acronym for CODer/DECoder
signifying the pulse coding/decoding function of the
device
159
Speech Coding - 2
Originally, CODEC functions were managed by
separate devices, each performing the function
necessary for PCM communication such as, sampling,
quantization, A/D, D/A, filtering, companding, etc.
Presently, these function are integrated into a single
chip e.g. Intel’s 2913
CODECS form the digital interface for most telephone
lines all over the world
At the exchange each analog signal from the local
telco is converted using an 8-bit -law or A-law codec,
with a standardized sampling rate of 8000 times per/s
 For max voice frequency  3400 Hz, Nyquist criterion is
satisfied
160

Speech Coding - 3
 This results in a data rate of 64 kbps for each voice link
 At the exchange, a number of these 8-bit data words from
different phone sources are multiplexed into a frame (32 for E-
type and 24 for A-type systems)
 They are then sent using either baseband or bandpass
signaling methods over the national and international exchange
See Digital Communications by Andy
Bateman
 They are then sent using
either baseband or
bandpass signaling
methods
 In order to keep pace with
the codec sampling rate, a
new frame must be
constructed and sent
every 1/8000 sec (see fig.)
161
Characteristics of Speech Signal - 1
Speech waveform have a number of useful properties
that can be exploited when designing efficient coders
1. Nonuniform probability
distribution of speech amplitude
2. Nonzero autocorrelation between
successive speech samples
3. Non-flat nature of the speech
spectra
4. Existence of voiced and unvoiced
segments in speech
5. Quasi-periodicity of voice speech
signals
6. Speech signals are essentially
bandlimited
(also see Fig. 13.18,
page 836)
Power spectrum
162
Characteristics of Speech Signal - 2
The most basic property of speech waveform that is
exploited in speech encoders is that they are
essentially bandlimited
A finite bandwidth means that it can be sampled at a
finite rate and reconstructed completely provided that
fs  2fmax (Nyquist criteria)
163
Hierarchy of Speech Coders
Speech Coders
Source Coders
Waveform Coders
Linear Predictive Coders
Frequency
Domain
Time
Domain
Vocoders
Nondifferential Differential
PCM ADPCM
Delta
CVSDM APC
Adaptive Transform Coding
Subband Coding
164

Coding Techniques for Speech - 1
“The goal of all speech coding systems is to
transmit speech with the highest possible quality
using the least possible channel capacity”
Speech coders differ widely in their approach to
achieve this objective
They all employ quantization & exploits different
properties of speech signal
165
Waveform Coding
A) Time Domain
 Designed to represent the time domain characteristics of
speech signal
 For high bit rates (16 - 64 kbps) it is sufficient to just sample
and quantize the time domain voice waveform, e.g., Differential
Pulse Code Modulation (DPCM)
 Differential Pulse Code Modulation (DPCM)
 In DPCM, the difference between successive samples are
encoded rather than the samples themselves
 Since difference b/w samples are expected to be smaller than the
samples themselves, fewer bits are required to represent the
difference
 because most signals sampled at Nyquist rate or faster exhibit
significant correlation between successive samples
166
 i.e., average change in successive samples is relatively
small
 Speech signals fall into this group because samples of
speech signals is very strongly correlated from one sample
instant to the next
Antialiasing
Filter
Sampler
Prediction
Filter
+ Quantizer
Digital Communication Channel
Regeneration
Circuit
Prediction
Filter
DAC
+
+
+ Analog
Input
Signal
Analog
Input
Signal
-
DPCM
Signal
+
DPCM Block Diagram
167
Hence exploiting this redundancy will result in better
performance
This is the concept behind DPCM
A refinement to this general approach is to predict
the current samples based on the previous sample
DPCM quantizes the difference of one sample and
the predicted value of the next sample (this is
usually much less than the absolute value of the
samples)
In practice, DPCM is implemented using a
prediction scheme that exploits the correlation
between successive samples
168

Instead of quantizing & coding sample values, as in
PCM, an estimate is made (with linear prediction
filter) for the next sample value based on previous
sample
 In DPCM, the error at the output of a prediction filter is
quantized, rather than the voice signal itself
 It is assumed that the error of the prediction filter is much
smaller than the actual signal itself
DPCM Issues
 Linear prediction filter is usually just a feed forward finite-
duration impulse response (FIR) filter
 The filter coefficients must be periodically transmitted
 While DPCM works well on speech, it does not work well
for modem signals
169
Adaptive PCM (APCM) and Adaptive DPCM
(ADPCM):
Many sources are quasi-stationary in nature such
that the variance and the ACF of the source vary
slowly with time
The efficiency and performance of PCM can be
improved by exploiting the slowly time-varying
statistics of the source
A simple implementation is to use a uniform
quantizer that varies its step size according to the
past signal samples
Such techniques are known as APCM and ADPCM
170
Unlike PCM, APCM and ADPCM however exploit
the redundancies present in the speech signal
 because adaptive quantizers vary the step size between
quantization levels depending on whether speech is
“loud” or “soft”
Since the speech samples are highly correlated, it
means that the variance of the difference between
adjacent speech amplitude is smaller than the
variance of the signal itself
In ADPCM, the quantization resolution can be
changed on the fly
ADPCM allows speech to be encoded at 32 kb/s
 This is used in the – DECT
171
 Delta Modulation (-mod):
 In communication systems application, bandwidth is limited
 A given transmission channel (wires-pairs, coaxial cables,
optical fibers, microwave links, and others) represents a
finite spectral resource
 Hence, developing spectrally efficient (reduced bandwidth)
signaling technique is important
 This is the motivation for Delta Modulation (DM)
 If a quantizer of a DPCM is restricted to 1 bit (i.e. 2 levels
only ±q), then the resulting scheme is called DM
 In other words, DM is a special case of DPCM where
there are only two quantization levels
 Delta modulation can be implemented with an extremely
simple 1 bit quantizer
172

Adaptive Delta Modulation
In conventional DM, both quantization and slope
overload noise is a problem
The exploitation of signal correlation in DPCM suggest
that oversampling a signal will increase the correlation
between samples
This can be overcome by oversampling (i.e., keeping
the DM size small and sampling at many times the
Nyquist rate)
It is an extreme case of DPCM in which signal is
oversampled and R = 1 bit/sample
Adaptive Delta Modulation at 16 kbits/sec can produce
reasonable quality speech
173
B) Frequency Domain
Spectral Waveform Coders manipulates the spectral
characteristics of speech waveform
Frequency domain samples are represented
according to their perceptual criteria
Subband Coding (SBC) is an example of spectral
waveform coding
174
Subband Coding
Human ear cannot detect quantization distortion at
all frequency equally well
Human perceptions of speech quality depend on
the frequency band
Subband coders filter the speech signal into
multiple bands using Quadrature Mirror Filters
(QMF) or Discrete Fourier Transform (DFT)
That is, the speech is divided into many smaller
bands and then encode each subband separately
according to some perception criteria
175
Band splitting is used to exploit the fact that individual
bands do not all contain signals with the same energy
This permits the accuracy of quantizer to be reduced in
bands with very low energy and very high energy
 Higher MSE may be tolerated at very low and very high
frequencies
Band splitting can be done in many ways (equally or
unequally) using a bank of filters
Each subband is sampled at a bandpass Nyquist rate
(lower than the sampling rate) and then encoded with
different accuracy based on perception criteria
Filtered signals are quantized using standard PCM
(different R for each signal)
176

Adaptive Transform Coding
Signal samples are grouped into frames and
encoded into number of bits proportional to its
perception significance
Correlated time samples are transformed into
(hopefully) uncorrelated frequency domain samples
using FFT or Discrete Cosine Transform
This is a more complex technique which involves
block transformations of input segment of the
speech signal
177
Source Coding (Model-Based Encoding)
For low bit rate voice encoding it is necessary to
mathematically model the voice and transmit the
parameters associated with the model
This type of coding attempts to replicate a model of
the process by which speech was constructed
178
A) Linear Predictive Coding (LPC)
Linear Predictive Coding (LPC) uses a prediction
algorithm for synthesis of the desired signal
Human speech is modeled as noise (air from lungs)
exciting a linear filter (throat, vocal cords, and mouth)
The excitation sequence and filter coefficients are
quantized by a linear prediction speech encoder
LPC quantizes excitation sequence, filter coefficients
and filter gain and transmits them to receiver
Prediction Filter X
Excitted
Sequence
Filter Coefficients
Filter Gain
Output Speech
179
 Vector quantization is frequently used in this technique
 In LPC, speech is divided into frames of approximately 20 ms
 Linear predictive coding is similar to DPCM with the following
exceptions:
 prediction filter is more complex
 more taps in the FIR filter
 filter coefficients are transmitted more frequently
 once every 20 milliseconds
 The error signal is not transmitted directly
 The error signal can be considered as a type of noise
 Instead the statistics of the “noise” are transmitted
– Power level
– Whether voiced (vowels) or unvoiced (consonants)
 This is where big savings (in terms of bit rate) comes from
180

B) Vocoder (voice coders)
Vocoders are coding devices that extract significant
components of a speech waveform, exploiting speech
redundancies, to achieve low bit rate transmission
Most vocoding techniques are based on linear
predictive coding
Vector Sum Excited Linear Prediction (VSELP)
Employed in U.S. Digital Cellular (IS-136) standard
Uses 20 ms frames
Each frame is represented with 159 bits (Total data
rate is  8 kbps)
A two stage vector quantizer is used to quantize the
excitation sequence
181
Some bits (like filter gain) are much more important for
perpetual quality than others. These are protected by
error correction coding
RPE-LTP
Regular Pulse Excited Long Term Prediction
Used in GSM (European Digital Cellular)  13 kbps
QCELP
Qualcomm Code Excited Linear Predictive Coder
Used in IS-95. (US Spread Spectrum Cellular)
Variable bit rate (full, half, quarter, eighth)
Original full rate was 9.6 kbps
Revised standard (QCELP-13) uses 14.4 kbps
182
Comparison of Speech Coding Standards
 References for Speech Coding Techniques:
 N. S. Jayant, “Coding Speech at Low Bit Rates,” IEEE
Spectrum, August 1986.
 N. S. Jayant, et. al., “Coding of Speech and Wideband
Audio,” AT&T Technical Journal, October 1990.
this article is more technical than the first, but still very
readable
Type Rate
(kb/s)
Complexity
(MIPS)
Delay
(ms)
Quality
PCM 64 0.01 0 High
ADPCM 32 0.1 0 High
Subband 16 1 25 High
VSELP 8 ~100 35 Fair
Theory ~1 ? ? High
183
The bit rate produced by the voice coder can be
reduced at a price
Increased hardware complexity
Reduced perceived speech quality
Tradeoff: Voice Quality vs. Bit Rate
(1)
(5)
(4)
(3)
(2)
Unsatisfactory
Poor
Fair
Good
Excellent
1.2 24
16
9.6
4.8
2.4 32 64
Waveform coders
Vocoders
Communications
quality
Toll quality
Bit Rate (kbps)
Perceived
Speech
Quality
184

Image and Video Coding
1000x1000 pixel image with 8 bits for each of 3 colors
requires 24 Mbits to encode
Video requires ~ 20 frames/second
Compression standards vital for any hope of digital
video
JPEG: Image compression of 20:1 or more
MPEG: Video compression of 100:1 or more
Reference:
P. H. Ang, et. al., “Video Compression Makes Big
Gains,” IEEE Spectrum, October 1990
185
Digital-To-Digital Conversion
(Line Coding)
In this section, we see how we can represent digital
data by using digital signals.
The conversion involves three techniques: line coding,
block coding, and scrambling.
Line coding is always needed; block coding and
scrambling may or may not be needed.
Digital-To-Digital Conversion
Line coding is the process of converting digital data
to digital signals. We assume that data, in the form of
text, numbers, graphical images, audio, or video, are
stored in computer memory as sequences of bits.
Line Coding - 1
Line coding and decoding

Signal Element Vs Data Element
In data communications, our goal is to send data
elements.
A data element is the smallest entity that can represent
a piece of information: this is the bit.
In digital data communications, a signal element carries
data elements.
A signal element is the shortest unit (timewise) of a
digital signal. In other words, data elements are what
we need to send; signal elements are what we can
send. Data elements are being carried; signal elements
are the carriers.
Line Coding - 2
Let r be the number of data elements carried by each
signal element. Figure below shows several situations
with different values of r.
Line Coding - 3
Signal element versus data element
 Data Rate Vs Signal Rate
 Data rate defines the number of data elements (bits) sent in
1s. The unit is bits per second (bps).
 Signal rate is the number of signal elements sent in 1s. The
unit is the baud.
 The data rate is sometimes called the bit rate; the signal rate
is sometimes called the pulse rate, the modulation rate, or
the baud rate.
 Relationship of data rate & signal rate (bit rate & baud rate).
 This relationship, of course, depends on the value of r. It also
depends on the data pattern C. If we have a data pattern of all 1s
or all 0s, the signal rate may be different from a data pattern of
alternating 0s and 1s.
Line Coding - 4
 A signal is carrying data in which one data element is encoded
as one signal element ( r = 1). If the bit rate is 100 kbps, what is
the average value of the baud rate if c is between 0 and 1?
 Solution
 We assume that the average value of c is 1/2 . The baud rate is
then
Example

Although the actual bandwidth of a digital signal is
infinite, the effective bandwidth is finite.
we can say that the bandwidth (range of frequencies)
is proportional to the signal rate (baud rate). The
minimum bandwidth can be given as
 We can solve for the maximum data rate if the bandwidth of the
channel is given.
Line Coding - 5
 The maximum data rate of a channel (see Chapter 3) is
Nmax = 2 × B × log2 L (defined by the Nyquist formula).
Does this agree with the previous formula for Nmax?
 Solution
 A signal with L levels actually can carry log2L bits per level. If each
level corresponds to one signal element and we assume the average
case (c = 1/2), then we have
Example
Output of the A/D converter is a set of binary bits
 which are abstract entities that have no physical definition
We use pulses to convey a bit of information, e.g.,
To transmit over a physical channel, bits must be
transformed into a physical waveform
Baseband systems transmit data using many kinds of
pulses
Before signals are applied to the modulator, it may be
put into several different waveforms
Transmitter - 1
1
0
t
f(t) t
f(t)
T
T
1
-1
195
A line coder or baseband binary transmitter
transforms a stream of bits into a physical waveform
suitable for transmission over a channel
There are many types of waveforms. Why? 
performance criteria!
Each line code type have merits and demerits
The choice of waveform depends on operating
characteristics of a system such as
Modulation-demodulation requirements
Bandwidth requirement
Synchronization requirement
Receiver complexity, etc.,
Transmitter - 2
196

 Baseline Wandering
 In decoding a digital signal, the receiver calculates a running
average of the received signal power. This average is called
the baseline.
 The incoming signal power is evaluated against this baseline
to determine the value of the data element.
 A long string of 0s or 1s can cause a drift in the baseline
(baseline wandering) and make it difficult for the receiver to
decode correctly.
 A good line coding scheme needs to prevent baseline
wandering.
Goals of Line Coding (qualities to look for) - 1
197
 DC Components
 When the voltage level in a digital signal is constant for a
while, the spectrum creates very low frequencies.
 These frequencies around zero, called DC (direct-current)
components, present problems for a system that cannot
pass low frequencies or a system that uses electrical
coupling (via a transformer).
 For example, a telephone line cannot pass frequencies
below 200 Hz. Also a long-distance link may use one or
more transformers to isolate different parts of the line
electrically.
 For these systems, we need a scheme with no DC
component.
198
Self-synchronization
To correctly interpret the signals received from the
sender, the receiver's bit intervals must correspond
exactly to the sender's bit intervals. If the receiver clock
is faster or slower, the bit intervals are not matched and
the receiver might misinterpret the signals.
The ability to recover timing from the signal itself
 i.e., self-clocking (self-synchronization)
- ease of clock lock or signal recovery for symbol synch.
Long series of ones and zeros could cause a problem
199
Low probability of bit error
Receiver needs to be able to distinguish the waveform
associated with a mark (or 1) from a space (or 0)
BER performance
 relative immunity to noise
Error detection capability
 enhances low probability of error
Transparency
property that any arbitrary symbol or bit pattern can be
transmitted and received, i.e., all possible data
sequence should be faithfully reproducible
200

Spectrum suitable for the channel
Spectrum matching of the channel
 e.g. presence or absence of DC level
In some cases DC components should be avoided
The transmission bandwidth should be minimized
Power Spectral Density (PSD)
Particularly it’s value at zero
 PSD of code should be negligible at the frequency near zero
Transmission bandwidth
Should be as small as possible
201
Summary of Major Line Codes - 1
Categories of Line Codes
1. Polar - send pulse or negative of pulse
2. Unipolar - send pulse or a “0”
3. Bipolar (a.k.a. Alternate Mark Inversion (AMI), pseudoternary)
 Represent 1 by alternating signed pulses
Generalized Pulse Shapes
1. NRZ - pulse lasts entire bit period
2. RZ - pulse lasts just half of bit period
3. Manchester Line Code
 Send a 2- pulse for either 1 (highlow) or 0 (lowhigh)
4. HS ( Half Sine)
202
Combined category and generalized pulse shapes
 Polar NRZ
 Wireless, radio, satellite applications (bandwidth efficient)
 Unipolar NRZ
 Turn the pulse ON for a ‘1’, leave the pulse OFF for a ‘0’ in
entire bit period
 For noncoherent communication where receiver can’t decide
the sign of a pulse
 fiber optic communication often use this signaling format
 Unipolar RZ
 RZ signaling has both a rising and falling edge of the pulse
 This can be useful for timing and synchronization purposes
203
 Bipolar RZ
 Alternate between positive and negative pulses to send a ‘1’
 This alternation eliminates the DC component
 desirable for many channels that cannot transmit DC components
 Generalized Grouping
 Non-Return-to-Zero: NRZ-L, NRZ-M NRZ-S
 Return-to-Zero: Unipolar, Bipolar, AMI
 Phase-Coded: bi--L, bi--M, bi--S, Miller, Delay Mod.
 Multilevel Binary: dicode, doubinary
 There are many other variations of line codes (see Fig. 2.22,
page 87 for more)
204

205
 NRZ = Non-Return-to-Zero
 RZ = Return-to-Zero
 AMI = Alternate Mark Inversion
Line Coder
Input Xn is the output of the A/D converter
or a sequence of values that is a function of the data bit
Output is given by
where
an = symbol mapping function
f(t) = pulse shape function
Tb = bit period (Tb=Ts/n for n bit quantizer)
These values are determined by the type of line code
that is being used
s t a f t nT
n b
n
( ) ( )
 



Line Coder
n
X s t
( )
206
Commonly Used Line Codes - 1
1. Unipolar NRZ
Unipolar NRZ is defined by unipolar mapping
The pulse shape for unipolar NRZ is:
where Tb is the bit period
a
A X
X
n
n
n

 

R
S
T
,
,
when
when
1
0 0
f t
t
T
NRZ
b
( ) ,

F
HG I
KJ
 Pulse Shape
1 0
0 1
1 1
A
3Tb
0 Tb 2Tb 5Tb
4Tb
207
208
Unipolar NRZ
Compared with its polar counterpart, this scheme is
very costly
The normalized power (power needed to send 1 bit
per unit line resistance) is double that for polar NRZ
For this reason, this scheme is normally not used in
data communications today

2. Polar Line Codes
A Polar line code uses the antipodal mapping
Polar NRZ uses NRZ pulse shape
Polar RZ uses RZ pulse shape
a
A X
A X
n
n
n

 
 
R
S
T
,
,
when
when
1
0
A
3Tb
0
Tb
2Tb 5Tb
4Tb
1 0
0 1
1 1
A
3Tb
0
Tb
2Tb 5Tb
4Tb
-A
-A
Polar RZ
Polar NRZ
where Xn is the nth data bit
209
Polar NRZ-L and NRZ-I
210
nonreturn to zero- level; nonreturn to zero- invert
NRZ-L and NRZ-I both have an average signal rate of N/2 Bd.
NRZ-L and NRZ-I both have a DC component problem.
In NRZ-L the level of the voltage determines the value of the bit.
In NRZ-I the inversion or the lack of inversion determines the value of the bit.
Polar NRZ-L and NRZ-I
 Baseline Wandering is a problem for both variations, it is
twice as severe in NRZ-L. If there is a long sequence of 0s
or ls in NRZ-L, the average signal power becomes skewed.
The receiver might have difficulty discerning the bit value. In
NRZ-I this problem occurs only for a long sequence of 0s. If
somehow we can eliminate the long sequence of 0s, we can
avoid baseline wandering. We will see shortly how this can
be done.
 The synchronization problem (sender and receiver clocks
are not synchronized) also exists in both schemes. Again,
this problem is more serious in NRZ-L than in NRZ-I. While a
long sequence of 0s can cause a problem in both schemes,
a long sequence of ls affects only NRZ-L.
211
3. Bipolar Line Codes
A space is mapped to '0' & a mark is alternately
mapped to -A and +A
Also called pseudoternary or AMI
Either RZ or NRZ pulse shape can be used
a
A X
A X
X
n
n
n
n

  
  

R
S
|
T
|
,
,
,
when and last mark -A
when and last mark +A
when 0
1
1
0
1 0
0 1
1 1
A
3Tb
0 Tb
2Tb 5Tb
4Tb
Bipolar
RZ
-A
212

Polar Biphase: Manchester Line Codes
Uses antipodal mapping and split-phase pulse shape
f t
t
T
T
t
T
T
b
b
b
b
( ) 

F
H
GG
I
K
JJ
 

F
H
GG
I
K
JJ

4
2
4
2
A
-A
1 0
1
1
0 1
213
 In Manchester and differential Manchester encoding, the transition at
the middle of the bit is used for synchronization.
 The minimum bandwidth of Manchester and differential Manchester is
2 times that of NRZ.
 The Manchester scheme overcomes several problems associated with NRZ-L, and
differential Manchester overcomes several problems associated with NRZ-I.
 First, there is no baseline wandering. There is no DC component because each bit
has a positive and negative voltage contribution.
 The only drawback is the signal rate. The signal rate for Manchester and differential
Manchester is double that for NRZ. The reason is that there is always one transition
at the middle of the bit and maybe one transition at the end of each bit.
214
 The bipolar scheme was developed as an alternative to NRZ. It has the
same signal rate as NRZ, but there is no DC component.
 The NRZ scheme has most of its energy concentrated near zero frequency,
which makes it unsuitable for transmission over channels with poor
performance around this frequency. The concentration of the energy in
bipolar encoding is around frequency N/2.
215
Bipolar schemes: AMI and pseudoternary
Bipolar encoding (a.k.a multilevel binary), three levels are used: positive,
zero, and negative.
 mBnL Multilevel Scheme:
 In the schemes, a pattern of m data elements is encoded as
a pattern of n signal elements in which 2m ≤ Ln.
 E.g., Multilevel: 2B1Q scheme (two binary, one
quaternary).
 It uses data patterns of size 2 and encodes the 2-bit patterns
as one signal element belonging to a four-level signal. In this
type of encoding m = 2, n = 1, and L = 4 (quaternary).
 The average signal rate of 2B1Q is S = N/4. This means that
using 2B1Q, we can send data 2 times faster than by using
NRZ-L. However, 2B 1Q uses four different signal levels,
which means the receiver has to discern four different
thresholds.
216

217
 The idea is to encode a pattern of 8 bits as a pattern of 6 signal
elements, where the signal has 3 levels (ternary).
218
Multilevel: 8B6T scheme eight binary, six ternary
 The 3 possible signal levels are represented as -, 0, and +.
 The first 8-bit pattern 00010001 is encoded as the signal pattern -0-
0++ with weight 0; the second 8-bit pattern 01010011 is encoded as -
+ - + + 0 with weight +1. The third bit pattern should be encoded as +
- - + 0 + with weight +1.
 To create DC balance, the sender inverts the actual signal. The
receiver can easily recognize that this is an inverted pattern because
the weight is -1. The pattern is inverted before decoding.
 In this scheme, we can have 28 = 256 different data patterns and 36 =
478 different signal patterns. There are 478 - 256 = 222 redundant
signal elements that provide synchronization and error detection.
 Part of the redundancy is also used to provide DC balance. Each
signal pattern has a weight of 0 or +1 DC values.
 That is, there is no pattern with the weight -1.
 To make the whole stream DC-balanced, the sender keeps track of
the weight. If two groups of weight 1 are encountered one after
another, the first one is sent as is, while the next one is totally
inverted to give a weight of -1.
219
Multilevel: 8B6T scheme eight binary, six ternary
The minimum bandwidth is very close to 6N/8.
The average signal rate of the scheme is theoretically
220
Multitransition: MLT-3 scheme
1. If next bit is 0, there is no transition.
2. If next bit is 1 and the current level is not 0, the next level is 0.
3. If the next bit is 1 and the current level is 0, the next level is the
opposite of the last nonzero level.

 One scheme that maps one bit to one signal element.
 The signal rate is the same as that for NRZ-I, but with greater
complexity (three levels and complex transition rules).
 It turns out that the shape of the signal in this scheme helps to
reduce the required bandwidth.
 Let us look at the worst-case scenario, a sequence of 1 s. In
this case, the signal element pattern +V0 -V0 is repeated every
4 bits.
 A nonperiodic signal has changed to a periodic signal with the
period equal to 4 times the bit duration.
 This worst-case situation can be simulated as an analog signal
with a frequency one-fourth of the bit rate. In other words, the
signal rate for MLT-3 is one-fourth the bit rate.
221
Multitransition: MLT-3 scheme
222
Summary of line coding schemes
Summary of Line Codes
NRZ level (or change)
"1" represented by one level
"0" represented by other level
NRZ-L
+V
-V
+V
+V
+V
+V
+V
+V
+V
+V
+V
+V
+V
-V
-V
-V
-V
-V
-V
-V
-V
-V
0
-V
0 T 2T 3T 6T 7T 8T 9T 10T 11T
4T 5T
Decode
NRZ
Delay
Modulation
Bi-o-S
Bi-o-M
Bi-o-L
RZ-AMI
Bipolar
RZ
Unipolar
RZ
NRZ-S
NRZ-M
Decode
RZ
NRZ Mark
"1" represented by a change in level
"0" represented by no change in level
NRZ Space
"1" represented by no change in level
"0" represented by a change in level
Unipolar RZ
"1" represented by a 1/2-bit wide pulse
"0" represented by no pulse condition
Bipolar RZ
"0's" & "1's" represented by opposite level polar
pulses that are half-bit wide
RZ AMI
"0" represented by no signal; successive "1's"
represented by equal amplitude alternating pulses
Bi-phase Level (Manchester II + 180)
"1" represented by a "10"
"0" represented by a "01"
Bi-phase Mark (Manchester I)
A transition at beginning of every bit period
"1" represented by a 2nd transition 1/2 bit period later
"0" represented by no 2nd transition
Bi-phase Space
A transition at beginning of every bit period
"1" represented by a no 2nd transition
"0" represented by a 2nd transition one-half bit period
later
Delay Modulation
A "1" to "0" or "0" to "1" changes polarity;
otherwise a zero is sent.
Decode NRZ
A "1" to "0" or "0" to "1" transition produces a half
duration polarity change; otherwise a zero is sent.
Decode RZ
A "1" represented by a transition at the midpoint of a
bit interval; a "0" is represented by no transition
unless it is followed by another zero; In this case, a
transition is placed at the end of the bit period.
1 1
0 0
0
0
1
1 0
1 1
223
Power Spectral Density (PSD) of Line Codes - 1
Ts = symbol duration (Ts= Tb for binary, Ts= kTb for M-ary)
f(t) = symbol pulse shape
an = a set of Random Variables representing data bits (voltage level of data)
 Average PSD of a line code is given by
where R(k) is the Autocorrelation (AC) Function of the data
sequence at the encoder output
 For Autocorrelation please see Section 1.4
 Correlation is a matching process
 AC is the matching of a signal with the delayed version of itself
s t a f t nT
n s
n
( )  


 a f
2
2
( )
( ) ( ) s
j fkT
s
k
s
X f
G f R k e
T



 
  
 

Line Coder
n
X s t
( )
224

AC is denoted as RXX(t1,t2) or RXX(t, t+) or RXX()
AC of a random process X(t) is given by
It follows that
Value of RX(t1, t2) when t1 = t2 = t is the average power
of X(t), i.e.,
Reading Assignment: Section 1.4
     
  

*
1 2 1 2
1 2 1 2 1 2 1 2
,
, ; ,
XX
R t t E X t X t
x x f x x t t dx dx
 
 
 
  
  
R t t E X t X t
XX 1 2 2 1
, *
a f a f a f

R t t E X t
XX ,
a f 
 
2
0
225
Average PSD of a line code (cont’d)
where
Pi = probability of getting (anan+k)i
M = # of positive values of anan+k
R k E a a k
a a P
n n k
n n k i i
i
M
( ) , , , ,
*
 






0 1 2
1

b g
226
“Brute-force” Method
Model s(t) as a Wide Sense Stationary (WSS) random
process
Find Autocorrelation Function (ACF) of s(t)
this step can be tricky & cumbersome!
Apply Wiener-Khintchine theorem to get PSD
PSD of a RP X(t), GX(f), is the Fourier transform of the ACF
Shortcut Method for Finding PSD of a Line Code
Assume equiprobable & independent data symbols
Polar line codes
X(f) = Fourier Transform of the pulse shape
2
2
( ) ( )
s
b
A
G f X f
T

How to Compute PSD of Line Codes - 1
227
Unipolar line codes
Bipolar line codes
Unipolar Line Codes with NRZ Pulse Shapes
If the pulse shape is NRZ, then
Thus
2
2 1
( ) ( ) 1
4
s
n
b b b
A n
G f X f f
T T T



 
 
  
  
 
 
 
 
2
2 2
( ) ( ) sin
s b
b
A
G f X f fT
T


( ) 0for when 0
b
n
X f f n
T
  
 
2
2
( ) ( ) 1 ( )
4
s
b
A
G f X f f
T

 
How to Compute PSD of Line Codes - 2
228

 Find the PSD of x(t) – Unipolar NRZ
 Possible levels = A, 0
 Assume that values are equally likely to occur with probability Pi = 0.5
For k=0:
1 0
0 1
1 1
A
3Tb
0 Tb 2Tb 5Tb
4Tb
Example 24
1( ) , 0 1
b
x t A t T binary
   
0 ( ) 0, 0 0
b
x t t T binary
   
k = 0 k  0
anan anan+k
00 00
11 01
10
11
 
   
   
2
1
1 2
1 2
2
1 1
2 2
(0)
0 0
2
i
n n i
i
n n n n
R p
a a
p p
a a a a
A
A A

 
 
    
 
229
For k  0:
Hence,
Example 24 Solution
 
       
       
4
1
1 2 3 4
1 2 3 4
1 1 1 1
4 4 4 4
2
( )
0 0 0 0
4
i
n n k i
i
n n k n n k n n k n n k
R k p
a a
p p p p
a a a a a a a a
A A A A
A


   
 
   
       
   

2
2
, 0
2
( )
, 0
4
A
k
R k
A
k




 
 


230
But
Applying the formula
Example 24 Solution
 
b
b
b b
sin fT
t
T
T fT



 

 
 
   
 
 
 
2 2
2
2
2
2
2
, 0
2
2 0
2 2
0
1
( )
( )
2
4
2
4
b
b
b
b b
j fkT
s
k
b
j fkT
b
b
k
b
j fkT
b
b
k k
b
j fkT j fkT
b
b
k k
b
G X R k e
f f
T
sin fT
T R k e
fT
sin fT
A T
e
fT
sin fT
A T
e e
fT



 











 

 
 
 
 
 
 
   

 
   
 
 
   
  
 
   
 
 
2
2
4
2
A
A

231
Using the fact that
we can write
Since
we have
Example 24 Solution
 
 
2
2
2
1
4
b
j fkT
b
b
s
k
b
sin fT
A T
G e
f
fT





   
  
   
 
 
 
2 1
b
b
j fkT k
T
k k
b
f
e Fourier Series
T


 
 

 
 
 
 
 
2
2
1
1
4
b
k
b
b
T
s
k
b
b
sin fT
A T f
G f
T
fT





   

 
    
 
 
  0 @ , 0
b
b k
T
b
sin fT
f k
fT


  
 
   
2
2
1
1
4
b
b
s
b
b
sin fT
A T
f
G f
T
fT



   

    
 
 
232

 Find the PSD of x(t) – Unipolar RZ
 This is the same as Unipolar NRZ except for pulse duration
of Tb/2 instead of Tb
 Hence
0
x t
2( )
3Tb 4Tb
2Tb
Tb
A
1 1
0 1 0
Example 25
2
1
2
, 0
( ) 1
0,
b
b
T
T
b
A t
x t binary
t T
  
 

 

0 ( ) 0, 0 0
b
x t t T binary
   
 
 
2
2
2
b
b
T
b
T
sin
T f
X f
f


 
  
 
 
   
2
2
2
2
1
1
16
b
b
b
T
n
b
T
s T n
b
sin
A T f f
G f
T
f





   

 
    
 
 
233
Find the PSD of x(t) – NRZ-L
(Left as an exercise. Please do)
0
x t
3( )
A 1 1
0 1
1 1 0 0
-A
Example 26
234
Comparison of Line Codes - 1
Self-synchronization (SS)
SS codes are good for error detection and correction
 Manchester codes have built in timing info because they
always have a zero crossing in the center of the pulse
 Polar RZ codes tend to be good because the signal level
always goes to zero for the 2nd half of the pulse
 NRZ signals do not have good SS capabilities
Error probability
Polar codes perform better (more energy efficient) than
Unipolar or Bipolar codes
Channel characteristics
Requires PSD of the line codes to determine channel
matching characteristics
235
Power Spectral Density comparison:
Different pulse shapes are used
 to control the spectrum of the transmitted signal
– (no DC value, bandwidth, etc.)
 guarantee transitions every symbol interval to assist in
symbol timing recovery
After line coding, the pulses may be filtered or shaped
to further improve there properties such as
 Spectral efficiency
 Immunity to Inter-symbol Interference (ISI)
Distinction between Line Coding and Pulse Shaping is
not easy
236

237
238
DC Components
Unipolar NRZ, polar NRZ, and unipolar RZ all have DC
components
Bipolar RZ and Manchester NRZ do not have DC
components
First Null Bandwidth
Unipolar NRZ, polar NRZ, and bipolar all have 1st null
bandwidths of Rb = 1/Tb
Unipolar RZ has 1st null BW of 2Rb
Manchester NRZ also has 1st null BW of 2Rb, although
the spectrum becomes very low at 1.6Rb
239
Summary
Timing Error
Detection
Average
Power
Peak
Power
First Null
Bandwidth
AC
coupled
Transparent
Unipolar NRZ Difficult No 2 4 f0 No No
Unipolar RZ Simple No 1 4 2f0 No No
Polar NRZ Difficult No 1 1 f0 No No
Polar RZ Rectify No 1/2 1 2f0 No No
Bipolar NRZ Difficult No 2 4 2f0 Yes No
Bipolar RZ Simple No 1 1 2f0 Yes Yes
Dipolar NRZ Rectify Yes 1 4 f0 Yes No
Dipolar RZ Difficult Yes 2 4 f0/2 Yes No
HDB3 Rectify Yes 1 4 f0 Yes Yes
CMI Simple Yes - - 2f0 Yes Yes
240

Generation of Line Codes
Transmitter:
 The FIR filter realizes the different pulse shapes
 Baseband modulation with arbitrary pulse shapes can be detected by
 correlation detector
 matched filter detector (this is the most common detector)
ROM
Make
Impulse
h[n] = p[n]
0 -1
1 +1
binary
bits
an anp[n]
an [n]
s[n]
impulse train which
represents the data
pulse shape defined by impulse
response of FIR filter
N
5N
3N
2N
4N
0
1 1 1 1
0 0
N 5N
3N
2N 4N
0
1 1 1 1
0 0
241
Pulse Shaping
Inter-symbol Interference
Baseband Communication System
Baseband Communication System:
We have been considering the transmitter side
Transmitted signal is created by the line coder
according to
where an is the information sequence & g(t) is pulse shape
s t a g t nT
n b
n
( ) ( )
 



A/D
Converter
Line
Coder
Channel
Analog Input To Receiver
an s t
( )
Transmiter
A/D
Converter
Line
Coder
Channel
Input
an s t
( )
Transmiter
Decoder
A/D
Converter
Receiver
Output
243
Problems with Line Codes - 1
A) Line codes are not bandlimited
absolute bandwidth, B, is infinite
power outside the 1st null bandwidth is not negligible
 i.e., power in the sidelobes can be quite high
 This can cause Adjacent Channel Interference (ACI)
If transmission channel is bandlimited, then high freq
components will be cut off
 High freq components correspond to sharp transition in
pulses
 Hence, the pulse will spread out
 If pulse spreads out into adjacent symbol period, then
inter-symbol interference (ISI) occurred
244

B) Inter-symbol Interference (ISI)
 ISI occurs when a pulse spreads out in such a way that
it interferes with adjacent pulses at the sample instant
 Causes
1. Channel induced distortion which spreads or disperses
the pulses
2. Multipath effects (echo)
3. Due to improper filtering (@ Tx and/or Rx), the received
pulses overlap one another thus making detection
difficult
245
Illustration of ISI
 Assume polar NRZ line code
 Input data stream and bit superposition
 Tb
 Tb
Tb
0
0
Tb
 Tb
Tb
0
 Tb Tb
0
data 1
data 0
input output
1 0
0 1
1 1
A
3Tb
0 Tb 2Tb 5Tb
4Tb
3Tb
0 Tb 2Tb 5Tb
4Tb
246
Channel output is the sum of the contributions from
each bit
 Some Notes on ISI
 ISI can occur whenever a non-bandlimited line code is
used over a bandlimited channel
 ISI can occur only at the sampling instants
 Overlapping pulses will not cause ISI if they have zero
amplitude at the time the signal is sampled
1 0
0 1
1 1
A
3Tb
0 Tb 2Tb 5Tb
4Tb
3Tb
0 Tb 2Tb 5Tb
4Tb
247
Strategies for Eliminating ISI - 1
Nyquist in the 1940’s, studied the problem of ISI
He suggested that by carefully manipulating the filtering
characteristics of the channel (Tx and/or Rx), ISI can be control
Recall filter Characteristics
 A filter is a freq selective device used to limit the spectrum of signal to
some band of interest
 Filters take an input waveform and modify the freq spectrum to produce
an output waveform
 Filters are energy storing elements used as frequency discriminator
Filter Classifications
Ideal Filter:
Filter is not physically realizable, only used for problem solving
X(f)
B
-B f
A
 Has a constant passband
 Perfect rejection
 No transition region
248

 Filter functions are implied in their respective names,
e.g., a LPF passes all freqs in the neighborhood of zero
LPF
-f1 f
A
f1
-f1
f1
-f1 f1
-f1 f1
-f2 f2
f2
-f2
H f
( )
H f
( )
H f
( )
H f
( )
f
f
f
HPF 
BPF 
BSF 
 Low-Pass Filter (LPF)
 High-Pass Filter (HPF)
 Band-Pass Filter (BPF)
 Band-Stop Filter (BSF)
249
Ideal Filters
 For the ideal low-pass filter transfer function with bandwidth Wf =
fu hertz can be written as:
Ideal low-pass filter
(1.58)
Where
(1.59)
(1.60)
( )
( ) ( ) j f
H f H f e 


1 | |
( )
0 | |
u
u
for f f
H f
for f f





0
2
( ) j ft
j f
e e 
 


Ideal Filters
 The impulse response of the ideal low-pass filter:
0
0
1
2
2 2
2 ( )
0
0
0
( ) { ( )}
( )
sin 2 ( )
2
2 ( )
2 sin 2 ( )
u
u
u
u
j ft
f
j ft j ft
f
f
j f t t
f
u
u
u
u u
h t H f
H f e df
e e df
e df
f t t
f
f t t
f nc f t t

 










 






 



Ideal Filters
 For the ideal band-pass filter
transfer function
 For the ideal high-pass filter
transfer function
Ideal band-pass filter Ideal high-pass filter

Realizable Filters
 The simplest example of a realizable low-pass filter; an RC filter
( )
2
1 1
( )
1 2 1 (2 )
j f
H f e
j f f

 

 
   
 
Frequency response of a typical filter is shown below:
Such a filter is characterized by three regions:
1.Passband:
 freqs in this band are transmitted with little or no attenuation
2.Stopband:
 the freqs in this band are completely rejected
3.Transition band (roll off):
 the gain of the freqs gradually falls off
H f
( )
0 707
. ( ) max
H f
H f
( ) max
f2
f1
Passband
Transition
Band
Transition
Band
Stop Band
Stop Band
1/2-power
bandwidth, B
Skirt of the filter
f
254
Realizable Filters
Phase characteristic of RC filter
Realizable Filters
 There are several useful approximations to the ideal low-pass filter
characteristic and one of these is the Butterworth filter
 Butterworth filters are
popular because they
are the best
approximation to the
ideal, in the sense of
maximal flatness in the
filter passband.
2
1
( ) 1
1 ( / )
n n
u
H f n
f f
 


Nyquist suggested that the overall channel filter
transfer function (TF) must have a transition region
“Nyquist frequency response”
This TF should have a transition band between
passband & stopband and symmetric about a freq
equal to 0.5 x 1/Ts
Point of symmetry
1 1
2 s
fs T
 
Attn
Frequency
257
Avoiding ISI
Use line code that is absolutely bandlimited
 Can’t actually do this (but can approximate)
 Would require Sa(.) or sinc(.) pulse shape
Use a line code that is zero during adjacent sample
instants
 It is ok for pulses to overlap somewhat, as long as there is no
overlap at the sample instants
 Question: Is there pulse shapes that don’t overlap during
adjacent sample instants?
 Answer: Yes, e.g., Raised-Cosine Rolloff pulse
Use a filter at the receiver to “undo” the distortion
introduced by the channel
 This is known as “Equalization”
258
Baseband Communication System Model - 1
hT(t) = Impulse response of the transmitter
hC(t) = Impulse response of the channel
hR(t) = Impulse response of the receiver
s t a h t nT T n
T
n T
n
s
b
( ) ,
 
 
 


where
 
( ) ( ) ( ),
1
where ( )= ( ) ( ),
n T
n
T C s
s
r t a g t nT n t h t
g t h t h t T
f


   

 
y t a h t nT n t
h t h t h t h t n t n t h t h t
n e
n
e
e T C R e R C
( ) ( )
( ) ( ) ( ) ( ), ( ) ( ) ( ) ( )
 
 
     
 


where
Transmitter
HT(f)
Receiver
HR(f)
Channel
HC(f)
+
n(t)
s(t) y(t)
r(t)
t = kT
x(t)
T = k/Tb
259
Note that he(t) is the equivalent impulse response of the
receiving filter
To recover the information sequence {an}, the output y(t)
is sampled at t = kT, k = 0, 1, 2, …
The sampled sequence is
or equivalently
 h0 is an arbitrary constant
 
( ) ( )
n e e
n
y kT a h kT nT n kT


  

y a h n h a a h n
k n k n
n
k o k n k n
n n k
k
      




 

,
Desired symbol scaled
by gain parameters ho
ISI terms - effect of other symbols at
the sampling instants t = kT
noise term
where h h kT n n kT k
k o k o
    
( ), ( ), , , ,
0 1 2 
260

Generally, the optimum filter at the Rx is matched to
the received pulse he(t)
If the received signal is matched, then
By proper design of transmitting and receiving filters, it
is possible to satisfy the condition that he(kT - nT) = 0
for n  k
This will eliminate the ISI term
2
2 2 2
( )
( ) ( ) ( )
o
R C T
h h t dt
H f df H f H f df


 
 
 
 
 
261
Signal Design for Bandlimited Channel
Zero ISI
To remove ISI, it is necessary and sufficient to make
the term he(kT - nT) = 0 for n  k and h0  0
This means that
A pulse will produce zero ISI if it satisfies the following
condition:
Nyquist studied this problem many years ago
 
,
( ) ( )
n n k
y kT h a a h kT nT n kT
o k n e e

 
   

h
e
nT
n
n
( )
,
,



R
S
T
1 0
0 0
h
e
t at t kT k
( )    
0 0
262
A pulse will produce zero ISI at sampling instants if
provided that its Fourier Transform satisfy
For channel bandwidth B, HC(f)  0, |f| > B and He(f) =
0 for |f| > B
H f H
e
f
n
T
T
n
( )  
FH IK 



h
e
nT
n
n
( )
,
,



R
S
T
1 0
0 0
Nyquist first method for zero ISI - 1
263
Case I: Sampling at above Nyquist rate:
 H(f) consist of non-overlapping replicas separated by fs = 1/T
In this case, elimination of ISI is not possible. Why?
 we cannot design He(f) to ensure that H(f)  T
T
B
or
T
B
 
F
H
I
K
1
2
1
2
B
B
 
B fs
fs
H f
( )
 fs
2 fs 0
B fs

2 fs
f
264

Case II: Sampling at Nyquist rate:
 In this case, the pulses touch and almost begin to overlap
 There exist one He(f) for which H(f)  T
 Pulse shape that satisfy this criteria is Sa(.) or Sinc(.)
function, e.g.,
T
B
or
T
B
 
FH IK
1
2
1
2
B
B fs
0 2 fs
 fs
2 fs
H f
( )
H
e
f B
f B
f B B
f
B
h
e
t c
t
T
( )
,
( ) sin



 FH IK

R
S
|
T
|
  FH IK
1
2
0
1
2 2
h
e
t c
t
T
c Bt
( ) sin sin
 FH IK   
2
265
 The smallest value of T for which transmission with zero
ISI is possible is
 Problems with Sa(.) or Sinc(.) function
 It is not possible to create Sinc pulses due to
1.Infinite time duration
2.Sharp transition band in the frequency domain
 Sa(.) pulse shape can cause ISI in the presence of timing
errors
 signal is not sampled at exactly the bit instant, then ISI will occur
We seek a pulse shape that
 Has a more gradual transition in the frequency domain
 Is more robust to timing errors
 Yet still satisfies Nyquist’s first condition for zero ISI
T
B
 1
2
266
Case III: Sampling at below Nyquist rate
In this case, pulses touch and overlap
There are many He(f) for which H(f)  T
T
B
or
T
B
 
F
H
I
K
1
2
1
2
fs
0
2 fs 2 fs
 fs
H f
( )
f
267
Raised Cosine Pulse - 1
 For fs > 2B, a particular pulse spectrum that has a desirable
spectral properties is the Raised Cosine (RC) spectrum
 The following pulse shape satisfies Nyquist’s method for zero
ISI
 The Fourier Transform of this pulse shape is
where  is the roll-off factor that determines the bandwidth
(0 1)
h
e
t
t
T
t
T
t
T
t
T
c
t
T
t
T
t
T
( )
sin cos
sin
cos


 FH IK 






e j e j e j
1 4 1 4
2 2
2
2 2
2
H
e
f
T f
T
T T
f
T T
f
T
f
T
( )
,
cos ,
,

 

 

F
H
I
K
L
NM O
QP

 



R
S
|
|
|
T
|
|
|
0
1
2
2
1
1
2
1
2
1
2
0
1
2



  

268

 BW occupied beyond 1/2T is called excess bandwidth (EB)
 EB is usually expressed as a %tage of the Nyquist frequency,
e.g.,
  = 1/2 ===> excess bandwidth is 50 %
  = 1 ===> excess bandwidth is 100 %
 RC filter is used to realized Nyquist filter since the transition
band can be changed using the roll-off factor
 The sharpness of the filter is controlled by the parameter 
 When  = 0 this corresponds to an ideal rectangular pulse
 B occupied by a RC filtered signal is increased from its min
value
to actual modulation bandwidth
B
Ts
min 
1
2
B B
 
 
min 1 
269
The Nyquist pulse shape can now be written as
with Fourier Transform
This is equivalent to equation 3.78, p. 139 in your text
h
e
t f Sa f t
f t
f t
( )
cos( )


L
NM O
QP
2 2
2
1 4
0 0 2


a f a f


where f
R
T
b
0
2
1
2
 
H
e
f
f f
f f
f
f f B
f B
( )
,
cos ,
,




F
HG I
KJ
L
NM O
QP  

R
S
|
T
|
|
1
1
2
1
2
0
1
1
1
a f

f f B
1 0
2
 
f B f
   0
H f
f W W
f W W
W W
W W f W
f W
( )
,
cos ,
,

 
 

F
HG I
KJ   

R
S
|
T
|
1 2
4
2
2
0
0
2 0
0
0
 a f
270
Comparatively
 A RC rolloff pulse shape is defined in this case by the rolloff
factor
 where fo is the 6 dB bandwidth of the pulse
 f1 and f are related to the pulse bandwidth B (or W) as follows
where absolute bandwidth theoretical minimum BW
excess bandwidth =
-
rolloff factor,
=
,
- ,
W
W W
W
R
T
r
W W
W
r
o
s
o
o
o
  
 

2
1
2
0 1

f W W f W W
1 0 0
2
   
, 
=
r
f
f
W W
W

0
0
0


 fo  f1 fo
f1 B
B
f
f
0 5
.
10
.
He f
( )
f B f f f f
 
   
0 1 0
,
Also see Fig. 3.17
271
Note that the bandwidth of a RC pulse shape is a
function of the bit rate and the rolloff factor
or solving for bit rate yields the expression
This is the max transmitted bit rate when an RC pulse
shape with rolloff factor  is used over a baseband
channel with bandwidth B
 This means that to achieve zero ISI, it is necessary
sometimes to reduce the symbol rate below the Nyquist rate,
for practically realizable filters
0 0 0 0
0
, 1 ( 1)
f
f B f B f f f f
f


 
 
        
 
 
R
B
b 

2
1 
272

273
Root RC rolloff Pulse Shaping
Later, we will show that the noise is minimized at the
receiver by using a matched filter
If the transmit filter is H(f), then the receive filter should
be H*(f)
The combination of transmit and receive filters must
satisfy Nyquist’s first method for zero ISI
Transmit filter with the above response is called the
root raised cosine-rolloff filter
Root RC rolloff pulse shapes are used in many
applications such as IS-54 and IS-136
H
e
f H f H f H f H
e
f
( ) ( ( ) ( )
   
( ) )
274
Practical Issues with Pulse Shaping - 1
 Like the Sa(.) pulse, RC rolloff pulses extend infinitely in time
 However, a very good approximation can be obtained by
truncating the pulse
 Can make h(t) extend from -3Tb to +3Tb
 RC rolloff pulses are less sensitive to timing errors than Sa(.)
pulses
 Larger values of  are more robust against timing errors
 Sample Applications:
 US Digital Cellular (IS-54/136) uses root RC rolloff pulse
shaping with = 0.35
 IS-95 uses pulse shape that is slightly different from RC
rolloff shape
 European GSM uses Gaussian shaped pulses
275
Practical Issues with Pulse Shaping - 2
Implementation of Raised Cosine Pulse:
Can be digitally implemented with an FIR filter
Analog filters such as Butterworth filters may also
be used
Practical pulses must be truncated in time
Truncation leads to sidelobes - even in RC
pulses
Sometimes a “square-root” raised cosine spectrum
is used at Tx and Rx
This has to do with matched filtering
276

EYE Diagram - 1
Effect of ISI and noise in digital communication can be
viewed on an oscilloscope from an eye diagram
 Width = time interval over which received signal can be sampled
 Height = defines the noise margin of the system
 Sensitivity to timing error = rate of closure of the eye
 Diagram displays y(t) on vertical with horizontal sweep rate set to fs = 1/Ts
277
ISI causes:
 the eye to close thereby reducing the margin of error
 distorts the position of the zero crossing, thereby causing the
system to be more sensitive to synchronization error
 Effect of timing error is seen as a skewing of the eye diagram
and a closing of the eye due to the received symbol stream no
longer being sampled at the point of zero ISI
 The addition of noise affects the timing recovery circuitry and also causes a
general closing of the eye
 Noise may occasionally causes full 'eye-closure'
EYE Diagram - 2
278
With no
bandwidth
limitation
With
bandwidth
limitation
EYE Diagram - 3
279
Eye Diagrams for Raised Cosine Filtered Data - 1
Small :
As  is reduced, the eye opening narrows, requiring the
accuracy of symbol timing to be even more exact
‘overshoot’ caused by filtering is greater for small 
 This increases the peak-to-mean ratio of the data energy
 Increases peak signal handling requirement of the
modulator/demodulator
A benefits of small  is greater bandwidth efficiency
280

Large :
Simpler filter
 fewer stages (or taps), hence easier to implement with
less processing delay
Less signal overshoot, resulting in lower peak to mean
excursions of the transmitted signal
Less sensitivity to symbol timing accuracy – wider eye
opening
 = 0 corresponds to Sa(.) function
Eye Diagrams for Raised Cosine Filtered Data - 2
281
Controlling ISI
Partial Response Signaling
Duobinary Signaling
Controlled ISI
To achieve zero ISI, we have seen that it is necessary to
transmit at below the Nyquist rate
Is it possible to relax condition on zero ISI and allow for
some amount of ISI in order to achieve fs > 2B?
The idea behind this is to introduce some controlled
amount of ISI instead of trying to eliminate it
ISI that we introduce is deterministic (or controlled) and
hence we can take care of it at the receiver
How do we do this?
 Controlled amount of ISI is introduced by combining a
number of successive binary pulses prior to transmission
 Since the combination is done in a known way, the receiver
can be designed to correctly recover the signal
283
Partial Response Signaling (PRS) - 1
A.k.a Doubinary signaling, Correlative coding, Polybinary
PRS is a technique that deliberately introduces some
amounts of ISI into the transmitted signal in order to ease the
burden on the pulse-shaping filters
It removes the need to strive at achieving Nyquist filtering
conditions, and high rolloff factors
This strategy involves two key operation
 Correlative filtering
 Digital precoding
Correlated filtering purposely introduces some ISI, resulting
in a pulse train with higher & correlated amplitude sequences
Nyquist rate no longer applies since the correlated
symbols are no longer independent
 Hence higher signaling rate can be used
284

The transfer function H(f) is equivalent to the Tap
Delay line
x t a t kT
k
k
( )  
  

y t a h t kT h t F H f
k
k
( ) , ( ) ( )
 
 
  
where 1
Digital
Precoding
Regenerator
H(f)
Impulse
Generator
ak
a k
' x t
( ) y t
( )
ak
T
T T
T
C1
Cn-2
C0
Cn-1 Cn
+
LPF @
B = R/2
x t
( )
y t
( )

285
Since h(t) = sinc(t/T) and R=1/T, the overall impulse
response is
and
where
y t a
k
c
n
c
t
T
n k k
a c
t
T
k
n
N
k k
( ) sin sin
  
FH IK

R
S
T
U
V
W
   
FH IK


0
k
a c a c a c a c a
o k k N k N n k n
n
N
     
  

 1 1
0

h t c
n
c
t
T
n
n
N
( ) sin
 
FH IK

0
286
Partial Response signaling changes the amplitude
sequence ak  a+
k
a+
k has a correlated amplitude span of N symbols
since each a+
k depends on the previous N values of ak
Also, when ak has M levels, a+
k sequence has M+ > M
levels
A whole family of Partial Response Signaling (PRS)
methods exists
Lets look at a few specific cases of PRS
287
Duobinary Signaling - 1
Simplest form of PRS with M = 2, N = 1, Co = C1 = 1
Input sequence is combined with a 1-bit delayed
version of itself and then pulse-shaped
Duobinary Encoder
x
a
a
k
k
k

R
S
T
1
0
,
,
if symbol = 1
if symbol = 0
+
Delay
T
xk
l q yk
xk1
H1

1
2T
1
2T
0
He f
( )
t kT


1
2T
1
2T
H2
0
288
y x x
k k k
  1

Each incoming pulse is added to the previous pulse
The bit or data sequence {yk} are not independent
Each yk digit caries with it the memory of the prior
digit
It is this correlation between digit that is considered
the controlled ISI which can be easily removed at
the receiver
Impulse Response of Duobinary Signal:
H f e j fT
1
2
1
( )    
H f
T f T
2
1
2
0
( )
,
,


R
S
T otherwise
289
From
it can be shown that (exercise show this)
H f H f H f e T T e e e
T fT e
j fT f
e
j fT j fT j fT j fT
T
( ) ( ) ( )
cos( ) ,
,
    



R
S
T
  
1 2
2
1
2
1
2
0
   
 
b g b g
else
h t
t T
t T
t T T
t T T
t T
t T
t T
t T T
c
t
T
c
t T
T
T t T
t T t
e( )
sin( / )
/
sin( ( ) / )
( ) /
sin( / )
/
sin( / )
( ) /
sin sin
sin( / )
( )
 


 

 













e j e j
2
H f T e e e
e
j fT j fT j fT
( )    
  
b g
290
Impulse response h(t) for the duobinary scheme is
simply the sum of two sinc waveforms, delayed by one
bit period w.r.t each other:
Duobinary signaling can be interpreted as adjacent
pulse summation followed by rectangular low pass
filtering
Encoder takes a 2 level waveform and produces a 3
level waveform

1
2T
1
2T
0
He f
( )

1
2T
1
2T
0
arg ( )
He f
f
f


2

2
Amplitude Response Phase Response
291
Duobinary Decoding:
The role of the receiver is to recover xk from yk
Transmitted signal (assuming no noise) is
xk can assume one of 2 values A, depending on
whether the k-th bit is 1 or 0
Since yk depends on xk and xk-1, yk can have 3 values
(no noise)
+
Delay
T
Decision
Circuit

xk
yk

xk1

yk
t kT

Duobinary Decoder
 A A
, ,
0
a f
-
y x x
k k k
  1
292

In general, (M-ary transmission), PRS results in 2M-
1 output levels
Detection involves subtracting xk-1 decisions from yk
digits such that
Decision rules is
y
k
A
A


R
S
|
T
|
2
0
2
,
,
,
if the kth and (k -1)th bits are 1's
if the kth and (k -1)th bits are different
 
x y x
k k k
  1
0, decide that opposite of previous
ˆ ˆ
ˆ
2, decide that 1
ˆ
k k
k
k
x x
y
x



  

293
The detection process is the reverse of the transmitter
process
Major drawback
 once errors are made, they tend to propagate through the
system
+
Delay
T
xk
l q
xk1
H1
-
Delay
T
Decision
Circuit

xk
yk

xk1

yk
t kT

Duobinary Decoder
LPF
Duobinary Encoder
A Duo-binary Baseband System
294
Advantage:
Duobinary signaling permits transmission at the
Nyquist rate without the need for linear phase,
rectangular shaped LPF
Disadvantages:
There is no one-to-one mapping between the
original binary digits and detected ternary symbol
(2  3)
Require more power
Ternary nature of duobinary signal requires about
3 dB greater SNR compared to ideal signaling
(i.e, binary) for a given PB
295
Decoding process, xk = yk-xk-1, results in errors
propagation, Why?
output data bits are decoded using previous data
bit. If previous bit is in error, then the new output
will be in error, and so on
–i.e., errors will propagate through the system
It is ineffective for AC coupled signal
AC coupling means that zero and low fred. data
are rejected
The PSD has substantial values at zero making it
unsuitable for AC coupled transmission
296

Note:
Problem 3 can be solved with a technique known
as precoding
Problem 4 can be solved with a technique known
as modified duobinary
297
Duobinary Transfer
Function and pulse
shape
(a) Cosine Filter
(b) Impulse
response of the
cosine filter
298
Composite pulses arising from like and unlike
combinations of input impulse pair
299
Duobinary waveform arising from an example binary
sequence
300

Duobinary Precoding - 1
A precoder consist of an exclusive-OR gate &
feedback through a one unit delay
The binary stream wk is applied to the input of the
duobinary filter with output yk
y w w
x w w
k k k
k k k
 
  

 
1
1 1
c h
1
1, if either or is 1
0,
k k
x w
w
k otherwise




Delay
T
+
w x w
k k k
  1
xk Duo-binary
Encoder
wk
wk1
yk
Delay
T
wk
wk1
y w w
k k k
  1
xk wk-1 wk wk+wk-1
0 0 0 0
0 1 1 2
1 0 1 1
1 1 0 1
Conversion
rule
301
The basic idea of precoding is that from the data
sequence {xk}, a new sequence {wk} (precoded
sequence) is generated
Unlike basic duobinary, precoding is nonlinear
The transmitted signal amplitude
At the receiver, the decoding decision rule is:
i.e,
y
k
a if w
a if w
a w
k k
k k
k k

  
 
R
S
T   
1 0
1 1
2 1
,
,
0, 2 1 1 mod 2
ˆ ˆ
2
1, 0
k
k k
k
if y
x x y
k if y
 
  
   
  


0, decide that 1
ˆ
2, decide that 0
ˆ
k
k
k
x
y
x



 

Duobinary Precoding - 2
302
 In general, (M-ary transmission), PRS results in 2M-1 output
levels
y
k
A
A


R
S
|
T
|
2
0
2
,
,
,
if the kth and (k -1)th bits are different
+
Delay
T
xk
l q
xk1
H1
-
Delay
T
Decision
Circuit

xk
yk

xk1

yk
t kT

Duobinary Decoder
LPF
Duobinary Encoder
Summary of Duobinary Baseband System - 1
303
Detection involves subtracting xk-1 decisions from yk
digits such that
Decision rules is
Decision rules if precoding is used

, 
, 
y
x
x
k
k
k


  
R
S
T
0 0
2 1
decide that
decide that
Summary of Duobinary Baseband System - 2
 
x y x
k k k
  1
1
0, decide that opposite of prior decoded value
ˆ ˆ
ˆ
2, decide that 1
ˆ
k k
k
k
x x
y
x

 


  

304

Modified Duobinary Signaling - 1
Also called class 4 signaling
Problem #4 (i.e, large DC value of duobinary PSD) can
be addressed by this signaling techniques
The encoder involves a two-bit delay, causing the ISI to
spread over two symbols (correlation span of 2 binary
digits)
Here again, we find that a 3 level signal is generated
Similarly
y x x
k k k
  2
H f e j fT
1
4
1
( )     H f
T f
T
2
1
2
0
( )
,
,


R
S
|
T
| otherwise
+
Delay
2T
xk
l q yk
xk2
H1
 1
2T
1
2T
H2
0
-
T
305
 From
it can be shown that (exercise show this)
 Spectrum shows a null @ zero but is still strictly bandlimited to 1/2T
H f H f H f e T T e e e
jT fT e f
e
j fT j fT j fT j fT
j fT
T
( ) ( ) ( )
sin( ) ,
,
    


R
S
T
  

1 2
4 2 2 2
2 1
2
1
2 2
0
   


b g b g
else
H f T e e e
e
j fT j fT j fT
( )    
2 2 2
  
b g
h t
t T
t T
t T T
t T T
t T
t T
t T
t T T
T t T
t T t
e( )
sin( / )
/
sin( ( ) / )
( ) /
sin( / )
/
sin( / )
( ) /
sin( / )
( )
 


 













2
2
2
2
2
2
306
Similar to basic duobinary, error propagation
necessitates the use of a precoding which is
implemented in a similar manner
Delay
2T
+
Delay
2T
k
x k
w
2
k
w 
 
k
x
2
k
x 
k
y
2
H
1
2T
 1
2T
1
H
307
 For consistency, lets characterize the PRS systems
Characterization of PRS Systems - 1
D D D
wk 

wk 

wk
yk
xk yk ŷk
x̂k 

duo

Mod.duo


x̂k
1, Duobinary
2, Modified Duobinary


 

, Duobinary
2 , Modified Duobinary
T
D
T

 

308

 Duobinary:
a) Without Precoding: (wk = xk)
b) With Precoding:
1
1
:
ˆ ˆ ˆ
:
k k k
k k k
Code y x x
Decode x y x


 
 
 
1
1 1
1
: k k k
k k k k
k k
Code w x w
y w w w
x w

 

 
   

ˆ
1, 0
ˆ
ˆ
0, 2
k
k
k
if y
Decode x
if y


 
 

309
 Modified Duobinary:
a) Without Precoding: (wk = xk)
b) With Precoding:
2
2
:
ˆ ˆ
:
ˆ
1, 1
ˆ
:
0,
k k k
k k k
k
k
Code y x x
Decode x y x
y
Output sequence x
else


 
 


 

 
2
2 2
2
: k k k
k k k k
k k
Code w x w
y w w w
x w

 

 
   

ˆ
1, 2
ˆ
:
ˆ
0, 0
k
k
k
if y
Decode x
if y
 

 


310
Example: (Duobinary Coding)
Example: (Duobinary Coding)
Find the output sequence of duobinary signaling
system if the input data sequence is 1 1 0 0 0 1 0 1
0 0 1 1 1
a) without precoding, b) with precoding
 Example: (Duobinary Coding)
Examples
311
Multipath Channels - 1
Have already seen that bandlimited channel induce ISI
A good strategy was to pick a pulse shape that was
bandlimited and thus was not distorted by the
channel
It is also possible for a channel that is not bandlimited
to cause ISI, e.g., the multipath channel
h t t t
c
( ) ( ) ( )
   
   
1 2
Difused
Component
Transmitter
Direct Ray
Specular
Component
Antenna
Gain Pattern
Receiver
312

If the direct path has time delay 1 and the reflected
path has time delay 2 (2 > 1) then the impulse
response of the channel is
The channel’s frequency response
A plot of the magnitude response will not be flat!
Because the magnitude response is not flat, the signal
will undergo distortion, possibly resulting in ISI
It is therefore possible to encounter ISI even when the
channel itself has an infinite bandwidth
So, how do we handle this problem?
  1 2
2 2
1 2
( ) ( ) ( ) j f j f
c
H f F t t e e
   
     
     
Multipath Channels - 2
313
Equalization
Equalization - 1
Nyquist filtering and pulse shaping schemes assumes
that the channel is precisely known and its
characteristics do not change with time
However, in practice we encounter channels whose
frequency response are either unknown or change
with time
e.g., each time we dial a phone #, the communication
channel will be different because the communication
route will be different
But, when connection is made, the channel becomes
time-invariant (land line only)
The characteristics of such channels are not known a
priori
315
Equalization - 2
Examples of time-varying channels are radio channels
These channels are characterized by time-varying
frequency response characteristics
To compensate for channel induced ISI and other
distortions, we use a process known as Equalization
a technique of correcting the frequency response of
the channel
The filter used to perform such a process is called an
equalizer
Channel
hC
(t)
Receiver
hR
(t)
Transmiter
hT
(t)
Equalizer
hEQ
(t)
+
Noise
n(t)
316

Equalization - 3
Since HR(f) is matched to HT(f), we usually worry
about HC(f)
Goal is to pick the frequency response Heq(f) of the
equalizer such that
with amplitude
and phase
( )
1
( ) ( ) 1 ( )
( )
j f
c
c eq eq
c
H f H f H f e
H f
 
  
( )
1
( )
H f
eq
c
H f

( ) ( )
eq c
f f
 

317
1. It can be difficult to determine the inverse of the channel
response
 If the channel response is zero at any frequency, then the
inverse is not defined at that frequency
 Rx generally does not know what the channel response is
 Channel changes in real time, so realistic equalization must be
adaptive
2. The equalizer can have an infinite impulse response even if
the channel has a finite impulse response
 The impulse response of the equalizer must usually be
truncated
3. The equalizer can actually enhance the noise in the channel
 Nonlinear equalization techniques are available that
minimize the amount of noise enhancement
Problems with Equalization
318
Equalization Techniques or Structures
Three Basic Equalization Structures
Linear Transversal Filter
 Simple implementation using Tap Delay Line or FIR filters
 FIR filter has guaranteed stability (although adaptive
algorithm which determines coefficients may still be
unstable)
Decision Feedback Equalizer
 Extra step in subtracting estimated residual error from
signal
Maximal Likelihood Sequence Estimator (Viterbi)
 “Optimal” performance
 High complexity and implementation problem (not heavily
used)
319
Linear Transversal Equalizer - 1
This is simply a linear filter with adjustable parameters
Parameters are adjusted on the basis of the measurement of
channel characteristics
A common choice for implementation is the transversal filter
(Tap Delay Line (TDL)) or the FIR filter with adjustable tap
coefficient
 Total number of taps = 2N+1
 Total delay = 2NT = 2N
C-N+1 CN-2
C-N
CN-1 CN
Algorithm for
coefficient adjustment
xk 
  yk
 


1
k
x 
320

N is chosen sufficiently large so that equalizer spans
length of the ISI
Assuming the ISI is limited to a finite # of samples, say
L, then 2N+1 > L
Output yk of the equalizer in response to the input
sequence {xk} is
where cn is the weight of the nth tap
y c x
k n k n k N N
n N
N
    

, , ,
2 2

321
Ideally, we would like the equalizer to eliminate ISI
resulting in
But this cannot be achieved
However, the tap gains can be chosen such that
There are two types of such equalizer (i.e., linear
equalizers)
y
k
k N
k 

   
R
S
T
1 0
0 1 2
,
, , , ,

y
k
k
k 


R
S
T
1 0
0 0
,
,
322
Preset Equalizer:
Transmits a training sequence that is compared at the
receiver with a locally generated sequence
Requires an initial training sequence
Differences between sequences are used to update the
coefficient cn
Time varying channel can change the sequence, since
the coefficients are fixed
Adaptive Equalizer:
Equalizer adjust itself periodically during transmission of
data
The tap weights constitute the adaptive filter coefficient
323
The 2 techniques can be combined into a robust
equalizer
In this case, there are two modes of operation
Training Mode
For the training mode, a known sequence is
transmitted and a synchronized version is
generated at the receiver
Decision-Directed Mode
When training mode is complete, the adaptive
algorithm is switched on
The tap weights are then adjusted with info from
training mode
324

The impulse response of the transversal filter is
If x(t) is the signal pulse corresponding to
then the equalized output signal is
2
( ) ( )
( )
N
eq n
n N
N
j fn
eq n
n N
h t c t n
H f c e  
 


 


  
y t c
n
x t n
n N
N
( ) ( )
 



X f H f H f H f
T C R
( ) = ( ) ( ) ( )
325
Nyquist zero ISI condition implies that
Since there are 2N+1 coefficients, we may express in
matrix form as
where
x = (2N+1)  (2N+1) matrix with elements x(kT - n)
c = (2N+1) column coefficient vector
y = (2N+1) column vector
 
( )
1, 0
0, 1, 2, ,
k
N
n
n N
y y kT
k
c x kT n
k N





  
 
   
 
326
y xc

Since this design forces the ISI to be zero at sampling
instants t = kT, the equalizer is called Zero-Forcing
Equalizer (ZFE)
Thus we obtain a set of (2N+1) linear equations for
ZFE
In the figure,  is chosen as high as T
 = T  Symbol-spaced equalizer;
 < T  Fractional-spaced equalizer
327
Survey of Equalizers
Types
Structures
Algorithms
Equalizer
Linear Nonlinear
DFE
ML Symbol
Detector
MLSE
Transversal Lattice Transversal
Channel Estimator
Transversal Lattice
 Zero Forcing
 LMS
 RLS
 Fast RLS
 Square Root RLS
 LMS
 RLS
 Fast RLS
 Square Root RLS
 LMS
 RLS
 Fast RLS
 Square Root RLS
 Gradient RLS  Gradient RLS
328

Example: Equalization
Problem
Example: Equalizer/Equalization
Example: Equalization
Example: Equalizer/Equalization
Problem
Examples: (Equalizer/Equalization)
329
‡ Decision Feedback Equalizer
 A Decision-Feedback Equalizer (DFE) is a nonlinear equalizer
that employs previous decisions to eliminate the ISI caused by
previously detected symbol
 It consists of a feed forward section a feedback section and a
detector connected together as shown
 The filters are usually fractionally spaced FIR with adjustable tap coefficients
 The detector is a symbol-by-symbol detector
 Note
 ‡  self study
Feedforward
Filter
Detector
Feedback
Filter
output
data
Input from
matched
filter
+
-
m
z
ˆm
z
330
‡ Maximum Likelihood Sequence Detector (MLSD)
 This technique provides an algorithm for searching through the trellis for the
ML signal path
 A trellis is a schematic used to represent signal waveforms with memory,
e.g., the trellis for duobinary PRS is given by
 For binary, this trellis contains 2 states corresponding to 2 possible input values
 Since the duobinary have memory of length L = 1, the number of states is S = 2L
 In general, for M-ary, the number of trellis states is S = ML
 Maximum Likelihood Sequence Detector selects the most probable path through
the trellis upon observing the received sequence y(kT)
 In general each node in the trellis will have M incoming paths and M metrics
 Search through the trellis for the minimum distance may be performed sequentially
using Viterbi algorithm - beyond the scope of this class!


1
2
t0
1
-1
1/2
new data bit/received signal level
1/2
1/2


1
2


1
2
t T
 t T
2 t T
3

1
0
1
0
1
0
1
0
1
0
1
0
331
Module 3
Baseband Communication System

Noise in Communication System
Transmitter Channel Receiver
y0(t)
n(t)
yi(t)
Si
, Ni
x(t)
m(t)
s(t)
S0
, N0
input output
PT
Noise on Communication Systems
In the process of communication, noise arises in
various forms
m(t) is corrupted in the transmitter by thermal noise due
to the presence of electronic devices (e.g., Audio
Amplifier)
c(t) is not a pure sine wave - in fact, it contains
harmonic distortions
s(t) experiences multiplicative noise in the process of
being transmitted thru the channel due to turbulence in
the air, reflection, refractions, multipath etc.
s(t) also suffers from additive noise during transmission
(passing automobiles, static electricity, lightning, power
lines, sunspots, etc)
thermal and short noise at the receiver
334
Noise Modeling - 1
All these different noise components degrade the
performance of communications system
Among these types of noise, the additive noise is the
most annoying
usually contains most power and is of most interest
in many applications
+ r(t)
s(t)
n(t)
(noise)
(modulated signal ) (received signal )
335
Noise Modeling - 2
In the channel, the signal experience attenuation, time
delay (precisely known) and additive noise
Most disturbances, interference, attenuation, etc., are
usually classified as noise
The most important type of noise that occur in
communications system is said to be “white noise”,
n(t)
Usually n(t) is assumed to be Additive, White and a
Gaussian Noise (AWGN) with power spectral density
Gn(f)
336

 White Noise is a random process having a flat
(constant) power spectral density Gn(f), over the entire
frequency range
 white because it is analogous to white light
 assumed to be a Gaussian random process
 usually additive in nature
 Hence this type of noise is commonly called Additive,
White and Gaussian (AWGN) with power spectral
density such that 0
( )
2
n
N
G f 
White Noise and Filtered Noise - 1
(f)
Gn
f
2-sided power spectral density of noise
0
0
2
N
337
This type of noise is wideband and cannot be expressed
in terms of quardrature components
However, in most communications systems operating at
carrier frequency fc, the bandwidth of the channel B (or
W), is small compared to fc  narrowband systems
In such situations, it is mathematically convenient to
represent the white noise process in terms of the
quadrature components
 Accomplished by passing signal plus noise at the receiving
terminal through an ideal BPF having a passband as
(f)
Gn
f
fc
-fc
0
2
N
0
B B
White Noise and Filtered Noise - 2
338
Signal-to-Noise Ratio (SNR) - 1
SNR is the figure of merit for evaluating the performance
of analog communications systems
 A certain signal m(t) (or x(t)) is transmitted with power PT
 s(t) is corrupted by additive noise n(t) during transmission
 Channel may also attenuate (and/or distort) the signal
 At receiver, we have a signal mixed with noise
 Signal and noise power at the receiver input are Si and Ni
 Receiver processes the signal (filters, demodulation, etc.)
to yield the desired signal power So, plus noise power No
y0(t)
n(t)
yi(t)
Si
, Ni
x(t)
m(t)
s(t)
S0
, N0
input output
PT
339
Assume that:
 Noise n(t) is zero-mean Gaussian with PSD Gn(f) = N0/2 or
η/2
Noise is uncorrelated with s(t)
Hence output power is
The output signal-to-noise ratio (SNR) is
2 2 2
0 0 0
0 0
( ) ( ) ( )
 
     
     
 
E y t E s t E n t
S N
Signal-to-Noise Ratio (SNR) - 2
0 0
( ) ( ) ( )
o
y t s t n t
 
2
0
0
0 2
0
0 0
( )
( )
E s t
S S
SNR
N N E n t
 
   
  
 
 
   
340

In baseband systems, signal is transmitted w/o modulation
and we also assume that channel is distortionless, hence
This mode of communication is used in short-haul links
over a pair of wires or coaxial cable
Although this mode of communication is not widely used,
their study is important because many of the basic
concepts can be carried over to modulated systems
Also, baseband systems are used as benchmark for
comparing the performance of analog systems
A baseband Communication System Model
Baseband System Model - 1
LPF Channel LPF
+
m(t)
o
S
0
N
i
N
i
S
T
S
)
(t
n Noise
input
)
(t
yD
)
( f
H p )
( f
HC )
( f
Hd
limits m(t) eliminates out-
of-band noise
   
0 0 d
x t x t t
 
341
Assumes:
m(t) is zero-mean, wide sense stationary random
process bandlimited to B Hz
Assume that the channel is distortionless with unit gain,
Signal-to-noise ratio is then given as
Therefore, for a baseband system,
0
, 2 ( )
B
i T n
S P N where N G f df
   
o
o
N
S
SNR 

Power
Noise
Power
Signal
Mean
0
i
S
S
SNR
N N B
b b
 
 
 
   
   
This is used as a
standard for making
comparisons of the
various analog
modulation schemes
Baseband System Model - 2
342
Receiver output SNR does not depend on the gain, gR
However, channel gain or losses will affect the output
2
T T T X
S g x g S
 
Channel
L
gT LPF
gR
( )
n t
( )
x t
X
S R
S
( )
R
x t
T
S
0 0
( ) ( )
x t n t

0 0
S N

Receiver
2 T
R R
S
S x
L
 
2
0 0 R R
S x g S
    0
0 R
output
g N B
N 
0
R
S S
N N B
o
  
 
 
0
T
S S
N LN B
o
  
 
 
With Gain - 1
343
Therefore
Larger value of SNR is desirable
This can be achieved by simply increasing PT
However, this is usually not possible since in practice,
(PT)max is limited by other considerations such as
 FCC (NCC) rule; transmitter cost; channel capacity;
interference with other channels, and so on
In practice, it is more convenient to deal with received
signal power Si instead of PT
With Gain - 2
0 i
S S

2
0 0 0
0
2
( ) ( )
B B
n
B B
N
N E n t G f df df N B
 
   
   
 
0
0 0
i
S S
S
N N B

   
 
 
344

Detection of Binary Signal in
Gaussian Noise
Binary Signal Transmission - 1
In a binary commun. system, binary data (0’s, 1’s) are
transmitted by means of 2 signal waveform s0(t) & s1(t)
0  s0(t), 0  t  Tb
1  s1(t), 0  t  Tb
Assumptions:
data bits 0 & 1 are equally probable (each has
probability 0.5)
0 and 1 are mutually independent
The channel corrupts the signal by adding noise,
denoted by n(t)
n(t) is assumed to be Additive White Gaussian
Noise with PSD N0/2 W/Hz
where Tb = 1/Rb
346
Binary Signal Transmission - 2
The received signal waveform is expressed as
 r(t) = si(t) + n(t), i = 0, 1; 0  t  Tb
Receiver is to determine whether a ‘0’ or a ‘1’ was
transmitted
Analysis that follow will assume that the filtering
operation is linear
linear input  linear output
Gaussian input  Gaussian output
347
Detection of Binary Signal in Gaussian Noise - 1
Recovery of signal at the receiver consist of 2 parts
Signal correlator or Matched filter
reduces received signal to a single variable z(T)
z(T) is called the test statistics
Detector (or decision circuit)
 compares the z(T) to some threshold level 0, i.e.,
where H1 and H0 are the two possible binary hypothesis
1
0
0
( )
H
H
z T 


 ( )
si t
n t
( )
z t
( )
t T

x h(t)
si t
( )
r t
( )
(AWGN)
z T
( )
1
H
0
H
348

Signal correlator and detector processes are
independent
Once r(t) is transformed to z(T), the shape of the
waveform is no longer important
This means that any kind of transmitter waveform
transforms to z(T) for detection purposes
Hence, detection for baseband and bandpass are the
same
A particular detector that minimizes the probability of
error is known as the maximum likelihood detector
That is, it minimizes the cost of making an error
Detection of Binary Signal in Gaussian Noise - 2
349
Maximum Likelihood Detector (MLD) - 1
The concept of maximum likelihood detector is based
on Statistical Decision Theory
It allows us to
 formulate hypothesis that characterizes the transmission
 test the hypothesis
 formulate the decision rule that operates on the data
 optimize the detection criterion
The formulation of this topic requires the knowledge of
probability (in particular Bayes’ rules) and random
variables
For a binary data stream there are two types of decision
 Soft decision (multi-level)
 Hard decision (2 level)
350
Hard decision is more common than soft decision
 Decides immediately whether the signal is 0 or 1
 Uses either Bayes decision criterion or Newman-Pearson
criterion
Digital 0
000
Matched
Filter
8-level 3-bit
quantization
Combined Soft decision/
error control decoding
S & H
Matched
Filter
Binary
quantization
Error control
soft decision hard decision
a) Soft decision Receiver
hard decision
Digital 1
010 100 110
000 010 100 110
1
0
b) Hard decision Receiver
soft decision
hard decision
S & H
351
Each soft decision contains
Information about the most likely transmitted signal
000 to 011  0
100 to 111  1
Information about the likelihood of a decision
Soft decisions are converted to hard decisions by
some algorithm
Let T be the length of time it takes to transmit one bit
of data
0
1
( ), 0 for a binary 0
( )
( ), 0 for a binary 1
s t t T
s t
s t t T
 

 
 

352

At the output of the demodulator
where ai(t) is the signal component & noise n is zero mean
Gaussian
At the sampling instant t = T
For simplicity we will drop the index such that z = ai + n
0 0
1 1
( ) ( ) ( ), 0 for a binary 0
( )
( ) ( ) ( ), 0 for a binary 1
z t a t n t t T
z t
z t a t n t t T
   

 
   

0 0
1 1
( ) ( ) ( ), 0 for a binary 0
( )
( ) ( ) ( ), 0 for a binary 1
z T a T n T t T
z T
z T a T n T t T
   

 
   

353
z(T) is known as decision variable or test statistics
and it is a random process corrupted by noise
Assume that pdf of z0(T) and z1(T) are Gaussian with
equal likelihood, and with 0 = a0, 1 = a1
a0
Region 0
Likelihood of s0
Region 1
Likelihood of s1
Decision
Line
P[z|s1 sent]
P[z|s0 sent]
Pe(s0)
a0
 o
p z s z a
( | ) exp
0
0
0
0
2
1
2
1
2
 

F
HG I
KJ
L
NM O
QP
  
p z s z a
( | ) exp
1
1
1
1
2
1
2
1
2
 

F
HG I
KJ
L
NM O
QP
  
Minimum error criterion  


 0
0 1
2
0
a a
354
This is an averaging operation
It makes sense because the logical point is halfway
between the two voltage levels representing each
symbol
Questions:
How do we implement this averaging operation?
How do we choose the threshold, 0?
Hypothesis:
H0: r(t) = s0(t) + n(t)  “0” sent
H1: r(t) = s1(t) + n(t)  “1” sent
355
Definitions of Probabilities:
P[s0], P[s1]  a priori probabilities
 These probabilities are known before transmission
P[z]
 probability of the received sample
p(z|s0), p(z|s1)
 conditional pdf of received signal z, conditioned on the class si
P[s0|z], P[s1|z]  a posteriori probabilities
 After examining the sample, we make a refinement of our previous
knowledge
P[s1|s0], P[s0|s1]
 wrong decision (error)
P[s1|s1], P[s0|s0]  correct decision
356

Decision Rule:
 Acquiring information at the receiver about the transmitted
signal involves making decisions
 We must decide which of the set of hypothesis best
describes the received signal
 This involves uncertain (error in judgment)
 If the signals we are trying to detect do not overlap, we
can make a decision without error
 On the contrary, we need some rules to help classify the
received signal once they fall in the overlap region
 A set of rules known as decision rules allow us to decide
( )
ˆi
z t
0

z T
( )
1
H
0
H
357
1. Bayes’ decision criterion:
It formulates the problem of making a decision
under conditions of uncertainty by selecting the
hypothesis with the greatest a posteriori probability
This scheme assumes that some errors are more
costly than others
Hence, it assigns cost (weighting factors) that
reflect the risk involved
This is the most widely applied decision rule in
communications
Types of Decision Rules - 1
358
2. Maximum a posteriori (MAP) criterion:
Decide that the received signal belongs to the class
with the maximum a posteriori probabilities, i.e.,
maximize P(si|z)
It equivalently examines the pdf conditioned on
each signal class (p(z|s0), p(z|s1)) and choose the
maximum
For the received signal za, the likelihood that za
belongs to s1 or s2 corresponds to the circled point
on the pdf
The decision criterion is based on the likelihood of
P[z|si], i = 0,
359
3. Newman-Pearson (N-P) criterion
Makes no assumption on the a priori source
statistics (requires only a posteriori probabilities)
Widely used in pulse detection in Gaussian noise as
in Radar applications where the source probabilities
(presence or absence of a target) is unknown
fix probability of false alarm
minimize probability of error
maximize probability of correct decision
360

4. Min-max criterion
Also in this criterion, the a priori probability is not
known
Since P(H1) is unknown, the rule maximizes the risk
with respect to P(H1) and minimizes the risk with
respect to P(H0)
361
Bayes’ Decision Criterion - 1
Recall that the Bayes equation is given by
where
Recall from probability theory that
In communications, we can interpret the Bayes’
equation as a description of an experiment involving
a received sample, and
a statistical knowledge of the signal classes to
which the received sample may belong
1
[ ] [ | ] [ ]
M
i i
i
P z P z s P s

 
[ | ] [ ]
[ | ] , 0,2, , 1
[ ]
i i
i
P z s P s
P s z i M
P z
  

[ | ] [ ] ( | ) [ ]
i i i
P s z P z p z s P s

362
That is,
si denote the ith transmitted signal class from a set
of M classes
zj denotes the jth sample of the received signal
Hence, we can write the Bayes equation in terms of
the pdf
where
( | ) [ ]
[ | ] , 0,2, , 1
( )
i i
i
p z s P s
P s z i M
p z
  

1
( ) ( | ) [ ]
M
i i
i
p z p z s P s

 
363
By examining a particular received sample zj, it is
possible to find likelihood that zj belongs to class si
This means that after the experiment, we will refine our
knowledge by computing the a posteriori probability
Note that the terms a priori and a posteriori imply
“cause to effect” and “effect to cause,” respectively
Assume that
Pdf of z0(T) and z1(T) are Gaussian with equal
likelihood, having mean values of a0 and a1 respectively
a0 and a1 are mutually independent
Noise n0 is independent zero mean AWGN with PSD No
364

In this case, for binary signal
The last equation corresponds to making a decision
based on the comparison of received signal to some
threshold level
1 0
[ | ] [ | ] decision rule
P s z P s z
 

0 0
1 1 ( | ) [ ]
( | ) [ ]
( ) ( )
p z s P s
p z s P s
p z p z
 
 1 1 0 0
( | ) [ ] ( | ) [ ]
p z s P s p z s P s


0
1
0 1
[ ]
( | )
( ) likelihood ratio test (LRT)
( | ) [ ]
P s
p z s
L z
p z s P s

 

365
The right-hand side (RHS) is called the likelihood ratio
When the two signals, s0(t) and s1(t), are equally likely,
i.e., P[s0] = P[s1] = 0.5, then the decision rule becomes
In terms of the Bayes criterion, it implies that the cost of
both types of error is the same
This type of decision rule is called the maximum a
posteriori (MAP) criterion (or minimum error
criterion)
0
1
0 1
[ ]
( | )
( ) likelihood ratio test (LRT)
( | ) [ ]
P s
p z s
L z
p z s P s

 

1
0
( | )
( ) 1 max likelihood ratio test
( | )
p z s
L z
p z s

 

366
Substituting the pdfs
2
0
0 0
0 0
1 1
: ( | ) exp
2 2
z a
H p z s
  
 
 

 
 
 
 
 
2
1
1 1
1 1
1 1
: ( | ) exp
2 2
z a
H p z s
  
 
 

 
 
 
 
 
2
1
2
1
1 1
2
0
0
2
0
0
1
1
( )
exp
2
( | ) 2
( ) 1 1
1
1
( | )
( )
exp
2
2
z a
p z s
L z
p z s
z a

 

 
 
 
 
 
 
 
 
 
 
 
 
2 2
0 1
 
 
2 2
1 0 1 0
2 2
0 0
( ) ( )
exp 1
2
z a a a a
 

 
 

   
 
367
Taking the log of both sides will give
Hence
where z is minimum error criterion and 0 is optimum threshold
2 2
1 0 1 0
2 2
0 0
( ) ( )
ln{ ( )} 0
2
z a a a a
L z
 
  
   

2 2
1 0 1 0 1 0 1 0
2 2 2
0 0 0
( ) ( )( )
2 2
z a a a a a a a a
  
   

 

2
0 1 0 1 0
2
0 1 0
( )( )
2 ( )
a a a a
z
a a


 

 
1 0
0
( )
2
a a
z 

 

368

For antipodal signal, s1(t) = - s0(t)  a1 = - a0
This means that if received signal was positive, s1(t)
was sent, else s0(t) is sent
0
z 

369
Probability of Error - 1
Error will occur if
s1 is sent  s0 is received
 P[H0|s1] = P[e|s1]
s0 is sent  s1 is received
 P[H1|s0] = P[e|s0]
The total probability of error is the sum of the errors
0
1 1
[ | ] ( | )
P e s p z s dz


 
0 0
0
[ | ] ( | )
P e s p z s dz


 
2
1 1 0 0
1
0 1 1 1 0 0
( , ) [ | ] [ ] [ | ] [ ]
[ | ] [ ] [ | ] [ ]
B i
i
P P e s P e s P s P e s P s
P H s P s P H s P s

  

 
 o
ao a1
0 1
 o
ao a1
0 1
See pp. 121~122 & section
B.2
370
If signals are equally probable
Hence, PB, is probability that an incorrect hypothesis is
made
Think of PB as the area under the tail of either of the
conditional distributions, p(z|s1) or p(z|s2), i.e.,
 
0 1 1 1 0 0
1
0 1 1 0
2
[ | ] [ ] [ | ] [ ]
[ | ] [ | ]
B
P P H s P s P H s P s
P H s P H s
 
 
 
1
1 0
0 1 1 0
2
[ | ]
[ | ] [ | ]
B
by symmetry
P P H s
P H s P H s
 


1 0 0
0 0
2
0
0
0 0
( | ) ( | )
1 1
exp
2 2
B
P p H s dz p z s dz
z a
dz
 

  
 

 
 
 
 

 
  
 
 
 
371
 This equation cannot be evaluated in closed form
 This is the famous Q-function or complementary error
function
 Hence,
1 0
0
2
B
a a
P Q


 
  
 
1 0
2
0
0
0 0
0
0
0
2
( )/ 2
1
1 1
exp
2 2
( )
,
1
exp *
2 2
B
a a
z a
P dz
z a
u du dz
u du A


  






 
 

 
  
 
 
 

  
 
   

  
 
372

Pe is minimized by choosing h(t) or H(f) such that optimum
threshold 0 is minimized
That is
A note on the Q(x) - complementary (co) error function
Equivalent Definitions
For large arguments (x large), Q function 
2
2
0 1 0 1
0 0
( ) ( ) [ ( ) ( )]
2 4
a t a t a T a T
or
 
 
 
2
1
( ) exp
2 2
x
Q x
x 
 
 
 
 
1
( ) e
2 2
x
Q x rfc
 
  
 
   
e 2 2
rfc Q
x x

373
Correlator
Correlator-Type Receiver - 1
The correlator cross-correlates r(t), with the 2
possible transmitted symbols s0(t) and s1(t)
Output for either z0 or z1 is given by
This cross-correlation process basically computes the
projection of r(t) into 2 basis functions s0(t) and s1(t)
 The outputs z0 and z1 are then feed to the Threshold Detector
x
Threshold
Detector
r t
( ) s t
0
( )
t T

z T
0
( )
 ( )
s t
i
x
z T
1
( )
s t
1
( )
()

z dt
T
0
()

z dt
T
0
z t
0
( )
z t
1
( )
0 0
0
( ) ( ) ( )
T
z T r t s t dt
  1 1
0
( ) ( ) ( )
T
z T r t s t dt
 
375
The detector compares z1 and z0 and decides that
 1 was transmitted if z1 > z0
 0 was transmitted if z1 < z0
So when s1(t) is transmitted,
PB = P[z0 > z1] = P[n0 > E + n1] = P[n0 -n1> E]
Let x = n0 - n1
 
   
2
2 2 2
0 1 0 1 0 1
2 2
0 0
0
2
2 ( ) 2
4 2
n
E x zeromean
E x E n n E n E n E n n
N N
E n t 
 
 
     
    
     
 
 
 
   
 
   
0, orthogonal
Noise Variance
376

Hence PB is
2
2
2
0
1
exp
2
2
1
exp
2
2
E
x
x
E
x
P dx
B
x
dx
E
Q
N

 



 
 
  
 
 
 
 

 
 
  
 
E
377
Other Forms of the Correlator
Form 1:
 A similar procedure can be used to derive the PB
Form 2:
 observe the correlating signal given by s1(t)-s0(t)
x
r t
( )
s t s t
1 0
( ) ( )

t T

 ( )
s t
i
()

z dt
T
0
x
r t
( )
s t
0
( )
t T

 ( )
s t
i
x

s t
1
( )
()

z dt
T
0
()

z dt
T
0
-
+
z t
0
( )
z t
1
( )
z T
( )
378
Suppose there are many signals si(t), i = 0, 2, …, M-1,
the received signal can be correlated using a bank of
correlators
x
Selects
si(t) with
the
max zi(t)
r t
( )
s t
0
( )
t T

z T
0
( )
 ( )
s t
i
x
z T
1
( )
s t
1
( )
()

z dt
T
0
()

z dt
T
0
x
z T
M1( )
s t
M1( )
()

z dt
T
0


379
Matched Filter

Matched Filter Receivers - 1
A matched filter is a linear filter that optimizes the
SNR for a symbol
i.e., maximizes the SNR at the output for a given
transmitted symbol waveform
Given r(t) = s(t) + n(t) at the input, we want to find the
filter characteristics h(t) or H(f) that maximizes the
output SNR
r t
( )
t T

 ( )
s t
i
z t
( ) z T
( )
h t s T t
b
( ) ( )
 
+
s t
( )
n t
( )
381
A filter that is matched to the waveform s(t), has an
impulse response
h(t) = s(Tb-t), 0  t  Tb
Notice that h(t) is a delayed version of the mirror
image (rotated on the t = 0 axis) of the original signal
waveform
E.g.,
s t
( ) s t
( )
 h t s T t
b
( ) ( )
 
Tb Tb
0 0 0
Tb
t
t
t
signal image signal delayed by Tb
382
This is a causal system
a system is causal if before an excitation is applied
at time t = T, the response is zero for - < t < T
Signal waveform at the output of the matched filter is
If we sample z(t) at t = Tb, we obtain
Hence the sampled output of the filter at time t = T is
exactly the same as the output of the correlator
0
0
( ) ( ) ( )
( ) ( )
t
t
b
z t r h t d convolution
r s T t d
  
  
  

  

0
( ) ( ) ( ) ( )
Tb
b
b
z T z t r s d
t T   
  

383
Important Property of Matched Filter:
If s(t) is corrupted by an AWGN, the filter with impulse
response h(t) maximizes the SNR
To prove this let
r(t) = si(t) + n(t), t0  t  t0+Tb , i= 0,1
S(f) = Fourier Transform of s(t)
H(f) = Transfer function of the filter h(t)
For MF, we want to determine h(t) or H(f) that
maximizes output SNR
384

Time Domain Analysis:
h(t) y(t)
r(t)
0
0 0
0 0
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
t
t t
T T
s n
y t r h t d
s h t d n h t d
sampleatt T
s h T d n h T d
y T y T
  
     
     
 

   
 
 
   
 
 
2
2
( )
( )
s
T n
y T
S
E
N y T
  
 
   
 
noise variance
385
But
 
2
2
0
( ) ( ) ( )
T
n
E E
y T n h T d
  
 

  
  
 
 
 
2
0 0
0
0 0
2
0
0
( ) ( )
( ) ( )
( )
( ) ( )
2
( )
2
T T
n
T T
T
E E h T h T t dtd
n n t
y T
N
h T h T t dtd
t
N
h T t dt
 

  

  
   
 
  

 
 

386
The noise variance depends on the PSD of the
noise and the energy in the impulse response, h(t)
 We can maximize this expression by holding the
denominator constant and then optimizing the numerator
From Cauchy-Schwarz inequality, we know that
with equality when x(t) = ky(t), k = constant
2 2
0 0
2 2
0 0
0 0
( ) ( ) ( ) ( )
( ) ( )
2 2
T T
T T
T
s h T d h s T d
S
N N
N h T t dt h T t dt
     
   
 
 
     
 
 
   
 
   
2 2 2
( ) ( )
( ) ( ) dt dt
x t y t
x t y t dt
 

 

 
   

 
387
Hence by replacing x(t) = h(t), y(t) = s(T-t)
It is clear here that SNR is maximum when h(t) =
ks(T-t)
2 2 2
2
0 0
0
2 2
0 0
0
0
( ) ( ) 2
( )
( )
2
2
T T
T
T
T
k s T t dt s T d
S
s t dt
N N
N k s T t dt
E
N
 
  
 
    
 
  


0
2 2
0 0
2
0
2
( ) ( )
( )
T T
N T
T
h d s T d
S
N h T t dt
   
 
 
  
 
  

388

Frequency Domain Analysis:
Since z(t) = a(t) + n0(t), where a(t) is the signal
component, we can write
Therefore
but the denominator is the noise variance
 
1 2
( ) ( ) ( ) ( ) ( ) j f
i i i
t
a t FT H f S f H f S f e df




  
 
2
2
( )
( )
T
a t
S
N E n t
  
 
 
 
0
2
2
2
2
( ) (0) ( )
( ) ( )
( )
n ny
nx
N
E n t R G f df
H f G f df
H f df






  
 
 
2
( ) ( ) ( )
ny nx
G f H f G f

eqn. 1.53
eqn. 1.42
389
Substituting
The numerator is of the form
 where Y(f) = S(f)ej2ft
If written with Cauchy-Schwartz inequality we have
0
2
2
2
2
( ) ( )
( )
j ft
N
T
H f S f e dt
S
N H f df






  
 
  
2
( ) ( )
H f Y f df



2 2 2
2 2
( ) ( ) ( ) ( )
( ) ( )
H f Y f df H f df Y f df
H f df S f df
  
  
 
 
 
  
 
 
Equality holds iff H(f) = KY*(f)
max at 0
390
Hence
But
2
2
2
0
2
2
2
0
2
0
( ) ( )
2
( )
( ) ( )
2
( )
2
( )
j ft
T
H f S f e df
S
N
N H f df
H f df S f df
N H f df
S f df
N

 



  
 





  
 
  

 


 
2 2
( ) ( )
S f df s t dt E Energyof thesignal
 
 
  
 
Parsaval’s theorem
391
Hence
This is the maximum SNR
It depends on signal energy E and noise PSD
Does not depend on signal waveform
When the signal is matched, it means that the transfer
function achieves the equality condition, i.e.,
This also means that the optimum choice of H(f) is
2
0 0
2 2
( )
T
E
S
S f df
N N
N


   

 
 
0 max
0
2
max
T
E
S
N
N
 
    
 
 
2
0 ( ) ( ) ( ) j fT
H f H f kS f e 


 
392

This implies that
If the signal is real, then
Thus
 
 
 
 
   
 
1 2 2
2 ( )
2 ( )
( ) ( ) ( )
( )
( )
j fT j ft
j f T t
j f T t
h t F H f kS f e e df
kS f e df
kS f e df
ks T t ks T t
 



  


   


  

 
  
 
 
   
   
 
ks T t ks T t

 
  
 
 
 
( )
h t ks T t
 
393
Similarly, bank of Matched filters is used to receive
several signals
The impulse response of the M matched filters are
given by
where sk(t) are the set of basis function
( ) ( ),
k k b
h t ks T t o t T
   
Selects
si(t)
with
the
max zi
(t)
r t
( )
t T

z T
0
( )
 ( )
s t
i
z T
1
( )
h t s T t
b
( ) ( )
 
0
z T
M1( )


h t s T t
b
( ) ( )
 
1
h t s T t
M b
( ) ( )
 
1
z t
0
( )
z t
1
( )
z t
M1( )
394
Summary of Matched Filters
A Matched filter is a detection filter that optimizes the
output SNR
r t
( )
t T

 ( )
s t
i
z t
( ) z T
( )
h t s T t
b
( ) ( )
 
+
s t
( )
n t
( )
Selects
si(t)
with
the
max zi(t)
r t
( )
t T

z T
0
( )
 ( )
s t
i
z T
1
( )
h t s T t
b
( ) ( )
 
0
z T
M1( )


h t s T t
b
( ) ( )
 
1
h t s T t
M b
( ) ( )
 
1
z t
0
( )
z t
1
( )
z t
M1( )
395
Correlator vs. Matched Filter - 1
The functions of the correlator and matched filter
are the same
Comparing (a) and (b) have
From (a)
x
r t
( )
s t
( )
t T

 ( )
s t
i
()

z dt
T
0
z t
( )
r t
( )
t T

 ( )
s t
i
z t
( ) z T
( )
h t s T t
b
( ) ( )
 
+
s t
( )
n t
( )
(a)
(b)
0
( ) ( ) ( )
T
z t r t s t dt
 
0
( ) ( ) ( ) ( )
T
z t z T s r d
t T   
  

396

From (b):
But
At sampling instant t = T, we have
This is the same result obtained in (a)
Hence
z T z T
( ) '( )

h t s T t h t s T t s T t
( ) ( ) ( ) [ ( )] ( )
         
  
   
z
z t r s T t d
t
'( ) ( ) ( )
  
0
z t r t h t r h t d r h t d
t
'( ) ( ) ( ) ( ) ( ) ( ) ( )
   
z  
z


     
0
' '
0
0
( ) ( ) ( ) ( )
( ) ( )
T
t T
T
T
z t z r s T T d
r s d
  
  

   


Correlator vs. Matched Filter - 2
397
Examples
Example Signal to Noise Ratio
Example Correlator Output
Example Matched Filter
398
#Generalized One Dimensional Signals - 1
One Dimensional Signal Constellation
-A +A
so s1
M=2
0
-A +A
s1 s2
M=4
0
-3A
so
+3A
s3
-5A -3A
s1 s2
M=8
0
-7A
so
-A
s3
+3A +5A
s5 s6
+A
s4
+7A
s7
E A A A
avg   
2 2
2
2
E A A A A A
avg     
9 9
4
5
2 2 2 2
2
E A A A A A A A A A
avg         
49 25 9 9 25 49
8
21
2 2 2 2 2 2 2 2
2
399
Binary Baseband Orthogonal Signals
 Binary Antipodal Signals
 Binary Orthogonal Signals
2-Dimensional Signal Constellation
An example:
+A
+A
s1
s0
1
E A A A
avg   
2 2
2
2
2
-A +A
so
s1
0
1
E A A A
avg   
2 2
2
2
1( )
t
 
1 2 0
( ) ( )
t t dt
o
T

z
2
( )
t
1
T
T
1
T
 1
T
T t
2
T
2
T
400

 Generalization to M-ary Orthogonal Signals
TimeDomain Signal Space
s t A t s A
s t A t s A
s t A t s A
s t A t s A
0 1 0
1 2 1
2 3 2
3 4 3
0 0 0
0 0 0
0 0 0
0 0 0
( ) ( ) ( , , , )
( ) ( ) ( , , , )
( ) ( ) ( , , , )
( ) ( ) ( , , , )
 
 
 
 




where {1(t), 2(t), 3(t) 4(t)}
are a set of orthonormal basis
functions
TimeDomain Signal Space
s t A t s A
s t A t s A
s t A t s A
s t A t s A
s t A t s A
s t A t s
0 1 0
1 2 1
2 3 2
3 4 3
4 5 4
5 6 5
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
( ) ( ) ( , , , , , , , )
( ) ( ) ( , , , , , , , )
( ) ( ) ( , , , , , , , )
( ) ( ) ( , , , , , , , )
( ) ( ) ( , , , , , , , )
( ) ( ) (
 
 
 
 
 
 





 0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
6 7 6
7 8 7
, , , , , , , )
( ) ( ) ( , , , , , , , )
( ) ( ) ( , , , , , , , )
A
s t A t s A
s t A t s A
 
 


M=8
M=4
where {1(t), 2(t), 3(t)
4(t), 5(t), 6(t), 7(t)
8(t)} are a set of
orthonormal basis
functions
401
where {1(t), 2(t), 3(t) … M-1(t)} are a set of orthonormal basis functions
0 1 0
1 2 1
2 3 2
3 4 3
1 1
Time Domain Signal Space
( ) ( ) ( , 0, 0, 0, 0, 0, 0, 0)
( ) ( ) (0, , 0, 0, 0, 0, 0, 0)
( ) ( ) (0, 0, , 0, 0, 0, 0, 0)
( ) ( ) (0, 0, 0, , 0, 0, 0, 0)
( ) ( ) (0, 0, 0, 0, 0, 0, 0, , )
M M M
s t A t s A
s t A t s A
s t A t s A
s t A t s A
s t A t s A





 
 
 
 
 
 
 

General M
(M is a power of 2)
402
Constellation is a method of representing the symbol
states of modulated bandpass signals in terms of their
amplitude and phase
That is, a geometric representation of signals
Three common types of binary signals:
Antipodal
 Two signals are said to be antipodal if one signal is the
negative of the other  s1(t) = - s0(t)
 Signal have equal energy with signal point on the real line
so s1
0
1 E E E E
avg   
2
E
 E
Most Common Signal Constellations - 1
403
On-Off Keying
Are one dimensional signals either ON or OFF
with signaling points falling on the real line
With OOK, there are just 2 symbol states to map
onto the constellation space
–a(t) = 0 (no carrier amplitude, giving a point at
the origin)
–a(t) = A cosct (giving a point on the positive
horizontal axis at a distance A from the origin)
so
s1
0
1 E E E
avg   
0
2 2
E
0
404

Orthogonal
Requires a 2 dimensional geometric
representation since there are 2 linearly
independent functions s1(t) and s0(t)
Typically, the horizontal axis is taken as a
reference for symbols that are In-phase with the
carrier cosct, and the vertical axis represents
the Quadrature carrier component, sinct
so
s1
0
E E E E
avg   
2
E
E
405
Maximum Likelihood Receiver
(derivation will be given in class)
406
Probability of Error
Probability of Error for Binary Signals - 1
Unipolar Baseband Signaling
For s1(t):
   
 
   
 
 
   
1 1 1 0
0 0
1 1 1 0
0 0
2
1
0
2
( ) ( ) | ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) 0 0 0
T T
T T
T
a T E z T s t E r s d r s d
E s n s d s n s d
E s d
A T
     
       
 
  
 
   
 
 
   
  

1
0
( ) , 0 , 1
( ) 0, 0 , 0
s t A t T for binary
s t t T for binary
  
  
A
t
T 3T 5T
1 1
1
0
0
x
r t
( )
s t s t
1 0
( ) ( )

t T

 ( )
s t
i
()

z dt
T
0
z t
( )
z T
( )
z T o
( )

r(t) = s(t) + n(t)
408

For s0(t):
 
 
 
   
 
 
 
 
0 0
1 0
0 0
0 1 0 0
0 0
2
0 1 0
0 0
( ) ( ) | ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) 0 ( ) 0
0
T T
T T
T T
a T E z T s t
E r s d r s d
E s n s d s n s d
E s s d s d
     
       
    

 
 
   
 
 
   
   

2
1 0
0
2 2
a a A T


  
409
Also:
 2 2
1 0
0
( ) ( )
T
d
E s t s t dt A T
  

0
2
d
E
P Q
B N
 
  
 
2
0
0 0
2
A T
P Q Q
B N N
  
 
   
 
   
0
b
E
P Q
B N
 
  
 
410
Bipolar Signaling (antipodal)
1
0
( ) , 0 , 1
( ) , 0 , 0
  
   
A
t
T
3T 5T
1 1
1
0
0
-A
x
r t
( )
s t
0
( )
t T

 ( )
s t
i
x

s t
1
( )
()

z dt
T
0
()

z dt
T
0
-
+
z t
0
( )
z t
1
( )
z T
( )
z T o
( )

1 0 1 0 0
( ) ( ) ( ) 0 0
z t z t z t a a 
      
E A A dt A T
d
T
  
z  
2
0
2
2
P Q
E
N
Q A T
N
Q
E
N
b
d
o o
b
o

F
HG I
KJ 
F
HG I
KJ 
F
HG I
KJ
2
4
2
2
2
411
Unipolar (orthogonal) Bipolar (antipodal)
P Q
E
N
b
b
o

F
HG I
KJ
2
P Q
E
N
b
b
o

F
HG I
KJ
 Bipolar signals require a
factor of 2 increase in
energy compared to
Unipolar
 Since 10log102 = 3 dB, we
say that bipolar signaling
offers a 3 dB better
performance than Unipolar 0 2 4 6 8 10 12 14 16 18 20
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
Eb/No (dB)
P
robability
of
Bit
Error
Othogonal
Antipodal
Q
E
N
b
o
F
HG I
KJ
Q
E
N
b
o
2
F
HG I
KJ 3-dB
412

Comparing BER Performance
For the same received signal to noise ratio, antipodal
provides lower bit error rate than orthogonal
0 2 4 6 8 10 12 14 16 18 20
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
Eb/No (dB)
Probability
of
Bit
Error
Othogonal
Antipodal
  
78 10 4
.
 

9 2 10
2
.
 For Eb/No = 10 dB
 Pb,orthogonal = 9.2x10-2
 Pb, antipodal = 7.8x10-4
413
Examples
Example
Probability of Error
Example
Probability of Error
414
Digital Baseband
Communication System
Baseband Communication Systems
A baseband signal x(t) with bandwidth B is a signal for
which X(f) is non-zero for |f|  B and for which X(f) = 0 for
|f| > B (PSD concentrated near DC)
A baseband communication system transmits
information using a baseband signal
Here the transmitter is simply a line coder (w/pulse
shaping function) that maps the sequence of bits an onto a
line code signal s(t)
A/D
Converter
Line
Coder
Channel
Analog Input To Receiver
an s t
( )
Transmiter
B
-B 0
X(f)
f
416

Problems with Baseband Communication - 1
Most channels require that the baseband signal be
shifted to a higher frequency
Since antenna size is inversely proportional to the
center frequency fc, this is difficult to realize
 Problems:
 Higher frequencies allow for the use of smaller antennas -
size versus 
 For speech signal f = 3 kHz   = 105
 Antenna size w/o modulation  = 105 m = 60 miles -
practically unrealizable
 This is evident that efficient antenna of realistic physical
size is needed for radio communication system
f
c


417
Most channels are shared by several transmitters at
the same time
Shifting each user to different freq the channel can
be divided into freq slots
Frequency Division Multiple Access (FDMA)
Thus we must look at the process of shifting a
baseband signal to higher frequency
This process is called Carrier Wave Modulation
418
Solution is to use Bandpass Communication Systems
 A bandpass signal has non-negligible spectrum only about
some carrier frequency fc >> 0
 i.e., x(t) with bandwidth B is a signal for which X(f) is non-
zero at some region about  fc and for which X(f) = 0
elsewhere
 Note: the bandwidth of a bandpass signal is the range of
positive frequencies for which the spectrum is non-zero
 Usually, the bandwidth of bandpass signal is twice the
bandwidth of the baseband signal used to create it
 Effective transmission of baseband information signal
usually requires the use of a bandpass signal
f
fc
X(f)
0
-fc
B
419
In a bandpass digital communication system, the
bit stream an is first converted to a baseband line code
m(t) by a line coder and is then converted to a
bandpass signal s(t) by a modulator
Baseband signals m(t) may be transformed into
bandpass signals s(t) through the process of
modulation
A/D
Converter
Line
Coder
Channel
Analog
Input
To
Receiver
an s t
( )
Transmiter
Modulator
m t
( )
420

We need some additional analytical tools to handle
bandpass signals
3 Major ways of Representing Bandpass Signals
Magnitude and Phase (M&P) Representation
In-phase and Quadrature (I&Q) Representation
Complex Envelope Representation
421
‡ Representation of Bandpass Signals
1. Magnitude and Phase (M & P)
Any bandpass signal can be represented as:
 R(t)  0 is real valued signal representing the magnitude
 (t) is a real valued signal representing the phase
This representation is easy to interpret physically, but
often is not mathematically convenient
In this form, modulated signal can represent information
through changing three parameters of the signal namely:
 Amplitude R(t): as in Amplitude Shift Keying (ASK)
 Phase (t): as in Phase Shift Keying (PSK)
 Frequency d(t)/dt: Frequency Shift Keying (FSK)
 
( ) ( )cos ( )
c
s t R t t t
 
 
2. In-phase and Quadrature (I & Q) Representation
Any bandpass signal can also be represented as
 x(t) is a real-valued signal called In-phase (I)
 y(t) is a real-valued signal called Quadrature (Q)
This is often a convenient form which
 Emphasizes the fact that two signals may be
transmitted within the same bandwidth
 Closely parallels the physical implementation of
the Tx/Rx
( ) ( )cos( ) ( )sin( )
c c
s t x t t y t t
 
 
Relationship Between M & P and I & Q Forms:
 To transform from M&P to I&Q
x(t) = R(t)cos(t), y(t) = R(t)sin(t)
To transform from I&Q to M&P
I and Q portions of the signal are orthogonal
Look at the correlation between I & Q portions
2 2
( ) ( ) ( )
R t x t y t
  ( ) tan
( )
( )
t
y t
x t
 L
NM O
QP
1
   
 
   
 
0
0
0
( )cos ( )sin
1
( ) ( ) sin sin
2
1
( ) ( ) 0
sin sin 2
0
2
T
c c
T
c c c c
T
c
x t t y t tdt
x t y t dt
t t t t
x t y t dt
t
 
   





 
 
 


3. Complex Envelope (CE) Representation
Any bandpass signal can also be represented as
where g(t) = complex envelope - complex-valued signal
S(t) is convenient in many instances for analysis. Why?
 Compact
 Easy to manipulate without recourse to trig. identities
 Relationship: Complex Envelope and M&P Forms
 To transform from CE to M&P:
R(t) = |g(t)|, (t) = g(t)
 To transform from M&P to CE:
g(t) = R(t)ej(t)
 
( ) Re ( )exp( )
c
s t g t j t


 Relationship: CE and I & Q Forms
 To transform from CE to I&Q:
x(t) = Re[g(t)], y(t) = Im[g(t)]
s(t) = Re[g(t)ejt] = Re[(x(t)+jy(t)).(cosct+jsinct)]
= x(t)cosct - y(t)sinct
 Relationship between Spectral Representations
 Assume that
 Fourier Transform (Deterministic Signals):
( ) Re ( )
j t
c
s t g t e

 
  
S f G f fc G f fc
( ) ( ) ( )
     
1
2
 Power Spectral Density (Random Signals):
 Relationship: Power and Envelope of Bandpass
 Power of bandpass signal is one half of power in
complex envelope:
G f G f f G f f
s g c g c
( ) ( ) ( )
    
1
4
2
(0)
1 1 1
( ) (0)
2 2 2
s s
g g
G R
g t R G

  
Bandpass Modulation & Demodulation - 1
Format Multiplex
Channel
Encoder
Source
Encoder
Spread
Format Demultiplex
Channel
Decoder
Source
Decoder
Despread
Performance
Measure Bits or
Symbol
To other
destinations
From other
sources
Digital
input
Digital
output
Source
bits
Source
bits
Channel
bits
Carrier and symbol
synchronization
Channel
bits

mi
l q
mi
l q

Pe
Multiple
Access
Waveforms
Multiple
Access
Modulate
Demodulate
&
Detect
428

Bandpass Modulation & Demodulation - 1
 Bandpass Modulation shifts the spectrum of a baseband
signal so that it becomes a bandpass signal
 Why Modulate? (a review)
 signals propagate well through the atmosphere
 allows many signals w/different carrier freqs to share the
spectrum
 is used to place signals at desired freq band for signal
processing
 Info signal must conform to limitation of it’s channel
 is used to map digital data sequence into waveform
Message
source
Signal
transmission
encoder
Signal
transmission
decoder
Decoder
Channel
Modulator
m t
( ) si
s t
i( )
Carrier Wave
x t
( ) x 
m
Transmitter Receiver
429
Aspects of Conversion
430
r = bits/signal = log2( L )
L = number of levels (signal elements)
N = bps
S = signals/sec (baud)
c = 1 for broadband (WAN digital-to-analog)
c = ½ for baseband (LAN digital-to-digital)
cN
S
r

Digital Modulation
431
Digital Modulation Schemes
Basic Digital Modulation Schemes:
Amplitude Shift Keying (ASK)  not commonly used
Frequency Shift Keying (FSK)  very useful
Phase Shift Keying (PSK)  very useful
For Binary signals (M = 2), we obtain BASK, BPSK,
BFSK, BAPK
For M > 2, many variations of the above techniques
exit usually classified as M-ary Modulation/detection,
e.g., MPSK
432

Most Common Digital Nodulation
433
MOdulation and DEModulation - 1
MODEM
NONCOHERENT
COHERENT
BINARY M-ary HYBRID BINARY M-ary HYBRID
ASK
(OOK)
FSK
(MSK)
PSK
ASK
FSK
PSK
(QPSK,
OQPSK)
APK(QAM)
ASK
FSK
DPSK
CPM
ASK
(OOK)
FSK
DPSK
CPM
(Phase info
required)
(No Phase info
required)
434
MOdulation and DEModulation - 1
Analysis or Method of Approach:
Modulation Process
 Mathematical Signal Representation
Power Spectral Density of the modulated signal
 Bandwidth of the System
Detection Processes
Performance of the system
 Error Probability
435
Amplitude Shift Keying

In amplitude shift keying, the amplitude of the carrier
signal is varied to create signal elements.
Both frequency and phase remain constant while the
amplitude changes.
Amplitude Shift Keying - 1
Modulation Process
 Also called ON-OFF Keying (OOK))
 In ASK, amplitude of carrier is switched between 2 (or more)
levels according to the digital data
 “1s” & “0s” are represented by two amplitude levels A1 & A0
x
m t
( )
A t
o
cos( )

s t
( )
Baseband Data Modulated bandpass Signal
OOK Modulator
Product modulator or
ON-OFF switch
0 T 3T
438
Implementation of binary ASK
 Analytical Expression:
where Ai = peak amplitude
 Hence,
0
cos( ), 0 1
( )
0, 0 0
i
A t t T binary
s t
t T binary
  

 
 

2
0 0 0
2
0
0
2
( ) cos( ) 2 cos( ) 2 cos( )
2 cos( )
cos( )
rms rms
E
T
s t A t A t A t
V
P t P
R
t
  


  
  

0
2
0 , 1
0 , 0
cos( ),
( )
0,
E
t T binary
T
t T binary
t
s t
  
 


 


440

 Generally, we can write
where
We may also write
This can be used to derive the transmitter for ASK:
2
0
( ) , 0,1,2, ..., 1
T
i i
E s t dt i M
  

0
2 ( )
0,1, 2,..., 1
( ) cos( ), 0 ,
i
E t
i i M
T
s t t t T
   
   
1 0 0 1
( ) ( )cos( ),
c t T binary
s t A m t t
   
 
0 0 0
( ) 0, t T binary
s t  

Ac
x
line coder
cos( )
c
t
m t
( )
Xn
441
Power Spectral Density (PSD)
From the given signal
The PSD can be found using
To evaluate this we must first find the PSD of the
complex envelope m(t)
Using the fact that m(t) is a unipolar NRZ line code
given by
( ) ( )cos
c c
s t A m t t


 
2
( ) ( ) ( )
2
s M c M c
A
G f G f f G f f
   
2, 1
( ) ( ),
0, 0
n n
n
for binary
m t a f t nT a
for binary



  
 

442
With and using the general expression for
PSD of a unipolar line code, we obtain
Note:
The spectrum of a digitally modulated signal depends
on the baseband data format used to represent the
digital data
 
 
 
 
2
2
2
2 1
4
2
1
2
2
2
2
( ) ( ) 1 ( )
( ) 1 ( )
( ) ( )
g c
c
A
c
T T
A
c
T T
A
c
G f F f f
TSa fT f
f TSa fT

 
 
 
 
 
2
c
A A

443
It can be seen that the bandwidth B of ASK modulated
signal is twice that occupied by the source baseband
stream
2
Tb f R
c b

f R
c b
 2
impulse
444

Bandwidth of ASK
 Bandwidth B, of ASK can be found from its power spectral
density
 B is twice that of unipolar NRZ line code used to create it, i.e.,
 This is the null-to-null bandwidth of ASK
 If raised cosine rolloff pulse shaping is used, then
 Spectral efficiency of ASK is half that of a baseband unipolar
NRZ line code
 This is because the quadrature component is wasted
 95% energy bandwidth
B r R W r R
b b
    
( ) ( )
1 1
2
1
B
T
R
b
b
 
3 3
B R
T
b
b
 
2 2
445
1) Low Pass Filter Receiver
Coherent detection requires the phase information
A coherent detector mixes the incoming signal with a
locally generated carrier reference
Multiplying r(t) by the receiver LO (say cos(ct)) yields a
signal with a baseband component plus a component at
2fc
x LPF
r t
( )
cos( )
t
t T

 ( )
s t
i
z T
( )
Receivers - Demodulators & Detectors
Coherent Receiver - 1
446
Passing this signal through a low pass filter
eliminates the high frequency component
An integrator can be used in place of the LPF
The output of the LPF is sampled once per bit
period
This sample z(T) is applied to a decision rule
–z(T) is called the decision statistic
447
2) Matched Filter Receiver
MF receivers are very common approach in
signal detection in most bandpass data
modems
r t
( )
t T

 ( )
s t
i
z t
( ) z T
( )
h t s T t
b
( ) ( )
 
448

3) Correlator Receiver
4) Quasi-coherent Square-law Receiver
x
r t
( )
s t s t
1 0
( ) ( )

t T

 ( )
s t
i
()

z dt
T
0
z t
( )
z T
( )
r t
( )
t T

 ( )
s t
i
()

z dt
T
0
z t
( )
z T
( )
( )2
449
Does not require a phase reference info at the receiver
If we do not know the phase and frequency of the carrier,
we can use a non-coherent technique to recover signal
1) Envelope Detector:
Non-Coherent Receiver - 1
LPF
r t
( )
t T

 ( )
s t
i
z t
( )
z T
( )
Rectifier
BPF
@ fo
Envelope Detector
 The simplest implementation
of an envelope detector
comprises a diode rectifier
and smoothing filter
450
2) Square-law Detector:
 If quadrature versions of the modulated carrier signal are
available then we may use the following receiver
 Noncoherent reception of OOK is popular in fiber optics
r t
( )
t T

 ( )
s t
i
z t
( ) z T
( )
( )2
()
( / )
( / )

z 

dt
T n
T n
1 2
1 2
I
Q
Non-Coherent Receiver - 2
451
BASK effectively uses unipolar signal source and the
performance depends on whether coherent or non-
coherent detection is used
Error analysis is similar for both cases
For both cases,
 s(t) is exactly the same as in both cases
 For coherent detection n(t) is Gaussian, however for
noncoherent detection n(t) is no longer Gaussian due to
the squaring operation
 Because of this squaring, the optimal threshold is not
necessarily halfway between the 2 possible values of s(t)
Derivation given in class
Probability of Error (Bit Error Rate)
( ) ( ) ( )
r t s t n t
 
452

Derivation
Probability of Error (Bit Error Rate) - ASK
453
Example 39: ASK
A binary ASK communication system employs
rectangular pulses of duration Tb and amplitude A to
transmit digital information at a rate R = 105 bps. If the
PSD of the AWGN is N0/2, where N0 = 10-2 W/Hz,
determine the value of A that is required to achieve the
probability of error of PB = 10-6
454
Frequency Shift Keying
Frequency Shift Keying (FSK) - 1
In frequency shift keying (FSK), the frequency of the
carrier signal is varied to represent data.
The frequency of the modulated signal is constant for
the duration of one signal element, but changes for the
next signal element if the data element changes.
Both peak amplitude and phase remain constant for all
signal elements.
456

457
Modulation Process:
 The instantaneous carrier freq is switched b/w 2 or
more levels according to the baseband digital data
data bits select a carrier at one or more freqs
the data is encoded in the freq
 FSK conveys the data using distinct carrier freqs to
represent symbol states
 Important property = amplitude of the modulated
wave is constant
458
Analytical Expression
Can also be expressed as
where
 
2
( ) cos , 0,1, , 1
i
E
T
s t t i M
i
 
   

0
( ) ( ) )
t
i d
t t m d
    

 
  
 
0
( ) ( )
i i d
d
f t f f m t
dt

  
 
0
2
( ) cos 2 2 , 0,1, , 1
i
E
s t f t i ft i M
T
 
    

1,
i i
i o
f f f
f f i f

  
  
Analog form
freq offset
459
 Generally, MFSK may be used to transmit k = log2M bps
waveforms
 f determines the degree to which we can discriminate among
M possible signals
 As a measure of similarity (or dissimilarity) between a pair of
signal waveforms, a correlation coefficient ij, is used
   
   
1
2
1
1 1
( ) ( )
cos 2 2 cos 2 2
cos cos 4 2 ( )
2 ( )
sin 2 ( )
2 ( )
T
i j
o
T
o o
o
T T
o
o o
E
s
E
s
E T
s
T T
s t s t dt
ij
f t i ft f t j ft dt
dt f t i j ft dt
i j ft
i j fT
i j fT

   
 



 
    

    
 
 
 

 
0 since fo >> 1/T
460

 Note:
 ij, is orthogonal when f is a multiple of 1/2T
 Minimum of ij, = - 0.217 @ f = 0.715/T



ij
i j fT
i j fT



sin ( )
( )
2
2


-0.217
0 715
.
Tb
1
Tb
3
2Tb
2
Tb
1
2Tb
f
1

ij
461
Binary FSK - 1
2 different freqs, f1 and f2 = f1 + f are used to transmit
binary data
Data is encoded in the freqs
That is, m(t) is used to select between 2 freqs
f1 is the mark freq, and f2 is the space freq
s t A t
o c
( ) cos( )
 
 
1 1 s t A t
c
1 2 2
( ) cos( )
 
 
0 0 0
1 1 1
2
( ) cos(2 ), 0
2
( ) cos(2 ), 0
E
s t f t t T
T
E
s t f t t T
T
 
 
   
   
f1
f2
462
Binary Orthogonal Phase FSK
When 0 an 1 are chosen so that 1(t) and 2(t) are
orthogonal, i.e.,
form a set of k = 2 basis orthonormal basis functions
s1
so
0
A
A
1( )
t
2( )
t
1
0
1
1 1
2 2 2
2
( ) cos( )
2
( ) cos( )
E
t t
T
E
t t
T
  
  
 
 
1 2
( ) ( ) 0
t t
 

 

Binary FSK - 2
463
 For NRZ Pulse Shape:
 We need to look at two cases
1.Continuous Phase: 1 = 2
2.Non-continuous Phase: 1  2
   
 
   
 
1 2 1 2
1 2 1 2
sin 2 sin
2sin
sin 4 2 sin 2
2sin 2
b b b
ij
b
b b b
b
T T T
T
fT fT fT
T
      


      
 
        


        


1 2
1 1 2 2
( ) ( )
2
cos( )cos( )
t t dt
ij
E
t t dt
T
  
   




 
  

Binary FSK - 3
464

Discontinuous Phase FSK
Phase discontinuities occur at symbol boundaries
 
1 2

Phase Discontinuities
1 1 1 1
0
0
Binary FSK - 4
465
Requiring 2 oscillators adds to the system
complexity and cost
Because there are 2 different fc it is difficult to use
complex envelope notation
This makes analysis difficult
Discontinuities in phase of s(t) at switching instants
result in undesirable spectral characteristics
Corresponds to high sidelobe levels which could
cause adjacent channel interference
Discontinuous-phase FSK is not used much in
practice
Binary FSK - 5
466
Continuous Phase FSK
No Phase Discontinuities
1 1 1 1
0
0
 
0 1

Frequency Modulator @ fo
m(t) BFSK
ON-OFF Level
Encoder
m(t) BFSK
X
x
+
 
1
2
1
2
( ) cos
t f t
Tb

 
2
2
2
2
( ) cos
t f t
Tb

Binary FSK - 6
467
Implementation of BFSK
Binary FSK - 7
468

A continuous-phase FSK (CPFSK) signal is
represented by:
Df is the frequency deviation constant
m(t) is a digital line code
Usually polar, either with or without pulse shaping
CPFSK is an FM signal with digital line code modulating
signal
CPFSK is much more common than discontinuous phase
FSK
 Unless otherwise specified, FSK will usually mean CPFSK
 
( ) cos ( )
cos( ( ) )
c c
t
c o f
i
s t A t
A t D m d

  


  
Binary FSK - 8
469
Peak Frequency Deviation
Thus:
Modulation Index
 minimum value of h for which the 2 possible signals do not
interfere with one another is h = 0.5
 CPFSK with h = 0.5 is called minimum shift keying (MSK)
 GSM uses MSK with Gaussian pulse shapes (GMSK)
1 ,
c
f f f
  2 ,
c
f f f
  1 2 2
f f f
  
2
2
f
h fT
R

  
2
f
D
f

 
Binary FSK - 9
470
Other FSK Modulation Methods
Vector or Quadrature
 FSK requires the generation of 2 symbols, one at a frequency
(c + 1) and one at a frequency (c – 1)
 To generate a freq. shift of  1 at modulator output , the I and
Q inputs need to be fed with  cos1 and 1sin respectively
 This approach is now frequently used to generate some of the
more elaborate filtered CPFSK formats in cellular handsets
See Fig. 4.24
Binary FSK - 10
471
Representation of Continuous Phase FSK
Magnitude and Phase
Complex Envelope Notation
Quadrature Notation
Alternate Representation for CPFSK
Because frequency is the time rate of change of the
phase, we can represent a bandpass signal as
( ) cos( ( ) )
t
c f
x t A D m d
 

 
g t A jD m d
c f
t
( ) exp( ( ) )
 z  
( )
( ) ( )
c
t
f
R t A
t D m d
  


 
( ) sin( ( ) )
t
c f
y t A D m d
 

 
Binary FSK - 11
472

Hence for CPFSK
If m(t) is polar NRZ (and A = 1)
( ) ( )
( ) ( )cos ( )cos 2
2
c c
d t d t
s t R t t R t f t
dt dt
 
 

   
   
   
   
   
   
   
( )
( ) ( )
c f
d t
R t A and D m t
dt

 
( )
( ) cos 2 cos(2 )
2
c c c i
D m t
f
s t A f t A f t
dt
 

 
 
 
 
  
 
 
 
1
2
, when ( ) 1
2
, when ( ) 1
2
f
c
i
f
c
D
f f m t
f
D
f f m t



  


 
   


Binary FSK - 12
473
PSD of CPFSK
Because complex envelope g(t) is a nonlinear function of
m(t), an exact expression for the PSD is difficult to obtain
A good approximation for s(t) can be found by considering
FSK to be the sum of 2 OOK signals
1 ( )
( ) cos(2 ( ) )
2
1 ( )
cos(2 ( ) )
2
c c
c c
m t
s t A f f t
m t
A f f t



 
  
 
 

 
  
 
 
 This approximation can be used to find the PSD
 Result is that the null-to-null bandwidth is
B f
r
Tb
 

2 1
2

e j
Binary FSK - 13
474
Clearly, the overall bandwidth occupied by the FSK
signal depends f
An FSK system using continuous phase transitions will
have much lower side-lobe energy than the
discontinuous case
Binary FSK - 14
475
 Sunde's FSK
 Sunde's FSK arises when the spacing between the 2
symbol frequencies is made exactly equal to the symbol
rate
 The spectrum uniquely contains 2 discrete spectral lines
at the two symbol frequencies in addition to a broad
spectral spread
 These spectral lines may be used in coherent FSK
detector as the source of carrier references, often
extracted using a PLL
Minimum Shift Keying (MSK)
MSK employs symbol spacing of one half the
symbol rate
Binary FSK - 15
476

 It produces a smooth spectrum with narrow main lobe
and reduced side-lobe energy
 This narrow symbol spacing means that MSK is spectrally
efficient (more than BASK and BPSK, and about QPSK)
 The price to be paid for this excellent performance is
more complexity in the generation and detection process
compared with Sunde's FSK
 Bandwidth is minimized when h = 0.5 (i.e. for MSK)
3
2
b
B r R
 
 
 
 
Binary FSK - 16
477
Detection of FSK:
Coherent
 Coherent detection of FSK is similar to that for ASK but in this
case there are 2 detectors tuned to the 2 carrier frequencies
 Drawback of using Sunde's FSK
 The bandwidth of the FSK signal is approximately 1.5 to 2
times that of an optimally filtered ASK or PSK binary signal
Binary FSK - 17
 Recovery of fc in receiver
is made simple if the
frequency spacing
between symbols is made
equal to the symbol rate
(Sunde’s FSK)
478
 The following configurations can be used for detecting FSK
signal
x
r t
( )
s t s t
1 0
( ) ( )

t T

 ( )
s t
i
()

z dt
T
0
z t
( )
z T
( )
x
Threshold
Detector
r t
( ) s t
0
( )
t T

z T
0
( )
 ( )
s t
i
x
z T
1
( )
s t
1
( )
()

z dt
T
0
()

z dt
T
0
z t
0
( )
z t
1
( )
h(t) = s(Tb-t)
t T

y t
( ) z T
( )
h(t) = s(Tb-t)
r t
( )
t T

 ( )
s t
i
y t
( ) z T
( )
+
Binary FSK - 18
479
Noncoherent
BPF/Envelope Detector:
 Pass the signal through 2 BPF tuned to the 2 frequencies
and detect which has the larger output averaged over a Ts

r(t)
BPF
Tuned @ f1
BPF
Tuned @ f2
Envelope
Detector
Envelope
Detector
Sampler
Time
Sync
Binary FSK - 19
480

Phase Locked Loop (PLL)
Zero-Crossing:
 One simple digital method involves counting the zero-
crossings of the carrier during a symbol and hence
directly estimating the frequency on a symbol-by-symbol
basis
Quadrature Receiver
Alternate BFSK demodulator is shown in Fig. 4.16
Binary FSK - 20
481
Probability of Error Performance for FSK
 (see derivation in class handout)
Coherent
Noncoherent
 Coherent orthogonal BFSK performance is identical to coherent ASK
 Eb/N0 penalty of noncoh. detection is only about 1 dB lower
 Note:noncoherent FSK performance is not nearly as bad as ASK
P Q
E
N
b
b
o

F
HG I
KJ
P
E
N
b
b
o
 
F
H
I
K
1
2 2
exp
Binary FSK - 21
482
Derivation
Probability of Error (Bit Error Rate) - FSK
483
Example 40: FSK
If a system's main performance criterion is bit error
probability, which of the following two modulation
schemes would be selected for an AWGN channel?
Show computations.
Binary noncoherent orthogonal FSK with Eb/NO = 13 dB
Binary coherent PSK with Eb/NO = 8 dB
484

Phase Shift Keying
Phase Shift Keying (PSK) - 1
In PSK, the phase of the carrier signal is switched
between 2 or more phases in response to the
baseband digital data
The info is contained in the instantaneous phase of
the carrier
For binary PSK, phase states of 0o and 180o are used
Waveform:
486
487
Analytical expression can be written as
where
g(t) = transmitting signal pulse shape
A = amplitude of the signal
 = carrier phase
Range of the carrier phase can be determined using
For a rectangular pulse, we obtain
 
( ) ( )cos , 0 , 1,2,...,
i o i
s t Ag t t t T i M
 
    
2 ( 1) 2
i i
i i
or
M M
 
 

 
2
( ) , 0 ; and assume
g t t T A E
T
   
488

We can now write the analytical expression as
 the carrier phase changes abruptly at the beginning of each
signal interval while the amplitude remains constant
 
 
0
2 1
2
( ) cos , 0 , 1,2,...,



    
s
i
i
E
M
T
s t t t T i M
Constant envelope
carrier phase changes abruptly at
the beginning of each signal interval
t
4T
3T
2T
T
0
180-phase
shift
0-phase
shift
-90-phase
shift
489
Also can be written as
For M-ary phase modulation M = 2k, where k is the # of
info bits per transmitted symbol
In an M-ary system, one of M  2 possible symbols, s1(t),
…, sm(t), is transmitted during each Ts-second signaling
interval
The mapping or assignment of k info bits into M = 2k
possible phases may be done in many ways, e.g. for M = 4
 
 
2 1
2
2 ( 1) 2 ( 1)
2
( ) cos
cos cos sin sin
c
c c
i
E
M
T
i i
E
M M
T
s t t
i
t t

 

 

 
 
 
 
 
490
It is also possible to transmit data encoded as the
phase change (phase difference) between
consecutive symbols
This technique is known as Differential PSK (DPSK)
There is no non-coherent detection equivalent for PSK
Q
I
Q
I
01
00
10
11
10
01
11
00




0
2
3
2
, , ,
3 5 7
, , ,
4 4 4 4
   
 
491
M_ary Constellations
M k MPSK
BPSK
QPSK
PSK
PSK



2
2
4
8 8
16 16
 
E
E
E
E














00
10
11
01
000
001
011
010
110 100
101
111
M=8
M=4
E








000
001
011
010
110
100
101
111
M=8
E




00
01
11
M=4
10
492

Binary Phase Shift Keying (BPSK) - 1
Is also called Phase Reversal Keying (PRK)
For BPSK, M = 2 and o = 0, 1 = 
 i.e.,2 carrier phases at o= 0 and 1 = rad are used to transmit data
There is a 1800 ( radian) phase shift
the two phases are separated by 180o
 Thus, binary phase modulated signal may be viewed as 2
quadrature carrier with amplitude depending on transmitted
phase of each signal
 
0
2
( ) cos , 0

 s
c
E
T
s t t for binary
 
 
1
2
2
( ) cos
cos 1
 

 
 
s
s
c
c
E
T
E
T
s t t
t for binary
493
In your text, BPSK modulated signal is also written as
where m(t) is the message waveform
Representation of BPSK
The complex envelope of an OOK signal is:
Complex envelope is entirely real
Complex envelope is equivalent to polar NRZ signaling
Imaginary portion of corresponds to Q component
 
2
( ) ( ) cos 2 
 
s
c c
E
T
s t m t f t
( ) Re ( ) c
j t
v t g t e

 
   where g t
binary
binary
( )
,
,


R
S
T
1 1
1 0
494
The magnitude and phase of an OOK signal are:
The in-phase and quadrature components are:
where y(t) = 0 and no Q component
( ) 1, constant envelope
0, binary1
( )
, binary0
where R t
t




 

0
( ) ( )cos( ( ))
i i
s t R t t t
 
 
0 0
( ) ( )cos( ) ( )sin( )
s t x t t y t t
 
 
495
The entire quadrature component is not used
 This means that half the bandwidth is wasted
 BPSK requires twice as much bandwidth as the polar line
code used to create it
 If y(t) can be used, then loss in spectral efficiency is
recovered
I-component is just the polar NRZ signal
If the second BPSK is transmitted as the Q-component,
then we have QPSK (quadrature PSK) signal
1, binary1
( )
1, binary0
x t

 


496

PSK Generation (Modulators)
The simplest means of realizing BPSK is to switch the sign
of fc with data signal, causing a 0° or 180° phase shift
This method is not too good because of the difficulty in
implementing bandpass high frequency, high Q filters
Data stream may be pre-shaped at baseband prior to
modulation  Because the modulation process
is linear, the baseband filter
shape is imposed directly onto
the bandpass modulating signal
497
Transmitters for PSK (Modulators)
Product modulators
Switching modulators
Differential encoding
Receivers for PSK (Demodulators)
 Coherent Receiver
 Maximum Likelihood Detector
 Square Law Detector
 Correlator Detector or Costas Loop
 Noncoherent Receiver
 Differential PSK
498
Modulation/Transmitter Process
Product modulators
Switching modulators
Differential encoding
 
 
2 1
2
2 ( 1) 2 ( 1)
2
( ) cos
cos cos sin sin

 

 

 
 
 
 
 
s
s
c
c c
i
E
M
T
i i
E
M M
T
s t t
i
t t
499
Power Spectral Density of PSK
or
In fact, a BPSK signal can be viewed as an ASK signal with
the carrier amplitudes as +A and –A (rather than +A and 0)
P f E f f T
f f T
f f T
f f T
b c b
c b
c b
c b
a f a f
a f
a f
a f



F
HG I
KJ  
 
F
HG I
KJ
L
N
MM
O
Q
PP

2
2 2
sin sin
 
P f c f f T c f f T
A T
b
b
A T
b
b
C C
( ) sin ( ) . sin ( ) .
     
2
2
2 2
2
2
025 025
e in s e in s
2
2
Bandwidth R
T
 
 Bbpsk signal is identical to Bbask
assuming the same degree of
pulse shaping
500

Receiver for PSK (Demodulators)
 Coherent Receiver
1. Low Pass Filtering
2. Maximum Likelihood Detector (matched filter &
correlator)
3. Square Law Detector
4. Correlator Detector/Costas Loop
501
1) Low Pass Filtering
 Incoming data signal is mixed with a locally generated
carrier reference, and the difference component is selected
at the output
 Multiplying r(t) by receiver LO (say Accos(ct)) yields 2
components: a baseband component & a component at
2fc
 LPF eliminates the high frequency component (@ 2fc )
 The output of the LPF is sampled once per bit period
 The sampled value z(T) is applied to a decision rule
x LPF
r t
( )
cos( )
t
t T

 ( )
s t
i
z T
( )
502
2. Matched Filter
3. Correlator receiver
4. Quasi-coherent square-law receiver
r t
( )
t T

 ( )
s t
i
z t
( ) z T
( )
h t s T t
b
( ) ( )
 
x
r t
( )
s t s t
1 0
( ) ( )

t T

 ( )
s t
i
()

z dt
T
0
z t
( )
z T
( )
r t
( )
t T

 ( )
s t
i
()

z dt
T
0
z t
( )
z T
( )
( )2
503
 Noncoherent Receiver
 There is no “noncoherent PSK” because non-
coherency implies no phase information
 With no phase, there is no PSK
 Instead, we use a pseudo noncoherent technique
known as Differential PSK (DPSK)
504

Probability of Error for BPSK
(see derivation in class notes or see class handout)
Probability of Error (Bit Error Rate) - FSK
505
0
2 b
B
E
P Q
N
 
  
 
Examples
Example
 Suppose that the binary PSK is used in transmitting info over
AWGN channel with power spectral density of N0/2 = 10-10
watts/Hz and Eb=A2T/2. Determine the signal amplitude
required to achieve an error probability of 10-6 if the data rate is
(a) 10 kbps, (b) 1Mbps
Example
 Find the expected number of bit errors made in one day by the
following continuously operating coherent BPSK receiver. The
data rate is 5000 bits/s. The input digital waveforms are s1(t) =
Acos(w0t) and s2(t) = -Acos(w0t) where A = 1 mV and the single-
sided noise power spectral density is N0 = 10-11 W/Hz. Assume
that signal power and energy per bit are normalized relative to a
1 ohm resistive load.
506
Binary Differential PSK - 1
 Binary DPSK is regarded as the noncoherent version of BPSK
 Data is encoded in phase shift between successive symbols
rather than the actual value of the phase
 The Basic Idea:
 If ak = 0 then shift carrier phase by 180o
 If ak = 1 then no shift in carrier phase
 Differential BPSK looks just like BPSK except that the phase shift are in
a different place
1 0 0 1 1 1 0 0
D-BPSK
BPSK
ak
507
This requires differential encoding of the data
The idea is to come up with an encoding/decoding
scheme that will give the same decoded output
regardless of whether the received data is inverted
In DPSK, the carrier phase of the previous data bit can
be used as a reference
508

Differential Data Encoding:
 The 1-bit delay can be realized very simply using a clocked shift register
1 0 1 1 1 1 0 1
D-BPSK
dk
1 0 0 1 1 1 0 0
ak
Delay
Ts
dk
dk1
d
d a
d a
k
k k
k k



R
S
T


1
1
0
1
,
,
ak
ak dk dk
1
0 0 1
0 1 0
1 0 0
1 1 1
509
If ak = 1, leave dk unchanged w.r.t. the previous bit
If ak = 0, change dk w.r.t. the previous bit
The encoded sequence {dk} is used to phase-shift a
carrier with phase angle 0 and  representing symbols 1
and 0 respectively
This encoding process is efficient since it does not
introduce any extra data bits and hence does not
affect the throughput
510
Differential Data Decoding:
The differential decoding process is equally simple to
implement using a 2nd exclusive-nor gate and a 1-bit
delay
Delay
Ts
dk
dk1

ak
EX-NOR
511
Drawback of Differential Encoding/Decoding:
When single bit errors occur in the received data
sequence due to noise, they tend to propagate as
double bit errors
Since the decoder is comparing the logic state of
current bit with previous bit, and if the previous bit is
in error, the next decoded bit will also be in error
Delay
Ts
dk
dk1

ak
EX-NOR
Delay
Ts
dk
dk1
ak
01101100 01111100
Error
512

DPSK Modulation:
DPSK combines two basic operations at the
transmitter
Differential encoding of the binary data, and
modulation
513
Demodulation of DPSK
 Exercise:
 Draw a matched filter implementation of the optimum
detector
Delay
T
r
k
r
k1
x
r t
( )
t T

 ( )
s t
i
()

z dt
T
0
z t
( )
z T
( )
Suboptimum Detector
x
r t
( ) cos0t
t T

 ( )
s t
i
( )

z dt
T
0
z t
( )
z T
( )
x ( )

z dt
T
0
z t
( ) x
x
T
T
+
sin0t
Optimum Detector
See Fig. 4.17 (b)
514
BER Performance for DPSK
 Theoretical performance for CPSK & DPSK is shown for an AWGN channel
 BER for CPSK is exactly the same as that derived for bipolar baseband
transmission
r t s t n t
( ) ( ) ( )
 
Delay
T
x t
( )
r
k1
x LPF
r t
( )
t T

 ( )
s t
i
y t
( )
z T
( )
x t A t n t A t T n t T
o o b b
( ) cos ( ) cos ( ) ( )
    
 
y t const A n t A n t T n n t T
c c b s s b
( ) ( ) ( ) ( )
      
P Z T
B b
 
Pr ( ) 0
k p
0
1
exp
2
b
B
E
P
N
 
 
 
 
515
 In general, DPSK performs less than BPSK because the errors
tend to propagate due to correlation between bit waveforms
 BPSK performs about 3 dB better than DPSK
 The difference decreases with increasing Eb/No
 Differentially Encoded PSK (DEPSK)
Sometimes, differentially encoded PSK is coherently
detected (see section 4.7.2)
In this case, the probability of error is
0 0
2 2
2 1
b b
B
E E
P Q Q
N N
 
   
 
 
   
   
 
516

Example 43 - DPSK
a) The bit stream 11011100101 is to be
transmitted using DPSK. Determine the
encoded sequence, the transmitted phase
sequence and the detected sequence.
517
M-ary Modulation
M-ary Digital Communications
 In M-ary signaling scheme, we may send one of M = 2k possible
symbols, s1(t), s2(t), … , sM(t) during each interval Ts
 We refer to each M-ary message sequence as a character or
symbol
 The rate at which M-ary symbols are transmitted through the channel
is called the Baud Rate
 M-ary signals may be generated by changing the Amplitude,
Frequency or Phase of the carrier in M discrete steps resulting to the
following:
 M-ary PSK
 M-ary ASK
 M-ary FSK
 Another way of generating M-ary signals is to combine different
methods of modulation into a hybrid form e.g.,
 Amplitude Phase Keying (APK)  ASK + PSK
519
Abbreviation Descriptive Names
 MASK M-ary Amplitude Shift Keying
 MQAM M-ary Quadrature Amplitude Modulation
 MFSK M-ary Frequency Shift Keying
 MPSK M-ary Phase Shift Keying
 M = 4
 QPSK Quadrature Phase Shift Keying
 /4 QPSK /4 Quadrature Phase Shift Keying
 OQPSK Offset Quadrature Phase Shift Keying
 DQPSK Differential QPSK
 /4 DQPSK /4 Differential QPSK
 M > 4 MPSK (e.g, 8-PSK, 16-PSK, 64-PSK, etc., )
 DMPSK Differential MPSK
 MSK Minimum Shift Keying
 DMSK (GMSK) Differential MSK (Gaussian MSK)
 MAPK M-ary Amplitude Phase Keying
M-ary Modulation Types – Partial List
520

Modulation Type Applications
 FM (analog)  AMPS
 MSK  CT2
 GMSK  GSM, DCS 1800, CT3, DECT, HIPERLAN-1
 QPSK  NADC (CDMA) - base transmitter
 OQPSK  NADC (CDMA) - mobile transmitter
 4-DQPSK  NADC (TDMA), PDC, PHP (Japan)
 /4-DQPSK  N. A. TDMA, PHS
 QPSK/OQPSK  CDMA One
 QAM  IEEE 802.11 (5.7 GHz), HIPERLAN-2
 GFSK  Bluetooth, IEEE 802.11-FHSS)
 DPSK  IEEE 802.11-DSSS
 CCK  IEEE 802.11-DSSS
Practical Modulation Schemes
521
M-ary vs. Binary
Each symbol in an M-ary alphabet can be related to a
unique sequence of k-bits
where M is the size of the alphabet
Any digital system that transmits k bits in Ts seconds using
bandwidth efficiency of
 Any digital system will become bandwidth efficient if its BTb is
increase
2
log 1
/ /
b
B
s b
R M
bits s Hz
B BT BT
   
2
2
log
log
1 1
b
s
s s
s
b
b s
R
k M bits
R
T T M s
T
T
R k kR
  
  
2
2 log
k
M k M
  
522
Quadrature PSK (QPSK)
Quadrature PSK (QPSK) - 1
QPSK (4PSK) is just 2 BPSK
arranged in phase-
quadrature, each operating
at half the bit rate of the
original bit stream
It transmits 2-bit of info using
4 states of phases
 2 bits are transmitted per
modulation symbol 2Tb=Ts)
The I and Q channels are
aligned and phase transition
occur once every Ts = 2Tb
seconds with a maximum at
180 degrees
524

Example QPSK encoding
General expression:
Also can be written as
2-bit
information

00 0
01 /2
10 
11 3/2
2 2 ( 1)
( ) cos 2 , 1,2,3,4 0
QPSK o s
E i
s
M
T
s
s t f t i t T

 
 
    
 
 Each symbol
corresponds to two bits
2 2 ( 1) 2 ( 1)
( ) cos cos sin sin
b
i c c
b
E i i
s t t t
T M M
 
 
 
 
 
 
 
525
The signals are:
 
1
2
cos
s
s
c
E
T
s t


   
2
2
2
2
cos sin
s
c c
s
E
E s
T T
s
s t t

 
  
   
3
2 2
cos cos
s s
c c
s s
E E
T T
s t t
  
  
   
4
3
2 2
2
cos sin
s s
c c
s s
E E
T T
s t t

 
  
1,3
2,4
2
2
( ) cos2 , 0 180
( ) sin 2 , 90 270
o o
s
o
s
o o
s
o
s
E
T
E
T
s t f t shift of and
s t f t shift of and
 
 
  
 

(see next slide for
illustration)
526
E
10
01
11
00
s0
s1
s2
s3
527
In terms of basis functions
we can write sQPSK(t) as
 With this expression, the constellation diagram can easily be
drawn
 For example:
   
1 1
2 2
( ) cos 2 ( ) sin 2
o o
s s
T T
t f t and t f t
   
 
 
1 2
2 ( 1) 2 ( 1)
( ) cos ( ) sin ( )
QPSK s s
i i
E E
M M
s t t t
 
 
 
   
 
   
s
E
00
10
11
01
2 s
E
00
10
11 01
I
Q
I
Q
3 5 7
, , ,
4 4 4 4
   
 
3
0, , ,
2 2
 
 

528

QPSK Modulator:
 Source data is first split into 2 data streams (often by allocating alternate bits
to the upper and lower modulator)
 with each data stream runs at half the rate of the input data stream
 Think of m1 & m2 as bit stream that modulates the quadrature carriers
 In QPSK the Tx is 2 BPSK Transmitters arranged in phase-quadrature, each
operating at half the bit rate of the original bit stream
Serial-to-
Parrallel
Converter
X
X

90o
~
I
Q
R
R
s
b

2
R
R
s
b

2
R T
b b
 1
m2
1
1



R
S
T
m1
1
1



R
S
T
A
ot
2
sin
A t
o
cos
A
ot
2
cos
A
o
m t t
2 2( )cos
A
o
m t t
2 1( )sin
R
R
s
b

2
m(t)
529
SystemView
0
0
2.5e-3
2.5e-3
5.e-3
5.e-3
7.5e-3
7.5e-3
10.e-3
10.e-3
12.5e-3
12.5e-3
-1.5
-500.e-3
500.e-3
1.5
Amplitude
Time in Seconds
Modulated QPSK (t22)
530
QPSK Demodulator:
QPSK receiver is composed of 2 BPSK receivers
one that locks on to the sine carrier and
the other that locks onto the cosine carrier
x
Compare
Z1 and Z0
r t
( ) 1
( )
t
t T

z T
1
( )
 ( )
s t
i
x
z T
0( )
2
( )
t
()

z dt
T
0
()

z dt
T
0
z t
1
( )
z t
0( )
 
2 ( ) sin
t A t
o

 
1( ) cos
t A t
o

z t s t t dt A t A t dt
A T
L
T
o o
T s
o
s s
1 1 1
0 0
2
2
( ) ( ) ( ) cos cos
  
z z
  
a fa f 
z t s t t dt A t A t dt
o
T
o o
T
s s
( ) ( ) ( ) cos sin
  
z z
1 2
0 0 0
  
a fa f
531
532

Implementation of QPSK
533
534
Phase Diagrams:
In QPSK phase transition between all the states are
possible
Since transition through the origin is possible (phase shift
of p), the signal envelope can pass through zero
momentarily
 This could lead to errors or signal loss during transmission



s1




s2
s3
s4
45o
Phasechanges: 0 90 180
, ,
 
o o
535
Offset QPSK

Offset QPSK - 1
 Offset Quadrature Phase Shift Keying (OQPSK), also called
staggered QPSK (SQPSK) is a modified version of QPSK
 Recall that in QPSK, the bit transition in I- & Q-channels occur
simultaneously
 However, in OQPSK, I-channel (or Q-channel) bit stream is
offset by one bit period relative to Q-channel (or I-channel) prior
to modulation  Notice that the I and Q channels
are not aligned
 This misalignment implies that
only one phase transition can
occur once every Ts = Tb sec with
a maximum at 90o
 Q-channel: even bits, mI(t)
 I-channel: odd bits, mQ(t)
537
 Offset between I and Q means that transition is potentially
possible every Tb sec
 OQPSK can be used to achieve a non-zero envelope in the
modulated signal
Offset QPSK - 2
538
 For OQPSK, symbol transition across the origin (phase
changes of 180o) is prohibited (Compare this to QPSK)
 OQPSK is a constant envelope modulation scheme that is
attractive for systems using nonlinear transponders, e.g.,
satellite communication
 Unlike QPSK, signal transition do not
pass through the origin
QPSK OQPSK
Offset QPSK - 3
539
Differential QPSK (DQPSK)

Differential QPSK (DQPSK)
For M = 4, the PSK signal can be considered as 2 BPSK
signals using sint and cost as carriers
 The 4-phases can then be differentially encoded by encoding 2
BPSK signals differentially as discussed
 i.e., DQPSK modulator uses same differential data encoder for
each parallel data stream as binary DPSK counterpart
It employs the same principle of using a 1 symbol delayed
version of the received symbol stream to act as the
reference for demodulation
541
/4 QPSK - 1
Another variant of QPSK which is now widely used in
majority of digital radio modems is the /4 QPSK format
It is so called because the 4 symbol set is rotated by /4 or
45o at every new symbol transition
The reason for this rotation is to ensure that the
modulation envelope of the QPSK signal never passes
through zero
450 450
Symbol 1 Symbol 2 Symbol 3
Time
/4 rotating symbol set
542
The fact that the modulation envelope does not pass
through zero is important for the design of radio power
amplifiers
Comparing the vector diagrams for QPSK and /4 QPSK,
this property is clearly evident
Since envelope never goes through zero, /4 QPSK
mitigates spectral spreading caused by system nonlinearity
/4-QPSK differs from QPSK in that I-Q phases of 0 & /2 &
those of –/4 & /4 are alternatively changed every Ts sec
Qk
Ik
/4 QPSK - 2
543
/4-QPSK is a compromise between QPSK and QPSK
It performs better in multipath environment
It is possible to differentially encode /4 QPSK  /4-
DQPSK
/4-QPSK is widely used because it can be noncoherently
detected
/4-QPSK Mapping:
Data bits
mI, mQ
Phase shift 
T=2mTs
Phase shift 
T=(2m+1)Ts
00 -3/4 
01 3/4 /2
10 -/4 -/2
11 /4 0
180
90
/ 4 135
Modulation Max pahsechange
o
QPSK
o
OQPSK
o
QPSK
  
/4 QPSK - 3
544

Generalized M-ary Differential PSK - 1
For the case of M > 2, the signal can also be differentially
encoded using the phase comparisons
Increasing M > 4 allows further improvements in
bandwidth efficiency, but the additional symbol states are
no longer orthogonal
 they do not lie on the sine or cosine axis of constellation diagram
Error Probability Performance:
BER is difficult to compute
Symbol error probability for general M-ary PSK is given
by
P M Q
E
N M
E
s
o
( ) sin

F
HG I
KJ
2
2 
545
For differentially coherent detection of DPSK it is given
by
P M Q
E
N M
E
s
o
( ) sin

F
HG I
KJ
2
2
2

Generalized M-ary Differential PSK - 2
546
M-ary ASK
 Generation and detection process is scaled up, requiring multi-level symbol
mapping and comparison
 Detection of MASK is performed with the same methods employed with
binary ASK for either coherent or non-coherent detection
 MASK is not practically useful because of
 its relatively poor BER performance
 its sensitivity to any gain variations in the
channel
 its need for reasonable linearity in the
transceiver processing
 Only BASK is usually used in practice
547
Orthogonal M-ary FSK - 1
Recall that in M-ary FSK we have M transmitted
signals si(t), i = 1,2, …, M having waveforms
A minimum frequency separation is required
Modulator is same as BFSK and individual
frequencies are separated by 1/2Ts
For coherent MFSK, the Rx consist of bank of M-
correlators or MF
 
2
( ) cos 2 2 , 1,2, , ,0
i o s
E
s
T
s
s t f t i ft i M t T
 
     

1
,
i i o
i
f f f f f i f

     
   
1 1 2
1
, log
2
i i i i s b
s
or f f where T T M
T Ts

   
    
548

MFSK is good for reliable data transmission in the
presence of high levels of noise
549
Probability of Error Performance:
Unlike M-ary ASK, M-ary FSK is important because of
its increased noise immunity compared to binary FSK
   
( ) 1 1
E
E kE
s b
N N
o o
P M M Q M Q
   
   
   
   
(eqn.
3.122)
P M
k
M
k
kE
k N
M
E
N
M
k
E
k N
M E
N
E
k
k
M
s
o
s
o
k
k
M
s
o
s
o
( )
( )
( )





FH IK 

F
H
I
K
 
F
H
I
K 
 FH IK 
F
H
I
K
  
F
H
I
K




1
1
1
1
1 1
1
2 2
1
1
1
2
exp
( )
exp exp
( )
exp
coherent
noncoherent
550
 The Eb/N0 required for error-free transmission will thus
approach the Shannon-Hartley limit of -1.6 dB
 M-ary FSK is a very effective modulation technique in
applications where the optimum performance in noise is
required
 for example in deep space missions where the path loss is so great
 As the number of symbol states
increases, the symbol averaging time
becomes very large, reducing the
effect of noise to almost zero
551
Quadrature Amplitude
Modulation

Quadrature Amplitude Modulation - 1
The most commonly used combination of amplitude
and phase signaling is the Quadrature Amplitude
Modulation (QAM)
Some books regard it as an extension of the QPSK
since it consist of two independent amplitude-
modulated carrier in quadrature. i.e.,
where ai and bi are amplitude levels obtained by
mapping k-bit sequence into amplitudes, or
where g(t) is the signal pulse shaping function
2
( ) [ cos sin ]
i i o i o
E
s t a t b t
T
 
 
( ) ( )[ cos sin ]
i i o i o
s t g t a t b t
 
 
553
It is sometimes regarded as M-ary APK with
constraints put on the amplitude and phase
where
In this case, both the amplitude and phase can be
varied
Any combination of M1-level amplitude and M2-level
phase can be used in the construction of QAM
1 2
2
( ) cos[ ], 1,2, , , 1,2, ,
i i o j
E
s t V t i M j M
T
 
   
 
1 2 2 1 2
2 , 2 , log ,
m n
M M m n M M
   
 
b
s
R
R
m n


554
Any combination of M1-level amplitude and M2-level
phase can be used in the construction of QAM
QAM waveform can be represented as a linear
combination of 2 orthogonal signals 1(t) and 2(t)
where
In vector notation:
1 2
( ) ( ) ( )
i i i
s t A t B t
 
 
1 2
2 2
( ) cos[ ], ( ) sin[ ]
o o
T T
s s
t t t t
   
 
   
1 2
, , ,
i i i i i i i
s s s A E B E a b
 
  
 
555
Using the vector representation, we can realize an L-
by-L matrix representing the coordinates of (ai, bi)
where
( 1, 1) ( 3, 1) ( 1, 1)
( 1, 3) ( 3, 3) ( 1, 3)
{ , }
( 1, 1) ( 3, 1) ( 1, 1)
L L L L L L
L L L L L L
a b
i i
L L L L L L
       
 
 
       
 

 
 
          
 


   

L M

556

The 4-QAM and 8-QAM constellations
557
16-QAM constellations
558
MQAM Modulator:
 Then each branch is applied to a DSB-SC AM modulator
 The output of both quadrature is added to yield an MQAM
signal
 Although the modulator above is for 16QAM, it is good for any
M-ary QAM by changing the level shifter
 A serial-to-parallel
converter divides the
incoming data stream
into two bit stream each
at one-half the rate
559
Conventional M-ary QAM Modulation
Data
Slicer
2-to-L-level
converter
2-to-L-level
converter
Premod
LPF
Premod
LPF
LO
Phase
split
X
X
+ BPF IF AMP
fb
2
fb
2
fb
f
L
b
2
1
2
log
90o
0o
DSB-SC
AM Mod
DSB-SC
AM Mod
I
Q
560

Correlator Receiver Structure:
With this receiver, any QAM signal can be recovered
with only two correlators
The output of the correlators give a point on the signal
constellation
x
Threshold
and
Decision
Logic
r t
( )
2
T ot
cos
t T

 ( )
s t
i
x
( )

z dt
T
0
2
T ot
sin
( )

z dt
T
0
561
M-ary QAM Demodulation:
This demodulator uses I & Q
remodulation of the received signal
It can be used to demodulate any MQAM signal by changing
the level shifter
Level shifter can be implemented by A/D flash decoder
consisting of M-1 comparators each which is set at various
M-threshold levels
Their output are sampled and applied to parallel-to-serial
converter
562
QAM Signal Constellation
Signal space diagram (constellation) is very important
in QAM
This is because any combination of M1-level amplitude
and M2-level phase (or amplitude) can be used to
construct M=M1M2 QAM signal
QAM allows the signal vectors to be placed anywhere
on the constellation plane
Usually, signal points are placed at equally spaced
distance
A particular constellation gives rise to different
probability of error
563
16-QAM Constellations
 Type I QAM (Star Constellation)
 C. R. Cahn, 1960
 Type II QAM Constellation
 J. C. Hancock and R. W. Lucky
 Type III QAM Constellation
 Compopiano & Glazer, 1962; J. Salz, J. R. Sheenhan, & D.J. Paris 1971
Q
I
I I
Q
Q
Type I Type II Type III
16 QAM (8, 8) 16 QAM (4, 12) 16 QAM (4, 8, 4)
564

Since we desire min radial distance, but max
separation between points the square constellation is
easier to implement and has a slightly better
probability of error performance
Type I and Type II constellations are not preferred for
Gaussian channels
need higher energy to achieve the same min
distance compared to Type III
565
QAM is used by high-speed wireline modems
Allows data rates of 9,600 bps and above over
ordinary telephone lines
9,600 bps modem uses 16-QAM or 32-QAM (V.22
and V.32)
14.4 kbps uses 128-QAM
28.8 kbps uses 512-QAM
566
Comparing the constellation diagrams of M-ary QAM
with M-ary PSK we can see that the spacing between
symbol states for QAM is greater than that for PSK
This is because PSK constellation are restricted to
symbol states of equal amplitude and thus on a circle
equidistant from the origin
The larger spacing between symbols for QAM means
that the detection process should be less susceptible
to noise
567
General Decision Rule for M-ary - 1
 Once a point on the signal constellation plane is determined for
the received signal, a decision can be made
 The decision rule is to pick the signal point that is closest to
the received point
 The distance between the signal point and the received point is
a function of the noise in the environment during the symbol
interval
 If the noise has moved the received point closer to a different
signal point, then the receiver will make an error
d
d
x
If the receiver
calculates this point
R
S
T
Then, it will pick the symbol
corresponding to this signal point
R
S
T
568

 Thus, for decision purposes, we partition the signal
constellation diagram into decision regions
min Euclidean distance amongst phasors gives rise to noise
immunity
 the min distance between any pair of signal vectors is
Minimum phase rotation amongst constellation points
 determines the phase jitter immunity
 resilience against clock recovery imperfections & channel
phase rotations
Peak-to-average phase power ratio
 robustness against nonlinear distortion of power amplifier
d s s E a a b b
ij i j i j i j
     
1
2
2 2
b g b g
n s
569
 In a special case where amplitudes take discrete values (2i-1-
M)d, constellation is rectangular
 Min separation between signal points determines PE(M)
 Energy of signal depends on the radial distance from origin to
signal point
 desire minimum radial distance, but max separation between
points
I
Q
E
d
d
 2( )
t
1( )
t
s2
s1
d d E
min  2
( , ) cos tan
a b a b t
b
a
o
  
FH IK

2 2 1

570
BER Performance for QAM - 1
 The exact performance of QAM depends on the shape of a
particular signal constellation diagram
 For a rectangular constellation, the probability of correct
detection is
 Hence the probability of error is given by
 This probability of error is exact for M = 2k, k is even
 That is, a rectangular QAM (Type III) can only be
implemented when k = 2M (even)
 Odd-bit constellations add complexities to the CODEC
 
1 3
( 1)
2 1
M
E
av
M M N
o
where P Q 

 
   
 
 
2
( ) 1
C M
P M P
 
 
2
( ) 1 ( ) 1 1
E C M
P M P M P
    
571
 A general performance of coherent QAM (even or odd), the
symbol error probability can be bounded as (M > 4)
 Eav is the average energy per bit
 k is the number of bits per symbol
2
3 3
( ) 1 1 2 4 , 1
( 1) ( 1)
av b
E
o o
E kE
P M Q Q k
M N M N
 
   
    
 
   
 
   
 
I
Q
d
d
d
5-bit QAM Constellation
• For this odd-bit constellation root of
M is not an integer
• It is not possible to gray encode
572

 The improvement of 16-QAM over 16-PSK comes from the
noise immunity capability of QAM
 However, the design requirements of QAM is more complicated
needing to handle both amplitude and phase
1 2
3log
2(1 ) 2
2
log ( 1)
2
( )
B
E
M
M b
M N
M o
P M Q



 
  
 

573
Offset QAM Modulation
Data
Slicer
2-to-L-level
converter
2-to-L-level
converter
Premod
LPF
Premod
LPF
LO
Phase
split
X
X
+ BPF IF AMP
fb
2
fb
2
fb
90o
0o
DSB-SC
AM Mod
DSB-SC
AM Mod
I
Q
Half
Symbol
Delay
Variants of QAM - 1
   
 
2 2 1
( ) 2 cos 2 1 sin
k k
k k
c c
s t a h t kT t a h t k T t
 

 
   
    
   
574
Superposed M-ary QAM Modulation
Be able to use built-in PSK MODEMs in realization
Less efficient than the conventional QAM
implementation
Data
input
QPSK
Modulator
Serial-to-2x2 bit
paralel converter
LO
+ BPF IF AMP
QPSK
Modulator
Variants of QAM - 2
575
Variable Rate QAM Modulations
QAM transmission over Rayleigh Fading Channel
 Burst error due to deep fades
 Varying the modulation levels in response to fading
conditions
Suitable for data transmission
Variable QAM constellation
QPSK
32-level Star
QAM
16 Star QAM
Type 1
2-level QPSK
BPSK
64-level Star
QAM
Variants of QAM - 3
576

Trellis-Coded Modulation - 1
Combined coding and modulation scheme
TCM achieves coding gain without BW expansion and reduction
of effective information rate
Both power and bandwidth efficient
In power limited environment:
 Use error correcting code  increases power efficiency
 Requires higher rate  higher bandwidth
In bandwidth limited environment:
 Choose higher-order modulation  increases spectral
efficiency
 Larger signal power is needed for the same signal separation
TCM combines the choice of higher-order modulation with
convolutional code
TCM achieves coding gain without BW expansion and reduction
of effective information rate
577
TCM is classified into two basic types
 Lattice type MPAM and MQAM
 Better power efficiency
 Constant amplitude type  MPSK
 Lower power efficiency, better over satellite channel
 Observations:
 We can use coding gain without BW expansion
 Coding and modulation are not separate entities
 Demodulation and decoding in single step
 Performance is governed by “free Euclidean” distance not
free hamming distance of the code
 Optimization of TCM is based on the “free Euclidean”
distance
 Detection is based on “soft decision”
Part 5: Digial Bandpass Communication
Trellis-Coded Modulation - 2
578
Summary list of Digital MODEM - 1
 Binary Modulation Schemes
 Amplitude Shift Keying or ON-OFF Keying
 Coherent and Noncoherent
 Frequency Shift Keying (FSK) or Continuous-Phase FSK
 Coherent and Noncoherent
 Phase Shift Keying (PSK)
 Coherent and Differential PSK
 M-ary (multi-level) Modulation Schemes
 M-ary Amplitude Shift Keying (MASK)
 M-ary Frequency Shift Keying (MFSK)
 M-ary Phase Shift Keying (MPSK)
 QPSK, Differential QPSK, OQPSK, /4 PSK and /4 QPSK
 M-ary Amplitude Phase Keying (MAPK)
 Quadrature Amplitude Modulation (MQAM)
579
 Minimum Shift Keying (MSK) or Fast Frequency Shift Keying
 Differential MPSK (MPSK)
 Differential Encoded MPSK (DEMPSK)
 Differential MSK (DMSK)
 Gaussian MSK (GMSK)
 Superposed QAM (SQAM)
 /4 Differential PSK
 Quadrature Partial Response (QPR)
 Sinusoidal Frequency Shift Keying (SFSK)
 Comparison of Modulation Schemes
 For practical application, the choice of digital MODEM depends on:
 bandwidth efficiency,
 power efficiency,
 error performance,
 Complexity of implementation, and
 Cost
580

 Probability of symbol error or Probability of bit error is related to:
Power efficiency
Bandwidth efficiency (spectral efficiency)
 The performance of modulation schemes is summarized based on BER and
complexity
 Usually transmitted power and complexity increases with increase in
bandwidth efficiency
 The linear or nonlinear nature of the channel also affect the choice of
digital MODEM
 Lastly, but not the least, government regulations also affect the choice of
digital MODEM
 A desirable characteristics of any modulation scheme is the simultaneous
conservation of bandwidth and power
This has lead to the combination of coding and modulation (also known
as Trellis Coded Modulation)
581
Power Efficiency - 1
 Definition:
 Power Efficiency (), is a measure of how much received
power is needed to achieve a specified bit error rate
 Power efficient modulation schemes requires less power for
satisfactory BER
  is a function of signal-to-noise ratio (SNR)
 In the computation of , it is assumed that:
 All modulation levels occur with equal probability, 1/M
 Gray encoding is used to map the information bits into levels
 Differential encoding may be employed
 Power efficient modems are not bandwidth efficient (next 2
slides)
 Power efficient schemes are more appropriate for satellite &
mobile communications
582
Power efficient modulation schemes include:
BPSK (or equivalently DSB-SC-AM in analog
system)
QPSK and 4-QAM
Assuming both I- and Q-channel is an unfiltered
balanced NRZ bit stream
BPSK and QPSK
 is 2 b/s/Hz theoretical (1.5 ~ 1.8 b/s/Hz
practical)
Low Eb/No for good error probability performance
Relatively simple hardware design
583
Summary of Power efficient modulation
More appropriate for satellite communications
systems
BPSK and QPSK
Requires less power for satisfactory BER
They are not bandwidth efficient modulation
Expressed in terms of SNR for required BER
Power efficient:
If a Pe= 10-8 requires an Eb / No < 14 dB
584

Bandwidth Efficiency - 1
Definition:
Bandwidth efficiency () is the ratio of the bit rate to
channel bandwidth expressed in bit per second per
hertz (b/s/Hz)
It is also called “Spectral Efficiency”
The primary objective of spectrally efficient modulation
is to maximize the bandwidth efficiency
With data rate denoted as R, and the channel
bandwidth by B, then Bandwidth Efficiency  is given
as
2
1
log 2 / /
b
b
R
M bits s Hz
B BT
   
585
 In theory, BT  1 (for role-off-factor,  = 0)
 In practice,  > 0,  out-of-band emission constraints
imposed by FCC spectrum regulation
 T is well defined, but B is not - hence  of a digitally
modulated signal depends on the definition adopted for B
Capacity of a digital communication system is directly
related to 
The max possible bandwidth efficiency is
 Note that binary systems are more power efficient, but
less spectral efficient than M-ary systems
max 2
log 1
C S bps
B N Hz

 
  
 
 
586
Note that bandwidth efficient modem are not power
efficient
Spectrally efficient modems include:
M-ary QAM
 In theory,  = 4, 6, & 8 b/s/Hz for 16-, 64-, and 256-
QAM, respectively
But in practice we have, 2.5-3.5, 4.5-5, & 5-6, respectively
 Available Eb/No > 30 dB
Usually, in spectral efficient modulation, the common
carrier band is subdivided into channels of width B
4-, 6-, 11-GHz bands in the USA have channel
bandwidths of 20, 30, and 40 MHz, respectively
More appropriate for digital microwave radio
587
Summary of Bandwidth efficient modulation
More appropriate for microwave radio
M-ary level schemes (MPSK, MQAM) (M > 4)
Can transmit more information bit / BW
They are not power efficient modulation
Expressed in terms of Rb/B (b/s/Hz)
Spectral Efficiency:
If spectral efficiency > 2 b/s/Hz
588

Spectral Efficiency Plane
589
How do I compare one modulation format to another?
Bandwidth of Coherent Binary Modulation Schemes
Comparison of some PSK Modulation Schemes
Modulation
Scheme
Required
Eb/No
Min Channel B for
ISI free signaling
Max 
(bits/s/Hz)
Required
CNR
BPSK 10.6 dB Rb 1 10.6 dB
QPSK 10.6 dB 0.5Rb 2 13.6 dB
8-PSK 14.0 dB 0.33Rb 3 18.8 dB
16-PSK 18.3 dB 0.25Rb 4 24.3 dB
Rectangular Pulses Raised Cosine
ASK 2/T (1+r)/T
FSK 4/T 2(1+r)/T
PSK 2/T (1+r)/T
Pb = 10-6
Comparison of Digital MODEM - 1
590
 Bandwidth Efficiency of some Modulation Schemes

Modulation
Scheme
EbNo
(dB)
Bandwidth Efficiency,  Immunity to
Nonlinearity
Implementation
Complexity
Nyquist Null-to-Null
BPSK 9.6 dB 1.0 0.5 D (worst) a (simple)
QPSK 9.6 dB 2.0 1.0 C a
OQPSK 9.6 dB 2.0 1.0 B c
MSK 9.6 dB N/A 2/3 A (best) d (complex)
M-ary System Bandwidth Efficiency bits/s/Hz
PSK, QAM
Coherent FSK Assuming frequency separation of Rs/2
Noncoherent FSK , Assuming frequency separation of 2Rs/2
1 log
2
2
M
2log2
3
M
M 
log
2
2
M
M
Pb = 10-5
591
Bandwidth Efficiency of M-ary PSK
M 2 4 8 16 32 64

(bits/s/Hz)
0.5 1.0 1.5 2.0 2.5 3.0
Bandwidth Efficiency of M-ary FSK
M 2 4 8 16 32 64

(bits/s/Hz)
1.0 1.0 0.75 0.5 0.3125 0.1875
592

Modulation
Scheme
Eb/No
(dB)
Bandwidth,
B
Equipment
Complexity
Comments
coh. ASK
noncoh. ASK
coh. FSK
noncoh. FSK
14.45
15.33
10.60
18.33
Moderate
Major
Minor
Major
 Rarely used;
8.45
2Rb
2Rb
2Rb
 2Rb
 Seldom used
 Performance does not justify complexity

 Used for slow speed data transmission
 Poor utilization of power and bandwidth

 Used for high speed data transmission
 Better overall performance but requires
complex equipment

Minor
coh. PSK 2Rb
Major
9.30
 Most commonly used in medium speed
data transmission
 Error tend to occur in pairs

Differential
PSK 2Rb
 0 0

 0 0

 0 0

 0 0

 o b
A T
 2 4
/
 o A
 /2 P P
eo e
 1
Assuming PB 

10
6
593
Complexity of Modulation Schemes
Complexity High
APK
M-ary PSK
QPR
CPFSK - optimal detection
MSK
OQPSK
QAM, QPSK
BPSK
Low
OOK - envelope detection
DQPSK
DPSK
CPFSK -discriminator detection
FSK - noncoherent detection
IEEE 1979
594
Probability of Error Calculations
Probability of Error Calculation - 1
What is the difference between symbol error and bit error?
Probability of symbol error vs. Probability of bit error?
 One important parameter of communication systems is the
SNR or the Eb/No defined as:
Also, p. 158 of your textbook defines
where S = Average message signal power
N = Noise variance NoW
W = Bandwidth
R = Rate
Generally,
 b
b
o
b
o o o o
E
N
ST
N
S
RN
SW
RN W
S
N W
W
R
S
N
W
R
     
e j e j
 b
b
o
b
o o b o
E
N
A T
N
A
N T
A
N W
   
2 2 2
1
( / )
E
M
E
b avg

1
2
log
596

Symbol Energy and SNR per Symbol:
Consider the signals sm, m = 1, 2, …, M
Assume symbols are equiprobable
Energy of signal m is Em: (called Es if Em is equal for all m)
Average energy per symbol
Average power
Average SNR per symbol
1
( ) , 1,2, ,
m M
P s m M
  
1
1
, ( equalforall )
M
av m s m
m
M
E E E if E m

 

1
, where the symbol rate is
T
av
av
E
T
P 
( if equal for all )
av s
m
o
o
E E
S
N
N
N
E m
 
 
 

597
Bit Energy and SNR per Bit
 Bit rate is R
 Average Energy per bit
 Average Power
 Average SNR per bit
Bit Rate
Symbol Rate
RT M
  log2
2
(called if is equal for all )
log
av
bav b s m
E
E E E E m
M
 
( if is equal for all )
av av b m
P E R E R E m
 
E
N
E
M N
E
N
E
M N
E m
bav av
o
b
o
s
o
m
0 2 2


 

log
(
log
if is equal for all )
598
Error Probability:
Probability of bit error is Pb
Probability of symbol error is Pe or PM or P(M)
We can compare modulation schemes in terms of the
Eb/No required to achieve a specified or Pe or Pb
Generally,
Relation between Pe and Pb for Orthogonal Signals
 Since the Euclidean distance between any 2 signals is
the same, there is no benefit to Gray coding
 When a symbol error occurs, each of the (M-1) remaining
symbols is chosen with probability (1/M-1)
P
M
P M
b e

1
2
log
( )
599
 The number of symbols conveying an incorrect bit in one
of the log2M positions is M/2
 The probability of having an incorrect bit in any one of the
log2M positions is
 The probability of bit error is
Relation between Pe and Pb and for PAM, QAM, PSK
 Assume that Gray coding is used, then the most probable
symbol errors cause exactly one bit error each, since
each symbol encodes bits:
 Hence
P
M
M
P
b e


2 1
( )
M
M
2
1
1


P
M
P M
b e

1
2
log
( )
600

Probability of Symbol Error for M-ary Orthogonal
Signals
Coherent Exact
Coherent Union Bound
Noncoherent Exact
All Cases
P
M
P
b M


1
2 1
( )
E
E
M
b
s

log2
P e y dy
M
y
M E
N
x
s
o
   





z
z
1
2
1 1
2
1
1
2
2
2
2
2
 
c h e j
exp
P M Q
E
N
M
s
o
 
F
HG I
KJ
 
1
P
M
n n
nE
n N
M
n
n
M s
o
 

F
HG I
KJ 


F
HG I
KJ



 ( ) exp
( )
1
1
1 1 1
1 1
601
Probability of Error of Modulation Schemes - 1
Modulation PM (coherent) Pb (coherent) Pb (noncoherent)
 Baseband Systems
 Antipodal
 Orthogonal
 Bandpass Systems
 BASK (OOK)
 BFSK
 BPSK
 
0
Es
N
Q
2 b
o
E
Q
N
 
 
 
0
2 b
E
Q
N
 
 
 
2
0
1 exp
8
2
A
N
 

 
 
0
1 exp
2 2
b
E
N
 

 
 
0
b
E
Q
N
 
 
 
0
b
E
Q
N
 
 
 
0
b
E
Q
N
 
 
 
0
s
E
Q
N
 
 
 
 
2Es
No
Q
0
1 exp
2
b
E
N
 

 
 
602
 QPSK
 OQPSK
 DPSK
 MASK
 MFSK
 MPSK
   
0 0
2
2 1
b b
E E
N N
Q Q

 
   
1 exp
2
b
o
E
N

0
2
2 sin
s
E
Q
N M

 
 
 
 
2
0
2 log
2 sin
E M
b
N M
Q 

2 1 6 2
2 1
( ) log
( )
M
M
MEb
M No
Q


F
H
I
K
 
2
0
Eb
N
Q
 
0
( 1) b
kE
N
M Q


 
0
4
2 sin
Es
N M
Q 

0
2
2 s
E
Q
N
 
 
 
 
2
0
Eb
N
Q
0
2Es
Q
N
 
 
 
2
1
2
e
Eb
No
M 


2( 1)
2
( 1)
E
M b
M M No
Q


 
 
 
 
0
( 1)
kEb
N
M Q
 
603
 MDPSK
 /4QPSK
 MQAM
 MSK
 GMSK
2
0 2
Q
Es
N M
sin 
e j
 2
2
0
Q
Es
N M
sin 
e j
 
4
3
1 0
Q
kEs
M N
( )
e j  FH IK
 

2 1 1
2
3 2
2 1
2
0
( )
log
log
( )
M
M
M
M
Eb
N
Q
 
0
2 b
E
N
Q 
 
0
2 b
E
N
Q
0.68
 
 
0
2 s
E
N
Q
604

Digital Communications
 Resource Sharing Techniques
 Duplexing
 Multiplexing Techniques
 Frequency Division Multiplexing
(FDM)
 Time Division Multiplexing (TDM)
 Code Division Multiplexing (CDM)
 Wavelength Division Multiplexing
(WDM)
 What is Multiple Access?
Module 5
Multiplexing & Multiple Access
 Multiple Access Techniques
 Frequency Division Multiple
Access (FDMA)
 Time Division Multiple Access
(TDMA)
 Practical TDMA Systems
 Code Division Multiple Access
(CDMA)
 How CDMA Works
 Practical CDMA Systems
 Hybrid Multiple Access
Techniques
 Multiple Access Techniques
 Frequency Division Multiple
Access (FDMA)
 Time Division Multiple Access
(TDMA)
 Practical TDMA Systems
 Code Division Multiple Access
(CDMA)
 How CDMA Works
 Practical CDMA Systems
 Hybrid Multiple Access
Techniques
Resource Sharing Techniques - 1
Duplexing (Review – Read Section 9.1)
Multiplexing Techniques (self study)
Frequency Division Multiplexing (FDM)
Time Division Multiplexing (TDM)
Code Division Multiplexing (CDM)
Wavelength Division Multiplexing (WDM)
Multiple Access Techniques
Frequency Division Multiple Access (FDMA)
Time Division Multiple Access (TDMA)
Code Division Multiple Access (CDMA)
Direct Sequence CDMA
Other Multiple Access Techniques
Resource Sharing Techniques - 2
Since the RF spectrum is a finite and limited resource,
it is necessary to share the available resources
between users
Forma
t
Channel
Encoder
Source
Encoder
Format
Channel
Decoder
Source
Decoder
Bits or
Symbol
To other
destinations
From other
sources
Digital
input
Digital
output
Source
bits
Source
bits
Channel
bits
Carrier & symbol
synchronization
Channel
bits

mi
l q
mi
l q
Tx
Rx
Performance
Measure

Pe
Modulate
Demodulate
&
Detect
Spread
Multiple
Access
Waveforms
Multiple
Access
Despread
Demultiplex
Multiplex
Duplexing Techniques - 1
 A technique commonly used in many
radio and telecommunication between a
pair of users – Tx and Rx
 Simplex
 Info is transmitted in one and only
one pre-assigned direction
 Half Duplex
 Transmission of information in only
one direction at a time
 Uses simplex operation both end
 Full Duplex
 Simultaneous transmission and reception of info in both directions
 In general, duplex operation require 2 frequencies
 May be achieved by simplex operation of 2 or more simplex at both ends
 Duplexing can be implemented in either Frequency or Time domain
 Frequency Division Duplexing (FDD) & Time Division Duplexing (TDD)
Terminal
A
Terminal
B
Terminal
A
Terminal
B
Terminal
A
Terminal
B
Full-duplex
Half-duplex
Simplex

 Frequency Division Duplexing (FDD)
 Multiplexes the Tx and Rx in one time slot in which
transmission and reception is on 2 different frequencies
 It provides simultaneous transmission channels for
mobile/base station
 i.e. each channel has a Forward and a Reverse
frequency
 At the base station, separate transmit and receive antennas
are used to accommodate the two separate channels
 At the mobile unit, a single antenna (with duplexer) is used
to enable transmission and reception
 To facilitate FDD, sufficient frequency isolation of the
transmit and receive frequencies is necessary
 FDD is used exclusively in analog mobile radio systems
Time Division Duplexing (TDD)
 Multiplexes the Tx & Rx in one frequency at different time slots
 A portion of the time is used to transmit and a portion is used to
receive
 TDD is used, for example, in a simple 2-way radio where a button
is pressed to talk and released to listen
 If the data rate from the base station >> the end-user’s data rate,
it is possible to use buffer-and-burst transmission (giving the
appearance of full duplex)
 TDD is only possible for digital transmission
Time Division Duplexing
Time
Amplitude
T T
R R
Multiplexing Techniques
 Multiplexing (sometimes called channelization) is the process of
simultaneously transmitting several information signals using a
single communication channel
 Commonly used to separate different users such that they share the
same resource without interference
 Communication recourses are allocated a priori and allocated resources are
fixed
 Only one pair of transceivers are required
Three major kinds
Frequency Division
Multiplexing
Time Division Multiplexing
Code Division Multiplexing
Frequency Division Multiplexing (FDM)
 In Frequency Division Multiplexing (FDM), the available
bandwidth is divided into non-overlapping frequency slots
 Each message is assigned a frequency slot within the available
band
 Signals are translated to different frequency band using
modulation and then added together to form a baseband signal
 The signals are narrowband and frequency limited
 FDM can be used for either digital or analog transmission
Frequency Band 1
Frequency Band 2
Frequency Band N
f0
f2
fN-2
fN-1
f1
f3
Time
Frequency

Time Division Multiplexing (TDM)
 Digitized info from several sources are multiplexed in time and transmitted
over a single communication channel
 The communication channel is divided into frames of length Tf
 Each frame is further segmented into N subinterval called slots, each with
duration Ts = Tf/N, where N is the number of users
 Each user is assigned a slot (or channel) within each time frame
 TDM is used to combine several low bit rate signals to form a high-rate
signal to be transmitted over a high bit rate medium
 Individual message signals need not have the same rate, or same type of
signal since each channel is independent of one another
 TDM is usually used for digital communication and cannot be used in analog
communication
 Different combining techniques are shown below
Slot
1
Slot
2
Slot
N
s1 s2 sk . . .
FRAME
. . .
Sync word Information or data word
s1 s2
. . .
Slot
N
. . .
Code Division Multiplexing (CDM)
 CDM is a multiplexing method
where multiple users are
permitted to transmit
simultaneously on the same
time and same frequency
 In CDM system, users time
share a higher-rate digital
channel by overlaying a higher-
rate digital sequence on their
transmission
 Each user is assigned distinct
code sequence (or waveform)
 This technique may be viewed
as a combination of FDM and
TDM using some sort of code
Signal 1
Signal 3
Signal 2
Frequency
Time
Signal 2
Signal 1
Signal 3
Signal 1
Signal 3
Signal 2
Slot 1 Slot 2 Slot 3
Band 1
Band 2
Band 3
Code Division Multiplexing
Wavelength Division Multiplexing (WDM)
In optics, the process of using laser source, repeater
amplifier, and optical detector to independently
modulated light carriers to be sent over a single fiber is
known as WDM
 Each individual light carrier could support data rates of up to
10 Gbps with users time multiplexed onto the channel
 WDM thus offers the possibility of several hundreds of gigabits
transmission over a single fiber and also bi-direction
transmission over the same fiber
 This process has been very
difficult until recently
 fc of light with sufficient
spectral stability is required
and was not available until
recently
What is Multiple Access?
Definition:
Multiple Access (MA) techniques
are multiplexing protocols that allow
more than a pair of transceivers
to share a common medium
i.e., the simultaneous use of a
channel by more than one user
Allocation of resources
 not defined a priori
 not necessarily fixed
Each user’s signal must be kept
uniquely distinguishable from other
users’ signals, to allow private
communications on demand
Users can be separated many ways:
physically: on separate wires by
arbitrarily defined “channels”
established in frequency, time, or
any other variable imaginable

Multiple Access Techniques
Multiple Access can be implemented in:
Frequency Division Multiple Access
 A user’s channel is a private frequency -
uses different frequencies for different users
Time Division Multiple Access (TDMA)
 A user’s channel is a specific frequency, but
it only belongs to the user during certain
time slots in a repeating sequence
 That is, same frequency is used but
different time for different users
Code Division Multiple Access (CDMA)
 Each user’s signal is a continuous unique
code pattern buried within a shared signal,
mingled with other users’ code patterns
 If a user’s code pattern is known, the
presence or absence of their signal can be
detected, thus conveying information
 Uses same frequencies and time but
different codes (3G wireless systems)
Space Division Multiple Access (SDMA)
Uses spot beam antennas to separate radio signals by
pointing at different users with different spot beam, e.g.,
ACTS
Multiple Access Protocol
Contention
(Random Access)
Contentionless
(Scheduling Access)
CDMA
Fixed
Assigned
Demand
Assigned
FDMA
TDMA
Polling
Token Passing
Repeated Random
Access
ALOHA
Slotted ALOHA
Random Access
w/reservation
Implicit
Explicit
 Demand Access Multiple
Access (DAMA)
Uses dynamic
assignment protocol
(allocates resources on
request)
 Random Access Multiple
Access (RAMA)
 Hybrid Multiple
Accesses
 Time Division CDMA, Time
Division Frequency
Hopping, FDMA/CDMA,
etc.
FDMA - 1
FDMA is the oldest and most familiar method of radio
communication
used since 1890 in broadcasting, two-way radio, and
cellular systems
Individual frequencies (private frequencies) are
assigned to individual users on demand for the
duration of their call
1 2 n
B
FRAME
Guard band (at the edges & between) to
minimize crosstalk

FDMA - 2
 Distant users are far enough that they cause no interference
 When the call is finished, the channel is released and available for a
new call
 If the transmission path deteriorates, the controller switches the
system to another channel
 FDMA is the method used in the original cellular systems
 “AMPS” Advanced Mobile Phone System
 Although technically simple to implement, FDMA is wasteful of BW
 Channel is assigned to a single conversation whether or not
somebody is speaking
 It cannot handle alternate forms of data, only voice is permissible
 Used extensively in the early telephone and wireless multi-user
communication systems
 FDMA is the most commonly used access protocol especially for
satellite communication

FDMA - 3
In a cluster, each user is assigned a portion of the
available bandwidth
Let
Ndata = number of data channel
Nctl = number of control channel
Total Bandwidth
Number of Channels
 
,
s data ctl
N N N
or N  
2
s s c g
B N B B
 
2
s g
s
c
B B
N N
B

 
2
s data c ctl c g
B N B N B B
  
data c s
N B B
 
Channel
1
Channel
2 ...... Channel
Ns
Bs
Bc
Bg
MHz
FDMA - 4
Number of channels/cluster
Number of channels/cell
Number of data channels/cluster
Number of data channels/cell
/
2
s g
ch cluster
c
B B
N
B


/
/
ch cluster
ch cell
N
N
N

/ / /
data cluster ch cluster ctl cluster
N N N
 
/
/
data cluster
data cell
N
N
N

 We can also determine the # of control channels per cluster of
cell in a similar manner
 Number of calls per hour per cell (where t is the trunk
efficiency)
/
number of calls per hour
ch cluster
calls t
N
N
N

  
FDMA - 5
Average number of
users per hour per cell
Spectral Efficiency
FDMA Capacity
number of calls/hour/cell
average # of calls/user/hour
user
N 
    
2
# of data channel/cluster
chls/MHz/km
sytem BW
data / cluster
cluster s cell
N
A B N A
  

 
BW available for data transmission
1
sytem bandwidth
data c
FDMA
s
N B
B
   
s
s c g
B
C
N B B


Channel
1
Channel
2 ...... Channel
Ns
Bs
Bc
Bg
MHz
Guard Bands
623
TDMA - 1
In TDMA, each user has a specific
frequency but only during an
assigned time slot
The freq is used by other users
during other time slots
Available time is divided into frames
of equal duration
 In each time slot, only one user is allowed to either transmit or
receive
 Number of time slots/ frame is a design parameter depending
on requirements (e.g., modulation, bandwidth, data rate, etc.)
 In TDMA, bitstream are broken into frames, frames broken into
slots and slots are assigned to users

TDMA - 2
 Forward and Reverse channels are duplexed within time domain (TDD) or
frequency domain (FDD)
 Slots contain data, error check, guard, synchronization training, and control
bits
 TDMA transmits data in a “buffer-and-burst” technique and hence
transmission is not continuous
 low battery consumption is achieved, and simplification of handoff
process is achievable
 Transmission from users are interlaced into cyclic time structure
 TDMA requires very high data rate compared to FDMA and hence
equalization is not required
Control Bits
Slot 1
Trail Bits
Information Data
Slot 2 Slot 3 Slot N
Guard Bits
Information Data
Trail Bits Sync. Bits
One TDMA Frame
TDMA/FDD
Also see Fig. 9.4
TDMA - 3
 Illustration of TDMA Transmission
 Each earth station
is assigned a time
slot in a repetitive
time frame
 Over the length
of the time slot-
the earth station
occupies the
entire bandwidth
of the
transponder
TDMA Operation
TDMA - 4
TDMA Systems
TDMA can operate in wideband or narrowband
 Wideband TDMA (W-TDMA)
– entire freq spectrum is available to any individual user
 Narrowband TDMA (N-TDMA)
– total available freq spectrum is divided into subbands, with each
subband operating as a TDMA system
– A user only uses the allocated subband
– Both frequency and time are partitioned
Basic Frame Structure
 Let
– Bs = Bt = total spectrum allocation
– Bg = guard band
TDMA - 5

TDMA - 6
–Bc = Channel bandwidth of individual user
–N = frequency reuse factor
–Nu = number of subbands
–Ld = number of information data symbols in
each slot
–Ls, = the total number of symbols in each slot
1 2 ...... (N-1)slot
Trailer
Preamble
Tf
T1 sec
Nslot
3
T2
TNslot
p
 t

TDMA - 7
s u slot
N N N
 
1, for W-TDMA
2
, for N-TDMA
s g
u
c
B B
N
B



 


u slot
cell
N N
N
N


u slot
cell
f
N N
N
s N



For voice communication with talk spurt
(on) state and silence (off) state
Nslot = m in your textbook
Overhead bits per frame
where
 bOH =overhead bits per frame
 Nr = # of reference burst per frame
 br = # of overhead bits per frame
 bp = # of overhead bits per preamble in each time slot
 bg = # of equivalent bits in each guard time interval
Total number of traffic bits per frame
Frame efficiency
TDMA - 8
OH r r t p t g r g
b N b N b N b N b
   
T f
b T R
 where R = channel bit rate
1 100%
OH
f
T
b
b

 

 
 
 
Total number of bits per frame
Information bit burst rate, Rb+
Spectral Efficiency of TDMA
TDMA Capacity
TDMA - 9
T 0
b H
b b
 
frame
b b
slot
T
R R
T
 
, for W-TDMA
2
, for N-TDMA
f p t d
f s
f p t s g
d
f s s
T L
T L
T B B
L
T L B
 

 
 




 
  
  


traffic frame b
f f
slot slot b
T T R
C
T T R
  
  

TDMA - 10
Advantages:
 No inter-modulation impairment
 Since TDMA uses one carrier at a time
 No interference from other simultaneous transmissions
 TDMA’s technology separates users in time ensuring that they will not
experience interference from other simultaneous transmissions
 Flexibility
 TDMA can be easily adapted for the transmission of data or voice
 Variable rates
 TDMA offers the ability to carry data rates of 64 kbps to 120 Mbps
(expandable in multiples of 64 kbps)
 This enables operators to offer PCS (fax, voice-band data, and SMS,
etc.), as well as bandwidth-intensive applications – multimedia and
videoconferencing
 Bandwidth efficient protocol
 TDMA uses bandwidth more effectively because no frequency guard
bands are required between channels
 Low power consumption
 since transmission is bursty and non-continuous
 i.e, TDMA provides the user with extended battery life and talk time since
the mobile is only transmitting a portion of the time (from 1/3 to 1/10)
during conversations
 Guard time between time slots may be used to accommodate
 clock instability
 delay spread
 transmission (or propagation) delays and pulse spreading
 Achieves selectivity in time domain, and selectivity is simpler than FDMA
 TDMA devices can be mass produced by VLSI giving rise to low cost
 TDMA offers the possibility of a frame monitoring of signal strength (or BER)
to enable better handoff strategies
 Ideal for digital communications
 TDMA is also the most cost-effective technology for upgrading a current
AMPS analog system to digital
TDMA - 11
 Ideal for satellite on-board processing
 TDMA is the only technology that offers an efficient utilization of hierarchical cell
structures offering pico-, micro-, and macro-cells
 Hierarchical cell structures allow coverage for the system to be tailored to support
specific traffic and service needs
 By using this approach, system capacities of more than 40-times AMPS can be
achieved in a cost-efficient way
 Because of its inherent compatibility with FDMA analog systems, TDMA allows
service compatibility with the use of dual-mode handsets
Disadvantage
 In TDMA, each user has a predefined time slot. However, users roaming
from one cell to another are not allotted a time slot
 Thus, if all the time slots in the next cell are already occupied, a call
might well be disconnected
 Likewise, if all the time slots in the cell in which a user happens to be in are
already occupied, a user will not receive a dial tone
 TDMA is subjected to multipath distortion because of its sensitivity to timing
 Even at thousandths of seconds, these multipath signals cause problems
 Overall TDMA is more complex and costly compared to FDMA
TDMA - 12
Practical TDMA Systems
IS-54 and IS-136 (TDMA)
IS-54: The original TDMA format, intended for use
within existing AMPS systems
 These systems use TDMA by dividing a 30-kHz channel into 3 time
slots, enabling 3 different users to occupy it at same time
 IS-54 provides a 3-fold increase in traffic capacity relative to AMPS,
given the same bandwidth allocation
 This effectively triples the capacity of the system (freq reuse)
 A second phase of the IS-54 standard provides for 6 (instead of 3)
TDMA user channels in each 30 kHz radio channel
 IS-136: Enhanced TDMA with special control channels to
allow short message service, battery life extension, other
features 6 timeslots, three users occupy in rotation

GSM (Groupe Special Mobile)
GSM standard was developed as a Pan-European
digital cellular standard to replace six incompatible
analog cellular systems then in use in different
geographic areas
GSM standard is similar to IS-54, employing TDMA, but
with 8 timeslots (7 or 8 users occupy in rotation), and
with RF carriers spaced 200 kHz apart
Japanese Digital Cellular
Please note that TDMA is well understood, commonly
employed, and is an efficient media access technique
CDMA
Each user’s signal is a continuous unique code pattern
buried within a shared signal, mingled with other users’
code patterns
If a user’s code pattern is known, the presence or absence
of their signal can be detected, thus conveying information
All CDMA users occupy same frequency at the same time!
Time and frequency are not used as discriminators
CDMA operates by using coding to discriminate between
users - instead of using freq or time slots
Each user is assigned a unique PN code sequence
The assigned code is uncorrelated with the data
Because the signals are distinguished by digital codes,
many users can share the same bandwidth simultaneously
i.e., signals transmitted in same frequency & same time
The PN code used for spreading must have
low cross-correlation values and
be unique to every user
Each user is a small voice in a roaring crowd - but with a
uniquely recoverable code
CDMA technology focuses primarily on the “DSSS”
technique
How CDMA Works – An Analogy
4 speakers are simultaneously giving presentation, each
with different language -- Arabic, Chinese, English & Hindu
CDMA
Principles
OF
English
Chinese
Hindu
Arabic English
Major
 You are in the audience, and English is your native language

How CDMA Works
You only understand the words of the English speaker
and tune out the Arabic, Chinese, and Hindu speakers
You hear only what you know and recognize
This is the general idea of CDMA systems
Multiple users share the same frequency band at the
same time, yet each user can only recognize his or her
own code
This technique allows numerous phone calls to be
simultaneously transmitted in one radio frequency band
 Coded conversations are encoded/decoded for each user
A signal correlated with a given PN code and
decorrelated with the same PN code returns the original
signal
Universal Frequency Reuse
Uses one universal cell frequency reuse pattern
improves the capacity of the system
Ease of freq management is also found in
DS/CDMA
Power Control
Reverse Link (from mobile unit to base station)
link is designed to be asynchronous and is
susceptible to the “near-far” problem
In order to remedy this, the use of power control is
employed
Characteristic of DS/CDMA
Effective use of the power control will ensure that
power control must be accurate and fast enough to
compensate for fading
Forward Link (from base station to mobile unit)
Link does not suffer much from near-far problem
since all cell signals can be received at the mobile
with equal power
When at excessive intercell interference, the power
control can be applied by increasing the power to the
mobile
Characteristic of DS/CDMA
1. In CDMA, a signal is spread into a larger freq band
than is needed to represent it - the redundancy gives
error resilience, and the wideband frequency combats
multipath effects because of frequency diversity
2.Cell-reuse patterns are no longer strictly necessary
3.CDMA is described as having a universal one-cell
reuse pattern
In Summary

1.Voice Activities Cycles
 CDMA is the only technique that succeeds in taking
advantage of the nature of human conversation
 In CDMA, all the users are sharing one radio channel
 The human voice activity cycle is 35%, the rest of the time
we are listening
 Because each channel user is active just 35% of the entire
cycle, all others benefit with less interference in a single
CDMA radio channel
2.Improved call quality, with better and more consistent sound
as compared to other systems
3.No Equalizer Needed
 When the transmission rate is much higher than 10 kbps in
both FDMA and TDMA, an equalizer is required
 On the other hand, CDMA only needs a correlator, which is
cheaper than the equalizer
Advantages of CDMA
4.No Hard Handoff
 In CDMA, every cell uses the same radio
 This feature avoids the process of handoff from one freq to another while
moving from one cell to another
5.No Guard Time in CDMA
 TDMA requires the use of guard time between time slots
 guard time does occupy the time interval for some info bits
 This “waste” of bits does not exists in CDMA, because guard time is not
needed in CDMA technique
6.Less Fading
 Less fading is observed in the wide-band signal while propagating in a
mobile ratio environment
7.Capacity Advantage
 Given correct parameters, CDMA can have as much as four times the
TDMA capacity; and twenty times FDMA capacity per channel/cell
8.No frequency management or assignment needed
 In both, TDMA and FDMA, the frequency management is always a critical
 Since there is only one channel in CDMA, no frequency management is
needed
Advantages of CDMA
9.Enhanced privacy
 CDMA signals resistant to interception or jamming
10.Soft Capacity
 Because in CDMA all the traffic channels share a single radio channel,
we can add one additional user so the voice quality is just slightly
degraded
11.Coexistence
 Both systems, analog and CDMA can operate in two different spectra,
with no interference at all
12.Simplified system planning through the use of the same frequency in
every sector of every cell
 Improved coverage characteristics, allowing for the possibility of fewer
cell sites
13.Increased talk time for portables
14.Bandwidth on demand
Advantages of CDMA
1.Capacity not well defined
The capacity of CDMA systems is not well defined. The
effective (Eb/No) formula demonstrates the interference-
limited nature of the system, but more than one factor in that
formula is affected by the number of users, making it hard to
gauge how performance degrades as a function of users
2. The Near-Far Problem
 Effect is present when an
interfering Tx is much closer
to Rx than the intended Tx
 Assume there are 2 users,
one near the base and one
far from the base as shown
 CDMA interference comes
mainly from nearby users Near-Far effect illustrated
Disadvantages of CDMA

Although the cross-correlation between codes A and B is low, the correlation between
the received signal from the interfering Rx and code A can be higher than the
correlation between the received signal from the intended Rx and code A
In CDMA, stronger received signal levels raise the noise floor at the base station
demodulators for the weaker signals, thereby decreasing the probability that weaker
signals will be received
The result is that proper data detection is not possible
To help eliminate the “Near-Far” effect, power control is used
 Base Station (BS) rapidly samples the signal strength of each mobile and
then sends a power change command over the forward link
 This sampling is done 800 times per second and can be adjusted in 84
steps of 1 dB
The purpose of this is so that the received powers from all users are roughly
equal
That is, when a mobile unit is close to a BS, its power output is lower
 the mobile unit transmits only at the power necessary to maintain
connection
This solves the problem of a nearby subscriber overpowering the BS receiver
and drowning out the signals of far away subscribers
An extra benefit of power control is extended battery life
Disadvantages of CDMA
IS-95 (cdmaOne)
After the development of the IS-54 standard,
Qualcomm, a San Diego-based company, developed a
new digital cellular system design utilizing Code Division
Multiple Access (CDMA)
This is known as IS-95
Unlike IS-54, which utilizes the same 30-kHz (same as
AMPS), IS-95 uses a SS signal with 1.2288 MHz
spreading bandwidth
 a frequency span equivalent to 41 AMPS channels
IS-95 has been shown to theoretically offer greater
traffic capacity than TDMA
CDMA2000
Practical CDMA Systems
CDMA Performance - 1
CDMA System Analysis
Users are identified by unique code sequence
Let
 K = number of users
 dk = kth users baseband data sequence with amplitude 1
 ak = kth users spreading code sequence with amplitude 1
 Please note that ak(t) and dk(t) are completely independent
   
b
b
k ki ki T b
i i
b
t iT
d s s P t iT
t
T

 
   
 
 
 
   
c
c
k kl kl T c
l l
c
t lT
a a a P t lT
t
T



 
   
 
 
 
CDMA Transmitter
 First the data symbols dk(t) are spread into ak(t)dk(t)
 Then spread signal is modulated (usually by PSK)
 Notice that
N=PG = Gp = number of chips per data symbol = processing gain
 Hence, resulting spread spectrum signal can be written as
x
Baseband
BPF
PN Code
Generator
Data signal Transmitted Signal
xk(t)
Chip Clock
~
ak(t)
dk
(t)
ak(t)dk(t)
Modulator
 
c
Acos t

1
c
c
f
T
 b
b c
c
T
T NT N
T
  
 
( ) cos 2
b c
k c ki kl c
i l c
t iT lT
s t A s a f t
T
 
 
 
 
  
   
 
 
 

where fc = carrier frequency,  = carrier phase
We can simplify the expression above and use
where , Pk = k-th user power
     
( ) 2 cos 2
k k k k c k
s t P a d f t
t t  
 
2 b
k
b
E
P
T

2 s
k
s
E
or P
T

 The Channel Model
 channel output is  
1
( ) kl
L
j
kl
k kl
l
t
h t e 

 


 
   
       
     
1
1
( )
- - -
2 cos
- -
2 cos
kl
k
k
L
j
k k kl
k kl k k c k
l
L
kl kl
k kl k k c kl
l
t
y t h s d
t t t
P a d t e d
t t
P a d t

 

  
    
 
  







 
 
 
 

where
Asynchronism
kl k kl c kl
    
   
Let L be the number of resolvable paths which is
assumed to satisfy the condition
1
m
c
T
L
T
 
 
 
 
•Tm = maximum delay spread
•Tc = chip period
CDMA Receiver
Signal is first demodulated and then despread
The signal is despread by the same amount through a
cross-correlation by locally generated PN sequence
 i.e., demodulation accomplished by remodulating w/spreading
code
 involves correlation of the received signal with the delayed
version of the spreading signal (despreading operation)
 In other words, the received signal is multiplied again by a
synchronized version of the PN code
 
0
b
T
dt

 kl
ŝ
( )
k d
a t T

 
r t
2 cos( )
k c k
P t
 

Demodulator
 
y t Decision
Device
CDMA system model (k-th user)
Notice that the despreading operation is similar to the
spreading operation
X
 
k
a t
 
k
d t X  +
r(t)
(t)
n
X
k
a (t-τ) c k
Acos(ω t+ )

X ( )

z0
T
dt
kl
s (t)
ˆ
c
Acos(ω t+ )
k

PN signal
Generator
Channel
Receiver
Transmitter

CDMA system model (K active users)
 Using a simplified diagram, can determine the received signal
X
1
a (t)
1
d (t) X 1
X
K
a (t)
(t)
K
d X 1
+
+
 r t
( )
 
n t
( )
n t
( )
X
(t-τ)
a
k
c c
A cos(ω t+ )

X ( )

z0
T
dt (t)
k
ŝ
c 1
cos(ω t+ )

c K
cos(ω t+ )

 
     
1
1 1
( ) ( )
- -
2 cos 2 ( )
K
k k
k
K L
kl kl
k kl k k c kl
k l
r t y n t
t
t t
P a d f t n t
 
  

 
 

  
 
 Assuming user #1 is our reference user.
 Assume that bit zero is transmitted and is being detected
(i.e., i = 0)
 Substituting
     
( 1)
1 1 cos 2
i Tb
c
iTb
z r a f t dt
t t 

 
     
1 1
0
cos 2
Tb
c
z r a f t dt
t t 
 
         
     
1 1
0
1 1
1
0
- -
2 cos cos
cos
K L Tb
kl kl
i k kl k k c c kl
k l
Tb
c
t t
z P a a d t t
t
n a t dt
t t
 
   

 
 
  
 
 What is Spread Spectrum?
 Significance of Spreading
 Basic Characteristics of SS System
 Classifications/ Benefits/Applications of Spread Spectrum
 Direct Sequence Spread Spectrum
 Summary of Direct Sequence Techniques
 Frequency Hopped Spread Spectrum
 Direct Sequence vs. Frequency Hopping
Module 6
Spread Spectrum (SS)
659
What is Spread Spectrum? - 1
 Spread Spectrum (SS) is a modulation technique where the
bandwidth of the transmitted signal is made to be greater than
the Bmin required for transmission
 The data is scattered (spread) across the available frequency
band in a pseudo random pattern
 The idea behind SS is to transform a signal with bandwidth B
into a noise-like signal of much larger bandwidth Bss
660

 Spreading Action
 At the transmitter, the baseband signal m(t), is usually
spread by a pseudo-noise (PN) code sequence p(t)
 Spreading is achieved by modulating the original
signal with a pseudo-random code sequence p(t)
 The code sequence p(t) is independent of the data
sequence m(t)
 In Spreading the signal
 The original signal is embedded in noise (see fig.)
 Power of spread signal = Power of original signal
 Total power is the area under the spectral density
curve (see fig.)
661
signals with equivalent total power may have
either a large signal power concentrated in a
small area or a small signal power spread over a
large area
Typically, power of SS signal is spread between
10-30 dB
i.e., power is spread over 10-1000 times original
power
Make signal resistant to noise, interference, and
snooping
Increases the probability of correct reception
662
Despreading
At the receiver, the received signal r(t) is despread by the
same amount
 by cross-correlating r(t) by a locally generated version of the PN
sequence p(t)
Cross-correlating with the correct sequence recovers the
original data
 Is evident from Shannon's capacity equation
 Observe the effect of increasing the bandwidth B
 If B is increased, we may decrease SNR without decreasing
capacity
2
log 1
S
C B
N
 
 
 
 
C = channel capacity in bits
B = bandwidth in hertz
S = signal power
N = noise power
663
Processing gain (PG or Gp) or “spreading factor” is
defined as
Gp is the improvement gained by spreading the BW
Gp determines the # of users that can be allowed in a system
Gp determines the amount of multipath effect reduction
Gp determines the difficulty of jamming or detecting a signal
Gp may be viewed as performance increase achieved by
spreading
 It can be used to describe the signal fidelity gained at the
cost of bandwidth expansion
Spread Bandwidth
Information Bandwidth
ss
p
B
PG G
B
  
Significance of Spreading - 1
664

It is through Gp that increased system performance
is achieved without requiring a high SNR
Gp (# of chips per data symbol ) can also be written
as
For SS systems, it is advantageous to have Gp as
high as possible
G
T
T
R
R
B
R
p
s
c
c
s
ss
s
  
2
Significance of Spreading - 2
665
 Carrier is unpredictable (pseudo-random noise) and
is wideband
BW of the transmitted signal must be greater then the
BW of the data signal
BW of transmitted signal must be determined by some
function that is independent of the message and is
known to the receiver
Despreading involves cross correlation of the received
signal with a synchronously generated replica of the
wideband carrier
Basic Characteristics of SS System
666
Direct Sequence Spread Spectrum (DS-SS)
Signal is modulated a 2nd (or 1st) time using a
wideband spreading signal/code
Frequency Hopping Spread Spectrum (FH-SS)
fc is randomly switched from one band to another
during radio transmission according to some specified
algorithm
Time Hopping Spread Spectrum (TH-SS)
The signal hope within a particular time frame
Only one time slot in a frame is modulated
Multi-Carrier Spread Spectrum (MC-SS)
Different carriers are used to transmit the signal
Classifications of Spread Spectrum - 1
667
Hybrid Forms of Spread Spectrum
These techniques implement SS in different ways, but
implementations requires:
Signal spreading by means of a code
Synchronization between pairs of users is required
Ensure that some signals do not overwhelm others
(power control)
Uses source and channel coding to optimize
performance
Direct Sequence and Frequency Hopping techniques
are the two most popular SS techniques
Classifications of Spread Spectrum - 2
668

Anti-jam (AJ) capability (especially narrow-band (NB)
jamming)
AJ capability is due to the unpredictable nature of the carrier
signal
Since NB interference affects only a small portion of the
spectrum, it is difficult to jam the entire spectrum
Because of the difficulty to jam or detect SS signals, the first
applications were in the military
Covert operation or low probability of intercept (LPI)
LPI can be achieved with high Gp and unpredictable fc
When power is spread thinly and uniformly in freq domain,
detection by surveillance receiver is difficult
Benefits and Applications of SS - 1
669
Multiple-access capability
SS systems are used for random and multiple
access systems
Users can start their transmission at an arbitrary
time without worrying about channel saturation
Multipath protection
SS implies a reduction of multipath effects, hence a
reduction in fading
i.e., high time resolution is attained by the correlation
detection of wide-band signals
670
Secure communications
SS systems achieves privacy due to unknown
random codes
Since code is unknown to a hostile user,
detection is difficult
Cryptographic capabilities result when the data
cannot be distinguished from the carrier to an
unauthorized observer
In this case, SS carrier is like a key in a cipher
system
A system using indistinguishable data and SS
carrier modulation is a form of privacy system
671
Low power spectral density (PSD)
Spreading over a large frequency-band reduces the
PSD, while Gaussian Noise level increases
This improved the spectral efficiency in some
special circumstances
Interference limited operation
Performance is limited by interference rather than
noise
Transmitter-receiver pairs using independent
random carriers can operate in the same BW with
minimal co-channel interference
672

Definition:
K = number of users, k = 1, 2, …, K
m(t) = user data signal with bit duration, Tb
p(t) = spreading code sequence (pulse or symbol of the PN
code) or “chip” with duration Tc
Note that Tc << Tb
each Tb is coded into a spreading sequence of Gp chip durations
Both m(t) and p(t) has amplitude ± 1 (anti-podal or polar)
In DS-SS, m(t) is directly multiplied by p(t)
B= bandwidth of data signal m(t)
Bss = bandwidth of spread signal s(t)
Note that Bss >> B
Please note that m(t) and p(t) are completely independent
Direct Sequence Spread Spectrum - 1
673
DS-SS Transmitter
DS-SS
Modulation
Multipath
Channel
Diversity
Receiver
Narrowband
Data Out
Narrowband
Data In
Spreading
Process
Data Bits
m(t)
Code Sequence, p(t)
Spread Signal
s(t)
1 1 0 1
1 0 0 0
1
0 0
0
1
0
1 0
1
Tb
+1
-1
m(t)
p(t)
m(t) x p(t)
-1
+1
+1
-1
1 0
1
1
0
1
0
1
1
0
1 0
1
1
0
1 0
chip
Spreading
Process
Data Bits
m(t)
PN Code Sequence
p(t)
Spread Signal
s(t)
2
b c ss
p
c b b
T R B
G
T R R
  
674
For example in IS-95, we have
b
c
p
T
T
G

Gp
Tb
Tc




Processing Gain
Bandwidth Expansion Factor
Code Length
Gp
Tb
Tc



=
9.6x103
12288 106
128
.
1 1 0 1
1 0 0 0
1
0 0
0
1
0
1 0
1
Tb
+V
-V
m(t)
p(t)
m(t) x p(t)
-V
+V
+1
-1
1 0
1
1
0
1
0
1
1
0
1 0
1
1
0
1 0
chip
Tc
IS-95
675
Each Tb is coded into a sequence of Gp chips
 This increases the rate by a factor of Gp
 Each binary chip can change with probability 0.5 in Tc sec.
First the data symbols m(t) are spread into p(t)m(t)
Then spread signal is modulated (usually by MPSK)
We must have
x
Baseband
BPF
PN Code
Generator
X
Message Transmitted
Signal
Sss(t)
Chip Clock
~
LO @ fc
 
2
( ) ( ) ( )cos 2
s
ss c
s
E
s t m t p t f t
T
 
 
1
;
c
c
f
T

b p c
b
c
T G T
T
G
T

 
676

DS-SS Receiver
r(t) is first demodulated and then despread
Demodulation is accomplished in part by re-modulation
with a PN spreading (coherent detection)
 The correlation of r(t) with the delayed version of the p(t)
(despreading operation)
 
0
b
T
dt

 m̂
( )
d
p t T

 
r t
2 cos( )
c
P t
 

Demodulator
 
y t Decision
Device
677
fc of DS is fixed, but m(t) is spread out into a much larger BW (at
least 10 times) by using PN code sequence
Both m(t) and Sss(t) signal use same amount of transmit power
However, the PSD of Sss(t) is much lower than that of m(t)
As a result, it is more difficult to detect the presence of
Sss(t)
In this case, the power density of m(t) is 10 times higher
than Sss(t), assuming the spreading ratio is 10
If there is an interference or jammer in the same band, it will be
spread out during the spreading operation
Hence, its impact is greatly reduced
i.e, the offending jammer's power is reduced by at least 90%
At the Rx the spread signal Sss(t), is despread in a similar
manner to recover m(t)
Summary of Direct Sequence Techniques
678
Frequency Hopped Spread Spectrum - 1
FH is the repeated switching of fc from one band to
another during transmission
Radio signal hops from one fc to another at a specific
hopping rate and sequence that appears to be random
(see animated)
 Overall BW required for FH is much wider than that required to
transmit the same info using only one fc
 Each fc and its associated sidebands must stay within a defined BW
 The fi(t) output of the Tx
jumps from one value to
another based on the
pseudo-random input
from the code generator
679
Typically, each fc is chosen from a set of 2k frequencies
spaced  Tb
The # of discrete frequency determines the BW of the
system
Gp is directly dependent on # of available freq choices for
a data rate
PN code does not directly modulate the data, but is used
to control the hopping sequence of fc
p t
( )
2P ct
cos( )

m t
( )
680

Minimum time required to change the frequency is
dependent on
the chip rate
the amount of redundancy used,
the distance to the nearest interference source
 Other FH transmitters will be
using different patterns, which
usually will be on non-interfering
freqs
 At Rx, FH is removed by mixing
with a local oscillator signal
which is hopping synchronously
with received signal

m t
 
r t
( )
681
To successfully jam a hopper, either the entire band
must be saturated with noise or jamming source must
be able to track the hopping sequence
Neither of these scenarios is likely to occur
naturally, and they are quite difficult to achieve
intentionally
FH-SS enjoys jamming & multipath immunity, as in
DS-SS
If data cannot be received on a particular channel due
to fading, hopper moves to an unfaded channel and
retransmits the data
FH is less effected by the “Near-Far” problem
682
FH sequences have only a limited number of “hits”
with each other
This means that if a near interferer is present, only
a number of “frequency-hops” will be blocked
instead of the whole signal
Usually FH is accomplished by multiple frequency
code selected FSK
Obtaining a high Gp is hard because of the
requirement that a frequency synthesizer be able
perform fast-hopping over fc
The faster the hopping-rate the higher the Gp
683
FH may be classified as either fast or slow
Slow FH is when the hopping rate is less than the
data rate
single hop per symbol bit
Fast FH is the converse
multiple hops per symbol bit
Hopping sequence is designed for allowing
orthogonality in cells and
minimum correlation with respect to intercell
interference
The motivation and advantages of FH is similar to that
of DS system
684

 Processing Gain
 FH does not spread the signal  no processing gain from spreading
 Power Usage
 FH requires more power to achieve same SNR compared to DS
 Synchronization
 Communication in FH is more difficult to synchronize compared to the DS
since both time and fc need to be in tune
In DS, only the timing of the chips needs to be synchronized since the
carrier fc is fixed
 Latency Time
 FH spend more time to search the signal to lock to it (longer latency time)
 DS radio can lock-in the chip sequence in just a few bits
 Usually, to make the initial synchronization possible, the hopper will park
at a fixed fc before hopping. If the jammer happens to locate at the same fc
as the parking fc, the hopper will not be able to hop at all!
 And once it hops, it will be very difficult, if not impossible to re-synchronize
if the Rx ever lost sync
Direct Sequence vs. Frequency Hopping - 1
685
 Complexity and Cost
 FH is usually more costly and more complicated than the DS because it
needs extra circuits for hopping and synchronizing
 Performance in Multipath
 FH performs better than DS in multipath fading environment
FH does not stay at the same fc and a null at one fc is usually not a null
at other fc (survives multipath environment better)
 Capacity
 FH can usually carry more data than the DS since FH is completely
narrowband at all times
 Interference Rejection Capability
 FH reduces its impact by avoiding the jammer and DS reduces its impact
by spreading or diluting the effect of the jammer (net effect is the same)
 Application
 Hence FH is more popular for voice than data communication because of
their higher error tolerance
686
Potable Comparison
Direct Sequence Frequency Hopper
Easy and Simple Complicated
Use Lower Power Use Higher Power
Short Latency Time Long Latency Time
Quick Lock-In Slow Lock-In
Short Indoor Range Long Indoor Range
Low Data Rate High Data Rate
Better for multipath channel
Less susceptible to
jamming
687

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