Combined Lecture 01-20.pdf

Principle Of
Communication Systems
Lecture 1
Instructor: Dr. Moazzam Islam
Tiwana,
Room 329 Academic Block 1,
moazzam.phd@gmail.com

2
Course Literature
Textbook:
• Analog and Digital Communication, (3rd Edition) by
B. P. Lathi, Oxford Printing Press
Reference Books:
• Communication Systems, (3rd Edition) by Simon Haykin, John Wiley &
Sons
• Analog and Digital Communication Systems, (6th Edition) by Leon W.
Couch II, Prentice Hall, 2001

3
Pre-requisites
• Signals and Systems

4
Marks Distribution
Theory Assessment:
Sessional I 10 Marks
Sessional II 15 Marks
Quizzes 15 Marks
Assignments 10 Marks
Terminal Exam 50 Marks
Total 100 Marks
Midterm 25 Marks

5
Course Objectives
• To introduce principles of analog and digital
communication systems and methods used in
modulating and demodulating signals in order to
carry information from a source to a destination

6
Communication
• Main purpose is to transfer information from a
source to destination (sink) via a channel or a
medium.

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9
Communication System
Increases with length
• Multipath effects
• Doppler Shift
• Noise
• Contact switches
• Lightning
• Engine ignition

10
• A source originates a message, such as a human voice, a
television picture.
• The message is converted by an input transducer into an
electrical waveform (baseband signal).
• The transmitter modifies the baseband for efficient
transmission.
• The channel is a medium such as a coaxial cable, an optical
fiber, a radio link.
• The receiver processes the signal received to undo
modifications made at the transmitter and the channel.
• The output transducer convert the signal into the original form

12
Digital and Analog Sources and
Systems
Basic Definitions:
• Analog Information Source:
An analog information source produces messages which are defined on a
continuum. (E.g. :Microphone)
• Digital Information Source:
A digital information source produces a finite and discrete set of possible
messages. (E.g. :Keyboard)
t
x(t)
t
x(t)
Analog Digital

Digital Transmission
13
• An analog signal is converted to a digital signal by means
of an analog to digital (A/D) converter.
• The signal m(t) is first sampled in the time domain.
• The amplitude of the signal samples ms(kT) is partitioned
into a finite number of intervals (quantization).

Analog to Digital Conversion
14
Sampling Theorem:
If the highest frequency in the signal spectrum is B, the signal can
be reconstructed from its samples taken at a rate not less than 2B
samples per second.

15
Digital and Analog Sources and
Systems
• A digital communication system transfers information from a
digital source to the intended receiver (also called the sink).
• An analog communication system transfers information from
an analog source to the sink.
• A digital waveform is defined as a function of time that can have a
discrete set of amplitude values.
• An Analog waveform is a function that has a continuous range of
values.

Digital Systems
• Digital signal are more robust to noise
• Advantages:
– Cheap electronic circuitry
– Immunity to noise
– Advanced signal processing (error correction,
equalization etc…)
…continued 3
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Riaz Hussain (rhussain@comsats.edu.pk) CIIT-IBD-EE EEE351 PCS Lecture-
01

17
Bandwidth
• Bandwidth of a channel is the range of
frequencies it can transmit with reasonable
fidelity. e.g. if a channel can transmit a signal
whose frequencies range from 0 to 4000 Hz then B
= 4KHz

18
Signal to Noise Ratio (SNR)
• It is important in two ways
– First Increasing the Signal Power will reduces the effect
of Noise on it. Larger SNR allows transmission over
longer distance.

19
Tradeo* Between S and B
• Telephone channels have limited Bandwidth but a
lot of Power
• Space vehicles have infinite bandwidth but power
is limited.

20
Channel Capacity
• Shannon Equation helps us in finding the
capacity of the channel
• ‘C’ is also known as rate of information
(bits/sec)

21
Modulation
• Modulation is a technique in which message signal is transmitted to
the receiver with the help of carrier signal.
• For modulation we change carrier’s amplitude, frequency or phase according
to message
Message signal
Message signal

22
Modulation (cont)
• The basic idea here is to superimpose the message signal
in analog form on a carrier which is a sinusoid of the form
Acos(wt + )
• There are three quantities that can be varied in proportion
to the modulating signal: the amplitude, the phase, and
frequency.
• The first scheme is called Amplitude Modulation and the
second two are called Angle Modulation schemes

23
Why Modulate
• Antenna size is a major concern
• For example in case of a wireless channel antenna size is inversely proportional to
the center frequency, this is difficult to realize for baseband signals.
– For speech signal with frequency f = 3 kHz  =c/f=(3x108)
/(3x103
)
– Monopole antenna size without modulation /4=105
/4 meters = 15 miles -
practically unrealizable
– Same speech signal if amplitude modulated using fc=900MHz will require an
antenna size of about 8cm.
– This is evident that efficient antenna of realistic physical size is needed for radio
communication system

24
Why Modulate (Cont.)
• Simultaneous Transmission of several Signals
– Frequency Division Multiplex (FDM)

Comparative Analysis of Analog and Digital Communication
25
Message Signals
Wireless
Channel
Recovered Messages
Transmitter
Receiver
Modulated Signal
DeMod
DeMod
DeMod
Modulator
Modulator
Modulator
MUX
(FDM)
DEMUX
(Tuner)
Analog Communication: Transmitter and Receiver
Once a particular modulated signal has
been isolated, the demodulator converts
the carrier variation of amplitude or
angle back into a baseband signal
voltage

Introduction to Signals
Chapter 02
Lecture 02

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Signals
• A signal is a set of information or data.
Examples
– a telephone or television signal,
• We deal exclusively with signals that are
functions of time.

Instantaneous Power Of A
Signal
Instantaneous power across a resistor is
where is the voltage and is the current across the resistor.
– In communication systems, power is normalized
by assuming , hence instantaneous power
can be represented by
where is either a voltage or a current signal.
( )
p t R
2
( ) 2
( ) ( )
v t
R
p t i t R
= =
( )
v t ( )
i t
1
R = W
2
( ) ( )
p t x t
=
( )
x t

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Energy of Signals
• The signal energy Eg of g(t) is defined (for
a real signal) as
• In the case of a complex valued signal
g(t), the energy is given by

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Average Power of Signals
• For signals with infinite duration as shown
below, total energy is infinite
• A more meaningful quantity is Average
Power is defined as
• For complex signals

Units of Signal Energy and Power
• Signal Energy:
• Signal Power:
– Logarithmic scale
• Convenient notation to deal with decimal points and
zeros
• Particularly, when signal power is very large or very
small
• Convention:
– dBw
– dBm
– Example:
9/21/20 6
Riaz Hussain (rhussain@comsats.edu.pk) CIIT-IBD-EE EEE351
PCS Lecture-02
Joules
Watts
[10𝑙𝑜𝑔!"𝑃] 𝑑𝐵𝑤 ;
[30 + 10𝑙𝑜𝑔!"𝑃] 𝑑𝐵𝑚 ;
−30 dBm represents: 𝑃 =∶ 10#$
𝑊𝑎𝑡𝑡𝑠

Example 2.1
• Measure the signal
a)
b)
Signal power is square of rms value so,
rms value:
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PCS Lecture-02
!
"
;

Example 2.2
• Measure the power of periodic signals
a)
b)
c)
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PCS Lecture-02
𝑔 𝑡 = 𝐴𝐶𝑜𝑠 𝜔!𝑡 + 𝜃 ;
𝑔 𝑡 = 𝐶"𝐶𝑜𝑠 𝜔"𝑡 + 𝜃" + 𝐶#𝐶𝑜𝑠 𝜔#𝑡 + 𝜃# ;
𝑔 𝑡 = 𝐷𝑒$%&!'
;

1.2 Classification Of Signals
1. Deterministic and Random Signals
• A signal is deterministic means that there is no uncertainty with
respect to its value at any time.
• Deterministic waveforms are modeled by explicit mathematical
expressions, example:
• A signal is random means that there is some degree of
uncertainty before the signal actually occurs.
• Random waveforms/ Random processes when examined over a
long period may exhibit certain regularities that can be described
in terms of probabilities and statistical averages. E.g. mean, rms,
variance
x(t) = 5Cos(10t)

2. Periodic and Non-periodic Signals
• A signal x(t) is called periodic in time if there exists a constant
T0 > 0 such that
(1.2)
t denotes time
T0 is the period of x(t).
0
x(t) = x(t + T ) for - < t <
¥ ¥

3. Continuous and Discrete Signals
• An continuous signal x(t) is a continuous function of time;
that is, x(t) is uniquely defined for all t
• A discrete signal x(kT) is one that exists only at discrete
times; it is characterized by a sequence of numbers defined
for each time, kT, where
k is an integer
T is a fixed time interval.
A discrete signal
Continuous signals

4. Energy and Power Signals
• x(t) is classified as an energy signal if, and only if, it has
nonzero but finite energy (0 < Ex < ∞) for all time, where:
(1.7)
• An energy signal has finite energy but zero average power.
• Signals that are both deterministic and non-periodic are
classified as energy signals
T/2
2 2
x
T / 2
E = x (t) dt = x (t) dt
lim
T
¥
®¥ - -¥
ò ò

• Power is the rate at which energy is delivered.
• A signal is defined as a power signal if, and only if, it has
finite but nonzero power (0 < Px < ∞) for all time, where
(1.8)
• Power signal has finite average power but infinite energy.
• As a general rule, periodic signals and random signals are
classified as power signals
4. Energy and Power Signals
T/2
2
x
T /2
1
P = x (t) dt
T
lim
T
®¥ -
ò

21/09/2020 EEE 352 14
Unit Impulse Function
• The Unit Impulse function or Dirac function
is defined as
• Multiplication of a function by an impulse
)
(
)
0
(
)
(
)
( t
t
t d
f
d
f =

21/09/2020 EEE 352 15
• Delayed impulse
• Sampling property of the Unit Impulse
Function
)
(
)
(
)
(
)
( T
t
T
T
t
t -
=
- d
f
d
f
)
0
(
)
(
)
( f
d
f =
ò
¥
¥
-
dt
t
t
)
(
)
(
)
( T
dt
T
t
t f
d
f =
-
ò
¥
¥
-

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Unit Step function
• Unit step function u(t), defined as

Correlation
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PCS Lecture-02
• The correlation coefficient of two signals is
given as:

Correlation Coefficient for
complex signals
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Correlation: Examples
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PCS Lecture-02
• Example 2.6:

Correlation: Application to Signal Detection
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PCS Lecture-02
Correlation: Measure of degree of similarity
• Used in:
– Radar, Sonar, Digital Communication, Electronic
Warfare, etc.
• Example:
– Let’s transmit a signal pulse 𝑔 𝑡 to detect a target, if
• Pulse reflected => Target is present
• Else => No target
– Challenge:
• Detect the heavily attenuated pulse in the presence of noise
– Solution:
• Use the correlation of the transmitted pulse with received pulse

Correlation: Application to Signal Detection
9/21/20 23
PCS Lecture-02
– Received Signal:
– Key to target detection is orthogonality between 𝜔 𝑡 and
𝑔 𝑡 − 𝑡@ i.e.
– To detect a find the correlation between 𝑧 𝑡 and the
delayed pulse 𝑔 𝑡 − 𝑡@
– When 𝑡@ is unknown:
• We use a bank of N correlators each using a different delay 𝜏#
𝒓𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒂𝒏𝒅
𝒂𝒕𝒕𝒆𝒏𝒖𝒂𝒕𝒊𝒐𝒏 𝒍𝒐𝒔𝒔
𝑷𝒓𝒐𝒑𝒂𝒈𝒂𝒕𝒊𝒐𝒏 𝒅𝒆𝒍𝒂𝒚
(RTT)
𝑪𝒂𝒑𝒕𝒖𝒓𝒆𝒔:
Noise and
Interference
𝑷𝒖𝒍𝒔𝒆 𝑬𝒏𝒆𝒓𝒈𝒚
𝑻𝒉𝒓𝒆𝒔𝒉𝒐𝒍𝒅𝒊𝒏𝒈
𝑰𝒅𝒆𝒏𝒕𝒊𝒇𝒚, 𝒑𝒆𝒂𝒌 𝒄𝒐𝒓relation at correct delay t0

Autocorrelation Functions
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PCS Lecture-02
The correlation of a signal with itself
The autocorrelation function 𝜓A 𝜏 of a function 𝑔 𝑡
Is the measure of similarity of a signal with its own displaced
version
Autocorrelation provides valuable information about the signal
spectrum

Energy of the Orthogonal Signals
9/21/20 25
PCS Lecture-02
Proof

Trignometric Fourier Series
• Used to represent a periodic function as the sum of
cosine and sine functions.
• Initially designed to solve heat equation.
• Widely used in solving mathematical and physical
problems, especially linear differential equations with
constant coefficients.
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Existence of the Fourier Series
• Weak Dirichlet condition
• Function has finite number of maxima and minima, and
finite number of discontinuties in one time period.
• If the function satisfies above two conditions then strong
Dirichlet condition exists.
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Trignometric Fourier Series
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Compact Trignometric Fourier
Series
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Examples 2.7, 2.8 and 2.9

Exponential Fourier Series
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Examples 2.10 and 2.11

Lecture 03
Analysis and Transmission of Signal – I
- Fourier Transform and Properties
- Modulation Property
- Convolution and Properties
- Bandwidth of Product of Signals
Dr. Ghufran Shafiq
EEE 351
Principles of Communication Systems
Fall 2020
Department of Electrical and Computer Engineering
COMSATS University Islamabad

Aperiodic Signal Representation by Fourier
Integral
2
 We have learned Fourier Series for Periodic Signals
 The Concept can be extended for Aperiodic Signals
Aperiodic Signal
Periodic Signal

Convolution
19
 Remember Cross Correlation Function?
 Convolution: Effect of LTI system on a
Signal
Note time reversal of w(t)

Properties of Convolution
20
 Time Convolution Property
if
Time Convolution→ Frequency Multiplication
 Frequency Convolution Property
Time Multiplication→ Frequency Convolution

Properties of Convolution
21
 Time Convolution Property Proof

Bandwidth of product of two signals
22
 Bandwidth of a signal
 Difference between highest and
lowest frequency in acceptable
magnitude range
 If g1(t) and g2(t) have bandwidths
B1 and B2, then bandwidth of
g1(t)g2(t) is
B1 + B2

Effect of Time-Derivation and Integration on
Fourier Transform
23
For Proofs, see example 3.14 and section 3.3.7

Example – Fourier
Transform Using
Differentiation
Property
24
1st derivative
2nd derivative

Example – Fourier
Transform Using
Differentiation
Property
25
1st derivative
2nd derivative

Important Fourier Transform Operations
26

Lecture 04
Analysis and Transmission of Signal – II
- Signal Transmission Through LTI System
- Distortion-less Transmission
- Ideal Filters and Practical Approximations
- Signal Distortion over a Channel
- Energy Spectral Density and Power Spectral Density
Dr. Ghufran Shafiq
EEE 351
Principles of Communication Systems
Fall 2020
Department of Electrical and Computer Engineering
COMSATS University Islamabad

Signal transmission through a LTI System
2
 Consider a linear time invariant (LTI) system
 Assume the input signal is a Dirac function δ(t). Call
the observed output h(t)
d(t) h(t)
LTI
System
 h(t) is called the unit impulse response function
 With h(t), we can relate the input signal to its output
signal through the convolution formula:

Signal transmission through a LTI System
3
 If
 Then using time convolution property of convolution
)
(
)
( 
G
t
g 
)
(
)
( 
H
t
h 
)
(
)
(
)
( t
h
t
g
t
y 
= )
(
)
(
)
( 

 H
G
Y =
 H(ω) is called the system frequency response or the
spectral response

Distortion-Less Transmission
4
 What is the required behavior of an ideal transmission line?
 The output signal from an ideal transmission line may have some time delay
and different amplitude than the input
 It must have no distortion—it must have the same shape as the input

Ideal Distortion-less Transmission
5
 Output signal in time domain
 Output signal in frequency domain
 Channel’s Frequency Response
)
(
)
( d
t
t
kg
t
y −
=
d
jwt
e
kG
Y −
= )
(
)
( 

d
jwt
ke
H −
=
)
(

Ideal Distortion-less Transmission
6
 Channel’s Frequency Response
d
jwt
ke
H −
=
)
(
( ) k
w
H =
( ) d
h wt
w −
=

For distortion-less transmission magnitude of frequency
transfer function must be constant and phase response
must be linear

Intuitive Explanation of Distortion-less
Transmission
7
( ) d
h wt
w −
=

 Suppose a signal composed of various sinusoids
being passed through a distortionless system
 Output signal is the input signal multiplied by k
and delayed by td
 System’s frequency response should be such
that each component suffers the same
attenuation k
 Each component undergoes same delay of td
seconds
 Time delay is negative of slope of the system phase
response
( ) k
w
H =
Constant Time
Delay→ Linear
Phase

Ideal filters
8
 Distortionless Transmission of certain band of
frequencies
 Suppresses unwanted frequency components
 Linear Phase, Constant Magnitude
 Ideal Lowpass Filter
Lowpass
Highpass
Bandpass

Ideal filters
9
 Not physically realizable because?
 Non-causal impulse response
 Everlasting Response
Impulse
Response
Frequency
Response

Practical (realizable) filters
10
 Several useful approximations to ideal low-pass filter
 One Approximation: Butterworth Filter
 Popular because they provide almost equal (flat) gain in passband
 Where B is the -3dB cutoff Frequency

Practical (realizable) filters
11
 Frequency (Magnitude) spectrum of Butterworth Filter

Comparison: Ideal and Butterworth Filter
12

Signal Distortion over a Communication
Channel
13
 Linear Distortion
 Distortion by Channel Nonlinearities
 Multipath Distortion
 Fading Channels

Linear Distortion
14
 Consider LTI Channel
 Distortion caused by non-ideal magnitude, phase or both (recall
distortionless transmission)
 Spreading or dispersion of signal will occur

Linear Distortion (Example)
15

Distortion caused by Channel Nonlinearities
16
 Consider a nonlinear channel where output is a nonlinear function of
input
( )
g
f
y =
( ) ( ) ( ) ( ) ......
3
3
2
2
1
0 +
+
+
+
= t
g
a
t
g
a
t
g
a
a
t
y
 We know that if the Bandwidth of g(t) is B Hz then Bandwidth of gk(t) =
kB Hz
 Expansion with McLaurin’s series Example

Distortion caused by Channel Nonlinearities
17
 Output spectrum spreads
beyond input spectrum
 Addition of new Frequencies
 High Efficiency Power Amplifiers
for Transmission are non-linear

Multipath Effects
18
 Transmitted signal arrives at receiver from
two or more paths
 Consider two paths: i) unity gain and delay of
td, ii) gain of α and a delay of (td + Δt)

Multipath Effects
19
 Destructive Interference
when w = nπ/Δt (n odd), cos wΔt = -1
( ) 1
,
,
0 
 
when
w
H
 Constructive Interference
when w = nπ/Δt (n even), cos wΔt = 1

Fading Channel
20
 Till now we assumed channel characteristics to be constant with
time
 In practice, these characteristics can vary with time
 Changes with season, time of day, hour to hour changes, weather
 One way to reduce fading is to use Automatic gain Control (AGC)

Signal Energy and Energy Spectral Density
21
 Parseval’s Theorem
 Signal Energy can also be determined from its Fourier Transform
 Changing order of integration

Parseval’s Theorem (Example)
22

Energy Spectral Density (ESD)
23
 Characterizes distribution of
signal’s energy in Frequency
Domain
 Useful in communication systems
when evaluating the signal and noise

Energy of Modulated Signals
24
 Recall, AM signal is
( ) ( ) t
w
t
g
t 0
cos
=

 Its Fourier Transform is
( ) ( ) ( )
 
0
0
2
1
w
w
G
w
w
G
w −
+
+
=

 The ESD can be found as
( ) ( ) ( )2
0
0
4
1
w
w
G
w
w
G
w −
+
+
=


Energy of Modulated Signals
25
 If w0≥2πB, then G(w+w0) and G(w-w0) are
nonoverlapping and
 Observe that the area under
modulated signal is half the area
under baseband signal
( ) ( ) ( )
 
2
0
2
0
4
1
w
w
G
w
w
G
w −
+
+
=

( ) ( ) ( )
 
0
0
4
1
w
w
w
w
w g
g −

+
+

=

g
E
E
2
1
=


Time Autocorrelation function and Energy
Spectral Density
26
 Recall, for a real signal the autocorrelation function g(t) is
defined as
 Autocorrelation is an Even Function



−
+
= dt
t
g
t
g
g )
(
)
(
)
( 


 Note that
)
(
)
( 


 −
= g
g

Time Autocorrelation function and Energy
Spectral Density
27
 Fourier Transform of Autocorrelation Function
 Therefore ESD is the Fourier Transform of Autocorrelation Function
  2
)
(
)
( 

 G
g =

)
(
)
( 

 Ψ

g

ESD of Input and Output
28
 If g(t) and y(t) are the input and the corresponding output of LTI
system, then
 Therefore ESD of output is |H(ω)2| times ESD of input

Signal Power and Power Spectral Density (PSD)
29
 Recall, the power Pg of a real signal g(t) is given by
−

→
= 2
2
2
)
(
1
lim
T
T
T
g dt
t
g
T
P
 We take a truncated signal gT(t)
 The integral on the right hand side will be the energy of the
truncated signal, thus
T
E
P T
g
T
g 
→
= lim

Signal Power and Power Spectral Density (PSD)
30
 From Parseval’s Theorem
 The power of signal is given as
( ) ( ) dw
w
G
dt
t
g
E T
T
gT
2
2
2
1




−


−
=
=

T
E
P T
g
T
g 
→
= lim
( )
dw
T
w
G
P T
T
g
2
lim
2
1



−

→
=

 where
( )
( )2
lim
T
w
G
w
S T
T
g 
→
=
 Sg(w) is the Power Spectral
Density of Power Signal, Which
is actually the time average of
ESD
( ) ( )dw
w
S
dw
w
S
P g
g
g 




−
=
=
0
1
2
1


( )df
f
S
P g
g 

=
0
2

Time Autocorrelation [Power Signals]
31
 Time Autocorrelation Function of Power Signals

−

→
+
=

2
2
)
(
)
(
1
lim
)
(
T
T
T
g d
t
g
t
g
T



 Similar to Energy signals, Autocorrelation Function of Power
signals is Even
)
(
)
( 
 −

=
 g
g

Time Autocorrelation [Power Signals]
32
 As ESD is Fourier transform of autocorrelation function for Energy
Signals )
(
)
( 

 Ψ

g
 Similarly, for power signals
T
gT
T
g
)
(
lim
)
(




→
=

 Because
  2
)
(
)
( 

 T
gT G
=

  )
(
)
(
lim
)
(
2


 g
T
T
g S
T
G
=
=



→
 So

PSD of INPUT and Output Signals
33
 Fourier Transform of Output is given as
 then
 Since
 So
)
(
)
(
)
( 

 G
H
Y =
)
(
)
(
lim
)}
(
{
2


 g
T
T
g S
T
G
=
=



→
2
2
2
)
(
)
(
)
( 

 G
H
Y =
)
(
)
(
)
(
2


 g
y S
H
S =

PSD of Modulated Signals
34
 The modulated signal can be represented by
 Its Fourier Transform
t
t
g
t 0
cos
)
(
)
( 
 =
 
)
(
)
(
4
1
)
( 0
0 




 −
+
+
= g
g S
S
S
g
P
P
2
1
=


Lecture 03
1
Amplitude Modulation
Dr. Mohammad kaleem
Lecture 05
03/10/2022 EEE 352 2
Lecture Outline
 Modulation
 Why Modulation
 AM Modulation
 Types of AM Modulations
 Comparison of AM Modulations
 Frequency Mixer
 Modulator Vs Mixer
 Examples
 Quiz
 Summery
1
2

Lecture 03
2
Modulation
The process by which some characteristics of a
carrier wave is varied in accordance with an
information-bearing signal.
• Two Types of Communication:
– Baseband Communication
– Passband Communication
Baseband designates the band of
frequencies of the source signal. e.g.
Audio Signal (4 kHz), Video (4.3 MHz)
10/3/2022 5
Why Modulation?
10/3/2022 4
• To
– use the range of frequencies more suited to the
medium
– allow the number of signals to be transmitted
simultaneously (Frequency Division Multiplexing)
– reduce the size of antennas in wireless links
Modulation causes a shift in the range of
frequencies in a signal
3
4

Lecture 03
3
Continuous-wave Modulation
10/3/2022 5
• Amplitude modulation
• Frequency modulation
• Phase modulation
• AM modulation family
– Double sideband-suppressed carrier (DSB-SC)
– Double sideband-Transmitted carrier (DSB-TC)
– Single sideband (SSB)
– Vestigial sideband (VSB)
10/3/2022 6
A carrier wave whose amplitude is varied in
proportion to the instantaneous amplitude of a
modulating voltage
Carrier Signal: or
cos(2 ) cos( )
c c
f t t
 
Modulating Message Signal: or
( ) : cos(2 ) cos( )
m m
m t f t t
 
The AM Signal: ( ) ( )cos(2 )
AM c
s t m t f t


5
6

Lecture 03
4
03/10/2022 EEE 351 7
AMPLITUDE MODULATION:
DOUBLE SIDE BAND (DSB)
• Modulating signal, base band signal, information
signal
• Carrier signal:
• with Spectrum
)
(t
m )
(
M
)
cos(
)
( c
ct
t
c 
 

 
)
(
)
(
5
.
0
)
( c
c
C 





 



Fourier Transform of
Cosine.
03/10/2022 EEE 352 8
DSB (cont)
• Modulation is the product of the base band with
the carrier
 
)
(
)
(
2
1
)
cos(
)
( c
c
c M
M
t
t
m 



 



Identity Equation: Multiplication of
Two Input signals produces
cosA*cosB = cos (A+B)+cos (A-B)
7
8

Lecture 03
5
03/10/2022 EEE 352 9
DSB (cont)
• DSB-SC (Suppressed Carrier) modulation
simply shifts the frequency contents of m(t) to
the carrier frequency
Wc >= 2∏B must
03/10/2022 EEE 352 10
USB & LSB
9
10

Lecture 03
6
03/10/2022 EEE 352 11
Demodulation
• To demodulate we multiply the signal by a
• Therefore the FT of this signal is
• If we lowpass filter this signal we recover
)
cos( t
wc
  







 )
2
cos(
2
1
2
1
)
(
)
(
cos
)
(
)
cos(
)
cos(
)
( 2
t
w
t
m
t
w
t
m
t
w
t
w
t
m c
c
c
c
 
)
2
(
)
2
(
4
1
)
(
2
1
)
2
cos(
2
1
2
1
)
( c
c
c w
M
w
w
M
w
M
t
w
t
m 










 
)
(
2
1
t
m
03/10/2022 EEE 352 12
Demodulation (cont)
• We need a carrier of exactly the same frequency
and phase as the carrier used for modulation:
Synchronous Detection or Coherent Detection
11
12

Lecture 03
7
03/10/2022 EEE 352 13
Demodulation (cont)
Comparison of AM Modulations Schemes
Parameters DSB-FC DSB-SC SSB VSB
Carrier Suppression NA Yes Yes NA
Side Band
Suppression
NA NA One side
band
suppressed
One side band
Partially
Suppressed
BW 2fm 2fm fm (fm+fv)
Transmission
Efficiency
less Better than
DSB-FC
Best Lower than SSB
Applications Broadcasting Broadcasting P-t-P TV Transmission
Power Transmission Pc+Pusb+Pl
sb
Pusb+Plsb Plsb OR
Pusb
Plsb+Pv+Pusb
Complexity Simple Simple Most
Complex
Less Complex
than SSB
13
14

Lecture 03
8
03/10/2022 EEE 352 15
Frequency Mixer or converter
• We wanted to change the modulated signal from wc to wI
• The product x(t) is
• Down conversion if we select
t
t
t
m
t
x mix
c 
 cos
cos
)
(
2
)
( 
   
 
t
t
t
m
t
x mix
c
mix
c 


 


 cos
cos
)
(
)
(
I
c
mix 

 

   
 
t
t
t
m
t
x I
c
c
I
c
c 




 




 cos
)
(
cos
)
(
)
(
   
 
t
t
t
m
t
x I
c
I 

 

 2
cos
)
cos
)
(
)
(
03/10/2022 EEE 352 16
Frequency Mixer (cont)
• Up conversion if we select
I
c
mix 

 

   
 
t
t
t
m
t
x I
c
c
I
c
c 




 




 cos
)
(
cos
)
(
)
(
   
 
t
t
t
m
t
x I
c
I 

 

 2
cos
)
cos
)
(
)
(
Note: Most commercial AM radio receivers receives RF = 560 – 1640KHz,
But this is shifted to an intermediate frequency (IF) which is 455KHz band
For the purpose of processing. This is normally done before demodulation.
15
16

Lecture 03
9
03/10/2022 EEE 352 17
Frequency Mixer (cont)
03/10/2022 EEE 352 18
Modulator Vs Mixer
Modulator acts as a Multiplier. It will produce the Product of two
Inputs (Base band + Carrier). If this device is used at Tx side we
Call it Modulator.
But at receiver side O/P tune circuit selects the required frequency. The
Same modulator at the Rx side is called Mixer.
Both Modulator and Mixer act like a Multiplier (Multiplying two input signals)
17
18

Lecture 03
10
Example: The modulating signal 20 cos(2∏ * 10 3 ∗ t) is used to modulate a carrier
Signal 40 cos(2∏ * 10 4 ∗ t). Find the modulation index, Percentage modulation,
Frequencies of sidebands components and their amplitudes. What is the
Bandwidth of the modulated signal?
Solution:
Given m(t) = Am = 20 cos(2∏ * 10 3 ∗ t)………I
Ac(t) = Ac = 40 cos(2∏ * 10 4 ∗ t)………..II
Required,
m = ? , %m =? , fusb = ? , Flsb = ? , BW = ?
Formula, Am = Am cos Wm t………….III
Ac = Ac cos Wc t ……………IV
m = Am / Ac
BW = 2 fm
If we compare equation I and III, we can find
Am = 20, Wm = 2∏ * 10 3
Therefore, fm = 1000Hz = 1KHz
Similarly, by comparing equations II and IV
Ac = 40, Wm = 2∏ * 10 4
Fc = 10,000Hz, = 10KHz
Example
03/10/2022 EEE 352 20
Where m = Am / Ac = 20 / 40 = 0.5, So % m = 50%,
Now the frequencies are calculated as
fusb = fc+fm = 10KHz + 1KHz = 11 KHz
flsb = fc-fm = 10KHz – 1KHz = 9 KHz
Amplitude of each side band = (m Ac/ 2) = (0.5 * 40)/2 = 10 Volts
Finally the BW of Modulated signal = 2fm = 2 * 1KHz
= 2 KHz
19
20

Lecture 03
11
03/10/2022 EEE 352 21
03/10/2022 EEE 352 21
Example
Riaz Hussain (rhussain@comsats.edu.pk) CIIT-IBD-EE EEE351 PCS Lecture-05
Quiz
03/10/2022 EEE 352 22
Which of the following Amplitude Modulation schemes require minimum
Band width (most efficient)
 Conventional AM
 Single Side band (SSB)
 Double side band suppressed carrier (DSB- SC)
 Vestigial Side band (VSB)
21
22

Lecture 03
12
03/10/2022 EEE 352 23
Summery
 Modulation
 Why Modulation
 AM Modulation
 Types of AM Modulations
 Comparison of AM Modulations
 Frequency Mixer
 Modulator Vs Mixer
 Examples
03/10/2022 EEE 352 24
23
24

Lecture 03
1
Chapter 04
Mohammad kaleem
Lecture 06
Lecture Outline
03/10/2022 EEE 351 2
 DSB-TC / DSB- FC
 Modulation Index (example)
 Power Efficiency
 Power Efficiency calculation of Tone Modulation
 De Modulation of AM Signals
 Rectifier Detector
 Envelope Detector
 DSB-TC Simple Radio Receiver
 Fox Hole Radio
 Example
 Summery
1
2

Lecture 03
2
03/10/2022 EEE 351 3
AMPLITUDE MODULATION with
Transmitted Carrier (DSB-TC)
• In this case we send the carrier with the signal
• We can think as the modulating signal to be










Sidebands
c
Carrier
c
AM t
t
m
t
A
t 

 cos
)
(
cos
)
( 

t
t
m
A
t c
DC
a
with
signal
ulating
AM 
 cos
)
(
)
(
mod















03/10/2022 EEE 351 4
AM (cont)
• The spectrum of this signal is
   



 



 




 



 

spectrum
Carrier
c
c
spectrum
SC
DSB
c
c
AM w
w
w
w
A
w
w
M
w
w
M
w )
(
)
(
2
1
)
(
)
(
2
1
)
( 











3
4

Lecture 03
3
03/10/2022 EEE 351 5
03/10/2022 EEE 351 6
A is not large enough
to satisfy the condition
5
6

Lecture 03
4
03/10/2022 EEE 351 7
AM (cont)
• A is large enough that . The demodulation
can be achieved by a simple envelope detector
If m(t) >= 0, for all t, then A = 0, will still satisfy the above condition. But if
m(t) is not ≥ 0, for all t i.e.; m(t) does take on negative values over some range of t.
• Let’s consider the peak value of to be . Then the
condition for envelope detection of AM signal is
• Which is equivalent to
0
)
( 
 t
m
A
)
(t
m p
m
0
)
( 
 t
m
A
p
m
A 
In terms of Amplitude
03/10/2022 EEE 351 8
Modulation Index
• We define the modulation index as
• Therefore we can see that if we want to maintain the
condition
• We have
A
mp


p
m
A 
1
0 
  Condition for envelope detection
7
8

Lecture 03
5
03/10/2022 EEE 351 9
Example 4.4
(Tone modulation: modulating signal
is pure sinusoid (or tone) )
03/10/2022 EEE 351 10
Example 4.4 (cont)
Sketch ФAM (t) at
μ = 0.5, 1
Here, B = UA
B = A
B = μA
2B = A
9
10

Lecture 03
6
03/10/2022 EEE 352 11
Percentage Modulation
Under modulated (<100%) 100% modulated
Envelope Detector
Can be used
Envelope Detector
Gives Distorted signal
Over Modulated (>100%)
A < mp
μ > 1
03/10/2022 EEE 351 12
Sideband and Carrier Power
• The advantage of envelope detection in AM has its price
• The carrier Power is (see example 2.2)
• The sidebands Power is half of modulating signal
(baseband)








sidebands
c
carrier
c
AM t
t
m
t
A
t 

 cos
)
(
cos
)
( 

2
2
A
Pc 
~~~~~~~
2
)
(
2
1
t
m
Ps 
11
12

Lecture 03
7
03/10/2022 EEE 351 13
Sideband and Carrier Power (cont)
• The sideband power is the useful power and the Carrier
Power is the power wasted
• We define the Power Efficiency as
%
100
*
)
(
)
(
)
(
2
1
2
)
(
2
1
~~~~~~~
2
2
~~~~~~~
2
~~~~~~~
2
2
~~~~~~~
2
t
m
A
t
m
t
m
A
t
m
P
P
P
TotalPower
r
UsefulPowe
s
c
s








03/10/2022 EEE 351 14
Sideband and Carrier Power (cont)
For the special case of tone modulation
then its power is
then
The max value when (100% modulation) is
t
A
t
m m

 cos
)
( 
 
2
)
(
2
~~~~~~~
2 A
t
m


 
 
 
 
%
100
*
2
%
100
*
2
2
%
100
*
2
2
1
2
2
2
1
2
2
2
2
2
2
2
2












A
A
A
A
A
A
1

 %
33


13
14

Lecture 03
8
03/10/2022 EEE 351 15
Demodulation of AM Signals
• We do not need a local generated carrier in this case
• If we have under-modulation then we can use
1. Rectifier detection
2. Envelope detection
03/10/2022 EEE 351 16
Rectifier detector
15
16

Lecture 03
9
03/10/2022 EEE 351 17
Rectifier detector (cont)
• If the AM wave is applied to diode and resistor circuit.
The negative part of the AM is suppressed. This is like
saying that we have half wave rectified the AM
Mathematically
  















 .....
5
cos
5
1
3
cos
3
1
cos
2
2
1
cos
)
(
' t
t
t
t
t
m
A
v c
c
c
c
R





  terms
other
t
m
A
vR


 )
(
1
'

03/10/2022 EEE 351 18
Rectifier detector (cont)
• If we pass this voltage thru a LPF we get
• If we use a capacitor, we block the DC and we obtain
 
)
(
1
t
m
A
v filtered 


 
)
(
1
t
m
vout


17
18

Lecture 03
10
03/10/2022 EEE 351 19
Envelope detector
Key Points (Detectors)
03/10/2022 EEE 351 20
 Both rectifier detector and envelope detector consists of a half wave rectifier
followed by LPF.
 Rectifier detector is a synchronous demodulator while envelope detection is a non-
Linear operation.
 The LPF of rectifier detector is designed to separate m(t) from terms such as
m(t)*cos Wct. It does not depend on the value of µ.
 The time constant RC of the LPF for envelope detector does depends on the value
of µ.
19
20

Lecture 03
11
DSB-TC Simple Radio receiver
03/10/2022 EEE 351 21
Fox Hole Radio Receiver
03/10/2022 EEE 351 22
21
22

Lecture 03
12
Fox Hole Radio Receiver
03/10/2022 EEE 351 23
Example
03/10/2022 EEE 351 24
An AM transmitter radiates 9kW of power, when the carrier is un modulated and
10.125kW when it is modulated sinusoidally. Calculate the modulation index and
Percentage of modulation index.
If an other sine wave, corresponding to 40% modulation is simultaneously
transmitted, then calculate the total radiated power.
Solution:
Total power
Where Pt = 10.125, Pc = 9KW, m = ?
m = 0.5
Case II: Now it is modulated by another sinusoidal wave, with 40% modulation.
Therefore, m = 0.4 (divided by 100)
For modulation by Multiple signals
23
24

Lecture 03
13
Example
03/10/2022 EEE 351 25
mtnew
mtnew
mtnew = 0.64
03/10/2022 EEE 351 26
03/10/2022 26
Summery
 DSB-TC / DSB- FC
 Modulation Index (example)
 Power Efficiency
 Power Efficiency calculation of Tone Modulation
 De Modulation of AM Signals
 Rectifier Detector
 Envelope Detector
 DSB-TC Simple Radio Receiver
 Fox Hole Radio
 Example
 Summery
25
26

Lecture 03
14
03/10/2022 EEE 351 27
03/10/2022 27
27

SSB and VSB Modulation
Techniques

ØDSB has two sidebands, USB and LSB
ØSSB has one half of the bandwidth of DSB
2

SSB (Cont.)
3
SSB signal can be coherently (synchronously) demodulated by
multiplying it by cos ωct (exactly like DSB-SC). Example of
USB demodulation is:
SSB-SC

SSB (Cont.)
Φ LSB (ω) = M+ ( ω + ωc ) + M- ( ω - ωc )
ΦLSB (t) = m+ (t) e -jωct + m- (t) e jωct
Substitute with
8

SSB (Cont.)
Hilbert transform
11

Ø DSB is passed through BPF to eliminate undesired band
Ø Most commonly used
Ø To obtain USB, the filter should pass all components
above ωc and attenuate all components below ωc
Difficult to design sharp cutoff filter
Voice spectrum
300 Hz 17

Generation of SSB signals (Cont.)
Difficult to build
18

Envelope detection of SSB with carrier (SSB+C)
If ‘A’ is large enough m(t) can be recovered by
envelop detection
19

Envelope detection of SSB with carrier
(SSB+C) (Cont.)
20

Amplitude modulation: Vestigial Sideband (VSB)
ØDue to the difficulties in generating SSB signals
VSB is used
ØVSB is asymmetric system
ØVSB is a compromise between DSB and SSB
ØVSB is easy to generate and its bandwidth is only 25 -
33% greater than SSB
21

Use of VSB in television broadcast
27

Carrier Acquisition
ØIn any technique from amplitude modulation techniques,
the local oscillator must be in synchronous with the
oscillator that is used in transmitter side.
Why?
ØConsider DSB-SC case:
Ø.
Ø.
Ø.
2

Carrier Acquisition (Cont.)
Ø.
Filtered by LPF
Ø.
3

Carrier Acquisition (Cont.)
4
Attenuation to the message
Ø.
Ø.
Beating effect
Solution
Quartz Crystal Oscillator?
Difficult to built
at high frequencies
Phase Locked Loop (PLL)

Phase Locked Loop (PLL)
Free running frequency
Ø.
Ø.
Ø.
Suppressed by LPF
5

PLL (Cont.)
If the frequencies are different
ØSuppose the frequency of the input sinusoidal
signal is increased from ωc to ωc + k
ØThis means that the incoming signal is
ØThus the frequency increase in incoming signal
causes θi to increase to θi + kt
Increasing θe, and vice versa
7

Carrier Acquisition in DSB-SC
(Cont.)
9

Carrier Acquisition in SSB-SC (Cont.)
11

Angle Modulation (Cont.)
Ø Two techniques are possible to transmit m(t)
by varying angle θ of a carrier.
8

Example (Cont.)
20
• This scheme is called Frequency Shift
Keying (FSK) where information digits are
transmitted by keying different frequencies.

Example (Cont.)
• The derivative ṁ(t) is zero except at points
of discontinuity of m(t) where impulses of
strengths ±2 are present.
22

Example (Cont.)
24
• This scheme is called Phase Shift Keying (PSK) where
information digits are transmitted by shifting the carrier
phase.
• The phase difference (shift) is π.

Bandwidth of Angle Modulated Waves (Cont.)
Ø.
Ø.
Ø.
2

Bandwidth of Angle Modulated Waves (Cont.)
Ø.
Ø.
Ø.
3

Narrowband Angle Modulation (Cont.)
Compare with AM modulated signal
Ø.
Ø.
5

Can not be ignored
Very complicated analysis
8
In practical FM,

WBFM (Cont.)
Simple way to analyze the problem
m(t)
Ø m(t) is band limited to B Hz
Ø m(t) is approximated by pulses of constant
amplitudes (cells),
ØFM analysis of constant amplitude is easier
9
(Staircase)

WBFM (Cont.)
m(t) ≈ Pulse interval ≤ 1 / 2B Sec
ØThe FM signal of one of these cells starting at t = tk
1/2B Sec
Sinusoidal
10

WBFM (Cont.)
ØThe FM spectrum of consists of the sum
of the Fourier transforms of the sinusoidal pulses.
11

WBFM (Cont.)
ØThe min. and max. amplitude of modulating signals
are -mp and mp
the min. frequency
the max. frequency
the Bandwidth
12

WBFM (Cont.)
Ø.
Ø.
ØDo not forget, this value is calculated for
ØFrom earlier analysis:
ØFor NBFM,
13
The Peak Frequency Deviation:

WBFM (Cont.)
14
For a truly wideband case,
A better bandwidth estimate is:

Generation of FM Waves
20
• Direct FM Method.
• Indirect FM Method.

NBFM Generation
21
Recall:
The output of the this NBFM has some
amplitude variations.
A nonlinear device designed to limit the
amplitude of a bandpass signal can
remove most of this distortion.

NBFM Generation (Cont.)
• Bandpass Limiter: used to remove amplitude
variations in FM wave.
22
• The input-output
characteristics of the
hard limiter.

NBFM Generation (Cont.)
• Hard limiter input and the corresponding output.
23
• Hard limiter
output with
respect to θ

Demodulation of FM Signals
• The simplest demodulator is an ideal differentiator
followed by an envelop detector.
27
envelop

Demodulation of FM Signals (Cont.)
28
envelop detector can be used to recover m (t )

FM Demodulation using PLL
29
Free running frequency

Introduction:
Binary data can be transmitted using a number of different types of pulses.
The choice of a particular pair of pulses to represent the symbols 1 and 0 is
called Line Coding and the choice is generally made on the grounds of one
or more of the following considerations:
– Presence or absence of a DC level.
– Power Spectral Density- particularly its value at 0 Hz.
– Bandwidth.
– BER performance (this particular aspect is not covered in this lecture).
– Transparency (i.e. the property that any arbitrary symbol, or bit,
pattern can be transmitted and received).
– Ease of clock signal recovery for symbol synchronisation.
– Presence or absence of inherent error detection properties.

Introduction:
After line coding pulses may be filtered or otherwise shaped to further
improve their properties: for example, their spectral efficiency and/ or
immunity to intersymbol interference. .

Different Types of Line
Coding

Unipolar signalling (also called on-off keying, OOK) is the type of line
coding in which one binary symbol (representing a 0 for example) is
represented by the absence of a pulse (i.e. a SPACE) and the other binary
symbol (denoting a 1) is represented by the presence of a pulse (i.e. a
MARK).
There are two common variations of unipolar signalling: Non-Return to
Zero (NRZ) and Return to Zero (RZ).

Unipolar Non-Return to Zero (NRZ):
In unipolar NRZ the duration of the MARK pulse (Ƭ ) is equal to the duration (To) of
the symbol slot.
1 0 1 0 1 1 1 1 1 0
V
0

In unipolar NRZ the duration of the MARK pulse (Ƭ ) is equal to the duration (To) of
the symbol slot. (put figure here).
Advantages:
– Simplicity in implementation.
– Doesn’t require a lot of bandwidth for transmission.
Disadvantages:
– Presence of DC level (indicated by spectral line at 0 Hz).
– Contains low frequency components. Causes “Signal Droop” (explained later).
– Does not have any error correction capability.
– Does not posses any clocking component for ease of synchronisation.
– Is not Transparent. Long string of zeros causes loss of synchronisation.

Figure. PSD of Unipolar NRZ

When Unipolar NRZ signals are transmitted over links with either
transformer or capacitor coupled (AC) repeaters, the DC level is removed
converting them into a polar format.
The continuous part of the PSD is also non-zero at 0 Hz (i.e. contains low
frequency components). This means that AC coupling will result in
distortion of the transmitted pulse shapes. AC coupled transmission lines
typically behave like high-pass RC filters and the distortion takes the form
of an exponential decay of the signal amplitude after each transition. This
effect is referred to as “Signal Droop” and is illustrated in figure below.

-V/2
V/2
1 0 1 0 1 1 1 1 1 0
V
0
-V/2
V/2
0
1 0 1 0 1 1 1 1 1 0
V
0
-V/2
V/2
0
Figure Distortion (Signal Droop) due to AC coupling of unipolar NRZ signal

Return to Zero (RZ):
In unipolar RZ the duration of the MARK pulse (Ƭ ) is less than the duration (To) of the symbol
slot. Typically RZ pulses fill only the first half of the time slot, returning to zero for the second
half.
1 0 1 0 1 1 1 0 0 0
V
0
To
Ƭ

Unipolar Return to Zero (RZ):
Advantages:
– Presence of a spectral line at symbol rate which can be used as
symbol timing clock signal.
Disadvantages:
– Presence of DC level (indicated by spectral line at 0 Hz).
– Continuous part is non-zero at 0 Hz. Causes “Signal Droop”.
– Occupies twice as much bandwidth as Unipolar NRZ.
– Is not Transparent

Unipolar Return to Zero (RZ):
PSD of Unipolar RZ

In conclusion it can be said that neither variety of unipolar signals is
suitable for transmission over AC coupled lines.

In polar signalling a binary 1 is represented by a pulse g1(t) and a binary 0 by
the opposite (or antipodal) pulse g0(t) = -g1(t). Polar signalling also has NRZ
and RZ forms.
1 0 1 0 1 1 1 1 1 0
+V
-V
0
Figure. Polar NRZ

In polar signalling a binary 1 is represented by a pulse g1(t) and a binary 0 by
the opposite (or antipodal) pulse g0(t) = -g1(t). Polar signalling also has NRZ
and RZ forms.
+V
-V
0
Figure. Polar RZ
1 0 1 0 1 1 1 0 0 0

PSD of Polar Signalling:
Polar NRZ and RZ have almost identical spectra to the Unipolar NRZ and RZ. However,
due to the opposite polarity of the 1 and 0 symbols, neither contain any spectral
lines.
Figure. PSD of Polar NRZ

PSD of Polar Signalling:
Polar NRZ and RZ have almost identical spectra to the Unipolar NRZ and RZ. However,
due to the opposite polarity of the 1 and 0 symbols, neither contain any spectral
lines.
Figure. PSD of Polar RZ

Polar Non-Return to Zero (NRZ):
Advantages:
– No DC component.
Disadvantages:
– Is not transparent.

Polar Return to Zero (RZ):
Advantages:
Disadvantages:
– Does not posses any clocking component for easy synchronisation. However,
clock can be extracted by rectifying the received signal.
– Occupies twice as much bandwidth as Polar NRZ.

Bipolar Signalling is also called “alternate mark inversion” (AMI) uses three
voltage
levels (+V, 0, -V) to represent two binary symbols. Zeros, as in unipolar, are
represented by the absence of a pulse and ones (or marks) are represented
by
alternating voltage levels of +V and –V.
Alternating the mark level voltage ensures that the bipolar spectrum has a
null at DC
And that signal droop on AC coupled lines is avoided.
The alternating mark voltage also gives bipolar signalling a single error
detection
capability.
Like the Unipolar and Polar cases, Bipolar also has NRZ and RZ variations.

Figure. BiPolar NRZ
1 0 1 0 1 1 1 1 1 0
+V
-V
0

PSD of BiPolar/ AMI NRZ Signalling:
Figure. PSD of BiPolar NRZ

BiPolar / AMI NRZ:
Advantages:
– Occupies less bandwidth than unipolar and polar NRZ schemes.
– Does not suffer from signal droop (suitable for transmission over AC coupled
lines).
– Possesses single error detection capability.
Disadvantages:
– Is not Transparent.

Figure. BiPolar RZ
1 0 1 0 1 1 1 1 1 0
+V
-V
0

PSD of BiPolar/ AMI RZ Signalling:
Figure. PSD of BiPolar RZ

BiPolar / AMI RZ:
Advantages:
– Occupies less bandwidth than unipolar and polar RZ schemes.
lines).
– Clock can be extracted by rectifying (a copy of) the received signal.
Disadvantages:
–Is not Transparent.

HDBn is an enhancement of Bipolar Signalling. It overcomes the transparency
problem encountered in Bipolar signalling. In HDBn systems when the
number of
continuous zeros exceeds n they are replaced by a special code.
The code recommended by the ITU-T for European PCM systems is HDB-3
(i.e. n=3).
In HDB-3 a string of 4 consecutive zeros are replaced by either 000V or
B00V.
Where,
‘B’ conforms to the Alternate Mark Inversion Rule.
‘V’ is a violation of the Alternate Mark Inversion Rule

The reason for two different substitutions is to make consecutive Violation
pulses
alternate in polarity to avoid introduction of a DC component.
The substitution is chosen according to the following rules:
1. If the number of nonzero pulses after the last substitution is odd, the
substitution pattern will be 000V.
2. If the number of nonzero pulses after the last substitution is even, the
substitution pattern will be B00V.

1 0 1 0 0 0 0 1 0 0 0 0
B 0 0 V 0 0 0 V

PSD of HDB3 (RZ) Signalling:
Figure. PSD of HDB3 RZ
The PSD of HDB3
(RZ) is similar to the
PSD of Bipolar RZ.

HDBn RZ:
Advantages:
– Occupies less bandwidth than unipolar and polar RZ schemes.
lines).
– Clock can be extracted by rectifying (a copy of) the received signal.
– Is Transparent.
These characteristic make this scheme ideal for use in Wide Area Networks

In Manchester encoding , the duration of the bit is divided into two halves.
The voltage
remains at one level during the first half and moves to the other level during
the
second half.
A ‘One’ is +ve in 1st half and -ve in 2nd half.
A ‘Zero’ is -ve in 1st half and +ve in 2nd half.
Note: Some books use different conventions.

Figure. Manchester Encoding.
1 0 1 0 1 1 1 1 1 0
+V
-V
0
Note: There is always a transition at
the centre of bit duration.

PSD of Manchester Signalling:
Figure. PSD of Manchester

The transition at the centre of every bit interval is used for synchronization at the
receiver.
Manchester encoding is called self-synchronizing. Synchronization at the receiving end
can be achieved by locking on to the the transitions, which indicate the middle of the bits.
It is worth highlighting that the traditional synchronization technique used for unipolar,
polar and bipolar schemes, which employs a narrow BPF to extract the clock signal
cannot be used for synchronization in Manchester encoding. This is because the PSD of
Manchester encoding does not include a spectral line/ impulse at symbol rate (1/To).
Even rectification does not help.

Manchester Signalling:
Advantages:
lines).
– Easy to synchronise with.
– Is Transparent.
Disadvantages:
– Because of the greater number of transitions it occupies a significantly large
bandwidth.
– Does not have error detection capability.
These characteristic make this scheme unsuitable for use in Wide Area Networks.
However, it is widely used in Local Area Networks such as Ethernet and Token Ring.

Reference Text Books
1. “Digital Communications” 2nd Edition by Ian A. Glover and Peter M. Grant.
2. “Modern Digital & Analog Communications” 3rd Edition by B. P. Lathi.
3. “Digital & Analog Communication Systems” 6th Edition by Leon W. Couch, II.
4. “Communication Systems” 4th Edition by Simon Haykin.
5. “Analog & Digital Communication Systems” by Martin S. Roden.
6. “Data Communication & Networking” 4th Edition by Behrouz A. Forouzan.
38

LECTURE 19
1. DETECTION OF BINARY SIGNALS IN GAUSSAIN NOISE
2. MATCHED FILTER AND CORRELATOR
3. MAXIMUM LIKELIHOOD RECEIVER

NOISE IN COMMUNICATION SYSTEMS
• Thermal noise (cannot be eliminated).
• Statistics of Thermal noise are well-known.
• Thermal noise is described by a zero-mean Gaussian random process,𝑛 𝑡
• Its PSD is flat;hence,it is called white noise.
Probability density function [w/Hz]
Power spectral
density
Autocorrelation
function

EFFECT OF NOISE
0 2 4 6 8 10 12 14 16 18 20
-2
-1
0
1
2
0 2 4 6 8 10 12 14 16 18 20
-2
-1
0
1
2
0 2 4 6 8 10 12 14 16 18 20
-2
-1
0
1
2

DETECTION OF BINARY SIGNAL IN GAUSSIAN
NOISE
• For any binary channel,the transmitted signal over a symbol interval (0,T) is:
• The received signal r(t) degraded by noise n(t) and possibly degraded by the impulse
response of the channel hc(t),is
where n(t) is assumed to be zero meanAWGN process
• For ideal distortionless channel where hc(t) is an impulse function and convolution with
hc(t) produces no degradation,r(t) can be represented as:
1
2
( ) 0 1
( )
( ) 0 0
i
s t t T for a binary
s t
s t t T for a binary
=
2
,
1
)
(
)
(
*
)
(
)
( =
+
= i
t
n
t
h
t
s
t
r c
i
T
t
i
t
n
t
s
t
r i =
+
= 0
2
,
1
)
(
)
(
)
(

RECEIVER FUNCTIONALITY
• Step 1:Waveform-to-sample transformation
• Filter followed by a sampler
• At the end of each symbol duration T, pre-detection point yields a sample z(T), called test
statistic
where ai(T) is the desired signal component,and no(T) is the noise component
• Step 2: Detection of symbol
• Compares the z(T) to some threshold level 0 , i.e.,
where H1 and H2 are the two possible binary hypothesis
0
( ) ( ) ( ) 1,2
i
z T a T n T i
= + =
1
2
0
( )
H
H
z T

LIKELIHOOD OF S1 & LIKELIHOOD OF S2
• Test Statistic :
• Assume that input noise is a Gaussian random process and receiving filter is linear
• pdf of noise
• Conditional pdfs
• pdf of the random variable z(T ), given that symbol s1 was transmitted:
• pdf of the random variable z(T ), given that symbol s2 was transmitted:
0
( ) ( ) ( ) 1,2
i
z T a T n T i
= + =
−
=
2
0
0
0
0
2
1
exp
2
1
)
(
n
n
p
2
1
1
0
0
1 1
( | ) exp
2
2
z a
p z s
−
= −
2
2
2
0
0
1 1
( | ) exp
2
2
z a
p z s
−
= −

THE MATCHED FILTER
• A linear (receiving) filter designed to maximize instantaneous SNR at t = T,
• Objective is to find the filter transfer function H0(f) that maximizes the above equation.
• We can express the signal ai(t) at the filter output as
where S(f) is the Fourier transform of input signal s(t)
• Output noise power can be expressed as:

THE MATCHED FILTER CONTD…
• Combining signal power and noise power into SNR equation,we get
• Find that value of H(f) = H0(f ) for which the maximum (S/N)T is achieved
• Now according to Schwarz’s Inequality:
* indicates

• Associate H(f) with f1(x) and S(f) ej2𝜋fT with f2(x) to get:
• Putting it into the SNR equation,we get
Thus (S/N)T depends on input signal energy
and power spectral density of noise and
NOT on the particular shape of the waveform

• Equality for maximum SNR holds for optimum filter transfer function H0(f),such that:
• h(t) is a delayed version of the mirror image of the original signal waveform

CORRELATION REALIZATION OFTHE MATCHED
FILTER
• Matched filter output:
• Putting the Matched filter h(t),we get
• When t =T,

Detection of Binary Signals
MAXIMUM LIKELIHOOD DETECTOR
OPTIMUM THRESHOLD

we are following one of the two formats described here. Often, we have to
provide some descriptive words (e.g., the rightmost bit is the earliest bit) to avoid
confusion.
Mathematical relationships often have built-in features guaranteeing the
proper alignment of time events. For example, in Section 3.2.3, a matched filter is
defined as having an impulse response h(t) that is a delayed version of the time-
reversed copy of the signal. That is, h(t) ! s(T # t). Delay of one symbol time T is
needed for the filter to be causal (the output must occur in positive time). Time re-
versal can be thought of as a “precorrection” where the rightmost part of the time
plot will now correspond to the earliest event. Since convolution dictates another
time reversal, the arriving signal and the filter’s impulse response will be “in step”
(earliest with earliest, and latest with latest).
126 Baseband Demodulation/Detection Chap. 3
Figure 3.8 Equivalence of matched
filter and correlator. (a) Matched fil-
ter. (b) Correlator.
needed for the filter to be causal (the output must o
versal can be thought of as a “precorrection” where
plot will now correspond to the earliest event. Sinc
time reversal, the arriving signal and the filter’s imp
(earliest with earliest, and latest with latest).
126 Baseband De
Fi
fil
te
Sklar_Chapter_03.indd 126
Matched Filter and Correlator
§ Matched filter reduces the received signal to a single variable !(#), after which the detection of
symbol is carried out.
§Matched Filter Correlator
§The concept of maximum likelihood detector is based on Statistical Decision Theory
§It allows us to
§ formulate the decision rule that operates on the data
§ optimize the detection criterion
filter follows the receiving filter; it is only needed for those systems where channel-
induced ISI can distort the signals. The receiving filter and equalizing filter are
shown as two separate blocks in order to emphasize their separate functions. In
most cases, however, when an equalizer is used, a single filter would be designed to
incorporate both functions and thereby compensate for the distortion caused by
both the transmitter and the channel. Such a composite filter is sometimes referred
to simply as the equalizing filter or the receiving and equalizing filter.
Figure 3.1 highlights two steps in the demodulation/detection process. Step 1,
the waveform-to-sample transformation, is made up of the demodulator followed
by a sampler. At the end of each symbol duration T, the output of the sampler, the
predetection point, yields a sample z(T), sometimes called the test statistic. z(T)
has a voltage value directly proportional to the energy of the received symbol and
inversely proportional to the noise. In step 2, a decision (detection) is made regard-
ing the digital meaning of that sample. We assume that the input noise is a Gaus-
sian random process and that the receiving filter in the demodulator is linear. A
linear operation performed on a Gaussian random process will produce a second
Gaussian random process [2]. Thus, the filter output noise is Gaussian. The output
of step 1 yields the test statistic
(3.3)
Sklar_Chapter_03.indd 108 8/22/08 6:18:32 PM
8/22/08 6:18:32 PM
signals. Since z(T) is a voltage signal that is proportional to the energy of th
ceived symbol, the larger the magnitude of z(T), the more error free will be th
cision-making process. In step 2, detection is performed by choosing the hypo
that results from the threshold measurement
where H1 and H2 are the two possible (binary) hypotheses. The inequality rela
ship indicates that hypothesis H1 is chosen if z(T) ! ", and hypothesis H2 is c
if z(T) # ". If z(T) $ ", the decision can be an arbitrary one. Choosing H1 is eq
lent to deciding that signal s1(t) was sent and hence a binary 1 is detected. Sim

Probabilities Review
§ % &' , % &) à a priori probabilities These probabilities are known before transmission
§ %(!) à Probability of the received sample
§ % ! &' , % ! &) à conditional pdf of received signal z, conditioned on the class si
§ % &' ! , % &) ! à a posteriori probabilities. After examining the sample, we make a
refinement of our previous knowledge
§ % &' &) , % &) &' à wrong decision (error)
§ % &' &' , % &) &) à correct decision

3.1 Signals and Noise 10
0
the conditional pdfs p(z|s1) and p(z|s2) can be expressed as
(3.5
and
(3.6
These conditional pdfs are illustrated in Figure 3.2. The rightmost conditional pd
p(z|s1), called the likelihood of s1, illustrates the probability density function of th
random variable z(T), given that symbol s1 was transmitted. Similarly, the leftmos
conditional pdf, p(z|s2), called the likelihood of s2, illustrates the pdf of z(T), give
that symbol s2 was transmitted. The abscissa, z(T), represents the full range of pos
sible sample output values from step 1 of Figure 3.1.
After a received waveform has been transformed to a sample, the actua
shape of the waveform is no longer important; all waveform types that are trans
formed to the same value of z(T) are identical for detection purposes. Later it i
shown that an optimum receiving filter (matched filter) in step 1 of Figure 3.1 map
all signals of equal energy into the same point z(T). Therefore, the received signa
energy (not its shape) is the important parameter in the detection process. This i
why the detection analysis for baseband signals is the same as that for bandpas
Figure 3.2 Conditional probability density functions: p(z|s1) and p(z|s2).
How to Choose the threshold *?
The output of the sampler is
where ai(T) is the desired signal component, and n0(T) is the noise component. To
simplify the notation, we sometimes express Equation (3.3) in the form of z ! ai "
n0. The noise component n0 is a zero mean Gaussian random variable, and thus
z(T) is a Gaussian random variable with a mean of either a1 or a2 depending on
whether a binary one or binary zero was sent. As described in Section 1.5.5,
the probability density function (pdf) of the Gaussian random noise n0 can be
expressed as
(3.4)
where # 2
0 is the noise variance. Thus it follows from Equations (3.3) and (3.4) that
(3.5)
and
(3.6)
These conditional pdfs are illustrated in Figure 3.2. The rightmost conditional pdf,
p(z|s1), called the likelihood of s1, illustrates the probability density function of the
random variable z(T), given that symbol s1 was transmitted. Similarly, the leftmost
conditional pdf, p(z|s2), called the likelihood of s2, illustrates the pdf of z(T), given
that symbol s2 was transmitted. The abscissa, z(T), represents the full range of pos-
After a received waveform has been transformed to a sample, the actual
shape of the waveform is no longer important; all waveform types that are trans-
formed to the same value of z(T) are identical for detection purposes. Later it is
shown that an optimum receiving filter (matched filter) in step 1 of Figure 3.1 maps
all signals of equal energy into the same point z(T). Therefore, the received signal
energy (not its shape) is the important parameter in the detection process. This is
the waveform-to-sample transformation, is made up of the demodulator followed
by a sampler. At the end of each symbol duration T, the output of the sampler, the
predetection point, yields a sample z(T), sometimes called the test statistic. z(T)
has a voltage value directly proportional to the energy of the received symbol and
inversely proportional to the noise. In step 2, a decision (detection) is made regard-
ing the digital meaning of that sample. We assume that the input noise is a Gaus-
sian random process and that the receiving filter in the demodulator is linear. A
linear operation performed on a Gaussian random process will produce a second
Gaussian random process [2]. Thus, the filter output noise is Gaussian. The output
of step 1 yields the test statistic
(3.3)
ar_Chapter_03.indd 108
ar_Chapter_03.indd 108 8/22/08 6:18
8/22/08 6:18
T) is a voltage signal that is proportional to the energy of the re-
he larger the magnitude of z(T), the more error free will be the de-
ocess. In step 2, detection is performed by choosing the hypothesis
the threshold measurement
(3.7)
2 are the two possible (binary) hypotheses. The inequality relation-
at hypothesis H1 is chosen if z(T) ! ", and hypothesis H2 is chosen
T) $ ", the decision can be an arbitrary one. Choosing H1 is equiva-
that signal s1(t) was sent and hence a binary 1 is detected. Similarly,

§ We choose threshold to minimize error probability using Likelihood Ratio test or the Maximum
a posteriori (MAP) criterion:
§ ……………….. 1
§ Problem is that a posteriori probabilities are not known.
§ Solution: Use Bayes’ theorem:
§ ……………. 2
§where
n0(T). The time T is the symbol duration. At each kT, where k is an integer, the re-
ceiver uses a decision rule for deciding which signal class has been received. For
ease of notation, Equation (B.5) is sometimes written simply as z ! ai " n0, where
the functional dependence on T is implicit.
B.2.2 The Likelihood Ratio Test and the Maximum
A Posteriori Criterion
A reasonable starting point for establishing the receiver decision rule for the case
of two signal classes is
(B.6)
Equation (B.6) states that we should choose hypothesis H1 if the a posteriori prob-
ability P(s1!z) is greater than the a posteriori probability P(s2!z). Otherwise, we
should choose hypothesis H2.
We can replace the posteriori probabilities of Equation (B.6) with their
equivalent expressions from Bayes’ theorem [Equation (B.4)], yielding
(B.7)
We now have a decision rule in terms of pdfs (likelihoods). If we rearrange Equa-
tion (B.7) and put it in the form
(B.8)
The optimum decision strategy then is as follows: If a head, zH, is received, choose hy
pothesis HH (that the other side is also a head). If a tail, zT , is received, choose hy
pothesis HT (that the other side is also a tail).
B.1.2 Mixed Form of Bayes’ Theorem
For most communication engineering applications of interest, the possible value
of the received samples are continuous in range, because of the additive Gaussia
noise in the channel. Therefore, the most useful form of Bayes’ theorem contains
continuous- instead of a discrete-valued pdf. We shall rewrite Equation (B.3) t
emphasize this change:
(B.4
Here, p(z! si) is the conditional pdf of the received continuous-valued sample,
conditioned on the signal class, si.
coin hypothesis.
B.1 Bayes' Theorem 109
optimum decision strategy then is as follows: If a head, zH, is received, choose hy-
esis HH (that the other side is also a head). If a tail, zT , is received, choose hy-
esis HT (that the other side is also a tail).
xed Form of Bayes’ Theorem
communication engineering applications of interest, the possible values
eived samples are continuous in range, because of the additive Gaussian
he channel. Therefore, the most useful form of Bayes’ theorem contains a
us- instead of a discrete-valued pdf. We shall rewrite Equation (B.3) to
e this change:
(B.4)
! si) is the conditional pdf of the received continuous-valued sample, z,
ed on the signal class, si.
(b) Conditioned on the two-headed-coin hy-
pothesis. (c) Conditioned on the two-tailed-
coin hypothesis.
yes' Theorem 1095

Substituting (2) in (1), we get
Re-arranging above equation gives
(B.6)
quation (B.6) states that we should choose hypothesis H1 if the a posteriori prob-
ility P(s1!z) is greater than the a posteriori probability P(s2!z). Otherwise, we
ould choose hypothesis H2.
uivalent expressions from Bayes’ theorem [Equation (B.4)], yielding
(B.7)
e now have a decision rule in terms of pdfs (likelihoods). If we rearrange Equa-
on (B.7) and put it in the form
(B.8)
en the left-hand ratio is known as the likelihood ratio and the entire equation is
ten referred to as the likelihood ratio test. Equation (B.8) corresponds to making
decision based on a comparison of a measurement of a received signal to a
reshold. Since the test is based on choosing the signal class with maximum a pos-
(B.6)
(B.7)
(B.8)
then the left-hand ratio is known as the likelihood ratio and the entire equation is
often referred to as the likelihood ratio test. Equation (B.8) corresponds to making
a decision based on a comparison of a measurement of a received signal to a
threshold. Since the test is based on choosing the signal class with maximum a pos-
teriori probability, the decision criterion is called the maximum a posteriori (MAP)
criterion. It is also called the minimum error criterion, since on the average, this cri-
terion yields the minimum number of incorrect decisions. It should be emphasized
that this criterion is optimum only when each of the error types are equally harmful
or costly. When some of the error types are more costly than others, a criterion that
(B.6)
(B.7)
(B.8)
then the left-hand ratio is known as the likelihood ratio and the entire equation is
often referred to as the likelihood ratio test. Equation (B.8) corresponds to making
a decision based on a comparison of a measurement of a received signal to a
threshold. Since the test is based on choosing the signal class with maximum a pos-
teriori probability, the decision criterion is called the maximum a posteriori (MAP)
criterion. It is also called the minimum error criterion, since on the average, this cri-
terion yields the minimum number of incorrect decisions. It should be emphasized
that this criterion is optimum only when each of the error types are equally harmful
or costly. When some of the error types are more costly than others, a criterion that
incorporates relative cost of the errors should best be employed [1].
1098 Fundamentals of Statistical
ability P(s1!z) is greater than the a posteriori probabilit
We can replace the posteriori probabilities of Eq
equivalent expressions from Bayes’ theorem [Equation (B
We now have a decision rule in terms of pdfs (likelihoods
then the left-hand ratio is known as the likelihood ratio a
often referred to as the likelihood ratio test. Equation (B.8
a decision based on a comparison of a measurement o
threshold. Since the test is based on choosing the signal cl
teriori probability, the decision criterion is called the maxi
criterion. It is also called the minimum error criterion, sinc
terion yields the minimum number of incorrect decisions.
that this criterion is optimum only when each of the error
or costly.When some of the error types are more costly th
incorporates relative cost of the errors should best be emp
Fundamentals of Statistical Decision Theory App. B
(B.6)
B.6) states that we should choose hypothesis H1 if the a posteriori prob-
z) is greater than the a posteriori probability P(s2!z). Otherwise, we
ose hypothesis H2.
an replace the posteriori probabilities of Equation (B.6) with their
expressions from Bayes’ theorem [Equation (B.4)], yielding
(B.7)
ve a decision rule in terms of pdfs (likelihoods). If we rearrange Equa-
nd put it in the form
(B.8)
t-hand ratio is known as the likelihood ratio and the entire equation is
ed to as the likelihood ratio test. Equation (B.8) corresponds to making
based on a comparison of a measurement of a received signal to a
ince the test is based on choosing the signal class with maximum a pos-
ability, the decision criterion is called the maximum a posteriori (MAP)
is also called the minimum error criterion, since on the average, this cri-
s the minimum number of incorrect decisions. It should be emphasized
erion is optimum only when each of the error types are equally harmful
hen some of the error types are more costly than others, a criterion that
s relative cost of the errors should best be employed [1].

§ If a prior probabilities are selected such that both symbols are equal likely, then likelihood ratio
test becomes
§This is known as maximum likelihood ratio test because we are selecting the hypothesis that
corresponds to the signal with the maximum likelihood.
§In terms of the Bayes’ criterion, it implies that the cost of both types of error is the same
§Plugging the likelihood values of s1 and s2 in likelihood ratio test, we have
B.2.3 The Maximum Likelihood Criterion
Very often there is no knowledge available about the priori probabilities of the hy-
potheses or signal classes. Even when such information is available, its accuracy is
sometimes mistrusted. In those instances, decisions are usually made by assuming
the most conservative a priori probabilities possible; that is, the values of the a pri-
ori probabilities are selected so that the classes are equally likely. When this is
done, the MAP criterion is known as the maximum likelihood criterion, and Equa-
tion (B.8) can be written as
(B.9)
Notice that the maximum likelihood criterion of Equation (B.9) is the same as the
maximum likelihood rule that was described in Example B.3.
B.3 SIGNAL DETECTION EXAMPLE
B.3.1 The Maximum Likelihood Binary Decision
The pictorial view of the decision process in Example B.3 dealt with triangular-
shaped probability density functions as a convenient example. Figure B.4 illustrates
the conditional pdfs for the binary noise-perturbed output signals, z(T) ! a1 " n0
and z(T) ! a2 " n0 from a typical receiver. The signals a1 and a2 are mutually inde-
pendent and are equally likely. The noise n0 is assumed to be an independent
Gaussian random variable with zero mean, variance #2
0, and pdf given by
(B.11)

Re-arranging gives
Taking natural log both sides will yield log-likelihood ratio
(B.11)
where a1 is the receiver output signal component when s1(t) is sent and a2 is the out-
put signal component when s2(t) is sent. The inequality relationship described by
Equation (B.11) is preserved for any monotonically increasing (or decreasing)
transformation. Therefore, to simplify Equation (B.11), we take the natural loga-
rithm of both sides, resulting in the log-likelihood ratio.
(B.12)
When the classes are equally likely,
(B.11)
(B.12)

3.1 Signals and Noise 10
expressed as
(3.4
where # 2
0 is the noise variance. Thus it follows from Equations (3.3) and (3.4) tha
(3.5
and
(3.6
These conditional pdfs are illustrated in Figure 3.2. The rightmost conditional pd
p(z|s1), called the likelihood of s1, illustrates the probability density function of th
random variable z(T), given that symbol s1 was transmitted. Similarly, the leftmo
conditional pdf, p(z|s2), called the likelihood of s2, illustrates the pdf of z(T), give
that symbol s2 was transmitted. The abscissa, z(T), represents the full range of po
After a received waveform has been transformed to a sample, the actu
shape of the waveform is no longer important; all waveform types that are tran
formed to the same value of z(T) are identical for detection purposes. Later it
shown that an optimum receiving filter (matched filter) in step 1 of Figure 3.1 map
all signals of equal energy into the same point z(T). Therefore, the received sign
energy (not its shape) is the important parameter in the detection process. This
why the detection analysis for baseband signals is the same as that for bandpa
Figure 3.2 Conditional probability density functions: p(z|s1) and p(z|s2).
1100 Fundamentals of Statistical Decision Theory App. B
(B.12)
so that
(B.13)
For antipodal signals, s1(t) ! "s2(t) and a1 ! "a2; thus, we can write
(B.14)
Therefore, the maximum likelihood rule for the case of equally likely antipodal sig-
nals compares the received sample to a zero threshold, which is tantamount to de-
ciding s1(t) if the sample is positive, and s2(t) if the signal is negative.

M-ary Detection – Correlator Design

M-ary Detection – Matched Filter Design

Example: Matched Filter Design
s(t)
A
-A
t
T
T/2
s(-t)
A
-A
t
-T -T/2
s(T-t) = h(t)
A
-A
t
T
T/2
Flip
Delay

Example: Matched Filter Ouput
A
-A
t
T
T/2
s(t) h(t)
A
-A
t
T
T/2
*
A
-A
+
T
T/2
s(+)
t1
h(t1- +)

A
-A
+
T
T/2
s(+)
h(t1- +)
A
t
T
T/2
Output
3T/2 2T
A
-A
T
T/2
s(+)
h(t2- +)
A
-A
T
s(+)
h(t3- +)
T/2
, -. −- . 0# = −-)
2/2
5/)
6
−-)
2/2
T
T/2
Output
3T/2 2T

A
-A
T
s(+)
h(t3- +)
T/2
−-)
2/2
T
T/2
Output
3T/2 2T
A
-A
T
s(+)
h(t4- +)
T/2
A
-A
T
s(+)
h(t5- +)
T/2 , -)
. 0# + , -)
. 0#
5
5/)
= -)
2
5/)
6
−-)
2/2
T
T/2 3T/2 2T
-)
2

Output
A
T
s(+)
h(t6- +)
A
-A
T
s(+)
h(t7- +)
T/2
−-)
2/2
T
T/2 3T/2 2T
-)
2
T/2
y(t)

Example: Correlator Ouput
A
-A
t
T
T/2
r(t)
A
-A
t
T
T/2
s(t)
T
T/2 3T/2 2T
-)
2
Same value at t=T as in the case
of the matched filter

LECTURE 21
Binary Signal Detection: Probability of Error, Optimizing Error Performance,
Error Performance of Binary Signaling
EEE 352
Principles of Communication System
Dr. Ghufran Shafiq

ERROR PROBABILITY
• Error in Binary Decision Making
• Detected s2 when s1 was sent P(e|s1) = P(H2|s1)
• Detected s1 when s2 was sent P(e|s2) = P(H1|s2)
• Total Error Probability
• With Equal Priors [P(s1) = P(s2)]

ERROR PROBABILITY
• Error in Binary Decision Making
𝑃𝐵 = න
𝛾0=
𝑎1+𝑎2
2
∞
𝑝 𝑧 𝑠2 𝑑𝑧
𝑃𝐵 = න
𝛾0=
𝑎1+𝑎2
2
∞
1
𝜎0 2𝜋
exp −
1
2
𝑧 − 𝑎2
𝜎0
2
𝑑𝑧
𝑷𝑩 = 𝑸(
𝒂𝟏−𝒂𝟐
𝟐𝝈𝟎
) [Complementary error function]
• Approximated by Tables or other approximations
• If x>3
𝑄 𝑥 ≈
1
𝑥 2𝜋
exp −
𝑥2
2

OPTIMIZING ERROR PERFORMANCE
• Probability of error with optimal threshold
𝑷𝑩 = 𝑸(
𝒂𝟏 − 𝒂𝟐
𝟐𝝈𝟎
)
• Minimize PB → maximize
𝒂𝟏−𝒂𝟐
𝟐𝝈𝟎
or
𝒂𝟏−𝒂𝟐
𝝈𝟎
𝟐
• Matched Filter maximizes SNR for given known signal: S/N = 2E/N0
• Consider the filter is matched to the input difference signal [s1(t)-s2(t)]
• Output SNR of Matched Filter
𝑆
𝑁 𝑇
=
𝑎1−𝑎2
2
𝜎0
2 =
2𝐸𝑑
𝑁0
𝐸𝑑 = ‫׬‬
0
𝑇
𝑠1(𝑡) − 𝑠2(𝑡) 2𝑑𝑡
𝐸𝑑
2𝑁0
=
𝑎1−𝑎2
2𝜎0
𝑃𝐵 = 𝑄(
𝐸𝑑
2𝑁0
)

PROBABILITY OF ERROR AND SIGNAL
CORRELATION
• Cross-correlation: Measure similarity between two signals
𝝆 =
𝟏
E𝒃
න
𝟎
𝑻
𝒔𝟏 𝒕 𝒔𝟐 𝒕 𝒅𝒕
𝐸𝑑 = න
0
𝑇
𝑠1(𝑡) − 𝑠2(𝑡) 2
𝑑𝑡
𝐸𝑑 = න
0
𝑇
𝑠1
2
𝑡 𝑑𝑡 + න
0
𝑇
𝑠2
2
𝑡 𝑑𝑡 − 2 න
0
𝑇
𝑠1 𝑡 𝑠2 𝑡 𝑑𝑡
Since 𝐸𝑏 = ‫׬‬
0
𝑇
𝑠1
2
𝑡 𝑑𝑡 = ‫׬‬
0
𝑇
𝑠2
2
𝑡 𝑑𝑡
𝐸𝑑 = 𝐸𝑏 + 𝐸𝑏 − 2𝜌𝐸𝑏 = 2𝐸𝑏 1 − 𝜌
Thus 𝑃𝐵 = 𝑄(
𝐸𝑑
2𝑁0
) becomes
𝑃𝐵 = 𝑄(
𝐸𝑏 1 − 𝜌
𝑁0
)

ORTHOGONAL SIGNALING
• Orthogonal Binary Signaling: zero cross correlation between signals
න
𝟎
𝑻
𝒔𝟏 𝒕 𝒔𝟐 𝒕 𝒅𝒕 = 𝟎
𝑃𝐵 = 𝑄(
𝐸𝑏 1 − 𝜌
𝑁0
) = 𝑄(
𝐸𝑏
𝑁0
)

ANTIPODAL SIGNALING
• Antipodal Binary Signaling: anti-correlated 𝒔𝟐 𝒕 = - 𝒔𝟏 𝒕
𝑃𝐵 = 𝑄(
𝐸𝑏 1 − 𝜌
𝑁0
) = 𝑄(
2𝐸𝑏
𝑁0
)

EXAMPLE 3.2: MATCHED FILTER
DETECTION FOR ANTIPODAL SIGNALS
𝑃𝐵 = 𝑄(
2𝐸𝑏
𝑁0
)

• Unipolar Signaling (Orthogonal Signaling)
• For s1(t), the received signal r(t) will be
• The signal component of correlator output
• Similarly for s2(t), r(t) = n(t) and a2(T) will be 0
• Thus optimum threshold is
𝛾0 =
𝑎1 + 𝑎2
2
=
𝐴2
𝑇 + 0
2
=
1
2
𝐴2𝑇
ERROR PROBABILITY PERFORMANCE
OF UNIPOLAR SIGNALING
𝑠1 𝑡 = 𝐴 0 ≤ 𝑡 ≤ 𝑇 for binary 1
𝑠2 𝑡 = 0 0 ≤ 𝑡 ≤ 𝑇 for binary 0
𝑟 𝑡 = 𝑠1 𝑡 + 𝑛 𝑡 = 𝐴 + 𝑛(𝑡)

• Energy of Difference Signal Ed
𝐸𝑑 = න
0
𝑇
𝑠1(𝑡) − 𝑠2(𝑡) 2𝑑𝑡
𝐸𝑑 = 𝐴2
𝑇
• Thus Bit-error performance becomes
𝑃𝐵 = 𝑄(
𝐸𝑑
2𝑁0
) = 𝑄(
𝐴2𝑇
2𝑁0
)
• Since Eb = 𝐴2𝑇/2 (avg. bit energy for equally likely s1 and s2), thus
𝑃𝐵 = 𝑄(
𝐸𝑏
𝑁0
)
OF UNIPOLAR SIGNALING

• Bipolar Signaling (Antipodal Signaling)
• Test statistic z(T): difference of correlator outputs
• For antipodal signals, 𝑎1 = −𝑎2, thus 𝛾0 = 0
• The energy of difference signal is
𝐸𝑑 = න
0
𝑇
𝑠1(𝑡) − 𝑠2(𝑡) 2𝑑𝑡 = න
0
𝑇
2𝐴 2𝑑𝑡 = 4𝐴2𝑇
• Thus the bit-error performance becomes
𝑃𝐵 = 𝑄(
𝐸𝑑
2𝑁0
) = 𝑄(
4𝐴2𝑇
2𝑁0
) = 𝑄(
2𝐴2𝑇
𝑁0
) = 𝑄(
2𝐸𝑏
𝑁0
)
OF BIPOLAR SIGNALING
𝑠1 𝑡 = 𝐴 0 ≤ 𝑡 ≤ 𝑇 for binary 1
𝑠2 𝑡 = −𝐴 0 ≤ 𝑡 ≤ 𝑇 for binary 0
𝑧 𝑇 = 𝑧1 𝑇 − 𝑧2(𝑇)
𝐸𝑏 = 𝐴2
𝑇

LECTURE 22
Reduction of Inter-symbol Interference (ISI)
EEE 352
Principles of Communication System
Dr. Ghufran Shafiq

RECAP: DIGITAL BASEBAND
DETECTION
2
Noise
(AWGN)
Distortion
Interference
Pulse Shaping
Decreased
SNR
Matched
Filter /
Correlator
Receiver

RECAP: DISTORTION-LESS
TRANSMISSION
3
( ) d
h wt
w −
=
ƒ Suppose a signal composed of various
sinusoids being passed through a
distortionless system
ƒ Output signal is the input signal multiplied by
k and delayed by td
ƒ System’s frequency response should be such
that each component suffers the same
attenuation k
ƒ Each component undergoes same delay of
td seconds
ƒ Time delay is negative of slope of the system
phase response
( ) k
w
H =
Constant Time
Delay→ Linear
Phase

ISI EFFECTS: BAND-LIMITED FILTERING OF
CHANNEL
• ISI due to filtering effect of the communications channel
• Channels behave like band-limited filters
)
(
)
(
)
( f
j
c
c
c
e
f
H
f
H =
Non-constant amplitude
Amplitude distortion
Non-linear phase
Phase distortion

• Various filters and reactive circuit elements throughout the system
(Transmitter, Channel, Receiver) affect the actual signal
• Signal distortion in channel:
• Wired medium has distributed reactance's that distort the pulses
• Wireless medium has multipath and fading that behave like undesirable filters
• Let H(f) be the transfer function of all the filtering effects and given as:
(3.77)
Ht(f), Hc(f), Hr(f) characterizes transmitter, channel and receiver filters
respectively.

• Intersymbol Interference effects worst during the detection
process at the receiver
• Merely ISI can result in a transfer of one symbol in another
and hence increase the error probability at the receiver
How to reduce or
cancel this effect?

HOW TO REDUCE OR CANCEL ISI?
• Use a filter such that the overall response creates ZERO interference at the
sampling times (Nyquist Pulse)
ℎ(𝑡) = ቊ
1, 𝑡 = 0
0, 𝑡 = 𝑛T
• Reduce ISI, while minimizing bandwidth
H(f)
1
0 1/2T
-1/2T 0
T
2T
3T
h(t) h(t-T)
W = 1/2T Hz

HOW TO REDUCE OR CANCEL ISI?
H(f)
1
0 1/T
-1/T 0
T
2T
3T
h(t) h(t-T)
H(f)
1
0 1/4T
-1/4T 0
T
2T
3T
h(t) h(t-T)
W > 1/2T Hz
W < 1/2T Hz

MINIMUM BANDWIDTH
• Minimum bandwidth: Wmin = 1/2T
• Since T = 1/Rs (Symbol Rate)
• Therefore
Wmin = 1/2/Rs = Rs/2
H(f)
1
0 1/2T
-1/2T 0
T
2T
3T
h(t) h(t-T)
W = 1/2T Hz

13
NYQUIST BANDWIDTH CONSTRAINT
• Nyquist bandwidth constraint:
• The theoretical minimum required system bandwidth to detect Rs [symbols/s]
without ISI is Rs/2 [Hz].
• Equivalently, a system with bandwidth W=1/2T=Rs/2 [Hz] can support a maximum
transmission rate of Rs = 2W=1/T [symbols/s] without ISI.
• Bandwidth efficiency, R/W [bits/s/Hz] :
• An important measure in DCs representing data throughput per hertz of bandwidth.
• Showing how efficiently the bandwidth resources are used by signalling techniques.
Hz]
[symbol/s/
2
2
2
1
=
W
R
W
R
T
s
s

14
PULSE SHAPING TO REDUCE ISI
• Goals and trade-off in pulse-shaping
• Reduce ISI
• Efficient bandwidth utilization
Compact signaling spectrum gives us higher allowable data rate or greater number of
users that can simultaneously be served.
Thus our goal is to reduce the required system bandwidth as much as possible and
Nyquist has provided us with the basic limitation
• Robustness to timing error (small side lobes)
• Problems with Nyquist Filter (Pulse)
• Infinite Impulse Response
• Sensitive to timing error (Large side-lobes)
• Solution: Use Raised Cosine Filter

15
THE RAISED COSINE FILTER
• Raised-Cosine Filter
• No ISI at the sampling time
−
−
−
+
−
=
W
f
W
f
W
W
W
W
W
W
f
W
W
f
f
H
|
|
for
0
|
|
2
for
2
|
|
4
cos
2
|
|
for
1
)
( 0
0
0
2
0
Excess bandwidth: 0
W
W − Roll-off factor
0
0
W
W
W
r
−
=
1
0 r
2
0
0
0
0
]
)
(
4
[
1
]
)
(
2
cos[
))
2
(sinc(
2
)
(
t
W
W
t
W
W
t
W
W
t
h
−
−
−
=

RAISED COSINE FILTER
• W, is absolute bandwidth, Wo =1/2T represents the minimum Nyquist
bandwidth for the rectangular spectrum.
• W-Wo is termed the “excess bandwidth”, which means additional bandwidth
beyond the Nyquist minimum (for rectangular spectrum W=Wo=1/2T)
• Roll-off factor is defined to be r=(W-Wo)/Wo,
where 0≤ r ≤ 1.
−
−
−
+
−
=
W
f
W
f
W
W
W
W
W
W
f
W
W
f
f
H
|
|
for
0
|
|
2
for
2
|
|
4
cos
2
|
|
for
1
)
( 0
0
0
2
0

17
THE RAISED COSINE FILTER
2
)
1
(
Baseband sSB
s
R
r
W +
=
|
)
(
|
|
)
(
| f
H
f
H RC
=
0
=
r
5
.
0
=
r
1
=
r
1
=
r
5
.
0
=
r
0
=
r
)
(
)
( t
h
t
h RC
=
T
2
1
T
4
3
T
1
T
4
3
−
T
2
1
−
T
1
−
1
0.5
0
1
0.5
0 T T
2 T
3
T
−
T
2
−
T
3
−
s
R
r
W )
1
(
Passband DSB +
=
Wo
-Wo

ROOT RC ROLLOFF PULSE SHAPING
• We saw earlier that the noise is minimized at the receiver by using a matched
filter
• This means that the transmit and the receive filter should be the same.
• The combination of transmit and receive filters must give an over all response
of Raised Cosine Pulse Shaping for zero ISI
• The impact of channel Hc(f) is cancelled used equalizer
• 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 (Digital Advanced Mobile Phone Systems)
• Note: HRRC(f) must be truncated to finite time duration,
)
(
)
(
)
(
)
(
)
(
)
( f
H
f
H
f
H
f
H
f
H
f
H RC
RRC
RRC
RRC
RC =
=
=

Lecture 22
Eye Diagram and Equalizer

Eye Diagram
n An eye diagram is obtained by superimposing the actual waveforms for
large numbers of transmitted or received symbols
q Perfect eye pattern for noise-free transmission of a binary signal (1’s and
0’s) with two digital waveforms
q
Tx Rx
channel

Concept of Eye diagram Mask. Waveform must not intrude into the shaded regions.

Lecture 6 5
Eye pattern
Actual eye patterns used to visually estimate the effect of noise and Intersymbol
Symbol Interference and bit error rate
time scale
amplitude
scale
Noise margin
Sensitivity to
timing error
Distortion
due to ISI
Timing jitter

Lecture 6 6
Example of eye pattern:
Binary-PAM,
n Perfect channel (no noise and no ISI)

Lecture 6 7
Binary-PAM,
n AWGN (Eb/N0=20 dB) and no ISI

Lecture 6 8
Binary-PAM,
n AWGN (Eb/N0=10 dB) and no ISI

n Eye pattern for 5-level PAM (PAM-5), as used to operate gigabit Ethernet
over 4 unshielded twisted pairs:

Eye Diagrams for Raised Cosine Filtered Data
n As roll of r is reduced, the eye opening dramatically narrows, requiring the
accuracy of symbol timing to be even more exact
n Benefits of small roll of r
q Maximum bandwidth efficiency achieved

11
Equalization
n Used to remove effect of ISI
n ISI due to filtering effect of the communications
channel (e.g. wireless channels)
)
(
)
(
)
( f
j
c
c
c
e
f
H
f
H =
Non-constant amplitude
Amplitude distortion
Non-linear phase
Phase distortion

12
Example of eye pattern with ISI:
Binary-PAM with SRRC pulse shaping
n Non-ideal channel and no noise
)
(
7
.
0
)
(
)
( T
t
t
t
hc +
=

13
Binary-PAM, SRRC pulse shaping
n AWGN (Eb/N0=20 dB) and ISI
)
(
7
.
0
)
(
)
( T
t
t
t
hc +
=

14
Binary-PAM, SRRC pulse …
n AWGN (Eb/N0=10 dB) and ISI
)
(
7
.
0
)
(
)
( T
t
t
t
hc +
=

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