Simulation of Wireless Communication Systems

Modeling of Wireless Communication
Systems using MATLAB

Dr. B.-P. Paris
Dept. Electrical and Comp. Engineering
George Mason University

last updated September 23, 2009

©2009, B.-P. Paris Wireless Communications 1

Approach

This course aims to cover both
theory and practice of wireless commuication systems, and
the simulation of such systems using MATLAB.
Both topics are given approximately equal treatment.
After a brief introduction to MATLAB, theory and MATLAB
simulation are pursued in parallel.
This approach allows us to make concepts concrete and/or
to visualize relevant signals.
In the process, a toolbox of MATLAB functions is
constructed.
Hopefully, the toolbox will be useful for your own projects.
Illustrates good MATLAB practices.


Outline - Prologue: Just Enough MATLAB to ...

Prologue: Learning Objectives

User Interface

Working with Vectors

Visualization


Outline - Part I: From Theory to Simulation

Part I: Learning Objectives

Elements of a Digital Communications System

Digital Modulation

Channel Model

Receiver

MATLAB Simulation


Outline - Part II: Digital Modulation and Spectrum

Part II: Learning Objectives

Linear Modulation Formats and their Spectra

Spectrum Estimation in MATLAB

Non-linear Modulation

Wide-Band Modulation


Outline - Part III: The Wireless Channel

Part III: Learning Objectives

Pathloss and Link Budget

From Physical Propagation to Multi-Path Fading

Statistical Characterization of Channels


Outline - Part IV: Mitigating the Wireless Channel

Part IV: Learning Objectives

The Importance of Diversity

Frequency Diversity: Wide-Band Signals


User Interface Working with Vectors Visualization

Part I

Prologue: Just Enough MATLAB to ...



Prologue: Just Enough MATLAB to ...

MATLAB will be used throughout this course to illustrate
theory and key concepts.
MATLAB is very well suited to model communications
systems:
Signals are naturally represented in MATLAB,
MATLAB has a very large library of functions for processing
signals,
Visualization of signals is very well supported in MATLAB.
MATLAB is used interactively.
Eliminates code, compile, run cycle.
Great for rapid prototyping and what-if analysis.



Outline


User Interface


Visualization



Learning Objectives

Getting around in MATLAB
The user interface,
Getting help.
Modeling signals in MATLAB
Using vectors to model signals,
Creating and manipulating vectors,
Visualizing vectors: plotting.



Outline


User Interface


Visualization



MATLAB’s Main Window



MATLAB’s Built-in IDE



MATLAB’s Built-in Help System

MATLAB has an extensive built-in help system.
On-line documentation reader:
contains detailed documentation for entire MATLAB system,
is invoked by
typing doc at command line
clicking “Question Mark” in tool bar of main window,
via “Help” menu.
Command-line help provides access to documentation
inside command window.
Helpful commands include:
help function-name, e.g., help fft.
lookfor keyword, e.g., lookfor inverse.

We will learn how to tie into the built-in help system.



Interacting with MATLAB
You interact with MATLAB by typing commands at the
command prompt (» ) in the command window.
MATLAB’s response depends on whether a semicolon is
appended after the command or not.
If a semicolon is not appended, then MATLAB displays the
result of the command.
With a semicolon, the result is not displayed.
Examples:
The command xx = 1:3 produces
xx =
1 2 3
The command xx = 1:3; produces no output. The variable
xx still stores the result.
Do use a semicolon with xx = 1:30000000;



Outline


User Interface


Visualization



Signals and Vectors
Our objective is to simulate communication systems in
MATLAB.
This includes the signals that occur in such systems, and
processing applied to these signals.
In MATLAB (and any other digital system) signals must be
represented by samples.
Well-founded theory exists regarding sampling (Nyquist’s
sampling theorem).
Result: Signals are represented as a sequence of numbers.
MATLAB is ideally suited to process sequences of
numbers.
MATLAB’s basic data types: vectors (and matrices).
Vectors are just sequence of numbers.



Illustration: Generating a Sinusoidal Signal

The following, simple task illustrates key beneﬁts from
MATLAB’s use of vectors.
Task: Generate samples of the sinusoidal signal
π
x (t ) = 3 · cos(2π440t − )
4
for t ranging from 0 to 10 ms. The sampling rate is 20 KHz.
Objective: Compare how this task is accomplished
using MATLAB’s vector function,
traditional (C-style) for or while loops.



Illustration: Generating a Sinusoidal Signal

For both approaches, we begin by deﬁning a few
parameters.
This increases readability and makes it easier to change
parameters.

%% Set Parameters
A = 3; % amplitude
f = 440; % frequency
phi = -pi/4; % phase
7
fs = 20e3; % sampling rate
Ts = 0; % start time
Te = 10e-3; % end time



Using Loops
The MATLAB code below uses a while loop to generate
the samples one by one.
The majority of the code is devoted to “book-keeping” tasks.

Listing : generateSinusoidLoop.m
%initialize loop variables
tcur = Ts;
kk = 1;

17 while( tcur <= Te)
% compute current sample and store in vector
tt(kk) = tcur;
xx(kk) = A*cos(2*pi*f*tcur + phi);

22 %increment loop variables
kk = kk+1;
tcur = tcur + 1/fs;
end



Vectorized Code

Much more compact code is possible with MATLAB’s
vector functions.
There is no overhead for managing a program loop.
Notice how similar the instruction to generate the samples
is to the equation for the signal.
The vector-based approach is the key enabler for rapid
prototyping.

Listing : generateSinusoid.m
%% generate sinusoid
tt = Ts : 1/fs : Te; % define time-axis
xx = A * cos( 2*pi * f * tt + phi );



Commands for Creating Vectors

The following commands all create useful vectors.
[ ]: the sequence of samples is explicitly speciﬁed.
Example: xx = [ 1 3 2 ] produces xx = 1 3 2.
:(colon operator): creates a vector of equally spaced
samples.
Example: tt = 0:2:9 produces tt = 0 2 4 6 8.
Example: tt = 1:3 produces tt = 1 2 3.
Idiom: tt = ts:1/fs:te creates a vector of sampling times
between ts and te with sampling period 1/fs (i.e., the
sampling rate is fs).



Creating Vectors of Constants

ones(n,m): creates an n × m matrix with all elements equal
to 1.
Example: xx = ones(1,5) produces xx = 1 1 1 1 1.
Example: xx = 4*ones(1,5) produces xx = 4 4 4 4 4.
zeros(n,m): creates an n × m matrix with all elements equal
to 0.
Often used for initializing a vector.
Usage identical to ones.
Note: throughout we adopt the convention that signals are
represented as row vectors.
The ﬁrst (column) dimension equals 1.



Creating Random Vectors

We will often need to create vectors of random numbers.
E.g., to simulate noise.
The following two functions create random vectors.
randn(n,m): creates an n × m matrix of independent
Gaussian random numbers with mean zero and variance
one.
Example: xx = randn(1,5) may produce xx = -0.4326
-1.6656 0.1253 0.2877 -1.1465.
rand(n,m): creates an n × m matrix of independent
uniformly distributed random numbers between zero and
one.
Example: xx = rand(1,5) may produce xx = 0.1576
0.9706 0.9572 0.4854 0.8003.



Addition and Subtraction
The standard + and - operators are used to add and
subtract vectors.
One of two conditions must hold for this operation to
succeed.
Both vectors must have exactly the same size.
In this case, corresponding elements in the two vectors are
added and the result is another vector of the same size.
Example: [1 3 2] + 1:3 produces 2 5 5.
A prominent error message indicates when this condition is
violated.
One of the operands is a scalar, i.e., a 1 × 1 (degenerate)
vector.
In this case, each element of the vector has the scalar added
to it.
The result is a vector of the same size as the vector operand.
Example: [1 3 2] + 2 produces 3 5 4.



Element-wise Multiplication and Division
The operators .* and ./ operators multiply or divide two
vectors element by element.
One of two conditions must hold for this operation to
succeed.
Both vectors must have exactly the same size.
In this case, corresponding elements in the two vectors are
multiplied and the result is another vector of the same size.
Example: [1 3 2] .* 1:3 produces 1 6 6.
An error message indicates when this condition is violated.
One of the operands is a scalar.
In this case, each element of the vector is multiplied by the
scalar.
The result is a vector of the same size as the vector operand.
Example: [1 3 2] .* 2 produces 2 6 4.
If one operand is a scalar the ’.’ may be omitted, i.e.,
[1 3 2] * 2 also produces 2 6 4.



Inner Product
The operator * with two vector arguments computes the
inner product (dot product) of the vectors.
Recall the inner product of two vectors is defined as
N
x ·y = ∑ x (n ) · y (n ).
n =1

This implies that the result of the operation is a scalar!
The inner product is a useful and important signal
processing operation.
It is very different from element-wise multiplication.
The second dimension of the first operand must equal the
first dimension of the second operand.
MATLAB error message: Inner matrix dimensions
must agree.
Example: [1 3 2] * (1:3)’ = 13.
The single quote (’) transposes a vector.


Powers
To raise a vector to some power use the .^ operator.
Example: [1 3 2].^2 yields 1 9 4.
The operator ^ exists but is generally not what you need.
Example: [1 3 2]^2 is equivalent to [1 3 2] * [1 3 2]
which produces an error.
Similarly, to use a vector as the exponent for a scalar base
use the .^ operator.
Example: 2.^[1 3 2] yields 2 8 4.
Finally, to raise a vector of bases to a vector of exponents
use the .^ operator.
Example: [1 3 2].^(1:3) yields 1 9 8.
The two vectors must have the same dimensions.
The .^ operator is (nearly) always the right operator.



Complex Arithmetic
MATLAB support complex numbers fully and naturally.
√
The imaginary unit i = −1 is a built-in constant named i
and j.
Creating complex vectors:
Example: xx = randn(1,5) + j*randn(1,5) creates a
vector of complex Gaussian random numbers.
A couple of “gotchas” in connection with complex
arithmetic:
Never use i and j as variables!
Example: After invoking j=2, the above command will
produce very unexpected results.
Transposition operator (’) transposes and forms conjugate
complex.
That is very often the right thing to do.
Transpose only is performed with .’ operator.



Vector Functions

MATLAB has literally hundreds of built-in functions for
manipulating vectors and matrices.
The following will come up repeatedly:
yy=cos(xx), yy=sin(xx), and yy=exp(xx):
compute the cosine, sine, and exponential for each element
of vector xx,
the result yy is a vector of the same size as xx.
XX=fft(xx), xx=ifft(XX):
Forward and inverse discrete Fourier transform (DFT),
computed via an efﬁcient FFT algorithm.
Many algebraic functions are available, including log10,
sqrt, abs, and round.
Try help elfun for a complete list.



Functions Returning a Scalar Result

Many other functions accept a vector as its input and
return a scalar value as the result.
Examples include
min and max,
mean and var or std,
sum computes the sum of the elements of a vector,
norm provides the square root of the sum of the squares of
the elements of a vector.
The norm of a vector is related to power and energy.
Try help datafun for an extensive list.



Accessing Elements of a Vector

Frequently it is necessary to modify or extract a subset of
the elements of a vector.
Accessing a single element of a vector:
Example: Let xx = [1 3 2], change the third element to 4.
Solution: xx(3) = 4; produces xx = 1 3 4.
Single elements are accessed by providing the index of the
element of interest in parentheses.



Accessing Elements of a Vector
Accessing a range of elements of a vector:
Example: Let xx = ones(1,10);, change the ﬁrst ﬁve
elements to −1.
Solution: xx(1:5) = -1*ones(1,5); Note, xx(1:5) = -1
works as well.
Example: Change every other element of xx to 2.
Solution: xx(2:2:end) = 2;;
Note that end may be use to denote the index of a vector’s
last element.
This is handy if the length of the vector is not known.
Example: Change third and seventh element to 3.
Solution: xx([3 7]) = 3;;
A set of elements of a vector is accessed by providing a
vector of indices in parentheses.



Accessing Elements that Meet a Condition
Frequently one needs to access all elements of a vector
that meet a given condition.
Clearly, that could be accomplished by writing a loop that
examines and processes one element at a time.
Such loops are easily avoided.
Example: “Poor man’s absolute value”
Assume vector xx contains both positive and negative
numbers. (e.g., xx = randn(1,10);).
Objective: Multiply all negative elements of xx by −1; thus
compute the absolute value of all elements of xx.
Solution: proceeds in two steps
isNegative = (xx < 0);
xx(isNegative) = -xx(isNegative);
The vector isNegative consists of logical (boolean) values;
1’s appear wherever an element of xx is negative.



Outline


User Interface


Visualization



Visualization and Graphics

MATLAB has powerful, built-in functions for plotting
functions in two and three dimensions.
Publication quality graphs are easily produced in a variety
of standard graphics formats.
MATLAB provides ﬁne-grained control over all aspects of
the ﬁnal graph.



A Basic Plot

The sinusoidal signal, we generated earlier is easily plotted
via the following sequence of commands:
Try help plot for more information about the capabilities of
the plot command.
%% Plot
plot(tt, xx, ’r’) % xy-plot, specify red line
xlabel( ’Time (s)’ ) % labels for x and y axis
ylabel( ’Amplitude’ )
10 title( ’x(t) = A cos(2pi f t + phi)’)
grid % create a grid
axis([0 10e-3 -4 4]) % tweak the range for the axes



Resulting Plot

x(t) = A cos (2π f t + φ)
4

2
Amplitude

0

−2

−4
0 0.002 0.004 0.006 0.008 0.01
Time (s)



Multiple Plots in One Figure

MATLAB can either put multiple graphs in the same plot or
put multiple plots side by side.
The latter is accomplished with the subplot command.
subplot(2,1,1)
plot(tt, xx ) % xy-plot
ylabel( ’Amplitude’ )
12 title( ’x(t) = A cos(2pi f t + phi)’)

subplot(2,1,2)
plot(tt, yy ) % xy-plot
17 ylabel( ’Amplitude’ )
title( ’x(t) = A sin(2pi f t + phi)’)



Resulting Plot
x(t) = A cos(2π f t + φ)
4

2
Amplitude

0

−2

−4
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
Time (s)
x(t) = A sin(2π f t + φ)
4

2
Amplitude

0

−2

−4
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
Time (s)



3-D Graphics

MATLAB provides several functions that create high-quality
three-dimensional graphics.
The most important are:
plot3(x,y,z): plots a function of two variables.
mesh(x,y,Z): plots a mesh of the values stored in matrix Z
over the plane spanned by vectors x and y.
surf(x,y,Z): plots a surface from the values stored in
matrix Z over the plane spanned by vectors x and y.
A relevant example is shown on the next slide.
The path loss in a two-ray propagation environment over a
ﬂat, reﬂecting surface is shown as a function of distance
and frequency.



1.02
1.01
1
0.99 2
10
0.98
0.97
0.96
0.95 1
10
Frequency (GHz)
Distance (m)

Figure: Path loss over a ﬂat reﬂecting surface.



Summary

We have taken a brief look at the capabilities of MATLAB.
Speciﬁcally, we discussed
Vectors as the basic data unit used in MATLAB,
Arithmetic with vectors,
Prominent vector functions,
Visualization in MATLAB.
We will build on this basis as we continue and apply
MATLAB to the simulation of communication systems.
To probe further:
Read the built-in documentation.
Recommended MATLAB book: D. Hanselman and
B. Littleﬁeld, Mastering MATLAB, Prentice-Hall.


Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation

Part II

Introduction: From Theory to Simulation



Introduction: From Theory to Simulation

Introduction to digital communications and simulation of digital
communications systems.
A simple digital communication system and its theoretical
underpinnings
Introduction to digital modulation
Baseband and passband signals: complex envelope
Noise and Randomness
The matched ﬁlter receiver
Bit-error rate
Example: BPSK over AWGN, simulation in MATLAB



Outline



Digital Modulation

Channel Model

Receiver

MATLAB Simulation



Learning Objectives

Theory of Digital Communications.
Principles of Digital modulation.
Communications Channel Model: Additive, White Gaussian
Noise.
The Matched Filter Receiver.
Finding the Probability of Error.
Modeling a Digital Communications System in MATLAB.
Representing Signals and Noise in MATLAB.
Simulating a Communications System.
Measuring Probability of Error via MATLAB Simulation.



Outline



Digital Modulation

Channel Model

Receiver

MATLAB Simulation



Source: produces a sequence of information symbols b.
Transmitter: maps bit sequence to analog signal s (t ).
Channel: models corruption of transmitted signal s (t ).
Receiver: produces reconstructed sequence of information
ˆ
symbols b from observed signal R (t ).

b s (t ) R (t ) ˆ
b
Source Transmitter Channel Receiver

Figure: Block Diagram of a Generic Digital Communications System



The Source
The source models the statistical properties of the digital
information source.
Three main parameters:
Source Alphabet: list of the possible information symbols
the source produces.
Example: A = {0, 1}; symbols are called
bits.
Alphabet for a source with M (typically, a
power of 2) symbols: A = {0, 1, . . . , M − 1}
or A = {±1, ±3, . . . , ±(M − 1)}.
Alphabet with positive and negative symbols
is often more convenient.
Symbols may be complex valued; e.g.,
A = {±1, ±j }.



A priori Probability: relative frequencies with which the source
produces each of the symbols.
Example: a binary source that produces (on
average) equal numbers of 0 and 1 bits has
1
π0 = π1 = 2 .
Notation: πn denotes the probability of
observing the n-th symbol.
Typically, a-priori probabilities are all equal,
1
i.e., πn = M .
A source with M symbols is called an M-ary
source.
binary (M = 2)
ternary (M = 3)
quaternary (M = 4)



Symbol Rate: The number of information symbols the source
produces per second. Also called the baud rate R.

Closely related: information rate Rb indicates
the number of bits the source produces per
second.
Relationship: Rb = R · log2 (M ).
Also, T = 1/R is the symbol period.

Bit 1 Bit 2 Symbol
0 0 0
0 1 1
1 0 2
1 1 3
Table: Two bits can be represented in one quaternary symbol.



Remarks

This view of the source is simplified.
We have omitted important functionality normally found in
the source, including
error correction coding and interleaving, and
mapping bits to symbols.
This simplified view is sufficient for our initial discussions.
Missing functionality will be revisited when needed.



Modeling the Source in MATLAB
Objective: Write a MATLAB function to be invoked as:
Symbols = RandomSymbols( N, Alphabet, Priors);

The input parameters are
N: number of input symbols to be produced.
Alphabet: source alphabet to draw symbols from.
Example: Alphabet = [1 -1];
Priors: a priori probabilities for the input symbols.
Example:
Priors = ones(size(Alphabet))/length(Alphabet);
The output Symbols is a vector
with N elements,
drawn from Alphabet, and
the number of times each symbol occurs is (approximately)
proportional to the corresponding element in Priors.



Reminders

MATLAB’s basic data units are vectors and matrices.
Vectors are best thought of as lists of numbers; vectors
often contain samples of a signal.
There are many ways to create vectors, including
Explicitly: Alphabet = [1 -1];
Colon operator: nn = 1:10;
Via a function: Priors=ones(1,5)/5;
This leads to very concise programs; for-loops are rarely
needed.
MATLAB has a very large number of available functions.
Reduces programming to combining existing building
blocks.
Difﬁculty: ﬁnd out what is available; use built-in help.



Writing a MATLAB Function

A MATLAB function must
begin with a line of the form
function [out1,out2] = FunctionName(in1, in2, in3)
be stored in a file with the same name as the function name
and extension ’.m’.
For our symbol generator, the file name must be
RandomSymbols.m and
the first line must be
function Symbols = RandomSymbols(N, Alphabet, Priors)



Writing a MATLAB Function

A MATLAB function should
have a second line of the form
%FunctionName - brief description of function
This line is called the “H1 header.”
have a more detailed description of the function and how to
use it on subsequent lines.
The detailed description is separated from the H1 header by
a line with only a %.
Each of these lines must begin with a % to mark it as a
comment.
These comments become part of the built-in help system.



The Header of Function RandomSymbols

function Symbols = RandomSymbols(N, Alphabet, Priors)
% RandomSymbols - generate a vector of random information symbols
%
% A vector of N random information symbols drawn from a given
5 % alphabet and with specified a priori probabilities is produced.
%
% Inputs:
% N - number of symbols to be generated
% Alphabet - vector containing permitted symbols
10 % Priors - a priori probabilities for symbols
%
% Example:
% Symbols = RandomSymbols(N, Alphabet, Priors)



Algorithm for Generating Random Symbols
For each of the symbols to be generated we use the
following algorithm:
Begin by computing the cumulative sum over the priors.
Example: Let Priors = [0.25 0.25 0.5], then the
cumulative sum equals CPriors = [0 0.25 0.5 1].
For each symbol, generate a uniform random number
between zero and one.
The MATLAB function rand does that.
Determine between which elements of the cumulative sum
the random number falls and select the corresponding
symbol from the alphabet.
Example: Assume the random number generated is 0.3.
This number falls between the second and third element of
CPriors.
The second symbol from the alphabet is selected.



MATLAB Implementation

In MATLAB, the above algorithm can be “vectorized” to
work on the entire sequence at once.
CPriors = [0 cumsum( Priors )];
rr = rand(1, N);

for kk=1:length(Alphabet)
42 Matches = rr > CPriors(kk) & rr <= CPriors(kk+1);
Symbols( Matches ) = Alphabet( kk );
end



Testing Function RandomSymols
We can invoke and test the function RandomSymbols as
shown below.
A histogram of the generated symbols should reﬂect the
speciﬁed a priori probabilities.
%% set parameters
N = 1000;
Alphabet = [-3 -1 1 3];
Priors = [0.1 0.2 0.3 0.4];
10
%% generate symbols and plot histogram
Symbols = RandomSymbols( N, Alphabet, Priors );
hist(Symbols, -4:4 );
grid
15 xlabel(’Symbol Value’)
ylabel(’Number of Occurences’)



Resulting Histogram
400

350

300
Number of Occurences

250

200

150

100

50

0
−4 −3 −2 −1 0 1 2 3 4
Symbol Value



The Transmitter
The transmitter translates the information symbols at its
input into signals that are “appropriate” for the channel,
e.g.,
meet bandwidth requirements due to regulatory or
propagation considerations,
provide good receiver performance in the face of channel
impairments:
noise,
distortion (i.e., undesired linear ﬁltering),
interference.
A digital communication system transmits only a discrete
set of information symbols.
Correspondingly, only a discrete set of possible signals is
employed by the transmitter.
The transmitted signal is an analog (continuous-time,
continuous amplitude) signal.


Illustrative Example
The sources produces symbols from the alphabet
A = {0, 1}.
The transmitter uses the following rule to map symbols to
signals:
If the n-th symbol is bn = 0, then the transmitter sends the
signal
A for (n − 1)T ≤ t < nT
s0 (t ) =
0 else.
If the n-th symbol is bn = 1, then the transmitter sends the
signal

for (n − 1)T ≤ t < (n − 1 )T

 A 2
s1 (t ) = −A for (n − 1 )T ≤ t < nT
2
0 else.




Symbol Sequence b = {1, 0, 1, 1, 0, 0, 1, 0, 1, 0}

4

2
Amplitude

0

−2

−4
0 2 4 6 8 10
Time/T



MATLAB Code for Example

Listing : plot_TxExampleOrth.m
b = [ 1 0 1 1 0 0 1 0 1 0]; %symbol sequence
fsT = 20; % samples per symbol period
A = 3;
6
Signals = A*[ ones(1,fsT); % signals, one per row
ones(1,fsT/2) -ones(1,fsT/2)];

tt = 0:1/fsT:length(b)-1/fsT; % time axis for plotting
11
%% generate signal ...
TXSignal = [];
for kk=1:length(b)
TXSignal = [ TXSignal Signals( b(kk)+1, : ) ];
16 end



MATLAB Code for Example

Listing : plot_TxExampleOrth.m

%% ... and plot
plot(tt, TXSignal)
20 axis([0 length(b) -(A+1) (A+1)]);
grid
xlabel(’Time/T’)



The Communications Channel
The communications channel models the degradation the
transmitted signal experiences on its way to the receiver.
For wireless communications systems, we are concerned
primarily with:
Noise: random signal added to received signal.
Mainly due to thermal noise from electronic components in
the receiver.
Can also model interference from other emitters in the
vicinity of the receiver.
Statistical model is used to describe noise.
Distortion: undesired ﬁltering during propagation.
Mainly due to multi-path propagation.
Both deterministic and statistical models are appropriate
depending on time-scale of interest.
Nature and dynamics of distortion is a key difference to
wired systems.



Thermal Noise
At temperatures above absolute zero, electrons move
randomly in a conducting medium, including the electronic
components in the front-end of a receiver.
This leads to a random waveform.
The power of the random waveform equals PN = kT0 B.
k : Boltzmann’s constant (1.38 · 10−23 Ws/K).
T0 : temperature in degrees Kelvin (room temperature
≈ 290 K).
For bandwidth equal to 1 MHz, PN ≈ 4 · 10−15 W
(−114 dBm).
Noise power is small, but power of received signal
decreases rapidly with distance from transmitter.
Noise provides a fundamental limit to the range and/or rate
at which communication is possible.



Multi-Path
In a multi-path environment, the receiver sees the
combination of multiple scaled and delayed versions of the
transmitted signal.

TX RX



Distortion from Multi-Path
5
Received signal
“looks” very
different from
Amplitude

transmitted signal.
0
Inter-symbol
interference (ISI).
Multi-path is a very
serious problem
−5 for wireless
0 2 4 6 8 10
systems.
Time/T



The Receiver
The receiver is designed to reconstruct the original
information sequence b.
Towards this objective, the receiver uses
the received signal R (t ),
knowledge about how the transmitter works,
Speciﬁcally, the receiver knows how symbols are mapped to
signals.
the a-priori probability and rate of the source.
The transmitted signal typically contains information that
allows the receiver to gain information about the channel,
including
training sequences to estimate the impulse response of the
channel,
synchronization preambles to determine symbol locations
and adjust ampliﬁer gains.


The Receiver

The receiver input is an analog signal and it’s output is a
sequence of discrete information symbols.
Consequently, the receiver must perform analog-to-digital
conversion (sampling).
Correspondingly, the receiver can be divided into an
analog front-end followed by digital processing.
Modern receivers have simple front-ends and sophisticated
digital processing stages.
Digital processing is performed on standard digital
hardware (from ASICs to general purpose processors).
Moore’s law can be relied on to boost the performance of
digital communications systems.



Measures of Performance

The receiver is expected to perform its function optimally.
Question: optimal in what sense?
Measure of performance must be statistical in nature.
observed signal is random, and
transmitted symbol sequence is random.
Metric must reﬂect the reliability with which information is
reconstructed at the receiver.
Objective: Design the receiver that minimizes the
probability of a symbol error.
Also referred to as symbol error rate.
Closely related to bit error rate (BER).



Summary

We have taken a brief look at the elements of a
communication system.
Source,
Transmitter,
Channel, and
Receiver.
We will revisit each of these elements for a more rigorous
analysis.
Intention: Provide enough detail to allow simulation of a
communication system.



Outline



Digital Modulation

Channel Model

Receiver

MATLAB Simulation



Digital Modulation

Digital modulation is performed by the transmitter.
It refers to the process of converting a sequence of
information symbols into a transmitted (analog) signal.
The possibilities for performing this process are virtually
without limits, including
varying, the amplitude, frequency, and/or phase of a
sinusoidal signal depending on the information sequence,
making the currently transmitted signal on some or all of the
previously transmitted symbols (modulation with memory).
Initially, we focus on a simple, yet rich, class of modulation
formats referred to as linear modulation.



Linear Modulation
Linear modulation may be thought of as the digital
equivalent of amplitude modulation.
The instantaneous amplitude of the transmitted signal is
proportional to the current information symbol.
Speciﬁcally, a linearly modulated signal may be written as
N −1
s (t ) = ∑ bn · p (t − nT )
n =0

where,
bn denotes the n-th information symbol, and
p (t ) denotes a pulse of ﬁnite duration.
Recall that T is the duration of a symbol.



Linear Modulation

Note, that the expression
N −1

bn s (t )
s (t ) = ∑ bn · p (t − nT )
× p (t ) n =0

is linear in the symbols bn .
Different modulation formats are
∑ δ(t − nT ) constructed by choosing appropriate
symbol alphabets, e.g.,
BPSK: bn ∈ {1, −1}
OOK: bn ∈ {0, 1}
PAM: bn ∈ {±1, . . . , ±(M − 1)}.



Linear Modulation in MATLAB
To simulate a linear modulator in MATLAB, we will need a
function with a function header like this:
function Signal = LinearModulation( Symbols, Pulse, fsT )
% LinearModulation - linear modulation of symbols with given
3 % pulse shape
%
% A sequence of information symbols is linearly modulated. Pulse
% shaping is performed using the pulse shape passed as input
% parameter Pulse. The integer fsT indicates how many samples
8 % per symbol period are taken. The length of the Pulse vector may
% be longer than fsT; this corresponds to partial-response signali
%
% Inputs:
% Symbols - vector of information symbols
13 % Pulse - vector containing the pulse used for shaping
% fsT - (integer) number of samples per symbol period



Linear Modulation in MATLAB

In the body of the function, the sum of the pulses is
computed.
There are two issues that require some care:
Each pulse must be inserted in the correct position in the
output signal.
Recall that the expression for the output signal s (t ) contains
the terms p (t − nT ).
The term p (t − nT ) reﬂects pulses delayed by nT .
Pulses may overlap.
If the duration of a pulse is longer than T , then pulses
overlap.
Such overlapping pulses are added.
This situation is called partial response signaling.



Body of Function LinearModulation

19 % initialize storage for Signal
LenSignal = length(Symbols)*fsT + (length(Pulse))-fsT;
Signal = zeros( 1, LenSignal );

% loop over symbols and insert corresponding segment into Signal
24 for kk = 1:length(Symbols)
ind_start = (kk-1)*fsT + 1;
ind_end = (kk-1)*fsT + length(Pulse);

Signal(ind_start:ind_end) = Signal(ind_start:ind_end) + ...
29 Symbols(kk) * Pulse;
end



Testing Function LinearModulation
Listing : plot_LinearModRect.m

%% Parameters:
fsT = 20;
Alphabet = [1,-1];
6 Priors = 0.5*[1 1];
Pulse = ones(1,fsT); % rectangular pulse

%% symbols and Signal using our functions
Symbols = RandomSymbols(10, Alphabet, Priors);
11 Signal = LinearModulation(Symbols,Pulse,fsT);
%% plot
tt = (0 : length(Signal)-1 )/fsT;
plot(tt, Signal)
axis([0 length(Signal)/fsT -1.5 1.5])
16 grid
ylabel(’Amplitude’)



Linear Modulation with Rectangular Pulses

1.5

1

0.5
Amplitude

0

−0.5

−1

−1.5
0 2 4 6 8 10
Time/T



Linear Modulation with sinc-Pulses
More interesting and practical waveforms arise when
smoother pulses are used.
A good example are truncated sinc functions.
The sinc function is defined as:
sin(x )
sinc(x ) = , with sinc(0) = 1.
x
Specifically, we will use pulses defined by
sin(πt /T )
p (t ) = sinc(πt /T ) = ;
πt /T

pulses are truncated to span L symbol periods, and
delayed to be causal.
Toolbox contains function Sinc( L, fsT ).



A Truncated Sinc Pulse
1.5

1
Pulse is very smooth,
0.5
spans ten symbol
periods,
is zero at location of
0 other symbols.
Nyquist pulse.
−0.5
0 2 4 6 8 10
Time/T



Linear Modulation with Sinc Pulses
2

Resulting waveform is
1 also very smooth; expect
good spectral properties.
Amplitude

0 Symbols are harder to
discern; partial response
signaling induces
−1 “controlled” ISI.
But, there is no ISI at
symbol locations.
−2
0 5 10 15 20 Transients at beginning
Time/T and end.



Passband Signals

So far, all modulated signals we considered are baseband
signals.
Baseband signals have frequency spectra concentrated
near zero frequency.
However, for wireless communications passband signals
must be used.
Passband signals have frequency spectra concentrated
around a carrier frequency fc .
Baseband signals can be converted to passband signals
through up-conversion.
Passband signals can be converted to baseband signals
through down-conversion.



Up-Conversion

A cos(2πfc t )

sI (t ) The passband signal sP (t ) is
× constructed from two (digitally
modulated) baseband signals, sI (t )
sP ( t ) and sQ (t ).
+ Note that two signals can be
carried simultaneously!
This is a consequence of
sQ (t ) cos(2πfc t ) and sin(2πfc t ) being
×
orthogonal.

A sin(2πfc t )



Baseband Equivalent Signals
The passband signal sP (t ) can be written as
√ √
sP (t ) = 2 · AsI (t ) · cos(2πfc t ) + 2 · AsQ (t ) · sin(2πfc t ).

If we deﬁne s (t ) = sI (t ) − j · sQ (t ), then sP (t ) can also be
expressed as
√
sP (t ) = 2 · A · {s (t ) · exp(j2πfc t )}.

The signal s (t ):
is called the baseband equivalent or the complex envelope
of the passband signal sP (t ).
It contains the same information as sP (t ).
Note that s (t ) is complex-valued.



Illustration: QPSK with fc = 2/T

Passband signal (top):
2 segments of sinusoids
1 with different phases.
Amplitude

0 Phase changes occur
−1
at multiples of T .
−2
Baseband signal
0 1 2 3 4 5 6 7 8 9 10
Time/T (bottom) is complex
2 1
valued; magnitude and
1.5
phase are plotted.
0.5
Magnitude is constant
Magnitude

Phase/π

1

0
(rectangular pulses).
0.5

0 −0.5
0 5 10 0 5 10
Time/T Time/T



MATLAB Code for QPSK Illustration

Listing : plot_LinearModQPSK.m
%% Parameters:
fsT = 20;
L = 10;
fc = 2; % carrier frequency
7 Alphabet = [1, j, -j, -1];% QPSK
Priors = 0.25*[1 1 1 1];
Pulse = ones(1,fsT); % rectangular pulse

12 Symbols = RandomSymbols(10, Alphabet, Priors);
Signal = LinearModulation(Symbols,Pulse,fsT);
%% passband signal
tt = (0 : length(Signal)-1 )/fsT;
Signal_PB = sqrt(2)*real( Signal .* exp(-j*2*pi*fc*tt) );



MATLAB Code for QPSK Illustration
Listing : plot_LinearModQPSK.m
subplot(2,1,1)
plot( tt, Signal_PB )
grid
22 xlabel(’Time/T’)
ylabel(’Amplitude’)

subplot(2,2,3)
plot( tt, abs( Signal ) )
27 grid
ylabel(’Magnitude’)

subplot(2,2,4)
32 plot( tt, angle( Signal )/pi )
grid
ylabel(’Phase/pi’)



Frequency Domain Perspective
In the frequency domain:
√
2 · SP (f + fc ) for f + fc > 0
S (f ) =
0 else.
√
Factor 2 ensures both signals have the same power.

SP (f ) S (f )
√
2·A
A

f f
− fc fc − fc fc



Baseband Equivalent System

The baseband description of the transmitted signal is very
convenient:
it is more compact than the passband signal as it does not
include the carrier component,
while retaining all relevant information.
However, we are also concerned what happens to the
signal as it propagates to the receiver.
Question: Do baseband techniques extend to other parts
of a passband communications system?



Passband System

√ √
2A cos(2πfc t ) 2 cos(2πfc t )

sI (t ) RI (t )
× NP ( t ) × LPF

sP ( t ) RP ( t )
+ hP (t ) +

sQ (t ) RQ (t )
× × LPF

√ √
2A sin(2πfc t ) 2 sin(2πfc t )



Baseband Equivalent System
N (t )

s (t ) R (t )
h (t ) +

The passband system can be interpreted as follows to yield
an equivalent system that employs only baseband signals:
baseband equivalent transmitted signal:
s (t ) = sI (t ) − j · sQ (t ).
baseband equivalent channel with complex valued impulse
response: h(t ).
baseband equivalent received signal:
R ( t ) = RI ( t ) − j · RQ ( t ) .
complex valued, additive Gaussian noise: N (t )


Baseband Equivalent Channel
The baseband equivalent channel is deﬁned by the entire
shaded box in the block diagram for the passband system
(excluding additive noise).
The relationship between the passband and baseband
equivalent channel is

hP (t ) = {h(t ) · exp(j2πfc t )}

in the time domain.
Example:

hP ( t ) = ∑ ak · δ(t − τk ) =⇒ h(t ) = ∑ ak · e−j2πf τ c k
· δ(t − τk ).
k k



Baseband Equivalent Channel

In the frequency domain

HP (f + fc ) for f + fc > 0
H (f ) =
0 else.

HP (f ) H (f )

A A

f f
− fc fc − fc fc



Summary
The baseband equivalent channel is much simpler than the
passband model.
Up and down conversion are eliminated.
Expressions for signals do not contain carrier terms.
The baseband equivalent signals are easier to represent
for simulation.
Since they are low-pass signals, they are easily sampled.
No information is lost when using baseband equivalent
signals, instead of passband signals.
Standard, linear system equations hold:

R (t ) = s (t ) ∗ h(t ) + n(t ) and R (f ) = S (f ) · H (f ) + N (f ).

Conclusion: Use baseband equivalent signals and
systems.


Outline



Digital Modulation

Channel Model

Receiver

MATLAB Simulation



Channel Model

The channel describes how the transmitted signal is
corrupted between transmitter and receiver.
The received signal is corrupted by:
noise,
distortion, due to multi-path propagation,
interference.
Interference is not considered in detail.
Is easily added to simulations,
Frequent assumption: interference can be lumped in with
noise.



Additive White Gaussian Noise

A standard assumption in most communication systems is
that noise is well modeled as additive, white Gaussian
noise (AWGN).
additive: channel adds noise to the transmitted signal.
white: describes the temporal correlation of the noise.
Gaussian: probability distribution is a Normal or Gaussian
distribution.
Baseband equivalent noise is complex valued.
In-phase NI (t ) and quadrature NQ (t ) noise signals are
modeled as independent (circular symmetric noise).



White Noise
The term white means specifically that
the mean of the noise is zero,

E[N (t )] = 0,

the autocorrelation function of the noise is

ρN (τ ) = E[N (t ) · N ∗ (t + τ )] = N0 δ(τ ).

This means that any distinct noise samples are
independent.
The autocorrelation function also indicates that noise
samples have infinite variance.
Insight: noise must be filtered before it can be sampled.
In-phase and quadrature noise each have autocorrelation
N0
2 δ ( τ ).



White Noise

The term “white” refers to the spectral properties of the
noise.
Speciﬁcally, the power spectral density of white noise is
constant over all frequencies:

SN (f ) = N0 for all f .

White light consists of equal components of the visible
spectrum.



Generating Gaussian Noise Samples

For simulating additive noise, Gaussian noise samples
must be generated.
MATLAB idiom for generating a vector of N independent,
complex, Gaussian random numbers with variance
VarNoise:
Noise = sqrt(VarNoise/2) * ( randn(1,N) + j * randn(1,N));
Note, that real and imaginary part each have variance
VarNoise/2.
This causes the total noise variance to equal VarNoise.



Toolbox Function addNoise

We will frequently simulate additive, white Gaussian noise.
To perform this task, a function with the following header is
helpful:
function NoisySignal = addNoise( CleanSignal, VarNoise )
% addNoise - simulate additive, white Gaussian noise channel
%
% Independent Gaussian noise of variance specified by the second
5 % input parameter is added to the (vector or matrix) signal passed
% as the first input. The result is returned in a vector or matrix
% of the same size as the input signal.
%
% The function determines if the signal is real or complex valued
10 % and generates noise samples accordingly. In either case the total
% variance is equal to the second input.



The Body of Function addNoise

%% generate noisy signal
% distinguish between real and imaginary input signals
if ( isreal( CleanSignal ) )
25 NoisySignal = CleanSignal + ...
sqrt(VarNoise) * randn( size(CleanSignal) );
else
NoisySignal = CleanSignal + sqrt(VarNoise/2) * ...
( randn( size(CleanSignal) ) + j*randn( size(CleanSignal) ) );
30 end



4

2
Amplitude

0

−2

−4

−6
0 5 10 15 20
Time/T
Es
Figure: Linearly modulated Signal with Noise; N0 ≈ 10 dB, noise
20
variance ≈ 2, noise bandwidth ≈ T .



Multi-path Fading Channels

In addition to corruption by additive noise, transmitted
signals are distorted during propagation from transmitter to
receiver.
The distortion is due to multi-path propagation.
The transmitted signal travels to the receiver along multiple
paths, each with different attenuation and delay.
The receiver observes the sum of these signals.
Effect: undesired ﬁltering of transmitted signal.



Multi-path Fading Channels

In mobile communications, the characteristics of the
multi-path channel vary quite rapidly; this is referred to as
fading.
Fading is due primarily to phase changes due to changes in
the delays for the different propagation paths.
Multi-path fading channels will be studied in detail later in
the course.
In particular, the impact of multi-path fading on system
performance will be investigated.



Outline



Digital Modulation

Channel Model

Receiver

MATLAB Simulation



Receiver

The receiver is responsible for extracting the sequence of
information symbols from the received signal.
This task is difﬁcult because of the signal impairments
induced by the channel.
At this time, we focus on additive, white Gaussian noise as
the only source of signal corruption.
Remedies for distortion due to multi-path propagation will
be studied extensively later.
Structure of receivers for digital communication systems.
Analog front-end and digital post-processing.
Performance analysis: symbol error rate.
Closed form computation of symbol error rate is possible for
AWGN channel.



Matched Filter
It is well known, that the optimum receiver for an AWGN
channel is the matched filter receiver.
The matched filter for a linearly modulated signal using
pulse shape p (t ) is shown below.
The slicer determines which symbol is “closest” to the
matched filter output.
Its operation depends on the symbols being used and the a
priori probabilities.

R (t ) ˆ
b
T
× 0 (·) dt Slicer

p (t )



Shortcomings of The Matched Filter

While theoretically important, the matched filter has a few
practical drawbacks.
For the structure shown above, it is assumed that only a
single symbol was transmitted.
In the presence of channel distortion, the receiver must be
matched to p (t ) ∗ h(t ) instead of p (t ).
Problem: The channel impulse response h(t ) is generally
not known.
The matched filter assumes that perfect symbol
synchronization has been achieved.
The matching operation is performed in continuous time.
This is difficult to accomplish with analog components.



Analog Front-end and Digital Back-end
As an alternative, modern digital receivers employ a
different structure consisting of
an analog receiver front-end, and
a digital signal processing back-end.
The analog front-end is little more than a ﬁlter and a
sampler.
The theoretical underpinning for the analog front-end is
Nyquist’s sampling theorem.
The front-end may either work on a baseband signal or a
passband signal at an intermediate frequency (IF).
The digital back-end performs sophisticated processing,
including
digital matched ﬁltering,
equalization, and
synchronization.



Analog Front-end

Several, roughly equivalent, alternatives exist for the
analog front-end.
Two common approaches for the analog front-end will be
considered brieﬂy.
Primarily, the analog front-end is responsible for converting
the continuous-time received signal R (t ) into a
discrete-time signal R [n].
Care must be taken with the conversion: (ideal) sampling
would admit too much noise.
Modeling the front-end faithfully is important for accurate
simulation.



Analog Front-end: Low-pass and Whitening Filter

The first structure contains
a low-pass filter (LPF) with bandwidth equal to the signal
bandwidth,
a sampler followed by a whitening filter (WF).
The low-pass filter creates correlated noise,
the whitening filter removes this correlation.

Sampler,
rate fs

R (t ) R [n] to
LPF WF
DSP



Analog Front-end: Integrate-and-Dump
An alternative front-end has the structure shown below.
Here, ΠTs (t ) indicates a ﬁlter with an impulse response that
is a rectangular pulse of length Ts = 1/fs and amplitude
1/Ts .
The entire system is often called an integrate-and-dump
sampler.
Most analog-to-digital converters (ADC) operate like this.
A whitening ﬁlter is not required since noise samples are
uncorrelated.

Sampler,
rate fs

R (t ) R [n ] to
ΠTs (t )
DSP



Output from Analog Front-end
The second of the analog front-ends is simpler
conceptually and widely used in practice;
it will be assumed for the remainder of the course.
For simulation purposes, we need to characterize the
output from the front-end.
To begin, assume that the received signal R (t ) consists of
a deterministic signal s (t ) and (AWGN) noise N (t ):

R (t ) = s (t ) + N (t ).

The signal R [n] is a discrete-time signal.
The front-end generates one sample every Ts seconds.
The discrete-time signal R [n] also consists of signal and
noise
R [n ] = s [n ] + N [n ].



Consider the signal and noise component of the front-end
output separately.
This can be done because the front-end is linear.
The n-th sample of the signal component is given by:
1 (n+1)Ts
s [n ] = · s (t ) dt ≈ s ((n + 1/2)Ts ).
Ts nTs

The approximation is valid if fs = 1/Ts is much greater than
the signal band-width.
Sampler,
rate fs

R (t ) R [n ] to
ΠTs (t )
DSP



The noise samples N [n] at the output of the front-end:
are independent, complex Gaussian random variables, with
zero mean, and
variance equal to N0 /Ts .
The variance of the noise samples is proportional to 1/Ts .
Interpretations:
Noise is averaged over Ts seconds: variance decreases with
length of averager.
Bandwidth of front-end ﬁlter is approximately 1/Ts and
power of ﬁltered noise is proportional to bandwidth (noise
bandwidth).
It will be convenient to express the noise variance as
N0 /T · T /Ts .
The factor T /Ts = fs T is the number of samples per
symbol period.



Receiver Performance
Our primary measure of performance is the symbol error
probability.
For AWGN channels, it is possible to express the symbol
error probability in closed form for many digital modulation
formats.
To ﬁx ideas, we employ the following concrete signaling
format:
BPSK modulation; symbols are drawn equally likely from
the alphabet {1, −1},
Pulse shaping:
2 2πt
p (t ) = · (1 − cos( )) for 0 ≤ t ≤ T .
3 T

This is a smooth, full-response pulse; it is referred to as a
raised cosine pulse.


BPSK Modulation with Raised Cosine Pulses

2

1
Amplitude

0

−1

−2
0 2 4 6 8 10
Time/T



Signal Model

In each symbol period, the received signal is of the form:

R (t ) = b · A · p (t ) + N (t ) for 0 ≤ t < T .

where
b ∈ {1, −1} is a BPSK symbol (with equal a priori
probabilities),
A is the amplitude of the received signal,
p (t ) is the raised cosine pulse, and
N (t ) is (complex) white Gaussian noise with spectral height
N0 .



Performance of Matched Filter Receiver

With these assumptions, the symbol error rate is

2A2 T 2Es
Pe,min = Q( ) = Q( ).
N0 N0

Es = A2 T is the symbol energy.
We used the fact that p2 (t ) dt = T .
∞ 1
Q(x ) = x
√ exp(−y 2 /2) dy is the Gaussian error
2π
function.
No receiver can achieve a symbol error rate smaller than
Pe,min .



Suboptimum Receivers
We discussed earlier that it is often not possible to use an
exact matched ﬁlter in practice.
Practical receivers can often be modeled as matching with
a pulse shape g (t ) that is (slightly) different than the
transmitted pulse shape p (t ).
Such receivers are called linear receivers; note that the
matched ﬁlter is also a linear receiver.
Example: Assume that we are using the
integrate-and-dump front-end and sample once per bit
period.
This corresponds to matching with a pulse
1
T for 0 ≤ t ≤ T
g (t ) =
0 else.



Performance with Suboptimum Receivers

The symbol error rate of linear receivers can be expressed
in closed form:
2ρ2 Es
g
Pe = Q( )
N0
where
ρg captures the mismatch between the transmitted pulse
p (t ) and the pulse g (t ) used by the receiver.
Speciﬁcally

p (t )g (t ) dt
ρ2 =
g .
p2 (t ) dt · g 2 (t ) dt



Performance with Suboptimum Receivers
For our example with raised cosine pulses p (t ) and
rectangular pulses g (t ), we ﬁnd

ρ2 = 2/3
g

and
4Es
Pe = Q ( ).
3N0
Comparing with the error probability for the matched ﬁlter:
The term 3Ns replaces 2Es .
4E
N0
0
Interpretation: to make both terms equal, the suboptimum
receiver must spend 1.5 times more energy (1.7 dB loss in
performance).



Performance Comparison

Symbol Error Probability 100

10−2

10−4

10−6
0 2 4 6 8 10
Es /N0 (dB)



MATLAB Code for Performance Comparison

Listing : plot_Q.m
%% Set parameters
EsN0dB = 0:0.1:10; % range of Es over N0 values on dB scale
EsN0lin = dB2lin( EsN0dB ); % convert to linear scale

7 %% compute Pe
PeMf = Q( sqrt( 2 * EsN0lin ) );
PeSo = Q( sqrt( 4/3 * EsN0lin ) );

%% plot
12 semilogy( EsN0dB, PeMf, EsN0dB, PeSo)
xlabel( ’E_s/N_0 (dB)’)
ylabel( ’Symbol Error Probability’ );
grid



Summary

We introduced the matched ﬁlter as the receiver that
minimizes the probability of a symbol error.
Practical considerations lead to implementing receivers
with simple analog front-ends followed by digital
post-processing.
Integrate-and-dump front-ends found in typical A-to-D
converters were analyzed in some detail.
Computed the error probability for the matched ﬁlter
receiver and provided expressions for the error rate of
linear receivers in AWGN channels.



Outline



Digital Modulation

Channel Model

Receiver

MATLAB Simulation



MATLAB Simulation

Objective: Simulate a simple communication system and
estimate bit error rate.
System Characteristics:
BPSK modulation, b ∈ {1, −1} with equal a priori
probabilities,
Raised cosine pulses,
AWGN channel,
oversampled integrate-and-dump receiver front-end,
digital matched ﬁlter.
Measure: Bit-error rate as a function of Es /N0 and
oversampling rate.



System to be Simulated

Sampler,
N (t )
rate fs

bn s (t ) R (t ) R [n] to
× p (t ) × h (t ) + ΠTs (t )
DSP

∑ δ(t − nT ) A

Figure: Baseband Equivalent System to be Simulated.



From Continuous to Discrete Time

The system in the preceding diagram cannot be simulated
immediately.
Main problem: Most of the signals are continuous-time
signals and cannot be represented in MATLAB.
Possible Remedies:
1. Rely on Sampling Theorem and work with sampled
versions of signals.
2. Consider discrete-time equivalent system.
The second alternative is preferred and will be pursued
below.



Towards the Discrete-Time Equivalent System
The shaded portion of the system has a discrete-time input
and a discrete-time output.
Can be considered as a discrete-time system.
Minor problem: input and output operate at different rates.

Sampler,
N (t )
rate fs

× p (t ) × h (t ) + ΠTs (t )
DSP

∑ δ(t − nT ) A



Discrete-Time Equivalent System
The discrete-time equivalent system
is equivalent to the original system, and
contains only discrete-time signals and components.
Input signal is up-sampled by factor fs T to make input and
output rates equal.
Insert fs T − 1 zeros between input samples.

N [n ]

bn R [n ]
× ↑ fs T h [n ] + to DSP

A



Components of Discrete-Time Equivalent System

Question: What is the relationship between the
components of the original and discrete-time equivalent
system?

Sampler,
N (t )
rate fs

× p (t ) × h (t ) + ΠTs (t )
DSP

∑ δ(t − nT ) A



Discrete-time Equivalent Impulse Response
To determine the impulse response h[n] of the
discrete-time equivalent system:
Set noise signal Nt to zero,
set input signal bn to unit impulse signal δ[n],
output signal is impulse response h[n].
Procedure yields:

1 ( n + 1 ) Ts
h [n ] = p (t ) ∗ h(t ) dt
Ts nTs

For high sampling rates (fs T 1), the impulse response is
closely approximated by sampling p (t ) ∗ h(t ):

h[n] ≈ p (t ) ∗ h(t )|(n+ 1 )Ts
2



Discrete-time Equivalent Impulse Response
2

1.5

1

0.5

0
0 0.2 0.4 0.6 0.8 1
Time/T

Figure: Discrete-time Equivalent Impulse Response (fs T = 8)



Discrete-Time Equivalent Noise

To determine the properties of the additive noise N [n] in
the discrete-time equivalent system,
Set input signal to zero,
let continuous-time noise be complex, white, Gaussian with
power spectral density N0 ,
output signal is discrete-time equivalent noise.
Procedure yields: The noise samples N [n]
are independent, complex Gaussian random variables, with
zero mean, and
variance equal to N0 /Ts .



Received Symbol Energy
The last entity we will need from the continuous-time
system is the received energy per symbol Es .
Note that Es is controlled by adjusting the gain A at the
transmitter.
To determine Es ,
Set noise N (t ) to zero,
Transmit a single symbol bn ,
Compute the energy of the received signal R (t ).
Procedure yields:
2
Es = σs · A2 |p (t ) ∗ h(t )|2 dt

2
Here, σs denotes the variance of the source. For BPSK,
2 = 1.
σs
For the system under consideration, Es = A2 T .


Simulating Transmission of Symbols

We are now in position to simulate the transmission of a
sequence of symbols.
The MATLAB functions previously introduced will be used
for that purpose.
We proceed in three steps:
1. Establish parameters describing the system,
By parameterizing the simulation, other scenarios are easily
accommodated.
2. Simulate discrete-time equivalent system,
3. Collect statistics from repeated simulation.



Listing : SimpleSetParameters.m
% This script sets a structure named Parameters to be used by
% the system simulator.

%% Parameters
7 % construct structure of parameters to be passed to system simulator
% communications parameters
Parameters.T = 1/10000; % symbol period
Parameters.fsT = 8; % samples per symbol
Parameters.Es = 1; % normalize received symbol energy to 1
12 Parameters.EsOverN0 = 6; % Signal-to-noise ratio (Es/N0)
Parameters.Alphabet = [1 -1]; % BPSK
Parameters.NSymbols = 1000; % number of Symbols

% discrete-time equivalent impulse response (raised cosine pulse)
17 fsT = Parameters.fsT;
tts = ( (0:fsT-1) + 1/2 )/fsT;
Parameters.hh = sqrt(2/3) * ( 1 - cos(2*pi*tts)*sin(pi/fsT)/(pi/fsT));



Simulating the Discrete-Time Equivalent System

The actual system simulation is carried out in MATLAB
function MCSimple which has the function signature below.
The parameters set in the controlling script are passed as
inputs.
The body of the function simulates the transmission of the
signal and subsequent demodulation.
The number of incorrect decisions is determined and
returned.

function [NumErrors, ResultsStruct] = MCSimple( ParametersStruct )



Simulating the Discrete-Time Equivalent System
The simulation of the discrete-time equivalent system uses
toolbox functions RandomSymbols, LinearModulation, and
addNoise.

A = sqrt(Es/T); % transmitter gain
N0 = Es/EsOverN0; % noise PSD (complex noise)
NoiseVar = N0/T*fsT; % corresponding noise variance N0/Ts
Scale = A*hh*hh’; % gain through signal chain
34
%% simulate discrete-time equivalent system
% transmitter and channel via toolbox functions
Symbols = RandomSymbols( NSymbols, Alphabet, Priors );
Signal = A * LinearModulation( Symbols, hh, fsT );
39 if ( isreal(Signal) )
Signal = complex(Signal);% ensure Signal is complex-valued
end
Received = addNoise( Signal, NoiseVar );



Digital Matched Filter

The vector Received contains the noisy output samples from
the analog front-end.
In a real system, these samples would be processed by
digital hardware to recover the transmitted bits.
Such digital hardware may be an ASIC, FPGA, or DSP chip.
The first function performed there is digital matched
filtering.
This is a discrete-time implementation of the matched filter
discussed before.
The matched filter is the best possible processor for
enhancing the signal-to-noise ratio of the received signal.



Digital Matched Filter

In our simulator, the vector Received is passed through a
discrete-time matched ﬁlter and down-sampled to the
symbol rate.
The impulse response of the matched ﬁlter is the conjugate
complex of the time-reversed, discrete-time channel
response h[n].

R [n ] ˆ
bn
h∗ [−n] ↓ fs T Slicer



MATLAB Code for Digital Matched Filter

The signature line for the MATLAB function implementing
the matched ﬁlter is:
function MFOut = DMF( Received, Pulse, fsT )

The body of the function is a direct implementation of the
structure in the block diagram above.
% convolve received signal with conjugate complex of
% time-reversed pulse (matched filter)
Temp = conv( Received, conj( fliplr(Pulse) ) );
21
% down sample, at the end of each pulse period
MFOut = Temp( length(Pulse) : fsT : end );



DMF Input and Output Signal
DMF Input
400

200

0

−200

−400
0 1 2 3 4 5 6 7 8 9 10
Time (1/T)
DMF Output
1500

1000

500

0

−500

−1000
0 1 2 3 4 5 6 7 8 9 10
Time (1/T)



IQ-Scatter Plot of DMF Input and Output
DMF Input
300

200
Imag. Part

100

0

−100

−200

−800 −600 −400 −200 0 200 400 600 800
Real Part
DMF Output

500
Imag. Part

0

−500

−2000 −1500 −1000 −500 0 500 1000 1500 2000
Real Part



Slicer
The final operation to be performed by the receiver is
deciding which symbol was transmitted.
This function is performed by the slicer.
The operation of the slicer is best understood in terms of
the IQ-scatter plot on the previous slide.
The red circles in the plot indicate the noise-free signal
locations for each of the possibly transmitted signals.
For each output from the matched filter, the slicer
determines the nearest noise-free signal location.
The decision is made in favor of the symbol that
corresponds to the noise-free signal nearest the matched
filter output.
Some adjustments to the above procedure are needed
when symbols are not equally likely.



MATLAB Function SimpleSlicer
The procedure above is implemented in a function with
signature
function [Decisions, MSE] = SimpleSlicer( MFOut, Alphabet, Scale )

%% Loop over symbols to find symbol closest to MF output
for kk = 1:length( Alphabet )
% noise-free signal location
28 NoisefreeSig = Scale*Alphabet(kk);
% Euclidean distance between each observation and constellation po
Dist = abs( MFOut - NoisefreeSig );
% find locations for which distance is smaller than previous best
ChangedDec = ( Dist < MinDist );
33
% store new min distances and update decisions
MinDist( ChangedDec) = Dist( ChangedDec );
Decisions( ChangedDec ) = Alphabet(kk);
end



Entire System

The addition of functions for the digital matched ﬁlter
completes the simulator for the communication system.
The functionality of the simulator is encapsulated in a
function with signature
function [NumErrors, ResultsStruct] = MCSimple( ParametersStruct )

The function simulates the transmission of a sequence of
symbols and determines how many symbol errors occurred.
The operation of the simulator is controlled via the
parameters passed in the input structure.
The body of the function is shown on the next slide; it
consists mainly of calls to functions in our toolbox.



Listing : MCSimple.m
38 Signal = A * LinearModulation( Symbols, hh, fsT );
if ( isreal(Signal) )
Signal = complex(Signal);% ensure Signal is complex-valued
end
Received = addNoise( Signal, NoiseVar );
43
% digital matched filter and slicer
MFOut = DMF( Received, hh, fsT );
Decisions = SimpleSlicer( MFOut(1:NSymbols), Alphabet, Scale );

48 %% Count errors
NumErrors = sum( Decisions ~= Symbols );



Monte Carlo Simulation

The system simulator will be the work horse of the Monte
Carlo simulation.
The objective of the Monte Carlo simulation is to estimate
the symbol error rate our system can achieve.
The idea behind a Monte Carlo simulation is simple:
Simulate the system repeatedly,
for each simulation count the number of transmitted
symbols and symbol errors,
estimate the symbol error rate as the ratio of the total
number of observed errors and the total number of
transmitted bits.



Monte Carlo Simulation

The above suggests a relatively simple structure for a
Monte Carlo simulator.
Inside a programming loop:
perform a system simulation, and
accumulate counts for the quantities of interest
43 while ( ~Done )
NumErrors(kk) = NumErrors(kk) + MCSimple( Parameters );
NumSymbols(kk) = NumSymbols(kk) + Parameters.NSymbols;

% compute Stop condition
48 Done = NumErrors(kk) > MinErrors || NumSymbols(kk) > MaxSy
end



Confidence Intervals

Question: How many times should the loop be executed?
Answer: It depends
on the desired level of accuracy (confidence), and
(most importantly) on the symbol error rate.
Confidence Intervals:
Assume we form an estimate of the symbol error rate Pe as
described above.
ˆ
Then, the true error rate Pe is (hopefully) close to our
estimate.
Put differently, we would like to be reasonably sure that the
ˆ
absolute difference |Pe − Pe | is small.



Confidence Intervals
More specifically, we want a high probability pc (e.g.,
ˆ
pc =95%) that |Pe − Pe | < sc .
The parameter sc is called the confidence interval;
it depends on the confidence level pc , the error probability
Pe , and the number of transmitted symbols N.
It can be shown, that

Pe (1 − Pe )
sc = zc · ,
N
where zc depends on the confidence level pc .
Specifically: Q (zc ) = (1 − pc )/2.
Example: for pc =95%, zc = 1.96.
Question: How is the number of simulations determined
from the above considerations?


Choosing the Number of Simulations
For a Monte Carlo simulation, a stop criterion can be
formulated from
a desired confidence level pc (and, thus, zc )
an acceptable confidence interval sc ,
the error rate Pe .
Solving the equation for the confidence interval for N, we
obtain
N = Pe · (1 − Pe ) · (zc /sc )2 .

A Monte Carlo simulation can be stopped after simulating N
transmissions.
Example: For pc =95%, Pe = 10−3 , and sc = 10−4 , we
find N ≈ 400, 000.



A Better Stop-Criterion
When simulating communications systems, the error rate is
often very small.
Then, it is desirable to specify the conﬁdence interval as a
fraction of the error rate.
The conﬁdence interval has the form sc = αc · Pe (e.g.,
αc = 0.1 for a 10% acceptable estimation error).
Inserting into the expression for N and rearranging terms,
Pe · N = (1 − Pe ) · (zc /αc )2 ≈ (zc /αc )2 .

Recognize that Pe · N is the expected number of errors!
Interpretation: Stop when the number of errors reaches
(zc /αc )2 .
Rule of thumb: Simulate until 400 errors are found
(pc =95%, α =10%).


Listing : MCSimpleDriver.m
9 % comms parameters delegated to script SimpleSetParameters
SimpleSetParameters;

% simulation parameters
EsOverN0dB = 0:0.5:9; % vary SNR between 0 and 9dB
14 MaxSymbols = 1e6; % simulate at most 1000000 symbols

% desired confidence level an size of confidence interval
ConfLevel = 0.95;
ZValue = Qinv( ( 1-ConfLevel )/2 );
19 ConfIntSize = 0.1; % confidence interval size is 10% of estimate
% For the desired accuracy, we need to find this many errors.
MinErrors = ( ZValue/ConfIntSize )^2;

Verbose = true; % control progress output
24
%% simulation loops
% initialize loop variables
NumErrors = zeros( size( EsOverN0dB ) );
NumSymbols = zeros( size( EsOverN0dB ) );



Listing : MCSimpleDriver.m
for kk = 1:length( EsOverN0dB )
32 % set Es/N0 for this iteration
Parameters.EsOverN0 = dB2lin( EsOverN0dB(kk) );
% reset stop condition for inner loop
Done = false;

37 % progress output
if (Verbose)
disp( sprintf( ’Es/N0: %0.3g dB’, EsOverN0dB(kk) ) );
end

42 % inner loop iterates until enough errors have been found
while ( ~Done )
NumErrors(kk) = NumErrors(kk) + MCSimple( Parameters );
NumSymbols(kk) = NumSymbols(kk) + Parameters.NSymbols;

47 % compute Stop condition
Done = NumErrors(kk) > MinErrors || NumSymbols(kk) > MaxSymbol
end



Simulation Results
−1
10

−2
10
Symbol Error Rate

−3
10

−4
10

−5
10
−2 0 2 4 6 8 10
Es/N0 (dB)



Summary

Introduced discrete-time equivalent systems suitable for
simulation in MATLAB.
Relationship between original, continuous-time system and
discrete-time equivalent was established.
Digital post-processing: digital matched ﬁlter and slicer.
Monte Carlo simulation of a simple communication system
was performed.
Close attention was paid to the accuracy of simulation
results via conﬁdence levels and intervals.
Derived simple rule of thumb for stop-criterion.



Where we are ...
Laid out a structure for describing and analyzing
communication systems in general and wireless systems
in particular.
Saw a lot of MATLAB examples for modeling diverse
aspects of such systems.
Conducted a simulation to estimate the error rate of a
communication system and compared to theoretical
results.
To do: consider selected aspects of wireless
communication systems in more detail, including:
modulation and bandwidth,
wireless channels,
advanced techniques for wireless communications.


Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation

Part III

Digital Modulation and Spectrum



Digital Modulation and Spectrum

Digital modulation formats and their spectra.
Linear, narrow-band modulation (including B/QPSK, PSK,
QAM, and variants OQPSK, π/4 DQPSK)
Non-linear modulation (including CPM, CPFSK, MSK,
GMSK)
Wide-band modulation (CDMA and OFDM)
The use of pulse-shaping to control the spectrum of
modulated signals.
Spectrum estimation of digitally modulated signals in
MATLAB.



Outline








Learning Objectives

Understand choices and trade-offs for digital modulation.
Linear modulation: principles and parameters.
Non-linear modulation: beneﬁts and construction.
The importance of constant-envelope characteristics.
Wide-band modulation: DS/SS and OFDM.
Visualization of digitally modulated signals.
Spectra of digitally modulated signals.
Closed-form expressions for the spectrum of linearly
modulated signals.
Numerical estimation of the spectrum of digitally modulated
signals.



Outline








Linear Modulation
We have already introduced Linear Modulation as the
digital equivalent of amplitude modulation.
Recall that a linearly modulated signal may be written as
N −1
s (t ) = ∑ bn · p (t − nT )
n =0

where,
bn denotes the n-th information symbol, and
p (t ) denotes a pulse of ﬁnite duration.
T is the duration of a symbol.
We will work with baseband equivalent signals throughout.
Symbols bn will generally be complex valued.



Linear Modulation
Objective: Investigate the impact of
the parameters
alphabet from which symbols bn
are chosen,
pulse shape p (t ), bn s (t )
symbol period T × p (t )

on signals in the time and frequency
domain.
Note: We are interested in the ∑ δ(t − nT )
properties of the analog signals
produced by the transmitter.
Signals will be signiﬁcantly
oversampled to approximate
signals closely.



Signal Constellations

The inﬂuence of the alphabet
from which symbols bn are 1000

chosen is well captured by the
500
signal constellation.
The signal constellation is

Imag. Part
0

simply a plot indicating the
location of all possible −500

symbols in the complex plane.
−1000
The signal constellation is the
−1500 −1000 −500 0 500 1000 1500
noise-free output of the Real Part

(digital) matched ﬁlter.



Characteristics of Signals Constellations
Three key characteristics of a signal constellation:
1. Number of symbols M in constellation determines
number of bits transmitted per symbol;

Nb = log2 (M ).

2. Average symbol energy is computed as
M
1
Es =
M ∑ |bk |2 · A2 |p (t )|2 dt;
k =1

we will assume that |p (t )|2 dt = 1.
3. Shortest distance dmin between points in constellation
has major impact on probability of symbol error.
Often expressed in terms of Es .



Example: BPSK
A

BPSK: Binary Phase
Shift Keying
Nb = 1 bit per
symbol,
Imaginary

0 Es = A2 , √
dmin = 2A = 2 Es .
Recall that symbol error
rate is √
Pe = Q (√ 2Es /N0 ) =
−A
Q (dmin / 2N0 ).
−A 0 A
Real



Example: QPSK
A

QPSK: Quaternary
Phase Shift Keying
Imaginary

0
Nb = 2 bits per
symbol,
Es = A2 ,
√ √
dmin = 2A = 2Es .

−A
−A 0 A
Real



Example: 8-PSK
A

8-PSK: Eight Phase
Shift Keying
Nb = 3 bit per
Imaginary

0 symbol,
Es = A2 , √
dmin = (2 − 2)A =
√ √
(2 − 2) Es .
√
2− 2 ≈ 0.6
−A
−A 0 A
Real



Example: 16-QAM

3A
16-QAM: 16-Quadrature
2A Amplitude Modulation
Nb = 4 bit per symbol,
A
Es = 10TA2 ,
√
dmin = 2A = 2 Es .
Imaginary

0
5

−A
Note that symbols don’t all
have the same energy; this is
−2A
potentially problematic.
−3A
−3A −2A −A 0 A 2A 3A
16-QAM is not commonly used
Real
for wireless communications.



Pulse Shaping

The discrete symbols that constitute
the constellation are converted to
analog signals s (t ) with the help of
pulses. bn s (t )
× p (t )
The symbols bn determine the
“instantaneous amplitude” of the;
while the pulses determine the
∑ δ(t − nT )
shape of the signals.
We will see that the pulse shape has
a major impact on the spectrum of
the signal.



Full-Response Pulses
Full-response pulses span exactly one symbol period.

2

1.5

1

0.5

0
0 0.2 0.4 0.6 0.8 1
Time/T



Partial-Response Pulses

Partial-response pulses are of duration longer than one
symbol period.
The main benefit that partial-response pulse can provide
are better spectral properties.
The extreme case, is an infinitely long sinc-pulse which
produces a strictly band-limited spectrum.
On the negative side, partial-response pulses (can)
introduce intersymbol-interference that affects negatively
the demodulation of received signals.
The special class of Nyquist pulses avoids, in principle,
intersymbol interference.
In practice, multi-path propagation foils the benefits of
Nyquist pulses.



Partial-Response Pulses
A (truncated) sinc-pulse is a partial-response pulse.

1.5

1

0.5

0

−0.5
0 2 4 6 8 10
Time/T



Raised Cosine Nyquist Pulses

The sinc-pulse is a special case in a class of important
pulse shapes.
We will see that raised cosine Nyquist pulses have very
good spectral properties and meet the Nyquist condition
for avoiding intersymbol interference.
Raised cosine Nyquist pulses are given by

sin(πt /T ) cos( βπt /T )
p (t ) = ·
πt /T 1 − (2βt /T )2

The parameter β is called the roll-off factor and determines
how quickly the pulses dampens out.



Raised Cosine Nyquist Pulses
1.5
β=0
β=0.3
β=0.5
β=1

1

0.5

0

−0.5
0 1 2 3 4 5 6 7 8 9 10
Time/T



Visualizing Linearly Modulated Signals
To understand the time-domain properties of a linearly
modulated signal one would like to plot the signal.
However, since baseband-equivalent signals are generally
complex valued, this is not straightforward.
Plotting the real and imaginary parts of the signal
separately does not provide much insight.
Useful alternatives for visualizing the modulated signal are
Plot the magnitude and phase of the signal; useful because
information is generally encoded in magnitude and phase.
Plot signal trajectory in the complex plane, i.e., plot real
versus imaginary part.
Shows how modulated signal moves between constellation
points.
Plot the up-converted signal (at a low IF frequency).



MATLAB function LinearModulation

All examples, rely on the toolbox function
LinearModulation to generate the
baseband-equivalent transmitted signal.
19 % initialize storage for Signal
LenSignal = length(Symbols)*fsT + (length(Pulse))-fsT;
Signal = zeros( 1, LenSignal );

% loop over symbols and insert corresponding segment into Signal
24 for kk = 1:length(Symbols)
ind_start = (kk-1)*fsT + 1;
ind_end = (kk-1)*fsT + length(Pulse);

Signal(ind_start:ind_end) = Signal(ind_start:ind_end) + ...
29 Symbols(kk) * Pulse;
end



Example: QPSK with Raised Cosine Nyquist Pulses
1.5

1

Magnitude
0.5

0
5 6 7 8 9 10 11 12 13 14 15
Time/T

1

0.5
Phase/π

0

−0.5

−1
5 6 7 8 9 10 11 12 13 14 15
Time/T

Figure: Magnitude and Phase; QPSK, Raised Cosine Nyquist Pulse
(β = 0.5)



1.5

1

0.5
Amplitude

0

−0.5

−1

−1.5
5 6 7 8 9 10 11 12 13 14 15
Time/T

Figure: Up-converted signal, fc = 2/T ; QPSK, Raised Cosine
Nyquist Pulse (β = 0.5)




1

0.5
Imaginary

0

−0.5

−1

−1.5 −1 −0.5 0 0.5 1 1.5
Real

Figure: Complex Plane Signal Trajectory; QPSK, Raised Cosine



Constant Envelope Signals

It is desirable for digitally modulated signals to have
constant magnitude.
This permits the use of non-linear power amplifiers:
Non-linear power amplifiers are more energy efficient;
more of the energy they consume is used to amplify the
signal.
Non-linear: the gain of the amplifier varies with the
magnitude of the input signal.
This leads to non-linear distortions of the signal — the pulse
shape is altered.
Constant envelope signals do not experience distortion
from non-linear power amplifiers.



Towards Constant Envelope Signals

The preceding examples show that it is not sufﬁcient for
the symbols to have constant magnitude to create constant
envelope signals.
In particular, abrupt phase changes of 180o lead to signal
trajectories through the origin of the complex plane.
To reduce the variation of the signal magnitude, one can
encode symbols such that 180o phase changes are
eliminated.
Generally, it is necessary to encode multiple symbols at the
same time.
We refer to such encoding strategies as linear modulation
with memory.



Example: OQPSK
OQPSK stands for Offset QPSK.
One can think of OQPSK as a modulation format that
alternates between using two constellations:
each of the two alphabets contains two symbols — one bit
is transmitted per symbol period T ,
in odd-numbered symbol periods, the alphabet
A = {1, −1} is used, and
in even-numbered symbol periods, the alphabet
A = {j, −j } is used.
Note that only phase changes of ±90o are possible and,
thus, transitions through the origin are avoided.
Despite its name, OQPSK has largely the same √
characteristics as BPSK. (Nb = 1, Es = A2 , dmin = 2 Es )



Toolbox function OQPSK

The toolbox function OQPSK accepts a vector of BPSK
symbols and converts them to OQPSK symbols:
function OQPSKSymbols = OQPSK( BPSKSymbols )

%% BPSK -> OQPSK
% keep odd-numbered samples, phase-shift even numbered samples
OQPSKSymbols = BPSKSymbols;
21 % phase shift even samples
OQPSKSymbols(2:2:end) = j*BPSKSymbols(2:2:end);



Using the Toolbox Function OQPSK

Calls to the toolbox function OQPSK are inserted between
the function RandomSymbols for generating BPSK
symbols, and
the function LinearModulation for creating
baseband-equivalent modulated signals.
Symbols = RandomSymbols(Ns, Alphabet, Priors);
13 Symbols = OQPSK(Symbols);



Example: OQPSK with Raised Cosine Nyquist Pulses
1.5

1

Magnitude
0.5

0
5 6 7 8 9 10 11 12 13 14 15
Time/T

1

0.5
Phase/π

0

−0.5

−1
5 6 7 8 9 10 11 12 13 14 15
Time/T

Figure: Magnitude and Phase; OQPSK, Raised Cosine Nyquist Pulse
(β = 0.5)



1.5

1

0.5
Amplitude

0

−0.5

−1

−1.5
5 6 7 8 9 10 11 12 13 14 15
Time/T

Figure: Up-converted signal, fc = 2/T ; OQPSK, Raised Cosine



1

0.8

0.6

0.4

0.2
Imaginary

0

−0.2

−0.4

−0.6

−0.8

−1
−1 −0.5 0 0.5 1
Real

Figure: Complex Plane Signal Trajectory; OQPSK, Raised Cosine



QPSK vs OQPSK

1

0.8
1
0.6

0.4
0.5

0.2
Imaginary

Imaginary
0 0

−0.2

−0.5
−0.4

−0.6
−1
−0.8

−1
−1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.5 0 0.5 1
Real Real



OQPSK with Half-Sine Pulses

An important special case 1

arises when OQPSK is used 0.8
in conjunction with pulses of
the form 0.6

πt
p (t ) = sin( ) for 0 ≤ t ≤ 2T . 0.4

2T
0.2

Note, that these pulse span
0
two symbol periods. 0 0.2 0.4 0.6 0.8 1
Time/T
1.2 1.4 1.6 1.8 2



1

0.8

0.6

0.4

0.2
Imaginary

0

−0.2

−0.4

−0.6

−0.8

−1
−1 −0.5 0 0.5 1
Real

Figure: Complex Plane Signal Trajectory; OQPSK with Half-Sine
Pulses




The resulting modulated signal is a perfectly constant
envelope signal.
Very well suited for energy efﬁcient, non-linear ampliﬁers.
It can be shown that the resulting signal is equivalent to
Minimum Shift Keying (MSK).
MSK will be considered in detail later.
Relationship is important in practice to generate and
demodulate MSK signals.



Differential Phase Encoding

So far, we have considered only modulation formats where
information symbols are mapped directly to a set of
phases.
Alternatively, it is possible to encode information in the
phase difference between consecutive symbols.
Example: In Differential-BPSK (D-BPSK), the phase
difference between the n-th and (n − 1)-th symbol period is
either 0 or π.
Thus, one bit of information can be conveyed.



Differential Phase Encoding
Formally, one can can represent differential encoding with
reference to the phase difference ∆θn = θn − θn−1 :

∆θn = φ0 · xn .

where
φ0 indicates the phase increment (e.g., φ0 = π for D-BPSK)
xn is drawn from an alphabet of real-valued integers (e.g.,
xn ∈ {0, 1} for D-BPSK)
The symbols generated by differential phase encoders are
of the form bn = exp(jθn ), where

θn = θn−1 + ∆θn = θ0 + ∑n =1 ∆θk
k
= θ0 + φ0 ∑n =1 xn
k



Toolbox function DPSK

The toolbox function DPSK with the following function
signature provides generic differential phase encoding:
1 function DPSKSymbols = DPSK( PAMSymbols, phi0, theta0 )

The body of DPSK computes differentially encoded
symbols as follows:
%% accumulate phase differences, then convert to complex symbols
Phases = cumsum( [theta0 phi0*PAMSymbols] );
DPSKSymbols = exp( j*Phases );



Example: π/4-DQPSK

An important modulation format, referred to as
π/4-DQPSK results when we choose:
φ0 = π/4, and
xn ∈ {±1, ±3}.
Note that with this choice, the phase difference between
consecutive symbols is ±π/4 or ±3π/4.
Phase differences of π (180o do not occur — no transitions
through origin.
The resulting signal has many of the same characteristics
as QPSK but has less magnitude variation.
Signal is also easier to synchronize, since phase transitions
occur in every symbol period.



Generating π/4-DQPSK Signals

Using our toolbox function, π/4-DQPSK are easily
generated.
To begin, we deﬁne the appropriate alphabet of symbols for
differential encoding:
Alphabet = -3:2:3; % 4-PAM

Then, the baseband-equivalent signal is generated via calls
to the appropriate toolbox functions.
Symbols = DPSK(Symbols, pi/4);
14 Signal = LinearModulation(Symbols,Pulse,fsT);



π/4-DQPSK with Raised Cosine Nyquist Pulses.
1

0.5
Imaginary

0

−0.5

−1

−1.5 −1 −0.5 0 0.5 1 1.5
Real

Figure: Complex Plane Signal Trajectory; π/4-DQPSK with Raised
Cosine Nyquist Pulses (β = 0.5)



Spectrum of Linearly Modulated Signals

Having investigated the time-domain properties of linearly
modulated signals, we turn now to their spectral properties.
Technically, the modulated signals are random processes
and the appropriate spectral measure is the power spectral
density.
Modulated signals are random because the information
symbols are random.
Speciﬁcally, we compute the power spectral density of the
baseband-equivalent signals.
The computation of the power spectral density for a
modulated signal is lengthy and quite involved.
Focus on highlights of results.



Spectrum of Linearly Modulated Signals
The power spectral density is greatly simpliﬁed with the
following (reasonable) assumptions:
Symbols have zero mean, and
are uncorrelated.
Full-response pulse shaping.
Then, the power spectral density depends only on the
Fourier transform of the pulse-shape and is given by

Ps (f ) = Es · |H (f )|2 .

Note, that the power spectral density does not depend on
the modulation format!
We will rely on numerical techniques to ﬁnd the spectrum
of signals for which this expression does not apply.



Example: Rectangular Pulse

Assume, a rectangular pulse is used for pulse shaping:

1 for 0 ≤ t < T
p (t ) =
0 else.

Then, with the assumptions above, the power spectral
density of the transmitted signals equals

sin(πfT ) 2
Ps (f ) = Es · ( ) .
πfT



Example: Rectangular Pulse

0

−10

Spectral characteristics:
−20

Normalization: Es = 1
−30
Zero-to-zero Bandwidth:
PSD (dB)

−40
2/T
Side-lobe decay ∼ 1/f 2
−50

Smoother pulses provide
−60
much better spectrum.
−70
−10 −8 −6 −4 −2 0 2 4 6 8 10
Normalized Frequency (fT)



Summary

Detailed discussion of linearly modulated signals and their
simulation in MATLAB.
Pulse-shaping: full and partial response,
Modulation formats with and without memory.
Constant envelope characteristics
Rationale for constant envelope signals (non-linear power
ampliﬁers)
Improving the envelope characteristics through offset
constellations.
Power Spectral Density of Modulated Signals
Closed form expressions for simple cases.
Next: numerical estimation of power spectral density.



Outline









Closed form expressions for the power spectral density are
not always easy to obtain, particularly for
Partial-response pulse-shaping,
non-linear modulation formats.
Objective: Develop a simple procedure for estimating the
power spectral density of a digitally modulated signal.
Start with samples of a digitally modulated waveform,
Estimate the PSD from these samples.
With a little care, the above objective is easily achieved.



First Attempt

One may be tempted to estimate the spectrum as follows:
Generate a vector of samples of the signal of interest:
N random symbols,
fs T samples per symbol period,
for the modulation format and pulse shape of interest.
Compute the Discrete Fourier Transform (DFT) of the
samples:
This is easily done using the MATLAB function fft.
A smoothing window improves the estimate.
Estimate the PSD as the squared magnitude of the DFT.
This estimate is referred to as the periodogram.



Periodogram
Question: How should the number of symbols N and the
normalized sampling rate fs T be chosen?
Normalized sampling rate fs T determines the observable
frequency range:
Observable frequencies range from −fs/2 to fs /2.
In terms of normalized frequencies fT , highest observable
frequency is fs T /2 · 1/T .
Chose fs T large enough to cover frequencies of interest.
Typical: fs T = 20
Number of Symbols N determines the frequency
resolution.
The PSD will be sampled N times for each frequency
interval of length 1/T .
Frequency sampling period: 1/(Nfs T ).
Typical: N = 100


Toolbox function Periodogram
function Ps = Periodogram( Signal )
% Periodogram - Periodogram estimate for PSD of the input signal.
%
% Input:
5 % Signal - samples of signal of interest
%
% Output:
% Ps - estimated PSD
%
10 % Example:
% Ps = Periodogram( Signal )

%% compute periodogram
% window eliminates effects due to abrupt onset and end of signal
15 Window = blackman( length(Signal) )’;
Ps = fft( Signal.*Window, length(Signal) );
% swap left and right half, to get freqs from -fs/2 to fs/2
Ps = fftshift( Ps );
% return squared magnitude, normalized to account for window
20 Ps = abs(Ps).^2 / norm(Window)^2;



Example: Rectangular Pulses
10

0

−10

−20
PSD (dB)

−30

−40

−50

−60

−70
−10 −8 −6 −4 −2 0 2 4 6 8 10

Figure: Periodogram estimate of PSD for rectangular pulse-shaping



Improving the Periodogram Estimate

The periodogram estimate is extremely noisy.
ˆ
It is known that the periodogram Ps (f ) has a Gaussian
distribution and
ˆ
is an unbiased estimate E[Ps (f )] = Ps (f ),
ˆ
has variance Var[Ps (f )] = Ps (f ).
The variance of the periodogram is too high for the estimate
to be useful.
Fortunately, the variance is easily reduced through
averaging:
Generate M realizations of the modulated signal, and
average the periodograms for the signals.
Reduces variance by factor M.



Variance Reduction through Averaging

The MATLAB fragment below illustrates how reliable
estimates of the power spectral density are formed.
for kk=1:M
% generate signal
20
% accumulate periodogram
Ps = Ps + Periodogram( Signal );

end
25
%average
Ps = Ps/M;



Averaged Periodogram
5

0

−5

−10

−15
PSD (dB)

−20

−25

−30

−35

−40

−45

−50
−10 −8 −6 −4 −2 0 2 4 6 8 10

Figure: Averaged Periodogram for a QPSK signal with Rectangular
Pulses; M = 500.



Remaining Issues

The resulting estimate is not perfect.
Near the band-edges, the power spectral density is
consistently over-estimated.
This is due to aliasing.
To improve the estimate, one can
increase the normalized sampling rate fs T .
For pulses with better spectral properties, aliasing is
signiﬁcantly reduced.
Additionally,
center of band is of most interest — to assess bandwidth,
simplicity of estimator is very attractive.



Spectrum Estimation for Linearly Modulated Signals

We apply our estimation technique to illustrate the spectral
characteristics of a few of the modulation formats
considered earlier.
Comparison of BPSK and π/4-DQPSK with rectangular
pulses,
Inﬂuence of the roll-off factor β of raised cosine pulses,
OQPSK with rectangular pulses and half-sine pulses
(MSK).



BPSK and π/4-DQPSK Spectrum
5
BPSK
0 π/4−DQPSK

−5

−10

−15
PSD (dB)

−20

−25

−30

−35

−40

−45

−50
−10 −8 −6 −4 −2 0 2 4 6 8 10

Figure: Spectrum of BPSK and π/4-DQPSK modulated signals with
rectangular pulses: Spectrum depends only on pulse shape.



Raised Cosine Pulses - Inﬂuence of β
0 β=0
β=0.3
−10 β=0.5
β=1
−20

−30

−40
PSD (dB)

−50

−60

−70

−80

−90

−100
−4 −3 −2 −1 0 1 2 3 4

Figure: QPSK modulated signals with Raised Cosine Pulse Shaping.
Width of main-lobe increases with roll-off factor β. Side-lobes are due
to truncation of pulses.



OQPSK with Rectangular and Half-Sine Pulses
Rectangular
0 Half−sine

−10

−20
PSD (dB)

−30

−40

−50

−60

−70
−10 −8 −6 −4 −2 0 2 4 6 8 10

Figure: OQPSK modulated signals with rectangular and half-sine
pulses (MSK). Pulse shape affects the spectrum dramatically.



Summary

A numerical method for estimating the spectrum of digitally
modulated signals was presented.
Relies on the periodogram estimate,
variance reduction through averaging.
Simple and widely applicable.
Applied method to several linear modulation formats.
Conﬁrmed that spectrum of linearly modulated signals
depends mainly on pulse shape; constellation does not
affect spectrum.
Signiﬁcant improvements of spectrum possible with
pulse-shaping.
Partial-response pulses (in particular, raised cosine Nyquist
pulses) have excellent spectral properties.



Outline









Linear modulation formats were investigated and two
characteristics were emphasized:
1. spectral properties - width of main-lobe and decay of
side-lobes;
2. magnitude variations - constant envelope characteristic is
desired but difﬁcult to achieve.
A broad class of (non-linear) modulation formats will be
introduced with
1. excellent spectral characteristics - achieved by eliminating
abrupt phase and amplitude changes;
2. constant envelope characteristic.
More difﬁcult to demodulate in general.



Reminder: Analog Frequency Modulation
We will present a broad class of non-linear modulation
formats with desirable properties.
These formats are best understood by establishing an
analogy to analog frequency modulation (FM).
Recall: a message signal m (t ) is frequency modulated by
constructing the baseband-equivalent signal
t
s (t ) = A · exp(j2πfd m (τ ) dτ ).
−∞

Signal is constant envelope,
signal is a non-linear function of message m (t ),
instantaneous frequency: fd · m (t ), where fd is called the
frequency deviation constant.



Continuous Phase Modulation
The class of modulation formats described here is referred
to as continuous-phase modulation (CPM).
CPM signals are constructed by frequency modulating a
pulse-amplitude modulated (PAM) signal:
PAM signal: is a linearly modulated signal of the
information symbols bn ∈ {±1, . . . , ±(M − 1)}:

N
m (t ) = ∑ hn · bn · pf (t − nT ).
n =0

CPM signal: FM of m (t )
t
s (t ) = A · exp(j2π m (τ ) dτ ).
0



Parameters of CPM Signals
The following parameters of CPM signals can be used to
create a broad class of modulation formats.
Alphabet size M: PAM symbols are drawn form the
alphabet A = {±1, . . . , ±(M − 1)}; generally M = 2K .
Modulation indices hn : play the role of frequency
deviation constant fd ;
Often hn = h, constant modulation index,
periodic hn possible, multi-h CPM.
Frequency shaping function pf (t ): pulse shape of PAM
signal.
Pulse length L symbol periods.
L = 1: full-response CPM,
L > 1: partial response CPM.
Normalization: pf (t ) dt = 1/2.



Generating CPM Signals in MATLAB
We have enough information to write a toolbox function to
generate CPM signal; it has the signature
function [CPMSignal, PAMSignal] = CPM(Symbols, A, h, Pulse, fsT)

The body of the function
calls LinearModulation to generate the PAM signal,
performs numerical integration through a suitable IIR ﬁlter
%% Generate PAM signal, using function LinearModulation
PAMSignal = LinearModulation( Symbols, Pulse, fsT );

%% Integrate PAM signal using a filter with difference equation
30 % y(n) = y(n-1) + x(n)/fsT and multiply with 2*pi*h
a = [1 -1];
b = 1/fsT;
Phase = 2*pi*h*filter(b, a, PAMSignal);

35 %% Baseband equivalent signal
CPMSignal = A*exp(j*Phase);



Using Toolbox Function CPM

A typical invocation of the function CPM is as follows:
%% Parameters:
fsT = 20;
Alphabet = [1,-1]; % 2-PAM
6 Priors = ones( size( Alphabet) ) / length(Alphabet);
Ns = 20; % number of symbols
Pulse = RectPulse( fsT); % Rectangular pulse
A = 1; % amplitude
h = 0.75; % modulation index
11
Signal = CPM(Symbols, A, h, Pulse,fsT);



Example: CPM with Rectangular Pulses and h = 3/4
1.5

1

Magnitude
0.5

0
5 6 7 8 9 10 11 12 13 14 15
Time/T

1

0
Phase/π

−1

−2

−3

−4
5 6 7 8 9 10 11 12 13 14 15
Time/T

Figure: Magnitude and Phase; CPM, M = 2, h = 3/4, full-response
rectangular pulses.



1.5

1

0.5
Amplitude

0

−0.5

−1

−1.5
5 6 7 8 9 10 11 12 13 14 15
Time/T

Figure: Up-converted signal, fc = 2/T ; M = 2, h = 3/4,
full-response rectangular pulses.



1

0.8

0.6

0.4

0.2
Imaginary

0

−0.2

−0.4

−0.6

−0.8

−1
−1 −0.5 0 0.5 1
Real

Figure: Complex Plane Signal Trajectory; M = 2, h = 3/4,
full-response rectangular pulses.



Excess Phase
The excess phase or instantaneous phase of a CPM signal
is given by
t
Φ(t, b ) = 2π 0 m (τ ) dτ
t
= 2π 0 ∑N=0 hn · bn · pf (τ − nT ) dτ
n
N t
= 2π ∑n=0 hn · bn · 0 pf (τ − nT ) dτ
= 2π ∑N=0 hn · bn · β(t − nT ),
n
t
where β(t ) = 0
pf (τ ) dτ.

pf (t ) β (t )

1/2T 1/2
t t
T 2T T 2T



Excess Phase
The excess phase
N
Φ(t, b ) = 2π ∑ hn · bn · β(t − nT )
n =0

explains some of the features of CPM.
The excess phase Φ(t, b ) is a continuous function of time:
due to integration,
Consequence: no abrupt phase changes,
expect good spectral properties.
Excess phase Φ(t, b ) has memory:
phase depends on all preceding symbols,
similar to differential encoding but with continuous phase
changes.



Excess Phase for Full-Response CPM
For full-response CPM signals, the excess phase for the
n-th symbol period, nT ≤ t < (n + 1)T , can be expressed
as
n −1
Φ(t, b ) = πh ∑ bk + 2πhbn β(t − nT ).
k =0

The two terms in the sums are easily interpreted:
1. The term θn−1 = πh ∑n−1 bk accounts for the accumulated
k =0
phase from preceding symbols.
We saw a similar term in connection with differential phase
encoding.
2. The second term, 2πhbn β(t − nT ), describes the
(continuous) phase change due to the current symbol bn .
This term goes from 0 to πh over the current symbol period
Similar analysis is possible for partial response CPM.


CPFSK

Continuous-Phase Frequency Shift Keying (CPFSK) is a
special case of CPM.
CPFSK is CPM with full-response rectangular pulses.
The rectangular pulses cause the excess phase to change
linearly in each symbol period.
A linearly changing phase is equivalent to a frequency shift
relative to the carrier frequency).
Speciﬁcally, the instantaneous frequency in the n-th symbol
period equals fc + bn · (πh)/(2T ).
Consequently, information is encoded in the frequency of
the signal (FSK).
Additionally, the phase is guaranteed to be continuous.



Phase Tree for Binary CPFSK
4h

3h

2h

h
Excess Phase/π

0

−h

−2h

−3h

−4h
0 0.5 1 1.5 2 2.5 3 3.5 4
Time/T

Figure: Phase Tree for a binary CPFSK signal; CPM with
bn ∈ {1, −1}, full-response rectangular pulses. Slope of phase
trajectories equals frequency offset.



Example: MSK

Minimum Shift Keying (MSK) is CPFSK with modulation
index h = 1/2.
Instantaneous frequency: fc ± 1/4T .
Transmitted signals in each symbol period are of the form

s (t ) = A cos(2π (fc ± 1/4T )t + θn ).

The two signals comprising the signal set are orthogonal.
Enables simple non-coherent reception.
Orthogonality is not possible for smaller frequency shifts
(⇒ minimum shift keying).



MSK: Magnitude and Phase
1.5

Magnitude 1

0.5

0
5 6 7 8 9 10 11 12 13 14 15
Time/T

1

0
Phase/π

−1

−2

−3
5 6 7 8 9 10 11 12 13 14 15
Time/T

Figure: Magnitude and Phase; MSK.



MSK: Up-converted Signal
1.5

1

0.5
Amplitude

0

−0.5

−1

−1.5
5 6 7 8 9 10 11 12 13 14 15
Time/T

Figure: Up-converted signal, fc = 2/T ; MSK.



MSK: Complex Plane Signal Trajectory
1

0.8

0.6

0.4

0.2
Imaginary

0

−0.2

−0.4

−0.6

−0.8

−1
−1 −0.5 0 0.5 1
Real

Figure: Complex Plane Signal Trajectory; MSK.



MSK

MSK is in many respects a nearly ideal waveform for
wireless communications:
constant envelope,
easily generated,
easy to demodulate,
either non-coherently, or
coherently via interpretation as OQPSK with half-sine
pulses.
Except: spectrum could be a little better.
Remedy: use smoother pulses
Gaussian MSK (GMSK).



GMSK
To improve the spectral properties, while maintaining the
other benefits of MSK, one can lowpass-filter the PAM
signal before FM.
A filter with impulse response equal to a Gaussian pdf is
used to produce a Gaussian MSK (GMSK) signal.
Equivalently, frequency shaping with the following pulse:
1 t /T + 1/2 t /T − 1/2
pf (t ) = (Q ( ) − Q( )),
2T σ σ
where,
ln(2)
σ=
2πBT
and
1 ∞
Q (x ) = √ exp(−z 2 /2) dz.
2π x



Frequency-Shaping Pulse for GMSK
The parameter BT is called the time-bandwidth product of
the pulse.
It controls the shape of the pulse (Typical value: BT ≈ 0.3).
The toolbox function GaussPulse generates this pulse.

BT=0.3
BT=0.5
BT=1
0.5

0.4

0.3

0.2

0.1

0
0 1 2 3 4 5 6 7 8 9 10
Time/T



GMSK: Magnitude and Phase
1.5

Magnitude 1

0.5

0
5 6 7 8 9 10 11 12 13 14 15
Time/T

1

0
Phase/π

−1

−2

−3
5 6 7 8 9 10 11 12 13 14 15
Time/T

Figure: Magnitude and Phase; GMSK (BT = 0.3).



GMSK: Up-converted Signal
1.5

1

0.5
Amplitude

0

−0.5

−1

−1.5
5 6 7 8 9 10 11 12 13 14 15
Time/T

Figure: Up-converted signal, fc = 2/T ; GMSK (BT = 0.3).



GMSK: Complex Plane Signal Trajectory

0.8

0.6

0.4

0.2
Imaginary

0

−0.2

−0.4

−0.6

−0.8

−1 −0.5 0 0.5 1
Real

Figure: Complex Plane Signal Trajectory; GMSK (BT = 0.3).



GMSK compared to MSK
Both MSK and GMSK are constant envelope signals.
GMSK has smoother phase than MSK.
In particular, direction changes are smooth rather than
abrupt.
Due to use of smooth pulses.
Expect better spectral properties for GMSK.
For GMSK, phase does not change by exactly ±π/2 in
each symbol period.
GMSK is partial response CPM; effect is akin to ISI.
Complicates receiver design; sequence estimation.
For BT 0.25, effect is moderate and can be safely
ignored; use linear receiver.
Next, we will compare the spectra of MSK and GMSK.



Spectrum of CPM Signals

The periodogram-based tools for computing the power
spectral density of digitally modulated signals can be
applied to CPM signals.
We will use these tools to
compare the spectrum of MSK and GMSK,
assess the inﬂuence of BT on the spectrum of a GMSK
signal,
compare the spectrum of full-response and partial
response CPM signals.



Spectrum of MSK and GMSK
0 MSK
GMSK (BT=0.3)

−20

−40
PSD (dB)

−60

−80

−100

−120
−10 −8 −6 −4 −2 0 2 4 6 8 10

Figure: Spectrum of MSK and GMSK modulated signals.



Inﬂuence of BT on GMSK Spectrum
20
BT=0.2
BT=0.3
BT=0.5
0
BT=1

−20

−40
PSD (dB)

−60

−80

−100

−120
−5 −4 −3 −2 −1 0 1 2 3 4 5

Figure: Spectrum of GMSK modulated signals as a function of
time-bandwidth product BT .



Spectrum of Partial Response CPM Signals
20
L=1
L=2
10 L=3
L=4

0

−10
PSD (dB)

−20

−30

−40

−50

−60
−5 −4 −3 −2 −1 0 1 2 3 4 5

Figure: Spectrum of CPM signals as a function of pulse-width;
rectangular pulses spanning L symbol periods, h = 1/2.



Summary
CPM: a broad class of digitally modulated signals.
Obtained by frequency modulating a PAM signal.
CPM signals are constant envelope signals.
Investigated properties of excess phase.
MSK and GMSK:
MSK is binary CPM with full-response rectangular pulses
and modulation index h = 1/2.
Spectral properties of MSK can be improved by using
smooth, partial response pulses: GMSK.
Experimented with the spectrum of CPM signals:
GMSK with small BT has very good spectral properties.
Smooth pulses improve spectrum.
Partial-response pulses lead to small improvements of
spectrum.
Demodulating CPM signals can be difﬁcult.


Outline








Wide-band Signals
Up to this point, we considered modulation methods
generally referred to as narrow-band modulation.
The bandwith of the modulated signals is approximately to
the symbol rate.
In contrast, spread-spectrum modulation produces signals
with bandwidth much larger than the symbol rate.
Power spectral density is decreased proportionally.
Useful for co-existence scenarios or for low probability of
detection.
OFDM achieves simultaneously high data rate and long
symbol periods.
Long symbol periods are beneﬁcial in ISI channels.
Achieved by clever multiplexing of many narrow-band
signals.



Direct-Sequence Spread Spectrum
Conceptually, direct-sequence spread spectrum (DS/SS)
modulation is simply linear modulation using wide-band
pulses.

1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8
DS/SS Pulse
−1 Narrowband Pulse
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time/T



Direct-Sequence Spread Spectrum
The effect of employing a wide-band pulse is that the
spectrum of the transmitted signal is much wider than the
symbol rate.
Signal bandwidth is determined by bandwidth of pulse.
The purpose of spreading is to distribute signal power over
a larger bandwidth, to achieve:
channel sharing with other narrow-band and wide-band
users,
robustness to jamming or interference,
low probability of detection.
Wide-band-pulse is generally generated from a
pseudo-random sequence.
Spreading sequence has M chips per symbol.
Spreading sequence may be periodic or not.
Then, bandwidth increase is M-fold.


Spectrum of DS/SS Signals
5
BPSK
0 DS/SS

−5

−10

−15
PSD (dB)

−20

−25

−30

−35

−40

−45

−50
−20 −15 −10 −5 0 5 10 15 20

Figure: Spectrum of a DS/SS signal; pseudo-random spreading
sequence of length M = 15. Same data rate narrow-band waveform
(BPSK with rectangular pulses) shown for comparison.



Demodulating DS/SS Signals

Principally, demodulation of DS/SS signals employs a
matched ﬁlter for the wide-band pulse.
Practically, the receiver front-end consists of an A-to-D
converter operating at the chip rate (or higher).
The subsequent digital matched ﬁlter for the Spreading
sequence is called a de-spreader.
In additive, white Gaussian noise a DS/SS waveform
performs identical to a narrow-band waveform with the
same symbol energy.
However, in environments with interference a DS/SS
waveform is more robust.
This includes interference from other emitters and ISI.



OFDM

Orthogonal Frequency Division Multiplexing (OFDM) is a
modulation format that combines the beneﬁts of
narrow-band and wide-band signals.
Narrow-band signals are easy to demodulate.
Wide-band signals can support high data rates and are
robust to multi-path fading.
In essence, OFDM signals are constructed by clever
multiplexing of many narrow-band signals.
Computationally efﬁcient via FFT.



Digital Synthesis of a Carrier Modulated Signal
To start, assume we wanted to generate the samples for a
linearly modulated signal:
symbol for the m-th symbol period bm (k ),
N samples per symbol period,
rectangular pulses,
up-conversion to digital frequency k /N (physical frequency
fs · k /N).
The resulting samples in the m-th symbol period are

sm,k [n] = bm (k ) · exp(j2πkn/N ) for n = 0, 1, . . . , N − 1.

If these samples were passed through a D-to-A converter,
we would observe a signal spectrum
centered at frequency fs · k /N, and
bandwidth 2fs /N.



Combining Carriers
The single carrier occupies only a small fraction of the
bandwidth fs the A-to-D converter can process.
The remaining bandwidth can be used for additional
signals.
Speciﬁcally, we can combine N such signals.
Carrier frequencies: fs · k /N for k = 0, 1, . . . N − 1.
Resulting carriers are all orthogonal.
Samples of the combined signal in the m-th symbol period
are
N −1 N −1
sm [ n ] = ∑ sm,k [n] = ∑ bm (k ) · exp(j2πkn/N ).
k =0 k =0

The signal sm [n] represents N orthogonal narrow-band
signals multiplexed in the frequency domain (OFDM).


Constructing an OFDM Signal

The expression
N −1
sm [ n ] = ∑ bm (k ) · exp(j2πkn/N ).
k =0

represents the inverse DFT of the symbols bm (k ).
Direct evaluation of this equation requires N 2
multiplications and additions.
However, the structure of the expression permits
computationally efﬁcient construction of sm [n] through an
FFT algorithm.
MATLAB’s function ifft can be used.



Constructing an OFDM Signal
The above considerations suggest the following procedure
for constructing an OFDM signal from a sequence of
information symbols b:
1. Serial-to-parallel conversion: break the sequence of
information symbols b into blocks of length N;
denote the k -th symbol in the m-th block as bm (k ).
2. Inverse DFT: Take the inverse FFT of each block m;
The output are length-N blocks of complex signal samples
denoted sm [n].
3. Cyclic preﬁx: Prepend the ﬁnal L samples from each block
to the beginning of each block.
Cyclic protects against ISI.
4. Parallel-to-serial conversion: Concatenate the blocks to
form the OFDM signal.



Properties of OFDM Signals
The signal resulting from the above construction has the
following parameters properties:
Bandwidth: fs (e.g., 10 MHz)
Number of subcarriers: N, should be a power of 2 (e.g.,
256)
Length of Preﬁx: L, depends on properties of channel
(e.g., 8)
Number of blocks: M (e.g., 64)
Subcarrier bandwidth: fs/N (e.g., 40 KHz)
Frame duration: M · (N + L)/fs (e.g., 1.7 ms)
Number of symbols in frame: M · N (e.g., 16,896)
Baud rate: N/(N + L) · fs (e.g. 9.7 MHz)
OFDM signals are not constant envelope signals.
Signals look like complex Gaussian noise.



OFDM Signals in MATLAB

The toolbox contains functions for modulating and
demodulating OFDM signals.
Their respective signatures are:
1. function Signal = OFDMMod( Symbols, NCarriers, LPrefix )

function SymbolEst = OFDMDemod( Signal, NCarriers, LPrefix, N

2. The bodies of these functions reﬂect the algorithm above.



Toolbox Function OFDMMod
%% Serial-to-parallel conversion
NBlocks = ceil( length(Symbols) / NCarriers );
% zero-pad, if needed
33 Blocks = zeros( 1, NBlocks*NCarriers );
Blocks(1:length(Symbols)) = Symbols;
% serial-to-parallel
Blocks = reshape( Blocks, NCarriers, NBlocks );

38 %% IFFT
% ifft works column-wise by default
Blocks = ifft( Blocks );

%% cyclic prefix
43 % copy last LPrefix samples of each column and prepend
Blocks = [ Blocks( end-LPrefix+1 : end, : ) ; Blocks ];

%% Parallel-to-serial conversion
Signal = reshape( Blocks, 1, NBlocks*(NCarriers+LPrefix) ) * ...
48 sqrt(NCarriers); % makes "gain" of ifft equal to 1.



Toolbox Function OFDMDemod
%% Serial-to-parallel conversion
32 NBlocks = ceil( length(Signal) / (NCarriers+LPrefix) );
if ( (NCarriers+LPrefix)*NBlocks ~= length(Signal) )
error(’Length of Signal must be a multiple of (NCarriers+LPrefix)’
end
% serial-to-parallel
37 Blocks = reshape( Signal/sqrt(NCarriers), NCarriers+LPrefix, NBlocks )

%% remove cyclic prefix
% remove first LPrefix samples of each column
Blocks( 1:LPrefix, : ) = [ ];
42
%% FFT
% fft works column-wise by default
Blocks = fft( Blocks );

47 %% Parallel-to-serial conversion
SymbolEst = reshape( Blocks, 1, NBlocks*NCarriers );
% remove zero-padding
SymbolEst = SymbolEst(1:NSymbols);



Using the OFDM functions

The code fragment below illustrates a typical use for the
OFDM functions in the toolbox.

Listing : ’MCOFDM.m’
%% simulate OFDM system
Signal = A * OFDMMod( Symbols, Nc, Lp );
39 Received = addNoise( Signal, NoiseVar );

% OFDM Receiver
MFOut = OFDMDemod( Received, Nc, Lp, NSymbols );
Decisions = SimpleSlicer( MFOut, Alphabet, Scale );



Monte Carlo Simulation with OFDM
−1
10

−2
10
Symbol Error Rate

−3
10

−4
10

−5
10
−2 0 2 4 6 8 10
E /N (dB)
s 0

Figure: Monte Carlo Simulation with OFDM Signals; BPSK Symbols,
256 subcarriers, preﬁx length 8.



Where we are ...

Considered wide variety of digital modulation techniques.
Linear modulation with pulse shaping,
Non-linear modulation: CPM and CPFSK, including MSK
and GMSK,
Wide-band modulation: DS/SS and OFDM.
Spectral properties of digitally modulated signals.
Numerical estimation of the spectrum via the periodogram.
Evaluation of spectrum for various representative
modulation formats.
The importance of constant envelope characteristics.
Next: characterizing the mobile channel.


Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels

Part IV

The Wireless Channel



The Wireless Channel
Characterization of the wireless channel and its impact on
digitally modulated signals.
From the physics of propagation to multi-path fading
channels.
Statistical characterization of wireless channels:
Doppler spectrum,
Delay spread
Coherence time
Coherence bandwidth
Simulating multi-path, fading channels in MATLAB.
Lumped-parameter models:
discrete-time equivalent channel.
Path loss models, link budgets, shadowing.



Outline







Learning Objectives

Understand models describing the nature of typical
wireless communication channels.
The origin of multi-path and fading.
Concise characterization of multi-path and fading in both
the time and frequency domain.
Doppler spectrum and time-coherence
Multi-path delay spread and frequency coherence
Appreciate the impact of wireless channels on transmitted
signals.
Distortion from multi-path: frequency-selective fading and
inter-symbol interference.
The consequences of time-varying channels.



Outline







Path Loss

Path loss LP relates the received signal power Pr to the
transmitted signal power Pt :

Gr · Gt
Pr = Pt · ,
LP

where Gt and Gr are antenna gains.
Path loss is very important for cell and frequency planning
or range predictions.
Not needed when designing signal sets, receiver, etc.



Path Loss

Path loss modeling is “more an art than a science.”
Standard approach: ﬁt model to empirical data.
Parameters of model:
d - distance between transmitter and receiver,
fc - carrier frequency,
hb , hm - antenna heights,
Terrain type, building density, . . ..



Example: Free Space Propagation
In free space, path loss LP is given by Friis’s formula:
2 2
4πd 4πfc d
LP = = .
λc c

Path loss increases proportional to the square of distance d
and frequency fc .
In dB:
c
LP (dB ) = −20 log10 ( ) + 20 log10 (fc ) + 20 log10 (d ).
4π

Example: fc = 1GHz and d = 1km

LP (dB ) = −146 dB + 180 dB + 60 dB = 94 dB.



Example: Two-Ray Channel
Antenna heights: hb and hm .
Two propagation paths:
1. direct path, free space propagation,
2. reﬂected path, free space with perfect reﬂection.
Depending on distance d, the signals received along the
two paths will add constructively or destructively.
Path loss:
2 2
1 4πfc d 1
LP = · · .
4 c sin( 2πchd hm )
fc
b

For d hb hm , path loss is approximately equal to:
2
d2
LP ≈
hb hm
Path loss proportional to d 4 is typical for urban
environment.©2009, B.-P. Paris Wireless Communications 288


Okumura-Hata Model for Urban Area
Okumura and Hata derived empirical path loss models
from extensive path loss measurements.
Models differ between urban, suburban, and open areas,
large, medium, and small cities, etc.
Illustrative example: Model for Urban area (small or
medium city)

LP (dB ) = A + B log10 (d ),

where
A = 69.55 + 26.16 log10 (fc ) − 13.82 log10 (hb ) − a(hm )
B = 44.9 − 6.55 log10 (hb )
a(hm ) = (1.1 log10 (fc ) − 0.7) · hm − (1.56 log10 (fc ) − 0.8)



Signal and Noise Power
Received Signal Power:
Gr · Gt
Pr = Pt ·
,
LP · LR
where LR is implementation loss, typically 2-3 dB.
(Thermal) Noise Power:
PN = kT0 · BW · F , where

k - Boltzmann’s constant (1.38 · 10−23 Ws/K),
T0 - temperature in K (typical room temperature,
T0 = 290 K),
⇒ kT0 = 4 · 10−21 W/Hz = 4 · 10−18 mW/Hz =
−174 dBm/Hz,
BW - signal bandwidth,
F - noise ﬁgure, ﬁgure of merit for receiver (typical value:
5dB). ©2009, B.-P. Paris Wireless Communications 290


Signal-to-Noise Ratio

The ratio of received signal power and noise power is
denoted by SNR.
From the above, SNR equals:

Pt Gr · Gt
SNR = .
kT0 · BW · F · LP · LR

SNR increases with transmitted power Pt and antenna
gains.
SNR decreases with bandwidth BW , noise ﬁgure F , and
path loss LP .



Es /N0

For the symbol error rate performance of communications
system the ratio of signal energy Es and noise power
spectral density N0 is more relevant than SNR.
Pr
Since Es = Pr · Ts = Rs and N0 = kT0 · F = PN /BW , it
follows that
Es B
= SNR · W ,
N0 Rs
where Ts and Rs denote the symbol period and symbol
rate, respectively.



Es /N0

Thus, Es /N0 is given by:

Es Pt Gr · Gt
= .
N0 kT0 · Rs · F · LP · LR
in dB:
E
( Ns )(dB ) = Pt (dBm) + Gt (dB ) + Gr (dB )
0
−(kT0 )(dBm/Hz ) − Rs(dBHz ) − F(dB ) − LR (dB ) .



Outline







Multi-path Propagation
The transmitted signal
propagates from the
transmitter to the receiver
along many different paths.
These paths have different
path attenuation ak ,
path delay τk , TX RX
phase shift φk ,
angle of arrival θk .
For simplicity, we assume
a 2-D model, so that the
angle of arrival is the
azimuth.
In 3-D models, the
elevation angle of arrival
is an additional parameter.


Channel Impulse Response

From the above parameters, one can easily determine the
channel’s (baseband equivalent) impulse response.
Impulse Response:
K
h (t ) = ∑ ak · ejφ k · e−j2πfc τk · δ(t − τk )
k =1

Note that the delays τk contribute to the phase shifts φk .



Received Signal

Ignoring noise for a moment, the received signal is the
convolution of the transmitted signal s (t ) and the impulse
response
K
R (t ) = s (t ) ∗ h (t ) = ∑ ak · ejφ k · e−j2πfc τk · s (t − τk ).
k =1

The received signal consists of multiple
scaled (by ak · ejφk · e −j2πfc τk ),
delayed (by τk )
copies of the transmitted signal.



Channel Frequency Response
Similarly, one can compute the frequency response of the
channel.
Direct Fourier transformation of the expression for the
impulse response yields
K
H (f ) = ∑ ak · ejφ k · e−j2πfc τk · e−j2πf τk .
k =1

For any given frequency f , the frequency response is a sum
of complex numbers.
When these terms add destructively, the frequency
response is very small or even zero at that frequency.
These nulls in the channel’s frequency response are typical
for wireless communications and are refered to as
frequency-selective fading.


Frequency Response in One Line of MATLAB

The Frequency response
K
H (f ) = ∑ ak · ejφ k · e−j2πfc τk · e−j2πf τk .
k =1

can be computed in MATLAB via the one-liner
HH = PropData.Field.*exp(-j*2*pi*fc*tau) * exp(-j*2*pi*tau’*ff);

Note that tau’*ff is an inner product; it produces a matrix
(with K rows and as many columns as ff).
Similarly, the product preceding the second complex
exponential is an inner product; it generates the sum in the
expression above.



Example: Ray Tracing
1300

1250

1200

1150
Transmitter

1100

1050
y (m)

1000

950 Receiver

900

850

800

750
450 500 550 600 650 700 750 800 850 900 950
x (m)

Figure: All propagation paths between the transmitter and receiver in
the indicated located were determined through ray tracing.



Impulse Response
−5
x 10
4

3

Attenuation
2

1

0
0.8 1 1.2 1.4 1.6 1.8 2
Delay (µs)

4

2
Phase Shift/π

0

−2

−4
0.8 1 1.2 1.4 1.6 1.8 2
Delay (µs)

Figure: (Baseband equivalent) Impulse response shows attenuation,
delay, and phase for each of the paths between receiver and
transmitter.



Frequency Response
−78

−80

−82

−84
|Frequency Response| (dB)

−86

−88

−90

−92

−94

−96

−98
−5 −4 −3 −2 −1 0 1 2 3 4 5
Frequency (MHz)

Figure: (Baseband equivalent) Frequency response for a multi-path
channel is characterized by deep “notches”.



Implications of Multi-path

Multi-path leads to signal distortion.
The received signal “looks different” from the transmitted
signal.
This is true, in particular, for wide-band signals.
Multi-path propagation is equivalent to undesired filtering
with a linear filter.
The impulse response of this undesired filter is the impulse
response h(t ) of the channel.
The effects of multi-path can be described in terms of both
time-domain and frequency-domain concepts.
In either case, it is useful to distinguish between
narrow-band and wide-band signals.



Example: Transmission of a Linearly Modulated Signal
Transmission of a linearly modulated signal through the
above channel is simulated.
BPSK,
(full response) raised-cosine pulse.
Symbol period is varied; the following values are
considered
Ts = 30µs ( bandwidth approximately 60 KHz)
Ts = 3µs ( bandwidth approximately 600 KHz)
Ts = 0.3µs ( bandwidth approximately 6 MHz)
For each case, the transmitted and (suitably scaled)
received signal is plotted.
Look for distortion.
Note that the received signal is complex valued; real and
imaginary part are plotted.



2
Transmitted
Real(Received)
1.5 Imag(Received)

1

0.5
Amplitude

0

−0.5

−1

−1.5

−2
0 50 100 150 200 250 300
Time (µs)

Figure: Transmitted and received signal; Ts = 30µs. No distortion is
evident.



2
Transmitted
Real(Received)
1.5 Imag(Received)

1

0.5
Amplitude

0

−0.5

−1

−1.5

−2
0 5 10 15 20 25 30 35
Time (µs)

Figure: Transmitted and received signal; Ts = 3µs. Some distortion is
visible near the symbol boundaries.



2
Transmitted
Real(Received)
1.5 Imag(Received)

1

0.5
Amplitude

0

−0.5

−1

−1.5

−2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Time (µs)

Figure: Transmitted and received signal; Ts = 0.3µs. Distortion is
clearly visible and spans multiple symbol periods.



Eye Diagrams for Visualizing Distortion
An eye diagram is a simple but useful tool for quickly
gaining an appreciation for the amount of distortion present
in a received signal.
An eye diagram is obtained by plotting many segments of
the received signal on top of each other.
The segments span two symbol periods.
This can be accomplished in MATLAB via the command
plot( tt(1:2*fsT), real(reshape(Received(1:Ns*fsT), 2*fsT, [ ])))

Ns - number of symbols; should be large (e.g., 1000),
Received - vector of received samples.
The reshape command turns the vector into a matrix with
2*fsT rows, and
the plot command plots each column of the resulting matrix
individually.


Eye Diagram without Distortion
0.5

Amplitude 0

−0.5
0 10 20 30 40 50 60
Time (µs)

1

0.5
Amplitude

0

−0.5

−1
0 10 20 30 40 50 60
Time (µs)

Figure: Eye diagram for received signal; Ts = 30µs. No distortion:
“the eye is fully open”.



Eye Diagram with Distortion
2

1

Amplitude 0

−1

−2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Time (µs)

2

1
Amplitude

0

−1

−2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Time (µs)

Figure: Eye diagram for received signal; Ts = 0.3µs. Signiﬁcant
distortion: “the eye is partially open”.



Inter-Symbol Interference

The distortion described above is referred to as
inter-symbol interference (ISI).
As the name implies, the undesired ﬁltering by the channel
causes energy to be spread from one transmitted symbol
across several adjacent symbols.
This interference makes detection mored difﬁcult and must
be compensated for at the receiver.
Devices that perform this compensation are called
equalizers.



Inter-Symbol Interference
Question: Under what conditions does ISI occur?
Answer: depends on the channel and the symbol rate.
The difference between the longest and the shortest delay
of the channel is called the delay spread Td of the channel.
The delay spread indicates the length of the impulse
response of the channel.
Consequently, a transmitted symbol of length Ts will be
spread out by the channel.
When received, its length will be the symbol period plus the
delay spread, Ts + Td .
Rule of thumb:
if the delay spread is much smaller than the symbol period
(Td Ts ), then ISI is negligible.
If delay is similar to or greater than the symbol period, then
ISI must be compensated at the receiver.



Frequency-Domain Perspective
It is interesting to compare the bandwidth of the transmitted
signals to the frequency response of the channel.
In particular, the bandwidth of the transmitted signal relative
to variations in the frequency response is important.
The bandwidth over which the channel’s frequency
response remains approximately constant is called the
coherence bandwidth.
When the frequency response of the channel remains
approximately constant over the bandwidth of the
transmitted signal, the channel is said to be ﬂat fading.
Conversely, if the channel’s frequency response varies
signiﬁcantly over the bandwidth of the signal, the channel
is called a frequency-selective fading channel.



Example: Narrow-Band Signal
−75

−80


−85

−90

−95

−100

−5 −4 −3 −2 −1 0 1 2 3 4
Frequency (MHz)

Figure: Frequency Response of Channel and bandwidth of signal;
Ts = 30µs, Bandwidth ≈ 60 KHz; the channel’s frequency response
is approximately constant over the bandwidth of the signal.



Example: Wide-Band Signal
−75

−80


−85

−90

−95

−100

−5 −4 −3 −2 −1 0 1 2 3 4
Frequency (MHz)

Figure: Frequency Response of Channel and bandwidth of signal;
Ts = 0.3µs, Bandwidth ≈ 6 MHz; the channel’s frequency response
varies signiﬁcantly over the bandwidth of the channel.



Frequency-Selective Fading and ISI
Frequency-selective fading and ISI are dual concepts.
ISI is a time-domain characterization for signiﬁcant
distortion.
Frequency-selective fading captures the same idea in the
frequency domain.
Wide-band signals experience ISI and
frequency-selective fading.
Such signals require an equalizer in the receiver.
Wide-band signals provide built-in diversity.
Not the entire signal will be subject to fading.
Narrow-band signals experience ﬂat fading (no ISI).
Simple receiver; no equalizer required.
Entire signal may be in a deep fade; no diversity.



Time-Varying Channel

Beyond multi-path propagation, a second characteristic of
many wireless communication channels is their time
variability.
The channel is time-varying primarily because users are
mobile.
As mobile users change their position, the characteristics
of each propagation path changes correspondingly.
Consider the impact a change in position has on
path gain,
path delay.
Will see that angle of arrival θk for k -th path is a factor.



Path-Changes Induced by Mobility
Mobile moves by ∆d from old position to new position.
distance: |∆d |
angle: ∠∆d = δ
Angle between k -th ray and ∆d is denoted ψk = θk − δ.
Length of k -th path increases by |∆d | cos(ψk ).

k -th ray k -th ray

|∆d | sin(ψk )
|∆d | cos(ψk )
ψk
∆d
Old Position New Position



Impact of Change in Path Length
We conclude that the length of each path changes by
|∆d | cos(ψk ), where
ψk denotes the angle between the direction of the mobile
and the k -th incoming ray.
Question: how large is a typical distance |∆d | between
the old and new position is?
The distance depends on
the velocity v of the mobile, and
the time-scale ∆T of interest.
In many modern communication system, the transmission
of a frame of symbols takes on the order of 1 to 10 ms.
Typical velocities in mobile systems range from pedestrian
speeds (≈ 1m/s) to vehicle speeds of 150km/h( ≈ 40m/s).
Distances of interest |∆d | range from 1mm to 400mm.



Question: What is the impact of this change in path length
on the parameters of each path?
We denote the length of the path to the old position by dk .
Clearly, dk = c · τk , where c denotes the speed of light.
Typically, dk is much larger than |∆d |.
Path gain ak : Assume that path gain ak decays inversely
−
proportional with the square of the distance, ak ∼ dk 2 .
Then, the relative change in path gain is proportional to
(|∆d |/dk )2 (e.g., |∆d | = 0.1m and dk = 100m, then path
gain changes by approximately 0.0001%).
Conclusion: The change in path gain is generally small
enough to be negligible.




Delay τk : By similar arguments, the delay for the k -th path
changes by at most |∆d |/c.
The relative change in delay is |∆d |/dk (e.g., 0.1% with the
values above.)
Question: Is this change in delay also negligible?



Relating Delay Changes to Phase Changes

Recall: the impulse response of the multi-path channel is
K
h (t ) = ∑ ak · ejφ k · e−j2πfc τk · δ(t − τk )
k =1

Note that the delays, and thus any delay changes, are
multiplied by the carrier frequency fc to produce phase
shifts.



Relating Delay Changes to Phase Changes

Consequently, the phase change arising from the
movement of the mobile is

∆φk = −2πfc /c |∆d | cos(ψk ) = −2π |∆d |/λc cos(ψk ),

where
λc = c/fc - denotes the wave-length at the carrier
frequency (e.g., at fc = 1GHz, λc ≈ 0.3m),
ψk - angle between direction of mobile and k -th arriving
path.
Conclusion: These phase changes are signiﬁcant and
lead to changes in the channel properties over short
time-scales (fast fading).



Illustration
To quantify these effects, compute the phase change over
a time interval ∆T = 1ms as a function of velocity.
Assume ψk = 0, and, thus, cos(ψk ) = 1.
fc = 1GHz.
v (m/s) |∆d | (mm) ∆φ (degrees) Comment
1 1 1.2 Pedestrian; negligible
phase change.
10 10 12 Residential area vehi-
cle speed.
100 100 120 High-way speed;
phase change signiﬁ-
cant.
1000 1000 1200 High-speed train or
low-ﬂying aircraft;
receiver must track
phase changes.



Doppler Shift and Doppler Spread
If a mobile is moving at a constant velocity v , then the
distance between an old position and the new position is a
function of time, |∆d | = vt.
Consequently, the phase change for the k -th path is
∆φk (t ) = −2πv /λc cos(ψk )t = −2πv /c · fc cos(ψk )t.

The phase is a linear function of t.
Hence, along this path the signal experiences a frequency
shift fd ,k = v /c · fc · cos(ψk ) = v /λc · cos(ψk ).
This frequency shift is called Doppler shift.
Each path experiences a different Doppler shift.
Angles of arrival θk are different.
Consequently, instead of a single Doppler shift a number of
shifts create a Doppler Spectrum.


Illustration: Time-Varying Frequency Response

−70

|Frequency Response| (dB) −80

−90

−100

−110

−120

−130
200
150 5
100
0
50

Time (ms) 0 −5
Frequency (MHz)

Figure: Time-varying Frequency Response for Ray-Tracing Data;
velocity v = 10m/s, fc = 1GHz, maximum Doppler frequency
≈ 33Hz.



Illustration: Time-varying Response to a Sinusoidal
Input
−80

Magnitude (dB)
−100

−120

−140
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time (s)

10

0
Phase/π

−10

−20

−30
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time (s)

Figure: Response of channel to sinusoidal input signal; base-band
equivalent input signal s (t ) = 1, velocity v = 10m/s, fc = 1GHz,
maximum Doppler frequency ≈ 33Hz.


Doppler Spread and Coherence Time
The time over which the channel remains approximately
constant is called the coherence time of the channel.
Coherence time and Doppler spectrum are dual
characterizations of the time-varying channel.
Doppler spectrum provides frequency-domain
interpretation:
It indicates the range of frequency shifts induced by the
time-varying channel.
Frequency shifts due to Doppler range from −fd to fd , where
fd = v /c · fc .
The coherence time Tc of the channel provides a
time-domain characterization:
It indicates how long the channel can be assumed to be
approximately constant.
Maximum Doppler shift fd and coherence time Tc are
related to each through an inverse relationship Tc ≈ 1/fd .


System Considerations
The time-varying nature of the channel must be accounted
for in the design of the system.
Transmissions are shorter than the coherence time:
Many systems are designed to use frames that are shorter
than the coherence time.
Example: GSM TDMA structure employs time-slots of
duration 4.6ms.
Consequence: During each time-slot, channel may be
treated as constant.
From one time-slot to the next, channel varies signiﬁcantly;
this provides opportunities for diversity.
Transmission are longer than the coherence time:
Channel variations must be tracked by receiver.
Example: use recent symbol decisions to estimate current
channel impulse response.


Illustration: Time-varying Channel and TDMA
−80

−90

−100
Magnitude (dB)

−110

−120

−130

−140

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (s)

Figure: Time varying channel response and TDMA time-slots;
time-slot duration 4.6ms, 8 TDMA users, velocity v = 10m/s,
fc = 1GHz, maximum Doppler frequency ≈ 33Hz.



Summary
Illustrated by means of a concrete example the two main
impairments from a mobile, wireless channel.
Multi-path propagation,
Doppler spread due to time-varying channel.
Multi-path propagation induces ISI if the symbol duration
exceeds the delay spread of the channel.
In frequency-domain terms, frequency-selective fading
occurs if the signal bandwidth exceeds the coherence
band-width of the channel.
Doppler Spreading results from time-variations of the
channel due to mobility.
The maximum Doppler shift fd = v /c · fc is proportional to
the speed of the mobile.
In time-domain terms, the channel remains approximately
constant over the coherence-time of the channel.


Outline







Statistical Characterization of Channel
We have looked at the characterization of a concrete
realization of a mobile, wire-less channel.
For different locations, the properties of the channel will
likely be very different.
Objective: develop statistical models that capture the
salient features of the wireless channel for areas of
interest.
Models must capture multi-path and time-varying nature of
channel.
Approach: Models reﬂect correlations of the time-varying
channel impulse response or frequency response.
Time-varying descriptions of channel are functions of two
parameters:
Time t when channel is measured,
Frequency f or delay τ.



Power Delay Profile
The impulse response of a wireless channel is
time-varying, h(t, τ ).
The parameter t indicates when the channel is used,
The parameter τ reflects time since the input was applied
(delay).
Time-varying convolution: r (t ) = h(t, τ ) · s (t − τ )dτ.
The power-delay profile measures the average power in
the impulse response over delay τ.
Thought experiment: Send impulse through channel at
time t0 and measure response h(t0 , τ ).
Repeat K times, measuring h(tk , τ ).
Power delay profile:
K
1
Ψh (τ ) = ∑ |h(tk , τ )|2 .
K + 1 k =0



Power Delay Proﬁle

The power delay proﬁle captures the statistics of the
multi-path effects of the channel.
The underlying, physical model assumes a large number
of propagation paths:
each path has a an associated delay τ,
the gain for a path is modeled as a complex Gaussian
random variable with second moment equal to Ψh (τ ).
If the mean of the path loss is zero, the path is said to be
Rayleigh fading.
Otherwise, it is Ricean.
The channel gains associated with different delays are
assumed to be uncorrelated.



Example
1 2

0.9 1.5

|h(τ)|2
0.8 1

Power Delay Profile 0.7 0.5

0.6
0
0 2 4 6
0.5 Delay τ (µs)

1
0.4

0.3 0.5

Phase of h(τ)
0.2 0

0.1 −0.5

0 −1
0 2 4 6 0 2 4 6
Delay τ (µs) Delay τ (µs)

Figure: Power Delay Proﬁle and Channel Impulse Response; the
power delay proﬁle (left) equals Ψh (τ ) = exp(−τ/Th ) with Th = 1µs;
realization of magnitude and phase of impulse response (left).



RMS Delay Spread
From a systems perspective, the extent (spread) of the
delays is most signiﬁcant.
The length of the impulse response of the channel
determines how much ISI will be introduced by the channel.
The spread of delays is measured concisely by the RMS
delay spread Td :
∞ ∞
2 (n ) (n )
Td = Ψh (τ )τ 2 dτ − ( Ψh (τ )τdτ )2 ,
0 0

where ∞
(n )
Ψh = Ψh / Ψh (τ )dτ.
0
Example: For Ψh (τ ) = exp(−τ/Th ), RMS delay spread
equals Th .
In urban environments, typical delay spreads are a few µs.


Frequency Coherence Function
The Fourier transform of the Power Delay Spread Ψh (τ ) is
called the Frequency Coherence Function ΨH (∆f )

Ψh (τ ) ↔ ΨH (∆f ).

The frequency coherence function measures the
correlation of the channel’s frequency response.
Thought Experiment: Transmit two sinusoidal signal of
frequencies f1 and f2 , such that f1 − f2 = ∆f .
The gain each of these signals experiences is H (t, f1 ) and
H (t, f2 ), respectively.
Repeat the experiment many times and average the
products H (t, f1 ) · H ∗ (t, f2 ).
ΨH (∆f ) indicates how similar the gain is that two sinusoids
separated by ∆f experience.



Coherence Bandwidth
The width of the main lobe of the frequency coherence
function is the coherence bandwidth Bc of the channel.
Two signals with frequencies separated by less than the
coherence bandwidth will experience very similar gains.
Because of the Fourier transform relationship between the
power delay proﬁle and the frequency coherence function:
1
Bc ≈ .
Td
Example: Fourier transform of Ψh (τ ) = exp(−τ/Th )
Th
ΨH (∆f ) = ;
1 + j2π∆fTh
the 3-dB bandwidth of ΨH (∆f ) is Bc = 1/(2π · Th ).
For urban channels, coherence bandwidth is a few 100KHz.


Time Coherence
The time-coherence function ΨH (∆t ) captures the
time-varying nature of the channel.
Thought experiment: Transmit a sinusoidal signal of
frequency f through the channel and measure the output at
times t1 and t1 + ∆t.
The gains the signal experiences are H (t1 , f ) and
H (t1 + ∆t, f ), respectively.
Repeat experiment and average the products
H (tk , f ) · H ∗ (tk + ∆t, f ).
Time coherence function measures, how quickly the gain
of the channel varies.
The width of the time coherence function is called the
coherence-time Tc of the channel.
The channel remains approximately constant over the
coherence time of the channel.



Example: Isotropic Scatterer
Old location: H (t1 , f = 0) = ak · exp(−j2πfc τk ).
At new location: the gain ak is unchanged; phase changes
by fd cos(ψk )∆t:
H (t1 + ∆t, f = 0) = ak · exp(−j2π (fc τk + fd cos(ψk )∆t )).

k -th ray k -th ray

|∆d | sin(ψk )
|∆d | cos(ψk )
ψk
∆d
Old Position New Position



Example: Isotropic Scatterer

The average of H (t1 , 0) · H ∗ (t1 + ∆t, 0) yields the
time-coherence function.
Assume that the angle of arrival ψk is uniformly distributed.

This allows computation of the average (isotropic scatterer
assumption:

ΨH (∆t ) = |ak |2 · J0 (2πfd ∆t )



Time-Coherence Function for Isotropic Scatterer
1

0.5
Ψ (∆t)
H

0

−0.5
0 50 100 150 200 250 300
Time ∆t (ms)

Figure: Time-Coherence Function for Isotropic Scatterer; velocity
v = 10m/s, fc = 1GHz, maximum Doppler frequency fd ≈ 33Hz. First
zero at ∆t ≈ 0.4/fd .



Doppler Spread Function

The Fourier transform of the time coherence function
ΨH (∆t ) is the Doppler Spread Function Ψd (fd )

ΨH (∆t ) ↔ Ψd (fd ).

The Doppler spread function indicates the range of
frequencies observed at the output of the channel when
the input is a sinusoidal signal.
Maximum Doppler shift fd ,max = v /c · fc .
Thought experiment:
Send a sinusoidal signal of
The PSD of the received signal is the Doppler spread
function.



Doppler Spread Function for Isotropic Scatterer

Example: The Doppler spread function for the isotropic
scatterer is
|ak |2 1
Ψd (fd ) = for |f | < fd .
4πfd 1 − (f /fd )2



Doppler Spread Function for Isotropic Scatterer
7

6

5

4
Ψd(fd)

3

2

1

0
−40 −30 −20 −10 0 10 20 30 40
Doppler Frequency (Hz)

Figure: Doppler Spread Function for Isotropic Scatterer; velocity
v = 10m/s, fc = 1GHz, maximum Doppler frequency fd ≈ 33Hz. First
zero at ∆t ≈ 0.4/fd .



Simulation of Multi-Path Fading Channels

We would like to be able to simulate the effects of
time-varying, multi-path channels.
Approach:
The simulator operates in discrete-time; the sampling rate
is given by the sampling rate for the input signal.
The multi-path effects can be well modeled by an FIR
(tapped delay-line)filter.
The number of taps for the filter is given by the product of
delay spread and sampling rate.
Example: With a delay spread of 2µs and a sampling rate of
2MHz, four taps are required.
The taps should be random with a Gaussian distribution.
The magnitude of the tap weights should reflect the
power-delay profile.




Approach (cont’d):
The time-varying nature of the channel can be captured by
allowing the taps to be time-varying.
The time-variations should reﬂect the Doppler Spectrum.



The taps are modeled as
Gaussian random processes
with variances given by the power delay proﬁle, and
power spectral density given by the Doppler spectrum.

s [n ]
D D

a0 (t ) × a1 ( t ) × a2 (t ) ×

r [n ]
+ +



Channel Model Parameters

Concrete parameters for models of the above form have
been proposed by various standards bodies.
For example, the following table is an excerpt from a
document produced by the COST 259 study group.

Tap number Relative Time (µs) Relative Power (dB) Doppler Spectrum
1 0 -5.7 Class
2 0.217 -7.6 Class
3 0.512 -10.1 Class
.
. .
. .
. .
.
. . . .
20 2.140 -24.3 Class



Channel Model Parameters
The table provides a concise, statistical description of a
time-varying multi-path environment.
Each row corresponds to a path and is characterized by
the delay beyond the delay for the shortest path,
the average power of this path;
this parameter provides the variance of the Gaussian path
gain.
the Doppler spectrum for this path;
The notation Class denotes the classical Doppler spectrum
for the isotropic scatterer.
The delay and power column specify the power-delay
proﬁle.
The Doppler spectrum is given directly.
The Doppler frequency fd is an additional parameter.



Toolbox Function SimulateCOSTChannel

The result of our efforts will be a toolbox function for
simulating time-varying multi-path channels:
function OutSig = SimulateCOSTChannel( InSig, ChannelParams, fs)

Its input arguments are
% Inputs:
% InSig - baseband equivalent input signal
% ChannelParams - structure ChannelParams must have fields
11 % Delay - relative delay
% Power - relative power in dB
% Doppler - type of Dopller spectrum
% fd - max. Doppler shift
% fs - sampling rate



Discrete-Time Considerations
The delays in the above table assume a continuous time
axis; our time-varying FIR will operate in discrete time.
To convert the model to discrete-time:
Continuous-time is divided into consecutive “bins” of width
equal to the sampling period, 1/fs.
For all paths arriving in same “bin,” powers are added.
This approach reﬂects that paths arriving closer together
than the sampling period cannot be resolved;
their effect is combined in the receiver front-end.
The result is a reduced description of the multi-path
channel:
Power for each tap reﬂects the combined power of paths
arriving in the corresponding “bin”.
This power will be used to set the variance of the random
process for the corresponding tap.



Converting to a Discrete-Time Model in MATLAB

%% convert powers to linear scale
Power_lin = dB2lin( ChannelParams.Power);

%% Bin the delays according to the sample rate
29 QDelay = floor( ChannelParams.Delay*fs );

% set surrogate delay for each bin, then sum up the power in each bin
Delays = ( ( 0:QDelay(end) ) + 0.5 ) / fs;
Powers = zeros( size(Delays) );
34 for kk = 1:length(Delays)
Powers( kk ) = sum( Power_lin( QDelay == kk-1 ) );
end



Generating Time-Varying Filter Taps
The time-varying taps of the FIR filter must be Gaussian
random processes with specified variance and power
spectral density.
To accomplish this, we proceed in two steps:
1. Create a filter to shape the power spectral density of the
random processes for the tap weights.
2. Create the random processes for the tap weights by
passing complex, white Gaussian noise through the filter.
Variance is adjusted in this step.
Generating the spectrum shaping filter:
% desired frequency response of filter:
HH = sqrt( ClassDoppler( ff, ChannelParams.fd ) );
% design filter with desired frequency response
77 hh = Persistent_firpm( NH-1, 0:1/(NH-1):1, HH );
hh = hh/norm(hh); % ensure filter has unit norm



Generating Time-Varying Filter Taps
The spectrum shaping filter is used to filter a complex
white noise process.
Care is taken to avoid transients at the beginning of the
output signal.
Also, filtering is performed at a lower rate with subsequent
interpolation to avoid numerical problems.
Recall that fd is quite small relative to fs .

% generate a white Gaussian random process
93 ww = sqrt( Powers( kk )/2)*...
( randn( 1, NSamples) + j*randn( 1, NSamples) );
% filter so that spectrum equals Doppler spectrum
ww = conv( ww, hh );
ww = ww( length( hh )+1:NSamples ).’;
98 % interpolate to a higher sampling rate
% ww = interp( ww, Down );
ww = interpft(ww, Down*length(ww));
% store time-varying filter taps for later use



Time-Varying Filtering
The final step in the simulator is filtering the input signal
with the time-varying filter taps.
MATLAB’s filtering functions conv or filter cannot be used
(directly) for this purpose.
The simulator breaks the input signal into short segments
for which the channel is nearly constant.
Each segment is filtered with a slightly different set of taps.

while ( Start < length(InSig) )
EndIn = min( Start+QDeltaH, length(InSig) );
EndOut = EndIn + length(Powers)-1;
118 OutSig(Start:EndOut) = OutSig(Start:EndOut) + ...
conv( Taps(kk,:), InSig(Start:EndIn) );

kk = kk+1;
Start = EndIn+1;



Testing SimulateCOSTChannel

A simple test for the channel simulator consists of
“transmitting” a baseband equivalent sinusoid.
%% Initialization
ChannelParameters = tux(); % COST model parameters
6 ChannelParameters.fd = 10; % Doppler frequency

fs = 1e5; % sampling rate
SigDur = 1; % duration of signal

11 %% generate input signal and simulate channel
tt = 0:1/fs:SigDur; % time axis
Sig = ones( size(tt) ); % baseband-equivalent carrier

Received = SimulateCOSTChannel(Sig, ChannelParameters, fs);



Testing SimulateCOSTChannel
1.8

1.6

1.4

1.2
Magnitude

1

0.8

0.6

0.4

0.2

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time (s)

Figure: Simulated Response to a Sinusoidal Signal; fd = 10Hz,
baseband equivalent frequency f = 0.



Summary

Highlighted unique aspects of mobile, wireless channels:
time-varying, multi-path channels.
Statistical characterization of channels via
power-delay proﬁle (RMS delay spread),
frequency coherence function (coherence bandwidth),
time coherence function (coherence time), and
Doppler spread function (Doppler spread).
Relating channel parameters to system parameters:
signal bandwidth and coherence bandwidth,
frame duration and coherence time.
Channel simulator in MATLAB.



Where we are ...

Having characterized the nature of mobile, wireless
channels, we can now look for ways to overcome the
detrimental effects of the channel.
The importance of diversity to overcome fading.
Sources of diversity:
Time,
Frequency,
Space.
Equalizers for overcoming frequency-selective fading.
Equalizers also exploit freqeuncy diversity.


The Importance of Diversity Frequency Diversity: Wide-Band Signals

Part V

Mitigating the Impact of the Wireless Channel



Mitigating the Impact of the Wireless Channel

Description and analysis of techniques to overcome the
detrimental inﬂuence of the wireless channel through various
forms of diversity.
The importance of diversity.
Sources of diversity: time, frequency, and space.
Equalization: overcoming ISI and exploiting diversity



Outline






Learning Objectives

The Importance of Diversity.
BER performance in a Rayleigh fading channel without
diversity.
Diversity to the rescue ...
What is diversity?
Acceptable performance in Rayleigh fading channels
requires diversity.
Creating and Exploiting Diversity.
spatial diversity through multiple antennas,
frequency diversity through wide-band signaling.
Equalization



Outline







We have taken a detailed look at the detrimental effects of
time-varying multi-path channels.
Question: How do time-varying multi-path channels affect
the performance of mobile, wireless communications
channels?
In particular, how is the symbol error rate affected by these
channels?
Example: Analyze the simple communication system
analyzed before in a time-varying multi-path environment.
BPSK
raised-cosine pulses,
low data rate, i.e., narrow-band signals.



System to be Analyzed
The simple communication system in the diagram below
will be analyzed.
Focus on the effects of the multi-path channel h(t ).
Assumption: low baud rate, i.e., narrow-band signal.

Sampler,
N (t )
rate fs

× p (t ) × h (t ) + ΠTs (t )
DSP

∑ δ(t − nT ) A



Assumptions and Implications
For our analysis, we make the following speciﬁc
assumptions:
1. Narrow-band signals:
The symbol period T is assumed to be much smaller than
the delay spread of the channel.
Implication: ISI is negligible; ﬂat-fading channel.
The delay spread of our channel is approximately 2µs;
choose symbol period T = 40µs (Baud rate 25KHz).
2. Slow fading:
The duration of each transmission is much shorter than the
coherence-time of the channel.
Implication: the channel remains approximately constant for
each transmission.
Assuming a Doppler frequency of 30Hz, the coherence time
is approximately 20ms;
transmitting 60 symbols per frame leads to frame durations
of 60 · 40µs = 2.4ms.



Implications on Channel Model
With the above assumptions, the multi-path channel
reduces effectively to attenuation by a factor a.
a is complex Gaussian (multiplicative noise),
The magnitude |a| is Rayleigh distributed (Rayleigh fading).
Short frame duration implies a is constant during a frame.

Sampler,
N (t )
rate fs

× p (t ) × × + ΠTs (t )
DSP

∑ δ(t − nT ) A a



Modiﬁcations to the Receiver

For the narrow-band channel model, only a minor
modiﬁcation to the receiver is required.
The effective impulse response of the system is a · p (t ).
Hence, the receiver should match with a · p (t ) instead of
just p (t ).
Most importantly, this reverses any phase rotations
introduced by the channel.
Problem: The channel attenuation a is unknown and must
be estimated.
This is accomplished with the help of a training sequence
embedded in the signal.
To be discussed shortly.



Instantaneous Symbol Error Rate
Assuming that we have successfully determined, the
channel attenuation a, the symbol error rate for a given a is
easily determined.
The attenuation a changes the received energy per symbol
to |a|2 · Es .
Consequently, the instantaneous symbol error rate for our
BPSK system is

2|a|2 · Es
Pe (a) = Q( ).
N0

Note that for each frame, the system experiences a
different channel attenuation a and, thus, a different
instantaneous symbol error rate.


Average Symbol Error Rate
The instantaneous symbol error rate varies from frame to
frame and depends on a.
The average symbol error rate provides a measure of
performance that is independent of the attenuation a.
It is obtained by averaging over a:

Pe = E[Pe (a)] = Pe (a) · PA (a) da,

where PA (a) is the pdf for the attenuation a.
For a complex Gaussian attenuation a with zero mean and
2
variance σa :
1 SNR
Pe = (1 − ),
2 1 + SNR
2
where SNR = σa · Es /N0 .


Symbol Error Rate with Rayleigh Fading
0
10
Rayleigh
AWGN
−1
10

−2
10
Symbol Error Rate

−3
10

−4
10

−5
10

−6
10
0 1 2 3 4 5 6 7 8 9 10
E /N0 (dB)
s

2
Figure: Symbol Error Rate with and without Rayleigh Fading; σa = 1



Conclusions

The symbol error rate over a Rayleigh fading channel is
much worse than for an AWGN channel.
Example: To achieve a symbol error rate of 10−4
on an AWGN channel, Es /N0 ≈ 8.2dB is required;
on a Rayleigh fading channel, Es /N0 ≈ 34dB is required!
The poor performance results from the fact that the
probability that channel is in a deep fade is signiﬁcant:

1
Pr(|a|2 Es /N0 < 1) ≈ .
Es /N0

Question: What can be done to improve performance?



Diversity
Diversity refers to the ability to observe the transmitted
signal over multiple, independent Rayleigh fading
channels.
Example: Multiple receiver antennas
Assume the receiver is equipped with L separate antennas
and corresponding receiver front-ends.
The antennas are spaced sufﬁciently to ensure that the
channels from transmitter to each of the receiver antennas
is independent.
Antenna spacing approximately equal to the wavelength of
the carrier is sufﬁcient.
Then, the receiver observes L versions of the transmitted
signal,
each with a different attenuation al and a different additive
noise Nl (t ).



Diversity Receiver

Question: How should the L received signals be
processed to minimize the probability of symbol errors.
Maximum-Ratio Combining:
Assume again that the channel attenuation al are known
and that the noise PSD is equal on all channels.
Then, the optimum receiver performs
matched filtering for each received signal: the l-th received
signal is filtered with filter al∗ · p ∗ (−t ).
The L matched filter outputs are added and fed into the
slicer.
Note, since the matched filter includes the attenuation al , the
sum is weighted by the attenuation of the channel.



Symbol Error Rate
The instantaneous error probability for given channel gains
a = {a1 , a2 , . . . , aL } is
¯

¯
Pe ( a ) = Q ( 2||a||2 Es /N0 ),
¯

where ||a||2 = ∑L =1 |ak |2 .
¯ k
The average error probability is obtained by taking the
expectation with respect to the random gains a ¯
1−µ L L L−k 1 + µ k −1
Pe = ( ) ·∑ ( ) ,
2 k =1
k −1 2
where
SNR 2
µ= and SNR = σa · Es /N0 .
1 + SNR



Symbol Error Rate with Receiver Diversity
0
10
L=1
−1 L=2
10 L=3
L=4
−2 L=5
10
AWGN
−3
10
Symbol Error Rate

−4
10

−5
10

−6
10

−7
10

−8
10

−9
10
0 2 4 6 8 10 12
E /N0 (dB)
s

Figure: Symbol Error Rate with Diversity over a Rayleigh Fading
2
Channel; σa = 1, AWGN channel without diversity.



Conclusions
The symbol error rate with diversity is much better than
without.
Example: To achieve a symbol error rate of 10−4
on an AWGN channel, Es /N0 ≈ 8.2dB is required;
without diversity on a Rayleigh fading channel,
Es /N0 ≈ 34dB is required!
with 5-fold diversity on a Rayleigh fading channel,
Es /N0 ≈ 5dB is required!
The improved performance stems primarily from the fact
that all L channels are unlikely to be in a deep fade at the
same time.
Performance better than an AWGN channel (without
diversity) is possible because diversity is provided by the
receiver.
multiple receiver antennas exploit the same transmitted
energy - array gain.


Symbol Error Rate with Transmitter Diversity
If diversity is provided by the transmitter, e.g., the signal is
transmitted multiple times, then each transmission can only
use symbol energy Es /L.
0
10

−1
10

−2
10

−3
10
Symbol Error Rate

−4
10

−5
10

−6
10

−7 L=1
10
L=2
L=3
−8 L=4
10
L=5
AWGN
−9
10
0 2 4 6 8 10 12
E /N (dB)
s 0



MATLAB Simulation

Objective: Perform a Monte Carlo simulation of a
narrow-band communication system with diversity and
time-varying multi-path.
Approach: As before, we break the simulation into three
parts
1. System parameters are set with the script ﬁle
NarrowBandSetParameters.
2. The simulation is controlled with the driver script
MCNarrowBandDriver.
3. The actual system simulation is carried out in the function
MCNarrowBand.

All three ﬁles are in the toolbox.



MATLAB Simulation

The simulation begins with the generation of the
transmitted signal.
New facet: a known training sequence is inserted in the
middle of the frame.
InfoSymbols = RandomSymbols( NSymbols, Alphabet, Priors );
% insert training sequence
45 Symbols = [ InfoSymbols(1:TrainLoc) TrainingSeq ...
InfoSymbols(TrainLoc+1:end)];
% linear modulation



MATLAB Simulation
To simulate a diversity system, L received signals are
generated.
Each is sent over a different multi-path channel and
experiences different noise.
Each received signal is matched ﬁltered and the channel
gain is estimated via the training sequence.

% loop over diversity channels
for kk = 1:L
52 % time-varying multi-path channels and additive noise
Received(kk,:) = SimulateCOSTChannel( Signal, ChannelParams, fs);
Received(kk,:) = addNoise( Received(kk,:), NoiseVar );

% digital matched filter, gain estimation
57 MFOut(kk,:) = DMF( Received(kk,:), hh, fsT );
GainEst(kk) = 1/TrainLength * ...
MFOut( kk, TrainLoc+1 : TrainLoc+TrainLength) * TrainingSeq’;



MATLAB Simulation

The ﬁnal processing step is maximum-ratio combining.
The matched ﬁlter outputs are multiplied with the conjugate
complex of the channel gains and added.
Multiplying with the conjugate complex of the channel gains
reverses phase rotations by the channel, and
gives more weight to strong channels.
61
% delete traning, MRC, and slicer
MFOut(:, TrainLoc+1 : TrainLoc+TrainLength) = [ ];
MRC = conj(GainEst)*MFOut;
Decisions = SimpleSlicer( MRC(1:NSymbols), Alphabet, ...



Simulated Symbol Error Rate with Transmitter
Diversity
−1
10

−2
10

−3
10
Symbol Error Rate

−4
10

−5
10

−6
10 Simulated L=2
L=1
L=2
−7 L=3
10
L=4
L=5
−8
AWGN
10
0 2 4 6 8 10 12
E /N (dB)
s 0

Figure: Symbol Error Rate with Diversity over a Rayleigh Fading
2
Channel; σa = 1, simulated system has diversity of order L = 2.



Summary

The strong, detrimental impact of mobile, wireless
channels on the error rate performance of narrow-band
systems was demonstrated.
Narrow-band system do not have inherent diversity and are
subject to flat Rayleigh fading.
To mitigate Rayleigh fading, diversity is required.
Quantified the benefits of diversity.
Illustrated diversity through antenna (spatial) diversity at the
receiver.
A complete system, including time-varying multi-path
channel and diversity was simulated.
Good agreement between theory and simulation.



Outline






Frequency Diversity through Wide-Band Signals
We have seen above that narrow-band systems do not
have built-in diversity.
Narrow-band signals are susceptible to have the entire
signal affected by a deep fade.
In contrast, wide-band signals cover a bandwidth that is
wider than the coherence bandwidth.
Beneﬁt: Only portions of the transmitted signal will be
affected by deep fades (frequency-selective fading).
Disadvantage: Short symbol duration induces ISI; receiver
is more complex.
The beneﬁts, far outweigh the disadvantages and
wide-band signaling is used in most modern wireless
systems.



Illustration: Built-in Diversity of Wide-band Signals
We illustrate that wide-band signals do provide diversity by
means of a simple thought experiments.
Thought experiment:
Recall that in discrete time a multi-path channel can be
modeled by an FIR ﬁlter.
Assume ﬁlter operates at symbol rate Ts .
The delay spread determines the number of taps L.
Our hypothetical system transmits one information symbol
in every L-th symbol period and is silent in between.
At the receiver, each transmission will produce L non-zero
observations.
This is due to multi-path.
Observation from consecutive symbols don’t overlap (no ISI)
Thus, for each symbol we have L independent
observations, i.e., we have L-fold diversity.



Illustration: Built-in Diversity of Wide-band Signals
We will demonstrate shortly that it is not necessary to
leave gaps in the transmissions.
The point was merely to eliminate ISI.
Two insights from the thought experiment:
Wide-band signals provide built-in diversity.
The receiver gets to look at multiple versions of the
transmitted signal.
The order of diversity depends on the ratio of delay spread
and symbol duration.
Equivalently, on the ratio of signal bandwidth and coherence
bandwidth.
We are looking for receivers that both exploit the built-in
diversity and remove ISI.
Such receiver elements are called equalizers.



Equalization

Equalization is obviously a very important and well
researched problem.
Equalizers can be broadly classified into three categories:
1. Linear Equalizers: use an inverse filter to compensate for
the variations in the frequency response.
Simple, but not very effective with deep fades.
2. Decision Feedback Equalizers: attempt to reconstruct ISI
from past symbol decisions.
Simple, but have potential for error propagation.
3. ML Sequence Estimation: find the most likely sequence
of symbols given the received signal.
Most powerful and robust, but computationally complex.



Maximum Likelihood Sequence Estimation

Maximum Likelihood Sequence Estimation provides the
most powerful equalizers.
Unfortunately, the computational complexity grows
exponentially with the ratio of delay spread and symbol
duration.
I.e., with the number of taps in the discrete-time equivalent
FIR channel.




The principle behind MLSE is simple.
Given a received sequence of samples R [n], e.g., matched
ﬁlter outputs, and
a model for the output of the multi-path channel:
r [n] = s [n] ∗ h[n], where
ˆ
s [n] denotes the symbol sequence, and
h[n] denotes the discrete-time channel impulse response,
i.e., the channel taps.
Find the sequence of information symbol s [n] that
minimizes
N
D2 = ∑ |r [n] − s[n] ∗ h[n]|2 .
n



The criterion
N
D2 = ∑ |r [n] − s[n] ∗ h[n]|2 .
n

performs diversity combining (via s [n] ∗ h[n]), and
removes ISI.
The minimization of the above metric is difﬁcult because it
is a discrete optimization problem.
The symbols s [n] are from a discrete alphabet.
A computationally efﬁcient algorithm exists to solve the
minimization problem:
The Viterbi Algorithm.
The toolbox contains an implementation of the Viterbi
Algorithm in function va.


MATLAB Simulation

A Monte Carlo simulation of a wide-band signal with an
equalizer is conducted
to illustrate that diversity gains are possible, and
to measure the symbol error rate.
As before, the Monte Carlo simulation is broken into
set simulation parameter (script VASetParameters),
simulation control (script MCVADriver), and
system simulation (function MCVA).



MATLAB Simulation: System Parameters

Listing : VASetParameters.m
Parameters.T = 1/1e6; % symbol period
Parameters.fsT = 8; % samples per symbol
Parameters.Es = 1; % normalize received symbol energy to 1
Parameters.EsOverN0 = 6; % Signal-to-noise ratio (Es/N0)
13 Parameters.Alphabet = [1 -1]; % BPSK
Parameters.NSymbols = 500; % number of Symbols per frame

Parameters.TrainLoc = floor(Parameters.NSymbols/2); % location of t
Parameters.TrainLength = 40;
18 Parameters.TrainingSeq = RandomSymbols( Parameters.TrainLength, ...
Parameters.Alphabet, [0.5 0.5]

% channel
Parameters.ChannelParams = tux(); % channel model
23 Parameters.fd = 3; % Doppler
Parameters.L = 6; % channel order



MATLAB Simulation
The ﬁrst step in the system simulation is the simulation of
the transmitter functionality.
This is identical to the narrow-band case, except that the
baud rate is 1 MHz and 500 symbols are transmitted per
frame.
There are 40 training symbols.

Listing : MCVA.m
41 % transmitter and channel via toolbox functions
InfoSymbols = RandomSymbols( NSymbols, Alphabet, Priors );
% insert training sequence
Symbols = [ InfoSymbols(1:TrainLoc) TrainingSeq ...
InfoSymbols(TrainLoc+1:end)];
46 % linear modulation
Signal = A * LinearModulation( Symbols, hh, fsT );



MATLAB Simulation

The channel is simulated without spatial diversity.
To focus on the frequency diversity gained by wide-band
signaling.
The channel simulation invokes the time-varying multi-path
simulator and the AWGN function.
% time-varying multi-path channels and additive noise
Received = SimulateCOSTChannel( Signal, ChannelParams, fs);
51 Received = addNoise( Received, NoiseVar );



MATLAB Simulation
The receiver proceeds as follows:
Digital matched ﬁltering with the pulse shape; followed by
down-sampling to 2 samples per symbol.
Estimation of the coefﬁcients of the FIR channel model.
Equalization with the Viterbi algorithm; followed by removal
of the training sequence.

% is long enough so that VA below produces the right number of symbols
MFOut = zeros( 1, 2*length(Symbols)+L-1 );
Temp = DMF( Received, hh, fsT/2 );
57 MFOut( 1:length(Temp) ) = Temp;

% channel estimation
MFOutTraining = MFOut( 2*TrainLoc+1 : 2*(TrainLoc+TrainLength) );
ChannelEst = EstChannel( MFOutTraining, TrainingSeq, L, 2);
62
% VA over MFOut using ChannelEst



Channel Estimation
Channel Estimate:

h = (S S)−1 · S r,
ˆ

where
S is a Toeplitz matrix constructed from the training
sequence, and
r is the corresponding received signal.

TrainingSPS = zeros(1, length(Received) );
14 TrainingSPS(1:SpS:end) = Training;

% make into a Toepliz matrix, such that T*h is convolution
TrainMatrix = toeplitz( TrainingSPS, [Training(1) zeros(1, Order-1)]);

19 ChannelEst = Received * conj( TrainMatrix) * ...
inv(TrainMatrix’ * TrainMatrix);



Simulated Symbol Error Rate with MLSE Equalizer
−1
10

−2
10

−3
Symbol Error Rate 10

−4
10

−5
10

−6
10 Simulated VA
L=1
L=2
−7 L=3
10
L=4
L=5
−8
AWGN
10
0 2 4 6 8 10 12
E /N (dB)
s 0

Figure: Symbol Error Rate with Viterbi Equalizer over Multi-path
Fading Channel; Rayleigh channels with transmitter diversity shown
for comparison. Baud rate 1MHz, Delay spread ≈ 2µs.



Conclusions
The simulation indicates that the wide-band system with
equalizer achieves a diversity gain similar to a system with
transmitter diversity of order 2.
The ratio of delay spread to symbol rate is 2.
comparison to systems with transmitter diversity is
appropriate as the total average power in the channel taps
is normalized to 1.
Performance at very low SNR suffers, probably, from
inaccurate estimates.
Higher gains can be achieved by increasing bandwidth.
This incurs more complexity in the equalizer, and
potential problems due to a larger number of channel
coefﬁcients to be estimated.
Alternatively, this technique can be combined with
additional diversity techniques (e.g., spatial diversity).


More Ways to Create Diversity

A quick look at three additional ways to create and exploit
diversity.
1. Time diversity.
2. Frequency Diversity through OFDM.
3. Multi-antenna systems (MIMO)



Time Diversity
Time diversity: is created by sending information multiple
times in different frames.
This is often done through coding and interleaving.
This technique relies on the channel to change sufﬁciently
between transmissions.
The channel’s coherence time should be much smaller than
the time between transmissions.
If this condition cannot be met (e.g., for slow-moving
mobiles), frequency hopping can be used to ensure that the
channel changes sufﬁciently.
The diversity gain is (at most) equal to the number of
time-slots used for repeating information.
Time diversity can be easily combined with frequency
diversity as discussed above.
The combined diversity gain is the product of the individual
diversity gains.


OFDM

OFDM has received a lot of interest recently.
OFDM can elegantly combine the beneﬁts of narrow-band
signals and wide-band signals.
Like for narrow-band signaling, an equalizer is not required;
merely the gain for each subcarier is needed.
Very low-complexity receivers.
OFDM signals are inherently wide-band; frequency
diversity is easily achieved by repeating information (really
coding and interleaving) on widely separated subcarriers.
Bandwidth is not limited by complexity of equalizer;
High signal bandwidth to coherence bandwidth is possible;
high diversity.



MIMO
We have already seen that multiple antennas at the
receiver can provide both diversity and array gain.
The diversity gain ensures that the likelihood that there is
no good channel from transmitter to receiver is small.
The array gain exploits the beneﬁts from observing the
transmitted energy multiple times.
If the system is equipped with multiple transmitter
antennas, then the number of channels equals the product
of the number of antennas.
Very high diversity.
Recently, it has been found that multiple streams can be
transmitted in parallel to achieve high data rates.
Multiplexing gain
The combination of multi-antenna techniques and OFDM
appears particularly promising.


Summary
A close look at the detrimental effect of typical wireless
channels.
Narrow-band signals without diversity suffer poor
performance (Rayleigh fading).
Simulated narrow-band system.
To remedy this problem, diversity is required.
Analyzed systems with antenna diversity at the receiver.
Veriﬁed analysis through simulation.
Frequency diversity and equalization.
Introduced MLSE and the Viterbi algorithm for equalizing
wide-band signals in multi-path channels.
Simulated system and veriﬁed diversity.
A brief look at other diversity techniques.


Simulation of Wireless Communication Systems

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