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

%% symbols and Signal using our functions
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


Simulation of Wireless Communication Systems

Simulation of Wireless Communication Systems

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