SIM UNIVERSITY
SCHOOL OF SCIENCE AND TECHNOLOGY
APPLICATION OF ZERO-FORCING
EQUALIZER IN DIGITAL
COMMUNICATIONS SYSTEM
STUDENT
: ZAW HTET AUNG (J0704960)
SUPERVISOR
: DR LU LIRU
PROJECT CODE : JAN2010/ENG/0061
A project report submitted to SIM University
in partial fulfilment of the requirements for the degree of
Bachelor of Engineering in Electronics
November 2010
Application of Zero-forcing equalizer
in digital communications system
Zaw Htet Aung (J0704960)
ABSTRACT
Equalization is a technique to compensate for the effect of the channel which causes
distortion in transmitted signal. Different kinds of equalizers are used depending upon the
application of the system and upon the kind of communication channel. These
applications range from acoustic echo cancellers to video de-ghosting systems. The
purpose of an equalization system is to compensate for transmission-channel distortion
such as a signal affected as frequency-dependent phase or as amplitude attenuation.
Besides correcting for channel frequency-response anomalies, the equalizer can cancel
the effects of multipath signal components. They may require significantly longer filter
spans than simple spectral equalizers, but the principles of operation are essentially the
same.
The literature in current project is mainly concerned in equalization of the
transmitted signal which has been distorted due to ISI (inter-symbol interference). This
project aims at studying and simulation of ZF (zero-forcing) equalization technique. The
capability of ZF equalizer in handling the ISI effect in Rayleigh channel environment for
QPSK signal has been discussed through practical approach. Simulation results shows
that ZF equalization can greatly improve the bit error rate of a system as compared to
system without using the same equalizer. The effect has been observed at different effects
of ISI.
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Application of Zero-forcing equalizer
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ACKNOWLEDGEMENT
Apart from the efforts of me, the success of this project depends largely on the
encouragement and guidelines of others. I take this opportunity to express my gratitude to
the people who have been instrumental in the successful completion of this project.
I would like to express my sincere gratitude to my project supervisor, Dr Lu Liru
for her excellent guidance, invaluable suggestions and enthusiastic encouragements along
the way from the beginning till end of this project. Without her encouragements and
guidance, this project would not have materialized.
Finally, yet importantly, I would like to express my heartfelt thanks to my
beloved parents for their blessings, my friends for their help and wishes for the successful
completion of this project.
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LIST OF CONTENTS
ABSTRACT ................................................................................................................................................. I
ACKNOWLEDGMENT ................................................................................................................................ II
TABLE OF CONTENTS ............................................................................................................................... III
TABLE OF FIGURES ................................................................................................................................... V
PART - I .................................................................................................................................................... 1
1 INTRODUCTION ................................................................................................................................ - 1 1.1
1.2
1.3
PROJECT REPORT ORGANIZATION ..................................................................................................- 1 OBJECTIVES ...................................................................................................................................- 2 PROJECT SCOPE..............................................................................................................................- 2 -
2 LITERATURE REVIEW ........................................................................................................................ - 3 2.1 FADING CHANNEL MODELS ...........................................................................................................- 3 2.1.1
Fading Channels.................................................................................................................. - 3 Fast fading channel ........................................................................................................................... - 4 Slow Fading channel ......................................................................................................................... - 4 Frequency-Selective Fading Channel ................................................................................................. - 5 Flat-Fading Channel .......................................................................................................................... - 5 2.1.2
Fading Models .................................................................................................................... - 5 Rayleigh Fading Model...................................................................................................................... - 5 Rician Fading Model.......................................................................................................................... - 7 2.2 EQUALIZATION ..............................................................................................................................- 9 2.2.1
Linear Equalization ............................................................................................................. - 9 2.2.2
Principle of ISI ................................................................................................................... - 10 2.2.3
FUNCTIONS OF A LINEAR EQUALIZER ................................................................................................- 11 2.2.4
Decision Feedback Equalization ........................................................................................ - 11 2.2.5
Adaptive Equalization ....................................................................................................... - 13 3 TECHNIQUES FOR ELIMINATION OF ISI ............................................................................................ - 14 3.1 INTER SYMBOL INTERFERENCE (ISI)......................................................................................................- 14 3.2 MITIGATING ISI USING OFDM ...........................................................................................................- 15 3.2.1
Orthogonality of Sub-Channel Carriers ............................................................................. - 17 3.2.2
Serial to Parallel Conversion ............................................................................................. - 17 3.2.3
Modulation with the Inverse FFT ...................................................................................... - 18 3.2.4
Cyclic Prefix Insertion ........................................................................................................ - 18 3.2.5
Parallel to Serial Conversion ............................................................................................. - 18 3.2.6
OFDM MODELING ............................................................................................................. - 18 3.3 ZERO-FORCING EQUALIZATION FOR ELIMINATING ISI EFFECT........................................................................ 21
4 IMPLEMENTING ZERO-FORCING EQUALIZATION ............................................................................ - 22 4.1 ISI EFFECT AND EQUALIZATION............................................................................................................- 23 4.1.1
Eye Diagram in context to ISI ............................................................................................ - 26 4.1.2
Raised Cosine in context to ISI .......................................................................................... - 29 -
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4.3
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ZERO FORCING ALGORITHM ...............................................................................................................- 32 ADVANTAGE AND DISADVANTAGE OF ZERO-FORCING EQUALIZATIION ............................................................. 34
5 SIMULATIONS AND CONCLUSIONS .................................................................................................. - 35 5.1
5.1
5.3
EFFECT OF ISI ON THE RECEIVED SIGNAL ................................................................................................- 38 EFFECTS OF FILTER AND WAVEFORM ANALYSIS ........................................................................................... 43
CONCLUSION AND FUTURE WORK .......................................................................................................- 46 -
PART-II ............................................................................................................................................... - 47 CRITICAL REVIEW AND REFLECTIONS ................................................................................................. - 47 REFERENCES ........................................................................................................................................... 49
APPENDIX .............................................................................................................................................. 51
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LIST OF FIGURES
FIGURE 2-1: RAYLEIGH DISTRIBUTED PDF GRAPH ............................................................... 7
FIGURE 2-2: RICIAN DISTRIBUTED PDF GRAPH .................................................................... 8
FIGURE 2-3: THE PRESENCE OF ADDITIVE NOISE IN LINEAR EQUALIZATION OF AN FIR
CHANNEL ................................................................................................................... 10
FIGURE 2-4: ACK FILTER AND A DECISION DEVICE IN
A DECISION FEEDBACK EQUALIZER ............................................................................ 12
FIGURE 3-1: ISI REPRESENTATIONS IN TIME DOMAIN (COURTESY OF NATIONAL
INSTRUMENTS [6]) ..................................................................................................... 16
FIGURE 3-2: REPRESENTATION OF REDUCED ISI WITH LOW SYMBOL RATE (OFDM)
(COURTESY OF NATIONAL INSTRUMENTS [6]) ............................................................... 17
FIGURE 3-3: FOURIER REPRESENTATION OF OFDM SYSTEM (COURTESY OF NATIONAL
INSTRUMENTS [6]) ..................................................................................................... 17
FIGURE 3-4: BLOCK DIAGRAM OF AN OFDM TRANSMITTER ............................................. 18
FIGURE 3-5: OFDM TRANSMISSION SYSTEM...................................................................... 20
FIGURE 4-1: EYE DIAGRAM OF A QPSK SIGNAL WITH NO ISI............................................. 25
FIGURE 4-2: EYE DIAGRAM OF A QPSK SIGNAL WITH ISI .................................................. 26
FIGURE 4-3: CASE 1 NON-OVERLAPPING SPECTRUM ........................................................... 27
FIGURE 4-4: CASE 1 OVERLAPPING SPECTRUM ................................................................... 27
FIGURE 4-5: RAISED-COSINE SPECTRUM ............................................................................ 30
FIGURE 4-6: TIME DOMAIN FUNCTION OF THE RAISED-COSINE SPECTRUM ......................... 31
FIGURE 5-1: BER COMPARISON WITHOUT ZF EQUALIZER UNDER THE EFFECT OF ISI ........ 36
FIGURE 5-2: BER COMPARISON USING ZF EQUALIZER UNDER THE EFFECT OF ISI ............ 37
FIGURE 5-3: BER COMPARISON WITH AND WITHOUT ZF EQUALIZER UNDER ISI .............. 38
FIGURE 5-4: BER COMPARISON BY SETTING THE ISI CO-EFFICIENT FREQ CUT=50 ............. 39
FIGURE 5-5: BER COMPARISON BY SETTING THE ISI CO-EFFICIENT FREQ CUT=100 .......... 40
FIGURE 5-6: BER COMPARISON BY SETTING THE ISI CO-EFFICIENT FREQ CUT=200 .......... 41
FIGURE 5-7: BER COMPARISON BY SETTING THE ISI CO-EFFICIENT FREQ CUT=600 .......... 42
FIGURE 5-8: FILTER RESPONSE IN FREQUENCY DOMAIN ..................................................... 43
FIGURE 5-9A: COMPARISON OF TRANSMITTED SYMBOLS BEFORE FILTER AND RECEIVED
DEMODULATED SYMBOLS AFTER FILTER AND NO ZERO-FORCING EQUALIZER UNDER
SMALL ISI EFFECT ...................................................................................................... 44
FIGURE 5-9B: COMPARISON OF TRANSMITTED SYMBOLS BEFORE FILTER AND RECEIVED
DEMODULATED SYMBOLS AFTER FILTER AND ZERO-FORCING EQUALIZER UNDER
SMALL ISI EFFECT ...................................................................................................... 44
FIGURE 5-10A: COMPARISON OF TRANSMITTED SYMBOLS BEFORE FILTER AND RECEIVED
DEMODULATED SYMBOLS AFTER FILTER AND NO ZERO-FORCING EQUALIZER UNDER
HIGH ISI EFFECT ......................................................................................................... 45
FIGURE 5-10B: COMPARISON OF TRANSMITTED SYMBOLS BEFORE FILTER AND RECEIVED
DEMODULATED SYMBOLS AFTER FILTER AND ZERO-FORCING EQUALIZER UNDER HIGH
ISI EFFECT ........................................................................................................................................ 46
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Application of Zero-forcing equalizer
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PART - I
CHAPTER 1
1 INTRODUCTION
In the design of large and complex digital systems, it is often necessary to have one
device communicate digital information to and from other devices. One advantage of
digital information is that it tends to be far more resistant to transmitted and interpreted
errors than information symbolized in an analog medium. This accounts for the clarity of
digitally-encoded telephone connections, compact audio disks, and for much of the
enthusiasm in the engineering community for digital communications technology.
However, digital communication has its own unique pitfalls, and there are multitudes of
different and incompatible ways in which it can be sent. The three main parts of the
telecommunication system are transmitter, receiver and the channel. Our main focus will
be imparted to the receiver in this section called equalizer. The equalizer is used to
estimate the transmitted bits/symbols in such a way that it eliminates the effect of
channel. The zero-forcing equalizer will be used in this context. The main objective of
zero-forcing equalizer is to eliminate the effect of ISI (inter-symbol interference). We
will discuss ISI in more dept in forth-coming chapters. Zero-forcing unlike MMSE is
useful in mitigating the ISI effect rather than induced noise in the signal. Comparison of
using zero-forcing algorithm as equalizer will be done against using a simple algorithm
for equalization.
1.1 Project Report Organization
The report has been organized into a number of sections. Chapter 2 deals with mitigating
ISI using different techniques of digital communication. Different channel estimation
techniques that are based on pilot arrangement were investigated in [2]. The performance
behavior of BER using optimized pilot symbols are compared with equal power case of
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pilot and information symbols using Rayleigh-faded channel environment [5]. Chapter 2
deals with the different possible channel effects and the compensation techniques like
equalization and possible equalization techniques which is responsible for mitigating the
channel effect. The effect of channel in our project case causes ISI, that’s why the chapter
3 is introduced which deals with all the possible techniques used for elimination of ISI.
Chapter 4 deals with the channel estimation using zero-forcing and effect of Rayleigh
channel upon the signal. In short, chapter 4 shows how the Zero-forcing is useful as
compared to traditional technique of using equalization. Chapter 5 deals with the
simulated results and observations of this project.
1.2 Objectives
The main objective of this project is to study the ZF equalization technique. By studying
equalization techniques in Rayleigh fading channel, we can understand the real world
communication channels in MATLAB scenario. In this report, we will get a deep insight
into the ZF equalization technique and will simulate this technique in Rayleigh fading
environment under the digital modulation technique of QPSK. By comparing the bit error
rate of communication system using ZF equalization and without equalization, the
system’s performance will be discussed.
1.3 Project Scope
The project shall include the following main tasks.
a. Studying different fading channel models
b. Modeling Rayleigh fading channel
c. Studying different equalization techniques
d. Studying ZF equalization technique
e. BER comparison of ZF equalized system with non-equalized system.
f. BER effect upon non-ZF equalized system for different ISI effects.
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Application of Zero-forcing equalizer
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CHAPTER 2
2 LITERATURE REVIEW
The literature of this project is focused on two main subjects. The first main thing is the
fading channel effect upon the transmitted signal. The second main subject is the
compensation effect for the above mentioned channel-effect called the equalization.
Besides the channel effect the main disorder is the ISI (inter-symbol interference), which
will be explained in the next chapter. Let us now explain these subjects one-by-one in the
following text:
2.1
Fading channel models
Majority of wireless systems exhibit the mobility characteristics. To take the mobility in
consideration for the sake of design is very much important. When signal traverse from
transmitter to the receiver, the wireless channel experience fluctuations in time randomly.
These random fluctuations are caused by the transmitter, receiver or due to motion of the
surrounding objects. Due to these fluctuations, the design of a reliable and stable system
is difficult to be achieved.
The time-varying behavior of the channel limits the performance of a wireless
communication channel system. That’s why the wireless communication channel is
designed differently than a wired communication channel. The complexity in the wireless
and mobile communication channel increases due to motion of non-stationary objects.
This is due to the complex behavior of the channel, that we take the fading channel
models in case of mobile and wireless communication system.
2.1.1 Fading Channels
Fading is actually the fluctuation in transmitted signal upon its phase, amplitude and
multi-path delays, properties of the signal over a short duration of time interval. In other
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Application of Zero-forcing equalizer
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words, fading is caused by interference of multiple versions of a same signal when the
signal arrives to the receiver after reflecting from multiple obstacles. The phase and
amplitude of these versions of the same signal combine constructively or destructively to
cause changes in amplitude and phase of the composite signal.
The fading channels are thus classified into following types and subtypes:
1) Based upon Doppler spread of the channel

Fast fading channel

Slow fading channel
2) Based upon multi-path delay of the channel

Frequency-selective fading channel

Flat fading channel
These fading models will be explained one by one in the following text:
Fast fading channel
Fast fading occurs when the delay constraint of a fading channel is relatively large than
the coherence time of the channel. The coherence time of a channel is the measure of its
Doppler spread, where Doppler spread is referred to as the spread due to Doppler shift of
a moving object. The phase and amplitude of the transmitted signal varies significantly in
the allowed period.
Slow Fading channel
Slow fading occurs when the delay constraint of a fading channel is relatively small than
the coherence time of the channel. The phase and amplitude of the transmitted signal is
considered approximately constant in the allowed period. One of the important reasons
behind the slow fading occurrence is the effect of shadowing. The shadowing occurs
when a large obstacle in between transmitter and receiver blocks the main path. The
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Application of Zero-forcing equalizer
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amplitude change due to shadowing is normally modeled by using a log-distance path
loss model.
Frequency-Selective Fading Channel
Frequency-selective fading occurs, when the signal band-width is larger than the
coherence bandwidth of the signal. These channels are dispersive in nature because the
signal energy of each symbol is spread-out in time. Due to this reason, the adjacent
transmitted symbols interfere with each-other, which are termed shortly as ISI, which
stands for inter-symbol interference. For compensating ISI, equalizers are deployed at the
receiving end. Frequency-Selective fading channel is one which is incurred in our project
case. Hence understanding the concept of frequency-selective is focused in the forthcoming chapters. In our project case, we have taken the channel effect to be the
frequency-selective fading.
Flat-Fading Channel
Flat-fading occurs, when the signal band-width is larger than the coherence bandwidth of
the signal. These channels are non-dispersive in nature. Usually these channels are not
affected by ISI.
2.1.2 Fading Models
There are many fading models that are used for the distribution of attenuation. Two of the
important models include Rayleigh fading model and Rician fading model. These models
will be explained in more detail in the following sub sections.
Rayleigh Fading Model
The Rayleigh Fading model is taken as the only channel model in this project. The
Rayleigh fading channel refers to the fact that any obstruction may ban the direct wave
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and the receiver only receives the reflected waves. Let us consider a transmitted wave
with the center frequency fc. The wave may reaches to the receiver with m different
channel paths. The received signal consists of sum of m different components in addition
with Gaussian distributed noise as given:
m
r(t)  ∑ai cos( 2f c t  i )  z i ( t)
(2.1)
i 1
where
ai : Amplitude of the transmitted signal
f c : Center frequency of the transmitted signal
φi : Respective phase shift that is incurred by the respective channel path
zi (t) : Gaussian distributed noise
From above discussion, it is obvious that Rayleigh fading is caused due to the multipath reception of the signals. The mobile receiver tends to receive a large amount of
scattered and reflected signals. As due to the wave cancellation, the moving received
antenna see the instantaneous power to be a random variable, which depends upon the
location of the received antenna.
In the equation 2.1, if fc=0 (i.e. by considering the stationary receiving station), then
it can easily be stated that the received signal r (t) becomes
m
r(t) = ∑ai cos(φ i ) + z i (t)
(2.2)
i =1
This converts the distribution to Gaussian one. The Rayleigh distribution corresponds
to the probability density function (PDF) of
p(~
r) =
~
r
~
r exp(- 2 )
2σ s
2
σs
~
r ≥0
(2.3)
r represent the envelope of the received signal r (t). Where the normalized value
And ~
of the PDF in equation 2.3 is achieved by using σ s2 =1. The normalized plot of Rayleigh
distribution is shown in figure 2.1
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Rayleigh distribution
0.7
0.6
0.5
PDF
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Random variable
6
7
8
Figure 2-1: Rayleigh distributed PDF graph
Rician Fading Model
When there is at-least one direct path between the transmitter and receiver system, then
the received signal follows the random characteristics of the Rician fading model. The
examples of Rician fading in wireless communication are that of a satellite and cellular
mobile communication channels. Rician fading model case arises when there is no
obstacle in the direct path between the transmitter and receiver. Such a path is referred to
as the line-of-sight (LOS). The received signal amplitude in case of the LOS wave is
constant with no fading at all. All the reflected waves are i.i.d random signals. These
reflected waves are called the scattered component. If we can see the reflected
components, we will realize that they follow the Gaussian random process. Such a
Gaussian process has a mean of zero with unity variance, where the envelope follows the
Rayleigh distribution.
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The probability density function (pdf) of the rician distributed signal can be written
as
~r 2  A 2
~r A 
 ~
) I 0 ( 2 ) 
 r exp(2 s2
 ~

~
p( r )  
 r ≥0
2
s




(2.4)
The terms in equation 2.4 can be defined as
A: Amplitude of dominant signal in direct path
I0 (.): Zero order Bessel function
σ 2 : Signal Power
K can be considered as the Rician factor and its relation is K 
A2
2 s2
The Rician distribution for different values of K-factor is given in figure 2-2.
Rician distribution
0.5
k 0.5
k 2
k 4
0.45
0.4
0.35
PDF
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
Random variable
Figure 2-2: Rician distributed PDF graph
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7
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Equalization
There are a number of procedures presented to counter the outcome of Multipart
promulgation. Space diversity, frequency diversity, channel equalization and amplitude
equalization are the most commonly used. Among these, the Space diversity and
frequency diversity need a bandwidth overhead, which is not eagerly obtainable in nearly
all common classification. In analogue broadcasting systems, these signal diversity
procedures were used and have been modified to digital systems without any difficulty
that go through extremely discerning interference. The amplitude equalizers are intended
to level the acknowledged spectrum to endorse the spectral form. An amplitude equalizer
is frequently used in combination with space diversity or frequency, which can endow
with enough equalization for unambiguous channels. Though, to effectively illustrate the
possessions of all channel forms, the channel equalizer is accepted, whether it is in
minimum and non-minimum phase.
The degraded data sequences are anticipated or restructured by the channel
equalizer from a set of acknowledged symbols. To advance the symbol error rates
equalizers have been implemented in telephone and mobile communication structures and
inside equalization the linear FIR filters are used. In past loading coils were used instead
to advance the voice communication in telephone structures. Magnetic hard disk storage
and optical recording are the equalization’s new applications which have take place that
operate to regain data when adjoining signals obstruct. The FIR approach categorizes the
acknowledged class sets to their preferred productivity by means of a linear function of
the filter inputs.
2.2.1 Linear Equalization
A renowned receiver system for mitigating inter-symbol interference (ISI) is the linear
equalization. Least squares error cost function or mean square error cost functions are
usually minimized for the computation of linear equalizers.
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Figure
2-3: The
presence
of additive
Figure -3igure
2-3:
The presence
of noise in Linear Equalization of an FIR channel
We can calculate the zero-forcing equalizer, if we identify the noise first-order and
second-order statistics and channel impulse response and the input. A supposition that
will be used all over this effort is that the input is self-sufficient and identically
disseminated, among unit variance and zero-mean and the noise is white Gaussian
therefore the recipient can utilize this information. Basically at all times, the noise
variance and the channel impulse response is anonymous at the recipient side. In the
cases where the noise second order statistics and the channel impulse response are
unrevealed at the receiver, is to presume those using training data and, then, use the
guesstimate as if they were the right amount are an ordinary loom headed for the sketch
of the ZF equalizer.
2.2.2 Principle of ISI
If the channel response (or the channel transfer function) for a specific channel is H(s)
then the input signal is multiplied by the reciprocal of this. This is intended to remove the
effect of channel from the received signal, in particular the ISI.
The zero-forcing equalizer removes all ISI, and is ideal when the channel is noiseless.
However, when the channel is noisy, the zero-forcing equalizer will amplify the noise
greatly at frequencies f where the channel response H(j2πf) has a small magnitude (i.e.
near zeroes of the channel) in the attempt to invert the channel completely. A more
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balanced linear equalizer in this case is the minimum mean-square error equalizer, which
does not usually eliminate ISI completely but instead minimizes the total power of the
noise and ISI components in the output.
2.2.3 Functions of a linear equalizer
Linear equalizers are normally useful on channels with a comparatively plane frequency
response, where the ISI is not harsh. Though, when there are null in the frequency range
of the acknowledged signal, a linear equalizer execute badly, in its effort to reverse the
channel frequency response, a linear filter extensively increases the noise at the locality
of the void (Null). The deprived presentation of linear equalizers in channels with harsh
ISI limits their utilization in wireless channels, which frequently have spectral nulls.
Nonlinear equalizers suggest considerable performance development in these cases.
Alternatively, we exercise the training facts & figures, and straightforwardly
calculate at the recipient the LS optimal equalizer with no mediator calculation of the
channel impulse feedback and the noise second-order figures, in the DLSE (direct least
square equalization) approach. We note down that LS equalizer cannot utilize the
information concerning the input second-order figures that is obtainable at the recipient
side.
2.2.4 Decision Feedback Equalization
As we already have conferred it before that the principle of an equalizer is to fight ISI
and to recuperate recent spread symbol from the measurements of the entered string. For
achieving this assignment, a linear equalizer utilizes existing and preceding
measurements. This is for the reason that preceding measurements is full of information
that is interrelated with the ISI phrase in existing acknowledged symbol and as a result,
they can assist in guesstimate the intrusion term and eliminate its end product. Certainly
if achievable, it would be favorable to use the previous symbols themselves with the
intention of cancelling their end product from existing symbol.
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Decision feedback equalizers (DFE) try to execute this approach and are for that
reason better matched for channels with well-defined ISI. In adding up to use an FIR
filter in the feed-forward path, like in linear equalization, a DFE utilizes a reaction filter
with the intention of feed back preceding conclusion and utilize them to minimize ISI.
The DFE formation is exposed in Fig 3.1, for guesstimate a belated edition of s (i ) , with
the shift purpose of the feed-forward and feedback filter represented by {F(z), B(z)},
correspondingly. It is observed from the figure that the input to the feedback filter

approaches from the production of the decision device, symbolized by s(i  ) . The

objective of this device is to plot the estimator s(i  ) , which is achievable by joining the
productivity of the feed-forward filter and feedback filter, to the contiguous position in
the symbol collection. Now in linear equalization, the feed-forward filter decreases ISI by
endeavoring to strength the collective scheme C ( z ) F ( z ) to be nearby to C ( z ) F ( z )  z   .
In most cases this objective is difficult to achieve, particularly for channels with
prominent ISI, and C ( z ) F ( z ) will have a significant impulse response series. The
function of the decision feedback filter in a DFE implementation is to utilize preceding
conclusion with the aim of cancelling this irregular ISI.
Figure 2-4: A feed-forward filter, a feedback filter and a decision device in a
decision feedback equalizer
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2.2.5 Adaptive Equalization
Mobile fading channels frequently change their conduct with time so they are unreliable.
To attain minimum BER at the recipient side the equalizer must continually follow these
alterations and regulate the equalizer taps consequently. In plain terminology the
equalizer have to be adaptive.
The universal working approaches of an adaptive equalizer consist of equalizer
channel tracking and equalizer training. Equalizer training is completed by transporting
identified bits and in view of that the equalizer taps are set and the channel is followed by
using recognized bits. The transmitter launched the predetermined length training
sequence; the receiver regulates equalizer coefficients properly in anticipation of the
characters in the training sequence are acknowledged with no ISI and this is prior to the
recognition of the training sequence by the receiver. When the training is completed the
equalizer coefficients stayed in the most favorable setting throughout data transmission.
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CHAPTER 3
3 TECHNIQUES FOR ELIMINATION OF ISI
This chapter deals with the effect of ISI (inter-symbol interference) and the possible
mitigating techniques for its elimination. Mathematical model of an OFDM system has
also been explained in this section to help understand the concept of ISI and its
elimination. The symbol/bit error probability analysis is introduced which is encountered
in a signal during its travel from transmitter to the receiver.
Digital communication systems admit each channel to operate at a specific
frequency and bandwidth. The reason behind this idea is to place more and more signal
contents in a limited bandwidth channel. In this chapter, concentration has been imparted
on how to utilize the frequency spectrum more effectively by introducing the concept of
OFDM. OFDM is introduced just to make the elimination of ISI effect more
understandable. In very last section of the chapter the equalization technique is
introduced just to introduce the concept of ISI elimination for this project report.
3.1
Inter Symbol Interference (ISI)
Inter-symbol interference (ISI) can be caused due to the multi-path characteristics of a
wireless communication channel in a single carrier system. An electromagnetic signal
passes through different multi-paths when transmitted from a transmitting antenna. This
causes the received signal to be consisted of multiple copies of the same symbol. Figure
2.4 shows the effect of multi-channel effect that causes the replica of different amplitudes
of a symbol.
The above mentioned disorder can cause distortion in the received signal. In the
given scenario, the direct-path signal component arrives at the earliest time, but the
reflected path components are attenuated and arrive a bit late. These reflected path
components are undesired because they interfere with the direct-path component signal.
The reflected signal components are attenuated by using a pulse-shaping filter, and hence
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both the started and ending reflected components are eliminated. At high data-rate, the
ISI effect is much more significant.
3.2
Mitigating ISI Using OFDM
In OFDM, the symbol duration gets longer and hence ISI effect becomes very
un-significant. Besides the mitigation of ISI effect, OFDM can also increase the
throughput of the system.
Figure 3-1: ISI representations in time domain (Courtesy of National Instruments [6])
OFDM is a modification to frequency division multiplexing in which a single channel is
comprised of multiple sub carriers on adjacent frequencies. The sub carriers in OFDM
case are overlapped just for maximizing the spectral efficiency. Obviously the
overlapping adjacent channels can interfere with one another.
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In figure (shown below), the time-domain representation of the OFDM received
signal is shown. In short, the OFDM system provides a reduced ISI with high throughput.
Figure 3-2: Representation of reduced ISI with low symbol rate (OFDM) (Courtesy of National
Instruments [6])
From the above discussion, we note that the time required for the reflected signals
to attenuate is same, whether the system is OFDM or not. The only difference is that, In
OFDM system the small percentage of the whole symbol period is affected from the
reflected signal components.
However, sub-carriers in an OFDM system are orthogonal to each-other so that they
must overlap without interference. As a result, OFDM system maximizes the spectral
efficiency without causing channel interference. The frequency domain of an OFDM
system is represented in the figure (shown below).
Figure 3-3: Fourier representation of OFDM system (Courtesy of National Instruments [6])
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3.2.1 Orthogonality of Sub-Channel Carriers
In OFDM, the frequency spectrum can be efficiently utilized by overlapping the adjacent
sub-carriers. These sub-carriers are partially overlapped without being interfered with the
adjacent sub-carriers. It is due to the reason that maximum power of each sub-carrier
corresponds directly with the minimum power of each adjacent channel.
It is obvious that the OFDM channels are different from band-limited FDM
channels in its way to apply the pulse shaping filter. With FDM systems, a sync-shaped
pulse is applied in the time domain to shape each individual symbol and prevent ISI.
With OFDM systems, a sync-shaped pulse is applied in the frequency domain for each
channel. This causes each sub-carrier to remain orthogonal to its subsequent sub-carrier.
For transmitting different sub-carriers along a single channel, an OFDM
communications system can be used to perform several steps, which is described in figure
(shown below):
Serial to
Parallel
Digital
Modulation
IFFT
+
CP
RF
Amplifier
D/A
converter
fc
Figure
diagram
an OFDM
Transmitter
Figure2-4
3-4: BlockBlock
diagram
of anofOFDM
Transmitter
3.2.2 Serial to Parallel Conversion
In OFDM system, each channel is divided into multiple sub-carriers. The sub-carriers
make optimal use of the frequency spectrum and also require high processing by
transmitter and the respective receiver. Once the data is divided among sub-carriers, each
sub-carrier is then modulated such that they behave as an individual channel. The
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receiver performs reverse process to divide the incoming signal into respective subcarrier and then demodulate them individually before reconstructing the original data.
3.2.3 Modulation with the Inverse FFT
The data modulation into a complex waveform is accomplished at the IFFT stage. The
modulation scheme here is considered independent of the specific channel and it can be
chosen based upon the channel requirements. It is thus possible for each individual subcarrier to have a different modulation mechanism. The purpose of introducing the IFFT
stage here is for modulating each sub-channel onto the specific sub-carrier.
3.2.4 Cyclic Prefix Insertion
As wireless communication system is incurred by multi-path channel reflections, hence
cyclic prefix is used for reducing ISI. In cyclic prefix case the replica of first part of a
symbol is appended to the last. As a result the multi-path components of the signal are
faded in order to reduce its interference with the subsequent symbols.
3.2.5 Parallel to Serial Conversion
Once cyclic prefix has been added to the sub-carrier channels, they must be transmitted
as one signal. Thus, the parallel to serial conversion stage is the process of summing all
sub-carriers and combining them into one signal. As a result, all sub-carriers are
generated perfectly simultaneously.
3.2.6 OFDM Modelling
The general OFDM system is shown in figure 3.5.
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Figure 3-5: OFDM transmission system
The channel under test is the frequency-selective and a set of sub-carriers are modulated
by information and pilot symbols. A single antenna is considered both at transmitter and
receiver. At the receiver end the channel is estimated using pilot symbols. Let’s assume
the channel to be frequency-selective and time-invariant over a block of OFDM symbols.
On receiving end after demodulation, the received signal at nth sub-carrier pilot symbol
can be written as
y[n]  p H (n) s(n)  (n) ,
n  Ip
(3.1)
Where Ip represents the sub-carriers that carry the pilot symbols, p is the
transmitted power per pilot symbol, H (n ) is the frequency response of the channel at the
nth sub-carrier, s(n) is the transmitted pilot symbol (where n  Ip ) and  (n) is the
additive white Gaussian noise (AWGN) with zero mean and variance of No/2. The block
OFDM symbol index in eq 3.1 is omitted.
The received data that corresponds to the information symbols can be explained
as:
y[n]
 s H (n) s(n)  (n) ,
n  Is
(3.2)
Where Is represents the sub-carriers that carry the information symbols, s is the
transmitted power per information symbol. Let us assume that number of sub-carriers are
N, and considering the size of Ip to be | Ip |=P. For our convenience it has been assumed
that | Is | = N-P, where it is also possible to have | Is | ˂ N-P. The null sub-carriers are just
inserted for spectrum shaping. It is also assumed to have information symbols taken from
M-QAM constellations. The frequency-selective channel is considered to be Rayleighfading and the channel impulse-response to be h =[ h (0), h (1),……, h ( L -1)],
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corresponding to a single receive-antenna, where L denotes the number of taps, i.e,
h ( l ), 
l ε [0, L -1], are uncorrelated Gaussian random variables having mean of
zero. It is assumed that the channel have a power delay profile with variance  h2 ( l ) . The
L 1
channel is normalized and hence

2
h
( l ) =1. we define L N matrix
l 0
[ F ] ,n =exp( j 2 (l  1)( n  1) / N ), if fn is the nth column of F, then H (n ) = fnH h, is a
Gaussian random variable with the mean zero and variance of 1. We have the average
SNR per pilot symbol as
p / N 0 , and the average SNR per information symbol to be
s / N 0 . Where the AWGN variable
w(n ) is assumed to be uncorrelated, n .
Let’s assume a set of pilot sub-carriers is given by Ip =  ni  where i ranges from
~
1 through P. We assume h =[ H (n1), H (n2) …….. H (np)]T , which is comprised of
channel frequency response for pilot sub-carriers. As long as definition of frequency
response matrix for pilot positions is concerned we define Fp as Fp = [fn1 , fn2,,……, fnp]T. It
~
is thus obvious about FFT pair that h = FpHh. In addition we can determine the P  1
vector
y=
~
p D( sp)h  w
(3.3)
If we further simplify equation 3.3, we will get the the vector y as
yp
=
p D( sp)FpH h  w
(3.4)
In the above equation (3.4), the vector y can be defined as y
= [ y( n1 ), y( n2 ),..... y( n p )] T , which is a vector of size P  1, and it contains the received
pilot data per block, where the pilot data symbols can be defined as s p =
[ s( n1 ), s( n2 ),.....s( n p )] T and the noise vector w can also be defined as w =
[ w( n1 ), w( n2 ),.....w( n p )] T . Both the vector s p and w are the vectors of size P  1.
The channel estimate can be determined from the equation 3.5 as
ĥ  Gyp =h +
(3.5)
where G  ( p F p D H ( s p ) D( s p ) F pH ) 1 (  p D( s p ) F pH ) H
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Application of Zero-forcing equalizer
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3.3
Zaw Htet Aung (J0704960)
Zero-Forcing Equalization for eliminating ISI effect
Beside OFDM, one can eliminate the effect of ISI using a simple technique at receiving
end. This effect is called zero-forcing equalization. Zero Forcing Equalizer refers to a
linear equalization algorithm type used in communication model to invert the frequency
response of the channel. Details of this method will be explained further in the following
chapter.
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CHAPTER 4
4 IMPLEMENTING ZERO-FORCING EQUALIZATION
This chapter is all about the Zero-forcing equalization for eliminating the ISI (intersymbol-interference effect), which is the main theme of concern in this project.
The Zero-Forcing Equalizer has an inverse channel effect to the received signal, for
restoring the signal before the channel. It is not useful for practical applications.
A channel may have a frequency response F (f) as shown in figure A.
Figure A: Channel frequency response
F ( f ) causes frequency dependent gain and phase rotation. A zero-forcing equalizer
inverts the frequency response, by calculating C (f) =1/F (f) as shown in figure B.
Figure B: Zero forcing equalizer: The inverse
of the channel frequency response
Ideally, the combination of channel and equalizer gives a flat frequency response and
linear phase as shown in Figure C.
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Figure C: Zero forcing equalizer: Inverse of the channel
frequency response
In reality, zero-forcing equalization does not work in most applications, for the following
reasons:
Even though the channel impulse response has finite length, the impulse response
of the equalizer needs to be infinitely long. The channel may have zeroes in its frequency
response that cannot be inverted At some frequencies, may be very small. To
compensate, grows very large. As a consequence, any noise added after the channel gets
boosted by a large factor and destroys the overall signal-to-noise ratio. The third item is
often the most important one.
If the channel response (or the channel transfer function) for a specific channel is
H(s) then the input signal is multiplied by the reciprocal of this. This is intended to
remove the effect of channel from the received signal, in particular the inter-symbol
interference (ISI).
The zero-forcing equalizer removes all ISI, and is ideal when the channel is
noiseless. However, when the channel is noisy, the zero-forcing equalizer will amplify
the noise greatly at frequencies f where the channel response H (j2πf) has a small
magnitude (i.e. near zeroes of the channel) in the attempt to invert the channel
completely. A more balanced linear equalizer in this case is the minimum mean-square
error equalizer, which does not usually eliminate ISI completely but instead minimizes
the total power of the noise and ISI components in the output.
4.1 ISI effect and Equalization
The aim of equalization is to mitigate the error probability. The optimum solution for the
channel equalization method is based on ML (maximum-likelihood) sequence detection
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criterion. The MLSE (maximum likelihood sequence estimation) for a channel with ISI
has a computational complexity that increases exponentially with the size of channel time
dispersion, such a large complexity is prohibitively expensive to implement. Therefore
suboptimum channel equalization approaches are used to compensate for the ISI.
The all-pass assumption made in-case of AWGN channel model is not always
practical. Due to the dispersive nature of the frequency spectrum, we sometime filter out
the transmitted signal toits limited bandwidth so that efficient utilization of the frequency
resource can be achieved. Many practical communication channels are bandpass and
sometime they respond differently to inputs with a different frequency spectrum due to
their dispersive nature. We need to refine the simple AWGN model to accurately
represent such types of practical communication channels. One such commonly
employed refinement is the dispersive channel model (shown below):
r (t )  u  hc (t )  n(t )
(4.1)
where u is the transmitted signal, h(t) is the impulse response of the channel, and
n(t ) n(t) is the AWGN noise with. We have modeled the dispersive characteristic of the
channel by the linear filter hc (t ) . The most common dispersive channel is the
bandlimited-channel for which the channel impulse response hc (t ) is equivalent to an
ideal lowpass filter. Such a lowpass filter smears the transmitted signal time causing the
effect of a symbol to spread along the adjacent symbols after a sequence of symbols is
transmitted. The resulting interference, called intersymbol interference, affects the
performance of the whole communication system. There are two other ways to decrease
the unwanted effects of ISI. The first method in this context is to design bandlimited
transmission pulses which mitigates the effect of ISI. In our project case, We have
described such a design for a simple case of bandlimited channels. The non-ISI pulses
obtained in this case are called the Nyquist pulses. The second method to filter the
received signal to cancel the ISI introduced is by the channel impulse response. This
approach is generally known as equalization.
To understand what ISI is, let us consider the transmission of a sequence of
symbols with the basic waveform u (t ) . To send the n th symbol bn , we send bnu (t  nT ) ,
where T is the symbol interval. Therefore, the transmitted signal is
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 b u(t  nT )
(4.2)
n
n
Based on the dispersive channel model, the received signal is given by
r (t )   bn v(t  nT )  n(t )
(4.3)
n
where v(t )  u  hc (t ) is the received waveform for the symbol. If a single symbol b0 is
transmitted, the optimal demodulator is the one that employs the matched filter, i.e., we
can pass the received signal through the matched filter v~ (t )  v(t ) and then sample the
MF (matched filter) output at time t  0 to obtain the decision statistic. When a sequence
of symbols is transmitted, we can still employ this matched filter to perform
demodulation. A reasonable strategy is to sample the matched filter output at time
t  mT to obtain the decision statistic for the symbol bm . At t  mT , the matched filter
output is
z m   bn v  v~ (mT  nT )  nm
n
(4.4)
2
 bm v   bn v  v~ (mT  nT )  nm
nm
where nm is the zero mean Gaussian variable with variance N 0
v
2
2
. The first term in
eqn. 4.4 is the desired signal contribution duo to the symbol bm and the second term
contains contribution from the other symbols. These unwanted contributions from other
symbols are called ISI (inter-symbol interference).
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Figure 4-1: Eye diagram of a QPSK signal with no ISI
4.1.1 Eye Diagram in context to ISI
Suppose v (t ) is considered to be time-limited, i.e. v(t )  0 except for 0  t  T . Then one
can easily say that v  v~ (t )  0 except for  T  t  T . Therefore, v  v~ (mT  nT )  0 for
all n  m and there is no ISI. As a result, the demodulation strategy above can be
interpreted as matched filtering for each symbol. Unfortunately, a timelimited waveform
is never bandlimited. Therefore, for a band-limited channel, v (t ) and, hence v  v~ (t ) are
not time-limited and thus inter-symbol interference is, in general, present. One way to
observe and measure the effects of ISI is to study the eye diagram of the received symbol.
The effects of ISI and other noises can be observed on an oscilloscope displaying the
output of the MF (matched filter) on the vertical input with the horizontal sweep rate set
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at multiples of 1 . Such a display is called eye diagram. For illustration, let us consider
T
the basic waveform u (t ) is the rectangular pulse pT (t ) and binary signaling is employed.
The eye diagrams for the cases where the channel is all pass (no ISI) and lowpass (ISI
present) are shown in figure 4.1 and 4.2, respectively. The effects of ISI are to cause a
reduction in the eye opening by reducing the peak as well as causing ambiguity in the
timing information.
Figure 4-2: Eye diagram of a QPSK signal with ISI
A careful observation on eqn. 4.4 reviews that it is possible to have no ISI even if the
v (t ) is bandlimited i.e., the basic pulse shape u (t ) and the channel is bandlimited. More
precisely, letting x(t )  v  v~ (t ) we can rewrite the decision statistics z m in eqn. 4.4 as:
zm
 bm x(0)   bn x(mT  nT )  nm
(4.5)
nm
There is no ISI if the Nyquist condition is satisfied:
c
x(nT )  
0
for n  0
for n  0
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(4.6)
Application of Zero-forcing equalizer
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Figure 4-3: Case 1 non-overlapping spectrum
Figure 4-4: Case 1 overlapping spectrum
Where c is some constant and, without loss of generality, we can set c =1. The Nyquist
condition in this form is not very helpful in the design of the ISI-free pulses. It turns out
that it is more illustrative to restate the Nyquist condition in frequency domain. To do so,
first let
x (t ) 

 x(nT ) (t  nT )
(4.7)
n  
Taking Fourier transform,
X ( f ) 
1
T

n
 X(f T)
(4.8)
n  
Where X  ( f ) is the Fourier transform of x(t ) . The Nyquist condition in (4.6) is
equivalent to the condition x (t )   (t ) or X  ( f )  1 in the frequency domain. Now, by
employing (4.8), we get

n
 X(f T) T
(4.9)
n  
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This is the equivalent Nyquist condition in frequency domain. It says that the folded
spectrum of x(t ) has to be flat for not having ISI.
When the channel is bandlimited to W Hz, i.e., X ( f )  0 for f  W , the Nyquist
condition has the following implications:

1
 2W
Suppose that the symbol rate is so high that T
. Then, the folded spectrum

n
 X(f T)
n  
looks like the one in figure 3. There are gaps between copies of
X(f). No matter how X(f) looks, Nyquist condition cannot be satisfied and ISI is
inevitable.

1
 2W
Suppose that the symbol rate is slower so that T
. Then copies of X(f) can

just touch their neighbors. The folded spectrum
T
X( f )  
0
for
n
 X(f T)
n  
is flat if and only if
f W
for otherwise
(4.10)
The corresponding time domain function is the sinc pulse
 T
x(t )  sinc t
(4.11)
We note that the sinc pulse is not timelimited and is not causal. Therefore, it is not
physically realizable. A truncated and delayed version is used as an approximation. The
1
 2W
critical rate T
above which ISI is unavoidable is known as Nyquist rate.
4.1.2 Raised Cosine in context to ISI
1
 2W
Suppose that the symbol rate is even slower so that T
. Then, copies of X(f)

overlap with their neighbors. The folded spectrum
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n
 X(f T)
n  
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Application of Zero-forcing equalizer
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different choices of X(f). An example of is shown in fig 4. Therefore, we can design an
ISI free pulse shape which gives a flat folded spectrum.
When the symbol rate is below the Nyquist rate, a widely used ISI free spectrum is the
raised cosine spectrum (fig 4)
1
2T
1
1 
for
 f 
2T
2T
1 
for
f 
2T

T


T 
1   
T 
X ( f )   1  cos
 f 

 
2T 
2 

0

0 f 
for
(4.12)
Where 0    1 is called the roll off factor. It determines the excess bandwidth beyond
1
2T . The corresponding time domain function (fig 6) is:
x(t ) 
 T
sin t
t

cos t
1  4 2 t
T
T

2
T2
when   0 , it reduces to the sinc function. We note that for   0 , x(t) decays as
while for   0 , x(t) (sinc pulse) decays as
1
(4.13)
1
t3
t . Hence, the raised cosine spectrum gives
a pulse that is much less sensitive to timing errors than the sinc pulse. Just like all other
bandlimited pulses, x(t) from the raised cosine spectrum is not timelimited. Therefore,
truncation and delay is required for realization.
Finally, recall that x(t) is the overall response of the transmitted pulse passing through
the bandlimited channel and the receiving filter. Mathematically,
X( f )  V( f )
2
 H c ( f )U ( f )
2
(4.14)
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Figure 4-5: Raised-cosine spectrum
where U ( f ) , V ( f ) and H c ( f ) are the Fourier transform of
u(t ), v(t ) and hc (t ) respectively. Given that an ISI-free spectrum X ( f ) is chosen, we can
employ (4.14) to obtain the simple case of a band-limited channel, i.e., the channel does
not introduce any distortion within its pass-band, we can simply choose U ( f ) to be
X ( f ) . Then the Fourier transform the transfer function of the matched filter is also
X ( f ) . For example, if the raised-cosine spectrum is chosen, the resulting ISI-free pulse
u(t) is called the square-root raised-cosine pulse. Of course, suitable truncation and delay
are required for physical realization.
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Figure 4-6: Time domain function of the raised-cosine spectrum
4.2 Zero Forcing Algorithm
In first insight let us consider different parameters used in zero-forcing equalization. Let
H E (z) be the equalizing circuit filter. The LTI filter with transfer function H E (z) is
considered to be the ZF equalizer. The only way to remove the ISI is to choose H E (z)
such that the output of the equalizer gives back the estimated output, i.e., Iˆk  I k for all k.
The filter transter function needs to be specified such that it becomes the multiplicative
inverse of the channel response G(z) i.e, H E ( z )  1
G( z )
. This method is what we call the
zero-forcing equalization as the ISI component is forced to zero. It must be noted that the
impulse response hE ,k need to be an infinite length sequence.
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We note that the effect of the equalizing filter on the noise is neglected in the
development of the zero-forcing equalizer above. In reality, noise is always present.
Although the ISI component is forced to zero, there may be a chance that the equalizing
filter will greatly enhancing the noise power and hence the error performance of the
resulting receiver will still be poor. To see this, let us evaluate the signal-to-noise ratio at
the output of the zero-forcing equalizer when the transmission filter HT ( f ) is fixed and
the matched filter is used as the receiving filter, i.e.,
H R ( f )  H T* ( f ) HC* ( f )
(4.14)
In this case, it is easy to see that the digital filter H (z ) is given by

He
j 2fT

N
 0
2T


n  
n 
n

HT  f  H C  f  
T 
T

2
(4.15)
and the PSD of the colored Gaussian noise samples nk in Figure 4.6 is given by

 nk e
j 2fT

N
 0
2T


n  
n 
n

HT  f  H C  f  
T 
T

2
(4.16)
Hence, the noise-whitening filter HW (z ) can be chosen as
H W (e j 2fT ) 
1
(4.17)
H (e j 2fT )
N
and then the PSD of the whitened-noise samples n~k is simply 0
2
.As a result, the
overall digital filter G (z ) in Figure 4.14 is
G (e j 2fT )  H (e j 2fT ) H W (e j 2fT )  H (e j 2fT )
(4.18)
Now, we choose the zero-forcing filter H E (z) as
H E (e j 2fT ) 
1

G (e j 2fT )
1
H (e j 2fT )
(4.19)
Since the zero-forcing filter simply inverts the effect of the channel on the original
information symbols I k , the signal component at its output should be exactly I k . If we
model the I k as iid random variables with zero mean and unit variance, then the PSD of
the signal component is 1 and hence the signal energy at the output of the equalizer is just
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Application of Zero-forcing equalizer
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
1
2T
1
2T
df 
1
. On the other hand, the PSD of the noise component at the output of the
T
equalizer is

1
2T
1
2T
Zaw Htet Aung (J0704960)
2
N0
H E (e j 2fT ) . Hence the noise energy at the equalizer output is
2
2
N0
H E (e j 2fT ) df . Defining the SNR as the ratio of the signal energy to the noise
2
energy, we have

N
SNR   0
2


2 1




n
n





2T
H
f

H
f

df







T
C
 12T n
T 
T  


  

1
1
(4.20)
Notice that the SNR depends on the folded spectrum of the signal component at the input
of the receiver. If there is a certain region in the folded spectrum with very small
magnitude, then the SNR can be very poor.
In this project we simulated the ZF algorithm in Matlab®. Following describes the
sequence of operation performed during the simulation.

A sequence of 5000 bits are generated i.e. QPSK modulated bits. These signals
were generated using pskmod command of Matlab®.

Simulation was done for signal-to-noise ratio that ranges from 0dB to 30dB.

Now by using the a combination of randn() command (real plus imaginary parts)
the Rayleigh channel was created.

For each SNR, 5000 bits are transmitted and are convolved with the Rayleigh
channel and then AWGN noise is added to the received signal. As stated by the
equation (4.1) i.e. r (t )  u  hc (t )  n(t ) , where hc (t ) is impulse response of
Rayleigh channel.

Now the received signal r(t) is passed through a ZF equalizer. i.e from equation
(4.53) the equalizer taps are found.
4.3 Advantage and disadvantage of Zero-Forcing equalization
The main advantage of the ZF equalizer is that it is very much useful in eliminating the
effect of ISI from the received signal. A disadvantage of this technique is that the noise
power induced in the signal cannot be removed by ZF equalization.
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CHAPTER 5
5 SIMULATIONS AND CONCLUSIONS
The simulation of the project was done in MATLAB. Some difficulties arose during the
implementation. To de-modulate the QPSK signal at receiver end from the modulated
signal was the first challenge during the implementation process. It was then corrected,
when the error was calculated by comparing the modulated signal (QPSK) at transmitting
end with the signal at receiving end before demodulation. The second difficulty arose
when it was required to add the ISI effect in the signal. The ISI was added to the signal
by applying a filter the coefficient of which was a cut frequency. This frequency was
decreased for creation of the ISI effect. The ISI effect was then adjusted by adjusting the
coefficient of the applied filter.
The simulation of the project is comprised of a number of comparisons. We have
compared the effect of received signal by applying the zero-forcing equalizer with and
without the effect of ISI. The analytical result of the Rayleigh-faded channel and AWGN
(Additive White Gaussian Noise) are also compared with the simulated results.
In first scene, we will look towards the effect of ISI on the received signal with
some noise without applying the ZF equalizer at receiver. The graph obtained in this case
is shown below:
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Figure 5-1: BER comparison without ZF equalizer under the effect of ISI
If we look at the second graph, we see the simulated result of the received signal affected
by ISI along with some noise effect when the effect of ZF equalizer is applied at the
receiving end.
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Figure 5-2: BER comparison using ZF equalizer under the effect of ISI
If we combine both of the above mentioned graphs, we see that by applying the ZF
equalizer the BER vs SNR result is better than without applying the same equalizer. The
graph shown below differentiates the result.
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Figure 5-3: BER comparison with and without ZF equalizer under ISI
The above graph shows it clearly that by applying the ZF equalizer, the bit-error-rate
decreases.
5.1 Effect of ISI on the received signal
Let us explain the effect of ISI on the received signal. As we know that the ISI is a sort of
disturbance just like noise that affects the bit-error-rate of the received signal. If the effect
of ISI is increased, the disturbance in the received signal is increased and hence the biterror-rate is increased.
For creating the effect of ISI in received signal while using matlab tool, we apply
a filter whose co-efficient is freq (frequency) cut. The freq cut is designed in such a way
that if increase its value the ISI effect decreases. Let us consider the 1st case in which the
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value of freq cut is 50. The graph (shown below) shows the effect of BER vs SNR for
freq cut=50.
Figure 5-4: BER comparison by setting the ISI co-efficient freq cut=50
Now let us take the second case in which the value of co-efficient freq cut = 100. The ISI
effect is decreased. The graph (shown below) shows the effect of ISI on the received
signal at freq cut=100.
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Figure 5-5: Effect Comparison by setting the ISI co-efficient freq cut=100
If the value of ISI co-efficient is further increased to freq cut=200, we can see that the
effect of ISI is further decreased. The graph (shown below) shows the effect of ISI on the
received signal for freq cut=200.
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Figure 5-6: BER comparison by setting the ISI co-efficient freq cut=200
Similarly for freq cut=600, the figure shown below shows that both the graphs (with and
without applying ZF effect) are approximately coincides with each-other.
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Figure 5-7: BER comparison by setting the ISI co-efficient freq cut=600
The above mentioned results of all the figures shows us that the graph of BER vs SNR
for both the cases (with and without applying ZF effect) shows the same result at the zero
ISI effect. From this theory, it has been concluded that the ISI affects the bit-error-rate
just like the noise. And the zero-forcing equalizer is much useful for mitigating the effect
of the bit-error-rate introduced by ISI. The other equalizers like MMSE and RLS etc do
not usually eliminate the ISI effect completely but instead minimizes the total power of
noise.
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5.2 Effects of filter and waveform analysis
Since we have implemented a filter to create ISI effects in the simulations of system
model, it would be advisable for us to show of the response of that particular filter and I
discuss about its waveforms.
Figure 5-8: Filter response in frequency domain
The above figure shows the frequency response of the filter in frequency domain. The
number of transmitted bits is 300 and filter coefficient (cut of frequency) is 100.
Here, waveform analysis, ISI and equalizer effects will be discussed further. For
analysis, we will consider two different scenarios: one is a typical rural area, where the
effect of ISI is not much severe (the filter effect or coefficient (cut of frequency) is huge)
and the other is an urban area, where the effect of ISI is severe (the filter effect is small).
Choosing of proper signal to noise ratio (SNR) is also to be considered for simulation.
We should not choose the big number such as 50 or more because SNR 50 means the
signal power is much more than the noise power or t the noise power is negligible. When
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the effect of noise is negligible (or zero), then no matter if we use the ZF equalizer or a
simple equalizer, we will get the received signal without errors. To see the equalizer and
ISI effects clearly, we will transmit 50 bits and set our simulation parameters as below.
For a typical rural area: SNR=5, and frcut=500 (very small ISI effect)
Figure 5-9a: Comparison of transmitted symbols before filter and received demodulated symbols
after filter and no zero-forcing equalizer under small ISI effect
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Figure 5-9b: Comparison of transmitted symbols before filter and received demodulated symbols
after filter and zero-forcing equalizer under small ISI effect
From the above two figures, you will observe that by setting negligible ISI effect, the
number of errors in both the files will be same because with no ISI effect both the
equalizers (ZF and the simple one) behave as if they were same equalizers. In this case,
the erroneous effect is only due to noise.
For a urban area: SNR=5, and frcut=20 (high ISI effect)
Figure 5-10a: Comparison of transmitted symbols before filter and received demodulated symbols
after filter and no zero-forcing equalizer under high ISI effect
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Figure 5-10b: Comparison of transmitted symbols before filter and received demodulated symbols
after filter and zero-forcing equalizer under high ISI effect
Again from the above two figures, you will see that without zero-forcing equalizer,
simulation shows more erroneous received bits, while with zero-forcing, simulation
creates lesser number of erroneous received bits. In this case the erroneous effect is due
to noise as well as due to ISI.
5.3 Conclusion and Future Work
From the above mentioned simulations, we can conclude that the effect of ISI can be
decreased (mitigated) using a specialized equalizing technique called zero-forcing. The
above figures show that the effect of ISI causes an increase of bit-error-rate. By applying
the zero-forcing equalizer, the bit-error-rate is decreased. The section 5.1 shows a list of
figures representing the effect of ISI on the bit-error-rate. The last figure shows the
negligible effect of ISI from which we can conclude that without applying the ZF
equalizer, the effect is same as when applying the ZF equalizer.
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To further enhance the process of removing ISI effect, more work is already
accomplished like OFDM modulation technique that can mitigate the effect of ISI to a
large extent (explained in chapter 3). Other techniques like feed-back equalizers are also
helpful in elimination of ISI effect. More research work is already in progress for the
elimination of ISI, like modification of feed-back zero-forcing equalization techniques by
applying advanced mathematical algorithms.
PART-II
Critical Review and Reflections
Distortion in a channel causes ISI to occur in a transmitted signal and this causes a
serious problem when we are dealing with a communication wireless channel. Different
techniques like OFDM or equalization is used to combat the effect of ISI. Main causes of
ISI are due to scattering effect in wireless channels when the coherent bandwidth of the
channel is less than the bandwidth of the transmitted signal. To combat with the severity
of ISI, the powerful equalizers are used at the receiving end. There are several equalizers
that are used to combat the effect of ISI, but the most common of these equalizers is
Zero-Forcing (ZF) equalizer. Due to outset of the project, an intensive literature research
on linear equalization was thus carried out. Many research papers and majority of ebooks available in libraries and on internet were used to for clearing the perception of
equalization and most especially on ZF equalization. After spending almost a month or
above, I was able to visualize the concepts.
In the second phase of the project, I was able to prepare the initial report for my
project, which was comprised of investigation of objectives, project-background,
proposed approaches to be employed and skills review. As initial stage plays an
important role for the successful end of the product, the proposed approaches were thus
thoroughly analyzed and selected.
In the project work, a simple communication system was studied as the first step.
In this study the main focus was on different techniques that are used for combating the
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effect of ISI. In the second phase the equalization was discussed in more detail. After
that, the characteristics of the channel were discussed, as the channel plays an important
role in any communication system. Different channel models were glanced over and
special focus was given to Rayleigh fading channel model. Then a simple communication
system employing Rayleigh fading channel in Matlab® was simulated. As the Rayleigh
channel was applied in the simulation model, that’s how the understanding of Rayleigh
channel was improved too much. Then I studied ZF equalization, and implemented it in a
communication system with only considering AWGN channel (without taking into
account Rayleigh fading channel). After successful simulation with AWGN channel I
implemented the same in Rayleigh fading channel environment.
From this project, I have learnt many aspects of the communication system in
perspective of Rayleigh fading channels and equalization. Applying the equalization
algorithms at the receiver was far most good experience. My existing skills such as
MATLAB programming, problem-solving techniques, research methodologies, analytical
applications, project and time management and technical report writing are improved too.
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References
[1]
Y. G. Li, “pilot-symbol-aided channel estimation for OFDM in wireless systems”,
IEEE Trans. Veh. Technol, Vol.49, pp. 1207-1215, July 2000
[2]
S. Coleri, M. Ergen, A. Bahai, “Channel estimation techniques based on pilot
Arrangement in OFDM systems”, IEEE Trans. Broadcasting, vol.48, pp. 467-478,
Mar, 2001.
[3]
S. Ohno and G. B. Giannakis, “Optimal training and redundant precoding for
block transmissions with application to wireless OFDM”, IEEE Trans. Commun,
Vol.50, pp.2113-2123, Dec. 2002
[4]
S. Chennakeshu and J. B Anderson, “Error rates for Rayleigh fading multichannel reception of MPSK signals”, IEEE Trans. Commun, vol. 43, pp. 338-346,
1995
[5]
X. Cai and G. B. Giannakis, “Error probability minimizing pilots for OFDM with
m-psk modulation over rayleigh-fading channels”, IEEE Trans. Veh. Technol, Vol.
53, pp. 146-155, Jan 2004
[6]
Developer Zone by National Instruments,
website: http://zone.ni.com/devzone/cda/tut/p/id/3740
[7]
Ramjee prasad , “OFDM for wireless communications systems” , volum 1. India,
prentice-Hall, 2004.
[8]
Eldering, C., Sylla, M., Eisenach, J., ‘Is there a Moore’s Law for bandwidth?’
IEEE Communications Magazine, vol.37, issue 10, pp.117-121, 1999
[9]
Harte, N., ‘Segmental Phonetic Models and features for speech Recognition’, Phd
Thesis, Dept. of Electrical and Electronic Engineering, Queen’s University of
Belfast, 1999.
[10]
Kalouptsidis, N., Theodoridis S., ‘Adaptive systems identification and signal
processing algorithms’, Prentice hall, 1993.
[11]
Siller C., ‘Multipath propagation’, IEEE communications magazine, vol. 22,
no.2, pp.6-15, Feb.1984.
[12]
Qureshi S., ‘Adaptive Equalization’, Proceeding of the IEEE, vol.73, no.9, pp1349-1387, Sept. 1985.
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Application of Zero-forcing equalizer
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[13]
Zaw Htet Aung (J0704960)
McLaughlin, S., ‘Shielding light on the future of SP for optical recording’, IEEE
Signal Processing magazine, vol.15, no.4, pp.83-94, July 1998.
[14]
A. Sayed, ‘Fundamentals of Adaptive Filtering’ New York: Wiley, 2003.
[15]
C. R. Johnson, Jr and W. A. Sethares, ‘Telecommunication Breakdown’
Eaglewood Cliffs, NJ: Prentice-Hall, 2004.
[16]
J.G. Proakis, ‘Digital Communications’ 3rd ed. New York: McGraw-Hill, 1995.
[17]
R. H. Clarke, “A statistical theory of mobile-radio reception,” Bell Sys. Tech. J.,
vol. 47, no. 6, pp. 957-1000, July-Aug. 1968.
[18]
A. Papoulis, Probabiulity, Random Variables, and Stochastic Processes,
McGraw-Hill, 1st Edition, 1965.
[19]
P. A. Bello, “ Charcterization of randomly time variant linear channels,” IEEE
Trans. Commun. Syst., vol. CS-11, no. 4, pp. 360-393, Dec. 1963.
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APPENDIX
Code for BER comparison without Zero-Forcing Equalizer under ISI
% BER comparison without ZF under ISI
% ZF refers to Zero-forcing
% ISI refers to Intersymbol Interference
clear all
clc;
close all;
format long g
warning off
N=10; % Number of reflected symbols i.e 10 copies of the same
transmitted symbols
M=4; % Modulation index
snr_size=30;
SNR_dB=[0:snr_size];
bits=5000;
% Total number of transmitted bits (information bits).
There is no concept of transmitted pilot bits in this code
theta = [0:M-1]; %0,1,2,3 random theta point
syms x;
theoryBer_Ray=[];
% Generation of qpsk signals
Phase = randsrc(N,bits,theta); %Generaion of random bits (with values
b/w 0 and 3)
m = pskmod(Phase,M);
%modulation of qpsk random signal (0to-1<0, 1-to-1<90, 2-to-1<180, 3-to-1<270)
j=sqrt(-1);
%imaginary number
% preserving the old data style in several cumulative matrices of the
size: (N X bits)
m_noise= ones(N,1)*(randn(1,bits)+randn(1,bits)*j)/sqrt(2);
effect with size N*bits
% the sigma_n coefficient will be applied later
% Noise
m_h= ones(N,1)*( ( randn(1,bits) ) + j*(randn(1,bits)) );
%
Rayleigh channel effect (with size N*bits)
m_r=
m_h.*m;
% Impact of channel effect upon the transmitted
modulated symbols with size N*bits
%-----------------------------------------------------%fft of the noise and the signal without noise
m_r_ch=[];
m_r_no=[];
for i=1:100
m_r_ch=cat(3,m_r_ch,m_r');
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m_r_no=cat(3,m_r_no,m_noise');
end
m_r_ch= fft( reshape(permute(m_r_ch,[3,1,2]),[100*bits,N]));
m_r_no= fft( reshape(permute(m_r_no,[3,1,2]),[100*bits,N]));
%
%
%
high frequency
filter response: (frcut/T )/sqrt((frcut/T)^2+x^2 )
where T- is the time interval between the signals
( here any other filter could be implemented )
frcut= 150;
% coefficient to adjust the ISI effect:i.e if the
effective cut off frequency
%of the channel is increased,the ISI effect disappears
nfrcut=bits*frcut;
freq=
nfrcut./sqrt(([1:100*bits]'*ones(1,N)).^2+nfrcut^2); % change
this formula in case another filter needs to be applied
%it must have the format: F(100*bit,1)*ones(1,N)
freq=freq/sqrt(mean(abs(freq(:,1)).^2)); %normalizing the filter
%------------------------------end filter frequency response
%---------------- applying filter in frequency domain, reversing to
time with ifft
%---------------- to get the noisless part (with ISI) of the signal and
noise
m_r_isi= ifft( m_r_ch.*freq ); m_r_isi=m_r_isi(100:100:end,:)';
%ISI
applied to signal without noise
m_r_noi= ifft( (m_r_no).*freq ); m_r_noi=m_r_noi(100:100:end,:)'; %ISI
applied to noise
%m_r_noi=m_r_noi/ sqrt(mean(abs(m_r_noi(1,:)).^2)); %normalizing noise
to 1, sigma_n to be multiplied later
% this one will be needed if noise level is counted after ISI is
applied
%--------------%
m_r_nozf= ifft( m_r_no./freq ); m_r_nozf=
m_r_nozf(100:100:end,:)';
% zf applied on pure noise (without isi)just in case one needs it
to model noise added by
% hardware already AFTER the ISI
%----------------------------------------------------------END ADDED
BLOCK
%---------------------------------------------------------for ZF=0:1 %
ZF cycle line
for snr=0:(snr_size-1)
snr_linear=(10)^(snr/10);
value of SNR (not log)
sigma_n=sqrt(1/snr_linear);
% Absolute or linear or anti-log
for i=1:bits
%------
%
equivalent of the commented out
lines:
b=m(:,i);
% Transmitted symbols (not bits)
h=m_h(1,i);
% Channel effect
noise=m_noise(1,i);
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if(ZF==0)
% ISI affected, ZF not applied
r=m_r_isi(:,i)+
m_r_noi(:,i) *sigma_n;
else
% ZF applied:
r=m_r(:,i)+ noise*sigma_n;
end
%END ADDED BLOCK--------------------------%%
WIthout ZF equalization
est_zf=h.\r;
hat_zf=zeros(N,1);
hat_zf(1:N,1)=pskdemod(est_zf,M);
% qpsk demodulation
hat_zf(1:N,1)=pskmod(hat_zf,M);
% qpsk modulation (for
converting the symbols again to 1<x form)
% Note: The purpose of doing demodulation 1st is to convert the nonexact
% angles of the received modulated symbols as:
%
%
%
2
%
3
Any symbol with distorted angle b/w -45 and +45 deg is converted to 0
Any symbol with distorted angle b/w +45 and 135 deg is converted to 1
Any symbol with distorted angle b/w 135 and -135 deg is converted to
Any symbol with distorted angle b/w -135 and -45 deg is converted to
% After demodulation, the exact values (0,1,2,3) are converted again to
% modulated form. This time, the values of angles are exact i.e
1<0,1<90,1<180, and 1<270.
e_zf=0;
for k=1:N
if hat_zf(k,1)~=b(k,1)
% Forced estimation is compared
to the sent signal vector
e_zf=e_zf+1; % error is added, if both signals (sent
and received) are not equal
end
end
pe_zf(i)=e_zf/k;
% Probability of error
%%
end
% end for (for i=1:bits)
pe_snr_zf(snr+1)=mean(pe_zf);
%%
%AWGN BER
theoryBer_AWGN(snr+1) = erfc(sqrt(10.^(snr/10))*sin(pi/M));
%Rayeligh + AWGN BER
theoryBer_Ray = [theoryBer_Ray pi^-1 *
0.75*double(int((1+(10.^(SNR_dB(1,snr+1)/10)*(sin(pi/M)^2)/(sin(x)^2)))
^-1, 0, (pi-(pi/M))))];
%%
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end
% end for (for snr=0:30)
if(ZF==0)
semilogy([0:snr_size-1],[pe_snr_zf],'g-*')
else if(ZF==1)
a=[];
% hold on
% semilogy([0:snr_size-1],[pe_snr_zf],'y-*')
end % end for the above mentioned if condition
%---------------------------------------end
% end for for ZF=0:1
hold on
semilogy([0:snr_size-1],[theoryBer_AWGN],'b-*')
hold on
semilogy([0:snr_size-1],[theoryBer_Ray(1:snr_size)],'m-*')
legend('Rayleigh simulation without ZF','AWGN analytical','Rayleigh
analytical');
title('BER comparison without ZF under ISI')
xlabel('Signal to noise ratio (SNR),dB'), ylabel('Bit Error Rate
(BER)')
axis([0 snr_size 10^-4 1])
end
% end ZF cycle
Code for BER comparison with Zero-forcing Equalizer under ISI
%
%
%
BER comparison with ZF under ISI
ZF refers to Zero-forcing
ISI refers to Intersymbol Interference
clear all
clc;
close all;
format long g
warning off
N=10; % Number of reflected symbols i.e 10 copies of the same
transmitted symbols
M=4; % Modulation index
snr_size=30;
SNR_dB=[0:snr_size];
bits=5000;
% Total number of transmitted bits (information bits).
There is no concept of transmitted pilot bits in this code
theta = [0:M-1]; %0,1,2,3 random theta point
syms x;
theoryBer_Ray=[];
% Generation of qpsk signals
Phase = randsrc(N,bits,theta);
b/w 0 and 3)
%Generaion of random bits (with values
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m = pskmod(Phase,M);
%modulation of qpsk random signal
to-1<0, 1-to-1<90, 2-to-1<180, 3-to-1<270)
j=sqrt(-1); %imaginary number
(0-
% preserving the old data style in several cumulative matrices of the
size: (N X bits)
m_noise= ones(N,1)*(randn(1,bits)+randn(1,bits)*j)/sqrt(2);
effect with size N*bits
% the sigma_n coefficient will be applied later
% Noise
m_h= ones(N,1)*( ( randn(1,bits) ) + j*(randn(1,bits)) );
%
Rayleigh channel effect (with size N*bits)
m_r=
m_h.*m; % Impact of channel effect upon the transmitted
modulated symbols with size N*bits
%-----------------------------------------------------%fft of the noise and the signal without noise
m_r_ch=[];
m_r_no=[];
for i=1:100
m_r_ch=cat(3,m_r_ch,m_r');
m_r_no=cat(3,m_r_no,m_noise');
end
m_r_ch= fft( reshape(permute(m_r_ch,[3,1,2]),[100*bits,N]));
m_r_no= fft( reshape(permute(m_r_no,[3,1,2]),[100*bits,N]));
%
%
%
high frequency
filter response: (frcut/T )/sqrt((frcut/T)^2+x^2 )
where T- is the time interval between the signals
( here any other filter could be implemented )
frcut= 150; % coefficient to adjust the ISI effect:i.e if the
effective cut off frequency
%of the channel is increased,the ISI effect disappears
nfrcut=bits*frcut;
freq=
nfrcut./sqrt(([1:100*bits]'*ones(1,N)).^2+nfrcut^2);
% change this formula in case another filter needs to be applied
%it must have the format: F(100*bit,1)*ones(1,N)
freq=freq/sqrt(mean(abs(freq(:,1)).^2)); % normalizing the filter
%------------------------------end filter frequency response
%---------------- applying filter in frequency domain, reversing to
time with ifft
%---------------- to get the noisless part (with ISI) of the signal and
noise
m_r_isi= ifft( m_r_ch.*freq ); m_r_isi=m_r_isi(100:100:end,:)'; %ISI
applied to signal without noise
m_r_noi= ifft( (m_r_no).*freq ); m_r_noi=m_r_noi(100:100:end,:)'; %ISI
applied to noise
%m_r_noi=m_r_noi/ sqrt(mean(abs(m_r_noi(1,:)).^2)); %normalizing noise
to 1, sigma_n to be multiplied later
% this one will be needed if noise level is counted after ISI is
applied
%---------------
- 55 -
Application of Zero-forcing equalizer
in digital communications system
Zaw Htet Aung (J0704960)
%
m_r_nozf= ifft( m_r_no./freq ); m_r_nozf=
m_r_nozf(100:100:end,:)';
% --- zf applied on pure noise (without isi)
% just in case one needs it to model noise added by
% hardware already AFTER the ISI
%----------------------------------------------------------END ADDED
BLOCK
%---------------------------------------------------------for ZF=0:1 %
ZF cycle line
for snr=0:(snr_size-1)
snr_linear=(10)^(snr/10);
value of SNR (not log)
sigma_n=sqrt(1/snr_linear);
%------
% Absolute or linear or anti-log
% magnitude of noise
for i=1:bits
% equivalent of the commented out
lines:
b=m(:,i);
h=m_h(1,i);
noise=m_noise(1,i);
if(ZF==0)
% ISI affected, ZF not applied
r=m_r_isi(:,i)+
m_r_noi(:,i) *sigma_n;
else
% ZF applied:
r=m_r(:,i)+ noise*sigma_n;
end
%END ADDED BLOCK--------------------------%%
WIthout ZF equalization
est_zf=h.\r;
hat_zf=zeros(N,1);
hat_zf(1:N,1)=pskdemod(est_zf,M);
hat_zf(1:N,1)=pskmod(hat_zf,M);
e_zf=0;
for k=1:N
if hat_zf(k,1)~=b(k,1)
% Forced estimation is compared
to the sent signal vector
e_zf=e_zf+1; % error is added, if both signals (sent
and received) are not equal
end
end
pe_zf(i)=e_zf/k;
% Probability of error
%%
end
% end for (for i=1:bits)
pe_snr_zf(snr+1)=mean(pe_zf);
%%
%AWGN BER
- 56 -
Application of Zero-forcing equalizer
in digital communications system
Zaw Htet Aung (J0704960)
theoryBer_AWGN(snr+1) = erfc(sqrt(10.^(snr/10))*sin(pi/M));
%Rayleigh + AWGN BER
theoryBer_Ray = [theoryBer_Ray pi^-1 *
0.75*double(int((1+(10.^(SNR_dB(1,snr+1)/10)*(sin(pi/M)^2)/(sin(x)^2)))
^-1, 0, (pi-(pi/M))))];
%%
end
% end for (for snr=0:30)
if(ZF==0)
semilogy([0:snr_size-1],[pe_snr_zf],'w')
else if(ZF==1)
semilogy([0:snr_size-1],[pe_snr_zf],'y-*')
end % end for the above mentioned if condition
%---------------------------------------end
% end for for ZF=0:1
hold on
semilogy([0:snr_size-1],[theoryBer_AWGN],'b-*')
hold on
semilogy([0:snr_size-1],[theoryBer_Ray(1:snr_size)],'m-*')
title('BER comparison with ZF under ISI')
xlabel('Signal to noise ratio (SNR),dB'), ylabel('Bit Error Rate
(BER)')
legend('','AWGN analytical','Rayleigh analytical','Rayleigh simulation
with ZF');
axis([0 snr_size 10^-4 1])
end % end ZF cycle
Code for BER comparison with and without Zero-forcing equalizer
under ISI
% BER comparsion with and without ZF under ISI
% ZF refers to Zero-forcing
% ISI refers to Intersymbol Interference
clear all
clc;
close all;
format long g
warning off
N=10;
% Number of reflected symbols i.e 10 copies of the same
transmitted symbols
M=4;
% Modulation index
snr_size=30;
SNR_dB=[0:snr_size];
bits=5000;
% Total number of transmitted bits (information
bits). There is no concept of transmitted pilot bits in this code
theta = [0:M-1];
syms x;
theoryBer_Ray=[];
%0,1,2,3 random theta point
- 57 -
Application of Zero-forcing equalizer
in digital communications system
Zaw Htet Aung (J0704960)
% Generation of qpsk signals
Phase = randsrc(N,bits,theta);
%Generaion of random bits
(with values b/w 0 and 3)
m = pskmod(Phase,M);
%modulation of qpsk random
signal (0-to-1<0, 1-to-1<90, 2-to-1<180, 3-to-1<270)
j=sqrt(-1);
%imaginary number
% preserving the old data style in several cumulative matrices of the
size: (N X bits)
m_noise= ones(N,1)*(randn(1,bits)+randn(1,bits)*j)/sqrt(2);
effect with size N*bits
% the sigma_n coefficient will be applied later
% Noise
m_h= ones(N,1)*( ( randn(1,bits) ) + j*(randn(1,bits)) );
% Rayleigh
channel effect (with size N*bits)
m_r=
m_h.*m;
% Impact of channel effect upon the transmitted
modulated symbols with size N*bits
%-----------------------------------------------------%fft of the noise and the signal without noise
m_r_ch=[];
m_r_no=[];
for i=1:100
m_r_ch=cat(3,m_r_ch,m_r');
% Concatenating a 3rd dimension
(with size 100) with the the received matrix (N*bits)
m_r_no=cat(3,m_r_no,m_noise'); % Concatenating a 3rd dimension
(with size 100) with the the noise matrix (N*bits)
end
m_r_ch= fft(
converting 3D
m_r_no= fft(
converting 3D
reshape(permute(m_r_ch,[3,1,2]),[100*bits,N])); %Again
mesh of received symbols into 2D matrix and taking fft
reshape(permute(m_r_no,[3,1,2]),[100*bits,N])); %Again
mesh of noise into 2D matrix and taking fft
frcut= 150; % coefficient to adjust the ISI effect:i.e if the
effective cut off frequency
%of the channel is increased,the ISI effect disappears
nfrcut=bits*frcut;
freq=nfrcut./sqrt(([1:100*bits]'*ones(1,N)).^2+nfrcut^2); % change this
formula in case another filter needs to be applied
%it must have
the format: F(100*bit,1)*ones(1,N)
freq=freq/sqrt(mean(abs(freq(:,1)).^2)); %normalizing the filter
%------------------------------end filter frequency response
%---------------- applying filter in frequency domain, reversing to
time with ifft
%---------------- to get the noiseless part (with ISI) of the signal
and noise
- 58 -
Application of Zero-forcing equalizer
in digital communications system
Zaw Htet Aung (J0704960)
m_r_isi= ifft( m_r_ch.*freq );
% Converting the received symbols
matrix(with the effect of ISI)into time domain
m_r_isi=m_r_isi(100:100:end,:)'; % ISI applied to signal without
noise
m_r_noi= ifft( (m_r_no).*freq );
m_r_noi=m_r_noi(100:100:end,:)'; % ISI applied to noise
%m_r_noi=m_r_noi/ sqrt(mean(abs(m_r_noi(1,:)).^2)); %normalizing noise
to 1, sigma_n to be multiplied later
% this one will be needed if noise level is counted after ISI is
applied
%--------------------%
m_r_nozf= ifft( m_r_no./freq ); m_r_nozf=
m_r_nozf(100:100:end,:)';
% zf applied on pure noise (without isi)just in case one needs it
to model noise added by
% hardware already AFTER the ISI
%----------------------------------------------------------END ADDED
BLOCK
%---------------------------------------------------------for ZF=0:1 %ZF cycle line
for snr=0:(snr_size-1)
snr_linear=(10)^(snr/10);
value of SNR (not log)
sigma_n=sqrt(1/snr_linear);
%------
% Absolute or linear or anti-log
% magnitude of noise
for i=1:bits
% equivalent of the commented out lines:
b=m(:,i);
% Transmitted symbols (not bits)
h=m_h(1,i);
% Channel effect
noise=m_noise(1,i);
if(ZF==0)
%ISI affected,ZF not applied
r=m_r_isi(:,i)+
m_r_noi(:,i) *sigma_n;
symbols are affected by the ISI as well as noise
else
% ZF applied:
r=m_r(:,i)+ noise*sigma_n;
end
%END ADDED BLOCK--------------------------%%
% The transmitted
WIthout ZF equalization
est_zf=h.\r;
hat_zf=zeros(N,1);
hat_zf(1:N,1)=pskdemod(est_zf,M); % qpsk demodulation
hat_zf(1:N,1)=pskmod(hat_zf,M);
% qpsk modulation (for
converting the symbols again to 1<x form)
% Note: The purpose of doing demodulation 1st is to convert the nonexact
% angles of the received modulated symbols as:
- 59 -
Application of Zero-forcing equalizer
in digital communications system
%
%
%
2
%
3
Zaw Htet Aung (J0704960)
Any symbol with distorted angle b/w -45 and +45 deg is converted to 0
Any symbol with distorted angle b/w +45 and 135 deg is converted to 1
Any symbol with distorted angle b/w 135 and -135 deg is converted to
Any symbol with distorted angle b/w -135 and -45 deg is converted to
% After demodulation, the exact values (0,1,2,3) are converted again to
% modulated form. This time, the values of angles are exact i.e 1<0,
1<90,1<180, and 1<270.
e_zf=0;
for k=1:N
if hat_zf(k,1)~=b(k,1)
to the sent signal vector
e_zf=e_zf+1;
(sent and received) are not equal
end
end
pe_zf(i)=e_zf/k;
%%
end
% Forced estimation is compared
% error is added, if both signals
% Probability of error
% end for (for i=1:bits)
pe_snr_zf(snr+1)=mean(pe_zf);
%%
%AWGN BER
theoryBer_AWGN(snr+1) = erfc(sqrt(10.^(snr/10))*sin(pi/M));
%Rayleigh + AWGN BER
theoryBer_Ray = [theoryBer_Ray pi^-1 *
0.75*double(int((1+(10.^(SNR_dB(1,snr+1)/10)*(sin(pi/M)^2)/(sin(x)^2)))
^-1, 0, (pi-(pi/M))))];
%%
end % end for (for snr=0:30)
if(ZF==0)
semilogy([0:snr_size-1],[pe_snr_zf],'g-*')
else if(ZF==1)
hold on
semilogy([0:snr_size-1],[pe_snr_zf],'y-*')
end % end for the above mentioned if condition
%---------------------------------------end
% end for ZF=0:1
hold on
semilogy([0:snr_size-1],[theoryBer_AWGN],'b-*')
hold on
semilogy([0:snr_size-1],[theoryBer_Ray(1:snr_size)],'m-*')
legend('Rayleigh simulation without ZF','AWGN analytical','Rayleigh
analytical','Rayleigh simulation with ZF');
title('BER comparison with and without ZF under ISI')
xlabel('Signal to noise ratio (SNR),dB'), ylabel('Bit Error Rate
(BER)')
- 60 -
Application of Zero-forcing equalizer
in digital communications system
Zaw Htet Aung (J0704960)
axis([0 snr_size 10^-4 1])
end % end ZF cycle
Code for filter response in frequency domain
% code for the filter response in frequency domain
clc
clear all
close all
bits=300;
% Number of transmitted bits
N=10;
frcut= 100; % filter coefficient
nfrcut=bits*frcut;
freq=
nfrcut./sqrt(([1:100*bits]'*ones(1,N)).^2+nfrcut^2);
freq=freq/sqrt(mean(abs(freq(:,1)).^2)); % --normalizing the filter...
freq=freq.^2;
plot(freq) % Also see the plot with command stem(freq)
axis([0 nfrcut 0 1.4])
title('Filter response in frequency domain'),xlabel('Frequency
axis'),ylabel('Power')
Code for comparing transmitted symbols before filter and received
demodulated symbols after filter and no zero-forcing equalizer
% Waveform showing of the transmitted bits before the filter and
% the demodulated received symbols after the filter
% The ZF equalizer is not applied in this case
clear all
clc;
close all;
format long g
warning off
snr=input('Enter the value of snr ');
N=1;
M=4; % Modulation index
snr_size=30;
SNR_dB=[0:snr_size];
bits=50;
% Total number of transmitted bits (information bits).
theta = [0:M-1];
syms x;
theoryBer_Ray=[];
% Generation of qpsk signals
Phase = randsrc(N,bits,theta);
subplot(211)
stem(Phase)
- 61 -
Application of Zero-forcing equalizer
in digital communications system
Zaw Htet Aung (J0704960)
xlabel('n'), ylabel('Amplitude')
title('Waveform of transmitted symbols before passing through
filter')
m = pskmod(Phase,M);
j=sqrt(-1);
% preserving the old data style in several cumulative matrices of the
size: (N X bits)
%
m_noise= ones(N,1)*(randn(1,bits)+randn(1,bits)*j)/sqrt(2);
the sigma_n coefficient will be applied later
m_h= ones(N,1)*( ( randn(1,bits) ) + j*(randn(1,bits)) );
m_r=
m_h.*m;
% this might rather be called m_b, but it's all right
%-----------------------------------------------------%fft of the noise and the signal without noise
m_r_ch=[];
m_r_no=[];
for i=1:100
m_r_ch=cat(3,m_r_ch,m_r');
m_r_no=cat(3,m_r_no,m_noise');
end
m_r_ch= fft(
m_r_no= fft(
%
%
%
reshape(permute(m_r_ch,[3,1,2]),[100*bits,N]));
reshape(permute(m_r_no,[3,1,2]),[100*bits,N]));
high frequency
filter response: (frcut/T )/sqrt((frcut/T)^2+x^2 )
where T- is the time interval between the signals
( here any other filter could be implemented )
frcut= 20;
i.e.
% coefficient to adjust the ISI effect: when frcut->inf,
% if the effective cut off frequency of the channel is
increased
% the ISI effect disappears
nfrcut=bits*frcut;
freq=
nfrcut./sqrt(([1:100*bits]'*ones(1,N)).^2+nfrcut^2); % change
this formula in case another filter needs to be applied
%it must have the format: F(100*bit,1)*ones(1,N)
freq=freq/sqrt(mean(abs(freq(:,1)).^2)); % --normalizing the filter...
%------------------------------end filter frequency
response
%---------------- applying filter in frequency domain, reversing to
time with ifft
%---------------- to get the noisless part (with ISI) of the signal and
noise
m_r_isi= ifft( m_r_ch.*freq );
m_r_isi=m_r_isi(100:100:end,:)'; %isi applied to signal without noise
m_r_noi= ifft( (m_r_no).*freq );
m_r_noi=m_r_noi(100:100:end,:)'; %isi applied to noise
ZF=0
- 62 -
Application of Zero-forcing equalizer
in digital communications system
Zaw Htet Aung (J0704960)
snr_linear=(10)^(snr/10);
value of SNR (not log)
sigma_n=sqrt(1/snr_linear);
%------
% Absolute or linear or anti-log
for i=1:bits
% equivalent of the commented out
b=m(:,i);
h=m_h(1,i);
noise=m_noise(1,i);
lines:
% ISI affected, ZF not applied
r=m_r_isi(:,i)+
m_r_noi(:,i) *sigma_n;
%END ADDED BLOCK--------------------------est_zf=h.\r;
%
hat_zf1=zeros(N,1);
hat_zf1(i)=pskdemod(est_zf,M);
stem(hat_zf1(i))
hold on
%
%
%%
end
% end for (for i=1:bits)
subplot(212)
stem(hat_zf1)
xlabel('n'), ylabel('Amplitude')
title('Waveform of demodulated received symbols after passing through
filter')
Code for comparing transmitted symbols before filter and received
demodulated symbols after filter and zero-forcing equalizer
% Waveform showing of the transmitted bits before the filter and
% the demodulated received symbols after the filter
% The ZF equalizer is applied in this case
clear all
clc;
close all;
format long g
warning off
snr=input('Enter the value of snr ');
N=1;
M=4;
% Modulation index
snr_size=30;
SNR_dB=[0:snr_size];
bits=50;
% Total number of transmitted bits (information bits).
theta = [0:M-1];
syms x;
theoryBer_Ray=[];
% Generation of qpsk signals
Phase = randsrc(N,bits,theta);
subplot(211)
- 63 -
Application of Zero-forcing equalizer
in digital communications system
Zaw Htet Aung (J0704960)
stem(Phase)
xlabel('n'), ylabel('Amplitude')
title('Waveform of transmitted symbols before passing through
filter')
m = pskmod(Phase,M);
j=sqrt(-1);
% preserving the old data style in several cumulative matrices of the
size: (N X bits)
%
m_noise= ones(N,1)*(randn(1,bits)+randn(1,bits)*j)/sqrt(2);
the sigma_n coefficient will be applied later
m_h= ones(N,1)*( ( randn(1,bits) ) + j*(randn(1,bits)) );
m_r=
m_h.*m;
% this might rather be called m_b, but it's all right
%-----------------------------------------------------%fft of the noise and the signal without noise
m_r_ch=[];
m_r_no=[];
for i=1:100
m_r_ch=cat(3,m_r_ch,m_r');
m_r_no=cat(3,m_r_no,m_noise');
end
m_r_ch= fft( reshape(permute(m_r_ch,[3,1,2]),[100*bits,N]));
m_r_no= fft( reshape(permute(m_r_no,[3,1,2]),[100*bits,N]));
%
%
%
high frequency
filter response: (frcut/T )/sqrt((frcut/T)^2+x^2 )
where T- is the time interval between the signals
( here any other filter could be implemented )
frcut= 20;
i.e.
% coefficient to adjust the ISI effect: when frcut->inf,
% if the effective cut off frequency of the channel is
increased
% the ISI effect disappears
nfrcut=bits*frcut;
freq=
nfrcut./sqrt(([1:100*bits]'*ones(1,N)).^2+nfrcut^2); % change
this formula in case another filter needs to be applied
%it must have the format: F(100*bit,1)*ones(1,N)
freq=freq/sqrt(mean(abs(freq(:,1)).^2)); % --normalizing the filter...
%------------------------------end filter frequency
response
%-----applying filter in frequency domain, reversing to time with ifft
%------ to get the noiseless part (with ISI) of the signal and noise
m_r_isi= ifft( m_r_ch.*freq );
m_r_isi=m_r_isi(100:100:end,:)'; %isi applied to signal without noise
m_r_noi= ifft( (m_r_no).*freq );
m_r_noi=m_r_noi(100:100:end,:)'; %isi applied to noise
ZF=1;
snr_linear=(10)^(snr/10);
value of SNR (not log)
% Absolute or linear or anti-log
- 64 -
Application of Zero-forcing equalizer
in digital communications system
Zaw Htet Aung (J0704960)
sigma_n=sqrt(1/snr_linear);
for i=1:bits
%------ % equivalent of the commented out
b=m(:,i);
h=m_h(1,i);
noise=m_noise(1,i);
lines:
% ZF applied:
r=m_r(:,i)+ noise*sigma_n;
%END ADDED BLOCK--------------------------est_zf=h.\r;
%
%
%
hat_zf1=zeros(N,1);
hat_zf1(i)=pskdemod(est_zf,M);
stem(hat_zf1(i))
hold on
%%
end
% end for (for i=1:bits)
subplot(212)
stem(hat_zf1)
xlabel('n'), ylabel('Amplitude')
title('Waveform of demodulated received symbols after passing through
filter')
- 65 -