Project Report

Bit Error Rate Performance Analysis of a Modern Channel Codec in Gaussian Channel Using Higher Modulation Schemes.


ACKNOWLEDGEMENT

I indebted to my supervisor Dr Li X Zhang and wish to express my heartfelt gratitude to her

for supervising and directing me all through the duration of this work. Her helpful comments

and suggestions as well as her critical reviews will not be forgotten. I am most grateful to my

accessor Mohsen Razavi for assisting and encouraging me all the way. Lastly I also wish to

thank all the students who helped me in some way or the other, I am very greatful.


ABSTRACT

The evolutions of wireless communication systems and networks in recent years have been

explosive.

In this work, the performance of a wireless communication system is investigated. This is

done by analysing the bit error rate performance of a modern channel codec in Gaussian

channel using higher modulation schemes. Also, the bit error rate performance for the

simulated and theoretical results of digitally modulated communication channels was

evaluated compared and analysed along with the use of a convolutional encoder.

Furthermore, this report investigates an advanced algorithm called soft output Viterbi

algorithm which is used in the decoding of the convolutionally encoded bits. Modulation

schemes such as BPSK, QPSK, 16-QAM and 64-QAM are used. Large coding gains were

observed with the use of a convolutional encoder in comparison to the uncoded system, with

as much as 4dB coding gain recorded in the QPSK system with convolutional code over the

uncoded system. Simulation and theoretical results of the various modulation schemes used

are presented.


LIST OF FIGURES AND TABLES

FIGURES

Fig 1-3: Block Diagram of a Digital Communication System

Fig 2-1: Block diagram for a communication link

Fig 4-1: Constellation Diagram for BPSK Modulation Scheme

Fig 4-2: Theoretical and Simulated BER against SNR in AWGN for BPSK

Fig 4-3: Constellation Diagram for QPSK Modulation Scheme

Fig 4-4: Theoretical and Simulated BER against SNR in AWGN for QPSK

Fig 4-5: Constellation Diagram for 16 QAM Modulation Scheme

Fig 4-6: Theoretical and Simulated BER against SNR for 16 QAM in AWGN

Fig 4-7: Constellation Diagram for 64 QAM

Fig 4-8: Theoretical and Simulated BER against SNR for 64 QAM

Fig 5-1: BPSK with Convolutional Code and theoretical and Simulated BER against SNR in

AWGN

Fig 5-2: QPSK with Convolutional Code and theoretical and Simulated BER against SNR in

AWGN

Fig 5-3: 16QAM with Convolutional Code and theoretical and Simulated BER against SNR in

AWGN

Fig 5-4: 64QAM with Convolutional Code and theoretical and Simulated BER against SNR in

AWGN


TABLE

Table 4-1: Constellation Mapping for 16 QAM

LIST OF ABBREVIATIONS AND SYMBOLS

Abbreviations

ADC Analogue to Digital Converter

AM Amplitude Modulation

ASK Amplitude Shift Key

AWGN Additive White Gaussian Noise

b/s bits per second

BER Bit Error Rate

BPSK Binary Phase-Shift Key

CW Continuous-Wave

DAC Digital to Analogue Converter

dB Decibels

DC Direct Current

DCS Digital Communication System

FM Frequency Modulation

FSK Frequency Shift Key

Hz Hertz

IF Intermediate Frequency

ISI Intersymbol Interference

MLD Maximum Likelihood Decoding

PM Phase Modulation

PSK Phase Shift Keying

RF Radio Frequency

RSC Recursive Systematic Convolutional code.


QPSK Quadrature Phase-Shift Key

16QAM 16 Quadrature Amplitude Modulation

64QAM 64 Quadrature Amplitude Modulation

SNR Signal-to-Noise Ratio

TCM Trellis Coded Modulation


Table of Contents ACKNOWLEDGEMENT ......................................................................................................... 3

ABSTRACT ............................................................................................................................... 4

LIST OF FIGURES AND TABLES ......................................................................................... 5

LIST OF ABBREVIATIONS AND SYMBOLS ...................................................................... 6

1. INTRODUCTION ............................................................................................................... 10

1.1 Evolution of Communication ......................................................................................... 10

1.2 Digital Communication .................................................................................................. 11

1.3 Fundamentals of Digital Communication ...................................................................... 12

1.4 Project Objectives .......................................................................................................... 13

1.5 Report Structure ............................................................................................................. 15

2. PROCESS DESIGN ............................................................................................................ 15

2.1 Block Diagram ............................................................................................................... 15

3. CHANNEL MODEL ........................................................................................................... 16

3.1 Additive White Gaussian Noise ..................................................................................... 16

3.2 Channel Capacity ........................................................................................................... 18

4. MODULATION SCHEMES ............................................................................................... 18

4.1 Modulation Overview .................................................................................................... 18

4.2 Detection and Demodulation .......................................................................................... 20

4.3 PSK Modulation Scheme ............................................................................................... 21

4.3.1 BPSK Modulation Scheme ...................................................................................... 21

4.3.2 QPSK Modulation Schemes .................................................................................... 23

4.4 Quadrature Amplitude Modulation ................................................................................ 25

4.4.1 16 QAM ................................................................................................................... 27

4.4.2 64 QAM ................................................................................................................... 29


4.5 Comparisons of BER in AWGN for Different Modulation Schemes ............................ 31

5. CHANNEL ENCODING ..................................................................................................... 32

5.1 Overview of Channel Coding ......................................................................................... 32

5.2 Convolutional Coding .................................................................................................... 32

5.3 Viterbi Decoding Algorithm .......................................................................................... 35

5.4 Modulation Schemes with Convolutional Code ............................................................. 36

5.5 Discussion of results ...................................................................................................... 40

6. CONCLUSION ................................................................................................................... 40

References ................................................................................................................................ 42

Bibliography ............................................................................................................................ 45

APPENDIX A : UNCODED BPSK SIMULATION OVER AWGN ..................................... 46

APPENDIX B: UNCODED QPSK SIMULATION OVER AWGN ...................................... 47

APPENDIX C: UNCODED 16QAM IN AWGN ................................................................. 49

APPENDIX D: UNCODED 64QAM IN AWGN ................................................................... 51

APPENDIX E: CODED BPSK IN AWGN ............................................................................. 53


1. INTRODUCTION

1.1 Evolution of Communication

Communication fundamentally described as the transmission of information from one point

to another, has been an important source in the rapid development of the world today. The

radical growth of the wireless communication segment of the communication industry has not

only captured the attention of the public but also that of the media.

In ancient times, information was transmitted wirelessly through the use of signals such as

smoke signals, flag signals, signal flares or torch signals. To convey these messages over

long distances, observation stations were built on hilltops and along roads and the messages

were passed from station to station. Overtime, these forms of communication began to

evolve.

The advancement into modern communication systems came in form of the telegraph

developed by Samuel Morse in 1837. This brought about a complete revolution in long-

distance communications when the words “What hath God wrought” were transmitted by

Morse’s telegraph between Washington D.C. and Baltimore, Maryland in 1844. The

invention of the telephone in 1875 by Alexander Graham Bell brought about further

improvements in communication systems. He illustrated the possibility of real-time

transmission of speech by electrical encoding and replication of sound with his invention.


This paved way for many other inventors as in 1864, James Clerk Maxwell predicted the

existence of radio waves and Heinrich Hertz went on to experimentally established the

existence of radio waves in 1887. Oliver Lodge demonstrated the possibility of wireless

communications over a short distance (150 yards), and then Guglielmo Marconi

demonstrated the first radio transmission from Cornwall, England to Signal Hill,

Newfoundland 1700 miles away across the Atlantic [1]. With this, radio communications was

born.

1.2 Digital Communication

Initially, radio systems transmitted analogue signals but today most radio systems transmit

digital signals composed of binary bits. The primary reason for this move to digital

communications is because digital communications provides better quality communication

and the capability for information to be more efficient and reliable. The binary bits are

obtained directly from the data signal or by conversion of the analogue signals to digital

signals by digitization. Claude Shannon in 1948 laid the theoretical foundation for digital

communications in his paper titled “A Mathematical Theory of Communication.” He went on

to dispute the general conception that the increasing rate of information transmission over a

channel would increase the probability of error. He illustrated that provided the transmission

rate was below the channel capacity; the probability of error could be reduced [2].


1.3 Fundamentals of Digital Communication

Fig 1-3: Block Diagram of a Digital Communication System [3]

Fig 1-3 shows the block diagram of a typical digital communication system and signal flow

through it. From the signal source to the transmitter, the signal undergoes a series of

transformations and also the same from the receiver to the sink. After the signal is received, it

is formatted. This converts the received signal into digital symbols as signals are processed in

this form within a digital modulation system [4]. Analogue to digital conversion and

compression of the signal (removal of redundant bits) occurs in the source encoder.

Encrypting the signal allows for security and channel coding reduces the probability of error

or the signal-to-noise ratio requirement at the expense of bandwidth. Multiplexing and

multiple accesses combine signals that may have originated from different sources [5]. The

process whereby symbols are converted to waveforms suitable with the transmission channel

is called modulation. Up until the modulator, the baseband signal is characterized by a

sequence of digital symbols. After modulation occurs, the message signal takes the form of a

digital waveform. Frequency spreading produces a signal prone to interference. The lower

blocks are the reverse process of the upper blocks. From reception of the message signal until

the demodulation process where it takes the form of a bit stream, the received message signal

appears as a digital waveform [6].


Despite the various digital communication system processes, there are three basic elements to

every digital communication system are the formatting, modulation and demodulation. The

process can be grouped into transmitter, the channel and the receiver. The transmitter

converts the message signal received into a suitable digital waveform for transmission over

the channel. Due to channel imperfections, the transmitted message signal is distorted by

noise during transmission and the received message signal is a corrupted version of the

transmitted message signal. A measure used to evaluate how much a signal has been

corrupted by noise is the SNR [7]. It compares the level of a signal to the level of background

noise and is defined the ratio of the average signal power to the average noise power. The

higher the ratio, the less obvious the background noise is. SNR is expressed in decibels (dBs).

The number of bit errors is due to the distortion of the transmitted signal as a result of noise

in the communication channel and can be defined as the number of received bits of data that

have been altered due to distortion in the channel. The number of bit errors divided by the

total number of transferred bits during a particular time interval is called the bit error rate

(BER) and is usually expressed as a percentage. The smaller the BER, the more reliable the

communication system is.

Two principal resources are exploited in a communication system and they are the

transmitted power and the channel bandwidth, hence communication channels can be

classified as either band limited or power limited. The transmitted power is the average

power of the transmitted signal while the channel bandwidth is defined as the band of

frequencies allocated for the transmission of the message signal [8].

Radio technology accelerated at a fast rate through the years to enable transmissions over

larger distances with better quality, less power and smaller devices thereby enabling public

radio communications and wireless networking. Radio waves are generally utilized by

antennas of appropriate size with wavelengths ranging from hundreds of meters to about one

millimetre. They are used for transmission of information through modulation. Television,

mobile phones and wireless networking all use radio waves.

1.4 Project Objectives

This project looks into some blocks in the digital communication system. These are the

formatting block, the channel encode, modulate, demodulate, and the channel decode blocks.


The overall aim of this project is to analyse a modern channel codec using the convolutional

coding technique and higher modulation schemes over a gaussian channel (AWGN Channel).

The project objectives are to understand

(a) Noise and its effect during data transmission.

(b) Maximum likelihood decoding using soft decision (SD).

(c) The correction of errors caused by noise by adding controlled amount of redundancy

using a convolutional codec.

(d) Viterbi convolutional decoding algorithm.

The project is divided into four parts

(a) Generation of additive white Gaussian noise (AWGN) and evaluation of its effects on a

data communication system

(b) Study of the performances of the different modulation schemes in the presence of AWGN

without error correction codes

(c) Comparison of the performances of the different modulation schemes in the presence of

AWGN

(d) Understand the effects of a (2, 1, 4) convolutional codec on the performance of a

communication system in the presence of AWGN.

(e) Comparison of the performance of the uncoded scheme and the coded scheme in terms of

coding gain.

The four modulation schemes to be implemented are BPSK, QPSK, 16QAM, and 64QAM.

The performances of the four schemes are compared in terms of plots of the bit error rate

(BER) against signal-to-noise ratio (SNR).

This projects aims to help understand the need to reduce errors caused by imperfection of the

channels through the use of error correction coding. For the completion of this project,

simulations are to be created using matlab to generate a random sequence data generator, the

different modulation schemes to be used, a Gaussian noise generator, a soft decision

demodulator, and a (2, 1, 4) convolutional encoder and its corresponding soft-output viterbi

decoder (SOVA).


1.5 Report Structure

The report is intended to provide a progressive description of the project background,

including the work done, accompanied by the results concluded in the end.

Chapter 2 covers the process design. It also emphasizes the most important blocks for DCS.

Chapter 3 discusses the channel model. It introduces AWGN noise and the channel capacity.

Chapter 4 considers the various modulation techniques and demodulation and deals with the

BER and SNR.

Chapter 5 introduces channel encoding and emphasizes on convolutional coding. Also it

analyses the performance of a DCS by comparing the graphs gotten for the BER for the

various modulation techniques used.

Chapter 6 is the conclusion.

2. PROCESS DESIGN

2.1 Block Diagram

Figure 2-1 shows the process design considered in this work. At the transmitter, input binary

data are randomly generated and convolutionally encoded before being passed into the

modulator. In the modulator, the encoded binary data (codeword sequence) are transformed

into signal waveforms by modulation (BPSK, QPSK, 16QAM and 64QAM). The signal

waveforms are passed through the channel model (AWGN) where it is corrupted by

randomly generated gaussian noise.


Figure 2-1: Block diagram for a communication link

At the receiver end, the reverse of what occured in the transmitting end ensues. The signal is

received at the receiving end corrupted with noise and demodulation takes place. The

demodulator converts a set of time ordered random variables into a code sequence and passes

it to the convolutional decoder. The decoder is then tasked with producing an estimate of the

original randomly generated input binary data.

3. CHANNEL MODEL

3.1 Additive White Gaussian Noise

Noise in the communication channel arises from different sources such as shot noise, black

body radiation from the earth and warm objects and thermal or Johnson noise. Thermal noise

is present in all communication systems and is caused by the thermal motion of electrons in

all dissipative components e.g. resistors and wires. It is a natural source of noise in the

channel that cannot be eliminated. Thermal noise described as a zero-mean Gaussian random

process has three important characteristics which are

Convolutional

Encoder Modulator

Convolutional

Decoder De-Modulator

AWGN

Channel

Input bits

Output bits


o Additive: This term implies that the noise is added or superimposed on the signal and

there is no multiplicative mechanism [9, 12].

o White: When the noise power has a uniform spectral density, it is referred to as white

noise. Here the power spectral density is flat for all frequencies because a thermal

noise source initiates an equal amount of noise power per unit bandwidth at all

frequencies [10].

o Gaussian: A Gaussian process, n(t) is a random function whose value (n) at any

arbitrary time t is statistically characterized by the Gaussian probability density

function p(n) [10].

P(n) = [11]

Where

σ2 = noise variance

A random signal is often represented as the sum of a Gaussian noise random variable and a

dc signal.

z = a + n Equation 3.1.2 [11]

Where

z = random signal

a = dc component

n = Gaussian noise random variable [11]

Therefore the thermal noise process in communication receivers is modelled as an AWGN

noise process. It consists of a linear addition of white noise with a uniform spectral density

and a Gaussian distribution of amplitude. The noise power spectral density of N0/2 is constant

at all frequencies up to about 1012Hz and has zero mean. AWGN noise is memoryless and

affects each transmitted symbol independently [12].


3.2 Channel Capacity

The increasing demand for wireless communications makes it important to determine the

capacity limits of communication channels. The channel capacity of a communication

channel controls the maximum data rate that can be transmitted in a given time with

negligible amount of bit error.

Assuming an AWGN channel is considered with input/output relationship and a channel

bandwidth B and a transmit power P, the channel SNR is a constant given by

Where No is the power spectral density of the noise.

According to Shannon’s information capacity theorem, the capacity of this AWGN channel

free from intersymbol interference is defined by

C = B log2 (1 + SNR) b/s [13]

Where

C = information capacity of the channel measured in bits per seconds

B = channel bandwidth

SNR = signal-to-noise ratio measured at the channel output

This theorem illustrates that data can be transmitted over an AWGN channel of bandwidth B

at the maximum rate of C bits per seconds with arbitrarily small probability of error provided

that an encoding system of sufficiently high complexity is employed [14].

4. MODULATION SCHEMES

4.1 Modulation Overview

Modulation refers to the modification of given signals (usually baseband signals) at low

operating frequencies into the higher operating frequencies with a particular transmission

characteristic. It is the process of conveying a message signals inside another signal that can

be physically transmitted over a radio channel. It involves the transformation of a baseband


message signal into a passband signal and is performed by altering some parameters of a

signal already at the required carrier frequency in accordance with the information to be

relayed. These parameters include the amplitude, phase and the frequency. The signal being

modified is referred to as the carrier signal and the signal doing the modification is called the

information signal. Varying the phase and the amplitude of the signal forms the basis of all

modulation schemes [15].

Modulation processes can be categorized into two methods.

o Continuous-Wave Modulation (CW): In this modulation process, a sinusoidal wave is

used as the carrier. For example, when the amplitude of the carrier is varied in

accordance with the message signal, it is referred to as amplitude modulation [16].

Other forms of this type of modulation include frequency modulation, angle

modulation and phase modulation. Passband data transmission modulates high

frequency sinusoidal carriers and can be called digital CW modulation.

o Pulse Modulation: Here, the carrier consists of a periodic sequence of rectangular

pulses [17].

In recent times, digital modulation techniques have taken over from analogue modulation

techniques. Examples of these modulation techniques include

o PSK: digital data modulates the phase of the carrier

o ASK: digital data modulates the amplitude of the carrier

o FSK: digital data modulates the frequency of the carrier

A unique pattern of binary bits are usually assigned equally to each of the phases, frequencies

and amplitudes in the above modulation techniques. The symbol represented by a particular

phase, frequency or amplitude comprises of this number of bits. In any M-ary system, the

alphabet M represents the total number of symbols and M = 2N. Each symbol represents a

message consisting of N bits [18]. The modulation alphabet M can be represented on a

constellation diagram for modulation schemes where the frequency of the modulated signal is

constant. The constellation diagram is a representation of a signal modulated by digital

modulation scheme. The symbols are represented as complex numbers and are viewed as

points on the complex plane. The in-phase signal and quadrature phase signals are substituted

for the real part (cosine carrier signal) and the imaginary part (sine carrier signal)


respectively. They are commonly called the I-axis and the Q-axis. The effect of noise affects

these components independently.

4.2 Detection and Demodulation

At the receiver, the transmitted signal across the AWGN channel becomes distorted as a

result of noise added to the signal. The received signal becomes disrupted by noise and is

expressed as

[19]

Where

r(t) = received signal

si(t) = transmitted signal

n(t) = zero mean Gaussian noise process

The receiver tries to undo the deterioration introduced by the channel and to demodulate the

received signal. It does this in two processes. First it detects the received signal by reducing it

to a single number with a signal component and noise component and finds the

mean value.

[20]

In the second step, a decision is made as to which signal was transmitted by comparing

to a threshold. The demodulator proceeds to convert the set of random variables into an

identical message sequence, similar to the originally input message sequence. However, the

demodulator can be configured in two ways.

o To give a hard output: In this form, the output of the demodulator is quantized on two

levels to give either a zero or a one.

o To give a soft output: In this form, the demodulator is implemented to give output

values greater than two quantization levels. It provides more information as compared

to hard output which is used to recover the message sequence with a better error

performance than that of hard output. Eventually, every message decision out of the

demodulator must be a hard one. [21]


4.3 PSK Modulation Scheme

This is a digital modulation scheme that transmits data by modulating the phase of a carrier

wave. Digital data is represented by a finite number of phases with each phase assigned a

unique pattern of binary bits. This unique pattern of bits each forms the symbol that is

represented by the particular phase. PSK modulation schemes can be represented on a

constellation diagram where the amplitude of each point along the in-phase axis is used to

modulate a cosine wave and the amplitude along the quadrature axis to modulate a sine wave.

The constellation points are chosen such that they are positioned with uniform angular

spacing round a circle so all the points can be transmitted with the same energy. BPSK and

QPSK modulations schemes are two examples of Modulation schemes that alter their phases.

The next two sub-chapters will consider BPSK and QPSK modulation schemes.

4.3.1 BPSK Modulation Scheme

Binary Phase Shift Key, the simplest form of PSK is a modulation scheme that uses two

phases separated by 180o or π radians, to relay information across the channel by adjusting

the phases of the carrier signal as needed. It is also be referred to as 2PSK or phase reversal

key (PRK) because of its antipodal state (i.e. states separated by ф = 180o). The carrier signal

is adjusted between these two phases. It has been stated that in any M-ary system, the

alphabet M represents the total number of symbols and the total number of bits per symbol

can be found from the mathematical expression M = 2N, where N is the number of bits. For

the BPSK modulation system, M = 2, thus N =1. This implies that the constellation diagram

for BPSK consists of two symbols and can only modulate at one bit per symbol. The

constellation diagram is shown in Fig 4-1.


Fig 4-1: Constellation Diagram for BPSK Modulation Scheme [22]

The general form for BPSK is expressed as

[23]

Where fc = frequency of the carrier wave.

Using the general form above, binary data can be transmitted with outputs of two phases (0

and π radians). For example to transmit a bit 1, the signal s1(t) is sent and is declared as

[24]

When the symbol is changed, the phase of the carrier is changed by π radians. Hence to

transmit a bit 0, the signal s0(t) is relayed where

= [25]

From the equations above, it is observed that a bit 1 is modulated into a +1 and a bit 0 is

modulated into a -1.

The theoretical BER of BPSK in AWGN is expressed below

[26]

The Matlab scripts used in this project generated random bits of 0’s and 1’s before

modulating them digitally (BPSK modulation) into -1’s and +1’s, respectively. At the

receiver end, the received soft values (also known as the channel observation), was directly

fed into the detection scheme, where soft values below 0 were detected as bit 0 while soft

values above 0 detected as bit 1. (If there we to be a SOVA decoder involved, then the

received soft values are fed directly into the SOVA, and the output of the SOVA is then

detected digitally as explained above).


The plot of the BER against the SNR is obtained to analyse the performance of BPSK

modulation in a digital communication system. Fig 4-2 shows the curve obtained.

Fig 4-2: Theoretical and Simulated BER against SNR in AWGN for BPSK

BPSK modulation is one of the most potent of all the Phase Shift Key forms because it takes

an extremely high level of noise for the demodulator to make an incorrect decision. Also its

ease of demodulation, power efficiency, low cost of implementation and straight forward

carrier recovery are further reasons for its potency. For these reasons, applications such as

power line communications, cable modems, deep space telemetry and GPS still use BPSK

modulation schemes. [27]

4.3.2 QPSK Modulation Schemes

QPSK modulation scheme is also known as 4-PSK consists of BPSK modulation on both the

in-phase and quadrature components of the signal. In a QPSK modulation scheme, two

successive bits in a data sequence are grouped together. Consequently, the data rate is


doubled as compared to that of the BPSK modulation scheme or the bit rate is reduced. This

reduction is bit rate, causes a direct reduction in the bandwidth of the channel. The reason for

this reduction is that in a communication system, the channel bandwidth depends on the bit

rate and when two or more bits are combined on a symbol, the bit rate is reduced. This

reduction in the bit rate also causes a reduction in the frequency of the carrier required. Thus,

as a result of the grouping of bits in symbols, the transmission channel bandwidth is reduced.

Also, the BER for QPSK modulation scheme is exactly the same as the BER for BPSK

because QPSK maintains the same data-rate of BPSK but at half the bandwidth of BPSK.

QPSK uses four points (symbols) on the constellation diagram spaced equally around a circle.

It undergoes four changes in phase and as a result can encode two bits per symbol because 4

= 22. When the symbol is changed to the next symbol, the phase of the carrier is changed by

π/4 radians (45 degrees). A symbol error will occur if the received signal is displaced across

either of the axis. The probability of displacement is the same in each direction.

Fig 4-3: Constellation Diagram for QPSK Modulation Scheme

The general form for QPSK is shown below

[28]


This yields four phases’ π/4, 3π/4, 5π/4, and 7π/4.

In this project, the randomly generated bits were paired into two’s with each pair digitally

modulated. At the receiver end, the received soft values pertaining to the transmitted bits are

then demodulated using soft demodulation. In the soft demodulator, the received soft bits are

firstly detected and converted into 1’s and 0’s (corresponding to the received bit as at the

transmitter). For example if soft bits [-0.45 2.32] is received, this is converted into a [0 1]

bits. Euclidean distance between the received soft value and their corresponding received bits

of 0’s and 1 is then evaluated for symbols whose input bits are 0’s or a 1 for the entire QPSK

constellation scheme. The minimum Euclidean distance for each comparison is then selected

and subtracted from each other, resulting to the soft output from the demodulator.

Fig 4-4: Theoretical and Simulated BER against SNR in AWGN for QPSK

A typical QPSK modulation scheme is used in applications such as UMTS, terrestrial radio

and geostationary satellite communications because the bandwidth efficiency is doubled

without power penalty and is also it is a very power efficient modulation scheme [29].

4.4 Quadrature Amplitude Modulation

Quadrature Amplitude Modulation (QAM) is a form of modulation scheme that conveys two

analogue message signals or two digital bit streams by modulating the amplitudes of two


carrier waves using the amplitude shift keying digital modulation scheme. Two physically

dependent message signals are applied to two separate product modulators that are supplied

with two carrier waves of the same frequency but differing in phase by 90o [30]. They are

called quadrature carriers or components as a result of this difference. The transmitted signal

consists of a sum of these two products modulator outputs and they occupy the same

bandwidth. The resulting signal after the combination is a combination of both phase-shift

keying and amplitude-shift keying. At the receiver output, separation of the two message

signals occurs; hence it can be viewed as a bandwidth conservation scheme [31].

QAM conveys data by changing some aspect of a carrier signal in response to a data signal.

The amplitude of two waves, 90 degrees out-of-phase with each other (in quadrature) are

changed (modulated or keyed) to represent the data signal. Amplitude modulating two

carriers in quadrature can be equivalently viewed as both amplitude modulating and phase

modulating a single carrier.

In QAM, constellation points are arranged with vertical and horizontal spacing in a squared

grid. 16 QAM, 64 QAM, 128 QAM and 256 QAM are some of the most common forms of

QAM. With the increasing number of constellation points, the points get closer together and

it is possible to transmit more bits per symbol. However, due to the close proximity of these

points to each other, they are more susceptible to noise thus resulting in a higher BER.

Hence, a higher order QAM has a higher data rate but relays data less accurately that lower

order QAM.

The QAM signal waveform is expressed as

[32]

Where Am =

AI and AQ are information bearing signal amplitudes of the quadrature carriers and is the

phase modulation. The next section of this report focuses on the 16 QAM and the 64 QAM.


4.4.1 16 QAM

In 16QAM modulation, 16 points are present on the constellation diagram. It includes four in-

phase values and four quadrature values resulting in 16 different states for the signal. 4 bits

per symbol can be transmitted consisting of two bits for the in-phase component and two bits

for the quadrature component. Different ways exists of building up the signal constellation,

however rectangular signal constellations have the noticeable advantage that they can be

generated as two PAM signal impressed on the phase quadrature carriers and can easily be

demodulated. The constellation diagram is shown in Fig 4-5 below

Fig 4-5: Constellation Diagram for 16 QAM Modulation Scheme [33]

As was mentioned earlier, in 16 QAM, the four bits transmitted can be split into I and Q

components. Assuming the first and second bits b0and b1 respectively represents the I

component, and the third and fourth bits b2 and b3 represents the Q components, a table of

the I and Q components can be drawn [ Table 4-1]. If for instance an input sequence of 0000,

1101 is to be modulated, the two bits for the Q component are 00 and 11 while the two bits

for the I component are 00 and 01. From the table, it is observed that when modulated, 00 and

11 on the Q component modulate to -3 and +1 respectively while 00 and 01 on the I

component modulates to -3 and -1. Thus when combined, 0000 and 1101 is modulated to

(-3, -3) and (+1, -1).


b0b1 I b2b3 Q

00 -3 00 -3

01 -1 01 -1

11 +1 11 +1

10 +3 10 +3

Table 4-1: Constellation Mapping for 16 QAM

Performance of 16 QAM in AWGN in the case of BER can be expressed as

[34]

In this project, the randomly generated bits were paired into four’s with each pair digitally

modulated. At the receiver end, the received soft values pertaining to the transmitted bits are

then demodulated using soft demodulation. In the soft demodulator, the received soft bits are

firstly detected and converted into 1’s and 0’s (corresponding to the received bit as at the

transmitter). For example if soft bits [-0.45 2.32 1.45 -0.33] is received, this is converted into

a [0 1 1 0] bits. Euclidean distance between the received soft value and their corresponding

received bits of 0’s and 1 is then evaluated for symbols whose input bits are 0’s or a 1 for the

entire 16QAM constellation scheme. The minimum Euclidean distance for each comparison

is then selected and subtracted from each other, resulting to the soft output from the

demodulator.


Fig 4-6: Theoretical and Simulated BER against SNR for 16 QAM in AWGN

4.4.2 64 QAM

In 64QAM modulation, 64 points are present on the constellation diagram. It includes eight

in-phase values and eight quadrature values resulting in 64 different states for the signal. 6

bits per symbol can be transmitted consisting of three bits for the in-phase component and

three bits for the quadrature component. The constellation diagram is shown in Fig 4-7.


Fig 4-7: Constellation Diagram for 64 QAM [35]

Performance of 64 QAM in AWGN in the case of BER can be expressed as

[36]

Randomly generated bits were paired into six’s with each pair digitally modulated. At the

receiver end, the received soft values pertaining to the transmitted bits are then demodulated

using soft demodulation. In the soft demodulator, the received soft bits are firstly detected

and converted into 1’s and 0’s (corresponding to the received bit as at the transmitter). For

example if soft bits [-0.45 2.32 1.45 -0.33 0.33 1.90] is received, this is converted into a [0 1

1 0 1 1] bits. Euclidean distance between the received soft value and their corresponding

received bits of 0’s and 1 is then evaluated for symbols whose input bits are 0’s or a 1 for the

entire 64QAM constellation scheme. The minimum Euclidean distance for each comparison

is then selected and subtracted from each other, resulting to the soft output from the

demodulator.


Fig 4-8: Theoretical and Simulated BER against SNR for 64 QAM

4.5 Comparisons of BER in AWGN for Different Modulation Schemes

From the graphs of the BER against SNR obtained, comparisons can be made for the

different modulation schemes. It is observed that for the same data rate for BPSK and QPSK

modulation schemes, the BER are exactly the same but QPSK modulates at half the

bandwidth of BPSK. Also, it is noticed that as the number of constellation point increases, the

data rate increases but modulation schemes with a high number of constellation points are

more prone to error than those with lower constellation points. This increase in the BER,

results from the compact nature of the points on the constellation diagram for high

modulation schemes.


5. CHANNEL ENCODING

5.1 Overview of Channel Coding

The channel encoder aims to protect the source data against errors introduced by noise and

other types of distortion [37]. Channel encoding can be segmented into two study areas which

are waveform coding and structured sequences. Waveform coding deals with the

transformation of the waveform into better waveforms to enable detection process less

susceptible to errors. Structured sequences have to do with transformation of data sequences

into better sequences having structured redundancy (redundant bits). The redundant bits are

utilized at the receiver used for the detection and correction of errors [38, 37].

Two types of channel coding are

1 Block Coding: In this form of coding, the encoder accepts information in successive k-bit

blocks. The successive k-bit blocks are independent hence the encoder has zero-memory. The

encoder adds (n-k) redundant bits derived from logical operations on the k information bits to

form n-bit codewords [39].

2. Convolutional Coding: The encoder accepts information bits as a continuous sequence and

needs a memory of length m because the successive m bits are dependent. The encoder

operates on the incoming sequence using a sliding window of length m bits.

This report focuses on the convolutional coding type of channel coding.

5.2 Convolutional Coding

Convolution coding described by 3 integers n, k and K, is a form of error-correcting code

where the code rate of the code is k/n (information per coded bit). Redundant bits are

generated using modulo-2 convolutions. k is the information bit symbol to be encoded, and

this is transformed into an n-bit symbol. K the constraint length of the convolutional code,

expressed in bits is defined as the number of shifts over which a single message bit can

influence the encoder output [40]

An important characteristic of convolutional codes that differentiates it from block codes is

the availability of a memory in the encoder. Another characteristic of convolutional coding is

that the encoder must deal with sequences as the codeword cannot be separated unlike in

block coding where codewords are produced on a block by block basis. In an encoder with an


m-stage shift register, the memory of the encoder equals m message bits and K=M+1 shift are

required for a message bit to enter the shift register and finally come out. The memory order

of the code M=K-1.

How does it work? Each message sequence m is transformed into a unique codeword

U = G(m), where G is the generator polynomial. At each unit of time, k-bits are shifted into

the first k stages of the register. The bits in the memory register are shifted k stages to the

right and the outputs of the n adders are sequentially sampled to obtain the binary code

symbols or coded bits. In convolutional coding, a given k-tuple within m does not uniquely

define its assiociated n-tuple within U since the encoding of each k-tuple is not only a

function of that k-tuple but is also a function of the K-1 input k-tuples that preceeds it [41]

The modulator uses these code symbols to decide the waveforms to be transmitted over the

channel. Shifts register implements sliding windows and the input data are presented a block

at a time to k shift registers in parallel through a demultiplexer [42]. Output of the shift

registers are combined by n sets of weighted modulo-n adders.

A convolutional encoder can be represented in a couple of ways.

• Connection representation [43]

• Polynomial representation

• State Diagram

• Tree Diagram

• Trellis Diagram

The connection representation is the easiest to understand but the most important are the State

diagram, tree diagram and the trellis diagram.

For the implementation of the tree diagram, each branch of the tree represents an input

symbol with the corresponding pair of output binary symbols indicated on the branch. An

input 0, specifies the upper part of the branch, and a 1 represents the lower branch. A specific

path in the tree is traced from the left to the right in accordance with the input message

sequence to get its coded form. The corresponding coded symbols on the branches of that

path constitute the input message sequence. However, a primary setback in the tree diagram

is the size of the tree. The number of branches on the tree increases as a function of 2L where


L represents the number of bits in the message input sequence. The causes an exponential

increase in the size of the tree with the length of the message sequence and quickly becomes

unrestrained. This increase in size leads to a large amount of redundancy in the diagram and

it is observed that after a time t = K, the tree structure repeats itself. The reason being for this

that generally in the implementation of tree diagrams, the tree structure repeats itself after K

branching and from there onwards, the upper and lower halves of the trees are identical [44].

Since the upper and lower halves of the tree are identical and all succeeding paths will be

indistinguishable, they can be merged to reduce its size. The merging of these halves of the

tree diagram results in the trellis diagram.

The trellis diagram provides a more controllable encoder than the tree diagram and is very

important in convolutional coding as it is used for the decoding process. The concept for the

implementation of the trellis diagram is almost the same as that for the tree diagram but in

this, the output generated by an input bit 0 is represented by a straight solid line and the

output generated by an input bit 1 is represented by dashed lines. At each unit of time the

trellis requires 2K-1 nodes to represent the 2K-1 possible encoder states [45].

A convolutional encoder may take up a finite number of states between which it moves at

each block period according to the input block. The state diagram is a directed graph showing

the states and the allowed transitions between them. Branch labels indicate data input and

code output for each state transition in the form data/code.

A convolutional encoder may be systematic or non-systematic. A systematic convolutional

encoder is one in which the information bits and the parity bits are separated while in a non-

systematic convolutional code, the information bit and the parity bits cannot be separated.

Also, a convolutional code can be recursive or non-recursive. A non-recursive convolutional

code is one that has only a feed forward system but no feedback. A recursive convolutional

code employs a feedback mechanism (feeding one or more of the tap outputs in the shift

register back to the input) [46].


5.3 Viterbi Decoding Algorithm

The concept for decoding a convolutional code is the Maximum Likelihood Decoding (MLD)

concept. According to this concept, the decoder chooses a particular received codeword

sequence corrupted with noise as the transmitted sequence if the likelihood is greater than all

the other possible likelihoods of the other transmitted sequences. The viterbi decoding

algorithm is the decoder used in decoding convolutional codes and it performs MLD.

Viterbi decoding algorithm was discovered by viterbi in 1967 [47]. It uses the trellis diagram

for the decoding process. This algorithm involves calculating a measure of similarity or

distance between the received signal at a particular time instant and all the trellis paths

entering each state at that time instant. The decoder does this and advances deeper into the

trellis eliminating trellis paths not considered for maximum likelihood choice from the trellis

diagram. When two paths enter the same state, the chosen path is called the surviving path

[48].

Two types of viterbi decoding include:

1. Hard output-decision Viterbi Algorithm (HODA): finds a path through the trellis that

most likely resembles the received code sequence and deletes the other paths. This

closest path here is found in terms of Hamming distance. Also, the output of the

demodulator here is quantized to two levels, 0 and 1 and fed into the decoder.

2. Soft output-decision Viterbi Algorithm (SOVA): Here, Euclidean distance metric is

used instead of hamming distance. The received bit used in the Euclidean distance

metric is processed by multi-bit quantization. This method provides the decoder with

more information than is provided in the hard decision case.

Every message decision out of the decoder must be a hard decision however; soft-decision

viterbi algorithm provides the decoder with more information which helps for better

recovering of the message sequence with a better error performance as compared to that of

hard decision decoding.

This report shall discuss a Recursive Systematic Convolutional code (RSC). This is a

convolutional code that has both feedfoward generator and feedback generator and where the

input and parity bits are easily separable. The reason for making the convolutional codes

recursive is to make the internal state of shift register depend on past outputs. This affects the

behaviour of the error patterns (a single error in the systematic bit produces an infinite


number of parity errors), with the results that a better performance of the overall coding

strategy is attained [49].

The feedfoward generator in octal is 15 but translates to 1101 in binary. The feedback

generator is 13 in octal and translates to 1011 in binary. The convolutional encode has an

input of one (i.e. it takes a message sequence one at a time) and an output of two. This gives

it a code rate of ½.

5.4 Modulation Schemes with Convolutional Code

QPSK with Convolutional Code

In this project, randomly generated information bits are firstly encoded with the

Convolutional encoder. The encoded bits are then digitally modulated using the BPSK

modulation scheme. The modulated signals are then transmitted in a Gaussian channel. At

the receiver end, the received soft values are then directly fed into the SOVA decoder. The

SOVA decoder then outputs soft values which it believes is the originally transmitted symbol.

The output from the SOVA is then sent into a detector to where soft values lower than zero is

assigned a 0 and soft values above zero is assigned a 1. The detected bits are then compared

with the originally generated random bits for the computation of the bit error rate (BER) at

that particular signal to noise ratio.

Fig 5-1: BPSK with Convolutional Code and theoretical and Simulated BER against SNR in

AWGN


QPSK with Convolutional Code


Convolutional encoder. The encoded bits are then digitally modulated using the QPSK

modulation scheme. The modulated signals are then transmitted in a complex Gaussian

channel. At the receiver end, the received soft values are then sent into the QPSK soft

demodulator as explained in the case of the uncoded QPSK system. The output from soft

demodulator is then fed into the SOVA decoder. The SOVA decoder then outputs soft values

which it believes is the originally transmitted symbol.





Fig 5-2: QPSK with Convolutional Code and theoretical and Simulated BER against SNR in

AWGN


16 QAM with Convolutional Code


Convolutional encoder. The encoded bits are then digitally modulated using the 16QAM


channel. At the receiver end, the received soft values are then sent into the 16QAM soft








Fig 5-3: 16QAM with Convolutional Code and theoretical and Simulated BER against SNR

in AWGN


64 QAM with Convolutional Code


Convolutional encoder. The encoded bits are then digitally modulated using the 16QAM


channel. At the receiver end, the received soft values are then sent into the 16QAM soft








Fig 5-4: 64QAM with Convolutional Code and theoretical and Simulated BER against SNR

in AWGN


5.5 Discussion of results

Coding Gain

This is defined as the reduction in the required SNR to achieve a specified probability. It is

the measure in the difference between the SNR ratio levels between the coded and the

uncoded system required to reach the same BER levels when used with error correcting

coding.

The performance curves obtained from the four different modulation schemes have been

investigated. If for instance, the uncoded BPSK modulation scheme with is observed, it is

noticed that at a BER of 10-2 for instance, the corresponding SNR value is approximately

4dB. While that of the coded BPSK modulation scheme is approximately 2dB at the same

BER. From this, it is illustrated that the coded scheme gives at least 2 dB coding gain over

the uncoded case. The importance of coding gain in communication systems cannot be

overemphasised due to the merit it gives in communication systems. First, coding gain

ensures less power is required in the transmission system while maintaining the systems

transmission reliability. This in turn brings a lot of financial savings to the communication

providers.

Also, the slight degradation in performance between the different coded systems illustrates

the trade-off between data security and error propagation due to noise. In generating the BER

plots shown, it was observed that degradation decreases as the amount of noise in the system

falls.

6. CONCLUSION This report has described the design and simulation of digital communication systems using

higher modulation schemes in AWGN channel. The performance of digital communication

systems have been analysed using a plot of the BER against SNR. This report also shows that

the use of channel coding is very important in today’s digital communication systems. First,

channel coding increases the reliability of the transmission system. Secondly, power (energy)


savings is done with the use of channel coding to the advantage of the communication service

providers and also making the world a greener place to live.


References [1] Haykin S. 4ed. Communication Systems. New Jersey: John Wiley & Sons, 2001.

[2] Haykin S. 4ed. Communication Systems. New Jersey: John Wiley & Sons, 2001.

[3] Bernard S. A Structured Overview of Digital Communications. IEEE Communications

Magazine, August 1983, Fig 1, p.5.

[4] Bernard S. 2ed. Digital Communications: Fundamentals and Applications. New Jersey:

Prentice Hall, 2001.













[12] Li X Zhang. ELEC 5446, Coding and Modulation for Digital Communication System

Lecture Notes, University of Leeds UK, 2009, not published.









[18] Proakis J. 4ed Digital Communications. New York, McGraw Hill, 2000







[22]














[32] Santoso T. B. Developing on Simulation Programming of 16QAM in the AWGN Channel

[online]. [Accessed 2nd May 2011]. Available from: http://lecturer.eepis-

its.edu/~tribudi/LN_suzuki_fukawa_lab/finalr7.pdf


[33] DSP Log Signal Processing for Communication [online]. [Accessed 3rd May 2011].

Available from: http://www.dsplog.com/2008/06/05/16qam-bit-error-gray-mapping/

[34] Harada H and Prasad R: Simulation and Software Radio for Mobile Communications.

Boston, Artech House Publishers.

[35] [online]. [Accessed 10th May 2011]. Available from

http://www.atmovermpeg.com/images/64-qam-constellation-diagram.gif

[36] Harada H and Prasad R: Simulation and Software Radio for Mobile Communications.

Boston, Artech House Publishers.

[37] Li X Zhang. ELEC 2430, Channel Coding. Lecture Notes, University of Leeds UK,

2010, not published.



[39] Li X Zhang. ELEC 2430, Channel Coding. Lecture Notes, University of Leeds UK,

2010, not published.






















Bibliography


APPENDIX A : UNCODED BPSK SIMULATION OVER AWGN clear all EbN0db = input(' Please enter Eb/N0 in dB : default [2.0] '); % This is a loop to help with the determination of the Eb/N0 in dB values if isempty(EbN0db) % the default values here are 0 to 10 dB EbN0db = [0:1:10]; end no_frame = input(' Please enter number of frames to be transmitted : '); % This is a loop to help with the determination of the number of frames used in the simulation if isempty(no_frame) % the default values here are 1 frame no_frame = [1]; end for nEN = 1:length(EbN0db) % this is the loop for the execution of the Eb/N0 values rate = 1 ; eb_n0_dB = EbN0db eb_n0_linear = 10^(eb_n0_dB(nEN)/10); noiseStdDev_bpsk = 1/sqrt(2*rate*eb_n0_linear); err = zeros(1,1) ; % help for computing the Bit error rate errs(nEN,1:1) = zeros(1,1) ; % help for computing the Bit error rate nferrs(nEN,1:1) = zeros(1,1) ; % help for computing the Bit error rate for nframe = 1:no_frame % this is the loop for the execution of the number of frames nos_bits = 5000 ; % this is the number of bits per frame x= round(rand(1,nos_bits)); % random data generation tx = 2*(x-0.5); % BPSK modulation before transmitting signal to -1 or +1 rx = tx + noiseStdDev_bpsk*randn(1,5000); % received BPSK modulated signal in an AWGN channel % Estimate the info. bits xhat = (sign( rx)+1)/2 ; % estimation of what the receivd bit is err = length(find(xhat~=x)) ; % Number of bit errors errs(nEN) = errs(nEN) + err ; % accumulation of the number of bits in errors end


ber(nEN) = errs(nEN)/(nos_bits*nframe) ; %calculate the bit error rate (BER) end % end of the Eb/N0 value loop. xx= [EbN0db] ; semilogy(xx,ber(1:nEN),'b'); %plot the graph hold on; % Theoretical BER %xx= [EbN0db] ; %YY= 0.5*erfc (sqrt(10.^(EbN0db/10)));

APPENDIX B: UNCODED QPSK SIMULATION OVER AWGN clear all; tic ; EbN0db = input(' Please enter Eb/N0 in dB : default [2.0] '); if isempty(EbN0db) EbN0db = [2.0]; end no_frame = input(' Please enter number of frames to be transmitted : '); if isempty(no_frame) no_frame = [1]; end k_Symbol_qpsk = 2; P_av_qam = 1; for nEN = 1:length(EbN0db) en = 10^(EbN0db(nEN)/10); noiseStdDev_qpsk = sqrt( P_av_qam /(k_Symbol_qpsk * en)); err = zeros(1,1) ; errs(nEN,1:1) = zeros(1,1) ; nferrs(nEN,1:1)= zeros(1,1) ; for nframe = 1: no_frame nos_bits = 5000; x= round(rand(1,nos_bits)); [AA GG] = size(x) ;


% QPSK Modulate U = GG/2 ; for sym_No2 = 1 : U qamSymbol(sym_No2,:) = x(1,(sym_No2 - 1)*2 + 1 : (sym_No2 - 1)*2 + 2); inputSeq2(sym_No2,:) = enc_qpsk2(qamSymbol(sym_No2,:)); end inputSeqI2 = inputSeq2(:,1); inputSeqQ2 = inputSeq2(:,2); %----------------------------------------------------------- % Channel %----------------------------------------------------------- real_sig = inputSeqI2 ; imag_sig = inputSeqQ2 ; % Add AWGN noise noiseI_qam = randn(1, U ) * noiseStdDev_qpsk; noiseQ_qam = randn(1, U ) * noiseStdDev_qpsk; receiveI_qam = real_sig + (noiseI_qam)'; receiveQ_qam = imag_sig + (noiseQ_qam)'; %----------------------------------------------------------- % Demodulate %----------------------------------------------------------- despread_symbol = [reshape(receiveI_qam,U,1) reshape(receiveQ_qam,U,1)]; for output_No = 1 : U [output_qam_sd(output_No,:)] = sd_hd_qpsk(despread_symbol(output_No,:)); output_r_sd = reshape(output_qam_sd(output_No,:), 2, 1); outputSymbolsd1((output_No - 1)*2 + 1 : (output_No - 1)*2 + 2,:) = output_r_sd; end outputSymbolsd = outputSymbolsd1' ; r( 1,: ) = outputSymbolsd ; % Estimate the info. bits xest = (sign( r)+1)/2 ; % Number of bit errors in current iteration


err = length(find(xest~=x)) ; errs(nEN) = errs(nEN) + err ; end ber(nEN) = errs(nEN)/(nos_bits*nframe) % end %nEN xx= [EbN0db] ; semilogy(xx,ber(1:nEN),'b'); hold on; % Theoretical BER %YY= 0.5*erfc (sqrt(10.^(EbN0db/10))); %semilogy(xx,YY); toc ;

APPENDIX C: UNCODED 16QAM IN AWGN clear all; tic ; EbN0db = input(' Please enter Eb/N0 in dB : default [2.0] '); if isempty(EbN0db) EbN0db = [2.0]; end no_frame = input(' Please enter number of frames to be transmitted : '); if isempty(no_frame) no_frame = [1]; end k_Symbol_16qam = 4; P_av_qpsk = 5; for nEN = 1:length(EbN0db) L_c=1; en = 10^(EbN0db(nEN)/10); noiseStdDev_qpsk = sqrt( P_av_qpsk /(k_Symbol_16qam * en)); err = zeros(1,1) ; errs(nEN,1:1) = zeros(1,1) ; nferrs(nEN,1:1)= zeros(1,1) ; for nframe = 1: no_frame


nos_bits = 5000; x= round(rand(1,nos_bits)); [AA GG] = size(x) ; % 16QAM Modulate U = GG/4 ; for sym_No2 = 1 : U qamSymbol(sym_No2,:) = x(1,(sym_No2 - 1)*4 + 1 : (sym_No2 - 1)*4 + 4); inputSeq2(sym_No2,:) = enc_16qam2(qamSymbol(sym_No2,:)); end inputSeqI2 = inputSeq2(:,1); % take everything for the 1st column inputSeqQ2 = inputSeq2(:,2); % take everything for the 2nd column %----------------------------------------------------------- % Channel %----------------------------------------------------------- real_sig = inputSeqI2 ; imag_sig = inputSeqQ2 ; % Add AWGN noise noiseI_qam = randn(1, U ) * noiseStdDev_qpsk; noiseQ_qam = randn(1, U ) * noiseStdDev_qpsk; receiveI_qam = real_sig + (noiseI_qam)'; receiveQ_qam = imag_sig + (noiseQ_qam)'; %----------------------------------------------------------- % Demodulate %----------------------------------------------------------- despread_symbol = [receiveI_qam receiveQ_qam]; for output_No = 1 : U [output_qam_sd(output_No,:)] = sd_hd_16qam(despread_symbol(output_No,:)); output_r_sd = reshape(output_qam_sd(output_No,:), 4, 1); outputSymbolsd1((output_No - 1)*4 + 1 : (output_No - 1)*4 + 4,:) = output_r_sd; end


L_all = outputSymbolsd1' ; % Estimate the info. bits xhat = (sign(L_all)+1)/2 ; % Number of bit errors in current iteration err = length(find(xhat(1:5000)~=x)) ; errs(nEN) = errs(nEN) + err end ber(nEN) = errs(nEN)/(nos_bits*nframe) % end %nEN xx= [EbN0db] ; semilogy(xx,ber(1:nEN),'*'); hold on; %Theoretical BER %YY= ((3/8)*erfc (sqrt((2/5)*(10.^(EbN0db/10)))))-((9/64)*(erfc (sqrt((2/5)*(10.^(EbN0db/10))))).^2); %semilogy(xx,YY); toc ;

APPENDIX D: UNCODED 64QAM IN AWGN clear all; tic ; EbN0db = input(' Please enter Eb/N0 in dB : default [2.0] '); if isempty(EbN0db) EbN0db = [2.0]; end no_frame = input(' Please enter number of frames to be transmitted : '); if isempty(no_frame) no_frame = [1]; end niter =1 ;


k_Symbol_qpsk = 6; P_av_qpsk = 21; for nEN = 1:length(EbN0db) L_c=1; en = 10^(EbN0db(nEN)/10); noiseStdDev_qpsk = sqrt( P_av_qpsk /(k_Symbol_qpsk * en)); err = zeros(1,1) ; errs(nEN,1:1) = zeros(1,1) ; nferrs(nEN,1:1)= zeros(1,1) ; for nframe = 1: no_frame nos_bits = 5004; x= round(rand(1,nos_bits)); [AA GG] = size(x) ; % 16QAM Modulate U = GG/6 ; for sym_No2 = 1 : U qamSymbol(sym_No2,:) = x(1,(sym_No2 - 1)*6 + 1 : (sym_No2 - 1)*6 + 6); inputSeq2(sym_No2,:) = enc_64qam2(qamSymbol(sym_No2,:)); end inputSeqI2 = inputSeq2(:,1); inputSeqQ2 = inputSeq2(:,2); %----------------------------------------------------------- % Channel %----------------------------------------------------------- real_sig = inputSeqI2 ; imag_sig = inputSeqQ2 ; % Add AWGN noise noiseI_qam = randn(1, U ) * noiseStdDev_qpsk; noiseQ_qam = randn(1, U ) * noiseStdDev_qpsk; receiveI_qam = real_sig + (noiseI_qam)'; receiveQ_qam = imag_sig + (noiseQ_qam)';


%----------------------------------------------------------- % Demodulate %----------------------------------------------------------- despread_symbol = [receiveI_qam receiveQ_qam]; for output_No = 1 : U [output_qam_sd(output_No,:)] = sd_hd_64qam(despread_symbol(output_No,:)); output_r_sd = reshape(output_qam_sd(output_No,:), 6, 1); outputSymbolsd1((output_No - 1)*6 + 1 : (output_No - 1)*6 + 6,:) = output_r_sd; end L_all = outputSymbolsd1' ; % Estimate the info. bits xhat = (sign(L_all)+1)/2 ; % Number of bit errors in current iteration err = length(find(xhat(1:length(x))~=x)) ; errs(nEN) = errs(nEN) + err end ber(nEN) = errs(nEN)/(nos_bits*nframe) % end %nEN xx= [EbN0db] ; semilogy(xx,ber(1:nEN),'*'); hold on; YY= ((7/24)*erfc (sqrt((1/7)*(10.^(EbN0db/10)))))-((49/384)*(erfc (sqrt((1/7)*(10.^(EbN0db/10))))).^2); Semilogy (xx YY) toc ;

APPENDIX E: CODED BPSK IN AWGN

Project Report

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