i

DS-CDMA systems using Chaotic complex

Bernoulli Spreading codes and Turbo Codes Submitted in partial fulfilment of the requirements

Of the degree of

Bachelor of Technology

Submitted By:

Snehal Chipade (114254)

Piyush Meshram (114243)

SUPERVISOR

Dr. S. Anuradha

(Assistant Professor)

ELECTRONICS AND COMMUNICATION ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY

WARANGAL

2015

ii

APPROVAL SHEET

This Project Work entitled by DS-CDMA systems using Chaotic complex Bernoulli Spreading

codes and Turbo Codes by Snehal Chipade and Piyush Meshram is approved for the Degree

of Bachelor of Technology, Electronics and Communication Engineering.

Examiners:

Supervisor:

Dr S. Anuradha

(Assistant Professor Department of Electronics and communication Engineering)

Chairman:

Dr. T. Kishore Kumar

(Head, Department of Electronics and communication Engineering)

Date: _____________________________

Place: _____________________________

iii

Declaration

We declare that this written submission represents our ideas in our own words and where other

ideas or words have been included, we have adequately cited and referenced the original sources.

We also declare that we have adhered to all principles of academic honesty and integrity and have

not misrepresented or fabricated or falsified any idea / data / fact / source in our submission. We

understand that any violation of the above will be a cause for disciplinary action by the Institute

and can also evoke penal action from the sources which have thus not been properly cited or from

whom proper permission has not been taken when needed.

_________________

Snehal Chipade

(UG114254)

_________________

Piyush Meshram

(UG114243)

Date:

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Certificate

This is to certify that the project work entitled “DS-CDMA systems using Chaotic complex

Bernoulli Spreading codes and Turbo Codes” is a bonafide record of work carried out by

“Snehal Chipade (UG114254) and Piyush Meshram (UG114243) ” submitted to the faculty

of “Electronics and Communications Engineering Department”, in partial fulfilment of the

requirements for the award of the degree of Bachelor of Technology in “Electronics and

Communications Engineering” at National Institute of Technology, Warangal during the academic

year 2014-2015.

Dr . T .Kishore Kumar Dr . S. Anuradha

Head of the Department Assistant Professor

Department of Department of

Electronics and Communication Engineering Electronics and Communication Engineering

NIT Warangal NIT Warangal

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Specifications

Modulation Technique Binary phase shift keying

Data Rate 9600 bps without FEC

4800 bps with FEC

Number of bits transmitted per user 10000

Number of users 30 (max.)

Signal to Noise ratio 8 decibels

Spreading sequences Gold sequences

Chaotic Bernoulli sequences

Bit Error Rate 10-4

Spreading factor 1023

Channels Additive white Gaussian noise channel

Rayleigh fading channel

Multiple access interference model

Channel coding Turbo coding

1/3 RSC convolution coding - 2

Trellis structure- poly2trellis(4, [13 15], 13)

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Abstract

The most important goal of spreading spectrum communication system is to protect

communication signals against interference and exploitation of information by unintended

listeners. In fact, low probability of detection and low probability of intercept are two important

parameters to increase the performance of the system. In Direct Sequence Code Division Multiple

Access (DS-CDMA) systems, these properties are achieved by multiplying the data information in

spreading sequences. Proper correlation function and more freedom of choice of appropriate

codeswith chaotic sequences results in numerous applications for constructing spreading codes. In

this paper, use of two-dimensional Bernoulli chaotic sequences is proposed as spreading codes.

The performance of a multi-user DS-CDMA system will be evaluated by applying these sequences

under Additive White Gaussian Noise (AWGN), Rayleigh fading channel and Multiple access

interference model. Simulation results indicate improvement of the performance in comparison

with conventional spreading codes like Gold codes as well as similar complex chaotic spreading

sequences. Besides, error control coding using turbo coders gives significant improvement in

performance of the system. Turbo coding is most popular technique for error control.

vii

Content

Specifications ………………………………………………………..v

Abstract ……………………………………………………………...vi

List of Figures ……………………………………………………….ix

List of Tables ………………………………………………………..xi

Abbreviation Notation and Nomenclature ………………………….xii

1. Introduction

1.1 Overview ………………………………………………………...…...1

1.2 Objective……………………………………………………………...2

1.3 Outline……………………………………………….……………......3

2. Literature review ………………………………………..…………4

3. Spread spectrum and multiple access system

3.1 Multiple access techniques …………………………………………..7

3.2 CDMA system ..………………………………………………………8

3.3 Spread spectrum modulation technique ………..…………………...10

4. Spreading code for CDMA system

4.1 Overview …………………………………………………................13

4.2 Factors affecting choice of spreading code …………………………13

` 4.3 Pseudo random sequences …………………….…………………….15

5. Introduction to chaotic sequence

5.1 Overview..…………………………………………………...............18

5.2 Chaotic system ……………………………………………………...18

5.2 Chaotic sequence ……..…………………………………………….19

viii

6. Error control coding technique

6.1 Introduction . …………………..…...………………………………23

6.2 Error control coding technique…….……………..……………… ...24

7. Multiple access interference

7.1 Motivation for MAI characterization ……..…………..…………… 34

7.2 Gaussian approximation ….………………...……………………… 35

7.3 Discrete analysis of MAI ..……………………………….…………36

8. Results ………………………………………..………………..…37

9. Conclusion ……………………………………………………..…44

Literature cited ...………………………………………………...….45

Acknowledgement ………………………………………………….47

ix

List of Figures

Fig. 3.1 Working of FDMA and TDMA …………………………………………………..... 8

Fig. 3.2 Working of CDMA system…………………………………………………………. 8

Fig. 3.3 CDMA System…………………………………………………………… …………9

Fig. 3.4 Block diagram of a spread spectrum system ………………………………………..10

Fig. 4.1 m-sequences generator …………………………………………………………… 16

Fig. 4.2 Gold code sequence generator configuration. ………………………………………17

Fig. 5.1 Block diagram of DS-CDMA transmitter with complex spreading sequence ………21

Fig 6.1 Digital Communication Model ………………………………………………….…...24

Fig 6.2 Convolutional Encoder with a=1 n=2and r=1/2 …………………………………..…28

Fig 6.3 Turbo Code …………………………………………………………………………. 28

Fig.6.4 Decoding of Concatenated Code with feedback …………………………………… 29

Fig. 6.5.A) No Systemic Non Recursive convolutional code with polynomial 13,15……….30

Fig. 6.5.B) Recursive system recursive convolution (RSC) code with polynomial

13(recursivity), 15(parity) ………………………………………………….. 30

Fig. 6.6 A Turbo code with component code 13,15. ………………………………………. 30

Fig. 6.7 A Turbo decoder……………………………………………………… …………….32

Fig. 8.1 Auto- correlation vs time…………………………………………………… ………37

Fig. 8.2 Cross-correction vs time ……………………………………………………………..37

Fig.8.3 Auto correlation of gold and Bernoulli sequences………………………………… .. 38

Fig. 8.4 Cross – correlation of gold and Bernoulli sequence ……………………..…………. 38

x

Fig. 8.5 Performance comparison of Gold and Bernoulli sequence in AWGN channel ……..39

Fig. 8.6 Performance Comparison of Gold and Bernoulli sequence in AWGN channel

with MAI ….…………………………………………………………………………39

Fig. 8.7 performance comparison of gold and Bernoulli sequence in Rayleigh

fading channel. ………………………………………………………………………40

Fig. 8.8 Comparison of Gold and Bernoulli sequence with and without turbo coding………40

Fig. 8.9 Turbo Code Performance for different number of iteration in AWGN channel

without MAI ……………………………………………………………………...…41

Fig. 8.10 Turbo Code Performance for different number of iteration in AWGN channel

with MAI……...………… …………………………………………………………...41

Fig. 8.11 Turbo Code Performance for different number of iteration in Rayleigh

fading channel… ……………………………………………………………………..42

Fig. 8.12 BER vs No. of users in Gold and Bernoulli’s sequence …...………………………42

xi

List of Tables

Table 8.1 BER comparison for Gold and Bernoulli sequences……………………………... 43

Table 8.2 BER comparison for different channel models ……………………………….43

Table 8.3 BER comparison for different number iterations in turbo decoder . ……………...43

xii

Abbreviation Notation and Nomenclature

ACF auto-correlation function

AWGN additive white Gaussian noise

BER bit error ratio

CCF cross-correlation function

CDMA code division multiple access

CIR carrier to interference ratio

CLB chip level based

DS direct sequence

ISI inter symbol interference

LFSR linear feedback shift register

MAI multiple access interference

MUD multi user detection

PN pseudo noise

PSD power spectral density

BPSK binary phase shift keying

QPSK quadrature phase shift keying

SNR signal to noise ratio

SS spread spectrum

TDL tapped-delay-line

FEC Forward error correction

RSC recursive systematic coding

1

Chapter 1

Introduction

1.1 Overview

Code Division Multiple Access (CDMA) cellular network is a promising wireless

technology and it has been in the focus of academic research since many years. In comparison

with Time Division Multiple Access (TDMA) and Frequency Division Multiple Access

(FDMA), CDMA is found to be more attractive for wireless access for its numerous advantages

like frequency reuse and multiuser detection. The performance of CDMA based wireless

systems is largely based on the characteristics of user specific spreading codes. These codes

spread the original information signal to a large bandwidth and also provide unique identity to

the various channels originating from base station as well as mobile station. At the receiver

side, the original information signal is recovered by correlating the received signal with the

synchronized replica of the spreading code. Thus spreading codes play an important role in

CDMA system to spread the user signals and distinguish among different users.

Multiple Access Interference (MAI) generally occurs in CDMA system due to non-

orthogonality between spreading codes and so it restricts the capacity of CDMA systems. The

MAI arising due to cross-correlation between the spreading codes assigned to different users.

A family of spreading codes for which cross- correlation function is zero at zero time shifts is

known as orthogonal codes. Orthogonal spreading can only be used if all the users are

synchronized, as the cross-correlation value between sequences at different time shifts is not

zero. MAI terms cannot be made zero as it is impossible to design spreading codes that have

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zero cross correlation with all shifts of other spreading codes. It is desired to make MAI terms

as minimum as possible by minimizing the magnitude of cross correlation.

Inter Symbol Interference (ISI) is another limiting factor in CDMA system which is a

function of autocorrelation values of the used spreading codes. In order to minimize the ISI

effect Auto-Correlation Function (ACF) of a spreading code should be impulsive type i.e. peak

value at zero time shift and minimum values at other shifts to distinguish the desired user at

receiver side from other users producing MAI.

1.2 Objective

Code Division Multiple Access (CDMA) is used as a multiple access technique in

telecommunications radio system that can transport multimedia traffic at high data rates. The

communications researchers have studied CDMA and are further developing it. In a CDMA

system availability of more number of codes, minimum magnitude of cross correlation,

impulsive peak auto correlation and BER performance plays a major role. So it is needed to

generate spreading code such that it support more number of user, good auto-correlation

property, good BER performance with better cross-correlation property which eliminate

undesirable effect like ISI, MAI by reducing the chance of false synchronization. Hence it is

desirable to have spreading codes which possess both impulsive Auto Correlation Function

(ACF) and all zero Cross Correlation Function (CCF) characteristics with large number of code

sets. Forward error control technique with turbo encoders and decoders is further applied to

achieve higher performance.

1.3 Problem Statement

a) To design chaotic Bernoulli sequences which gives better performance than

conventional gold sequences.

b) To apply error control techniques with turbo codes to improve BER.

3

1.4 Outline

Chapter 2 gives the literature review of the project.

Chapter 3 introduces different multiple access techniques and gives precise idea of DS-CDMA

system.

Chapter 4 introduces different spreading techniques used in DS-CDMA system. M-sequences

and gold sequences are discussed.

Chapter 5 introduces to chaotic system and sequences. Chaotic Bernoulli sequences are

discussed and importance of Bernoulli sequences due to its auto-correlation properties is

described.

Chapter 6 explains different error control coding techniques in brief. Turbo encoders and

decoders are introduced.

Chapter 7 gives MAI model used for simulation.

Chapter 8 include simulation results.

4

Chapter 2

Literature Review

In spreading spectrum systems, main information is spread in a wide bandwidth. One of

the main methods to spread the spectrum is Direct Sequence Code Division Multiple Access

(DS-CDMA) in which information of all users will be sent simultaneously at the same

frequency band. The spectrum will be spread by using the sequences with specialized features

which are called spreading sequences. Indeed, for each user, a particular sequence will be sent

instead of each information bit. The receiver uses the same spreading sequence to detect the

received data. As a result of spreading, the transmitted signal occupies more bandwidth than

the original message. Properties of spreading sequences are mainly determined by their auto-

correlation and cross-correlation.

The code should be orthogonal or quasi-orthogonal to each other in different delays, to

use in multipath channels ideally. The most commonly used sequences in DS-CDMA are

maximal length (m-sequence) and Gold, which are implementable by Linear Feedback Shift

Registers (LFSRs). In recent decades, using chaotic sequences as spreading codes has been of

great interest. Chaotic sequences, despite the deterministic nature, show the irregular and

pseudo-random behaviour. These sequences can be produced in arbitrary length and they are

entirely non-periodic. Due to extreme sensitivity of chaotic sequences in initial conditions,

numerous chaotic sequences with desired properties can be created. Generated chaotic

sequences are almost orthogonal with the low cross-correlation value that made them desirable

to be used in spread spectrum systems.[9]

5

Non-orthogonality between spreading codes in CDMA system users causes MAI which

consequently restricts the capacity of CDMA systems. Many strategies have been proposed to

moderate the effect of MAI, for example, multiuser detection can be employed to use multiple

user information for better detection of each user signal rather than using single conventional

detection. Here the conventional DS-CDMA detector follows a single user detection strategy

in which each user is detected separately without regard for the other users. So a better strategy

is multiuser detection, where information about multiple users is used to improve detection of

each individual user.

Unlike the conventional pseudo-noise sequences, chaotic sequences can be made by

simple recursive equations. These sequences with different characteristics and parameters have

been used in several references.[3][4][5][6].

In [6], it has been shown that chaotic sequences with exponentially vanishing auto-

correlation function can improve the performance of spreads spectrum systems.

In [11] and [2], chaotic sequences with negative autocorrelation at lag of 1 have been

studied as a near optimal spreading sequence.

Complex chaotic sequences are introduced in [3] and [4], and performance of the DS-

CDMA system based on them has been analyzed. The spread spectrum system can have better

performance in terms of cross-correlation and the average Bit Error Rate (BER), using two-

dimensional or complex sequences. This is due to more flexibility to select real and imaginary

parts of each sequence.

In [3], the complex IKEDA map is used to generate two dimensional chaotic sequences.

In a multi-user asynchronous DS-CDMA systems (such as the reverse link of cellular systems),

each user is affected by interference of other users. The average interference can be described

by the auto-correlation function of codes. So by designing the sequences with appropriate auto-

correlation functions, the proper spreading codes sequences can be produced. This idea has

been used to proper design of one-dimensional chaos spreading sequences in [10].

When digital data is transmitted through a communication channel, errors are embedded

6

into it. Error Detection Coding is used to detect and correct the errors in the received

information [8]. Most popular ECC is Turbo Coding. Turbo Codes are also known as Parallel

Concatenated codes and the decoding complexity is small for the dimension of code, whereas

the code length possible is very long. The bounds of Shannon’s Limit also become achievable

for all practical purposes because the decoding complexity becomes small [1]. Use of turbo

coding under MAI channel model is given in [7].

Here, the idea of utilizing two Bernoulli chaotic sequences to form two-dimensional

(complex) sequences is presented. This project shows that correlation properties of new

sequences are better than conventional gold sequences. We have used turbo coding technique

to further achieve better performance. Additionally, according to the new complex sequences,

it is expected that spreading codes with a better cross-correlation than the one-dimensional

codes can be selected.

7

Chapter 3

Spread Spectrum and multiple access system

3.1 Multiple Access Techniques

The main task of the communication system designer is to make the best use of the

system resources. The challenge is to make the most efficient use of the Radio Frequency (RF)

bandwidth. Frequency Division Multiple Access (FDMA), Time Division Multiple Access

(TDMA) and Code Division Multiple Access (CDMA) are the three major access techniques,

used to share the available bandwidth in a mobile communication system to provide group of

users in one RF channel. CDMA technique is a wideband system and it comes under spread

spectrum multiple access technique. Figure 3.2 (a) shows the working of FDMA, in which every

user communicate over an individual channel over the whole period of time. FDMA is often

referred to as the first generation system. In TDMA system, more than one user can share the

same channel at the same time as shown in Figure 3.2 (b). This multiple access method is used

in the Global System for Mobiles (GSM) system, which was the first digital cellular standard

for voice communications and is termed as the second generation system.

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Fig.3.1 Working of FDMA and TDMA system

Figure 3.2 shows the working of CDMA system. In CDMA system all users share a

common channel in time and frequency. All users transmit continuously over the full channel

bandwidth but the users are separated only in space.

Fig.3.2 Working of CDMA system

3.2 CDMA System

CDMA is a multiple access technique that uses spread-spectrum modulation by each

accessing party with its own unique spreading code, with all accessing parties sharing the same

spectrum. Spread spectrum modulation is accomplished by means of PN (Pseudo Noise) code.

This is shown in the Figure 3.4 as given below. Here, Ci (t) and di (t) represent the PN code and

information sequence used for ith user where, i = 1, 2… n. Here the information sequence is

modulated by the wideband spreading signal thereby spreading the information signal spectrum

to a substantially greater bandwidth prior to transmission. It is important to recognize that

CDMA can only be accomplished by spread-spectrum modulation, while spread spectrum

modulation does not mean CDMA.

9

In CDMA each user is provided with an individual, distinctive PN code. If these codes

are uncorrelated with each other, then within the same mobile cell, ‘n’ independent users can

transmit signal at the same time and in the same radio bandwidth. The receivers de-correlate or

de-spread the information and regenerate only the desired data sequence di (t). The concept of

a CDMA system has been illustrated in Figure 3.4. If ‘n’ mobile units are transmitting at the

same time, the base station receiver will have ‘n’ spectrally overlapped and time overlapped

signals. The terms base station and mobile station could be interchanged. If the received power

of all these signals is same as ‘Ps’ then any one of the desired signals will be interfered by ‘n-

1’ other equal powered CDMA signals. Thus at the RF receiver port, the desired carrier to

interference (C/I) ratio will be 1/ (n-1) i.e. C/I will be in negative dB value. This negative C/I

is caused by self-interference, that is, by the ‘n-1’ other Direct Sequence (DS) Spread Spectrum

(SS) carriers that simultaneously occupy the same bandwidth. In the de-spreading and

demodulation process this negative RF wideband C/I is converted into a narrowband positive

baseband signal-to-interference (S/I) component. The baseband S/I have to be sufficiently large

to lead to a relatively low probability of error ‘Pe’. The S/I is frequently designed to be several

decibels higher than the baseband signal-to-noise ratio (S/N).

Fig.3.3 CDMA System

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3.3 Spread Spectrum Modulation Techniques

Spread spectrum techniques employ a transmission bandwidth that is much greater than the

minimum required signal bandwidth. While the system is very bandwidth inefficient for a single

user, the advantage of spread spectrum is that many users can simultaneously use the same

bandwidth without significantly interfering with one another. In a multiple-user, Multiple

Access Interference (MAI) environment, spread spectrum systems become very bandwidth

efficient.

The process of spread spectrum is realized in practice by modulating or multiplying the

data sequence by means of a code sequence, which will not be dependent on the data sequence

in any way. The same spreading code is used at the receiver end to retrieve the original message

signal from the spread-spectrum modulated signal by de-spreading the received signal that may

be corrupted by some other jamming signal. This de-spreading operation at the receiving end

must be performed in synchronism with the spreading operation at the transmitter to allow

perfect detection of the data sequence. The generic block diagram of a spread spectrum system

has been presented in Fig. 3.4

Fig.3.4 Block diagram of a spread spectrum system

In fact, spread spectrum modulation technique possesses some unique characteristics

that have made it beneficial in many applications. In a ground-based mobile radio environment,

it can be used to provide multi-path rejection of the transmitted signal. As independent users

access the channel bandwidth by means of different spreading codes, the multiple-access

technique is generally called Code Division Multiple Access (CDMA).

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In order to enhance the bandwidth of the message signal, spreading codes are

judiciously used in such a system. Binary message sequence {mk} and code sequence {ck} can

be represented by different types of line code. Normally, they are represented by polar non-

return-to-zero form where binary ‘1’ is characterized by a rectangular pulse of unit positive

amplitude and a rectangular pulse of unit negative amplitude symbolize a binary ‘0’.

Accordingly, two types of time varying signal, m (t) and c (t), can be defined in accordance

with information sequence and spreading sequence respectively. At the transmitting end, the

spread-spectrum modulated signal s (t) is generated in accordance with the following equation:

s (t) = c (t).m (t) (1)

The message signal m (t) is a narrowband signal and the spreading signal c (t) is of

wideband in nature. Due to the effect of multiplication, the product signal s (t) has a wideband

spectrum. The process of multiplication has the effect of chopping the data sequence m (t) into

a number of small time increments, called chips. The signal s (t) is used to modulate a high

frequency carrier signal in case of pass band communication.

The received signal r (t) contains the transmitted signal and additive interference signal

that is directed towards the receiver in order to disrupt the communication. Thus the received

signal can be written as:

r (t) = s (t) + n (t) = c (t).m (t) + n(t) (2)

where, n (t) denotes the narrowband noise signal.

In order to retrieve the information bearing signal m(t) from the noise corrupted received

signal r(t), the presence of the spreading signal in perfect synchronism with that of the

transmitter is a necessary criterion at the receiver. In fact, at the receiving end both spreading

and de-spreading operations are done simultaneously on two components of the received signal.

The first component s (t) is wideband in nature, whereas n (t) is a narrow band jamming signal.

The effect of the spreading signal c (t) at the receiving end is to de-spread the wideband signal

s (t) while spreading the narrowband signal n (t) as in transmitter. For the proper reception of

the information sequence, the receiver must operate synchronously with the transmitter.

Receiver of a baseband transmission model consists of a multiplier, followed by a low pass

12

filter. In order to facilitate the coherent detection, presence of a locally generated carrier signal

is an additional mandatory requirement in the receiver of such a system. The output of the

multiplier, x (t), is given by:

x(t) = r(t).c(t) = s(t).c(t) + n(t).c(t) = m(t).c2(t) + n(t).c(t) (3)

Thus the multiplier at the receiver multiplies the spreading signal c (t) twice by the data

signal m (t) and once by the undesired interference signal n (t). In case of polar non- return to

zero line coding, the amplitude of c (t) can only attain two values, namely ±1. Thus irrespective

of the bit pattern present in the spreading sequence, the value of c2(t) is always equal to 1.

Therefore, the multiplier signal can be rewritten as:

x (t) = m (t) + n (t).c (t) (4)

So, the multiplier reproduces the data signal m (t) at its output, except the term n (t).c

(t). This corresponds to a wideband signal and thus can be eliminated from the multiplier output

by using a low pass filter. The filtering operation is actually realized with the help of an

integrator that calculates the area under the signal x (t) for the entire bit duration. Depending

upon the integrator output, a decision is being made. If the magnitude of this value is greater

than zero, a decision is made about the transmission of a binary ‘1’ during this interval. On the

other hand, the receiver decides in favor of the symbol ‘0’ if the integrator output is less than

zero. Occasionally, it may assume a value, which is exactly equal to zero. In such a case, a

random guess is done to determine which symbol was actually transmitted. The bandwidth of

such a filter is so chosen that it can accommodate entire power of the information signal.

Whereas, most of the power of the unwanted signal is filtered out by the low pass filter. Thus,

using this particular type of modulation the actual information signal can be recovered at the

receiver. In this manner, the process of spread spectrum combats against the narrowband

interference signal.

13

Chapter 4

Spreading codes for CDMA system

4.1 Overview

In a spread-spectrum system, spreading code plays a significant role as far as the system

performance is concerned. Thus the choice of the spreading code is very crucial as it strongly

dictates the system BER performance. The ultimate selection of a code family will entirely

depend upon the requirement of the communication system. The huge family of CDMA

spreading codes can be divided broadly into two categories, namely non-orthogonal spreading

code and orthogonal spreading code.

4.2 Factors Affecting Choice of Spreading Codes

In a spread-spectrum system, spreading code plays a significant role as far as the system

performance is concerned. Thus the choice of the spreading code is very crucial as it strongly

dictates the system BER performance. The ultimate selection of a code family will entirely

depend upon the requirement of the communication system. The huge family of CDMA

spreading codes can be divided broadly into two categories, namely non-orthogonal spreading

code and orthogonal spreading code.

14

4.2.1 Auto-Correlation

Auto Correlation Function (ACF) is a measure of the similarity between a spreading

code {an} and its time shifted replica. For a code sequence {an}, it is mathematically expressed

as:

𝐶(𝐾) = ∑ 𝑎𝑛𝑎𝑛+1𝑁𝑛=1 (1)

Ideally, this autocorrelation function should be impulsive type i.e. peak value at zero time

shifts and zero values at all other shifts. This is required at receiver side to differentiate the

desired user from other users producing MAI.

Hence, spreading codes should be chosen such that it ensures highest possible peak value

at zero time shifts and lower correlation peaks at non-zero time shifts (side lobes). For example,

PN codes possess ideal impulsive ACF characteristics but these codes are non-orthogonal.

4.2.2 Cross-correlation

Cross-correlation is the degree of agreement between two different spreading code

sequences {an} and {bn}. Mathematically it is expressed as:

𝑅(𝑘) = ∑ 𝑎𝑛𝑁𝑛=1 𝑏𝑛+𝑘 (2)

Cross-correlation function (CCF) actually indicates the correlation between the desired

code sequence and the undesired ones at the receiver. Therefore, in order to mitigate the effect

of MAI at the receiver, the cross-correlation value must be zero at all time shifts. Hence it is

desirable to have a code family consisting of spreading codes which possess both impulsive

ACF and all zero CCF characteristics. But unfortunately no such code family exist which

possess both the required characteristics simultaneously.

4.2.3 Variable Spreading factors

The aim of the next generation wireless system is to provide multimedia services not

just voice or low data rate services. Multimedia services like video conferencing, MMS, etc

require support for variable data rate services instead of fixed rate data services. For this

purpose, more than one data channel with variable data rates will be required by each user of

15

the CDMA system. Thus, for ensuring constant chip rate after spreading of variable rate data

channels, spreading codes with variable spreading factors are required.

4.2.4 Size of code set and length of code

In order to support increasing demand of users over the same bandwidth in the CDMA

system; code family must be very large. It is difficult to construct a large code family consisting

of codes which possess the desired characteristics. Such desirable codes are few in number.

Therefore the spreading codes should be optimized properly as per system requirement. The

length of the spreading code should also be large enough to maintain the noise like property of

the spread signal. This will also ensure safety against eavesdroppers.

4.2.5 Peak to Average power Ratio (PAPR)

In multi-carrier CDMA system high PAPR of the transmitted signal is a major problem.

In this system, depending on input data, summation of sub-carriers may produce a signal with

large or small amplitude. Due to nonlinear amplification of the transmitted signal with high

PAPR the BER degradation may be caused. This PAPR is a function of the ACF and CCF of

the selected codes in the multi-carrier CDMA systems. So, spreading codes with low PAPR

value must be selected.

4.3 Pseudo-Random Sequences

A pseudo-random (PN) sequence is a code sequence of 1’s and 0’s whose autocorrelation

has properties similar to those of white noise. Some of the popular PN sequences are Maximal

length shift register sequences (m-sequences), gold sequences etc.

4.3.1. Maximal length shift register Sequence (m-sequence)

Maximal length shift register sequences are by definition, the longest codes that can be

generated by a given shift register or a delay element of a given length. In binary shift register

sequence generators, the maximum length sequence is 2n-1 chips, where n is the number of

stages in the shift register. A shift register sequence generator consists of a shift register

16

working in conjunction with appropriate logic, which feeds back a logical combination of the

state of two or more of its stages to input. The output of a sequence generator, and the contents

of its n stages at any sample (clock) time, is a function of the outputs of the stages fed back at

the preceding sample time.

Linear feedback shift registers (LFSR) can be implemented in two ways. The Fibonacci

implementation consists of a simple shift register in which a binary-weighted modulo-2 sum of

the taps is fed back to the input. (The modulo-2 sum of two 1-bit binary numbers yields 0 if the

two numbers are identical and 1 if they differ: 0+0=0, 0+1=1, 1+1=0.)

Fig.4.1 m-sequence generator

For any given tap, weight gi is either 0, meaning "no connection," or 1, meaning it is fed

back. Two exceptions are g0 and gm, which are always 1 and thus always connected. Note that

gm is not really a feedback connection, but rather is the input of the shift register. It is assigned

a feedback weight for mathematical purposes. The Galois implementation consists of a shift

register, the contents of which are modified at every step by a binary-weighted value of the

output stage.

4.3.2. Gold sequences

For CDMA applications, m-sequences are not optimal. For CDMA, we need to

construct a family of spreading sequences, one for each which, in which the codes have well-

defined cross-correlation properties. In general, m-sequences do not satisfy the criterion. One

popular set of sequences that does are the Gold sequences. Gold sequences are attractive

because only simple circuitry is needed to generate a large number of unique codes.

17

A Gold sequence is constructed by the XOR of two m-sequences with the same

clocking. Figure shows the schematic for Gold code generation.

Fig.4.2 Gold code sequence generator configuration

To achieve increased capacity, at an expense of altering the correlation properties

slightly, a pair of m-sequences may be used to generate a set of Gold sequence, which have the

property that the cross-correlation is always equal to –1, when the phase offset is zero. Non-

zero phase offset produces a correlation value from one of the three possible values. In this

work a pair of specially selected m-sequences (where m = 5) is taken, and performing the

modulo-2 sum of the two sequences for each of the L=2m-1 cyclically shifted version of one

sequence relative to the other sequence

18

Chapter 5

Introduction to Chaotic Sequences

5.1 Overview

In the past few decades, there has been a great deal of interest in the study of non-linear

dynamical system from which chaos developed. The diverse applications of chaos to various

areas are growing. However, not until the past ten years that chaos is of great interest in

communication and more research are undergoing in either theory or practice.

The most significant feature of the chaotic system is its sensitively dependence on its initial

condition. It is properly illustrated by the finding of Professor E.N. Lorenz, teaching

Meteorology at MIT. In 1961, Prof. Lorenz attempted to solve a much-simplified model and

finally he did succeed in simulating real weather patterns for weather predictions. However,

something drew his attention: when he slightly changed the initial conditions in the model, the

resulting weather patterns changed completely after a very short period. He discovered the fact

that very simple differential equations could possess sensitive dependence on initial conditions.

5.2 Chaotic System

A chaotic dynamical system is an unpredictable, deterministic and uncorrelated system that

exhibits noise-like behavior through its sensitive dependence on its initial conditions, which

generates sequences similar to PN sequence. The chaotic dynamics have been successfully

employed to various engineering applications such as automatic control, signals processing and

watermarking. Since the signals generated from chaotic dynamic systems are noise-like, super

19

sensitive to initial conditions and have spread and flat spectrum in the frequency domain, it is

advantageous to carry messages with this kind of signal that is wide band and has high

communication security. Then, numerous engineering applications of secure communication

with chaos have been developed.

5.3 Chaotic Sequence

A chaotic sequence is non-converging and non-periodic sequence that exhibits noise-like

behaviour through its sensitive dependence on its initial condition. Chaotic systems have

sensitive dependence on their initial conditions. A large number of uncorrelated, random-like,

yet deterministic and reproducible signals can be generated by changing initial value. These

sequences so generated by chaotic systems are called chaotic sequences. Chaotic sequences are

real valued sequences. Since the spreading sequence in a Chaotic Spread Spectrum (SS) is no

longer binary, the application of the chaotic sequences in DS-CDMA is thus limited. A further

attempt to transform continuous values to binary ones by using digital encoding technique is

therefore used to adopt it in DS-CDMA. Some criteria are performed. Moreover, since chaotic

dynamical system is a deterministic system, disguishing modulation as noise would be easily

made upon its random-like behavior. The use of chaotic sequences for spectral spreading in a

direct-sequence spread spectrum system (DS/SS) has been shown to provide several advantages

over conventional binary sequences, particularly pseudo-noise sequences which are frequently

used in digital communication.

Different chaotic sequences with proper auto-correlation features for DS-CDMA systems

are presented. Generally, chaotic sequences are generated by recursive maps and have particular

properties. One of these properties is extreme sensitivity to initial values, as for the two initial

values that are very close together, after a few steps, the output values will have a large relative

distance from each other.

5.3.1 Chaotic Bernoulli Sequence with negative auto-correlation

In a DS-CDMA system, the average interference seen by user k from user i is obtained as

follows

20

𝑟 𝑘 ,𝑖 = 2𝑁2 + 4 ∑ 𝐴 𝐾(𝑙 )

𝑁−1

𝑙=1

𝐴 𝑖(𝑙 ) + ∑ 𝐴 𝑘(𝑙 )𝐴 𝑖( 𝑙 + 1)

𝑁−1

𝑙=1−𝑁

(1)

Using Equation (5.1), the average Signal to Noise Ratio (SNR) at the output of correlation

receiver for 𝑖𝑡ℎuser in AWGN channel is as follows

Ak(l) = {

∑ 𝑐𝑘(𝑛)

𝑐𝑘(𝑛+1)𝑁−1−𝑙

𝑛=0 (0 ≤ 𝑙 ≤ 𝑁 − 1)

∑ 𝑐𝑘(𝑛−𝑙)

𝑐𝑘(𝑛) 𝑁−1+𝑙

𝑛=0 (1 − 𝑁 ≤ 𝑙 ≤ 0)

0 (|𝑙| > 𝑁)

(2)

where N is the spreading factor and Ak(.) is the auto- correlation function of real

spreading sequence [{ck(n)} for n=0 to (N-1)] used for Kth user and is defined as follows

𝑆𝑁𝑅𝑖 = {1

6𝑁3∑ 𝑟𝑘,𝑖 +

𝑁0

2𝐸𝑏

𝐾

𝑘=1,𝑘≠𝑖

}

−1

(3)

Where K is the number of active users, N0/2 is the power spectral density of white noise

and Eb is signal energy of ith user for each bit. In other words, we can say that system quality is

directly related to the auto-correlation functions. Different chaotic sequences with proper auto-

correlation features for DS-CDMA systems are presented. Generally, chaotic sequences are

generated by recursive maps and have particular properties. One of the properties is extreme

sensitivity to initial values, as for the two initial values that are very close together, after a few

steps, the output values will have a large relative distance from each other. One of the maps

used to generate chaotic sequence is the Bernoulli map, which is defined as follows

Ʈ(xn+1)= {2𝑥𝑛 0 ≤ 𝑥 ≤ 0.5

2𝑥𝑛 − 1 0.5 ≤ 𝑥 ≤ 1 (4)

By choosing the initial point x0 €(0.1), a real valued chaotic sequence is{ xn= Ʈn(x0) }

obtained, in which means that the function Ʈ is applied for n times. To construct a spreading

code, a binary converter function is used for converting the continuous values (xn) to +1 values.

Binary converter function is defined as follows

B(x) = Θ 1

8

(𝑥) − Θ 3

8

(𝑥) + Θ 3

4

(𝑥) (5)

where Θ𝑡(𝑥) is the threshold function as follows:

21

Θ 𝑡 (𝑥) = {−1 (𝑥 < 𝑡)1 (𝑥 ≥ 𝑡)

(6)

It is also possible to use other binary functions. Average auto-correlation function for the

binary sequence is defined as follows

C(l)= E[𝐵(𝜏(𝑥))𝐵(𝜏𝑙(𝑥))] = ∫ B(τ(x))B(𝜏𝑙(𝑥))f(x)dx (7)

where f(.) is the Invariant Density Function for τ(x) and the integral is calculated over

interval [0,1]. It is shown for one-dimensional Bernoulli binary sequences, constructed by the

binary function B(.), that f (x) = 1 and E[B(τ1( x))] = 0. If all active users use the Bernoulli

sequences generated by this way, and the initial values of sequences are statistically

independent of each other, the average interference parameter (AIP) for each user can be written

as follows

𝑟 = 2𝑁2 + 4 ∑ (𝑁 − 𝑙)2𝑁−1𝑙=1 𝐶(𝑙) + 2 ∑ (𝑁 − 𝑙)(𝑁 − 𝑙 + 1)𝐶(𝑙)𝑐(𝑙 − 1)𝑁−1

𝑙=1 (8)

Normalized AIP is defined as R=limn-->∞ 𝑟

2𝑁2 and can be supposed as a criterion for

spreading code performance. It is clear that for a completely random sequence, C(l) is in form

of

𝐶(𝑙) = {

1 (𝑙 = 0)

𝜖 (𝑙 = 1)0 (𝑙 > 2)

(9)

5.3.2 Two-dimensional Bernoulli spreading sequence

In case of using two-dimensional or complex sequences, there is more freedom of choice of

sequences with appropriate cross-correlation properties. The two different Bernoulli sequences

(with different initial conditions), as real and imaginary parts of a two-dimensional sequence is

used

Fig.5.1 Block diagram of DS-CDMA transmitter with complex spreading sequence

22

The two generated Bernoulli sequences by Bernoulli map with different initial

conditions x, y ϵ [0,1] and binary function B(.). Assume that the initial conditions x and y are

statistically independent. The complex binary sequence is defined as follows

{1

√2(𝐵(𝜏𝑛(𝑥)) + 𝑗𝐵(𝜏𝑛(𝑦)))}

0

𝑁−1

(10)

Factor of 1

√2 has been considered to have the same power as real sequence. Average

autocorrelation function is obtained as follows

𝐶(𝑙) = 1

2𝐸 [(𝐵(𝜏(𝑥)) + 𝑗𝐵(𝑦)))(𝐵(𝜏𝑙(𝑥)) − 𝑗𝐵(𝜏𝑙(𝑦)))]

= 1

2∫(𝐵(𝜏(𝑥)) + 𝑗𝐵(𝑦)))(𝐵(𝜏𝑙(𝑥)) − 𝑗𝐵(𝜏𝑙(𝑦)))𝑓(𝑥)𝑓(𝑦)𝑑𝑥𝑑𝑦 (11)

where f(.) is invariant density function for τ and the integral is calculated over interval

[0,1]. As said before, it is shown that f(x) = 1and 𝐸[(𝜏𝑙(𝑥))]=0 for one-dimensional Bernoulli

binary sequence constructed byB(.). Because of the deterministic nature of theBernoulli map

and the independence of x and y, it can be written

C(l) = 1

2(𝐸[𝐵(𝜏(𝑥))𝐵(𝜏𝑙(𝑥))] + [𝐵(𝜏(𝑦))𝐵(𝜏𝑙(𝑦))]) (12)

23

Chapter 6

Error Control Coding Techniques

6.1 Introduction

In information theory and coding theory, error detection and correction or error

control are techniques that enable reliable delivery of digital data over

unreliable communication channels. Many communication channels are subject to channel

noise, and thus errors may be introduced during transmission from the source to a receiver.

Error detection techniques allow detecting such errors, while error correction enables

reconstruction of the original data in many cases. The general idea for achieving error detection

and correction is to add some redundancy (i.e., some extra data) to a message, which receivers

can use to check consistency of the delivered message, and to recover data determined to be

corrupted. Error-detection and correction schemes can be either systematic or non-systematic:

In a systematic scheme, the transmitter sends the original data, and attaches a fixed number

of check bits (or parity data), which are derived from the data bits by some deterministic

algorithm. If only error detection is required, a receiver can simply apply the same algorithm to

the received data bits and compare its output with the received check bits; if the values do not

match, an error has occurred at some point during the transmission. In a system that uses a non-

systematic code, the original message is transformed into an encoded message that has at least

as many bits as the original message.

Good error control performance requires the scheme to be selected based on the

characteristics of the communication channel. Common channel models include memory-less

models where errors occur randomly and with a certain probability, and dynamic models where

errors occur primarily in bursts. Consequently, error-detecting and correcting codes can be

generally distinguished between random-error-detecting/ correcting and burst – error -

24

detecting/ correcting. Some codes can also be suitable for a mixture of random errors and burst

errors. The positions of the error control coding and decoding are shown in the transmission

model

Fig 6.1 Digital Communication Model

6.2 Error Control Coding Techniques

Error control coding aims at developing methods for coding to check the correctness of

the bit stream transmitted. The bit stream representation of a symbol is called the code word of

that symbol. Different error control mechanisms:

Linear Block Codes

Repetition Codes

Convolution Codes

Turbo Codes

6.2.1 Linear Block Codes

A code is linear if two codes are added using modulo-2 arithmetic produces a third

code word in the code. Consider a (n, k) linear block code. Here,

1. n represents the code word length

2. k is the number of message bit

3. n−k bits are error control bits or parity check bits generated from message using an

appropriate rule.

25

We may therefore represent the code word as:

𝐶𝑖 = { 𝑏𝑖 , 𝑖 = 0,1,2, … . . , 𝑛 − 𝑘 − 1

𝑚𝑖+𝑘−1 , 𝑖 = 𝑛 − 𝑘, 𝑛 − 𝑘 + 1, … 𝑛 − 1 (1)

The (n − k) parity bits are linear sums of the k message bits.

bi = p0im0 + p1im1 + · · · + pk−1,imk−1 (2)

where the coefficients are

𝑝𝑖𝑗 = {1, 𝑖𝑓 𝑏𝑖 𝑑𝑒𝑝𝑒𝑛𝑑𝑠 𝑜𝑛 𝑚𝑗

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (3)

We define the 1-by-k message vector, or information vector, m, the 1-by-(n − k) parity

vector b, and the 1-by-n code vector c as follows:

m = [m0,m1, · · · ,mk−1]

b = [b0, b1, · · · , bn−k−1] (4)

n = [n0, n1, · · · , nn−1]

We may thus write simultaneous equations in matrix equation form as

b = Mp

where P is a k by n − k matrix.

c matrix can be expressed as a partitioned row vector in terms of the vectors m and b as

follows

c = [b : m]

c = m[P : Ik]

where Ik is a k-by-k identity matrix. Now, we define k-by-n generator matrix as

G = [P : Ik]

c = mG

Closure property of linear block codes: Consider a pair of code vectors ci and cj

corresponding to a pair of message vectors mi and mj respectively.

ci + cj = miG + mjG

= (mi + mj) G (5)

26

The modulo-2 sum of mi and mj represents a new message vector. Correspondingly, the

modulo-2 sum of ci and cj represents a new code vector. Suppose,

H = [In−k|PT ]

HGT = [In−k : PT] [𝑃𝑇

𝐼𝑛−𝑘]

= PT + PT

= 0

Post multiplying the basic equation c = mG with HT , then using the above result we get

cHT = mGHT

= 0

• A valid code word yields the above result.

• The matrix H is called the parity-check matrix of the code.

• The set of equations specified in the above equation are called parity-check-equations.

6.2.2 Repetition codes

This is the simplest of linear block codes. Here, a single message bit is encoded into a

block of n identical bits, producing an (n, 1) block code. This code allows variable amount of

redundancy. It has only two code words - all-zero codeword and all-one codeword.

Example, Consider a linear block code which is also a repetition code. Let k = 1 and

n = 5. From the analysis done in linear block codes

G = [1111: 1]

The parity check matrix takes the form

𝐻 = [

1 00 1

0 00 0

: 1: 1

0 00 0

1 00 1

: 1: 1

] (6)

6.2.3 Cyclic Codes

Cyclic property: Any cyclic shift of a code word in the code is also a code word. Cyclic

codes are well suited for error detection. Certain set of polynomials are chosen for this purpose.

Properties of polynomials:

1. Any polynomial B(x) can be divided by a divisor polynomial C(x) if B(x) is of higher

degree than C(x).

27

2. Any polynomial B(x) can be divided once by C(x) if both are of same degree.

3. Subtraction uses exclusive OR.

Cyclic Redundancy Check Codes:

Let M(x) be the original message polynomial of kth degree. The following steps are followed in

getting the code word in CRC.

1. Multiply the message polynomial M(x) by xn−k.

2. Divide xn−kM(x) by the generator polynomial G(x), obtaining the remainder B(x).

3. Add B(x) to xn−kM(x), obtaining the code polynomial C(x).

6.2.4 Convolution codes

Convolutional codes are commonly described using two parameters-

1. The code rate

2. The constraint length

The code rate, k/n, is expressed as a ratio of the number of bits into the convolutional

encoder (k) to the number of channel symbols output by the convolutional encoder (n) in a

given encoder cycle. The constraint length parameter, K, denotes the "length" of the

convolutional encoder, i.e. how many k-bit stages are available to feed the combinatorial logic

that produces the output symbols. Closely related to K is the parameter m, which indicates how

many encoder cycles an input bit is retained and used for encoding after it first appears at the

input to the convolutional encoder. The m parameter can be thought of as the memory length

of the encoder i.e. number of memory registers.

Constraint length, L = k (m-1) (7)

Convolutional codes are widely used as channel codes in practical communication

systems for error correction. The encoded bits depend on the current k input bits and a few past

input bits. The main decoding strategy for convolutional codes is based on the widely used

Viterbi algorithm. As a result of the wide acceptance of convolutional codes, there have been

several approaches to modify and extend this basic coding scheme. Trellis coded modulation

(TCM) and turbo codes are two such examples. In TCM, redundancy is added by combining

coding and modulation into a single operation. This is achieved without any reduction in data

rate or expansion in bandwidth as required by only error correcting coding schemes.

28

A simple convolutional encoder is shown in Fig.6.2. The information bits are fed in

small groups of k-bits at a time to a shift register. The output encoded bits are obtained by

modulo-2 addition (EXCLUSIVE-OR operation) of the input information bits and the contents

of the shift registers which are a few previous information bits.

Fig. 6.2 A convolutional encoder with k=1, n=2 and r=1/2

If the encoder generates a group of ‘n’ encoded bits per group of ‘k’ information bits,

the code rate R is commonly defined as R = k/n. In Fig.6.2, k = 1 and n = 2. The number, K of

elements in the shift register which decides for how many code words one information bit will

affect the encoder output, is known as the constraint length of the code. For the present example,

K = 3.

6.2.5 Turbo Codes

Turbo codes are error-correcting codes with performance close to the Shannon theoretical limit

[SHA]. These codes have been invented at ENST Bretagne (now TELECOM Bretagne),

France, in the beginning of the 90's [BER]. The encoder is formed by the parallel concatenation

of two convolutional codes separated by an interleaver or permute. An iterative process through

the two corresponding decoders is used to decode the data received from the channel

Fig 6.3 Turbo code

29

Each elementary decoder passes to the other soft (probabilistic) information about each bit of

the sequence to decode. This soft information, called extrinsic information, is updated at each

iteration.

6.2.1.1 The genesis of turbo codes

The invention of turbo codes finds its origin in the will to compensate for the

dissymmetry of the concatenated decoder of Figure 6.3. To do this, the concept of feedback - a

well-known technique in electronics – is implemented between the two component decoders

(figure 6.4).

Fig 6.4: Decoding the concatenated code with feedback

The use of feedback requires the existence of Soft-In/Soft-Out (SISO)

decoding algorithms for both component codes. As the SOVA algorithm was already available

at the time of the invention, the adoption of convolutional codes appeared natural for both

codes. For reasons of bandwidth efficiency, serial concatenation is replaced with parallel

concatenation. Actually, parallel concatenation combining two codes with rates R1 and R2 gives

a global rate Rp equal to:

Rp=R1R2/(1−R1)(1−R2) (8)

This rate is higher than that of a serially concatenated code, which is:

30

Rs=R1*R2 (9)

for the same values of R1 and R2 , and the lower these rates, the larger the difference. Thus, with

the same performance of component codes, parallel concatenation offers a better global rate,

but this advantage is lost when the rates come close to unity. Furthermore, in order to ensure a

sufficiently large dmin for the concatenated code, classical non-systematic non-recursive

convolutional codes (Figure 6.5.a) have to be replaced with recursive systematic convolutional

(RSC) codes (Figure 6.5 b).

Fig6.5: (A) Non-systematic non-recursive convolutional code with polynomials 13, 15.

(B) Recursive systematic convolutional (RSC) code with polynomials 13 (recusivity), 15

(parity)

Fig 6.6: A turbo code with component codes 13, 15

31

What distinguishes both codes is the minimum input weight wmin . The input weight w is

the number of "1" in an input sequence. Suppose that the encoder of Figure 6.5 a is initialized

in state 0, then fed with an all-zero sequence, except in one place (that is,w=1). The encoder

will retrieve state 0 as soon as the fourth 0 following the 1 will appear at the input. We have

then wmin=1. In the same conditions, the encoder of Figure 6.5 b needs a second 1 to retrieve

state 0. Without this second 1, this encoder will act as a pseudo-random generator, with respect

to its output Y . So, wmin=2 and this property is very favorable regarding dmin when parallel

concatenation is implemented. A typical turbo code is depicted in Figure 6.4. The data are

encoded both in the natural order and in a permuted order by two RSC codes C1 and C2 that

issue parity bits Y1 and Y2. In order to encode finite-length blocks of data, RSC encoding is

terminated by tail bits or has tail-biting termination. The permutation has to be devised carefully

because it has a strong impact on dmin. The natural coding rate of a turbo code is R=1/3 (three

output bits for one input bit). To deal with higher coding rates, the parity bits are punctured.

For instance, transmitting Y1 and Y2 alternately leads to R=1/2.

The original turbo code [BER] uses a parallel concatenation of convolutional codes. But

other schemes like serial concatenation of convolutional codes [BEN] or algebraic turbo codes

[PYN] have since been studied. More recently, non-binary turbo codes have also been proposed

[DOU].

6.2.1.2 Turbo-decoding

Decoding the code of Figure 6.3 by a global approach is not possible, because of the

astronomical number of states to consider. A joint probabilistic process by the decoders

of C1 and C2, has to be elaborated. Because of latency constraints, this joint process is worked

out in an iterative manner in a digital circuit. Turbo decoding relies on the following

fundamental criterion: when having several probabilistic machines work together on the

estimation of a common set of symbols, all the machines have to give the same decision, with

the same probability, about each symbol, as a single (global) decoder would.

To make the composite decoder satisfy this criterion, the structure of Figure 6.3 is

adopted. The double loop enables both component decoders to benefit from the whole

redundancy. The term turbo was given to this feedback construction with reference to the

principle of the turbo-charged engine.

32

Fig 6.6: A turbo decoder

The components are SISO decoders, permutation (π) and inverse permutation (π−1)

memories. The node variables of the decoder are Logarithms of Likelihood Ratios (LLR). An

LLR related to a particular binary datum d is defined as:

LLR (d) =ln (Pr (d=1) Pr(d=0)) (10)

The role of a SISO decoder is to process an input LLR and, thanks to local redundancy

(i.e. y1 for DEC1, y2 for DEC2), to try to improve it. The output LLR of a SISO decoder may

be simply written as

LLRout (d) =LLRin (d) +z (d) (11)

Where z (d) is the extrinsic information about d, provided by the decoder. If this works

properly, z (d) is most of the time negative if d=0, and positive if d=1. The composite decoder

is constructed in such a way that only extrinsic terms are passed by one component decoder to

the other. The input LLR to a particular decoder is formed by the sum of two terms: the

information symbols (x) stemming from the channel and the extrinsic term (z) provided by the

other decoder, which serves as a priori information. The information symbols are common

inputs to both decoders, which is why the extrinsic information must not contain them. In

addition, the outgoing extrinsic information does not include the incoming extrinsic

information, in order to cut down correlation effects in the loop. There are two families of SISO

33

algorithms, those based on the SOVA [BAT] [HAG], the others based on the MAP (also called

BCJR or APP) algorithm [BAH] or its simplified versions. Turbo decoding is not optimal. This

is because an iterative process has obviously to begin, during the first half-iteration, with only

a part of the redundant information available (either y1 or y2). Fortunately, loss due to sub-

optimality is small (less than 0.5dB).

34

Chapter 7

Multiple Access Interference

7.1 Motivation for Multiple Access Interference Characterization

The ever increasing demand for world-wide multiple access wireless communications

drives the need to maximize the current system capability and transmission capacity. Many

current systems employ direct-sequence spread-spectrum (DSSS) techniques to enable multiple

access capability. A clearer understanding of how multiple users effect overall system

performance, through characterization of the multiple access interference (MAI), enables more

efficient use of current systems and better designs for future systems. Although a large body of

research exists on how MAI impacts direct-sequence spread-spectrum multiple access

(DS/SSMA) performance, most of this work relies on analytical approximations. Additionally,

there are numerous factors that can affect MAI contributions in a system between workload and

environment (e.g., number of simultaneous transmitters, type of multiple access coding, and

code length). Current approximations only account for a limited number of these factors and

can require extensive recalculation when factors are changed or added. The state of existing

research into MAI in DS/SSMA systems leaves open a need for a representative system model

that is easily modified to account for different factors.

This need lends itself to characterization of MAI in DS/SSMA systems through

simulation. Simulation of DS/SSMA system performance provides two main benefits. First,

simulation allows for rapid testing of the effects that numerous factors can have on system

performance and such factors can be changed or others added relatively easily. Second,

simulation enables verification of future approximations through a vehicle that more closely

35

represents the actual, physical communication system.

7.2 Gaussian Approximation of MAI

In one class of algorithm, the MAI term is approximated by a Gaussian random variable.

From (2) the MAI term consists of the sum K-1 discrete binary random variable. From the

center limit theorem, as the number of user k increases, the distribution of the MAI term

approaches that of a Gaussian. Each single user binary probability distribution consists of the

coded bit probability 𝑃𝑟(𝑑𝑡(𝑘)

) = 1 and 𝑃𝑟(𝑑𝑡(𝑘)

) = −1 , k ε {1,….,k} . The mean variance of

these binary distribution for the kth user, k ε {1,….,k} are calculated as

𝑚𝑡(𝑘)

= 𝑃𝑟(𝑑𝑡(𝑘)

= 1) − 𝑃𝑟(𝑑𝑡(𝑘)

= −1) (1)

𝑣𝑡2 (𝑘)

= 1 − [ 𝑃𝑟(𝑑𝑡(𝑘)

= 1) − 𝑃𝑟(𝑑𝑡(𝑘)

= −1)]2 (2)

The addition of these binary distribution results in the MAI term ρ ∑ 𝑑𝑡(𝑖)𝐾

𝑖=1 for the

kth user having Gaussian distribution with mean and variance:

µ𝑡 ,𝑀𝐴𝐼(𝑘)

= ρ ∑ [𝑃𝑟(𝑑𝑡(𝑘)

= 1) − 𝑃𝑟(𝑑𝑡(𝑘)

= −1)]𝐾𝑖=1

= ρ ∑ 𝑚𝑡(𝑖)𝐾

𝑖=1𝑖≠𝑚

(3)

𝜎𝑡,𝑀𝐴𝐼2(𝑘)

= 𝜌2 ∑(1 − [𝑃𝑟(𝑑𝑡(𝑘)

= 1) − 𝑃𝑟(𝑑𝑡(𝑘)

= −1)]2)

𝐾

𝑖=1𝑖≠𝑘

= 𝜌2 ∑ 𝑣𝑡2 (𝑘)𝐾

𝑖=1𝑖≠𝑚

(4)

36

The probability distribution ƥ (𝑟𝑡(𝑘)

| 𝑑𝑡(𝑘)

= 𝑑), d ∈ {-1, 1} for the kth user is derived as a

Gaussian random variable with mean µ𝑡(𝑘)

= 𝑑 + µ𝑡,𝑀𝐴𝐼(𝑘)

and variance 𝜎𝑡2(𝑘)

= 𝜎2 + 𝜎𝑡,𝑀𝐴𝐼2 (𝑘)

7.3 Discrete Analysis of MAI

The continuous Gaussian approximation of the MAI receiver becomes accurate when the

number of users become large. Thus, a different approach is needed to determine the probability

distribution of the MAI when the number of user is small. the MAI term ρ ∑ 𝑑𝑡(𝑖)𝐾

𝑖=1𝑖≠𝑚

for user k

is discrete quantity hat can only take on K different values since each coded bit 𝑑𝑡(𝑖)

can only

take on values {-1,1} .As a result , the MAI term can be represented as a random variable with

a binomial distribution .this probability distribution is obtained for user k by convolving the

other k-1 sets of bit probability to produce the set of probability values

{𝑃𝑡,𝑚(𝑘)

}={𝑃𝑟 (∑ 𝑑𝑡(𝑖)𝐾

𝑖=1𝑖≠𝑚

= 𝑚)}, where m ∈{-(K-1),-(K-3) …,(K-3),(K-1)}. The value of the

metric is determined by convolving the distribution of the MAI and the Gaussian noise. The

Convolution of these two distribution results in

ƥ (𝑟𝑡(𝑘)

| 𝑑𝑡(𝑘)

= 𝑑) =1

√2ᴨ𝜎2∑ 𝑒𝑥𝑝 [−

(𝑟𝑡(𝑘)

−𝑑−𝑝𝑚)2

2𝜎2]𝑚 × 𝑃𝑡,𝑚′

(𝑘) (5)

where m’ ∈ {-(k-1),-(k-3)… (k-3), (k-1)} and d ∈ {-1, 1}.

37

Chapter 8

Results

DS-CDMA system was studied with two types of spreading sequences. Conventional

gold sequence performance was compared with new chaotic sequences generated using

Bernoulli map. It is observed that the performance of chaotic sequences is better than Gold

sequences. Correlation properties o DS-CDMA system was studied under two channel

environments. AWGN channel and Rayleigh fading channel. MAI effect on performance is

also studied and corresponding BER is plotted. Also, performance of DS-CDMA system was

studied for different number of users. It is observed that the BER decreases with increase in

number of users.

Fig.8.1 Auto- correlation magnitude vs time Fig.8.2 cross-correction magnitude vs time

-1500 -1000 -500 0 500 1000 1500-5

0

5

10

15

20

25

30

time shift

magnitude

auto-correlation

auto-correlation of bernoulli sequences

-1500 -1000 -500 0 500 1000 1500-2

0

2

4

6

8

10

12

14

16

time shift

magnitude

cross-correlation

cross-correlation of bernoulli sequences

38

Fig. 8.3 Auto-correlation comparison of gold and Bernoulli sequences

Fig. 8.4 Cross-correlation comparison of gold and Bernoulli sequences

-150 -100 -50 0 50 100 150-10

0

10

20

30

40

50

60

70

time shift

mag

nitu

de

auto-correlation

auto-correlation of gold sequences

auto-correlation of bernoulli sequences

-150 -100 -50 0 50 100 150-10

0

10

20

30

40

50

60

time shift

magnitude

cross-correlation

cross-correlation of gold sequences

cross-correlation of bernoulli sequences

39

Fig.8.5 Performance comparison of gold and Bernoulli sequences in AWGN channel

Fig.8.6 Performance comparison of gold and Bernoulli sequences in AWGN channel with

MAI

0 2 4 6 8 10 12 14 1610

-4

10-3

10-2

10-1

100

SNR, dB

Bit E

rror

Rate

Performance of CDMA

bernoulli

gold sequence

0 2 4 6 8 10 12 14 1610

-4

10-3

10-2

10-1

100

SNR, dB

Bit E

rror

Rate

Performance of CDMA

bernoulli

gold sequence

40

Fig.8.7 Performance comparison of gold and Bernoulli sequences in Rayleigh fading channel

Fig.8.8 Comparison of gold and Bernoulli sequences with and without turbo coding

0 5 10 15 20 2510

-3

10-2

10-1

100

SNR, dB

Bit E

rror

Rate

Performance of CDMA

bernoulli

gold sequence

0 2 4 6 8 10 12 14 1610

-3

10-2

10-1

100

SNR, dB

Bit E

rror

Rate

Performance of CDMA

without turbo coding gold seq.

with turbo coding gold seq.

without turbo coding bernoulli seq.

with turbo coding bernoulli seq.

41

Fig.8.9 Turbo code Performance for different number of iterations in AWGN channel without

MAI

Fig.8.10 Turbo code Performance for different number of iterations in AWGN channel with

MAI

0 2 4 6 8 10 1210

-4

10-3

10-2

10-1

100

SNR, dB

Bit E

rror

Rate

Performance of CDMA

without turbo coding

with turbo coding itr=1

with turbo coding itr=2

with turbo coding itr=3

with turbo coding itr=5

with turbo coding itr=10

0 1 2 3 4 5 6 7 8 9 1010

-4

10-3

10-2

10-1

100

SNR, dB

Bit E

rror

Rate

Performance of CDMA

without turbo coding

with turbo coding itr=1

with turbo coding itr=2

with turbo coding itr=3

with turbo coding itr=5

with turbo coding itr=10

42

Fig.8.11 Turbo code performance for different number of iterations in Rayleigh fading

channel

Fig.8.12 BER vs number of users for Gold and Bernoulli sequences

0 2 4 6 8 10 12 1410

-4

10-3

10-2

10-1

100

SNR, dB

Bit E

rror

Rate

Performance of CDMA

without turbo coding

with turbo coding itr=1

with turbo coding itr=2

with turbo coding itr=3

with turbo coding itr=5

with turbo coding itr=10

43

Table 8.1 BER comparison for Gold and Bernoulli sequences

SNR 0 1 2 3 4 5 6 7 8

Gold

sequences

0.0724 0.1936 0.1412 0.1354 0.1244 0.0948 0.0736 0.0589 0.0360

Bernoulli

sequences

0.0724 0.0582 0.0364 0.0231 0.0100 0.0043 0.0018 7.2e-04 2e-04

Table 8.2 BER comparison for different channel models

SNR 0 1 2 3 4 5 6 7 8

AWGN

channel

0.0724 0.0582 0.0364 0.0231 0.0100 0.0043 0.0018 7.2e-04 2e-04

AWGN

channel

with

MAI

0.1256 0.0993 0.0798 0.582 0.0332 0.0257 0.0112 0.0063 0.0026

Rayleigh

fading

channel

0.3204 0.3045 0.2810 0.2597 0.2386 0.2025 0.1836 0.1583 0.1244

Table 8.3 BER comparison for different number iterations in turbo decoder

SNR Without

Turbo

coding

With Turbo Coding (iterations)

1 2 3 5 10

0 1.29e-01 1.105e-01 9.85e-02 9.81e-02 9.73e-02 9.54e-02

1 9.72e-02 8.37e-02 7.44e-02 5.32e-02 4.19e-02 3.23e-02

2 6.84e-02 4.95e-02 3.55e-02 2.51e-02 1.75e-02 1.45e-02

44

Chapter 9

Conclusion

In this project, two-dimensional Bernoulli chaotic sequences are introduced. They can

be used to generate complex spreading sequences for utilizing in DS-CDMA system. Amongst

the complex sequences it is much easier to choose a proper set of sequences for multi-user DS-

CDMA systems. Complex two-dimensional sequence is obtained by combining two one-

dimensional Bernoulli sequences with different initial conditions. Employing the new

sequences can improve the performance of DS-CDMA systems compared with the

conventional spreading codes as well as similar complex spreading codes, such as the gold

codes. The performance improvement is due to the following reasons

the proper auto-correlation function

more freedom of choice of appropriate codes among complex codes

which results in decrease of BER and increase in cellular capacity.

It is also concluded that turbo codes provide better error performance and sufficient

coding gain as compared with system without turbo coding. Increase in number of iterations

improves BER performance but computation time increases too. Turbo coding shows

significant advantage in case of channel with MAI consideration as well as for fading channel.

45

Literature cited

[1] Claude, B., & Alan, G.. “Near Optimum Error Correcting Coding and Decoding: Turbo

Codes”, IEEE Transactions on Communications, Vol. 44, No. 10, 1996.

[2] CongL. & Shaoqian L., “Chaotic spreading sequences with multiple access performance better

than random sequences”. IEEE Transactions on Circuits and Systems: Fundamental Theory and

Applications, 47, 3, 2000.

[3] Kurian A. P., Puthusserypady S., & Htut S. M., “Performance enhancement of DS/CDMA

system using chaotic complex spreading sequence”. IEEE Transactions on Wireless

Communications, 4, 3, 2005.

[4] Mazzini G., Setti G., & Rovatti R., “Chaotic complex spreading sequences for asynchronous

DS-CDMA—Part I: System modeling and results”. IEEE Transactions on Circuits and SystemsPart

I, 44, 937–947. 1997.

[5] Rahnama N., & Talebi S., “Performance comparison of chaotic spreading sequences generated

by two different classes of chaotic systems in a chaos-based direct sequence-code division multiple

access system”. IET Communications, 7(10), 1024–1031, 2013.

[6] Rovatti R., Mazzini G., &Setti G., “On the ultimate limits of chaos-based asynchronous DS-

CDMA-I: Basic definitions and results”. IEEE Transactions on Circuits and Systems, 51, 1336–

1347, 2004.

[7] Stienstra D. & Khandani A. K., “Iterative multi-user turbo-code receiver for ds-cdma”. 2005

[8] Todd K. Moon, “Error Correction Coding Mathematical Methods and Algorithms.”

[9] Tsuneda A., “Design of binary sequences with tunable exponential autocorrelations and run

statistics based on one-dimensional chaotic maps”. IEEE Transactions on Circuits and systems-I,

52 (2), 454–462, 2005.

[10] Tsuneda A. & Miyazaki Y. “Performance evaluation of LFSR-based spreading sequences

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[11] Tsuneda A. & Miyazaki, Y., “Binary spreading sequences with negative auto-correlation

based on chaos theory and gold sequences for application to asynchronous DS/CDMA

46

communications”. IEICE Transactions on Fundamentals of Electronics, Communications and

Computer Sciences, E93.A(11), 2307–2311, 2010.

[12] G.V.Reddy, “Performance Evaluation of different DS-CDMA receivers using chaotic

sequences”, ethesis.nitrkl.ac.in/18/, 2007.

[13] Stavroulakis P. “Chaos applications in telecommunications”. Hoboken, NJ: CRC Press.

ISBN-13: 978-0849338328, 2005.

47

Acknowledgements

We consider it as a great privilege to express our deep gratitude to many respected

personalities who guided, inspired and helped us in the successful completion of our project.

We would like to express our deepest gratitude to our guide Dr. S Anuradha, Assistant

Professor, Department of Electronics and Communication Engineering, National Institute of

Technology, Warangal, for her constant supervision, guidance, suggestions and invaluable

encouragement during this project.

We are grateful to Dr. T Kishore Kumar, Head of Department of Electronics and

Communication Engineering, National Institute of Technology, Warangal, for his moral support

to carry out this project.

We are very thankful to Sri. M V Raghunadh and Dr. P Srihari Rao, Project

Coordinators for their continuous support throughout the year.

We are very thankful to Project Evaluation Committee, for their strenuous efforts to

evaluate our projects.

We wish to thank all the staff members in the department for their kind cooperation and

support given throughout our project work. We are also thankful to all of my friends who have

given valuable suggestions and help in all stages of the development of the project.

Snehal Chipade (114254)

Piyush Meshram (114243)