Dipanwita Thesis

8/8/2019 Dipanwita Thesis

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A Thesis Project Report on

Performance of Different M-ary modulation and

Coherent Phase modulation over Wireless FadingChannel

Submitted in the Partial Fulfilment of the Requirement for the Award of Degree of

MASTER OF TECHNOLOGY

INELECTRONICS & TELECOMMUNICATION

By

Name:Dipanwita Biswas

Roll No:751001

GuidedbyAsst.Prof.S.S.SINGH,

Dept.ofE&TC,KIITUNIVERSITY.BHUBANESWAR.Mr.A.CHANDRA,

Dept.ofECE,NIT.DURGAPUR.


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KALINGA INSTITUTE OF INDUSTRIAL

TECHNOLOGY, UNIVERSITY

B H U B A N E S W A R , O R I S S A , I N D I A

Ce r t i f i c a t e This is to certify that the thesis paper entitled Performance of

Different M-ary modulation and Coherent Phase modulation over Wireless Fading

Channel being submitted by Dipanwita Biswas bearing Roll No:751001, in partial fulfillment of the requirement for the award of the final semester of Master of Technology in Telecommunication &

Engineering, is a bonafide work carried out at Department of

Electronics & Telecommunication, KIIT University under my/our

supervision.

Prof.S.S.Singh Mr.A.Chandra Prof.A.K.SenAsst.Prof.E&TC Lecturer Professor.E&TCCo-ordinator(M-Tech) Dept of ECE, HOD of Dept.E&TCDept of E & TC, NIT, Durgapur. Dept of E&TC,KIIT University KIIT University


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ACKNOWLEDGEMENTS

I would like to express my gratitude to Asst. Prof. S. S. SINGH, Department of

Electronics and Telecommunication for his immense help. During my thesis tenure, I

had the great fortune and honour to discuss with him on problems and for his valuable

suggestion, guidance and for kind co-operation.

I would like to thank my thesis supervisor, Mr. A. Chandra, Department of

Electronics and Communication Engineering, NIT Durgapur, for providing me with

the right balance of guidance and independence in my research. I am greatly indebted

to him for his full support, constant encouragement and advice both in technical and

non-technical matters valuable suggestion, guidance and for kind co-operation

through out the bringing up of the Thesis peper.

I would like to express my gratitude to Prof. A. K. Sen., Head of the

Department, all my Teachers and my friends for their Co-operation.

DIPANWITA BISWAS

Department of Electronics andTelecommunication Engineering,

KIIT University hibaneswar,, B

Orissa


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Abstract

Demand for mobile and personal communications is growing at a rapid pace, both in

terms of the number of potential users and the introduction of new high-speed services.

Meeting this demand is challenging since wireless communications are subject to three major

constraints: a complex and harsh fading channel, a scarce usable radio spectrum, and

limitations on the power and size of handheld terminals. Another major problem, which is

very much in concern for wireless communication system in recent years is the

Synchronization mismatch of the three basic parameters as Phase, Frequency and Time

between the transmitter and receiver. Therefore it is the primary need to study the modulation

and demodulation schemes of the signals, and there performance in various environment as

Additive White Gaussian Noise channel and Fading channel and how the signal quality can

be improved.

M-ary Modulation schemes, one of most efficient digital data transmission systems

have been discussed. It focuses on basic M-ary modulation schemes like MPSK, MFSK,

MDPSK, and MQAM. This paper gives the representation ofM-ary modulation methods,

their geometrical representation and representation of Basis functions and mainly it deals with

the probability of error calculation for all the above stated M-ary modulation schemes. A

brief discussion about the Error performance of the BPSK, and BFSK with the probability of

error calculation over Rayleigh, Rician and Nanagami-q (Hoyt) channel, how the Signal to

Noise ratio will effected by these Fading channels. The performance of the M-ary modulated

signals are very much degraded by these Fading channels and a large amount of spectrum and

power is wasted at the receiving end to receive these signals. Therefore, effective spectral and

power efficient fading mitigation techniques are required. Diversity techniques play a vital

role in supporting such high speed connections over radio channels by mitigating the

detrimental effects of multiuser interference and multipath fading impairments. The Selection

Combining diversity and the Maximal Ratio Combining diversity techniques are anglicised in

this thesis, to mitigate the effect of fading on BPSK and BFSK and a comparison was donebetween the BPSK and BFSK to show which one give us the better result.

The Synchronization mismatch of the Phase between the transmitter and receiver is

discussed in a brief space. In case of wireless communication it is not possible to estimate the

correct Phase of the propagating wave; therefore it is difficult at the receiver end to receive

the signals with the correct phase as transmitted by the transmitter. The performance of the

wave is degraded very much at the receiver. This thesis examined the problem of wrong

phase estimation of the BPSK and QPSK signals, and then compared with DPSK and

DMPSK where no phase estimation is need.


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Contents

Acknowledgements iii

Abstract iv

1 Introduction 1-5

1.1 Motivation and Background 3

1.2 Thesis Objective 3

1.3 Thesis Outline 4

2 M-ary Modulation Schemes 6-33

2.1 Introduction 6

2.2 Applications 62.3 General Description of The M-ary scheme 7-12

2.3.1 Geometric representation 8

2.3.2 Base band representation 8

2.3.3 Gray coding 9

2.3.4 M-ary Pulse Amplitude Modulation 9

2.3.5 M-ary Phase Shift Keying 9

2.36 Differential M-ary Phase Shift Keying 10

2.3.7 M-ary Frequency Shift Keying 10

2.3.8 M-ary Quarature Amplitude Modulation 11

2.4 Constellation Diagrams of Different M-ary Schemes 12-15


2.4.2 M-ary Phase Shift Keying 12


2.4.4 M-ary Quadrature Amplitude Modulation 14

2.5 Receiver Structures 15-17

2.6 Error Performance 17-26


2.6.2 M-ary Phase Shift Keying 192.6.3 Differential M-ary Phase Shift Keying 20


2.6.5 M-ary Quadrature Amplitude Modulation 25

2.7 Alternate Error Expressions 26-29

2.8 Results 29-33

3 Fading Channel Modelling and Antenna

Diversity Techniques 34-45

3.1 Introduction 34


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List of Figures:

2.1 Constellation diagram of MPAM 12

2.2 Constellation diagram of MPSK 13

2.3 Constellation diagram of MFSK 13

2.4 Constellation diagram of MQAM 14

2.5 Constellation diagram of 16-QAM with respectiveamplitude and phase 14

2.6 Receiver structure of coherent detection of signals 15

2.7 Coherent detection of MPAM 15

2.8 Coherent detection of MPSK 16

2.9 DMPSK detection 16

2.10 Coherent detection of MQAM 16

2.11 Coherent detection of MFSK 17

2.12 Decision region for representing andis

js 19

2.13 Angle between two vectors 21

2.14 Decision boundary representation 21

2.15 Symbol error probability curve of MPAM 30 2.16 Symbol error probability curve of MPSK 30

2.17 Symbol error probability curve of MFSK 31

2.18 Symbol error probability curve of MQAM 31

2.19 MPSK curves for different values of M 32

2.20 MFSK curves for different values of M 32

2.21 MQAM curves for different values of M 33

3.1 Constellation diagram of BPSK 36

3.2 Constellation diagram of BFSK 37

3.3 Symbol error probability curves of BPSK over AWGNand Rayleigh fading channel 40

3.4 Symbol error probability curves of BFSK over AWGNand Rayleigh fading channel 41

3.5 Symbol error probability curves of BPSK over AWGNand Rician fading channel 41

3.6 Symbol error probability curves of BFSK over AWGNand Rician fading channel 42

3.7 Symbol error probability curves of BPSK and BFSK overRician fading channel 42


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3.8 Symbol error probability curves of BPSK over AWGNand Nakagami-q (Hoyt) fading channel 43

3.9 Symbol error probability curves of BPSK over Rayleigh,Rician and Nakagami-q (Hoyt) fading channel 43

3.10 Bit error probability curves of BPSK over Rayleighfading channel and after Selection combining. 47

3.11 Bit error probability curves of BFSK over Rayleighfading channel and after Selection combining. 48

3.12 Bit error probability curves for increasing numberof Antennas of BPSK 48

3.13 Bit error probability curves of BPSK with Rayleighfading and Selection combining, Maximal ratio combining. 49

4.1 Baseband constellation diagram of BPSK. 51

4.2 PDF curve of PSK with additive noise and imperfect phase () 52

4.3 Symbol error rate curves of BPSK with imperfect phaseover AWGN 56

4.4 symbol error rate curvers of BPSK with and without phaseerror over Rayleigh fading channel. 57

4.5 symbol error rate curves of BPSK with and without phaseerror over Rayleigh fading channel. (Harmite method) 57

4.6 comparison of BPSK with phase error and DPSK overAWGN channel 58

4.7 symbol error curves of BPSK with out phase error and DPSKover AWGN 59

4.8 symbol error rate curves of BPSK with out phase error andDPSK over Rayleigh fading. 60

4.9 comparison of BPSK with phase error and DPSK over Rayleighfadind channel 60

4.10 Bit error rate curves of the QPSK without phase error andwith phase error over AWGN channel 61

4.11 Bit error rate curves of QPSK with out phase error and

pi/4DQPSK over AWGN channel 62

4.12 BER curves of QPSK with phase error and pi/4DQPSKover AWGN channel 62

4.13 BER curves of QPSK with phase error and pi/4DQPSKover Rayleigh channel 63


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Chapter 1

1 INTRODUCTION

Today communication enters our daily lives in so many different ways that it is very

easy to overlook the drawbacks of this system. The telephones at our hands, the radio and

televisions in our living rooms, the computer terminals with access to the Internet in our

homes and offices, and our newspapers are all capable of providing rapid communication

from every corner of the globe. The ability to communicate with people on the move has

evolved remarkably since Guglielmo Marconi [23]first demonstrated radios ability to

provide continuous contact with ships sailing the English Channel. That was 1897, and since

then new wireless communications methods and services have been enthusiastically adopted

by people throughout the world. At present, more than 1 billion people pay a monthly

subscription for wireless telephone service, and the wireless communications industry has

been experiencing phenomenal annual growth rates exceeding 50% over the past several

years.

The wireless revolution was triggered and is being sustained by several important

factors: advances in microelectronics, high-speed intelligent networks, positive user response

and an encouraging regulatory climate worldwide .Beyond the arena of mobile

communications, there are numerous wireless applications including Wireless local area

networks (WLANs), Bluetooth, Local Multipoint Distribution Systems (LMDS), satellite

communications and radiofrequency identification (RFID) operating at frequencies extending

into the millimetre-wave regime (>30 GHz). The Bluetooth standard offers fast and reliable

digital transmissions of both voice and data over the globally available 2.4 GHz ISM

(Industrial, Scientific and Medical) band. LMDS, the next big broadband radio access system

provides broadband telecommunications access in the local telephone exchange operate at a

very high frequencies of the order of 30GHz. Current spectrum allocations for cellular and

PCS systems are concentrated at frequency bands around 900 MHz and 1.8 GHz; future

allocations for PCS systems are expected around 2.4 GHz and 5.8 GHz. The move to higher

(millimeter-wave) frequencies has been motivated by the need for more and more bandwidth

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for multimedia applications such as wireless cable TV and high-speed internet access, and

by increased overcrowding of lower frequency bands. To meet this increasing demand, new

wireless techniques and architectures must be developed to maximize capacity and quality of

service (QoS) without a large penalty in the implementation complexity or cost. This provides

many new challenges to engineers involved with system design, one of which is ensuring the

integrity of the data is maintained during transmission.

In wireless system, signal strength is changed rapidly over a small travel distance or

time interval, the reason behind this is multipathpropagation of the signal. Multipath in radio

channel creates small scale fading. Due to the fading effects the signal received by the mobile

at any point in the space may consists of a large number of plane waves having randomly

distributed amplitude, phase, and angles of arrival.

The degradation of transmission quality due to channel fading cannot be simply

overcome by increasing the transmitted signal power. This because, even with high

transmitted power, when the channel is in deep fading, the instantaneously received SNR per

bit can still be very low, so there is a high probability of transmission error during the deep

fading. Therefore to solve this problem or to mitigating the effect of fading, diversity is

applied.

The signals which are transmitted from the transmitter and propagating through the

space, then received by the receiver is suffered from another problem, known as the

synchronizationproblem. The synchronization problem is the most discussed issue in recent

time in the field of wireless communication. The mismatch of certain parameters at the

transmitter and receiver is called synchronization problem. The parameters are amplitude,

frequency and phase, any one of these parameters if not tracked perfectly at the receiver the

signal is distorted.

1.1 MOTIVATIONAND BACKGROUND

In recent years, we are experiencing huge growth rates in wireless and mobile

communication system. Increasing mobility awareness in society and the world wide

deregulation of former monopolized markets. While traditional communication paradigms

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deal with fixed networks, mobility raises a new set of questions, techniques, and solutions.

For many countries, wireless communication is the only solution due to the lack of an

appropriate fixed communication infrastructure. All these make wireless communication

system so much popular and create ever-increasing demand to understand the development

and possibilities of wireless communication. There are many new and exciting systems

currently being developed in research labs, more mobile devices, the merging of classical

voice and data transmission technologies and the extension of todays Internet applications

onto mobile and wireless devices. 4G wireless system is coming, the demand for the higher

data rate and so that spectrum bandwidth is increasing simultaneously. Therefore it is

necessary to increase the number of base-station and the coverage area. To give a strong and

high data rate signal, the number of cell, micro cell would be increased and the frequency

reuse should be maximized. But the allocated area and spectrum is limited or restricted. The

result is to increasing the rate of interference of the spectrums, cross talk, and the performance

degradation of the wireless signals. So it is the primary need for todays scenario that to

achieve the higher data rate with these limited spectrum bandwidth and improve the

performance of the signals.

1.2 THESIS OBJECTIVES

The objective of this thesis is to study the M-ary digital modulation schemes on the

basis of the error performance that are very important in to-days communication scenario.

The demand for higher data rate and better bandwidth efficiency is increased day by day, but

the total bandwidth allocation is limited. Therefore it is very much necessary to study the

modulation schemes which give us the better result. M-ary modulation schemes achieve

betterbandwidth efficiency than other modulation techniques and give higher data rate. We

also study the constellation diagrams of the M-ary modulated signals, modulations with large

constellations have higher data rates for a given signal bandwidth, but will have higher error

rates which require more transmission power to maintain a given Quality of service as

determined by the communication service.

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Next we study that how multipath and fading effect the modulated signals, and the

degradation of the symbol error rate due to these multipath and different fading channel. In

this section signal degradation due to Rayleigh, Rician and Nakagami-q (Hoyt) is to be

discussed. All these study is very much important, if it is considered for the wireless

environment and every one known that todays world is wireless. Because of that the study of

the propagation behaviour over the wireless channel is very much important. Another

necessary study is to how diversity will improve the degradation of the faded signals. By

applying diversity to the receiver end the version of the incoming signal which gives the best

SNR can be achieved.

For the wireless communication system another threat of the signals is the

synchronization mismatch. The main objective is to study the degradation of the errorrates

due to the imperfect phase estimation of the coherently detected signals at the receiver end

and to show BPSK and QPSK signals how affected due to the phase synchronization

mismatch.

1.3 THESIS OUTLINE

The thesis begins with an overview of M-ary digital modulation schemes. It is

discussed, about the various application of the M-ary modulation scheme, in all most all field

of the communication system in the Section 2 of Chapter 2. A brief description about the

general form the M-ary modulation schemes, what will be the geometric representation and

baseband representation of a signal, what is grey coding in the Section 2.3. Section 2.4 is

about the constellation diagrams of the M-ary modulated signals. The receiver structures

diagrams are depicted in Section 2.5. The error performance of the signals is discussed in

detailed in Section 2.6. Section 2.7 is trying to give the alternative expressions of the

probability of the symbol error rates over AWGN channel. Finally Section 2.8 depicts all the

simulation results.

Chapter 3is devoted to the performance analysis of BPSK and BFSK over thefadingchannels

and mitigating the effect of fading by the diversity techniques. Section 3.1 will give a brief

introduction about the fading, how a signal will be faded and what is Multipath effects. In

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Section 3.2, we discussed error performance of the BPSK and BFSK over Rayleigh , Rician,

and Nakagami-q (Hoyt) fading channels. The output results of the simulation of the error

performance are showed in Section 3.3. Section 3.4 is about the diversity, how diversity will

improve the performance of the faded BPSK and BFSK signals, why diversity is so

important. In this thesis only selection combining (SC) and maximal ratio combining (MRC)

,diversity techniques are discussed. Finally Section 3.4 will display the simulation results.

Chapter 4 is about the error performance analysis of the BPSK and QPSK signals with the

imperfect phase estimation. Section 4.1 will give a introduction about the phase error and how

will it effected the signals. In Section 4.2, we calculate the probability of error due to the

phase degradation for BPSK signal over AWGN and Rayleigh fading environment in Section

4.3. Hermite method of integration is applied for the calculation of the Probability of error

over Rayleigh fading channel in Section 4.4. Section 4.5 is for the simulation results.

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Chapter 2

M-ary Modulation Schemes

2.1 INTRODUCTIONAdvancement in very large-scale integration (VLSI) and digital signal processing

(DSP) technology have made digital modulation more cost effective than analog modulation.

Digital modulation offers many advantages such as greater noise immunity and robustness to

channel impairments, easier multiplexing of various forms of information and greater

security. Moreover, digital transmissions accommodate digital error control codes, support

complex signal conditioning and processing techniques such as source coding, encryption,

equalization and diversity combining to improve the performance of the overall

communication link.

2.2 APPLICATIONS

Digital modulation schemes are classified into two large categories: constant envelope

and non-constant envelope. Under constant envelope class, there are three subclasses:FSK,

PSK,and CPM. Under non-constant envelope class, there are three subclasses: ASK, QAM,

and other non-constant envelope modulations. Among the listed schemes, ASK, PSK, and

FSK are basic modulations, and MSK, GMSK, CPM, MHPM, and QAM, etc. are advanced

schemes. The advanced schemes are variations and combinations of the basic schemes [1]

The generic non-constant envelope schemes, such as ASK and QAM, are

generally not suitable for systems with nonlinear power amplifiers. However QAM,

with a large signal constellation, can achieve extremely high bandwidth efficiency .M-

ary PAM has got applications in variety of areas such as communication and radar

systems, inter-vehicle communication, satellite communication [2] and location

among satellite formation [3-5], low power communication applications [6] and radar

systems [7-8].M-ary QAM has got too many practical applications which require

higher data rates like ADSL, modems, digital CATV applications [9], HDTV systems

[10].

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The PSK schemes have constant envelope but discontinuous phase transitions from

symbol to symbol. M-ary PSK schemes were used in quasi-optical wireless array applications

[11], compressed Image communication in mobile fading channel [12], space applications

[13], Tracking and Data Relay Satellite System (TDRSS) [14], telemetry with high

performance wireless MEMS strain-sensing applications [15] communication systems like

TDMA [16] and land mobile satellite communication links [17].

M-ary FSK modulation is widely applied in signal-to-noise ratio (SNR) or, power limited

systems such as deep space probes [18], satellites and space telemetry where link capacity may

be enhanced at the cost of required transmission bandwidth.

2.3 GENERALDESCRIPTIONOFTHE M-ARYSCHEME

M-ary digital signal set is represented as

( ) cos(2 )i i i i s t A f t = + (2.1)

For M-ary digital PAM;

iA is the Amplitude of the input signal, where i =1,2,3. 1M ,given by

Ai= (2m-1-M)d.

wherem=1,2,3......8, dis the distance between two signal points. i =0;

andc

i c

nf f

T= = ;

where cn is the fixed integer, Tis the symbol period which is n times the bit period.

For M-ary digital PSK;

i is the phase of the input signal, which is given by

i =2( 1)i

M

; i =1,2,3. 1M .

andc

i c

nf f

T

= = ; iA A= ;

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For M-ary digital FSK;

ifis the frequency of the input signal, which is given by

( )c

i

n i

f T

+= ; i =1,2,3. 1M .

and i =0; iA A= ;

In M-ary signaling schemes one of Msignals s0(t),s1(t), ..sM-1(t) are transmitted

during each signaling interval of duration Ts. These signals are generated by changing the

amplitude, phase (or) frequency of the carrier in the M-discrete steps. In M-ary modulation,

n=log2Mdata bits are represented by a symbol, where M=2n.So the bandwidth efficiency is

increased to n times.

2.3.1Geometric representation

The essence of geometric representation of signals is to represent any set of Menergy

signals { ( )}is t as linear combinations ofNorthogonal basis functions, where N M . That

if there is a set of real valued energy signals s 1(t),s2(t), ..sM(t), each of duration Tseconds,

then

1

0( ),

1,2,......

N

i ij j

j

t T s s t

i M

=

= =

where the coefficients of the expansion are defined by

0

1,2,...( ) ( ) ,

1, 2,....

T

ij i j

i M s s t t dt

j N

== =

The real valued basic functions 1 2( ), ( ),..., ( )Nt t t are orthogonal, by which we mean

0

1 if i=j( ) ( )

0 if i j

T

i j ijt t dt = =

Where ij is the kronecker delta. The first condition of the above equation states that each

basis function is normalized to have unit energy. The second condition states that the basis

functions 1 2( ), ( ),..., ( )Nt t t are orthogonal with respect to each other over the interval

0 t T .

2.3.2Base band representation

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Let, the transmitted signal s(t) be a narrowband signal cantered around

frequencyfc, such that

{ }tf2j ce)t(s~)t(s =

where, )t(s~

is called the baseband/low-pass equivalent or, complex envelope ofs(t). For an

unmodulated carrier ( )tf2cosA)t(s c= , the complex envelope )t(s~ is simplyA.

2.3.3 Gray coding

The bits sequence to signal mapping could be arbitrary provided that the mapping is one-to-one. However, a method called Gray coding is usually used in signal assignment in MPSK.

Gray coding assigns n-tuples with only one-bit difference to two adjacent signals in the

constellation. When an M-ary symbol error occurs, it is more likely that the signal is detected

as the adjacent signal on the constellation, thus only one of the n input bits is in error.

2.3.4M-ARY PULSE AMPLITUDE MODULATION

A M-ary digital PAM signal set is represented as

2 f

( ) Re[ ]ci t

i i s t A e

= cos2i c A f t = , i=1,2,...,M, 0 t T (2.2)

whereAi denote the set ofMpossible amplitudes.(M=2n).Ai=(2m-1-M)d,m=0,1,2M-1

.where 2d= mnd is the distance between adjacent amplitudes.

2.3.5M-ARY PHASE SHIFT KEYING

A M-ary digital PSK signal set is represented as

( ) cos(2 )i c i s t A f t = + , i=1,2,.M., 0 t T (2.3)

where 2( 1) /i i M = .

There are two orthogonal basic function 1(t) & 2(t),contained in the expansion ofSi(t).the

above expression(equation 2)can be written as

i i c i c( ) cos cos2 sin sin2 t s t A f t A f =

i1 1 i2 2( ) ( ) s t s t = + (2.4)

where

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1 1

0

( ) ( ) ( ) cos

T

i i s i s t s t t dt E = =

2 20

( ) ( ) ( ) sinT

i i s i

s t s t t dt E = =

where

21

2s E A T = is the symbol energy of the signal. The phase related tosi1andsi2as

2

1

tan iii

s

s =

2.3.6DIFFERENTIAL M-ARY PHASE SHIFT KEYING

In differential MPSK, the M-ary information is contained in the phase

transitions rather than in the phase value. For example, in QPSK constellation ,there

are four possible phase transition(0, &2 )and two bits (a 4-ary symbol)are

required to choose one of them. In the general MPSK case, there are M possible

phase transitions (( 1)22 40, , ,............,

MM M M

) & log2Mbits are required to

choose one of them.

First, lets start with the expression for the transmitted signal

( )i

s t during interval 1i n i +

( ) cos(2 )i c i

s t A f t = + + (2.5)

where 0 is the intermediate frequency, st is the sampling time, is the unknown carrier

phase, & i is the differentially modulated phase given by

1i i i = + (2.6)

The phase shift i depends on the input symbol, { }0,1,......., 1i M so that the

data are encoded on one of the Mpossible phase transition.

2.3.7M-ARY FREQUENCY SHIFT KEYING

There are M signals with different frequency to represent the M messages. in this

modulation scheme the information of the transmitted signal is carried by the frequency. The

expression for the ith

signal is

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( ) cos[ ( ) ]i c s t A n i t T

= + , i=1,2,M-1. 0 t T (2.7)

Carrier frequency2

cc

nf

T= .where nc is the fixed integer.where T is the symbol period

which is n times the bit period. If the initial phases are the same for all i, for the signals to be

orthogonal, the frequency separations between any two of them must be m/2T for coherent

case and m/T for non-coherent case. Thus the minimum separation between two adjacent

frequencies is 1/2Tfor orthogonal case and 1/Tfor non-coherent case. These are the same as

those of the binary case. Usually a uniform frequency separation between two adjacent

frequencies is chosen for MFSK.

2.3.8M-ARY QUADRATURE AMPLITUDE MODULATION

Quadrature amplitude modulation (QAM) is such a class of non-constant envelope

schemes that can achieve higher bandwidth efficiency than MPSK with the same average

signal power. In MPAM scheme, signals have the same phase but different amplitude and in

MPSK scheme; signals have same amplitude but different phase or varying phase. Now

MQAM is a scheme, where signals have varying amplitude and varying phase. A MQAM

signal set is represented as

( ) cos(2 )i i c i s t A f t = + i=1,2M. (2.8)

whereAiis the amplitude and i is the phase of the ith

signal in the M-ary signal set.

( ) cos(2 )i i c i s t A f t = + , i=1,2M

1 22 2

i c i c A cos f t A sin f t = (2.9)

where 1 cosi i iA A = 2 sini i iA A =

2 2

1 2i i i A A A= +

Equation (8) can be written as a linear combination of two orthogonal function

i1 1 i2 2( ) ( ) s t s t = +

where,

1

2( ) cos(2 )ct f t

T = 0 t T

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22

( ) sin(2 )ct f tT

= 0 t T

and 1 1cos

2 2

S S

i i i i

E E s A A = =

2 2sin

2 2

S Si i i i

E E s A A = =

2.4 CONSTELLATION DIAGRAMS OF DIFFERENT M-ARY SCHEMES

A constellation diagram is a representation of a signal modulated by a digital

modulation scheme. It displays the signal as a two-dimensional scatter diagram in the

complex plane at symbol sampling instants. In a more abstract sense, it represents the possible

symbols that may be selected by a given modulation scheme as point in the complex plane.

Measured constellation diagrams can be used to recognize the type of interference and

distortion in a signal.

2.4.1 M-ARY PULSE AMPLITUDE MODULATION:

MPAM signals are one dimensional and represented by ( ) ( )i is t s f t = , where ( )f t is

defined as the unit energy signal.

Fig.2.1-constellation diagram for MPAM illustrated for M=8

Construction of the above diagram can be done from the following equationAi=(2m-

1-M)d . where m=1,2,8,as M=8.and The Euclidean distance between any pair of

points is

2d= ( ) 2mn i jd s s= , where mnd is the Euclidean distance between any pair of points.

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2.4.2 M-ARY PHASE SHIFT KEYING:

The MPSK signal constellation is two-dimensional. Each signal ( )is t is represented by a

point ( )1, 2i is s in the coordinates presented by ( ) ( )1 2t , t . Polar coordinates of the signal are

( , )iE . The Mpoints are equally spaced on the circle of radius and centre at origin.

Gray coding is usually used in signal assignment in MPSK. Here a 8-PSK constellation

diagram is represented.

2.4.3 M-ARY FREQUENCY SHIFT KEYING:

Here a 2-FSK and a 3-FSK constellation diagrams are re presented.

Fig.2.3-constellation diagram for MFSK illustrated for (a) M=3 and (b) M=3

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min2

sd E= , where sE is the energy of the corresponding signals as s1(t),s2(t)ors3(t).And

number of orthogonal signals depend on how many signal is considered. In case of 2-FSK ,it

is 2 ( ( ) ( )1 2t , t ).

2.4.4 M-ARY QUADRATURE AMPLITUDE MODULATION:

Fig.2.4 constellation diagram for MQAM, illustrated for M=16

Fig.2.5 constellation diagram of 16-QAM with respective amplitude and phase

This constellation is known as square constellation. With an even number of bits per symbol,

may have

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whereL is a positive integer. Under this condition, an M-ary QAM square constellation can

always be viewed as the Cartesian product of one dimensional L-ary PAM constellation with

it self. HereL=4,so it is a figure of a Cartesian product of the 4-PAM constellation with itself.

A few of the other constellations offer slightly better error performance, but with a much

more complicated system implementation, like star constellation.

2.5 RECEIVER STRUCTURES

The type of modulation and detection (coherent or non-coherent) determines the structure of

the decision circuits and hence the decision variable, denoted byz. The decision variable, z,

is compared with M-1 thresholds, corresponding to Mdecision regions for detection purposes.

Fig.2.6 Receiver structure for coherent detection of signals.

Figure (2.6) depicts the generalized receiver structure for the coherent detection of the signals,

figure (2.7) to figure (2.11) depicts the modified suitable versions of the figure (2.6) for the

detection of the a particular modulation signal as MPAM, MPSK, MDPSK, MQAM and

MFSK.

2cos(2 )cf t

T

Fig.2.7 coherent detection of MPAM

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Fig.2.8coherent detection of MPSK

Fig.2.9MDPSK detection

2cos(2 )cf t

T

2

sin(2 )cf tT

Fig.2.10 coherent detection of MQAM

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Fig.2.11 coherent detection of MFSK.

2.6ERROR PERFORMANCE

The error performance, which is discussed in this thesis, is mainly caused due to the

channel noise. Channel noise is introduced anywhere between the transmitter output and the

receiver input.

Modulation schemes are chosen or designed according to channel characteristic in order

to optimize their performance. Additive white Gaussian noise (AWGN) channel is a universal

channel model for analyzing modulation schemes. In this model, the channel does nothing but

add a White Gaussian noise to the signal passing through it. Then the received signal is

represented as

( ) ( ) ( )r t s t n t = + (2.10)

where r(t)is the received signal, s(t) is the transmitted signal, and n(t) is the

additive white Gaussian noise. We calculating error performance which caused by

n(t) with zero mean andN0/2 as variance.

2.6.1M-ARY PULSE AMPLITUDE MODULATION:

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The error probability of the coherent detection for an M-ary PAM with equal amplitude

spacing can be derived as follows. Assuming AWGN channel with two side noise probability

of noise ofN0/2,

0 0( ) ( ) [ ( ) ( )] ( )

T T

i ir r t t dt s t n t t dt s n = = + = + (2.11)

where n is Gaussian noise with zero mean and a variance ofNo/2.thus ris Gaussian

with mean si and variance N0/2.Assuming si is transmitted, a symbol error occurs when the

noise n exceeds in magnitude one-half of the distance between two adjacent levels. The

average symbol error probability for all equally likely amplitude levels is

1

2

ij

s r i

dM p p r s

M

= | |

(2.12)

where dij is the distance between adjacent signal levels, as siand sj. and ijd can be

written as 2ij i jd s s d = =

Thus

( )

2

0

0

1 1 2 xN

s r i d

M M

p p r s d e dxM M N

= | | =

2

2

20

2

0

21 2 2( 1)

2

x

Nd

dM Me dx Q

M M N

= =

(2.13)

The symbol error probability can be expressed in terms of the average energy or power

of the signals. The average energy of the signals is

2

1 1

1 1M M

avg i i

i i

E E AM M= =

= = (2.14)

2 22 2

1

2 2

( 1)1 1(2 1 )

3

1( 1)

3

M

i

M M di M d

M M

M d

=

= =

=

(2.15)

Therefore

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2

0

2

0

62( 1)

( 1)

62( 1)( 1)

avg

s

avgs

EMp Q

M M N

or

p TMp QM M N

=

=

(2.16)

where avgavg

Ep

T=

In plotting the probability of error for M -ary signals such as M -ary PAM, it is

customary to use the SNR per bit as the basic parameter. Since bkTT = and .log2 Mk = thus

( ) ( )2

2

0

2 1 (6 log )1

bavs

M M EP QM M N

= (2.17)

2.6.2 M-ARY PHASE SHIFT KEYING:

Fig.2.12 decision region for representing is and js

( )21

,M

s i j

j

p p s s=

(2.18)

Assume that 2 ( , )i jp s s pair wise error probability in that if a dada transmission system

uses only a pair of signals is , js then 2 ( , )i jp s s is the probability of the receiver mistaking js

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for is . In figure (2.12) the decision boundary is represented by the bisector that is

perpendicular to the line joining the point is and js .when is is sent and if the observation

vector x lies on the side of the bisector where js lies, an error is made.

2 ( , )i jp s s =p(x is close to js than is / is is sent)

2

2

00

1exp

ijd

vdv

NN

=

(2.19)

where ijd is the Euclidean distance between is and js .

ijd = i js s| |

From the complementary error function, can write

22( ) exp( )u

erfc u z dz

= (2.20)

now0

vz

N= , then we find equation(2.19)

2

0

1( , )

2 2

ij

i j

d p s s erfc

N

=

(2.21)

Substituting equation(2.21) into equation(2.18)

1 0

1

2 2

Mij

s

j

d p erfc

N=

(2.22)

Now from the constellation diagram it can be said

12 18d 2 sind E

M

= , and as ( )( ) 2 2erfc u Q u= therefore equation (2.18) becomes

0

2sins

s

Ep Q

N M

(2.23)

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2.6.3 DIFFERENTIAL M-ARY PHASE SHIFT KEYING:

Fig.2.13 angle between two vectors.

Fig.2.14 decision boundary representation

If i-i-1=i=k is the value of the phase communicated in the ith

transmission

interval, then a correct decision made when falls in the kth wedge defined by the

angular interval ,k kM M

+

.then we can write

1 2 1( / ) mod2 ( )

( )

s r i i

M

r k X

M

p c k pM

p f x dxM

= ( )

= @ (2.24)

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where kx = , analytical evaluation of symbol error probability reduces to find the

probability density function of x.

From [19], we get the result ,in which 1st

expressed in the form of a convolution integral

based on probability density functions of 1 and 2 and then evaluating this integral using the

method of characteristics function with the result

2

00

0

1(sin )[1 (1 cos sin )]

2

( ) exp{ (1 cos sin )}

0 otherwise

s

sX

Ex

N

E f x x x d

N

x

+ +

=

(2.24)

Since this probability density function is an even function ofx and independent of the

particular phasek transmitted in the ith

interval, then ( / )s p c k is independent of k and hence

the average symbol error probability is given by

1

1( ) ( / ) 1 ( )

= ( ) ( )

=2 ( )

MM

s sMk

M

M

M

p E p E k f d M

f d f d

f d

=

= =

(2.25)

We can generalize the equation(2.24) from the reference of [20],then write from case 1,with

1 M

= & 2 =

,for equal signal condition(=0),uncorrelated noise.

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2

2 2

2

2

2 2

2

( ) ( ) ( )

exp{ [1 cos cos ]}sin2

=0- 41 cos cos

exp{ [1 cos cos ]}sin2

=4

1 cos cos

X

M

f x dx F F M

At

MMdt

tM

At

MM dt

tM

=

(2.26)

Since

2

22

ASNR

= then

2

22

0 0

(log )2

s sE EA MN N

= = (2.27)

Substituting equation (2.27) into equation (2.26) , the symbol error probability of

equation(2.25) becomes

2

0

2

exp{ [1 cos cos ]}sin

( )2

1 cos cos

s

s

Et

N MM p E dt

tM

=

(2.28)

This is in the form of a single integral of simple function. Many approximations

have been found for the symbol error probability performance.

Fleck & Trabka given that for a larges

o

E

N

2

0

exp( )

( ) 14 [ ]8

ss p E erfc E

N

= + + (2.29)

where = 2 sin( )s

o

E

N M

Arthurs and Dym gives

0

( ) ( sin )

2

ss

E p E erfc

N M

(2.30)

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2.6.4 M-ARY FREQUENCY SHIFT KEYING:

From figure (2.12) we can say that the receiver decides in favour of symbol s1, if

the received signal point represented by the observation vector x lies near s 1. An error

occurs when x lies nearer to a symbol other than s1 as 1 1j s s j .

ep = 1-p(x is close to s1/s1 is sent) (2.31)

Because of symmetry, the average symbol error is same as the equation (2.29)

s ep p=

12

1 1 1 2 11 ( / ) ( / )s

j p s s p s s ds ds

= (2.32)

Assuming AWGN channel with zero mean and a variance0

2s

NE then

2

11 1

00

( )1( / ) exp s

ss

s Ep s s

N EN E

=

(2.32)

and

2

1

00

1( / ) exp

j

j

ss

sp s s

N EN E

=

(2.34)

Substituting equation (2.34) and equation (2.33) into equation (2.32) and normalizing the

variables we obtain

2

0

21 11 exp{ }

22

ss

Ep x

N

=

(2.35)

For equal energy, equiprobable signal set, the symbol error probability is

( )( )0

22

111 exp 1 ( )

22

sEN

M

s

x p Q x dx

=

(2.36)

Now if we expand the term ( ) 11 ( ) MQ x get an infinite series containing infinite

range integrals.

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( ) 11 ( ) 1 ( 1) ( ) ............MQ x M Q x +;

We only take the first term and neglecting other higher power as they given us a negative

result, we can write

( )

[ ]02

2

11 exp 1 ( 1) ( )

22

sEN

s

x p M Q x dx

=

(2.37)

( ) ( )0 0

2 22 2

1 ( 1)1 exp exp ( )

2 22 2

s sE EN Nx xM

dx Q x dx

= +

(2.38)

( )0

22

1( 1) exp ( )

22

sENx

M Q x dx

=

(2.39)

If the signal set is not symmetrical, an upper bound has been obtained, as

02

ij

s

i j

dp Q

N

(2.40)

As the signal set for MFSK is equal energy and orthogonal, distances between any two

adjacent signals are equal, 2ij sd E=

Therefore0

( 1) ss

E p M Q

N

(2.41)

2.6.5 M-ARY QUADRATURE AMPLITUDE MODULATION:

For square QAM constellation with M=2n, where n is even .QAM constellation is

equivalent to two PAM signal on quadrature carriers, each having L M= signal points. A

QAM symbol is detected correctly only when two PAM symbol are detected correctly. Thus

the probability of correct detection of a QAM symbol is

2(1 )c M

p p= (2.42)

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where Mp is symbol error probability of a M -ary AM with one-half the average

power of the QAM signal, from equation (2.16) we have

0

62( 1)( 1)

avg

MEMp Q

M NM =

(2.43)

where0

avgE

N is the average SNR per symbol. The symbol error probability of

square MQAM is

2 21 (1 ) 2s M M M p p p p= = (2.44)

At high SNR,

0

34( 1)2

( 1)

avg

s M

EM p p Q

M NM

=

(2.45)

Therefore

2

0

31 1 2

( 1)

avg

s

Ep Q

M N

(2.46)

or

2

2(3log )( ) 1 1 2( 1)

e

Mp Q

M

(2.47)

2.7 ALTERNATE ERROR EXPRESSION

The classical definition of complementary error function, Gaussian probability integral

function/ Q function, Marcums Q function and incomplete gamma function has the argument

(or, one of the arguments) in the integration limit, not in its integrand. This makes

computation of these functions quite difficult and when one needs to evaluate any integration

involving them (this is most common while calculating average symbol error probability) the

person cannot possibly use the method of exchanging sequence of integration in the

expressions involving multiple integrals. Alternate representations of these functions have the

integration limits independent of function arguments. This feature greatly simplifies the

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evaluation process. Also integrations are defined over a finite range which makes the

numerical calculation easy.

Gaussian Q-function:

The one dimensional Gaussian Q-function ( )xQ is defined as the complement of

cumulative distribution function (cdf) corresponding to the normalized Gaussian Random

variable X . This function can be given by

( ) dyyxQ

x

=

2exp

2

12

This representation suffers from two disadvantages. This relation requires truncation of

the upper infinite limit when using numerical integral evaluation or algorithmic techniques.

The presence of the augment of the function as the lower limit of the integral poses analytical

difficulties when argument depends on other random parameter that require statistical

averaging over their probability distributions.

( )xQ can also be defined as

( )

dx

xQ

=

2

0

2

2

sin2exp1

where this is defined only for 0x

Complementary error function and Q function

From [21-(7.4.11)]

We have, ( )xaerfcex

dtxt

e axat

2

2

20

22

=

+

0,0; >> xa .

Let, 1=x ; then the integral representation of ( )erfc become

( )( )21

2

0

2

1

a te

erfc a dt t

+

=+

With a change of variable = cott ,

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( ) 20

csc

2

2 aerfc a e d

=

which gives,

( )

=2

0

sin2

2

2

dezerfc

z

0; z

Writing the same in terms of Q function,

( )

=2

0

sin2 2

2

1

dezQ

z

0; z .

In these alternate representations, the integrand contains a term 2sin whose maximum

value occurs at 2 = . By replacing the integrand by its maximum value, we obtain the well

known Chernoff bound,

( )2zezerfc 0; z

and

( )2

2

21

z

ezQ

0; z

Further from [22-(3.468.1)]

we have,

( )uerfcu

dxuxx

e

u

x2

222 4

2

=

0; >u

With a change of variable cscux = ,

( ) ( )( ) ( ) =0

4

csc2

cotcsccotcsc

422

duuu

euuerfc

u

which gives,

( )

=4

0

sin22 2

2

1

dezQ

z

0; z

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From these alternative expressions of the Q and erfc function we can give an alternative

expression for the error probabilities of the MPAM,MPSK,MFSK,MQAM and MDPSK

For MPAM equation (2.17) can be written as

222 20

3(log )2( 1)( ) exp

( 1)sine MPAM

MMp d

M M

=

(2.48)

For MPSK equation (2.23) can be written as

2 20

2 sin1

( ) expsin

e MPSK Mp d

=

(2.49)

For MFSK equation (2.39) and be written as

220

( 1)( ) exp

2sine MFSK

Mp d

=

(2.50)

For MQAM equation (2.47)becomes

2

2220

3(log )2( ) 1 1 exp

2( 1)sine MQAM

Mp d

M

(2.51)

2.8RESULTS

The graphical analysis for the performance of M -ary modulation schemes over AWGN

channel are given below in Figure (2.15) to Figure (2.21).These graphs were plotted by taking

SNR in dB on the abscissa and probability of error on ordinate. Figure (2.15) to figure (2.18)

showed that the simulated and theoretical curves of the symbol error rate of the modulation

schemes almost matched each other or the simulated curves satisfied the theoretical symbol

error rates. Figure (2.19) to figure (2.21), from these figures it is observed that as the value of

M increases the probability of error is also increases.

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Fig.2.15 symbol error probability (simulation and theoretical) curve of MPAM (illustrated for M=4)

MPSK

Fig.2.16 symbol error probability (simulation and theoretical) curve of MPSK (illustraed forM=8)

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Fig.2.17 symbol error probability (simulation and theoretical) curve of MFSK (illustraed forM=8)

Fig.2.18 symbol error probability (simulation and theoretical) curve of MQAM (illustraed forM=16)

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Fig.2.19 symbol error probability curves of MPSK (illustrated for M=4,M=8 and M=16)

Fig.2.20symbol error probability curves of MFSK (illustrated for M=4,M=8 and M=16)

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Fig.2.21 symbol error probability curves of MQAM (illustrated for M=4,M=8 and M=16)

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Chapter 3

Fading Channel Modelling and Antenna Diversity Techniques

3.1 INTRODUCTION

In wireless communication system, propagation path or the transmission path between

the transmitter and receiver may be ofLine-of-sight path orNon-line-of-sight path. In case of

line-of-sight, the path between the transmitter and the receiver is direct. Whereas in non-line-

of-sight, the path is not direct, rather it is obstructed by some building, trees, mountains or

foliage, even-if some times earths curvature acts as a obstacle. Non-line-of-sight propagation

is mainly noticed in urban areas, where the density of obstacle is more. Due to the presence of

these obstacles in the path, the propagating wave may be reflected, diffracted, orscattered.

All there are the causes of or the creator of the non-line-of-sight path. So the transmitted

signal propagates through multiple path, with each having associated delay. And, the receiver

receives multiple copies of the transmitted signal. Interaction between these waves creates

multipath fading at a certain location and the strength of the waves decreases as the distance

between the transmitter and receiver increases. Therefore a wireless channel is random in

nature, and it is very difficult to predict the nature of the transmitted signal, like wired

channel which is stationary andpredictable. For all these reason it is all most impossible to

choose a single channel that would model all the propagation environment.

Because of the random nature of the wireless channels, a number of statistical models

has been developed to represent the channel conditions. Propagation models have

traditionally focused on predicting the average received signal strength at a given distance

from the transmitter, as well as the variability of the signal strength in close spatial proximity

to a particular location.Large-scalepropagation models are the models that predict the mean

signal strength for an arbitrary transmitter-receiver (T-R) separation, thereby estimating the

radio coverage of a transmitter [23]. In Small-scale propagation models, the propagating wave

fluctuates rapidly over a very short travel distance.

This chapter mainly study the error performance of the signals in Small-scale-fading

channel or in more detail about the signal which propagated through Slow-Flat-fading

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channel. ( has the constant gain and linear phase response over a bandwidth, which is greater

than the bandwidth of the transmitted signal ). The Rayleigh, Rician and Nakagami are the

most commonly used statistical models to represent small-scale fading phenomenon while

empirical models like the Hata, Okamura models predict the mean signal strengths

(representing the large-scale-fading effects) by estimating the path-loss component through

availablemeasurements [23].

3.2 ERROR PERFORMANCE

3.2.1Rayleigh Fading

In a stationary flat and slow fading channel ,two conditions can be find out (a) the delay

spread introduced by the multipath propagation environment is negligible in respect to the

symbol interval. From this we can say that the effect of the channel can be represented by a

complex gain ( ) exp[ ( )]t j t ,where ( )t is amplitude fading and ( )t is the phase

distortion. For a transmitted signal ( )is t ,with symbol interval sT ,the received signal in the

fading channel is

( ) ( ) exp[ ( )] ( ) ( )i

r t t j t s t n t = + (3.1)

Now the second condition is (b) that it is possible for the receiver to estimate ( )t and

remove it. So for the following BER calculation we assume ( ) 0t = .

In a cellular system the effect of propagation path loss and shadowing on the received

signal is compensated for by power control and the received signals experience only

multipath Rayleigh fading. In other words, the received signal energy is attenuated by a factor

2 ,that is 2E . In the following analysis we take /SNR bit =

) ( ) ( )b b p p x f x dx

( = (3.2)

where is the average received SNR/bit with respect to2

2

2

( )

= E( )

x f x dx

=

(3.3)

where (.)f is the probability density function of the amplitude fading . For Rayleigh

fading channel , follows a Rayleigh distribution with probability density function

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2

2 2exp , 0

( ) 2

0 0

xxx

f x

x

=


45/76

This expression is useful to simplify the error evaluation process. Also integrations are

defined over a finite range which makes the numerical calculation easy.

In case of Rayleigh fading,

2 2

( ) 2E = . Now in the present case of BPSK22Y = . From equation (3.5), getting the value of ( )bp x as

2( ) ( 2 )b

p x Q x

= (3.7)

Now substituting the equation(3.7)and equation(3.4) into equation(3.2).

therefore2

2

2 20

( ) ( 2 ) exp( )2

1= 1

2 1

b

X

xx p Q x d

=

+

(3.8)

3.2.1.2 ANALYSIS OF BFSK

Binary data are represented by two signals with different frequencies in BFSK.

Typically these two frequencies are for 0 and 1, the signals are

1 1

2 2

( ) cos 2 , 0 t T, for 1

( ) cos 2 , 0 t T, for 0

s t A f t

s t A f t

= = (3.9)

Fig.3.2: constellation diagram of BFSK

From equation (2.39) the symbol error probability of a BFSK signal can be expressed as

0

ors sE p Q p QN

(3.10)

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where0

E

N =

Alternative Expression of the Error:

220

1( ) expsin

e BPSK p d

= (3.11)

From the previous discussion of BPSK, the probability of error of BFSK in Rayleigh

fading channel is

( / )

0

( ) (1/ 2) ( 2).(1/ )b

P erfc e d

=

+=

21

21

(3.12)

3.2.2Rician Fading

The probability density function (PDF) of instantaneous signal to noise ratio (SNR)

under Rician fading is non-central chi-square distributed with n=2 degrees of freedom,

22

0 002 2 2 2( ) exp .exp .2 2

z

xxx f x I

=

(3.13)

Where 0 is 0 ( )t at any t.2

0 is the power of the LOS component and 0(.)I is the

zero-order modified Bessel function of the first kind. The Rician fading channel has an

important parameter called the K factor. It is defined as,

2

0

22z

K

@

therefore equation (3.13) can expressed as

( )( ) ( )

+

+

+=

KKI

KK

Kf

12

1exp

10 ; 0 (3.14)

3.2.2.1 ANALYSIS OF BPSK

MGF method: The probability of error for coherent detection of BPSK over Rician

channel is given by the following equation,

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( )

( ) ( )

2 2

2 2

0

1 sin1( ) exp

1 sin 1 sinb

K KaP d

K K

+= + + + +

(3.15)

The value can be obtained only through numerical integration.

3.2.2.2ANALYSIS OF BFSK

MGF method: Applying MGF method, the probability of error of BFSK over Rician

channel is also calculated and the calculation giving the following equation,

( )

( ) ( )

2 2

2 2

0

1 sin1 (1/ 2)( ) exp

1 sin (1/ 2) 1 sin (1/ 2)b

K KP d

K K

+= + + + +

(3.16)

3.2.3Nakagami-q (Hoyt) Fading

Nakagami-q (Hoyt) Fading is used to model fading environments more severe than

Rayleigh fading. This distribution is not frequently used in practice because it only applies in

the case where there is not a strong wave arriving at the receiver, and when at the same time

the in- phase and quadrature components of the received signal have different variances or are

correlated. This situation is usually, but not always, limited to the case of the Non-LineOf-

Sight (NLOS) propagation [24].

The Probability density function of instantaneous signal to noise ratio (SNR) under Hoyt

(Nakagami-q) fading is

( ) ( ) ( )2

2 42

02 2

1 11exp

2 4 4

q qqf I

q q q

+ + =

; 0 ; 10 q (3.17)

or, equivalently

( )

=

p

pI

ppf

1exp

10

(3.18)

where ( )222

1

4

q

qp

+=

10 p

and ( )0I designates the zeroth-order modified Bessel function of the first kind and is

the instantaneous fading amplitude.

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3.2.3.1ANALYSIS OF BPSK

1 22 2

2 4

0

1 2( ) 1

sin sinb

pP d

= + +

(3.19)

3.3 RESULTS

Fig. 3.3 symbol error probability (simulation and theoretical) curves of BPSK over AWGN and

Rayleigh fading channel

From this fig it is observed that the amplitude fading severally degrades the

transmission performance. In AWGN channel, for large value of ,the probability of error

decreases exponentially with respect to2 . On the other words in a Rayleigh fading channel,

the probability of error decreases linearly with respect to the . The smaller the bp value

required, the worse the performance degradation.

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Fig. 3.4 symbol error probability (simulation and theoretical) curves of BFSK over AWGN and

Rayleigh fading channel

It clear from the above two figures (fig 3.3 and fig 3.4) that the BFSK signal is much more

effected in the Rayleigh fading channel than BPSK.

Fig. 3.5 symbol error probability (simulation and theoretical) curves of BPSK over AWGN and Rician

fading channel

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Fig. 3.6 symbol error probability (simulation and theoretical) curves of BFSK over AWGN and Rician

fading channel

It is observed from figure 3.5 and figure 3.6 that signal strength degradation is less in

case of Rician fading channel than the Rayleigh fading channel.

Fig. 3.7 symbol error probability (simulation and theoretical) curves of BPSK and BFSK over Rician

fading channel

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Fig. 3.8 symbol error probability (simulation and theoretical) curves of BPSK over AWGN

and Nakagami-q (Hoty) fading channel

Fig. 3.9 symbol error probability curves of BPSK over Rayleigh, Rician and Nakagami-q (Hoty)

fading channel

The figure 3.9 Showed that the performance of BPSK is better in Rician fading channel

than other two fading channel, because in Rician channel the NLOS signals are also consider.

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3.4 MITIGATING THE EFFECT OF FADING BY DIVERSITY

Diversity combining devotes the entire resources of the array to service a single user.

Specifically, diversity schemes enhance reliability by minimizing the channel fluctuations due

to fading. The central idea in diversity is that different antennas receive different versions of

the same signal. The chances of all these copies being in a deep fades is small. These schemes

therefore make most sense when the fading is independent from element to element and are of

limited use (beyond increasing the SNR) if perfectly correlated (such as inLOS conditions).

Independent fading would arise in a dense urban environment where the several multipath

components add up very differently at each element.

Diversity is a commonly used technique in wireless systems to combat channel fading,

due to the following reasons [25];

(1) The degradation of the transmission quality due to fading cannot simply overcome by

increasing the transmitted power. Even with high transmitted power, when the

channel is in deep fad, the instantaneously received SNR per bit can still be very low,

is resulting a high probability of transmission error.

(2) The power of the reverse link in a wireless system is limited by the battery capacity in

hand-held subscriber units. With diversity the required transmitted power will be

reduced.

(3) Diversity limited the interferences, thus it support addition of more user and hence

increases the capacity of the system.

The physical model assumes the fading to be independent from one element to the next.

Each element, therefore, acts as an independent sample of the random fading process. Our

goal here is to combine these independent samples to achieve the desired goal of increasing

the SNR and reducing the BER. N elements in the receiving antenna array we receive N

independent copies of the same signal. It is unlikely that all N elements are in a deep fade. If

at least one copy has reasonable power, one should conceivably be able to adequately process

the signal.

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3.4.1SELECTION COMBINING:

As each element is an independent sample of the fading process, the element with the

greatest SNR is chosen for further processing. Let l be the received SNR per bit of the lth

channel at any instant, with mean ( )l cE = for {1,2,....., }l L . As l is proportional to

2

l , then l follows an exponential distribution with parameter c ,

1exp , 0

( )

0 otherwise

l c c

xx

f x

=

(3.20)

Now for selection diversity,

1 2max{ , ,....., }

L =

For 0,x has the following cumulative distribution function (cdf)

1 2

( ) ( )

=p( x .... )L

f x p x

x x

=

(3.21)

1

( )L

l

l

p x=

= with independently faded channel (3.22)

0

( )l

Lx

f z dz = with identically distributed fading (3.23)

1 exp( )

L

c

x =

(3.24)

And for 0x < , the cumulative distribution function of is ( ) 0f x = . The probability

density function of is then

( )( ) exp 1 exp( ) , x 0

L

c c c

df x L x xf x

dx

= = (3.25)

BPSK

Selection diversity improves the degradation of the faded BPSK signal and getting the

improved version of the signal as a result. First calculate the SER,

0

( ) ( )e s

p p f x d

= (3.26)

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where ( )sp is the SER in AWGN, which can express as

1( ) ( )

2s

p erfc = (3.27)

And can get the expression of ( )f x from equation (3.25), now substituting the

equation (3.27) and into equation (3.26),

1

11

( 1) 12 1

L Lk

ekk

Lp

k

+=

= + + (3.28)

BFSK

For calculating the symbol error performance of the BFSK signal, we using the SER of

BFSK in AWGN channel ( )sp .

( ) 1/ 2) ( 2s

p erfc = (3.29)

now substituting the equation(3.25) and equation (3.29) into equation (3.26), getting

( )1

0

1( 1) 1

12 2 1

Lk

e

k

Lp

k k

=

= + + +

(3.30)

3.4.2 MAXIMAL RATIO COMBINING:

In the above formulation of selection diversity, we chose the element with the best

SNR. This is clearly not the optimal solution as fully (L 1) elements of the array are

ignored. Maximal Ratio Combining (MRC) obtains the weights that maximize the output

SNR, i.e., it is optimal in terms of SNR.

The SNR per bit at the output of the combiner for thethk symbol is ,

2

10

Lb

k lkl

E

N

=

=

(3.31)

where0

bE

Nis the SNR for the AWGN channel with 1lk = andL=1. in Rayleigh fading

environment the lk s are identically independently distributed Rayleigh random variables

with parameter2

. Therefore , k follows a chi-square distribution with 2L degrees of

freedom. The probability density function is

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1 exp( / )( ) ; 0

( 1)!

L

c

L

c

x x f x x

L

=

(3.32)

where2

02 /

c bE N = is the average SNR per bit in each diversity channel.

BPSK:

Substituting equation (3.29) and equation (3.32) into equation (3.26), gives

1 1

1

[0.5(1 )] [0.5(1 )]L L l

L l

el

l

p +

=

= +

(3.33)

where1

c

c

=

+

3.5 RESULTS

Fig.3.10 Bit error probability curves of BPSK over Rayleigh fading channel and after Selection

combining.

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Fig.3.11 Bit error probability curves of BFSK over Rayleigh fading channel and after Selection

combining.

It is observed from the figure (3.9) and figure (3.10) selection combining technique

improves the symbol error performance. Signal to noise ratio of the BPSK and BFSK is much

better for a low symbol error rate. It is also clear that the performance of BFSK is better than

the BPSK.

Fig.3.12 Bit error probability curves for increasing number of Antennas of BPSK

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This figure (3.11) shows that the Selection diversity can greatly improved the Bit error rate

performance. The degree of improvement is more significant when number of Antenna (i.e L)

is increased from 1 to 2 then it is when further increased from 2 to 4 and to 8.

Fig.3.13 Bit error probability curves of BPSK with Rayleigh fading and Selection combining,

Maximal ratio combining.

It is clear from the figure (3.12) that with Maximal ratio combing technique the degree of

improvement of the Bit error rate performance of BPSK signal is much better than the

Selection combining technique for the same number of receiver antennas.(i.e.L=2).

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Chapter 4

Coherent Phase modulation with Imperfect Phase

Estimation

4.1 INTRODUCTION

Jitter is an unwanted variation of one or more characteristics of a periodic signal in

electronics and telecommunications. Jitter may be seen in characteristics such as the interval

between successive pulses, or the amplitude, frequency, or phase of successive cycles. Jitter is

a significant factor in the design of almost all communications links.

In digital data communication there is a hierarchy of synchronization problems to be

considered. First, assuming that a carrier-type system is involved, there is the problem of

carrier synchronization which concerns the generation of a reference carrier with a phase

closely matching that of the data signal. This reference carrier is used at the data receiver to

perform a coherent demodulation operation, creating a baseband data signal. Next comes the

problem of synchronizing a receiver clock with the baseband data-symbol sequence. This is

commonly called bit synchronization, even when the symbol alphabet happens not to be

binary. In this thesis only the carrier synchronization is considered [26].

In previous section, it is assumed implicitly that the same clock controlled both the

transmitter and receiver operations. This means that the corresponding events in the

transmitter and receiver are synchronous. In M-ary schemes, most demodulation schemes are

coherent; they make use of the phase information of the carrier. Optimum demodulation

requires then a local carrier at the receiver side whose frequency and phase are in perfect

agreement with that of the transmitted signal. So, two pairs of ideal identical oscillators at the

transmitter and receiver sides could ensure the synchronization. But in practice, the signal

emitted by a pair of oscillators with the same nominal frequency will start drifting from each

other because of their physical inability [27].

Two pairs of oscillators which are used at the transmitter and receiver is suffering from

various factors, like due to temperature variation, device non-linearity, ageing,power supply,

ripple and impulse. For all these reasons the performance of the oscillators degrades.

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The degradation of the modulated signal due to imperfect phase can be improved by

sending a pilot tone with the transmitting signal. But for this arrangement, the transmitted

power should, need to be increased, and for the pilot tone the effective bandwidth will

decreases, that is not desirable. During the demodulation the signal power associated with the

pilot tone may result in undesired DC component with practical filters. This may further

degrade the decision process or post problems to the receiver amplifier. Carrier recovery

technique, as PLL can be used. If the signal is distorted enough in the channel, operation of

PLL is suffered from instability and non-linear operations. Also the inherent noise of the PLL

there can be a phase mismatch. In case of wire communication, the phase of the transmitted

signal can be estimated and take corresponding steps to repair the errors. But in wireless

communication, it is very difficult to predict the phase of the incoming signal.

4.2 DEGRADATIONDUETO PHASE ERRORIN AWGN CHANNEL

Average error probability is,

( ) ( ) ( )p e p e p d

= (4.1)

Where 2( ) 221( )

2

m

p e

=

and

( ) is the probability of Bit error rate due to the incorrect phasep e

4.2.1 CALCULATING ( )p e FOR BPSK:

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Fig.4.1 baseband constellation diagram of BPSK.

From Baseband Constellation of BPSK , Baseband Equivalent Signals are,

exp( 0)sE j For 1 And

exp( )sE j For 0

Considering 1 to be sent from the transmitter for a given phase distortion ,the equivalent

received baseband signal is,

( )exp (0 )sE j + (4.2)

The projection on Real axis (which the decision device (DD) use for comparison with

threshold=0) is therefore

cos( )sE

Now the DD makes the correct decision when

cos( ) 0sE >

In the presence of noise the inequality changes to

cos( ) ( ) 0s E n t + >%

Where ( )n t% is the baseband equivalent noise following aGaussian PDF with mean zero and

2 0

2n

N = .

The total projection is also thus Gaussian in nature having mean of cos( )sE and

variance.

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Fig.4.2PDF curve of PSK with additive noise and imperfect phase ( )

2 0

2n

N = . So the probability of taking a correct decision is

( )0 cos( )

cos( ) ( ) 0s

r s

n

E p E n t Q

+ > = %

(4.3)

i.e,

0

0

0

cos( )( )

2

2=Q cos( )

2=1-Q cos( )

s

s

s

E p c Q

N

E

N

EN

=

(4.4)

Thus the probability of erroneous detection

0

( / ) 1 ( / )

2=Q coss

p e p c

E

N

=

(4.5)

Which is dependent on the variable . Therefore average error probability

2

220

2 ( )1( ) Q cos exp

22

s mE p e d N

=

(4.6)

Assuming m =0 i.e no biasness and using the alternate expression forQ(x).

i.e22

0

1( ) exp

2sin

zQ z d

=

(4.7)

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2 22

2 22 00

cos1 1( ) exp exp

sin 22

sE

p e d d N

=

(4.8)

Interchanging the order of integral,

2 22

2 2200

cos1 1( ) . exp exp

sin 22

sE

p e d d N

=

(4.9)

For small ( )4

x x

,we can approximate cos(x)with the first two terms its a infinite series

representation ,

i.e

2

cos( ) 12

xx ;

2

2

2

1 1cos cos(2 )

2 2

41 11

2 2 2

=1-

= +

+

;

2 2

2 2 220 00

1 1 1( ) . exp expsin 2 sin2

s sE E p e d d N N

=

(4.10)

2

2002 2

0

sin /exp

sinsin 2

s

s

Ed

NE

N

=

(4.11)

2

220

1 1exp

sin1e

p da

;(4.12)

Where,

2

sina

=

From the [28] we can derive another expression of the probability of error due to phase

degradation, from equation(4.5)

0

1( / ) cos

2

bE

p e erfcN

=

( )1 cos2

erfc = (4.13)

Expanding the right-hand side into a Taylor series yields

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( ) 21

( / ) exp( )2

p e erfc

= +

(4.14)

Now from equation(4.1)

( ) 21

( ) exp( )2

e p erfc

= +

(4.15)

4.3 PROBABILITYOFERRORDUETOPHASEJITTERIN RAYLEIGHFADINGCHANNEL

The probability of error rate is given by the following expression

0( ) ( )e e p p f d

= (4.16)

where ( )ep is the bit error rate of a BPSK signal in AWGN channel

and

1( ) expf

=

(4.17)

Therefore,

2

0 0

1( ) exp exp( )exp

2e p erfc d

= +

(4.18)

21

1 12 1 1

ep

= + + (4.19)

4.4 HERMITEMETHODOFINTEGRATION

By applying the Hermite [29] method of integration, the new form of probability of

error is given by

2

1

( ) ( )n

x

e i i

i

p e f x dx w f x

=

= = (4.20)

Where

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( ) 2

0

2( ) cosb

E f x p e Q

N

= =

(4.21)

After the calculation and interchanging the limits is replaced by 2 ( )x t

Therefore,

( )21

1cos ( 2 ( )) ( )

2

n

e

i

p erfc x t w t =

= (4.22)

Where n=9, and getting the value of ( )x t and ( )w t from the table of[b] corresponding n=9.

( )

1

1 1( ) 1

2 1

n

e fading

i

t p w t

t=

= +

(4.23)

Where,

( )22 cos 2 ( )t x t=

4.5 RESULTS

Fig.4.3 Symbol error rate curves of BPSK with imperfect phase over AWGN

From figure (4.3) it is observed that the symbol error rate is increased, when a prominent

phase error is present for the same value of signal to noise ratio in AWGN channel. Therefore

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we can say that the imperfect phase estimation in the receiver degrades signal performance

effectively.

Fig.4.4 symbol error rate curvers of BPSK with and without phase error over Rayleigh

fading channel.

This curve shows that derived error expression is indeed an upper bound when is

small(From simulation


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Fig.4.5 symbol error rate curves of BPSK with and without phase error over Rayleigh fading

channel. (Harmite method)

In case of figure (4.4) the BER curve dose not give satisfactory result for the high signal to

noise ration, the theoretical and the simulation curves dose not match with each other. And, it

is only applicable for the low SNR values. Therefore, the Hermite method of integration is

applied and it gives a good output result for the high SNR values. It is also observed from the

figure (4.4) and figure (4.5) that the imperfect phase estimation effects the bit error rate in

fading channel.

4.5 COMPARISONOF MPSK (WITHPHASEERROR) WITH MDPSK.

As discussed in the introduction part, PSK signals suffered from the phase distortion or, it

is not possible to predict the exact phase by the receiver. Now we compare PSK with its

counter part DPSK ,for which no oscillator matching or synchronized carrier phase reference

is need. Form figure (4.6) to figure (4.13) it is observed that after a threshold the differential

PSK signals showed better results than the PSK.

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Fig.4.6 comparison of BPSK with phase error and DPSK over AWGN channel

Fig.4.7 symbol error curves of BPSK with out phase error and DPSK over AWGN.

The figure (4.6) and figure (4.7) shows that the performance of BPSK over AWGN

channel is much better when there is no phase error is present (i.e. when 0 = ). But

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with the presence of phase error (i.e. when 20o

) the performance will be

decreased as the phase error is increased.

Fig.4.8 symbol error rate curves of BPSK with out phase error and DPSK over Rayleigh fading

Fig.4.9 comparison of BPSK with phase error and DPSK over Rayleigh fading channel.

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From figure (4.8) it is concluded that the performance of the BPSK is better than the DPSK

over Rayleigh Fading channel, but when we considering Phase jitter or SER due to phase

degradation (figure.4.9), after a certain point of High SNR (43dB) it shows worst error

performance than DPSK, and we get a clear cross over point after which BPSK gives

more SER for high SNR.

Fig.4.10 Bit error rate curves of the QPSK without phase error and with phase error

over AWGN channel.

This figure (4.10) shows that the Bit error rate of the QPSK is increased over the AWGN

channel when the phase error is considered, for the same value of the SNR.

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Fig.4.11 Bit error rate curves of the QPSK withour phase error and pi/4DQPSK over AWGN channel.

Fig.4.12 BER curves of the QPSK with phase error and pi/4DQPSK over AWGN channe.

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From figure (4.11) and figure (4.12) it can be said that the QPSK without phase error

is showed a better probability of error performance in case of the AWGN channel. In

case of the imperfect phase estimation QPSK performance is increasingly degreded

as, the value of phase is increased ( 25o

).

Fig.4.13 BER curves of the QPSK with phase error and pi/4DQPSK over Rayleigh channe

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Chapter 5

Conclusions and Future Work

5.1 CONCLUSION

The focus of this thesis is on the mathematical analysis of the M-ary modulation

schemes over AWGN channel and different Fading channel environment. More specifically,

it focuses on the performance analysis of the coherent phase modulation with imperfect phase

estimation.

This thesis begins with an elaborate discussion of the M-ary modulation schemes as, M-

ary Amplitude modulation, M-ary Phase shift keying, M-ary Frequency shift keying, M-ary

Quadrature Amplitude modulation, Differential M-ary phase shift keying with there

Geographic and Baseband representation. It mainly contains the analysis of the probability of

error of the M-ary signals over Additive White Gaussian Noise (AWGN) channel. From the

simulation curves and the mathematical analysis of the signals it is observed that as the

number of signals or number ofMincreases (M0, 1, 2,.,M) the error probability or more

clearly the probability of Symbol error rate is increased. It was also observed that the Binary

Frequency Shift Keying is giving the least probability of error over AWGN channel. All this

analysis is very much important for the basic analysis of any communication system, as the

digital M-ary modulation offers much more advantages than other modulation schemes.

In chapter 3, the analysis of Binary Phase Shift Keying (BPSK) and Binary Frequency

Shift Keying (BFSK) is done over Rayleigh, Rician and Nakagami-q (Hoyt) fading channels.

Nakagami-q (Hoyt) Fading is used to model fading environments more severe than Rayleigh

fading, and the signal error rate degradation least in case of Rician Fading channel. The

probability of error is increased in case of BFSK signal in Fading channel, and it showed

worst error performance than the BPSK signal, which is the reveres case if consider the

performance over the AWGN channel.

Finally chapter 4, mainly discussed about effect of imperfect estimation of the Phase at

the receiver in case of the Coherent Phase modulation signals. Here the probability of error

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rate is analyzed for the BPSK and QPSK signal with the imperfect phase estimation over

AWGN and Rayleigh Fading channels. As we go through the simulation curves it is observed

that as the phase error was increased the probability of the error rate is increased

simultaneously. Then BPSK and QPSK signals are compared with the Differential Phase

Shift Keying (DPSK) and DQPSK signals. The result showed that over AWGN and Rayleigh

Fading channel the probability of error of the BPSK and QPSK without any phase error is

better than the DPSK and QPSK. But when the analysis is done with the phase error

consideration for the BPSK and QPSK, they gave worse performance than the DPSK and

QPSK. Therefore it is concluded that the DPSK and DQPSK is used more effectively in

wireless communication systems when ever the other modulation schemes suff

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