Analysis of Switched Diversity Receivers

over κ-µ and η-µ Fading Channels

A thesis submitted in partial fulfilment of the requirements for the

award of the degree of

Master of Technology

Submitted by Supervisor

Kalpant Pathak Dr. Pravas Ranjan Sahu

Roll. No. 11410240

Depatrment of Electronics and Electrical Engineering

Indian Institute of Technology Guwahati

Guwahati-781039, Assam, India

CERTIFICATE

This is to certify that the work contained in this thesis entitled “Analysis of Switched

Diversity Receivers over κ-µ and η-µ Fading Channels” by Kalpant Pathak,

Roll no: 11410240, has been carried out at Department of Electronics & Electrical

Engineering, Indian Institute of Technology Guwahati under my supervision and that it

has not been submitted elsewhere for a degree.

August 13, 2014 (Pravas Ranjan Sahu)

Associate Professor

Department of EEE

i

Acknowledgement

I feel it as a great privilege in expressing my deepest and most sincere gratitude to

my supervisor, Dr. Pravas Ranjan Sahu for the valuable suggestions and guidance during

the course of the thesis, without which this work would not have seen the light of the day.

I would also like to thank the Head of the Department and the other faculty members

for their kind help in carrying out this work. I am very grateful to the non-teaching

staff of the department who has always been my side from the very beginning of my

work. Thanks go out to all my friends at the Image Processing Lab. They have always

been around to provide useful suggestions, companionship and created a peaceful research

environment. My special Thanks to Pravin Kumar, Sushant Kumbhar, Pawan Kumar

and Sandip Gupta for their timely help in all respects during my M.Tech course. Special

Thanks to my parents for their tremendous support and love all through.

ii

Abstract

In wireless communication, multipath fading results in severe degradation in perfor-

mance of the system. In fading, received signal level fluctuates rapidly degrading the bit

error rate (BER) performance.

Diversity combining is the key technology used in 3G/4G wireless communication

systems to mitigate fading. In recent years, multiple diversity combining techniques

have been developed and analyzed over different fading channels. These combining tech-

niques have different implementation complexities and performances depending upon the

channel conditions. Among all known diversity combining techniques, switched diversity

receivers have least implementation complexity as they don’t require channel estimation

at the receiver.

The intent of this thesis is to analyze switch and stay combining (SSC), switch and

examine combining (SEC) and switch and examine combining with post examining se-

lection (SECps) over κ-µ and η-µ fading channels using moment generating function

(MGF) approach. A novel mathematical expression for the MGF of the receiver output

signal-to-noise ratio is derived and BER for binary phase shift keying (BPSK) and binary

frequency shift keying (BFSK) modulation schemes are obtained using it. The evaluated

results are plotted for channel parameters of interest and effect of fading on the combiner

performance is studied.

iii

Contents

1 Introduction 1

1.1 Wireless Communication System . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Statistical Models for Fading Channels . . . . . . . . . . . . . . . 4

1.2 Effect of Fading on System Performance . . . . . . . . . . . . . . . . . . 5

1.3 Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Diversity Combining Techniques . . . . . . . . . . . . . . . . . . . 8

2 Literature Review and Motivation 12

2.1 Diversity Combining Schemes . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Selection Combining . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Switch and Stay Combining . . . . . . . . . . . . . . . . . . . . . 13

2.1.3 Switch and Examine Combining . . . . . . . . . . . . . . . . . . . 14

2.2 κ-µ and η-µ Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 BER Analysis 18

3.1 The κ-µ Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 The η-µ Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 The η-µ Distribution: Format 1 . . . . . . . . . . . . . . . . . . . 19

3.2.2 The η-µ Distribution: Format 2 . . . . . . . . . . . . . . . . . . . 19

3.3 Probability of Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1 Moment Generating Function of Output SNR . . . . . . . . . . . 20

3.3.2 Probability of Bit Error Analysis . . . . . . . . . . . . . . . . . . 24

4 Numerical and Simulation Results 26

iv

5 Conclusion and Future Work 38

Appendices 39

A Solution of I(s) in Equations 3.6, 3.9 and 3.12 39

B Solution of I1(s) in Equation 3.12 40

v

List of Figures

1.1 A typical urban/suburban mobile radio environment . . . . . . . . . . . . 2

1.2 Comparison between Rayleigh faded and AWGN Channel . . . . . . . . . 6

1.3 Block diagram of Space diversity combiner . . . . . . . . . . . . . . . . . 7

1.4 Block diagram of Maximal Ratio Combiner . . . . . . . . . . . . . . . . . 8

1.5 Block diagram of Equal Gain Combiner . . . . . . . . . . . . . . . . . . . 9

1.6 Block diagram of Selection Combiner . . . . . . . . . . . . . . . . . . . . 10

1.7 Block diagram of Dual branch Switch and Stay Combiner . . . . . . . . . 10

4.1 BER performance of SSC receiver over κ-µ fading channel for BPSK and

BFSK modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 BER performance of SSC receiver over η-µ fading channel for BPSK and

BFSK modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 BER performance of SEC receiver over κ-µ fading channel for L=3 for

BPSK and BFSK modulations . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 BER performance of SEC receiver over η-µ fading channel for L=3 for

BPSK and BFSK modulations . . . . . . . . . . . . . . . . . . . . . . . . 30

4.5 BER performance of SEC receiver over κ-µ fading channel for L=5 for

BPSK and BFSK modulations . . . . . . . . . . . . . . . . . . . . . . . . 31

4.6 BER performance of SEC receiver over η-µ fading channel for L=5 for

BPSK and BFSK modulations . . . . . . . . . . . . . . . . . . . . . . . . 32

4.7 Comparison of BER performance over Rayleigh fading by varying γT . . 33

4.8 BER performance of SECps receiver over κ-µ fading channel for L=2 for

BPSK and BFSK modulations . . . . . . . . . . . . . . . . . . . . . . . . 35

4.9 BER performance of SECps receiver over η-µ fading channel for L=2 for

BPSK and BFSK modulations . . . . . . . . . . . . . . . . . . . . . . . . 36

vi

4.10 Comparison of BER performance of SC, SEC|L=2/SSC and SECps com-

bining schemes over Rayleigh fading . . . . . . . . . . . . . . . . . . . . . 37

vii

Chapter 1

Introduction

In past few years, wireless communication has played an important role in information

technology as information can be transmitted without the need of dedicated link between

transmitter and receiver unlike wired communication, where a dedicated link/channel

exist between transmitter and receiver. Compared to wired communication systems,

wireless systems introduce a very interesting feature ‘mobility’.

In any kind of communication, wired or wireless, there are some parameters like

bandwidth, transmitted power, data rate etc. which decide the reliability of a system.

The one which optimizes all of them is said to be a perfect system. In recent years, lots

of research has been done on both kinds of communication so that a reliable system can

be designed with high bandwidth, low transmitted power, high data rates and low bit or

symbol error probability.

1.1 Wireless Communication System

In a wireless communication system, data is transmitted in the form of electromagnetic

waves using antennas. When signal propagates through the wireless media, phenomena

such as reflection and scattering through buildings, trees etc. and refraction through the

edges causes the signal to follow multiple paths having different path loss factors and

different delays. Thus, at the receiver, the received signal consist of multiple copies of

same information bearing signal having different amplitudes and different phases arising

due to different path lengths.

Figure 1.1 shows a typical urban/suburban mobile radio environment. In the figure,

1

NLOS

LOS

NLOS

BS

MS

Figure 1.1: A typical urban/suburban mobile radio environment

the direct path between the transmitter and the receiver is called line-of-sight (LOS) path,

whereas, the path corresponding to reflected signal is called non line-of-sight (NLOS)

path.

These multipaths have different phases corresponding to different path-delays, so that

they interfere at the receiver either constructively or destructively resulting in variation in

signal-to-noise ratio. In addition, mobility introduces time variation in channel response,

i.e. if a very short pulse is transmitted, the received signal appears as a train of pulses

due to presence of multipaths. Secondly, as a result of time varying response, if same

procedure is followed multiple times, a change is observed in the received pulse train

over time, which will include changes in the sizes of individual pulses, changes in relative

delays among the pulses and often, changes in the number of pulses observed.

Hence, the equivalent low-pass time varying impulse response of the channel can be

modeled as [1]:

c(τ ; t) =∑i

αi(t)e−j2πfcτi(t)δ[t− τi(t)] (1.1)

where, αi(t) and τi(t) are time varying attenuation factor and path delay for ith path

respectively.

For a transmitted signal sl(t) = 1 the received signal for the case of discrete multipath

is given by [1]:

rl(t) =∑i

αi(t)e−jθi(t)

where, θi(t) = 2πfcτi(t).

The αi(t) and τi(t) associated with different signals vary at different rates and in

2

random manner. So, received signal rl(t) can be modeled as a random process. For

large number of paths, central limit theorem can be applied and rl(t) can be modeled

as complex-valued Gaussian random process i.e. c(τ ; t) is also a complex-valued random

process in t variable [1].

1.1.1 Fading

Due to time varying nature of phases and amplitudes of the received signals, the

multipaths add constructively or destructively at the receiver resulting in fluctuation in

the received signal. This amplitude variations in the received signal is termed as fading.

There are some parameters which define the fading characteristics of the channel.

These parameters are:

Coherence Time

Coherence time, Tc of the channel measures the period of time over which two samples

of channel response taken at same frequency but at different time instants are correlated.

The coherence time is also related to Doppler spread, fd by [1]:

Tc ≃1

fd

Coherence Bandwidth

Coherence bandwidth, fc of the channel measures the frequency range over which two

samples of channel response taken at same time instants but at different frequencies are

correlated. The coherence bandwidth is related to maximum delay spread, τmax by [1]:

fc ≃1

τmax

Depending upon the parameters described above, fading can be classified into four

groups:

1.1.1.1 Slow and Fast Fading

In slow fading, the symbol time duration Ts is smaller than the channel’s coherence

time, so that multiple symbols undergo same fading. Whereas, in fast fading, the symbol

time duration Ts is larger than the channel’s coherence time, hence, fading decorrelates

from symbol to symbol [2].

3

1.1.1.2 Frequency-Flat and Frequency-Selective Fading

Fading is said to be frequency-flat, if signal bandwidth is less than channel’s coher-

ence bandwidth fc, i.e. all spectral components of the signal undergo same fading. In

frequency-selective fading, signal bandwidth is greater than channel’s coherence band-

width, so different spectral components of the signal are affected by different fading [2].

Frequency selectivity results in inter-symbol interference (ISI).

1.1.2 Statistical Models for Fading Channels

As wireless channel impulse response is modeled as a random process, multiple dis-

tributions have been developed to model the fading channel.

For large number of scatterers, central limit theorem can be applied and channel re-

sponse c(τ ; t) can be modeled as a complex-valued Gaussian random process. When the

impulse response c(τ ; t) is modeled as zero-mean complex random process, the envelope

|c(τ ; t)| is Rayleigh distributed and phase ∠c(τ ; t) is uniformly distributed. If there are

fixed scatterers in addition to randomly moving scatterers, c(τ ; t) can be modeled as

non-zero mean complex Gaussian process so that envelope |c(τ ; t)| is Rice distributed [1].

Different fading models are described below.

1.1.2.1 Rayleigh Model

Rayleigh model is used when there is no LOS component present in the received

signal. The probability density function (PDF) of the envelope α ≥ 0 of channel impulse

response is given as [2]:

pα(α) =2α

Ωe−

α2

Ω , α ≥ 0

where, Ω = E[α2] is the mean-square value of α.

1.1.2.2 Nakagami-n (Rice) Model

Nakagami-n distribution is also called Rice distribution. This is suitable for the

propagation paths having one strong LOS path and many weaker NLOS paths. The pdf

is given as [2]:

pα(α) =2(1 + n2)e−n

2α

Ωe−

(1+n2)α2

Ω I0

(2nα

√1 + n2

Ω

), α ≥ 0

4

where, 0 ≤ n ≤ ∞ is the Nakagami-n fading parameter and is related to Ricean factor

‘K’ by K=n2. I0(·) is bessel’s function of first kind and order zero [3].

Nakagami-n distribution includes the Rayleigh distribution as a special case (n=0)

and spans from Rayleigh fading (n=0) to no-fading (n=∞).

1.1.2.3 Nakagami-q Model

The PDF of Nakagami-q distributed RV is given as [2]:

pα(α) =(1 + q2)α

qΩe− (1+q2)2α2

4q2Ω I0

((1− q4)α2

4q2Ω

), α ≥ 0

where, 0 ≤ q ≤ 1 is Nakagami-q fading parameter. This fading distribution includes

one-sided Gaussian (q=0) and Rayleigh (q=1) as special cases.

1.1.2.4 Nakagami-m Model

The PDF of Nakagami-m distributed random variable (RV) is given as [2]:

pα(α) =2mmα2m−1

ΩmΓ(m)e−

mα2

Ω , α ≥ 0

where, Γ(·) is gamma function [3] and 1/2 ≤ m ≤ ∞ is Nakagami-m fading parameter.

This fading model includes one-sided Gaussian (m=1/2), Rayleigh (m=1) fading and

no-fading (m=∞) as special cases. For m < 1, it closely approximates Hoyt distribution,

and this mapping is given by [2]:

m =(1 + q2)2

2(1 + 2q4), m ≤ 1

Similarly, for m > 1, it closely approximates the Nakagami-n (Rice) distribution and this

mapping is one-to-one and given by [2]:

m =(1 + n2)2

1 + 2n2, n ≥ 0

1.2 Effect of Fading on System Performance

Fading, in wireless communication systems, results in poor BER performance as com-

pared to non-fading environment in presence of additive white Gaussian noise (AWGN).

This performance degradation arises due to amplitude variation of the received signal

because of fading. Figure 1.2 shows the BER curves for Rayleigh faded wireless channel

5

0 5 10 15 20 2510

−4

10−3

10−2

10−1

Input average SNR (γ) in dB

Pro

babi

lity

of e

rror

, Pe

Performance of BPSK over fading and non−fading environments

Rayleigh FadingAWGN Channel

Figure 1.2: Comparison between Rayleigh faded and AWGN Channel

and non-fading AWGN channel over BPSK modulation scheme. From the figure it can

be observed that, to have an BER of 10−3, SNR needed for non-fading channel is ≃ 7dB,

where as for Rayleigh faded channel, it is ≃ 23dB. In such case, the transmitted power

for Rayleigh faded channel will be ≈ 40 times that of AWGN channel.

To mitigate the effects of fading on the system and to improve performance of the

system, Diversity techniques are used at the receiver end.

1.3 Diversity

In a typical mobile radio environment, it is observed that if two or more radio channels

are sufficiently separated in frequency, time or space, then the fading in various channels

is more or less independent. Diversity, is the key technology that uses this independence

of channels to mitigate the effect of fading [4]. The main object of diversity techniques is

to use several replicas of transmitted signal to improve system performance. There are

various ways to obtain independently faded channels:

6

Receiving Antenna

1

2

N

Combiner

(MRC/EGC

/SC/SSC)

Output

Figure 1.3: Block diagram of Space diversity combiner

• Frequency Diversity: The message is simultaneously transmitted over different

carrier frequencies such that the separation between frequencies should be larger

than channel’s coherence bandwidth.

• Time Diversity: Same message is transmitted over different time slots such that

the time separation between adjacent transmissions should be greater than channel’s

coherence time.

• Space Diversity: Desired message is transmitted/received through multiple trans-

mit/receive antennas separated by at least half of the carrier wavelength. In the

case of shadowing, the separation should be at least 10 carrier wavelength.

• Angle Diversity: The message is received simultaneously using multiple antennas

pointing in different directions.

The multiple replicas of the transmitted signal obtained using any of the method

mentioned above, is then combined efficiently to improve the SNR at the detector input.

Figure 1.3 shows a space diversity technique with a combiner, which combines the multiple

signals received on different antennas. There are different types of diversity combining

techniques, which are discussed in next section:

7

SumDetector

1

2

N

Adaptive Control

1G

2G

NG

1g

2g

Ng

Figure 1.4: Block diagram of Maximal Ratio Combiner

1.3.1 Diversity Combining Techniques

There are different types of diversity combining techniques used in practice [2], which

are as follows:

1.3.1.1 Maximal Ratio Combining

In maximal ratio combining technique, the received multiple faded copies of the trans-

mitted signal are co-phased. The co-phased signal copies are weighted individually in

proportion to their strength to maximize SNR at the output of the combiner. Assuming

the received signal SNR at the input of the combiner is γi, i = 1, 2, ..N, the output SNR

can be shown to be [5]:

γMRC =N∑i=1

γi

The MRC operation requires estimation of phase and amplitude of each received input

branch signal. Hence, the complexity of implementation is high.

1.3.1.2 Equal Gain Combining

Different weights for each branch may not be convenient as it may increase the com-

plexity of the receiver as in the case of MRC. So it is convenient to set all the gains to

unity, while cophasing all signals before combining [2]. This technique of combining is

called Equal Gain Combining.

8

Co-Phased

And

Sum

Detector

1

2

N

Adaptive Control

1g

2g

Ng

Figure 1.5: Block diagram of Equal Gain Combiner

For EGC, the output SNR is given as [2]:

γEGC =

(∑Ni=0 α

2i

)Es∑N

i=0 PNi

where, αi is the fading amplitude for ith copy of the transmitted signal.

1.3.1.3 Selection Combining

In selection combining (SC), the system chooses the received signal having maximum

SNR out of all copies of signals received. In this scheme the output SNR can be given as

[2]:

γSC = maxγ1, γ2, ..., γN

1.3.1.4 Switch and Stay Combining

The switch and stay combining (SSC) technique discussed here is presented in [2]

and also shown in Figure 1.7. In this system, there are only two copies of fading signals

are used. The combiner has only two antennas to receive fading signals. The received

signal is fed as shown in Figure 1.7. In this scheme the received SNR γ1 at antenna L1

is compared with a predefined threshold γT . Switching occurs to the input branch L2

if γ1 < γT . And it again switches to first branch if γ1 > γT . It may happen that after

switching the input SNR γ2 at L2 is less than γT or even less than γ1, in such case the

switch will still be connected to L2 until the SNR of first branch becomes greater than

γT . Switching from branch L2 to branch L1 is done in similar manner.

9

Select

Maximum

of N

branches

Detector

1

2

N

1g

2g

Ng

Figure 1.6: Block diagram of Selection Combiner

Receiver Estimator

Comparator

Switch

Logic

Control

Present

Threshold

Data

1L 2

L1g

2g

Figure 1.7: Block diagram of Dual branch Switch and Stay Combiner

The SSC output SNR, γSSC can be given as:

γSSC =

γ1, if γ1 ≥ γT

γ2, otherwise

1.3.1.5 Switch and Examine Combining

Unlike SSC combining scheme, switch and examine combining (SEC) adds the ben-

efit of having multiple branches at the receiver, especially when they are independent

and identically distributed (i.i.d.) or equicorrelated and identically distributed. In SSC

scheme, receiver switches between the best two paths, adding a path does not improve

the performance unless the added path is better than at least one of the best two ones.

10

In SEC combining scheme, the receiver starts examining from the first path. If first

path is acceptable, it continues to receiver from it, else, it switches and examines the

next available path. This process continues until an acceptable path is found or all paths

have been examined. In the latter case, the receiver stays on the last examined path [6]

or selects the best path for reception [7].

11

Chapter 2

Literature Review and Motivation

A number of works have been reported on the analysis of switched diversity schemes

over different fading channel models. Selection combining, switch-and-stay combining

and multibranch switch-and-examine combining schemes have been extensively studied

over different fading channels. Also, different switching algorithms have been suggested

to improve the system performance. Different research works regarding various switched

combining schemes are listed in following section:

2.1 Diversity Combining Schemes

2.1.1 Selection Combining

• D. G. Brennan in [4] has analyzed the SC, MRC and EGC combining schemes.

In this paper, some departures from ideal conditions, such as non-Rayleigh fading

and partially coherent signals are also considered. Also, merits of predetection

and postdetection combining is discussed. These schemes have been analyzed over

Rayleigh fading channel and performance comparison has been presented.

• M. A. Blanco in [8], has studied the performance of non coherent binary frequency

shift keying (NC-BFSK) with dual selection combiner over independent and iden-

tically distributed Nakagami-m fading channels.

• T. Eng et. al. in [9], has compared different diversity combining techniques over

Rayleigh fading channels. In this paper, for coherent reception, the authors have

compared performance of MRC, SC and a generalization of SC, where two/three

12

signals with largest amplitudes are coherently combined and these techniques are

denoted by second/third order SC (SC2 and SC3). Also, same techniques are inves-

tigated for non coherent reception, with EGC and non coherent SC2 and SC3. It

has been shown that SC2 and SC3 techniques have better performance than conven-

tional SC scheme and under certain conditions they may approach the performance

of EGC or MRC. Also, for non coherent reception it is shown that SC2 and SC3

perform better than EGC for higher BER values.

• In year 1997, O. C. Ugweje and V. A. Aalo in [10], have analyzed the selection com-

bining scheme over correlated Nakagami fading channels and the effect of branch

correlation and fading parameter on the performance of the system has been stud-

ied. Also, the joint distribution of the combiner output is obtained and analysis

has been done for both coherent and non coherent demodulation schemes. Differ-

ent graphs have also been shown to understand the effect of fading parameter and

branch correlation. In the paper, it has been concluded that for dual selection di-

versity system, branch correlation does not influence the BER significantly whereas,

as Nakagami fading parameter m is increased, the BER performance improves sig-

nificantly at higher SNRs.

• In [11], M. K. Simon and M. S. Alouini have analyzed dual selection combining over

correlated Rayleigh and Nakagami-m fading channels. Also, they have presented

the expressions for the outage probability and average error probability performance

of dual SC receiver over correlated Rayleigh fading in closed form, in particular for

binary differential phase shift keying (BDPSK) and the results are also extended

for Nakagami-m fading.

2.1.2 Switch and Stay Combining

Switch and stay combining (SSC) has been a popular combining scheme as it is the

least complex combining scheme among all combining schemes. A number of works have

been reported on the analysis of SSC scheme. Some of them are listed below:

• In [12], A. A. Abu-Dayya and N. C. Beaulieu have studied the SSC scheme over

independent and correlated Ricean fading channels using a discrete time model.

Also, in the paper, BER for non coherent binary frequency shift keying is derived.

13

In addition, an optimum switching threshold, which gives minimum BER, has also

been obtained for independent fading case. The obtained performance is then com-

pared to conventional selection combining scheme and effect of fading severity on

BER and optimum threshold is investigated. For independent channels, it has been

shown that for given Rice parameter K, optimum threshold increases with the SNR

and for fixed SNR, optimum threshold increases with K.

• In [13], A. A. Abu-Dayya and N. C. Beaulieu have analyzed the performance of SSC

scheme over slow and flat Nakagami-m fading channels. In addition, expression

for optimum threshold, which gives minimum BER, is obtained. Also, the effect

of fading parameter and branch correlation on BER performance have also been

studied.

• In [14], Y. C. Ko et. al. has analyzed dual SSC scheme over Rayleigh, Nakagami-m

and Nakagami-n fading channels using moment generating function (MGF) ap-

proach. In this paper, effect of unequal average SNR, branch correlation and im-

perfect channel estimation has also been studied. Also, the authors have derived

the closed form expression for MGF of SSC output SNR, γSSC considering all three

fading channels and using these MGF expressions, closed form expressions of prob-

ability of error has been obtained.

2.1.3 Switch and Examine Combining

Unlike SSC, SEC adds the benefit of having multiple branches at the receiver and

hence, performs better than SSC. Some important works reported on the performance

analysis of conventional SEC, modified SEC and generalized SEC are listed below:

• In [6], H. Yang and M. S. Alouini have analyzed both SSC and SEC diversity com-

bining schemes over Rayleigh fading channels. This work presents generic formulas

for cumulative distribution function (CDF), probability density function (PDF)

and moment generating function (MGF) of combiner output. Using these formulas,

closed-form expressions for outage probability and average bit error rate have been

obtained and a comparison between the two schemes has been shown. It is observed

that, for increasing number of branches, SSC does not improve the performance as

14

long as the branches are identically distributed, whereas, SEC improves the BER

with the increase in number of branches.

• In [15], G. C. Alexandropoulos et. al. has studied the performance of SEC diver-

sity scheme over arbitrarily correlated and not necessarily identically distributed

Nakagami-m fading channels. The authors have obtained analytical expressions

for distribution of SEC output SNR for constant correlation model. For integer

and half integer values of m, under the assumption of most general case of cor-

relation, analytical expressions for the distribution of output SNR is derived for

L ≤ 3. Whereas, for L > 3 analytical approximations are presented. Also, with the

help of graphs, it is shown that with increasing branch correlation, the performance

degrades.

• In [16], A. M. Magableh and M. M. Matalgah have analyzed the BER performance

and ergodic capacity of multibranch SEC combining over Weibull fading channels.

The authors used the expressions of the PDF of combiner output SNR to evaluate

the BER performance and ergodic capacity. The Weibull distribution considered in

this paper is a generalized fading distribution, which can be used to model mobile

radio environment operating in 800/900 MHz frequency range [2]. This fading

distribution includes Rayleigh and exponential distribution as special cases.

• In 2006, in [7], H. C. Yang and M. S. Alouini has presented a new kind of switch-

and-examine combining known as SEC with post examining selection (SECps) and

analyzed its performance over i.i.d. Rayleigh fading channels. In this technique,

when no diversity path is acceptable, the receiver selects the best diversity branch

for reception instead of choosing it randomly as in the case of conventional SEC.

Using such combining schemes, author have shown that the performance can be

improved. Authors have also shown that, using appropriate switching threshold

SECps can have same error performance as that of SC with much less path estima-

tions on average.

In past few years, some more generalized fading distributions such as κ-µ and η-µ distri-

butions have also been developed and performance of wireless communication systems is

also analyzed over these channels. In the following section, some of the research works

done on such fading channels are presented:

15

2.2 κ-µ and η-µ Fading Channels

The diversity combining schemes have been extensively studied over some general

fading distributions such as Rayleigh, Ricean, Nakagami-m, and Weibull etc. In 2007, in

[17], M. D. Yacoub has suggested κ-µ and η-µ fading channels to model the mobile radio

environment more accurately. These fading distributions are used to model nonhomoge-

neous fading environment. The research works done on κ-µ and η-µ fading channels are

listed below:

• In [17], M. D. Yacoub has presented the κ-µ and η-µ fading channels. These fading

models considers a signal composed of clusters of multipath waves. Within any one

cluster, the phases of scattered waves are random and have similar delay times with

delay-time spreads of different clusters being relatively large.

In the case of κ-µ fading model, all scattered components within each cluster have

identical powers but within each cluster, a dominant component is found. Whereas,

the η-µ fading distribution has two formats. In format-1, it is assumed that, within

each cluster the in-phase and quadrature components of scattered waves are inde-

pendent and have different powers, while in format-2, the in-phase and quadrature

components of scattered waves have identical powers but they are correlated with

each other. In the paper, it is shown that these fading models also accommodate

Hoyt, Nakagami-m, Rayleigh and Ricean distribution as their special cases.

In the paper, the author has presented the derivations of PDF and CDF for both

κ-µ and η-µ distributions. Also, author stated the usability of these fading mod-

els. The κ-µ distribution is suited for line-of-sight propagation model and includes

Rayleigh, Ricean and Nakagami-m as special cases, whereas the η-µ distribution

is best suited for non line-of-sight propagation model and includes Rayleigh, Hoyt

and Nakagami-q as special cases. Further, the author has also presented various

estimators for fading parameter estimation.

The results presented in the paper are also field verified and it is observed that in

a wide sense, Nakagami-m can be thought of as a mean distribution of κ-µ and η-µ

distributions.

• In [18], N. Y. Ermolova has derived the closed form expressions for MGF of η-

µ and κ-µ fading distributions. Also, using these expressions, BER is evaluated

16

numerically for BPSK modulation scheme.

• In [19], N. Y. Ermolova has used the derived MGF expressions in [18] to evaluate

certain integrals useful in obtaining the BER expressions for BPSK and rectangular

quadrature amplitude modulation (QAM) signalling schemes. Also, the effect of

fading parameters on the system is studied.

• In [20], authors have analyzed the L-branch SC receiver over κ-µ and η-µ fading

channels. Authors have analytically obtained the moments of SC receiver output

SNR and bit error rate for binary, coherent and non-coherent modulation schemes.

Also, the numerical results are verified by comparing them with simulation results.

• In [21], authors have studied the performance of L-branch MRC receiver over κ-µ

and η-µ fading channels. Authors have derived a highly accurate approximation

to the average symbol error rate (ASER) expression for QAM modulation scheme.

Also, the results are verified using computer simulations.

2.3 Motivation

Switched diversity combining schemes such as SSC, SEC and SECps are less complex

diversity combining schemes as they don’t require channel estimation at the receiver and

also, they reduce the switching rate required among the available diversity branches.

Though these diversity combining schemes have already been examined over different

fading channels including Rayleigh, Ricean, Nakagami [2], Weibull [16] etc., analysis over

nonhomogeneous fading distributions such as κ-µ and η-µ is not available in literature.

In this thesis work, we have analyzed SSC, SEC and SECps diversity combining

schemes over κ-µ and η-µ fading channels for BPSK and coherent BFSK modulation

schemes and studied the effect of channel parameters on bit error rate performance of the

system.

17

Chapter 3

BER Analysis

For bit error rate analysis of SSC, SEC and SECps combining schemes, the channel is

assumed to be slow and flat fading with κ-µ or η-µ distribution. For transmitted signal

s(t) with symbol energy Es, the complex low pass equivalent of the received signal at ith

(i=1,2...L) path over the symbol duration Ts second can be given as ri(t) = αiejϕis(t) +

ni(t), where ni(t) is the complex additive white Gaussian noise (AWGN) having two

sided power spectral density N0/2, random variable (RV) ϕi is instantaneous phase and

RV αi is fading envelope which is either κ-µ or η-µ distributed. The PDFs of κ-µ and

η-µ distributed RVs are given in following sections.

3.1 The κ-µ Distribution

The probability density function (PDF) of κ-µ distributed RV is given as [17]

pακ−µ(αi) =2µαµi

κµ−12 eµκ

(1 + κ

Ωi

)µ+12

e−µ(1+κ)

Ωiα2i Iµ−1

2µ

√κ(1 + κ)

Ωi

αi

, (3.1)

where Ωi = E[α2i ], E[·] is the expectation operator, κ > 0 and µ > 0 are the parameters

of the distribution and Iv(·) is the modified Bessel function of the first kind and vth

order [3]. The parameter κ is the ratio of the power due to dominant components to the

total power due to scattered components and µ is the number of multipath clusters. This

distribution includes Rice (µ=1 and κ=K), Nakagami-m (κ→0 and µ=m), Rayleigh (µ=1

and κ→0) and one sided Gaussian distribution (µ=0.5 and κ→0) fading distributions as

special cases. This distribution is better suited for line-of-sight propagation.

18

3.2 The η-µ Distribution

The probability density function (PDF) of η-µ distributed RV is given as [17]

pαη−µ(αi) =4√πµµ+

12hµα2µ

i

Γ(µ)Hµ− 12Ω

µ+ 12

i

e− 2µh

Ωiα2i Iµ− 1

2

(2µH

Ωi

α2i

), (3.2)

where Γ(·) is the gamma function, and h and H are functions of the parameter η defined

for two formats in next subsections. µ denotes the number of multipath clusters. This

fading distribution includes Hoyt (η=q2,µ=0.5), Nakagami-m (η=1,µ=m/2), Rayleigh

and one sided Gaussian distribution as special cases.

3.2.1 The η-µ Distribution: Format 1

In this format 0 < η < ∞ is the power ratio of the in-phase and quadrature compo-

nents of the scattered-waves of each multipath. In this case, h = 2+η−1+η4

and H = η−1−η4

.

Within 0 < η ≤ 1, we haveH ≥ 0, on the other hand, within 0 < η−1 ≤ 1, we haveH ≤ 0.

Because Iv(−z) = (−1)vIv(z), the distribution is symmetrical around η=1. Therefore, as

far as the envelope(or power) distribution is concerned, it is sufficient to consider η only

within one of these ranges. In Format 1, H/h = (1− η)/(1 + η) [17].

3.2.2 The η-µ Distribution: Format 2

In this format −1 < η < 1 is the correlation coefficient between the in-phase and

quadrature components of the scattered waves of each cluster. In this case, h = 11−η2 and

H = η1−η2 . Within 0 ≤ η < 1, we have H ≥ 0, on the other hand, within −1 < η ≤ 0, we

have H ≤ 0. Because Iv(−z) = (−1)vIv(z), the distribution is symmetrical around η=0.

Therefore, as far as the envelope(or power) distribution is concerned, it is sufficient to

consider η only within one of these ranges. In Format 2, H/h = η [17].

3.3 Probability of Error Analysis

The probability of error for a digital communication system in a fading channel with

AWGN can be given as [2]

Pe =1

π

π/2∫0

Mγ

(ψ

sin2 θ

)dθ, (3.3)

19

where Mγ(·) is the MGF of the receiver output SNR, ψ is the modulation parameter i.e.,

ψ=1 for BPSK, ψ=0.5 for BFSK, and ψ=0.715 for BFSK with minimum correlation [22].

Thus, for the BER analysis of the SSC, SEC and SECps receivers under consideration in

κ-µ / η-µ fading channels applying Equation 3.3, we need to obtain the MGF of γSSC ,

γSEC and γSECps i.e. MγSSC(s), MγSEC

(s) and MγSECps(s). Expressions for the MGFs are

derived in the following subsection.

3.3.1 Moment Generating Function of Output SNR

The MGF of an RV x is defined as

M(s) = E[e−sx] =

∫ ∞

−∞e−sxp(x)dx

where, p(x) is the PDF of RV x. The MGF of the output SNR of SSC, SEC and SECps

combining schemes is derived in following subsections.

3.3.1.1 Switch and Stay Combining

If γSSC denotes the SNR per symbol of the combiner output and γT denotes the

predetermined switching threshold, to derive the MGF of SSC output SNR, we first

derive the CDF of the output SNR, PγSSC(γ) in terms of CDF of individual branch SNR,

Pγ(γ) as [2]

PγSSC(γ) =

Pγ(γT )Pγ(γ), γ < γT

Pγ(γ)− Pγ(γT ) + Pγ(γ)Pγ(γT ), γ ≥ γT(3.4)

Differentiating PγSSC(γ) with respect to γ, we get the PDF of SSC output SNR in terms

of CDF Pγ(γ) and the PDF pγ(γ) of the individual branch SNR as [2]

pγSSC(γ) =

Pγ(γT )pγ(γ), γ < γT

(1 + Pγ(γT ))pγ(γ), γ ≥ γT(3.5)

Using the above PDF, an expression for the MGF of γSSC in a fading channel can be

obtained from the formula [2]

MγSSC(s) = Pγ(γT )Mγ(s) +

∞∫γT

pγ(γ)e−sγdγ

︸ ︷︷ ︸I(s)

, (3.6)

20

where Pγ(γT ) is the cumulative distribution function (CDF) of κ-µ / η-µ fading distribu-

tion and pγ(γ) is the probability density function (PDF) of the SNR γ = Es

N0α2.

3.3.1.2 Switch and Examine Combining

The CDF of the output SNR of SEC receiver for L i.i.d branches, is given as [2]

PγSEC(γ) =

[Pγ(γT )]L−1Pγ(γ), γ < γT∑L−1

j=0 [Pγ(γT )]j[Pγ(γ)− Pγ(γt)] + [Pγ(γT )]

L, γ ≥ γT(3.7)

Differentiating PγSEC(γ) with respect to γ, we get the PDF of SEC output SNR in

terms of CDF Pγ(γ) and the PDF pγ(γ) of the individual branch SNR as [2]

pγSEC(γ) =

[Pγ(γT )]L−1pγ(γ), γ < γT∑L−1

j=0 [Pγ(γT )]jpγ(γ), γ ≥ γT

(3.8)

using the above PDF, an expression for the MGF of γSEC for L-branch SEC in a fading

channel can be obtained from the formula [2]

MγSEC(s) = [Pγ(γT )]

L−1Mγ(s) +L−2∑j=0

[Pγ(γT )]j

∫ ∞

γT

pγ(γ)e−sγdγ︸ ︷︷ ︸

I(s)

, (3.9)

where L is the number of diversity branches, Pγ(γT ) is the cumulative distribution func-

tion (CDF) of κ-µ / η-µ fading distribution and pγ(γ) is the probability density function

(PDF) of the SNR γ = Es

N0α2.

3.3.1.3 Switch and Examine Combining with post-selection

The CDF of the output SNR of L-branch SECps receiver in terms of PDF of individual

branch SNR is given as [2]

PγSECps(γ) =

[Pγ(γ)]L, γ < γT

1−∑L−1

j=0 [Pγ(γT )]j[1− Pγ(γ)], γ ≥ γT

(3.10)

Differentiating Equation 3.10 with respect to γ, we can obtain the PDF of SECps output

SNR in terms of CDF Pγ(γ) and the PDF pγ(γ) of the individual branch SNR as [2]

21

pγSECps(γ) =

L[Pγ(γ)]L−1pγ(γ), γ < γT∑L−1

j=0 [Pγ(γT )]jpγ(γ), γ ≥ γT

(3.11)

Using the above PDF, an expression for the MGF of γSECps for L-branch SECps in a

fading channel can be obtained

MγSECps(s) = L

∫ γT

0

[Pγ(γ)]L−1pγ(γ)e

−sγdγ︸ ︷︷ ︸I1(s)

+L−1∑j=0

[Pγ(γT )]j

∫ ∞

γT

pγ(γ)e−sγdγ︸ ︷︷ ︸

I(s)

, (3.12)

where L is the number of diversity branches, Pγ(γ) is the cumulative distribution func-

tion (CDF) of κ-µ / η-µ fading distribution and pγ(γ) is the probability density function

(PDF) of the SNR γ = Es

N0α2.

In order to findout the MGF of ouput SNR for SSC, SEC and SECps diversity com-

bining we need to find the Pγ(γT ), Mγ(s) and I(s) in Equations 3.6 and 3.9. These

quantities are derived below:

For κ-µ and η-µ fading scenarios an expression for pγ(γ) is given as [18]

pγκ−µ(γ) =µ

eµκ

(1 + κ

γ

)µ+12 (γ

κ

)µ−12e−

µ(1+κ)γ

γIµ−1

(2µ

√κ(1 + κ)γ

γ

)(3.13)

pγη−µ(γ) =2√πhµ

Γ(µ)

(µ

γ

)µ+ 12 ( γ

H

)µ− 12e−

2µhγγ Iµ− 1

2

(2µHγ

γ

)(3.14)

Further, an expression for Pγ(γT ) (CDF) of κ-µ distribution is given as [17]

Pγκ−µ(γT ) = 1−Qµ

(√2κµ,

√2(1 + κ)µγT

γ

), (3.15)

where Qm(α, β) is the Marcum-Q function defined as [23]

Qm(α, β) =1

αm−1

∞∫β

xme−x2+α2

2 Im−1(αx)dx.

and Pγ(γT ) of η-µ distribution is given as [17]

Pγη−µ(γT ) = 1− Yµ

(H

h,

√2hµγTγ

), (3.16)

where Yv(a, b) function is defined as [17]

Yv(a, b) =2

32−v√π(1− a2)v

av−12Γ(v)

∫ ∞

b

x2ve−x2

Iv− 12(ax2)dx.

22

with −1 < a < 1 and b ≥ 0.

The MGF of individual branch SNR, Mγ(s) for κ-µ and η-µ distribution is given as

[18]

Mγκ−µ(s) =

(µ(1 + κ)

µ(1 + κ) + sγ

)µexp

(µ2κ(1 + κ)

µ(1 + κ) + sγ− µκ

)(3.17)

Mγη−µ(s) =

(4µ2h

(2(h−H)µ+ sγ)(2(h+H)µ+ sγ)

)µ(3.18)

To obtain an expression for MγSSC(s) in Equation 3.6, MγSEC

(s) in Equation 3.9 and

MγSECps(s) in Equation 3.12 we need to solve the second term integral I(s). A solution

for I(s) has been obtained in Appendix A, which is given as below.

Iκ−µ(s) =∞∑n=0

µ2n+µκn(1 + κ)n+µ

n!Γ(n+ µ)eκµ[µ(1 + κ) + sγ]n+µΓ

(n+ µ,

(µ(1 + κ)

γ+ s

)γT

), (3.19)

Iη−µ(s) =2√πhµ

Γ(µ)

∞∑n=0

µ2n+2µH2n

n!Γ(n+ µ+ 0.5)(γs+ 2µh)2n+2µΓ

(2n+ 2µ,

(2µh

γ+ s

)γT

),

(3.20)

where Γ(·, ·) is the upper incomplete Gamma function [24].

Thus, putting Equations 3.15, 3.17 and 3.19 in Equations 3.6 and 3.9, expressions for

MγSSCκ−µ(s) and MγSECκ−µ

(s) can be obtained and on putting Equations 3.16, 3.18 and

3.20 in Equations 3.6 and 3.9, expressions for MγSSCη−µ(s) and MγSECη−µ

(s) can be ob-

tained.

Further, to obtainMγSECps(s) in Equation 3.12 for κ-µ fading channel, we have to find

the integral in the first term of Equation 3.12, which is solved for L = 2 and κ-µ fading

channel in Appendix B and thus the MGF MγSECps(s) for κ-µ fading channel is given as

MγSECps(s)|κ−µ =2Mγ(s)− [3− Pγ(γT )] ·

∫ ∞

γT

pγ(γ)e−sγdγ−

2√πµ

e2µκ

∞∑j=1−µ

∞∑n=0

(−1)n

n!

((1 + κ)γT

γ

)µ−j+12(1

κ

)µ−j−12

× csc(π4(2j + 2µ+ 1)

)(2µ(1 + κ)

γ+ s

)nγnT

×G1,34,6

[4µ2κ(1 + κ)γT

γ

∣∣∣∣∣ −(µ−j−1

2+ n), 0, 1

2, 14

j+µ−12

,− j+µ−12

, j−µ+12

, µ−1−j2

, 14,−(µ−j+1

2+ n)] ,

(3.21)

where, Gm,np,q [·] is the Meijer’s G-function [24].

23

3.3.2 Probability of Bit Error Analysis

The probability of error can be obtained by using the derived MγSSC(s), MγSEC

(s)

and MγSECps(s) in Equation 3.3.

3.3.2.1 SSC and SEC combining

The expressions of probability of error, Pe for SSC and SEC combining schemes are

given below.

Pe|SSC =Pγ(γT )

π

π/2∫0

Mγ

(ψ

sin2 θ

)dθ +

1

π

π/2∫0

I

(ψ

sin2 θ

)dθ (3.22)

Pe|SEC =[Pγ(γT )]

L−1

π

π/2∫0

Mγ

(ψ

sin2 θ

)dθ +

L−2∑j=0

[Pγ(γT )]j · 1π

π/2∫0

I

(ψ

sin2 θ

)dθ (3.23)

The first term in Equations 3.22 and 3.23, i.e. 1π

∫ π/20

Mγ

(ψ

sin2 θ

)dθ represents the proba-

bility of error with no-diversity and can be solved in a closed form as [19]

peκ−µ =Γ(µ+ 1

2)(µ(1 + κ))µ

√ψγ

2√πeµκΓ(µ+ 1)[µ(1 + κ) + ψγ]µ+

12

× Φ1

(µ+

1

2, 1;µ+ 1;

µ(1 + κ)

µ(1 + κ) + ψγ,

µ2κ(1 + κ)

µ(1 + κ) + ψγ

)(3.24)

where Φ1(·) is a confluent hypergeometric function of two variables, which can be imple-

mented by using its finite integral representation [25]

Φ1(a, b; c;x1, x2) =Γ(c)

Γ(a)Γ(c− a)

∫ 1

0

eux2ua−1(1− u)c−a−1(1− ux1)−bdu

and

peη−µ =Γ(2µ+ 1

2)

2√πΓ(2µ+ 1)

(4µ2h(2(h+H)µ+ ψγ)−1

(2(h−H)µ+ ψγ)

)µ× F1

(1

2, µ, µ; 2µ+ 1;

2(h−H)µ

2(h−H)µ+ ψγ,

2(h+H)µ

2(h+H)µ+ ψγ

)(3.25)

where F1(·) is an Appell’s hypergeometric function [25] that has a finite integral repre-

sentation

F1(a, b1, b2; c; x1, x2) =Γ(c)

Γ(a)Γ(c− a)

∫ 1

0

ua−1(1− u)c−a−1

2∏i=1

(1− uxi)−bidu

while the evaluation of the second term of Equations 3.22 and 3.23 requires a numerical

integration.

24

3.3.2.2 SECps combining

The expressions of probability of error, Pe for SECps combining scheme is given below.

Pe|SECps =L

π

∫ π/2

0

I1

(ψ

sin2 θ

)dθ +

L−1∑j=0

[Pγ(γT )]j · 1π

∫ π/2

0

I

(ψ

sin2 θ

)dθ (3.26)

The numerical integration of the above equation is quite complex. So, simulation results

of BER analysis of SECps over κ-µ and η-µ fading channels are obtained and plotted in

next chapter.

25

Chapter 4

Numerical and Simulation Results

The derived expression for Pe in Equation 3.22 is numerically evaluated for SSC

combining and is plotted in Figure 4.1 for BPSK (ψ=1) and BFSK (ψ=0.5) signaling

over κ-µ fading channels. The graph is plotted with respect to average SNR per input

branch (γ) for γT=10 dB and different values of κ and µ. In the figure, the curves for

κ=0 correspond to Nakagami-m fading with µ=m [17]. Thus, the curve for κ=0 and µ=1

is for the well-known Rayleigh fading distribution [17]. Also, the curves for µ=1 is for

Rice distribution with κ as rice parameter K [17]. It can be observed from the figure that

for a given µ, Pe improves with increase in κ. It is because of the increase in the power

of the dominant components of the waves over the scattered components of waves of the

fading model. Further, comparing the BER for BPSK and BFSK it can be observed that

BFSK has a poor performance (≃3 dB) as expected.

In Figure 4.2, BER is numerically evaluated for SSC combining and is plotted for

BPSK and BFSK signalling over η-µ fading channels. In the figure, the curves for η=1

correspond to Nakagami-m fading with µ=m/2 [17]. Thus, the curve for η=1 and µ=0.5

is for the well-known Rayleigh fading distribution [17]. Also, curves for µ=0.5 correspond

to Hoyt distribution with Hoyt parameter q2=η [17]. For a given µ, Pe improves with

increase in η.

Figures 4.3 and 4.4 show the BER performance of SEC combining for κ-µ and η-

µ distributions, respectively, for L=3. It can be observed that BER performance has

improved over dual-SSC receiver as three branches are involved for receiving the signal.

From the Figures 4.1 and 4.3, it can be observed that for BPSK over Rayleigh fading,

to have an BER of ≈ 10−2, an average SNR of 11.3dB is required in SSC receiver, while

26

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Pro

babi

lity

of

erro

r, P

e

κ=0κ=1κ=2Simulation

γT = 10 dB

BPSK (ψ=1)

Rayleigh

µ = 1

µ = 2

µ = 4

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Pro

babi

lity

of

erro

r, P

e

κ=0κ=1κ=2Simulation

µ = 2

µ = 1

Rayleigh

µ = 4

γT = 10 dB

BFSK (ψ=0.5)

Figure 4.1: BER performance of SSC receiver over κ-µ fading channel for BPSK and

BFSK modulations

27

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Prob

abili

ty o

f er

ror,

Pe

η=0.1η=0.3η=1Simulation

γT = 10 dB

BPSK (ψ=1)

Rayleigh

µ = 0.5

µ = 1

µ = 2

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Pro

babi

lity

of

erro

r, P

e

η=0.1η=0.3η=1Simulation

γT = 10 dB

BFSK (ψ=0.5)

Rayleigh

µ = 0.5

µ = 1

µ = 2

Figure 4.2: BER performance of SSC receiver over η-µ fading channel for BPSK and

BFSK modulations

28

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Prob

abili

ty o

f er

ror,

Pe

κ=0κ=1κ=2Simulation

γT = 10 dB

BPSK (ψ=1)

Rayleigh

µ = 1

µ = 2

µ = 4

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Prob

abili

ty o

f err

or, P

e

κ=0κ=1κ=2Simulation

γT = 10 dB

BFSK (ψ=0.5)

Rayleigh

µ = 1

µ = 2

µ = 4

Figure 4.3: BER performance of SEC receiver over κ-µ fading channel for L=3 for BPSK

and BFSK modulations

29

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Prob

abili

ty o

f er

ror,

Pe

η=0.1η=0.3η=1Simulation

γT = 10 dB

BPSK (ψ=1)

Rayleigh

µ = 0.5

µ = 1

µ = 2

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Prob

abili

ty o

f er

ror,

Pe

η=0.1η=0.3η=1Simulation

γT = 10 dB

BFSK (ψ=0.5)

Rayleigh

µ = 0.5

µ = 1

µ = 2

Figure 4.4: BER performance of SEC receiver over η-µ fading channel for L=3 for BPSK

and BFSK modulations

30

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Pro

babi

lity

of

erro

r, P

e

κ=0κ=1κ=2Simulation

γT = 10 dB

BPSK (ψ=1)

Rayleigh

µ = 1

µ = 2

µ = 4

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Prob

abili

ty o

f er

ror,

Pe

κ=0κ=1κ=2Simulation

γT = 10 dB

BFSK (ψ=0.5)

Rayleigh

µ = 4

µ = 1

µ = 2

Figure 4.5: BER performance of SEC receiver over κ-µ fading channel for L=5 for BPSK

and BFSK modulations

31

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Prob

abili

ty o

f er

ror,

Pe

η=0.1η=0.3η=1Simulation

γT = 10 dB

BPSK (ψ=1)

Rayleigh

µ = 0.5

µ = 2

µ = 1

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Pro

babi

lity

of

erro

r, P

e

η=0.1η=0.3η=1Simulation

γT = 10 dB

BFSK (ψ=0.5)

Rayleigh

µ = 0.5

µ = 1

µ = 2

Figure 4.6: BER performance of SEC receiver over η-µ fading channel for L=5 for BPSK

and BFSK modulations

32

0 2 4 6 8 10 12 14 16 18 2010

−4

10−3

10−2

10−1

100

γ in dB

Pro

babi

lity

of

erro

r, P

e

No DiversitySSCSEC (L=5)

γT = 5 dB

BPSK (ψ=1)Rayleigh Fading

γT = 10 dB

Figure 4.7: Comparison of BER performance over Rayleigh fading by varying γT

10dB is required in SEC receiver with L=3, i.e. SEC receiver with L=3 has better

performance (≃1.3dB) than dual-SSC receiver. Similarly, Figures 4.5 and 4.6 show the

BER performance of SEC receiver over generalized fading channels for L=5. Again,

improvement can be observed for SECL=5 over SECL=3 and SSC. It can be observed from

the Figure 4.5, for BPSK over rayleigh fading the required SNR to have an BER of ≈ 10−2

is now reduced to 8.5dB, which is 1.5dB below than that of the SECL=3 and 2.8dB below

than that of the dual-SSC receiver. Similarly, for η-µ fading channels, improvement in

BER can be observed with increasing number of branches. These results are also verified

for special cases of κ-µ and η-µ distributions using simulation.

In Figure 4.7, comparison of BER improvement is done for different combining schemes

over Rayleigh fading channels for different values of γT . It can be observed that, in SSC

receiver for BPSK over Rayleigh fading with γT=5dB, BER of 10−2 can be observed

at γ=9dB, while with γT=10dB, same BER can be observed at γ=11.3dB. Similarly, for

SECL=5 receiver with γT=5dB, γ=5dB is required for BER ≈ 10−2, while with γT=10dB,

γ=8.5dB is required for the same BER. This is because for higher γT probability of

switching to the next branch reduces and most of the time switch will be connected to

33

the first branch and for lower values of γT , most of the time switch will rest on the last

branch.

Similarly, Figures 4.8 and 4.9 show the BER performance of SECps combining for

κ-µ and η-µ distributions, respectively, for L=2 and γT=2dB. From the figures, it is

observed that the performance is better than SSC and SEC|L=2. For γT=2dB, to have an

BER of 10−2, the SNR required for SECps is ≈ 7.5dB, whereas, ≈ 8.5dB is required for

conventional SEC|L=2, i.e. SECps has 1dB better performance than that of conventional

SEC|L=2 or SSC over Rayleigh fading channel. Further, it can be observed that BFSK has

3dB poor performance as compared to BPSK as expected. Again, the effect of channel

parameters on BER performance can be observed, i.e. for fixed µ, increasing κ or η gives

better BER performance as expected. Also, for fixed κ or η, increasing µ improves the

BER performance.

In Figure 4.10, it is shown that for γ << γT , as both the diversity branches will

be below threshold, the SECps performs same as conventional selection combining (SC),

whereas for γ >> γT , the average SNR of the current branch will fall below the threshold

very occasionally, so in such case it performs as no-diversity has been used. For medium

range of average SNR, the SECps has almost same performance as that of conventional

SEC.

34

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

100

γ in dB

Pro

babi

lity

of e

rror

, Pe

κ=0κ=1

µ = 2

γT = 2 dB

BPSK

µ = 1

µ = 4

Rayleigh

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

100

γ in dB

Pro

babi

lity

of e

rror

, Pe

κ=0κ=1

µ = 2µ = 1

µ = 4

Rayleigh

γT = 2 dB

BFSK

Figure 4.8: BER performance of SECps receiver over κ-µ fading channel for L=2 for

BPSK and BFSK modulations

35

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

100

γ in dB

Pro

babi

lity

of e

rror

, Pe

η=0.1η=0.3

µ = 1

γT = 2 dB

BPSK

µ = 2 µ = 0.5

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

100

γ in dB

Pro

babi

lity

of e

rror

, Pe

η=0.1η=0.3

µ = 1

µ = 0.5

µ = 2

γT = 2 dB

BFSK

Figure 4.9: BER performance of SECps receiver over η-µ fading channel for L=2 for

BPSK and BFSK modulations

36

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

γ in dB

Pro

ba

bili

ty o

f e

rro

r, P

e

SEC−ps(L=2)SEC(L=2)/SSCSC(L=2)

BPSK (ψ = 1)Rayleigh ChannelγT = 2 dB

Figure 4.10: Comparison of BER performance of SC, SEC|L=2/SSC and SECps combining

schemes over Rayleigh fading

37

Chapter 5

Conclusion and Future Work

The BER performance of switch and stay combining (SSC), switch and examine com-

bining (SEC) and switch and examine combining with post-examining selection (SECps)

over κ-µ and η-µ fading channels for BPSK and BFSK modulation schemes using MGF

based approach has been presented. Mathematical expressions for the MGF of SSC and

SEC output SNR has been derived for both fading distributions and MGF of SECps

output SNR has been derived for κ-µ fading channels, and using it numerical results for

the BER has been obtained.

The numerically evaluated BER results are plotted for different parameters, and the

results are compared with simulation results for the special cases of κ-µ and η-µ distri-

butions for the system under consideration.

The presented work can be extended for generalized switch and examine combin-

ing (G-SEC) and other antenna selection techniques over other non-homogeneous fading

channels too. The G-SEC combining schemes have been studied extensively for Rayleigh,

Ricean and Nakagami fading channels. However, these schemes have not been analyzed

over non-homogeneous fading channels such as κ-µ and η-µ fading channels.

38

Appendix A

Solution of I(s) in Equations 3.6, 3.9

and 3.12

From Equation 3.6,

I(s) =

∞∫γT

pγ(γ)e−sγdγ. (A.1)

Putting Equation 3.13 in Equation A.1, we get

I(s) =

∞∫γT

β exp

(−µ(1 + κ)γ

γ− sγ

)Iµ−1

(2µ

√κ(1 + κ)γ

γ

)dγ, (A.2)

where

β =µ(1 + κ)

µ+12 γ

µ−12

κµ−12 exp(µκ)γ

µ+12

.

Using [24, (8.445)] for modified Bessel function in Equation A.2, we get

I(s) = β∞∑n=0

µ2n+µ−1

n!Γ(n+ µ)

(√κ(1 + κ)

γ

)2n+µ−1 ∞∫γT

γn+µ−1 exp

(−µ(1 + κ)γ

γ− sγ

)dγ

the integration term in above equation can be represented as upper incomplete gamma

function Γ(·, ·) and thus, I(s) can be expressed as in Equation 3.19.

39

Appendix B

Solution of I1(s) in Equation 3.12

For L=2 branches, I1(s) can be written as

I1(s) = 2

∫ γT

0

Pγ(γ)pγ(γ)e−sγdγ (B.1)

where, Pγ(γ) is the CDF of individual branch SNR, and for κ-µ distribution, is given by

[17]

Pγ(γ) = 1−Qµ

(√2κµ,

√2(1 + κ)µγ

γ

)(B.2)

and, pγ(γ) is the PDF of the SNR of κ-µ distributed channel, and is given by Equa-

tion 3.13. Further, on substituting Pγ(γ) in Equation B.1, I1(s) can be written as

I1(s) = 2 ·

[∫ γT

0

pγ(γ)e−sγdγ −

∫ γT

0

Qµ

(√2κµ,

√2(1 + κ)µγ

γ

)pγ(γ)e

−sγdγ

]or

I1(s) = 2·

Mγ(s)−∫ ∞

γT

pγ(γ)e−sγdγ︸ ︷︷ ︸

I(s)

−∫ γT

0

Qµ

(√2κµ,

√2(1 + κ)µγ

γ

)pγ(γ)e

−sγdγ︸ ︷︷ ︸I2(s)

(B.3)

where, I(s) has already been calculated in Appendix A, and Mγ(s) is given in Equa-

tion 3.17. Now, I2(s) can be written as

I2(s) = θ ·∫ γT

0

Qµ

(√2κµ,

√2(1 + κ)µγ

γ

)γ

µ−12 e−(

µ(1+κ)γ

+s)γIµ−1

(2µ

√κ(1 + κ)µγ

γ

)dγ

40

where,

θ =µ

eµκ

(1 + κ

γ

)µ+12(1

κ

)µ−12

using the identity [26],

QM(α, β) = e−(α2+β2)/2

∞∑j=1−M

(α

β

)jIj(αβ)

I2(s) can now be written as

I2(s) = θe−κµ∞∑

j=1−µ

(1 + κ

κγ

)−j/2 ∫ γT

0

γ(µ−j−1)/2e−(2µ(1+κ)

γ+s)γIj

(2µ

√κ(1 + κ)µγ

γ

)

×Iµ−1

(2µ

√κ(1 + κ)µγ

γ

)dγ

again, using the identity [27]

Iµ(√z)Iv(

√z) =

√π csc

(1

4π(2µ+ 2v + 3)

)G1,2

3,5

[z

∣∣∣∣∣ 0, 12, 14

µ+v2,−µ+v

2, µ−v

2, v−µ

2, 14

],

−µ− v − 1 /∈ N

also, using the series representation of exponential function and putting γγT

= u, I2(s)

can now be written as

I2(s) = λ

∞∑1−µ

∞∑n=0

(−1)n

n!g(j)

(2µ(1 + κ)

γ+ s

)nγ(µ−j+1

2+n)

T

×∫ 1

0

u(µ−j−1

2+n)G1,2

3,5

[4µ2κ(1 + κ)γTu

γ

∣∣∣∣∣ 0, 12, 14

j+µ−12

,− j+µ−12

, j−µ+12

, µ−1−j2

, 14

]︸ ︷︷ ︸

I3(s)

du (B.4)

where,

λ =µ√π

e2µκ

(1 + κ

γ

)µ+12(1

κ

)µ−12

and

g(j) =

(1 + κ

κγ

)−j/2

csc(π4(2j + 2µ+ 1)

)using the identity of Meijer-G function [28]∫ 1

0

x−α(1− x)α−β−1Gm,np,q

[zx

∣∣∣∣∣ apbq]dx = Γ(α− β)Gm,n+1

p+1,q+1

[z

∣∣∣∣∣ α, apbq, β

]

41

I3(s) can be solved as

I3(s) = G1,34,6

[4µ2κ(1 + κ)γT

γ

∣∣∣∣∣ −(µ−j−1

2+ n), 0, 1

2, 14

j+µ−12

,− j+µ−12

, j−µ+12

, µ−1−j2

, 14,−(µ−j+1

2+ n)] (B.5)

now, putting Equation B.5, λ and g(j) in Equation B.4, I2(s) can be found. Further, on

putting I(s), I2(s) and Mγ(s), I1(s) can be found.

Using the derived I1(s), the MGF of output SNR of SECps scheme, MγSECps(s) can be

found and is given in Equation 3.21.

42

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45