P E R F O R M A N C E ANALYSIS OF V I T E R B I D E C O D I N G IN R A Y L E I G H F A D I N G W I T H

C H A N N E L E S T I M A T I O N E R R O R S by

Dingyi L i n

B . S c , Zhongshan University, Guangzhou, China , 1998

A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F

T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F

M A S T E R O F A P P L I E D S C I E N C E

in

T H E F A C U L T Y O F G R A D U A T E S T U D I E S

(Electrical and Computer Engineering)

T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A

A p r i l 2006

© Dingyi L i n , 2006

Previous studies on the effect of channel estimation errors on the bit error rate ( B E R ) performance of V i terb i decoding (VD) concern various types of fading channels with additive white Gaussian noise ( A W G N ) , modulation and interleaving schemes. Pairwise error probabilities ( P E P ) have been derived using Laplace transform. Studies of V D on fading channels with impulsive noise and perfect channel estimation are also available in the literature.

In this thesis, the B E R performance of unquantized V D wi th B P S K is analyzed for a frequency-nonselective slow Rayleigh fading channel with A W G N and Gaussian distributed channel estimation errors. Closed-form expressions for the P E P are de­rived. Upper bound and lower bound on the B E R are obtained. It is shown that channel estimation errors have the same effect on B E R as channel noise. Computer simulation results show that the upper bound is fairly tight.

In practice, the channel might be estimated using pilot symbols, together with various interpolation filters. It is shown that the channel estimation error variances are usually unequal for different data symbol positions. The B E R performance of V D optimized for unequal estimation error variances is compared with that of V D optimized for equal variances using computer simulation.

The B E R performance of V D in Rayleigh fading and impulsive noise with channel estimation error is also studied. The optimal metrics are derived for V D in Rayleigh fading with Gaussian channel estimation errors, for Laplacian noise and Gaussian mixture noise. The B E R performances of V D ' s for various scenarios are compared.

Contents

Abstract ii

Contents iii

List of Figures v

List of Symbols ix

List of Abbreviations xii

Acknowledgements xiii

Chapter 1 Introduction 1

1.1 Motivat ion and Goal 2

1.2 Thesis Outline 3

Chapter 2 Review of Related Works 4

2.1 Convolutional Codes and the Vi terb i Decoder 4

2.2 Fading Channel Models 5

2.3 Pi lot Symbol Assisted Modulat ion 7

2.4 Impulsive Noise 13

2.5 Review of Related Results 14

Chapter 3 V D in Rayleigh Fading with A W G N 17

3.1 System Model 17

3.2 Variances of Est imation Errors in a Frame 20

3.3 Performance of V D optimized for unequal estimation error variances . 24

3.4 Upper and Lower Bounds 28

Chapter 4 V D in Rayleigh Fading with Impulsive Noise 47

4.1 Laplacian Noise 47

4.2 Gaussian Mixture Noise 54

Chapter 5 Conclusion 59

Bibliography 61

Appendix A Derivation of the Conditional pdf's

pRelRi](Re[ri]\E£[Hi] = Re[hi], Si = Sj) andplmlRi](Im[ri]\Im[Hi] = Im[hi], Si =

Si) 65

Appendix B Derivation of Optimal Bit Metric for Laplace Noise . . 68

Appendix C Derivation of Optimal Bit Metric for Mixture Gaussian

Noise 72

List of Figures

2.1 Pi lot symbols in P S A M 7

2.2 Fi lter coefficients of Gaussian interpolation for M = 100 9

2.3 Relative positions of the pilot symbols and the interpolated frame in

Gaussian interpolation 10

2.4 Fi lter coefficients of 4-tap ideal low-pass filter for M — 100 11

2.5 Relative positions of the pilot symbols and the interpolated frame in

4-tap ideal low-pass filter 12

2.6 Gaussian, Laplace and Gaussian mixture pdf's. A l l pdf's have a vari ­

ance of 1 14

2.7 Gaussian, Laplace and Gaussian mixture pdf's on Log scale. A l l pdf's

have a variance of 1 15

3.1 System model 18

3.2 Discrete-time representation of the system model 19

3.3 Variances of estimation error for linear interpolation at different data

symbol positions 21

3.4 Variances of estimation error for Gaussian interpolation at different

data symbol positions 22

3.5 Variances of estimation error for 10-tap Sine filter at different data

symbol positions 23

3.6 Variances of estimation error for 10-tap Wiener filter at different data

symbol positions 24

3.7 Comparison of B E R of V D ' s with unequal variance metric and equal

variance metric for convolutional code (33,25,37) 27

3.8 Upper and lower bounds on the B E R of unquantized V D for convo­

lutional code (133,171) as a function of S N R for different values of

E S R 35

3.9 Upper and lower bounds on the B E R of unquantized V D for convo­

lutional code (133,171) as a function of S N R for E S R = -co d B and

E S R = -10 d B 36

3.10 Upper and lower bounds on the B E R of unquantized V D for convo­

lutional code (133,171) as a function of E S R for different values of

S N R 37

3.11 Upper and lower bounds on the B E R of unquantized V D for convo­

lutional code (33,25,37) as a function of S N R for different values of

E S R 38

3.12 Upper and lower bounds on the B E R of unquantized V D for convo­

lutional code (33,25,37) as a function of E S R for different values of

S N R 39

3.13 Upper and lower bounds on the B E R of unquantized V D for convolu­

tional code (133,171) for E S R = 1 / S N R 39

3.14 Upper and lower bounds on the B E R of unquantized V D for convolu­

tional code (33,25,37) for E S R = 1 / S N R 40

3.15 B E R of unquantized V D for convolutional code (133,171) as a function

of S N R 41

3.16 B E R of unquantized V D for convolutional code (133,171) as a function

of E S R '42

3.17 Approximate B E R of unquantized V D for convolutional code (133,171)

as a function of S N R for E S R = - 8 d B 43

3.18 B E R of unquantized V D for convolutional code (33,25,37) as a function

of S N R 43

3.19 B E R of unquantized V D for convolutional code (33,25,37) as a function

of E S R 44

3.20 Region for (upperbound - simulation)/simulation < 0.1 for convolu­

tional code (133,171) 45

3.21 Region for (upperbound - simulation)/simulation < 0.1 for convolu­

tional code (33,25,37) 46

4.1 Histogram of the channel estimation error in Rayleigh fading and Lapla -

cian noise with 10-tap Wiener filter, compared wi th Gaussian his­

togram with the same variance 48

4.2 B E R performance of V D o p t , V D E u c i i d and V D s q u a r e d Euclid, in Rayleigh

fading with Laplacian noise and Gaussian channel estimation error, as

a function of S N R 49

4.3 B E R performance of V D o p t , V D E u c i i d and V D s q u a r e d Euclid, in Rayleigh

fading with Laplacian noise and Gaussian channel estimation error, as

a function of E S R 50

4.4 B E R performance of V D s q u a r e d Euclid in Rayleigh fading with Laplacian

noise and A W G N 51

4.5 B E R performance of V D E u c i i d in Rayleigh fading with Laplacian noise

and A W G N 52

4.6 B E R performance of V D o p t in Rayleigh fading wi th Laplacian noise

and A W G N 53

4.7 B E R of V D in Rayleigh fading with Gaussian mixture noise and Gaussian

channel estimation error, for e = 0.05, crj/a^ = 21 and crj/a^ = 181,

as a function of S N R 55

4.8 B E R of V D in Rayleigh fading with Gaussian mixture noise and Gaussian

channel estimation error, for e = 0.05, cr]/a^ = 21 and o^ja^ = 181,

as a function of E S R 56

4.9 B E R of V D ' s with optimal metrics in Rayleigh fading with Gaussian

mixture noise, Laplacian noise and A W G N , and Gaussian channel es­

t imation error, as a function of S N R 57

4.10 B E R of V D ' s with optimal metrics in Rayleigh fading with Gaussian

mixture noise, Laplacian noise and A W G N , and Gaussian channel es­

t imation error, as a function of S N R 58

List of Abbreviations

A W G N Addit ive Whi te Gaussian Noise

B E R B i t Error Rate

B S C Binary Symmetric Channel

E S R Error-to-Signal Ratio

F I R Finite Impulse Response

i . i .d . Independent identically distributed

ISI Inter-Symbol-Interference

M L Maximum-l ikel ihood

pdf Probabil ity Density Function

P S A M Pi lot Symbol Assisted Modulat ion

P E P Pairwise Error Probabil ity

R P Random Process

rv Random Variable

S N R Signal-to-Noise Ratio

T C M Trellis-Coded Modulat ion

V A Vi terb i Algor i thm

V D Vi terb i Decoder or Decoding

List of Symbols

ddj The number of codewords of weight d that correspond to information

sequences of weight j

bd The total number of nonzero information bits associates with codewords

of weight d

c Complex channel gain sequence

Cj Complex channel gain

i?e[cj] Real part of q

Im[ci] Imaginary part of q

dfree Free distance

ti Channel estimation error

Re[ei] Real part of

Im[ei] Imaginary part of

fd M a x i m u m Doppler frequency

f(t) Unit-energy basis function

h Estimated channel gain sequence

hi Estimated channel gain

Re[hi] Real part of hi

Im[hi] Imaginary part of hi

M Frame length

M{ri\ hi, Si) B i t metric

M (r| h, s) Pa th metric

rii Channel noise

Re[rii} Real part of rii

Im[ni] Imaginary part of

n(t) Channel noise

Pi Channel estimation at pilot symbol position

P;, B i t error rate

Pd The probability that a weight d path accumulates a higher metric than

the all-zero path

qt Coefficient of interpolation filter

Ti Received Signal

i?e[rj] Real part of Ti

Imfri] Imaginary part of r ,

ri(t) Received signal

T Symbol duration

s Channel symbol sequence

Si Channel symbol

si(t) Transmitted signal

x Information bit sequence

Xk Information bit

y Codeword sequence

Hi Encoded bit

z Decoded bit sequence

Zk Decoded bit

o% Variance of i?e[cj] or Irn[ci]

a\ Variance of -Re[e;] or Im[ei]

Variance of Re[hj\ or Im[hi]

Variance of impulsive noise

Variance of i?e[nj] or Im[rii]

Variance of Re[ri\ or Im[ri]

Variance of background Gaussian

Variance of Laplace distribution

Impulsive index

Acknowledgements

I am grateful to my supervisor Prof. C y r i l Leung, who has given me constant guidance,

invaluable advice, and the fundamental knowledge in the field, and who has set up an

example of being attentive and pursuing academic excellence. I also appreciate the

efforts of many professors who have been reviewing my thesis, including Prof. Lutz

Lampe and Prof. Hussein Alnuweiri .

I would like to express my heartfelt gratitude to my parents, Shiquan L i n and

Zhian L i u , parents in-law, Haoran W u and Haoying Shi , for their patient and constant

support, physically and mentally; and to my wife, L i y u n W u , who has been always

on my side, providing love, care and encouragement.

Finally , to al l my friends and relatives who care about me, thank you!

This work was partially supported by the Natura l Sciences and Engineering Re­

search Counci l Grant OGP0001731.

D I N G Y I ( C H R I S ) L I N

THE UNIVERSITY OF BRITISH COLUMBIA

April 2006

Chapter 1

Introduction

During the past decade, much of the growth in the telecommunications industry has

been in the wireless sector. For cellular wireless communications, the th ird generation

(3G) standard has been developed and systems are currently being deployed around

the world. The 3 G systems are capable of supporting circuit-switched and packet

data at 144 kbps for high mobility (vehicular) traffic, 384 kbps for pedestrian traffic

and 2 Mbps for indoor traffic [1]. This data rate enables many new applications,

e.g., personal applications that combine entertainment and information, multimedia

message services, mobile access to intranet and extranet, etc.

For fixed broadband wireless access, I E E E 802.16 standards have been developed

to offer an alternative to cable network access, e.g., optical fibre, cable modem and

digital subscriber line (DSL) . The standards cover the medium access control and

physical layers for the frequency range 2-11 G H z and 10-60 G H z . O n the commercial

side, W i M A X (world interoperability for microwave access) technical working groups

have been established to develop a set of system profiles, protocol implementation

conformance statements, etc., to handle some of the shortcomings of the I E E E 802.16

standards [2]. W i M A X systems are able to cover a large geographical area up to a

radius of 50 k m and to deliver a large bandwidth up to 72 Mbps to end-users.

For wireless L A N , I E E E 802.11a/b/g standards compliant devices are widely used

and support data rates up to 54 Mbps. More recently, the I E E E 8 0 2 . l l n standard is

being developed and some of its proposals include speeds up to 540 Mbps [3].

In many of these wireless communication systems, convolutional codes and Vi terb i

decoders (VD) are used to deliver reliable communications. For example, they have

been incorporated in the I E E E 802.11a [4] and I E E E 802.16 [5] standards, and 3 G P P

(3rd Generation Partnership Project) specification [6] for channel coding and decod­

ing.

1.1 Motivation and Goal

In order for the V D to efficiently recover the transmitted signal in a fading environ­

ment, the complex fading channel gain has to be estimated. Depending on the design

and implementation, there exist various sources of error that can make the estimated

channel gain different from the actual channel gain. For systems that use pilot sym­

bols, the estimate of the complex channel gain for each data symbol is produced

by interpolating the received pilot symbols. In digital implementations, quantiza­

t ion introduces errors in the channel gain estimation. Thermal noise is inevitable in

any hardware implementation. It is thus important to evaluate the effect of channel

estimation errors on the performance of V D .

In previous related papers on V D in fading with channel estimation errors, it

was assumed that the channel estimation errors had equal variances. It was shown

in [7], which was on uncoded pilot-symbol-assisted modulation ( P S A M ) with Wiener

filter, that the channel estimation error variances were unequal. It is interesting to

compare the performance of V D with optimal metric for unequal variances and that

with optimal metric for equal variances in unequal variances environment.

Although additive white Gaussian noise ( A W G N ) is commonly assumed in com-

munications system studies, it has been found that in many situations [8-15], the noise

exhibits an impulsive nature. This motivates an investigation of the performance of

V D in impulsive noise.

The objectives of this thesis are (1) to analyze the impact of channel estimation

errors on the B E R performance of V D , (2) to compare the B E R performance of V D

with optimal metric for unequal variances and that wi th optimal metric for equal

variances in unequal variances environment, (3) to develop an expression that links

the channel estimation error and the B E R performance of V D , (4) to derive optimal

metrics for V D in impulsive noise with channel estimation errors.

1.2 Thesis Outline

The thesis is organized as follows. In Chapter 2, the V D , fading channel, pilot symbol

assisted modulation ( P S A M ) and impulsive noise models are introduced, and existing

results are reviewed. In Chapter 3, the system model used in the thesis is introduced,

and the variance of the estimation error in P S A M is examined. Closed-form expres­

sions for upper bounding and lower bounding the B E R of V D in Rayleigh fading

with B P S K and A W G N are then obtained and compared with simulation results. A

B E R comparison of the V D with optimal metric for unequal variances and that with

optimal metric for equal variances is provided using simulation results. In Chapter 4,

optimal metrics are derived for V D in Rayleigh fading with Gaussian channel estima­

tion errors, for Laplace noise and Gaussian mixture noise. B E R results are obtained

using simulation. In Chapter 5, the main findings of the thesis are summarized and

directions for further research are presented.

Chapter 2

Review of Related Works

2.1 Convolutional Codes and the Viterbi Decoder

Convolutional codes were invented in 1955 [16]. They differ from block codes in that

they transform the whole information sequence, regardless of its length, into one single

codeword rather than segmenting the sequence into blocks of fixed length. There are

three main types of decoding algorithms for convolutional codes, namely sequential

decoding, feedback decoding and Vi terb i decoding. The Vi terb i algorithm (VA) is

the only one which provides maximum-likelihood ( M L ) decoding.

The V A makes use of the trellis structure of convolutional codes to decode the

received codewords. O n every reception of branch symbols, the V A computes the

branch metrics and adds them to the current survivors' metrics; it then compares

the metrics of paths entering the same node and selects the path with lowest metric

as the survivor. Finally , the path with the lowest metric (highest probability) is the

decoded path and the corresponding codeword is the decoded codeword.

A detailed discussion of convolutional codes and the V D can be found in [17,

chapters 11 and 12].

2.2 Fading Channel Models

W h e n a signal is transmitted in a wireless mobile environment with time-varying mul -

t ipath propagation, the pass-band received signal can be modelled as a superposition

of multiple attenuated and delayed copies of the transmitted signal [18, chapter-14]:

x(t) = ^ a n ( t ) s [ t - rn{t)} (2.1) n

where s(t) is the transmitted signal, an(t) is the attenuation of the n t h path and

rn(t) is the delay of the n t h path. When these copies of the transmitted signal add

constructively, the received signal strength is high; when they add destructively, the

received signal strength is low. The fluctuations in the amplitude of the received

signal are termed signal fading.

For some channels, such as the tropospheric scatter channel, where the pass-band

received signal can be viewed as consisting of a continuum of mult ipath components,

the pass-band received signal can be expressed in an integral form:

/

oo a(r;t)s(t - r ) d r (2.2)

•CO

where a(r; t) is the time-varying band-pass channel impulse response. The equivalent

low-pass received signal can be expressed as

/

CO a{r-t)e-^^Sl{t-r)dT (2.3)

-co

where si(t) is the equivalent low-pass transmitted signal. Thus the equivalent low-pass

channel impulse response is

c (r ; t ) = a(r;t)e-j27TfcT. (2.4)

The coherence bandwidth and the coherence time are two important parameters

that characterize the fading channel. The coherence bandwidth ( A / ) c is defined as

the bandwidth over which the frequency correlation function is above a specific value

[19], say 0.5. When the coherence bandwidth ( A / ) c is much greater than the signal

bandwidth, the channel is said to be frequency-nonselective: al l frequency components

of the signal are altered by the channel in the same way. The equivalent low-pass

received signal can then be expressed in a product form, instead of a convolution [18]:

n(t) = C(0;t)Sl(t) (2.5)

where C(0;t) represents the time-varying transfer function C(f;t) at / = 0 which

can be expressed in a complex form C(0;t) — r(t)e^^ where r(t) is the amplitude of

C (0 ; t ) .

The coherence time ( A i ) c is defined as the time over which the time correlation

function is above a specific value [19], say 0.5. When the coherence time ( A t ) c is

much greater than one signal interval, the channel is said to be slowly fading, and

r{t)e^^ can be regarded as constant over a signal interval. Then the equivalent

low-pass received signal can be further simplified as

n(t) = re^siit), 0<t<T. (2.6)

Three distributions are commonly used to model the envelope of the channel

impulse response c(r;t), namely Rayleigh, Ric ian and Nakagami-m distributions. If

C(0;£) is modelled as a complex-valued Gaussian random process (RP) with zero-

mean, its envelope R — |C(0;£)| at any instant t has a Rayleigh distribution:

pR(r) = -2e-r2'2°\ r > 0 (2.7)

where cr2 is the variance of the real or imaginary part of C(0;£).

If C(0;t) is a nonzero-mean complex-valued Gaussian R P with an expected am­

plitude of 5, and the variance of the real or imaginary part of C (0 ; t) is cr2, the envelope

R of C (0 ; t) has a Ric ian distribution:

PR(T) = y2e^2+s2)/^Io(^), r>0 (2.8)

6

where IQ(X) represents the zeroth order modified Bessel function of the first kind.

The envelope R is also commonly modelled by the Nakagami-m distribution:

' « < r > - F R Q " ^ ' ^ - r s 0 <2'9>

where 0 = E(R2), m — E^2_a^, and T(x) represents the G a m m a function. The

parameter m is usually larger than or equal to 1/2 [20].

More detailed discussions of fading channel models can be found in [18, chapter

14] and [21, chapter 1].

2.3 Pilot Symbol Assisted Modulation

Pilot symbol assisted modulation ( P S A M ) is one of the techniques that can be used to

estimate the channel gain. Compared to the pilot tone assisted modulation ( P T A M ) ,

P S A M is less complex in implementation, occupies less bandwidth and has lower

peak-to-average power ratio [22].

frame of M symbols

1

next frame

1

data symbols

pilot symbols

Figure 2.1: Pi lot symbols in P S A M

A t the transmitter, pilot symbols are periodically inserted into the data stream

as illustrated in F ig . 2.1. We assume that the pilot symbol is the first symbol in a

frame. The pilot sequence is usually pseudo random and is known to the receiver. The

receiver then extracts the channel information at the pilot positions, and calculates

the channel gains at the data positions by interpolation.

There are several kinds of interpolation filters [7,23,24]. We assume that one

pilot symbol is inserted for every M — 1 data symbols, thus M can be considered as

the length of a frame. Let pi denote the estimated channel gain at the pilot position

of the zth frame and q^j) denote the zth-tap filter coefficient that is used to calculate

the estimated channel gain for the jth data symbol in a frame. We also assume that

the interpolation filter uses the L nearest pilot symbols. The channel gain estimate

at the jth data symbol of the L^-Jth frame is

where [ x\ denotes the largest integer < x.

Zeroth-order Interpolation

When the fading process is changing very slowly and the channel gain is assumed to

be constant in a frame, zeroth-order interpolation [23] can be used. The zeroth-order

interpolation filter has only one tap (L = 1) and

Linear Interpolation

Linear interpolation [23,25] is another simple interpolation method. Its filter has only

two taps where

L (2.10)

i = i

(2.11)

(2.12)

(2.13)

1.2

0.8

0.6

o 0.4

0.2

0

-0.2

-0 .4

I 1 1 1 I 1 1 1 1

_ q2

q3

V. -

N \ S

s

s \

\ s

1 1 1 1 1 1 i 1 1 10 20 30 40 50 60 70 80 90 100

Samples

Figure 2.2: Fi lter coefficients of Gaussian interpolation for M = 100

Gaussian Interpolation The Gaussian interpolation filter [23] has three taps, where

3 M

M = i - ( T 7 + l ) 3

2 "M M

(2.14)

(2.15)

(2.16)

The values of qi, q2 and q3 are plotted in F ig . 2.2 for M = 100. The relative positions

of the pilot symbols pi, p 2 and P3 are plotted in F ig . 2.3.

interpolated frame

f Figure 2.3: Relative positions of the pilot symbols and the interpolated frame in Gaussian interpolation

Ideal Low-pass Filter

A n ideal low-pass filter can be implemented in time domain with a truncated Sine

function [24] where

qiU) = Sine{i-l^±±\-jj) l<i<L (2.17)

and

Sinc(x) = sin(iTx)

TtX (2.18)

or in frequency domain with fast Fourier transform ( F F T ) [26]. The filter coefficients

of a 4-tap ideal low-pass filter are plotted in F ig . 2.4 for M = 100. The relative

positions of the pilot symbols pi, p2, PJ, and p^ are plotted in F ig . 2.5.

Wiener Filter

Although the ideal low-pass filter is commonly used in interpolating the deterministic

signals, it can not minimize the mean squared estimation error in estimating the

stochastic process. In order to minimize the mean squared error, a Wiener filter can

be used [7].

The Wiener filter provides a solution to the following problem [27, chapter 5].

Let u> = [u>(0) ui(l) • • -u(L — 1)] denote the complex coefficients of an finite impulse

response (FIR) filter, u = [u(0) u(l) • • • u(L — 1)] denote the input to the filter where

u(n) are samples of a wide sense stationary process, v(L — 1) denote the filter output

Figure 2.4: Fi lter coefficients of 4-tap ideal low-pass filter for M = 100

at time L — 1, L-l

u ( L - i ) = £ y ( i ) u ( i ) , (2.19) i=0

d(L — 1) denote the desired signal at time L — 1, and

e(L-l) = d(L - 1) - v{L - 1) (2.20)

denote the estimation error. The coefficients of the F I R filter are chosen to minimize

the mean squared estimation error E{ee*}.

It was shown in [27, chapter 5] that the filter coefficients are

p R " 1 (2.21)

where p denotes the cross-correlation vector of d(L — l) and vector u, and R 1 denotes

the inverse of the covariance matrix of u. The minimum mean squared estimation

t

interpolated frame

P4

Figure 2.5: Relative positions of the pilot symbols and the interpolated frame in 4-tap ideal low-pass filter

error is

E{ee*} = E{dd*} - E{vv*}. (2.22)

W h e n the mean squared estimation error is minimized, the estimation error e is

uncorrelated with every filter input u(z) and the filter output v(L — 1).

In the case of interpolation, the input to the filter is a vector of channel esti­

mates at the L nearest pilot positions. Then for every data symbol position, a set of

optimized filter coefficients is obtained through (2.21), and the filter output is the in ­

terpolated value at that data symbol position, which has the minimum mean squared

estimation error.

Limit on Frame Length

Let T denote the symbol duration and M denote the length of a frame. Assuming

that one pilot symbol is transmitted per frame, then the pilot symbol rate (or channel

sampling rate) is Let fa denote the maximum Doppler frequency, then the power

spectrum of the fading channel has a bandwidth of 2fd [21, chapter 1]. According to

the sampling theorem, the channel sampling rate must be at least 2/^, so that the

length of a frame must not exceed ^j^f [7].

2.4 Impulsive Noise

Impulsive noise has been investigated by many researchers for characterizing man-

made R F noise, low-frequency atmospheric noise and underwater acoustic noise [8,

chapter 3]. As opposed to Gaussian noise, the probability density function (pdf) of

impulsive noise tends to have heavier tails, which means that large deviation from the

mean is more likely. Some commonly used impulsive noise models, e.g., generalized

Gaussian noise, generalized Cauchy noise, mixture noise and Middleton class A noise,

are described in [8]. In this thesis, we focus on Laplacian noise and Gaussian mixture

noise.

Laplacian noise has been suggested to be a fairly accurate noise model at ex­

tremely low frequency [9], and discussed for signal detection in [10] and Vi terb i de­

coding in [14]. Laplacian noise is a special case of generalized Gaussian noise, and

the pdf of a zero mean Laplace random variable (rv) with variance 2f52 is

1 Inl PN(U) = ^ P ( - j ) - (2-23)

Mixture noise is another k ind of widely used impulsive noise model. Its pdf is

pN(n) = {l-e)r1{n) + ei(n), (2.24)

where the impulsive index e is a small positive constant, r)(n) is a Gaussian pdf

representing the background Gaussian noise, and i(n) is some other pdf with a heavier

tai l that represents the impulsive noise. W h e n i(n) is also a Gaussian function, we

have Gaussian mixture noise with pdf

1 n2 1 n2

pN{n) = (1 - € ) - _ _ e x p ( - — ) + e ^— exp(-—^) (2.25)

where a2 is the variance of the background Gaussian noise and a] is the variance of

the impulsive noise. The ratio CF2/CF2 is usually in the range (20,10000) [8]. The total

variance of a Gaussian mixture rv is (1 — e)a2 + eaj. It is also noted that a smaller

e implies a heavier ta i l in pdf. Gaussian mixture noise has been used as the noise

model for multiuser detection in [28] and for signal detection in [29].

The pdf's of the Gaussian, Laplace and mixture Gaussian rv's with zero mean

and variance 1 are plotted in F ig . 2.6 and F ig . 2.7. To better compare the tails of

three distributions, F ig . 2.7 is plotted on log scale.

Figure 2.6: Gaussian, Laplace and Gaussian mixture pdf's. A l l pdf 's have a variance of 1.

2.5 Review of Related Results

The B E R performance of V D on the binary symmetric channel (BSC) , A W G N channel

and general memoryless channels is studied in [30], which provides upper bounds on

the B E R using the generating function approach. A B E R lower bound for the V D is

derived in [17, chapter 12] from the genie-aided approach which enables the decoder

Figure 2.7: Gaussian, Laplace and Gaussian mixture pdf's on Log scale. A l l pdf's have a variance of 1.

to choose between the correct codeword and the codeword that is at distance dfree

away from the correct codeword.

Many researchers have studied the V D in various practical systems. In [31]

and [32] the V D was examined for trellis-coded modulation ( T C M ) in uncorrelated

Rayleigh fading and A W G N . The optimal metric was derived for imperfect channel

estimation, which in the case of P S K can be shown to be identical to that for perfect

channel estimation. The pairwise error probability ( P E P ) was derived by the method

of Laplace transform, and provided in terms of residues. This method was extended

to uncorrelated Ric ian fading in [33]. A similar method was used in [34] to study the

V D for finite-depth interleaved convolutional codes in Rayleigh fading. The P E P for

B P S K was given in terms of residues with interpolation filter coefficients and inter­

leaving depth as parameters. In [35] the V D was studied for convolutional codes in

spatially and temporally correlated Rayleigh fading D S - C D M A systems with channel

estimation error. A n upper bound on B E R was derived for perfect channel estimation

by means of upper bounding the P E P , and an approximate B E R was provided for

imperfect channel estimation. In [36] and [37] the V D was considered for convolu­

tional codes in uncorrelated Ric ian block fading and Rayleigh block fading, where

the channel gain is constant over a frame. The P E P and the union bound on B E R

were provided for both perfect and imperfect channel estimation. In [38] the V D was

studied for T C M in correlated Rayleigh fading with imperfect phase reference and

an approximate P E P was given. In [7] the uncoded P S A M with Wiener filter was

studied. It was shown that the variances of the estimation error were unequal. In

al l these studies, unequal variances of the channel estimation error have not been

considered in V D , and the final results for P E P were not in closed form.

Whi le A W G N was assumed in the above papers, the following ones were using

impulsive noise models in the study of V D . The V D for combating inter-symbol-

interference (ISI) was studied in [11] and [12], for Cauchy noise with unknown para­

meters and for a-stable noise respectively. In [11] per-survivor-processing was used

to estimated the noise parameters; while in [12] a penalty function was incorporated

in the metric. O n the other hand, an approximation to the log likelihood function

was used as the metric for the V D in simplified class A noise in [13]. Opt imal met­

rics for the V D in Laplacian noise, Cauchy noise and Logistics noise were derived

in [14]. Moreover, the B E R performance of V D in direct-sequence spread-spectrum

multiple-access system in Ric ian fading with perfect channel estimation is discussed

in [15], including Gaussian mixture noise and inter-user interference. To the best of

our knowledge, the V D with imperfect channel estimation in impulsive noise has not

been studied so far.

Chapter 3

V D in Rayleigh Fading with

A W G N

3.1 System Model

The system model used in this thesis is shown in F ig . 3.1. For clarity, we use up­

percase letters to denote rv's and the corresponding lowercase letters to denote their

sample values. The information bit sequence x = (x 0 , £i , • • •) is first convolutionally

encoded. Then the encoded bit sequence y = (yoiZ/i, • • •) is mapped into channel

symbol sequence s = (so,si , • • •)• We assume that s* = +1 is transmitted if yi = 0

and Si = —1 is transmitted if ^ = 1. During the ith signal interval, Si is multiplied

by a unit-energy basis function f(t) and the resulting waveform, Si(t), where

si(t) = Si-f(t), 0<t<T (3.1)

is transmitted over the channel.

We assume that the channel is a frequency-nonselective, slow Rayleigh fading

Convolutional Encoder

B P S K

Encoded Bit to Channel Symbol

Mapper

Si fit) fit)

Channel Gain Estimator

hi

Viterbi Decoder

Modulator

Demodulator

Figure 3.1: System model

channel. The received signal is

rt(t) = Ci • si(t) + n(t)

= Ci-Si-f(t)+n(t), 0<t<T (3.2)

where Cj denotes the complex channel gain during the ith. interval and is a sample of

a zero-mean complex-valued Gaussian rv C; with a real or imaginary part variance of

OQ\ the real and imaginary parts of C j are assumed to be independent; n(t) denotes

the low-pass A W G N and is a sample function of a zero-mean complex-valued Gaussian

R P with a real or imaginary part variance of a%.

The received signal is input to a correlation demodulator, whose output is

rT U = ri(t)f*(t)dt

f • Si • f(t)f*(t)dt+ [Tn(t)r(t)dt Jo Jo

(3.3)

where /*(£) denotes the complex conjugate of fit). Lett ing

rT rii - n(t)f*(t)dt (3.4)

Zk

Convolutional Encoder

Encoded Bit to Channel Symbol

Mapper

Channel Gain Estimator

hi

Viterbi Decoder

Discrete-time Fading Channel

Figure 3.2: Discrete-time representation of the system model

(3.3) becomes

7*2 — ' Si ~\~ Tli (3.5)

where nj is a sample of a zero-mean complex-valued Gaussian rv, JVj, with a real or

imaginary part variance of a\. The rv Nt is independent of d or Si. The real and

imaginary parts of Ni are assumed to be independent.

A discrete-time model of the system is shown in F ig . 3.2. A channel gain estimator

provides an estimation of Cj for the V D . The estimated channel gain hi is assumed

to have the form

hi Ci (3.6)

where the channel estimation error is a sample of a zero-mean complex-valued

Gaussian rv Ei, wi th a real or imaginary part variance of o\ . The real and imaginary

parts of Ei are independent. For simplicity, we assume that Ei is independent of C,.

Al though the estimated channel gain and the channel estimation error are affected by

A W G N at the pilot symbol position, they are independent of iV, at the data symbol

position. The V D then uses the estimated channel gain sequence h = (ho, hi,...) to

decode the received sequence r = (rn,7"i> • • •)• The decoded bit sequence is denoted

by z = (z0,zl,...).

For convenience, we define S N R and E S R as S N R = (JQ/G2N and E S R = O\I<J'2

C

[39].

3.2 Variances of Estimation Errors in a Frame

Depending on the type of interpolation used, the variances of the estimation error

are usually different at different data symbol positions. We assume that the received

symbol at the pilot symbol position has the form in (3.5), and the channel gain p; at

the pilot symbol position is obtained by

Pi = — Si

= Ci + ^. (3.7) Si

For linear interpolation as defined in (2.12) and (2.13), the interpolated channel gain

is

M , ) - ( i - - L ) ( c + =I)4(«> + S )

There are two sources of channel estimation error: A W G N and the fading process.

W h e n the A W G N is the dominant source of the channel estimation error, the estima­

tion error is

eU) « <3J»

and its variance

4, = { ( 1 - ^ ) 2 + (^)2}^' j = l - M - l . (3.10)

For Gaussian interpolation, the interpolated channel gain is

W ) = ~ ^ ) C l + (1 - j^)C2 + + ^ ) C 3

AiJL _ j_\VI + n - + + -?-)— (3 ID

+ 2 ( M * M }S l

+ { 1 M>} s2 + 2{ M* + M} s3' 1 j

the estimation error is

<1) - - ^ - M ^ - ^ - ^ - ^ M ^ + M ^ ' ( 3 - 1 2 )

and its variance

Real Part Imaginary Part

40 60 80 Samples

100

Figure 3.3: Variances of estimation error for linear interpolation at different data symbol positions

Simulations are performed to measure the variance of the real and imaginary

parts of the estimation error at different data symbol positions. The results for

0.3

Samples

Figure 3.4: Variances of estimation error for Gaussian interpolation at different data symbol positions

linear interpolation are plotted in Fig . 3.3. The frame length M is 100 symbols, the

fdT product values are 0.001, 0.002, 0.003, 0.004 and 0.005. The noise variance a\

is 0.158, corresponding to a channel gain variance UQ = 0.5 and S N R = 5 d B . A

reference curve corresponding to (3.10) is also plotted. The figure shows that the

variances of the estimation error are not constant. When the frame length M is much

smaller than jJZr' e , § ' ' ^ 0 f ° r fd^ = 0.001, the variances are very close to those

predicted by (3.10).

A similar plot for Gaussian interpolation is shown in F ig . 3.4. The reference

curve is obtained using (3.13).

A plot for a 10-tap Sine filter is shown in F ig . 3.5. It can be seen that for M <

2jjri the curves are much flatter than those for linear and Gaussian interpolation,

and the variances of estimation error are very close to the variance of the A W G N .

0.24

0.22

O 0.18

.§ 0.16

Real Part Imaginary Part

fdT=0.005

40 60 Samples

100

Figure 3.5: Variances of estimation error for 10-tap Sine filter at different data symbol positions

The curves change negligibly as fdT changes from 0.001 to 0.004.

Variances for a 10-tap Wiener filter are plotted in F ig . 3.6. We notice that when

M < Tjjrjy, the variances are essentially equal for all data symbol positions. The

Wiener filter provides a substantial performance improvement over the Sine filter in

terms of the estimation error variance, especially when M <C jf^f- For example,

when fdT = 0.01, the estimation error variance of the 10-tap Wiener filter is about

6 d B lower than that of the 10-tap Sine filter. When fdT = 0.04, the difference is

smaller. The estimation error variance of the 10-tap Wiener filter is about 1.7 d B

lower than that of the 10-tap Sine filter.

20 40 60 Samples

80 100

Figure 3.6: Variances of estimation error for 10-tap Wiener filter at different data symbol positions

3.3 Performance of V D optimized for unequal es­

timation error variances

We assume that the channel gains for consecutive received symbols are uncorrelated

and that the V D performs unquantized decoding. The uncorrelated condition can

be achieved by using channel-symbol interleaving [17, section 10.4.4]. Let i?e[c*] and

Im[ci] denote the real and imaginary parts of the channel gain a; Re[ci\ and ira[cj]

are samples of independent identically distributed (i.i.d.) zero-mean Gaussian rv's

with variance OQ; Z2e[Ci] and Im[Ci] are independent. Let i?e[nj] and Im[rii] denote

the real and imaginary parts of the A W G N n*; i?e[n;] and Im[rii] are samples of i . i .d .

zero-mean Gaussian rv's with variance a%; Re[Ni] and Im[Ni] are independent. Let

Re[ri] and Im[ri) denote the real and imaginary parts of r j . From (3.5) we have

Re[ri] = Re[ci\ • Si + Re[ni] (3.14)

Im[ri] = Im[ci] • Si + Irn[rii\. (3.15)

Thus given Re[ri] and Im\ri] are samples of i . i .d . zero-mean Gaussian rv's wi th

variance a\ where

4 = 4 + 4 ; (3.16)

Re[Ri] and Im[Ri] are independent.

The real and imaginary parts of the channel estimation error, i.e., Re[ei] and

Im[ei] are samples of i . i .d . zero-mean Gaussian rv's with variance a%.. From (3.6) we

have

Re[hi] = Re[ci] - Re[ei] (3.17)

Im[hi] = Im[ci] — Im[ei}. (3.18)

Thus Re[hi] and Im[hi] are samples of i . i .d . zero-mean Gaussian rv's with variance

ajf. where

4 , = 4 + 4 ; (3-19)

Re[H~i] and Im[Hi] are independent. Given Re[hi], Im[hi] and S j , i?e[r j ] and i m [ r j ] are

samples of i . i .d . Gaussian rv's with means, mRe[Ri]\ReiHi}=Re[hi} and mlm[Ri]\Im[Hi\=:im[hi},

variance, 4e[i?1]|*e|//i]=Ke[/ll] a n d 4m[fli]|/m[/fi]=/m[hi]» a n d conditional pdf's as derived

in Appendix A . The joint conditional pdf is

PReiRi}im{Ri}(R4ri\^ Imlri} I Re[Hi] = Re[hi], Im[Hi] = Im[h^,Si = Si)

= Pite[/y (fle[ri]|J?e[#i] = i?e[ / i*] , $ = s )

•Pjm[fli](^[rj]|/m[ffi] = i m [ / i * ] , Si = st). (3.20)

We assume that a codeword of length L is transmitted. Let h denote the es­

timated channel gain sequence, s denote the transmitted sequence corresponding to

the codeword and r denote the received sequence. The V D selects a sequence s that

maximizes the value of p R(r| H = h, S = s). The negative log likelihood function is

logp R (r| H = h,S = s) L - l

= l°gPRciRi]imiRi](Re[ri\i / m N I Re[Hi] = Re[hi], Im[Hi] = Im[hi], St = Si)

- (Re[rt] - SiRelhi}^ / (2(0% - ^) exp

i=0

exp /m[rj - sjmm^j / (2(0% -

L - l

= £ ^ ^Re[ri] - SiRe[hi]^f- ) + ( Imfri] - Silm[hi]^f-

i=0

\

- log

1 ) (3.21)

Thus the optimal bit metric for unequal variances is

2

M(ri,hi,Si) = Re[ri] - SiRelhil^j + ^Irn[ri} - Silm[hi]^f-

- m

- l o g —

It can be simplified as

(3.22)

M(ruhi,Si) = (-Re[ri]Re[hi\si - Im[ri]Im[hi]si). (3.23)

For equal variances, the optimal bit metric for the V D in Rayleigh fading with channel

estimation error and A W G N , for P S K and Q A M is derived in [31] as

M{ri,hi,Si) = \Re\n} - RelhiYs^j + [lm[ri] - Im[hi]§i^J . (3 .24)

B y further removing the common terms, it is simplified as

M(ri,hi,Si) = -Re[ri]Re[hi]si - Im[ri]Im[hi]si. (3.25)

A n d the path metric is

L-l

M ( r ,h , s ) = ^2M(ri,hi,Si). (3.26) i=0

The equal variances metric is independent of the actual estimation error variance

value, and is the same as that for perfect channel estimation. We also refer to (3.24)

as squared Euclidean distance metric.

LU CD

10

10

10

10"

10

10 -6

Unequal variances metric + Equal Variances metric

SNR (dB)

Figure 3.7: Comparison of B E R of V D ' s with unequal variance metric and equal variance metric for convolutional code (33,25,37)

Simulation was performed to compare the B E R of the V D with optimal metric

for unequal variances ( V D u n e q u a i ) and that of the V D with optimal metric for equal

variances ( V D e q u a i ) . Here we use a convolutional code whose generator sequence in

octal form is (33,25,37) [17]. This is a rate | code with a free distance dfree of 12 [40]

and is used in 3 G P P systems [6] .The implementation of the simulation is based on the

MATLAB® example [41, function vitdec] which is capable of convolutional encoding

and Vi terb i decoding in an A W G N channel. The ability to decode in frequency-

nonselective slow fading channel is added to the V D . The simulation program is also

built as a standalone binary file to save MATLAB® license usage and to accelerate

program execution.

We assume that o\. is changing according to (3.10) which corresponds to the

2-tap linear interpolation, and adjacent samples of are independent. This can

be achieved by interleaving. F ig . 3.7 shows the simulation results for V D u n e q u a i and

V D e q u a ] . It can be seen that the two V D ' s have more or less the same B E R perfor­

mance. This finding agrees with that in [7].

3.4 Upper and Lower Bounds

Derivation of Expressions

The generating function (weight enumerator) of a convolutional code [17, chapter 11]

has the form

oo oo

T(X,Y) = Y,Y,ad<iXdYi (3-27) d=l i = l

where a^j is the number of codewords of weight d that correspond to information

sequences of weight j. B y taking the partial derivative of the generating function

w.r.t. Y and setting Y to 1, we have

dT(X, Y) dY

where

y=i = X > * d (3-28)

d=l

oo

&d = ^J-a-dj (3-29)

is the total number of nonzero information bits associated with codewords of weight

d.

Without loss of generality, we assume that the all-zero codeword is transmitted.

A n upper bound on the B E R of the V D for a (n, k) convolutional code is obtained

in [30] as -. oo

Pb < T Y\ bdPd (3.30) k 4 i

where djree is the free distance of the convolutional code, {bd} are the coefficients

from (3.29), Pd is the P E P that a weight d path accumulates a higher metric than

the all-zero path. A lower bound on the B E R of the V D is obtained in [17, chapter

12],

pb > \PdSree. (3.31)

We assume that the variances of the channel estimation error are equal over a

frame, and use the metric in (3.24), so that

d-1 d-1

Pd = P{J2M(Ri>Hi'Si = - 1 ) > J 2 M ( R " H i ^ i = 1^ i=0 i=0 d-1

= P{J2(-MPi}Re[Hi) ~ ImiRijImlHi}) i=0

d-1 > ^(ReiRilReiHi] + J m ^ / m ^ ] ) }

i=0 d-1

= PiY^iReiRilRelHi] + Im[Ri]Im[Hi\) < 0}. (3.32) i=0

Although a numerical evaluation of Pd is given in [31] as a sum of residues of

the Laplace transform of a cumulative pdf, the effect of the channel estimation error

is not clear. Here we derive a closed form expression for Pd, by first deriving the

conditional probability

P j^0Re[^]#e [# i ] + /m[f l i ] /m[tf i ] ) < 0\{Re[Hi] = Reiki}}, {Im[Hi\ = J m ^ ] } J ,

then averaging over {Re[Hi}} and {Im[Hi\}, as follows.

Let

d-l

ud = ^{Re[hif + Im[hif). (3.33) i=0

Since { i?e[/ij]} and {Im[hi}} are samples of i . i .d . Gaussian rvs with zero-mean and

variance a\ as derived in (3.19), ud is a sample of a chi-square-distributed rv with 2d

degrees of freedom. The pdf of the corresponding rv, Ud, is

p " > d ) = ^ 2 ^ r ( d ) u ' " l e " " / 2 ^ ' U d ~ ° - ( 3 - 3 4 )

Given Re[hj\, Im[hi] and S j , Re[ri\ and i ra [r j ] are samples of Gaussian rvs with

means, m^R^iuiH^R^hi] and mim[Ri]\Im[Hi]=Im[hi], and variance, vRe[Ri]lRe[Hi]=Re[hi]

and o'|m[/ei]|/m[f/i]=/m[/lj]) a s derived in Appendix A . Assuming that all-zero codeword

is transmitted, Si = 1, then mRe[Ri]\Re[Hi]==Re[hi], m / m[H i]|/m[// i]=/m[/ l i], crL[Ki]|Ke[//i]=fle[/ii]

a n d <7/m[fli]|/m[Hi]=/m[hi] become

2 ^He[Ri]|He[//i]=He[/ii] = i?e[ / l i ] - j - (3.35)

^/m[Ki]|/m[Hi]=/m[/ii] = / m f / l j ] — (3.36) 4

CrHe[Ki]|fie[/fi]=fie[^] = a f i ~~ ~2~ (3.37)

<T/m[Ri]|/m[//i]=/m[fti] = ~ ~2~- (3.38)

Let

Re[gi] = Re[ri]Re[hi]

Im[gi] = Im[ri]Im[hi\.

Thus, given Re[hi], Re[gi] is a sample of a Gaussian rv with mean

(3.39)

(3.40)

mRe[Gi]\Re[Hi]=Re[hi] = R&[hi) • "T.«e[Hj]|fle[Hi]=fle[/ii]

•H

and variance

0'iie[Gi]|/te[ifi]=fle[/ii] — Re[hi] • Ofle[/ei]|fle[//i]=.Re[fc;

i i >

similarly, given Im[hi], Im[gi] is a sample of a Gaussian rv with mean

^/m[Gi]|/m[Hi]=fle[/ii] = Im[hi] • m/ T n[ f i i]|/m[// i] = / m[ h i]

= Im[hi] 2UC

and variance

(3.41)

(3.42)

(3.43)

Let d-1

<Tlm[Gi]\Im[Hi]=Iin[hi] ~ I™\hif • 0']m[Ri\\Im[Hi]=Jm[hi

= Im[hi]2[o*R

'H

wd = '^(Re[ri]Re[hi\ + Im[ri]Im[hi}). t=0

(3.44)

(3.45)

Thus given {i?e[/ii]} and {Im[hi]}, % = 0, ...,d - 1, wd is a sample of a Gaussian rv

with mean 2 d-1

mwd ?f J2(Re[hi]2 + Im[hi

" i=0

(3.46)

and variance

d - l

°ia = ri-5)E(*]'+/mw') 'H 4

2 °C

j=0

( a | - - ^ K (3.47)

The conditional probability is then

d - l

P{J^(/?e[i2 i]i2e[// ' i] + Im[Ri]Im[Hi]) < 0 i=0

| {Re[Hi] = Reiki}}, {Im[Hi] = Imfa]}}

= P{Wd < 0 | {Re[Hi\ = Reiki}}, {Im[Hi\ = Jm[^]}} 1 / 1 mwd eric —= 2 V\/2 cr w d

= 2 4 2 J ( 3 - 4 8 )

where erfc(-) denotes the complementary error function.

We then average (3.48) over Ud in (3.34). B y using [42, (8.250) and (8.253)], the

complementary error function can be expressed as an infinite sum:

P , - r - erfc [ 4 / ^ 1 - ud-le-u^dud

P d ~ Jo 2 e r f C i V 2 ( a ^ - a ^ ) j a * W ( ^

7o \ 2 ^ F ^ f } \*{?\O*H-O%<J*H)] ( 2 f c - l ) ( f c - l ) !

a f 2 ^ ( d ) U d 6 m d

2 ^ 2 - r ( d ) 7 0 d

i 0 0

l _ r <#2«T(d) 7 0

2

( 4 ^ fc-1/2 n

V*F£fV ^ 1 2 ( ^ - 4 ^ ) / (2fc - l ) (* - l ) fe=i /•oo

B y using [42, (3.351)], the integrals in (3.49) can be simplified as

1 1 p< - r 5 f 2 % 5 ) ( " - 1 ) ! ( 2 4 ) d

1 oo , 4 N fe-1/2 .,

V^tC \2(°>4H - °h°l) J ( 2 f c - l ) ( f c - l ) !

1 -(k + d - h (2a2H)k+d-1/2. (3.50)

ajf2dT{dy 2'

B y using the properties of the G a m m a function in [42, (8.339), (8.338) and (8.331)]

and letting k' = k — 1,( (3.50) can be further simplified as:

2 y/i T(d) V a%a% - a'c

oo /• 4 N k—1

(fc + d - ^ ) ! (2fc- l ) ( f c - 1)P 2

i r a + d) / 4 2 ^ r (d) V 4 4 - <4

r ( § ) ~ r ( i + fc')ra + d + fc') r - 4 1 r ( j ) - T(l + k')T^ + d + k') f - 4 1

' r ( | ) r ( i + r ( f + A-) 1 4 4 - 4 J

1 r (± + d) 2 ^ T(d) V 4 4 - 4

2 ' 2 ' -'2'o%a%-o*c

1 r a + d) 2 A r ( d ) V ( ^ + ^ ) ( ^ + 4 ) - ^

2 ' 2 ' 2 ' ( 4 + 4 ) ( 4 + 4 ) - 4

1 r ( i + d) / 1 2 • r (d) V ( 4 / 4 + 4 / 4 x 4 / 4 + 4 / 4 ) - 4 / 4

f n 3 - 1 \ 2' 2 + d ; 2 ; ( 4 / 4 + 4 / 4 ) ( 4 / 4 + 4 / 4 ) - 4 / 4

1 _ r ( ^ + d) 2 0 F - T ( d ) y (1 + ESR)(1 + l / S N R ) - 1

f1 1 3 —1 1 ' F {2' 2 + d ; 2 ! (1 + ESR)(1 + l / S N R ) - 1 J ' ( 3 " 5 1 )

where F{-} denotes the hypergeometric function [43, (15.1.1)]. Substituting Pd into

(3.30) and (3.31), we obtain an upper bound and a lower bound on the B E R of the

V D .

It might be noted that Pd in (3.32) is the probability that a sum of products of

correlated zero mean Gaussian rv's is less than zero. One solution is given in [18,

Appendix B] for the probability that a sum of products of correlated nonzero mean

Gaussian rv's is less than zero. However, if applied to zero mean Gaussian rv's, the

expression has a zero denominator. A small modification yields the following result.

d - l / \ / \ k / \ 2d-l

where

vi y/(crh + a2N)((rh + <72E)-crh

^/(1 + 1/SNR)(1 + E S R ) + 1 (3.53) 7(1 + 1 /SNR)(1 + E S R ) - 1

It might also be noted that the metric in (3.26) is a weighted sum of the received

signals, which bears similarity to the decision variable in maximum-ratio-combining

( M R C ) . A n expression for the P E P for M R C with B P S K is given in [44] as

where

7 = (o*c + o%){a* + ol) - of ( 3 ' 5 5 )

A l l three expressions for Pd, i.e., (3.51), (3.52) and (3.54), show that the channel

estimation error and A W G N have exactly the same effect on the B E R of the V D . It

is thus equally important to improve channel gain estimation accuracy as S N R .

SNR (dB)

Figure 3.8: Upper and lower bounds on the B E R of unquantized V D for convolutional code (133,171) as a function of S N R for different values of E S R

Numerical Results

The upper and lower bounds are plotted for two convolutional codes (133,171) and

(33,25,37). The first is a rate \ code with a free distance djree of 10 [40] and is used

in I E E E 802.11a [4] and 802.16 [5] standards; the second was introduced previously

in section 3.3. The coefficients {bd} in (3.28) are known [45]. For our purposes, they

are obtained using the MATLAB® function distspec, up to d = 25. For d > 25, the

bdPd term becomes very small and can therefore be neglected from the upper bound

and lower bound calculations. Since the hypergeometric function is not available in

MATLAB®, the value of Pd is calculated using Maple®. Computer simulations were

also used to validate the upper and lower bounds.

F ig . 3.8 shows the upper and lower bounds on the B E R of the unquantized V D

for the convolutional code (133,171) as a function of S N R , for different values of E S R .

V 1 i ' I l

• ^ - ^ A ESR = -10 dB

-

_ ^ 7 Q ESR = -°° dB _

upper bound

\ x.

X \ .

— — lower bound X ^ \

i 0 5 10 15 20 25

SNR (dB)

Figure 3.9: Upper and lower bounds on the B E R of unquantized V D for convolutional code (133,171) as a function of S N R for E S R = -oo d B and E S R = -10 d B

The B E R performance of the V D degrades rapidly with increasing E S R , especially

when E S R is large. For example, for a target B E R of 1 0 - 2 and using the upper

bounds, there is about 1.7 d B S N R degradation for E S R = -6 d B relative to E S R =

-10 d B , while there is about 8.5 d B S N R degradation for E S R = -2 d B relative to E S R

= -6 d B . We also notice the B E R floors of the V D when channel estimation errors

exist. The error floors increase with increasing E S R . Moreover, the gap between the

upper and lower bounds increases slightly with increasing E S R .

F ig . 3.9 shows the upper and lower bounds on the B E R of the V D with per­

fect channel estimation ( E S R = -oo dB) and E S R = -10 d B for the convolutional

code (133,171) as a function of S N R . It can be seen that, when there is no channel

estimation error, the B E R of the V D does not have an error floor with increasing

S N R .

F ig . 3.10 shows the upper bounds and the lower bounds on the B E R of the V D for

-25 -20 -15 -10 -5 0 ESR (dB)

Figure 3.10: Upper and lower bounds on the B E R of unquantized V D for convolu­tional code (133,171) as a function of E S R for different values of S N R

the convolutional code (133,171) as a function of E S R , for different values of S N R . It

can be seen that the B E R of the V D degrades rapidly with increasing E S R , especially

when S N R is small. Comparing with F ig . 3.8, we notice that these two graphs are

exactly the same except that one is the mirror image of the other, as implied by the

expression of P4.

Fig . 3.11 and F ig . 3.12 show the upper and lower bounds on the B E R of the

unquantized V D for the convolutional code (33,25,37). The bounds are similar to

those in F ig . 3.8 and F ig . 3.10, except that the bounds for the convolutional code

(33,25,37) are lower than those for (133,171). Comparing their upper bounds in

F ig . 3.10 and F ig . 3.12, for the same E S R of -25 d B and same B E R of 10~ 6 , the

unquantized V D requires a S N R of 6 d B for the convolutional code (133,171) while

only 4 d B is needed for the convolutional code (33,25,37). The reason for this is

that the convolutional code (33,25,37) has a larger dfree than the convolutional code

SNR (dB)

Figure 3.11: Upper and lower bounds on the B E R of unquantized V D for convolu­tional code (33,25,37) as a function of S N R for different values of E S R

(133,171).

F ig . 3.13 and F ig . 3.14 show the upper and lower bounds for E S R = 1 /SNR,

which corresponds to P S A M with Sine filter, for the convolutional codes (133,171)

and (33,25,37) respectively. It can be observed that in this case, there is no B E R

floor.

Figure 3.13: Upper and lower bounds on the B E R of unquantized V D for convolu-tional code (133,171) for E S R = l / S N R

SNR (dB)

Figure 3.14: Upper and lower bounds on the B E R of unquantized V D for convolu­tional code (33,25,37) for E S R = 1 / S N R

tr m 10"

10

— upper bound

— lower bound

+ simulation

10 15 SNR (dB)

20 25

Figure 3.15: B E R of unquantized V D for convolutional code (133,171) as a function of S N R

Computer simulations are used to validate the upper and lower bounds. F ig . 3.15

and Fig . 3.16 show the computer simulation results for the convolutional code (133,171)

compared with the theoretical bounds. The bounds agree with the simulation results

very well. A s S N R increases and E S R decreases, i.e., the variances a2N and a\ de­

crease, the upper bounds are very close to the simulation B E R results.

A n approximate B E R of the V D is provided in [17, chapter 12] as

Ph « bd. Pa, •

B y setting S N R to oo in (3.51), the B E R floor is thus

(3.56)

P = h P rb udfree

rdSree |SNR=oo bdfree bdfreT{\+dfree)

• F 1 1 3 - 1 1

(3.57) 2 ^ • ^ ( d / r e e ) ^ / E S R ' 1 2 ' 2 ' ^ e e ; 2 ; E S R ]

The approximate B E R in (3.56) for the convolutional code (133,171) and E S R =

-8 d B is plotted in F ig . 3.17, along with the upper bound, the lower bound and the

-25 -20 -15 -10 -5 0 ESR (dB)

Figure 3.16: B E R of unquantized V D for convolutional code (133,171) as a function of E S R

simulation results. It can be seen that the B E R estimate given by (3.56) is very good

at high SNR ' s .

F ig . 3.18 and F ig . 3.19 show the simulation results, along with the upper bounds,

lower bounds and approximate B E R ' s for the convolutional code (33,25,37). Both

graphs show very good correspondence of the upper bounds and approximate B E R

with the simulation results.

In order to assess the tightness of the upper bounds on the B E R of the unquan­

tized V D for convolutional code (33,25,37), the simulation points where (upperbound

- simulation)/simulation < 0.1 are plotted in F ig . 3.20 and F ig . 3.21 as asterisks. It

can be observed that, for the convolutional code (133,171), when E S R < -8 d B and

S N R > 8 d B , the upper bounds are within 10% of the simulation B E R ; for the con­

volutional code (33,25,37), when E S R < -4 d B and S N R > 4 d B , the upper bounds

are within 10% of the simulation B E R .

SNR (dB)

Figure 3.17: Approximate B E R of unquantized V D for convolutional code (133,171) as a function of S N R for E S R = -8 d B

25 SNR (dB)

Figure 3.18: B E R of unquantized V D for convolutional code (33,25,37) as a function of S N R

-15 -10 ESR (dB)

Figure 3.19: B E R of unquantized V D for convolutional code (33,25,37) as a function of E S R

CO T3, DC CD LU

-15

10 15 20 SNR (dB)

30

Figure 3.20: Region for (upperbound - simulation)/simulation < 0.1 for convolutional code (133,171)

CO

tr C/) LU

10 15 20 SNR (dB)

Figure 3.21: Region for (upperbound - simulation)/simulation < 0.1 for convolutional code (33,25,37)

V D in Rayleigh Fading with

Impulsive Noise

In this chapter, the V D in Rayleigh fading with channel estimation errors is stud­

ied, for Laplacian noise and Gaussian mixture noise. First , the optimal metrics are

derived, and then their B E R performances are studied using simulation.

4.1 Laplacian Noise

In this section, the B E R performance of V D in Rayleigh fading with Laplacian noise

and channel estimation error is studied. The system model is the same as in F ig . 3.2,

except that the real and imaginary parts of the noise rv iV* now have independent

Laplacian distributions defined in (2.23). Since the estimated channel gain is a sum of

rv's as in (2.10), according to Central L i m i t theorem, its pdf approaches a Gaussian

pdf as the interpolation filter grows longer. To facilitate our analysis, the channel

estimation error E{ is thus assumed to be Gaussian distributed and has the same

variance for al l values of i. A histogram of the channel estimation error in Rayleigh

fading and Laplacian noise with a 10-tap Wiener filter is plotted in F ig . 4.1, and

T 1 1 1 I I I

— — Gaussian histogram actual histogram

- 2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Figure 4.1: Histogram of the channel estimation error in Rayleigh fading and Lapla ­cian noise with 10-tap Wiener filter, compared with Gaussian histogram with the same variance

compared with the histogram of a Gaussian rv with the same variance. It can be seen

that the Gaussian distribution provides a good approximation.

The optimal bit metric for Laplacian noise without fading is given in [14] as

M(vi, hi, Si) = \Re[ri] - s*| + \Im[ri] - st\, (4.1)

whereas the optimal metric for fading with perfect channel estimation is

M{n, hi, Si) = \Re[ri] - Re[ci]si\ + \Im{ri} - Irn[ci]si\. (4.2)

We refer to (4.2) as Euclidean distance metric.

The optimal bit metric for Rayleigh fading with channel estimation error is de­

rived in Appendix B as

M{ru hu Si) = f(Re[n], Re[hi], st) + f(Im[ri], Im[hi], Si), (4.3)

4 6 SNR (dB)

Figure 4.2: B E R performance of V D o p t , V D E u c i i d and V D s q u a r e d Euclid, in Rayleigh fading wi th Laplacian noise and Gaussian channel estimation error, as a function of S N R

where

f(ri,hi,Si)

, f f-ria% + 2hisiPal - log < exp{ 2 2 )

•erfc(

0(*2E + 4)

Si(3aEac\/(J2E + a%

•erfc{ rj0a% + rif3a2

c - hiSi(5a2c + o\o2

c, (4.4)

Simulations were performed for three V D ' s (1) V D o p t wi th the optimal metric in

(4.3), (2) V D E u c i i d wi th Euclidean distance metric in (4.2) and (3) V D s q u a r e c i Euclid with

squared Euclidean distance metric in (3.24), for the convolutional code (33,25,37).

-6 - 4 ESR (dB)

Figure 4.3: B E R performance of V D o p t , V D E u c i i d and V D s q u a r e d Euclid, in Rayleigh fading with Laplacian noise and Gaussian channel estimation error, as a function of E S R

The results in F ig . 4.2 show that V D o p t has the best performance as expected.

A t low E S R , the B E R of VDEuciid approaches that of V D o p t ; the performance of

VD S q U a r ed Euclid is about 1 d B worse than that of V D o p t at target B E R of 10~ 6 . This

can be explained by examining the optimal metric and the Euclidean distance metric.

4 6 SNR (dB)

= <

Figure 4.4: B E R performance of V D s q u a r e c l E u c l i d in Rayleigh fading with Laplacian noise and A W G N

Lett ing a\ = 0,

Pi(Re[ri], Re[hi], §i)

— \og{exp(2Re[hi]si) • erfc(oo) + exp(—2Re[hi]si) • erfc(—oo)},

Re[ri] - Re[hi]si < 0

— log{exp(2Re[hi]si) • er/c(—oo) + exp(—2Re[hi]si) • erfc(oo)},

Re[ri] - Re[hi]si > 0

2Re[hi]si - log 2, Rein] - Re[hi]§i < 0

-2Re[hi]si - log 2, Re[r^ - Re[hi]si > 0

p2{Im[ri}, Irn[hi\, st)

2Im[hi]si — log 2, Im[ri] — Im[hi]si < 0

—2Im[hi]si — log 2, Im[ri] — Im[hi]si > 0

(4.5)

(4.6)

10° Euclid, dist. metric in Lap. noise Euclid, dist. metric in AWGN

rr LU

m

N 10

- 7

0 2 4 6 8 10 SNR (dB)

Figure 4.5: B E R performance of V D E u c i i d in Rayleigh fading with Laplacian noise and

Substituting (4.5) and (4.6) into (4.3), the metric only differs from the Euclidean

distance metric by a constant. So that when channel estimation error is small, the

optimal metric is approaching the Euclidean distance metric.

A t high E S R , where the channel estimation error dominates, the performance of

V D o p t and V D s q u a r e c j Euclid are almost the same, while the performance of VDEuciid is

slightly worse.

F ig . 4.3 shows the B E R performance of the V D ' s as a function of E S R . A t low

S N R , VDEuciid almost has the same performance as that of V D o p t ; at high S N R ,

V D s q u a r e d Euclid and V D o p t have the same performance.

It is also interesting to show how V D s q u a r e d Euclid, V D s q u a r e d Euclid and V D o p t per­

form in A W G N compared with Laplacian noise. The B E R performance of V D s q u a r e d Euclid

A W G N

optimal metric for Lap. noise in

J I L

0 2 4 6 8 10 SNR (dB)

Figure 4.6: B E R performance of V D o p t in Rayleigh fading with Laplacian noise and A W G N

is plotted in F i g 4.4. It is shown that at low E S R , i.e., when Laplacian noise domi­

nates, the performance of V D s q u a r e c j Euclid is noticeably degraded by Laplacian noise;

while at high E S R , their performances are similar. The B E R performances of V D E u c ! i d

and V D o p t are plotted in F ig . 4.5 and F ig . 4.6. It can be seen that there is perfor­

mance difference in A W G N and Laplacian noise, for both VDEuciid and V D o p t . The

difference at low E S R is larger than that in high E S R . However, the difference of

VDEuciid and V D o p t at low E S R is much smaller than that of V D s q u a r e d Euclid-

4.2 Gaussian Mixture Noise

In this section, the B E R performance of V D in Rayleigh fading with Gaussian mixture

noise and Gaussian channel estimation errors is analyzed. The system model is the

same as in F ig . 3.2, except that the pdf's of the real and imaginary parts of the noise

rv Ni are independent Gaussian mixture distributed as defined in (2.25).

The optimal bit metric is derived in Appendix C as

where

M{ru hh §i) = - \ogpRe[Ri](Re[ri\ | Re[Hi] = Re[ht}, St = §i)

- logPim[Ri}{Im[ri\ | Im[Hi] = Im[hi], Si = st)

PRe[Ri}{Re[ri) I Re[Hi] = Re[hi],Si = §i)

exp - (Rein] - s.ReM^rf / ( 2 (a£ + a? - ^ ) )

^2^(4 + ffa «E+aC1

exp + e -

(ite[r<] - s ^ h i } ^ ) 2 / ( 2 (a£ + a] - ^ )

Pim{Ri](I™[ri] | Im[Hi] = Im[hi],Si = 54)

exp ( 1 - c ) -

- (lm[n] - « ] ^ ) J / ( % ° h + ^ " ^ ) )

exp +e-

- {lm[ri\ - silrnih,]^)2 J (2(a% + aj - ^ )

(4.7)

(4.8)

(4.9)

Simulations were performed for different Gaussian mixture noise parameters. The

convolutional code (33,25,37) was used. Samples of the Gaussian mixture noise are

generated according to the empirical transform method [46, section 7.2.2.2].

The B E R of the V D with optimal metric in Rayleigh fading with Gaussian mix­

ture noise and Gaussian channel estimation error is plotted in F ig . 4.7, for e = 0.05,

15 SNR (dB)

Figure 4.7: B E R of V D in Rayleigh fading with Gaussian mixture noise and Gaussian channel estimation error, for e = 0.05, a]/a2 = 21 and aj/a2 = 181, as a function of S N R

_ 2 i a n d af/a2 — 181. The pdf's of the Gaussian mixture noise are plotted

in F ig . 2.7. It is shown in F ig . 4.7 that at low S N R , the V D with aj/a2 = 181 has

better performance, and the performance difference increases as E S R decreases. This

is because at low S N R , the threshold over which a noise sample can cause error is low,

say, 1 in F ig . 2.7, and the probability of a noise sample has magnitude larger than 1

is lower wi th tf/tf = 181 than wi th a]/a2 = 21. A t high S N R and high E S R , the

V D ' s in the two environments have the same performance. This is because the effect

of high channel estimation error masks the effect of the Gaussian mixture noise, even

though the pdf of the noise rv with crf/a2 — 181 has a heavier ta i l than that wi th

aj/tf = 21. For high S N R and low E S R , we expect that the V D with a]/a* = 181

wi l l have better performance than that in a]/a2 = 21. Unfortunately, the B E R is too

10"

c f /CT2=181

of / (^=21

- 6 - 4 ESR in dB

Figure 4.8: B E R of V D in Rayleigh fading with Gaussian mixture noise and Gaussian channel estimation error, for e = 0.05, crj/cr2, = 21 and a]/a2 — 181, as a function of E S R

low to be verified by simulation. F ig . 4.8 shows the B E R of the V D as a function of

E S R . A s expected, the channel estimation error causes the B E R performance of the

V D to degrade.

It is also interesting to compare the performance of the V D ' s in Gaussian mixture

noise, Laplacian noise and A W G N . The results with optimal metrics are plotted in

F ig . 4.9. Similar observation to those for F ig . 4.7 can be made. For E S R = 0 , the V D

in Gaussian mixture noise has better performance at low S N R , because of its lower

probability of having large magnitude noise; while at high S N R , al l three V D ' s have

the same performance, because the channel estimation error dominates. W h e n E S R

is low, the V D in Gaussian mixture noise sti l l has better performance at low S N R ;

while at high S N R , the V D in Gaussian mixture noise has worse performance because

Figure 4.9: B E R of V D ' s with optimal metrics in Rayleigh fading with Gaussian mixture noise, Laplacian noise and A W G N , and Gaussian channel estimation error, as a function of S N R

of the heavy ta i l of the Gaussian mixture pdf. The B E R of the V D ' s as a function of

E S R are plotted in F ig . 4.10. As E S R increases, the B E R decreases, as expected.

<r LU

-6 - 4 ESR (dB)

Figure 4.10: B E R of V D ' s with optimal metrics in Rayleigh fading with Gaussian mixture noise, Laplacian noise and A W G N , and Gaussian channel estimation error, as a function of S N R

Conclusion

In previous related research, the P E P of V D on fading channels with channel gain

estimation errors and A W G N was either obtained approximately, or using Laplace

transform. It was generally assumed that the channel estimation errors associated

with different data symbols had equal variances. Channel estimation errors have not

been considered in the study of V D in fading and impulsive noise.

In this thesis, the impact of Gaussian distributed channel estimation errors on the

B E R of the unquantized V D in frequency-nonselective slow Rayleigh fading was stud­

ied, for A W G N , Laplacian noise and Gaussian mixture noise. The main contributions

are summarized as follows.

• The B E R performances of two V D ' s (1) V D u n e q u a i wi th optimal metric for un­

equal channel estimation error variances and (2) V D e q U a i with optimal metric

for equal channel estimation error variances were compared, in an unequal vari ­

ances environment. It is shown that when the frame length M is less than jj^f,

the V D e q u a i has more or less the same performance as V D u n e q U a i -

• Closed-form expressions of the P E P were derived for V D with B P S K in Rayleigh

fading with A W G N and Gaussian channel estimation errors. Upper and lower

bounds on the B E R are then obtained. It is found that channel estimation errors

affect the B E R performance of V D in a way identical to A W G N . Computer

simulation results for two commonly used convolutional codes (133,171) and

(33,25,37) show close agreement with the upper bounds.

The optimal metric was derived for V D in Rayleigh fading with Laplacian noise

and Gaussian channel estimation errors. The B E R performances of the V D ' s

with optimal metric ( V D o p t ) , Euclidean distance metric ( V D E u c i i d ) and squared

Euclidean distance metric ( V D s q u a r e d Euclid) in Rayleigh fading and Laplacian

noise were compared, using simulation. It is shown that when Laplacian noise

is dominant, the B E R of V D o p t and VDEuciid are almost the same; when channel

estimation errors dominate, the B E R of V D o p t and V D s q u a r e d E u ciid are similar.

V D o p t , V D E u c i i d , VD s q u a red Euclid were compared for Laplacian noise versus A W G N

It is found that the performance difference between the two noise models i n ­

creases as the channel estimation error decreases. The performance difference

is typically smallest for V D o p t and largest for V D s q u a r e d Euclid-

The optimal metric is derived for V D in Rayleigh fading with Gaussian mixture

noise and Gaussian channel estimation errors. The B E R performance of the V D

with optimal metric is obtained for different noise parameter values. For fixed

S N R and E S R values, the B E R decreases as the noise becomes more impulsive.

Among related topics for future study are the following:

V D in which quantization is applied.

Extension of the thesis work to other modulation, channel fading, and channel

estimation error models.

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Derivation of the Conditional pdf's

PRe[R^Re^^\Re\.H^ = Re\hii Si = Si)

and

P i m [ R l P m V i ] \ I m [ H i \ = Im[hi], Si = si)

In this appendix, we derive the conditional p d f s p^e[/j i](i?e[rj] | Re[Hi] = Re[hi\, Si =

Si) and PimiRi](Im[ri\ I Im[Hi\ = Im[hi],Si = s,), which are used in obtaining the

expression for M(r{\ hi, s,) in section 3.3 and Pa in section 3.4.

The covariance, C, of Re[Ri) and Re[Hi], given Si = Si, is

C = EiiReiRj-EiRelRiWiRelH^-EiRelHi}})}

= E {{Re[Ci]Si + Re[Ni]){Re[Ci] - Re[Ei})}

= E{Re[Ci]2Si + Re[Ci]Re[Ni] - Re[Ei]Re[Ci]si - Re[Ei\Re[Ni}}

= E{Re[Ci]2Si} + E{Re[Ci}Re[Ni}} - E{Re[Ei}Re[Ci]Sl} - E{Re[Ei]Re[Ni]}

= cr2csi ( A . l )

where E{-} denotes the expectation operation. The correlation coefficient, pi, of

Re[Ri) and Re[Hi], given Si = Si, is

A Pi =

Thus we have

O'RdHi

- ^ L . (A.2) O~RO-HI

PRe[Ri}(Re[ri] I Re[Hi) = Re[hi],Si =

PRe[Ri]Re[Hi](Re[ri], Re[hi] | Si = Si) PRelHi](Re[hi])

exp 2(1 -P?)

fle[rj]2 _ Q » Jfe[rj]fle[/tj] , fle[fc

exp

y/2«(l - p 2 ) ^

VMi - P\)°\

exp - ( i ? e N - ^ g ^ ) 2 / ( 2 ( l - p 2 ) 4 )

exp

v ^ i - P ? K

^ • / (2(1 - ( ^ ) 2 K )

exp - ( f e h ] - Sifie[A,] f ^ ) 2 1 (2(o% - |-))

Given = s,, i?e[#j] = i?e[/ij], i?e[i?j] is a Gaussian R V with mean

^e[«i]|He[Wi]=fle[/i<] = Sj .Re[ / ii ] -y-

and variance

<JRe[-Ri]|fte[/fi]=.Re[/ii] = a R 'Hi

(A.3)

(A.4)

(A.5)

The conditional pdf, mean and variance of Im[Rj\ can be similarly obtained as:

Plm[Ri](Im[ri] I Im[Hi\ = Im[hi],Si = Si)

exp Im[ri] - sjm[hi}^j / ( 2 ( 4 - ^ ) (A.6)

rmim.[Ri)\Im\Hi}=Im[hi) = SiIm[hi]—£- (A.7)

aIm[Ri]\Im[Hi]=Im[hi = a R „2 • a Hi

(A.8)

Derivation of Optimal Bit Metric

for Laplace Noise

The optimal bit metric is derived for the V D in Rayleigh fading with Laplace noise

and channel estimation error.

In the following derivation, Re[] is dropped for convenient, and all upper case and

lower case letters represent the real parts of their corresponding RV ' s and samples.

The pdf of Hi conditioned on d = Q is

PHi(hi | Ci = Ci) = —exp( \ 2% ), (B . l )

the pdf of Rt conditioned on d = Cj and Si = Sj is

PRM I d = a,Si = Si) = jpexP(-r* ^"")' (B-2)

the pdf of Ci is

1 c 2

P^ici) =-7=^-exp{--\). (B.3)

Given Ci = Cj , Ri and Hi are independent. Thus the joint pdf of Ri and Hj conditioned

on Si = Si is

PRi,Hi{ri,hi | Si = Si)

/

oo PRiin I Ci = Ci, Si = §i) • pHi{hi I Ci = Ci) • pCi(ci)dci

-oo

——> oo ^ P V ^ / l ^ w £

._^ e x p (__| ) d Q

1 , r , - c ^ _ (a - hj)2 _

W*E*c J-oo e X P [ P 2 a | 2o*c)a€i

+ I r e x v ( - C i S i ~ r i - ( C i " ^ - -$-)dc-

4nPaEac J-oo 2o\ 2a2c a2

E (3 2a2E p

+ T J T - e ^ ( - i - A ) + c A - | ) + ( - A + ^ ) ) d c , 4irP<JE<Tc Jn/si 2aE 2aC °E P 2aE P

(B.4)

Lett ing A = + E>\ = ~^t~ '% and C\ = -£jr + ^, the first integral of

(B.4) becomes

/

TilSi exp(-Ac2 - BiCi - Ci)dCi

•00

= expi-d + %) exp(-(VAa + ^=)2)dci. (B.5)

Lett ing \[A~Ci + ^= = U, dcj = ^ d £ ; , (B.5) becomes 2Ari + B1a,;

- ^ eM-C^^erM-2-^^!) (B.6) 2vC4 4 A y 2SiVC4

where erfc(-) denotes the complementary error function.

Lett ing B2 = + % and C 2 = t tV - %, the second integral of (B.4) becomes ^ E P E P

/ e x p ( - A c 2 - S 2 C j - C2)dCi J ri/Si

* , n B\ t (2Ari + B2Sj^ exp(-C2 + -—)erfc{— — ) . (B.7) 2 ^ 4 4 A ; 2siVA

Thus

PRi,Hi(ri,hi | Si = §i)

1 f-h2p2 - 2rj0<j% - 2nl3a2c + 2hCsjPo2

c + s2a2Ea2

c^ ~-eXP\ 0 / 3 2 / _ 2 i _ 2 -\ / 4 / 3 ^ v ^ | T ^ " V 2P2(a2

E + a2c)

Si^E^cy/^s + erg. - / i 2 / ? 2 + 2ri0a% - 2n(3a2

c - 2hiSx(3o2c + s2a2

Ea2c.

•-exP( 0 0 2 / - 2 , _ 2 N ) 4/5v / 2T\/a| + a2c y y 2f5\o\ + o2

c)

crfcC^^ + T i f 3 ( j 2 c ~ + g < g g g g ) (B 8)

Using (3.19), the conditional probability

PRi(ri | /fj = hit Si = Si)

= PRi,Hi(ri,hi | Si = §i)

PHM J _ -2rj/?a| - 2rj/3gg. + 2hjSj(ia2

c + •§2ct|ct27 .

4 / ? ^ 2/?2(ai + cT27) j

CTfc(~Ti^a'E ~ r i / 9 g g + hi§i^ac + Ha\ac^ Sif3o-EvcVaE + ac

1 +2n/?a| - 27-i/frg; - 2hiSi/3a2c + s2a2

Ea2c

+ A(3eXP[ W\o\ + o2c) }

Si0<TE<7cy/aE + CTc

Following the analysis in section 3.3, the optimal bit metric is

M(r{, hi, §i) = - logpRe[Ri](Re[ri] | Re[Hi] = Re[hi], Si = Si)

- logpIm{Ri](Im[ri] | Im[Hi] = Im[hi\, Si = s{), (B.10)

and by removing all common terms, it is simplified as

M(rh hu §i) = f(Re[n], Re[hi],Si) + f(Im[n], Im[hi\, s{), ( B . l l )

where

f(ri,hi,Si)

, f ,-riaE + 2hisiPo-2c

• e r / c ( — ,

Si(3aE(Tcy'crE + °c r i t r | - 2hCsil5u2

c

+exP\ 2 , 2^ )

v2 , „n„1 _u.z.R„2 , ^ 2 ^ 2 crfcf E + ^ ~ + a ^ a g ) I (B 12)

Derivation of Optimal Bit Metric

for Mixture Gaussian Noise

The optimal bit metric of the V D in Rayleigh fading with mixture Gaussian noise

and channel estimation error is derived.

In the following derivation, Re[ ] is dropped for convenient, and all upper case and

lower case letters represent the real parts of their corresponding RV ' s and samples.

The pdf of Hi conditioned on Ci = Ci is

PHiihi | Ci = Ci) = —exp(-^ C l ^ ), (C . l )

the pdf of Ri conditioned on Q = q and Si = ii is

PRiin | Ci = Ci,Si = Si)

! - £ / (n-CiSi)2 e (n - c ^ ) 2

exp( —7 ) + exp( —2 ), (C.2) V27rav 2cr2 V27ra/ 2a]

the pdf of Ci is

1 c 2

PCM) = nr- EXP(-7T2-)- ( C - 3 )

Given d = Cj , Ri and Hi are independent. Thus the joint pdf of Ri and Hi conditioned

on Si = §i is

PRi,Hi(ri,hi | Si = Si)

/

oo PRiiXi | Ci = Ci, S, = §i) • pHi(hi | Q = a) • pCi{ci)dci

-oo oo oo

-f{ J —oo k

1 - e , fa - Ci§i)2

exP( ) + oo

1 2ixo~n

2TT

1

2tf 1

27T<7/ ezp( z - 5 ) 2a)

2a\ 1

exp(-2ol

)dci

•ea;p( TTO ) —oo v >?7

,2

2a» 1 « r f - ^ >

v / _ e x p ( - ^ ) d c l

+e 27TCT/ 2 4 2a|

(C.4)

It can be seen that the first integral is actually a joint pdf of two correlated

Gaussian RV ' s , which has been derived in Appendix A , only requires replacing 0 N

with a2. This is also true for the second integral, only requires replacing aff wi th a).

B y modifying (A.3), the conditional pdf is

PRi{n I Hi = hi, Si = Si)

exp ( 1 - 6 ) -

r 2 - ^ ^ ) 2 / ( 2 ( 4 + 4 - ^ ) )

exp +e-

S ^ ^ ) 2 / ( 2 ( ^ + ^ 2 - ^ ) )

Following the analysis in section 3.3, the optimal bit metric is

(C.5)

M(ri,hi,Si) = - logpjfc[^](.Re[rj] | Re[Hi) = #e[^], 5, = Sj)

- logpIm[Ri](Im[ri] I Jm[#i] = Im[hi], St = S j ) . (C.6)