頻率選擇性衰減通道之結合通道估計與錯誤更正的四相相位偏移調變碼之設計

國

立

交

通

大

學

電信工程研究所

碩

士

論

文

頻率選擇性衰減通道之結合通道估計與

錯誤更正的四相相位偏移調變碼之設計

QPSK-modulated Code Design for Combined Channel

Estimation and Error Correction on a Frequency-selective

Fading Channel

研 究 生：陳詒欣

指導教授：陳伯寧

頻率選擇性衰減通道之結合通道估計與

錯誤更正的四相相位偏移調變碼之設計

QPSK-modulated Code Design for Combined Channel

Estimation and Error Correction on a Frequency-selective

Fading Channel

研 究 生：陳詒欣 Student：Yi-Hsin Chen

指導教授：陳伯寧 博士 Advisor：Dr. Po-Ning Chen

國 立 交 通 大 學

電信工程研究所

碩 士 論 文

A Thesis

Submitted to Institute of Computer and Information Science College of Electrical Engineering and Computer Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in

Computer and Information Science

June 2011

Hsinchu, Taiwan, Republic of China

頻率選擇性衰減通道之結合通道估計與

錯誤更正的四相相位偏移調變碼之設計

學生：陳詒欣

指導教授

陳伯寧 博士

國立交通大學電信工程研究所碩士班

摘

要

QPSK-modulated Code Design for Combined Channel

Estimation and Error Correction on a

Frequency-selective Fading Channel

Student: Yi-Hsin Chen kkkAdvisor: Po-Ning Chen

Institute of Communications Engineering National Chiao Tung University

Abstract

In this thesis, we design and examine the QPSK-modulated codes for combined channel estimation and error protection over tfrequency-selective channels. We found that when QPSK modulation is considered, phase information of the channel coefficients is more es-sential than the respective amplitude information, particularly for the blind receiver we are interested in. Under the assumption that the unknown phases are synchronized among differ-ent channel taps, we establish a close-form-expressed union bound for the error performance and later use it as a criterion to search for the optimal code design. Our simulations show that the QPSK-modulated codes can provide an acceptable improvement over the BPSK-modulated codes.

Acknowledgements

I acknowledge Dr. Po-Ning Chen and Dr. Chia-Lung Wu for their encouragement and guid-ance. Without their advice, this work would not have been possible. I would also like to thank my dear laboratory mates. Finally, I give the great respect to my family for their continuing support.

Contents

1.2 Acronyms and Notations . . . 3

2 Preliminaries 4 2.1 System Model . . . 4

2.2 Code Design Criterion . . . 6

2.3 Code Searching Algorithm . . . 9

3 A Channel with Synchronized Phases Among Channel Taps: The Modified

System Model 13

3.1 Impacts of Phase Distortions and Amplitude Distortions . . . 14 3.2 PDFs of the Channel Coefficients of the Modified System Model . . . 16 3.3 Approximate PDFs of the Channel Coefficients of the Modified System Model 17 3.4 The Decoding Criterion of the Modified System Model . . . 19

4 The Simulation Results 21

4.1 System Settings . . . 21 4.2 General Remarks . . . 38

5 Conclusion Remarks and Future Work 39

List of Figures

2.1 The modified system model . . . 11

3.1 E[X4

n] with respect to fh1,h2 via Monte Carlo simulation. . . 19

4.1 The BERs of BPSK-modulated codes using decoder with no/phase/amplitude distortion information. The channel simulated is the Rayleigh-fading channel with P = 1. Here, the codeword lengths examined are N = 4, 8, 12, respec-tively. . . 24 4.2 The BERs of BPSK-modulated codes using decoder without/with phase

dis-tortion information. The channel simulated is the Rayleigh-fading channel with P = 1. Here, the codeword lengths examined are 4, 8, 12. . . 25 4.3 The BERs of QPSK-modulated codes using decoder with no/phase/amplitude

phase distortion information. The channel simulated is the Rayleigh-fading channel with P = 1. Here, the codeword lengths examined are N = 4, 8, 12. . 26 4.4 The BERs of QPSK-modulated codes using decoder without/with phase

dis-tortion information. The channel simulated is the Rayleigh-fading channel with P = 1. Here, the codeword lengths examined are N = 4, 8, 12. . . 27

4.5 The BERs of BPSK-modulated codes using decoder without/with phase dis-tortion information. The channel simulated is the multi-path Rayleigh-fading channel with P = 2. Here, the codeword lengths examined are N = 4, 8, 12. . 28 4.6 The BERs of BPSK-modulated codes using decoder without/with phase

dis-tortion information. The channel simulated is the multi-path Rayleigh-fading channel with P = 2. Here, the codeword lengths examined are N = 4, 8, 12. . 29 4.7 The BERs of QPSK-modulated codes using decoder without/with phase

dis-tortion information. The channel simulated is the multi-path Rayleigh-fading channel with P = 2. Here, the codeword lengths examined are N = 4, 8, 12. 30 4.8 The BERs of QPSK-modulated codes using decoder without/with phase

dis-tortion information. The channel simulated is the multi-path Rayleigh-fading channel with P = 2. Here, the codeword lengths examined are N = 4, 8, 12. 31 4.9 The BERs of BPSK-modulated and QPSK-modulated codes using decoder

without any information on the channels. The channel simulated is the multi-path Rayleigh-fading channel with P = 2, for which the phases of two channel taps are independent. Here, the codeword lengths examined are N = 4, 8, 12. 32 4.10 The BERs of BPSK-modulated and QPSK-modulated codes using decoder

without any information on the channels. The channel simulated is the multi-path Rayleigh-fading channel with P = 2, for which the phases of two channel taps are independent. Here, the codeword lengths examined are N = 4, 8, 12. 33

4.11 The BERs of BPSK-modulated and QPSK-modulated codes using decoder with the information that the phases of two channel taps are synchronized. The channel simulated is the multi-path fading with P = 2, for which the phases of two channel taps are also synchronized. Here, the codeword lengths examined are N = 4, 8, 12. . . 34 4.12 The BERs of BPSK-modulated and QPSK-modulated codes using decoder

with the information that the phases of two channel taps are synchronized. The channel simulated is the multi-path fading with P = 2, for which the phases of two channel taps are also synchronized. Here, the codeword lengths examined are N = 4, 8, 12. . . 35 4.13 The BERs of the (N, N/2) and (N, N/4) BPSK-modulated codes using

de-coder with the information that the phases of two channel taps are synchro-nized. The channel simulated is the multi-path fading with P = 2, for which the phases of two channel taps are also synchronized. Here, the codeword lengths examined are N = 4, 8, 12. . . 36 4.14 The BERs of the (N, N), (N, N/2), (N, N/4) QPSK-modulated codes using

decoder with the information that the phases of two channel taps are synchro-nized. The channel simulated is the multi-path fading with P = 2, for which the phases of two channel taps are also synchronized. Here, the codeword lengths examined are N = 4, 8, 12. . . 37

Chapter 1

Introduction

1.1

Overview

Traditionally, in a communication system over a fading environment, channel estimation, channel equalization and error correction are carried out at the receiver in sequence. The good performance of this typical system then relies heavily on an accurate estimate of channel coefficients. As such, a known training sequence of a nontrivial length has to be employed. However, even with a long training sequence, it might still be hard to yield an acceptably good estimate for channel coefficients under severe channel fades.

In 2002, Skoglund et al. [1] proposed combining channel estimation and equalization with the error correction and found the optimal non-linear binary code for channels with unknown parameters by computer search. Simulations showed that the computer searched code outperforms the Golay (23, 12) code extended with one known pilot by 1.3 dB at WER = 10−2_{. At such background, this thesis will continue the quest along this research}

line, and focus on the code design based on BPSK/QPSK modulations for combined channel estimation and error correction over a fading channel.

In particular, we attempt to design the QPSK modulated codes, rather than the BPSK modulated codes as in [1], because we wish to improve the transmission rate. The

pre-liminary simulations however indicate that the receiver may need to know the phases in order to achieve an acceptable performance for QPSK modulated codes. Yet, we found that by adding a simple assumption such that the phases among different channel coeffi-cients are identical although they are unknown, the performance of the blind receiver can be significantly improved. The problem of this simple assumption is that it does not give a close-form-expressed union bound that can be conveniently used as the criterion for code search. An approximation will then be proposed and showed to be useful in code search.

The remaining parts of the thesis are organized as follows. Chapter 2 introduces the background knowledge about the code design for combined channel estimation and error correction, and gives a modified system model used in subsequent chapters. Chapter 3 assumes that the phases among different channel coefficients are synchronized (i.e., identical), and derives the optimal blind decoding criterion for this modified system model. Chapter 4 presents and discusses simulation results. Chapter 5 concludes the thesis and remarks on the possible future work.

1.2

Acronyms and Notations

The acronyms and some common identifiers used in this thesis are listed as follows. BER bit error rate

WER word error rate ML maximum-likelihood SNR signal-to-noise ratio

N block length of a code, i.e., the number of symbols in a codeword K number of bits in an information sequence to be encoded

P number of channel paths or taps

L L = N + P − 1 is the sum of a codeword length plus the minimum guard interval without interblock interference

The following notations are used in this thesis.

Symbol Meaning

v a vector (The following notations are simple representative examples. Similar notations are applied to other alphabets.)

vk the k-th component of a vector v

kvk2 _{the norm of a vector v}

X a matrix

xk,` the element of a matrix X at row k and column `

XT transpose of a matrix X

XH Hermitian transpose of a matrix X IL the L × L identity matrix

Chapter 2

Preliminaries

In this chapter, we provide the background knowledge about the code design for combined channel estimation and error correction. Specifically, Section 2.1 introduces the typical system model considered in the non-coherent system literature. Section 2.2 talks about the design criterion that is usually adopted in the literature. Section 2.3 gives the algorithm that will be used for code searching. Section 2.4 concludes this chapter by presenting a modified system model that will be used in the subsequent chapters.

2.1

System Model

The system model introduced in this section is precisely the one adopted in [1] except that the code symbols are not restricted to {−1, +1} but can be extended to {−1, +1, −ı, +ı}. In the system considered therein, a length-N codeword b = [b1, . . . , bN]T of an (N, K) block

code C is transmitted through a quasi-static block fading channel of memory order (P − 1); hence, the channel coefficients can be expressed by a P -by-1 vector h. Other than the fading effect, the system suffers an additive white Gaussian noise. This gives a system model as

where B,            b1 0 · · · 0 ... b1 . .. ... bN ... ... 0 0 bN . .. b1 ... ... ... ... 0 0 · · · bN            L×P with every bi ∈ {±1, ±ı}, (2.1)

n is an L × 1 zero-mean complex Gaussian vector with covariance matrix σ2_I

L, and L =

N + P − 1. Here, we adopt the notational convention by denoting the L-by-L identity matrix by IL.

Note that as similar to [1], (P − 1) should be regarded as an upper bound of the true memory order and is the only information that both the transmitter and receiver priorly know. In addition, the channel coefficient vector h, although unknown to both transmitter and receiver, remains constant during the transmission of a codeword. Under these two assumptions, the optimal decoder is the so-called joint maximum-likelihood (JML) decoder:

(ˆb, ˆh) = arg min

(b,h)∈C×CPky − Bhk

2_,

where C is the set of all codewords, and C consists of all complex numbers.

For a given codeword b, it can be derived that the optimal estimation ˆh that minimizes ky − Bhk2 _{is equal to}

ˆ

h= (BHB)−1BHy,

where superscript “H” denotes the Hermitian transpose operation. Hence, the JML estima-tor of the transmitted codeword is given by

ˆ b = arg min b_∈C ky − B(B H_B)−1_BH_yk2 = arg min b_∈C ky − PByk 2_{, (2.2)}

process. Notably, PB and B (equivalently, b) are not one to one correspondence unless the

first bit is fixed. For this reason, we will fix the first bit as b1 = 1 in our code design.

2.2

Code Design Criterion

In this thesis, we attempt to search an (N, K) code C by simulated annealing, which mini-mizes the union bound of the average block error probability Pe defined by

Pe , 2−K X b_∈C Pr ˆb 6= b b transmitted  = 2−KX i∈J Pr ˆb 6= b(i) b(i) transmitted  ,

where, for convenience, we denote by J the set of indices of codewords in C. The union bound of Pe, which we adopt in this thesis, is given by

Pe ≤ 2−K X i∈J X j∈J ,j6=i pj|i, (2.3)

where pj|i is the pairwise error probability (PEP) defined by

pj|i , Pr ˆb = b(j)   b(i) transmitted  . (2.4)

The PEP, according to (2.2), can be represented as

pj|i = Pr ky(i) − PB(j)y(i)k2 < ky(i) − PB(i)y(i)k2

+1

2 Pr ky(i) − PB(j)y(i)k

2

= ky(i) − PB(i)y(i)k2
, (2.5)

where PB(i) = B(i)(BH(i)B(i))−1BH(i) is the projection matrix onto the column space of

B(i) and the received vector y(i) = B(i)h + n is now complex Gaussian distributed with mean my(i) = B(i)mh and covariance matrix Sy(i) = B(i)ShBH(i) + σn2IL. Since y(i) has

density and since our code book satisfies PB(i) 6= PB(j) for i 6= j, the second term in (2.5)

is equal to zero when i 6= j. Thus,

By defining Q(j, i) , PB(j) − PB(i), the PEP in (2.6) can be rewritten as

pj|i = Pr yH(i)Q(j, i)y(i) > 0
. (2.7)

It is reasonable to assume that σ2

n > 0; hence, the real and symmetric matrix Sy(i) is

positive definite and can be factorized to Sy(i) = S1/2y (i)S1/2y (i). We then consider the real

and symmetric L × L matrix

S1/2_y (i)Q(j, i)S1/2_y (i),

and decompose it into sum of outer products of orthonormal eigenvectors {q_n}L

n=1 with

eigenvalues as multiplicative coefficients {λn}Ln=1, i.e.,

S1/2_y (i)Q(j, i)S1/2y (i) = L

X

n=1

λnqnqTn. (2.8)

Without loss of generality, we index these eigenvalues in descending order, namely,

λ1 ≥ λ2 ≥ · · · ≥ λL. (2.9)

By this, together with (2.8), we get

yH(i)Q(j, i)y(i) = S−1/2_y (i)y(i)
H

S1/2_y (i)Q(j, i)S1/2_y (i) S−1/2_y (i)y(i)
= L X n=1 λn  qT_nS−1/2_y (i)y(i) 2 = L X n=1 λn|Xn|2, (2.10) where Xn , qTnS−1/2y (i)y(i)

is complex Gaussian distributed with mean

and variance σ2

Xn = 1. This leads to that {|Xn|

2_}L

n=1are noncentral χ2-variables. Because the

eigenvectors {q_n}L

n=1 are orthonormal, {Xn}Ln=1 are independent and identically distributed

random variables. With (2.7) and (2.10), we conclude that pj|i is the probability that the

weighted sum of these noncentral χ2_{-variables is larger than zero.}

We can further re-formulate (2.10) and obtain

¯ L

X

l=1

λl· χ2(2kl; ηl), (2.11)

where ¯L is the number of distinct eigenvalues in {λn}Ln=1, kl is the multiplicity of the

eigen-value λl, and χ2(2k; η) stands for a noncentral χ2-variable with 2k degrees of freedom and

noncentrality parameter η. Condition (2.9) immediately gives λ1 > λ2 > · · · > λ_L¯.

Denote for convenience the random variable

Yi,j , yH(i)Q(j, i)y(i)

with its characteristic function φi,j(t) = E[eı tYi,j]. Since Yi,j consists of ¯L independent χ2

-variable, φi,j(t) is given by

φi,j(t) = ¯ L Y l=1 1 − 2ıλlt
−kl/2 exp ı ¯ L X r=1 ηrλrt 1 − 2ıλrt ! . (2.12)

We then note that pj|i can be expressed in terms of φi,j(t) according to

pj|i= 1 2π Z ∞ 0 Z ∞ −∞

φi,j(t)e−ıtydt



dy. (2.13)

In general, (2.13) dose not have a closed-form formula. Nevertheless, when a codeword is transmitted over the Rayleigh fading channel with mh = 0, the closed-form expression

exists. By assuming that λp > 0 ≥ λp+1 and letting q, P ¯ L l=1kl, (2.13) can be solved [3] as pj|i= p X l=1 1 (kl− 1)!  ∂kl−1 ∂xkl−1Fl(x)  x=λl , (2.14)

where

Fl(x) = xq−1

Y

1≤r≤ ¯L,r6=l

(x − λr)−kr.

Consequently, under Rayleigh fading channel with mh = 0, from [3], the design criterion

(2.3) can be evaluated via (2.14).

2.3

Code Searching Algorithm

In this subsection, we will introduce the simulated annealing algorithm and later will apply it for code searching.

A typical simulated annealing algorithm follows the below procedure:

Choose initial code J and initial temperature T . REPEAT

REPEAT

Chose another code J0_.

Set ∆ = (J0_{) − (J ).}

IF (∆ < 0)

THENSet J = J0_.

ELSE With probability p, set J = J0_.

UNTIL (Reach a certain number of energy drops or too many iterations.) Set T = αT.

UNTIL (Reach the targeted freezing temperature.)

The detail of the above algorithm, specifically for our code search, is given below.

in alphabetical order.

• This system is heated initially and then cooled down, for which the initial temperature is T = 107 _{and the targeted freezing temperature is 10}−7_.

• In the inner loop, a random code perturbation is implemented. The new code J0 _is

almost the same as J . The only difference between them is that one of the codewords in J , drawn in random, will be replaced by a randomly picked word originally outside J .

• The energy function (·) of this system is the union bound in (2.3) without the multi-plicative constant 2−K_{, i.e.,}

(J ),X

i∈J

X

j∈J ,j6=i

pj|i,

which can be evaluated via (2.14).

• When ∆ < 0, the code J0 _{performs better (i.e., has a smaller union bound) and hence}

the new code J0 _{will substitute the original one J ; such case will be referred to as}

A_{-perturbation.}

To avoid falling into a local minima, replacing the old code J by the new code J0

is still conducted with probability p = exp(−∆_T ) when ∆ ≥ 0. Such a replacement when it is conducted will be referred to as B-perturbation.

• The inner loop ends if more than 5 A -perturbations occur, or more than 500 B-perturbations is reached.

• Following [1], we set α = 0.995.

We conclude the subsection by pointing out that after the code is selected, the encoder will map the uniformly distributed information messages to the codewords in alphabetical order.

In other words, the first information sequence in its own alphabetical order will correspond to the first code word also in its respective alphabetical order, the second information sequence will be mapped to the second codeword, etc. Simulations show that such simple mapping will result in a BER that is comparible to the best mapping obtained by simulated annealing [4]; hence, we adopt this simple alphabetical-order-based mapping for ease of our system implementation.

2.4

A Modified System Model Presumed in This

The-sis

In reality, the channel coefficients h = [h1, h2, . . . , hP]T may not be totally independent. As

an example for a channel with two taps as shown in Figure 2.1, their phases (that incur from their delays in signal traveling over two different major paths) should have constant differ-ence. So the traditionally convenient assumption that h1 and h2 are independent complex

variables may not be reasonable. In particular, such assumption would make the noncoherent detection of the QPSK signals somehow infeasible. Thus, we modify the system model intro-duced in Section 2.1 in this subsection (and also in the sequel) by assuming that the phases of all channel taps have constant difference while their amplitudes remain independent. For simplicity, we further assume these differences are all zero.

Transmitter · · · , bn−1, bn, · · · Receiver · · · , yn−1, yn, · · · -Path 1 H j Path 2

Figure 2.1: The modified system model

convenience of its description, we let P = 2. With the new setting, the reception at time n can be obtained as

yn = h1· bn+ h2· bn−1

= a1eıθ· bn+ a2eıθ · bn−1. (2.15)

Notice that h1 = a1eıθ and h2 = a2eıθ are no longer independent Gaussian distributed

even if θ is assumed independent of both a1 and a2. In our analysis, these parameters are

assumed to be unknown constants during the transmission of a codeword. Nevertheless, in later simulations, a1 and a2 will be assumed independent Rayleigh distributed, and θ that is

Chapter 3

A Channel with Synchronized Phases

Among Channel Taps: The Modified

System Model

In this chapter, we will start from Section 3.1, which points out by a simple example that for a blind receiver for QPSK modulations, the phase distortion affects the performance more than the amplitude distortion does. This leads to a straightforward inference that the receiver may need to know the phases in order to yield an acceptable performance. Yet, continuing the discussion along this line, we found that by adding a mild assumption such that the phases among different channel coefficients are synchronized (i.e., identical), the performance of a blind receiver can already be adequately improved even without knowing (estimating) the exact value of these phases. It thus suffices to consider the modified system model introduced in Section 2.4.

Since channel coefficients of the modified system model are no longer independent to each other, some discussion on the statistics of the modified system model becomes necessary, which will be given in Section 3.2. To find the best code design based on the minimization of the union bound, an evaluable formula for the pairwise error rate is needed, which requires an approximate statistics of the modified system model; this is the focus of Section 3.3. We

end this chapter by deriving the optimal blind decoding criterion for the modified system model in Section 3.4.

3.1

Impacts of Phase Distortions and Amplitude

Dis-tortions

In this section, we will demonstrate that the phase distortion will affect the performance more than the amplitude distortion does. For simplicity and clarity, we let P = 1. Cases with larger P should have similar behavior; hence, we omit them.

First, we consider the case that the phase distortion ϑ , eı(∠h) _{= e}ıθ _{is unknown while}

the amplitude distortion hm , |h| is known at the receiver, where h = |h|eıθ = hmϑ is the

complex-valued channel coefficient. Since hm is known at the receiver, the joint

maximum-likelihood (JML) decoder becomes

(ˆb, ˆh) = arg min

(b,ϑ)∈C×C:|ϑ|=1ky − Bhmϑk

2_{, (3.1)}

where C is the set of all codewords, and C consists of all complex numbers. Denoting the L-by-1 matrix Bhm by A, we get

ky − Aϑk2 = (y − Aϑ)H(y − Aϑ)

= kyk2− yHAϑ − ϑ∗AHy+ ϑ∗AHAϑ = kyk2− ϑyHA− ϑyHA
∗+ |ϑ|2AHA = kyk2− 2ReϑyH_{A + A}H_A.

This results in that

ˆ ϑ = arg min ϑ∈C:|ϑ|=1ky − Aϑk 2 should satisfy ˆ ϑyHA= |yHA|.

I.e., ˆ ϑ = A H_y |AH_y| = BHy |BH_y|. (3.2)

Via (3.2), the decoder in (3.1) turns to ˆ b = arg min b_∈C ky − Bhm ˆ ϑk2 = arg min b_∈C y− Bhm BHy Nhm 2 (3.3)

Next, consider the alternative case that the amplitude distortion hm is unknown but the

phase distortion ϑ is known at the receiver. Similarly, denoting Bϑ by D, we get ky − Dhmk2 = (y − Dhm)H(y − Dhm)

= kyk2− (yHD+ DHy)hm+ kDk2h2m. (3.4)

Hence, the estimate of the amplitude distortion, which minimizes (3.4) subject to hm > 0,

is given by ˆhm =  yH_{D+ D}H_y 2kDk2 + =  y H_{Bϑ + ϑ}∗_BH_y 2kBϑk2 + =  e ıθ_yH_{B+ e}−ıθ_BH_y 2N + , (3.5)

where {x}+ _{, max{x, 0}, and the last step follows from |ϑ|}2 _{= 1 and kBk}2 _{= N. Following}

(3.5), we obtain ˆ b = arg min b_∈C ky − Bϑˆhmk 2 = arg min b_∈C y− 1 2NB(e ı2θ_yH_{B+ B}H_y) 2 . (3.6)

With the availability of (3.3) and (3.6), we proceed to compare the resultant performances of the two decoders. Simulation shows that knowing only the amplitude distortion has no

performance gain. In other words, the knowledge of the phase distortion is more critical for a blind receiver. Notably, as will be shown in subsequent chapter, (3.6) will lead to a performance improvement over the criterion in (2.2).

For a practical system, it may be too idealistic to assume that the phase distortion is known or can be accurately estimated at the receiver unless a certain non-trivial implemen-tation cost for channel estimation is permissible. We then found that by assuming that the phases of different channel taps are synchronized as the modified system model has assumed in Section 2.4, an evident improvement over (2.2) can be obtained. This somewhat justi-fies the necessity of the provision of the modified system model. As aforementioned, the respective decoding criterion for the modified system model will be given in Section 3.4.

3.2

PDFs of the Channel Coefficients of the Modified

System Model

In this section, the probability density functions (PDFs) of the channel coefficients of the modified system model introduced in Section 2.4 are discussed.

In this scenario, the simplest case shall be P = 2 because we need at least two channel taps to synchronize their phases. Assume that the amplitude distortion hm1 and hm2 are independent Rayleigh distributed with mean

q

πτ2

2 , where τ2 = σ2

2 , and the phase distortion

θ is uniformly distributed over [−π, π) and independent of both hm1 and hm2. By these assumptions, the marginal PDfs of these parameters can be written as follows.

fhm1 (u) = u τ2e −u2 2τ 2, u ≥ 0 fh_m2(v) = v τ2e −_{2τ 2}v2 _{, v ≥ 0} fθ(θ) = 1 2π, −π ≤ θ ≤ π

When denoting the two channel coefficients as h1 = x1+ ıy1 and h2 = x2+ ıy2, we get

x1 = hm1cos θ1,

y1 = hm1sin θ1, x2 = hm2cos θ2, y2 = hm2sin θ2,

where θ1 = θ2 = θ with probability one. By the independence of hm1, hm2 and θ, we derive

fh1,h2(ue ıθ1_{, ve}ıθ2_{) = f} hm1,hm2,θ1,θ2(u, v, θ1, θ2) = u τ2e −u2 2τ 2 · v τ2e −v2 2τ 2 · 1 2πδ(θ1− θ2). (3.7) Through the Jacobian transformation, we get

fx1,y1,x2,y2(x1, y1, x2, y2) = 1 2πτ4e −x 2 1+y 2 1+x 2 2+y 2 2 2τ 2 δ  tany1 x1 − tany2 x2  . (3.8)

Since (3.8) does not lead to a close-form formula for the pair-wise error rate for the modified system model in Section 2.4, we will seek an approximate to it in the next section.

3.3

Approximate PDFs of the Channel Coefficients of

the Modified System Model

When the channel coefficients are zero-mean independent Gaussian distributed, the pairwise error rate exhibits a close-form expression as have been derived in (2.14). In order to use this close-form expression, a zero-mean Gaussian approximate fh1,h2 to (3.8) is needed.

A straightforward approximate to fh1,h2 in (3.7) is to make the zero-mean independent Gaussian fh1,h2 having the same covariance matrix as fh1,h2. We then derive based on fh1,h2

that E [h1h∗2] = Z C Z C ab∗fh1,h2(a, b)dadb = Ehm1 · e jθ_{· h} m2 · e −jθ = E [hm1 · hm2] = E [hm1] E [hm2] ; (3.9)

hence, the covariance matrix due to fh1,h2 is given by

Sh =  σ2 πσ2 4 πσ2 4 σ 2  . (3.10)

Accordingly, we adopt fh1,h2 ∼ N(0, Sh) as the approximate PDF to fh1,h2.

Since the pairwise error probability (PEP) in Section 2.2 is determined by the statistics of Xn, we examine how well fh1,h2 approximate fh1,h2 by analyzing the moments of Xn in the sequel. Apparently, the second moments of Xn due to both fh1,h2 and fh1,h2 are the same. We then proceed to compare the fourth moment of Xn.

From [5], an elementary result for a zero-mean complex Gaussian random variable Z is that

EZ2m_{ = m! · E}m_|Z|2_{ .}

Therefore, the fourth moment of Xn from (2.10) with respect to the zero-mean Gaussian

fh1,h2 is

EX4

n = 2, (3.11)

since E[Xn] = 0 and E[Xn2] = 1 under PDF fh1,h2. The fourth moment of Xn with respect to fh1,h2 however does not have a close-form formula. Thus, we numerically obtain it via Monte Carlo simulation and summarize the results in Figure 3.1.

We can then conclude from the figure that the fourth moments of the precise fh1,h2 and the approximate fh1,h2 are almost equal for N = 4, 6 and 8. Therefore, the PEP formula

2 4 6 8 10 12 14 1 1.5 2 2.5 3 3.5 4 E_b/N₀ in dB E[X n 4] E[X_n4] for N=4 E[X n 4 ] for N=6 E[X_n4] for N=8 Figure 3.1: E[X4

n] with respect to fh1,h2 via Monte Carlo simulation.

derived in Section 2.2 should be a good approximate to the true PEP when the precise PDF fh1,h2 of the channel coefficients are used.

3.4

The Decoding Criterion of the Modified System

Model

We end this chapter by providing the optimal decoding criterion for the modified system model.

Consider without loss of generality the case of P = 2. The channel coefficient vector h= [h1 h2]T of the modified system model is given by

h= hmϑ, (3.12)

where hm = [hm1 hm2]

T _{is amplitude distortion vector and ϑ , e}ı(∠h1) = eı(∠h2) is phase distortion.

By the formula below, min (b,h)∈C×CPky − Bhk 2 _{= min} (b,hm,ϑ)∈C×C2×[−π,π):hm≥0 ky − Bhk2 = min b_∈C h_m,∈Cmin2_:h m≥0 min ϑ∈[−π,π)ky − Bhk 2_,

we can find the minimizers for hm and ϑ in sequence as follows.

ˆ ϑ = (hH_mBHBhm)−1hHmBHy, ˆhm1 = ( _L X n=1 (y_n∗gn,1+ yngn,1∗ ) − ˆhm2 L X n=1 (gn,1g∗n,2+ gn,1∗ gn,2) ), N, ˆhm2 = ( _L X n=1 (y_n∗gn,2+ yngn,2∗ ) − ˆhm1 L X n=1 (gn,1g∗n,2+ gn,1∗ gn,2) ), N,

where gi,j represents the element located at ith row and jth column of the matrix G = B ˆϑ.

With these minimizers, the optimal decoding criterion of the modified system model is straightforwardly given by

arg min

b_∈C ky − Bˆhm

ˆ ϑk2.

Chapter 4

The Simulation Results

In this chapter, we will illustrate the simulation results in Section 4.1, and remarked them in Section 4.2.

4.1

System Settings

In this section, we present the performances of codes designed in the previous chapters. In our simulations, other than the requirement that the channel taps satisfy E[hhH] = (1/P )IP, where IP denotes the P -by-P identity matrix, we will focus on three kinds of

channel setting:

1. The channel is a Rayleigh-fading channel with P = 1.

2. The channel is a Rayleigh-fading channel with P = 2, in which h is a zero-mean, independent and identically distributed complex Gaussian random vector. Note that in this setting, the phases of the two channel taps are independent. This system model is the same as the one used in [1] and [4].

3. The channel is the one we have mentioned in Section 2.4. The channel taps h becomes dependent because the phases of the two channel taps are required to be the same.

The amplitude of h remains independent.

Three kinds of decoders will be examined in our simulations:

1. The decoder does not know both the amplitude distortions and the phase distortions. 2. The decoder does not know the amplitude distortions but knows the phase distortions. 3. The decoder does not know the phase distortions but knows the amplitude distortions. 4. The decoder does not know the amplitude distortions and the phase distortions, but

knows that phases distortions are synchronized.

Figures 4.1–4.4 use codes designed in Sections 2.2 and 2.3. Figures 4.1 and 4.2 (respec-tively, Figures 4.3 and 4.4) illustrate the performances of the first ,the second decoders, and the third decoders for codes based on BPSK (respectively, QPSK) modulations that are transmitted over the first channel model, i.e., P = 1

Figures 4.5–4.8 give the performances of codes designed for the second system model, i.e. P = 2. Likewise, Figures 4.5 and 4.6 (respectively, Figures 4.7 and 4.8) show the performances of the first and the second decoders for codes based on BPSK (respectively, QPSK) modulations.

Figures 4.9 and 4.10 compare the BPSK-based and QPSK-based codes, which are de-signed for and transmitted over the second P = 2 system model but decoded using the second decoder.

Figures 4.11 and 4.12 compare the BPSK-based and QPSK-based codes, which are de-signed for the third P = 2 system model but are transmitted over the third P = 2 channel. The decoder in these two figures is the third one.

for BPSK-based and QPSK-based codes, the performances are summarized in Figures 4.13 and 4.14.

6 8 10 12 14 16 18 10−2 10−1 100 BPSK P=1 N=4 E b/N0 (dB) BER with no info. with phase info. with amp. info.

6 8 10 12 14 16 18 10−2 10−1 100 BPSK P=1 N=8 E b/N0 (dB) BER with no info. with phase info. with amp. info.

6 8 10 12 14 16 18 10−2 10−1 100 BPSK P=1 N=12 E b/N0 (dB) BER with no info. with phase info. with amp. info.

Figure 4.1: The BERs of BPSK-modulated codes using decoder with no/phase/amplitude distortion information. The channel simulated is the Rayleigh-fading channel with P = 1. Here, the codeword lengths examined are N = 4, 8, 12, respectively.

5 10 15 10−2

10−1 100

BPSK P=1 Without Phase Info.

E b/N0 (dB) BER N=4 N=8 N=12 5 10 15 10−2 10−1 100

BPSK P=1 With Phase Info.

E_b/N₀ (dB)

BER

N=4 N=8 N=12

Figure 4.2: The BERs of BPSK-modulated codes using decoder without/with phase distor-tion informadistor-tion. The channel simulated is the Rayleigh-fading channel with P = 1. Here, the codeword lengths examined are 4, 8, 12.

6 8 10 12 14 16 18 10−2 10−1 100 QPSK P=1 N=4 E b/N0 (dB) BER with no info. with phase info. with amp. info.

6 8 10 12 14 16 18 10−2 10−1 100 QPSK P=1 N=8 E b/N0 (dB) BER with no info. with phase info. with amp. info.

6 8 10 12 14 16 18 10−2 10−1 100 QPSK P=1 N=12 E b/N0 (dB) BER with no info. with phase info. with amp. info.

Figure 4.3: The BERs of QPSK-modulated codes using decoder with no/phase/amplitude phase distortion information. The channel simulated is the Rayleigh-fading channel with P = 1. Here, the codeword lengths examined are N = 4, 8, 12.

5 10 15 10−3

10−2 10−1 100

QPSK P=1 Without Phase Info.

E b/N0 (dB) BER N=4 N=8 N=12 5 10 15 10−3 10−2 10−1 100

QPSK P=1 With Phase Info.

E b/N0 (dB) BER N=4 N=8 N=12

Figure 4.4: The BERs of QPSK-modulated codes using decoder without/with phase distor-tion informadistor-tion. The channel simulated is the Rayleigh-fading channel with P = 1. Here, the codeword lengths examined are N = 4, 8, 12.

0 5 10 15 10−3

10−2 10−1 100

BPSK P=2 N=4 Without/With Phase Info.

E b/N0 (dB) BER without info. with info. 0 5 10 15 10−3 10−2 10−1 100

BPSK P=2 N=8 Without/With Phase Info.

E b/N0 (dB) BER without info. with info. 0 5 10 15 10−3 10−2 10−1 100

BPSK P=2 N=12 Without/With Phase Info.

E

b/N0 (dB)

BER

without info. with info.

Figure 4.5: The BERs of BPSK-modulated codes using decoder without/with phase dis-tortion information. The channel simulated is the multi-path Rayleigh-fading channel with P = 2. Here, the codeword lengths examined are N = 4, 8, 12.

0 5 10 15 10−3

10−2 10−1 100

BPSK P=2 Without Phase Info.

E b/N0 (dB) BER N=4 N=8 N=12 0 5 10 15 10−3 10−2 10−1 100

BPSK P=2 With Phase Info.

E b/N0 (dB) BER N=4 N=8 N=12

Figure 4.6: The BERs of BPSK-modulated codes using decoder without/with phase dis-tortion information. The channel simulated is the multi-path Rayleigh-fading channel with P = 2. Here, the codeword lengths examined are N = 4, 8, 12.

0 5 10 15 10−3

10−2 10−1 100

QPSK P=2 N=4 Without/With Phase Info.

E b/N0 (dB) BER without info. with info. 0 5 10 15 10−3 10−2 10−1 100

QPSK P=2 N=8 Without/With Phase Info.

E b/N0 (dB) BER without info. with info. 0 5 10 15 10−3 10−2 10−1 100

QPSK P=2 N=12 Without/With Phase Info.

E

b/N0 (dB)

BER

without info. with info.

Figure 4.7: The BERs of QPSK-modulated codes using decoder without/with phase dis-tortion information. The channel simulated is the multi-path Rayleigh-fading channel with P = 2. Here, the codeword lengths examined are N = 4, 8, 12.

0 5 10 15 10−3

10−2 10−1 100

QPSK P=2 Without Phase Info.

E b/N0 (dB) BER N=4 N=8 N=12 0 5 10 15 10−3 10−2 10−1 100

QPSK P=2 With Phase Info.

E b/N0 (dB) BER N=4 N=8 N=12

Figure 4.8: The BERs of QPSK-modulated codes using decoder without/with phase dis-tortion information. The channel simulated is the multi-path Rayleigh-fading channel with P = 2. Here, the codeword lengths examined are N = 4, 8, 12.

0 5 10 15 10−3 10−2 10−1 100 E b/N0 (dB) BER BPSK, N=4 QPSK, N=4 0 5 10 15 10−3 10−2 10−1 100 E b/N0 (dB) BER BPSK, N=8 QPSK, N=8 0 5 10 15 10−3 10−2 10−1 100 E b/N0 (dB) BER BPSK, N=12 QPSK, N=12

Figure 4.9: The BERs of BPSK-modulated and QPSK-modulated codes using decoder with-out any information on the channels. The channel simulated is the multi-path Rayleigh-fading channel with P = 2, for which the phases of two channel taps are independent. Here, the codeword lengths examined are N = 4, 8, 12.

0 5 10 15 10−3

10−2 10−1 100

BPSK P=2 Unsynchronized Phase Among Channel Taps

E b/N0 (dB) BER N=4 N=8 N=12 0 5 10 15 10−3 10−2 10−1 100

QPSK P=2 Unsynchronized Phase Among Channel Taps

E b/N0 (dB) BER N=4 N=8 N=12

Figure 4.10: The BERs of BPSK-modulated and QPSK-modulated codes using decoder without any information on the channels. The channel simulated is the multi-path Rayleigh-fading channel with P = 2, for which the phases of two channel taps are independent. Here, the codeword lengths examined are N = 4, 8, 12.

0 5 10 15 10−3 10−2 10−1 100 E b/N0 (dB) BER BPSK, N=4 QPSK, N=4 0 5 10 15 10−3 10−2 10−1 100 E b/N0 (dB) BER BPSK, N=8 QPSK, N=8 0 5 10 15 10−3 10−2 10−1 100 E b/N0 (dB) BER BPSK, N=12 QPSK, N=12

Figure 4.11: The BERs of BPSK-modulated and QPSK-modulated codes using decoder with the information that the phases of two channel taps are synchronized. The channel simulated is the multi-path fading with P = 2, for which the phases of two channel taps are also synchronized. Here, the codeword lengths examined are N = 4, 8, 12.

0 5 10 15 10−3

10−2 10−1 100

BPSK P=2 Synchronized Phase Among Channel Taps

E b/N0 (dB) BER N=4 N=8 N=12 0 5 10 15 10−3 10−2 10−1 100

QPSK P=2 Synchronized Phase Among Channel Taps

E b/N0 (dB) BER N=4 N=8 N=12

Figure 4.12: The BERs of BPSK-modulated and QPSK-modulated codes using decoder with the information that the phases of two channel taps are synchronized. The channel simulated is the multi-path fading with P = 2, for which the phases of two channel taps are also synchronized. Here, the codeword lengths examined are N = 4, 8, 12.

0 5 10 15 10−3 10−2 10−1 100 BPSK E b/N0 (dB) BER (4,2) (4,1) 0 5 10 15 10−3 10−2 10−1 100 BPSK E b/N0 (dB) BER (8,4) (8,2) 0 5 10 15 10−3 10−2 10−1 100 BPSK E b/N0 (dB) BER (12,6) (12,3)

Figure 4.13: The BERs of the (N, N/2) and (N, N/4) BPSK-modulated codes using decoder with the information that the phases of two channel taps are synchronized. The channel simulated is the multi-path fading with P = 2, for which the phases of two channel taps are also synchronized. Here, the codeword lengths examined are N = 4, 8, 12.

0 5 10 15 10−3 10−2 10−1 100 QPSK E b/N0 (dB) BER (4,4) (4,2) (4,1) 0 5 10 15 10−3 10−2 10−1 100 QPSK E b/N0 (dB) BER (8,8) (8,4) (8,2) 0 5 10 15 10−3 10−2 10−1 100 QPSK E b/N0 (dB) BER (12,12) (12,6) (12,3)

Figure 4.14: The BERs of the (N, N), (N, N/2), (N, N/4) QPSK-modulated codes using decoder with the information that the phases of two channel taps are synchronized. The channel simulated is the multi-path fading with P = 2, for which the phases of two channel taps are also synchronized. Here, the codeword lengths examined are N = 4, 8, 12.

4.2

General Remarks

Figures 4.1, 4.3, 4.5 and 4.7 clearly illustrate that knowing the phase information at the receiver can yield evident performance gain, but knowing the amplitude information at the receiver yield no gain. For example, it can be seen from Figures 4.1 and 4.3 that when N is fixed as 12, we can obtain respectively 1 dB and 1.2 dB performance gain by providing phase information to the receiver at BER= 10−2_{. When the channel becomes of two paths,}

the performance gains are reduced down to 0.8 and 1 dB as shown from Figures 4.5 and 4.7, respectively. A side observation is that QPSK-modulated codes will yield a little more performance gain than BPSK-modulated codes when the phase information is additionally provided to the receiver.

Next, we compare the performance dfferences between BPSK and QPSK modulated codes when the codes are transmitted over the Rayleigh fading channel (Figure 4.9) and its corresponding modified channel model (Figure 4.11) in which the phases of different channel taps are assumed to be synchronized. By considering the codeword length N = 12 and channel memory order P = 2, the performance difference is about 0.2 dB at BER=10−2 _in

Figure 4.9, and about 0.5 dB at BER=10−2_{in Figure 4.11. This infers that when the phases}

of the channel taps are synchronized, QPSK modulation is more favored.

Finally, we examine the performances of codes with different code rates in Figures 4.13 and 4.14. By fixing the energy possessed by each information bit, we obtain that (N, N/2) codes have the best performance. The reasons are as follows. For code rates smaller than one half, each code bit may have insufficient energy to combat noises, which degrades the performance. For code rates larger than one half, the “distance” between PB pairs decreases;

Chapter 5

Conclusion Remarks and Future Work

In this thesis, we derived an approximation close-form formula for the union bound of the error performance for QPSK-modulated codes transmitted over a frequency-selective fading channel and demodulated by a blind receiver. Based on the criterion, we then searched for good QPSK-modulated codes for combined channel estimation and error protection by computers. Simulations however show that the blind receiver of QPSK-modulated codes may need to know the phase information of the channel taps in order to obtain an acceptable coding gain over the BPSK-modulated codes.

By adding a simple assumption that phases among different channel taps are synchronized even if they are unknown, a significant improvement in performance can then be observed. To be specific, our simulations show that QPSK-modulated codes perform 0.5 dB better than BPSK-modulated codes when BER = 10−2 _{and codeword length N = 12.}

Several issues can be further studied. The first one is to find a systematic code design for our simulated-annealing-based computer-searched codes. The second issue is to establish a decoding scheme for such codes with low decoding complexity. Other than these two, further analysis of the bizarre effect of code rates as have been reported in Section 4.2 could also be worth of further study. Extensions of the QPSK modulations to QAM modulation in order

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