高速移動單輸入單輸出/多輸入多輸出正交分頻多工系統的子載波間干擾消除:低複雜度演算法及效能分析

國 立 交 通 大 學

電信工程學系

博 士 論 文

高速移動單輸入單輸出/多輸入多輸出正交

分頻多工系統的子載波間干擾消除:

低複雜度演算法及效能分析

ICI Mitigation for High-mobility SISO/MIMO

OFDM Systems: Low-complexity Algorithms

and Performance Analysis

研 究 生：許 兆 元

指導教授：吳 文 榕

高速移動單輸入單輸出/多輸入多輸出正交分頻多工系

統的子載波間干擾消除:低複雜度演算法及效能分析

ICI Mitigation for High-mobility SISO/MIMO OFDM

Systems: Low-complexity Algorithms and Performance

Analysis

研 究 生：許兆元

Student:

Chao-Yuan

Hsu

指導教授：吳文榕 博士

Advisor:

Dr.

Wen-Rong

Wu

國立交通大學

電信工程學系

博士論文

A Dissertation

Submitted to Institute of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Communication Engineering

Hsinchu, Taiwan

高速移動單輸入單輸出/多輸入多輸出正交

分頻多工系統的子載波間干擾消除:低複雜

度演算法及效能分析

研究生：許兆元 指導教授：吳文榕 博士

國立交通大學

電信工程學系博士班

摘要

在正交分頻多工系統中，一個基本的假設是在一個正交分頻多工符元時間內通道 是靜止不變的。然而，在高速移動的環境下，這假設就不再成立了。因此會造成 子載波間干擾且使系統效能降低。強制歸零(zero-forcing, ZF)及最小均方差 (minimum mean square error, MMSE)等化器是兩個簡單的干擾消除方法。不幸

的，強制歸零等化器需要執行 N N× 子載波間干擾矩陣的反矩陣運算，此處

N

是 正交分頻多工系統的子載波數目。當子載波數目變大時，計算複雜度將會變得很 高。對於最小均方差等化器，除了子載波間干擾矩陣的反矩陣運算之外，還需要 一個矩陣與矩陣的乘法運算。這將使得最小均方差等化器的複雜度變得比強制歸 零等化器還高。在本論文中，我們首先提出一個低複雜度的強制歸零等化器來解 決單輸入單輸出正交分頻多工系統中的問題。主要的概念是探究子載波間干擾矩 陣的特殊結構及應用牛頓反矩陣疊代法。依據我們的演算法結構，快速傅立葉轉 換(fast Fourier transform, FFT)可以被結合到疊代過程中，進而使得複雜度

從 3

( )

O N 降到O N( log₂N)。此外，疊代次數約略為一或兩次。我們亦分析所提

方法的收斂行為及推導其訊號干擾雜訊比(signal to interference noise ratio, SINR)。對於最小均方差方法，我們首先改寫其數學表示式，使矩陣與矩陣的乘 法運算可以被避免。與先前提出的低複雜度強制歸零方法相似，我們也將探究子 載波間干擾矩陣的特殊結構及應用牛頓反矩陣疊代法來降低最小均方差方法中 反矩陣的高運算複雜度。在多輸入多輸出正交分頻多工系統中，強制歸零及最小 均方差等化器所需的複雜度問題將變得更難以解決。有鑑於此，我們將延伸在單 輸入單輸出正交分頻多工系統中所提出的演算法至多輸入多輸出正交分頻多工 系統中。這樣的延伸應用，使得所降低的複雜度比在單輸入單輸出正交分頻多工 系統中還大。模擬結果顯示，所提出的低複雜度強制歸零及最小均方差等化器效 能跟直接強制歸零(direct ZF)及直接最小均方差(direct MMSE)等化器的效能相 當，但是所需的複雜度卻是大幅降低。最後，我們也將所提的高速移動干擾消除 方法再延伸應用到正交分頻多工存取(OFDMA)上傳系統中並將此概念進一步應用 來消除子載波偏移所引起的干擾。模擬結果顯示，所提方法可以大幅降低所需複 雜度。

ICI Mitigation for High-mobility SISO/MIMO

OFDM Systems: Low-complexity Algorithms

and Performance Analysis

Student：Chao-Yuan Hsu Advisor：Dr. Wen-Rong Wu

Institute of Communication Engineering

National Chiao Tung University

Abstract

In orthogonal frequency-division multiplexing (OFDM) systems, it is generally assumed that the channel response is static in an OFDM symbol period. However, the assumption does not hold in high-mobility environments. As a result, intercarrier interference (ICI) is induced and the system performance is degraded. A simple remedy for this problem is the application of the zero-forcing (ZF) and minimum mean square error (MMSE) equalizers. Unfortunately, the direct ZF method requires the inversion of an N× ICI matrix, where N is the number of subcarriers. When N N

is large, the computational complexity can become prohibitively high. As for the direct MMSE method, in addition to an N× matrix inverse, it requires an extra N N× matrix multiplication, making the required computational complexity higher N

compared to the direct ZF method. In this dissertation, we first propose a low-complexity ZF method to solve the problem in single-input-single-output (SISO) OFDM systems. The main idea is to explore the special structure inherent in the ICI

matrix and to apply Newton's iteration for matrix inversion. With our formulation, fast Fourier transforms (FFTs) can be used in the iterative process, reducing the complexity from O N( 3) to O N( log₂N). Also, the required number of the iteration is typically one or two. We also analyze the convergence behavior of the proposed method and derive the theoretical output signal-to-interference-noise-ratio (SINR). For the MMSE method, we first reformulate the MMSE solution in a way that the extra matrix multiplication can be avoided. Similar to the ZF method, we then exploit the structure of the ICI matrix and apply Newton's iteration to reduce the complexity of the matrix inversion. For a multiple-input-multiple-output (MIMO) OFDM system, the required complexity of the ZF and MMSE methods becomes more intractable. We then manage to extend the proposed ZF and MMSE methods for SISO-OFDM systems to MIMO-OFDM systems. It turns out that the computational complexity can be reduced even more significantly. Simulation results show that the performance of the proposed methods is almost as good as that of the direct ZF and MMSE methods, while the required computational complexity is reduced dramatically. Finally, we explore the application of the proposed methods in mobility-induced ICI mitigation for OFDM multiple access (OFDMA) systems, and in carrier frequency offset (CFO) induced ICI mitigation for OFDMA uplink systems. As that in OFDM systems, the proposed methods can reduce the required computational complexity, effectively.

Acknowledgements

During the Ph.D. program, I would like to show my gratitude to many people. First, I would like to thank my advisor, Prof. Wen-Rong Wu, for his guidance and patience. He spends a lot of time in discussing the problems I encounter in my research, providing valuable suggestions, and teaching me how to write technical papers. In addition to academic research, Prof. Wu also helps me a lot in my daily life. I do learn a lot from Prof. Wu. I really appreciate what he has done for me. I have to say that without Prof. Wu’s help, I cannot complete my Ph.D. degree smoothly. I also appreciate the suggestions of the oral defense committee members.

Second, I am grateful to all the members in Prof. Wu’s lab. for their valuable discussion and help in academic research including Shou-Sheu Lin, Chun-Fang Lee, Yu-Tao Hsieh, Ren-Jr Chen, Hua-Lung Yang, Yinman Lee, Fan-Shuo Tseng, Hung-Dau Hsieh, Chun-Tao Lin, Pei-Ju Chung, and so on. Besides, I want to thank Shou-Sheu Lin for his encouragement and help during the period of the Ph.D. program. I would like to thank all my friends who ever encourage or help me, especially Kuo-Chun Huang and Nai-Fang Hsu.

Finally, I would like to show my deep gratitude to my family, especially my best-loved parents, Hua-Chung Hsu and Mei-Hua Lee, for their support and encouragement in the Ph.D. program period.

Contents

Abstract iii

Acknowledgements v

Contents vi

List of Tables x

List of Figures xiii

1 Introduction 1

1.1 ICI Problem . . . 1

1.2 ICI Mitigation . . . 3

1.3 Proposed Approach . . . 6

1.4 Organization of the Dissertation . . . 8

2 Mobility-induced ICI Mitigation for SISO-OFDM Systems 11 2.1 Signal Model . . . 11

2.2 ZF Method . . . 12

2.2.1 Proposed Newton-ZF Method . . . 13

2.2.2 Derivation of the Initial Matrix . . . 15

2.2.4 Performance Analysis . . . 22

2.2.5 Simulation Results . . . 28

2.3 MMSE Method . . . 36

2.3.1 Proposed Newton-MMSE Method . . . 37

2.3.2 Derivation of the Initial Matrix . . . 40

2.3.3 Complexity Analysis . . . 43

2.3.4 Simulations . . . 45

3 Mobility-induced ICI Mitigation for MIMO-OFDM Systems 51 3.1 Signal Model . . . 51

3.2 ZF Method . . . 53

3.2.1 Proposed Newton-ZF Method . . . 53

3.2.2 Derivation of the Initial Matrix . . . 55

3.2.3 Complexity Analysis . . . 57

3.2.4 Simulations . . . 59

3.3 MMSE Method . . . 64

3.3.1 Proposed Newton-MMSE Method . . . 64

3.3.2 Derivation of the Initial Matrix . . . 65

3.3.3 Complexity Analysis . . . 67

3.3.4 Simulations . . . 69

4 Mobility-induced ICI Mitigation for SISO/MIMO-OFDMA Systems 75 4.1 SISO-OFDMA Signal Model . . . 75

4.1.1 Simulations . . . 78

4.2 MIMO-OFDMA Signal Model . . . 83

4.2.1 Simulations . . . 86

5 CFO-induced ICI Mitigation for OFDMA Uplink Systems 93 5.1 Signal Model . . . 93

5.2 Previous Methods . . . 95

5.2.1 Conventional Method . . . 95

5.2.2 CLJL Method . . . 96

5.2.3 CLJL-PIC Method . . . 97

5.3 ZF Method . . . 98

5.3.1 Proposed Newton-ZF Method . . . 98

5.3.2 Pre-compensation Method . . . 102 5.3.3 Complexity Analysis . . . 102 5.3.4 Performance Analysis . . . 104 5.3.5 Simulations . . . 114 6 Conclusions 127 Bibliography 129

List of Tables

2.1 Complexity comparison among N-ZF, PSE, and direct ZF methods in a SISO-OFDM system. . . 22 2.2 Complexity of the initial matrix calculation for the N-ZF method in a

SISO-OFDM system. . . 22 2.3 Complexity comparison among N-ZF, PSE, and direct ZF methods in a

SISO-OFDM system (

and !

). . . 37 2.4 Complexity comparison between the N-MMSE and direct MMSE methods in a

SISO-OFDM system. . . 45 2.5 Complexity of the initial matrices calculation for the N-MMSE method in a

SISO-OFDM system. . . 45 2.6 Complexity comparison between the N-MMSE and direct MMSE methods in a

SISO-OFDM system ("# , and$&%('*)% ,+  $  )  + ). . . 50 3.1 Complexity comparison between N-ZF and direct ZF methods in a-
.

MIMO-OFDM system. . . 58 3.2 Complexity of the initial matrix calculation for the N-ZF method in a 
/

MIMO-OFDM system. . . 59 3.3 Complexity comparison between N-ZF and direct ZF methods in a-
.

MIMO-OFDM system (

and !

). . . 63 3.4 Complexity comparison between the N-MMSE and direct MMSE methods in a

0
1

3.5 Complexity of the initial matrices calculation for the N-MMSE method in a

0
2

MIMO-OFDM system. . . 69 3.6 Complexity comparison between the N-MMSE and direct MMSE methods in a

0
2 MIMO-OFDM system ("# , and$&%(')% ,+  $  )  + ). . . 74 4.1 Complexity comparison between direct ZF and N-ZF methods in a SISO-OFDMA

system ("#

, % 43

, and 5

). . . 83 4.2 Complexity comparison between direct MMSE and N-MMSE methods in a

SISO-OFDMA system ( , %('  %  43 , and -'   ! ). . . 83 4.3 Complexity comparison between direct ZF and N-ZF methods in a-
.

MIMO-OFDMA system ( , % 53 , and 5 ). . . 91 4.4 Complexity comparison between the N-MMSE and direct MMSE methods in a

6
7 MIMO-OFDMA system (8 , $&%('*)% ,+  $ 3 ) 3 + , and $ -')* ,+  $  )  + ). . . 91 5.1 Complexity comparison among proposed method, CLJL-PIC method, and

di-rect ZF method. . . 104 5.2 Complexity comparison among direct ZF method, CLJL-PIC method, and

pro-posed method when59:

and; 4:

. . . 121 5.3 Complexity comparison of the direct ZF method, the banded ZF method, and

the proposed method when53:

and; <9

List of Figures

2.1 An example of the structure of a banded initial matrix for =8

and % >

. The elements in the shaded area are non-zeros, while the others are zeros. . . . 16 2.2 An example of ?(@ for

"5

and% 

. Note that?(@ overlaps with?(@AB' and

?C@EDF' . . . 18

2.3 SINR analysis of N-ZF method (% G3

and #

) for case 1, where HI G3KJL

and SNR = 35 dB. . . 32 2.4 SINR analysis of N-ZF method (%

G3

and #

) for case 2, where HI G3KJL

and SNR = 35 dB. . . 32 2.5 BER comparison among one-tap FEQ, PSE, N-ZF (%

M3

and 8

), direct ZF, and direct MMSE methods in a SISO-OFDM system;HI = 0.1 and 16-QAM

modulation. . . 33 2.6 BER comparison among one-tap FEQ, N-ZF (%



and 

), direct ZF, and direct MMSE methods in a SISO-OFDM system; HI = 0.1 and 16-QAM

modulation. . . 33 2.7 BER comparison between the direct ZF and MMSE methods using the exact

Jakes and LTV channels in a SISO-OFDM system;HI = 0.1 and 16-QAM

mod-ulation. . . 34 2.8 BER comparison among one-tap FEQ, N-ZF (%



and 

), direct ZF, and direct MMSE methods in a SISO-OFDM system; HI = 0.2 and 16-QAM

2.9 BER comparison among one-tap FEQ, N-ZF (% N

and 

), direct ZF, and direct MMSE methods in a SISO-OFDM system; HI = 0 O 0.2, 16 QAM

modulation, and SNR = 30 dB. . . 35 2.10 Complexity comparison between N-ZF (%

P

, M

, and Q M

) and direct ZF methods in a SISO-OFDM system for various . . . 35 2.11 Complexity comparison between N-ZF (%

P

, M

, and Q SR

) and direct ZF methods in a SISO-OFDM system for various . . . 36 2.12 The structure of a circular band initial matrix is depicted for4

and% 

. The elements in the shaded area are non-zeros, while the others are zeros. . . . 41 2.13 The structure of?(@ is presented for

TS

and % P

. Note that ?(@ overlaps

?C@EDF' for

modulo-VU

(i.e., the relationship is circular). . . 42 2.14 BER comparison among the one-tap FEQ, N-MMSE ($&%'*)%

W+  $  )  + and $ -')* ,+  $  )  +

), and direct MMSE methods in a SISO-OFDM system;HI



3KJ3X

and 16-QAM modulation. . . 47 2.15 BER comparison among the one-tap FEQ, N-MMSE ($&%'*)%

W+  $  )  + and $ -')* ,+  $ X ) X +

), and direct MMSE methods in a SISO-OFDM system;HI



3KJL

and 16-QAM modulation. . . 48 2.16 BER comparison among one-tap FEQ, N-MMSE (%Y'

 %  # , -'   !X , andQ 5R

), and direct MMSE methods in a SISO-OFDM system;HI = 0O 0.2,

16 QAM modulation, and SNR = 30 dB. . . 48 2.17 Complexity comparison between N-MMSE (%Y'

 %  Z , -'   X , and Q !

) and direct MMSE methods in a SISO-OFDM system for various . . . 49 2.18 Complexity comparison between N-MMSE (%Y'

 %   , -'   <3 , and Q 5R

) and direct MMSE methods in a SISO-OFDM system for various . . . 49 3.1 BER comparison among two-tap FEQ, N-ZF (%



and 

), direct ZF, and direct MMSE methods in a 
[

MIMO-OFDM system; HI = 0.05 and

3.2 BER comparison among two-tap FEQ, N-ZF (% 

and 5

), direct ZF, and direct MMSE methods in a\
C

MIMO-OFDM system;HI = 0.1 and 16-QAM

modulation. . . 61 3.3 BER comparison among one-tap FEQ, N-ZF (%



and 

), direct ZF, and direct MMSE methods in a (


SISO-OFDM system; HI = 0 O 0.2, 16

QAM modulation, and SNR = 30 dB. . . 62 3.4 Complexity comparison between N-ZF (%

] , M , and Q M ) and direct ZF methods in a 0
1

MIMO-OFDM system for various . . . 62 3.5 Complexity comparison between N-ZF (%

] , M , and Q SR ) and direct ZF methods in a 0
1

MIMO-OFDM system for various . . . 63 3.6 BER comparison among the two-tap FEQ, N-MMSE ($&%Y')%

,+  $  )  + and $ -'*)* ,+  $  )  +

), and direct MMSE methods in a 6


MIMO-OFDM sys-tem; HI

43KJ3X

and 16-QAM modulation. . . 71 3.7 BER comparison among the two-tap FEQ, N-MMSE ($&%Y')%

,+  $  )  + and $ -'*)* ,+  $ <3 ) <3 +

), and direct MMSE methods in a 
[

MIMO-OFDM system; HI

53KJL

and 16-QAM modulation. . . 72 3.8 BER comparison among one-tap FEQ, N-MMSE (%Y'

 %  > , -'    <3 , and Q 8R

), and direct MMSE methods in a C
^

MIMO-OFDM system;

HI = 0O 0.2, 16 QAM modulation, and SNR = 30 dB. . . 72

3.9 Complexity comparison between N-MMSE (%Y'

 %  Z , -'   SX , and Q !

) and direct MMSE methods in a_
Y

MIMO-OFDM system for various . . . 73 3.10 Complexity comparison between N-MMSE (%Y'

 %   , -'   <3 , and Q 5R

) and direct MMSE methods in a_
Y

MIMO-OFDM system for various . . . 73

4.1 BER comparison among one-tap FEQ, direct ZF, and N-ZF (% `3

, 

) methods in a SISO-OFDMA system;H&I = {0.05, 0.02, 0.04, 0.03} and 16-QAM

modulation. . . 81 4.2 BER comparison among one-tap FEQ, direct ZF, and N-ZF (%

`3

, 

) methods in a SISO-OFDMA system;H&I = {0.05, 0.1, 0.04, 0.03} and 16-QAM

modulation. . . 81 4.3 BER comparison among one-tap FEQ, direct MMSE, and N-MMSE (%'

 %  S3 , -'   M

) methods in a SISO-OFDMA system; H&I = {0.05, 0.02,

0.04, 0.03} and 16-QAM modulation. . . 82 4.4 BER comparison among one-tap FEQ, direct MMSE, and N-MMSE (%'

 %  P3 , -'   a

) methods in a SISO-OFDMA system; H&I = {0.05, 0.1,

0.04, 0.03} and 16-QAM modulation. . . 82 4.5 BER comparison among one-tap FEQ, direct ZF, and N-ZF (%

`3

, 

) methods in a 6
7

MIMO-OFDMA system; H&I = {0.02, 0.05, 0.03, 0.04} and

16-QAM modulation. . . 89 4.6 BER comparison among one-tap FEQ, direct ZF, and N-ZF (%

`3

, 

) methods in a (
^

MIMO-OFDMA system; H&I = {0.04, 0.1, 0.08, 0.07} and

16-QAM modulation. . . 89 4.7 BER comparison among two-tap FEQ, direct MMSE, and N-MMSE (%'

 %  43 , -'   ! ) methods in a_


MIMO-OFDMA system;H&I = {0.02,

0.05, 0.03, 0.04} and 16-QAM modulation. . . 90 4.8 BER comparison among two-tap FEQ, direct MMSE, and N-MMSE (%'

 %  43 , -'   ! ) methods in a_


MIMO-OFDMA system;H&I = {0.04,

0.1, 0.08, 0.07} and 16-QAM modulation. . . 90 5.1 Theoretical average SINR for the proposed method, exact and approximated

(Case 1: CFOs = $ 3KJL )Wb 3KJc )Wb 3KJ3X ) 3KJc +

and SNR = 30 dB; Case 2: CFOs =

$ 3KJ:d ) 3KJ:d ) 3KJLX ) 3KJ: + and SNR = 15 dB). . . 120

5.2 Theoretical average SINR for the proposed method (exact). . . 120 5.3 Theoretical subcarrier SINR for the proposed method (exact; SNR = 30 dB). . . 122 5.4 BER performance comparison for conventional, CLJL, CLJL-PIC, proposed,

and direct ZF methods (16-QAM modulation, and CFOs =$ 3KJL )Wb 3KJc )Wb 3KJ3X ) 3KJc + ).122 5.5 BER performance comparison for conventional, CLJL, CLJL-PIC, proposed,

and direct ZF methods (64-QAM modulation, and CFOs =$ 3KJL )Wb 3KJc )Wb 3KJ3X ) 3KJc + ).123 5.6 BER performance comparison for conventional, CLJL, CLJL-PIC, proposed,

and direct ZF methods (16-QAM modulation, and CFOs =$ 3KJL )Wb 3KJ: )Wb 3KJ3X ) 3KJc + ).123 5.7 BER performance comparison for CLJL-PIC (e

R

), proposed, and direct ZF methods (16-QAM modulation, and CFO of the fourth user increases from 0 to 0.5). . . 124 5.8 BER performance comparison for CLJL-PIC (e

aR

), proposed (Q >

), and direct ZF methods (16-QAM modulation, CFOs = $

3KJL )Wb 3KJc )Wb 3KJ3X ) 3KJc + , and near-far scenario). . . 124 5.9 BER performance comparison for the proposed method with and without PC

(16-QAM modulation, and CFOs = $

3KJ:d ) 3KJ:d ) 3KJL ) 3KJ: + ). . . 125 5.10 BER performance comparison for the proposed method with PC (16-QAM

modulation and CFOs = $fb 3KJL ) 3KJR ) 3KJ: )Wb 3KJc + ). . . 125 5.11 BER performance comparison for the conventional, CLJL, banded ZF, and

di-rect ZF and proposed methods (16-QAM modulation, and CFOs = {0.1, -0.2, -0.05, 0.2, -0.3, 0, -0.1, 0.4, -0.3, 0.05, 0, -0.1, 0.05, -0.1, 0.3, 0.15}). . . 126

Chapter 1

Introduction

§ 1.1 ICI Problem

I

Nwireless single-carrier (SC) communication systems, data transmission occupies the whole

available bandwidth. Due to the multipath channel, a SC system usually suffers from the severe intersymbol interference (ISI). Consequently, a SC system requires a complicated time-domain equalizer to combat the ISI effect. Compared to a SC system, a conventional multi-carrier (MC) system divides the whole available bandwidth into many non-overlapped narrow subchannels and subcarriers are used simultaneously to transmit data on these narrow subchan-nels. Since each data stream is transmitted on a narrow subchannel, it is subject to little ISI which makes the channel equalizer simpler. Moreover, since the data streams are transmitted on independent subchannels, different modulation schemes can be used for the subchannels. Since the subchannels are non-overlapped in conventional MC systems, guard bands are re-quired between these subchannels to avoid inter-channel interference. Owing to the extra guard bands, the conventional MC system is bandwidth-inefficient.

To solve the problem, a bandwidth-efficient MC technique, called orthogonal frequency-division multiplexing (OFDM), was developed. The technique, dating back to the 1960’s, over-laps subchannels in an orthogonal way such that bandwidth efficiency can be greatly improved.

In 1971, Weinstein and Ebert used IDFTs/DFTs to perform the OFDM baseband modulation and demodulation instead of a bank of subcarrier oscillators [1]. This method provides an efficient digital implementation of OFDM systems. In 1980, Peled and Ruiz introduced the concept of cyclic prefix (CP) which fills the vacant guard interval with a cyclic extension of an OFDM symbol [2]. This results in a circular convolution between the transmit data and the channel response. With the CP, OFDM can convert a frequency-selective channel into a set of frequency non-selective channels, and only a one-tap frequency-domain equalizer is required for each subcarrier signal. This greatly reduces the complexity of the channel equalization in the OFDM receiver. Nowadays, OFDM is known to be an effective and successful technique to cope with the multipath channel effect in wireless communications [3]. Since all subcar-rier signals overlap orthogonally in the spectrum, an ideal OFDM system has no intercarsubcar-rier interference (ICI). Thus, OFDM can be easily developed as a frequency-division multiple ac-cess (FDMA) scheme. An OFDM-based FDMA system is generally referred to as an OFDMA system [4], [5]. In an OFDMA system, subcarriers are divided into exclusive groups, and each group is assigned to a user for simultaneous data transmission. The OFDM technique has been adopted in many systems, e.g. Asymmetric Digital Subscriber Line (ADSL), IEEE 802.11a/g, IEEE 802.16e-2005 [6], IEEE 802.16m, 3GPP Long Term Evolution (LTE), Digital Audio Broadcasting (DAB), and Terrestrial Digital Video Broadcasting (DVB-T).

For conventional OFDM systems, it is usually assumed that the channel is static during an OFDM symbol. However, in high-speed mobile environments, this assumption does not hold anymore. If the channel is time-variant in an OFDM symbol period, orthogonality will be destroyed. As a result, ICI is induced and the system performance is degraded. The behavior of mobility-induced ICI has been extensively investigated in the literature [7], [8], [9], [10], [11], [12]. In [7], [8], it is shown that the interference on a subcarrier mainly comes from neighboring subcarriers. Also, the interference level is proportional to the Doppler frequency.

Another factor that affects orthogonality in an OFDM system is carrier frequency offset (CFO). In OFDM systems, CFO is always present due to imperfect oscillators. In the presence

of CFO, the orthogonal property of an OFDM system is also destroyed and the ICI is induced, degrading the system performance significantly [13]. Different from the CFO-induced ICI in OFDM systems, CFO in OFDMA uplink systems causes not only the self-interference but also the multiuser interference (MUI), degrading the system performance even more severely [14], [15]. ICI mitigation has been studied by many researchers and this will be the focus of the dissertation.

§ 1.2 ICI Mitigation

Since an OFDM system is vulnerable to mobility and CFO, various techniques have been pro-posed to cope with these two kinds of ICI. First, we discuss the mobility-induced ICI problem. Two algorithms are well-known, namely, 1) the zero-forcing (ZF) method and 2) minimum mean square error (MMSE) method. Unfortunately, these methods require the inversion of an

=
^

ICI matrix, where is the number of subcarriers. Except for a matrix inversion, the MMSE method also needs to conduct an extra 


matrix multiplication. Thus, its com-putational complexity is even higher than that of the ZF method. The payoff for the higher complexity is its enhanced performance. If is large, the computational complexity of both algorithms can become prohibitively high. Systems with a lot of subcarriers are not uncommon in real-world applications. For example, for the application of DVB, the number of subcarriers can be as large as 8192. To solve the problem of a large ICI matrix inversion, a simpler ICI equalizer for the ZF method was developed in [16]. As mentioned, ICI on a subcarrier mainly comes from a few neighboring subcarriers. Thus, ICI from the other subcarriers can then be ignored. This method has good performance in low-mobility environments. In high-mobility environments, however, the number of insignificant ICI terms will be decreased and the com-putational complexity will be significantly increased.

Successive interference cancellation (SIC) and parallel interference cancellation (PIC) are two well-known multiuser interference (MUI) cancellation techniques in

code-division-multiple-access (CDMA) systems. Since the characteristic of ICI is similar to that of MUI, these methods can be directly applied to ICI mitigation in OFDM systems. A method combining the MMSE and SIC techniques was first proposed in [17]. Later, it was improved with a recursive method in [18], reducing the required complexity further. Although good performance can be achieved with these methods, the required complexity is still high and the time delay can be intolerably large. The PIC technique was then employed to solve the problem [19], [20], [21], [22], [23]. Although the processing delay is greatly reduced, the performance is discounted as well. Other approaches use transmitter frequency-domain coding or beamforming to reduce ICI or to en-hance the received signal-to-interference-noise-ratio (SINR). Interested readers may see [24], [25], [26], [27].

Apart from the processing in the frequency domain, some researchers also explore that in the time domain. In [28], a time-domain filtering technique maximizing the signal-to-ICI-plus-noise ratio was proposed for single-input-single-output(SISO)/multiple-input-multiple-output (MIMO) OFDM systems. One disadvantage of this method is that it requires matrix operations to solve a generalized eigenvalue problem. Another approach involves the use of a time-variant time-domain equalizer, making the time-variant channel less variant. Transferring the equalizer from time-domain to frequency-domain, one can obtain a frequency-domain per-tone equalizer (PTEQ). The PTEQ was originally proposed to deal with the insufficient CP problem in OFDM systems. Lately, it is extended to suppress ICI in SISO/MIMO-OFDM systems [29], [30], [31], [32], [33]. The PTEQ is well-known for its good performance; however, its implementation complexity and storage requirement can be high. In [34], a two-stage equalizer was proposed. In the first stage, a time-domain windowing technique is used to shorten the ICI response in the frequency domain. In the second stage, an iterative MMSE method is used to suppress the residual ICI. Although the windowing approach is simple, the iterative MMSE processing is not trivial. To further enhance the system performance, another approach called turbo equalization can be applied to mitigate ICI [35], [36], [37]. In [37], a block turbo MMSE method was proposed. The main feature is that this method uses the whole ICI matrix to obtain the MMSE

solution although it ignores some insignificant ICI terms.

Next, we discuss the CFO-induced ICI mitigation problem. For OFDM and OFDMA down-link systems, the CFO can be easily estimated and compensated in the receiver [38], [39]. How-ever, for OFDMA uplink systems, the problem is more involved. In the literature, various ICI mitigation methods have been proposed to solve the problem. One direct method is to estimate CFO in the base station and transmit the information back to mobile stations for CFO correc-tion. Another approach is to transmit redundant information in subcarriers such that ICI can be cancelled with a simple method in the receiver end. This approach is called the self-ICI-cancellation [24], [40], [41], [42], [43], [44]. However, these methods mentioned above will sacrifice the transmission rate.

Yet another viable approach eliminates the need for extra transmission overhead by com-pensating for ICI in the receiver. CFO compensation methods for OFDMA uplink systems have been reported [45], [46], [47], [48], [49], [50], [51]. The simplest method is to treat the CFO-induced ICI as that in OFDM systems and to compensate for ICI with a time-domain phase de-rotation operation for each user [45]. This approach can suppress self-ICI, but it does not take MUI into account. In [46], a post-FFT CFO compensation method was proposed, improving the performance of the phase de-rotation approach. Unfortunately, the MUI problem still remains. In [47], a scheme combining the method in [46] with the PIC technique was developed. Other PIC-related works can be found in [48], [49]. It is simple to observe that the CFO-induced ICI on a subcarrier mainly comes from neighboring subcarriers. Thus, the method in [50] modifies the CFO-induced ICI matrix into a banded matrix, and reduces the computational complexity of the ZF and MMSE methods. However, its performance may be compromised due to the sim-plification. Taking advantage of an interleaved-OFDMA structure, the authors in [51] proposed a method that divides the whole system into several smaller subsystems, after which the MMSE method was applied to the subsystems. This method has good performance, and it requires low computational complexity; however, it is only applicable to an ideal interleaved structure (i.e., uniform subcarrier-spacing for each user). The aforementioned methods were developed

for CFO-compensation. CFO estimation methods have also been reported for OFDMA uplink systems [52], [53], [54], [55].

§ 1.3 Proposed Approach

As mentioned, the main problem in the ZF and MMSE methods is the matrix inversion. Thus, how to conduct this operation efficiently becomes the main concern. It is found that some iterative methods can be much more efficient than the direct matrix inversion method. We first discuss the mobility-induced ICI problem. In [56], the Gauss-Seidel iteration was used to conduct the matrix inversion. However, it still needs a matrix inverse in its iterative process. Another method called operator-perturbation was recently proposed [57]. Similar to [56], this method also requires a matrix inverse in its iterations. Thus, the computational complexity for the methods in [56] and [57] is still high. In [58], it was discovered that the ICI matrix for a linear time-variant (LTV) channel model exhibits a special structure, allowing the application of fast Fourier transforms (FFTs) in the matrix inversion. The LTV channel model was proposed in [16] and its original purpose is for the time-variant channel estimation [59]. Exploiting this structure, a power-series expansion (PSE) method was proposed for the ICI matrix inversion [58], [60]. Although the PSE method can greatly reduce the computational complexity, it does not perform well in high-mobility environments.

In this dissertation, we propose low-complexity ZF and MMSE methods to solve the mobility-induced ICI problem in SISO/MIMO-OFDM(A). Similar to [58], we exploit the special struc-ture inherent in the LTV channel model. For the ZF method, we first develop a method that can implement Newton’s iteration for the ICI matrix inversion in SISO-OFDM systems. With our specially designed architecture, FFTs can be used in the iterative process, reducing the computational complexity effectively. We also propose a method for the calculation of initial values. With those values, Newton’s iteration can converge very fast, usually within a couple of iterations. Unlike the PSE method [58], our method works well even in high-mobility

en-vironments. Simulation results show that the performance of the proposed low-complexity ZF method can be as good as that of the direct ZF method. However, the required computational complexity is reduced from 



to 



. We also analyze the convergence behavior of the proposed low-complexity ZF algorithm and derive the theoretical output SINR. Using a new MIMO-OFDM system formulation, we then extend the proposed method to ICI mitigation in MIMO-OFDM systems. It is shown that in MIMO-OFDM systems, the computational com-plexity can be reduced even more significantly. For an g

g system, where g is the number

of transmit (receive) antennas, the proposed algorithm can reduce the computational complexity from  hg

*

to hg

  

.

As mentioned, the matrix inversion is the main obstacle in the ZF method, and some re-searchers try to use iterative methods to overcome this problem [56], [57], [58]. Although these methods can reduce the computational complexity of the ZF method, they are not applicable for the MMSE method. As mentioned above, the MMSE method has to conduct an extra 
7

matrix multiplication which cannot be avoided in these approaches. Using the basic idea in the proposed low-complexity ZF method, we further develop an efficient low-complexity MMSE method. The main contribution in the proposed ZF method is to develop an efficient ICI mit-igation scheme using Newton’s iteration. With the approach, we can use FFTs/IFFTs in the computation of the matrix inversion, dramatically reducing the computational complexity. The proposed MMSE algorithm inherits this property, and further eliminates the requirement of the




matrix multiplication. Simulation results show that the performance of the proposed low-complexity MMSE method is similar to that of the direct MMSE method. However, the re-quired computational complexity is reduced from



to 

!  

. We also extend the proposed method to ICI mitigation in MIMO-OFDM systems. For ang

g system, the

pro-posed algorithm can reduce the computational complexity from  hg * to  g    . It is simple to see that for MIMO-OFDM systems, the proposed method can reduce the compu-tational complexity even more significantly. Moreover, we apply the proposed ZF and MMSE methods to ICI mitigation in SISO/MIMO-OFDMA systems.

Next we discuss the CFO-induced ICI problem in OFDMA uplink systems. The ZF method is known to be a simple yet effective method for CFO compensation. However, it has to invert the CFO-induced ICI matrix whose dimension equals the number of subcarriers. As a result, the computational complexity can become prohibitively high when the number of subcarriers is large, a case commonly found in OFDMA systems. As we can see, this problem is similar to ICI mitigation in high-mobility environments. We then propose a low-complexity iterative ZF method to cope with the problem. Following the idea described above, we use Newton’s method to iteratively perform the matrix inversion. Taking advantage of the special structure of the CFO-induced ICI matrix, we develop a method that can implement Newton’s method with FFTs. With our specially designed initial matrix, the proposed iterative method can stop within two to three iterations. From simulation results, we find that the performance of the proposed method is similar to that of the direct ZF method. However, the required compu-tational complexity is reduced from 



to 

!F

. We also analyze the theoretical SINR enhancement of the proposed algorithm. Two approaches are used for the analysis; one is simple but approximated, and the other is complicated but exact. The issue of convergence is also discussed.

§ 1.4 Organization of the Dissertation

The rest of this dissertation is organized as follows. In Chapter 2, we first describe an LTV-based SISO-OFDM signal model. This signal model involves DFT/IDFT and diagonal matrices which can be used to develop low-complexity ICI mitigation algorithms. Since the PSE method also exploits the LTV-based SISO-OFDM signal model, it is briefly reviewed. Then we describe the proposed ZF method and present the complexity and performance analysis. Lastly, we show simulation results to corroborate the proposed algorithm. Except for the ZF method, we also consider the MMSE method. We reformulate the MMSE solution and extend the proposed ZF method to avoid the extra matrix multiplication and matrix inversion.

In Chapter 3, we focus on ICI mitigation in a MIMO-OFDM system. For this system, conducting the matrix inversion is more difficult since the dimension of the ICI matrix can be huge. To solve the problem, we derive a MIMO-OFDM signal model that allows the application of the proposed low-complexity ZF and MMSE methods developed in Chapter 2. It can be shown that the reduction in computational complexity is even more significant in MIMO-OFDM systems.

In Chapter 4, we further consider the ICI mitigation problem in an OFDMA uplink system. Based on the LTV channel model, we derive a SISO-OFDMA signal model from which the low-complexity algorithms developed in Chapter 2 can be applied. This signal model can be viewed as a generalized SISO-OFDM signal model. When the number of users is one, the SISO-OFDMA signal model is degenerated to the SISO-OFDM signal model. We also extend the model from SISO-OFDMA systems to MIMO-OFDMA systems. Chapters 2, 3 and 4 give a complete treatment for mobility-induced ICI mitigation in SISO/MIMO-OFDM(A) systems.

In Chapter 5, we discuss low-complexity algorithms for CFO-induced ICI mitigation in OFDMA uplink systems. We first describe the OFDMA uplink signal model that is composed of diagonal and DFT/IDFT matrices. Applying the low-complexity algorithms developed in Chapter 2, we obtain low-complexity CFO-induced ICI mitigation methods. We also propose a pre-compensation method to further enhance the performance of the low-complexity meth-ods. Complexity and performance analyses are also provided to verify the effectiveness of the proposed method.

In Chapter 6, we draw some conclusions for the dissertation and outline possible topics for further research.

Chapter 2

Mobility-induced ICI Mitigation for

SISO-OFDM Systems

§ 2.1 Signal Model

Consider a mobile OFDM system whose channel variation is large such that the mobility-introduced ICI cannot be ignored. It was shown in [59] that the LTV channel model can be used to approximate a time-variant channel for the normalized Doppler frequency up to 3i

, where the normalized Doppler frequency is defined as the maximum Doppler frequency divided by subcarrier spacing. Using the LTV channel model, we can approximate the time-variant channel in a specific OFDM symbol period as

jk ml n jporqkts l
j ' qk ) (2.1)

where l is the time index, jfk

ml

is the u th-tap channel response at time instant l ,

jvorqk

is its constant term, andj

'

qk

is its variation slope. We assume thatl is

3

at the midpoint of an OFDM symbol. Let w o `xjporqo ) jporq '*) JWJWJ ) jporqy AB'{zE| , wt' `xj ' qo ) j ' q '*) JWJWJ ) j ' qy AB'{zE| , } o 8~*Uh €w o , and }1' `~*Uh €wt' , where ~*Uh €

denotes a circulant matrix with the first column vector being  .

Also, we define‚t' Sx ƒb s  r„ ) ƒb s R r„ ) JWJWJ ) b  r„ zE| and…' 4†Uˆ‡‰ Š‚‹' , where

the notation, †Uˆ‡‰ €Œ

, denotes a diagonal matrix with the diagonal vector of Π. According to

(2.1), we can express the receive time-domain signal in the OFDM symbol (after CP removal) as   €} ots …'Ž}1' { s‘ ) (2.2) where and

are the receive and transmit time-domain>


signal vectors, respectively, and



is the noise vector (additive white Gaussian). Let ’ be a unitary discrete Fourier transform

(DFT) matrix with the property that ’’“

” y , where ” y is an •
– identity matrix. Moreover, let —  ˜ ’  , — Z=˜ ’  , —  ˜ ’  , — w o =˜ ’w o , — wt' ˜ ’wt' , — } o 4†Uˆ‡‰ — w o , and — }1' 4†Uˆ‡‰ — wt'

. Multiplying both sides of (2.2) by˜

’ , we can express

the receive signal in the frequency domain as

—   ˜ ’   ˜ ’7 €} o‹s …'Ž}1' ’ “ ’  s ˜ ’  N™ — } o‹s ’(…'š’ “ — }1'{›t—  s —  4œ  —  s —  ) (2.3) whereœž — } oBs ’(…'r’“ —

}1' is the so-called ICI matrix. Note that

œ

can also be rewritten as

œ ž — } oKs — …' — }1' , where — …'  ’(…'’“ Sx~*Uh m—‚‹' zE| and—‚‹'  „˜ 1

’“Ÿ‚‹'. Since the ICI

matrix is not a diagonal matrix, the ICI exists. If the channel is time-invariant, the frequency-domain ICI term, —

…' , will disappear. Thus, the signal model will become the traditional OFDM

signal model.

§ 2.2 ZF Method

Among the ICI mitigation methods, the simplest remedy for ICI is the ZF method. Denote the ZF equalized signal as ¡ £¢

. Then, we can obtain the equalized signal as‹ £¢¤`œ AB'

—



. From the above formulation, it is simple to see that direct implementation of the ZF method

will require high computational complexity if is large. Thus, it is a critical problem for the ZF method. Then, the PSE method was introduced in [58] to solve the problem. The idea is to expressœ AB' as œ AB'  ™,¥ ” y1s ’(…'š’ “ — }1' — } AB' o¦ — } o › AB'  — } AB' o ” y b§ AB' ) (2.4) where §  ’(…'’“ } and }  b — }1' — } AB' o . Next, ” y b§ AB'

is expanded with a power series and the high order terms are truncated, i.e.,

” y b^§ AB'¨ª©G« @E¬ o § @ , where ­ is the

highest order retained in the expansion. The convergence condition for this expansion is that

® § ®¯  [58], where ® § ®

indicates the p-norm of§ [60]. Finally, the equalized —

 , denoted as ±°v²³ , is equal to — } AB' o © « @E¬ ov´ @, where ´ @  § @ —  . Note that ´ @EDF'  § ´ @  ’(…'š’“µ } ´ @ . Thus, ´

@ can be recursively calculated. Also, with the special structure of § , FFTs/IFFTs can

be used to calculate ´

@. Although the computational complexity can be reduced effectively,

the performance of the PSE method is unsatisfactory in high-mobility environments. This will be verified from simulation results. In the following subsection, we will present the proposed method to solve the problem.

§ 2.2.1 Proposed Newton-ZF Method

As mentioned, the performance of the PSE method is unsatisfactory in high-mobility environ-ments. To solve the problem, we seek for a more flexible and powerful iterative method for matrix inversion. Specifically, we find Newton’s iteration is useful. Newton’s iteration is well-known for its fast convergence [61], [62], and it has been investigated extensively [63], [64], [65], [66]. Let ¶"· be the estimated matrix inverse of

œ

at the Q th iteration. The hQ

s



th Newton’s iteration can be described as follows:

¶·DF'  &” y b¶· œ  ¶·&)Q 53 )  )  ) JWJWJ )*¸ J (2.5) Let — ¹ · P” y b[¶· œ 

represent the estimation residual. Equation (2.5) implies that ®

” y b ¶· œ ®º»® ” y b5¶ o œ ® ƒ¼ for all Q . If ® ” y b4¶ o œ ®¯ 

convergence [67]. From (2.5), we can clearly see that Newton’s iteration requires matrix-to-matrix multiplications whose computational complexity is 

1

. Thus, the computational complexity is high. As a matter of fact, its complexity is even higher than that of the direct ZF method when Q is large. Thus, direct application of Newton’s iteration for matrix inversion is

not feasible. In what follows, we propose a method to solve the problem. Iterating (2.5), we obtain a sequence of matrices $<¶

o )r¶'*) JWJWJ )r¶· + . The relationship between¶ o

and¶· can be found straightforwardly in

¶·   ¼ AB' ½ ¾ ¬ o ~ · q ¾ Š¶ o œ ¾ ¶ o ) (2.6) where~ · q

¾ is the coefficient of the

¿ th summation term in (2.6). The expression in (2.6) can be

seen as an expansion form of Newton’s iteration, while that in (2.5) an iterative form. It turns out that to obtain a low-complexity algorithm, we have to use the expansion form. Assign~

·

q

¾ ’s

as coefficients of a polynomial function ofÀ , i.e.,H· €À

Á ~ · qo À o s ~ · q 'ŽÀ ' s JWJWJ s ~ · q  ¼ AB' À  ¼ AB' . Then, the polynomial H&·DF', €À

can be derived from H&· €À

as H·DF' €À = H· €À b!À x H· €À z  , whereH o €À ÂM

. This is to say that~

·

q

¾ can be recursively calculated. Note that our objective

is to obtain the equalized result¶"·—



, not the matrix inverse¶"· itself. Multiplying both sides

of (2.6) by—  and letting ·  ¶·—  andà ¾  Š¶ o œS ¾ ¶ o — 

, we have the equalized result as

 ·   ¼ AB' ½ ¾ ¬ o ~ · q ¾ à ¾ J (2.7) From the definition ofÃ

¾ , we can then have the following relationship:

à ¾ DF'  Š¶ o œS à ¾ J (2.8) As a result,Ã

¾ can be recursively calculated as well. Using this approach, we have transformed

matrix-to-matrix multiplications in (2.6) into matrix-to-vector multiplications in (2.7) and (2.8). To complete our low-complexity algorithm, we make use of the special structure inherent in the ICI matrix. From the foregoing derivation, we know that œÄ

— } ons ’(…'š’“ — }1' . Using

this structure, we can then rewrite (2.8) as à ¾ DF'  ™ ¶ o ¥ — } o‹s ’(…'š’ “ — }1' ¦ › à ¾  ¶ o ™ — } o à ¾ s ’(…'š’ “ ¥ — }1' à ¾ ¦ › J (2.9) Note that — } o , —

}1' , and …' are all diagonal matrices. If we further pose a constraint that¶

o

is a diagonal matrix, we can transform matrix-to-matrix operations into vector-to-vector and DFT/IDFT operations as shown in (2.9). As we know, DFTs/IDFTs can be efficiently imple-mented with FFTs/IFFTs whose complexity is 

GfF

. As a result, the computational complexity of the proposed algorithm is 

!fF

. The constraint on¶

o

may not always yield satisfactory performance in all scenarios. Instead of a diagonal matrix, we may let¶

o

be a low-bandwidth banded matrix. Let the

U

){u

th entry of a matrixÅ be denoted as Å

U

){u

. The banded matrix is defined as follows. Å

U ){u »Æ83 , if Ç U buÈÇ º % , and Å U ){u \83 , otherwise. Here, % is the bandwidth of the banded matrix. If %

]3

, the banded matrix is reduced to a diagonal matrix. If %

8

, the banded matrix will have three non-zero diagonal vectors. With this type of¶

o

, the computational complexity in (2.9) will only be increased slightly. For later simulations, we will only consider the cases of%

#3

and % 8

. It turns out that for % 8

, the performance of the proposed algorithm is good enough. For easy reference, we denote the proposed low-complexity ZF method as the Newton-ZF (N-ZF) method.

§ 2.2.2 Derivation of the Initial Matrix

In Subsection 2.2.1, we have proposed the N-ZF method to reduce the complexity of the direct ZF method using FFTs. However, we still have to determine the initial matrix¶

o

for the N-ZF method. A good initial matrix can reduce the number of iterations significantly and provide good mitigation performance. As known, the main function of ZF is to invert the ICI matrix, and in the ideal case, ” y

bɶ·

œÊPË y

, where Ë y

is an 


zero matrix. As a result, if

” y

b0¶

o œ 

can be made as close toË y

as possible, fast convergence in Newton’s algorithm can be obtained. Based on this idea, we propose to minimize the Frobenius norm of ” y

bԦ

o œ

i.e., ¶ o 5ÌÍrnÎ6ÏÐ Ñ ® ” y b^¶ œ ®  ¢ ) (2.10) where ® — ¹ ® ¢

means the Frobenius norm of —

¹

and¶ is a banded matrix with bandwidth % .

Before the derivation of the optimal solution in (2.10), we first observe a property in a banded matrix. Fig. 2.1 shows an example of a banded initial matrix for ]

and % `

. In the figure, only the data in the shaded area are zeros. Note that the number of the non-zero elements in each row may not be the same. For the 0th and the 7th rows, the number of the non-zero elements is 2. For the rest of rows, the number of the non-zero elements is R

. For a general case, the number of the non-zero elements in the U

th row first increases, remains the same, and finally decreases (as U

increases). Due to this property, we need to consider the three cases when solving (2.10). Define ¿@—

qk  œ U ){u , ÒÓ@ qk  ¶ o U ){u , and ‡ @ qk  © y AB' Ô ¬ o — ¿Õ @ q Ô — ¿ kƒq

Ô . Differentiating (2.10) with respect to ÒÖÕ

@

qk

and setting the result to zero, we can

×{ØtÙ

׃ØÁÚÜÛ¡Ý

׃؋Þ

ß\ØÂÚ

Figure 2.1: An example of the structure of a banded initial matrix for TS

and % Z

. The elements in the shaded area are non-zeros, while the others are zeros.

obtain the following equation: ?C@à»@ 5á @h) U‹43 )  ) JWJWJ ) b  ) (2.11)

whereà»@ consists of the non-zero elements in the

U

th row vector of the optimum¶

o

. ?C@, à»@,

andá

@ for the above-mentioned three cases are defined as follows:

1. ForU‹ %)% s  ) JWJWJ ) b  b‘% , ?C@  âã ã ã ä ‡ @AKå q @AKåçæ<æ<æ ‡ @AKå q @EDBå ... ... ... ‡ @EDBå q @AKåçæ<æ<æ ‡ @EDBå q @EDBå èêé é é ë ) (2.12) à»@ Mx ÒÓ@ q @AKåÁ)šÒÓ@ q @AKå±DF'*) JWJWJ )šÒÓ@ q @EDBå-z| ) (2.13) á @ Mx — ¿ Õ @AKå q @ ) ¿— Õ @AKå±DF' q @ ) JWJWJ ) ¿— Õ @EDBå q @ z | J (2.14) 2. ForU‹43 )  ) JWJWJ )%8b  , ?C@  ?Yåì 30í % s U ) 30í % s U{ ) (2.15) à»@ Sx ÒÓ@ qo )šÒÓ@ q '*) JWJWJ )šÒÓ@ q @EDBå-z | ) (2.16) á @ Sx — ¿ Õ orq @ ) ¿— Õ ' q @ ) JWJWJ ) ¿— Õ @EDBå q @ z | ) (2.17) whereîY U ' íU  ){u&' í u 

indicates a submatrix ofî , obtained from the

U

'th row to the

U



th row and from theu' th column to theu

 th column ofî . 3. ForU‹4 b‘%) b% s  ) JWJWJ ) b  , ?C@  ? y AB'{AKåì U b s  s % í£ %) U b s  s % í£ % ) (2.18) à»@ Sx ÒÓ@ q @AKåÂ)šÒÓ@ q @AKå±DF') JWJWJ )šÒÓ@ qy AB'ˆz | ) (2.19) á @ Sx — ¿ Õ @AKå q @ ) ¿— Õ @AKå±DF' q @ ) JWJWJ ) ¿— Õ y AB' q @ z | J (2.20) Note that ?C@ in the second case is an upper left submatrix of ?å in (2.12), while that in the

third case is a lower right submatrix of ?

y

0

A

A

₃ 7

A

5

A

4

A

A

₆ 1

A

A

₂ ï(ð0ñ

A

Figure 2.2: An example of ?(@ for

•>

and % ò

. Note that ?(@ overlaps with ?(@AB' and

?C@EDF' . solution for (2.10) byà6@  ? AB' @ á

@. For clearly understanding the structure of ?Y@, we show

an example in Fig. 2.2 for 8

and% a , where ?2 U ){u \8‡ @ qk

. From the figure, we can see that?

o

is the upper left 6


submatrix of?' . For UÓ] )  ) JWJWJ ) X

, the lower right 6
^

submatrix of?(@ is exactly the same as the upper left »
2

submatrix of?Y@EDF' . The lower right »


submatrix of?Yó is ?Yô . Using this property, we can obtain a recursive algorithm for fast

computation of? AB' @ . Since ‡ kƒq @ ‡ Õ @ qk

, ?C@ is a Hermitian matrix. For Uõ %)% s  ) JWJWJ ) b  b¤% , we can

further partition?(@ into the following form

?C@  â ä»ö @>÷W“ @ ÷ @aø(@ è ë ) (2.21)

and?C@EDF' into the following form

?C@EDF'  â ä ø(@ —Ã @EDF' — ᓠ@EDF' —ù @EDF' è ë ) (2.22)

where

ö

@ and—ù @EDF' are scalars,÷,@ and—ñ@EDF' are column vectors, and

ø

@ is a square matrix whose

dimension is smaller than that of ?(@ by one. Since ?(@ is a Hermitian matrix, we can write its

inverse as ? AB' @  â äYú @‚‹“ @ ‚ @ … @ è ë ) (2.23) where ú

@ is a scalar, ‚È@ is a column vector, and …@ is a square matrix with dimension smaller

than that of ? AB'

@ by one. From the block matrix inversion formula [68], we can obtain

?

AB' @EDF'

from? AB'

@ via the following formula

? AB' @EDF'  â ä ø AB' @ s —û @EDF' —û “ @EDF' — ü @EDF' —û @EDF' — ü @EDF' —û “ @EDF' — ü @EDF' — ü @EDF' è ë ) (2.24) where — ü @EDF'  —ù @EDF'‹b— ᓠ@EDF'Žø AB' @ — à @EDF' AB' ,—û @EDF'  b ø AB' @ —ñ@EDF' , and ø AB' @  …Y@vb ‚È@‚‹“ @ ú @ J (2.25) ForU‹43 )  ) JWJWJ )%8b 

,?C@EDF' includes?C@ as its submatrix. We then have

? @EDF'  â ä ? @ — à @EDF' — ᓠ@EDF' —ù @EDF' è ë J (2.26) Consequently,? AB'

@EDF' can be obtained by (2.24), whereø

AB' @  ? AB' @ . For U‹5 b%) JWJWJ ) b  ,

?C@ becomes a submatrix of?(@AB' given by ?C@AB'  â ä»ö @a÷W“ @ ÷@a?C@ è ë J (2.27) Thus, ? AB'

@ can be obtained with (2.25) as follows,

? AB' @  …Y@AB'‹b ‚È@AB'ƒ‚‹“@AB' ú @AB' J (2.28) Thus, we only have to conduct one matrix inversion, explicitly, i.e., ?

AB'

o

, and its dimension is

€%

s  Ó

€%

s 

To further reduce the complexity, we can make an approximation when calculating ‡

@

qk

. From the definition, we have ‡

@ qk  © y AB' Ô ¬ o — ¿Õ @ q Ô — ¿ kƒq

Ô . We can reduce the number of terms

in-cluded in the summation. We let ‡

@ qk ¨N© Ô&ýWþ ¿Õ— @ q Ô — ¿ kƒq Ô , where ÿ  ¯ U b5 ínU s ‹)
 ¯ u6b4 í u s ‹) 

, and is the number of one-sided ICI terms taken into considera-tion (3 º º »„ b  ). The notation ¯ UÖí u)  denotes a sequence of $ U b  @ y ) U s  b  @EDF'y  ) JWJWJ ){u»b  k y  + (U

andu are integers and

U

º

u ). With this approach,

‡

@

qk

is ap-proximately evaluated, so is?(@ in (2.11). The value of then determines the accuracy of the

solution in (2.11). A small can greatly reduce the complexity, but results in low accuracy of the solution. Recall that ICI on a subcarrier mainly comes from a few neighboring subcarriers. As a result, we can always find a small only affecting the final result slightly. For the deter-mination of , it depends on the value of ICI; the larger the ICI, the larger we should use. In our simulations, the largest we use is two.

As mentioned, if % !3

, ¶

o

will become a diagonal matrix. In this case, the initial values can be approximated as ÒÓ@ q @ ¨ — ¿Õ@ q @ © Ô&ýWþ Ç¿(@— q Ô Ç  ) (2.29) where ÿ  ¯ U b– íÜU s ‹)

. There is an interesting property in (2.29). If we only take the diagonal terms of the ICI matrix into account (i.e., 53

), the initial values will degenerate into the coefficients of the conventional one-tap frequency-domain equalizer (FEQ). If there is no ICI, Newton’s iteration with (2.29) will stop after initialization (Q

43

).

§ 2.2.3 Complexity Analysis

In Subsections 2.2.1 and 2.2.2, we have completed the derivation of the N-ZF method in SISO-OFDM systems. In this subsection, we will analyze the required computational complexity of the N-ZF method, and compare it with that of the PSE and direct ZF methods.

From (2.7) and (2.8), it is clear that the computational complexity of the N-ZF method mainly consists of the following three parts:

1. à ¾ iteration, where à ¾  Š¶ o œ  à ¾ AB' andà o  ¶ o —  , 2. Bandedœ construction, whereœž } o‹s ’(…'š’“Á}1' , 3. Banded¶ o calculation.

Since the diagonal and DFT/IDFT structures inœ 

, Ã

¾ can be obtained using (2.9). As a result,

we require x!F s  s  % s R„ Ž b–%1 €% s 

z complex multiplications (CMs) and

x!   s  % s „ Ž b4%1 €% s 

z complex additions (CAs). In addition, we need

x  s  % Ž b5%1 €% s  z CMs and x % b4%1 €% s  z CAs for à o  ¶ o —  and  real additions (RAs) for each ~

·

q

¾

Ã

¾ in (2.7). As to the construction of the banded

œ  , we require h s „ Ž CMs and h s „ Ž

CAs. For calculating¶

o

, we need to construct matrices?(@

forU‹53 )  ) JWJWJ ) b 

and they requirex ƒb  %  s % s : Ÿ% s s „ Ž s „R %  b „R %b : Ÿ%  b  Ÿ%6z CMs and x h s : Ÿ%–b  %  bY% Ž s „R % Ks  %  s „R %–b : Ÿ%  b  Ÿ%6z

CAs. For solvingà6@

 ? AB' @ á @, it requires x R„ Ž b d„&: z CMs, x „ Ž b £„&: z CAs, and  b 

real divisions (RDs) for the case of % ]

. For the PSE method, it can be obtained in the same way. Finally, we summarize the required computational complexity for the N-ZF method, the PSE method, and the direct ZF method [69] for a SISO-OFDM system in Table 2.1. We also summarize the computational complexity for calculating the initial matrix in the N-ZF method in Table 2.2.

Table 2.1: Complexity comparison among N-ZF, PSE, and direct ZF methods in a SISO-OFDM system.

Methods Real multiplications Real divi-sions Real additions Direct ZF   s   b '   s   s '{'  b   ó PSE : ­ !  s 9 ­ s  Ž : 9 ­ G  s : ­ s 9 Ž N-ZF (% 43 )  ·D  b :f ŽG   s x ·D  s  1b  z  R  ·DF' b  Ž5   s xX_
 ·DF' s  1b X z N-ZF (% # )  ·D  b :f Ž!  s x&: s R±
Ó ·D *s 9 z b &: ìb  ·D  b d  b  R  ·DF' b  Ž!  s &: s d1
/ ·DF' s RK Ž b &: 1b  ·D  b :fX

Table 2.2: Complexity of the initial matrix calculation for the N-ZF method in a SISO-OFDM system.

Methods Real multiplications Real divisions Real additions N-ZF (% # ) &: s &:f Ž b &: 2b d &: s Rd Ž b &: 6b :fX  b 

§ 2.2.4 Performance Analysis

For the proposed N-ZF algorithm, the iteration number is usually preset. Unlike other iterative algorithms, the convergence is not a concern here. The reason we can use a preset iteration number is due to the fast convergence property of Newton’s iteration and our good initial values. If the proposed algorithm converges, only a small number of iterations is necessary. On the

other hand, if the proposed algorithm diverges, the preset number of iterations will limit the performance degradation. As a matter of fact, even for divergence cases, we can still have improved SINRs if the iteration number is set properly. We will provide intuitive statements to explain why this is true. It turns out that the determination of the iteration number is simple and straightforward.

Now we start with the analysis of convergence behavior. After that, we will derive theoreti-cal SINRs the proposed algorithm can provide. We first perform the eigenvalue decomposition for — ¹ o as follows: — ¹ o  øø AB' ) (2.30) whereø Mx à o )Ãt') JWJWJ )à y

AB'ƒz is a matrix composed of eigenvectors of

— ¹ o , and !†Uˆ‡‰ x o )  '*) JWJWJ )  y AB'{zE| consists of eigenvalues,  @’s. We assume that Ç  @rÇNÇ  k Ç for U º u . Since — ¹ ·  — ¹ 

·AB' , then we can decompose

— ¹ · as — ¹ ·  ø ƒ¼ ø AB' J (2.31) If Ç  o Ç ¯  , then — ¹ · Ë y

as Q ¸ . Thus, we can have the convergence condition for

Newton’s iteration as F — ¹ o ¯  , where F — ¹ o

denotes the spectral radius of —

¹ o

; the spectral radius indicates the largest absolute value of all eigenvalues [62]. That is to say, for Newton’s iteration to converge, the amplitudes of all eigenvalues of —

¹ o

have to be smaller than one. For a moderate mobile speed, this condition holds for most cases. If not, the number of eigenvalues with amplitudes greater than one is small and their amplitudes does not deviate from one too much. These results can be easily observed from simulations though difficult to be proved theoretically. In what follows, we will first show that even for divergence cases, we may still benefit from Newton’s iteration. Letø

AB' Sx o )  '*) JWJWJ )  y AB'ƒzE| , Ç  @ŽÇ 5 forU‹43 )  ) JWJWJ )b  and Ç  @rÇ ¯  for U ì) s  ) JWJWJ ) b  . By definition, — ¹ · ” y b‘¶· œ . Then, we can

represent the ICI matrix as ¶· œž4” y b y AB' ½ @E¬ o   ¼ @ ñ@  | @ 4” y b ° AB' ½ @E¬ o  ƒ¼ @ ñ@  | @ b y AB' ½ k ¬ °  ƒ¼ k à k  | k J (2.32) As for¶· , we can reformulate it as

¶·  ” y1s ” y b^¶·AB' œS ¶·AB'  ” y1s — ¹ ·AB' ” y1s — ¹ ·A  ÜJWJWJ ” y1s — ¹ o ¶ o J (2.33) Using (2.31), we can further express¶ò· as

¶·  ø ” y1s   ¼ ø AB' ø ” y1s   ¼! ø AB' JWJWJ ø ” y2s  ø AB' ¶ o  y AB' ½ k ¬ o " ·AB' # @E¬ o  s  %$ k '& à k  | k ¶ o  y AB' ½ k ¬ o)( kƒq ·à k  | k ¶ o ) (2.34) where( kƒq · +* ·AB' @E¬ o  s  %$ k

. With (2.32) and (2.34), the ZF-equalized signal can be expressed as  ·  ¶· œ  —  s ¶·Ž—   —  b-, AB' ½ @E¬ o  ƒ¼ @ ñ@  | @ —  b y AB' ½ k ¬ ,  ƒ¼ k à k  | k —  s —  ·  —  b ° AB' ½ @E¬ o   ¼ @ ñ@  | @ —  b y AB' ½ k ¬ °   ¼ k à k  | k —  s y AB' ½ k ¬ o( kƒq ·,à k  | k — o ) (2.35) where—  ·  ¶·Ž— 

. Since the eigenvectors $&Ã

o )Ãt'*) JWJWJ )Ã y AB' +

span the -dimensional space, we can decompose—



and— o

using these eigenvectors. Let— 1 © y AB' . ¬ o ü . à . and — o  © y AB' . ¬ o0/ . à ., wherexü o ) ü '*) JWJWJ ) ü y AB'{zE|  ø AB' —  and x / o ) / '*) JWJWJ ) / y AB'ˆzE|  ø AB' — o

co-efficients in the decomposition of —



and— o

, respectively. Then, we can rewrite (2.35) as

 ·  —  b ° AB' ½ @E¬ o   ¼ @ ñ@  | @21 y AB' ½ . ¬ o ü . à .43 b y AB' ½ k ¬ °   ¼ k à k  | k 1 y AB' ½ . ¬ o ü . à .53 s y AB' ½ k ¬ o( kƒq ·,à k  | k 1 y AB' ½ . ¬ o / . à .43  —  b ° AB' ½ @E¬ o  ƒ¼ @ ü @mñ@vb y AB' ½ k ¬ °  ƒ¼ k ü k à kŸs y AB' ½ k ¬ o ( kƒq · / k à k J (2.36) Let · Mx6 · qo ) 6 · q '*) JWJWJ ) 6 · qy AB'{z | and — ^Sx — 6 o ) — 6 ') JWJWJ ) — 6 y AB'ƒz | . Thus, the ¿ th subcarrier signal

after equalization can be expressed as

6 · q ¾  — 6 ¾ s — H' q · q ¾ s — H  q · q ¾ s — H  q · q ¾ ) (2.37) where — H' q · q ¾  b © ° AB' @E¬ o   ¼ @ ü @ŠÃ±@Ž m¿ , — H  q · q ¾  b © y AB' k ¬ °   ¼ k ü k à k m¿ , and — H  q · q ¾  © y AB' k ¬ o ( kƒq · / k à k m¿

. From (2.37), we can see that the equalized signal suffers from three interference terms. For —

H'

q

·

q

¾ , it will become large when

Q increases; however, for

— H  q · q ¾ , it will become

small when Q increases. As for the noise term,

— H  q · q ¾ , its dependence on Q is not strong. As

mentioned, only a few eigenvalues’ amplitudes will be larger than one (i.e., is small) and their

amplitudes often do not deviate from one too much. Then, it is easy to see that the decreasing amount of — H  q · q

¾ will be larger than the increasing amount of

—

H'

q

·

q

¾ in the early iteration. Thus,

for divergence cases, the interference will decrease first and then increase as the iteration pro-ceeds. If we can stop the iteration before the overall interference increases, we can still have the performance gain even though the iteration diverges eventually. Additionally, we can in-crease % to make the initial matrix closer to the exact matrix inverse. By this way, may be

minimized and — H'

q

·

q

¾ will decrease, which makes the proposed method work better. Because of

the fast convergence property of Newton’s method, the number of iterations required is small. For example, it can be as small as one or two when %



. For divergent cases, the overall interference is still decreasing in the first one or two iterations.

Since the performance of an OFDM system depends on each subcarrier SINR, we will analyze the subcarrier SINR of the proposed algorithm in the sequel. From (2.3), we can express the equalized signal as ¶"·—

 87 ·*—  s ¶·Ž—  , where 7 ·  © ƒ¼ AB' ¾ ¬ o ~ · q ¾ Š¶ o œ ¾ DF' is the

equalized ICI matrix. Ideally, 7

· will be an identity matrix. Let —

  x —À o ) —À&'*) JWJWJ ) —À y AB'{zE| , 9  :; =< $BÇ — 6 @rÇ  + , 9  : > =< $BǗÀ,@rÇ  + (3 º U º b  ), and?  9  : > „ 9 

:; . The subcarrier SINR for the

proposed method withQ iterations in the

U

th subcarrier, denoted as@BACEDµ·

q @, can be shown as @BACED.· q @  < $BÇF · @ q @ — 6 @rÇ  + < $BÇ © y AB' GIH!J GLKH $ F · @ qk — 6 k Ç  + s < $BÇ © y AB' k ¬ o Ò · @ qk —À k Ç  +  9  :; ÇF · @ q @ Ç  9  :; © y AB' GIH!J GLKH $ ÇF · @ qk Ç  s 9  : > © y AB' k ¬ o ÇÒ · @ qk Ç   ÇF · @ q @ Ç  © y AB' GIH!J GLKH $ ÇF · @ qk Ç  s ? © y AB' k ¬ o ÇÒ · @ qk Ç  ) (2.38) whereF · @ qk M7 · U ){u andÒ · @ qk  ¶· U ){u

. For comparison, we also calculate the SINR in the

U

th subcarrier before equalization, denoted as@BACEDV@, as follows:

@BACEDì@  < $BÇ¿(@— q @— 6 @rÇ  + < $BÇ © y AB' GIH!J GLKH $ — ¿(@ qk — 6 k Ç  + s < $BǗÀ,@rÇ  +  9  :; Ç¿(@— q @rÇ  9  :; © y AB' GIH!J GLKH $ Ç¿(@— qk Ç  s 9  : >  Ç¿(@— q @rÇ  © y AB' GIH!J GLKH $ Ç¿(@— qk Ç  s ? J (2.39) As a result, we must calculate each element in7

· and¶· for@BACED.·

q

@. To make the following

derivation more compact, we rewrite the ICI matrix asœ  ©+N O ¬ o — … O — } O , where — … o 4” y and P!# . Sinceœž © N O ¬ o — … O — }

O , the equalized ICI matrix

7 · can be expanded as 7 ·   ¼ AB' ½ ¾ ¬ o ~ · q ¾RQ ST N ½ OIUWV J ¬F' ¶ o — … OIUWV J — } OIUWV J N ½ OIUWV  ¬F' ¶ o — … OIUWV  — } OIUWV  JWJWJ N ½ O UWVU ¬F' ¶ o — … O UWVU — } O UWVUYXZ [   ¼ AB' ½ ¾ ¬ o ~ · q ¾RQ ST N ½ OIUWVJ ¬ o N ½ OIUWV  ¬ o JWJWJ N ½ O UWVU ¬ o ¾ # \ ¬ o ¶ o — … OUWV] — } O UWV] XZ [   ¼ AB' ½ ¾ ¬ o ~ · q ¾RQ ST N ½ OIUWV J ¬ o N ½ OIUWV  ¬ o JWJWJ N ½ O UWVU ¬ o_^ ? ¾ XZ [ ) (2.40)