Abstract of thesis entitled

A Study of Channel Estimation for OFDM Systems and

System Capacity for MIMO-OFDM Systems

Submitted by

Zhou Wen

For the degree of Doctor of Philosophy

at the university of Hong Kong in July 2010

This thesis concerns about two issues for the next generation of wireless

communications, namely, the channel estimation for orthogonal frequency-division

multiplexing (OFDM) systems and the multiple-input multiple-output orthogonal

frequency-division multiplexing (MIMO-OFDM) system capacity.

For channel estimation for OFDM systems over quasi-static fading channels having

resolvable mulitipath number L, a novel fast linear minimum mean square error (LMMSE)

channel estimation method is proposed and investigated. The proposed algorithm deploys

Fourier transform (FFT) and the computational complexity is therefore significantly

reduced to O(Nplog2(Np)), as compared to that of O(Np3) for the conventional LMMSE

method, where the notation O() is the BachmannLandau function and Np is the number

of pilots for an OFDM symbol. The normalized mean square errors (NMSE) are derived in

closed-form expressions. Numerical results show that the NMSE is marginally the same

with that of the conventional LMMSE for signal to noise ratio (SNR) ranges from 0 dB to

25 dB. For channel estimation for OFDM systems over fast fading and dispersive channels,

a novel channel estimation and data detection method is proposed to reduce the

inter-carrier interference (ICI). A new pilot pattern composed of the comb-type and the

grouped pilot pattern is proposed. A closed-form expression for channel estimation mean

square error (MSE) has been derived. For SNR = 15 dB, normalized Doppler shift of 0.06,

and L = 6, both computer simulation and numerical results have consistently shown that

the ICI is reduced by 70.6% and 43.2%, respectively for channel estimation MSE and bit

error rate (BER). The pilot number per OFDM symbol is also reduced significantly by

92.3%, as compared to the comb-type pilot pattern.

A closed-form mathematic expression has been proposed for the capacity of the

closed-loop MIMO-OFDM systems with imperfect feedback channel. The lower threshold

of feedback SNR is derived. For L = 6, numerical results show that the lower threshold of

feedback SNR is proportional to antenna numbers N and system SNR. The increasing rate

of the feedback SNR threshold increases from 0.82 to 1.01 when N increases from 2 to 16.

The variance and mean of OFDM system capacity over Rayleigh channels and Ricean

channels have been respectively investigated that the closed-form expression for the

capacity variance has been proposed. The resultant system capacity variances over the two

channels are respectively evaluated by numerical method and also verified by computer

simulation. The joint probability density function (PDF) of two arbitrary correlated Ricean

random variables has also been derived in an integral form. Numerical results reveal that

the variance of OFDM system is proportional to SNR and inversely proportional to L for

the two channels respectively. For the same two respective channels, the variance

marginally increases with a linear rate of 0.166 bit2/dB and 0.125 bit2/dB, when L = 2 and

SNR ranges from 0 dB to 15 dB. The variance is reduced from 1.75 bit2 to 1.30 bit2 and

from 1.48 bit2 to 1.26 bit2, when SNR = 10 dB and L ranges from 2 to 4.

(Total words: 495)

A Study of Channel Estimation for OFDM Systems and

System Capacity for MIMO-OFDM Systems

by

Zhou Wen B. Eng., M. Eng., USTC, P. R. China

A thesis submitted in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy at the university of Hong Kong in July 2010.

i

Declaration

I declare that this thesis represents my own work, except where due

acknowledgement is made, and that it has not been previously included in a

thesis, dissertation or report submitted t to this University or any other

institution for a degree, diploma, or other qualifications.

Signature: ___________

Zhou Wen

ii

Acknowledgements

I would like to take this opportunity to express my gratitude to all the people who have

ever helped me in the thesis writing and the course of the research.

My sincere and hearty thanks and appreciations go firstly to my supervisor, Dr. W.H. Lam,

whose suggestions and encouragement have given me much insight into the research work.

It has been a great privilege and joy to study under his guidance and supervision. His

insightful observation and effective feedback inspired me during the research. Furthermore,

it is my honor to benefit from his personality and diligence, which I will treasure my whole

life.

I also gratefully acknowledge Prof. V.O.K. Li, Prof. G.L. Li, Prof. Y.C. Wu, Prof. S.C.

Chan, Prof. T.S. Ng and Prof. Agnes S.L. Lam for their interesting courses and helpful

discussions. I would like to thank the office staff and technical staff from the EEE

department for their helpful administrative and facility supports. Especially, Ms. Julie

Hungs readiness to help students is very impressive. I also appreciate the HKSAR

government for the studentship support to the study in the University of Hong Kong.

I am extremely grateful to all my friends and classmates who have kindly provided me

assistance and companionship in the process of preparing this thesis: Dr. Zhi Zhang, Dr.

Zhiqiang Chen, Dr. Mingxiang Xiao, Dr. Fei Mai, Mr. Xueyong Liu, Mr. Xiaoguang Dai,

Ms. Ziyun Shao, Mr. Ka-Chung Leung, Mr. Peng Zhang, Dr. Yanhui Geng, Ms. Qiong Sun,

Mr. Haoling Xiahou, Mr. Zhibo Ni, Mr. Jun Zhang, Mr. Xiaolei Sun, Mr. Chengwen Xing.

They have made the life during the past four years an enjoyable and memorable experience.

Finally, I wish to express my hearty gratitude to my parents, for their encouragements and

love in all my endeavors.

iii

Contents

Declarations......................................................................................................................... i

Acknowledgements............................................................................................................. ii

Contents.............................................................................................................................. iii

List of Figures.....................................................................................................................vii

Chapter 1: Introduction...................................................................................................... 1

1.1 Research motivation................................................................................................ 4

1.2 Organization and contributions of the thesis........................................................... 5

Chapter 2: OFDM systems and MIMO systems .............................................................. 9

2.1 Wireless Channel................................................................................................... 10

2.1.1 Large scale propagation ............................................................................. 11

2.1.2 Small scale propagation ............................................................................. 13

2.1.3 Typical wireless channel models................................................................ 17

2.2 OFDM systems ..................................................................................................... 20

2.2.1 Basic principles and characteristics for OFDM systems ........................... 21

2.2.2 Peak-to-Average (PAR) of OFDM systems ............................................... 30

2.2.3 Channel estimation for OFDM systems..................................................... 33

2.2.4 Synchronization of OFDM systems........................................................... 38

2.2.5 Advantages and disadvantages of OFDM systems .................................... 39

2.3 MIMO systems...................................................................................................... 40

2.3.1 Basic MIMO system model ....................................................................... 40

2.3.2 Functions of MIMO systems...................................................................... 42

2.3.3 Overview of Space Time codes.................................................................. 45

2.3.4 Capacity of MIMO systems ....................................................................... 52

iv

2.4 MIMO-OFDM systems......................................................................................... 54

2.5 Summary ............................................................................................................... 56

Chapter 3: Channel estimation for OFDM systems over quasi-static fading channels

............................................................................................................................................. 57

3.1 Introduction........................................................................................................... 58

3.2 System Model ....................................................................................................... 61

3.3 The Proposed Fast LMMSE Algorithm ................................................................ 63

3.3.1 Properties of the channel correlation matrix in frequency domain ............ 63

3.3.2 The proposed fast LMMSE channel estimation algorithm ........................ 65

3.3.3 Computational complexity comparison between the proposed method and

the conventional LMMSE method...................................................................... 69

3.4 Analysis of the Mean Square Error (MSE) of the Proposed Fast LMMSE

Algorithm.................................................................................................................... 70

3.4.1 MSE analysis of the conventional LMMSE algorithm.............................. 71

3.4.2 MSE analysis for the proposed fast LMMSE algorithm............................ 72

3.5 Numerical and Simulation Results........................................................................ 75

3.6 Conclusion ............................................................................................................ 81

Chapter 4: Channel estimation and data detection for OFDM systems over fast

fading channels.................................................................................................................. 87

4.1 Introduction........................................................................................................... 88

4.2 System Model ....................................................................................................... 91

4.3 The Proposed Channel Estimation and Data Detection ........................................ 92

4.3.1. The proposed pilot pattern ........................................................................ 92

v

4.3.2. Channel Estimation and data detection for the first M1 OFDM symbols of

each block ........................................................................................................... 94

4.3.3. Channel estimation and data detection for the last M2 OFDM symbols of

each block ........................................................................................................... 95

4.3.4. Summary of the proposed channel estimation and data detection ............ 98

4.4. Analysis of MSE of the proposed channel estimation method ............................ 99

4.4.1. MSE analysis of channel estimation for the first M1 OFDM symbols ... 100

4.4.2. MSE analysis of channel estimation for the last M2 OFDM symbols .... 103

4.4.3 MSE analysis of channel estimation for one OFDM block ..................... 105

4.5 Numerical and Simulation Results...................................................................... 106

4.6. Conclusion ......................................................................................................... 112

Chapter 5: MIMO-OFDM system capacity with imperfect feedback channel ......... 118

5.1 The open-loop and closed-loop capacity for MIMO Systems ............................ 119

5.1.1 MIMO system model ............................................................................... 119

5.1.2 MIMO system capacity............................................................................ 120

5.1.3 Numerical Results and discussion............................................................ 124

5.2 The closed-loop capacity with imperfect feedback channel for MIMO-OFDM

systems ...................................................................................................................... 127

5.2.1 System Model .......................................................................................... 128

5.2.2 Closed-Loop Capacity and Feedback SNR for MIMO-OFDM Systems 130

5.2.3 Numerical Results .................................................................................... 136

5.3 Summary ............................................................................................................. 142

Chapter 6: Capacity of OFDM systems over time and frequency selective fading

vi

channels............................................................................................................................ 144

6.1 Introduction......................................................................................................... 145

6.2 OFDM System Model ......................................................................................... 147

6.3 OFDM System Capacity ..................................................................................... 148

6.3.1 OFDM system capacity over Rayleigh fading channels .......................... 148

6.3.2 OFDM system capacity over Ricean fading channels ............................. 153

6.4 Numerical and Simulation Results...................................................................... 157

6.5 Conclusion .......................................................................................................... 161

Chapter 7: Conclusions and future works .................................................................... 167

7.1 Conclusions......................................................................................................... 167

7.2 Future works ....................................................................................................... 169

APPENDIX A: The derivation of the rank of channel frequency autocorrelation matrix

RHH in Chapter 3............................................................................................................... 170

APPENDIX B: The derivation of equation (3-20) in Chapter 3....................................... 171

APPENDIX C: The derivation of the joint PDF of two arbitrary correlated Ricean random

variables ............................................................................................................................ 173

Appendix D: List of Abbreviations................................................................................... 176

REFERENCES.................................................................................................................. 179

Publications....................................................................................................................... 191

vii

List of Figures

Fig 1.1: Organization of the thesis. ....................................................................................... 6

Fig 2.1: Path Loss, shadowing and multipath versus distance............................................ 11

Fig 2.2: The Doppler power spectrum function expressed by (2-4). .................................. 14

Fig 2.3: The multi-path effect between the transmitter and the receiver in wireless

communication........................................................................................................ 14

Fig 2.4: Time varying impulse response of a wireless channel, for the path number N = 3, 4,

and 5. ....................................................................................................................... 15

Fig 2.5: Four kinds of small scale propagations. ................................................................ 16

Fig 2.6: PDFs for Rayleigh fading with the variance 2 = 0.5, 2, and 5, respectively. ....... 17

Fig 2.7: PDFs for Ricean fading with Ricean factor Kr = 0 dB, 10 dB, and 20 dB,

respectively.............................................................................................................. 18

Fig 2.8: PDFs for Nagakami-m fading with m = 0.5, 1, and 10.......................................... 19

Fig 2.9: The continuous OFDM system model. .................................................................. 22

Fig 2.10: The waveform of ( )kG w ....................................................................................... 24

Fig 2.11: Equivalent transmitter for OFDM systems.......................................................... 25

Fig 2.12: Equivalent receiver for OFDM systems. ............................................................. 25

Fig 2.13: CP for an OFDM symbol..................................................................................... 26

Fig 2.14: SNRloss versus CP length. .................................................................................. 27

Fig 2.15: The inter-symbol interference of OFDM systems without CP. ........................... 28

Fig 2.16: Extraction of the data in frequency domain......................................................... 28

Fig 2.17: The discrete baseband OFDM system model. ..................................................... 29

Fig 2.18: The output power versus the input power for a power amplifier......................... 31

Fig 2.19: The power spectrum comparison between the input signal and the output signal

passing through an amplifier. .................................................................................. 32

Fig 2.20: Two kinds of pilot patterns (black dot: pilot, white dot: user data). .................... 34

Fig 2.21: Pilot-aided channel estimation for OFDM systems............................................. 35

Fig 2.22: The basic MIMO system model. ......................................................................... 40

Fig 2.23: Received signal after diversity operation. ........................................................... 43

viii

Fig 2.24: Diversity-multiplexing tradeoff, d*(r) versus r. .................................................. 44

Fig 2.25: The Alamouti STBC diagram for 22 MIMO systems. ...................................... 46

Fig 2.26: V-BLAST system diagram................................................................................... 49

Fig 2.27: The baseband MIMO-OFDM system model. ...................................................... 55

Fig 3.1: Baseband OFDM system. ...................................................................................... 61

Fig 3.2: Channel estimation based on comb-type pilots. .................................................... 62

Fig 3.3: The first row of the channel autocorrelation matrixp pH H

R , A . ............................. 82

Fig 3.4: The first row of the LMMSE matrix1

SNR + p p p pH H H HR R I with different SNRs.

................................................................................................................................. 83

Fig 3.5: Normalized Mean square error (NMSE) of channel estimation of LMMSE

algorithm versus that of the proposed fast LMMSE algorithm by computer

simulation and numerical method. .......................................................................... 83

Fig 3.6: NMSE of LMMSE algorithm with matched SNR and mismatched SNRs versus

SNR, by simulation and numerical method, respectively. ...................................... 84

Fig 3.7: NMSE of the proposed fast LMMSE algorithm with matched SNR and

mismatched SNRs versus SNR, by simulation and numerical method, respectively.

................................................................................................................................. 84

Fig 3.8: Bit error rate (BER) of the LS, LMMSE, the proposed fast LMMSE and perfect

channel estimation versus SNR............................................................................... 85

Fig 3.9: BER comparison between LMMSE channel estimation with matched SNR and

LMMSE channel estimation with designed SNRs.................................................. 85

Fig 3.10: BER comparison between the proposed fast LMMSE channel estimation with

estimated SNR and the proposed fast LMMSE channel estimation with designed

SNRs. ...................................................................................................................... 86

Fig 4.1: Pilot pattern (gray circle: user data, black circle: pilot). ..................................... 114

Fig 4.2: The normalized mean square error (NMSE) of channel estimation for the first M1

OFDM symbols, for df T = 0.01, 0.06 and 0.1, respectively. ................................. 114

Fig 4.3: The NMSE of channel estimation based on equi-spaced and grouped pilot pattern,

ix

for the polynomial order Q =1, 2, 3 and the normalized Doppler shift df T = 0.01 and

0.1, respectively..................................................................................................... 115

Fig 4.4: The NMSE of channel estimation based on grouped pilot pattern, for c =1, 2, 3 and

the normalized Doppler shift df T = 0.01 and 0.1, respectively.............................. 115

Fig 4.5: The NMSE of channel estimation based on grouped pilot pattern, for the number

of pilot groupsgroupN =18, 36, 72 and the normalized Doppler shift df T = 0.01 and

0.1, respectively..................................................................................................... 116

Fig 4.6: The NMSE of channel estimation for the proposed algorithm and LS algorithm by

numerical method and simulation at df T = 0.01 and 0.06, respectively. ............... 116

Fig 4.7: Bit error ratio (BER) of LS, the proposed algorithm and the algorithm in [29], for

normalized Doppler shift df T = 0.01 and 0.06, respectively. ................................ 117

Fig 5.1: The eigenmode transmission of MIMO systems. ................................................ 123

Fig 5.2: The MIMO system open-loop capacity versus the number of transmitter

antennas TN , for the number of receiver antennas 1RN = ...................................... 124

Fig 5.3: The MIMO system open-loop capacity versus the number of receiver antennas RN ,

for the number of transmitter antennas 1TN = ....................................................... 124

Fig 5.4: The capacities of the N by 1 MISO system, the 1 by N SIMO system, and the N by

N MIMO system as a function of N, for SNR = 5 dB........................................... 125

Fig 5.5: The open-loop and closed-loop capacity for MIMO systems, versus SNR. ....... 126

Fig 5.6: The closed-loop MIMO-OFDM system model. .................................................. 128

Fig 5.7: The open-loop and closed-loop system capacity for MIMO-OFDM systems

having different transmitter antenna and receiver antenna numbers. ................... 138

Fig 5.8: The capacity gain of the closed-loop capacity with imperfect feedback over that of

the open-loop capacity versus feedback channel SNR, for 4T RN N= = . .............. 140

Fig 5.9: The capacity gain of closed-loop capacity with imperfect feedback over that of the

open-loop capacity versus feedback channel SNR, for system SNR = 10 dB...... 140

Fig 5.10: The lower threshold of the feedback SNR versus the MIMO-OFDM system SNR,

x

for different antenna pairs. .................................................................................... 142

Fig 6.1: The PDF of the capacity at a certain subcarrier,,

( )i kC

f x in (6-36), for SNR = 0 dB,

5 dB, 10 dB, and 20 dB, respectively.................................................................... 162

Fig 6.2: The joint PDF of of 21| ( , ) |H i k and 22| ( , ) |H i k , 2 21 2| ( , )| ,| ( , )|

( , )H i k H i k

f x y ,for the coefficient

of equation (6-23), = 0.61................................................................................... 162

Fig 6.3: The coefficient of equation (6-24), , versus different subcarrier gap between

1k and 2k . ................................................................................................................ 163

Fig 6.4: The variance of OFDM system capacity for the number of channel paths L = 2, 4,

and 8, over the Rayleigh fading channel............................................................... 163

Fig 6.5 The variance of OFDM system capacity versus the CP of an OFDM symbol in unit

of sample point, over the Rayleigh fading channel............................................... 164

Fig 6.6: The variance of OFDM system capacity versus the number of subcarriers of one

OFDM symbol, for Rayleigh fading channels. ..................................................... 164

Fig 6.7: The variance of OFDM system capacity over Ricean fading channels for L = 2, 4,

8, respectively........................................................................................................ 165

Fig 6.8: The mean value of OFDM system capacity for Rayleigh fading channel and

Ricean fading channel, by numerical method. ...................................................... 165

Fig 6.9: The variance of OFDM system capacity for Rayleigh fading channel and Ricean

fading channel, by computer simulation and numerical method. ......................... 166

1

Chapter 1: Introduction

The research on wireless communication systems with high data rate, high spectrum

efficiency and reliable performance is a hot spot. There are several advanced

communication technologies or protocols proposed recently, including Orthogonal

frequency division multiplexing (OFDM) [1], multiple input multiple output (MIMO) [2],

Ultra-Wideband (UWB) technology [3], cognitive radio [4], World Interoperability for

Microwave Access (WiMAX) [92], and 3GPP Long Term Evolution (LTE) [92], [93].

OFDM is an efficient high data rate transmission technique for wireless communication.

OFDM presents advantages of high spectrum efficiency, simple and efficient

implementation by using the fast Fourier Transform (FFT) and the inverse Fast Fourier

Transform (IFFT), mitigration of inter-symbol interference (ISI) by inserting cyclic prefix

(CP) and robustness to frequency selective fading channel. MIMO is the use of multiple

antennas at both the transmitter and receiver to improve communication performance. It is

one of several forms of smart antenna technology. MIMO technology has attracted

attention in wireless communications, because it increases in data throughput without

additional bandwidth or transmit power. It achieves this by higher spectral efficiency and

link reliability or diversity. The combination of MIMO with OFDM technique is a

promising technique for the next generation wireless communication. A new protocol draft

employing the MIMO-OFDM as the physical layer technology, IEEE 802.11n, as an

amendment to IEEE 802.11 standards has been proposed [53]. Wireless LAN technology

2

has seen rapid advancements and MIMO-OFDM has gradually been adopted in its

standards. The following table shows the existing IEEE 802.11 WLAN protocols.

Table 1.1 Existing 802.11 WLAN Standards

IEEE Protocol Name 802.11b 802.11a 802.11g 802.11n

Standard Approved Sept. 1999 Sept. 1999 June 2003 Released in

2009 Available Bandwidth 83.5 MHz 580 MHz 83.5 MHz 83.5/580 MHz Frequency Band of

Operation 2.4 GHz 5 GHz 2.4 GHz 2.4/5 GHz Non-Overlapping

Channels (US) 3 24 3 3/24

Data Rate per Channel 111 Mbps 654 Mbps 154 Mbps 1600 Mbps

Modulation Type DSSS, CCK OFDM DSSS, CCK,

OFDM

DSSS, CCK,

OFDM, MIMO

UWB is a technology for transmitting data spread over a large bandwidth (usually larger

than 500 MHz) that shares among users. UWB was traditionally applied in

non-cooperative radar imaging. Most recent applications include sensor data collection,

precision locating, and tracking applications. The concept of cognitive radio was first

proposed by Dr. J. Mitola and Prof. G. Q. Maguire [4] in 1999 and was an extension to the

concept of software radio. Cognitive radio is an intelligent communication system that

could detect and track the communication environments. It would adjust the transmitter

and the receivers parameters adaptively according to the changes of environment

parameters such as the mobile velocity of the user, so that the system stability could be

3

ensured, the system performance could remain a good condition, and the spectrum

efficiency could be improved. WiMAX is a telecommunications protocol that provides

fully mobile Internet access. The name "WiMAX" was created by the WiMAX Forum,

which was founded in 2001. The forum refers to WiMAX as a standards-based technology

enabling the delivery of last mile wireless broadband access as an alternative to cable and

digital subscriber line (DSL). The basis of WiMAX is IEEE 802.16 standard which is

sometimes referred to as WiMAX equivalently. The current WiMAX revision is based

on IEEE 802.16e, which was approved in December 2005. The physical layer of WiMAX

adopts a lot of advanced technologies such as scalable orthogonal frequency division

multiplexing access (OFDMA), MIMO, adaptive antenna array and so on. Current

WiMAX that is based on the IEEE 802.16e protocol belongs to 3G family. Future WiMAX

is based on IEEE 802.16m, which has been submitted to the International

Telecommunication Union (ITU) for International Mobile Telecommunication Advanced

(IMT-Advanced) standardization. Future WiMAX, or the proposed WiMAX release 2, is

considered as a candidate of 4G family. LTE is the latest standard in the mobile

communication systems. The current generation of mobile communication system is

collectively known as 3G. Although LTE is often referred to as 4G, the first released LTE is

actually a 3.9G technology as it does not completely meet the 4G requirements. The main

advantages of LTE include high throughput, low latency, plug and play, a simple

architecture resulting in low power consumption, supporting seamless passing by base

stations with former wireless networks such as Global System for Mobile Communications

4

(GSM), Universal Mobile Telecommunications System (UMTS), and CDMA2000. LTE

also adopts OFDM and MIMO technologies in the physical layer. It uses a 2 by 2 MIMO

system as the basic configuration, that is, both the base station and the mobile end equip 2

antennas. The next step for LTE evolution is LTE Advanced and is currently being

standardized by 3rd Generation Partnership Project (3GPP) organization.

The thesis studies two issues: channel estimation for OFDM systems and

MIMO-OFDM system capacity. The chapter is organized as follows. Section 1.1 describes

the research motivation. Section 1.2 provides the thesis contributions and the overall

organization of the thesis.

1.1 Research motivation

Earlier OFDM systems such as the digital audio broadcasting (DAB) system in Europe

does not require channel estimation module. It only uses DPSK demodulation for the sake

of reducing the complexity of the receiver. However, with increasing demands of high data

transmission rate and reliable communication quality, channel estimation has become a

necessary part in the OFDM system. For example, the digital video broadcasting (DVB)

system adopts the channel estimation module. In a broadband wireless environment, the

channel is often time varying and frequency selective, which distorts the transmitted signal

significantly, so that accurate and real time channel estimation is the challenging topic in

the OFDM system. Channel state information can be used for the detection of the received

signal, improving the capacity of the system throughput by adjusting the modulation at the

5

transmitter through the feedback. Therefore, one issue of the thesis studies channel

estimation for OFDM systems over time varying and frequency dispersive fading

channels.

With the increasing number of mobile phone users and higher demands for wireless

services, future communication systems should have higher system capacity. MIMO

technique is a breakthrough of improving system capacity. Telatar [46] and Foschini [47]

have firstly formulated the system capacity of the MIMO systems assuming independent

and identically distributed fading at different antennas. They have proved that the MIMO

system capacity for n transmitter antennas and n receiver antennas increases linearly with

n at a fixed transmitter power. That is, MIMO systems can improve the system capacity

significantly without increasing the system bandwidth. A number of MIMO techniques

known as layered space time architectures or Bell Laboratories layered space time

(BLAST) architectures [5][8] have been proposed. Many studies on MIMO system

capacity have been conducted. Since the combination of MIMO with OFDM is a trend, a

lot of research work has been done on the MIMO-OFDM system capacity. However, the

research on MIMO-OFDM system capacity with imperfect feedback channel is not mature

and corresponding work is not much. Therefore, the second issue of the thesis is to study

the MIMO-OFDM system capacity with imperfect feedback channels.

1.2 Organization and contributions of the thesis

Firstly, we briefly describe the organization of the thesis. Chapter 1 gives a general

6

introduction, research motivation, organization, and contributions of the thesis. Chapter 2

describes the basic OFDM system model and MIMO system model. The wireless channel

model, the principles of OFDM and MIMO systems, the combination of MIMO and

OFDM, that is, MIMO-OFDM system is also introduced in Chapter 2. Next, as depicted in

Fig 1.1, the thesis begins with the first issue, that is, channel estimation. The proposed

channel estimation method in Chapter 3 is based on quasi-static fading channels and that in

Chapter 4 is based on fast fading channels. Then, the thesis switches to next issue, that is

system capacity. In Chapter 5, the MIMO-OFDM system capacity with imperfect feedback

channel is investigated. In Chapter 6, the capacity variances for OFDM systems over

Rayleigh and Ricean fading channels are derived, respectively. Finally, Chapter 7

concludes the thesis and discusses future research works.

Fig 1.1: Organization of the thesis.

7

Secondly, the major contributions of this thesis are summarized as follows.

9 A fast linear minimum mean square error (LMMSE) channel estimation method for

OFDM systems over slow fading channels has been proposed. Unlike the

conventional method, the channel state information is not needed in advance.

Almost the same performance with the conventional LMMSE channel estimation in

terms of the normalized mean square error (NMSE) of channel estimation and bit

error rate (BER) could be achieved for the proposed method. The computational

complexity can be reduced significantly since the proposed method replaces the

inverse operation with FFT operation. (Chapter 3)

9 A new pilot pattern and corresponding channel estimation method and data

detection for OFDM systems over fast fading channels have been proposed. The

proposed channel estimation and data detection based on the proposed pilot pattern

can eliminate inter-carrier interference (ICI) effect effectively. And the number of

required pilots is also reduced significantly, compared with the conventional least

square (LS) method. MSE analysis for the channel estimation based on the grouped

pilot pattern is provided, too. (Chapter 4)

9 The closed-loop MIMO-OFDM system capacity with imperfect feedback channel

has been formulated. We use the feedback SNR to measure the closed-loop capacity.

Since low feedback SNR may not yield positive gain of closed-loop capacity over

8

open-loop capacity, there exists a lower threshold of feedback SNR. The lower

thresholds for different antenna pairs are further investigated by numerical method.

(Chapter 5)

9 The variances of OFDM system capacity over Rayleigh fading channels and Ricean

fading channels have been derived. The effects of SNR, the number of channel

paths, power profile of fading channel, the delay of the channel on the variance

have been thoroughly investigated, for both multipath Rayleigh channels and

multipath Ricean channels. The joint PDF of two arbitrary correlated Ricean

random variables has been provided in an integral form. (Chapter 6)

9

Chapter 2: OFDM systems and MIMO systems

Since the thesis studies the channel estimation for OFDM systems and system capacity for

MIMO-OFDM systems, this chapter briefly introduces the background of OFDM systems

and MIMO systems.

OFDM is an effective technology which provides high spectrum efficiency, high data

transmission rate and is robust to multi-path fading [1]. OFDM has already been widely

put into practice in DAB system, DVB system and WLAN. MIMO systems which employ

multi-element antenna arrays at the transmitter and receiver ends are capable of high data

rate transmission. A number of MIMO techniques known as layered space time

architectures or BLAST architectures have been proposed [5]-[7], [9].

As OFDM technique can mitigate the ISI and transform the frequency selective fading

channel into a set of flat fading channels, the combination of MMO with OFDM technique

is a trend for future wireless communication. A new protocol draft, IEEE 802.11n, as an

amendment to IEEE 802.11 standards has been proposed and investigated [53]. The draft

proposes MIMO-OFDM as the physical layer technique and is to be approved by IEEE.

The chapter is organized as follows. Section 2.1 provides an introduction to wireless

channel in communication systems. Section 2.2 describes the basic OFDM system model,

principles, and related key technologies in OFDM systems. Section 2.3 introduces the

basic MIMO system model and briefly overviews the existing MIMO systems. Section 2.4

presents the MIMO-OFDM system model. The last section 2.5 summarizes the chapter.

10

2.1 Wireless Channel

Since there exist reflections, scattering, and diffraction in the transmission of

electromagnetic wave, the spatial environments such as the landscape of a city,

obstructions and so on will make complicated impacts on the transmission of

electromagnetic wave. There are two kinds of propagation, including the large-scale

propagation and the small scale propagation. The signal variations due to path loss or

shadowing occur over relatively large distances, this variation is referred to as the large

scale propagation effects. Path loss is a major component in the analysis and design of the

link budget of a telecommunication system. The small scale propagation refers to the

phenomena that the amplitude of the received wireless signal varies very fast in a short

time period or a short distance. The sources of small scale propagation include the Doppler

shift effect and multi-path effect. We begin with the introduction of the large scale

propagation.

11

2.1.1 Large scale propagation

Fig 2.1: Path Loss, shadowing and multipath versus distance.

Fig 2.1 plots the ratio of the received-to-transmit power in dB versus the distance for

the combined effects of path loss, shadowing, and multipath. Observe that the free space

path loss is linearly proportional to the log-distance between the transmitter and receiver.

The shadowing loss has slower variations compared to that of the multipath effect. The

large scale path loss model has many successful types including the free space path loss

model, the Hata model, and the Okumura model and so on. Most of them are obtained by

a combination of the analytical and empirical methods. We next briefly introduce the free

space path loss model as an example of path loss effect.

9 Free space path loss model: An example of path loss effect

Consider a signal transmitted through free space to a receiver located at distance d from

the transmitter. It is assumed that there does not exist any obstructions between the

12

transmitter and receiver. The channel model with this kind of transmission method is

called a LOS channel, and the corresponding received signal is named the LOS signal.

The average received power PR expressed in dB form is given by

PR = PT + 10 log10 (Gl ) + 20 log10 () 20 log10 (4) 20 log10 (d) dB (2-1)

where Gl is the product of the transmit and receive antenna gains in the LOS direction, PT

is the transmitter signal power, d is the distance between the transmitter and the receiver,

is the wave length in unit of meter.

9 Shadow fading

The signal fading due to shadowing from obstacles affecting the wave propagation is

referred to as the shadow fading. A signal will typically experience random variation

due to blockage from objects in the transmission path. Reflecting surfaces and

scattering objects may produce random variation of the received signal, too. Thus, a

model for the random attenuation due to these effects is also needed. Statistic models

have been used to characterize this attenuation since the parameters such as the

locations, sizes of the blocking objects or scattering objects are generally unknown. The

most common model for this additional attenuation is the log-normal shadowing

fading. This model has been demonstrated to accurately model the shadow fading in both

outdoor and indoor radio propagation environments. The path loss is a random

variable X with a log-normal distribution expressed by

210

2

(10 log )( ) exp , 022

x

xx

x up x xx

= > (2-2)

13

where = 10/ ln10, x is the mean value of xdB, xdB = 10 log10 x in unit of dB, and

xis the standard deviation of xdB , also in unit of dB.

2.1.2 Small scale propagation

The small scale fading of a signal is a more rapid fluctuation, which is usually caused by

constructive and destructive interference between two or more versions of the same signal

(multi-path effect) or Doppler effect due to moving terminals or surroundings objects. We

briefly introduce the multi-path effect and the Doppler shift effect.

2.1.2.1 Doppler shift effect

The Doppler shift effect, named after Austrian physicist Christian Doppler who proposed it

in 1842, is the change in frequency of a wave for an observer moving relative to the source

of the wave. When the mobile handset moves away from the base station, the carrier

frequency is decreased and rather than vice versa. The Doppler shift between the base

station and the mobile handset, fd, is expressed as

cos( )dvf = (2-3)

Where v is the velocity of the mobile handset, is the carrier wave length, is the angle

between the velocity of the mobile handset in the radial direction of the mobile handset

and the base station. Assuming that the arrival angles of the signal is uniformly distributed

in the range (-, ), the power spectrum density function (PSD) is given by [36]

12 2

( ) 1 ,av c c d c dd d

P f fS f f f f f ff f

= + (2-4)

14

where Pav is the average power of the received signal and fc is the carrier frequency.

c df f c df f+cf

( )S f

Fig 2.2: The Doppler power spectrum function expressed by (2-4).

Fig 2.2 shows the classic U-shape power spectrum. Note that no components fall outside

the interval [fc - fd, fc + fd] and the power of the transmitted signal spreads between (fc - fd)

Hz and (fc + fd) Hz.

2.1.2.2 Multi-path effect

Fig 2.3: The multi-path effect between the transmitter and the receiver in wireless

communication.

Multi-path effect refers to the propagation phenomenon that results in radio signals

15

reaching the receiving antenna by two or more paths. Causes of multi-path include

atmospheric ducting, ionospheric reflection and refraction, and reflection. Fig 2.3 shows

the multi-path effect in wireless communication.

0t t=

0t t=

0t t=

1t 1 2t +

1t 1 3t +

1t 1 4t +

( )a

( )b

( )c

Fig 2.4: Time varying impulse response of a wireless channel, for the path number N = 3, 4,

and 5.

Assume that the transmitted signal is an impulse function with a very short pulse width

expressed by 0( ) ( )S t a t= and the signal then pass through a wireless channel. At the

receiver, the received signal is given by

[ ]01

( ) ( )N

i ii

S t a S t t=

= (2-5) where ai is the magnitude of the i-th path and i is the excess delay of the i-th path, N is the

number of resolvable paths. As in Fig 2.4, the received signal is a series of pulses and the

number of pulses is the number of channel paths. Note that usually the amplitude ai is time

16

varying and it is a function of time t.

Fig 2.5: Four kinds of small scale propagations.

Thus, based on Doppler shift effect and multi-path effect, T. S. Rappaport divided

wireless fading channels into four categories, as shown in Fig 2.5. The first one is the fast

fading channel, referred to the kind of channel when the coherence time of the channel,

which is a function of Doppler shift, is small relative to the symbol duration of the

transmitted signal. The second is the slow fading channel, referred to the kind of channel

when the coherence time of the channel is large relative to the symbol duration of the

transmitted signal. The third one is the flat fading channel, referred to the kind of channel

when the coherence bandwidth of the channel, which is a function of the maximum delay

of the channel, is larger than the bandwidth of the signal. The fourth one is the frequency

selective fading channel, referred to the channel when the coherence bandwidth of the

17

channel is less than the bandwidth of the signal.

2.1.3 Typical wireless channel models

There are mainly three kinds of wireless channel models, as described as follows.

(1) Rayleigh fading channel

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2 = 0.52 = 22 = 5

Fig 2.6: PDFs for Rayleigh fading with the variance 2 = 0.5, 2, and 5, respectively.

Rayleigh fading is a reasonable model when there are many objects in the environment

that scatter the radio signal before it arrives at the receiver. The envelope of the channel

impulse response is Rayleigh distributed, as expressed by

2

2 2( ) exp , 02r rf r r

= (2-6)

where r is the amplitude of the channel impulse response. Fig 2.6 depicts several PDFs for

Rayleigh fading channel with the variance 2 = 0.5, 2, and 5. The PDF curve with larger

18

variance has a broader spreading range. Usually, the normalized autocorrelation function

of a Rayleigh fading channel is a 0-th order Bessel function of the first kind, as expressed

by

0( ) (2 )dR J f = (2-7) Where fd is the Doppler shift and 0 ( )J is the 0-th order Bessel function of the first kind. A

well known simulation model for the Rayleigh fading channel is the Jakes Model, firstly

proposed by William C. Jakes in 1975.

(2) Ricean fading channel

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

Ricean factor Kr = 0 dB

Ricean factor Kr = 10 dB

Ricean factor Kr = 20 dB

Fig 2.7: PDFs for Ricean fading with Ricean factor Kr = 0 dB, 10 dB, and 20 dB,

respectively.

Ricean fading occurs when one of the channel paths, typically a line of sight signal

(LOS), is much stronger than the others. In a Ricean fading channel, the amplitude gain is

19

characterized by a Ricean distribution, which is formulated by

2 2

02 2 2( ) exp 2d dr k rkrf r I

+ = (2-8)

Where r is the amplitude of the channel impulse response, 22 is the power of the non-LOS multi-path components, 2dk is the power of the LOS component, 0 ( )I is the

modified Bessel function the 0-th order. Fig 2.7 shows the PDFs for Ricean fading with

Ricean factor Kr = 0 dB, 10 dB, and 20 dB, respectively. The Ricean factor Kr is defined

by

Kr = 2dk /22 . (2-9)

(3) Nakagami fading channel

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

m = 1m = 0.5m = 10

Fig 2.8: PDFs for Nagakami-m fading with m = 0.5, 1, and 10.

Both Rayleigh and Ricean distributions can be obtained by using mathematics to

20

describe the physical properties of the channel models. However, a few channels can not

be characterized well by the two previous models. So a more general fading distribution

was created whose parameters could be adjusted to characterize more channel models. The

distribution is the Nakagami fading distribution, or Nakagami-m fading distribution, first

proposed by M. Nakagami, in 1960 [10]. The Nakagami fading distribution is given by

22 1 /2 0( ) ( )

0 0

mm mr Pm r e r

p r m Pr

=

21

about how to modulate/demodulate band signal by Discrete Fourier Transformation (DCT)

[1]. To suppress ISI, they proposed the empty guard interval between two adjacent

symbols, but the orthogonality between two subcarriers over a frequency selective channel

can not be ensured. Another major contribution was made by A. Peled and A. Ruiz [11],

who introduced the concept of CP in 1980, which ensured the orthogonality among

subcarriers of an OFDM symbol. The CP is copied from the end of the OFDM symbol and

it is transmitted followed by each OFDM symbol. When the length of CP is larger than

that of the impulse response of the fading channel, ICI could be avoided.

2.2.1 Basic principles and characteristics for OFDM systems

,0iX

, 1i NX

tje 0

tj Ne 1

,1iX ...

xi(t)tje 1

(a) The transmitter for continuous OFDM systems.

22

.

.

.

xi(t)

tje 0 )(sT

tje 1 )(sT

tj Ne 1 )(sT

T

T

,0iX

, 1i NX

,1iX

T

T

T

T

(b) The receiver for continuous OFDM systems.

(c) The frequency division for OFDM systems.

Fig 2.9: The continuous OFDM system model.

As shown in Fig 2.9(a), the transmitted signal xi(t) for OFDM systems is given by

23

1

,0

( ) ( )N

i i k kk

x t X g t iT

== (2-12)

where

2 [0, )( )

0 others

kj f t

ke t T

g t =

(2-13)

and

0 , 0,1,..., 1.kkf f k NT

= + = (2-14)

where ,i kX denotes the transmitted signal of the k-th subcarrier at the i-th OFDM symbol

interval, T is one OFDM symbol duration excluding the CP length, N is the number of

OFDM subcarriers, fk is the frequency of the k-th subcarrier, fo is the lowest frequency

which corresponds to the first subcarrier. For the baseband OFDM system, the lowest

frequency fo = 0.

The orthogonality between two subcarriers of an OFDM symbol

The orthogonality between two arbitrary subcarriers is described in the following.

9 Two arbitrary subcarriers are orthogonal in the time domain since the following

equation holds.

*( ) ( ) ( )k lg t g t dt T k l= (2-15) 9 Two arbitrary subcarriers are also orthogonal in the frequency domain.

The Fourier transform of gk(t) is expressed by

=)(wGk / 2sin( )2 jwTwT eW )2( kfw (2-16)

24

where denotes convolution. Fig 2.10 shows the waveform of Gk(w).

2 kf w

( )kG w

Fig 2.10: The waveform of ( )kG w .

The transmitted signals for the k1-th subcarrier and k2-th subcarrier in frequency domain

are therefore1,i k

X1( )kG w and 2,i kX 2 ( )kG w , respectively, and satisfy

1 2

1 2

1 2

*, ,

( )1 2, ,

1 2

( ) ( )

sin( 2 ) sin( 2 )42 2

0

k k

i k k i k l

j T f fk ki k i k

k k

X G w X G w dw

w f T w f TX X e dww f w f

= =

(2-17)

Thus, two arbitrary subcarriers are also orthogonal in the frequency domain.

Using FFT/IFFT to implement the transmitter modulation and receiver demodulation

When sampling an OFDM symbol at a sample rate N/Ts and assuming f0=0then the

transmitted signal can be expressed as

1

, , ( )0( )

s

N

i m i k k mt i Tk N

x X g t iT

= +== , m0,1,..., N-1. (2-18)

25

Equation (2-18) can be further derived as

{ }0 12 2, , N .0

IDFTm mNj f T j kN N

i m i k i kk

x e X e N X

=

= = , m0,1,..., N-1. (2-19)

After sampling, the transmitter shown in Fig 2.9(a) is equivalent to the following diagram.

,0iX,1iX

, 1i NX

,i mx

Fig 2.11: Equivalent transmitter for OFDM systems.

And the receiver shown in Fig 2.9(b) is also equivalent to the following diagram.

Fig 2.12: Equivalent receiver for OFDM systems.

Since FFT and IFFT are easy to implement, the computational complexity can be reduced

significantly compared to other systems.

Cyclic Prefix

A. Peled and A. Ruiz [11] firstly introduced the concept of cyclic prefix in 1980, which not

only eliminates ISI, but also ensures the orthogonality between two subcarriers of an

OFDM symbol. Cyclic prefix is a duplicate of last part of an OFDM symbol, shown as

26

following.

C y c lic p re fix (C P )

tim eA n O F D M sy m b o l

Fig 2.13: CP for an OFDM symbol.

The insertion of CP has two effects. On one hand it eliminates the interference between

OFDM symbols, however, on the other hand it also leads to the loss of SNR since it is

necessary to provide the transmitter more power.

(1) Loss of SNR:

At the receiver the SNR loss is given by

)1(log10 10 =lossSNR (2-20) where /CPT T = , TCP is the CP length and T is the OFDM symbol length. The longer the CP is, the greater lossSNR is, as shown in Fig 2.14. Observe that the SNRloss increases with

the increase of CP length and when 2.0

27

0 0.1 0.2 0.3 0.40.01

0.1

1

10

Relative length of CP

Loss

of S

NR

(dB

)

Fig 2.14: SNRloss versus CP length.

(2) Eliminate ISI:

Recall that a signal x(t) passes through the channel with impulse response h(t), the

received signal is given by

( ) ( )* ( )r t h t x t= (2-21)

If the channel has multiple paths, there will be ISI effect. The formula above in discrete

form is written as

1

0

L

m i m ii

r h x

=

= (2-22) where L is the number of paths. Fig 2.15 shows the ISI effect for OFDM systems without CP insertion. We set L = 2.

28

(n-1)-th frame

2,1 NnS 1,1 NnS 0,nS 1,nS 1, NnS 0,1+nS ...

n-th frame (n+1)-th frame

......0h

2,1 NnS 1,1 NnS 0,nS 1,nS 1, NnS 0,1+nS .........1h

2,1 NnS 1,1 NnS 0,nS 1,nS 1, NnS 0,1+nS .........2h

2,1 Nnr 1,1 Nnr 0,nr 1,nr 1, Nnr 0,1+nr .........

+

+

=

Fig 2.15: The inter-symbol interference of OFDM systems without CP.

Without CP insertion, ISI will occur for OFDM systems. However, the insertion of CP

could avoid ISI among OFDM symbols since the channel delay will be absorbed if the CP

length is equal to or larger than the maximum delay of the channel. Considering the CP

insertion, at the receiver the received signal should first remove CP and then perform FFT,

shown in Fig 2.16.

,i kY

Fig 2.16: Extraction of the data in frequency domain.

29

Summary of OFDM system operation

This section summarized the process of OFDM systems. Fig 2.17 depicts the discrete

baseband OFDM system model. The baseband transmitted OFDM signals can be written

as

( )1 ,0

( ) exp , 0N

i i n ii

x t X jw t t T

== (2-23)

where wi = 2i/T is the central frequency of the n-th subchannel and xi(t) is the i-th

corresponding transmitted symbol in time domain, T is the time duration of one OFDM

symbol excluding CP. Assuming the period of the sampling clock is Tsamp, then the discrete

time sequence at the transmitter is

( ),

1

, N ,0

( )

exp IDFT ( )

i n i samp

N

i n i samp i ni

x x nT

X jw nT X

=

== = (2-24)

H ( t , f)

,0iX

,1iX

, 1i NX

ID F T...

In se rt C P...

P a rra lle l to S e r ia l

tran s fo rm

.

.

.

f il te r ing

A W G N n o ise

,0iY

,1iY

, 1i NY

D F T...

C P re m o v in g

.

.

.

S / P...

f il te r in g

tra n sm itte r

R e c e iv e r

c h a n n e l

,0ix

, 1i Nx

,1ix

Fig 2.17: The discrete baseband OFDM system model.

30

The time sequence xi,n is then put into the CP insertion module. After parallel to serial

transformation and filtering, it is sent to the receiver via the wireless fading channel. The

received signal after DFT operation, Yi,k, is given by

, , , ,i k i k i k i kY X H W= + (2-25)

where wi,k denotes the AGWN with zero mean and variance 2w , Hi,k is the frequency

response of the radio channel at the k-th subcarrier of the i-th OFDM symbol. If the

channel frequency response Hi,k is known at the receiver, the estimated transmitted signal

Xi,k, ,i kX , is given by

, , ,/ , 0,1,..., 1.i k i k i kX Y H k N= = (2-26)

2.2.2 Peak-to-Average (PAR) of OFDM systems

The Peak-to-Average (PAR) is a major problem of OFDM systems [81]-[83]. As the

transmitted signal is a summation of N independent subcarriers, the envelope of the

transmitted signal is not a constant. The PAR of OFDM systems is defined as

( )( )

2n

2

maxPAR=10lg .n

n

x

E x

| | | | (2-27)

where xn is the transmitted OFDM signal in time domain. Since the power amplifier has a

limited linear range, the transmitted signal of OFDM systems will distort when passing

through a power amplifier. The power spectrum of the transmitted signal will extend to a

larger range so that the performance of OFDM systems will be reduced correspondingly.

31

Input power

Output power

P_in

P_out

Fig 2.18: The output power versus the input power for a power amplifier.

Fig 2.18 shows the output versus the input power for a general power amplifier. Observe

that when the input signal power is less than a value P_in, the output signal is linearly

proportional to the input signal. However, when the input power exceeds P_in, the output

signal will be compressed. To illustrate the spectrum leak of the input signal, we give an

example. Fig 2.19(a) is a typical power spectrum of band-limited transmitted signal. Fig

2.19(b) is the power spectrum of the output signal passing through an amplifier. The

energy leak effect is very obvious by comparing the two subplots.

32

-4 -3 -2 -1 0 1 2 3 4

-60

-50

-40

-30

-20

-10

0

10Spectrum at Input to Amplifier

Normalized Frequency (f/fsym)

Log

Mag

nitu

de (d

B)

(a) The power spectrum of a band-limited input signal.

-4 -3 -2 -1 0 1 2 3 4

-60

-50

-40

-30

-20

-10

0

10Spectrum at Output of Amplifier

Normalized Frequency (f/fsym)

Log

Mag

nitu

de (d

B)

SPECTRAL REGROWTH

(b) The power spectrum of the output signal passing through an amplifier.

Fig 2.19: The power spectrum comparison between the input signal and the output signal

passing through an amplifier.

There are several methods proposed to deal with the PAR problem of OFDM systems,

including clipping [82], Error-Control precoding, peak cancellation, PAR reduction codes

33

[83], symbol scrambling codes, and so on.

2.2.3 Channel estimation for OFDM systems

For wide-band wireless communication systems, the channel is time varying and

dispersive fading, which will distort the transmitted signal. Thus, the accurate and

real-time estimation of channel is a challenging task in OFDM systems.

The present channel estimation methods generally can be divided into two kinds. One

kind is based on the pilots and the other is blind channel estimation which does not use

pilots. Blind channel estimation methods do not use pilots and have higher spectral

efficiency. However, they often suffer from high computation complexity and low

convergence speed since they often need a large amount of receiving data to obtain some

statistical information such as cyclostationarity induced by the cyclic prefix. Blind channel

estimation methods are not suitable for applications with fast varying fading channels.

There are two classical pilot patterns for the pilot aided channel estimation methods,

which are the block-type pattern and the comb-type pattern, as the following Fig 2.20

shows.

34

(a) The block-type pilot pattern.

(b) The comb-type pilot pattern.

Fig 2.20: Two kinds of pilot patterns (black dot: pilot, white dot: user data).

The block-type pilot pattern is depicted in Fig 2.20(a). The pilots are inserted into all the

subcarriers of one OFDM symbol with a certain period. The block-type can be adopted in

slow fading channel, that is, the channel is static within a certain period of OFDM symbols.

35

The comb-type pilot pattern is plotted in Fig 2.20(b). It refers to that the pilots are inserted

at some specific subcarriers in each OFDM symbol. The comb-type is preferable in fast

varying fading channels, that is, the channel varies over two adjacent OFDM symbols but

remains static within one OFDM symbol. The comb-type pilot arrangement based channel

estimation has been shown to be more applicable since it can track fast varying fading

channels in comparison with the block-type one. When the fading channel can not be

viewed as static within an OFDM symbol, ICI occurs. The comb-type pilot pattern can not

eleminate ICI and the grouped pilot pattern was proposed by Song and Lim [29] to

suppress ICI. The grouped pilot pattern will be discussed in Chapter 4 in detail.

( )pY k

( )pX k

( )pH k ( )H k( )Y k

Fig 2.21: Pilot-aided channel estimation for OFDM systems.

Fig 2.21 depicts the basic pilot-aided channel estimation processing for OFDM systems.

Firstly, the pilot signal Yp (k) is extracted from the received signal after FFT operation Y(k),

where k is the index of subcarrier of one OFDM symbol. Secondly, the channel estimator

performs channel frequency response estimation at pilot subcarriers. There are some

channel estimation methods for this part such as LS, MMSE estimator and so on. Thirdly,

once the channel frequency response estimation at pilot subcarriers, ( )pH k , is obtained, the

36

estimator performs interpolation to obtain channel frequency response estimation at all

subcarriers. There are linear interpolation method, second-order polynomial interpolation

method [15], DFT based interpolation method and so on. Next, we briefly describe the LS

estimator and the MMSE estimator.

9 Least square (LS) estimator It is assumed that the channel frequency response at pilot subcarriers is

[ (0) (1) ( 1)]Tp p p p pH H H N= H " (2-28)

where Np is the number of pilots and the notation ()T denotes transpose. The extracted

pilot signal vector is given by

[ (0) (1) ( 1)]Tp p p p pY Y Y N= Y " (2-29)

Then the vector pY can be further expressed as

p p p p= +Y X H W (2-30)

where

(0) 0

0 ( 1)

p

p

p p

X

X N

= X %

pW is the Guassian noise vector at pilots positions and it is given by

Wp = [Wp(0) Wp(1)Wp(Np-1)]T (2-31)

Let [ (0) (1) ( 1)]Tp p p p pH H H N= H " denote the estimated channel response. According

the LS criterion, ( ) ( )Hp p p p p p p= e Y H X Y H X should achieve the minimum. By

differential operation to ep, we have that

37

,H

p ls ls p pQ=H X Y (2-32)

is enable ep to achieve the minimum value, where the matrix QLS is given by

1( )HLS p pQ= X X (2-33)

Therefore the frequency response estimation at pilot subcarriers is given by

1,

(0) (1) ( 1)(0) (1) ( 1)

T

p p p pp ls p p

p p p p

Y Y Y NX X X N

= = H X Y " (2-34)

9 Minimum Mean Square Error (MMSE) estimator

Minimum mean square error (MMSE) estimator has a better estimation accuracy

compared to LS estimator in terms of MSE. At the same MSE, the MMSE estimator has

about 5 to 12 dB SNR gain over that of LS estimator. However, the main shortcoming of

the MMSE estimator is the higher computation complexity. With the increase of operation

points, i.e., the number of subcarriers, its computational complexity will increase

exponentially. The literature [84] also proposed the LMMSE estimator through linear

approximation to the MMSE estimator, using the transmitted signal correlation matrix in

frequency domain. The MMSE estimation in frequency domain for the pilot

subcarriers, , p mmseH , is given by

( ), , , 11 2 1, , , ( )p p ls p ls p ls p p p p Hp mmse H H H H p ls H H H H w p p p lsR R R R = = +H H X X H (2-35) where , p lsH is the channel frequency response at pilots positions using LS algorithm and it

is expressed in (2-34). 2w is the variance of the AWGN noise, the cross correlation

matrices in (2-35) are given by

{ },p p

HH H p pR E= H H

38

{ }, , ,p p ls HH H p p lsR E= H H { }, , , , p ls p ls HH H p ls p lsR E= H H . (2-36)

where E() denotes expectation and the notation ()H denotes Hermitian transpose, Hp is

given by formula (2-34).

2.2.4 Synchronization of OFDM systems

The synchronization of the received signal is an important topic in telecommunications.

OFDM system is very sensitive to frequency offset and phase noise, so synchronization is

the key problem for OFDM systems. Lack of necessary synchronizer, the orthogonality

among sub channels of an OFDM symbol will be destroyed, which will introduce ICI. The

synchronization of OFDM system includes timing and frequency synchronization.

Timing synchronization includes symbol synchronization and sample clock

synchronization. The aim of symbol synchronization is to seek for an appropriate FFT

window. If a FFT window can not be found correctly, ISI will be introduced so that OFDM

system performance will be reduced. The aim of sampling clock synchronization is to

synchronize the local sample clock at the receiver and the sender clock at the transmitter. If

there is an error between the transmitter clock and the receiver clock, the ICI will occur,

and it will also affect the symbol synchronization so that worsens the system performance.

For single carrier communication systems, the carrier frequency offset only produce a

phase rotation of the transmitted signal. And the phase rotation can be recovered by

channel equalization. However, for OFDM systems, the frequency asynchrony between the

transmitter carrier and the receiver carrier will destroy the orthogonality between the sub

39

carriers within one OFDM symbol. ICI could also be introduced due to the carrier

frequency offset, which could seriously decrease the system performance. Thus, frequency

synchronization is an indispensable part in OFDM systems.

There are a lot of synchronization algorithms, which can be divided into three kinds,

that is, CP based method [87], Preamble based method [85], [86], and pilots based method

[88].

2.2.5 Advantages and disadvantages of OFDM systems

Generally speaking, the followings are the advantages of OFDM systems.

9 High spectrum efficiency.

9 Through CP insertion, ISI can be eliminated.

9 OFDM system adopts FFT/IFFT to implement modulation and demodulation so that

the computational complexity is low.

9 The receiver only needs to perform simple equalization.

9 High system capacity and high data rate.

And the disadvantages of OFDM system include:

9 The amplitude of OFDM varies greatly and has large PAR, which will result in the

distortion of signal. So it is necessary that the RF power amplifier has a wide linear

range to ensure that distortion of the transmitted signal is reduced as far as possible.

9 OFDM system is sensitive to frequency excursion, so it has strict requirement about

frequency synchronization.

40

2.3 MIMO systems

MIMO is an important breakthrough in wireless communication. MIMO technique was

first proposed by Marconi in 1908 and used to cancel the channel fading in order to

enhance the system performance. However, the signal processing costs were very high at

that time so that MIMO was not given proper due. Until the early 1990s, people began to

study MIMO technique again as the signal processing has been further developed.

2.3.1 Basic MIMO system model

Fig 2.22: The basic MIMO system model.

Fig 2.22 shows the basic MIMO system model with NT transmitter antennas and NR

receiver antennas, where H denotes the fading channel. MIMO systems can be divided

into two categories, shown as follows.

2.3.1.1 MIMO system with flat fading channels

When the fading channel is flat, that is, there is only one resolvable path, the impulse

response between the n-th transmitter antenna and the m-th receiver antenna, hm,n, is

41

expressed by

, ,( , ) ( ) ( )m n m nh t h t t = (2-37)

Thus, the MIMO system channel matrix is given by

, . ,( ) ( )R T R Tm n N N m n m n N Nh a jb = = +H (2-38)

where am,n and bm,n are Gaussian random variables with zero mean and unity variance. The

received signal Y, is given by

= +Y HX w (2-39)

where the received signal Y = 1 2 RT

Ny y y " , the transmitted signal X =

1 2 T

T

Nx x x " , the AWGN noise w = 1 2 RT

Nw w w " .

2.3.1.2 MIMO system with frequency selective fading channels

When the number of channel resolvable paths is larger than 1, the channel is frequency

selective. The impulse response between the n-th transmitter antenna and the m-th receiver

antenna, hm,n, is expressed by

1

, , ,0

( , ) ( ) ( )L

m n m n l ll

h t h t =

= (2-40) where L is the number of resolvable paths, l is the delay of the l-th path. And the received

signal Y at n-th time slot, is given by

1

0

L

l n l nl

== +Y H X w (2-41)

where the received signal Y = 1 2 RT

Ny y y " , the transmitted signal X =

42

1 2 T

T

Nx x x " , the AWGN noise w = 1 2 RT

Nw w w " , the channel matrix of the l-th path, , ,( ) R Tl m n l N Nh =H .

2.3.2 Functions of MIMO systems

MIMO systems can provide two types of gains, that is, spatial multiplexing gain and

diversity gain. For spatial multiplexing, a high rate signal is split into multiple lower rate

streams and each stream is transmitted from a different transmit antenna. The receiver

could separate these streams, provided that these data streams arrive at the receiver

antenna array with low correlation coefficient between each antenna pair. Spatial

multiplexing technique is very powerful for increasing channel capacity. On the other hand,

MIMO systems can be designed to realize diversity in order to deal with the signal fading.

The basic idea of diversity is to provide the receiver several independent fading versions of

the same signal, so that the probability of simultaneously experiencing deep fading for all

versions can be reduced. Fig 2.23 depicts a received signal after diversity operation.

Observe that the combined signal is enhanced in SNR, compared with the 1-th branch

signal and the 2-th branch signal.

43

Signal after diversity

Signal of 2-th branch

Signal of 1-th branch

Fig 2.23: Received signal after diversity operation.

In summary, a MIMO system could provide two types of gains: diversity gain and

spatial multiplexing gain. To increase a kind of gain will reduce the second gain with no

doubt. There is a fundamental trade off between how much of each type of gain any

MIMO system scheme could extract: higer spatial multiplexing gain comes at the price of

sacrificing diversity gain. L. Zeng and N. C. Tse [89] provided an optimal tradeoff curve

between diversity gain and multiplexing gain. To formulate their result, we firstly give the

following definition.

Definition 1: A MIMO scheme{C(SNR)} is said to achieve spatial multiplexing gain r and

diversity gain d if the data rate

44

( )limlog( )SNRR SNR r

SNR= (2-42)

and the average error probability

log ( )limlog( )

eSNR

P SNR dSNR

= (2-43)

So we have the following result presented in Theorem 1 [89].

Theorem 1: The optimal tradeoff curve d*(r) is given by the piecewise-linear function

connecting the points (k, d*(k)), k = 0,1,..., min(NR, NT), where

d*(k) = (NR-k)(NT-k). (2-44)

where d is the diversity gain defined in (2-43), r is the multiplexing gain defined in (2-42).

NR is the number of receiver antennas and NT is the number of transmitter antennas.

Fig 2.24: Diversity-multiplexing tradeoff, d*(r) versus r.

45

Fig 2.24 further plots the diversity-multiplexing tradeoff curve. Observe that the optimal

tradeoff curve intersects the r axis at *maxr = min(NR, NT), which means that the maximum

achievable spatial multiplexing gain *maxr is the total number of degrees of freedom provided

by the channel, as suggested by the ergodic capacity result. And at this point, no positive

diversity gain could be achieved. That is, as r approaches *maxr , the data rate approaches the

ergodic capacity and there is not protection against the randomness in the fading channel.

2.3.3 Overview of Space Time codes

Space time codes for MIMO systems are divided into three kinds, including the space time

block code (STBC) [9], space time trellis code (STTC) [12], and layered space time code

(LSTC) [6]-[7].

Spacetime trellis codes (STTCs) distribute a trellis code over multiple antennas and

multiple time-slots and provide both coding gain and diversity gain. Spacetime block

codes (STBCs) act on a block of data at a time and provide only diversity gain, but are

much less complex in implementation compared with the STTCs. Both the STBC systems

and STTC systems are designed to achieve the diversity gain.

The LSTC systems are used to improve the data transmission rate, that is, to acquire the

spatial multiplexing gain. LSTC systems were first proposed by G. J. Foschini [6] in 1996.

The Bell laboratories layered space time (BLAST) system was exploited by G. J. Foschini

at Lucent Technologies' Bell Laboratories. The system is the earliest MIMO system in

history. Multiple antennas are equipped at both the transmitter and the receiver in order to

46

exploit the many different paths in a highly scattering wireless environment. By careful

allocation of the data to be transmitted to the transmitting antennas, multiple data streams

were transmitted simultaneously and independently within a single frequency band. Their

results showed that the capacity of the system grew directly in line with the relatively

small value of the number of transmitter antennas and the number of receiver antennas.

However, the performance of anti-fading is not good enough. Next, we introduce two

classic space time codes for MIMO systems, that is, STBC and V-BLAST systems.

2.3.3.1 Space time block code (STBC) for MIMO systems

+ +

C hannel estim ator 1 C om bination

C hannel estim ator 2

M axim um L ikelihood detection

A W G N noise w 1

A W G N noise w 2

T ransm itter A ntenna 1

R eceiver A ntenna 1

T ransm itterA ntenna 2

R eceiver A ntenna 2

1 1h

1s*2s

2s*1s

1s 2s

1S 2S

1 1h h 22h 21h 12

h 12

1 1h h 21h 21 h 22 h 12 h 22

Y 1 Y 2

Fig 2.25: The Alamouti STBC diagram for 22 MIMO systems.

47

STBC was first proposed by Alamouti in 1998 and was further developed by V. Tarohk et.

al [91] according to the general orthogonal design principle. Since the column of the

coding matrix is orthogonal to the row of the coding matrix at the transmitter, the receiver

can perform maximum likelihood (ML) detection with low computational complexity.

Fig 2.25 shows the Alamouti STBC diagram, for NR = NT = 2, where NR is the number of

receiver antennas, NT is the number of transmitter antennas. At time t, two transmitted

signal S1 and S2 are sent out. At the next time t + T, the conjugate versions of the previous

signals are sent out. The superscript ()* in Fig 2.25 denotes conjugate. We use matrix to

represent the transmitted signal.

*1 2

*2 1

s ss s =

X (2-45)

Let X1 denote the first row of the matrix X and X2 denote the second row of the matrix X.

We have

1 2 0X X =i (2-46)

where ( )i denotes inner product of two vectors. That is, the two rows are orthogonal. Next,

the fading channel is assumed to be flat and static within two adjacent symbols, that is, hij(t)

= hij(t +T), for i = j = 1, 2. At time t, the received signals Y1(t) and Y2(t) are given by

1 11 1 21 2 1( ) ( )Y t h s h s w t= + +

2 21 1 22 2 2( ) ( )Y t h s h s w t= + + (2-47)

At time t + T, the received signals Y1(t+T) and Y2(t+T) are given by

* *1 11 2 21 1 1( ) ( )Y t T h s h s w t T+ = + + +

48

* *2 21 2 22 1 2( ) ( )Y t T h s h s w t T+ = + + + (2-48)

It is assumed that the channel estimators in Fig 2.25 are perfect. The output of the channel

estimator hij, i, j = 1, 2, is then put into the combiner. The combiner generates 1s and 2s , and

1s , 2s are given by * * * *

1 11 1 21 1 12 2 22 2* * * *11 11 1 21 2 1 21 11 2 21 1 1

* * * *12 12 1 22 2 2 22 12 2 22 1 2

2 2 2 2 *11 21 12 22 1 11 1

( ) ( ) ( ) ( )

( ( )) [ ( ) ( )]

( ( )) [ ( ) ( )]

(| | | | | | | | ) ( )

s h Y t h Y t T h Y t h Y t Th h s h s w t h h s h s w t Th h s h s w t h h s h s w t Th h h h s h w t

= + + + + += + + + + + + +

+ + + + + += + + + + + * * *21 1 12 2 22 2

2 2 2 211 21 12 22 1

( ) ( ) ( )

(| | | | | | | | )

h w t T h w t h w t Th h h h s w

+ + + += + + + +

(2-49)

* * * *2 11 1 21 1 12 2 22 2

2 2 2 2 * * * *11 21 12 22 2 11 1 21 1 12 2 22 2

2 2 2 211 21 12 22 2

( ) ( ) ( ) ( )

(| | | | | | | | ) ( ) ( ) ( ) ( )

(| | | | | | | | )

s h Y t T h Y t h Y t T h Y th h h h s h w t T h w t T h w t h w t Th h h h s w

= + + + += + + + + + + + +

= + + + +

(2-50)

Finally, the two signals of each branch, 1s , 2s , are put into the maximum likelihood

detector to generate the estimated transmitted signal, that is, 1s and 2s .

49

2.3.3.2 V-BLAST systems

Serial

to

parallel

TX

TX

TX

RX

RX

RX

RX

RX

1

TN

1

RN

channel

estimation

and

data

detection

Fig 2.26: V-BLAST system diagram.

The vertical Bell Laboratories layered space time (V-BLAST) architecture [7] is a typical

example of spatial multiplexing technique. As in Fig 2.26, in V-BLAST systems, each data

stream is extracted by a transmitter antenna independently and there is no coding cross

transmitter antennas. The fading channel is flat. Thus, the received signal is given by

= +Y HX w (2-51)

where the received signal Y = 1 2 RT

Ny y y " , the transmitted signal X =

1 2 T

T

Nx x x " , the AWGN noise w = 1 2 RT

Nw w w " , the channel matrix H is

given by

, . ,( ) ( )R T R Tm n N N m n m n N Nh a jb = = +H . (2-52)

The aim of data detection for V-BLAST systems is to detect the transmitted signal X from

the received signal Y. The algorithms of data detection for V-BLAST systems are divided

50

into four kinds. The followings give a brief overview of the four kinds of algorithms.

9 Maximum Likelihood (ML) detection

The estimated signal x for ML detection is given by

arg min= x

x Y HX (2-53)

The ML detection is to find the optimal x to satisfy formula (2-53) and the computational

complexity is very high although it can achieve the optimal performance in terms of BER.

9 Zero Forcing (ZF) detection

The estimated signal x for ZF detection is given by

+=x H Y (2-54) where ( H H+ = 1H H H) H is the pseudo inverse of H. ZF algorithm is very simple and easy

to implement, however, the performance is worse than ML algorithm.

9 MMSE detection

Firstly, the estimated channel matrix MMSEG is acquired by minimizing the cost function

2( ) {|| || } {( ) ( )}He E E= = H X HY X HY X HY (2-55)

By differentiating above formula and making the first order derivative zero, we have the

estimated channel matrix is given by

2 2 1( / )H HMMSE w X = +G H H I H (2-56) where 2X is the power of the transmitted signal X. Therefore, the detected signal X is given by

MMSE=X G Y (2-57)

51

where GMMSE is expressed in (2-56). MMSE detection minimizes the overall error due to

noise and mutual interference among different sub streams and achieves better

performance in comparison with ZF detection. However, the computational complexity is

higher than that of ZF detection.

9 Successive Interference Cancellation (SIC) detection

The key idea of SIC detection is to distinguish the transmitted signal of each sub stream by

successively canceling the interferences between two sub streams. The SIC algorithm can

be summarized as follows.

Step 1: Initialization: let =Y Y and G1 be the first row of the matrix GMMSE or H+, where

GMMSE is given by (2-56) and H+ is the pseudo inverse of H. Note that if G1 is

chosen from GMMSE, the SIC algorithm is also referred to as MMSE-SIC detection.

Similarly, if G1 chosen from H+, the SIC algorithm is referred to as ZF-SIC

detection.

Step 2: For i = 1, the first sub stream signal is detected by