適用於多重輸入多重輸出正交分頻多工之二階統計特性預先編碼盲式等化器

國立交通大學

國立交通大學

國立交通大學

國立交通大學

電子工程學系

電子工程學系

電子工程學系

電子工程學系

電子研究所碩士班

電子研究所碩士班

電子研究所碩士班

電子研究所碩士班

碩

碩

碩

碩

士

士

士

士

論

論

論

論

文

文

文

文

適用於多重輸入多重輸出正交分頻多工之二階統計特

適用於多重輸入多重輸出正交分頻多工之二階統計特

適用於多重輸入多重輸出正交分頻多工之二階統計特

適用於多重輸入多重輸出正交分頻多工之二階統計特

性預先編碼

性預先編碼

性預先編碼

性預先編碼盲式等化器

盲式等化器

盲式等化器

盲式等化器

Second-Order Statistics Based

Precoder-Blind Equalization for MIMO-OFDM

學生 ： 郭冠麟

指導教授 ： 李鎮宜 教授, 馮智豪 教授

中華民國九十七年七月

中華民國九十七年七月

中華民國九十七年七月

中華民國九十七年七月

性預先編碼盲式等化器

性預先編碼盲式等化器

性預先編碼盲式等化器

性預先編碼盲式等化器

Second-Order Statistics Based

Precoder-Blind Equalization for MIMO-OFDM

研 究 生：郭冠麟 Student：Kuan-Ling Kuo

指導教授：李鎮宜, 馮智豪 Advisor：C.-Y. Lee, C. C. Fung

國 立 交 通 大 學

電子工程學系 電子研究所 碩士班

碩 士 論 文

A Thesis

Submitted to Department of Electronics Engineering & Institute of Electronics College of Electrical and Computer Engineering

National Chiao Tung University in Partial Fulfillment of the Requirements

for the Degree of Master of Science

in

Electronics Engineering July 2008

Hsinchu, Taiwan, Republic of China

i

適用於多重輸入多重輸出正交分頻多工之二階統計特

適用於多重輸入多重輸出正交分頻多工之二階統計特

適用於多重輸入多重輸出正交分頻多工之二階統計特

適用於多重輸入多重輸出正交分頻多工之二階統計特

性預先編碼

性預先編碼

性預先編碼

性預先編碼盲式等化器

盲式等化器

盲式等化器

盲式等化器

學生：郭冠麟 指導教授：李鎮宜 教授, 馮智豪 教授

國立交通大學

國立交通大學

國立交通大學

國立交通大學

電子工程學系

電子工程學系

電子工程學系

電子工程學系

電子研究所碩士班

電子研究所碩士班

電子研究所碩士班

電子研究所碩士班

摘要

摘要

摘要

摘要

許多通道效應，諸如路徑衰減、遮蔽效應，以及多重路徑衰減等等，已知會 對無線通訊系統的效能造成破壞。特別對於多載波系統而言，多重路徑衰減會引 起符號間干擾和載波間干擾而嚴重破壞傳輸的穩定性。與通道雜訊不同的是，符 號間干擾和載波間干擾所造成的位元錯誤率，無法透過簡單地升高訊雜比加以解 決。因此，通道等化時常成為必需步驟以減少此類不良影響。 傳統的等化程序包含了兩個步驟：通道估計與通道等化。首先，序言符號和 導引符號等大量的訓練符號被應用於估計通道；等化器便可根據通道估計加以設 計。然而，由於大量訓練符號的使用，頻譜效率無疑將減少。除此之外，當通道 為快速衰減時，將需要傳送更多的導引符號，進一步地消耗稀少的傳輸資源。隨 著高資料傳輸率的需求成長，此類型的傳輸代價將成為未來通訊系統的瓶頸。 在此篇論文中，一個適用於多重輸入多重輸出多頻分工系統的二階統計特性 預先編碼盲式等化器被提出以減輕此類傳輸代價。被提出的技術可以等化有限脈 衝響應的多重輸入多重輸出通道，並且不需要一個獨立的通道估計步驟。相較於

ii 考慮，造成等化器的設計未被最佳化。所提出的架構在傳送端使用了正交預先編 碼器，以滿足傳送訊號必須是時間相關的可辨別條件。此外，編碼器被設計為可 減少接收端等化器的時間延遲和計算複雜度。在位元錯誤率與計算複雜度方面， 此架構能優於先前提出的二階統計特性預先編碼盲式等化器。模擬結果顯示，位 元錯誤率的效能也逼近一個有理想通道資訊的最小平方迫零法等化器。模擬結果 也顯示，所提出系統在 10 根天線與訊雜比 10dB 的情況下，通道容量可達到 25 bits/s/Hz 以上。

iii

Second-Order Statistics Based

Precoder-Blind Equalization for MIMO-OFDM

Student：Kuan-Ling Kuo Advisor：C.-Y. Lee, C. C. Fung

Department of Electronics Engineering and Institute of

Department of Electronics Engineering and Institute of

Department of Electronics Engineering and Institute of

Department of Electronics Engineering and Institute of

Electronics,

Electronics,

Electronics,

Electronics,

N

NN

National Chiao Tung University

ational Chiao Tung University

ational Chiao Tung University

ational Chiao Tung University

Abstract

Channel effects such as path loss, shadowing, and multipath fading have been known to hamper performance of wireless communication systems. In particular, multipath fading causes intersymbol interference (ISI) and intercarrier interference (ICI) in the case of multicarrier systems, if it is not handled properly, can severely degrade link reliability. Unlike additive channel noise, BER degradation from ISI and ICI cannot be prevented by simply increasing the SNR. Hence, channel equalization is often necessary in order to mitigate these detrimental effects.

Conventional equalization procedure usually involves two steps which includes channel estimation followed by equalization. Large number of training symbols, such as preambles and pilots, are first used to identify the channel. An equalizer can then be designed using the channel estimate. However, spectral efficiency will undoubtedly

iv

consuming scarce transmission resources. With the growing demand for higher data rate, this type of transmission overhead will be a bottleneck for future communication systems.

In this thesis, a second-order statistics based precoder-blind equalizer for MIMO-OFDM system is proposed to alleviate the problems of transmission overhead. The proposed technique can equalize FIR-MIMO channels directly without an explicit channel estimation step. This is an advantage over the traditional two-step approach, where the channel is first estimated followed by the design of the equalizer coefficients, because the second step will likely result in a suboptimal equalizer since errors in the channel estimates are not taken into account. The proposed scheme uses a set of orthogonal precoders at the transmitter such that the transmitted signal can be colored temporally thereby satisfying the identifiability condition previously proposed by Hua and Tugnait. Moreover, the precoders are designed to reduce latency and computational complexity of equalization at the receiver. The scheme is shown to outperform previously proposed SOS-based precoder-blind equalization schemes in terms of BER and computational complexity. Simulation results have shown that the BER performance is close to that of a least-squares zero-forcing equalizer with perfect channel knowledge. Simulation results have also shown that the capacity of the proposed scheme can be above 25 bits/s/Hz while 10 antennas are exploited at SNR of 10 dB.

v

Acknowledgement

Thanks Profs. Chen-Yi Lee and Carrson C. Fung for inspiring me to see the possible ways throughout the journey of my study, giving valuable guidance to integrate my thinking and providing abundant resources during the past two years. Thanks to them, I was able to develop the research skills to undertake and complete this work.

I also like to thank my lab-mates, Jui-Yuan Yu and Tsan-Wen Chen, for their great help to me. They were necessary companions for this thesis to come into being.

1 Introduction 1

1.1 Research Motivation . . . 1

1.2 Thesis Organization . . . 5

1.3 Publications . . . 6

2 Fundamentals of MIMO-OFDM Systems 7 2.1 OFDM System Model . . . 7

2.2 MIMO-OFDM Systems Model . . . 11

2.2.1 Concept of MIMO System . . . 12

2.2.2 MIMO-OFDM Model . . . 17

2.3 Frame Format . . . 19

3 Precoder-Blind Equalization Algorithm for MIMO-OFDM 21 3.1 SOS-Based Blind Identifiability and Equalizability Conditions for FIR-MIMO Systems . . . 21

3.1.1 Identifiability Conditions . . . 21

3.1.2 Equalizability Conditions . . . 22

3.2 SOS-Based Blind Channel Identification and Equalization . . . 23

3.2.1 Existing SOS-Based Periodic Precoding Channel Identification . . . 23

3.2.2 SOS-Based Precoder-Blind Equalization . . . 29 vi

3.3 Precoder Design . . . 33

3.3.1 Precoder Format . . . 33

3.3.2 Temporal Correlation Injection . . . 36

3.4 Results and Discussions . . . 39

3.4.1 Computational Complexity . . . 39

3.4.2 Implementation of Precoders by Memory-Based IFFT/FFT . . . . 40

3.4.3 BER performance . . . 45

3.4.4 Channel Capacity . . . 54

4 Conclusion and Future Work 56 4.1 Conclusion . . . 56

4.2 Future Work . . . 57

ADC Analog-to-Digital Converter AWGN Additive White Gaussian Noise BER Bit Error Rate

CFO Carrier Frequency Offset CP Cyclic Prefix

CSI Channel State Information DAC Digital-to-Analog Converter DMT Discrete Multitone

FFT Fast Fourier Transform FIR Finite Impulse Response HOS Higher-Order Statistics ICI Intercarrier Interference

IFFT Inverse Fast Fourier Transform ISI Intersymbol Interference

JD Joint Diagonalization LOS Line-Of-Sight

LS Least-Squares

LTI Linear Time Invariant

MIMO Multiple-Input and Multiple-Output

OFDM Orthogonal Frequency-Division Multiplexing viii

PAPR Peak-to-Average-Power Ratio QPSK Quadrature Phase Shift Keying SCO Sampling Clock Offset

SIMO Single-Input and Multiple-Output SISO Single-Input and Single-Output SNR Signal-to-Noise Ratio

SOS Second-Order Statistics STBC Space-Time Block Code STC Space-Time Code STO Sampling Timing Offset STTC Space-Time Trellis Code

Notations: Upper (lower) bold face letters indicate matrices (column vectors). Super-scriptH _{denotes Hermitian,} T _{denotes transposition. E[·] stands for expectation. diag(x)}

denotes a diagonal matrix with x on its main diagonal; IN denotes an N × N identity

matrix; 0M×N denotes an M × N all zero matrix.

2.1 Simplified block diagram of OFDM systems . . . 8

2.2 Cyclic prefix of an OFDM symbol . . . 8

2.3 Spectrum of an OFDM symbol . . . 9

2.4 A transmitter of the OFDM system . . . 10

2.5 A receiver of the OFDM system . . . 10

2.6 A block diagram of the Alamouti space-time encoder . . . 13

2.7 Schematic of the generic MIMO system . . . 14

2.8 MIMO-OFDM system . . . 20

3.1 Block diagram of equalization process with 2 receive antennas where JD represents joint diagonalization. . . 31

3.2 The transmitter of the precoder-blind equalizer system with 2 Tx/Rx an-tennas. . . 34

3.3 The receiver of the precoder-blind equalizer system with 2 Tx/Rx antennas. 35 3.4 A typical memory-based FFT architecture [36]. . . 42

3.5 Spectrum of the coloring precoders. . . 45

3.6 Comparison of BER performance between system without coloring and with coloring for 3-tap channel. . . 46

3.7 BER vs. different number of received OFDM symbols for 3-tap channel at SNR = 10, 14, 18 dB. . . 47

3.8 Comparison of BER vs. SNR with different algorithms for 3-tap channel. . 50

3.9 Comparison of BER vs. SNR with different algorithms for 7-tap channel. . 50

3.10 BER vs. SNR with different lag for 3-tap channel. . . 51

3.11 BER vs. SNR with different P for 3-tap channel. . . 53

3.12 BER vs. SNR with different γ for 3-tap channel. . . 53

3.13 Capacity vs. Number of antennas with different SNR. . . 55

3.14 Capacity per n vs. SNR with different n. . . 55

2.1 Summary of frame format . . . 19

3.1 Values for Ci,τp for L + q > NtP . . . 38

3.2 Values for Ci,τp for L + q ≤ NtP . . . 38

3.3 Values for Ci,τp when q = 2, L = 2, P = 2, Nt = 2 . . . 39

3.4 Performance comparison . . . 49

Chapter 1

Introduction

1.1

Research Motivation

In general communication environments, many surrounding objects such as vehicles, build-ing or trees may act as reflectors of transmitted signal. Reflected waves are produced with attenuated amplitude by these obstacles and arrive at the receiver from various directions with different delays. The sum of these multiple reflected waves forms the multipath channel, which causes various forms of distortion on the transmitted signal. For a nar-rowband system, the transmitted signal usually occupies a bandwidth smaller than the coherence bandwidth of channels. Therefore, all spectral components of the transmitted signal experience the same fading during transmission. This class of fading channel is called frequency flat, which requires no equalization since the fading attenuation is fixed over the entire transmission. However, with the insatiable need to transmit more data at any time and anywhere, next generation wireless communication systems, such as IEEE 802.11n and 802.16m, are required to support higher data rate than their predecessors, while the mobile terminals are undergoing faster mobility. To deliver systems that can op-erate reliably under such harsh conditions, researchers have also focused their attention on the wireless broadband communications systems, which aim to offer multimedia services 1

requiring data rates beyond 2 Mbps. This can be achieved by increasing the sampling rate of the information bearing signal. However, this will likely cause the bandwidth of the transmitted signal to be larger than the coherence bandwidth of the channel, which distorts the spectrum of the transmitted signal differently at different frequencies and causes ISI in the time domain. In this case, the channel is said to be frequency selective fading.

Since its invention in the 1980s [1], OFDM has become the method of choice for high data rate transmission over frequency selective fading channels. One of the major advan-tages of OFDM is that it can completely equalize (assuming accurate CSI) a frequency selective fading channel with relatively low computational complexity compared to other signaling schemes. This can be achieved as long as the length of the CP is not less than the order of the fading channel.

Data rate and link reliability can be increased through the use of MIMO techniques which have been recently proposed in the literature [2,3]. Analytic and simulation results have shown that system capacity and link reliability can be dramatically improved by means of spatial multiplexing and spatial diversity, respectively. Spatial multiplexing can be achieved by methods such as VBLAST [4] or linear precoding [5]. In the latter approach, the source data is split into multiple data streams and precoded by a precoder matrix, such as the right singular vector matrix of the channel, before each stream is launched to a different transmit antenna. When the data arrive at the receiver, the receiver can recover the transmitted signal by simply multiplying the signal by the left singular vector matrix of the channel, followed by multiplication of the singular values of the channel. Therefore, the maximum number of data streams is limited by min(Nt, Nr),

where Nt and Nr denote the number of transmit and receive antennas, respectively. On

the other hand, spatial diversity is achieved when a single data stream is transmitted using multiple antennas with the assumption that there is negligible spatial correlation 1.1 Research Motivation 2

Chapter 1: Chapter

among the transmit and receive antennas. It has been shown that techniques such as STTC and STBC can be used to achieve full diversity order, but at the expense of lower code rate or increase computational complexity. The motivation is that the signal will undergo independent fading in different spatial channels such that even if one signal is corrupted, it is unlikely that its duplicates will be similarly distorted; allowing recovery of the transmitted signal to be possible.

Recently, a vast amount of literature has been devoted to the study of MIMO-OFDM systems; with the hope that the marriage of the two technologies can increase system performance without incurring a large penalty in terms of computations for broadband communication systems [6]. However, large transmission overhead in the form preamble, signal pilot and guard interval severely hamper the performance of such systems. For example, in the IEEE 802.16-2004 standard, widely known as fixed WiMAX, it was shown in [7] that the overhead introduced by the physical layer alone can be more than 50%. This has made blind channel estimation and equalization techniques for MIMO-OFDM systems an attractive alternative because of their ability to increase capacity of MIMO communication systems without sacrificing spectral efficiency [8, 9].

Traditionally, blind channel estimation and equalization has been based on HOS of the received signal [10, 11]. Although these schemes are technically different, most of them apply higher-order cumulants of the observation to optimize certain criteria in order to identify the zeros of a nonminimum phase system, however, under many strict condi-tions of transmitted signal. Higher-order whiteness and higher-order weak stationarity of transmitted signal are both required for the exploitation of higher-order cumulants. Furthermore, the estimates of HOS usually converge slower than those of SOS such that more received symbols are required. Hence, much of the research effort has shifted toward using SOS after the seminal work by [12] and [13] since SOS based techniques can also es-timate and equalize nonminimum phase FIR channels at much lower latency than its HOS 1.1 Research Motivation 3

counterparts. SOS based techniques can basically be categorized into either deterministic or statistics based approach. A deterministic approach was proposed in [13] to estimate FIR-SIMO channels. In this case, the system input is treated as an arbitrary unknown deterministic signal. The basic idea is to exploit commutability of the convolution oper-ator using hk[n] ∗ xj[n] = hj[n] ∗ xk[n], where hk[n] and xk[n] denote the channel impulse

response from the transmit antenna to the kth _{receive antenna and the received signal of}

the kth _{receive antenna, respectively. For each pair of (k, j), a set of linear equations can}

be given to solve the channel responses. A statistical approach for oversampled FIR-SISO systems is presented in [12]. The cyclostationarity of the received signal is used to iden-tify channels via oversampling, and the channel matrix is assumed to have full column rank. The correlation matrices with lag equal to 0 and 1 are both required. The noise power estimation is therefore demanded since the estimation of the correlation matrix with lag equal to 0 is contaminated by channel noise. These previous results have led to the work by [14] which exploited the subspace method to estimate FIR-SIMO channels. Good performance in terms of mean squared error (MSE) can be achieved in high SNR condition. However, its performance degrades at a fast rate in low SNR condition, such as 0 − 10 dB. An FIR-MIMO extension of the subspace method was proposed in [15, 16] which suffers from the same problem as its SIMO counterpart in low SNR condition. The subspace channel estimation method requires the channel transfer function matrix, H(z), to be irreducible and column-reduced [17], which limited the application of SOS based methods to a narrow class of communication channels.

Recently, [18] has shown that a weaker condition for the identifiability of H(z) exists, where H(z) can be identified up to scaling and permutation ambiguity if H(z) is irreducible and the power spectral density matrix of the channel input signal is a diagonal matrix with distinct diagonal functions. [19] has proposed an algorithm for estimating H(z) un-der this weaker condition, but a direct equalization algorithm was never discussed. [20] 1.1 Research Motivation 4

Chapter 1: Chapter

has proposed a SOS based blind equalization algorithm which implicitly uses the identi-fiability conditions stated in [18] for flat fading channels, where the number of transmit antennas, Nt, has to be equal to the number of receive antennas, Nr. [21] also exploited

this condition by designing a novel SOS based channel estimation algorithm to estimate MIMO channels for OFDM based systems. The algorithm uses cyclic power spectral den-sity of the received signal to decouple the MIMO channels into parallel SISO channels for estimation. The technique requires the use of a precoder to inject cyclostationarity into the input bitstream. Although not stated in [21], but the precoder actually colors the signal such that MIMO channel equalization using SOS is possible. [22] has extended the SOS algorithm in [20] such that any FIR-MIMO channel H(z) can be equalized up to a scaling, phase, and block delay ambiguity given that the identifiability conditions in [18] are satisfied. This was accomplished by designing the blind FIR equalizer within the space-time precoder-equalizer system where redundancy is injected into the transmit-ted bitstream to make FIR-MIMO channel equalization possible using an FIR equalizer. In [22], the independently distributed input signal streams were colored using a set of low complexity filters to satisfy the power spectral density condition stated in [18] such that the algorithm in [20] can be extended to be applicable to ISI channels and general space-time systems. However, the precoder that was proposed was not optimally designed. As shown in the Section 3.4, this not only impacts the equalization performance, but also increases the computational complexity at the receiver.

1.2

Thesis Organization

In this thesis, we extended the precoder-blind equalization scheme to MIMO-OFDM spa-tial multiplexing system, and proposed a new set of precoder to perform direct channel equalization for MIMO-OFDM systems such that improved BER performance and lower receiver complexity can be achieved compared to that of [22]. In Chapter 2, we will 1.2 Thesis Organization 5

give a description of the system model for MIMO-OFDM, followed by a review of blind identification conditions, blind equalization conditions and the equalization algorithms in Chapter 3. A novel precoder design and a blind equalizer for MIMO-OFDM are pro-posed in Chapter 3, which directly equalize an MIMO-OFDM channel blindly without a explicit channel estimation step. Simulation results will show that the proposed algo-rithm has lower or comparable performance as other SOS-based algoalgo-rithms but with less computational complexity. The conclusion and future work will be discussed in Chapter 4.

1.3

Publications

Conference:

Kuan-Ling Kuo, Tsan-Wen Chen, Carrson C. Fung and Chen-Yi Lee, Second-Order Statistics Based prefilter-Blind Equalization for MIMO-OFDM, to be presented at the 14th _{Asia-Pacific Conference on Communications, Oct. 2008.}

Journal:

Kuan-Ling Kuo, Tsan-Wen Chen, Carrson C. Fung and Chen-Yi Lee, Second-Order Statistics Based Precoder-Blind Equalization for MIMO-OFDM, in preparation.

Chapter 2

Fundamentals of MIMO-OFDM

Systems

2.1

OFDM System Model

Multicarrier systems, such as OFDM, have often been used to combat against frequency selective fading channel effects. With OFDM, the data is transmitted in various narrow-band orthogonal channels. This is accomplished easily by the use of IFFT and FFT to modulate and demodulate the signal. A simplified block diagram of OFDM is shown in Figure 2.1, where fk denotes the subcarrier and ˜η(n) is the channel noise. On one hand,

IFFT and FFT are efficient in terms of spectrum usage and implementation because only one pair of oscillators for the I- and Q-path is required by the RF front-end instead of multiple oscillators to modulate different paths at different carriers. When the transmit-ted signal passes through the channel, ISI and ICI usually occur. CP is hence introduced to combat ISI and ICI. The CP, shown in Figure 2.2, is a copy of the tail part of a OFDM symbol, which is inserted between the current symbol and its preceding symbol. As long as the length of the CP is not less than the order of the fading channel, ISI can be avoided. This can be explained by noting the channel matrix, with the addition of the 7

CP of sufficient length, is now a circulant matrix such that it can always be diagonalized by a FFT matrix. Consequently, the frequency selective fading channel has now been converted into a frequency flat fading channel.

ISI channel H

f

_k P/S S/P

IFFT

S/P P/S

-f

_k

%

( )

_n

η

FFT

Figure 2.1: Simplified block diagram of OFDM systems

CP

T

g

T

s

Figure 2.2: Cyclic prefix of an OFDM symbol

With all its strengths, OFDM-based systems do suffer from problems such as high PAPR and high sensitivity to frequency synchronization. High PAPR is possible since the linear summation of independent phases from all subcarriers often lead to a constructive combination. A DAC and an ADC with high resolution, which result in large hardware complexity and cost, are thereby needed to transmit OFDM signal with high PAPR. Linearity range of power amplifiers is also required to be large to prevent saturation. This undoubtedly leads to the use of expensive linear power amplifier. In fact, a small amount 2.1 OFDM System Model 8

Chapter 2: Chapter

Figure 2.3: Spectrum of an OFDM symbol

of peak clipping is usually allowed to limit the PAPR [23]. Moreover, nonideal effects of frequency synchronization such as SCO, STO and CFO will also degrade performance of OFDM-based systems because they will induce ICI and ISI into the system.

Since the modulation and demodulation of OFDM are implemented efficiently by using IFFT and FFT operation, respectively, the output of the IFFT is given by

u[n] = M_X−1 m=0 s[m]ej2πMmn = M_X−1 m=0 s[m]φm[i],

where s[m] is the source data in the mth _{path. Similarly, the output of the FFT is given}

as y[n] = M_X−1 m=0 x[m]e−j2πMmn = M_X−1 m=0 x[m]ψm[i],

where x[m] is the received data in the mth _{path. M is the IFFT/FFT size.}

Figure 2.4 and Figure 2.5 show the transmitter with CP insertion and receiver with CP removal of an OFDM system, respectively. At the transmitter, the IFFT transforms source data from frequency-domain into time-domain. After a parallel-to-serial conversion, the CP is inserted to the information bearing signal before it is delivered over the channel. At the receiver, CP is first removed followed by FFT demodulation. Assumed the given channel is constant during the transmission of an OFDM symbol, the received signal x[n] 2.1 OFDM System Model 9

IFFT

P/S

S/P

v (M v) v M × −
     

0

I

I

CP insertion

Figure 2.4: A transmitter of the OFDM system

S/P

_FFT

P/S

CP

deletion

[0

_Mxv

I

_M

]

Figure 2.5: A receiver of the OFDM system

Chapter 2: Chapter can be expressed as

x[n] = u[n] ∗ h[n] + η[n],

where u[n] is the transmitted data, η[n] represents the channel noise, h[n] denotes the channel impulse response, and ∗ denotes linear convolution. Because of multipath chan-nels, orthogonality as shown in Figure 2.3 will be destroyed by ISI and ICI. However, as long as the length of the CP is not less than the order of h[n], ISI effect can be avoided. At the same time, linear convolution of u[n] and h[n] will become circular convolution be-cause of the insertion of CP. It is known that circular convolution in time domain results in multiplication in frequency domain when the channel is stationary. The demodulated signal y[k] in frequency domain at the receiver can then be written as

y[k] = h′[k]s[k] + η′[k],

which is the product of source data s[k] and subcarrier channel response h′_{[k] plus the}

noise η′[k] in frequency domain. Consequently, the orthogonality among subcarriers is maintained without ICI.

2.2

MIMO-OFDM Systems Model

The quality of a wireless link can be measured by three metrics: the transmission rate, the transmission range and the transmission reliability. Conventionally, the transmission rate may be increased by reducing the transmission range and reliability. By contrast, the transmission range may be extended at the cost of a lower transmission rate and reliability, while the transmission reliability may be improved by reducing the transmission rate and range [24]. However, with the advent of MIMO-OFDM systems, the three metrics mentioned above may be simultaneously improved as described in [24]. MIMO-OFDM communication systems have shown that an increased capacity, coverage and reliability is achievable with the aid of MIMO techniques. Although MIMO techniques can potentially 2.2 MIMO-OFDM Systems Model 11

be combined with any other modulation or multiple access schemes, MIMO-OFDM has attracted extensive research because high spectral efficiency and high reliability can be obtained with relatively low computational complexity.

2.2.1

Concept of MIMO System

MIMO is a technology which exploits the rich scattering environment at the transmitter and/or receiver such that when multiple spatial data streams are launched into the chan-nel, they will be distorted independently by the channel; thereby increasing the probability of recovering the transmitted data at the receiver. One of the motivation for MIMO is to significantly increase the data throughput without additional bandwidth or transmit power. This is achieved by taking advantages of the rich scattering environment sur-rounding the transmission terminal and by using spatial multiplexing, where a high-rate source data stream is split into multiple low rate data streams according to the number of transmit antennas. Each data stream is then emitted by a different transmit antenna in the same bandwidth. In a rich scattering environment, the transmitter is able to transmit the signal in parallel channels with distinguished non-zero eigenmode. The transmission from different transmit antennas are hence distinguished. Through spatial multiplexing, multiple data streams can be transmitted simultaneously over independent parallel chan-nels, and therefore the transmission rate and capacity will be increased. Consequently, MIMO techniques are often applied for high speed broadband wireless communications.

Considering the trade-off between capacity and reliability, it is possible to increase the link reliability by the use of transmit diversity. If a single data stream is transmitted by multiple transmit antennas, several observations of the same stream will be obtained by multiple receive antennas. Under a rich scattering environment, the spatial diversity can be maximized for the fixed number of antennas. In this case, fading effects of channels are reduced such that the overall system becomes more reliable. STC techniques such as 2.2 MIMO-OFDM Systems Model 12

Chapter 2: Chapter

STTC and STBC are usually applied to MIMO systems with transmit diversity. The basic idea of STC is transmitting multiple and redundant copies of a data stream to achieve transmit diversity. An example of STBC using Alamouti code [25] is shown in Figure 2.6, which has a simple two-branch transmit diversity. A block of two modulated symbols, u1

and u2, is encoded by a coding matrix. The encoder outputs are then transmitted in

Modulator

Encoder

[

u u

1 2

]

1 2 2 1

u

u

u

u

∗ ∗  

−

  

[

u u1 2

]

1

u

1

u

2 ∗  

=

−



u

2

u

2

u

1 ∗  

=



u

[

s s1 2

]

Ant 1

Ant 2

Figure 2.6: A block diagram of the Alamouti space-time encoder

two consecutive transmission periods from two transmit antennas, which are denoted as u1 = [u1−u∗2] and u2 = [u2u∗1], respectively. In the first transmission period, u1and u2 are

transmitted simultaneously from the first antenna and the second antenna, respectively. During the second period, −u∗

2 and u∗1 are transmitted from the first antenna and the

second antenna, respectively. It is clear that the encoding involves with both space and time domains. Furthermore, the inner product of u1 and u2 is zero. In other words,

u1· u2 = u1u∗2− u∗2u1 = 0,

such that the transmit sequences from two transmit antennas are orthogonal. The key feature of this scheme is that a full diversity gain can be achieved with a simple maximum-likelihood decoding algorithm at the receiver with perfect CSI. STBC with the number of transmit antennas greater than 2 based on orthogonal designs are discussed in [26]. Obviously, a trade-off between spatial multiplexing and spatial diversity for MIMO exists. 2.2 MIMO-OFDM Systems Model 13

For a MIMO system with the fixed number of antennas, data rate and link reliability cannot be optimized simultaneously under the same channel [27].

Ant 1

TX

Ant 2

Ant N

_t

h

11

h

NrNt

RX



( )

r N n 

( )

2 n 

%

( )

1

n

η

Figure 2.7: Schematic of the generic MIMO system

The schematic of a generic single-user MIMO system is illustrated in Figure 2.7, where Nt transmit antennas and Nr receive antennas are equipped at the transmitter and

re-ceiver, respectively. The following are conditions assumed for the MIMO system in Figure 2.7:

• The channel maintains invariant during the transmission of one frame, which means that channel is block or slow fading.

• The channel is frequency-flat fading, which means that spectrum of channel is con-stant over the whole bandwidth such that channel gain can be represented by a 2.2 MIMO-OFDM Systems Model 14

Chapter 2: Chapter complex number.

Under the above assumptions, the input/output relation of a narrowband, single-user MIMO system can be written as

x[n] = Hu[n] + η[n], where x[n] = [ x1[n] x2[n] · · · xNr[n] ] T , u[n] = [ u1[n] u2[n] · · · uNt[n] ] T , η[n] = [ η1[n] η2[n] · · · ηNr[n] ] T .

u[n] is an Nt× 1 transmit signal vector, x[n] is an Nr× 1 receive signal vector, H is the

Nr× Nt channel matrix, and η[n] is the Nr× 1 channel noise vector. The channel matrix

is given as H =           h11 h12 · · · h1Nt h21 h22 · · · h2Nt .. . ... . .. ... hNr1 hNr2 · · · hNrNt           ,

where hij = αij+ jβij = |hij|ejφij represents the complex gain of the channel from the jth

transmit antenna to the ith _{receive antenna. α}

ij and βij are the real part and imaginary

part of hij, respectively. φij is the phase angle of hij and |hij| is the magnitude. If αij

and βij are independent and Gaussian distributed random variables, then |hij| is Rayleigh

distributed, which leads to a Rayleigh flat-fading channel.

While the transmission bandwidth is larger than the coherent bandwidth of channels, the channel is considered to be frequency selective. Denoting a frequency selective fading channel impulse response from the jth _{transmit antenna to the i}th _{receive antenna as}

hij = [hij[0] hij[1] . . . hij[q]]T, where q is the channel order. The Nr × Nt MIMO

matrix can then be written as Hℓ =           h11[ℓ] h12[ℓ] · · · h1Nt[ℓ] h21[ℓ] h22[ℓ] · · · h2Nt[ℓ] .. . ... . .. ... hNr1[ℓ] hNr2[ℓ] · · · hNrNt[ℓ]           ,

where ℓ = 0, 1, . . . , q. The input/output relation of the frequency selective fading channel can then be written as

x[n] =

q

X

ℓ=0

Hℓu[n − ℓ] + η[n]. (2.1)

Compared with frequency flat fading attenuation, (2.1) is a linear superposition of the product of Hℓ and the transmit signal vector due to convolution. (2.1) can be rewritten

more compactly if we further define

H,           H0 H1 · · · Hq 0 · · · 0 0 H0 H1 · · · Hq 0 · · · 0 .. . . .. ... ... ... ... ... ... 0 · · · 0 H0 H1 · · · Hq           and ˇ x[n] , xT[n] xT[n − 1] · · · xT[n − L + 1]T , ˇ u[n] , uT_{[n] u}T_{[n − 1] · · · u}T_{[n − L − q + 1]}T _, ˇ η[n] , ηT[n] ηT[n − 1] · · · ηT[n − L + 1]T ,

where H is an NrL × Nt(L + q) MIMO frequency selective fading channel matrix. L is the

number of the received signal vector. Using these definitions, the input/output relation in (2.1) can be written as

ˇ

x[n] = Hˇu[n] + ˇη[n].

Chapter 2: Chapter

2.2.2

MIMO-OFDM Model

In this thesis, we shall consider a MIMO-OFDM system with Nt transmit antennas and

Nr receive antennas as shown in Figure 2.8. A single transmit antenna is employed by

each OFDM modulator including IFFT and CP insertion. Let s(i)_m,ℓ denotes the complex-valued data symbol transmitted on the mth _{tone in the ℓ}th _{OFDM symbol from the i}th

transmit antenna for i = 1, 2, . . . , Nt. Also, let K = M + v denote the overall OFDM

symbol length, where M is the size of the IFFT/FFT and v is the length of the CP. Then the transmitted signal ui[n] after CP insertion can be written as [21]

ui[n] = X ℓ g[n − ℓK] M−1_X m=0 s(i)_m,ℓej2π_Mm(n−ℓK)_,

where g[n] is a rectangular function rect[0,K−1][n] with

rect[T1,T2][n] =      1, n = T1, T1+ 1, . . . , T2. 0, otherwise.

Then the received signal at the kth _{receive antenna can be written as}

xk[n] = Nt X i=1 " X ℓ hk,i[ℓ]ui[n − ℓ] # + ηk[n], (2.2)

where ηk[n], for k = 1, 2, . . . , Nr, is the stationary additive white channel noise at the kth

receive antenna and hk,i[ℓ] is the discrete-time channel impulse response of the channel.

Defining x[n] _{, [ x}1[n] x2[n] · · · xNr[n] ] T , u[n] , [ u1[n] u2[n] · · · uNt[n] ] T , η[n] , [ η1[n] η2[n] · · · ηNr[n] ] T ,

as the spatial receive signal vector, transmit signal vector and the channel noise vector, respectively, then (2.2) can be written as

x[n] = X

ℓ

Hℓu[n − ℓ] + η[n], (2.3)

where [Hℓ]k,i = hk,i[ℓ] is the Nr× Nt channel matrix of order q, that is, H(z) = q X ℓ=0 Hℓz−ℓ

is the channel transfer function matrix.

Assuming L OFDM symbols are transmitted. Defining the convolution matrix

H,           H0 H1 · · · Hq 0 · · · 0 0 H0 H1 · · · Hq 0 · · · 0 ... ... ... ... ... ... ... ... 0 · · · 0 H0 H1 · · · Hq           , and ˇ x[n] , xT[n] xT[n − 1] · · · xT[n − L + 1]T , ˇ u[n] , uT_{[n] u}T_{[n − 1] · · · u}T_{[n − L − q + 1]}T _, ˇ η[n] , ηT[n] ηT[n − 1] · · · ηT[n − L + 1]T

as the spatiotemporal received signal vector, transmitted signal vector and noise vector, respectively, then (2.3) can be expressed as

ˇ

x[n] = Hˇu[n] + ˇη[n].

Chapter 2: Chapter

2.3

Frame Format

Conventionally, information in the physical layer is divided into frames for transmission, where each frame consists of a header followed by a payload. The header contains the preambles, which is a unique identifier that can be used for synchronization and channel estimation. The payload contains both actual data along with intermittent pilot symbols that are used to establish and maintain CSI during transmission. However, assuming there is no amplitude, phase and permutation ambiguity, the pilot symbols are not needed for the blind equalization system under consideration because equalization can be achieved by only utilizing the portion of the payload. Synchronization can also be achieved by a number of techniques such as [28], [29] and [30], and it is not considered in this thesis. The frame format contains 400 OFDM symbols with CP inserted between OFDM symbols. Table 2.1 shows a summary of frame format.

Table 2.1: Summary of frame format IFFT/FFT block size 64 Number of OFDM symbol in one frame 400

Number of data subcarriers 64 Number of pilot subcarriers 0

Length of cyclic prefix 1₄ OFDM symbol length Constellation mapper QPSK

Coded bits per subcarrier 2

ISI channel H  ( ) 1 n  Ant 1 IFFT P/S S/P IFFT P/S S/P Ant i S/P S/P FFT P/S FFT P/S ( ) v v Mv M × −         0 I I CP deletion [0_Mxv I_M] ( ) v v M v M × −         0 I I CP deletion [0_Mxv I_M] CP insertion CP insertion  ( ) r N n  S/P FFT P/S CP deletion [0_Mxv I_M] IFFT P/S S/P v(Mv) v M × −         0 I I CP insertion _{Ant N} t  ( ) i n 

Figure 2.8: MIMO-OFDM system

Chapter 3

Precoder-Blind Equalization

Algorithm for MIMO-OFDM

3.1

SOS-Based Blind Identifiability and

Equalizabil-ity Conditions for FIR-MIMO Systems

3.1.1

Identifiability Conditions

In this section, we will review the necessary and sufficient conditions for SOS-based blind identifiability as stated in [18]. Denoting the power spectral density matrix of u[n] as Suu(z) and let H(z) be the z-transform of the channel matrix H_ℓ as defined in Section

2.2.2. A FIR-MIMO system is blindly identifiable up to a permutation and scaling if (a) H(z) is (column) irreducible and (b) Suu(z) is diagonal with distinct diagonal (rational or

polynomial) functions. A polynomial matrix H(z) is said to be irreducible if the greatest common divisor of its Nt× Nt minors is 1. In other words, H(z) has full column rank

for almost all z except for z = 0. Two polynomial or rational functions are defined to be distinct if they differ from each other by more than one constant factor, and therefore different sets of zeros and poles are required by (b). Condition (b) implies that the input 21

signals are spatially uncorrelated from each other but temporally correlated with distinct power spectrums. However, methods to make transmitted signal satisfying condition (b) are not discussed in [18]. In this thesis, a low computational complexity precoder that can satisfy condition (b) is proposed to blindly equalize H(z). Details of its design will be discussed in Section 3.3.

Note that the conditions discussed in [18] are less stringent than those previously proposed in [15], where H(z) is required to be both irreducible and column-reduced, which put hasher restrictions on the class of FIR-MIMO channels that can be blindly estimated by using SOS of the received signal.

3.1.2

Equalizability Conditions

Traditionally, work done on blind channel identification and equalization has mainly fo-cused on the former; with the assumption that the equalizer can be designed based on channel estimates using different criteria such as zero-forcing or minimum mean-square error. However, as we have alluded before, it is often better to directly design the equal-izer because this will implicitly take into account the channel estimate error. This has led a number of researchers to wonder if the identifiability conditions stated in the previous section is necessary (and perhaps sufficient) for the design of a SOS based blind equalizer that can directly equalize FIR-MIMO channels. Surprisingly, [31] has recently shown that in fact the conditions for direct equalizability is much weaker than those of identifiability. In this section, we shall elaborate what these conditions are and discuss the similarities between our proposed design with that of [31].

In [31], it stated that if there exists a FIR filter with transfer function, W(z), such that

W(z)H(z) = PDΛ(z),

where P is a permutation matrix, D is a regular constant diagonal matrix, and Λ(z) is a 3.1 SOS-Based Blind Identifiability and Equalizability Conditions for FIR-MIMO

Systems

Chapter 3: Chapter

N × N regular diagonal matrix with diagonal entries being monic monomials, then H(z) is defined as equalizable. Based on this definition, it was proven in [31] that if H(z) is equalizable, then H(z) can be uniquely factorized (up to multiplication with a unitary matrix) as

H(z) = HI(z)HP(z), (3.1)

where HI(z) is a Nr × Nt irreducible matrix with deg(HI(z)) ≤ deg(H(z)) and HP(z)

is a Nt× Nt paraunitary matrix. We will show that (3.1) is consistent with the overall

system response of the proposed precoder-blind equalizer system in Section 3.2.2. Hence, unlike the identifiability conditions previously stated in the last section, the equalizability conditions do not require H(z) to be irreducible and column-reduced. This allows a larger class of channels which cannot be previously identified blindly by SOS based techniques, but can now be directly equalized blindly by SOS based methods. It was also shown in [32] that using SOS of the received signal is insufficient to equalize H(z) unless the transmitted signal is temporally colored. Such condition coincides with the identifiability conditions stated in [18], which motivates the development of our proposed precoder-blind equalizer algorithm.

3.2

SOS-Based Blind Channel Identification and

Equal-ization

3.2.1

Existing SOS-Based Periodic Precoding Channel

Identifi-cation

An existing SOS based blind channel identification scheme for MIMO-OFDM systems [21] is reviewed in this section, which will be compared with the presented algorithm in Section 3.4. Although the scheme proposed in [21] is a two-step equalizer system instead of 3.2 SOS-Based Blind Channel Identification and Equalization 23

direct equalization, it exploits precoding and SOS of received signal to estimate channel. Therefore, we choose this algorithm as comparison with the proposed one. The basic idea of [21] is to exploit periodic nonconstant-modulus precoding at the transmitter such that a separate identification of the individual scalar subchannels hi,j[ℓ] can be achieved by

cyclostationary statistics. Providing each transmit antenna with a different signature in the cyclostationary domain, the cyclic power spectrum matrices of all transmit antennas except one will be zero for a given cycle. Thereby, it is possible to identify the entire channel system on a column by column basis ( subchannel by subchannel basis ) up to a constant diagonal matrix of phase rotations.

The inherent ambiguity can be resolved using short training sequences in practice, and hence we usually assume that the ambiguity is known or solved. Altogether, periodic precoding serves to transform the whole MIMO channel identification into several scalar subchannel problems, and the redundancy introduced by the CP is used to blindly identify these scalar subchannels.

At the transmitter, the individual data streams are multiplied by P -periodic precoding sequences prior to transmission. The precoding sequences are required to be different for different transmit antennas. Using the notation in Section 2.2.2, the precoded transmitted signal of the ith _{transmit antenna can be written by}

ui[n] = X ℓ a(i)_ℓ g[n − ℓK] M_X−1 m=0 s(i)_m,ℓej2π_Mm(n−ℓK)_, i = 1, 2, . . . , Nt,

where a(i)_ℓ and s(i)_m,ℓ denote the precoding sequence and data symbol on the mth _{tone of the}

ℓth _{OFDM symbol from the i}th _{antenna, respectively. Noting that a}(i) ℓ = a

(i)

ℓ+P, which is

P -periodic. The entire ℓth _{OFDM symbol transmitted from the i}th_{antenna are multiplied}

by a(i)_ℓ before IFFT is applied. Since a(i)_ℓ is constant over the entire OFDM symbol, this multiplication can also be performed equivalently in the time-domain after the IFFT and parallel-to-serial conversion. Defining a Nr× Nr cyclic correlation matrix of the receive

Chapter 3: Chapter signal vector x[n] as

Rxx[n, τ ] = E



x[n]xH[n − τ ].

Assuming that channel noise is statistically independent of the source data s(i)_m,ℓ, it was shown in [21] that Rxx[n, τ ] = X ℓ Hℓ X r Ruu[n − ℓ, r]HH_{r−τ +ℓ}+ R_ηη[τ ], (3.2) where Rηη[τ ] = E  η[n]ηH[n − τ ], Ruu[n, τ ] = E  u[n]uH_{[n − τ ]} = diag{ruu(i)[n, τ ]} Nt i=1.

Since the precoding sequences a(i)_ℓ is P -periodic, it can be shown that

Ruu[n, τ ] = Ruu[n + P K, τ ], (3.3)

which shows that u[n] is a P K-periodic cyclostationary transmit signal vector. That is, each of the entries in u[n] is a scalar cyclostationary random process with cyclostationarity period P K. From (3.2) and (3.3), it can be shown that Rxx[n, τ ] = Rxx[n + P K, τ ] such

that x[n] is a cyclostationary receive signal vector with period P K as well.

Due to the P K-periodicity of Rxx[n, τ ] in n, the Fourier series coefficients with respect

to n can be expanded from Rxx[n, τ ]. The Fourier series coefficient matrices can be written

as e Rxx[k, τ ] = 1 P K P K−1_X n=0 Rxx[n, τ ]e−j 2π P Kkn, k = 0, 1, . . . , P K − 1.

The cyclic power spectral matrices can then be obtained by the use of z-transform with respect to τ , which is given by

Sxx[k, z) = X τ e Rxx[k, τ ]z−τ = H(zej_{P K}2πk_)S uu[k, z)HH(1/z∗) + S_ηη(z)δ[k], (3.4)

where Sηη(z) = X τ Rηη[τ ]z −τ_,

Suu[k, z) = diag{S_uu(i)[k, z)}N_i=1t

= X τ e Rxx[k, τ ]z−τ = X τ 1 P K P K−1_X n=0 Ruu[n, τ ]e−j 2π P Kknz−τ.

An example consisting a 2 × 2 MIMO-OFDM system with P = 4 was given in [21] to illustrate the effectiveness of the algorithm. From (3.4), it can be shown that

[Sxx[k, z)]_0,0 = H_0,0(zej 2π P Kk)S(0) uu[k, z)H ∗ 0,0(1/z∗) + H0,1(zej 2π P Kk)S(1) uu[k, z)H0,1∗ (1/z∗), (3.5) and [Sxx[k, z)]_1,1 = H_1,0(zej 2π P Kk)S(0) uu[k, z)H1,0∗ (1/z∗) + H1,1(zej 2π P Kk)S(1) uu[k, z)H ∗ 1,1(1/z∗). (3.6)

To separating the subchannels, the goal is to find cycles k1 6= 0 and k2 6= 0, which satisfy

Suu(0)[k1, z) = 0, Suu(1)[k1, z) 6= 0,

S(1)

uu[k2, z) = 0, Suu(0)[k2, z) 6= 0.

In [21], it was shown that the Fourier series coefficient matrix eRuu[k, τ ] can be written as

e

Ruu[k, τ ] = diag{ eR(i)_uu[k, τ ]}N_i=1t

= M P KδM[τ ]A (g,g)  τ, k P K  × diagnσi2Φ (i) P [k] oNt i=1, (3.7)

Chapter 3: Chapter where Φ(i)_P [k] = P_X−1 r=0 |a(i)r |2e−j 2π Prk, A(g,g)  τ, k P K  = 1 Me −j2π k P K K+τ −1 2 ×        sin(_{P K}πk(K−τ )) sin(π_{P K}k ) , 0 ≤ τ ≤ K − 1 sin(_{P K}πk(K+τ )) sin(π_{P K}k ) , −K + 1 ≤ τ < 0.

Since Sss(i)[k, z) = P_τRe(i)ss[k, τ ]z−τ, we have Sss(i)[k, z) = 0 if and only if eR(i)ss[k, τ ] = 0.

Therefore, the solution for k can be found by choosing k1 and k2 such that

Φ(0)_P [k1] = 0, Φ(1)P [k1] 6= 0,

Φ(1)_P [k2] = 0, Φ(0)P [k2] 6= 0.

Setting k to be k1 and k2in [Sx[k, z)]_0,0and [Sx[k, z)]_1,1and noting that Φ (i)

P [−k] = Φ (i)∗ P [k],

from (3.5) and (3.6) it follows that [Sxx[±k₁, z)]_0,0 = H_0,1  ze±jP K2πk1  S_uu(1)[±k1, z)H0,1∗ (1/z∗), (3.8) [Sxx[±k₁, z)]_1,1 = H_1,1  ze±jP K2πk1  Suu(1)[±k1, z)H1,1∗ (1/z∗), (3.9) [Sxx[±k₂, z)]_0,0 = H_0,0  ze±jP K2πk2  Suu(0)[±k2, z)H0,0∗ (1/z∗), (3.10) [Sxx[±k₂, z)]_1,1 = H_1,0  ze±jP K2πk2  S_uu(0)[±k2, z)H1,0∗ (1/z∗). (3.11)

It is clear that the subchannels H0,1(z) and H1,1(z) can be identified from (3.8) and (3.9),

respectively. Similarly, the subchannels H0,0(z) and H1,0(z) can be identified from (3.10)

and (3.11), respectively. As a result, the 2 × 2 MIMO channel identification problem is transformed into a identification problem of four scalar subchannels.

The algorithm previously proposed in [33] and [34] can be exploited to identify the individual scalar subchannels. The following discussion is restricted to the identifica-tion of H0,0(z) only because the remaining subchannels can be identified using the same

procedure. Starting from (3.10), it can be shown that [Sxx[k₂, z)]_0,0S_uu(0)[−k₂, z)H_0,0(ze−j 2π P Kk2) − [S_xx[−k 2, z)]0,0Suu(0)[k2, z)H0,0(zej 2π P Kk2) = 0. (3.12) From (3.12), straightforward manipulations show that

L_h0,0−1 X ℓ=0 [a(k2,−k2) x,u [n − ℓ]e j_{P K}2πk2ℓ_{− a}(−k2,k2) x,u [n − ℓ]e −j2π P Kk2ℓ]h 0,0[ℓ] = 0, (3.13) where a(k,ℓ)_x,u [n] =X j [ eRxx[k, j]]_(0,0)C_u(0)[ℓ, n − j] (3.14)

and Lh0,0 is the length of the impulse response of the subchannel H0,0(z). Because

OFDM-based systems require an upper bound on channel order to correctly choose the CP length, a safe estimate of Lh0,0 can be assumed to be available and known. From (3.7), we can

then obtain that eR(0)uu[k, τ ] 6= 0 for τ = −M, 0, M, and eR(0)uu[k, τ ] = 0 else. This implies

that

a(k,ℓ)x,u [n] = [ eRxx[k, n]]_0,0R(0)_uu[ℓ, 0]

+ [ eRxx[k, n − M]]_0,0R(0)_uu[ℓ, M]

+ [ eRxx[k, n + M]]_0,0R(0)_uu[ℓ, −M].

For the purpose of solving the subchannel H0,0, (3.13) is rewritten for convenience in

vector-matrix form as [T(k2,−k2) x,u D−k2 − T(−k2 ,k2) x,u D k2_{] h} 0,0 = 0, (3.15)

where T(k,ℓ)x,u is a the (4K + 3Lh0,0 − 6) × Lh0,0 Toeplitz matrix with the first row equal to

[a(k,ℓ)x,u [−2K − Lh0,0 + 3] 0 . . . 0]

and the first column equal to

[a(k,ℓ)_x,u [−2K − Lh0,0 + 3] . . . a

(k,ℓ)

x,u [2K + Lh0,0 − 3] 0 . . . 0]

T_.

Chapter 3: Chapter Moreover, D = diag{e−jP K2πℓ} L_h0,0−1 ℓ=0 , h0,0 = [h0,0[0] h0,0[1] . . . h0,0[Lh0,0 − 1]] T

and 0 represents a 4K + 3Lh0,0− 6 × 1 all-zero vector. The subchannel estimate ˆh0,0 can

then be found by solving the optimization problem given as ˆ h0,0 = arg min kh0,0k=1 [T(k2,−k2) x,u D −k2 _{− T}(−k2,k2) x,u D k2_{] h} 0,0 2. (3.16) The other subchannels H0,1(z), H1,1(z) and H1,0 can be estimated using the same

proce-dure as given by (3.8), (3.9), (3.11), respectively.

In fact, an estimation of the cyclostationary statistics eRxx[k, τ ] is exploited to replace

the ideal one, which is written as beRxx[k, τ ] = 1 T T−1 X n=0 x[n]xH_{[n − τ ]e}−j2π P Kkn. (3.17)

The estimation of a(k,ℓ)x,s [n], denoted by ˆa(k,ℓ)x,s [n], can then be obtained using (3.14) where

e

Rxx[k, τ ] is replaced by bRexx[k, τ ].

3.2.2

SOS-Based Precoder-Blind Equalization

To satisfy the blind identification conditions stated in Section 3.1, the transmitted signal vector ˇu[n] is assumed to be spatially uncorrelated but temporally correlated with distinct power. The notation used here is the same as that in Section 2.2.2. Without loss of generality, ˇu[n] can assume to have unit variance and zero mean. Define the correlation matrix of ˇu[n] as Ruˇu_ˇ[τ ] , E  ˇ u[n]ˇuH_{[n + τ ]}_{, then R} ˇ uu_ˇ[0] = I_NtL. The autocorrelation

matrix of ˇu[n] can be expressed as Ruˇu_ˇ[τ ] = E  ˇ u[n]ˇuH_{[n + τ ]} =      INtL, for τ = 0, diag (ρ1[τ ], . . . , ρNtL[τ ]) , for τ 6= 0,

where ρ1[τ ] 6= · · · 6= ρNtL[τ ] 6= 0. It is worth to note that distinct values for the

diag-onal elements of Ruˇu_ˇ[τ ] also satisfy conditions stated in Section 3.1.2, which makes the

correlation functions of all transmitted streams linearly independent. In this case, direct blind equalization can be achieved by the use of temporal SOS [31]. We further assume that ˇη[n] is white Gaussian distributed and is mutually uncorrelated with ˇu[n]. Then the autocorrelation matrix of the channel output ˇx[n] can be written as

Rxˇ_ˇx[τ ] =      HRuˇ_ˇu[0]HH + σ_ηˇ2_ˇ ηINr(L+q), for τ = 0, HRuˇ_ˇu[τ ]HH, for τ 6= 0, (3.18) where σ2 ˇ

ηˇη is the variance of the noise signal ˇη[n]. Defining ˇv[n] = Hˇu[n] as the channel

output vector ignoring channel noise and Rvˇv_ˇ[τ ], E  ˇ v[n]ˇv[n + τ ]H. Since Ruˇu_ˇ[0] = I_NtL, therefore Rvˇv_ˇ[0] = HHH.

Let W be a whitening matrix that whitens ˇv[n] such that EWˇv[n]ˇvH[n]WH = INr(L+q), where W = Σ−12 ˇ v Q H ˇ v with Σ− 1 2 ˇ

v being the square root inverse of the eigenvalue matrix of Rvˇv_ˇ[0], and Q_ˇv being

the eigenvector matrix of Rˇv_ˇv[0]. Then we can obtain

WRvˇv_ˇ[0]WH = E



Wˇv[n]ˇvH_[n]WH

= WHHH_WH

= INr(L+q). (3.19)

According to (3.19), the effective channel U = WH is a unitary matrix such that UH_{U =}

UUH _{= I.}

Chapter 3: Chapter

JD

U estimation

( )

0

xx

R

=

−12 H x x

W

Σ Q

R

zz

( )

τ

j

[

1 2

]

T

=

x

x

x

z

=

[

z

₁

z

₂

]

T

y

=

[

y

₁

y

₂

]

T

Figure 3.1: Block diagram of equalization process with 2 receive antennas where JD represents joint diagonalization.

Applying W to the received signal vector ˇx[n], we can obtain ˇz[n] = Wˇx[n]

= W [Hˇu[n] + ˇη[n]]

= Uˇu[n] + Wˇη[n]. (3.20) From (3.20), we see that U can be equalized by

U−1ˇz[n] = UH_{ˇz[n] = ˇ}

u[n] + UHWˇη[n]. (3.21) From (3.21), the problem of equalization becomes finding the unitary equalization matrix of U. Noting that U−1 _{= U}H_{, so the inversion of equalization matrix causing high}

compu-tational complexity can be replaced by Hermitian operation. Furthermore, U is consistent with the factorization (3.1) of equalizable channels stated in Section 3.1.2 because any unitary matrix is a paraunitary matrix. This implies that U is equalizable. Defining the correlation matrices for ˇz[n] and ˇη[n] as Eˇz[n]ˇzH_{[n + τ ]}_{and E}_η[n]ˇˇ

ηH[n + τ ], respec-tively. From (3.18), Eη[n]ˇˇ ηH[n + τ ] = 0, for τ 6= 0. Thus, the correlation matrix of 3.2 SOS-Based Blind Channel Identification and Equalization 31

ˇz[n] can be written as

Rˇz_ˇz[τ ] = URu_ˇu_ˇ[τ ]UH, for τ 6= 0. (3.22)

Hence, the equalizer U can be obtained by diagonalizing Rˇz_ˇz[τ ]. According to [22], we

can find U that equalizes frequency-selective channels if the source signal has different spectral energy. In addition, the chance of eigenvalue degeneracy can also be reduced by performing a joint diagonalization on a set of Rˇz_ˇz[τ ] with various τ 6= 0, i.e.

UH_R ˇ

z_ˇz[τ_j]U = diag (ρ₁[τ_p], ρ₂[τ_p], . . . , ρ_NtL[τ_p]) , for p = 0 ≤ t ≤ P,

where τ0, τ1, . . . , τP are non-zero time lags. Denoting the estimate of U as bU. To note that

b

U has a permutation and scalar ambiguity to U. This is corresponding to the property stated in Section 3.1.2, which shows the total blind equalization system is transparent if and only if the autocorrelation functions of system outputs are the same as some per-mutation of those of system inputs. The overall JD equalization process is illustrated in Figure 3.1 with 2 receive antennas.

The entire equalization process is summarized as follows:

Chapter 3: Chapter

1. Estimate the autocorrelation matrix of the received signal Rˇx_ˇx[0].

2. Compute the whitening matrix W by

W= Σ−12 ˇ x Q H ˇ x,

where Σˇv is the eigenvalue matrix of R_ˇxx_ˇ[0],

and Qxˇ is the eigenvector matrix of Rx_ˇx_ˇ[0].

3. Whiten the received vector ˇx[n] by W to obtain ˇz[n] = Wˇx[n].

4. Obtain a set of correlation matrices Rˇz_ˇz[τ_p] of the whitened vector ˇz[n] for p =

0, 1, . . . , P .

5. Perform JD on the set of Rˇz_ˇz[τ_p] for p = 0, 1, . . . , P to estimate the equalization

matrix U.

6. Equalize ˇz[n] by UH _{to obtain the estimate of ˇu[n].}

7. Remove the temporal correlation of the equalized signal by decoloring decoders after FFT.

3.3

Precoder Design

3.3.1

Precoder Format

If all the source data streams are uncorrelated, then the required temporal correlation property can be easily achieved by shaping the power spectral density of each data stream [18]. [22] proposed to use a set of low complexity precoders to color the source signal stream such that FIR blind equalization is possible at the receiver to equalize FIR-MIMO channels. However, the precoders in [22] were chosen arbitrarily without regards on its 3.3 Precoder Design 33

effects on BER performance. Moreover, no investigation was carried out about how the precoder can be used to reduce computational complexity at the receiver while sustaining equalization performance. In this thesis, a new set of precoders are proposed that will allow us to select a subset of {Rˇz_ˇz(τ_p)} such that it not only reduces the computational

complexity at the receiver, but it also does not impact the equalization performance compare to the case when the full set of autocorrelation matrices are used. As seen in Figure 3.2, the precoders are applied in the frequency domain (prior to IFFT) to all Nt transmit antennas of a MIMO-OFDM system. Compared with [22], the precoders

for OFDM-based systems can be simplified to multiplication over all subcarriers due to IFFT. The set of coloring precoders are denoted as {P0(z), P1(z), . . . , PNt−1(z)}, where

Pi(z) = diag (αi,0, αi,1, . . . , αi,M−1) for i = 1, 2, . . . , Nt.

Ant 1

IFFT

P/S S/P

IFFT

P/S S/P Ant N_t ( ) v v M v M × −       ! 0 I I ( ) v v M v M × −       ! 0 I I CP insertion CP insertion Coloring precoders , i m

α

β

i n,

Figure 3.2: The transmitter of the precoder-blind equalizer system with 2 Tx/Rx anten-nas.

αi,mis the real-valued multiplier coefficient of the mthpath of IFFT for the ithtransmit

antenna as illustrated in Figure 3.2. A scaling matrix is then applied in the time domain, 3.3 Precoder Design 34

Chapter 3: Chapter ISI channel H Blind Equalizer "

( )

1 n # "

( )

2 n # $ ( ) r N n % S/P S/P FFT P/S FFT P/S CP deletion [0_Mxv I_M] CP deletion [0_Mxv I_M] Decoloring decoders ,

1

i m

α

,

1

i n

β

Figure 3.3: The receiver of the precoder-blind equalizer system with 2 Tx/Rx antennas. which is given as

Si(z) = diag (βi,0, βi,1, . . . , βi,M−1) for i = 1, 2, . . . , Nt,

where βi,n is a scaling factor that is used to satisfy the distinct power condition. At the

receiver, the multiplicative inverse of the precoders is used to decolor the colored signal as shown in Figure 3.3. The proposed real-valued multiplier αi,m is formed with two parts.

The first part generates the orthogonality among different precoders, and the second part introduces temporal correlation to the transmitted signal. Since the performance of the joint diagonalization algorithm is based on spectral overlap of the source signals [20], this led to the use of orthogonal precoders. αi,m can be expressed as

αi,m= Oi(m) " 1 − P−1 X p=0 Ci,τpcos  2πmτp M # , (3.23) where Oi(m) is a function having only two possible values +1 and −1. Oi(m) can be

designed to generate orthogonality among different precoders by being assigned different shape for different precoders. Ci,τp determines the magnitude of corresponding cosine

term. Distinct values of Ci,τp must be used for various values of n and p in order to

satisfy the distinct power conditions in [18]. The number of cosine term can be decided arbitrarily by choosing P . Furthermore, different τp is used for different cosine terms with

τp = 0, 1, . . . , P − 1.

3.3.2

Temporal Correlation Injection

The reason for using cosine is because we can completely control how many autocorrelation matrices in {Rˇz_ˇz(τ_p)} we need in (3.23) for the joint diagonalization. Namely, the

auto-correlation matrices needed by joint diagonalization can be determined using this form of precoders at the transmitter. Once the needed autocorrelation matrices are known, the range for the lag can be dramatically reduced. Therefore, the latency and computational complexity at the receiver are both decreased. This can be seen by considering the inverse Fourier transform of cos2πmτp

M  , which is written as F−1  cos  2πmτp M  = M_X−1 m=0 cos  2πmτp M  ej2πmnM = 1 2 "_M−1 X m=0 ej2πm(n+τp )M + M−1_X m=0 ej2πm(n−τp )M # = 1 2(δ[n + τp] + δ[n − τp]) . (3.24) As a result, the output of the IFFT can be given by

ˇ u(n) ∗  1 −Ci,τ0 2 (δ[n + τ0] + δ[n − τ0])  = ˇu(n) − Ci,τ0 2 (ˇu[n + τ0] + ˇu[n − τ0]), (3.25) where ∗ denotes convolution and P = 1. From (3.25), it is apparent that temporal correlation of lag τ0 can be generated. The autocorrelation function of the ith transmit

Chapter 3: Chapter

antenna for τ0 can also be given as

ri[τ0] = E [ui[n]ui[n + τ0]] = E  ˜ si[n] − Ci,τ0 2 (˜si[n + τ0] + ˜si[n − τ0])   ˜ si[n + τ0] − Ci,τ0 2 (˜si[n + 2τ0] + ˜si[n])  = −Ci,τ0 2 E  ˜ s2i[n] + ˜s2i[n + τ0]  = −Ci,τ0E  ˜ s2i[n]  = −Ci,τ0,

which is completely controlled by the coefficient of the cosine term. Therefore, only Rˇz_ˇz(τ₀) will have to be used by the joint diagonalizer at the receiver. In fact, using the rest

of the Rˇz_ˇz(τ_p), ∀p 6= 0 will not improve the equalization performance. This will be verified

in the simulation results in the next section when we compare equalization performance of our proposed algorithm using different τp. Besides varying τp, the parameter P can

also be used to improve performance of the equalizer. This can be achieved by increasing the value of P such that more temporal correlation is added to the transmitted bitstream. However, as will be seen in Section 3.4, P cannot be increased indefinitely because the precoder will introduce too much amplitude variation into the bitstream which degrades the BER performance, even though a better estimation of U can be obtained.

According to the blind identification conditions stated in [18] and the MIMO-OFDM system model stated in Section 2.2.2, correlation matrix Ruu(τ ) must have distinct

diag-onal power. In other words, Ci,τp has to be assigned various values for different transmit

antennas and time index n. In order to satisfy this condition, we consider a set of co-efficients c0, c1, . . . , cw, where w = max{L + q − 1, NtP − 1}, for the value of Ci,τp. Let

the time index n be n = 0, 1, . . . , L + q − 1. Table 3.1 and 3.2 list the values for Ci,τp for

different values of p and at different time. As seen in the tables, the coefficients for the current time instant are a left-shifted version of those at the previous time instant. For example, when n = 0, C1,τ0 = c0, C1,τ1 = c1, . . . , CNt,τP −2 = cNtP−2, CNt,τP −1 = cNtP−1.