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TitleDesign and analysis of detection algorithms for MIMO wirelesscommunication systems

Advisor(s) Yuk, TTI; Cheung, SW

Author(s) Shao, Ziyun.;•µ —õ

.

Citation

Issued Date 2011

URL http://hdl.handle.net/10722/174460

RightsThe author retains all proprietary rights, (such as patent rights)and the right to use in future works.

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Design and Analysis of Detection Algorithms for

MIMO Wireless Communication Systems

by

SHAO, Ziyun

B.Sc.(Eng), M.Sc.(Telecom)

A thesis submitted in partial fulfillment

of the requirements for the degree ofDoctor of Philosophy

at the

Department of Electrical and Electronic Engineering

The University of Hong Kong

in

October 2011

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DECLARATION

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

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

dissertation or report submitted to this University or to any other institution for a

degree, diploma or other qualifications.

Signed

SHAO, Ziyun

October 2011

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To my beloved parents and husband

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I

Abstract of thesis entitled

Design and Analysis of Detection Algorithms for MIMO Wireless

Communication Systems

Submitted by

SHAO Ziyun

for the degree of Doctor of Philosophy

at The University of Hong Kong

in October 2011

The increasing demand for high-mobility and high data rate in wireless

communications results in constraints and problems in the limited radio spectrum,

multipath fading, and delay spread.

The multiple-input multiple-output (MIMO) system has been generally

considered as one of the key technologies for the next generation wireless

communication systems. MIMO systems which utilize multiple antennas in both

the transmit side and the receive side can overcome the abovementioned

challenges since they are able to increase the channel capacity and the spectrum

usage efficiency without the need for additional channel bandwidth.

The detection algorithm is a big bottleneck in MIMO systems. Generally, it is

expected to fulfill two main goals simultaneously: low computational complexity

and good error rate performance. However, the existing detection algorithms are

either too complicated or suffering from very bad error-rate performance.

The purpose of this thesis is to comprehensively investigate the detection

algorithms of MIMO systems, and based on that, to develop new methods which

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II

can reduce the computational complexity while retain good system performance.

Firstly, the background and the principle of MIMO systems and the previous work

on the MIMO decoding algorithms conducted by other researchers are thoroughly

reviewed. Secondly, the geometrical analysis of the signal detection is

investigated, and a geometric decoding algorithm which can offer the optimum

BLER performance is proposed. Thirdly, the semidefinite relaxation (SDR)

detection algorithms are extended to high-order modulation MIMO systems, and a

novel SDR detector for 256-QAM constellations is proposed. The theoretical

analysis on the tightness and the complexity are conducted. It demonstrates that

the proposed SDR detector can offer better BLER performance, while its

complexity is in between those of its two counterparts. Fourthly, we combine the

SDR detection algorithms with the sphere decoding. This is helpful for reducing

the computational complexity of the traditional sphere decoding since shorter

initial radius of the hyper sphere can be obtained. Finally, the novel

lattice-reduction-aided SDR detectors are proposed. They can provide

near-optimum error rate performance and achieve the full diversity gain with very

little computational complexity added compared with the stand-alone SDR

detectors.

Total words: 343

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III

ACKNOWLEDGEMENT

First of all, I would like to express my sincere gratitude to my supervisors, Dr.

S. W. Cheung and Dr. T. I. Yuk for their precious guidance and persistent

encouragement throughout my entire PhD study. They taught me academic

knowledge and research skills and enlightened my passion to explore the

unknown scientific world. The thesis would not have been completed without

their supports.

I would also like to thank Mr. Eric W.L. Ng,

Ms. Julie Hung and Ms. Lily Lo

in the Department of Electrical and Electronic Engineering for their kind help

during the past few years.

I truly appreciate the friendship of all teammates and friends in HKU for their

kind help, advice, guidance and encouragement, most notably Dr. Z. Zhang, Dr.

M. X. Xiao, Dr. F. Mai, Dr. X. G. Dai, Dr. W. Zhou, Dr. Z. Kong, Dr. K. C.

Leung, Mr. Y. F. Weng, Mr. Y. Y. Sun, Miss M. J. Mao, Miss L. Li, Mr. H. L.

Xiahou, and Mr. Z. B. Ni. In particular, many thanks to Dr. Dai who always had

taken seriously every question I asked him.

Finally, I am most grateful to my parents and my husband. Their selfless love,

continuous supports and encouragements throughout all these years are the most

precious thing to me. Without these, I could never get my work done well.

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IV

CONTENTS

ABSTRACT I

ACKNOWLEDGMENTS III

CONTENTS IV

CHAPTER 1 INTRODUCTION

1.1 MIMO Wireless Communication System...........................2

1.2 Detection Problem for MIMO Wireless Communication

Systems..........................................................................................8

1.3 Literature Review................................................................9

1.4 Motivation and Contribution of the Thesis .......................14

1.5 Thesis Outline ...................................................................15

CHAPTER 2 STATE-OF-THE-ART MIMO DETECTION

ALGORITHMS

2.1 Introduction .......................................................................17

2.2 Linear Decoders ................................................................17

2.3 Sphere Decoding ...............................................................19

2.4 Successive Interference Cancellation ...............................25

2.5 Lattice-Reduction Aided Detection...................................27

2.6 Summary ...........................................................................34

CHAPTER 3 GEOMETRIC DETECTION

ALGORITHMS

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V

3.1 Introduction .......................................................................35

3.2 Geometrical Analysis of Signal Decoding for MIMO

Channels ......................................................................................35

3.3 Ellipsoid-searching decoding algorithm ...........................40

3.4 Simulation Results ............................................................48

3.5 Summary ...........................................................................52

CHAPTER 4 MIMO DETECTION ALGORITHMS

BASED ON SEMIDEFINITE

RELAXATION

4.1 Introduction .......................................................................54

4.2 Convex Optimization Problems........................................54

4.3 Semidefinite Relaxation....................................................59

4.4 Semidefinite Relaxation Detection Algorithms for

Low-Order Modulation Systems.................................................62

4.5 Semidefinite Relaxation Detection Algorithms for

High-Order Modulation Systems................................................74

4.6 SDR-initiated Sphere detector ..........................................89

4.7 Summary ...........................................................................93

CHAPTER 5 LATTICE-REDUCTION-AIDED

SEMIDEFINITE RELAXATION

DETECTION ALGORITHMS

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VI

5.1 Introduction .......................................................................94

5.2 Lattice Reduction ..............................................................96

5.3 Lattice-Reduction-Aided SDR Detection .......................101

5.4 Simulation Results ..........................................................110

5.5 Discussion .......................................................................116

CHAPTER 6 CONCLUSIONS AND

RECOMMENDATIONS

6.1 Conclusions .....................................................................118

6.2 Recommendations...........................................................119

LIST OF FIGURES 121

LIST OF TABLES 124

ABBREVIATIONS 125

REFERENCES 127

PUBLICATIONS 138

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Chapter 1 Introduction

1

CHAPTER 1

INTRODUCTION

Wireless communications is the most vital field in digital communications.

The first wireless telegraph was developed by an Italian scientist Marconi, who

used radio waves to transmit telegraph messages without connecting wires over

the Bristol Channel in 1897. But the wireless communications did not provide

mobile services for people until 1960s when the AT&T Bell Labs proposed and

developed the Cellular Radio Network. In the recent two decades, the demand for

higher data rates and better quality has been increasing constantly with the

development of the wireless communications technologies. From the

second-generation (2G) mobile communications services which provide up to 115

Kbit/s data rates to the third-generation (3G) mobile communications services that

are able to provide peak data rates at 56 Mbit/s, people are expecting that the

speed of the fourth-generation (4G) mobile communications can reach up to

1Gbit/s.

However, the property of high-mobility and high data rate of the wireless

communications systems result in several challenges such as limited radio

spectrum, multipath fading, delay spread and so on. The multiple antenna systems

with multiple antennas in both the transmitter and receiver sides can overcome

these challenges. It can increase the channel capacity and spectrum usage

efficiency without the need of additional channel bandwidth. Such kind of system

is the so-called multiple-input multiple-output (MIMO) wireless communication

system.

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The MIMO technology has gone through a long history. It was firstly

proposed for application in wireless communication systems in the 1970s. In 1993,

Indian scientists Paulraj and Kailath introduced the idea of using spatial

multiplexing (SM) in MIMO system [1]. Since 1990s, the researchers in AT&T

Bell Lab have given a huge boost on MIMO technology. In 1995, Telatar showed

that the capacity of the MIMO systems in the fading channel conditions increases

linearly with the number of the transmit antennas and the receive antennas [2]. In

1996, Foschini proposed a diagonally-bell laboratories layered space-time

(D-BLAST) architecture for MIMO systems [3]. In 1998, Golden and other

researchers [4] built the laboratorial platform of MIMO system by using

vertical-bell laboratories layered space-time (V-BLAST) algorithm, where the

spectral efficiencies could reach 20-40bit/s/Hz at the indoor fading rates.

Up till now, MIMO technology has been widely considered as one of the key

technologies of the next generation wireless communication systems [5]. Some

mobile communications standards such as the 3G syetems, long-term-evolution

(LTE) and 4G have included the MIMO technology. The standard of wireless local

area network (WLAN) 802.11n recommends MIMO combined with orthogonal

frequency-division multiplexing (MIMO-OFDM). In many other wireless

communication research fields, such as ultra-wide-band (UWB) system and

cognitive radio (CR), researchers are considering to take MIMO technology into

consideration. Therefore, MIMO system is a promising solution to future wireless

communications and has become a very hot issue in both the academic and the

industrial fields.

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1.1 MIMO Wireless Communication System

1.1.1 MIMO System Structure

The general structure of MIMO system can be illustrated in Fig.1.1. It

consists of multiple transmit antennas and multiple receive antennas. With the

space-time coding (STC) [82-83], correlated data is transmitted by different

antennas and the information redundancy can improve the system error rate

performance. This type of MIMO system is the MIMO diversity system which

focuses on the reliability. There is another type of MIMO system called MIMO

multiplexing system, in which, different data streams are transmitted by different

antennas simultaneously so as to increase the transmission data rate.

Figure 1.1 Structure of MIMO system.

1.1.2 Space-Time Coding for MIMO Systems

In MIMO systems, space-time coding is an important method to improve the

spatial diversity and reliability. There are two main kinds of the space-time codes:

the space-time trellis code (STTC) and the space-time block code (STBC) [5]. The

STBC is more popular than the STTC since it has a simpler structure and can offer

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better performance. In 1998, Alamouti proposed a kind of STBC called Alamouti

code [6], which is able to provide full diversity gain with low complexity. Fig 1.2

illustrates an example for a 2 2× antennas system, the transmitted signal X is

given by:

*

1 2

*

2 1

x x

x x

⎛ ⎞−= ⎜ ⎟

⎝ ⎠X (1.1)

The column vectors of the matrix are orthogonal to each other, and sent by

different antennas during each time slot. At the receiver, the received signal is

separated by linear transformation and then decoded by maximum likelihood

decoding.

Figure 1.2 MIMO space-time block code system.

1.1.3 Spatial Multiplexing for MIMO Systems

In MIMO systems, different antennas transmitting the signal in parallel can

offer the multiplexing gain. This multiplexing gain is called spatial multiplexing,

which is unique in MIMO system. Spatial multiplexing can offer a linear increase

in data rate with the number of the transmit antennas and the receive antennas

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without in need of additional bandwidth and power. In the SM-MIMO system, the

data stream is split into several sub-streams, then modulated and transmitted by

different antennas simultaneously. With no doubt, this process could result in a

gain in data rate. The D-BLAST [3] is the first structure using MIMO

multiplexing. The data streams are multiplexed diagonally and transmitted in the

period of transmitting a block of signal. Then another V-BLAST [4, 7] structure is

proposed, which is more effective. The data streams are transmitted in parallel,

that is, the i th− data symbol is transmitted by the i th− antenna directly. The

BLAST structure can provide high multiplexing gain, and the capacity of such

system is increased linearly with the number of the transmit antennas and the

receive antennas [2]. In this thesis, we will focus on the SM-MIMO system.

1.1.4 MIMO System Model

The Spatial multiplexing MIMO system is illustrated in Fig.1.3, in which,

T N transmit antennas send the signal vectors to

R N receive antennas over a

wireless channel. At the transmit side, the user data stream is partitioned intoT

N

sub-streams and then sent by different transmit antennas. At the receive side, each

receive antenna receives signal vectors from all the transmit antennas.

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Figure 1.3 MIMO spatial multiplexing system.

Thus, the MIMO system is modeled as [5]:

11 12 11 1 1

2 2 221 22 2

1 2

T

T

R T T R R R T

N

N

N N N N N N N

h h hr x n

r x nh h h

r x nh h h

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

(1.2)

or

r = Hx + n (1.3)

where r is the R

N -dimensional received signal vector, and x is the

T N -dimensional signal vector in the transmit lattice. H denotes the channel

matrix, with elementsijh representing the transfer function from the j-th transmit

antenna to i-th receive antenna. n is the R N -dimensional additive noise. In this

thesis, the signal vector x is assumed to be a statistically independent variable

with zero mean and unit variance 2x 1σ = . Perfect channel knowledge is assumed

to be known to the receiver. In addition, the channel matrix H is assumed to be a

flat fading channel and all the entries in H are complex Gaussian and

independent. The noise is an independently and identically distributed (i.i.d.)

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zero-mean Gaussian noise vector with elements having a fixed variance 2nσ .

The complex transmission in (1.3) can be equivalently represented in real

matrix form as:

( )

( )

( ) ( )( ) ( )

( )

( )

( )

( )

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

Re H -Im HRe r Re x Re n= +

Im r Im x Im nIm H Re H (1.4)

with Re(•) and Im(•) being the real and imaginary parts of (•), respectively. The

real-valued representation is written as [5]:

r = H x + n (1.5)

The dimension of r and x are 2 R R M N = and 2T T M N = , respectively. H

becomes an R T M M × matrix. And the noise n is an - R M dimensional vector.

1.1.5 Channel Capacity

The channel capacity represents the maximum possible data rate that can be

reliably transmitted. In other words, the possible error rate should be acceptably

small [84-86]. In a flat fading channel, the channel capacity can be written as

2log detT

C N

ρ ⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦I Q (1.6)

,

,

H

R T

H T R

N N Q

N N

⎧ <= ⎨

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1.2 Detection Problem for MIMO Wireless Communication

Systems

1.2.1 MIMO Detection

Detection is the reverse process of the signal transmission. As mentioned

before, multiple transmit antennas send different signal symbols simultaneously,

the received signal vector is a superposition of all transmitted signal distorted by

the channel matrix and corrupted by the additive Gaussian noise. Although we

assume that the channel information is known to the receiver, the noise is random

and unknown. Thus, detection is aimed at finding the transmitted signal vector

based on the channel matrix and the received signal vector. Generally, the

detection algorithm needs to fulfill two main goals. One is low computational

complexity, and the other is good error rate performance. Unfortunately, these two

goals usually contradict each other: low computational complexity means bad

error rate performance, while good error rate performance results in high

computational complexity. In practice, a tradeoff has to be made between them.

1.2.2 Maximum Likelihood Decoding

The maximum likelihood (ML) decoding [74-75] is the optimum decoding

algorithm as it can provide the best error rate performance. It is formulated as:

2ˆ arg min ML

∈Ω

= −x

x r Hx (1.8)

It implies that ML calculates the Euclidean distance between the possible transmit

signal vectors and the received signal vector, and then choose the one which is

closest to the received vector as the solution. However, since all the possible

signal vector in the lattice space Ω should be considered, and their Euclidean

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distances away from the received vector have to be calculated, the complexity of

ML decoding increases exponentially with the number of transmit antennas and

polynomially with the size of the constellation [5]. Thus, in an SM-MIMO system,

the complexity of ML decoding is non-deterministic polynomial time hard

(NP-hard). This disadvantage makes the system impractical to be implemented.

Thus, detection has become one of the major challenges in MIMO system.

The main objective of designing a detection algorithm is that it provides good

error rate performance with low computational complexity. The key point is how

to balance the performance and complexity.

1.3 Literature Review

In this section, the existing MIMO detection methods will be simply

overviewed. Generally, the detection methods of MIMO system can be classified

into two types: linear detection algorithm and non-linear detection algorithm. ML

decoding is known as a non-linear detection algorithm, which performs the

exhausted search of the lattice space and provides the optimum error rate

performance. However, it is very complicated and impractical. Thus, the

low-complexity linear detectors are proposed.

Linear detection algorithms such as zero-forcing [8] and minimum

mean-square-error [9] estimation are sub-optimum methods in which the received

signal vector is multiplied by a transformation matrix to get an estimation vector,

and after that, the determination is carried out to get the final solution. In

zero-forcing (ZF), the received signal vector is multiplied by the generalized

inverse matrix of the channel matrix and then quantized to get the result. The

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performance of ZF is poor since the noise is enlarged by the generalized inverse

matrix. The minimum mean-square-error (MMSE) detector takes the noise

variance into account and minimizes the square error between the transmitted

signal vector and estimated vector. Thus it can provide better performance than ZF.

Although the linear detection algorithm has very low complexity, their error rate

performance is inferior.

The non-linear detection algorithms include sphere decoding (SD), successive

interference cancellation (SIC) detection, lattice reduction aided decoding (LRD),

semidefinite relaxation (SDR) detection, and so on. They are proposed to improve

the error rate performance.

Sphere decoding is firstly introduced to significantly reduce the average

decoding complexity [14], yet achieving the optimum performance as ML

decoding. After that, the SD has been further discussed in various publications

[15]-[17]. SD decoding is also a search-based algorithm like ML decoding. In ML

decoding, the search is conducted among the whole lattice structure, while in SD,

the searching process is confined inside a hyper sphere of certain radius centered

at the received signal point as illustrated in Fig. 1.4. The solution of ML decoding

is likely to locate inside the hyper sphere. To improve the searching efficiency,

some searching strategies are proposed, such as Fincke-Pohst (F-P) [18] and

Schnorr-Euchner (S-E) [19].

F-P sphere decoding is usually thought as the original sphere decoding

algorithm. The search begins at the root then calculates the weights of connected

branches and nodes. In the F-P strategy, the tree is traversed depth-first and from

left to right, which means enumerating the points located within a sphere along a

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branch until a node is encountered. For all other branches, the same process is

conducted until all nodes within the hyper sphere are discovered.

It has been shown that the S-E enumeration is more computational efficient

than the F-P enumeration [20]. The S-E strategy performs traversing the tree

depth-first too. But it calculates the branch weights and searches them in

increasing order. And after it obtains a lattice point, the radius of the hyper sphere

is reduced to be the distance between the received signal point and the lattice

point. Then the search process is restarted again with the new radius. As a result,

the lattice points visited is less and the searching process becomes faster.

Figure 1.4 Sphere decoding.

There are two major aspects to be improved for the sphere decoding. The first

one is to determine the initial radius of the hyper sphere. If the radius is too large,

there will be too many node weights need to be calculated. In the contrast, if it is

too small, it would be possible that no point is inside the hyper sphere, and the

search should be restarted again with a larger radius. Thus, a proper initial radius

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of the hyper sphere can reduce the complexity of SD. In [14] and [21], it uses the

noise variance and probability equation to define a proper initial radius. The

MMSE equalizer is also applied to set the initial radius [22]. Another aspect is the

searching strategy. In [23], a statistical pruning method that uses a set of bounds

based on the minimum metric of the current solution is proposed for S-E sphere

decoding. A preprocessing stage and a new ordering are engaged in the searching

method in [24]. With the new ordering, the nodes are expanded according to the

level and offset coefficients. These searching strategies provide higher

computational efficiency. Although SD is able to provide the BLER performance

of ML detection with less complexity, it has been proven that its expected

complexity is still exponential [25]. Thus it becomes impractical when the system

order is high and the SNR is low.

The successive interference cancellation detection [88] has the error rate

performance gain by sacrificing a certain complexity compared with the linear

detections. It detects the signal from the first transmit antenna instead of that of all

the transmit antennas, and then subtract its impact from the received signal vector.

After that, it detects the signal from the second transmit antenna. At this time, the

row of the channel matrix that is corresponding to the first antenna is deleted.

Thus, the dimension of channel matrix becomes ( )1 R T M M × − and the

dimension of the transmitted signal vector is reduced to 1T M − . The process

continues until all the elements of the signal vector are detected. Because the

detected signal has effect on the later signal to be detected, we need to firstly

detect the reliable signals in order to reduce the error propagations. Usually,

ordering is adopted to improve the performance. The authors in [10] proposed an

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ordered successive interference cancellation algorithm based on the maximum

signal to noise ratio (SNR). In [7, 11] it detects the signal with the maximum

signal to interference plus noise ratio (SINR) first, and the ordering obeys the

log-likelihood ratio rule in. ZF or MMSE methods are engaged to obtain an

estimate of the value in each dimension of the transmitted signal vector, which are

known as ZF-SIC and MMSE-SIC [12, 13], respectively.

Lattice reduction aided detectors [26]-[29] are developed in order to improve

the error rate performance of MIMO systems. They are combined with the linear

detection such as ZF and MMSE, and so are called LR-ZF and LR-MMSE [29].

In MIMO transmissions, if the column of the channel matrix is less correlated, the

transmit signal from different antennas is more independent and can be detected

more correctly. The lattice reduction technique is to find an optimal lattice basis of

a matrix which is more orthogonal and short compared with the original lattice

basis. In MIMO detection, it transforms the channel matrix H into new matrix

H by an unimodular matrix. The more orthogonal of the matrix H is, the more

significant improvement of the error rate performance will be obtained.

However, lattice reduction is also an N-P hard problem. The most popular

lattice reduction algorithm is the Lenstra–Lenstra–Lovász (LLL) algorithm [30]. It

is a polynomial time algorithm that operates iteratively and stops when the lattice

basis is obtained. It has been shown that in LLL algorithm, the average number of

iterations is bounded by a polynomial in the dimension of the lattice [31]. Then in

[32], researchers proposed a novel joint sorting and reduction algorithm that can

offer better diversity order and error rate performance than the traditional LLL at

the cost of a polynomial computation time. A complex LLL reduction algorithm is

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introduced in [33]. It can reduce the average complexity by almost half of the

conventional LLL and achieve full diversity in LR detection.

Recently, MIMO decoders using semidefinite relaxation approach have

attracted great attention. They are able to provide acceptable BLER performance

and feature polynomial worst-case complexity [34]. Since the ML decoding

problem has the optimum error rate performance, the SDR detection applies the

convex optimization toolbox such as SEDUMI [35] to solve the convex relaxation

of ML optimization problem, which is called semidefinite programming (SDP).

The SDR approach was firstly applied to detect binary phase-shift keying

(BPSK) and four quadrature amplitude modulation (4-QAM) signals [36]-[38]. It

has been shown that the SDR detector for BPSK can achieve full receive diversity

[39]. Then the extensions to different SDR techniques for 16-QAM signals had

been proposed, such as polynomial-inspired SDR (PI-SDR) [40],

bound-constrained SDR (BC-SDR) [41] and virtually-antipodal SDR (VA-SDR)

[42], all exhibit acceptable BLER performance and relatively low complexity. In

[43], it has been proved that there exists an equivalence among PI-SDR, BC-SDR

and VA-SDR for 16-QAM, all of them provide the same BLER performance.

1.4 Motivation and Contribution of the Thesis

MIMO system is a promising solution to the high data rate and high reliability

of the future wireless communications. It can increase the channel capacity and

spectrum usage efficiency without the need of additional channel bandwidth. The

detection design is one of the major challenges of MIMO systems, since there is

always a tradeoff between the computational complexity and the error rate

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performance. The thesis aims at investigating detection algorithms to reduce the

complexity while retain good system performance.

The main contributions of the thesis are as follows:

• Geometrically analyzing the signal detections of MIMO system, which is

another perspective of reconsidering the principle of the ML decoding. Based on

this, a geometric decoding algorithm is proposed, which can provide the optimum

error rate performance.

•

Giving the extension of the existing SDR detection algorithms to

high-order modulation MIMO system, and proposing a novel SDR detection

algorithm for 256-QAM MIMO system which could offer better performance than

its existing counterparts. The theoretical analysis on the tightness of the SDR

detection algorithms is also conducted.

•

Combining the SDR detection algorithms with the sphere decoding,

which can achieve the optimum BLER performance while retain acceptable

computational complexity.

• Proposed the lattice-reduction-aided semidefinite relaxation detection. It

is able to achieve the full diversity gain and improve the error rate performance

with a little complexity added.

1.5 Thesis Outline

This thesis is composed of six chapters. All references quoted within this

thesis are listed in the References part. The thesis outline is given as follows:

Chapter 1 is the introduction on the background information of the MIMO

system and literature review of the previous work. The research motivation and

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aims of this thesis are also reported.

Next in Chapter 2, the state-of-the-art of the MIMO detection algorithms are

investigated.

In Chapter 3, the geometric analysis of the signal detection is given. Then an

optimum ellipsoid-searching decoding algorithm is introduced and simulation

results are given.

After that, convex optimization and the interior point method are investigated

in Chapter 4. The semidefinite relaxation detection algorithms for low-order

modulation and high-order modulation system are given. Then the theoretical

analysis on the tightness and the complexity of SDR are also involved. In addition,

the semidefinite relaxation initiated sphere detector is given. At the end, the

simulation results are given.

Next in Chapter 5, two different lattice-reduction-aided semidefinite

relaxation detection algorithms are investigated with the simulation results.

Finally, the conclusion and recommendation are given in Chapter 6.

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Chapter 2 State-of-the-Art MIMO Detection Algorithms

17

CHAPTER 2

STATE-OF-THE-ART MIMO DETECTION

ALGORITHMS

2.1 Introduction

This chapter will be devoted to review the state-of-the-art of the MIMO

detection algorithms. Firstly, the linear decoders which include the zero-forcing

decoder and the minimum mean-square-error decoder are introduced. Linear

decoder has the advantage of extremely low computational complexity. However,

they suffer from the unsatisfactory block error rate (BLER) performance.

Secondly, the sphere decoding is elaborated. Similar to the ML decoding, sphere

decoding is another searching-based detector, and it can offer optimum BLER

performance. Nevertheless, its expected complexity is still very high, although it

has been dramatically reduced compared with the ML decoder. Thirdly, the

successive interference cancellation and the lattice reduction detection are

introduced. They both can be combined with the linear decoders, so as to improve

their BLER performance with certain complexity added.

2.2 Linear Decoders

As elaborated in Chapter 1, the complexity of ML decoder increases

exponentially with the number of jointly decoded symbols [38]. However, the

complexity of some sub-optimal decoders increases linearly with the number of

jointly decoded symbols, i.e. the number of symbols in one code block. These

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decoders are called linear decoders. The well-known linear decoders include the

zero-forcing (ZF) decoder and the minimum-mean-square-error (MMSE) decoder.

2.2.1 Zero-Forcing

Zero forcing (ZF) detection [8] is the simplest linear detection algorithm. It

forces the impact of the channel matrix to be zero and is given by:

( ) ( )1

*ˆ H H ZF Q Q

−⎡ ⎤= = ⎢ ⎥⎣ ⎦x H r H H H r (2.1)

where *H is the pseudo-inverse matrix [45] of the channel matrix H , the

superscript H represents the complex conjugate transpose, ( )Q i is quantization

to the constellation values. The decoder finds the value of ˆ ZF

x which is the closest

to 1( ) H H −H H H r . However, the noise signal in each stream becomes correlated to

each other by the matrix *H , which results in decoding error. Thus ZF detection

is of suboptimum performance.

The diversity order of ZF detection is 1 R T

M M − + . The full diversity gain

given by ML decoding is R

M . Thus, the performance of ZF is poor especially

when the number of antennas is large. The performance can be improved by the

minimum mean-square-error detection approach.

2.2.2 Minimum Mean-Square-Error

As suggested by its name, minimum mean-square-error (MMSE) [9] is an

approach to minimize the mean-square-error (MSE). The MMSE detection can be

written as:

( )( )1

2ˆ H H MMSE n

Q σ −

= +x H H I H r (2.2)

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where 2nσ denotes the noise power and I is the identity matrix. Compared with

the equation (2.1) of ZF, the difference lies in the term of 2nσ I . ZF decoding

separates the co-channel signals and cancels all the inter-symbol-interference (ISI).

However, this inevitably leads to noise enhancement. On the other hand, MMSE

detection attempts to minimize the overall errors which are caused by the noise,

and make balance between the ISI mitigation and noise enhancement. Assuming

that the noise 2nσ

is zero, the MMSE becomes the same with ZF. Generally,

MMSE decoding tends to provide better error rate performance than ZF decoding.

Both ZF and MMSE are classified as linear detection algorithm. Although

they have very low complexity, their error rate performances are still far away

from being satisfactory.

2.3 Sphere DecodingML decoding applies an exhaustive search process, in which, all the lattice

points of the constellation are visited, and then choose the one with the minimum

distance to the received point as the solution. Although it is an optimum decoding

algorithm, its computational complexity increases dramatically with the increase

in the number of the antennas and the constellation size [5]. Sphere decoding [18]

(SD) is proposed aiming at reducing the decoding complexity, while retaining the

optimal BLER performance. It searches only a subset of the lattice points that are

located inside a hyper sphere centered at the received signal vector. The lattice

points which are located outside the hyper sphere will not be taken into account.

Thus, the number of lattice points searched by the algorithm depends on the initial

radius of the hyper sphere. Although the points inside the hyper sphere are not

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searched exhaustively, the calculations are based on the branches in a tree which

are possible to lead to the final result.

There are two main problems to be solved in SD. One is how to determine the

initial radius. If the radius is too large, there will be a large number of points

inside the hyper sphere and the complexity will be too large. If the radius is too

small, there may be no point inside the sphere and the search process has to be

restarted. Usually, the ZF equalized result is taken to calculate the initial radius.

The other problem is how to tell which lattice points are located inside the sphere.

It is very difficult to identify whether a lattice point is located inside a hyper

sphere or not, but it is very easy to do so for a two-dimensional sphere by simply

checking whether the integer values of the lattice points lie in the interval of the

sphere. Inspired by this, for an N -dimensional sphere the points can be determined

from one dimension to the other successively. It means that for a -α

dimensional

point, if its ( )1α − − dimension values lie in the ( )1α − − dimensional sphere of a

certain radius, there will be a new interval for its thα − value to determine if it

lies in the -α dimensional sphere.

SD is aimed at finding out the solution ˆSD

x which is the same with ˆ ML

x that

has the minimum Euclidean distance from the received signal r . It searches the

lattice points x within aT

M − dimensional hyper sphere, which can be given by:

2d − ≤r Hx (2.3)

where d is the initial radius of the hyper sphere. The searching of the lattice

points is a kind of iterative algorithm. For simplicity, we have to separate the

original problem into several sub-problems. Thus, the channel matrix H is firstly

reduced into an upper triangular matrix by using the QR decomposition:

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0

⎡ ⎤= ⎢ ⎥

⎣ ⎦

RH Q (2.4)

[ ]

1,1 1,2 1,

2,2 2,

1 2 ,

0

0 0

, 0 0

0 0 0

0 0 0

T

T

T T

M

M

M M

r r r r r

r

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Q Q

…

…

…

…

…

…

where R R M M ×∈Q is an orthogonal matrix and T T M M ×∈R is an upper triangular

matrix. Decomposing Q by [ ]1 2=Q Q Q , where 1 R T M M ×∈Q and ( )2

R R T M M M × −∈Q ,

the Euclidean distance involved in the ML decoding can be written as:

2−r Hx

2

1 2[ , ]⎡ ⎤

= − ⎢ ⎥⎣ ⎦

Rr Q Q r

0

2

1

2

T

T

⎡ ⎤ ⎡ ⎤= −⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

RQr x

0Q (2.5)

2 2

1 2

T T = − +RxQ r Q r

Let1

T ′ =r Q r , (2.5) then becomes:

2 2

2

T +′ − x QR rr (2.6)

Thus, to minimize the Euclidean distance2

−r Hx is equivalent to minimize

2′ −r Rx in the sphere decoding.

Let2

2

2

T d d ′ = − Q r , (2.3) can be rewritten as:

2 2d ′ ′− ≤r Rx (2.7)

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2

1,1 1,1 1,1 1

2 2,2 2, 2 2

,

0

0 0 0

T

T

T T T T

M

M

M M M M

r r r r x

r r r xd

r xr

⎡ ⎤′⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥′ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ′− ≤⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥⎢ ⎥ ⎢ ⎥′ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

…

…

2

2

,

1

T M

i i j j

i j i

r r x d = =

⎛ ⎞′− ≤⎜ ⎟

⎝ ⎠∑ ∑ (2.8)

where,i jr is the ( ),i j th− element of R . The inequality equation (2.8) is then

expanded to be:

2 2 2 2

, 1 1, 1, 1 1 1 1,

1

( ) ( ) ... ( )T

T T T T T T T T T T T

M

M M M M M M M M M M M i i

i

r r x r r x r x r r x d − − − − −=

′ ′ ′ ′− + − − + + − ≤∑

(2.9)

It can be seen from the inequality equation (2.9) that there is only one unknown

qualityT M

x in the first term 2,

( )T T T T M M M M

r r x′ − . Similarly, there are two unknown

qualitiesT M

x and -1T M x in the second term2

1 1, 1, 1 1( )

T T T T T T T M M M M M M M r r x r x− − − − −′ − −

and so on. The necessary condition of inequality equation (2.9) is:

2 2

,( )

T T T T M M M M r r x d ′ ′− ≤ (2.10)

So the value ofT M

x can be solved at first. The boundary ofT M

x is:

, ,

T T

T

T T T T

M M

M

M M M M

r d r d x

r r

⎡ ⎤ ⎢ ⎥′ ′′ ′− +⎢ ⎥ ⎢ ⎥≤ ≤⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦

(2.11)

It is easy to find out the possible values ofT M

x by using (2.11). For example,

they are the odd numbers in the interval for QAM constellations. Usually, there

may be more than one possible value which are saved in the memory.

Secondly, one of the possible values ofT M

x is selected to solve1T M

x − by the

following inequality equation:

2 2 2

, 1 1, 1, 1 1( ) ( )

T T T T T T T T T T T M M M M M M M M M M M r r x r r x r x d − − − − −′ ′ ′− + − − ≤ (2.12)

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The boundary of1T M

x − is

2 2

, 1 1,

11, 1

2 2

, 1 1,

1

1, 1

( )

( )

T T T T T T T T

T

T T

T T T T T T T T

T

T T

M M M M M M M M

M M M

M M M M M M M M

M

M M

d r r x r r x x

r

d r r x r r x x

r

− −

−− −

− −

−

− −

⎡ ⎤′ ′ ′− − − + −⎢ ⎥ ≤⎢ ⎥⎢ ⎥

⎢ ⎥′ ′ ′− − + −⎢ ⎥≤⎢ ⎥⎣ ⎦

(2.13)

It can be seen from (2.13) that the signals for the previously detected

dimensions have been subtracted from the received signal.

If there is no possible value satisfying the inequality equation, the searching

process goes back to the previous step and then selects another possible value.

When all the element values of one lattice point are obtained, its Euclidean

distance away from the received point is calculated. The new distance is of course

smaller than the initial radius d ′ , so we replace it by the new distance in (2.9) and

restart the searching process.

The searching process works in an iterative way to update the radius until no

more vectors satisfy the inequality equation (2.9). The last vector is then taken as

the decoding solution.

In conclusion, the searching process is done from theT M th− dimension to

the 1st dimension. The intervals of x are calculated and the value of the each

element of x are then determined. When one lattice point is obtained, the radius

of the hyper sphere is reduced and the searching is restarted again. The searching

process goes on until only one branch is left, which is then taken as the final result.

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Figure 2.1 The tress search structure of sphere decoding.

The searching process to determine all the lattice points in a

T M − dimensional hyper sphere can also be illustrated by the tree search structure

shown in Fig. 2.1. Each level of the tree represents each dimension of the signal

vector x. The nodes in the figure are the values of the element of x. Take 16-QAM

constellation for example: a node emits four sub-nodes which are the values of -3,

-1, 1, 3 from the left to right. The black nodes denote the possible values that

have been visited. From the figure, we know that in theT

M th− dimension, the

possible values ofT M

x are -1 and 1. Then we select -1 to continue the searching

process down to the next dimension. It can be seen that there is one branch

emitted from the node =-1T M

x that is visited, which means all the element values

of one lattice point are obtained. Then we calculate the new radius and restart the

searching process. The search continues until no other branch can be found, the

lattice point corresponding to the last branch is selected as the final solution.

Although lots of new methods have been proposed to determine a proper

initial radius [14, 21], and many new searching strategies [22-24, 62-66] have

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been developed to improve the searching efficiency, sphere decoding still suffers

from some drawbacks. The expected complexity of SD increases with the number

of antennas and the size of the constellation [15]. So SD is not suitable for large

size MIMO system. In addition, the complexity of SD is not fixed, but changing

with the number of the nodes visited in the searching strategies, which makes it

impractical for hardware implementation. As a result, the sub-optimum successive

interference cancellation and lattice reduction aided detection are developed to

improve the performance of linear detection with a comparable low complexity.

2.4 Successive Interference Cancellation

The successive interference cancellation (SIC) [88] is a sub-optimum non-

linear equalizer. Its basic idea is to estimate the current symbol and subtract the

impact of those interfering symbols which have been detected earlier. So the

distortion on the current detecting symbol caused by the previous symbols is then

eliminated. The SIC method can be combined with the linear detection algorithms

such as ZF and MMSE to improve their error rate performance. However this will

cause additional complexity.

Firstly, we apply QR decomposition of the channel matrix H :

=H QR (2.14)

where Q is a unitary matrix and R is an upper triangular matrix. The MIMO

system model (1.5) can be transferred into:

= +r QRx n

T T = +Q r Rx Q n (2.15)

Let T ′ =r Q r and T ′ =n Q n , the i th− row of the ′r is:

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,

T M

i i j j i

j i

r R x n=

′ ′= +∑ (2.16)

which is composed of the symbols from i x to T M x . Also, the T M th− row of ′r

only involves the symbolT M

x :

,T T T t T M M M M M r R x n′ ′= + (2.17)

As a result, we could detect the symbolT M

x first by:

,

T

T

T T

M

M

M M

r x

R

′=「」 (2.18)

Then the ( )1T M th− − symbol 1t N x − will be detected next, where the interfering

symbolT M

x can be canceled by subtracting its impact:

1 ,

1

1, 1

T T T T

T

T T

M M M M

M

M M

r R x x

R

−

−

− −

′ −=「」 (2.19)

Thus, the i th− symbol is detected by:

1, 1 1

,

i i i i

i

i i

r R x

x R

+ + +′−

=「」 (2.20)

The SIC works from theT

M th− symbolT M

x upwards to the 1st symbol1

x till all

the symbols are detected in the end.

The drawback of the SIC detection is the error propagation caused when there

is wrong decision in one symbol. If the former detected symbol is wrong, the

latter ones have very large probability to suffer from errors. In order to solve this

problem, the methods of detection ordering are proposed [7, 10, 11]. The symbol

detection ordering is in accordance with the power of the symbols, which can be

determined by the SNR or the signal to interference plus noise ratio (SINR). That

means that the symbol which has large SNR or SINR will be detected first.

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Similarly, the detection ordering will improve the error rate performance but

increase the complexity.

2.5 Lattice-Reduction Aided Detection

2.5.1 Lattice Reduction

The lattice structures are very popular in field of wireless communications

especially the MIMO system. A lattice is composed by a discrete set of N

dimensional vectors which is denoted by N

L . Every lattice can be produced by a

linear combination of a set of independent integer vectors, which is called the

basis of the lattice given by { }1 2, , M b b b , M N ≤ . Any lattice has infinitely

many lattice bases, and the lattice basis composed of near orthogonal vectors

which are also with short lengths is the desirable one.

The procedure of determining a lattice basis with short and near orthogonal

vectors is termed as Lattice Basis Reduction [19]. To realize the reduction goal is

an N-P hard problem, whose running time varies exponentially with the

dimension of the lattice. A popular suboptimum reduction algorithm which is

called LLL algorithm [30] is proposed to reduce the complexity of Lattice Basis

Reduction.

In MIMO systems, for ill conditioned channel, the noise enhancement is large.

However, for orthogonal channel, there will be no noise enhancement. The lattice

reduction (LR) algorithm is employed to transform the original channel matrix

into a new channel matrix with much better channel condition [47, 76-81]. The

new channel matrix is composed of vectors with the shortest lengths or roughly

orthogonal to each other. The LR algorithm can be combined with the existing

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linear detectors such as ZF and MMSE detectors. The linear detectors with a

better channel conditioned matrix will achieve better BLER performances.

2.5.2 Analysis of MIMO Detections in Lattice Space

This section is to analyze the different detections including ZF decoding, SIC

detection and ML decoding in the perspective of the lattice space. Herein, we

assume a 2 2× MIMO system using 4-PAM modulation. The original transmit

signals lattice is shown in Fig. 2.2.

Figure 2.2 The original transmit signal lattice.

With the distortion resulted from the channel matrix, the received signal

lattice without Gaussian noise is indicated in Fig 2.3. The vectors [ ]1 2h h is the

basis of the channel matrix H [65].

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Figure 2.3 The received signal lattice.

Fig. 2.4 and Fig 2.5 show the decision regions of the ZF decoding and SIC

detection, respectively.

Figure 2.4 ZF decoding decision region.

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Figure 2.5 SIC detection decision region.

It can be seen that the ZF decoding transforms the received signal to the

transmitted signal space to make the decision, where the decision region becomes

parallelogram. The decision region of SIC detection is rectangular which is

different from that of the ZF decoding.

The ML decoding decision region as indicated in Fig. 2.6 is the best one,

which is formed by two roughly orthogonal bases. Any point in a decision region

is closer to the lattice point in its decision region than to other lattice points in

other regions. It thus inspires that if we make the decision of the signal in an

orthogonal or roughly orthogonal basis, then transform the decision value back to

the original signal space, the detection result will be much better than that in the

original basis.

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Figure 2.6 ML decoding decision region.

2.5.3 Lattice Reduction Aided Linear Detections

The lattice reduction for MIMO system aims at transforming the original

channel matrix H into a new one H which contains more orthogonal and shorter

basis vectors by multiplying a unimodular matrix T . The unimodular matrix has

only integer elements in it and its determinant equals to 1± . For linear detections,

the decoding performance is much better for the case of orthogonal matrix.

Herein, =H HT and let 1−=z T x , the MIMO system model of (1.5) can be

written as:

1= −= + + = +r Hx n HTT x n Hz n (2.21)

where Hx and Hz represent the same point in the lattice space.

In this equivalent system model, the zero-forcing detection is applied to

obtain [29]:

*

LR ZF − =z H r

( )* *== +H Hz n z + H n

1

= ZF −

T x (2.22)

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The new pseudo-inverse matrix *H tends to generate less noise enhancement than

the original pseudo-inverse matrix *H . The solution LR ZF −z is then quantitized

and multiplied by T, so as to recover the solution in the transmit signal lattice:

( )ˆ = LR ZF LR ZF Q− −z T z (2.23)

Figure 2.7 LR-ZF decision region.

Fig. 2.7 illustrates the decision region of LR-ZF, which is a parallelogram

with its sides parallel to the new channel basis vectors1

h and2

h . The

parallelogram is much less stretched than the parallelogram of ZF shown in Fig.

2.5. This explains why the LR-ZF detector can offer better BLER performance

than ZF detection.

In MMSE detection the noise term is taken into consideration. Apply MMSE

detection in the system model (2.21) to obtain [65, 76]:

( )1

2 H H H

LR MMSE nσ

−

− = +z H H IT T H r

1= MMSE

−T x (2.24)

Similarly, the final solution becomes:

( )ˆ = LR MMSE LR MMSE Q− −z T z (2.25)

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Another LR-MMSE detection for an extended channel matrix is introduced in

[29]. Define the extended channel matrix which is a ( )T R T M M M + × matrix:

ext

nσ

⎡ ⎤= ⎢ ⎥

⎣ ⎦

HH

I (2.26)

Define a ( ) 1T R M M + × extended received vector ext x :

=ext

⎡ ⎤⎢ ⎥⎣ ⎦

xx

0 (2.27)

The extended MMSE detector is:

( )1

H H

ext MMSE ext ext ext ext

−

− =z H H H r

1= ext ext −T x (2.28)

where =ext ext ext H H T .

This extended MMSE detector is in fact the same as the LR-MMSE detector

given in (2.24) since they both follow the MMSE detection scheme of (2.2) in the

system model (2.21) except that the lattice reduction is conducted on ext H instead

of H . Both of them can outperform the MMSE detector because the noise

enhancement is less when perform on H or ext H .

The diversity order of the lattice reduction aided linear detectors can reach the

full diversity of R M , which is the same as that of ML detection, and much better

than that of the linear detectors. There are significant BLER performance

improvements of LR-ZF and LR-MMSE compared with ZF and MMSE.

However, the performances of LR-ZF and LR-MMSE are still worse than ML

decoding due to the noise enhancement. The complexity of LR aided linear

detectors is a little larger than that of their corresponding linear detectors due to

the additional polynomial-time for preprocessing the LLL algorithm.

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2.6 Summary

Several existing detection algorithms for MIMO systems have been reviewed

thoroughly. As far as the comprehensive performance is concerned, they either

suffer from very high computational complexity, or rather bad error rate

performance. Thus, a lot more attentions should be paid on developing new

detection algorithms for MIMO systems, especially for the cases in which large

number of antennas and high level modulation are involved.

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Chapter 3 Geometric Detection Algorithms

35

CHAPTER 3

GEOMETRIC DETECTION ALGORITHMS

3.1 Introduction

Since the minimum Euclidean distance principle of ML decoding could result

in the optimum error rate performance, the purpose of this chapter is to introduce

another perspective of reconsidering this principle. In the hyper space spanned by

the transmit lattice points, the Euclidean distance involved in the ML decoding is

found to be related to a series of concentric hyper ellipsoids. Searching the lattice

point with the minimum Euclidean distance away from the received signal point is

equivalent to searching for the lattice point that lies on the surface of the smallest

possible hyper ellipsoid. Decoding algorithms following this perspective are often

termed as geometrical detection. In this chapter, the geometrical analysis of signal

decoding for MIMO channels is presented. Then, the proposed ellipsoid searching

decoding algorithm (ESA) [69] are elaborated. It is an add-on to the standard

suboptimal detection schemes, such as ZF or MMSE. Simulation results

demonstrate that it offers the optimum error rate performance and higher diversity

gains than the standard suboptimal detection schemes.

3.2 Geometrical Analysis of Signal Decoding for MIMO Channels

The Euclidean distance involved in the ML decoding can be rewritten as:

( )2

= f −x r Hx

( ) ( )1= T

c c

−− −x x Μ x x (3.1)

where the matrix ( )1

= T −

Μ H H . The vectorc

x is the result of ZF equalization

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which is determined by:

( )1

= T T c−

x H H H r

( )1

+ T T −

= x H H H n (3.2)

= +x n

where ( )1

T T −

=n H H H n .

Substituting (3.1) into (1.8) yields:

( ) ( )1ˆ argmin T

ML c c

−

∈Ω

= − −x

x x x Μ x x (3.3)

It can be seen from (3.2) and (3.3) that in the absence of noise, i.e., the

transformed Gaussian noise term ( )1

T T −

=n H H H n , both ZF equalization and

ML decoding result in the same correct solution. The reason why ML decoding

can offer much better performance than ZF equalization lies in the fact that the

transformed Gaussian noise has been minimized by the exhaustive search used in

ML decoding. However, the results of ZF equalization are directly distorted by the

transformed Gaussian noise n .

By using eigenvalue decomposition [72-73], the matrix M can be

decomposed into:

( )1

= =T T −

Μ H H VΛV (3.4)

where ( )1 2, ,..., T T T M M

M diag λ λ λ ×= ∈Λ , represents the T M eigenvalues

arranged in descending order, and 1 2= , , , T T

T

M M

N

×⎡ ⎤ ∈⎣ ⎦V V V V is the

corresponding eigenvector matrix.

The condition of the channel can be indicated by the channel condition

number which is defined as:

( ) 1= / T M λ κ λ H (3.5)

It has been proven [68] that the channel condition number ( )κ H has a profound

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impact on the error rate performances of the linear detections. If ( )κ H is low,

the error rate performances of the linear detections are very close to optimum.

However, if the ( )κ H is very high, the error rate performances become

unacceptably bad due to the considerable noise contamination n . In the

geometric point of view, the contour surfaces of the probability density function

(PDF) of the noise n are the hyper ellipsoids with their directions of axis

decided by the eigenvector matrix V . The direction of the i th− axis of the

hyper ellipsoids is given by the direction of the vector represented by the i th−

column of V , and the length of the i th− axis is proportional to the square root

of the corresponding eigenvalue.

Fig 3.1 illustrates the PDF of the received signal vector of ZF detections in

2 2× MIMO systems using BPSK modulation. The solid lines are the boundary

lines of the ZF decision regions and the dash lines are the boundary lines of the

ML decision regions. The channel condition number in Fig. 3.1 (a) is 1.5 which

can be considered as a good channel, while the channel condition number in Fig.

3.1 (b) is 7.5 which can be thought as a bad channel. In the case with good

channel condition, the boundary lines of ZF decision region is very likely close to

those of the boundary lines of ML decision region. So the performance of ZF

detection is good in this case. In the case with bad channel condition, the ML

decision regions are able to match to the PDF of the received signal vector, but the

ZF decision regions can not. The boundary lines of the ML decision region are

approximately orthogonal to the dominant principal axis which is corresponding

to the vector 1V . In general, the decision regions of ZF detector cannot have this

property since its boundary lines always go through the origin point.

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(a)

(b)

Figure 3.1 Probability density function of the received signal vector of ZF

detections in 2 2× MIMO systems.

(a) Case for good channel condition. (b) Case for bad channel condition.

Geometrically, the Euclidean distance function ( ) f x given in (3.1)

represents an elliptic paraboloid [70] in an 1T M + dimensional space with its

axis perpendicular to anT M dimensional subspace spanned by the signal vectors

in Ω . cx is the global minimum point of the elliptic paraboloid and is located

on the subspace spanned by the signal vectors in Ω as shown in Fig. 3.2. From

(3.1) it can be known that the function( )

f x reaches its minimum value atc

x

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if it is a continuous function, that is:

min( ) ( ) 0c f f = =x x (3.6)

The horizontal-cross section of the elliptic paraboloid (3.1) is aT M dimensional

hyper ellipsoid given by:

( ) 2 f a=x (3.7)

where 2a represents the height of the cross section above the T M dimensional

space as shown in Fig. 3.2. The length and the direction of the i-th semiaxis of the

hyper ellipsoid are given as ia λ and iV , respectively. With different value of

2a , a series of concentric hyper ellipsoids could be obtained and projected onto

the subspace spanned by the vectors as shown by the dash lines in Fig. 3.2. Thus,

searching the lattice point with the minimum Euclidean distance is equivalent to

searching for the lattice point that lies on the surface of the smallest possible

hyper ellipsoid.

Figure 3.2 Elliptic paraboloid with axis perpendicular to a subspace spanned bylattice points.

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3.3 Ellipsoid-Searching Decoding Algorithm

From section 3.2 we know that 2( ) f a=x represents a hyper ellipsoid

centered at the pointc

x , moreover, the length and the direction of its

i th− semi-axis are given asia λ and iV , respectively. By choosing different

values of a , a group of similar hyper ellipsoids can be obtained. Thus, the

solution of ML decoding must be located on a hyper ellipsoid which has the

minimum surface area among its similar hyper ellipsoids.

Figure 3.3 Elliptic paraboloid in 3-dimensional space.

Fig. 3.3 shows a two dimensional lattice point space (1 2 x x− plane) with

three lattice points Point 1, Point 2, and Point 3 as shown in the figure. With

different 2a , a group of similar hyper ellipsoids can be obtained, and their

projection onto the 1 2 x x− plane are ellipses which are all centered at the point

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1

c

−x = H r . For each lattice point, there exists an ellipse that passes through it. The

corresponding ellipse of the ML solution is the one that has the minimum area. As

shown in Fig. 3.3, Point 1 is taken to be the ML solution while Point 2 and Point 3

are not, since Point 1 is located on the inner-most ellipse which has the minimum

area.

However, finding the smallest hyper ellipsoid containing the solution signal

vector is not an easy task. If we use the largest hyper ellipsoid which contains all

the signal vectors, then the complexity will be the same as ML decoding. Here we

propose an ellipsoid-searching decoding algorithm that uses a small hyper

ellipsoid containing the solution symbol vector to start the search and then

identify all the symbol vectors inside. The ESA consists of the following 4 steps:

3.3.1 Start with Zero-Forcing Points

It is well known that the ZF decoding is a kind of linear equalization

algorithm. Although it can not offer very good performance like ML decoding, its

solution however usually lies in the neighborhood of the transmit signal point.

Thus we can consider choosing the hyper ellipsoid that goes through the ZF

solution to start the searching process. Firstly, the ZF equalized zf x is solved.

Then its corresponding 2 zf a is computed. The starting hyper ellipsoid is obtained

as:

( ) 2 zf zf f a=x (3.11)

3.3.2 Determine a Circumscribed Hyper Rectangle

After determining the hyper ellipsoid, the next key task is to identify whether

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there are any lattice points located inside this hyper ellipsoid. The axes of the

T M -dimensional rectangular coordinate system for the lattice point space are

denoted asiα - axes. Since the directions of the hyper ellipsoid’s semiaxes are not

in parallel with the axes of the coordinate system of the lattice point space, it is

rather complicated to directly use the surface equation (3.11) of the hyper

ellipsoid. Here we propose to use a circumscribed hyper rectangle as follows.

We set up a new T M -dimensional rectangular coordinate system with iα ′ -

axes ( 1,2,3, ..., T i M = ) being coincided with the i th− semiaxis of the hyper

ellipsoid and the origin coincided with the global minimum pointc

x . We use the

superscript prime to denote the variables in the new coordinate system. The

coordinates of the 2 T M apexes of the circumscribed hyper rectangle in this new

coordinate system are given by

1 2, ,...T

p p p pM k x x x′ ⎡ ⎤′ ′ ′=

⎣ ⎦

(3.12)

where 1,2,3,...2 T M p = , pj zf j

x a λ ′ = ± , and zf a is related to the hyper

ellipsoid given by (3.11). It can be easily shown that, by using coordinate

transformation, the coordinates of the 2 T M apexes in the original lattice point

space are:

( )T

T

p p c′= ⋅ +k V k x (3.13)

where V is the eigenvector matrix in (3.4), and it serves as the transformation

matrix:

11 21 31 1

12 22 32 2

1 2 13 23 33 3

1 2 3

, , ,T

T

T

T

T T

T

T T

M

M

M M

M

T M

M M M

v v v v

v v v v

v v v v

v v v v

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤= =⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥

⎢ ⎥⎣ ⎦

V V V V

…

(3.14)

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Thus the value of the i th− component of p

k can be obtained as

( )1

T M

pi qi pq ci

q

x v x x

=

′= +∑ (3.15)

whereci x is the i th− component of cx . Since pq zf q x a λ ′ = , the maximum and

minimum boundaries of the values of the each component in pk in the iα ′ - axes

can be expressed as:

_ max

1

T M

i ci qi zf q

q

x x v a λ =

= + ∑ (3.16a)

_ min

1

T M

i ci qi zf q

q

x x v a λ =

= − ∑ (3.16b)

Since the circumscribed hyper rectangle encloses the hyper ellipsoid, any

lattice point 1 2 ... T M s s s⎡ ⎤= ⎣ ⎦s inside the hyper ellipsoid satisfies:

_ min _ maxi i i x s x< < 1,2,3,..., T i M = (3.17)

It should be noted that this is not a sufficient condition for identifying the

lattice points lying inside the hyper ellipsoid.

From (3.17), we can obtain the possible value set { }1 2 3, , ,i i i iξ ε ε ε = of the

i th− element for the lattice points located inside the hyper ellipsoid. So the

search set becomes a larger hyper rectangle that encloses the circumscribed hyper

rectangle. For PAM and QAM, the elements of j

ξ are the odd numbers between

_ maxi x and _ mini x , and it can be easily shown that the number of elements is:

1

T M

i qi zf q

q

Num v a λ =

⎢ ⎥= ⎢ ⎥

⎣ ⎦∑ (3.18)

3.3.3 Narrow the Search Set into Ellipsoid

As mentioned before, the search set becomes a larger hyper rectangle and the

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number of lattice points inside is1,

T M

i

i i l

Num= ≠∏ . If there is any i Num equals zero,

then it means that there is no lattice point located inside the hyper ellipsoid. The

searching process will terminate and the zero forcing point chosen before is

considered as the solution.

Otherwise, assuming the possible value set ω ξ has the largest number of

elements among all the possible value sets, we form the combinations from the

other 1T M − possible value sets, and then substitute each of these combinations

into (3.11), to determine the lattice point elements of the possible value set ω ξ

that are located inside the hyper ellipsoid. In doing so, the number of

combinations that need to be considered is smaller and hence lesser computation

complexity. Denoting the k th− combination by:

1, 2, 1, 1, ,, , , , T k

k k k k M k ω ω ε ε ε ε ε − +⎡ ⎤= ⎣ ⎦Com (3.19)

1,1,2,...,

T M

j j j

k Numω = ≠= ∏

where , j k ε represents an arbitrary element of the set jξ .

Geometrically, the k Com is a line pierced through the hyper ellipsoid. The

intersection of the line and the hyper ellipsoid consists of two points, known as

max,k E and min,k E along the thω − axis. Hence, the corresponding possible value

set { }, ,1, ,2,, ,...k k k ω ω ω ζ ς ς = for the thω − element of the lattice points are the

odd numbers between max,k E and min,k E . Thus, any lattice point that is located

inside the hyper ellipsoid can be expressed as:

1, 2, 1, , , 1, ,, , , , , , T T

k k k d k k M k d k ω ω ω ε ε ε ς ε ε − +⎡ ⎤= ⎣ ⎦x (3.20)

1,2,...,k d n=

where k n is the number of the elements of ,k ω ζ fork Com .

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3.3.4 Calculate the Euclidean Distance

All the Euclidean distances of the signal vectors inside the hyper ellipsoid can

be calculated recursively. Let the Euclidean distance the signal vector ,d k x is

denoted as,d k μ . So the Euclidean distance +1,d k μ of the signal vector 1,d k +x can

be written as:

+1, , ,d k d k d k μ μ μ = + (3.21)

where 1, , 2d k d k ω + = +x x .

Substituting (3.21) into (3.1), we will get:

( )2

, ,4 4 T

d k i d k iμ = − −h r Hx h (3.22)

After all the Euclidean distances are calculated, the signal vector with the

minimum distance is then selected as the solution.

3.3.5 Examples

The following subsections will give two examples of the ESA in two

dimensional space and three dimensional space.

3.3.5.1 2-D lattice space

For a 2 2× 8-PAM MIMO system, the lattice set is a 2-dimensional space as

shown in Fig. 3.4, where it is assumed that the ellipse and its circumscribed

rectangle have been determined using our proposed method as described

previously. The semiaxes of the ellipse are in parallel with vectors 1V and 2V

with lengths 1 zf a λ and 2 zf a λ , respectively. The global minimum point cx

is marked by a triangle on the figure. The coordinates of the four apexes, A, B, C

and D, in the new coordinate system are given by ( )1 2, zf zf A a aλ λ = − − ,

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( )1 2, zf zf B a aλ λ = − + , ( )1 2, zf zf C a aλ λ = − , and ( )1 2, zf zf D a aλ λ = + , respectively.

Substituting these vectors into (3.15) yields the corresponding coordinates in the

lattice point space. From (3.16) the 1x coordinates of points A and D are chosen

as1_min x and 1_max x , respectively, and the 2x coordinates of points B and C

are chosen as 2_min x and 2_max x , respectively. Using (3.16), we can obtain a

possible set of values along each axis, i.e., two values {1, 3} along the1x -axis

and one value {1} along the 2x -axis. Since the number of values along the

1x -axis is larger than that along the 2x -axis, we substitute 2,1 1ε = into the hyper

ellipsoid equation (3.11). As shown in Fig. 3.5, the possible value along the

1x -axis is 1,1,1 3ς = , so the point [ ]1,1 3 1 T

=x is obtained. Since it is the only

point located inside the ellipse, it would be the final solution.

Figure 3.4 2-D lattice space example.

3.3.5.2 3-D lattice space

Here, we continue to consider the case of 3-dimensional lattice space, namely

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3 3× 8-PAM. Fig. 3.5 shows a 3-dimensional ellipsoid with its circumscribed

rectangle which has been set up by the method introduced in section 3.3.2. cx is

the center of the ellipsoid, whose semiaxes are aligned along vectors 1V , 2V ,

3V , with their lengths being 1 zf a λ , 2 zf a λ and 3 zf a λ , respectively. By

substituting the coordinates of the eight points A to H to (3.15) and (3.16), 1_min x

and1_max x , 2_min x and 2_max x , 3_min x and 3_max x , which are all marked as

dots are obtained. The possible set of values along 1x -axis is {1, 3, 5}, and the

possible set of values along the 2x -axis is {1, 3}. Along 3x -axis, the possible set

of value is {-1}. Since the number of possible values along the 1x -axis is the

largest compared to those along the other axes, we substitute

[ ]1 2,1 3,1, 1, 1ε ε ⎡ ⎤= = −⎣ ⎦Com and [ ]2

2,2 3,2, 3, 1ε ε ⎡ ⎤= = −⎣ ⎦Com into (3.11) to

determinemax,k E and min,k E along the 1x -axis. As shown in Fig. 3.5, the

possible value set 1,1ζ

along the 1x -axis is {1} for

1

Com and 1,2ζ

is {5} for2

Com , so the point [ ]1,1 1 1 1 T

= −x and the point [ ]1,2 5 3 1 T

= −x are

obtained. By calculating their corresponding Euclidean distance, it can be

concluded that the point1,2x that has a smaller Euclidean distance is taken as the

final solution.

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Figure 3.5 3-D lattice space example.

3.4 Simulation Results

The ESA algorithm for MIMO systems has been briefly introduced. It

contains three main steps: Firstly, determine the hyper ellipsoid. Secondly, find

out the probable value sets for each component of the lattice point that is located

inside the hyper ellipsoid. Finally, search for the ML solution. In the first step,

either ZF decoding or MMSE decoding can be selected for determining the hyper

ellipsoid. In the second step, we firstly determine a loose boundary for each

component of the lattice points that may be located in the hyper ellipsoid. Then,

by further shrinking the value set of theT M -th component, all the redundant

points can be discarded and the lattice points inside the hyper ellipsoid are exactly

detected.

Since the ESA algorithm strictly sticks to ML decoding, it can achieve the

same performance with ML decoding. The ML decoding searches the entire lattice

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points while the ESA algorithm only searches a small subset. The ESA algorithm

is assessed by means of the simulation results of the error rate performance. In the

simulations, we used 4-QAM, 16-QAM, 64-QAM in Rayleigh flat fading

channels with i.i.d. complex zero-mean Guassian noise. Fig. 3.6 illustrates the

BLER performance of ESA compared with ML decoding and ZF decoding using

4-QAM. Fig. 3.7 shows the BLER performance of ESA compared with ML

decoding ZF decoding using 16-QAM. And Fig. 3.8 shows the BLER

performance of ESA compared with ML decoding and ZF decoding using

64-QAM. It can be seen that the performances of ESA can achieve that of ML

decoding and are much better than ZF decoding.

Table 3.1 compares the complexity of ML decoding and ESA. The numbers

of lattice points visited by ML decoding and ESA for transmitting 16-QAM and

64-QAM constellations in 2 2×

to 4 4×

MIMO systems are indicated. It can be

observed that compared with the ML decoding, the number of lattice points

visited by the ESA is substantially reduced from 95.7% to 99.8%. The more

number of antennas and the higher level of modulation the system applies, the

greater complexity reduction the ESA can achieve.

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(a)

(b)

Figure 3.6 Comparison of BLER performance of ESA, ML decoding and ZF

using 4-QAM.

(a) 4 4× MIMO systems. (b) 6 6× MIMO systems.

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(a)

(b)

Figure 3.7 Comparison of BLER performance of ESA, ML decoding and ZF

using 16-QAM.

(a) 4 4× MIMO systems. (b) 8 8× MIMO systems.

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