基於幾何方法欠定多輸入多輸出系統之高效率解碼演算法

國 立 交 通 大 學

電信工程研究所

碩 士 論 文

基於幾何方法欠定多輸入多輸出系統之

高效率解碼演算法

Geometry Based Efficient Decoding Algorithm for

Underdetermined MIMO Systems

研 究 生：吳智湧 Student: Chih-Yung Wu

指導教授：李大嵩 博士 Advisor: Dr. Ta-Sung Lee

基於幾何方法欠定多輸入多輸出系統之

高效率解碼演算法

Geometry Based Efficient Decoding Algorithm for

Underdetermined MIMO Systems

研 究 生：吳智湧 Student: Chih-Yung Wu

指導教授：李大嵩 博士 Advisor: Dr. Ta-Sung Lee

國立交通大學

電信工程研究所

碩士論文

A Thesis

Submitted to Institute of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master

in

Communication Engineering

June 2010

Hsinchu, Taiwan, Republic of China

基於幾何方法欠定多輸入多輸出系統之

高效率解碼演算法

學生：吳智湧

指導教授：李大嵩 博士

Chinese Abstract

國立交通大學電信工程研究所

摘要

Geometry Based Efficient Decoding Algorithm for

Underdetermined MIMO Systems

Student: Chih-Yung Wu

Advisor: Dr. Ta-Sung Lee

English Abstract

Institute of Communication Engineering

National Chiao Tung University

Abstract

The design of high-performance and low-power consumption receiver is one of

the key issues of MIMO systems. The sphere decoding algorithm (SDA) is an

effective detector for MIMO systems. However, typical SDA fail to work in

underdetermined MIMO systems where the number of transmit antennas is larger than

the number of receive antennas. The generalized sphere decoder (GSD) had been proposed for underdetermined MIMO systems. However, its decoding complexity is

exponentially increasing with the antenna number difference. In this thesis, we

propose a decoder for underdetermined MIMO systems with low decoding complexity.

The proposed decoder consists of two stages: 1. Obtaining all valid candidate points

efficiently by slab decoder. 2. Finding the optimal solution by conducting the intersectional operations with dynamic radius adaptation to the candidate set obtained

from Stage 1. We also propose a reordering strategy that can be incorporated into the

proposed decoding algorithm to provide a lower computational complexity and

near-ML decoding performance for underdetermined MIMO systems. Simulations

Acknowledgement

I would like to express my deepest gratitude to my advisors, Dr. Ta-Sung Lee, for

his enthusiastic guidance and great patience. I learned a lot from his positive attitude in

many areas, especially in the training of oral presentation. I thank Chester senior

whose knowledge and experience have benefited me tremendously in my research.

Thanks are also offered to all members in the Communication System Design and

Signal Processing (CSDSP) Lab. Last but not least, I would like to show my sincere

Table of Contents

Chinese Abstract ...I

English Abstract ... II

Table of Contents ... IV

List of Figures ... VI

Acronym Glossary ...VII

Notations….. ... ...VIII

Chapter 1 Introduction... 1

Chapter 2 Underdetermined MIMO System Model ... 4

2.1 System Model ...5 2.2 Channel Capacity ...6 2.3 MIMO Diversity ...10 2.3.1 Receive Diversity...10 2.3.2 Transmit Diversity ... 11 2.4 Spatial Multiplexing... 11 2.5 MIMO Detection...13

2.6 Underdetermined MIMO Detection...15

2.6.1 Generalized Sphere Decoder (GSD) ...15

2.6.2 Slab Sphere Decoding (SSD) Algorithm ...17

Chapter 3 Proposed Geometry Based Decoding Algorithm with

Intersection of Candidate Sets ... 24

3.1 Efficient Search Method for Points in Slab ...25

3.2 Efficient Decoding Algorithm with Intersection of Candidate Sets...29

3.3 Computer Simulation ...34

3.4 Summary ...38

Chapter 4 Preprocessing of Proposed Decoding Algorithm ... 39

4.1 Preprocessing ...40

4.2 Alternative Approach of Preprocessing ...44

4.3 Computer Simulations ...48

4.4 Summary ...51

Chapter 5 Conclusions and Future Works ... 52

List of Figures

Fig. 2-1 MIMO system block diagram ...5

Fig. 2-2 Spatial multiplexing system ...12

Fig. 2-3 Encoding procedure of D-BLAST (n=3)...13

Fig. 2-4 Encoding procedure of V-BLAST (n=3) ...13

Fig. 3-1 An example of slab with 4-PAM and k = 2...28

Fig. 3-2 Block diagram of typical underdetermined decoding algorithms ...29

Fig. 3-3 Illustration of intersection of candidate sets ...31

Fig. 3-4 SER comparison of SSD and the proposed method using 16-QAM ...36

Fig. 3-5 Complexity comparison of SSD and the proposed method using 16-QAM...36

Fig. 3-6 SER comparison of SSD and the proposed method using 64-QAM ...37

Fig. 3-7 Complexity comparison of SSD and the proposed method using 64-QAM...38

Fig. 4-1 Geometrical diagram of slabs with different y...41

Fig. 4-2 Probability density function of y_m ...47

Fig. 4-3 The reduced percentage of Method I and Method II...48

Fig. 4-4 SER comparison of proposed method with and without preprocessing ...50 Fig. 4-5 Complexity comparison of proposed method with and without preprocessing 50

Acronym Glossary

AWGN additive white Gaussian noise

D-BLAST diagonal Bell laboratories layered space-time

EGC equal-gain combining

GSD generalized sphere decoder

GS-QRD Gram-Schmidt process-QR decomposition

Householder-QRD Householder process-QR decomposition

MIMO multiple-input multiple-output

ML maximum-likelihood

MRC maximum ratio combining

PDA plane decoding algorithm

SDA sphere decoding algorithm

SD spatial diversity

SE Schnorr and Euchner

SER symbol-error-rate

SIC

successive interference cancellation

SISO single-input-single-output

SLA Slab Decoding Algorithm

SM spatial multiplexing

SNR signal-to-noise ratio

SSD slab-sphere decoder

STC space-time code

V-BLAST vertical Bell laboratories layered space-time

Notations

γ ⋅ _{lower incomplete Gamma function}

( )

Γ • Gamma function

( )

P • regularized Gamma function

C channel capacity

H complex channel matrix

H real channel matrix

t

N number of transmit antennas

r

N number of receive antennas

P transmitted symbol power

n complex AWGN

n real AWGN

x complex transmitted signal vector x real transmitted signal vector y complex received signal vector y _{real received signal vector}

Chapter 1

Introduction

Recently, in order to satisfy the growing demands of the personal

communications, the design of next generation wireless communication systems goes

for supporting high data rate and high mobility. However, the link quality suffers from

frequency selective and time selective fading caused by multipath propagation in

wireless channels. Moreover, the quality and reliability of wireless communication are

degraded by Doppler shift and carrier frequency/phase. Beside, due to the limited

available bandwidth and transmitted power, the design challenge of wireless

communication systems becomes more difficult. Therefore, many innovative

techniques have been devised and extensively used in this field to improve the

reliability and the spectral efficiency of wireless communication links e.g. the coded

multicarrier modulation, smart antenna and multiple-input multiple-output (MIMO)

technology [1-4] and adaptive modulation [5], [6].

Among these technologies, MIMO is the most outstanding one. MIMO

technology involves the use of multiple antennas to improve link performance. There

are two major features of MIMO technologies: spatial multiplexing for increasing

data rate and spatial diversity for improve link quality. Spatial multiplexing offers a

same time. Spatial diversity provides diversity gain to mitigate fading effects by using

the multiple (ideally independent) copies of the transmitted signal in space, time and

frequency. They are usually trade-offs to each other and provide an effective and

promising solution while achieving high-data rate and reliable transmission.

The major MIMO signal detection schemes include linear detection, successive

interference cancellation (SIC) [7], [8] and the maximum-likelihood (ML) detection.

The advantages of the first two detection schemes are low decoding complexity and

easy implementation but their detection performances are non-optimal. ML detection

provides optimal detection performance but its complexity increases exponentially

with the size of constellation and the number of transmit antennas. Therefore, the

design of high-performance and low decoding complexity is the one of key issues of

MIMO designs. To reduce the complexity of the ML detector, the sphere decoding

algorithm (SDA) [9], [10] has received considerable attention as an efficient detection

scheme for MIMO systems. However, typical SDA fail to work in underdetermined

MIMO systems where the number of transmit antennas is larger than the number of

receive antennas.

To overcome the above drawbacks of typical SDA, the conventional generalized

sphere decoder (GSD), double-layer sphere decoder (DLSD) and slab sphere decoder

(SSD) are introduced in [11-15]. These decoders transform underdetermined systems

into overdetermined systems that can be solved by the SDA. Since the GSD performs

an exhaustive search on (N_t −N_r) dimensions for the ML solution, the decoding complexity is increasing with the size of constellation and the antenna number

difference. The DLSD uses the outer sphere decoder to find the valid candidate points

and then the inner sphere decoder uses those points to find the solution. The SSD uses

the geometry concept to find the valid candidate points to reduce the searching

in SSD.

In this thesis, our goal is to reduce the decoding complexity of the SSD without

degrading the decoding performance. The proposed decoder consists of two stages: 1.

Obtaining all valid candidate points efficiently by the slab decoder. 2. Finding the

optimal solution by conducting the intersectional operations with dynamic radius

adaptation to the candidate set obtained from Stage 1. We also propose a reordering

strategy that can be incorporated into the proposed decoding algorithm to provide a

lower computational complexity and near-ML decoding performance in

underdetermined MIMO systems.

The organization of this thesis is given as follows. In Chapter 2, the signal model

and typical detection schemes of overdetermined MIMO systems are introduced. In

Chapter 3, the proposed decoder is presented. Two preprocessing scheme for the

proposed decoder are suggested in Chapter 4. Simulation results are presented in both

Chapter 3 and Chapter 4. Finally, Chapter 5 gives the conclusion and future works of

Chapter 2

Underdetermined MIMO System

Model

Many MIMO technologies are implemented in order to achieve the multiplexing

gain and more diversity gain for wireless communication system. But in some

scenarios (e.g. a strong LoS signal at receiver), an overdetermined MIMO system can

be degraded into an underdetermined MIMO system. In underdetermined MIMO

systems, the well-known SDA fail, therefore we need some special decoding schemes

for these cases. In this chapter, those technologies of MIMO systems are introduced.

We first introduce the MIMO system model in Section 2.1. Section 2.2 introduces the

channel capacity. Second, the spatial diversity (SD) and the spatial multiplexing (SM)

techniques are introduced in Section 2.3 and Section 2.4, respectively. Finally, the

commonly used detection schemes for underdetermined MIMO systems including

generalize sphere decoder (GSD), and Slab-sphere decoder (SSD) will be given in

2.1 System Model

Consider a Gaussian MIMO system with N transmit antennas and _t N receive _r antennas as shown in Fig. 2-1. The relation between transmitted signal vector and

received signal vector can be written as

, = + y Hx n (2.1) where ₁, , ,₂ r 1 r N N y y y × ⎡ ⎤ = _{⎢⎣ ⎥⎦} ∈

y stands for the received signal vector,

[

]

1, , ,2 N_t Nt

x x x ×

= ∈

x stands for the transmitted signal vector, H stands for

the frequency-flat fading channel matrix:

1,1 1,2 1, 2,1 2,2 2, ,1 ,2 , , t t r t r r r t N N _{N N} N N N N h h h h h h h h h × ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ =_{⎢ ⎥} ∈ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ H (2.2)

where the elements of H are complex i.i.d. Gaussian random variables with normal

distribution CN(0,1) and n=

[

]

(

)

TX RX Channel matrix H



x



x



x



y



y



y

The complex equation in (2.1) can be rewritten into an equivalent real system by

real value decomposition as

, = + y Hx n (2.3) where { } { } { } { } { } { } [Re Im ] [Re Im ] [Re Im ] , T T M N T T M T T = ∈ = ∈ = ∈ y y y x x x n n n (2.4) and

{ }

{ }

{ }

{ }

Note the H is a size M× matrix where N M = ×2 N_r andN = ×2 N_t.

2.2 Channel Capacity

Channel capacity is the upper bound of data rates in bits per channel that can be

reliably transmitted over a communication channel. In other words, by channel coding

theorem, if the data rate of transmission is below the channel capacity, the transmitted

signals can be recovered with an arbitrarily small error probability. First, we introduce

the single-input-single-output (SISO) channel capacity. Second, the MIMO channel

capacity is introduced. The channel capacity is defined as [16]

( ) max I( ; ), p x C = X Y (2.6) where I( ; )X Y =H( )Y −H( | ),Y X (2.7)

entropy of Y and differential conditional entropy of Y with knowledge of X given,

respectively. In (2.6), it states that the mutual information is maximized with respect to

all possible transmitter statistical distributions p(x).

SISO Channel Capacity

For SISO systems, the ergodic capacity of a random channel can be defined as

( ):max I( ; ) ,1 p x P C E X Y = ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ = ⎨_⎪ ⎬_⎪ ⎪ ⎪ ⎩ ⎭ (2.8)

where P =E x k

{

}

the expectation of all channel realization, and the mutual information is equal to

2 2

log (1+γ h ). The channel capacity defined in (2.8) means that the maximum of

mutual information between X and Y of all statistical distribution on the X. From (2.8),

the SISO system ergodic capacity I( ; )X Y can be replaced by log (1₂ +γ h2) [16]

{

}

2

log (1 ) bits/sec/Hz,

C =E +γ h (2.9)

where γ =P σ2 is the average SNR at the receiver, P is the transmit power.

MIMO Channel Capacity

For a MIMO system with N transmit antennas and _t N receive antennas, the _r ergodic capacity of a random MIMO channel can be defined as [1]

( ) ( ):trmax_{xx t} I( ; ) , p x N C E X Y = ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ = ⎨_⎪ ⎬_⎪ ⎪ ⎪ ⎩ R ⎭ (2.10)

where R_xx =E

{ }

( ) _{2 2} I ; log det( _{Nr xx} H) . t P X Y E N σ ⎧ ⎡ ⎤⎫ ⎪ ⎪ ⎪ _{⎢ ⎥}⎪ = ⎨_{⎪ ⎢} + _⎥⎬_⎪ ⎪ ⎣ ⎦⎪ ⎩ ⎭ I HR H (2.11)

Substituting (2.11) into (2.10), we have

2 2

(max_xx) _t log det( r ) bits/sec/Hz.

H N xx tr N _t P C E N σ = ⎧ ⎡ ⎤⎫ ⎪ ⎪ ⎪ _{⎢ ⎥}⎪ = _{⎨ ⎢} + _⎥⎬ ⎪ ⎪ ⎪ ⎣ ⎦⎪ ⎩ ⎭ R I HR H (2.12)

When the channel knowledge is unknown to the transmitter, the optimal transmit

signals are chosen to be independent and equal power. With independent and uniform

power distribution, the covariance matrix of the transmit signal vector is then given by

r

xx = N

R I . As a result, the ergodic capacity of a MIMO system can be written as [1]

2 ₂

log det( _Nr H) bits/sec/Hz.

t P C E N σ ⎧ ⎡ ⎤⎫ ⎪ ⎪ ⎪ _{⎢ ⎥}⎪ = ⎨_{⎪ ⎢} + _⎥⎬_⎪ ⎪ ⎣ ⎦⎪ ⎩ ⎭ I HH (2.13)

By using the eigenvalue decomposition, the matrix product of HH can be H decomposed as HHH = ΛE E where E is an H N_r×N_r matrix which consists of the eigenvectors satisfying EEH =E EH =I_Nr and Λ = diag

{

}

2_{, if 1} , if 1 0, i i r i r r i N σ λ = ⎨⎧⎪⎪⎪_{⎪ + ≤ ≤}≤ ≤ ⎪⎪⎩ (2.14)

where σ is the ith squared singular value of the channel matrix H and _i2

( )

{

}

rank min _t, _r

r = H ≤ N N is the rank of the channel matrix. Then the capacity of

a MIMO channel can be rewritten as

2 2 2 ₂ 2 ₂ 1 log det( ) log det( ) log (1 ) bits/sec/Hz. r r H N t N t r i i t P C E N P E N P E N σ σ λ σ = ⎧ ⎡ ⎤⎫ ⎪ ⎪ ⎪ _{⎢ ⎥}⎪ = ⎨_{⎪ ⎢} + Λ _⎥⎬_⎪ ⎪ ⎣ ⎦⎪ ⎩ ⎭ ⎧ ⎡ ⎤⎫ ⎪ ⎪ ⎪ _{⎢ ⎥}⎪ = ⎨_{⎪ ⎢} + Λ _⎥⎬_⎪ ⎪ ⎣ ⎦⎪ ⎩ ⎭ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ = ⎨_⎪ + ⎬_⎪ ⎪ ⎪ ⎩ ⎭

∑

Note that the second equation holds due to the fact det

(

)

(

)

r t

N + = N +

I AB I BA

for matrices A∈ Nr×Nt _, B∈ N Nt× r _and

r

H

N

=

E E I . Eq. (2.15) shows that the capacity of a MIMO channel is made up by the sum capacities of r independent SISO

sub-channels with power gain λ for i = 1,2,…,r and transmit power _i P N_t individually.

When the channel knowledge is known to the transmitter, the capacity of a MIMO

channel is the sum of the capacities associated with the independent SISO channels and

is given by 2 ₂ 1 log (1 ) bits/sec/Hz, r i i i t P C E N γ λ σ = ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ = ⎨_⎪ + ⎬_⎪ ⎪ ⎪ ⎩ ⎭

∑

where γ_i =E x

{ }

{ }

_∑

maximize the mutual information as

1

2 2

1

max log (1 ) bits/sec/Hz,

r i t i r i i i t N P C E N γ γ λ σ = = = ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ = ⎨_⎪ + ⎬_⎪ ⎪ ⎪ ⎩ ⎭ ∑

∑

the optimal power allocation of the ith sub-channel is a water-filling solution given by

[1], [16] 2 opt _{for 1,2, , ,} i i m i r P σ γ μ λ ₊ ⎛ _⎞⎟ ⎜ _⎟ =⎜_⎜ − _⎟⎟ = ⎜⎝ ⎠ (2.18)

where μ is chosen to satisfy the constraint

∑

2.3 MIMO Diversity

Diversity techniques are widely used in MIMO systems to improve the reliability

of transmission without increasing the transmit power or bandwidth. There are many

diversity techniques such as space, frequency and time diversity. In this section the

space diversity is introduced, it is so called antenna diversity.

2.3.1 Receive Diversity

Receive diversity involves the receiver with multiple antennas. At the receiver,

multiple copies of the transmitted signal are received, which can be efficiently

combined with an appropriate signal processing algorithm. There are four main types

of combining techniques, include selection combing, switch combining, equal-gain

combining (EGC) and the maximum ratio combining (MRC).

1. Selection combining – The received signal with the best quality is chosen

and the choosing criterion is based on SNR.

2. Switch combining – Switch the received signal path to an alternative

antenna when the current received signal level falls below a given threshold.

3. EGC – It is a simple method since it does not require estimation of the

channel. The receiver simply combines the received signals from different

receive antennas with weights set to be equal.

4. MRC – It forms the output signal by a linear combination of all the

received signals and is the optimal combination technique which achieves

2.3.2 Transmit Diversity

Transmit diversity techniques provide more diversity benefits at the receiver with

multiple transmit antennas, has received much attention, especially in wireless cellular

systems. There are two broad categories of transmit diversity: the open loop schemes

and the closed loop schemes. In the open loop schemes, the transmitter transmits

signals without feedback information from receiver, e.g. Space-time code (STC). In the

closed loop schemes, the transmitter transmits signals with feedback channel

information from receiver, e.g. transmit beamforming.

2.4 Spatial Multiplexing

Spatial multiplexing is a transmission technique of MIMO wireless

communication systems which increases the channel capacity without additional power

or bandwidth, as shown in Fig. 2-2. In other words, spatial multiplexing means that

transmit independent and separately data signals on each transmitted antenna in order

to increase the channel capacity. If there are N antennas and _t N antennas of _r transmitter and receiver, respectively, the maximum spatial multiplexing order is

{

}

min _t, _r ,

D= N N (2.19)

if a linear receiver is used. This means that D data streams can be transmitted in parallel,

the data rate can be increased by D times in the ideal case. The practical multiplexing

gain is limited by correlation of channels, which means that some of the parallel

streams may have very weak channel gains. Two typical spatial multiplexing schemes,

Diagonal Bell Laboratories Layered Space-Time (D-BLAST)

Laboratories [2]. D-BLAST uses multiple antennas at both the transmitter and the

receiver. The encoder uses a space time arrangement that corresponds to a diagonal

layering. Fig. 2-3 show the encoding procedure for D-BLAST.

Vertical Bell Laboratories Layered Space-Time (V-BLAST)

suitable for practical situation. Therefore, a simplified version of the BLAST algorithm

is known as V-BLAST [17]. In V-BLAST system, independently encoded data streams

are transmitted from each transmit antenna simultaneously. The encoding procedure is

shown in Fig. 2-4.

Fig. 2-2 Spatial multiplexing system



x



x



x



y



y



y

Fig. 2-3 Encoding procedure of D-BLAST (n=3)

Fig. 2-4 Encoding procedure of V-BLAST (n=3)

2.5 MIMO Detection

In this section, we introduce the classification of MIMO detection schemes

including Zero-Forcing (ZF), Zero-Forcing Successive Interference Cancellation

(ZF-SIC) and Maximum-Likelihood (ML) detection. Assume that the received signal

is given by (2.3) . = + y Hx n (2.20) 0 0 0 3 3 3 1 1 1 4 4 4 2 2 2 5 5 5

0

0

0

α

β

γ

α

β

γ

α

β

γ

α

β

γ

α

β

γ

α

β

γ

⎡

⎤

⎢

⎥

⎢

⎥

= ⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

x

α

α

α

β

β

β

γ

γ

γ

⎡

⎤

⎢

⎥

⎢

⎥

= ⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

x

(1) Zero-Forcing (ZF)

The ZF scheme is a kind of linear detection that means the received signal y is

multiplied by a filter G _ZF

(

)

where H is the Moore-Penrose pseudo-inverse of H. The output vector after the † filter is as follows

† _.

=

y H y (2.22)

The ZF can remove the spatial interferences from the received signal; however, the

main drawback of ZF scheme is the resulting noise enhancement.

(2)

Successive Interference Cancellation (ZF-SIC)

Denote H as

1, , N_t .

⎡ ⎤

= ⎢_⎣ ⎥_⎦

H h h (2.23)

The received signal is given by

1 1 2 2 Nt Nt .

x x x

= + = + + + +

y Hx n h h h n (2.24)

The main idea of SIC is to cancel the detected symbol from the received signal to

improve primary ZF detection performance. The ZF filter is given in (2.21). The

decision statistics of the ith symbol is obtained as

† _,

i _i

x = H y (2.25)

where H†_i is the ith row of H . Taking hard decision to † x , the estimation symbol i i

s can be obtained. After s is detected, it is subtracted from the received signal to i

remove its influence, then updating the received signal as

' _,

i i

s

= −

where h is the ith column in the channel matrix H. Iterating the above procedure, _i the ZF-SIC solution can be achieved.

(3)

Maximum-Likelihood (ML) Detection

The well known optimal detection scheme of the MIMO systems is the ML

detection. The ML detection searches all possible combinations of transmitted

symbols via the following criterion:

ML 2 arg min_N , ∈ = − x x y Hx (2.27)

where N denotes the set of all possible transmitted symbol vectors. The computational complexity of an exhaustive searching algorithm for the ML solution

increases exponentially with N. Therefore, it is not easy to be implemented at the

receiver in practice which is the main drawback of this method.

2.6 Underdetermined MIMO Detection

The classification of underdetermined MIMO detection schemes (e.g. GSD and

SSD) are introduced in this section

2.6.1 Generalized Sphere Decoder (GSD)

In order to reduce the complexity of ML detection, the SDA was proposed to

achieve ML performance with low complexity. Hence, it is adopted on the receiver

design in recent years. But the SDA fails in the underdetermined MIMO systems and

underdetermined systems to overdetermined system that can be solved by SDA [19].

Consider a Gaussian MIMO system with N transmitted antennas and _t N _r received antennas, the received real signal can be formed as (2.3). The ML estimator

x of x is obtained by minimizing the Euclidean distance of y from the legal lattice points as

( )2

2

arg min arg min .

N N ρ

∈ ∈

= − = −

x x

x y Hx R x (2.28)

where = − −{ 3, 1, 1, 3} for 16-QAM cases, ρ = H HHT

(

)

M×M unitary matrix, and R is a M× upper triangular matrix corresponding to N

the QR-decomposition of H, i.e. H =QR . The matrix R can be represented by

[

]

=

R R R , where R₁ ∈ M M× is a upper triangular matrix and

2 ∈ M N M× −

R . Similarly, x can be represented by x= ⎣⎡x x_G, _G⎤_⎦T, where G and

G are the indices corresponding to the first M and the last N −M elements of the

x vector. The minimum distance corresponding to the ML estimator in (2.28) can be rewritten by ( )

[

]

where ρ =

[

]

The GSD checks all the valid constellation points whose squared Euclidean

distance calculated from (2.28) are smaller than a specified positive number C. It is

last equation in (2.29). The SDA will find if the squared minimum distance is less

than C. Otherwise, a failure of the SDA for the given x_G is declared and then the

G

x will be discarded.

If the candidate constellation point (x_G,x_G) is found within the sphere, the value of C will be updated and the algorithm continues to search the remaining points

for x_G. If no candidate constellation point (x_G,x_G) is found within the sphere, then the entire algorithm will repeat with a larger value of the radius C. The GSD based on

the exhaustive search over x_G and the SDA for every point of. Because of the exhaustive search over x_G , the complexity exponentially increase of the order

N −M .

2.6.2 Slab Sphere Decoding (SSD) Algorithm

To perform (2.28) efficiently, an algorithm is proposed in [18], [19] to solve a

search problem that finds all the lattice points satisfying

2 _C2

− ≤

y Hx (2.30)

for given a radius C (>0). Apparently, the point that is the closest to center of the

hypersphere y , is the ML decision point. By decomposing the channel using

QR-decomposition, Eq. (2.30) can be rewritten as

2 ' _{− ≤C}2_,

y Rx (2.31)

where y' =Q y . T

'

, , ,

M M M M M N N

C y ⎡r x r x ⎤ C

− ≤ −_⎢⎣ + + _⎥⎦≤ (2.32)

at the Mth layer. Eq. (2.32) involves N-M+1 dimensions for detection. Eq. (2.32) is

similar to a detection problem of a real-valued MISO system. First, we want to find

the constellation points falling inside this slab. There are two algorithms that can help

us find those constellation points, i.e., Plane Decoding Algorithm (PDA) [12] and

Slab Decoding Algorithm (SLA) [14], [15].

Plane Decoding

For a MISO system with k transmitted antennas where the inputs are independent

symbols, the received signal can be written as

1 1 k k ,

y =h x + +h x + (2.33) η

where x_k ∈ , h is the channel response and _n η ∼CN(0,σ2) stands for AWGN. ML estimation of the transmitted vector x=

[

]

( 1 )

(

)

the estimator means to find the point x ∈ k which is the closest to the hyperplane

P given as

1 1

: _{k k} .

P h x + +h x = (2.35) y

First, define X, X , _V X_PD as the sets of the points to be visited, the points that have been visited, and the points that are close to P in all dimensions, respectively.

Then, initialize them with X=X = _V

{ }

The main idea of the PDA is to find those candidates (X_PD) which are close to

Step 1: If X is empty, go to Step 5. Otherwise, we calculate

{

}

{

}

( )

( )

Step 2: If

{

}

the point x =x( )1 except that xj =aj where x is close to P in dimension-j. Then, if xj =x( )_j1 and then the point x( )1 is close to P in dimension-1,2,…,j and do:

z If j < k, update j = j + 1. Go to Step 1.

z If j = k, the point x( )1 is close to P in all dimensions and is stored in

PD

X . Next, discard x( )1 from the set X and reset j = 1. Go back to

Step 1 to check a new point in X.

Else, if xj ≠a( )_j1 , then discard x( )1 from the set X and reset j = 1. Go back to Step 1.

Step 3: If

{

}

the point x =x( )1 except that xj =aj where x is close to P in dimension-j. Then, if xj =x( )_j1 and then the point x( )1 is close to P in dimension-1,2,…,j and do:

z If j < k, update j = j + 1. Go to Step 1.

z If j = k, the point x( )1 is close to P in all dimensions and is stored in

PD

X . Next, discard x( )1 from the set X and reset j = 1. Go back to

Step 1 to check a new point in X.

Else, if xj ≠a( )_j1 , then discard x( )1 from the set X and reset j = 1. Go back to Step 1.

Step 4: If

{

}

two points x =x( )1 except that xj =aj and x =x( )1 except that

j j

x =a where x and x are close to P in dimension-j. Then, if

( )1

j _j

x =x and then the point x( )1 is close to P in dimension-1,2,…,j

and do:

z If j < k, update j = j + 1 and if x∉X_V then update X ={X,x } and X_V =

{

}

z If j = k, the point x( )1 is close to P in all dimensions and is stored in

PD

X . Next, discard x( )1 from the set X and reset j = 1. Go back to

Step 1 to check a new point in X.

If xj =x( )_j1 and then do:

z If j < k, update j = j + 1 and if x∉X_V then update X =

{

}

{

}

z If j = k, the point x( )1 is close to P in all dimensions and is stored in

PD

X . Next, discard x( )1 from the set X and reset j = 1. Go back to

Else, if x( )_j1 ≠a aj, j, then discard x( )1 from the set X and reset j = 1. Go back to Step 1.

Step 5: Each point x in X_PD, update

if 0.

k k k

x = −x ∀k h <

The PDA guarantees to achieve the ML solution only for the MISO systems. For

MIMO systems, we will need to find those points that fall inside the slab

[

]

C y h x h x C

− ≤ − + + ≤ (2.36)

The following algorithm is designed to accomplish this.

Slab Decoding

Obviously, although the X_PD does not contain all the lattice points that fall inside the slab in (2.36), the X_PD provides a useful starting point for slab detection. The procedures of SDA are summarized as follows:

Step 1: Sorting the points of X_PD according to their Euclidean distances. Therefore, ( ) ( ) ( )

{

}

( )

( )

( )

{

}

Step 3: For each point x∈X_{PD C;≤} 2, move away along each direction for

finding other points which Δy2( )x ≤C2_{. It is done by the following}

loop.

b. Compute ( ) 0 min jn , max , s u u d s ∈ ⎛ _⎞⎟ ⎜ = _⎜⎝ + _⎟⎟⎠

where d stands for the separation of every adjacent constellation.

If u( )_jn ≠u₀ and then do the following. z Set ( ) 0 n j u =u . z If 2

(

)

{

}

If u( )_jn ≠u₀ and then do the following. z Set ( ) 0 n j u =u . z If 2

(

)

{

}

d. If j < k, then update j = j + 1 and go back to b.

e. If j = k, then update n = n + 1 and j = 1. Then, go back to b.

f. If n = X_{PD C;≤} 2 , then all lattice points that fall inside the slab are

found.

The two algorithms can find all the lattice points satisfying (2.36) for a given C.

Each point of the set can be substituted into the original problem in (2.31), to obtain

2 2

1

G − G ≤C

y R x (2.37)

where y_G ∈ M−1, R₁∈ M− × −1 M 1 corresponds to the first M − columns 1 and rows of the R and x_G =

[

]

2.7 Summary

In this Chapter, we review the MIMO communication systems. In a rich

multi-path scattering environment, the MIMO system deliver significant performance

enhancement in terms of link quality and data rate. Spatial diversity is a key MIMO

technique which mitigates fading and is realized by providing the receiver with

multiple copies of the transmitted signal in space or time. Spatial multiplexing offer a

linear increase in data rate by transmitting independent data streams from the

individual transmit antennas.

Two popular detectors of underdetermined MIMO systems are GSD and SSD.

The complexity of GSD increases exponentially with the order of N −M . In order to reduce the complexity many algorithms have been proposed. The SSD uses a

geometrical approach to solve the problem, and it has lower complexity than existing

Chapter 3

Proposed Geometry Based Decoding

Algorithm with Intersection of

Candidate Sets

In Chapter2, we introduce the SSD which is a low complexity solution for

underdetermined systems; however, the SSD has some disadvantages. First, because

the two algorithms of SSD are independently and sequentially implemented, many

constellation points are multiply checked. Second, in the PDA of SSD, in order to find

the “X_PD”, we perform many searches in the same dimension. As a result there are too many candidate points that fall inside the slab such that the SDA is executed too

many times. For those reasons, its computational complexity is high.

In this chapter, we introduce the proposed a low complexity search method for

finding those points falling inside the slab and a low complexity decoding algorithm

without the use of SDA. The details of the proposed algorithm will be introduced in

Sections 3.1-3.2. The simulation results will be provided in Section 3.3 to show that

3.1 Efficient Search Method for Points in Slab

The search method is similar to the PDA and SLA. We use only one algorithm to

find those points that fall inside the slab, so no constellation points and dimensions

are multiply searched. First, performing QR-decomposition to the real channel matrix

H, we obtain

, =

H QR (3.1)

where Q∈ M N× and R is an M× upper triangular matrix. Substituting (3.1) N into (2.30), we have

2 ' _{− ≤C}2_,

y Rx (3.2)

where y' =Q y . If N>M, we will have T

'

, , ,

M M M M M N N

C y ⎡r x r x ⎤ C

− ≤ −_⎢⎣ + + _⎥⎦≤ (3.3)

at the M-th layer. Eq.(3.3) involves N-M+1 dimensions for detection. The algorithm

still aims to find those points that fall inside the slab.

First, define as ' , , 1 , , 1 , ' _{1, 1 1,2 1, 1 1, 1} 1 ' _{, , 1 ,} 0 Dist , 0 0 i i i i i i M i M i N _i i i i M i M i N i i i N M M M M M N M y r r r r r _x r r r r x y x r r r y + + + + + + + + + + + ⎡ ⎤ _{⎡ ⎤} ⎢ ⎥ _{⎢ ⎥}⎡ ⎤ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = _{⎢ ⎥− ⎢ ⎥⎢ ⎥} ⎢ ⎥ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎣ ⎦}⎢_⎣ ⎥_⎦ ⎢ ⎥ ⎣ ⎦ (3.4) where 1≤ ≤i M . Second, define as X , _in in V X , out V X , in V D , out V

D , X_VD the set of the points inside the slab, the sets of the points inside the slab to be visited, the sets of the points

of

out V

X and the points that have been visited, respectively. Then, initialize them

with

out V

X =X_VD=

{ }

out [0, 0, , 0] V

D = indicates that no dimension is visited in

the beginning.

The main idea of the algorithm is to find those point (X ) which fall into the _in slab (3.3). We design an algorithm to find the set X efficiently and the procedures _in of the search algorithm are summarized as follows.

Step 1: If in V X and out V

X are empty, go to Step 5.

Step 2: If

out V

X is empty, go to Step 4. Otherwise, we calculate

( )

( )

( )

where j is chosen from the unvisited dimension of

out V

D

corresponding to each x.

Step 3: For each value x ∈

{

}

( )1

=

x x where xj = , and x is inside the slab. x

z Update the jth element of

in V D corresponding to each x . If in V X ∉ x then

{

}

And the value a ∈

{

{

}

}

the points x =x( )1 where xj = , and x is outside the slab. a

z If x_j,out ∈ , Update a

{

}

out out,

V V

X = X x and the jth element of

out V

D corresponding to each x . If j is the last visited dimension

then discard x( )1 from the set

out V

X . Go to Step 2.

z Else, then discard x( )1 from the set

out V X , update

{

}

X = X x and the jth element of

out V

D corresponding to

each x . Go back to Step 2. Step 4: If

in V

X is empty, go to Step 1. Otherwise, we calculate

( )

( )

( )

where j is chosen from the unvisited dimension of

in V

D

corresponding to each x.

Step 5: For each value x ∈

{

}

( )1

=

x x where xj = , and x is inside the slab. x

z Update the jth element of

in V D corresponding to each x . If in V X ∉ x then update

{

}

X = X x , and compute the

M

Dist by yΔ .

And the value a ∈

{

{

}

}

the slab.

z Update

{

}

out out,

V V

X = X x and the jth element of

out V

D

corresponding to each x . If j is the last visited dimension then

discard x( )1 from the set

out V

X . Go to Step 4.

Step 6: Each point x in X falls in the slab, which is shown in Fig. 3-1. _in

The proposed search algorithm can efficiently find the valid candidate points

satisfying (3.3) for a given C. Typical underdetermined decoders use all those points

to find the near-ML solution by the SDA when the radius of constraints is not large. In

the above description, we know the decoding complexity is closely related to the

number of candidate points. However, the number of candidate points is still large

with large antenna number difference. Therefore, an efficient decoder by conducting

the intersectional operations with dynamic radius adaptation will be proposed in next

section.

Fig. 3-1 An example of slab with 4-PAM and k = 2

C C 1 1 2 2 h x +h x =y 1 x 2 x

3.2 Efficient Decoding Algorithm with

Intersection of Candidate Sets

The main idea of typical underdetermined decoding algorithms is to find a

candidate set in which the number of candidate points is as small as possible. The

SDA is then employed to find the ML solution. The conceptual diagram is illustrated

in Fig. 3-2. From Fig. 3-2, we know that the decoding complexity depends on the

number of candidate points, which is positively correlated to the constellation size

(e.g. 64-QAM) and antenna number difference (i.e. N_t −N_r).

Fig. 3-2 Block diagram of typical underdetermined decoding algorithms

We propose an efficient geometry based decoding algorithm, where (3.2) is

rewritten and expanded as the summation form as follows:

2 ' 1 1,1 1,2 1, 1 1, 1, ₁ 2 ₂ ' _{1, 1 1, 1,} 1 ' _{, ,} 0 0 0 0 0 0 M M N M M M M M N M N M M M N M y r r r r r _x x C r r r y x r r y − − − − − − ⎡ ⎤ _{⎡ ⎤} ⎢ ⎥ _{⎢ ⎥}⎡ ⎤ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥_{−⎢ ⎥}⎢ ⎥ _≤ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎣ ⎦}⎢_⎣ ⎥_⎦ ⎢ ⎥ ⎣ ⎦ Y Underdetermined decoding algorithm Assume H is know

Generate a candidate set x_G Use SDA to find ML

2 ₂ 1 ' M N i ij j i= ⎡⎢_⎣y − j i= r x ⎤⎥_⎦ ≤C

∑

∑

The total number of slab equations included in (3.5) is M. These equations can be

written with corresponding IDs as follows:

Slab equations ID Slab equations

Slab M − ≤C yM' −⎢⎡_⎣rM M M, x + +rM N N, x ⎥⎤_⎦≤C

Slab 2 − ≤C y2' −_⎣⎢⎡r x2,2 2 + +rM N N, x ⎤⎥_⎦ ≤C (3.6)

Slab 1 − ≤C y1' −_⎣⎡⎢r x1,1 1+ +rM N N, x ⎥_⎦⎤ ≤C,

In the SSD, only the last slab equation (Slab M) is used. However, the total available

number of slab equations is M in (3.6). Thus, the basic idea of the proposed method

is to utilize all the available slab equations to find the candidate set, instead of only

using Slab M.

Assume that C is large enough to include the ML solution and the index satisfies

i > j. Using Slabs i and j, we can find two candidate sets (i.e. x x ) which include i, j

the ML solution. Therefore, the ML solution must fall inside the intersection of the

two candidate sets (i.e. x ). This is illustrated in Fig. 3-3. After the slab intersection _G is performed, we will check the radius constraint of Slab i and j according to

2 , , 1 , , ₂ , , . 0 0 j j j j j j i j N j i i i i i N N x r r r r y x _C y r r x + ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎡ ⎤ _{⎢ ⎥⎢ ⎥} ⎢ ⎥ −_{⎢ ⎥⎢ ⎥} ≤ ⎢ ⎥ _{⎢ ⎥} ⎣ ⎦ ⎣ _{⎦ ⎢ ⎥} ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (3.7)

It is inefficient to perform slab intersection and constraint checking procedure

sequentially. Because the procedure requires large storage memory and many times of

intersection checking for candidate points, we use another approach to achieve the

same goal.

Fig. 3-3 Illustration of intersection of candidate sets

The above procedure can be executed by the following efficient approach: we

first find the candidate set x and then check (3.7) with the next Slab (e.g. Slab j). In i

other words, we omit the procedure of storing x and matching the points of the two j

candidate sets, which reduces the complexity and memory storage. Therefore, we will

find a new candidate set satisfying Slab j (or i) equation in (3.6) and (3.7). There are

two intersection scenarios:

Scenario 1: we find x and check (3.7) to expand the dimensions and i

discard the points which does not satisfy (3.7). After that, we

obtain a new candidate set.

Scenario 2: we first find x and check (3.7). If the point of j x does not j

satisfy (3.7), we discard it. After that, we obtain a new candidate

set.

i

x

j

x

G

x

In above Scenarios, we know that Scenario 1 has lower complexity than Scenario 2

because the average number of candidate points of x is smaller than that of i x . By j

this reason, we expect that the number of candidate points can be as small as possible

in the first candidate set just like Scenario 1. Therefore, we choose Slab M in (3.6) to

find the first candidate set. Similarly, in order to have a lower complexity in the next

procedure, we also expect that the number of points in the new candidate set can be as

small as possible. By this reason, we choose Slab M − to be next one to intersect 1

with Slab M, and so on. The proposed decoding algorithm guarantees to find the ML

solution after performing M − times of the above efficient approach. 1

Discussion of the radius (C)

Note that

2 ₂ 2

min N

C = n ∼χ (3.8)

is Chi-square distributed with N degrees of freedom. Its cumulative distribution

function is given by

(

)

(

)

(

)

where γ • is the lower incomplete Gamma function, ( ) Γ • is the Gamma function, ( )

and P • is the regularized Gamma function. In order to ensure the high efficiency ( )

of the decoder, the radius starts [10], [15] with

( )

C = σ F− Φ + Δi N (3.10)

where i = 0 , Δ and Φ are judiciously set, e.g., Δ =0.001and Φ =0.99. If no points can be found within C, update i in (3.10) by i = i + 1.

The main idea of the decoding algorithm is to perform intersection of more slabs

to achieve a lower decoding complexity. The procedure of the decoding algorithm is

summarized as follows.

Step 1: Set i=M − . Use Slab M equation 1

− ≤C y'_M −_⎣⎢⎡R_{M M M,} x + +R_{M N N,} x _⎦⎥⎤ ≤C,

to find the initial candidate set (x_i+1). The corresponding distance (Dist_i+1) can be obtained by the proposed search algorithm in Section 3.1.

Step 2: If i= , go to Step 4. The upper bound (1 x_i,up) and lower bound (x_i,low) of x corresponding to each candidate point of candidate set _i will be found by the radius constraint

' , , 1 , , 1 , ' _{1, 1 1, 1, 1 1, 1} 1 ' _{, , 1 ,} 0 , 0 0 i i i i i i M i M i N _i i i i M i M i N i i N M M M M M N M y r r r r r _x r r r r x y C x r r r y + + + + + + + + + + + ⎡ ⎤ _{⎡ ⎤} ⎢ ⎥ _{⎢ ⎥}⎡ ⎤ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥_{−⎢ ⎥}⎢ ⎥ _≤ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎢ ⎥}⎢ ⎥ ⎢ ⎥ _{⎣ ⎦}⎢_⎣ ⎥_⎦ ⎢ ⎥ ⎣ ⎦ and

(

)

(

)

(

)

(

)

∑

∑

Therefore, we can easily find Dist_i through

(

)

(

)

, 1

Dist_i = y_i −

_∑

Step 3: Using the upper and lower bounds corresponding to each candidate

x_i = ⎡_⎢⎣⎡_⎣⎢x_i,low,x_i,up⎥⎦⎤∩ ,x_i+1, x_2N⎤_⎥⎦.

Choose a candidate point with minimum distance from x_i and compute the ZF-SIC solution. It gives us a new radius (C_new). If

new

C <C, update it. Set i = i +1 and go to Step 2.

Step 4: Choose a candidate point with minimum distance from x₁ to be the

estimate of x.

Note: (1) If the initial candidate set (x_M ) is empty, increase C .

(2) If the candidate set (xi,1≤ ≤i M −1) is empty, use the ZF-SIC solution as the estimate of x.

3.3 Computer Simulation

In this section, computer simulations are conducted to evaluate the

symbol-error-rate (SER) and the decoding complexity. In order to compare the

complexity of the proposed decoding algorithm with other decoding algorithms, we

define the complexity weights of different operations according to [15]. The weight of

additions and subtractions is one if real and two if complex. Each of The

multiplications and divisions is counted one if the result is real and six if it is not real.

The total complexity of each simulated algorithm is the sum of the number of times of

each operation multiplied by its corresponding weight. The numerical results are

measured and averaged over 1000 independent channels for various average

signal-to-noise ratio (SNR). In addition, the calculation of complexity also includes

all preprocessing procedures (e.g. QR decomposition). We set Φ = 0.99 and