運用傳送功率分配於欠定多輸入多輸出系統達到公平傳輸速率

國 立 交 通 大 學

電信工程研究所

碩 士 論 文

運用傳送功率分配於欠定多輸入多輸出系

統達到公平傳輸速率

Achieving Fair Rate by Transmit Power Allocation

for Underdetermined MIMO Systems

研 究 生：溫振鵬 Student: Chen-Peng Wen

指導教授：李大嵩 博士 Advisor: Dr. Ta-Sung Lee

運用傳送功率分配於欠定多輸入多輸出系統達到公平

傳輸速率

Achieving Fair Rate by Transmit Power Allocation for

Underdetermined MIMO Systems

研 究 生：溫振鵬 Student: Chen-Peng Wen

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

運用傳送功率分配於欠定多輸入多輸出系統達

到公平傳輸速率

學生：溫振鵬

指導教授：李大嵩 博士

Chinese Abstract

國立交通大學電信工程研究所碩士班

摘要

在無線通訊系統中，多輸入多輸出技術不需額外頻寬及傳輸功率即能提高傳 輸速率及改善傳輸品質。然而，在多用戶多輸入多輸出上鏈系統的環境中，由於 部分用戶可能距離基地台較遠或被障礙物所阻擋，以致通道增益較小，導致傳輸 速率變差。在本篇論文中，吾人提出一個傳送功率分配法達到公平的傳輸速率。 吾人將一非線性最佳化問題轉換成適當形式，將之套用到現有的演算法中。此公 平化的機制可使得等效子通道增益較為均勻，因此等效通道矩陣會傾向於良置 （well-conditioned）。因為在欠定多輸入多輸出系統中，有效率的解碼器均基於 球體解碼器，因此通道矩陣有較小的條件數（condition number），可使得解碼的 複雜度降低。最後，吾人亦提出另一功用函數用於改善條件數，進一步降低運算 複雜度。模擬結果顯示吾人提出方法的有效性。

Achieving Fair Rate by Transmit Power Allocation

for Underdetermined MIMO Systems

Student: Chen-Peng Wen

Advisor: Dr. Ta-Sung Lee

English Abstract

Institute of Communication Engineering

National Chiao Tung University

Abstract

In wireless communication systems, multiple-input multiple-output (MIMO)

technology offers significant increases in data rate and link range without additional

bandwidth and transmit power. However, in uplink multi-user MIMO systems, some

users will suffer from small channel gains due to being far away from the base station

or blocked by obstacles in practical environments. This result in poor data rates for

those users. In this thesis, we propose a transmit power allocation scheme with fair

rate allocation for all users. We reformulate a nonlinear optimization problem to a

modified form which can be applied to the existing algorithms. The proposed fairness

scheme also leads to uniform sub-channel gains. Thus the equivalent channel matrix

will tend to be well-conditioned. Since efficient decoders of underdetermined MIMO

systems are based on sphere decoders, the decoding complexity can be reduced with a

smaller channel condition number. Finally, we also propose an alternative utility

function to improve the condition number, to further reduce the decoding complexity.

Acknowledgement

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

enthusiastic guidance and great patience, especially in the training of presentation. I

learned a lot from his positive attitude in many areas. 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 thanks to my family for their

Table of Contents

Chinese Abstract ...I

English Abstract ... II

Table of Contents ... IV

List of Figures ... VI

Acronym Glossary ... VIII

Notations ... IX

Chapter 1 Introduction... 1

Chapter 2 MIMO Systems... 4

2.1 System Model ...4 2.2 Channel Capacity ...6 2.3 MIMO Diversity ...9 2.3.1 Receive Diversity...9 2.3.2 Transmit Diversity ...10 2.4 Spatial Multiplexing...10

2.5 Underdetermined MIMO Decoder...12

2.5.1 GSD algorithm ...13

2.5.2 Slab Sphere Decoding (SSD) Algorithm ...14

2.5.3 Regularization Method...20

2.6 Summary ...21

Chapter 3 Proposed Transmit Power Allocation ... 22

3.2 Proposed Transmit Power Allocation for Fair Rates...24

3.3 Interior Point Algorithm...26

3.4 Computer Simulations ...29

3.5 Summary ...32

Chapter 4 Condition Number Discussion... 33

4.1 Condition Number Effect...34

4.2 Proposed Utility Function for Condition Number ...38

4.3 Computer Simulations ...42

4.4 Summary ...46

Chapter 5 Conclusions and Future Works ... 48

List of Figures

Fig. 2-1 MIMO system ...5

Fig. 2-2 Spatial multiplexing system ... 11

Fig. 2-3 Encoding procedure of D-BLAST (n=3)...12

Fig. 2-4 Encoding procedure of V-BLAST (n=3) ...12

Fig. 3-1 Example of uplink MU-MIMO system...23

Fig. 3-2 CDF of the number of iterations for interior-point algorithm...28

Fig. 3-3 The PDF of the ratio of minimum rate to maximum rate ...29

Fig. 3-4 Minimum rates versus reciprocal of channel gains...30

Fig. 3-5 Maximum rates versus reciprocal of channel gains ...31

Fig. 3-6 Sum rate comparison for fairness scheme...32

Fig. 4-1 Tree search example for 4-PAM showing sphere radius, tree levels and detection layers...35

Fig. 4-2 CDF of r for a 5₅₅ × matrix with different condition numbers ...36 5 Fig. 4-3 Complexity as a function of condition number...37

Fig. 4-4 CDF of condition number with different channel correlations ...38

Fig. 4-5 CDF of the condition number for SU-MIMO...39

Fig. 4-6 CDF of the condition number for MU-MIMO...40

Fig. 4-7 Comparisons of condition number for fairness scheme and determinant based scheme ...41

Fig. 4-8 Decoding complexity comparison using SSD at receiver. Transmitter has four antennas, and receiver has three antennas. ...42

Fig. 4-9 Decoding complexity comparison using SSD at receiver. Transmitter has six antennas, and receiver has five antennas. ...43

Fig. 4-10 Decoding complexity comparison using regularization method at receiver.

Transmitter has four antennas, and receiver has three antennas...44

Fig. 4-11 Decoding complexity comparison using regularization method at receiver.

Transmitter has six antennas, and receiver has five antennas. ...45

Acronym Glossary

D-BLAST diagonal Bell laboratories layered space-time

EGC equal-gain combining

GSD Generalized Sphere Decoding

MIMO multiple-input multiple-output

MISO multiple-input-single-output

ML maximum-likelihood

MRC maximum ratio combining

MU-MIMO multi-user multiple-input multiple-output

SDA sphere decoding algorithm

SIC successive interference cancellation

SIMO single-input-multiple-output

SISO single-input-single-output

SNR signal to noise ratio

SINR signal to interference plus noise ratio

SSD slab sphere decoding

STBC space-time block code

STC space-time code

STTC space-time trellis code

SU-MIMO single-user multiple-input multiple-output

Notations

( )⋅−1 _{inverse operator} ( )⋅ T _{transpose operator}

{ }

E ⋅ expectation operator

( )⋅ † _{Moore-Penrose pseudo-inverse operator}

C channel capacity

H channel matrix

r

M number of receive antennas

t

N number of transmit antennas

n noise vector

γ _{average SNR at the receiver}

x transmit signal vector

Chapter 1

Introduction

Next generation wireless communication systems are expected to provide users

with higher data rate services including video, audio, data and voice signals. The

rapidly growing demand for these services drives the wireless communication

technologies towards higher data rate, higher mobility and higher link quality.

However, the time-selective and frequency-selective fading in wireless channel

caused by multipath propagation, Doppler shifts and carrier frequency/phase drifts

severely affect the quality and reliability of wireless communication. Besides, the

available bandwidth and power are limited which makes the design of wireless

communication systems extremely challenging. Hence, recently there are many

innovative techniques that improve the reliability and the spectral efficiency of

wireless communication links. Some popular examples include the coded

multicarrirer modulation, smart antenna, in particular multiple-input multiple-output

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

MIMO technology involves the use of multiple antennas at the transmitter and

receiver to improve communication performance. The technology offers some

benefits that overcome the challenges posed by both the impairments in wireless

technology are the diversity gain and the spatial multiplexing gain. Diversity gain

mitigates fading by providing the receiver with multiple (ideally independent) copies

of the transmitted signal in space, time or frequency. Spatial multiplexing offers a

linear increase in data rate by transmitting multiple independent data streams within

the bandwidth of operation.

There are many signal detection schemes for MIMO systems such as linear

detection, successive interference cancellation (SIC) [7], [8] and the

maximum-likelihood (ML) detection. Both linear detection and the SIC schemes are

easy to be implemented but their detection performances are not optimal. The optimal

detection scheme is ML detection; however, the complexity of the ML detection

scheme grows exponentially with the size of the transmit symbol alphabet and the

number of transmit antennas. To reduce the complexity of ML detection, the sphere

decoding algorithm (SDA) is introduced in[9-12] to achieve the same performance as

ML detection with reduced complexity. The basic idea of SDA is to search the nearest

lattice point to the received signal vector within a given sphere radius. However, the

typical SDA fails to decode in underdetermined MIMO systems. Thus several

algorithms are proposed to decode the underdetermined MIMO systems, including

Generalized Sphere Decoding (GSD) algorithm [13], Slab Sphere Decoding (SSD)

algorithm [14-16], and Regularization Method [17].

In uplink multi-user MIMO (MU-MIMO) systems, if the number of users is

larger than the number of base station antennas, then it can be regarded as an

underdetermined MIMO system. Considering in the practical environments, some

users will suffer from small channel gains due to being far away from the base station

or blocked by obstacles. This results in poor data rates for those users. The

degraded..

In this thesis, our major goal is to achieve fair data rates for all users. We propose

a transmit power allocation to realize this. And we reformulate a nonlinear

optimization problem into a modified form which can be applied to the existing

algorithms. The proposed fairness scheme also leads to uniform sub-channel gains.

Thus the equivalent channel matrix will tend to be well-conditioned. We further

propose a determinant based utility function to improve the condition number. Thus

the complexity of the SDA based decoder can be reduced.

The remainder of the thesis is organized as follows. In Chapter 2, The signal

model of the MIMO systems is introduced first. Secondly, several algorithms for

decoding underdetermined MIMO systems are presented. In Chapter 3, the proposed

transmit power allocation scheme is developed. Discussion on the condition number

and the determinant based utility function will be described in Chapter 4. Simulation

results of the proposed methods are also illustrated in this chapter. Finally, we

summarize the contributions of our works and give some potential future works in

Chapter 2

MIMO Systems

In wireless communications, one can improve communication performance by

using multiple-input and multiple-output (MIMO) technology. MIMO offers

significant increases in data rate and link reliability without additional bandwidth or

transmit power. In this chapter, we give a review of MIMO systems. We first introduce

the MIMO system model in Section 2.1. Section 2.2 introduces the channel capacity.

Then, the spatial diversity and the spatial multiplexing techniques are introduced in

Section 2.3 and Section 2.4, respectively. The generalized sphere decoding (GSD)

algorithms have been studied as a solution to the ML detection for underdetermined

MIMO systems with reduced complexity. We will give an introduction of the GSD

algorithms in Section 2.5.

2.1 System Model

Figure 2-1 shows the typical multiple-input-multiple-output (MIMO) system

with N transmit antennas and _t M receive antennas. The frequency-flat fading _r channel matrix H can be written as

11 12 1 21 22 2 1 2 t t r t r r r t N N _{M N} M M M N h h h h h h h h h × ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ =_{⎢ ⎥}∈ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ H , (2.1)

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

zero-mean and unit variance. The relation between the transmitted signal vector and

received signal vector can be written as

y=Hx+n , (2.2) where ₁, , ,₂ r 1 Mr M y y y × ⎡ ⎤ = _{⎢ ⎥}∈ ⎣ ⎦

y and x= _⎢⎡_⎣x x1, , ,2 x_{Nt ⎦}⎤_⎥∈ Nt×1 are the

received signal vector and transmitted signal vector , respectively.

1 1, 2, , _Mr Mr

n n n ×

⎡ ⎤

= _{⎢⎣ ⎥⎦} ∈

n denotes the i.i.d. complex additive white Gaussian

noise (AWGN) vector with zero-mean and covariance matrix σ I . When2 M_r >N_t,

the system is called an overdetermined MIMO system. WhenM_r <N_t , it is called an underdetermined MIMO system.

Fig. 2-1 MIMO system

The complex MIMO system can be transformed into an equivalent real system.

By using the real-value decomposition, (2.2) can be written as

M r

y

1

x

2

x

Nt

x

1

y

2

y

H

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 and

{ }

{ }

{ }

{ }

Re Im Im Re T T M N T T × ⎡ ₋ ⎤ ⎢ ⎥ ⎢ ⎥ = ∈ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ H H H H H .

Note that the dimension of H is M× where N M = ×2 M_r and N = ×2 N_t.

2.2 Channel Capacity

Channel capacity is the highest rate in bits per channel use at which information

can be transmitted with an arbitrary probability of error. We first introduce the

single-input-single-output (SISO) channel capacity and then study the capacity of a

MIMO channel. Note that single-input-multiple-output (SIMO) and multiple-input-

single-output (MISO) channel are sub-sets of the MIMO case. The channel capacity is

defined as [19] ( ) max I( ; ), p x C = X Y (2.4) where I( ; )X Y =H( )Y −H( | ),Y X (2.5) is the mutual information between X and Y, H( )Y and H(Y X are the differential | )

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

to all possible transmitter statistical distributions p(x).

The ergodic capacity of a SISO system with a random complex channel gain h is

given by [19]

C =E

{

log (1₂ +γ h2)

}

bits/sec/Hz, (2.6) where γ =P/σ2 is the average SNR at the receiver, P is the transmit power and

E{x} is denotes the expectation over all channel realizations. For a MIMO system

with N transmit antennas and M receive antennas, the capacity of a random MIMO

channel is given by [1]

_{2 2} bits/sec/Hz

(max_xx) log det( ) ,

H M xx tr N P C E N σ = ⎧ ⎡ ⎤⎫ ⎪ ⎪ ⎪ ⎪ = ⎨_⎪ ⎢_⎢ + ⎥_⎥⎬_⎪ ⎣ ⎦ ⎪ ⎪ ⎩ ⎭ R I HR H (2.7)

where R_xx =E

{ }

xxH is the covariance matrix of the transmitted signal vector x . If the channel knowledge is unknown to the transmitter, the signals are chosen to be

independent and equal power. The covariance matrix of the transmit signal vector is

then given by R_xx =I . As a result, the ergodic capacity of a MIMO system can be _M written as [1]

C E log det(_{2 M} P_{2 xx} H) bits/sec/Hz,

N σ ⎧ ⎡ ⎤⎫ ⎪ ⎪ ⎪ ⎪ = ⎨_⎪ ⎢_⎢ + ⎥_⎥⎬_⎪ ⎣ ⎦ ⎪ ⎪ ⎩ I HR H ⎭ (2.8)

By using the eigenvalue decomposition, the matrix product of HH can be H decomposed as HHH = ΛE E , where E is an MH ×M matrix which consists of

the eigenvectors satisfying EEH =E EH =I and _M Λ = diag

{

λ λ₁, , ,₂ … λ_M

}

is a diagonal matrix with the eigenvalues λ ≥ on the main diagonal. Assuming that _i 0 the eigenvalues λ are ordered so that _i λ_i ≥λ_i+1, we have

2_{, if 1} if 1 0, i i i r r i M σ λ = ⎨⎧⎪⎪⎪_{⎪ + ≤ ≤}≤ ≤ ⎪⎪⎩ , (2.9)

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

( ) { }

rank min ,

r = H ≤ N M is the rank of the channel matrix. Then the capacity of a MIMO channel can hence be rewritten as

2 2 2 2

2 2

1

log det( ) log det( )

log det(1 ) bits/sec/Hz

H M M r i i P P C E E N N P E N σ σ λ σ = ⎧ ⎡ ⎤⎫ ⎧ ⎡ ⎤⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ⎨_{⎪ ⎢}⎢ + Λ _⎥⎥_⎪⎬= ⎨_⎪ ⎢_⎢ + Λ⎥_⎥⎬_⎪ ⎣ ⎦ ⎣ ⎦ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎧ ⎡ ⎤⎫ ⎪ ⎪ ⎪ ⎪ = ⎨_⎪ ⎢_⎢ + ⎥_⎥⎬_⎪ ⎣ ⎦ ⎪ ⎪ ⎩ ⎭

∑

I E E I , (2.10)

Note that the second equation holds due to the fact det

(

I_m +AB

)

=det

(

I_n +BA

)

for matrices A∈ m n× and B∈ m n× and E EH =I . (2.10) shows that the _M

capacity of a MIMO channel is made up by sum of the capacities of r SISO

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

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

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

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

∑

, (2.11)

where γ =_i E x

{ }

_i 2 for i=1,2,…,r is the transmit power in the ith sub-channel and satisfy 1 r i i N λ = =

∑

. Since the transmitter can access the spatial sub-channels, we can allocate variable power across the sub-channels to maximize the mutual information.

The optimal power allocation of the ith sub-channel is given by[1], [19]

2 opt _{for 1,2, ,} i i M i r P σ γ μ λ ₊ ⎛ _⎞⎟ ⎜ _⎟ =⎜_⎜ − _⎟⎟ = ⎜⎝ ⎠ , (2.12) where μ is chosen to satisfy the constraint

∑

r_i=1γ_iopt =N and ( )⋅ denotes the ₊

(2.12) is found iteratively through the water-filling algorithm [1], [19].

2.3 MIMO Diversity

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

of transmission without increasing the transmit power or sacrificing the bandwidth.

There are many diversity techniques such as time diversity, frequency diversity and

space diversity. In this section we focus on the space diversity that is so called antenna

diversity.

2.3.1 Receive Diversity

Receive diversity involves the use of multiple antennas at the receiver. 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). In the selection

combining, the received signal with the best quality is chosen and the choosing

criterion is based on SNR. Switch diversity switches the received signal path to an

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

EGC 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. MRC forms the output signal by a linear combination of all the

received signals and is the optimal combination technique which achieves the

2.3.2 Transmit Diversity

Transmit diversity techniques which provide 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. Space-time code (STC)

is an open loop scheme which jointly designs of channel coding and modulation to

improve system performance by providing both transmit diversity and coding gain.

STC can be classified into two categories, the space-time block code (STBC) and the

space-time trellis code (STTC).

2.4 Spatial Multiplexing

Spatial multiplexing is a transmission technique of MIMO wireless

communication systems which increases the transmission data rate without additional

bandwidth or power consumption. In the spatial multiplexing systems, N different data

streams are transmitted from different transmit antennas simultaneously or sequentially

and these data streams are separated and demutiplexed to yield the original transmitted

signals according to their unique spatial signatures at the receiver, as illustrated in Fig.

2-2.The separation of data streams at the receiver can be done possibly by the fact that

rich scattering multi-paths contribute to lower correlations between MIMO channel

coefficients and hence create a channel matrix with full rank and low condition number

to N unknowns from a linear system of M equations. In the following, two typical

Fig. 2-2 Spatial multiplexing system

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

The concept of layered space-time processing was introduced by Foschini at Bell

Laboratories [4]. D-BLAST uses multiple antennas at both the transmitter and the

receiver, and an elegant diagonally-layered coding sequence in which code blocks are

dispersed across the diagonals in space-time. The high-rate information bit stream is

first demultiplexed into N substreams, and each substream is encoded by a conventional

1-D constituent code. The encoders apply these coded symbols to the input to form a

semi-infinite matrix X of N rows to be transmitted. The encoding procedure is shown in

Fig. 2-3.

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

The D-BLAST algorithm suffers from certain implementation complexities which

is not suitable for practical implementation. Therefore, a simplified version of the

BLAST algorithm is known as V-BLAST. It is capable of achieving high spectral

efficiency while being relatively simple to be implemented. The coding procedure of

the V-BLAST can be viewed as there is an encoder on each transmit antenna. The

output coded symbols of each encoder are transmitted directly from the corresponding

antenna which is shown in Fig. 2-4.



1

x



2

x



n

x



1

y



2

y



m

y

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

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

2.5 Underdetermined MIMO Decoder

Maximum-likelihood (ML) detection complexity increases exponentially

depending on the number of transmit antennas and the constellation size. Hence, it is a

serious issue in designing the receiver in recent years. In order to reduce the

complexity of ML detection, the sphere decoding algorithms (SDA) [9-12] are

proposed to solve the problem and achieve the ML performance. But the SDA fails in

the underdetermined MIMO systems. There are several algorithms that can solve the

underdetermined problem, such as Generalized Sphere Decoding (GSD) [13]

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 Encoder Encoder Encoder Layer Time Antenna 0 1 2 0 1 2 0 1 2

α

α

α

β

β

β

γ

γ

γ

⎡

⎤

⎢

⎥

⎢

⎥

= ⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

x

Encoder α Encoder β Encoder γ Layer Time Antenna

Method [17]. We introduce these algorithms in this section.

2.5.1 GSD algorithm

Consider a MIMO system with N transmit antennas and _t M receive antennas. _r The received real signal can be written as (2.3):

.

= +

y Hx n

The ML estimator x of x is obtained by minimizing the Euclidean distance of y

from the legal lattice points can be represented as

( )2

2

arg min arg min .

N N ρ

∈ ∈

= − = −

x x

x y Hx R x (2.13)

where = ± ±

{

1, 3, ,±(2 -1)k

}

is the 4-QAM, 16-QAM, 64-QAM constellations fork =1,2, 3 , respectively. ρ = H HHT

(

T

)

−1y , Q is an M×M orthogonal matrix, and R is an M×N upper triangular matrix corresponding to the QR-decomposition of H, i.e. H=QR . The matrix R can be represented as

[

₁, ₂

]

=

R R R , where R₁ ∈ M M× is an upper triangular and R₂ ∈ M N M× − .

Similarly, x can be represented as x= ⎣⎡x x_G, _G⎤_⎦T, where G and G are the indices corresponding to the first M and the last M -N elements of the x . The minimum

distance corresponding to the ML estimator in (2.13) can be rewritten as

( )

[

]

2 2 1 2 2 1 2 1 arg min min min , min min , N N M M G G N M M G G G G G ρ ρ ρ − − ∈ ∈ ∈ ∈ ∈ − ⎛ _⎞⎟ ⎜ = ⎜_⎜ − − ⎟_⎟⎟ ⎝ ⎠ ⎛ _⎞⎟ ⎜ = ⎜_⎜ − ⎟_⎟⎟ ⎝ ⎠ x x x x x R x R R R x R x R x (2.14) where ρ =

[

R R₁, ₂

]

ρ−R x . _{2 G}

The GSD checks all legal constellation points in a sphere with radiusC . That

means we set the squared Euclidean distance in (2.14) to be smaller than a positive

number C . The problem can be solved by exhaustive search over 2 x and _G employing the SDA to compute the last equation in (2.14). The SDA algorithm finds

the valid candidates if the squared minimum distance is less than C . Otherwise, a 2 failure of the SDA for the given x_G is declared and then the x_G will be discarded.

If a candidate constellation point (x_G,x_G) is found within the sphere, the value

of C is updated and the algorithm continues to search the remaining points for 2 x_G. If no candidate constellation point is found within the sphere, then the entire

algorithm is repeated with a value larger than the original radius C . The GSD is

based on the exhaustive search over x_G and each searched point should follow the

SDA. Because of the exhaustive search over x_G, its complexity will exponentially

increase depending on the size of N−M.

2.5.2 Slab Sphere Decoding (SSD) Algorithm

To perform (2.13) efficiently, an algorithm is proposed in [9], [20] to solve a

search problem that finds all the lattice points satisfying

2 _C2

− ≤

y Hx (2.15) 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, (2.15) can be rewritten as 2

' _{− ≤C}2_,

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

− ≤C y_M' −_⎣⎢⎡r_{M M M,} x + +r_{M N N,} x ⎤_⎥⎦ ≤C, (2.17)

at the Mth layer. (2.17) involves N-M+1 dimensions for detection. (2.17) 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 and Slab Decoding

Algorithm.

Plane Decoding Algorithm

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 + η (2.18) where x_k ∈ , h is the channel response and _n η ∼CN(0,σ2) stands for AWGN. ML estimation of the transmitted vector x=

[

x₁, ,x_k

]

can be written as

( 1 )

(

)

2 1 1 , , arg min , k k ML k k x x ∈ y h x h x = − + + x (2.19)

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

P h x: _{1 1}+ +h x_{k k} = (2.20) 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

{ }

x( )1 where the (1) stands for the order of the vector in a set and j = 1.

The main idea of the PDA is to find those candidates (X_PD) which are close to P in all dimensions. The procedures of the PDA are summarized as follows:

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

{

: min s.t.

}

j B x a x x x x ∈ = >

{

: max s.t.

}

, j B x a x x x x ∈ = < where [ ] ( )

( )

1 1 B j j y x x h Δ = − x ( )

( )

1 [ ]1 [ ]1 [ ]1 1 1 2 2 k k y h x h x h x y Δ x = + + + −

Step 2: If

{

ak ≠ Φ ∧

}

{ak = Φ is not true, go to Step 3. Otherwise, we have 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

{

ak = Φ ∧

}

{ak ≠ Φ is not true, go to Step 4. Otherwise, we have the }

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

{

ak ≠ Φ ∧

}

{ak ≠ Φ is not true, go to Step 5. Otherwise, we have two }

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

where x and x are 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 and if x∉X_V then update X ={X,x and }

{

,

}

V V

X = X x . 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.

If xj =x( )_j1 and then do:

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

{

X,x and

}

{

,

}

V V

X = X x . Go to Step 2.

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 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_{1 1}+ +h x_{k k}

]

≤C, (2.21) The following algorithm is designed to accomplish this.

Slab Decoding Algorithm

Obviously, although the X_PD does not contain all the lattice points that fall inside the slab in (2.21), 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, ( ) ( ) ( )

{

1 2 3

}

sort _{, , ,...} PD PD PD PD X = x x x where Δy2

( )

x_PD( )i ≤ Δy2

( )

x( )_PDj if i ≤ . j Step 2: For a given C, find the set

( )

{

}

2 sort ; PD : PD C X _≤ = x∈X − ≤ ΔC y x ≤C

finding other points which Δy2( )x ≤C2_{. These newly found points}

are then added to X_{PD C;≤} 2. It is done by the following loop.

a. Initialize n = 1, and j = 1. Pick the nth point u( )n ∈X_{PD C;≤} 2.

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

(

( )n

)

2 y C Δ u ≤ , then X_{PD C;≤} 2 =

{

X_{PD C;≤} 2,u( )n

}

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

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

(

( )n

)

2 y C Δ u ≤ , then X_{PD C;≤} 2 =

{

X_{PD C;≤} 2,u( )n

}

.

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 (2.21) for a given C.

Each point of the set can be substituted into the original problem in (2.16), to obtain

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

[

x x₁, , ,₂ x_M−1

]

∈ M−1. SinceR is an upper triangular ₁ matrix with full rank, we can solve the problem by SDA directly. After the substitution

of all points, the ML solution can be found.

2.5.3 Regularization Method

Regularization method intends to transfer the underdetermined MIMO systems

to overdetermined MIMO systems. By doing this transformation, one can directly use

the SDA in a simple way. It first considers a constant modulus constellation, and

derives the algorithm. Then it shows how MIMO systems with non-constant modulus

constellations can be adapted so that the algorithm is applicable. The ML detection is

equivalent to ( )2 2 min min N N ρ ∈ − = ∈ − x y Hx x R x , (2.23)

where R is an upper triangular matrix such that R RT =H H . In the T overdetermined MIMO systems, i.e.M >N , H H is full rank. The SDA is T applicable due to the non-zero diagonal terms of R . However, for the

underdetermined MIMO systems, i.e.M <N , the Cholesky factor R of H H is H rank-deficient and only the first M rows of R are non-zero. Because the elements of

x are of constant modulus, that means the product αx x is a constant. We can get T

an equivalent minimization problem as

(

)

(

)

2 ₂ min min S H H H H H H N S α α ∈ ∈ − + ⎡ ⎤ = _⎢⎣ − − + + _⎥⎦ x x y Hx x y y y Hx x H y x H H I x . (2.24)

T

=

G D D, and D is an upper triangular matrix. By defining λ = G H y−1 H , (2.24) is equivalent to

min ( )2

S ∈

x D λ −X . (2.25)

(2.25) is an overdetermined case, thus can directly use SDA. If the constellation is not

constant modulus, the non-constant modulus constellation can be represented as

combination of constant modulus constellations. For example, q-QAM (q =2k) can be represented as a weighted sum of /2k QPSK constellations when k is an even number. That is, for w ∈q-QAM and w ∈_i QPSK, 0≤ <i k/2, we have

1 2 0 2 2 ( ) 2 k i i i z z − = =

_∑

.

2.6 Summary

In this chapter, we give a review of the MIMO communication systems.

Exploiting multi-path scattering, MIMO systems deliver significant performance

enhancements in terms of data rate and link quality. Spatial diversity is one of the

MIMO techniques which mitigates fading and is realized by providing the receiver

with multiple copies of the transmitted signal in space or time. MIMO systems offer a

linear increase in data rate through spatial multiplexing by transmitting multiple and

independent data streams without requiring additional bandwidth or transmit power.

The underdetermined MIMO systems can be solved by several algorithms. GSD

algorithm has to perform exhaustive search over (N−M) dimensions. The SSD checks all the points in a geometrical slab. The regularization method transfer the

Chapter 3

Proposed Transmit Power Allocation

In this chapter, we introduce the proposed transmit power allocation for

MU-MIMO systems. We aim to find a power allocation matrix such that all user data

rate will be close to each other. We choose the sum of logarithmic average user rates

as our utility function. We reformulate this nonlinear optimization problem to a

suitable form, thus the Interior-point method can be applied. The proposed method

can also be applied to single user MIMO (SU-MIMO) systems. The simulation results

shows that the proposed method provides fair data rate for all users. The Chapter is

organized as follows. In Section 3.1, we introduce that the special uplink MU-MIMO

can be regarded as the underdetermined MIMO system. The proposed transmit power

allocation is introduced in Section 3.2. The reformulation of the nonlinear

optimization problem and Interior-point algorithm are described in Section 3.3.

Section 3.4 contains the numerical results of the proposed method, and Section 3.5

3.1 Uplink MU-MIMO System

In the uplink scenario, if there are N users transmit the signal simultaneously, and

each user is equipped with one antenna. The base station has M antennas. When the

number of base station antennas M is larger than the number of users N, it can be

viewed as an underdetermined MIMO system. Therefore, the existing algorithms can

be used to decode the received signals. Fig. 3.1 is a practical example in uplink

MU-MIMO system. User 6 is blocked by a high building and User 11 is far away

from the base station. The channel gains are depending on the shadowing and distance

between the transmitter and receiver. In general, these two users will suffer from

small channel gains. Hence, User 6 and 11 will have lower data rates than the other

users. We aim to use a power allocation to let all users have fair rates. Thus User 6

and User 11 will achieve higher data rates.

Fig. 3-1 Example of uplink MU-MIMO system.

3.2 Proposed Transmit Power Allocation for

Fair Rates

In the uplink MU-MIMO systems, the users are independent and separated. They

cannot exchange the information to each other. Thus precoding techniques at the

transmitter cannot be applied in this case. However, we can use power allocation to

improve the performance.The transmit power allocation is proposed to allocate power

to different users. We incorporate the power allocation matrix

(

1, 2, , N

)

diag P P P

=

P into our system model, where P is the power _i

transmitted by the ith user. Thus, the received signal in (2.3) becomes

y=HPx+n , (3.1) and P_i ≤P_maxi is the power constraint for the ith user. The matrix HP in (3.1) can be regarded as the equivalent channel matrix. It can also be considered as matrix

P provides different gains to different columns of H . Assuming that the receiver has

the perfect channel state information (CSI). The maximum achievable rate for the ith

user is R =_i log (1₂ +SINR )_i , i =1,2, ,N (3.2) where

(

)

(

)

2 2 2 1 2 2 2 2 2 1 2 SINR_i i i Mi i j j Mj j j i h h h P Mσ h h h P ≠ + + + = +

_∑

+ + + and 2 σ is the noise

power. Here we treat the other users as interference at the receiver. By the concept of

[21], we choose the utility function U R R

(

₁, ₂, ,R_N

)

as

(

)

( )

( )

(

)

F 1, 2, , N ln 1 ln 2 ln N

U R R R = R + R + + R . (3.3) ∗

following optimization problem:

(

)

max U_F R R₁, ₂, ,R_N subject to P_i ≤P_maxi i =1,2, ,N (3.4) We choose the logarithm function in (3.3) because it provides better fairness for the

rate R of each user. For logarithm function, the larger input, the more suppressed _i output. This means that the different rates will be closer to each other. If we choose

the utility function as U_sum

(

R R₁, ₂, ,R_N

)

=R₁+R₂ + +R_N , i.e., maximizing the total sum rate, then the water filling algorithm [18] will be the solution.

The utility function in (3.3) can also be applied to SU-MIMO system. The

difference between SU-MIMO and MU-MIMO systems is the power constraint.

When the transmitter has N antennas and receiver has M antennas, the received signal

can be written as same as (3.1). The optimization problem in (3.4) becomes

(

)

max U_F R R₁, ₂, ,R_N subject to

∑

N_i=1P_i ≤P_max.

where P is the transmitted power of the ith antenna and _i P_max is the maximum transmit power. The achievable data rate will become fair for all users with the

proposed power allocation applied.

In [26] we know that the optimization problem in (3.4) cannot be solve

mathematically as a closed form. The reason is that the utility function is a very

complicated nonlinear function. Thus we need to reformulate the problem to a

suitable form which can be solved by the existing algorithms. Here we choose the

interior-point method [23], [24] to solve the optimization problem, since it is more

3.3 Interior Point Algorithm

In this Section, we reformulate the optimization problem in (3.4) to apply the

existing algorithms. We start from transforming our optimization problem to a

suitable form for the Interior-point method, and then give a brief algorithm of the

Interior-point method. We can regard (3.4) as the constrained nonlinear optimization

problem:

min f x subject to ( ) c_j( )x ≥0, j =1,2, ,N

where f( )x = −U_F

(

R R₁, ₂, ,R_N

)

_and c_j( )x = −P_j +P_{maxj are continuous and}

have continuous second partial derivatives. By introducing the slack variable

[

y y1, , ,2 yN

]

=

y , the problem can be converted to

( )

min f x subject to c x( )− =y 0, y≥0

where c x( )=

[

c₁( )x ,c₂( )x , ,c_N ( )x

]

T . The inequality constraint y≥0 can be incorporated into the objective function by adding a logarithmic barrier function. This

yields the minimization problem :

( ) ( ) min 1 N ln _i i fτ f τ y = = −

_∑

x x subject to c x( )− =y 0 (3.5) where τ > is the barrier parameter. Hence all the constraints are equalities. The 0 term 1 ln N i i y τ =

−

_∑

in (3.5) acts like a barrier that prevents any component y from _i becoming negative, since the logarithm function has no definition on the negative

values. We solve the problem (3.5), and obtain the optimal solution to the original

problem as τ → . The Lagrangian for the problem in (3.5) is 0

( ) [ ( ) ] 1 L( , , , ) N ln _i T i f y τ τ = = −

_∑

− − x yλ x λ c x y (3.6) and the KKT conditions for the problem in (3.4) are given by

( ) ( ) L T 0 x f ∇ = ∇ x −A x λ = , 1 L 0 y τ − ∇ = − Y e+λ = , ( ) L 0 ∇_λ =c x − =y , where ( )= ∇

[

c₁( ),∇c₂( ), ,∇c_N( )

]

T A x x x x

{

1 2

}

diag y y, , ,y_N = Y [1,1, ,1]T = e .

Once we reformulate the optimization problem in (3.4) to the form of (3.5), the

problem can be applied to Interior-point algorithm by using the Lagrangian function

in (3.6).

The interior-point algorithm can be briefly summarized as follows.

Step 1. Input an initial set

{

x y λ with ₀, ,_{0 0}

}

y₀ >0, λ₀>0, and an intitial barrier parameter τ . Set ₀ l = , 0

{

x y₀∗, ∗₀,λ₀∗

}

=

{

x y₀, ,₀ λ₀

}

, and initialize the outer-loop tolerance ε_outer.

Step 2. Set k = , 0 τ =τ_l, and initialize the inner-loop tolerance ε_inner. Step 3. Using the first and second derivatives to evaluate

{

Δx_k,Δy_k,Δλ _k

}

and α such that _k 1 1 k+1 k k k k k k k k k k k α α α + + ⎧⎪ _{= + Δ} ⎪⎪ ⎪⎪ _{= + Δ} ⎨⎪ ⎪⎪ _{= + Δ} ⎪⎪⎩ x x x y y y λ λ λ

will get a descent

direction for the objective function.

Step 4. If α_kΔx_k + α_kΔy_k + α_kΔλ_k <ε_inner,

set

{

x_l∗₊₁,y_l∗₊₁,λ_l∗₊₁

} {

= x_k+1,y_k+1,λ_k+1

}

and continue to Step 5; otherwise, set k = +k 1 and repeat from step 3.

Step 5. If x_l∗−x_l∗₊₁ + y_l∗−y_l∗₊₁ + λ − λ_l∗ _l∗₊₁ <ε_outer, output

{

x y∗, ,∗ λ∗

} {

= x y_l∗, ,∗_l λ_l∗

}

and stop; otherwise, calculate τ_l+1, set

{

x y0, ,0 λ0

}

=

{

x yl∗, ,l∗ λl∗

}

, l = +l 1, and repeat from Step 2.

The Interior-point algorithm is described by two loops. The two loops can prevent

finding the local minimum. This algorithm will be convergent by choosing

appropriate error tolerances.

14 16 18 20 22 24 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of iterations CDF 1000 Channels, (6,4) K_MU

Fig. 3-2 CDF of the number of iterations for interior-point algorithm

Fig. 3-2 is the CDF as a function of the number of iterations when the interior-point

algorithm is applied to solve our optimization problem. There are 1000 channel

realizations and M = , 4 N = . We choose the error tolerances to be 0.1. Fig. 3-2 6 shows that the range of the number of iterations is between 15 and 22. After solving

3.4 Computer Simulations

In this section, we simulate the proposed power allocation for MU-MIMO

systems. All the simulations are measured and averaged over 1000 independent

channels. Here M = and 4 N = . Fig. 3-3 shows the PDF of the ratio of 6 minimum rate to maximum rate. We can see that the ratio tend to approach to 1. That

also means the proposed power allocation will tend to uniform data rates.

Fig. 3-4 simulates the minimum user data rates versus the reciprocal of channel

gains in 1000 channel realizations. It compares the three schemes: waterfilling power

allocation, no power allocation, and the proposed fairness power allocation. Since the

channel gain 0.7 0.75 0.8 0.85 0.9 0.95 0 2 4 6 8 10 12 Min (R) / Max (R) PD F

1000 Channels, (6,5)

Fairness scheme

is inversely proportional to the square of distance between the transmitter and receiver,

the horizontal axis can be regarded as the user’s distance from the base station. In the

waterfilling power allocation scheme, the user may be turned off when the distance is

large enough. If we choose the fairness power allocation scheme, the smallest user

rate will be larger than the other two schemes. Fig. 3-5 simulates the maximum user

rates versus the reciprocal of channel gains in 1000 channel realizations. In the

waterfilling power allocation scheme, the user with best channel gain will be allocated

with the largest power, thus the achieved rates will be higher than the other schemes.

In the fairness power allocation scheme, we sacrifice the rate of the best user and

obtain more fair rates.

0 5 10 15 20 25 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 1/gain Ra te

Min user rate, 1000 channels, (6,4)

Fairness scheme No power control Waterfilling scheme

Fig. 3-6 compares the sum rate of four users with different schemes. In the

waterfilling power allocation scheme, the utility function is chosen to maximize the

sum rate of all users. Thus, the sum rate of the waterfilling scheme is always higher

than the other schemes. Although the sum rate of the proposed fairness scheme is

lower than the scheme without power allocation, it could obtain fair rates for all users.

0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1/gain Ra te

Max user rate, 1000 channels, (6,4)

Waterfilling scheme No power control Fairness scheme

-20 -10 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 SNR (dB) Ra te

Sum rate of 4 users

WF scheme

No power allocation Fairness scheme

Fig. 3-6 Sum rate comparison for fairness scheme

3.5 Summary

In this chapter we give a detailed description of the proposed transmit power

allocation. We reformulate the nonlinear optimization problem, and apply the

Interior-point method to solve it. In both SU-MIMO and MU-MIMO systems, after

the proposed transmit power allocation, all users will tend to have fair transmission

rates. This means that it can prevent the users with small channel gains from suffering

poor data rates. We compare the sum rate of the waterfilling power allocation and the

scheme without power allocation with the proposed power allocation. It’s a trade-off

Chapter 4

Condition Number Discussion

In Chapter 3, we have introduced the proposed transmit power allocation with

proportional fairness rates. This means that the sub-channel gains will be close to each

other with the proposed power allocation applied. And the equivalent channel matrix

will tend to be a well-conditioned channel matrix. In this chapter, we state that the

condition number of the equivalent channel matrix is statistically smaller by

observing the simulation results. If an underdetermined MIMO system is

well-conditioned, the decoding complexity of the executed SDA will be reduced.

Motivated by the fairness scheme, we propose a determinant based power allocation

to further reduce the condition number of the equivalent matrix. Thus the decoding

complexity of the underdetermined systems can be reduced with the proposed power

allocation applied. In Section 4.1, we explain why the decoding complexity can be

reduced with a smaller condition number. The determinant based utility function is

provided in Section 4.2. Section 4.3 shows the simulation results. Section 4.4

4.1 Condition Number Effect

Although SDA can reduce the decoding complexity of ML detection from

exponentially increasing to polynomially increasing, its complexity still grows heavily

when the condition number of the channel matrix is large. The condition number is

traditionally and calculated by taking the ratio of the maximum to minimum singular

values of the channel matrix. For MIMO systems, the channel condition number is

calculated from the instantaneous channel matrix without the need for stochastic

averaging. Small values for the condition number imply a well-conditioned channel

matrix while large values indicate an ill-conditioned channel matrix.

Consider an overdetermined MIMO systems with N transmit antennas and

M receive antennas. The idea of SDA is to check all the points in a hyper-sphere with

radius d . It finds the nearest point from the received signal to be the estimated signal.

That is,

2 2 2

arg min arg min

N N d

∈ ∈

= − = − <

x x

x y Hx y Rx , (4.1)

where H =QR , y=Q y , and R is the upper triangular matrix. Without loss of T generality, we let N =M . We can rewrite (4.1) as a summation form

(

)

2 ₂

1

M N

i ij j i= y − j i= r x <d

∑

∑

, start from the last equation and work backward. We expand the last equation as

N N N NN NN y d y d x r r − + < < . (4.2)

It means that all the constellation points satisfy (4.2) could be the candidate of N

x .

Fig. 4-1 Tree search example for 4-PAM showing sphere radius, tree levels and detection layers 1 1 N N ij j ij j i j i i j i i ii ii y d r x y d r x x r r = + = + − − + − < <

∑

∑

. (4.3)

From (4.2) and (4.3), we know that SDA is a tree search. Fig. 4-1 is a tree search

example for the 4-PAM constellation. We can search from Level 1 to Level N, and the

distance for all levels should be less than the radius d. Because SDA decodes the

transmit signal from the last layer x_N, the boundary of x_N should be as small as

possible. This will reduce the number of searching points, as well as the decoding

complexity.

In [22], we know that the r_NN of R depends on the condition number of the

channel matrix H . The lower the condition number is, the larger the r_NN is. Thus, we can obtain a lower decoding complexity when the system is well-conditioned.Fig.

4-2 shows the CDF of r₅₅ for a 5× matrix with different condition numbers. 5 We can see that when the condition number is less than 10, the value of r₅₅ will tend

to be larger.Fig. 4-3 shows the FLOPS (Floating Point Operation Per Second) of the 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r₅₅ CDF 1000 data, (5,5) CN>10 CN<10

Fig. 4-2 CDF of r₅₅ for a 5× matrix with different condition numbers 5

SDA versus condition number. The transmitter and receiver are both equipped with

four antennas, and the transmitter uses the QPSK modulation. We can see that when

the condition number is larger than 5 the complexity increase rapidly. Fig. 4-4 shows

the CDF (Cumulative Density Function) of the condition number with different

channel correlation. Assuming that there are 1000 channel realizations in a 2× 2 MIMO system. Let R be the correlation matrix of H . A useful measure of the _hh degradation in performance due to channel correlation, for a system with K diversity

branches, is provided by the Kth root of the determinant of the channel correlation

matrix, det

(

R_hh

)

1/K[24]. If det

(

R_hh

)

1/Kis smaller than 0.5, it can be regarded as high correlation. If det

(

R_hh

)

1/K is larger than 0.5, it can be regarded as low

condition number is larger than 10 dB. Fig. 4-4 shows that both channels with low

and high correlation are probable to be ill-conditioned channels. It also shows that a

MIMO system is more likely to have a lower condition number when the channel has

low correlation. 0 2 4 6 8 10 12 14 16 18 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 7.1x 10 4 Condition Number FLO P S QPSK, (4,4) SDA

0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Condition Number (dB) CDF

1000 Channels, (2,2)

low correlation high correlation

Fig. 4-4 CDF of condition number with different channel correlations

4.2 Proposed Utility Function for Condition

Number

In Chapter 3, we know that the proposed transmit power allocation results in

proportional fairness data rates. That means that the data rate for each user

2

log (1 SINR )

i i

R = + will be close to each other. The SINR in the log function will also be close to each other. When the data is transmitted from the ith user, the SINR at

the receiver can be written as

2 2 SINR Noise i i i j j j i P P ≠ = +

_∑

h h . h is the column i

vector of the channel matrix H . At the high SNR, the data rate will be