中 華 大 學

中 華 大 學 碩 士 論 文

題目：MIMO-OFDM 系統的最佳功率控制

Optimal Power Control of MIMO-OFDM Systems

系 所 別：電機工程學系碩士班 學號姓名：M09401029 李建億 指導教授：李 柏 坤 博士

中華民國 九十六 年 八 月

Optimal Power Control of MIMO-OFDM Systems

研 究 生：李建億 Student：Jian-Yi Li

指導教授：李柏坤 博士 Advisor：Dr. Bore-Kuen Lee

中華大學

電機工程學系碩士班

碩士論文

A Thesis

Submitted to Institute of Electrical Engineering Chung Hua University

In Partial Fulfillment of the Requirements For the Degree of

Master of Science In

Electrical Engineering August 2007

Hsin-Chu, Taiwan, Republic of China

中 華 民 國 九 十 六 年 八 月

MIMO-OFDM 系統的最佳功率控制

研 究 生：李建億 指導教授：李柏坤 博士

中華大學

電機工程學系碩士班

中文摘要

對於 MIMO-OFDM 的系統來說，我們的目標是在功率總和有限的情況 下，用現有的功率控制策略把信號的 SINR 極大化，或者把誤差給極小 化。然而，這些功率控制策略分配功率時，必需每個子通道都要計算 一次。隨著現代通訊技術的進步，OFDM 系統的子通道越來越多，這 樣的話計算量也會越來越沉重，為了盡可能的降低計算量，首先把系 統模型用轉移函數的方法重寫，然後再用 MMSE 策略去最佳化這個系

統，並把這最佳化問題轉成H_∞的最佳化問題，受限於 的功率限制。

我們再把轉移函數用狀態空間表示法去表示它，則 H2

H_∞的最佳化問題可 以再轉成線性的最佳化問題，而我們可以容易的用 MATLAB 的 BMI 工具箱去找到最佳化的解。透過模擬，我們可以看到當雜訊的變異量 小時，這個被提出來的演算法將會分配較多的功率給通道增益較好的 子通道，相反的，當雜訊的變異量大時，這演算法將會分配較多的功 率給通道增益較差的通道。

Student：Jian-Yi Li Advisor：Dr. Bore-Kuen Lee

Institute of Electrical Engineering Chung Hua University

Abstract

For MIMO-OFDM systems, the objective of the existing power control strategies is maximization of the signal to interference and noise ratio (SINR) or minimization of the bit error ratio (BER) subject to constrained total transmission power. However, these strategies need to compute power assignment law via a specific optimization method for each subcarrier. With the advance of modern communication technology, the OFDM system may have more and more subcarriers in the future. In this case, the computation burden will become more and more heavier. To reduce the possible computation burden, the system model is first rewritten by using transfer matrices. Then the constrained power control problem concerning MMSE optimization is transformed into an optimal optimization problem subject to an H ₂ power constraint. Using state-space representation of the transfer matrices, the proposed constrained optimization problem can be reformulated as an linear optimization problem subject to bilinear matrix inequalities (BMI), which can be easily solved by using available BMI toolbox under the numerical software MATLAB. Through simulation study, it is shown that when the noise variance is smaller, the proposed algorithm will allocate more power to those subcarriers with better channel gain. On the contrary, when the noise variance is larger, then more power will be distributed over those carriers with worse channel gain.

H_∞

H_∞

Acknowledgement

I would like to express my sincere gratitude to my advisor, Dr.

Bore-Kuen Lee, for his helpful advice, patient guidance, encouragement, and valuable support during the course of the research. I am obliged to Dr.

Chi-Kuang Hwang, and my classmates for their helpful discussions and comments.

Finally, I want to express my sincere gratitude for my parents for their encouragement and suggestions.

1 Introduction 5

2 System Model and Review of Power Control Strategy 8

2.1 System Model . . . 8

2.2 Review of Some Existing Power Control Strategies . . . 11

2.2.1 Uniform Power Control . . . 14

2.2.2 Water-Filling Power Control . . . 14

2.2.3 Maximization of The Harmonic Mean SINR (HARM) . . . 14

2.2.4 MMSE Power Control . . . 15

2.2.5 Maximization of The Minimum SINR(MAXMIN) . . . 15

2.2.6 MEPE (Minimal E¤ective Probability of Error) Power Control 16 3 Power Control Based on the Transfer Function Approach 18 3.1 Problem Formulation . . . 18

3.2 Solution to the Proposed Problem . . . 22

4 Simulation 30

1

2

5 Conclusions 35

A Appendix 1 37

A.1 SISO-OFDM Model . . . 37 A.2 MIMO-OFDM Model . . . 39

B Appendix 2 42

B.1 Derivation of the State-Space Representation in (3.13) . . . 42

2.1 The MIMO-OFDM system model. . . 9

4.1 Maximum eigenvalues vs subcarrier for the 2+2 MIMO channel. . . . 33 4.2 Sigma_v values vs H in…nite norm values. . . 33 4.3 Normalized power distribution over the subcarriers. . . 34

3

4 Abstract

For MIMO-OFDM systems, the objective of the existing power control strategies is maximization of the signal to interference and noise ratio (SINR) or minimization of the bit error ratio (BER) subject to constrained total transmission power. However, these strategies need to compute power assignment law via a speci…c optimization method for each subcarrier. With the advance of modern communication technol- ogy, the OFDM system may have more and more subcarriers in the future. In this case, the computation burden will become more and more heavier. To reduce the possible computation burden, the system model is …rst rewritten by using transfer matrices. Then the constrained power control problem concerning MMSE optimiza- tion is transformed into an optimal H₁optimization problem subject to an H2 power constraint. Using state-space representation of the transfer matrices, the proposed constrained H₁ optimization problem can be reformulated as an linear optimization problem subject to bilinear matrix inequalities (BMI), which can be easily solved by using available BMI toolbox under the numerical software MATLAB. Through simulation study, it is shown that when the noise variance is smaller, the proposed algorithm will allocate more power to those subcarriers with better channel gain. On the contrary, when the noise variance is larger, then more power will be distributed over those carriers with worse channel gain.

keywords: MIMO, OFDM, Power Control

Introduction

In the communication system, OFDM systems [13] have already been extensively used due to easy implementation of modulator and demodulator by using IFFT and FFT techniques. To further improve Quality-of-Service (QoS) of OFDM systems, the array antennas on the transmitter and the receiver have been considered to form multiple- input and multiple-output (MIMO) OFDM systems [12] in order to increase date rate, improve the signal to interference and noise ratio (SINR), and reduce the bit error ratio (BER). However, to counteract the time-varying wireless channel, power control of MIMO-OFDM system is also an important issue to ensure QoS subject to constrained total power of the transmission antennas.

In the literature, there are not a few existing power control methods for MIMO- OFDM systems. The uniform power control is the simplest method which gives the same power to each subcarrier, and on the contrary, the popular water-…lling method assigns more power to those subcarriers with better channel gain [1], [6], [8], [9]. The MMSE power control is another common method as proposed in [10].

5

6 Maximization of the minimum SINR (MAXMIN) method is discussed in [1] where it increases power for those subcarriers with worse channel gain and reduces the power for those with better channel channel. The objectives of the existing power control strategies [1][6][7][8][9][11] are maximization of the signal to interference and noise ratio (SINR) or minimization of the bit error ratio (BER) subject to constrained total transmission power. However, these strategies need to compute power assignment law via a speci…c optimization method for each subcarrier. With the advance of modern communication technology, the OFDM system may have more and more subcarriers in the future. In this case, the computation burden will become more and more heavier.

To reduce the possible computation burden for MIMO-OFDM power control de- sign, we shall …rst rewrite the system model by using the transfer function approach based on the MIMO-OFDM structure proposed in [1]. Then the constrained power control problem concerning MMSE optimization is transformed into an optimal H₁ optimization problem subject to an H2 power constraint. Using state-space represen- tation of the transfer matrices, the proposed constrained H₁ optimization problem can be reformulated as an linear optimization problem subject to bilinear matrix in- equalities (BMI), which can be easily solved by using available BMI toolbox under the numerical software MATLAB.

The remaining of this study is organized as follows: In Chapter 2, system model of the considered MIMO-OFDM system is described in Section 2.1. Review of some existing power control strategies is made in Section 2.2. In Chapter 3, we reformulate the power control problem by using the transfer function approach in Section 3.1

and the solution to the proposed is attacked by bilinear matrix inequality (BMI) optimization technique in Section 3.2. Simulation study is performed in Chapter 4.

Finally, we summarize our conclusions in Chapter 5.

Chapter 2

System Model and Review of Power Control Strategy

2.1 System Model

Consider an MIMO-OFDM communication system, which has M inputs and R out- puts as shown in Fig. 2.1. In the MIMO-OFDM system with N subcarriers, St(k) denotes the frequency-domain signal, which are transmitted by the k-th subcarrier via the M antennas in the t-th symbol interval, b (k) denotes the pre-beamforming vector, H_t;i;j(k) denotes the channel gain of the k-th subcarrier between the i-th transmitter antenna and the j-th receiver antenna for 1 i M and 1 j R. Yt(k) denotes the vector of the received signals on the k-th subcarrier of the R antennas at the t-th symbol time interval, and Vt(k) is a zero-mean additive noise vector. The channel gain Ht;i;j(k) is assumed invariant during a symbol time interval. We suppose that E jS^t(k)j² = 1 for all t and k. By the derivation in Appendix 1, the system model

8

Output Data Input Data

P/S S/P

Pre-beamforming

Post-beamforming b

b

0

N--1

N--1

a

a

0

DFT

DFT I^DFT

IDFT

CP ^&P^/S

CP &P/S

CPRemoveal& S/P

CPRemoveal& S/P N

N

N N

N

N

N

N N

N

1

1

R M

Figure 2.1: The MIMO-OFDM system model.

can be expressed as (2.1). [1]

Y_t(k) = H (k) b (k) S_t(k) + V_t(k) ; 0 k < N (2.1)

where the channel gain matrix H (k) for the k-th subcarrier is de…ned as H (k) = H (z)j_z=e^{j 2 k}_N 2 C^{R M};

H (z) = 2 66 66 66 66 66 4

H_t;1;1(z) H_t;1;2(z) H_t;1;M(z) H_t;2;1(z) H_t;2;2(z) H_t;2;M(z)

... ... . .. ...

H_t;R;1(z) H_t;R;2(z) H_t;R;M (z) 3 77 77 77 77 77 5

; (2.2)

10 H_t;i;j(z) = h_t;i;j(0) + h_t;i;j(1) z ¹+ + h_t;i;j(L) z ^L;

and fh^t;i;j(l)g^Ll=0 is the impulse response of the the channel between the j-th trans- mitter antenna and the i-th receiver antenna with L being the delay spread. For the convenience of further discussion, the pre-beamforming vector b (k) and the received signal Yt(k) are represented by

b (k) = 2 66 66 66 66 66 4

b₁(k) b₂(k)

...

b_M(k) 3 77 77 77 77 77 5

2 C^{M 1} (2.3)

Y_t(k) = 2 66 66 66 66 66 4

Y_t;1(k) Y_t;2(k)

...

Y_t;R(k) 3 77 77 77 77 77 5

2 C^{R 1} (2.4)

The received signal rt(k) after post-beamforming can be expressed as

r_t(k) = a^H(k) Y_t(k)

= a^H(k) H (k) b (k) S_t(k) + a^H(k) V_t(k) (2.5)

Let bS_t(k)be the estimated signal of St(k)by applying the decision operator dec f g = signfR [ ]g on r^t(k) as

Sb_t(k) = decfr^t(k)g

= dec a^H(k) Y_t(k) (2.6)

2.2 Review of Some Existing Power Control Strate- gies

In this section, we shall review some existing power control strategies for an MIMO- OFDM system [1]. If the channel state information (CSI), including H (k) and R_v(k) = E V_t(k) V_t^H(k) , is known at the transmitter and the receiver, the beam- forming vectors a (k) and b (k) can be designed to maximize the SIN Rk at the k-th subcarrier, which is de…ned as [1]

SIN R_k= En

a^H(k) H (k) b (k) S_t(k) ²o

E ja^H(k) V_t(k)j² = a^H(k) H (k) b (k) ²

a^H(k) R_v(k) a (k) : (2.7) In the SIN Rk function (2.7), one goal is to maximize the SIN Rk by adjusting the two unknown vectors a (k) and b (k) for k-th subcarrier. But it will cause cou- pling issue when we calculate a (k) and b (k) simultaneously, so one can …x b (k) and determine a (k) as a function of b (k) at …rst. One can de…ne a (k) = R

1

v2 (k) a (k), and the SIN Rk can be rewritten as

SIN R_k = a^H(k) H (k) b (k) ² a^H(k) R_v(k) a (k)

=

a^H(k)n R

H

v 2 (k) H (k) b (k) b^H(k) H^H(k) R

1

v 2 (k)o a (k)

a^H(k) a (k) (2.8)

Note that the length of a (k) does not a¤ect the SIN Rk value and thus we can choose ka (k)k = 1. The SINR^k value can then be reduced as

SIN R_k= a^H(k) M₁(k) a (k) (2.9)

where

M₁(k) = R

H

v 2 (k) H (k) b (k) b^H(k) H^H(k) R

1

v 2 (k) (2.10)

12 Let v (k) = R

H

v 2 (k) H (k) b (k). Then the symmetric matrix M1(k) can be repre- sented as M1(k) = v (k) v^T (k). As M1(k) is of rank 1, we have

M₁(k) v (k) =kv (k)k²v (k)

which implies kv (k)k² is the maximal eigenvalue of M1(k) with the corresponding eigenvector v (k). Therefore, to maximize SIN Rk with a given b (k), a (k) can be chosen as

a (k) = kR

1

v2 (k) H (k) b (k) (2.11)

where k can be chosen as any number and it does not a¤ect the value of the SIN Rk. With the de…nition a (k) = R

1

v2 (k) a (k), we have

a (k) = _kR_v¹(k) H (k) b (k) (2.12)

Then we can substitute a (k) into (2.7) to obtain the following SIN Rk expression

SIN R_k =

(R_v¹(k) H (k) b (k))^H H (k) b (k)

2

fRv¹(k) H (k) b (k)g^HR_v(k)fRv¹(k) H (k) b (k)g

= b^H(k) M₂(k) b (k) (2.13)

where

M₂(k) = H^H(k) R_v¹(k) H (k) (2.14)

With the assumption E jS^t(k)j² = 1, the total transmission power p (k) on the k-th subcarrier is equal to

p (k) =kb (k)k² = b^H(k) b (k) (2.15)

If the transmission power is constrained, then b (k) can be represented by

b (k) =p

p (k)u(k)

where u(k) is a unit vector.

Suppose that we want to maximize SIN Rk by choosing some optimal u(k). Then u(k) can be chosen as the unit eigenvector corresponding to the maximum eigenvalue

max(k) of M2(k) with

max(k) u(k) = H^H(k) R_v(k) ¹H (k) u(k)

ku(k)k = 1 (2.16)

The SIN R at the k-th subcarrier can be expressed as

SIN Rk = max(k) p (k) (2.17)

For the k-th subcarrier with power constraint p (k), the pre-beamforming vector b (k) representing the spatial distribution of the transmitted power has been deter- mined. However, how to assign power p (k) to each subcarrier has not been decided yet. Next, we shall review some existing power control strategies for assigning p (k).

First, assume that the CSI including the matrices H (k) and Rv(k) is already known. We consider a total transmission power constraint P0, i.e.,

N 1

X

k=0

kb (k)k² =

N 1

X

k=0

p (k) = P₀ (2.18)

With this constraint, six existing power control strategies are introduced in the following. [1]

14

2.2.1 Uniform Power Control

This is the simplest method in which the same power is given to each subcarrier with

p (k)_GEOM = P₀ N

a (k) = R_v¹(k) H (k) b (k)

b^H(k)H^H(k) R_v¹(k) H (k) b (k) (2.19)

It can maximize that geometric mean of the SINR, g fSINRg =

NY1 k=0

SIN R

1 N

k , subject to the power constraint (2.18).

2.2.2 Water-Filling Power Control

This is a common method in which more power is given to subcarriers with better SINR. The water-…lling power control strategy is expressed as [6]

p (k)_CAP = max 0; 1

max(k) a (k) = R_v¹(k) H (k) b (k)

1 + b^H(k)H^H(k) R_v¹(k) H (k) b (k) (2.20) where is a constant calculated to satisfy the power constraint (2.18).

2.2.3 Maximization of The Harmonic Mean SINR (HARM)

HARM is the another choice in which more power is given to subcarriers with worse SINR. Subject to the global transmission power constraint (2.18), its objective is to maximize the harmonic mean SIN RHARM of the SINR, where

SIN R_HARM = N PN 1

k=0 1 SIN Rk

= N

PN 1 k=0

1

max(k)p(k)

(2.21)

The HARM power control strategy is expressed as

p (k)_HARM = P₀ PN 1

i=0

1

max2 (i) p 1

max(k) SIN R_k;HARM = p₀

PN 1 i=0

1

max2 (i) p

max(k) a (k) = R_v¹(k) H (k) b (k)

b^H(k)H^H(k) R_v¹(k) H (k) b (k) (2.22)

2.2.4 MMSE Power Control

The MMSE power control strategy is presented as follows

p (k)_{M M SE} = max (

0;p

max(k)

1

max(k) )

a (k) = R_v¹(k) H (k) b (k)

1 + b^H(k)H^H(k) R_v¹(k) H (k) b (k) (2.23) where is a constant calculated to satisfy the power constraint (2.18). When P0 is low, the MMSE strategy is to give more power to those subcarriers with better SINR and it is possible that the strategy does not assign power to serious fading subcarriers.

When P0 is high enough, the MMSE strategy will give more power to subcarriers with worse SINR and the transmission power p (k)_HARM and p (k)_{M M SE} will be equal as P₀ ! 1; i.e.;

Plim0!1

p (k)_{M M SE}

p (k)_HARM = 1: (2.24)

2.2.5 Maximization of The Minimum SINR(MAXMIN)

Subject to the power constraint (2.18), the purpose of MAXMIN strategy is to make SINR of each subcarrier equal, i.e.,

max(i) p (i) = _max(j) p (j)

16 The MAXMIN power control strategy is given as

p (k)_{M AXM IN} = P₀ PN 1

i=0 1 max(i)

1

max(k)

SIN R_{k;M AXM IN} = P₀ PN 1

i=0 1 max(i) a (k) = R_v¹(k) H (k) b (k)

b^H(k)H^H(k) R_v¹(k) H (k) b (k) (2.25)

2.2.6 MEPE (Minimal E¤ective Probability of Error) Power Control

We de…ne the e¤ective probability of error Pe;ef f as the mean error probability averaged over all the subcarriers, i.e.,

P_{e;ef f} = 1 N

NX1 k=0

P_e(k) (2.26)

where Pe(k) is the error probability of the k-th subcarrier. Minimizing the upper bound of Pe;ef f as done in [1] under Gaussian assumptions on the noise and interfer- ence, the MEPE power control strategy is given by

p (k)_{M EP E} = 2 k_m

maxf0; log ( ^max(k) )g

max(k) a (k) = R_v¹(k) H (k) b (k)

1 + b^H(k)H^H(k) R_v¹(k) H (k) b (k) (2.27)

where is a constant calculated to satisfy the power constraint (2.18) and km is a constant depending on the signal constellation applied to the carriers. When BPSK is used, km = 2.

When P0is low, the MEPE strategy will give more power to subcarriers with better SINR and it is possible that no power is assigned to serious fading subcarriers. When

P₀ is high enough, the MEPE strategy will give more power to those subcarriers with worse SINR and the transmission power p (k)_{M AXM IN} and p (k)_{M EP E} will be equal as P0 ! 1

Plim0!1

p (k)_{M EP E}

p (k)_{M AXM IN} = 1: (2.28)

Chapter 3

Power Control Based on the Transfer Function Approach

3.1 Problem Formulation

As reviewed in the precious section, the existing power control strategies need to compute power assignment law via a speci…c optimization method for each subcarrier.

With the advance of modern communication technology, the OFDM system may have more and more subcarriers in the future. In this case, the computation burden will become more and more heavier. To reduce the possible computation burden, we shall rewrite the system model (2.1) by using transfer functions. The methodology is to compute the pre-beamforming vector b(k) and the post-beamforming vector a(k) for

18

0 k N 1 from the transfer functionsea (z) and eb(z) via

b(k) = eb (z)jz=e^jwk

a(k) = ea (z) jz=e^jwk;

with wk = ^{2 k}_N for 0 k N 1: By replacing b(k) and a(k) by ea (z) and eb(z), respectively, we shall …nd ea (z) and eb(z) via minimizing some performance index as de…ned in the sequel. The advantage of the proposed method is that by one-pass optimization calculation, we can obtain the transfer functionsea (z) and eb(z) and use them to directly compute b(k) and a(k) for each subcarrier.

Based on the proposed methodology, the system model (2.1) can be extended as

e

Y_t e^jw = H e^jw eb e^jw Se_t e^jw + eV_t e^jw er^t e^jw = ea^H e^jw Ye_t e^jw

Se^_t e^jw = dec er^t e^jw (3.1)

Compared with the system model in (2.1), the extended functions eY_t(e^jw) ; eS_t(e^jw) ; eV_t(e^jw) ; er^t(e^jw) ; and e^St(e^jw) are de…ned as

Y_t(k) = Ye_t e^jw j^w=wk

St(k) = Set e^jw j^w=wk

V_t(k) = Ve_t e^jw j^w=wk

r_t(k) = er^t e^jw j^w=wk

S^_t(k) = e^S_t e^jw j^w=wk

20 The received data error Et(e^jw)for the t-th symbol is expressed as

E_t e^jw = Se_t e^jw er^t e^jw

= [1 ea^H e^jw H e^jw eb e^jw ] eS_t e^jw ea^H e^jw Ve_t e^jw (3.2)

With the assumption E Se_t(e^jw)

2

= 1, the variance of Et(e^jw)is then given as

E Se_t e^jw er^t e^{jw 2} = 1 ea^H e^jw H e^jw eb e^{jw 2}

+ea^H e^jw Re_v e^jw ea e^jw (3.3)

where eR_v(e^jw) = En

Ve_t(e^jw) eV_t^H(e^jw)o

. Obviously, we have Rv(k) = eR_v(e^jw)j^w=wk: In this study, we assume that eR_v(e^jw) = ²_vI_R where IR is the identity matrix of dimension R R.

However, there is a di¤erent error under a di¤erent frequency w, and our goal is to …ndea (e^jw)and eb (e^jw) to minimize the largest error, i.e.,

min

ea(e^jw);eb(e^jw)

max

w2[0;2 )E Se_t e^jw er^t e^jw

2

= min

ea(e^jw);eb(e^jw)

max

w2[0;2 )

1 ea^H e^jw H e^jw eb e^{jw 2}+ ²_vea^H e^jw ea e^jw (3.4)

Recall that the H₁ norm of a stable rational function f (z) is given as

kf(z)kH₁ = sup

w2[0;2 )

f (e^jw)

By using the notations of the H₁norm of a transfer function k kH₁and the Euclidean norm of a vector k k, the optimization problem de…ned in (3.4) can be rewritten as

min

ea(z);eb(z)2H1

k (z)k²_H1 (3.5)

where the transfer matrix (z)is de…ned as

(z) = 1 +ea^T (z) H (z) eb (z)

vea (z)

On the other hand, the choice of eb (z)is related to the global power constraint as de…ned (2.18). Under the assumption that the number of the subcarriers is greatly larger than 1, i.e., N >> 1; then power constraint on eb (z) can be derived as

P₀ =

N 1

X

k=0

kb (k)k²

=

N 1

X

k=0

eb e^{jwk 2}

= N

2

N 1

X

k=0

eb e^{jwk 2}4w , 4w = 2 N ' N 1

2 Z 2

0

eb e^{jw 2}dw (3.6)

Recall that the H2 norm of a stable rational function f (z) is given as

kf(z)k²H2 = 1 2

Z 2 0

f e^{jw 2}dw

Therefore, the global power constraint can be rewritten as

eb(z) ²

H2

= P₀

N (3.7)

Based on the above discussion, the proposed transfer function approach is to solve the following optimization problem

min

ea(z);eb(z)2H1

k (z)k²_H1 (3.8)

subject to eb(z) ²

H2

= P₀

N (3.9)

22 Remark 1 The proposed optimization problem in (3.8) and (3.9) can be extended to

min

ea(z);eb(z)2L1

k (z)k²_L1 subject to

eb(z) ²

L2

= P₀ N

3.2 Solution to the Proposed Problem

The proposed problem formulated in the previous section can be solved by using the matrix inequality approaches, such as LMI (linear matrix inequality) and BMI (bilinear matrix inequality) provided the transfer matrices H (z), eb (z), and ea (z) are expressed by state-space representation. For the channel gain transfer matrix H(z), we have

H (z) = H₀+ H₁z ¹+ H₂z ²+ + H_Lz ^L

where H` = [ht;i;j(`)]1 i R; 1 j M for 0 ` L. With the impulse response matrices fH<sup>`</sup>g0 ` L, a state-space representation of H (z) is given by

x^H_k+1 = A_Hx^H_k + B_Hu_k

y_k^H = C_Hx^H_k + D_Hu_k (3.10)

where x^H_k 2 C^{LM 1}, uk 2 C^{M 1}, y^H_k 2 C^{R 1}, and

A_H = 2 66 66 66 66 66 4

0_{M M} 0_{M M} 0_{M M}

IM ...

. .. ...

I_M 0

3 77 77 77 77 77 5

2 C^{LM LM}; B_H = 2 66 66 66 66 66 4

I_M 0M M

...

0_{M M} 3 77 77 77 77 77 5

2 C^{LM M}

C_H = H1_{R M} H2_{R M} HL_{R M} 2 C^{R LM}; D_H = [H₀]2 C^{R M}

Note that due to multipath fading, the channel gain matrix H (z) is of FIR model.

However, to gain more freedom in the optimization problem, we chooseea (z) and eb(z) as transfer matrices which have the following state-space representations:

x^a_k+1 = A_ax^a_k+ B_au_k

y_k^a = C_ax^a_k+ D_au_k (3.11)

and

x^b_k+1 = A_bx^b_k+ B_bu_k

y_k^b = Cbx^b_k+ Dbuk (3.12)

where Aa 2 C^{na na}, Ba 2 C^{na 1}, Ca 2 C^{R na}, Da 2 C^{R 1}; A_b 2 C^{nb nb}; B_b 2 C^{nb 1}; C_b 2 C^{M nb}; and Db 2 C^{M 1}:

Based on the above state-space representations, we are ready to transform the H₁ cost function in (3.8) and the H2 constraint in (3.9) into matrix inequalities as done in the following. First, for the the H₁ cost function in (3.8), the transfer matrix

24 (z)can be represented by

(z) = D + C(zI A) ¹B (3.13)

where

A = 2 66 66 66 66 66 4

A^T_a C_a^TC_H C_a^TD_HC_b 0 0 A_H B_HC_b 0

0 0 A_b 0

0 0 0 A_a

3 77 77 77 77 77 5

; B = 2 66 66 66 66 66 4

C_a^TD_HD_b B_HD_b

B_b B_{a v}

3 77 77 77 77 77 5

C = 2 66 4

B_a^T D_a^TC_H D^T_aD_HC_b 0

0 0 0 C_a

3 77 5 ; D =

2 66 4

D_a^TD_HD_b 1 D_{a v}

3 77 5

The derivation of the above state-space model can be referred to Appendix 2. To solve ea (z) and eb(z) from the proposed problem, we shall need the following results.

Lemma 1 (Schur’s complement) Suppose that R (x) > 0. Then the matrix inequality

Q (x) S (x) R ¹(x) S^T (x) > 0:

is equivalent to 2

66 4

Q (x) S (x) S^T (x) R (x)

3 77

5 > 0 (3.14)

Lemma 2 (Bounded real lemma) For a transfer matrix F (z) = D+C (zI A) ¹B 2 H₁; then kF (z)k²₁ r² if and only if there exists a matrix X = X⁰; X > 0 such that [2]

2 66 4

A B C D

3 77 5

02 66 4

X 0 0 I

3 77 5

2 66 4

A B C D

3 77 5

2 66 4

X 0

0 r²I 3 77

5 < 0: (3.15)

Note that by Schur’s complement, the matrix inequality in (3.15) is equivalent to 2

66 66 66 66 66 4

X 0

0 r²I

A^TX C^T B^TX D^T XA XB

C D

X 0 0 I

3 77 77 77 77 77 5

> 0 (3.16)

By the above inequality, minimization of k (z)kH₁ is equivalent to

min

X;Aa;Ba;Ca;Da;Ab;Bb;Cb;Db

subject to 2

66 66 66 66 66 4

X 0

0 I

A^TX C^T B^TX D^T XA XB

C D

X 0 0 I

3 77 77 77 77 77 5

> 0 (3.17)

X > 0 (3.18)

where = r²: With the special state-space structure of (z) in (3.13), we shall accordingly impose the structure of X as

X = 2 66 66 66 66 66 4

X₁ 0 0 0

0 X₂ 0 0

0 0 X₃ 0

0 0 0 X₄

3 77 77 77 77 77 5

26 and the matrix inequality (3.17) can be rewritten as

2 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 4

X1 0 0 0 0

0 X₂ 0 0 0

0 0 X₃ 0 0

0 0 0 X4 0

0 0 0 0 I

Q^T

Q

X₁ 0 0 0 0 0

0 X2 0 0 0 0

0 0 X₃ 0 0 0

0 0 0 X₄ 0 0

0 0 0 0 I 0

0 0 0 0 0 I

3 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 5

> 0 (3.19)

where

Q = 2 66 66 66 66 66 66 66 66 66 4

X₁A^T_a X₁C_a^TC_H X₁C_a^TD_HC_b 0 X₁C_a^TD_HD_b 0 X₂A_H X₂B_HC_b 0 X₂B_HD_b

0 0 X₃A_b 0 X₃B_b

0 0 0 X₄A_a X₄B_{a v}

B^T_a D_a^TCH D_a^TDHCb 0 D_a^TDHDb 1

0 0 0 C_a D_{a v}

3 77 77 77 77 77 77 77 77 77 5

(3.20)

Note that in matrix Q, the term X1C_a^TD_HC_b is coupled by three decision variables including X1, Ca; and Cb:The same situation also applies to the term X1C_a^TD_HD_b. To remove these coupling, the matrix inequality (3.19) can be transformed, via a

congruent transformation, into the following one:

2 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 4

Q₁ 0 0 0 0

0 Q₂ 0 0 0

0 0 X₃ 0 0

0 0 0 X₄ 0

0 0 0 0 I

P^T

P

Q₁ 0 0 0 0 0

0 Q₂ 0 0 0 0

0 0 X₃ 0 0 0

0 0 0 X₄ 0 0

0 0 0 0 I 0

0 0 0 0 0 I

3 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 5

> 0 (3.21)

where Q1 = X₁¹, Q2 = X₂ ¹;and

P = 2 66 66 66 66 66 66 66 66 66 4

A^T_aQ₁ C_a^TC_HQ₂ C_a^TD_HC_b 0 C_a^TD_HD_b

0 A_HQ₂ B_HC_b 0 B_HD_b

0 0 X₃A_b 0 X₃B_b

0 0 0 X₄A_a X₄B_{a v}

B_a^TQ₁ D^T_aC_HQ₂ D_a^TD_HC_b 0 D^T_aD_HD_b 1

0 0 0 C_a D_{a v}

3 77 77 77 77 77 77 77 77 77 5

(3.22)

It is obvious that (3.21) is a bilinear matrix inequality.

Now we turn to transform the H2 constraint in (3.9) into matrix inequalities. It is well known [3] that for a transfer matrix eb (z) = D_b+ C_b(zI A_b) ¹B_b; its H2 norm

28 can be computed via

eb(z) ²

H2

= T r B^T_bL₀B_b+ D_b^TD_b = P₀

N (3.23)

where L0 is a positive de…nite matrix satisfying

A^T_bL₀A_b L₀+ C_b^TC_b = 0 (3.24)

With the facts in (3.23) and (3.24), computation of the H2 norm of eb (z) can be reformulated as follows [3]:

min subject to B_b^TY B_b <

A^T_bY A_b Y + C_b^TC_b < 0 + T rfDb^TD_bg = P₀

N (3.25)

Using the Schur’s complement to remove the coupling of decision variables in (3.25), the above optimization is equivalent to the following one.

min subject to 2

66 4

B^T_bY Y B_b Y

3 77

5 > 0 (3.26)

2 66 66 66 4

Y A^T_bY C_b^T

Y A_b Y 0

C_b 0 I

3 77 77 77 5

> 0 (3.27)

+ T rfDb^TDbg = P₀

N (3.28)

In summary, the proposed optimization problem can be solved from the following bilinear matrix inequality optimization problem to obtainea (z) and eb(z):

min + (3.29)

subject to

(3.21), (3.26), (3.27), (3.28), and Q₁; Q₂; X₃; X₄ > 0

where is the set of decision variables, i.e., =fX; A^a; B_a; C_a; D_a; A_b; B_b; C_b; D_bg :

Chapter 4

Simulation

In this section, several numerical results are presented. Here, we consider an MIMO- OFDM system which has 2+2 array antennas. In the MIMO-OFDM system, there are 64 subcarriers and the channel delay spread is 2. BPSK is used for the modulation scheme. In this simulations, we suppose that the CSI including the matrices H (k) and Rv(k) is already known and there are no interferences at the receiver. i.e., only additive white Gaussian noise. The channel gain transfer matrix is given by

H(z) = 2 66 4

0:2120 1:0078 0:2379 0:7420

3 77 5+

2 66 4

1:0823 0:3899 0:1315 0:08799

3 77 5 z ¹+

2 66 4

0:6355 0:4437 0:5596 0:9499

3 77 5 z ²

The maximum eigenvalues max(k)for the MIMO channel is shown in Fig. 4.1. Here we assume that the structures of eb (z) and ea (z) are given by

eb(z) = 2 66 4

b₁(z) b₂(z)

3 77

5 ; b¹(z) = ¹₀+

1 1z ¹

1 + ¹₁z ¹; b₂(z) = ²₀+

2 1z ¹ 1 + ²₁z ¹

ea (z) = 2 66 4

a₁(z) a₂(z)

3 77

5 ; a¹(z) = ¹₀+

1 1z ¹

1 + ¹₁z ¹; a₂(z) = ²₀+

2 1z ¹ 1 + ²₁z ¹

30

For di¤erent values of the noise variance ²_v; the optimal pre-beamforming transfer function eb (z) and the optimal post-beamforming transfer function ea (z) are listed in the following table. The relationship between the v value and the H₁ norm value k (z)k_H1 is illustrated in Fig. 4.2 and is also listed in Table 1. In Fig. 4.2, we can see that as v value is decreased, the H₁ norm value k (z)k_H1;i.e., the square root of the worst-case variance of the estimation error E Se_t(e^jw) ert(e^jw)

2

over all the frequency range, is also decreased. By using BPSK modulation, the BER performance under di¤erent values of v is also compared in Table 1. The BER performance is satisfactory.

v 1:0000 0:6310 0:3981 0:2512

b₁(z) 2:0206+3:9636z ¹ 1 1:7069z ¹

1:3639 6:3819z ¹ 1+4:7972z ¹

8:551 17:9587z ¹ 1 2:075z ¹

1:2381+2:3154z ¹ 1 4:0635z ¹

b₂(z) 0:889+14:0452z ¹ 1+9:2383z ¹

0:0198 8:9681z ¹ 1 5:833z ¹

2:8813+6:2224z ¹ 1 3:488z ¹

1:2711 8:1046z ¹ 1 8:7701z ¹

a1(z) 0:3897 0:9694z ¹ 1+8:0004z ¹

0:4605+1:7324z ¹ 1 8:5437z ¹

0:6186+0:0819z ¹ 1+8:9331z ¹

0:6363+2:6892z ¹ 1 6:1992z ¹

a₂(z) 1:3199+0:4743z ¹ 1 6:256z ¹

1:1515+2:7807z ¹ 1 7:5671z ¹

1:063+0:0283z ¹ 1 7:9455z ¹

0:4898 4:431z ¹ 1+7:6012z ¹

k (z)kH₁ 0:4876 0:3426 0:3163 0:2220

BER 0:0002 0 0 0

32

v 0:1585 0:1000 0:0631 0:0398

b₁(z) 0:7743+3:5888z ¹ 1 6:2249z ¹

0:0668+6:588z ¹ 1+3:9543z ¹

0:6668+1:5785z ¹ 1+7:2198z ¹

2:5736+3:0289z ¹ 1 4:1923z ¹

b₂(z) 0:0839+8:4572z ¹ 1+9:1101z ¹

1:2583+7:0154z ¹ 1 6:8267z ¹

2:0853+18:9966z ¹ 1+9:8608z ¹

0:2951+3:0626z ¹ 1+8:9557z ¹

a₁(z) 1:2223 4:2696z ¹ 1+9:2383z ¹

0:1541+0:1462z ¹ 1+6:1779z ¹

1:4048+3:3377z ¹ 1 8:0529z ¹

0:687+1:3839z ¹ 1+4:5756z ¹

a₂(z) 0:7028 1:8221z ¹ 1+3:0845z ¹

1:2424+6:8704z ¹ 1+7:7001z ¹

1:3041 1:0443z ¹ 1+6:4599z ¹

1:298+9:9451z ¹ 1 5:0121z ¹

k (z)kH₁ 0:1437 0:1226 0:1158 0:0969

BER 0 0 0 0

Table 1: Optimal eb (z) and ea (z) vs di¤erent v

Now we turn to discuss the power allocation strategy of the proposed algorithm.

Normalized power distribution over the subcarriers is shown in Fig. 4.3. Compared with Fig. 4.1, it is shown that when the noise variance is smaller, the proposed algorithm will allocate more power to those subcarriers with better channel gain, i.e., larger max(k) :On the contrary, when the noise variance is larger, then more power will be distributed over those carriers with worse channel gain.

0 10 20 30 40 50 60 70 2

2.5 3 3.5 4 4.5 5 5.5 6

Maximum eigenvalues vs Frequency

Subcarrier index (k)

Maximum eigenvalue

Figure 4.1: Maximum eigenvalues vs subcarrier for the 2+2 MIMO channel.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10^{- 2} 10^{- 1} 10⁰

MMSE with H∞ norm

σv

H∞

Figure 4.2: Sigma_v values vs H in…nite norm values.

34

0 10 20 30 40 50 60 70

0 0.2 0.4 0.6 0.8 1

Subcarrier index: k

Normalized Power distribution

σ_n=1 and 0.6310

0 10 20 30 40 50 60 70

0.4 0.5 0.6 0.7 0.8 0.9 1

Subcarrier index: k

Normalized Power distribution

σ_n=0.2512, 0.1585, and 0.1

σ_n=1 σn=0.6310

σ_n=0.2512 σ_n=0.1585 σn=0.1

Figure 4.3: Normalized power distribution over the subcarriers.

Conclusions

For MIMO-OFDM systems, the existing power control strategies need to compute power assignment law via a speci…c optimization method for each subcarrier. With the advance of modern communication technology, the OFDM system may have more and more subcarriers in the future. In this case, the computation burden will become more and more heavier. To reduce the possible computation burden, the system model is …rst rewritten by using transfer matrices. Then the constrained power control prob- lem concerning MMSE optimization is transformed into an optimal H₁optimization problem subject to an H2 power constraint. Using state-space representation of the transfer matrices, the proposed constrained H₁optimization problem can be reformu- lated as an linear optimization problem subject to bilinear matrix inequalities (BMI), which can be easily solved by using available BMI toolbox under the numerical soft- ware MATLAB. Through simulation study, it is shown that when the noise variance is smaller, the proposed algorithm will allocate more power to those subcarriers with better channel gain. On the contrary, when the noise variance is larger, then more

35

36 power will be distributed over those carriers with worse channel gain.

In thus study, the target of the power control strategy is the MMSE performance.

In the future, we can consider the power control strategy to maximize the SINR.

In this way, we can compare our proposed algorithm with the existing ones such as Water-…lling, Uniform, HARM, MMSE, MAXMIN and MEPE techniques.

Appendix 1

A.1 SISO-OFDM Model

In this section, we derive an equivalent discrete-time baseband signal model of the OFDM system. In an OFDM system with N subcarriers, St(k)denotes the frequency- domain signal before the modulator and is transmitted via the k-th subcarrier in the t-th symbol interval. The time-domain samples of the t-th OFDM symbol st(l) are produced by IDFT as

s_t(l) = 1 N

NX1 k=0

S_t(k)e^{j2 klN} ; l = 0; :::; N 1 (A.1)

Suppose the interval of an OFDM symbol without the guard interval is de…ned as T, then the sampling time can be de…ned as Ts = T =N: The cyclic pre…x is adopted as the guard interval, which is inserted in front of the N time-domain samples of a symbol to prevent the ISI and to maintain the orthogonality among subcarriers. The time-domain samples with the guard interval, denoted as s^g_t(l), in a total t-th OFDM

37

38 symbol can be expressed as

s^g_t (l) = s_t(l + N G)_N ; 0 l N + G 1 (A.2)

where G denotes the number of samples in the guard interval, and (n)_N denotes the remainder of n divided by N , ie.,(n mod N ). The guard interval is chosen to be larger then the multipath delay spread L of the channel, so the ISI can be eliminated.

As this waveform is transmitted over the multipath channel, the received sampling data y^g_t(l) at the l-th instant of the t-th OFDM symbol can be expressed as

y_t^g(l) = Xl d=0

s^g_t(l d) h^g_t (l; d) + XL d=l+1

s^g_{t 1}(l d + N + G) h^g_t (l; d) + v_t^g(l) (A.3)

where L denotes the maximum delay spread, v^g_t(l) represents the ambient channel noise, and h^g_t(l; d) denotes the equivalent discrete time channel response at position d and instant l. At the receiver end, the samples in guard interval are …rst removed to obtain the signal

y_t(l) = y^g_t (l + G) ; 0 l N 1

= XL

d=0

s_t(l d)_Nh_t(l; d) + v_t(l) (A.4)

The above signal is then fed into the DFT demodulator to obtain the following signal

Y_t(k) =

NX1 l=0

y_t(l) e ^{j2 klN} (A.5)

For further analysis, it is assumed that the multipath channel model is …xed within one symbol time interval, i.e., ht(l; d) = h_t(d). Note that due to the periodic property

e ^{j2 k(lN d)} = e ^{j2 k(lN d)N}, we have

s_t(l d)_N = 1 N

NX1 r=0

S_t(r) e^{j2 r(lN d)N}

= 1

N

NX1 r=0

S_t(r) e^{j2 r(lN d)} (A.6)

Then it follows that

Y_t(k) =

NX1 l=0

( _L X

d=0

s_t(l d)_Nh_t(d) + v_t(l) )

e ^{j2 klN}

= 1

N

NX1 l=0

XL d=0

NX1 r=0

S_t(r) e^{j2 r(lN d)}h_t(d) e ^{j2 klN} + V_t(k)

= 1

N

NX1 l=0

NX1 r=0

S_t(r) e ^{j2 (kN r)l} XL

d=0

h_t(d) e ^{j2 rdN} + V_t(k)

= 1

N

NX1 r=0

S_t(r)

NX1 l=0

e ^{j2 (kN r)l}

!

H_t(r) + V_t(k) (A.7)

where

V_t(k) =

NX1 l=0

v_t(l) e ^{j2 klN}

and

H_t(k) = XL d=0

h_t(d) e ^{j2 rdN} =

NX1 d=0

h_t(d) e ^{j2 kdN}

with the identi…cation ht(d) = 0 for L + 1 d N 1. As PN 1

l=0 e ^{j2 (kN r)l} = N (k r), we then have

Yt(k) = St(k) Ht(k) + Vt(k) (A.8)

A.2 MIMO-OFDM Model

In the previous section, the system model (A.8) is for SISO-OFDM systems. Here we shall consider modeling of an MIMO-OFDM system. Since the channel is linear, so