OFDM無線網路之合作通訊---子計畫四：合作式多使用者多輸入多輸出正交分頻多工系統(II)

行政院國家科學委員會專題研究計畫 成果報告

OFDM 無線網路之合作通訊--子計畫四:合作式多使用者多輸

入多輸出正交分頻多工系統(2/2)

研究成果報告(完整版)

計 畫 類 別 ： 整合型

計 畫 編 號 ： NSC 97-2219-E-009-005-

執 行 期 間 ： 97 年 08 月 01 日至 98 年 07 月 31 日

執 行 單 位 ： 國立交通大學電信工程學系（所）

計 畫 主 持 人 ： 吳文榕

計畫參與人員： 碩士班研究生-兼任助理人員：申沁寧

碩士班研究生-兼任助理人員：莊勝富

碩士班研究生-兼任助理人員：張閔堯

博士班研究生-兼任助理人員：林鈞陶

博士班研究生-兼任助理人員：曾凡碩

博士班研究生-兼任助理人員：許兆元

報 告 附 件 ： 出席國際會議研究心得報告及發表論文

處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 99 年 01 月 07 日

行政院國家科學委員會專題研究計畫成果報告

OFDM無線網路之合作通訊

Cooperative communication for OFDM-based wireless networks

子計畫(四)

合作式多使用者多輸入多輸出正交分頻多工系統

計畫編號：NSC 95-2219-E-009-005

執行計畫：97年8月1日至98年7月31日

主持人：吳文榕教授 國立交通大學電信系教授

Email:wrwu@faculty.nctu.edu.tw

I. Abstract

Existing precoder designs for an amplify-and-forward (AF) cooperative system often assume a linear receiver at the destination, and a precoder at the relay. The performance enhancement of such a system is then limited. In this project, we consider a nonlinear successive interference cancellation (SIC) receiver, and at the same time take the source precoder into consideration. Using the geometric mean decomposition (GMD), we propose a joint source/relay precoders design method, fully exploring information provided by direct and relay links. With our method, the design problem can be transformed to a standard scalar concave optimization problem, and a closed-form solution can be obtained. Simulations show that the proposed design can significantly enhance the performance of a MIMO AF cooperative system.

II. Introduction

Recently, the amplify-and-forward (AF)-based Multi-input-multi-output (MIMO)

cooperative communication (CC) system was proposed in [1]-[4]. With the aid of channel state information (CSI), the precoder can then be designed and applied, either for capacity enhancement [1], [2], or for link quality improvement [3], [4]. For analysis simplicity, these works only consider the design of the relay precoder. The works in [1], [3] and [4] even ignore the transmission of the direct link (channel link from source to destination). In addition, the receiver in the destination is assumed to be linear. To the best of our knowledge, the joint source/relay precoders design for AF-based MIMO-CC systems has not been reported in the literatures. Also, nonlinear receivers at the destination have not been addressed either.

In this project, we aim to propose a joint source/relay precoders design for a QR successive interference cancellation (QR-SIC) receiver. It is well known that when the QR-SIC receiver is adopted, the precoder design using the geometric mean decomposition (GMD) technique in the conventional MIMO system [5], [6] is asymptotically optimal. This motivates us

to consider the application of the GMD technique in our design. Given a channel matrix, one can use the GMD method to derive a precoder making the diagonal elements of the corresponding R matrix equal. However, unlike the conventional MIMO systems, the equivalent channel matrix in an AF-CC system now is a function of the relay precoder, so is the R matrix. Using the GMD approach, we can first derive the source precoder, and reduce the joint design problem to a relay precoder design problem. However, the optimization involves a highly nonlinear function, and a direct solution is difficult to obtain. We then propose a method simplifying the problem as a standard scalar concave optimization problem. With our method, a closed-form solution can be obtained. Simulation shows that the proposed scheme can significantly improve the BER performance as compared to existing schemes。

III. Proposed System Model and Problem Formulation

III-A. Precoders for AF system and QR-SIC receiver

We consider a simple three-node cooperative MIMO AF system (See Figure 1). Under this scenario, signals can be transmitted from the source to the destination (direct link), and from the source to the relay, and then the relay to the destination (relay link) [1], [2]. Let N,

R, and M denote the number of antennas at the

source, the relay, and the destination, respectively. Also, let all channels be flat-fading. The signals received from the source and the relay (at the destination) can be combined into a vector form as [1], [2]: SR H

#

Source: N antennas

#

Relay: R antennas Destination: M antennas SD H RD H

: First time slot : Second time slot

s

y_R F_R

R F_S

Fig. 1: Three nodes MIMO relay system with QR-SIC

receiver. ,1 ,2 : : : , D SD D S RD R R D RD R SR S = = ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ =_{⎢ ⎥} +_{⎢ + ⎥} ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ = + n H n H y F s _{H F n n} H F H HF s n     ，(1)

where is the relay precoder matrix;

, , and

are the channel matrices between the source and the relay, the source and the destination, the relay and the destination,

respectively; , , and

are the noise vector received at the destination in the first-phase, that at the destination in the second-phase, and that at the relay in the first-phase, respectively. Here, we

assume that , and

= R R R ∈ × F ^ R N SR ∈ × H ^ M N SD ∈ × H ^ M R RD ∈ × H ^ 1 ,1 M D ∈ × n ^ n_D,2 ∈^M×1 1 R R∈ × n ^ , L=N ≤R M R_nD,1 ,1 H,1 D D E ⎡_⎢⎣n n ⎤_⎥⎦ = , = = , and 2 n M σ I R_{nD,2 ,2} H_,2 D D E ⎡_⎢⎣n n ⎤_⎥⎦ 2 n M σ I H 2 R =E⎡⎢_⎣ R R⎤⎥_⎦=σn R R n n I L I , where

is a noise variance. Also, the elements of the signal vectors are i.i.d. with a zero-mean and a covariance matrix , where is the power transmitted on a symbol. With the above assumptions, the covariance matrix of the equivalent noise vector is given by

2 n σ 2 s =σs R 2 s σ QR-SIC ˆs

#

( ) 1/Ri i, − l D y SR H

#

Source: N antennas

#

Relay: R antennas Destination: M antennas SD H RD H

: First time slot : Second time slot

s y_R F_R R F_S ˆs QR-SIC

#

( ) 1/R( )i i, 1/Ri i, − l D y

2 2 H n n M H H n RD R R RD n M E σ σ σ ⎡ ⎤ = _{⎢⎣ ⎥⎦} ⎡ 2 ⎢ ⎥ = ⎢ ⎥ ⎢ + ⎥ ⎣ ⎦ R nn I 0 0 H F F H I n ⎤ ⎤ ⎥ ⎥ ⎥⎦ ⎤. (2)

Note that the equivalent noise vector is not white. To facilitate later analysis of QR-SIC receiver, we first apply a whitening operation to the receive vector. Let W be a whitening matrix. From (1), we can have

, (3) : D D S S = = + = + y Wy WHF s W HF s n   _

where and . Due to the whitening, we have =

= . From (2) in (3), we can then obtain the whitening matrix as = H WH n=Wn H E ⎡_⎢⎣nn ⎤_⎥⎦ H H E ⎡⎢_⎣Wnn W ⎥_⎦ 2 2 n M σ I . (4)

(

)

1/2 0 0 M H H RD R R RD M − ⎡ ⎢ ⎥ = ⎢ ⎢ ₊ ⎢⎣ I W H F F H I

The equivalent channel matrix after the whitening process can be reformulated as

(

)

12 SD H H RD R R RD M RD R SR − ⎡ ⎢ ⎥ ⎢ =_⎢ + ⎢⎣ H H H F F H I H F H  ⎤ ⎥ ⎥ ⎥⎦ (5)

From (3), we can see that an AF-CC system can be seen as a MIMO system with the channel matrix defined in (5). However, note that the effective channel matrix in (5) is a function of the relay precoder, and this is quite different from the scenario considered in MIMO systems. Since FR is unknown, FS is not directly

solvable when existing precoder design methods are applied.

It is well-known that nonlinear MIMO receivers can have better performance though

their complexity may be higher. In this paper, we mainly consider the QR-SIC receiver. In such an approach, the equivalent channel of the precoded system is first represented by the QR decomposition, i.e., HF S =QR, where Q is a

2M×2M orthogonal matrix, and R is a

2M×N upper triangular matrix. Equation (3) can then be rewritten as

. (6) H H D D = = + = + y Q y Q QRs Q Rs n     H_n N

Thus, the signal can be detected via a standard QR-SIC procedure.

III-B. Problem formulation

With the QR-SIC as the receiver, [5] and [6] propose a precoder design method such that diagonal elements of R in (6) can be made equal. This method is referred to as the geometric mean decomposition (GMD). It has been shown that [6] the GMD can minimize the block error rate (BLER), and also maximize the lower bound of channel’s free distance. In [5], the GMD detector was proved to be asymptotically optimal for high SNR, in terms of both channel throughput and bit error rate (BER) performance.

Due to its optimality, we then adopt the GMD method in our design. Let have a full rank, i.e., . It was shown in [5] and [6] that can be decomposed as

H ( ) rank H =N H , (8) H = H QRP  

where and are

unitary matrices; the upper triangular matrix has identical diagonal elements given by 2M×2M ∈ Q ^ P ∈^N N× 2M N× ∈ R ^ , for all , (9) 1/ , , 1 N N i i k k r σ = ⎛ _⎞⎟ ⎜ _⎟ = ⎜_{⎜ ⎟⎟} ⎜⎝

∏

H ⎠  i = "1, ,

where is the ith diagonal element in , and is the kth singular value of .

The precoder (at the source) in the GMD method is then determined as , i i r R ,k H P S T t H 0 σ > H , (10) S =α F 

where is a scalar designed to satisfy the power constraint, i.e.,

= . Here, P α

( )

(

H H

)

S S tr F E ss F 2 2 , sN P σ α ≤ S,T is the maximal

available power at the source. Thus, our design problem can then be formulated as

1/N , , , ₁ max . . S R N i i ii k k r αr α σ s = ⎛ _⎞⎟ ⎜ _⎟ = = ⎜_{⎜⎜⎝ ⎟⎟} ⎠

∏

H F F   ,

(

)

(

)

(

)

(

)

2 , 2 2 , , , S H H s S S S T R R R R H H H R s SR S S SR n R R R T tr P tr E tr P F P F F F y y F F H F F H I F α σ σ σ = ⎡ ⎤ ≤ _{⎢⎣ ⎥⎦} = + ≤  (11)

where PR,T is the maximal available power at the

relay. Note here that the cost function in (11) relates to singular values , of which is a complicated nonlinear function of the relay precoder , as shown in (5). A direct maximization of (11) is then difficult. In the next section, we will propose an effective method to solve the precoders and .

,i, σ_H i= "1, ,N H R F S F F_R

IV. Proposed Joint Source/Relay Precoders Design

IV-A. Proposed method

Taking a close look at (11), we see that the optimum at source is actually easy to obtain. From the first two constraints, we can obtain the optimum source precoder, denoted by , as

S F * S F , * 2 S T S s P N σ = F P . (12) 

Alternatively, the optimum , however, is much more difficult to obtain. Substituting into (11), the joint design problem can then be simplified to a relay precoder design problem, as shown below: R F * S F 1/N , , 2 1 , 2 , max . . R N S T k k s S T H H R SR SR n R R P s t N P tr P N σ σ σ = ⎛ _⎞⎟ ⎜ _⎟ ⎜ _⎟ ⎜ ⎟ ⎜⎝ ⎠ ⎛ ⎛ ⎞ ⎞_⎟ ⎜ ⎜ ⎟_{⎟ ⎟} ⎜ ⎜ + ⎟ ⎟≤ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜⎜ ⎟ ⎟⎟ ⎜ ⎝ ⎠ ⎝ ⎠

∏

H F F H H I F  R T . (13) Since singular values of are involved, a direct maximization of (13) may be difficult. We then propose an alternative cost function having the same optimum precoder , i.e.,

H * R F 1/N , * , 2 1 arg max R N S T R k k s P N σ σ = ⎛ _⎞⎟ ⎜ _⎟ = ⎜_{⎜ ⎟⎟} ⎜⎝

∏

H ⎠ F F  , (14) , (15) 2 , 1 = arg max R N k k σ = ⎛ _⎞⎟ ⎜ _⎟ ⎜ _⎟ ⎜ ⎟ ⎜⎝

∏

H ⎠ F  , (16)

(

)

arg max det

R H = F H H   where

(

)

1 H H H H H SD SD SR R RD H H RD R R RD M RD R SR − ⎡ =_⎢⎣ + ⎤ + ⎥ ⎦ H H H H H F H H F F H I H F H   _×

)

. (17)

The equality in (15) is due to the cost functions monotonically increasing property in ;

(16) follows = =

, where is the ith

eigenvalue of . With the cost function in (16), the solution becomes easier to work with.

, 1 N k k σ =

∏

H 2 , 1 N k k σ = ⎛ _⎞⎟ ⎜ _⎟ ⎜ _⎟ ⎜ ⎟ ⎜⎝

∏

H ⎠ , 1 H N k k λ =

∏

_{H H} 

(

det H H H λ_{H H} H _,i H H H 

The following lemma gives a hint regarding how (16) can be solved.

Lemma 1: Let be a positive

definite matrix, and be its th entry. Then, we have N N× ∈ M ^ ( , )i j M ij . (18) ( ) ( ) 1 det N . i i i = ≤

_∏

M M

The equality in (18) holds when M is a diagonal matrix [7]. It turns out that when is diagonalized, the cost function in (16) is then maximized. Unfortunately, from (17) we can see that is a summation of two separated matrices and one of them does not depend on , and the diagonalization is still difficult to conduct. The following lemma suggests a feasible way to overcome the problem.

H H H  H H H  R F

Lemma 2: Let and are two positive definite matrices, then [7] N N× ∈ A ^ N N× ∈ B ^ 1/ 2 1/2

det(A+B)=det( )det(A I_N +A− BA− )

(19)

From (16) and (19), we can see that if we let = and B = , we will have A H SD SD H H H H H SR R RD H F H

(

_{H H}

)

1 RD R R RD M − + H F F H I HRD RF HSR

)

H

(

* _{arg max det} R H R = F F H  (20) 1/2 1/2

= arg max det( )

R

N + − −

F I A BA

where is ignored since it is not a function of . From (20), we see that as long as is diagonalized, ( ) det A R F 1/2 1/2 − − A BA H H H will be

diagonalized. This suggests a precoder structure as described in next subsection.

IV-B. Optimal relay precoder design

Now, the optimization in (13) can be restated as follows ( )

(

)

(

(

)

(

)

)

1/2 1 1 , 2 , maxdet where . . is diagonal and . R H H H H N SD SD SR R RD H H H RD R R RD M RD R SR SD SD S T H H R SR SR n R R RT s t P tr P N σ − − − = + × + ⎛ ⎛ ⎞ ⎞_⎟ ⎜ ⎜ ₊ ⎟_{⎟ ⎟≤} ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎝ ⎠ ⎝ ⎠ F M M I H H H F H H F F H I H F H H H M F H H I F /2 R N (21)

The diagonalization requirement motivates us to consider the following singular value decomposition (SVD) ; (22) H RD = rdΣrd rd H U V , (23)

(

)

1/2 : H SR SR SD SD H sr sr sr − ′ = ′ ′ ′ = Σ H H H H U V

where and are the

left singular vectors of and , respectively; and

are the diagonal singular-value matrices of and , respectively;

and are the right singular vector matrices of and , respectively. To have a full diagonalization of M, it turns out that the optimum have the following structure

M M rd ∈ × U ^ R R sr′ ∈ × U ^ RD H H_SR′ M R rd × Σ ∈ \ R N sr′ × Σ ∈ \ RD H H_SR′ H R rd ∈ × V ^ H N sr′ ∈ × V ^ RD H HSR′ * R F , (24) H R = rdΣr sr′ F V U

where is a diagonal matrix with its ith diagonal element, , yet to be determined. Let and be the ith diagonal element of and , respectively. Substituting (22), (23) and (24) into (21) and taking the log operation to the cost function, we can rewrite (21) as: r Σ , r i σ , rd i σ σ′_{sr i,} rd Σ Σsr′

( ) , 2 2 , , , 2 , 1 _{1 , ,} , 2 2 , , , , 1 max ln 1 1 . . , , r i N r i rd i sr i p _{i N i r i rd i} N _{S T} r i sr i sr n R T r i i p p s t P p i i P p N σ σ σ σ σ ≤ ≤ ₌ = ⎛ _⎞⎟ ⎜ _⎟ ⎜ + ⎟ ⎜ _⎟ ⎜ + ⎟ ⎜⎝ ⎠ ⎛ _⎞⎟ ⎜ _{′ ′ + ⎟≤ ≥} ⎜ _⎟ ⎜ ⎟ ⎝ ⎠

∑

∑

D 0. ′ N (25) where and

. The cost function now is

reduced to a function of scalars. Since the cost function and the inequality constraints are all concave [8], (25) becomes a standard concave optimization problem. As a result, the optimal solutions , , can be solved by means of Karush-Kuhn-Tucker (KKT) conditions. After some tedious derivations, we can obtain 2 , , r i r i p =σ D_sr′ =

(

)

H H sr′ SD SD sr V H H V , r i p i = "1, ,

(

)

(

)

(

)

, , 2 2 2 2 , , , 2 , 2 2 2 4 2 , , , , ( , ) 1 1 1 1 ₂ , (26) 1 4 1 r i S T rd i sr i sr n sr i sr i rd i sr i rd i sr i p P i i N µ σ σ σ σ σ σ σ σ σ − + − ⎡ ⎢ ⎢ ⎢ = ⎢_⎢ ⎛_{⎜ ⎞⎟} ′ ′ + ⎟ ′ + ⎜ _⎟ ⎢ _⎜⎝ ⎟_⎠ ⎢⎣ ⎤ ′ + ⎥ ⎥ + − _⎥ ′ + ′ + ⎥ ⎥⎦ D

where [ ]y+ =max 0,( y and is chosen to ) satisfy the power constraint in (25). Substituting (26) into (24), we can then obtain the optimum relay precoder. Finally, substituting (24) into (5) and conducting the decomposition in (8), we can then obtain the optimum source precoder via (12).

µ

V. Simulations and Conclusions

We consider N=R=M=4 case. Assume that channel state information (CSI) of all links are known at all nodes, and perfect synchronization can be achieved. Furthermore, the elements in

each channel matrix are assumed to be i.i.d. complex Gaussian random variables with a zero mean and a same variance. We let the received SNR at each antenna of the relay in the first-phase, and that at each antenna of the destination in the second phase be 15 dB, and vary the received SNR at each antenna of the destination in the first-phase. Also, the modulation scheme is QPSK.

Fig. 2 shows the BER comparison for three un-precoded receiver schemes, the optimal relay precoder with MMSE receiver [4], and for two precoded QR-SIC receivers which are the conventional GMD precoder [5], [6] and the proposed precoding scheme. As shown in the figure, the relay-only precoded systems outperform the un-precoded ones (except for MMSE-OSIC). This is because the amplified signal from the relay can somewhat benefit the receiver. The proposed scheme has significant performance improvement compared to the other schemes. Particularly, it outperforms the conventional GMD approach since the proposed method not only makes the diagonal elements of

R in (6) equal but also maximizes the values,

yielding a higher received SNR for each transmitted symbol stream.

0 5 10 15 10-7 10-6 10-5 10-4 10-3 10-2 10-1 SNR (dB) BE R Un-precoded ZF Un-precoded MMSE Optimal linear relay precoder [4] Un-precoded MMSE-OSIC GMD at source Proposed joint precoder

Fig. 2. BER performance for proposed method and other

VI. References

[1] X. Tang and Y. Hua, “Optimal design of non-regenerative MIMO wireless relays,” IEEE Trans.

Wireless Communications, vol. 6, no. 4, pp. 1398-1407,

April 2007.

[2] O. Munoz-Medina, J. Vidal, and A. Agustin, “Linear transceiver design in nonregenerative relays with channel state information,” IEEE Trans. Signal Processing, vol. 55, no. 6, pp. 2593-2604, June 2007.

[3] N. Lee, H. Park, and J. Chun, “Linear precoder and decoder design for two-way AF MIMO relaying system,”

IEEE VTC 2008 Spring, pp. 1121-1125, May 2008.

[4] W. Guan and H. Luo, ”Joint MMSE transceiver design in non-regenerative MIMO relay systems,” IEEE Trans.

Communication Letters, vol. 12, no. 7, pp. 517-519, July

2008.

[5] Y. Jiang, J. Li, and W. W. Hager, “Joint transceiver design for MIMO communications using geometric mean decomposition,” IEEE Trans. Signal Processing, vol. 53, no. 10, pp. 3791-3803, Oct. 2005.

[6] J. K. Zhang, A. Kavcic, and K. M. Wong, “Equal-diagonal QR decomposition and its application to precoder design for successive-cancellation detection,” IEEE

Trans. Inf. Theory, vol. 51, no. 1. pp. 154-172, Jan. 2005.

[7] D.S. Bernstein, Matrix Mathematics, Princeton University Press 2005.

[8] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press 2004.

行政院國家科學委員會補助國內研究生出席國際學術會議報告

97 年 12 月 20 日

報告人姓名

林鈞陶

_{（科系所）}

就讀校院

■博士班研究生

國立交通大學電信工程研究所

□碩士班研究生

時間

會議

地點

2008/12/15—2008/12/17

國家：澳大利亞

城市：黃金海岸

本會核定

補助文號

會議

名稱

(中文) 訊號處理暨通訊系統國際研討會

(英文)

2nd International Conference on Signal Processing and

Communication Systems (ICSPCS’2008)

發表

論文

題目

(中文)使用展開投影技術之低複雜度多輸入多輸出偵測方法

報告內容應包括下列各項：

一、參加會議經過

這次會議我於台灣時間 2008 年 12 月 12 日晚上十點出發，於澳洲時間 2008 年 12 月 13 日早午九

時左右抵達澳洲布里斯本國際機場，由於會議地點黃金海岸距離機場約有兩小時車程，故抵達會

議地點時已近傍晚。前兩天主要是準備會議中口頭報告之內容，此會議於 12 月 15 日開始，第一

天早上主要是一些專題的演講，提供最新的相關領域資訊。下午開始則是進行論文口頭報告以及

海報展示。主要為通訊訊號處理相關的研究成果發表，與會者可以聽取學者發表他們近期的研究

成果，得知目前通訊系統在軟硬體上相關產業的發展現況，作為日後研究方向的參考。我的報告

也在 18 日下午順利完成，並於 19 日一早搭機返台。

二、與會心得

ICSPCS＇2008 的會議議程包含許多領域的研究，發表的論文大多是訊號處理的相關的軟硬體研究

成果，我大多是選擇參與關於通訊訊號處理方面的議程，基本上是關於 OFDM (orthogonal

frequency-division multiplexing) 及 MIMO (multiple input multiple output)系統的研究。

由於現今通訊系統都逐漸走向 MIMO-OFDM 的架構來提升傳輸速率，因此也有許多關於應用於

MIMO-OFDM 前置編碼的前瞻性研究在此會議中提出討論。吾人這次參加會議報告的主題是有關於

MIMO 系統中低複雜度訊號偵測方法，此次會議讓吾人思考日後研究方向或許可以更進一步地將前

置編碼應用於我們所提出之低複雜度接收機。此次報告我採用口頭報告方式，而其中報告過程順

暢，報告後有許多研究者提問並給予建議，對我們團隊的研究成果深感興趣。

三、考察參觀活動(無是項活動者省略)

四、建議

希望未來台灣也能爭取類似國際學術會議之主辦

五、攜回資料名稱及內容

ICSPCS’ 2008 光碟內有收錄被接受之論文以及隔年 ICSPCS’2009 之徵稿資訊

六、其他