合作式放大傳遞多輸入多輸出中繼系統之強健性Tomlinson-Harashima來源端與線性中繼端前置編碼設計

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(1)國立交通大學電信工程研究所碩士論文合作式放大傳遞多輸入多輸出中繼系統之強健性 Tomlinson-Harashima 來源端與線性中繼端前置編碼設計 Robust Tomlinson-Harashima Source and Linear Relay Precoders Design in Amplify-and-Forward MIMO Relay Systems. 研究生：張閔堯指導教授：吳文榕博士. 中. 華. 民. 國 99. 年. 7. 月.

(2) 合作式放大傳遞多輸入多輸出中繼系統之強健性 Tomlinson-Harashima 來源端與線性中繼端前置編碼設計 Robust Tomlinson-Harashima Source and Linear Relay Precoders Design in Amplify-and-Forward MIMO Relay Systems 研究生：張閔堯. Student：Min-Yao Chang. 指導教授：吳文榕博士. Advisor：Dr. Wen-Rong Wu. 國立交通大學電信工程研究所碩士論文. A Thesis Submitted to Institute of Computer and Information Science College of Electrical Engineering and Computer Science National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Master in Computer and Information Science July 2010 Hsinchu, Taiwan, Republic of China. 中華民國九十九年七月.

(3) 合作式放大傳遞多輸入多輸出中繼系統之強健性 Tomlinson-Harashima 來源端與線性中繼端前置編碼設計. 研究生：張閔堯. 指導教授：吳文榕博士. 國立交通大學電信工程研究所碩士論文摘要. 在現存的合作式放大傳遞多輸入多輸出中繼系統傳收機的設計，通常假設此系統可得到完美的通道狀態資訊(perfect CSI)。但在實際系統應用上，可能無法獲得完美的通道狀態資訊。基於非完美通道狀態資訊的考量下，強健性的傳收機設計在實際應用上是需要被考慮的。在本論文中，我們提出一種強健性的傳收機設計，此傳收機中的來源端使用 Tomlinson-Harashima 前置編碼 (THP)、線性中繼端前置編碼與最小均方錯誤 (minimum mean-squared error)接收機。當兩個前置編碼的組合及非完美通道狀態資訊被考慮進來時，傳收機的設計變得相當困難。為了克服這個設計上的困難，我們提出一種前置編碼結構與設計方法，使得原本的傳收機設計可轉換為凹曲線最佳化問題，由此導出解析解。我們在 TH 前置編碼後串接一個單位前置編碼。這個額外的前置編碼的功能不僅可以簡化最佳化的問題而且可改善整個系統的效能表現。由於最佳化的問題是由多個前置編碼組成，我們使用最初分解(primal decomposition)將原本的最佳化問題分解成次要問題(subproblem)與主要問題(master problem)。依序解決次要問題與主要問題，原本由兩個前置編碼構成的問題，可簡化成設計單一中繼端前置編碼的問題。但是要解決主要問題仍然相當困難，因此我們對這個主要問題提出一個最低界線，並經由一些操作將此問題轉換成凹曲線最佳化的形式。透過 Karush-Kuhn-Tucker(KKT)條件可以推導出解析解。模擬的結果顯示在完美/非完美的通道狀態資訊環境下，所提出的強健性傳收機設計在效能上的表現比現存的線性傳收機設計好。. i.

(4) Robust Tomlinson-Harashima source and linear relay precoders design in amplify-and-forward MIMO relay systems. Student:Min-Yao Chang. Advisor: Dr. Wen-Rong Wu. Institute of Communication Engineering National Chiao Tung University. Abstract. The. existing. transceiver. design. in. amplify-and-forward. (AF). multiple-input. multiple-output (MIMO) relay systems often assume the perfect channel state information (CSI) is available. In practice, the perfect CSI is not attainable and the robust design with considering imperfect CSI is applied. In this paper, we propose a robust transceiver design for the system with Tomlinson-Harashima (TH) precoder, a linear relay precoder and a minimum-mean-square (MMSE) receiver. Since two precoders and the imperfect CSI are involved, the robust design problem is difficult. To overcome the difficulty, we additionally cascade a unitary precoder after the TH precoder. The unitary precoder can not only simply the optimization problem but improve the performance of the system. With the precoders, we use the primal decomposition to divide the original optimization problem into a subproblem and a master problem. The subproblem can be solved and the two-precoder problem can be reduced to the problem composed of single relay precoder. However, the master problem is still difficult to solve. We then proposed a lower bound for the cost function and transfer the master problem to a convex optimization problem. A closed-form solution can then be obtained by Karush-Kuhn-Tucker (KKT) conditions. Simulations show the that the proposed transceiver design have better performance than existing linear transceiver with either perfect/imperfect CSI.. ii.

(5) Acknowledgement. 在完成本論文的過程中，感謝所有幫助過我的人。感謝我的指導教授 - 吳文榕教授，感謝文榕老師在學業上的指導及提供專業的研究建議，使我明白學術研究的基礎需要正確的技巧與態度；在遇到未來規劃上的困難時，文榕老師也能適時分享經驗及想法，這些都令我受益匪淺。感謝我的父母親與家人，他們在精神上對我的支持給我很大的勇氣與毅力完成學業。感謝 720 實驗室的所有同仁，兩年的研究所期間從他們的身上學到很多東西，也共同經歷了多事情。感謝凡碩、鈞陶、兆元、弘道、俊芳、其翰、南鈞、光敏等學長們，每當我遇到難題，請教他們總是可以獲得許多學習的機會，也時常與他們討論社會上或其他領域上的議題，讓我從他們的人生閱歷中得到一些知識。特別感謝凡碩學長，由於他在學術及生活上的經驗分享，我學到做人處事應有積極的態度與熱誠，並瞭解人生有許多難題需要耐心面對，也有許多樂趣等待發掘。感謝勝富、沁寧、瑞慶、綺瑩等同學，感謝你們的陪伴並一起度過兩年的研究生活。感謝所有這兩年中認識的老師、朋友們，與你們的互動讓我在各方面成長很多。感謝所有幫助過我的人，此為誌謝。. iii.

(6) Contents. Chinese abstract.........................................................................................................................i Abstract .....................................................................................................................................ii Acknowledgement....................................................................................................................iii Contents....................................................................................................................................iv List of Tables.............................................................................................................................v List of Figures ...........................................................................................................................v 1 Introduction ........................................................................................................................1 2 Joint MMSE Transceiver Desgin with Tomlinson-Harashima Source and Linear Relay Precoders ..................................................................................................................4 2.1 System model ...............................................................................................................4 2.2 Problem formulation.....................................................................................................7 2.3 Joint source and relay precoders design .......................................................................8 3 Robust Joint MMSE Transceiver Desgin with Tomlinson-Harashima Source and Linear Relay Precoders....................................................................................................15 3.1 System model .............................................................................................................15 3.2 Problem formulation...................................................................................................19 3.3 Robust joint source and relay precoder design...........................................................19 4 Simulation results and discussions..................................................................................29 4.1 Simulation Setup ........................................................................................................29 4.2 Simulation results and discussions .............................................................................30 5 Conclusions .......................................................................................................................37 Appendix .................................................................................................................................39 Appendix A : Optimal feed-back matrix B [37]...............................................................39 Appendix B: MSE in (3.7)................................................................................................40 Appendix C: To prove the cost function (3.22) is monotonically decreasing in α S ≥ 0 .41 Appendix D: To prove the property in (3.42)...................................................................43 Appendix E: Derivation of optimum solution in (3.53) ...................................................44 References................................................................................................................................46. iv.

(7) List of Tables. Table 2.1: Precoding loss γ p2 (in dB) of Tomlinson-Harashima Precoding [21]. ....................6 Table 3.1: Computational complexity of THP source and linear relay precoders (MMSE receiver) ..................................................................................................................27 Table 3.2: Computational complexity of robust THP source and linear relay precoders (MMSE receiver)....................................................................................................28. List of Figures. Figure 2.1: THP source and linear relay precoded AF MIMO relay system with MMSE receiver ................................................................................................................ 14 Figure 4.1: MSE performance comparison for existing precoded systems and proposed robust/non-robust TH source and linear relay precoded system. (All with MMSE receiver.) ...............................................................................................................32 Figure 4.2: BER performance comparison for existing precoded systems and proposed robust/non-robust TH source and linear relay precoded system. (All with MMSE receiver.) ...............................................................................................................33 Figure 4.3: MSE performance comparison for existing precoded systems and proposed robust/non-robust. TH. source. and. linear. relay. precoded. system.. ( δ = γ = 0 , σ e2 = 0.003 ) .............................................................................................33 Figure 4.4: BER performance comparison for existing precoded systems and proposed robust/non-robust. TH. source. and. linear. relay. precoded. system.. ( δ = γ = 0 , σ e2 = 0.003 ) .............................................................................................34 Figure 4.5: MSE performance comparison for existing robust/non-robust relay precoded systems and proposed robust/non-robust TH source and linear relay precoded. v.

(8) system. ( δ = γ = 0 , σ e2 = 0 / σ e2 = 0.003 )......................................................................34 Figure 4.6: BER performance comparison for existing robust/non-robust relay precoded systems and proposed robust/non-robust TH source and linear relay precoded system. ( δ = γ = 0 , σ e2 = 0 / σ e2 = 0.003 )......................................................................35 Figure 4.7: MSE comparison for proposed robust precoded system with different δ . ( γ = 0 , σ e2 = 0.002 ) ...................................................................................................35 Figure 4.8: MSE comparison for proposed robust precoded system with different σ e2 . ( δ = γ = 0 ) ..............................................................................................................................36. vi.

(9) Chapter 1 Introduction. In recent years, many works have been devoted to the study of cooperative communication due to its great potential to improve coverage, capacity and reliability of wireless link [1]-[3]. Due to shadowing, multipath fading, path losses observed in wireless channels, the link between a signal source and a destination may not be always reliable. In cooperative communication systems, relays are placed at some strong shadowing environments such that signals can be transmitted to the destination by a direct link and relay links. With the additional relay links, it provides the enhancement of diversity gain or capacity gain [1]-[18]. There are some relay strategies such as amplified-and-forward (AF), decode-and-forward (DF), and compress-and-forward(CF) [1],[2]. In AF, the relays receive signal from the source and retransmit it to the destination with signal amplification only. In DF, the relays decode the received signal, re-encode information bits, and then transmission the resultant signal to the destination. CF, being a compromise between AF and DF, estimates information bits, compress their information, and then transmit the modulated signal to the destination. In this thesis, we only consider the AF-based system since the AF strategy requires less implement complexity and smaller processing delay. Research in this subject has attracted a lot of attention. MIMO systems have been widely studied in the literature since it can enhance spatial diversity or multiplexing gain in the rich scattering environments. It is known that a precoder can be used in a MIMO system to further enhance the performance. Linear or non-linear precoder designs in the point-to-point MIMO system have been extensively studied [7]-[9]. In cooperative systems, multiple antennas can be equipped at the source, the relays, and the destination, resulting MIMO relay systems. Similar to conventional MIMO systems, the precoding operation can be conducted in a MIMO relay system [12]-[19]. In this thesis, we only consider the transceiver design in AF MIMO systems. For the AF MIMO relay system, the capacity bound of the systems have been derived in [12]. Apart from the capacity, the link quality is another criterion has been studied. In [15], a dual-hop single relay precoder in an AF MIMO system was designed for a minimum mean-square error (MMSE) receiver without considering direct link (source to destination). It. 1.

(10) has been shown in [15] that the joint design has a better performance than separate design scheme. In [16], a joint transceiver design for the multi-relay case has been discussed. In [18], a transceiver design has been proposed for a three-node AF MIMO system using MMSE criterion. This scheme takes both the direct and relay links into consideration, and uses a linear precoder at the source and another linear precoder at the relay. Recently, the joint source and relay precoders design for multiple transmission streams were studied in [17]-[20], where the source and relay precoders are jointly designed with the direct and relay links via MMSE [17], QR successive-interference-cancellation (SIC) [19], and MMSE-SIC [20], respectively. As well known, the nonlinear transceivers are superior to linear transceivers. To obtain better performance for the precoded system, we then focus on a nonlinear source precoder design in this thesis. There exist several nonlinear MIMO transceivers, e.g. the system with a Tomlinson-Harashima precoder (THP), and that with a decision-feed-back equalizer (DFE). As well known, the DFE at the destination estimate and cancel the interferences and it may cause error propagation in low SNR environments. We consider a precoded AF MIMO relay system in which the nonlinear THP is used at the source, a linear precoder at the relay and a MMSE receiver at the destination [22]. The THP can pre-cancel the known interference at the source and will not induce error propagation. It is widely used in point-to-point or multiuser MIMO systems [21]. In the first part of the thesis, we consider a transceiver design in AF MIMO relay system. The design uses the MMSE criterion and take both the direct and relay links into consideration. Since the THP is involved, the cost function becomes a highly nonlinear and complicated function of the source precoder and relay precoder. The optimization problem becomes very difficult. To solve the problem, we first cascade a unitary precoder with the THP. The specially designed unitary precoder not can only simplify the optimization problem but also improve the MMSE performance. Then, we use the primary decomposition method, decomposing the problem into a master problem and a subproblem. With our formulation, the subproblem problem, designing the unitary precoder and feed-back matrix in the THP, can be degenerated to the system in [24], and the solution is readily obtained. In the master problem, the cost function becomes a function of the relay precoder only. With some precoder structure, we can translate the optimization problem from matrix-valued into a scalar-valued optimization problem, and use Karush-Kuhn-Tucker (KKT) conditions to obtain a closed-form solution for the source and relay precoders. Most transceiver design in AF MIMO system assume that it knows perfect channel state information (CSI) of each link at each node [14]-[20]. In practice, the perfect CSI is not. 2.

(11) attainable due to channel estimation or quantization errors. For conventional MIMO systems, some works study the sensitivity of the MIMO precoder with respect to channel uncertainties [29], [30]. In [31]-[35], a robust design for the THP precoded point-to-point MIMO system has been studied. In [36], the design is extended to an AF MIMO relay system. In the design, the direct link is not taken into consideration. In the second part of this thesis, we study a robust AF MIMO transceiver design with the THP. The optimization problem is similar to that of perfect CSI. The only difference is to consider the estimation errors as extra noise sources. Still, we use the primal decomposition method to decompose the optimization problem into subproblem and master problem. To ease the optimization, we then propose a method that can translate the master problem to a standard scalar-valued concave optimization. The key idea is to apply some approximations for the cost function such that the optimization in the master problem can be solved. We then propose a relay precoding structure in the optimization. Though the structure is suboptimal method, however, it can translate the master optimization problem to a standard scalar-valued concave optimization problem. Finally, similar to the perfect CSI case, we can obtained the close-form solution of the relay and source precoder by using KKT conditions. The organization of the thesis is described as follows. In Chapter 2, we describe the proposed THP precoded AF MIMO relay system. In Chapter 3, we take the channel estimation error into consideration and propose a robust transceiver design. In Chapter 4, we present the simulation results and related discussions. Finally, we draw conclusions in Chapter 5.. 3.

(12) Chapter 2 Joint MMSE Transceiver Desgin with Tomlinson-Harashima Source and Linear Relay Precoders. In this chapter, we consider the MMSE transceiver design with a nonlinear Tomlinson-Harashima precoder (THP) at the source, a linear precoder at the relay in AF MIMO relay systems. Here, we assume that perfect CSIs of all channels are known at the destination. In Section 2.1, we first give the system model, while in Section 2.2 we formulate the design problem under the MMSE criterion. It is found that the MSE is a complicated function of the source and relay precoders, and the optimization problem is non-convex. Thus, the problem is difficult to solve. In Section 2.3, we propose a method translating the two-precoder design problem into a single-relay problem. By using this method, the optimization problem can be formulated as a convex optimization problem, and the close-form solution can then be obtained.. 2.1 System model We consider a typical three-node AF MIMO relay system with a THP. The block diagram of the system is shown in Figure 2.1. The system includes a THP precoder cascaded with a unitary precoder FS at the source, a linear precoder FR at the relay, and a MMSE receiver G at the destination. Here, we define the number of antenna at the source, the relay and the destination as N, R and M, respectively. The MIMO channels are assumed to be flat fading. In this cooperative system, we use a half-duplex relay protocol, which means it require a two-phase transmission for a data packet. Let’s start with the THP. The THP conducts a interference pre-cancelling operation characterized by a backward, strict low-triangular matrix B and a modulo operation MOD m ( i ) . Let the input signal vector be s ∈ ℂ N ×1 ; each element. of s = [s1, ⋯, sN ]. T. is a symbol mapped to a square m-QAM constellation where. Each QAM. {. {. symbol is drawn from the set A = sI + jsQ sI , sQ ∈ ±1,… , ± m. }} . The feed-back operation. conducted in THP may increase the transmit power, and it can be avoided by a modulo. 4.

(13) operation [21]. The modulo operation applied over both the real and image parts of the input x is expressed as: x + m  MODm (x ) = x − 2 m ⋅   2 m . (2.1). With B and the modulo operation, the elements of x can be expressed as: xk = sk −. k −1. ∑ B(k, l )xl. l =1. + ek. (2.2). where xk is the kth element of vector x, x ∈ ℂ N×1 , B(k , l ) is the (k,l) element of matrix B, and e = [e1, ⋯, eN ]T denotes the errors caused by the modulo operation. Then, (2.2) can be rewritten with the following matrix form as: x = C −1 v. (2.3). where C = B + IN is a lower triangular matrix and v = s + e [21]. The transmission in the cooperative system has two-phase [1]. In the first phase, the THP precoded signal x is passed through the cascaded unitary precoder FS , and subsequently send to the relay and the destination simultaneously. As we will show, an appropriate design of the additional unitary precoder will improve the performance of the MIMO relay system. In the second phase, the received signal at the relay is multiplied by the relay precoder. FR and the resultant signal is then transmitted to the destination. Therefore, the signal received at the destination after the two consecutive phases can be expressed as a vector form as [17]-[20], [22]:  HSD  yD :=   FS x + H F H  RD R SR   := H. n D,1   H F n + n  RD R R D ,2   . (2.4). := w. where H and w denote the equivalent channel matrix and the equivalent noise vector, respectively. In (2.4), x ∈ ℂN ×1 is the THP precoded signal vector (2.3); yD ∈ ℂ2M ×1 is the received signal vector at the destination; HSR ∈ ℂR ×N , HSD ∈ ℂM ×N and HRD ∈ ℂM ×R are the channel matrices of the source-to-relay, the source-to-destination, and. the. relay-to-destination channels, respectively. Note that these channel matrices are all assumed to be flat fading channel; nD,1 ∈ ℂM ×1 , nR ∈ ℂR ×1 , and nD,2 ∈ ℂM ×1 are the received noise vectors at the destination, at the relay in the first-phase, and at the destination in the second-phase. Here, we assume that N ≤ M such that sufficient degree of freedom for signal transmission can be assured.. 5.

(14) Note that if v can be estimated at the destination, s can be reconstructed by the modulo operation. We define the mean-square-error of the estimation as:. {. J = E GyD − v. 2. }.. (2.5). By minimizing the MSE, we can derive the optimum G. The signal elements of s is assumed to be independent each other and the variance is σ s2 . It has been shown in [21] that each. element. {. of. x. is. approximately. {. region A = sI + jsQ sI , sQ ∈ ±1,… , ± m. }}. i.i.d. and. distributed. in. the. uniformly. The approximation error becomes. small as the number of signal levels is large. Thus, it is valid only when m is sufficiently large. With the approximation, we have E xxH  = σ s2 IN , E  vvH  = σ s2CCH . The modulo operation in the THP may cause a transmit power penalty, called precoding loss. According [21], precoding loss for a two-dimentional m-ary square constellations can be calculated as: E  xk γ =  E  sk  2 p.  = m 2  m −1  2. (2.6). where xk , sk indicates the k-th element of signal vector x , s , respectively. We show the precoding loss for various m in Table 2.1. As we can see, the precoding loss is negligible and vanishes completely as m goes to infinity. It is recommended that at least m ≥ 16 should be used.. Table 2.1: Precoding loss. γ p2. (in dB) of Tomlinson-Harashima Precoding [21].. Taking the derivative the MSE with respect to G and setting the result to zero, we can obtain the optimum G as [24]:. {. }. ∂J ∂ H = E Tr ( Gy D − v )( Gy D − v )  H H   ∂G ∂G = G ⋅ E  y D y DH  − E  vy DH  = 0. (2.7). Then, the optimum G, denoted by Gopt, is given by. (. Gopt = σ s2CFSH HH σ s2 HFS FSH HH + Rw. 6. ). −1. (2.8).

(15) where Rw = E  wwH  is the covariance matrix of the equivalent noise. Note that the equivalent noise is colored. Denote the variance of the noise at the relay as σ n2,r , and that at the destination as σ n2,d . Substituting (2.8) into (2.5), we obtain the minimum MSE. To simplify the expression of the MSE, we consider the following leamma: Lemma 2.1: Matrix inversion lemma [42] A H ( AA H + I ) A = I − ( A H A + I ) −1. −1. (2.9). where A, I denote matrices with appropriate size. Using the lemma, we can rewrite the error matrix in (2.5) as. ( ) CH ɶ H HF ɶ )−1 CH = C (σ s−2IN + FSH H S. E = C σ s−2IN + FSH HH Rw−1HFS. −1. (2.10). and (2.5) becomes J min = Tr {E}. (2.11). where. ɶ = R −1/ 2H H w  σ n−,1d HSD ＝  σ 2 H F FH HH + σ 2 I n ,d M  n ,r RD R R RD. (. ).   −1/ 2 HRD FR HSR  . (2.12). is defined as the equivalent channel matrix after noise whitening. Note that the MSE is contributed by both the relay link and direct links. If we ignore the direct link, the MSE will be reduced to that in [24]. From (2.10) and (2.11), we see that the achievable minimum MSE is a complicated function of C , FS and FR . In the next section, we formulate the precoders design problem using the signal model derived in this section.. 2.2 Problem formulation For the MIMO relay system, two precoders are involved. Using (2.5), (2.8)-(2.12), we can formulate the precoders design problem as:. 7.

(16)     − 1 H ɶ H HF ɶ min Tr C σ s−2IN + FSH H C  S C, FS , FR   :=E ( C, FS , FR )   s.t.. (. ). (2.13). FS =α US ,C 1,C 2. where ɶ HH ɶ = σ −2 HH H + H n ,d SD SD. (. H H H HSR FRH HRD + σ n2,d IM σ n2,r HRD FR FRH HRD. ). −1. (2.14) HRD FR HSR. and C 1 , C 2 denote the power constraints at the source and the relay, respectively:. { : Tr {F (σ. }. {. } 2 2 H H H n ,r IR + σ s HSR FS FS HSR ) FR } ≤ PR,T .. C 1 : Tr E  FS xxH FSH  = σ s2 Tr FS FSH ≤ PS ,T , C2. R. (2.15). Here we let FS = α U S in which α is a scalar and U S is a unitary matrix. In next section, we show that the unitary structure can facilitate the derivation of the optimization problem and improve the performance of the MIMO rely system. From (2.13), it is apparent that both the cost function and the constraints are complicated function of FS and FR . Yet, the problem is non-convex. Solving such a problem is a very difficult problem, if not impossible. In the next section, we propose a method to overcome the problems.. 2.3 Joint source and relay precoders design Since a direct solution for optimum FS and FR in (2.13) is difficult, we use the primal decomposition method [42] such that the problem can be translated into a subproblem and a master problem and FS and FR can be solved separately. In the subproblem, the relay precoder is assumed to be known and the source precoder is solved as a function of the relay precoder. Then, in the master problem, the relay precoder is solved. The problem now can be re-formulated as. 8.

(17) min Tr {E} = min min Tr {E(C, FS , FR )}. C,FS , FR. FR. C, FS. s.t.. (. ɶ H HF ɶ E = C σ s−2IN + FSH H S FS =α US ,. {. ). −1. CH ,. }. C 1 : σ s2Tr FS FSH ≤ PS ,T ,. { (. (2.16). ) }. H C 2 : Tr FR σ n2,r IR + α 2σ s2 HSR HSR FRH ≤ PR,T .. In the subproblem, as mentioned, the optimum C and FS are derived as a function of. FR by assuming FR is given. Then the joint precoder design problem is reduced to the determination of FR which is the master problem. The unitary precoder FS is included for two reasons: (i) It can simply the solutoin of the relay precoder. (ii) By a proper design of US , the minimum MSE can have an amenable form, leading to a tractable optimization problem. Since FS =α US , the subproblem becomes the optimization of α , US and C , given as. min. ( ). ( ). C FR ,α US FR. Tr ( E(C, α US , FR )). s.t.. (2.17). α US ,C 1′,C 2. (. ɶ H HU ɶ E = C σ s−2IN + α 2 USH H S. ). −1. CH. where C1′ : Nα 2σ S2 ≤ PS ,T is obtained from C1 by setting FS to be unitary. If we fix US and C in (2.17) , we can find that the trace of MSE matrix is a decreasing function of α . So under the transmission power constraint, we can have the optimum α as αopt =. PS ,T. N σ s2. to. minimize the MSE. We substitute α opt into constraint C2 in (2.17) and find that C2 is not a function of the source precoder. So, we can just consider it in the master problem. Thus the subproblem becomes: −1    PS ,T H H  H H −2 ɶ HU ɶ  min Tr  C  σ s IN + U H C  S    C ( FR ), US ( FR ) N σ s2 S   . (2.18). With a known relay precoder, the problem (2.18) is similar to the THP design in the conventional MIMO system, and the optimum solution C, denote as Copt , has been solved as [24]. 9.

(18) Copt = DL−1. (2.19). where PS ,T H H   ɶ ɶ LL =  σ s−2 IN + 2 US H HU S  N σs  . −1. H. P   ɶ H HU ɶ is the Cholesky factorization of  σ s−2 IN + S ,T2 USH H S  N σs  . (2.20) −1. while D is a diagonal. matrix that scales each element on the main diagonal of C to unity. (The proof is summarized in Appendix A). Substituting (2.19), (2.20) into (2.18), we then have J min.   −1 H  PS ,T H H   −2 ɶ ɶ = Tr C σs IN + US H HUS  C  2     N σs  . =. (2.21). 2/N. N. 2. ∑ L (k, k ). k =1. N  ≥ N  ∏ L (k, k )  k =1. The inequality in (2.21) is obtained from the arithmetic-mean-geometric-mean (AM-GM) inequality, and the equality is held when L ( i, i ) = L ( j , j ) , ∀ i ≠ j . If U S is designed properly, the bound in (2.21) can be achieved. For this purpose, we first decompose U S as the form of U S = VH U′S. (2.22). ɶ , and U′ ∈ ℂN ×N is an unitary where VHɶ ∈ ℂN ×N is the left singular matrices of H S. matrix to be further specified. Note that this decomposition is always possible for any unitary matrix. Substituting (2.22) into (2.20), we can have (2.20) as −1. PS ,T   ′ LL = US′H  σ s−2IN + 2 Λ  US N σ  s  H. {. where Λ = diag λ ɶ , ⋯, λ ɶ H,1. H,N. }. (2.23). ɶ := D. , and λ ɶ , ⋯, λ ɶ H,1. H,N. ɶ HH ɶ . To are the eigenvalues of H 1. ɶ 2 which can be obtain U′S , we apply geometric mean decomposition (GMD) [26] on D expressed as ɶ 1/ 2 = QRPH D. (2.24). Q, P are some unitary matrices, and R is a upper triangular matrix with equal diagonal. elements. Letting U′S = P and substituting (2.24) in (2.23), we can have L = R H .. 10. The.

(19) lower bound in (2.21) is achieved since L ( i, i ) = L ( j , j ) , ∀ i ≠ j . So, the optimum FS , denote as FS ,opt , can be expressed as FS ,opt =. PS ,T Nσ e2. (2.25). VH P. From (2.23) and (2.24), we can have the resultant MSE 1/ N. J min.    N  N 1  = ∑ L(k , k )2 = N ∏  PS ,T −2  k =1 k =1  λ ɶ  H,k N σ 2 + σ s  s  . (2.26). Now, our problem becomes to minimize (2.26) in the master problem. Note that the equality of the right side of (2.26) is satisfied when the diagonal elements of L are all equal. With a propoer FS , we can not only minimize the MSE but also make the optimization problem more tractable. To proceed, let us consider the following equivalence: 1/ N.    N  1  min N ∏  P FR S , T 2 −   k =1 λ  Hɶ ,k N σ 2 + σ s    s. N  N   −2 PS ,T  ɶ HH ɶ  = max  σ s + det I H   N    PS ,T FR  N   . (2.27). N. P   Note that  σ s−2 S ,T  in (2.27) is a constant so we can reformulate the master problem as: N    N  ɶ HH ɶ max det  IN + H P  FR  S ,T  s.t.. (2.28). PS ,T    H  H C 2 : Tr FR  σ n2,r IR + HSR HSR  FR  ≤ PR,T . N    . The problem (2.28) is still difficult to solve because the cost function is a nonlinear function of FR and the problem is not convex either. To solve the problem, we propose a relay precoder structure such that a closed-form solution can be solved. Directly solving (2.28) is not feasible. We then use a lemma describe below: Lemma 2.2 [41]: Let M ∈ ℂN ×N be a positive definite matrix, then. det ( M ) ≤. N. ∏ M (i, i ). (2.29). i =1. where M(i, i ) denotes the ith diagonal element of M. Note that the equality in (2.29) holds. 11.

(20) ɶ HH ɶ , it turns out that when M is diagonal, the when M is a diagonal matrix. If we let M = H. maximization of the cost function becomes possible. To have the diagonalization, we need another lemma shown below: Lemma 2.3 [41]: Let A ∈ ℂN ×N be a positive matrix and B ∈ ℂN ×N , then. (. det ( A + B ) = det ( A ) det IN + A −1/ 2BA−1/ 2. ). (2.30). (. H H Form (2.30), we let B = HSR σ n2,r HRD FR FRH HHRD + σ n2,d IM FRH HRD. and A =. ). −1. HRD FR HSR. N H I + σ n−,2d HSD HSD we have the following equivalence: PS ,T N.  N  ɶ HH ɶ arg max det  IN + H P  FR  S ,T  H H ′H FRH HRD = arg max det  IN + HSR σ n2,r HRD FR FRH HRD + σ n2,d IM  FR. (. (. ′ := HSR N / PS ,T IN + where HSR. 1 − −2 H 2 σ n,d HSD HSD. ). ). −1. ′ HRD FR HSR. ). (2.31). . Note that det ( A ) is not the function of. FR , so we can ignore it. The optimization problem can be rewritten as max det ( M′ ) FR. s.t.. (. H H H M′ = IN + H′SR FR HRD. (. H σ n2,r HRD FR FRH HRD. C 2 : σ n2,r FR. 2 2. +. +. PS ,T N. σ n2,d IM FR HSR. ). −1. 2 2. ′ HRD FR HSR. ). (2.32). ≤ PR,T .. Taking a close look at (2.32), we can see that there exists a precoder structure for the relay precoder FR such that the diagonalization can be achieved. Consider the singular value. ′ : decomposition (SVD) on HRD and HSR HRD = Urd Σrd VrdH. (2.33). ′ = Usr ′ Σsr ′ Vsr′H HSR. (2.34). ′ , ′ ∈ ℂR ×R are left singular matrices of HRD and HSR where Urd ∈ ℂM ×M and Usr ′ ∈ ℝR ×N are the diagonal singular value matrices of respectively; Σrd ∈ ℝM ×R and Σsr. ′ , respectively; VrdH ∈ ℂR ×R and Vsr′H ∈ ℂN ×N HRD and HSR. are the right singular. ′ , respectively. If we let FR have the structure as the following matrices of HRD and HSR. 12.

(21) form, a full diagonalization of the matrix of the determinant in (2.32) can be achieved:. ′H FR,opt = Vrd Σr Usr. (2.35). where Σ r is a diagonal matrix with its ith diagonal element of σ r ,i . Let σ rd ,i and σ sr′ ,i ′ , respectively. With the relay precoder structure, be the ith diagonal element of Σrd and Σsr. the optimization in the master problem can finally be translated into a scalar-value concave optimization problem. Substituting (2.33), (2.34), (2.35) into (2.32), we can rewrite (2.32) as: 2  ′2,i pr ,iσ n2,d σ rd ,iσ sr max ∑ ln  1 + 2 2  pr ,i , 1≤i ≤ N i =1 pr ,iσ n2,r σ rd ,i + σ n ,d  s.t.. N. N.   . (2.36).  PS ,T 2  ′ ( i, i ) + σ n2,r  ≤ PR,T , pr ,i ≥ 0, σ sr′ ,i Dsr  N . ∑ pr ,i . i =1. where. (. ). H ′ (i, i ) stands for the ′ = Vsr′H N / PS ,T IN + HSD pr ,i = σ r2,i , Dsr HSD Vsr′ , and Dsr. ′ . It is apparent that the mast problem is now a scalar optimization ith diagonal element of Dsr. problem. And, since pr ,i ≥ 0 , the cost function (2.36) is concave [42]. To solve this problem, we apply the KKT conditions given by [42] and find the solution for pr ,i , i = 1, ⋯, N as:. pr ,i.   µ = +  2  PS ,T 2 −2 −2 2 2 ′ (i, i ) + σ n  (σ n,r σ n,d σ sr′ ,i + 1)  σ rd ,i  N σ sr′ ,i Dsr    4 1 σ n ,d 4 σ n4,r 2.  σ2  σ rd4 ,i  2 n,r 2 + 1   σ n,d σ sr′ ,i . −. 2 σ rd ,i.  σ n2,r  2  σ n,d. +.     ,  + σ sr′2,i    . 2 2 1 σ n,dσ sr′ ,i 1+ 2 σ n2,r. (2.37). where µ is chosen to satisfy the power constraint in (2.36). Substituting (2.37) into (2.35), ɶ in (2.12) can also be obtained. we can obtain the optimum relay precoder, and then H. Finally, the unitary source precoder can be derived by substituting (2.24) into (2.25) , and the matrix C can be obtained by (2.19).. 13.

(22) Figure 2.1: THP source and linear relay precoded AF MIMO relay system with MMSE receiver.. 14.

(23) Chapter 3 Robust Joint MMSE Transceiver Desgin with Tomlinson-Harashima Source and Linear Relay Precoders. As well known, the performance of transceiver design relies on the accuracy of channel state information (CSI). In the literature, most transceiver designs assume perfect CSI. Our design in Chapter 2 also assumes that the destination has perfect CSIs of the three links. However, for real-world implementation, perfect CSIs are usually not attainable due to channel estimation or quantization error. The performance of the transceiver designed with imperfect CSIs may be degraded seriously. In this chapter, we consider a robust nonlinear transceiver design in which the THP, the linear relay and the MMSE receiver are used at the source, the relay and the destination, respectively. The imperfect CSIs from both the relay and direct links are incorporated into design where channel estimation errors are modeled as Gaussian random variables. The design procedure is similar to that we have used in Chapter 2. The main idea is to use the primal decomposition and some approximations such that a close-form solution of the optimization problem can be obtained. In Section 3.1, we build the system model taking the channel uncertainty into consideration. In Section 3.2, we formulate the designing problem under the MMSE criterion. In Section 3.3, we propose a new approach to solve the problem in closed-form.. 3.1 System model We consider a three-node AF MIMO relay precoded system which has been presented in Chapter 2, as shown in Figure 2.1. With the two-phase transmission protocol, the signal from the source is transmitted to the relay and the destination simultaneously in the first phase. Then the received signal at the relay is multiplied by the relay precoder and send to the destination in the second phase. For simplicity, we let the signal notations are similar to those in Chapter 2, (2.1)-(2.4). In general, the actual channel matrix can be modeled as a summation of an estimated channel and an error matrix [36]. Thus, we have:.

(24) SR + ∆H HSR = H SR. 15. (3.1).

(25)

(26) RD + ∆H HRD = H RD. (3.2).

(27) SD + ∆H HSD = H SD. (3.3). where HSR ∈ ℂR ×N , HSD ∈ ℂM ×N and HRD ∈ ℂM ×R are the actual channel matrices of the source-to-relay, the source-to-destination, and the relay-to-destination links, respectively;.

(28) SR , H

(29) RD , and H

(30) SD are the estimated channel matrices of H , H , and H , H SR RD SD respectively. ∆HSR , ∆HRD , and ∆HSD are the corresponding channel errors matrices in which all the elements are assumed to be zero mean Gaussian random variables. An estimation channel matrix can be further decomposed into a product of three matrices. For example, ∆HSR can be express as: 2 1/ 2 ∆HSR = Σ1/ SR H i.i.d. Ψ SR. (3.4). where the elements of Hi.i.d. are independent and identically distributed (i.i.d) Gaussian random variables with zero-mean and unit variance; ΣSR ∈ ℂR ×R and ΨSR ∈ ℂN ×N are the row and column covariance matrices of ∆HSR , respectively [38]. From (3.4), it is clear that vec ( ∆HSR ) ~ CN ( 0NR ×1, ΣSR ⊗ ΨTSR ) , where CN ( m, C ) denotes a complex Gaussian random vector with mean m and covariance C [44]. Similarly, we can set the distribution of channel estimation error ∆HRD and ∆HSD as vec ( ∆HRD ) ~ CN ( 0RM ×1, ΣRD ⊗ ΨTRD ) and vec ( ∆HSD ) ~ CN ( 0NM ×1, ΣSD ⊗ ΨTSD ) . It is noteworthy that the expression of ΣSR , ΨSR , ΣRD , Ψ RD , ΣSD , ΨSD depend on specific channel estimation algorithms. For example, ΨSR = RT ,SR and ΣSR = σe2,sr RR,SR if we use the estimation method proposed in [29]. RT ,SR and RR,SR are the transmit and receive antenna correlation matrices; σ e2,sr is the source-relay link channel estimation error variance. And note that the other two matrices have the similar structures. Here, we assume all the channels are time-invariant and all second-order statistics - ΣSR , ΨSR , ΣRD , Ψ RD , ΣSD , ΨSD are known as a prior. At the destination, we can have a single received vector for the two-phase transmission:  HSD  yD :=   FS x +  HRD FR HSR  := H. n D,1   H F n + n  RD R R D ,2   . (3.5). := w. We use the MMSE receiver, that takes both the noise and the channel estimation error. 16.

(31) into account at the destination, to recover the transmitted signal. Define the MSE as:. {. J ( C, FS , FR , G ) = E ∆, w GyD − v. { {((. = E ∆, w Tr. 2. }. GHFS − C ) x + Gw. ) ((GHF. − C ) x + Gw. S. ). H. }}. (3.6). where G represents the equalization matrix and the subscript ∆ , w denote that the expectation is taken over both the channel estimation error and noise. With the same assumption in Chapter 2, we can have E xxH  = σ s2 IN and E  vvH  = σ s2CCH . Thus, we can rewrite the MSE in (3.6) as: J ( C, FS , FR , G ) =   TSD  2  Tr σ s G H 

(32)

(33) SR F FH H

(34) SD  HRD FR H  S S  . {. (. ). Tr FR TSR FRH Ψ RD ΣRD. {. }. }. {.

(35) CH − Tr σ 2CFH H

(36) H GH + Tr σ 2CCH − Tr σ s2GHF s s S S  σ 2 I n ,d M  + Tr G    0. {(.    GH  

(37) RD F T FH H

(38) HRD   +H R SR R  . H H

(39) SD F FH H

(40) SR

(41) RD H FRH H S S. }. 0

(42) RD F FH H

(43) RD σ n2,r Tr ( FR FRH Ψ RD ) ΣRD + σ n2,r H R R H. ) }. {.    GH  + σ n2,d IM   . (3.7). }.

(44) FH H

(45) H + R

(46) w + ∆err GH − Tr σ 2GHF

(47) CH − = Tr G σ s2 HF s S S S. {. H. }. {.

(48) GH + Tr σ 2CCH Tr σ s2CFSH H s. }. where

(49) SD  H 

(50)   H=

(51) RD F H

(52) SR  H R  . (. ). σ s2 Tr FS FSH ΨSD ΣSD  ∆err =   0  . (3.8).   2 2 H Tr FR (σ s TSR + σ n,r IR ) FR Ψ RD ΣRD +   2

(53) H H

(54) H  σ s HRD FR Tr FS FS ΨSR ΣSR FR HRD   0. (. ( (. ). ). nD,1    H H H

(55) H H 

(56) Rw = E  

(57)  n D,1 n R FR H RD + n D,2      HRD FR nR + n D,2   . (. ). (. ). H

(58) SD F FH H

(59) SD TSD = Tr FS FSH ΨSD ΣSD + H S S H.

(60) SR F FH H

(61) SR TSR = Tr FS FSH Ψ SR ΣSR + H S S. The derivation of the MSE in (3.7) can be found in Appendix B.. 17. ). (3.9). (3.10) (3.11) (3.12).

(62) Since the cost function is convex in G, we can find the minimum MSE in (3.7) by taking the gradient of J with respect to G and set the result to zero. Note that we keep FS, FR, and C fixed in the operation. Then we can obtain the optimum equalization matrix G, denote as Gopt , as H.

(63) Gopt = σ s2CFSH H. (. H.

(64) FH H

(65) +R

(66) w + ∆err σ s2 HF S S. ). −1. (3.13). Substituting (3.13) to (3.7), we then have the minimum MSE as. (. ). J C, FS , FR , Gopt ≜ J min ( C, FS , FR ). {. (. }. (. 

(67) H HF

(68) FH H

(69) H + σ −2 R

(70) w + ∆err = σ s2Tr CCH − σ s2Tr CFSH H s S S . )). −1.

(71) CH  . HF S . By using matrix inversion lemma A H ( AA H + I ) A = I − ( A H A + I ) −1. −1. (3.14). [42], (3.14) can. be further rewritten as:. J min ( C, FS , FR ) −1     H  H 

(72) HF

(73) FH H

(74) + σ −2 R

(75) w + ∆err  HF

(76) CH  = σ s2 Tr {CC H } − σ s2 Tr CFSH H  S S s S      R∆     −1   H  1 H 1  H H − − − −   2 H 2 H

(77) H

(78) H

(79) 2  2

(80) 2 2 = σ s Tr {CC } − σ s Tr C FS H R ∆ R ∆ HFS FS H R ∆ + I 2 M  R ∆ HFS C      A A AH  AH   . (. (. 

(81) H R −1 HF

(82) +I = σ s2 Tr C FSH H ∆ S N  = σ s2 Tr {E ( C, FS , FR )}. ). −1. ). (3.15).  CH  . where. (.

(83) w + ∆err R ∆ = σ s−2 R. ). 0  R ∆,1,1 = σ s−2   0 R ∆,2,2   . (3.16). and. (. ). R ∆,1,1 = σ n2,d IM + σ s2Tr FS FSH ΨSD ΣSD. ( (. (. ). ). (3.17). ). H

(84) SR F FH H

(85) SR R ∆,2,2 = Tr  FR σ s2 Tr FS FSH Ψ SR ΣSR + H + σ n2,r IR FRH Ψ RD  ΣRD S S   H H 2

(86) 2 H

(87) H

(88)

(89) RD F Tr F FH Ψ +σ s2 H R S S SR ΣSR FR HRD + σ n ,r HRD FR FR HRD + σ n ,d IM .. ( (. ). ). (3.18). From (3.16)-(3.18), we can see that the MSE in (3.15) is a complicate and nonlinear function. 18.

(90) of FS and FR .. 3.2 Problem formulation In Section 3.1, we construct the system model using the MMSE criterion. The objective is to find C , FS and FR so that the MSE (3.15) can be minimized. We now can formulate the joint precoder design problem as below:. (. 

(91) H R −1 HF

(92) +I min σ s2Tr C FSH H ∆ S N C, FS , FR  s.t. ). −1.  CH  . (3.19). C 1,C 2. where. {. }. C 1 : σ s2Tr FS FSH ≤ PS ,T ,. { (. ) } H

(93) SR F FH H

(94) SR σ n2,r Tr {FR FRH } + σ s2 Tr {FR ( Tr ( FS FSH ΨSR ) ΣSR + H ) FRH } ≤ PR,T , S S. H C 2 : E Tr FR σ n2,r IR + σ s2HSR FS FSH HSR FRH  =  . (3.20). In (3.20), C 1 and C 2 stand for the transmission power constraints at the source and the relay, respectively, C is a lower triangular matrix with unit diagonal, and R ∆ is the matrix specified in (3.16). From (3.20), we see that the cost function and constraints are functions of C , FS and. FR . As we can also see, the functions are complicated and the problem is not a convex optimization problem. Since imperfect CSIs of all links are involved, the problem becomes much more difficult than that in Chapter 2. In the next section, we propose a new approach to solve this problem.. 3.3 Robust joint source and relay precoder design Similar to that in Chapter 2, we also cascade an unitary precoder FS after the THP. The precoder FS can not only facilitate the optimization but improve the BER performance. The cost function is a nonlinear function of FS , FR and C . It is then difficult to find the optimum precoders simultaneously. We use the primal decomposition method so that the optimization problem can be translated into a subproblem and a master problem [42]. The procedure is first to split unknown variables into two group and the variables in the first group. 19.

(95) are treated as known constants. Then, the variables in second group are solved as the functions of the functions of the variables in first group (the subproblem), and the cost function is reduce to a function of the variables in the first group. Finally, the variables in first group are solved (the master problem). In this case, the subproblem is to find the optimal C and FS by letting FR be fixed, and the master problem is optimized for FR . For convenience, we rewrite the problem in (3.19) as:. {. }. {. }. min Tr E ( C, FS , FR ) = min min Tr E ( C, FS , FR ). C,FS , FR. FR. C, FS. s.t. FS =αS US , C 1′ : σ s2αS2 N ≤ PS ,T , C 2 : Tr. { (. FR σ n2,r IR. H

(96) SR H

(97) SR + σ s2αS2 H. +. σ s2αS2 Tr. (. ). ) }. (3.21). ΨSR ΣSR FRH.         H   2 H 2 2

(98)

(99)  = Tr FR σ n,r IR + σ s αS HSR HSR + Tr ( ΨSR ) ΣSR FR  ≤ PR,T       H    ′ ′

(100) SR H

(101) SR := H . ). (. (.

(102) R −1 HF

(103) +I where E (C, FS , FR ) = C FSH H ∆ S N H. ). −1. (. H

(104) ′SR := H

(105) SR H

(106) SR CH and H + Tr ( ΨSR ) ΣSR. ). 1/ 2. .. We let FS have the form as that shown in (3.21), where αS is a scalar and US ∈ ℂN ×N is a unitary matrix. The constraint C 1′ is obtained by substituting FS = α US into C 1 in (3.20). In the subproblem, we can observe that the transmitted power constraint at the relay, C 2 , is not the function of C or FS , so we can move it to the master problem. Let FS = α US , the cost function and the constraints can then become functions of C , α and US . Then the subproblem can be rewritten as:. 20.

(107) min. ( ). ( ). C FR ,α US FR. (. Tr E ( C, FS , FR ). ). s.t.. (. ɶ H HF ɶ E ( C, FS , FR ) = C FSH H + IN S FS =α SUS ,. ). −1. CH ,. (3.22). C 1′ : σ s2αS2 N ≤ PS ,T , H   

(108) ′SR H

(109) ′SR  FH  ≤ P C 2 : Tr FR  σ n2,r IR + σ s2αS2 H R R,T    . where E is now a function of C, αS , US , and ɶ HH ɶ := H

(110) H R −1 H

(111) H ∆. =. σ s2. H H

(112) SD. . (. H

(113) SR H FRH.

(114) SD (σ 2 I + σ 2α 2Tr ( Ψ ) Σ )−1  H  0 s s SD SD  n ,d M   −1  

(115) RD F H

(116) SR  0 ∆A + A )   H ( R   .

(117) HRD  H. −1 H H H

(118) SD = σ s2 σ n−,2d H (σ n2,d IM + σ s2αs2Tr ( ΨSD ) ΣSD ) H

(119) SD + H

(120) SR FRH H

(121) RD ( ∆A + A )−1 H

(122) RD FR H

(123) SR. (3.23). ). with.         H 2 2 2 H

(124)

(125)     ∆A = Tr FR σ s αS Tr ( ΨSR ) ΣSR + HSR HSR + σ n ,r IR FR Ψ RD ΣRD         H ′ ′

(126) SR H

(127) SR =H    . ). (. (3.24). H

(128) RD F Σ FH H

(129) RD +σ s2αS2 Tr ( Ψ SR ) H R SR R. H.

(130) RD F FH H

(131) RD + σ 2 I A = σ n2,r H R R n ,d M. (3.25). We first optimize αS by treating C , FR and U S as known entities. It is simple to see that the cost function in (3.22) is monotonically decreasing with αS (see the proof in Appendix C). So, the optimum value can be found as. αS ,opt =. PS ,T. N σ s2. (3.26). This corresponds to the largest αS under the constraint C 1′ in (3.22). Substituting (3.26) into (3.22). The subproblem can be re-written into a conventional point-to-point THP MIMO problem [24] given by −1   PS ,T   H ɶH ɶ H min Tr C  U H HU + I C  S N  C ( FR ), US ( FR ) N σ s2 S    . 21. (3.27).

(132) The optimum solution of C has been derived in [24] (see Appendix A), denote as Copt , as :. Copt = DL−1. (3.28). and  PS ,T H ɶ H ɶ  LL =  2 US H HUS + IN   N σs . −1. H. (3.29) −1. P . H HU. is the Cholesky factorization of  S ,T2 USH H + IN  ; L is a lower triangular matrix S  N σs . with real diagonal elements; D is a diagonal matrix scaling the diagonal elements of C to unity, that is D = diag ( L ( k , k ) ) , k = 1,… , N . Substituting (3.28) and (3.29) into (3.27), we can have the cost function as: −1   PS ,T   H ɶH ɶ H Tr C  U H HU + I C   2 S S N    N σ s . =. N. ∑ L(k, k )2. k =1.   ≥ N  ∏ L(k, k )2   k =1  N. 1. (3.30). N. which is a function of US . Note that L ( k , k ) means the kth diagonal element of L. The inequality in (3.30) is the arithmetic-geometric inequality (AGI) and the equality is held when. L ( i, i ) = L ( j , j ) , ∀i ≠ j . The next step is to find US so that the bound in (3.30) can be achieved. First, we decompose US as US = VHɶ US′. (3.31). ɶ and U′ ∈ ℂN ×N is a unitary where VHɶ ∈ ℂN ×N is the right singular matrices of H S. matrix to be determined later. Note that the decomposition can always be conducted for a unitary matrix. Substituting (3.31) in (3.30), we have:  PS ,T H ɶ H ɶ  LL =  2 US H HUS + IN   N σs . −1. H. −1. =. US′H.  PS ,T  ′  2 Λ + IN  US N σs   . (3.32). ɶ := D. {. }. where Λ = diag λHɶ ,1, ⋯, λHɶ ,N , λ ɶ. H,k. ɶ HH ɶ . To achieve the are the eigenvalues of H. ɶ 1/ 2 . Let D ɶ =D ɶ 1/ 2D ɶ 1/ 2 and D ɶ 1/ 2 is equality of the AGI, we apply GMD [26] method on D ɶ . Then we have: the square-root matrix of D. 22.

(133) ɶ 1/ 2 = QRPH D. (3.33). where Q and P are unitary matrices, and R is an upper triangular matrix with equal and real diagonal elements. Letting U′S = P and substituting (3.33) into (3.32), we can have LLH = RH R. (3.34). which indicates that L = R H , and the diagonal elements of L are real and all equal. Then the equality of AGI in (3.30) is held so that the lower bound is achieved. Therefore, the optimum FS , denoted as FS ,opt , can be expressed as FS ,opt =. PS ,T. N σ s2. (3.35). Vɶ P H. By substituting (3.28) and (3.35) into (3.15), the result MSE can be expressed as: 1. N N = σ s2 ∑ L(k, k )2 = σ s2N  ∏ L(k , k )2  k =1  k =1  N. J min.  PS ,T  1 = σ s2N ∏  + λ  2 H ɶ ,k k =1  N σ s  N. −1/ N. (3.36). Now, the problem becomes the minimization of (3.36), which is master problem. It is seen that the cost function now is a function of FR . To proceed, we consider the following equivalence: N.  PS ,T. ∏  N σ 2 λHɶ ,k. k =1 . s. PS ,T H     ɶ ɶ + 1  = det   IN + 2 H H   N σs   . (3.37). Substituting (3.37) into (3.36) , we can reformulate the problem as PS ,T H   ɶ ɶ max det  IN + 2 H H FR N σ s   s.t.. (3.38). H PS ,T ′   

(134) SR H

(135) ′SR  FH  ≤ P . C 2 : Tr FR  σ n2,r IR + H R R,T N    . ɶ HH ɶ is still complicate and difficult to deal with, we find Since the structure of H. another equivalent form as follows: PS ,T N σ s2. H. H ɶ HH ɶ =H

(136) ′′SR FH H

(137) RD

(138) RD F H

(139) ′′SR H ( ∆A + A )−1 H R R. where. 23. (3.39).

(140)

(141) ′′SR = H. −1  PS ,T PS ,T 

(142)  H  2

(143)

(144) HSR  IN + HSD  σ n ,d IM + Tr ( ΨSD ) ΣSD  HSD    N N σ n2,d N    . PS ,T. −1/ 2. (3.40). Then (3.38) can be rewritten as H  H

(145) ′′SR FH H

(146) RD

(147) RD F H

(148) ′′SR  max det  IN + H ( ∆A + A )−1 H R R FR   s.t. C 2 .. (3.41). Taking a close look at (3.41), we can find that ( ∆A + A ) is a complicated function of. FR . The master problem is difficult to solve, even using the numerical method [42]. To overcome the problem, we propose maximizing a lower bound of (3.41) instead of trying to maximize it directly. In this manner, we can have an amenable form and the optimization problem in the master optimization can be made easier. For this purpose, we consider the following property. Property: The utility function in (3.38) is lower bounded by H 

(149) ′′SR FH H

(150) HRD ( ∆A + A )−1 H

(151) RD F H

(152) ′′SR  det  IN + H R R   H 

(153) ′′SR FH H

(154) HRD ( ∆A′ + A )−1 H

(155) RD F H

(156) ′′SR  ≥ det  IN + H R R  . (3.42). where.         H

(157) SR H

(158) SR ∆A = Tr  FR  σ s2αS2 Tr ( ΨSR ) ΣSR + H + σ n2,r IR  FRH Ψ RD  ΣRD         H

(159) ′SR H

(160) ′SR =H    . ). (. (3.43). H

(161) RD F Σ FH H

(162) RD +σ s2αS2 Tr ( Ψ SR ) H R SR R. ∆A′ = PR,T λmax ( Ψ RD ) λmax ( ΣRD ) IM +. the equality of (3.42) is held when. PS ,T N. H

(163) RD F FH H

(164) RD Tr ( ΨSR ) λmax (ΣSR )H R R. (3.44). Ψ RD = βRD IR , ΣRD = γ RD IM ×M , ΨSR = βSR IN and. ΣSR = γ SR IR ×R with some scalars βRD , βSR , γ RD and γ SR . As we will see, the optimization with the lower bound is much easer. The derivation of the lower bound in (3.42) can be found in Appendix D. Using the lower bound of the utility function, we can reformulate the master optimization problem as:. 24.

(165) H  H −1

(166)

(167) ′′SR FH H

(168) RD

(169) ′′  max det  IN + H ∆A′ + A ) H ( RD FR H SR  R FR   s.t.. (3.45).

(170) RD F FH H

(171) RD , ( ∆A′ + A ) = α IM + β H R R H. H PS ,T ′   

(172) SR H

(173) ′SR  FH  ≤ P . C 2 : tr FR  σ n2,r IR + H R R,T N    . where. α = PR,T λmax ( Ψ RD ) λmax ( ΣRD ) + σ n2,d β =. PS ,T N. (3.46). Tr ( Ψ SR ) λmax (ΣSR ) + σ n2,r. (3.47). As we will see in latter development, ∆A′ in (3.45) is easier to handle compared to ∆A . Although the function in (3.45) is simplified, it is still a complicated function of FR . We now let FR have a specific structure such that the master optimization problem can be easier to solve. This relay precoder structure can transfer the matrix-valued optimization problem in (3.45) to a scalar-valued problem, though it is suboptimal. Similar the procedure in Chapter 2, we first use Lemma 2.2 and Lemma 2.3. Consider the following singular-value-decomposition (SVD):.

(174) RD = U

(175) rd Σ ɵ rd V

(176) rdH H. (3.48).

(177) ′′SR = U′′ Σ′′ V′′H H sr sr sr. (3.49).

(178) rd ∈ ℂM ×M and U′′ ∈ ℂR ×R are left singular matrices of H

(179) RD and H

(180) ′′SR , where U sr ′′ ∈ ℝR ×N are the diagonal singular value matrices of respectively; Σrd ∈ ℝM ×R and Σsr

(181) ′′SR , respectively; VH ∈ ℂR ×R and V′′H ∈ ℂN ×N are the right singular HRD and H rd sr

(182) ′′SR , respectively. Substitute (3.48) and (3.49) into (3.45), we can matrices of HRD and H. rewrite (3.45) as. det ( M′ ) s.t. C2. (.

(183) Σ

(184) rd α I + β Σ

(185) rd V

(186) F FH V

(187) rd Σ

(188) rd M′ = I N + Σ′′sr U′′sr FRH V M R R H rd. H rd. ). −1. (3.50).

(189) rd V

(190) rd F U′′ H Σ′′ Σ R sr sr. From Lemma 2.2, we see that if M′ is diagonalized, the utility function in (3.50) can be maximized. Note here that the result is not held when the power constraint in (3.50) is included. However, we use it to obtain a suboptimal solution. Let FR have a structure shown. 25.

(191) below:.

(192) rd Σ U′′H FR,opt = V r sr where. Σr. (3.51). is a diagonal matrix with ith diagonal element σ r ,i , i = 1, ⋯, κ ,. κ = min { N , R} whose value will be determined. The general structure for the relay precoder should have the form of. {F. R. FR = U r Σ r VrH , FR ∈ C2. }. with U r is a M × M unitary matrix. and Vr is a R × R unitary matrix. However, the optimum FR in (3.50) is very difficult to find. Here, we only consider a specific feasible set of FR in (3.51) simplifying the optimization problem. The next step is to transfer the matrix-valued problem to the scalar-valued one with the precoder structure of FR in (3.51). Let σ rd ,i and σ sr′′ ,i be the ith diagonal element of.

(193) rd Σ. and Σ′′sr , respectively. Substituting (3.51) into (3.50) and taking ln operation on the utility function, we can rewrite (3.50) as: 2  pr ,iσ sr′′2,iσ rd ,i max ∑ ln  1 + 2 pr ,i ,1≤ i ≤ κ i =1  + β pr ,iσ rd α ,i  s.t.. κ. κ. ∑ pr ,i (σ n2,r. i =1.     (3.52). + Dsr (i, i )) ≤ PR,T ,. pr ,i ≥ 0,. where. pr ,i = σ r2,i. and. Dsr =. PS ,T N. ). (. H

(194) SR H

(195) SR ′′H H ′′ + Tr ( ΨSR ) ΣSR Usr Usr. with. Dsr (i, i ). being the ith diagonal element of Dsr . Now, the utility function and the constraints are all functions of scalars. Since the utility function and the constraints are all concave for pr ,i ≥ 0 [42], we see that (3.52) is a standard concave optimization problem. As a result, we can find the optimum solution of pr ,i , i = 1, ⋯, κ by Karush-Kuhn-Tucker (KKT) conditions. The solution is given by:. pr ,i.    =   . µ. (. σ n2,r. + Dsr (i, i ) ). 2 σ rd ,i. β ⋅ α.  β   2 + 1   σ sr′′ ,i . 1. +. σ rd4 ,i. 4 2   β β + 1   α 2  σ sr′′2,i  2.   α+  2β − 2  σ rd ,i ( β + σ sr′′2,i )   . +. ασ sr′′2,i. (3.53). where µ is chosen to satisfy the power constraint in (3.52). The detail derivation can be. 26.

(196) found in Appendix E. Substituting (3.53) into (3.51), we can have the optimum relay precoder ɶ HH ɶ by (3.23). Subsequently, F and FR . After the optimum FR is found, we can obtain H S C can also be obtained by (3.35) and (3.28) with the same procedure described in Chapter 2.. In this chapter, we joint design the robust transceiver in an AF MIMO relay system in which a THP is used at the source, a linear precoder at the relay, and an MMSE receiver at the destination. Since the channel uncertainty has been taken into consideration, we can expect that the design will outperform that in Chapter 2. The price we pay is a more complicated design. The computational complexity of the non-robust/robust designs includes SVD, GMD, and matrix inversion operations, are mentioned in Chapter 2 and Chapter 3. The overall computational complexity and steps of the non-robust/robust designs, measured by FLOPs, are summarized in Table 3.1 and Table 3.2.. Table 3.1: Computational complexity of THP source and linear relay precoders (MMSE receiver). 27.

(197) Table 3.2: Computational complexity of robust THP source and linear relay precoders (MMSE receiver). 28.

(198) Chapter 4 Simulation results and discussions. 4.1 Simulation Setup In this section, we describe our simulation environment. We consider an AF MIMO relay system with N, R and M antennas at the source, the relay and the destination, respectively. We let N = R = M = 4 . The widely used exponential model [29] is chosen for the generation of the channel estimation error covariance matrices, which can be represented by. ΨSR = Ψ RD = ΨSD.  1 δ δ2 δ3   δ 1 δ δ2  = 2 δ δ 1 δ    δ 3 δ 2 δ 1  . (4.1). and.  1 γ γ2 γ3   2 γ 1 γ γ  2   ΣSR = ΣRD = ΣSD = σe 2 (4.2) γ γ 1 γ    γ 3 γ 2 γ 1   where δ and γ are the correlation coefficients of the row and column covariance matrices,. σ e2 denotes the estimation error variance. The resultant covariance matrices can be obtained

(199) SR , H

(200) RD from the channel estimation method proposed in [29]. The estimate channels, H

(201) SD , are generated base on the following distributions and H  1 − σ e2

(202) SR ~ CN  0 vec H , ΣSR ⊗ ΨTSR  (4.3) NR ×1 2 σe    1 − σ e2 T 

(203) vec HRD ~ CN  0MR ×1, Σ ⊗ Ψ (4.4) RD RD  σ e2    1 − σ e2

(204) SD ~ CN  0 vec H , ΣSD ⊗ ΨTSD  (4.5) MN ×1 2 σe   So that the relationships of the actual and estimation channels can be expressed as:. (. ). (. ). (. ). 29.

(205)

(206) SR + ∆H , H = H

(207) RD + ∆H