Robust Sum MSE Optimization for Downlink Multiuser MIMO Systems with Arbitrary Power Constraint: Generalized Duality Approach

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1 Robust Sum MSE Optimization for Downlin Multiuser MIMO Systems with Arbitrary Power Constraint: Generalized Duality Approach Tadilo Endeshaw Bogale, Student Member, IEEE and Luc Vandendorpe Fellow, IEEE arxiv:3.6433v [math.oc] 25 Nov 23 Abstract This paper considers linear minimum meansquare-error (MMSE) transceiver design problems for downlin multiuser multiple-input multiple-output (MIMO) systems where imperfect channel state information is available at the base station (BS) and mobile stations (MSs). We examine robust sum mean-square-error (MSE) minimization problems. The problems are examined for the generalized scenario where the power constraint is per BS, per BS antenna, per user or per symbol, and the noise vector of each MS is a zero-mean circularly symmetric complex Gaussian random variable with arbitrary covariance matrix. For each of these problems, we propose a novel duality based iterative solution. Each of these problems is solved as follows. First, we establish a novel sum average meansquare-error (AMSE) duality. Second, we formulate the power allocation part of the problem in the downlin channel as a Geometric Program (GP). Third, using the duality result and the solution of GP, we utilize alternating optimization technique to solve the original downlin problem. To solve robust sum MSE minimization constrained with per BS antenna and per BS power problems, we have established novel downlin-uplin duality. On the other hand, to solve robust sum MSE minimization constrained with per user and per symbol power problems, we have established novel downlin-interference duality. For the total BS power constrained robust sum MSE minimization problem, the current duality is established by modifying the constraint function of the dual uplin channel problem. And, for the robust sum MSE minimization with per BS antenna and per user (symbol) power constraint problems, our duality are established by formulating the noise covariance matrices of the uplin and interference channels as fixed point functions, respectively. We also show that our sum AMSE duality are able to solve other sum MSE-based robust design problems. Computer simulations verify the robustness of the proposed robust designs compared to the non-robust/naive designs. I. INTRODUCTION The spectral efficiency of wireless channels can be enhanced by utilizing multiple-input multiple-output (MIMO) systems. This performance improvement is achieved by exploiting the transmit and receive diversity. In [], a fundamental relation between mutual information and minimum The authors would lie to than the Region Wallonne for the financial support of the project MIMOCOM in the framewor of which this wor has been achieved. Part of this wor has been presented in the 45th Annual Conference on Information Sciences and Systems (CISS), Baltimore, Maryland, Mar. 2, and in the 22nd IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), Toronto, Canada, Sep. 2. Tadilo E. Bogale and Luc Vandendorpe are with the ICTEAM Institute, Université catholique de Louvain, Place du Levant 2, Louvain La Neuve, Belgium. {tadilo.bogale, luc.vandendorpe}@uclouvain.be, Phone: , Fax: mean-square-error (MMSE) has been derived for MIMO Gaussian channels. Furthermore, it has been shown that different transceiver optimization problems are equivalently formulated as a function of MMSE matrix, for instance, minimizing bit error rate, maximizing capacity etc [2] [4]. For these reasons, mean-square-error (MSE) based problems are commonly examined in multiuser networs. The linear MMSE transceiver design problems for multiuser MIMO systems can be examined in the uplin and/or downlin channels. In [5] and [6], sum MSE minimization problem is examined in the uplin channel. The authors of these papers exploit the fact that the sum MSE optimization problem in the uplin channel (after ran relaxation) can be formulated as a convex optimization problem for which global optimal solution can be obtained efficiently. It is well now that transceiver design problems in the uplin channel are better understood than that of the downlin channel. Due to this fact most literatures (also our current paper) focus on solving transceiver design problems in the downlin channel. In [7], the sum MSE minimization problem is addressed in the downlin channel. This paper has solved the latter problem directly in the downlin channel. In this channel, however, since the precoders of all users are jointly coupled, downlin MSE-based transceiver design problems have more complicated mathematical structure than their dual uplin problems [3], [8]. In [3] and [9], the sum MSE minimization constrained with a total base station (BS) power problem is considered in the downlin channel. The authors of these papers solve the downlin sum MSE minimization problem by examining the equivalent uplin problem, and then establishing the MSE duality between uplin and downlin channels. These two papers show that duality based approach of solving the downlin problem has easier to handle mathematical structure than that of [7]. Moreover, in some cases, duality solution approach can exploit the hidden convexity of the downlin channel MSE-based problems (for example minimization of sum MSE constrained with a total BS power problem [3], [8]). These duality are established by assuming that perfect channel state information (CSI) is available at the BS and mobile stations (MSs). However, due to the inevitability of channel estimation error, CSI can never be perfect. This motivates [] to establish the MSE duality under imperfect CSI for multiple-input singleoutput (MISO) systems. The latter wor is extended in [] for MIMO case. In [2], the MSE downlin-uplin duality has been established by considering imperfect CSI both at the BS

2 2 and MSs, and with antenna correlation only at the BS. This duality is examined by analyzing the Karush-Kuhn-Tucer conditions of the uplin and downlin channel problems. In [3], we have established three inds of MSE uplin-downlin duality by considering that imperfect CSI is available both at the BS and MSs, and with antenna correlation only at the BS. These duality are established by extending the three level MSE duality of [8] to imperfect CSI. In [4], we have shown that the three inds of MSE duality nown from [8] can be established for the perfect and imperfect CSI scenario just by transforming the power allocation matrices from uplin to downlin channel and vice versa. All of these MSE duality are established by assuming that the entries of the noise vector of each MS are independent and identically distributed (i.i.d) zero-mean circularly symmetric complex Gaussian (ZM- CSCG) random variables all with the same variance. However, in practice, MSs are spaced far apart from each other and the noise vector of each MS may include other interference Gaussian signals [2]. For these reasons, the noise variances of all MSs are not necessarily the same. This motivates [5] to exploit the three inds of MSE downlin-uplin duality nown in [8] and [4] for the scenario where the noise vector of each MS is a ZMCSCG random variable with arbitrary covariance matrix. However, still the wor of [5] exploits the MSE downlin-uplin duality by assuming that perfect CSI is available at the BS and MSs. Moreover, the duality of all of the aforementioned papers including [5] are able to solve total BS power constrained MSE or average mean-squareerror (AMSE)-based problems only. In [6] (see also [7] and [8]), duality based iterative solutions for rate and signalto-interference-and-noise ratio (SINR)-based problems with multiple linear transmit covariance constraints (LTCC) have been proposed. The problems of [6] are examined as follows. First, new auxiliary variables are incorporated to combine the LTCC into one constraint. Second, eeping the introduced auxiliary variables constant, the existing duality approach is applied to get the optimal covariance matrices. Third, eeping the covariance matrices of all users constant, the introduced variables are updated by sub-gradient optimization method. Fourth, the second and third steps are repeated until convergence. However, the aforementioned iterative algorithm has one major drawbac. For fixed introduced variables, if the global optimality of the downlin (dual uplin) problem is not ensured, this iterative algorithm is not guaranteed to converge. Besides this drawbac, the application of the latter duality for MSE-based problems is not clear. In a practical multi-antenna BS system, the maximum power of each BS antenna is limited [9]. Due to this, we developed duality based iterative algorithms to solve sum MSE-based constrained with each BS antenna power design problems for the perfect CSI scenario in [2]. On the other hand, channel estimation error is inevitable, and in some scenario allocating different powers to different users or sym- Note that for fixed introduced variables, the global optimum covariance matrices for the problems of [6] can be obtained. Thus, the duality-based iterative algorithm of this paper is guaranteed to converge for the problems of [6]. However, as will be clear later, since all of our problems are non-convex, the iterative algorithm of [6] is not guaranteed to converge for our problems. bols according to their channel conditions (to ensure fairness among users or symbols) has practical interest. This motivates us to first propose novel generalized sum AMSE duality and then we utilize our new duality results to examine the following sum MSE-based robust design problems. Robust sum MSE minimization with a per total BS (P), per BS antenna (P2), per user (P3) and per symbol (P4) power constraints, respectively. These problems are examined by considering that the BS and MS antennas exhibit spatial correlations, and the CSI at both ends is imperfect. The robustness against imperfect CSI is incorporated into our designs using stochastic approach [4]. Moreover, the noise vector of each MS is a ZMCSCG random variable with arbitrary covariance matrix. To the best of our nowledge, the problems P P4 are not convex. Furthermore, duality based solutions for these problems with our noise covariance matrix assumptions are not nown. In the current paper, we propose duality based iterative solutions to solve the problems. Each of these problems is solved as follows. First, we establish novel sum AMSE duality. Second, we formulate the power allocation part of the problem in the downlin channel as a Geometric Program (GP). Third, using the duality result and the solution of GP, we utilize alternating optimization technique to solve the original downlin problem. We have established novel downlin-uplin duality to solve P and P2. On the other hand, we have established novel downlin-interference duality to solve P3 and P4. For problem P, the current duality is established by modifying the constraint function of the dual uplin channel problem. And, for the problems P2 and P3 P4, our duality are established by formulating the noise covariance matrices of the uplin and interference channels as fixed point functions, respectively. The duality of this paper generalize all existing sum MSE (AMSE) duality. The main contributions of the current paper can thus be summarized as follows. ) We have established novel sum AMSE downlin-uplin and downlin-interference duality to solve the problems P P4 for the generalized scenario where the noise vector of each MS is a ZMCSCG random variable with arbitrary covariance matrix, the BS and MS antennas exhibit spatial correlations and the CSI at both ends is imperfect. The downlin-uplin duality are established to solve P and P2, whereas the downlin-interference duality are established to solve P3 and P4. For problem P, the current duality is established by modifying the constraint function of the dual uplin channel problem. And, for the problems P2 and P3 P4, our duality are established by formulating the noise covariance matrices of the uplin and interference channels as fixed point functions, respectively. Therefore, the current sum AMSE duality generalize the hitherto sum AMSE duality. 2) By employing the system model of [4] and [5], we formulate the power allocation part of our robust sum MSE minimization problems in the downlin channel as GPs. These GPs are formulated by modifying the GP formulation approach of [3] to our scenario. Consequently, our GP solution and duality results enable us

3 3 to solve the problems P - P4 by applying alternating optimization techniques (i.e., duality based iterative algorithms) of [3] and [4]. As will be clear later, the current duality based iterative algorithms can solve other sum MSE-based robust design problems. 3) In our simulation results, we have observed that the proposed duality based iterative algorithms utilize less total BS power than that of existing algorithms for the problems P2 P4. 4) We examine the effects of channel estimation errors and antenna correlations on the system performance. The remaining part of this paper is organized as follows. In Section II, multiuser MIMO downlin, and virtual uplin and interference system models are presented. In Section III, MIMO channel model under imperfect CSI is discussed. In Section IV, we formulate our robust design problems P - P4 and discuss the general framewor of our duality based iterative solutions. Sections V - IX present the proposed duality based iterative solutions for solving the problems P - P4. The extension of our duality based iterative algorithms to other problems is discussed in Section X. In Section XI, computer simulations are used to compare the performance of the proposed duality based iterative algorithms with that of existing algorithms, and the robust and non-robust/naive designs. Finally, conclusions are drawn in Section XII. Notations: The following notations are used throughout this paper. Upper/lower case boldface letters denote matrices/column vectors. The [X] (n,n), [X] (n,:), tr(x), X T, X H and E(X) denote the (n, n) element, nth row, trace, transpose, conjugate transpose and expected value of X, respectively. I n (I) is an identity matrix of size n n (appropriate size) and C(R) M M represent spaces of M M matrices with complex (real) entries. The diagonal and bloc-diagonal matrices are represented by diag(.) and bldiag(.) respectively. Subject to is denoted by s.t and (.) denotes optimal solution. The nth norm of a vector x is represented by x n. The superscripts (.) DL, (.) UL and (.) I denotes downlin, uplin and interference, respectively. d H H d 2 = d B H H 2 d K d d 2 d K d d K H H K (a) H d 2 H 2 V V V 2 V K H H 2 V 2 H 2K V K H 2KSK H K H KKSK H K (b) H S H 2S H KS H KK (c) n 2 n n K n H K W H W H 2 W H K T H n I n I S n I K H KSK n I KSK d d 2 d K d t H t H S t H K t H KSK ˆd ˆd S ˆd K ˆd KSK Fig.. Multiuser MIMO system model. (a) downlin channel. (b) virtual uplin channel. (c) virtual interference channel. II. SYSTEM MODEL In this section, multiuser MIMO downlin, and virtual uplin and interference system models are considered. The BS equipped with N transmit antennas is serving K MSs. Each MS has M antennas to multiplex S symbols. The total number of MS antennas and symbols are M = K M and S = K S, respectively. The entire symbol can be written in a data vector d = [d T,, d T K ]T, where d C S is the symbol vector for the th MS. In the downlin channel (see Fig..(a)), the BS precodes d C S into an N length vector by using the overall precoder matrix B = [B,, B K ], where B C N S is the precoder matrix for the th MS. The th MS uses a linear receiver W C M S to recover its symbol d as d DL =W H (H H Bd + n ) = W H (H H B i d i + n ) i= where n C M is the additive Gaussian noise at the th MS and H H CM N is the MIMO channel between the BS and th MS. Without loss of generality, we can assume that the entries of d are i.i.d ZMCSCG random variables all with unit variance 2, i.e., E{d d H } = I S, E{d d H i } =, i, and E{d n H i } =. The noise vector of the th MS is a ZMCSCG random variable with positive semi-definite covariance matrix R n C M M. To establish the sum AMSE duality, we model the virtual uplin and interference channels as shown in Fig..(b)-(c). The virtual uplin and interference channels are modeled by introducing precoders {V = [v,, v S ]} K and decoders {T = [t,, t S ]} K, where v s C M and t s C N,. In the uplin channel, it is assumed that n is a ZMCSCG random variable with diagonal covariance 2 We would lie to mention here that when the symbols are not white prewhitening operation can be applied before the transmitter (precoder) and the inverse operation can be performed after the receiver (decoder) [2].

4 4 matrix Ψ R N N = diag(ψ,, ψ N ). In the interference channel, it is assumed that the th user s sth symbol is estimated independently by t s C N. Moreover, {n I s, s}k (Fig..c) are also ZMCSCG random variables with covariance matrices { s R N N = diag(δ s,, δ sn ), s} K and the channels between the th transmitter and all receivers are the same (i.e., {H js = H, j, s} K ). In both the uplin and interference system models, the transmitters are the same (i.e., {V } K ). Furthermore, the channels between all transmitters and each receiver (one receiver in the case of uplin system model and S receivers in the case of interference system model) are the same (i.e., {H } K ). As will be clear later, the virtual uplin and interference system models differ on their noise covariance matrices (which is clearly seen from Fig..(b)-(c)) and the approach where the decoders are designed. In the case of downlin-uplin duality, all decoders of the uplin channel are designed with a common noise covariance matrix Ψ, whereas, in the case of downlin-interference duality, the th user sth symbol decoder of the interference channel t s is designed by employing its own noise covariance matrix s. III. CHANNEL MODEL Considering antenna correlation at the BS and MSs, we model the Rayleigh fading MIMO channels as H H = R /2 m HH w R/2 b,, where the elements of {HH w }K are i.i.d ZMCSCG random variables all with unit variance, and R b C N N and R m C M M are antenna correlation matrices at the BS and MSs, respectively [4], [2]. The channel estimation is performed on {H H w }K using an orthogonal training method [22]. Upon doing so, the th user true channel H H and its minimum-mean-square-error (MMSE) estimate Ĥ H can be related as (see Section II.B of [22] for more details about the channel estimation process) H H =ĤH + R /2 m EH wr /2 b = ĤH + E H, () where R m = (I M + σe 2 R m ), E H is the estimation error and the entries of E H w are i.i.d with CN (, σ2 e ). By robustness, we mean that {E H w }K are unnown but {ĤH, R b, Rm and σe 2 }K are nown. We assume that each MS estimates its channel and feeds the estimated channel bac to the BS without any error and delay. Thus, both the BS and MSs have the same channel imperfections. With these assumptions, the downlin instantaneous MSE (ξ DL ) and AMSE ( ξ DL ) matrices of the th user are given by ξ DL ξ DL =E d,n {(d =I S + W H (H H d DL DL )(d d ) H } B i B H i H + R n )W i= W H H H B B H H W =E E H w {ξ DL } = I S + W H Γ DL W W H ĤH B B H ĤW (2) where Γ DL = ĤH BBH Ĥ + σe 2 tr{r bbb H }R m + R n. The total sum AMSE ξ DL = K tr{ξdl } is given by ξ DL =S + tr{w H Γ DL W W H ĤH B B H ĤW }. (3) IV. PROBLEM FORMULATIONS Mathematically, the aforementioned robust sum MSE minimization problems can be formulated as P : P2 : P3 : min {B,W } K min {B,W } K ξ DL, s.t tr( B B H ) P max (4) ξ DL, s.t [ B B H ] n,n p n, n (5) min ξ DL, s.t tr{b B H } ˇp, (6) {B,W } K P4 : min ξ DL, s.t b H sb s ˇp s,, s (7) {B,W } K where P max, p n, ˇp and ˇp s are the maximum powers available at the BS, nth BS antenna, th user and th user sth symbol, respectively. Since, the problems P - P4 are not convex, convex optimization framewor can not be applied to solve them. To the best of our nowledge, duality based solutions for the problems P P4 are not nown. In the following, we present duality based iterative algorithm to solve each of these problems. For better exposition of our algorithms, let us explain the general framewor of downlin-uplin duality based approach for solving each of these problems as shown in Algorithm I. Algorithm I Initialization: For each problem, initialize {B } K such that the power constraint functions are satisfied with equality. Then, update {W } K by using minimum average-mean square-error (MAMSE) receiver approach, i.e., W = (Γ DL ) Ĥ H B,. (8) Repeat Uplin channel ) Transfer the total sum AMSE from downlin to uplin channel. 2) Update the receivers of the uplin channel {T } K using MAMSE receiver technique. Downlin channel 3) Transfer the total sum AMSE from uplin to downlin channel. 4) Update the receivers of the downlin channel {W } K by MAMSE receiver (8). Until convergence. One can notice that this iterative algorithm is already nown in [3], [4] and [4]. However, the iterative approach of these papers are limited to solve MSE-based constrained with a total BS power problems with {R n = σ 2 I} K. As will be clear later, the main challenge arises at step 3 of Algorithm I where

5 5 at this step, the approaches of these papers are not able to ensure the power constraints of P P4. In the following sections, we establish our novel sum AMSE downlin-uplin and downlin-interference duality. The downlin-uplin duality are able to maintain the power constraints of P and P2 at step 3 of Algorithm I. Thus, the latter duality can solve these two problems using Algorithm I. The sum AMSE downlin-interference duality are able to maintain the power constraint of each user and symbol at step 3 of Algorithm I. Hence, these duality can solve the problems P3 and P4 with Algorithm I. V. SUM AMSE DOWNLINK-UPLINK DUALITY TO SOLVE P In this section, we establish the sum AMSE downlinuplin duality to solve total BS power constrained robust sum MSE minimization problem (P). This duality can be established by assuming that Ψ = σ 2 I, where σ 2 >. Now, we transfer the sum AMSE from downlin to uplin channel and vice versa. To this end, we compute the total sum AMSE in the uplin channel as ξ UL =S + tr{t H Γ c T + σ 2 T H T where Γ c = K i= (ĤiV i V H i 2R{T H ĤV }} (9) Ĥ H i + σ 2 ei tr{r miv i V H i }R bi). A. Sum AMSE transfer (From downlin to uplin channel) The sum AMSE can be transferred from downlin to uplin channel by using a nonzero scaling factor β which satisfies V = βw, T = B/ β () where V = bldiag(v,, V K ) and T = [T,, T K ]. Substituting V and T of (9) by () and then equating ξ UL = ξ DL yields tr{b H ĤWW H Ĥ H B} + β σ 2 tr{b H B} + tr{b H ( σeitr{r 2 mi W i Wi H }R bi )B } = i= tr{w H Ĥ H BB H ĤW} + tr{w H R n W} + tr{w H (σetr{r 2 b BB H }R m )W } () where R n = bldiag(r n,, R nk ). It follows, β can be determined as β 2 = σ 2 tr{b H B}/tr{W H R n W}. (2) Thus, during the sum AMSE transfer from downlin to uplin channel, the following holds true P DL sum = tr{b H B} = σ 2 tr{vh R n V}. (3) B. Sum AMSE transfer (From uplin to downlin channel) For a given uplin channel precoder/decoder pairs (V/T), the sum AMSE can be transferred from uplin to downlin channel by using a positive β which satisfies B = βt, W = V/β. (4) Substituting these B and W in (3) and equating ξ UL = ξ DL, β can be determined as β 2 = tr{v H R n V}/tr{σ 2 T H T}. (5) As can be seen from (2) and (5), the scaling factors β and β do not depend on {σe 2 }K. This fact can be seen from ξ DL and ξ UL, after substituting {Γ DL }K and Γ c, where {σe 2 }K are amplified by the same factor. The downlin power is given by P DL sum = tr{bb H } = tr{β 2 TT H } = tr{v H R n V}/σ 2. (6) From (3) and (6), we can see that tr{bb H } = tr{v H R n V}/σ 2 = Psum DL is always satisfied. Consequently, the duality of this section maintains the power constraint of P during the sum AMSE transfer from uplin to downlin channel and vice versa. Therefore, with the duality of this section, the latter problem can be solved iteratively by Algorithm I. When {σ 2 =, {σe 2 = }K }, the duality of this section turns to that of in [5]. When R n = σ 2 I, our duality fits that of in [3]. Hence, the sum AMSE duality of this section generalizes all existing sum AMSE downlin-uplin duality. VI. SUM AMSE DOWNLINK-UPLINK DUALITY TO SOLVE P2 This duality is established to solve per BS antenna power constrained robust sum MSE minimization problem (P2). To establish this duality, we first compute the total sum AMSE in the uplin channel as ξ UL2 =tr{t H Γ c T + T H ΨT T H ĤV V H Ĥ H T} + S. (7) Now, we transfer the sum AMSE from downlin to uplin channel and vice versa. A. Sum AMSE transfer (From downlin to uplin channel) In order to use this sum AMSE transfer for a per BS antenna power constrained robust sum MSE minimization problem, we set the uplin channel precoder and decoder pairs as V = W, T = B. (8) Substituting (8) into (7) and then equating ξ DL = ξ UL2 yields tr{w H R n W } = tr{b H ΨB }, τ = N ψ n p n = p T ψ (9) n=

6 6 where τ = K tr{wh R nw }, ψ = [ψ,, ψ N ] T, p = [ p,, p N ] T, p n = tr{ bh n bn } and bh n is the nth row of B. The above equation shows that by choosing {ψ n } N n= appropriately, we can transfer any precoder/decoder pairs of the downlin channel to the corresponding decoder/precoder pairs of the uplin channel. However, here {ψ n } N n= should be selected in a way that P2 can be solved using Algorithm I. To this end, we choose ψ as τ p T ψ. (2) By doing so, the uplin channel will achieve lower sum AMSE compared to that of the downlin channel (ξ DL ξ UL2 ). As will be clear later, to solve (5) with Algorithm I, ψ should be selected ensuring (2). This shows that for P2, step of Algorithm I can be carried out with (8). To perform step 2 of Algorithm I, we update T of (8) by using the uplin MAMSE receiver which can be expressed as T =(Γ c + Ψ) ĤV = (A + Υ + Ψ) ĤW (2) where A = ĤWW H Ĥ H, Υ = K i= σ2 ei tr{r miw i Wi H}R bi and the second equality is obtained by applying (8) (i.e., V=W). The above expression shows that by choosing {ψ n > } N n=, (A + Υ + Ψ) is always invertible. Next, we transfer the total sum AMSE from uplin to downlin channel (i.e., we perform step 3 of Algorithm I) and show that the latter sum AMSE transfer ensures the power constraint of each BS antenna. B. Sum AMSE transfer (From uplin to downlin channel) For a given total sum AMSE in the uplin channel, we can achieve the same sum AMSE in the downlin channel by using a nonzero scaling factor ( β) which satisfies B = βt, W = V/ β. (22) In this precoder/decoder transformation, we use the notations B and W to differentiate with the precoder and decoder matrices used in Section VI-A. By substituting (22) into (3) (with B=B, W=W) and then equating the resulting sum AMSE with that of the uplin channel (7), β can be determined as N tr{v β H 2 R n V } = ψ n t H t n n, n= β N 2 = tr{w H R n W }/ ψ n t H t n n = n= τ N n= ψ n t H n t n (23) where t H n is the nth row of the MAMSE matrix T (2) which is given by [(A+Υ+Ψ) ] (n,:) HW and the second equality follows from (8). The power of each BS antenna in the downlin channel is thus given by b H n b n =tr{ β 2 t H n t n } = τ t H n t n N i= ψ i t H i t i p n, n (24) where b H n is the nth row of B. We can rewrite the above expression as where f n = τ p n ψ n t H n t n N i= ψ i t H i t i = τ p n ψ n f n, n (25) ψ n [(A + Υ + Ψ) ] (n,:) A([(A + Υ + Ψ) ] (n,:) ) H N i= ψ i[(a + Υ + Ψ) ] (i,:) A([(A + Υ + Ψ) ] (i,:) ) H. Let us assume that there exist {ψ n > } N n= that satisfy From (26), we get n= ψ n = f n, n. (26) ψ n = τ p ψ n t H t n n ψ n t H N n i= ψ i t H t ψ n p n = τ t n n N i i i= ψ i t H t i i N N ψ n t H n ψ n p n = τ t n N i= ψ i t H t = τ. (27) i i n= The above expression shows that the solution of (26) satisfies (27). Moreover, as { p n p n } N n=, the latter solution also satisfies (2). Therefore, for P2, by choosing {ψ n } N n= such that (26) is satisfied, step 3 of Algorithm I can be performed. Next, we show that there exists at least a set of feasible {ψ n > } N n= that satisfy (26). In this regard, we consider the following Theorem [23], [24]. Theorem : Let (X,. 2 ) be a complete metric space. We say that : X X is an almost contraction, if there exist κ [, ) and χ such that (x) (y) 2 κ x y 2 + χ y (x) 2, x, y X. (28) If satisfies (28), then the following holds true ) x X : x = (x). 2) For any initial x X, the iteration x n+ = (x n ) for n =,, 2, converges to some x X. 3) The solution x is not necessarily unique. Proof: See [23], [24] (Theorem.). Note that in [23] and [24], Theorem has been proven for a generalized complete metric space (X, d) instead of (X,. 2 ). By defining as (ψ) [f, f 2,, f N ] with {ψ n [ǫ, (τ ǫ N i=, i n p i)/ p n ]} N n= 3, it can be easily seen that (ψ ) (ψ 2 ) 2, ψ ψ 2 2 and ψ 2 (ψ ) 2 are bounded for any ψ, ψ 2 ψ. This shows the existence of κ and χ ensuring (28). Consequently, (ψ) is an almost contraction which implies ψ n+ = (ψ n ), with ψ = [ψ, ψ 2,, ψ N ] ǫ N, for n =,, 2, converges (29) where N is an N length vector with each element equal to unity. Thus, there exists {ψ n ǫ} N n= that satisfy (26) and its solution can be obtained using the above fixed point 3 Here ǫ should be very close to zero (for our simulation, we use ǫ = min( 6, {τ/ p n} N n= )).

7 7 iterations. Once the appropriate {ψ n } N n= is obtained, step 4 of Algorithm I is immediate and hence P2 can be solved iteratively using this algorithm. Note that when {σ 2 e = }K, the duality of this section turns to that of [2]. VII. SUM AMSE DOWNLINK-INTERFERENCE DUALITY TO SOLVE P3 This ind of duality is established to solve per user power constrained robust sum MSE minimization problem (P3). To establish this duality, it is sufficient to assume { s = = µ I N, s} K. With this assumption, the th user AMSE and total sum AMSE in the interference channel can be expressed as ξ I =tr{t H Γ c T + µ T H T + I S T H H V V H H H T } (3) ξ I = ξ I = tr{t H Γ c T T H HV V H H H T} + S+ tr{µ T H T }. (3) Lie in Section VI, here we transfer the sum AMSE from downlin to interference channel and vice versa. A. Sum AMSE transfer (From downlin to interference channel) Lie in Section VI, it can be shown that the sum AMSE can be transferred from downlin to interference channel by setting V = W and T = B. Substituting these V and T in (3), then equating ξ DL = ξ I yields tr{w H R n W } = tr{µ B H B } τ = p T µ where p = tr{b H B }, p = [ p,, p K ] T and µ = [µ,, µ K ] T. For this sum AMSE transfer, we also choose µ such that τ p T µ (32) is satisfied. This shows that step of Algorithm I can be carried out by choosing V = W and T = B. To perform step 2 of Algorithm I, we update each receiver of the interference channel by applying MAMSE receiver method which is given by t s =(Γ c + µ I N ) Ĥ v s = (A + Υ + µ I N ) Ĥ w s,, s. (33) The above expression shows that by choosing {µ > } K, invertibility of (A + Υ + µ I) can be ensured. B. Sum AMSE transfer (From interference to downlin channel) For a given total sum AMSE in the interference channel, we can achieve the same sum AMSE in the downlin channel by using a nonzero scaling factor ( ˇβ) which satisfies B = ˇβT, W = V/ ˇβ. By substituting B and W in (3) (with B=B, W=W) and then equating the resulting total sum AMSE with that of the interference channel (3), ˇβ can be determined as ˇβ 2 = tr{w H R n W }/ tr{µ T H T } τ = K µ tr{(a + Υ + µ I N ) 2 A } (34) where A = ĤW W HĤH and the second equality is derived by substituting {t s, s} K with the MAMSE receiver (33). The power of each user in the downlin channel is thus given by tr{ B BH } =tr{ ˇβ 2 T T H } (35) = τtr{(a + Υ + µ I N ) 2 A } K i= µ itr{(a + Υ + µ i I N ) 2 A i } ˇp. This inequality constraint can be expressed as µ ˇf, (36) where ˇf = τˇp tr{µ (A+Υ+µ I N ) 2 A } K. Lie in Section i= µitr{(a+υ+µiin ) 2 A i} VI, it can be shown that there exist {µ > } K that satisfy µ = ˇf,. (37) Moreover, (37) can be solved exactly lie that of (26) and the solution of (37) satisfies (32). As a result, one can apply the duality of this section to solve P3 by Algorithm I. VIII. SUM AMSE DOWNLINK-INTERFERENCE DUALITY TO SOLVE P4 This duality is established to solve per symbol power constrained robust sum MSE minimization problem (P4). To establish this duality, it is sufficient to assume { s = µ s I N, s} K. With the latter assumption, the th user sth symbol AMSE and total sum AMSE in the interference channel can be expressed as ξ I2 s = + t H sγ c t s + µ s t H st s 2R{t H sh v s } (38) S ξ I2 = ξ I2 s = tr{t H Γ c T T H HV V H H H T}+ s= S + S µ s t H st s. (39) s= Lie in Section VII, we now transfer the sum AMSE from downlin to interference channel and vice versa.

8 8 A. Sum AMSE transfer (From downlin to interference channel) The sum AMSE can be transferred from downlin to interference channel by setting V = W and T = B. Substituting these V and T in (39), equating ξ DL = ξ I2 and after some steps we get τ = ˇ p T µ, where ˇ p s = b H s b s, ˇ p = [ˇ p,, ˇ p S, ˇ p K,, ˇ p KSK ] T and µ = [ µ,, µ S,, µ K,, µ KSK ] T. For this sum AMSE transfer, we also choose µ such that τ ˇ p T µ (4) is satisfied. At this stage step of Algorithm I is performed. To reduce the sum AMSE in the interference channel (i.e., to perform step 2 of Algorithm I), we apply the MAMSE receiver for the th user sth symbol as t s =(Γ c + µ s I N ) Ĥ v s =(A + Υ + µ s I N ) Ĥ w s,, s. (4) As we can see, when { µ s >, s} K, (A + Υ + µ si N ) is always invertible. B. Sum AMSE transfer (From interference to downlin channel) For a given total sum AMSE in the interference channel, we can achieve the same sum AMSE in the downlin channel by using a nonzero scaling factor ˇβ which satisfies B = ˇβT and W = V/ ˇβ. By substituting these B and W in (3) (with B=B, W=W) and then equating the resulting total sum AMSE with that of the interference channel (39), ˇβ can be determined as S ˇβ 2 = tr{w H R n W }/ tr{ µ s t H st s } = s= τ K S s= µ str{(a + Υ + µ s I N ) 2 A s } (42) where the second equality is derived by substituting {t s, s} K with the MAMSE receiver (4) and A s = Ĥ w s w sĥh H. The power of the th user sth symbol in the downlin channel is thus given by tr{ b s bh s } = tr{ ˇβ2 t s t H s} (43) τtr{(a + Υ + µ s I N ) 2 A s } = K Si i= j= µ ijtr{(a + Υ + µ ij I N ) 2 A ij } ˇp s. This inequality constraint can also be expressed as µ s ˇ fs,, s (44) where ˇ fs τtr{ µ = s(a+υ+ µ si N ) 2 A s} ˇp K Si. Lie in s i= j= µijtr{(a+υ+ µijin ) 2 A ij} Section VI, it can be shown that there exist { µ s >, s} K that satisfy µ s = ˇ fs,, s. Thus, the feasible { µ s, s} K of (44) can be found exactly lie that of (26). As a result, with the duality of this section, we can solve P4 iteratively with Algorithm I. IX. A METHOD TO IMPROVE THE CONVERGENCE SPEED OF Algorithm I To increase the convergence speed of Algorithm I, a downlin power allocation step can be included inside Algorithm I. In [3], for fixed transmit and receive filters, the power allocation part of P with {R n = σ 2 I, R m = I} K has been formulated as a GP by employing the system model of [4]. This GP formulation is derived by extending the approach of [4] to imperfect CSI scenario. Moreover, in [4], we show that the system model of [4] is appropriate to solve any ind of total BS power constrained robust MSE-based problems using duality approach (alternating optimization). This motivates us to utilize the system model of [4] in the downlin channel only and then include the power allocation step (i.e., GP) into Algorithm I for each of the problems P P4. To this end, we decompose the precoders and decoders of the downlin channel as B =G P /2, W = U α P /2, (45) where P = diag(p,, p S ) R S S, G = [g g S ] C N S, U = [u u S ] C M S and α = diag(α,, α S ) R S S are the transmit power, unity norm transmit filter, unity norm receive filter and receiver scaling factor matrices of the th user, respectively, i.e., {gs H g s = u H s u s =, s} K. With this decomposition, our downlin system model (Fig..(a)) can be plotted as shown in Fig. 2. d d 2 d K = d P 2 G H H H H 2 H H K n 2 n n K U H U H 2 U H K α α 2 α K Fig. 2. Downlin multiuser MIMO system model 2. P 2 P 2 2 P 2 K For better exposition of the GP, we collect the powers of the entire symbol as P = bldiag(p,, P K ) = diag(p,, p S ), the overall filter matrix at the BS and users as G = [G,, G K ] = [g,, g S ] and U = bldiag(u,, U K ) = [u,, u S ], respectively, where g l C N (u l C M ) is the transmit (receive) filter of the lth symbol with g l 2 = u l 2 =. The scaling factors are staced as α = bldiag(α,, α K ) = diag(α,, α S ) and the AMSE in the downlin channel can be collected as ξ = [ξ DL,,, ξ DL K,S K ] T d d 2 d K = [ξ DL,, ξ DL S ] T = [{ξ DL l } S l= ]T (refer [4] and [3] for the details about (45) and the above descriptions). By applying (45) and the technique of [4] (see (23) of [4]), the AMSE of the lth symbol in the downlin channel can be expressed as ξ DL l =p l [(D + α 2 Φ T )p] l + p l αl 2 u H l R n u l (46)

9 9 where σef(j) 2 R/2 mf(j) u j 2 2 R /2 bf(j) g l 2 2 [Φ] l,j = + gl H Ĥu j 2, for l j, for l = j [D] l,l =α 2 l ( g H l 2α l R(u H l (47) Ĥu l 2 + σ 2 ef(l) R/2 mf(l) u l 2 2 R /2 bf(l) g l 2 2) Ĥ H g l ) + (48) and p = [p,, p S ] T. In (47) and (48), f(i) is the smallest such that m= S m i, where S m is the mth user total number of symbols. For example, if K = 2, M = 2 and S = 2 then σef() 2 = σ2 ef(2) = σ2 e, σef(3) 2 = σ2 ef(4) = σ2 e2, R bf() = R bf(2) = R b, R mf() = R mf(2) = R m, R bf(3) = R bf(4) = R b2 and R mf(3) = R mf(4) = R m2. Using the above ξ DL l, for fixed G, U and α, the power allocation part of problem P can be formulated as min {p l} S l= S l= ξ DL l, s.t S p l P max. (49) As ξ DL l is a posynomial (where {p l } S l= are the variables), (49) is a GP for which global optimality is guaranteed. Thus, it can be efficiently solved by using standard interior point methods with a worst-case polynomial-time complexity [25]. Lie in P, it can be shown that for fixed G, U and α, the power allocation parts of P2 P4 can be formulated as GPs. Our duality based iterative algorithm for each of these problems including the power allocation step is summarized in Algorithm II. Algorithm II Initialization: For P, set any σ 2 > (we use σ 2 = for our simulation). Then, initialize all the other parameters lie in Algorithm I. Repeat Uplin (Interference) channel ) For P transfer the total sum AMSE from downlin to uplin channel by (2) and for P2 P4 set V = W, T = B. Then, for P2 compute {ψ n } N n= using (25); for P3 compute {µ } K using (36) and for P4 compute { µ s, s} K using (44). 2) Update the MAMSE receivers of the uplin (interference) channel of P, P2, P3 and P4 using T = Γ c ĤV, (2), (33) and (4), respectively. Downlin channel 3) Transfer the total sum AMSE from uplin (interference) to downlin channel using (5), (23), (34) and (42) for P, P2, P3 and P4 respectively. 4) For each of the problems P P4, decompose the precoder and decoder matrices of each user as in (45). Then, formulate and solve the GP power allocation part. For example, the power allocation part of P can be expressed in GP form as (49). 5) For each of the problems P P4, by eeping {P } K constant, update the receive filters {U } K and scaling factors {α } K by applying downlin MAMSE receiver approach i.e., {U α = Γ ĤH G P } K DL, where Γ = ĤH GPGH Ĥ + l= σe 2 tr{r bgpg H }R m + R n. Note that in these expressions, {α } K are chosen such that each column of {U } K has unity norm. Then, compute {B, W } K using (45). Until convergence. Convergence: It can be shown that at each step of this algorithm, the sum AMSE of the system is nonincreasing [3], [4]. Thus, the above iterative algorithm is guaranteed to converge. However, since all of our problems are non-convex, this iterative algorithm is not guaranteed to converge to the global optimum. X. EXTENSION OF Algorithm II TO OTHER PROBLEMS In some cases, the power of each precoder entry needs to be constrained i.e,. b H sn b sn p sn,, s, n. This ind of power constraint has practical interest for coordinated BS systems where each BS is equipped with single antenna and the BS targets to allocate different powers to different symbols (see also [26], [27]). Mathematically, the latter problem can be formulated as P5 : min {B,W } K ξ DL, s.t b H snb sn p sn,, s, n. (5) It can be shown that this problem can be solved by Algorithm II with { s = diag(δ s,, δ sn ), s} K. Consider the following problem variations P6 : P7 : P8 : min {B,W } K s= S p s, s.t ξ DL ε t, tr( B B H ) P max (5) min {B,W } K s= S p s, s.t ξ DL ε t, [ B B H ] n,n p n, n (52) min {B,W } K s= S p s, s.t ξ DL ε t, tr{b B H } ˇp, (53) S P9 : min p s, {B,W } K s= s.t ξ DL ε t, b H sb s ˇp s,, s (54) S P : min p s, {B,W } K s= s.t ξ DL ε t, b H snb sn p sn,, s, n (55) where ε t is the total sum AMSE target. The power allocation parts of P6 P can be formulated as GPs. It is clearly seen that by modifying the power allocation step of P to that of P6, one can apply the solution approach of P to solve P6 (see also [4]). Next, we explain how the solution approach of P2 can be extended to solve P7. In the latter problem, each iteration should guarantee a non-increasing total

10 BS power. To ensure this non-increasing total BS power, (9) must be satisfied. This can be achieved just by modifying (24) to { b H n b n p n } N n=. This leads to the following fixed point function. where f n = τ p n ψ n = fn, n (56) ψ n [(A + Υ + Ψ) ] (n,:) A([(A + Υ + Ψ) ] (n,:) ) H N i= ψ i[(a + Υ + Ψ) ] (i,:) A([(A + Υ + Ψ) ] (i,:) ) H. Therefore, one can apply the solution approach of P2 to solve P7 by modifying the power allocation step of P2 to that of P7 and computing {ψ n } N n= with (56). The problems P8 P can be solved lie in P7. The details are omitted for conciseness. It is clearly seen that the analysis of this paper can be applied to solve other robust sum MSE-based constrained with groups of antennas, users or symbols power problems. XI. SIMULATION RESULTS In this section, we present simulation results for P P4. For all of our simulations, we tae K = 2, {M = S = 2} K and N = 4. The entries of {R b, R m } K are taen from a widely used exponential correlation model as {R b = ρ i j b, R m = ρ i j m }K, where ρ b(ρ m ) < and i, j N(M ),. All of our simulation results are averaged over randomly chosen channel realizations. It is assumed that σe 2 =., σe2 2 =.2, ρ b =., ρ b2 =.2, ρ m =.5, ρ m2 =.2, R n = σi 2 M and R n2 = σ2i 2 M2. The Signal-to-Noise ratio (SNR) is defined as P sum /Kσav 2 and is controlled by varying σav, 2 where P sum is the total BS power, σ2 2 = 2σ 2 and σav 2 = (σ 2 + σ2)/2. 2 For the computation of SNR, we used the P sum obtained from the perfect CSI design of the proposed algorithm. We compare the performance of our iterative algorithm (Algorithm II) with that of [7] for the robust, non-robust and perfect CSI designs. The nonrobust/naive design refers to the design in which the estimated channel is considered as perfect. Note that when {σe 2 = } K, the solution method of the latter paper turns to that of in [26]. A. Simulation results for problem P In this subsection, we compare the performance of our proposed algorithm with that of [7] when P max = P sum =. The comparison is based on the total sum AMSE and average symbol error rate (ASER) 4 of all users versus the SNR. Fig. 3.(a)-(b) show that Algorithm II and the algorithm in [7] achieve the same sum AMSE and ASER. Moreover, these figures show the superior performance of the robust design compared to that of the non-robust design. Next, we discuss the effects of antenna correlations factors {ρ b, ρ m } K on 4 For the sum AMSE design, ASER is also an appropriate metric for comparing the performance of the robust and non-robust designs [22]. QPSK modulation is utilized for each symbol. the system performance. For this purpose, we change the previous {ρ b, ρ m } K to ρ b =.4, ρ b2 =.5, ρ m =.7 and ρ m2 =.8 and plot the sum AMSEs of all designs in Fig. 3.(c). From Fig. 3.(a) and Fig. 3.(c), we can observe that the sum AMSE of non-robust, robust and perfect CSI designs increase as {ρ b, ρ m } K increase. This observation fits to that of [4] where P is examined for {R n = σ 2 I} K. B. Simulation results for P2 P4 In this subsection, we compare the performances of our proposed algorithm with that of [7] based on the total power utilized at the BS and the total achieved sum AMSE. We employ { p n = 2.5} N n= for P2, {ˇp = 5} K for P3, { ˇp s = 2.5, s} K for P4 and {ρ b, ρ m } K as in the first paragraph of Section XI. As can be seen from Fig. 4.(a)-(c), for P2, the proposed algorithm utilizes less total BS power than that of [7] for all designs. However, for P3 and P4, the proposed algorithm utilizes less total BS power than that of [7] for the robust designs only. Next, with the powers of Fig. 4.(a)-(c), we plot the total sum AMSEs of our algorithm and the algorithm in [7] as shown in Figs. 5.(a)-(c). These figures show that for the robust and non-robust designs, both of these algorithms achieve the same sum AMSE. To examine the effects of antenna correlation factors, we use {ρ b, ρ m } K as in Section XI-A. Again Fig. 6.(a)-(c), show that our proposed algorithms utilize less total BS power than that of [7] in all designs and in the robust design only, for P2 and P3 P4, respectively. With the powers of Fig. 6.(a)-(c), we plot the sum AMSE of our algorithm and that of [7] as shown in Fig. 7.(a)-(c). These figures also show that both of these algorithms achieve the same sum AMSE. In all figures of this section, the robust design outperforms the non-robust design and the improvement is larger for high SNR regions. This can be seen from the term Γ DL of (2) where, at high SNR regions, σ 2 is negligible compared to σe 2 tr{r bbb H }R m (the term due to channel estimation error). Thus, in the high SNR regions, since the non-robust design does not tae into account the effect of σe 2 tr{r bbb H }R m which is the dominant term, the performance of this design degrades. This implies that as the SNR increases, the performance gap between the robust and nonrobust design increases. Furthermore, when {ρ b, ρ m } K increases, the system performance degrades. This is because as {ρ b, ρ m } K increases, the number of symbols with low channel gain increases (this can be easily seen from the eigenvalue decomposition of R b (R m )). Consequently, for a given SNR value, the total sum AMSE also increases [4]. XII. CONCLUSIONS This paper considers sum MSE-based linear transceiver design problems for downlin multiuser MIMO systems where imperfect CSI is assumed to be available at the BS and MSs. These problems are examined for the generalized scenario where the constraint functions are per BS antenna, total BS, user or symbol power, and the noise vector of each MS is a ZMCSCG random variable with arbitrary covariance matrix. Each of these problems is solved as follows. First, we establish

11 novel sum AMSE duality. Second, we formulate the power allocation part of the problem in the downlin channel as a GP. Third, using the duality result and the solution of GP, we utilize alternating optimization technique to solve the original downlin problem. We have established downlin-uplin duality to solve per BS antenna (groups of BS antenna) and total BS power constrained robust sum MSE-based problems. And, we have established downlin-interference duality to solve per user (groups of users) and symbol (groups of symbols) power constrained robust sum MSE-based problems. For the total BS power constrained robust sum MSE-based problems, the current duality is established by modifying the constraint function of the dual uplin channel problem. On the other hand, for the robust sum MSE minimization constrained with each BS antenna and each user (symbol) power problems, our duality are established by formulating the noise covariance matrices of the uplin and interference channels as fixed point functions, respectively. We have shown that our sum AMSE duality are able to solve any sum MSE-based robust design problem. Computer simulations verify the robustness of the proposed design compared to the non-robust/naive design. REFERENCES [] D. P. Palomar and S. Verdu, Gradient of mutual information in linear vector Gaussian channels, IEEE Tran. Info. Theo., vol. 52, no., pp. 4 54, Jan. 26. [2] D. P. Palomar, A unified framewor for communications through MIMO channels, Ph.D. thesis, Technical University of Catalonia (UPC), Barcelona, Spain, 23. [3] S. Shi, M. Schubert, and H. Boche, Downlin MMSE transceiver optimization for multiuser MIMO systems: Duality and sum-mse minimization, IEEE Tran. Sig. Proc., vol. 55, no., pp , Nov. 27. [4] S. Shi, M. Schubert, and H. Boche, Rate optimization for multiuser MIMO systems with linear processing, IEEE Tran. Sig. Proc., vol. 56, no. 8, pp , Aug. 28. [5] S. Serbetli and A. Yener, Transceiver optimization for multiuser MIMO systems, IEEE Tran. Sig. Proc., vol. 52, no., pp , Jan. 24. [6] E. A. Jorswiec and H. Boche, Transmission strategies for the MIMO MAC with MMSE receiver: Average MSE optimization and achievable individual MSE region, IEEE Tran. Sig. Proc., vol. 5, no., pp , Nov. 23. [7] P. Ubaidulla and A. Chocalingam, Robust transceiver design for multiuser MIMO downlin, in Proc. IEEE Global Telecommunications Conference (GLOBECOM), Nov. 3 Dec. 4 28, pp. 5. [8] R. Hunger, M. Joham, and W. Utschic, On the MSE-duality of the broadcast channel and the multiple access channel, IEEE Tran. Sig. Proc., vol. 57, no. 2, pp , Feb. 29. [9] M. Schubert, S. Shi, E. A. Jorswiec, and H. Boche, Downlin sum- MSE transceiver optimization for linear multi-user MIMO systems, in Conference Record of the Thirty-Ninth Asilomar Conference on Signals, Systems and Computers, Oct. 28 Nov. 25, pp [] M. B. Shenouda and T. N. Davidson, On the design of linear transceivers for multiuser systems with channel uncertainty, IEEE Jour. Sel. Commun., vol. 26, no. 6, pp. 5 24, Aug. 28. [] P. Ubaidulla and A. Chocalingam, Robust joint precoder/receive filter designs for multiuser MIMO downlin, in th IEEE Worshop on Signal Processing Advances in Wireless Communications (SPAWC), Perugia, Italy, 2 24 Jun. 29, pp [2] D. Ding and S. D. Blostein, Relation between joint optimizations for multiuser MIMO uplin and downlin with imperfect CSI, in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas, NV, USA, Mar. 3 Apr. 4 28, pp [3] T. Endeshaw, B. K. Chalise, and L. Vandendorpe, MSE uplindownlin duality of MIMO systems under imperfect CSI, in 3rd IEEE International Worshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Aruba, 3 6 Dec. 29, pp [4] T. E. Bogale, B. K. Chalise, and L. Vandendorpe, Robust transceiver optimization for downlin multiuser MIMO systems, IEEE Tran. Sig. Proc., vol. 59, no., pp , Jan. 2. [5] T. E. Bogale and L. Vandendorpe, MSE uplin-downlin duality of MIMO systems with arbitrary noise covariance matrices, in 45th Annual conference on Information Sciences and Systems (CISS), Baltimore, MD, USA, Mar. 2. [6] R. Zhang, Y-C. Liang, and S. Cui, Dynamic resource allocation in cognitive radio networs, IEEE Sig. Proc. Mag., vol. 27, no. 3, pp. 2 4, May 2. [7] L. Zhang, Y-C. Liang, Y. Xin, R. Zhang, and H.V Poor, On Gaussian MIMO BC-MAC duality with multiple transmit covariance constraints, in IEEE International Symposium on Information Theory (ISIT), Jun Jul. 29, pp [8] L. Zhang, R. Zhang, Y-C. Liang, Y. Xin, and H.V Poor, On the Gaussian MIMO BC-MAC duality with multiple transmit covariance constraints, Sep. 28, 4v/. [9] W. Yu and T. Lan, Transmitter optimization for the multi-antenna downlin with per-antenna power constraints, IEEE Trans. Sig. Proc., vol. 55, no. 6, pp , Jun. 27. [2] T. E. Bogale and L. Vandendorpe, Sum MSE optimization for downlin multiuser MIMO systems with per antenna power constraint: Downlinuplin duality approach, in 22nd IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), Toronto, Canada, 4 Sep. 2. [2] T. Yoo, E. Yoon, and A. Goldsmith, MIMO capacity with channel uncertainty: Does feedbac help?, in Proc. IEEE Global Telecommunications Conference (GLOBECOM), Dallas, USA, 29 Nov. 3 Dec. 24, vol., pp. 96. [22] D. Ding and S. D. Blostein, MIMO minimum total MSE transceiver design with imperfect CSI at both ends, IEEE Tran. Sig. Proc., vol. 57, no. 3, pp. 4 5, Mar. 29. [23] V. Berinde, Approximation fixed points of wea contractions using the Picard iteration, Nonlinear Analysis Forum, vol. 9, no., pp , 24. [24] V. Berinde, General constructive fixed point theorems for Ciric-type almost contractions in metric spaces, Carpathian, J. Math, vol. 24, no. 2, pp. 9, 28. [25] S. Boyd and L. Vandenberghe, Convex optimization, Cambridge University Press, Cambridge, 24. [26] S. Shi, M. Schubert, N. Vucic, and H. Boche, MMSE optimization with per-base-station power constraints for networ MIMO systems, in Proc. IEEE International Conference on Communications (ICC), Beijing, China, 9 23 May 28, pp [27] T. E. Bogale, L. Vandendorpe, and B. K. Chalise, MMSE transceiver design for coordinated base station systems: Distributive algorithm, in 44th Annual Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, USA, 7 Nov. 2. Tadilo Endeshaw Bogale (S 9) was born in Gondar, Ethiopia. He received his B.Sc and M.Sc degree in Electrical Engineering from Jimma University, Jimma, Ethiopia and Karlstad University, Karlstad, Sweden in 24 and 28, respectively. From 24 to 27, he was with the Mobile Project Department, Ethiopian Telecommunications Corporation (ETC), Addis Ababa, Ethiopia. Since 29 he has been woring towards his PhD degree and as an Assistant Researcher with the ICTEAM institute, University Catholique de Louvain (UCL), Louvain-la-Neuve, Belgium. His research interests include robust (non-robust) transceiver design for multiuser MIMO systems, centralized and distributed algorithms, and convex optimization techniques for multiuser systems.

12 2 Luc Vandendorpe (M 93-SM 99-F 6) was born in Mouscron, Belgium, in 962. He received the Electrical Engineering degree (summa cum laude) and the Ph.D. degree from the Universit Catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 985 and 99, respectively. Since 985, he has been with the Communications and Remote Sensing Laboratory of UCL, where he first wored in the field of bit rate reduction techniques for video coding. In 992, he was a Visiting Scientist and Research Fellow at the Telecommunications and Traffic Control Systems Group of the Delft Technical University, The Netherlands, where he wored on spread spectrum techniques for personal communications systems. From October 992 to August 997, he was Senior Research Associate of the Belgian NSF at UCL, and invited Assistant Professor. He is currently a Professor and head of the Institute for Information and Communication Technologies, Electronics and Applied Mathematics. His current interest is in digital communication systems and more precisely resource allocation for OFDM(A)-based multicell systems, MIMO and distributed MIMO, sensor networs, turbo-based communications systems, physical layer security and UWB based positioning. Dr. Vandendorpe was corecipient of the 99 Biennal Alcatel-Bell Award from the Belgian NSF for a contribution in the field of image coding. In 2, he was corecipient (with J. Louveaux and F. Deryc) of the Biennal Siemens Award from the Belgian NSF for a contribution about filter-banbased multicarrier transmission. In 24, he was co-winner (with J. Czyz) of the Face Authentication Competition, FAC 24. He is or has been TPC member for numerous IEEE conferences (VTC, Globecom, SPAWC, ICC, PIMRC, WCNC) and for the Turbo Symposium. He was Co-Technical Chair (with P. Duhamel) for the IEEE ICASSP 26. He was an Editor for Synchronization and Equalization of the IEEE TRANSACTIONS ON COMMUNICATIONS between 2 and 22, Associate Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS between 23 and 25, and Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING between 24 and 26. He was Chair of the IEEE Benelux joint chapter on Communications and Vehicular Technology between 999 and 23. He was an elected member of the Signal Processing for Communications committee between 2 and 25, and an elected member of the Sensor Array and Multichannel Signal Processing committee of the Signal Processing Society between 26 and 28. Currently, he is an elected member of the Signal Processing for Communications committee. He is the Editor-in-Chief for the EURASIP Journal on Wireless Communications and Networing. L. Vandendorpe is a Fellow of the IEEE. Average SER (a) (b) (c) Fig. 3. Comparison of the proposed iterative algorithm (Algorithm II) and the algorithm in [7]. (a) In terms of total sum AMSE when ρ b =., ρ b2 =.2, ρ m =.5 and ρ m2 =.2. (b) In terms of ASER when ρ b =., ρ b2 =.2, ρ m =.5 and ρ m2 =.2. (c) In terms of total sum AMSE when ρ b =.4, ρ b2 =.5, ρ m =.7 and ρ m2 =.8. The non-robust/naive, robust and perfect CSI designs are denoted by (Na), (Ro), and (Pe), respectively.