Average Sum MSE Minimization in the Multi-User Downlink With Multiple Power Constraints

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1 1 Average Sum MSE Minimization in the Muti-User Downink With Mutipe Power Constraints Andreas Gründinger, Michae Joham, José Pabo Gonzáez-Coma, Luis Castedo, and Wofgang Utschick, Associate Institute for Signa Processing, Technische Universität München, Germany Department of Eectronics and System, University of A Coruña, Spain Emai: {gruendinger,joham,utschick}@tum.de and {jose.gcoma,uis}@udc.es Abstract We consider a sum mean square error (SMSE) based transceiver design for the muti-user downink with mutipe inear transmit power constraints. Since the muti-antenna transmitter has ony imperfect channe state information (CSI), the average SMSE is imized via an aternating optimization (AO). For fixed equaizers, the average SMSE imizing precoders are found via an upink-downink SMSE duaity. The precoder update therewith transforms to an upink max- average SMSE probem, i.e., a imum mean square error (MMSE) equaizer design and an outer worst-case noise search. This dua probem is optimay soved, e.g., by a gradient projection approach, and strong duaity is shown. Index Terms upink-downink sum MSE duaity; imperfect CSI; genera power constraints; per-antenna constraints I. INTRODUCTION Upink-downink duaity is an utmost usefu too to transform precoder optimizations in muti-antenna broadcast channes (BCs) to ess compex fiter designs and power aocations in the dua mutipe access channes (MACs). We focus on an average SMSE transceiver design via AO in this context, where the downink precoder update becomes a simpe MMSE fiter cacuation in the dua upink. This resut was reveaed in [1] using SINR upink-downink duaity (e.g., see [2]). Simpified versions of this duaity and AO approach were shown in [3] and depend on the required mean square error (MSE) term, i.e., SMSE, per-user MSE, or per-stream MSE. Note that these references impose a sum power constraint and perfect transmitter CSI (TxCSI). In contrast, we expore an AO transceiver design for imperfect TxCSI and extend the work in [4] to mutipe inear power constraints, incuding per-beam and per-antenna constraints as specia cases. This formuation corresponds to the scenario, where the antennas cannot negotiate on their payoad, e.g., either due to their distributed ocations or due to the radio-frequency hardware. A simiar probem was considered by Bogae and Vandendorpe in [5], [6]. They introduced an upink-downink duaity for the precoder design step within the AO based on an agorithmic argumentation and for specia types of power constraints, e.g., per-antenna and per-precoder constraints. In contrast, we derive the upink-downink duaity for the average SMSE precoder optimization probem concisey via Acknowedgment This joint work has been funded by DFG under grant Jo 724/1-2 and by Xunta de Gaicia, MINECO of Spain, and FEDER funds of the EU under grants 2012/287 and TEC C4-1-R. Lagrangian duaity theory and for generaized power constraints, which incude above constraints as specia cases. Due to strong duaity, the computationay attractive upink fiter and power aocation soutions ead to an optima precoder update for the ocay optima aternating convex search. This work sha give the main idea of the SMSE precoder optimization via (Lagrangian) upink-downink duaity. We show that the duaity hods for singe-antenna and mutiantenna receiver scenarios. Furthermore, we show that the average SMSE measure varies ony a itte for Gaussian channes if a per-antenna, per-array, or a sum power constraint are imposed in the optimization, even though the dynamic power range of each antenna is increasing in this order. We remark that simiar upink-downink duaity based transceiver designs are appicabe for other optimization probems as we, e.g., an average MSE baancing which we recenty presented in [7]. Moreover, even though the shown AO considers ony imperfect TxCSI, i.e., the receivers expoit perfect CSI to cacuate their equaizers, the method is appicabe when the receivers have aso imperfect CSI (e.g., see [6]) and the transmitter knows the receivers strategies. II. SYSTEM MODEL First, we consider a K-user vector downink with singeantenna receivers and an N-antenna transmitter. The k-th user s MSE in this system reads as MSE k =1 2 Re{fk h H k b k }+ f k 2 h H k b i 2 + f k 2 σk 2 (1) when mutuay independent zero-mean unit-variance data signas are ineary precoded with b i, i = 1,..., K, and sent over the channe h k C N. The receivers are subject to zero-mean additive noise with variance σk 2 and fiter the obtained signa with f k = v f k C. 1 As mentioned above, we consider imperfect TxCSI. The transmitter sha ony be aware of some channe statistics. For exampe, we may assume the error mode h k = h k + h k (2) with the estimated channe mean h k and the Gaussian error h k N C (0, C k ) with known covariance matrix C k. 1 Here, v is a normaization variabe for the precoder design in the AO.

2 2 A. Muti-Antenna Receivers In the second scenario, the receivers have mutipe antennas and support a muti-stream data transmission, such that the k-th user s MSE reads as MSE k =M k 2 Re{tr(Fk H Hk H B k )} (3) + Fk H Hk H B i 2 F+tr(Fk H Σ k F k ). To obtain (3), we assumed that M k independent zero-mean unit variance data signas are ineary precoded by B k and transmitted to the k-th receiver, which fiters the signas with F k = v F k. 1 If we additionay substitute b k = vec(b k ) and f k = vec(f k ) in (3), we obtain the equivaent beamformerike MSE representation [cf. (1)] MSE k =M k 2 Re{fk H (I Mk Hk H )b k } (4) + (I Mi Fk H Hk H )b i 2 2+fk H (I Mk Σ k )f k. Simiar to the vector channe case, the transmitter sha ony be aware of the channe statistics for the muti-antenna receiver scenario. The matrix equivaent of the channe mode in (2) is H k = H k + H k (5) with the channe mean H k and the Gaussian error h k = vec( H k ) N C (0, C k ) with covariance matrix C k. B. Transmit Power Limitations In either of the two scenarios, the transmitter sha be subject to genera power imitations of the form b H i A i, b i = A 1/2 i, b i 2 2 P, = 1,..., L (6) where A i, C N N, A i, 0 has the square root representation A i, = A H/2 and rank { L =1 A i,} = N. i, A1/2 i, Important exampes from wireess communications in vector channes are a sum power constraint, per-beam constraints, and per-antenna constraints. Depending on the imposed transmit power constraint(s), the matrices A i, have different forms: sum power: A i, =I N for a,..., K and L=1; per-beam: A i,i =I N and A i, =0 N N, i with L=K; per-antenna: A i, =e e T with L=N. Besides these specia cases, the formuation in (6) aso incudes power constraints per antenna array, e.g., per-array: A i, =bockdiag{0, A, 0} and L<N, where the matrices A 0 may have fu rank, that is, 1 rank{a i, } N. These per-array constraints are important for systems where severa muti-antenna transmitters, e.g., WiFi stations, cooperativey suppy severa mobie devices. We remark that the power constraints in (6) are aso suitabe for the scenario with mutipe streams per user. Then, the matrices A i, for the sum, per-beam, per-antenna, and perarray constraints are bockdiagona with repications of above matrices on the diagona, i.e., A i, = I Mk A i,. III. AVERAGE SUM MSE MINIMIZATION The goa is the imization of the average downink SMSE SMSE DL = K k=1 E[MSE k] for perfect RxCSI, imperfect TxCSI, and subject to above transmit power constraints, i.e., b,f SMSE DL s. t.: b H i A i, b i P, = 1,..., L (7) where we substitute b = [b T 1,..., b T K ]T and write f = [f 1,..., f k ] T or f = [f T 1,..., f T k ]T for the singe- or mutiantenna receiver scenario, respectivey, to simpify expositions. If the receivers have instantaneous CSI, they can empoy MMSE fiters. These fiters are f k,mmse = v f k,mmse = h H k b k K hh k b i 2 + σ 2 k for the case of vector channes resuting in the average SMSE [ ] h H k SMSE DL = K E b k 2 K hh k b. (9) i 2 + σk 2 k=1 Unfortunatey, this objective is non-convex in the precoders and contains an expectation whose cosed-form expression is ony known for very simpe channe modes, e.g., zeromean Gaussian channes [8], and whose evauation requires either numerica integration or Monte-Caro methods for other standard modes. The same properties hod for the matrix channes, where the MMSE equaizers read as (8) F k,mmse = v F k,mmse = Y 1 k HH k B k (10) with Y k = K HH k B ibi HH k + Σ k, and resut in the average SMSE expression SMSE DL = k=1 M k E [ tr ( B H k H k Y 1 k H H k B k )]. (11) Since SMSE DL in (11) is non-convex in B k, imizing the average SMSE directy w.r.t. the precoders is difficut if the cosed form MMSE equaizers are incorporated into (7). However, as the origina MSE expressions in (1) and (3) are convex in the precoders for fixed equaizers and vice versa, i.e., the MSEs and aso the SMSEs are biconvex functions [9] in b and f, an aternating convex search of the precoders and fiters converges to ocay optima transceivers in the mutiuser downink (cf. [10]). The foowing two steps woud be performed in each iteration of the aternating convex search for f and b if perfect CSI were avaiabe (cf. [1]): 1) The equaizers in f (from f = v f) are first found in the downink for fixed precoders in b as in (8) or (10). 2) Second, the downink precoders in b are optimized as equaizers in the dua upink system for given f. Note that the empoyed upink-downink duaity in Step 2 of the AO can reduce the compexity for impementation and computations. Otherwise, the precoder update requires a convex sover for incuding the generaized constraints. Aso average SMSE imizations can be based on upinkdownink duaity [4]. Then, the resuting precoders in b imize an upper bound for the imum achievabe average

3 3 SMSE for fixed f, which is reduced in each AO step and hence converges (cf. [9, Theorem 4.5]). For a further extension to the generaized power constraints in (6), we derive the upink formuations of the precoder design optimization in Step 2 of the AO concisey via Lagrangian duaity in what foows. IV. PRECODER DESIGN VIA UPLINK-DOWNLINK DUALITY If the downink equaizers f = v f are fixed up to v, the downink precoder optimization step within the AO can be written for both antenna scenarios in the generaized form b,v SMSE DL s. t.: b H A b P, =1,..., L (12) where the generaized average SMSE reads as SMSE DL = M 2 Re{v hh b} + v 2 b H Rb + v 2 σ 2 (13) and the generaized power constraints are equivaent to those in (7) if A = bockdiag{a 1,,..., A K, }. To obtain (13), we introduced the effective channe mean the effective noise variance and the second-order matrix h = E[Ĥ f], (14) σ 2 = E[ f H Σ f], (15) R = I M E[H F F H H H ], (16) which sha have fu-rank. 2 The expressions for the substitutes M, Ĥ, H, F, and Σ depend on the considered scenario: for singe-antenna receivers, we have M = K, Ĥ = bockdiag{h 1,..., h K }, H = [h 1,..., h K ], F = diag{ f 1,..., f K }, Σ = diag{σ 2 1,..., σ 2 K} (17) and for muti-antenna receivers, we have M = K k=1 M i, Ĥ = bockdiag{i M1 H 1,..., I MK H K }, H = [H 1,..., H K ], F = bockdiag{ F 1,..., F K }, Σ = bockdiag{i M1 Σ 1,..., I MK Σ K }. (18) Note that the expectations to compute h, R, and σ 2 require numerica evauations for the considered imperfect TxCSI, i.e., with known channe mean and error statistics. We computed the expected vaues in (14)-(16) via the corresponding sampe means for the simuations. Before deriving the dua upink probem of (12), we first present a prima soution based on consecutive power imizations. The power imization probem is required for proving strong (upink-downink) duaity in Section IV-B. 2 If R is degenerate, we can find an equivaent representation of (13) and the power constraints in (12) via repacing b by b = V b, where V is an orthonorma basis for the span of R. The SMSE imization (13) needs then to be performed w.r.t. b instead of b. A. Agorithmic Soution of the Prima Probem If the SMSE imizing v = b H h/(b H Rb + σ 2 ) is substituted in (13), the downink objective, which reads as SMSE DL = M h H b 2 b H Rb + σ 2, (19) becomes quasiconvex in b. The ower eve set S = {b C MN ε SMSE DL } is convex, i.e., ε SMSE DL has the convex second order cone (SOC) representation M ε [b H R H/2, σ] H 2 h H b, where w..o.g. we restrict hh k b k to be rea and positive. Since, moreover, the power constraints have the SOC form A 1/2 i, b 2 P, (12) can equivaenty be rewritten as b,ε ε s. t.: A 1/2 b 2 P, =1,..., L, [b H R H/2, σ] H 2 h H b M ε. (20) This probem may be soved in the downink via a bisection over ε [0, M], where it is tested in each iteration whether there exists a feasibe b that achieves the target ε. We test feasibiity using the stricty monotonicay decreasing function ψ DL (ε), which we define as the power factor that imizes the convex SOC power baancing probem b,α α2 s. t.: A 1/2 b 2 α P, =1,..., L, [b H R H/2, σ] H 2 h H b M ε. (21) Note that this function is the inverse function to ε DL (ψ) that denotes the imum of (20) if we substitute P with ψp, that is, ε DL (ψ DL (ε)) = ε. Therefore, if ψ DL (ε) < 1, the corresponding precoder b can attain the target ε without vioating the power constraints in (20), and if we finay achieve ψ DL (ε) = 1, the corresponding ε and b are the soutions of the SMSE imization in (20). B. Dua Upink SMSE Probem Based on Lagrangian duaity, we next show that the downink precoder design probem in (12) can be transformed into a dua upink max- average SMSE optimization. Theorem 1. The strongy dua upink SMSE optimization of the downink SMSE imization in (12) reads as max SMSE UL s. t.: σ 2 λ 1, µ P 1 (22) µ 0 u,λ 0 =1 where the upink SMSE is given by SMSE UL = M 2 λ Re{ h H u}+u H( λr+ µ A )u, =1 i.e., an inner equaizer design and power aocation and an outer worst-case noise covariance search w.r.t. the dua variabes µ = [µ 1,..., µ L ] T corresponding to the downink power constraints in (12). The strong duaity proof between the SMSE imizations in (12) and (22) is based on the downink power baancing probem in (21) and its dua upink SMSE formuation.

4 4 Lemma 1. The strongy dua upink max- probem to the downink power baancing in (21) can be written as max µ 0 u,λ 0 λσ2 s. t.: µ P 1, ε SMSE UL. (23) =1 In the appendix, this dua upink probem is derived via simiar steps as in [11], which is based on SINR constraints. Proof of Theorem 1. We show that the optima vaue of the prima SMSE imization in (12) is aso the optimum of the dua probem (22). Let ε denote the prima optima average SMSE vaue. Then, we know that the optimum vaue of (21) is one [cf. Subsection IV-A], which is aso the optimum of (23) due to zero duaity gap (cf. Lemma 1), i.e., ψ DL (ε) = ψ UL (ε) = 1. Finay, it remains to prove that ψ UL (ε) = 1 is an identifier to decare ε as the soution for (22). If ψ UL (ε) = 1, the corresponding power aocation λ = 1 σ 2 (24) and the corresponding variabes µ satisfy the SMSE constraint ε SMSE UL in (23) with equaity [cf. Appendix A]. Next, we prove that these parameters are aso optima for (22) in order to concude that the optimum of (22) is ε. To show that (24) is optima for (22), we substitute the SMSE imizing upink equaizer u SMSE = ( λ λr + ) µ A 1 h, (25) =1 which is an optimizer for both probems (23) and (22), into the upink SMSE. The resuting upink SMSE reads as [cf. (38)] u SMSE UL = M λ h H( λr + ) µ A 1 h, (26) =1 which is decreasing in λ 0. Therefore, the SMSE is imized by maximizing λ, which is upper bounded by σ 2 λ 1 in (22). This shows that (24) is aso optima for (22). Now, we prove that the optimizer µ for (23), which exacty meets ε, is aso the SMSE maximizer in (22). To prove achievabiity, we remark that µ is feasibe for (22), i.e., it satisfies the constraint L =1 µ P 1. Therefore, the optimum ε UL of (22) satisfies ε UL ε. For the converse, we assume that there is another SMSE maximizing µ for (22), which satisfies the constraint set and achieves an SMSE ε UL > ε for the power aocation in (24) and the equaizer in (25). However, since this µ woud aso be feasibe for (23), it contradicts the optimaity of λ for (23), which was assumed to achieve the SMSE target ε. Therefore, we have ε UL ε for a feasibe µ in (22). Together with the achievabiity resut, this converse proof indicates that µ is indeed an SMSE maximizer for (22) if and ony if it is an optimizer of (23) and that the max- SMSE vaue is ε for (22) if ψ UL (ε) = 1. If we know the optima u and λ, the downink variabes are obtained via a scaing of these variabes, i.e., (cf. [3]) b = βu, v = λ/ β. (27) The squared transformation factor reads as β = λσ 2 L =1 µ u H A u (28) and foows from inserting (27) into (13) and equating the SMSE terms, i.e., SMSE UL = SMSE DL. How to obtain u and λ together with the dua variabe µ is detaied next. C. Agorithmic Soution of the Dua Probem In contrast to the compex NM 1 compex-vaued beamformer search presented in Section IV-A, which is suited for the prima SMSE imization probem (12), we next present a semidefinite programg (SDP) soution and a gradient projection (GP) approach for the dua probem (22) that search ony for the L non-negative reas in µ. The soutions for the power aocation and the SMSE imizing equaizer for the inner imization in (22) are given in (24) and (25), respectivey. Inserting these expressions, the outer maximization in (22) transforms to the convex probem max M h H( ) R + σ 2 µ A 1 h L s. t.: µ P 1. µ 0 =1 =1 (29) Note that the constraint in (29) is tight at the optima point since the objective is eementwise increasing in µ. We may sove the worst-case noise search in (29) by means of a standard interior-point sover (e.g., CVX [12] with [13]). Using Schur s compement, (29) can be rewritten into the SDP probem formuation ] max ε ε,µ 0 s. t.: [ M ε hh h R+σ 2 L =1 µ A µ P 1 =1 0, (30) with a semidefiniteness constraint that is affine in µ. Aternativey, we find the optimizer of (29) iterativey, e.g., with a GP agorithm. The -th component of the gradient δ = [δ 1,..., δ L ] T is δ = σ 2 hh k X 1 A X 1 hk (31) where X = R + σ 2 L =1 µ A, and a GP step reads as µ (j+1) P C ( µ (j) + a j δ ) where a j denotes the step size in iteration j. The orthogona projection onto C = { µ R L + L =1 µ P 1 } can be expressed as P C (x) = [x γp] + (32) where p = [P 1,..., P L ] T, [ ] + performs an eementwise projection onto the non-negative orthant, i.e., [z] + = max(0, z) for a scaar z R, and γ is chosen to satisfy the constraint in (30) with equaity (cf. [14]). Bogae and Vandendorpe proposed a fixed-point (FP) update in [5] and [6] to find the optimizer µ, which eads to the

5 5 prima optima beamformer update in the AO. In our notation, this update rue reads as µ (j+1) pt µ (j) δ (j) µ (j) δ (j),t µ (j). (33) P The normaized power iteration may be motivated via the compementary sackness conditions for the power constraints in (12), i.e., µ P = µ b H A b for = 1,..., L, that need to be satisfied for any ocay optima KKT point of (12). V. NUMERICAL RESULTS The main simuation setup consist of an N antenna transmitter and K = 2 users, each equipped with either a singe antenna or M 1 = M 2 = 2 antennas for the muti-user vector and MIMO channe exampes, respectivey. The noise variances are fixed to σk 2 = 1 and Σ k = I M and the power is varied between 10 db and 20 db. We created the channes means for a Gaussian mode and for an exponentia channe mode. 3 For the previous mode, we have drawn channe mean reaizations h k and h k = vec( H k ) from a standard Gaussian distribution and the channes covariance matrices are C k = κ N I N and C k = κ M k N I M k N for vector channes and matrix channes, respectivey, with κ = 10 db. For each of the channes means, we have drawn another channe reaizations to compute the average SMSE for imperfect TxCSI and perfect RxCSI resuts with the aternating optimization for an accuracy of ɛ For the atter exponentia channe mode, we used an exponentia power profie, i.i.d. uniformy-distributed phase coefficients, and i.i.d. sow og-norma shadow-fading, i.e., h k = τ k t k where [t k ] i = e jφ i,k ρ i k, φ i,k [0, 2π) and n(τk db) N (0, 1) and C k = κτ k N diag(ρ 1 k,..., ρ N k ) for the vector channes. Due to the exponentia mode, the transmitter prefers to serve the k-th user with the k-th antenna eement, whie the shadow fading factors τ k are a measure for the users ink quaity. This mode is used to anayze mutispotbeam SatCom setups, where the antenna directivity pays an important roe (e.g., see [15]). An extreme case for the exponentia mode is ρ = 0. Each transmit antenna exacty serves one user if K = N and the channes are orthogona. Then, per-antenna constraints and a sum power constraint resut in the same SMSEs when τ k = τ for a k = 1,..., K. In contrast, per-antenna constraints resut in arger SMSEs than a sum-power constraint, e.g., when τ k τ j for at east one receiver pair k and j. For the numerica simuations we used ρ = 0.1. In Fig. 1, we potted the average achievabe imum SMSE for Gaussian channe mean reaizations in the vector and MIMO case and for the exponentia vector channe mode with N = 4. Four ines can be seen for each of these modes that correspond to a sum power imitation P (soid ine with square marks), per-array power constraints for antennas 1, 2 and 3, 4 with P = P/2 (dashed ine), and per-antenna constraints with P = P/N (dash-dotted ine). For the ines for the per-array and per-antenna constraint, we used the AO with 3 An exponentia power profie is amongst others used for anayzing interce effects in muti-spotbeam SatCom communications (e.g., see [15]). average MSE Gaussian imp. RxCSI per-antenna per-array sum power MIMO Gaussian exponentia power LP in db Figure 1: average SMSE vs. SNR for per-antenna, per-array, and a sum power constraint in a vector and MIMO BC with N = 4 transmit antennas and K = 2 users (with M k = 2 receive antennas per user) for κ = 10 db average MSE 10 0 Gaussian K = imp. RxCSI per-antenna per-array sum power exponentia N = 8 Gaussian N = power LP in db Figure 2: average SMSE vs. SNR for per-antenna, per-array, and a sum power constraint in a vector BC with N = 4 and K = 4 users and for N = 8 and K = 2 users for κ = 10 db the GP based dua beamformer update. The other beamformer update methods ead to the same SMSE performance curves. With the choices for P, the per-antenna constraints are stricter than the per-array constraints and the sum power constraint, resuting in an increased average SMSE. This is best visibe for the exponentia vector channe mode, but resuts in ony sight SMSE differences for the Gaussian vector mode. In the Gaussian MIMO channe mode, the mutistream equaizers additionay compensate for the mutipe transmit power restrictions with per-array and per-antenna constraints. In other words, the sum power imitation gives a tight ower bound for the achievabe average SMSE with perarray or per-antenna constraints, which require more compex beamformer designs. An upper bound is obtained by assug that the RxCSI equas the imperfect TxCSI (green soid ine). We further remark that the arger K (for constant N) the more ikey the Gaussian channe characteristics does not prefer certain (groups of) antennas and the tighter the SMSE with a sum power constraint approximates the SMSE with the stricter per-array or per-antenna constraints (see Fig. 2). However, if we increase N (for constant K), the difference between sum-power-constrained imum SMSE and the per-

6 6 empirica CDF N = 4 N = Gaussian K = 2 exponentia K = MIMO M k = 2 Gaussian K = argest antenna power in db Figure 3: empirica CDF of argest per-antenna power in db (normaized to P ) of the SMSE imizing resuts with a sum power constraint and κ = 10 db empirica CDF F (x) AO iterations 0.6 per-antenna per-array 0.4 sum-power prima SOCP 0.2 dua SDP dua GP dua FP [4] iteration number x Figure 4: empirica CDF of required number of outer and inner iterations of the average SMSE imization for the Gaussian channe mode with N = 4, K = 2, and κ = 10 db antenna and per-array constrained imum SMSEs increases for the exponentia channe mode but remains the same or even decreases for the Gaussian channe mode. Even though above SMSE resuts impy ony a or oss when we use more reaistic per-antenna or per-array constraints compared to the ess compex sum power constraints, we dramaticay decrease the dynamic range of the per-antenna or per-array gains. In Fig. 3, we depict the empirica CDF of the argest per-antenna power for the given vector channe modes when imizing the average SMSE with a sum power constraint. The antenna power in db is normaized such that an associated per-antenna power imitation P = P/N represents 0 db. For the vector BC with N = 4 and K = 2 users, a doube of the per-antenna bound P (i.e., 3 db) is surpassed in more than 20% of the channe reaizations for the exponentia and the Gaussian mode. This 20% bound decreases to about 2 db for an increasing number of receive antennas (or users) and fixed N = 4. This observation is in accordance with the concusion that the SMSE with a sum-power constraint we approximates the achievabe SMSE performance with per-antenna constraints for arge K (see aso Fig. 2). In contrast, the normaized 20% argest per-antenna powers of the sum-power constrained optimization increase when increasing the number of transmit antennas to N = 8. Whie the increase is ony to about 4 db for the Gaussian mode, 6 db are surpassed with the exponentia mode. Because it prefers to serve the two users with ony two of the eight antennas, the sum-power constraint SMSE optimization focuses most of the power to these two antennas. Since, moreover, the channes are subject to shadow fading, their gains are ikey to differ, which resuts in an unbaanced power distribution for the users main serving antennas. This unbaanced power is finay the reason for the increased SMSE gap between the sum power and the per-antenna constrained optimization resuts in Fig. 2. To obtain an impression about the average SMSE precoder optimization via AO and the compexity of the dua beamformer updates, we potted the empirica CDF for the required number of iterations unti convergence of the outer AO and the inner (dua) beamformer design for the average SMSE imization in the Gaussian channe setup with N = 4, K = 2, and κ = 10 db in Fig. 4. For the chosen setup, the number of AO iterations is amost independent with respect to the power constraints, i.e., four per-antenna constraints, two per-array constraint, or one sum power constraint, but the compexity of the inner beamformer updates increases with the number of power constraints L. Both interior-point based soution approaches (dashed ines), i.e., the prima beamformer update with a bisection over SOCPs and the dua worst-case noise SDP, require amost the same number of interior-point iterations for per-antenna and per-array constraints (see Fig. 4). The SOCP based interiorpoint probem has a sighty ower worst-case compexity than the dua SDP method, but a sequence of these SOCPs needs to be soved in contrast to one SDP. Therefore, we experienced the dua SDP based beamformer update to be much faster than the SOCP based beamformer update. Note that the worst-case compexity of the conic programs is increasing in the number of power constraints L [16]. The number of SOCs and nonnegative variabes is L + 1 in (21) and in (30), respectivey. For the GP approach with an Armijo step size rue and the fixed-point (FP) update from [6], the number of iterations and, therewith, the computationa compexity of the worst-case noise search increases from the per-array to the per-antenna constraint scenario (see Fig. 4). For exampe, whie the GP converges in ess than 20 iterations for 80% of the performed simuations with the two per-array constraints, more than 40 iterations are required for the per-antenna constraints. This effect is even more dramatic for the FP update rue, whose convergence speed is much sower than that of the GP with the Armijo step size rue. Especiay, when some power constraints are inactive at the optima point, the FP may require far more than a hundred iterations if convergence can be detected at a. Whie more than a hundred iterations are required in 10% of the simuations for per-array constraints, this number increases to about 19% for the per-antenna constraints. For this reason, we experienced that the GP based beamformer update is faster on average than the FP method [6], even though each FP iteration is ess compex than a GP iteration with an Armijo step size rue.

7 7 VI. CONCLUSION This paper provided an upink-downink duaity for the (average) SMSE based precoder design with generaized power constraints by means of an aternating optimization. Simiar to the standard SMSE imizing AO approach, the precoder update in each AO step is transformed to a dua power aocation and fiter design, which, however, aso incudes additionay a worst-case noise covariance search. Therewith, the compex search for an N M dimensiona precoder is reduced to a worst case noise search over L non-negative reas that are the dua variabes to the imposed power constraints. The soution is found via an SDP and a GP approach. The computationa compexity of these approaches is increasing with the number of power constraints. The simuations showed that the achievabe SMSE with an imposed sum power constraint can give a good ower bound approximation of the performance with per-array or perantenna power constraints for Gaussian channes. This good match of the SMSE is surprising since the dynamic range of the required per-antenna powers for the sum power constrained optimization is much arger compared to the restrictions with per-array or per-antenna constraints. A remarkabe difference in the SMSEs is visibe for the used exponentia channe mode, when the number of antennas is much arger than the number of users in the system and the channes are subject to different mutipicative fading, i.e. they have different norms. A. Proof of Lemma 1 APPENDIX Note that ξ = ξ( x 2 + z) with ξ 0 can be used as a Lagrangian mutipier for the convex SOC constraint x 2 z without oss of optimaity and strong duaity [17, Appendix A]. Hence, is the Lagrangian function of the convex power baancing probem in (21) can be written as L(α, b, λ, µ) = λσ 2 + α 2( ) 1 µ P (34) =1 + b H( Y λ M ε h h H) b where Y = λr + L =1 µ A and λ 0 and µ 0, = 1,..., L, are the mutipiers to the SMSE and power constraints in (21), respectivey. The dua objective resuts from the unconstrained imization of (34) w.r.t. α and b, i.e., g(λ, µ) = α,b L(α, b, λ, µ). Since α and b are unconstrained, g(λ, µ) uness µ P 1 (35) =1 and Y λ h h H M ε 0 N N. With Schur s compement [18, Appendix A.5.5], we can recast the atter condition as (cf. [11]) (M ε) λ h H Y 1 h 0. (36) Therewith, the dua probem of (21) reads as max µ 0,λ 0 λσ2 s. t.: µ P 1, ε M λ h H Y 1 h. (37) =1 The SMSE constraint upper bounds the objective in (37) since its right hand side is decreasing in λ when µ is fixed. In other words, λ satisfies the SMSE constraint with equaity if ε is attainabe. Hence, jointy reversing the maximization over λ into a imization and the direction of the inequaity in the SMSE constraint does not affect the soution. Since, moreover, M λ h H Y 1 h = u SMSE UL (38) with the SMSE imizing equaizer u given in (25), (37) is equivaent to the upink power imization in (23). 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