Mean Square Error Beamforg in SatCom: Upin-Downin Duaity with Per-Feed Constraints Andreas Gründinger, Michae Joham, Andreas Bartheme, and Wofgang Utschic Associate Institute for Signa Processing, Technische Universität München, Germany Emai: {gruendinger,joham,a.bartheme,utschic}@tum.de Abstract Baancing the per-user average mean square errors MSEs) of a sateite communication SatCom) system under perfeed imitations inear transmit power constraints is the focus of this wor. We propose an upin-downin duaity for this optimization via Lagrangian mutipier theory. Strong duaity is shown. Simpe fixed point methods are used for the upin power aocation and the worst-case noise covariance cacuation. Simuation resuts are used to compare the performance for the SatCom channe with that of a standard Gaussian channe mode. Index Terms upin-downin MSE duaity; imperfect CSI; MSE baancing; per-feed constraints; sateite communication I. INTRODUCTION Muti-spotbeam sateite communication SatCom) requires adaptive beamforg to manage the increasing interce interference in the downin [1], when narrowing down the ce width beow the 3 db coverage and increasing the frequency reuse in future systems. The goa is a reiabe data service provision proportiona to the receivers demands. The beamformer optimization must dea with the specia channe mode and needs to incorporate per-feed power imitations that differs from the standard sum power constraint in terrestria wireess communications. Chrisopouos et a. [] mode the per-feed imitations as genera inear power constraints. Additiona non-inear power constraints were incuded in [3] to represent saturation effects in the radio frequency ampifiers. Here, we restrict to inear per-feed constraints which is the basis for future wor on non-inear power imitations. Specia forms of per-feed constraints, e.g., per-antenna constraints, can aso be encountered in terrestria systems when power sharing is impossibe, e.g., due to a physica separation of the antennas. In [4], the ratios between achievabe and target signa-to-interference-and-noise-ratios SINRs) are maximized under such power restrictions. The soution is obtained via repeatedy soving reated second order cone SOC) programs using a convex optimization toobox, e.g., CVX [5]. However, the number of spotbeams and antennas in SatCom, that can meet a few hundreds, eads to a arge number of per-feed constraints. Therefore, this approach is not attractive and the use of upin-downin duaity is favorabe. Upin-downin duaity is an utmost usefu too for beamformer optimizations which are difficut to sove directy in the vector broadcast channe BC). The probem is transformed to a dua mutipe access channe MAC) probem with the same achievabe SINRs [6] or average) MSEs [7]. The precoder The research for this wor was supported by Deutsche Forschungsgemeinschaft DFG) under fund Jo 74/1-. design transates to an equaizer design and a power aocation. Fixed point methods for these tass are efficient, reiabe, and can be impemented without using optimization tooboxes [8]. Due to deays, shadowing, and scattering effects in SatCom, the channe state information CSI) at the transmitter sateite) is imperfect. Therefore, we design the beamformers according to a -max baancing of the average MSEs. Note that a ower bound for the rates is maximized via imizing MSEs. MSE duaities between the vector BC and the vector MAC were first reveaed using SINR duaity [9] for a sum power constraint. The SINR upin-downin duaity was extended to inear power constraints via Lagrangian mutipier theory, by Yu and Lan [10] for the power imization and in [11] for a max- baancing formuation. However, simiar average MSE duaities are missing to the best of our nowedge. Ony for the average sum MSE imization via aternating optimization, an upin-downin approach was presented by Bogae and Vandendorpe in [1]. We propose an upin-downin average MSE duaity with inear per-feed power constraints via Lagrangian mutipier theory. In the dua upin, the -max average MSE probem has a weighted sum-power constraint and incudes a search for the worst case noise covariance. Strong duaity can be shown via a reated power imization probem formuation. Expoiting duaity, we sove the -max average MSE baancing probem in the upin. Simuation resuts for a standard channe mode and for a SatCom channe mode are presented. II. DOWNLINK SYSTEM MODEL The downin received signas, e.g., from a SatCom scenario, read as y = h H t s + h H i t is i + n. The independent data signas s i N C 0,1) are ineary precoded with t i at the transmitter and sent over the channes h C N to the K mobie receivers. The additive noise at mobie i is n i N C 0,σi ), i = 1,...,K. The received signa is scaed with f C, i.e., ŝ =f y, such that the MSE E [ s ŝ ] for the -th receiver s signa estimate reads as MSE =1 Re{f hh t }+ f h H t i + f σ. 1) A. Linear Per-Feed Power Constraints The per-feed transmit power imitations are represented as t H i A i,t i = A 1/ i, t i P, = 1,...,L ) where A i, = A H/ i, A1/ i, 0 with ran { L A i,} = N.
Important exampes from terrestria communications are a sum power constraint, per-beam constraints, and per-antenna constraints. Depending on the imposed transmit power constraints), 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. Per-feed constraints are equa to per-antenna constraints if a singe horn antenna sends the output of a high-power ampifier to the refector at the sateite. However, when sma phased arrays form the feeds, the matrices A i, have ran{a i, } > 1. The per-feed constraints in ) are a first step towards more advanced power constraints for the beamformer design of muti-beam SatCom systems. For exampe, non-inear power constraints coud mode the saturation effects in the radio frequency ampifiers in SatCom cf. [3]). B. Channe State Information Modes Imperfect CSI at the transmitter is the second issue in SatCom. We assume nowedge of the first and second order moment of the channes h, i.e., h = E[h ], R = E[h h H ]. 3) The expressions for h and R resut from the empoyed fading mode. For exampe, the Gaussian modeh = h + h with h N C 0,C ) is common to represent fading effects and imited training and/or feedbac capabiities in terrestria systems. Therewith, the second moment is R = h hh +C. Muti-spotbeam sateite mobie channes foow the fading mode in [13] see aso [1]). The basis is Rician fading that modes the ine-of-sight characteristic of sateite channes, i.e., z = κ κ+1 z + 1 κ+1 z 4) with Rician factor κ, ine-of-sight component z, and compex random z. Since scatterers are mainy around the receivers and far from the transmitting sateite, we modeed z =w z and w N C 0,σw ) in previous wor e.g., [14]). Such a restriction is not imposed in this wor, i.e., z N C 0,C z ) may have a fu ran covariance matrix, e.g., C z = I N. According to Loo s mode [15], the ine-of-sight component is subject to a mutipicative og-norma distributed factor ξ, i.e., z = ξ ẑ where nξ ) Nm,σξ ). This factor comprises rain fading, shadowing, and the receivers mobiity. The resuting fading vector in 4) is distorted by the beam gain characteristic B = diagb 1,,...,b N, ), that depends on the reative position of the receivers to the spotbeam centers. Here, b j, = g j, e jψ j, contains the taperedaperture antenna gain [16] from ) antenna j to user with J1u g i, = i, ) u i, + 36 J3u i,), u where 3 J1 ) and J 3 ) are i, the first ind Besse functions of order one and three, respectivey, of u i, =.0713 sinθ i,) sinθ. The ange θ 3dB) i, is between beamcenter i and user as seen from the sateite and θ 3dB is the one-sided haf-power beamwidth. For an approximation of the sma phase shifts ψ j,, we assumed that the antennas are oriented in a pane orthogona to the centra beam, that is directed to Munich. The channe to mobie reads finay as cf. [3]) h = g FSL, B z 5) where g FSL, = ) λ 1 4π d modes the free space oss FSL) with waveength λ and atitude d. The moments of 5) are gfsl, κ h = κ+1 em +σξ / B ẑ, R = e σ ξ h hh + g 6) FSL, κ+1 B C z B H. III. AVERAGE MEAN SQUARE ERROR BALANCING We design the beamformers to imize the maximum average MSE. We remar that this is a conservative approach for average rate baancing. The rate r = og MMSE ) is a convex function of the imum MSE MMSE), i.e., the MSE with perfect CSI MMSE receive fiters f,mmse = h H t hh t. 7) i +σ Therewith, Jensen s inequaity provides the ower bound E[r ] og E[MMSE ]). Baancing the average MSEs with the fiters in 7), i.e., K E[MMSE ] = 1 E [ h H t / )] h H t i +σ, 8) remains difficut for imperfect transmitter CSI. The expectation in 8) is over a ratio of correated random parameters. Even though cosed form expressions may be found for this expectation, a direct imization of max E[MMSE ] w.r.t. the beamformers is sti a non-convex optimization probem. We resove this issue with a beamformer optimization that considers the receivers to have the same imperfect channe nowedge as the transmitter. Note that any receive fiter different from that in 7) resuts in an upper bound to the achievabe instantaneous MMSE with perfect receiver CSI. Therefore, aso an upper bound for the average MMSE in 8) is obtained. The joint imization of these MSE upper bounds resuts in the maximization of ower bounds for the achievabe rates. Whenever the average MSEs are baanced at a eve ˆε, the average rates are ensured to ie above og ˆε). If the receivers have the same imperfect CSI as the transmitter, the -max average MSE optimization with inear) per-feed constraints reads as f,t s.t.: max MSE A 1/ i, t i P, = 1,...,L where t = [t T 1,...t T K ]T, f = [f 1,...,f K ] T, and the average MSEs E[MSE ] are given by MSE =1 Re{ f h H t }+ f t H i R t i + f σ. 10) We provide a dua upin formuation to sove probem 9), which is then a receive fiter design and power aocation 9)
probem over a worst-case noise covariance matrix. This matrix may be found via a subgradient method, for exampe, and the power aocation is a simpe fixed point agorithm. IV. MSE UPPER BOUND MINIMIZATION By inserting the MMSE fiters for imperfect receiver CSI h f H,MMSE = t th i R, 11) t i +σ probem 9) becomes a quasiconvex program. The objective, i.e., max MMSE with h MMSE H = 1 t th i R, 1) t i +σ is the pointwise maximum of quasiconvex functions, since the ower eve set of the MMSE features a convex reformuation. Moreover, the per-feed power constraints in 9) are convex. To see this, we first reformuate 9) with 1) as ˆε,t ˆε s.t.: A 1/ i, t i P, = 1,...,L, h H t K 1 ˆε) R 1/ t i +σ, = 1,...,K 13) where we introduced the baancing eve ˆε [0,1] as a sac variabe for the maximum of the K average MMSEs andr 1/ is the square root matrix of R =R H/ R 1/. Since the average MMSEs in 1) and the power imitations are independent w.r.t. a phase shift of the beamformers, we restrict h H t to be rea and positive in 13). 1 The convex ower eve set representations for the MMSEs are then obtained via a square root operation on both sides of the inequaity as is seen in 14). A. Bisection Over Power Minimizations We may find the optimizers of 13) via soving a series of convex probems. This is simiar to SINR baancing with per-antenna constraints [4], where the baanced SINRs and the corresponding beamformers are found with a bisection. In each bisection step, a power imization in SOC form is soved. We rewrite the power imization corresponding to 13) as α,t α s.t.: Re{ h H t } 1 ˆε [t H I K R H/ ),σ ], Im{ h H t } = 0, = 1,...,K 1/ A t α P, = 1,...,L 14) where A 1/ are bocdiagona matrices with eements A 1/ i,, i = 1,...,K, and the joint MSE eve ˆε is fixed. Obviousy, the imumα ˆε) of 14) is stricty monotonicay decreasing in ˆε. Simiary, the imum ˆε P 1,...,P L ) of 13) is stricty monotonicay decreasing in α if P = α P and P > 0 is fixed. Therefore, a simpe ine search via 14), 1 Under this restriction, a possibe soutions of 9) resut from the possibe soutions of 13) via t i = ejφ i t i, φ i [0,π). e.g., a bisection, is abe to find the optimizers of 13) if it meets α ˆε) = 1 with a predefined accuracy. We used the discipined convex programg toobox CVX [5] to find α ˆε) and chec our numerica simuations. B. Upin-Downin MSE Duaity Aternativey, the average MSE baancing probem in 9) can be soved in the dua upin. As can be inferred from the proof of Proposition 1, the dua upin average MSE baancing optimization can be written as MSE i,ul max max µ 0 λ 0,u i s.t.: λ i σi µ P. 15) The average upin MSE that corresponds to user i reads as MSE i,ul = 1 λ i Re{ h H i u i} +u H i K λ R + =1 µ A i, )u i. 16) The upin power aocation vector λ = [λ 1,...,λ K ] T 0 comprises the dua variabes associated with the MMSE constraints in 13). The vectorµ = [µ 1,...,µ L ] T 0, that defines the worst-case noise covariance matrix L µ A i, in 16), contains the dua variabes for the per-feed constraints. Note that the optima fiters inu=[u T 1,...,uT K ]T of 15) are K u i = λ R + =1 ) µ A i, hi λi. 17) Inserting 17) into 16), 15) may be written as a max- MMSE baancing probem with the MMSEs K ) MMSE i,ul = 1 λ i hh i λ R + µ A i, hi, 18) =1 which are independent w.r.t. a common scaing of µ and λ. Proposition 1. The duaity gap between 9) and 15) is zero. Proof. To prove the strong duaity resut, we create an inverse power imization to the upin max- MSE baancing probem in 15). This power imization probem is moreover strongy dua to the convex power imization probem in 14). Therefore, the same transmit power is required to achieve the same MSE for a users in the upin and the downin. Since the power imization in 14) is again inverse to the downin MSE baancing probem in 9), the baanced MSE in the upin and the downin is the same. To find the upin power imization, we remar that 15) is independent w.r.t. a common scaing of µ and λ and the power constraint wi be satisfied with equaity in the optimum. Therefore, we may repace the sum power constraint in 15) by the two constraints L µ P 1 and λ iσi 1 Note that that the constraints in 14) may not be attainabe, e.g., when ˆε < K N K even if a R s are ran-one [17]. For matrices R with a ran arger than one, the attainabe ˆε can ie far beow this bound. We set α ˆε) to infinity in this cases.
without changing the soution. Keeping the former of the two constraints and changing the atter one to λ iσi P, the imum baanced upin MSE ˆε UL P) becomes a stricty monotonicay decreasing function in P 0. The corresponding inverse function reads as P ˆε) =max λ i σi 19) µ 0 λ 0,u s.t.: µ P 1, ˆε MSE i,ul, i = 1,...,K. It remains to show that 19) is dua to the convex optimization in 14). This proof directy foows the steps from Yu and Lan in [10]. Note that duaity can be based on the quadratic constraints from 13) instead of those in 14) since the resuting KKT conditions are equivaent [18, Appendix A]. Hence, we can write the Lagrangian function of 14) as ) Lα,t,λ,µ) = λ i σi +α 1 µ P 0) + t i Y i λ i 1 ˆε h i hh i ) t i where Y i = L µ A i, + =1 λ R. The dua objective resuts from the unconstrained imization of 0) w.r.t.αandt, i.e., gλ,µ) = α,t Lα,t,λ,µ). Since α and the t i s are unconstrained,gλ,µ) uness µ P 1 1) and Y i λi 1 ˆε h i hh i 0 N N. With Schur s compement [19, A.5.5], we can recast the atter condition as cf. [10]) 1 ˆε) λ i hh i Y i h i 0. ) Equivaence foows since Y i 0 N N, 1 ˆε > 0, and I N Y i Y i ) h i = 0 N as Y i λ i R i = λ i E[h i h H i ] λ i h i hh i. With ) and 1), the dua probem of 14) reads as λ i σi s.t.: µ P 1, 3) max µ,λ 0 1 ˆε λ i h H L i µ A i, + =1 λ, i = 1,...,K. R ) hi The right hand side of the MMSE constraint in 3) is positive and subineary monotonicay increasing in λ 0 when µ is fixed. In other words, these right hand sides define a standard interference function [0] that is parametrized in µ and there is a unique λ 0 that satisfies a MMSE constraints with equaity and imizes the objective if ˆε is attainabe. Hence, reversing the maximization over λ into a imization and the direction of the inequaity in the MMSE constraints does not affect the soution. Moreover, since ˆε 1 λ i hh i Y i h i = MMSE i,ul 4) as can be seen in 18), we indeed obtain the power imization formuation in 19). This proves that a soution of 15) resuts in a soution for 9). The downin beamformers and receive fiters foow from the upin fiters in 17) and powers in λ by cf. [7]) t i = β i u i, fi = λ i / βi, i = 1,...,K 5) when the baanced MSEs of 9) and 15) are equa, i.e., MSE i = MSE i,ul, i = 1,...,K. 6) Inserting 5) into the downin MSEs from 10), the equation system in 6) can be rewritten as Ψβ = Σλ 7) where Σ = diagσ1,...,σ K ), β = [β 1,...,β K ] T, and { i [Ψ] i,j = λ u H i R u i,+ L µ u H i A i,u i i = j, λ i u H j R iu j, i j. As Ψ is coumn-wise diagonay doant with positive diagona eements and non-positive off-diagona eements, its inverse exists and has non-negative eements. That means, we can sove 7) for positive β, and therewith cacuate the downin beamformers and fiters in 5). C. Iterative Upin MSE Baancing An iterative soution for the upin MSE baancing probem consists of two nested oops. The inner oop soves the power aocation and equaizer optimization) in the MAC, whie the outer oop contros the upin noise covariance cf. [11]). To baance the MSEs and satisfy the sum-power constraint in 15) with equaity, we use the gobay convergent update λ n+1) 1 ˆε n+1) i h H K i =1 λn) R + L µ, 8) A i, ) hi which foows from the constraint formuation in 3). The normaization with 1 ˆε n+1), where [cf. constraint in 15)] L ˆε n+1) 1 µ P σ i / h H i =1 λn) R + L µ A i, ) h, i is to expoit fu transmit power. This fixed-point update is ess compex than the eigenvector cacuation from [8], but more iterations are required unti convergence cf. [11]). To find µ, a subgradient projection method simiar to [1] can be empoyed in the outer oop. The -th component of the subgradient δ = [δ 1,...,δ L ] T is δ = P + th i A i,t i, where the t i s are the beamformers from the previous iteration. Therewith, a subgradient projection step reads as µ j+1) P C µ j) +a j δ ). The projection sha w..o.g. be onto the simpex C = { µ R L + L µ = 1 }, since a scaing of µ does not change the optimum of 15) and a j denotes the step size in iteration j. Another update rue, that is used in the iterature, is cf. [11]) µ µm) t H i P A i,t i, µ m+1) µ L µ, 9) which ensures strong duaity in the convergence point, i.e, µ P th i A ) i,t i = 0. We remar that 9) increases/decreases those µ that correspond to vioated/satisfied
average MSE 10 0 10 1 10 Europe, SatCom κ = 5 db κ = 15 db imp. CSI, SatCom per. CSI, SatCom imp. CSI, Gauss. per. CSI, Gauss. 10 0 10 0 30 SNR P /σ in db Figure 1: average MSE vs. SNR Parameter Vaue sateite configuration GEO; Ka-band; reuse 1 beamwidth θ 3dB in degree) 0. number of beams custer 7/Europe) 7/18 og-norma fading m /σξ [] 3.06 db/1.51 db max sateite antenna gain 5 dbi max user antenna gain 40 dbi base receive noise power; approx. FSL -118 dbw; 10 db SNR P /σ -10,...,30 db Tabe I: Lin budget Parameters in SatCom power constraints, respectivey, which is a necessary requirement for convergence to a oca maximizer. Both updates for µ converged to the same MSEs in our simuations. However, the fixed-point search in 9) is ess compex than the subgradient method cf. [11]). V. NUMERICAL RESULTS We computed resuts for a standard Gaussian fading mode and a SatCom mode. For the standard fading mode, the channe means h are drawn from a standard Gaussian distribution and scaed to have the same norm as the sateite channe means, and the covariances are C = 1 N I N. The main parameters for the considered SatCom scenarios are shown in Tabe I. Per-antenna constraints are imposed, i.e., one antenna per feed. The users are randomy paced within the 3 db area of the beams, i.e., N = K and one user per spotbeam. The baanced average MSEs are cacuated for a 7 ce system with 100 different user pacements and one user reaization for an 18 ce system that represents the coverage of Europe. In Fig. 1, the average) baanced MSEs are depicted vs. P /σ. For perfect CSI, the MSEs decrease unbounded whie the imperfect CSI curves MSE bound) saturate. The higher the Rician factor κ, the ower the saturation eve. For the SatCom channes with κ = 15 db, the muti-path scattering may be negected. No saturation is visibe in the given SNR regime. 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