Predistortion and Precoding Vector Assignment in Codebook-based Downlink Beamforming

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1 013 IEEE 14th Worshop on Signal Processing Advances in Wireless Communications (SPAWC Predistortion and Precoding Vector Assignment in Codeboo-based Downlin Beamforming Yong Cheng, Student Member, IEEE, and Marius Pesavento, Member, IEEE Communication Systems Group, Technische Universität Darmstadt, Mercstr. 5, 6483 Darmstadt, Germany Abstract This paper considers codeboo-based downlin beamforming proposing a linear adaptive channel predistortion method to close the performance gap between codeboo and noncodeboo based beamforming defined in wireless standards, e.g., LTE-A. The proposed channel predistortion method does not involve any additional signalling overhead. In our novel beamforming concept, we optimize the codeboo-based precoding vector assignment, the power allocation, and the channel predistortion matrix jointly to minimize the transmitted power of the base station (BS while guaranteeing the quality-of-service (QoS of the mobile stations (MSs and satisfying the smoothness conditions on the channel predistortion matrix. The joint channel predistortion and codeboo-based beamforming (PCB problem represents a non-convex mixed integer program (MIP. An alternating optimization algorithm (ATOA and an alternating feasibility search algorithm (AFSA are developed to approximately solve the PCB problem. Simulation results show that the proposed codeboo-based beamforming achieves significant reductions of the transmitted power of the BS and remarable increases of the percentage of feasible cases of all Monte Carlo runs, as compared to the standard codeboo-based beamforming. Numerical results also show that the proposed design performs very close to the non-codeboo based beamforming in the given settings and that the proposed ATOA and AFSA are close-to-optimal. Index Terms Codeboo-based Precoding, Channel predistortion, Non-convex Mixed Integer Program, Alternating Optimization, Alternating Feasibility Search I. INTRODUCTION Downlin beamforming (i.e., single-layer precoding represents an efficient transmission scheme in which multiple MSs are served on the same time-frequency resources and increases significantly the spectrum and energy efficiency of cellular networs [1] [4]. Both non-codeboo and codeboobased beamforming schemes are defined in modern wireless standards, e.g., LTE-A [5] [7]. Unlie in non-codeboo based beamforming where the normalized beamformers tae values on a continuous manifold, in codeboo-based beamforming the normalized beamformer of each MS is restricted to a predefined precoding vector contained in a finite codeboo [4] [6]. As a result, codeboo-based beamforming inherently exhibits performance loss in term of, e.g., increased transmitted power of the BS required to guarantee the QoS of the MSs, as compared to non-codeboo based beamforming (see Section VI. Enhancing the performance of codeboo-based beamforming to achieve that of non-codeboo based beamforming without introducing additional signaling overhead, e.g., reducing the The corresponding author. s: cheng, pesavento}@nt.tudarmstadt.de. This wor was supported in part by the LOEWE Priority Program Cocoon ( transmitted power of the BS while ensuring the QoS of the MSs, is therefore of great practical interest. We propose in this paper to incorporate a predistortion step in the standard codeboo-based beamforming [4] [6], which consists in linear predistortion of the downlin channel vectors. The channel predistortion step introduces additional degrees of freedom for beamformer design in the standard codeboo-based beamforming. As shall be discussed further in next section, the predistortion step involves minor modification in the transmission of the cell-specific reference signals and user payload data and therefore it does not introduce any additional signaling overhead and can straightforwardly be applied in current 3G and 4G standards. We consider QoSconstrained codeboo-based beamformer design, where the precoding vector assignment, the power allocation, and the predistortion matrix are jointly optimized to minimize the total transmitted power of the BS subject to individual QoS constraints of the MSs, with the QoS of a MS measured by its received signal-to-interference-plus-noise ratio (SINR [1] [3]. To ensure that the predistorted channels can be traced in the channel estimation procedure performed at the MSs, smoothness constraints are imposed on the predistortion matrix. The joint channel predistortion and codeboo-based beamforming (PCB problem can be cast as a non-convex mixed integer program (MIP, which remains a non-convex program even after fixing the integer constraints, and efficient optimal solutions for non-convex MIPs are not available in the literature [8]. To facilitate practical applications, we develop an alternating optimization algorithm (ATOA to approximately solve the PCB problem. The ATOA iterates between solving two subproblems: (i optimizing the precoding vector assignment and the power allocation under a fixed predistortion matrix, which represents a mixed integer linear program (MILP that can be efficiently solved using, e.g, the branch-and-cut (BnC method [8], and (ii optimizing the predistortion matrix and the power allocation under a fixed precoding vector assignment, which can be approximated by a second-order cone program (SOCP that can be efficiently solved using, e.g, the interiorpoint method [9]. Since either one of the two subproblems in the ATOA may become infeasible even when the original PCB problem is feasible, an alternating feasibility search algorithm (AFSA, which follows a similar procedure as the ATOA, is also developed to compute feasible solutions of the PCB problem which are used to initialize the ATOA. Convergence properties of the ATOA and AFSA are analyzed. Simulations results show that the joint channel predistortion /13/$ IEEE 445

2 013 IEEE 14th Worshop on Signal Processing Advances in Wireless Communications (SPAWC and codeboo-based beamformer design achieves significant reductions in the transmitted power of the BS and large increases in the percentage of feasible cases over all Monte Carlo runs, as compared to the standard codeboo-based beamforming. Numerical results also show that the performance of the proposed design is very close to that of the non-codeboo based beamforming in terms of both the total transmitted power of the BS and the percentage of feasible cases and that the ATOA and the AFSA yield near-optimal solutions of the PCB problem under the considered settings. II. SYSTEM MODEL AND PROBLEM STATEMENT Consider a downlin cellular system comprising a single BS equipped with M antennas and K single-antenna MSs. Denote h (t C M 1 and w (t C M 1 as the frequencyflat channel vector and the beamformer, respectively, of the th MS in the tth time-slot, K 1,,,K}. Note that we omit the time-slot index t in the sequel for succinctness of presentation when it is clear from the context. We study here the codeboo-based beamforming scheme as defined in, e.g., LTE-A [4] [6], where the normalized beamformers, i.e., w / w, of the th MS is chosen as one of the given precoding vectors in the beamformer codeboo B that consists of L unit-norm precoding vectors, i.e., B v 1, v,, v L }, with v l C M 1 and v l = 1, l L 1,,,L}. The codeboo-based beamforming naturally admits performance loss in terms of, e.g., increased transmitted power of the BS required to ensure the QoS of the MSs, as compared to non-codeboo based beamforming [4] [6]. To improve the performance of codeboo-based beamforming, e.g., to reduce the transmitted power of the BS, we propose here a predistortion step applying a common linear transformation of the channel vectors h, K} with the predistortion matrix G C M M. With the linear channel predistortion, the received signal y C at the th MS is given by ( y = G h xj +z = wj h x j +z (1 j=1 w j j=1 where x C represents the normalized data symbol, i.e., E x } = 1, of the th MS, z C denotes the white Gaussian noise at the th MS, with zero mean and variance σ, and the predistorted channel vector h is defined as h G h, K. ( It is important to point out that the channel predistortion ( can be embedded in the transmitter chains for the cell-specific reference signals and user payload data [6], i.e., instead of transmitting K j=1 w j x j, the BS transmits K j=1 w j G x j when channel predistortion is applied. The th MS directly estimates the predistorted channel vector h with the help of cell-specific reference signals that are transmitted by the BS, e.g., in every downlin subframe for a LTE-A system [6]. The th MS then uses the predistorted channel vector h for coherent data symbols detection. As a result, the proposed predistortion step introduces no additional signalling overhead. To ensure that the channel estimation procedure at the th MS, which is designed for the original channel process h (t}, is not adversely affected and can be equivalently performed on the predistorted channel process h (t}, we impose the following smoothness-constraints: G (th (t h (t ǫ (t, K (3 where ǫ (t denotes a small constant at time-slot t. Eq. (3 implies, with δ (t h (t h (t 1, that h (t h (t 1 = G (th (t h (t+h (t h (t 1+h (t 1 G (t 1h (t 1 ǫ (t+δ (t+ǫ (t 1, K. (4 Eq. (4 suggests that the norm of the change of the process h (t} in two consecutive time-slots is bounded and the parameter ǫ (t can be configured so that the th MS can successfully estimate the predistorted channel process h (t}. To model the precoding vector assignment, we introduce the binary variable b,l 0,1} to indicate with b,l = 1 that the lth precoding vector v l B is assigned to the th MS, and b,l = 0 otherwise. We further introduce the variable p,l 0 to model the power allocated to the lth precoding vector for the th MS. Since only one precoding vector shall be assigned to a MS in single-layer precoding, we have that b,l = 1, K (5 0 p,l Gv l b,l P (MAX, K, l L (6 w p,l b,l v l = p,l v l, K (7 =1 =1 K Gw = p,l Gv l P(MAX (8 where Eq. (6 implements the big-m method [8] to ensure that p,l Gv l = 0 and p,l = 0 when b,l = 0. Eq. (6 is automatically satisfied when b,l = 1 due to Eq. (8, which represents the per-bs sum-power constraint, with P (MAX denoting the maximum transmission power of the BS. To obtain Eq. (8, we have used the following fact w j G h L = p j,l v l G h, j, K (9 which holds because of Eqs. (5 (7. Assuming that the data symbols of the MSs are mutually independent and independent from the noise, the received SINR at the th MS under single-user detection, denoted by SINR, can then be expressed as [1] [4] w SINR G h j=1,j wj G h L = p,m v m G h L p j,l v l G h, K (10 K j=1,j where we have used Eq. (9 in the second line of Eq. (10. We consider here QoS-constrained channel predistortion and /13/$ IEEE 446

3 013 IEEE 14th Worshop on Signal Processing Advances in Wireless Communications (SPAWC beamformer design, where the precoding vector assignment and the power allocation b,l,p,l, K, l L}, and the predistortion matrix G are jointly optimized to minimize the transmitted power of the BS while guaranteeing the prescribed SINR target γ representing the QoS requirement of the th MS, K. The joint channel predistortion and codeboobased beamforming (PCB problem can be stated as min b,l,p,l,g} =1 p,l Gv l (11a s.t. (3, (5, (6, and (8 (11b SINR γ, K b,l 0,1}, K, l L. (11c (11d When the predistortion matrix is fixed to the identity matrix, i.e., when G = I, problem (11 reduces to the standard codeboo-based beamforming problem [4] [6], which is a MILP and can be efficiently solved using, e.g., the BnC method [8]. owever, when the predistortion matrixgis to be optimized, problem (11 represents a non-convex MIP due to the SINR constraints (11c, i.e., problem (11 remains a nonconvex program even after fixing the integer constraints (11d and it can not be efficiently solved to optimality [8]. III. AN ALTERNATING OPTIMIZATION ALGORITM We now from Eq. (10 that the terms p j,l v l G h, j, K, l L, in the SINR constraints (11c mae problem (11 a non-convex MIP. To deal with these terms, we propose in this section an alternating optimization approach. In each alternation, one MILP and one SOCP are solved. A. MILP with Fixed Predistortion Matrix We first consider optimizing the precoding vector assignment and the power allocation under a fixed predistortion matrix. Denote G (n 1 as the predistortion matrix used to initialize the nth alternation, e.g., choosing G (0 = I in the first alternation. With G = G (n 1, Eqs. (6 and (8 and the SINR constraints (11c can be respectively rewritten as 0 p,l G (n 1 v l b,l P (MAX, K, l L (1 p,l G (n 1 v l P (MAX (13 =1 j=1,j p j,l h G(n 1 v l γ p,m h G (n 1 v m, K. (14 With the fixed predistortion matrixg (n 1, the per-bs sumpower constraint (13 and the SINR constraints (14 become linear constraints in the variables p,l, K, l L} and therefore problem (11 reduces to the following MILP:,l,p(n,l } argmin b,l,p,l } =1 p,l G (n 1 v l (15a s.t. (5, (11d, and (1 (14 (15b which can be efficiently solved using, e.g., the BnC method [8]. If problem (15 is infeasible with G (0 = I in the first alternation, an alternating feasibility search algorithm (AFSA can be applied to compute a feasible initialization of the matrix G (0, which is further discussed in Section IV. B. SOCP with Fixed Precoding Vector Assignment We next consider optimizing the predistortion matrix and the power allocation under a fixed precoding vector assignment. In the nth alternation, for a fixed precoding vector assignment, i.e., for the given indicators,l, K, l L}, we define the auxiliary optimization vector u C M 1 as u,l Gv l, K (16 which is linear in the predistortion matrixg. We further define the auxiliary optimization variable q > 0 as q 1 / L p,l, K. (17 Eqs. (5, (6, and (10 together suggest that one and only one of the variables p,l, l L} is non-zero. ence, with Eqs. (16 and (17, the per-bs sum-power constraint (8 and the SINR constraints (11c can be respectively rewritten as =1 K p,l Gv l = j=1,j h u j =1 u q P (MAX (18 γ h u q, K. (19 Since the quadratic-over-linear function h u j / is jointly convex in the variables and u j, j, K [9], the SINR constraints (19 represent difference of convex (DC constraints [10], [11]. A common approach to deal with such DC constraints (19 is to linearize the right-hand-side (RS of Eq. (19 [10], [11]. Define respectively the constant variable q (n and the vector û (n in the nth alternation as q (n 1 / L û (n p (n,l, K (0,l G(n 1 v l, K. (1 The first-order Taylor expansion of the function h u /q, i.e., the linearization of the RS of the SINR ( constraints (19, at the point [ q (n,(û(n T ] T, denoted by ψ q (n,û(n, is given by [10]: ( ψ q (n,û(n } Re u R û (n (n / q h û (n /, q K ( ( q (n where Re } stands for the real part of a variable and the matrix R h h, K. Replacing ( the RS of Eq. (19 with the linear approximation ψ q (n,û(n, we obtain the following strengthened SINR constraints [10]: /13/$ IEEE 447

4 013 IEEE 14th Worshop on Signal Processing Advances in Wireless Communications (SPAWC j=1,j h u j γ ( ψ q (n,û(n, K. (3 Since the quadratic-over-linear functions can be implemented with second-order cone formulations [9], with the given precoding vector assignment,l, K, l L} and the strengthened SINR constraints (3, the PCB problem (11 can be approximated by the following SOCP: } G (n,q (n u,u(n argmin (4a G,q,u } q =1 s.t. (3, (18, and (3 (4b q > 0, K (4c which can be efficiently solved using, e.g., the interior-point method [9]. The power allocation p,l, K, l L} is optimized via the variables q, K} in problem (4. Init.: Set α and N (MAX, initialize G (0 according to the AFSA presented in Section IV, and set n = 1. Repeat: Step 1: Solving problem (15 obtaining the total transmitted power Υ (n K L =1 p(n,l G(n 1 v l. Step : Solving problem (4 obtaining the total transmitted power Φ (n K =1 u(n /q(n. Step 3: If Υ (n Φ (n < α, or if n = N (MAX, stop. Otherwise, update n n+1 and go bac to Step 1. Alg. 1: The alternating optimization algorithm (ATOA Since the point G (n 1, q (n,û(n, K} is a feasible solution of problem (4, problem (4 is feasible as long as problem (15 is feasible. The feasibility of problem (15 and that of the PCB problem (11 will be discussed in Section IV. The alternating optimization algorithm (ATOA iterates between solving problem (15 and problem (4 until the reduction of the total transmitted power of the BS is less than a predefined numerical tolerance α > 0, or until the maximum number of allowable alternations N (MAX is reached. Since the total transmitted power resulted from solving problems (15 and (4 does not increase and it is positive, the total transmitted power computed in the alternating process converges and hence the proposed ATOA converges (not necessarily to optimal solutions of problem (11. The computational complexity of the ATOA can easily be controlled by tuning the parameters α and N (MAX. The proposed ATOA is summarized in Alg. 1. IV. AN ALTERNATING FEASIBILITY SEARC ALGORITM As mentioned in the previous section, the simple choice of G (0 = I may result in an infeasible problem (15 in the first alternation in Alg. 1 even if the original PCB problem (11 is feasible. In this case, an alternating feasibility search algorithm (AFSA can be applied to compute a feasible initialization G (0. The AFSA follows a similar procedure as Alg. 1. owever, in Step 1 of the AFSA, instead of problem (15, the following MILP feasibility problem [8], [9],l,p(n,l,r(n} argmin r (5a b,l,p,l,r} s.t. (5, (11d, (1, and (13 ( K p j,l h G(n 1 v l γ j=1,j (5b p,m h G (n 1 v m +r, K (5c is solved using, e.g., the BnC method [8], with G (0 fixed to I. The optimal value r (n in (5a represents a measure of infeasibility [9]. In the case that r ( n is less than or equal to zero, or if r ( n is less than the prescribed infeasibility tolerance β > 0, the point G (n 1,,l,p(n,l, K, l L} is a feasible solution of the original PCB problem (11 [8], [9]. Accordingly, in Step of the AFSA, instead of problem (4, the following SOCP feasibility problem [9] G (n,q (n,u(n,s(n} argmin G,q,u,s} s.t. (3, (18, and (4c ( K h u j j=1,j γ ( ψ q (n,û(n s, (6a +s, K (6b (6c is solved using, e.g., the interior-point method [9]. If the measure of infeasibility s (n is less than β, then the point G (n,,l,p(n,l, K, l L} is a feasible solution of problem (11 [8] [10]. The AFSA algorithm iterates between solving problems (5 and (6 until a feasible solution of problem (11 is found, or until the reduction in the infeasibility measure, i.e., r (n s (n is less than β, or until the maximum number of allowable alternations N (MAX is reached. The proposed AFSA algorithm is summarized in Alg.. Init.: Set β and N (MAX, and set G (0 = I and n = 1. Repeat: Step 1: Solving problem (5. If r (n < β, stop. Step : Solving problem (6. If s (n < β, stop. Step 3: If r (n s (n < β, or if n = N (MAX, stop. Otherwise, update n n+1 and go bac to Step 1. Alg. : The alternating feasibility search algorithm (AFSA Similar to the proposed ATOA in Alg. 1, the developed AFSA in Alg. converges. owever, the proposed AFSA is not guaranteed to yield a feasible solution of the PCB problem (11 even when problem (11 is indeed feasible. In such cases, user admission control mechanisms may be applied to select a subset of the MSs to be served, which, however, is out of the focus of this paper and is left as future wor. We remar that the proposed alternating optimization approach represents a two-stage numerical method, namely in the first stage, the AFSA in Alg. is applied to compute a /13/$ IEEE 448

5 013 IEEE 14th Worshop on Signal Processing Advances in Wireless Communications (SPAWC feasible solution of problem (11, which is used to initialize the ATOA in Alg. 1 in the second stage. V. NUMERICAL RESULTS AND DISCUSSIONS We simulate a downlin system with K = 4 MSs, andm = 4 antennas and the maximum transmission power P (MAX = 15 db at the BS. We adopt the following channel model [7]: (i 3GPP LTE pathloss mode: PL = log 10 (d (in db, with d (in m denoting the BS-MS distance; (ii lognorm shadowing with zero mean, 8 db variance; (iii Rayleigh fading with zero mean and unit variance; and (iv transmit antenna power gain of 9 db, and a system bandwidth of 1.4 Mz giving the noise power σ = 143 db, K. The distances between the BS and the MSs are randomly generated in the interval [0.05,1] m. The codeboo with L = 16 unitnorm precoding vectors defined in LTE-A [5] is used. We choose α = 10, β = 10 5, and N (MAX = N (MAX = 3 in the simulations. For the smoothness constraints (3, we choose ǫ = ǫ h, with ǫ 0.1,0.,0.3,0.4}, K. The SINR targets of the MSs are chosen to be identical and are listed in the figures. As references, the standard codeboo-based beamforming corresponding to G = I and the non-codeboo based beamforming are also simulated. All simulation results are averaged over 000 Monte Carlo runs. Total transmitted power of the BS [db] Standard codeboo based Proposed design, ε = 0.1 Proposed design, ε = 0. Proposed design, ε = 0.3 Proposed design, ε = 0.4 Non codeboo based Minimum SINR requirements γ [db] Fig. 1: Total transmitted power of the BS vs. γ Fig. 1 displays the total transmitted power of the BS vs. the SINR requirementsγ. We observe from Fig. 1 that, compared to the standard codeboo-based beamforming, the proposed design, i.e., with the optimized predistortion matrix G, the transmitted power of the BS is significantly reduced, e.g. a reduction of 4.45 db at γ = 1 db with ǫ = 0.4. The proposed design yields total transmitted powers that are very close to the lower bounds set by the non-codeboo based beamforming, e.g., exceeding the lower bounds by less than 0.44 db for all considered values of γ with ǫ = 0.4. Fig. depicts the percentage of feasible cases over all the Monte Carlo runs vs. the SINR targetsγ. We observe that the percentage of feasible cases is tremendously increased with the proposed design as compared to the standard codeboo-based beamforming, e.g., an increase of 46% at γ = 1 db with ǫ = 0.4. We also see that the proposed design almost achieves the upper bounds of percentages of feasible cases set by the non-codeboo based beamforming with ǫ = 0.4. The fact that the proposed design almost achieves the lower bounds of transmitted powers of the BS and the upper bounds of percentages of feasible cases set by the non-codeboo based beamforming also proves that the alternation optimization approach, i.e., the ATOA in Alg. 1 and the AFSA in Alg., is close-to-optimal for solving the PCB problem (11. Percentage of feasible cases achieved Non codeboo based Proposed design, ε = 0.4 Proposed design, ε = 0.3 Proposed design, ε = 0. Proposed design, ε = 0.1 Standard codeboo based Minimum SINR requirements γ [db] Fig. : Percentage of feasible cases achieved vs. γ VI. CONCLUSION We have proposed an adaptive linear channel predistortion procedure that is applied on the channel vectorsh, K} to improve the performance of the standard codeboo-based beamforming, which does not involve additional signaling overhead and can straightforwardly be incorporated into current 3G and 4G cellular standards. The effectiveness of the proposed design is showcased via studying the PCB problem (11, which is solved approximately by the developed ATOA in Alg. 1 and the AFSA in Alg.. Simulation results clearly illustrate the improvement of the proposed design over the standard codeboo-based beamforming, e.g., significant reductions of the transmitted power of the BS, and that the proposed ATOA and AFSA are close-to-optimal for the PCB problem (11 with the considered settings. REFERENCES [1] M. Bengtsson and B. Ottersten, Optimal and Suboptimal Transmit Beamforming. In: andboo of Antennas in Wireless Communications, CRC Press, Aug [] M. Schubert and. Boche, Solution of the multi-user downlin beamforming problem with individual SINR constraints, IEEE Trans. Veh. Technol., vol. 53, no. 1, pp. 18 8, Jan [3] A. Gershman, N. Sidiropoulos, S. Shahbazpanahi, M. Bengtsson, and B. Ottersten, Convex optimization-based beamforming: From receive to transmit and networ designs, IEEE Signal Process. 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