Joint beamforming and power control in d

Joint Beamforming and Power Control in Downlink Multiuser MIMO Systems Shiny Abraham† , Dimitrie C Popescu† , and Octavia A Dobre‡ Department of Electrical and Computer Engineering, Old Dominion University, USA Faculty of Engineering and Applied Science, Memorial University of Newfoundland, Canada † ‡ and P = diag[p1 , , pK ] is a K × K diagonal matrix containing the transmit powers corresponding to distinct users in the system The transmitted signal x is received by the K user receivers through distinct MIMO channels characterized by matrices G1 , , GK of dimension Nk × N , and is corrupted by additive Gaussian noise vectors n1 , , nK with dimension Nk and covariance matrices Wk = E[nk n> k ], k = 1, , K Thus, the received signal by a given user k is given by Abstract— In this paper we discuss joint beamforming and power control for downlink multiuser MIMO systems with target signal-to-interference+noise ratios (SINR) constraints We derive necessary conditions for minimizing total interference at downlink receivers and present an algorithm which adapts the beamforming vectors and powers incrementally to meet specified SINR targets with minimum powers The proposed algorithm is illustrated with numerical examples obtained from simulations Index Terms— MIMO systems, downlink, beamforming, power control, quality of service I I NTRODUCTION MIMO wireless systems exploit spatial dimensions in wireless channels to provide increased capacity and diversity, as well as to mitigate interference [1] This is achieved through transmit beamforming and receiver combining techniques [2], [3] which take advantage of the significant diversity that is available in MIMO systems When users are expected to meet specified target SINRs at the receiver, beamforming is complemented with transmitter power control, and joint beamforming and power control problems have been discussed in [4], [5] A related problem was approached in the context of cellular MIMO systems in [6] where the proposed algorithm exploits network duality and is implemented using already existing algorithms In this paper we present a new algorithm for joint beamforming and power control in downlink multiuser MIMO systems with target SINR constraints at the receivers The proposed algorithm employs incremental updates for beamforming vectors and transmitted powers that are designed to reduce interference at downlink receivers subject to specified SINR and beamforming constraints II S YSTEM M ODEL AND P ROBLEM S TATEMENT We consider the downlink of a multiuser MIMO wireless system in which the base station is equipped with N transmit antennas and communicates with K active users We assume that a given user k has Nk receive antennas such that Nk ≤ N , ∀k The signal transmitted by the base station is expressed as x= K X √ b p s = SP1/2 b rk = Gk x + nk = Gk SP1/2 b + nk We assume that all channel matrices Gk are known by the base station transmitter and that they are fixed for the entire duration of the transmission Our goal in this setup is to derive an algorithm by which the base station transmitter performs joint beamforming and power adaptation such that a set of specified target SINRs {γ1 , , γK } is achieved by all users III SINR AND I NTERFERENCE E XPRESSIONS In order to decode the desired signal at a given receiver k we rewrite the received signal in equation (2) from the perspective of user k ⎛ ⎞ K X √ √ b p s ⎠ + nk (3) rk = Gk bk pk sk + Gk ⎝ | {z } =1, 6=k desired signal | {z } interference + noise (zk ) where the interference-plus-noise zk seen by user k has correlation matrix ⎛ ⎞ K X ⎝ s p s> ⎠ G> Zk = E[zk z> k ] = Gk k + Wk (4) =1, 6=k We note that being a correlation matrix Zk is symmetric and positive definite Thus, it can be diagonalized as Zk = Ek ∆k E> k and we can define the whitening transformation −1/2 (1) Tk = ∆k =1 E> k (5) In transformed coordinates equation (3) can be written as √ ˜rk = Tk rk = Tk Gk bk pk sk + Tk zk √ (6) ˜ k bk pk sk + wk = G where S = [s1 sK ] is the N × K matrix of unitnorm beamforming vectors, b = [b1 , , bK ]> is the Kdimensional vector of symbols transmitted to active users, 978-1-4244-4726-8/10/$25.00 © 2010 IEEE k = 1, , K (2) 444 RWS 2010 ˜ k = Tk Gk is the MIMO channel matrix seen where G by user k in new coordinates and wk = Tk zk is the equivalent white noise term with covariance matrix E[wk wk> ] = Tk Zk T> k = INk equal to the identity matrix We now apply the SVD to the transformed channel matrix to obtain ˜ k = Uk Dk Vk> G The decision variable for user k obtained by matched rk,inv and implies that the interference filtering is ˆbk = ˜s> kˆ experienced by user k is ˜ −2 sk (14) ik = ˜s> kD ˜ k and its corresponding SINR is γk = pk /ik IV J OINT B EAMFORMING AND P OWER C ONTROL F OR D OWNLINK MIMO S YSTEMS Since the base station transmitter has knowledge of all the user beamforming vectors, transmit powers, and MIMO channel matrices, we will obtain the beamforming and power update equations by solving the constrained minimization of the sum of interference functions, that is (7) Let us denote the rank of user k’s transformed MIMO ˜ k ) This is equal to the channel matrix ρk = rank(G number of non-zero singular values and satisfies ρk ≤ min(N, Nk ) The singular value matrix Dk may be partitioned as ∙ ¸ ˜k D 0ρk ×(Nk −ρk ) (8) Dk = 0N×ρk 0(Nk −ρk )×(Nk −ρk ) ⎧ ⎨ ⎩ ˜ k is a ρk ×ρk diagonal matrix containing the nonwhere D zero singular values and the zero matrices have appropriate dimensions We premultiply by U> k in equation (6) to obtain √ ¯rk = Dk Vk> sk bk pk + U> (9) k wk ˜s1 , , ˜sK p1 , , pK γk = γk∗ ⎫I = ⎬ ⎭ K X ik subject to k=1 and ˜s> sk = 1, k˜ k = 1, , K (15) In order to solve this constrained optimization problem we define the corresponding Lagrangian function L(˜s1 , , ˜sK , p1 , , pK , λ1 , , λK , ξ1 , , ξK ) = ¯ k = U> and we define ¯sk = Vk> sk and w k wk We can then rewrite equation (9) as √ ¯k ¯rk = Dk ¯sk bk pk + w (10) =I+ K X k=1 k ả X K pk k + ξk (˜s> sk − 1) k˜ ik k=1 (16) where λk and ξk are Lagrange multipliers associated the constraints in equation (15) The necessary conditions for minimizing the Lagrangian (16) are obtained by differentiating with respect to the corresponding variables, ˜sk , pk , and multipliers λk , ξk , k = 1, , K and by equating the corresponding partial derivatives to zero Differentiating with respect to the beamforming ˜sk leads to the eigenvalue/eigenvector equation ∂Lk ˜ −2˜sk = νk˜sk = =⇒ D (17) k ∂˜sk where νk is expressed in terms of the Lagrange multipliers as well as user power pk and beamforming vector ˜sk The exact expression of νk is not relevant and any eigenvector ˜ −2 will satisfy the necessary condition (17) However, of D k a meaningful choice for updating user k’s beamforming vector is the eigenvector xk corresponding to the minimum ˜ −2 since for a given power value pk this eigenvalue of D k minimizes the term ik corresponding to user k in the sum interference function I In order to avoid potential steep changes in the beamforming vector that may not be tracked by the receiver due to the minimum eigenvector being far away in signal space from the current beamforming vector, we will use an incremental update that adapts the user beamforming vector in the direction of the minimum eigenvector xk defined by: ˜sk (n) + mβxk (n) ˜sk (n + 1) = (18) k˜sk (n) + mβxk (n)k The partition (8) on the singular value matrix Dk induces the following partition on the transformed beamforming vector ¯sk corresponding to user k ∙ ¸ ¯sk1 ¯sk = (11) ¯sk2 where ¯sk1 has dimension ρk × and ¯sk2 has dimension (Nk − ρk ) × Taking into account the partitions in (8) and (11) we note that the last (N − ρk ) components in the ¯rk vector can be safely ignored as they will be equal to zero Furthermore, the last (N − ρk ) components ¯sk2 of the transformed beamforming vector should be set to zero ( ¯sk2 = 0(Nk −ρk )×1 ) in order to ensure that no transmitted signal energy will be wasted on those dimensions of the transformed channel matrix with zero singular values Thus, we focus only on those dimensions corresponding to strictly positive singular values and reduce dimensionality to the rank of the transformed channel matrix by taking the first ρk elements in ¯rk to obtain ˜ k ˜sk bk √pk + w ˜k ˆrk = [Iρk 0]¯rk = D (12) ˜ k = [Iρk 0]w ¯ k We invert the channel where ˜sk = ¯sk1 and w in equation (12) to obtain the equivalent expression ˜ −1ˆrk = ˜sk bk √pk + D ˜ −1 w ˜k ˆrk,inv = D (13) k | {z } | k{z } desired signal interf.+noise 445 where β is a parameter that limits how far in terms of Euclidian distance the updated beamforming vector can be from the old one and m = sgn[˜s> k (n)xk (n)] This update corresponds to an incremental interference avoidance update [7] that implies a decrease in the user interference function ik and an increase in the user SINR Upon adaptation of user k beamforming vector, the value of its corresponding effective interference function is given by the expression: ˜ −2 (n)˜sk (n + 1) i0k (n) = ˜sk (n + 1)> D k −2 > ˜ (n)˜sk (n) ≤ ˜sk (n) D k c) IF ρk < Nk append a zero vector of appropriate dimension and obtain the actual beamforming vector sk = Vk ˜sk d) Update user k’s power using equation (23) 3) IF change in sum interference function I is larger than specified tolerance then GO TO Step ELSE STOP: a fixed point has been reached Numerically, a fixed point of the algorithm is reached when the beamforming and power updates result in changes of the sum interference functions I that are smaller than the specified tolerance We note that the algorithm is guaranteed to converge to a fixed point since at each update the corresponding iterations go in the direction of a stationary point of the constrained optimization problem (15) with convex cost function and convex variable sets [8] We also note that, as it is the case with incremental algorithms in general, the convergence speed of the algorithm depends on the values of the beamforming and power increments specified by the algorithm constants β and μ V S IMULATIONS R ESULTS We illustrate the proposed algorithm for a system with N = 10 transmit antennas at the base station and K = active users with Nk = antennas each and white noise with covariance matrix Wk = 0.1I4 at all receivers The power matrix is initialized to P = 0.1I5 while the beamforming matrix S and the user channel matrices are initialized randomly The algorithm parameters are set to β = 0.02, μ = 0.01, = 0.02, and the target SINRs are initialized to γ ∗ = {5, 4, 3, 2, 1} In the first experiment we simulate the algorithm for fixed number of active users with variable target SINRs Once the beamforming vectors and powers that meet the specified targets are obtained using the proposed algorithm, user increases its target SINR from to 2.5 As a result of this change the algorithm starts updating user beamforming vectors and powers until a new fixed point that meets the new specified set of target SINRs is reached, when user decreases target SINR to 1.75 and initiates new updates for beamforming vectors and power The algorithm adjusts their values until a new fixed point is reached where the target SINRs are once again met for all users The variation of user SINRs and powers for this experiment are plotted in Figure from which we note that each time a change in the target SINR of user occurs there is a sharp change in all the users’ SINR and power values which is then compensated by the algorithm In the second experiment we start from the same initializations as before (including the same initial target SINRs) but after convergence to the fixed point where specified target SINRs are met, user becomes inactive Its corresponding beamforming vector and power are dropped from S and P matrices which determines the algorithm to update the beamforming vectors and powers for remaining users until a new fixed point is reached where the target (19) Differentiating now the Lagrangian (16) with respect to the multiplier λk we obtain another necessary condition for the constrained optimization problem (15) pk ∂Lk = − γk∗ = (20) ∂λk ik which indicates that, given the interference function ik the transmitted power corresponding to user k should match its target SINR, that is p∗k = γk∗ ik k = 1, , K (21) Thus, given the value of the effective interference function i0k (n) after beamforming update in equation (19) the power value matching the desired target SINR is ˜ −2 (n)˜sk (n + 1) p0k (n) = γk∗˜sk (n + 1)> D k (22) Since the value p0k (n) may not be close to the current power value pk (n) of user k and in order to avoid abrupt variations we will use a “lagged” power update given by pk (n + 1) = (1 − μ)pk (n) + μp0k (n) (23) with < μ < a suitably chosen constant We note that, the smaller the μ constant is, the more pronounced the lag in the power update is and the smaller the incremental power change will be The proposed algorithm for joint beamforming and power control in downlink multiuser MIMO systems uses the beamforming and power update equations (18) and (23) and is formally stated here: 1) Input Data: • Beamforming and power matrices S, P, downlink channel matrices Gk , target SINR values γk∗ , and noise covariance matrices Wk , k = 1, , K • Constants β, μ, and tolerance 2) FOR each user k = 1, , K DO a) Apply the whitening transformation (5) followed by the SVD (7), compute corresponding ˜ −2 (n), and determine its minimum eigenvecD k tor xk (n) b) Update user k’s transformed beamforming vector using equation (18) 446 10 user1 user2 user3 user4 user5 user1 user2 user3 user4 user5 SIR SIR 5 3 2 1 0.5 Updates 1.5 2000 4000 (a) SINR variation 12000 14000 16000 0.8 user1 user2 user3 user4 user5 0.7 0.6 user1 user2 user3 user4 user5 0.7 0.6 0.5 0.5 Power Power 8000 10000 Updates (a) SINR variation 0.8 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 6000 x 10 0 0.5 Updates 1.5 2000 4000 6000 8000 10000 Updates 12000 14000 16000 18000 x 10 (b) Power variation Fig (b) Power variation Tracking variable SINRs experiment Fig Tracking variable number of active users experiment ACKNOWLEDGMENT This work was supported in part by the National Sciences and Engineering Research Council of Canada R EFERENCES SINRs of active users γ ∗ = {5, 4, 3, 2} are satisfied Then, a new user becomes active in the system and its (randomly initialized) beamforming vector and power are added to the S and P matrices under new user with new target SINR equal to 0.5 This determines the algorithm to update again all beamforming vectors and powers until a new fixed point is reached where the target SINRs for all users are satisfied The variation of user SINRs and powers for this experiment are plotted in Figure from where we note that, similar to the previous experiment, each time a change in the number of active users in the system occurs there is a sharp change in all active users’ SINR and power values which is compensated by the proposed algorithm [1] C B Peel, Q H Spencer, A L Swindlehurst, and M Haardt, “An Introduction to the Multi-User MIMO Downlink,” IEEE Communications Magazine, pp 60–67, October 2004 [2] D Love and R W Heath Jr, “Equal Gain Transmission in MultipleInput Multiple-Output Wireless WSystems,” IEEE Transactions on Communications, vol 51, no 7, pp 1102–1110, July 2003 [3] S Thoen, L Van der Perre, B Gyselinckx, and M Engels, “Performance analysis of combined transmit-SC/receive-MRC,” IEEE Trans on Communications, vol 49, no 1, pp 5–8, January 2001 [4] F Rashid-Farrokhi, K J Liu, and L Tassiulas, “Joint Optimal Power Control and Beamforming in Wireless Networks using Antenna Arrays,” IEEE Transactions on Communications, vol 46, no 10, pp 1313–1324, October 1998 [5] ——, “Transmit Beamforming and Power Control for Cellular Wireless Systems,” IEEE Journal on Selected Areas in Communications, vol 16, no 8, pp 1437–1450, October 1998 [6] B.-Y Song, R L Cruz, and B Rao, “A Simple Joint Beamforming and Power Control Algorithm for Multiuser MIMO Wireless Networks,” in Proc 60th IEEE Vehicular Technology Conf – VTC 2004 Fall, vol 1, Los Angeles, CA, September 2004, pp 247–251 [7] D C Popescu and C Rose, Interference Avoidance Methods for Wireless Systems Kluwer Academic Publishers, 2004 [8] D Bertsekas, Nonlinear Programming Athena Scientific, 2003 VI C ONCLUSIONS In this paper we presented a new algorithm for joint beamforming and power control in downlink MIMO systems with target SINRs at the receivers The proposed algorithm uses incremental updates for beamforming vectors and powers that reduce the total interference in the system, and can be used for tracking changing target SINRs and/or variable number of active users 447 ... becomes inactive Its corresponding beamforming vector and power are dropped from S and P matrices which determines the algorithm to update the beamforming vectors and powers for remaining users... uses the beamforming and power update equations (18) and (23) and is formally stated here: 1) Input Data: • Beamforming and power matrices S, P, downlink channel matrices Gk , target SINR values... becomes active in the system and its (randomly initialized) beamforming vector and power are added to the S and P matrices under new user with new target SINR equal to 0.5 This determines the algorithm