MULTI-ANTENNA DOWNLINK PRECODING WITH INDIVIDUAL RATE CONSTRAINTS: POWER MINIMIZATION AND USER ORDERING Chi-Hang Fred Fung, Wei Yu, and Teng Joon Lim Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto 10 King’s College Road, Toronto, Ontario M5S 3G4, Canada Email: {cffung,weiyu,limtj}@comm.utoronto.ca ABSTRACT This paper addresses the problem of achieving a specified information rate for every user in a multi-user MIMO broad- cast channel. We propose two methods for finding the set of transmit covariance matrices that minimizes the transmit power subject to the dirty-paper coding rates equal to some specified values. The first method applies to the case with a fixed encoding order and each receiver has one antenna. The second applies to the general case without a fixed en- coding order and with time sharing taken into account, and has a complexity polynomial in the number of users. 1. INTRODUCTION For broadcast channels (BC), dirty-paper coding (DPC) has recently been shown to be capacity-achieving [1]. In this pa- per, we solve the problem of achieving a specified DPC rate for every user in a BC. We consider a MIMO BC consisting of a transmitter with n t transmit antennas and K users each equipped with n r receive antennas. The received signal for user i, 1 ≤ i ≤ K is y i = H i x + z i where H i ∈ C nr×nt represents user i’s channel, z i ∈ C nr×1 is additive white circularly symmetric Gaussian noise with zero mean and E{z i z j } = δ ij , and x ∈ C nt×1 is the trans- mitted signal. The DPC rates for user i with encoding order π(K),...,π(1) (meaning π(K) is encoded first) is R BC π(i) = 1 2 log   I + H π(i)  ∑ j≤i S π(j)  H H π(i)      I + H π(i)  ∑ j<i S π(j)  H H π(i)    (1) where S i is the transmit covariance matrix of user i. Given a target rate point ˜ R =( ˜ R 1 ,..., ˜ R K ), we want to find the transmit covariance matrices, S i , so that the total transmit power, tr( ∑ i S i ), is minimized while achieving the target rate point. We propose two novel methods for solving this problem. The first method applies to the case with a fixed encoding order and n r =1. The second applies to the general case without a fixed encoding order, with time sharing taken into account, and with n r ≥ 1. 2. SPECIAL CASE: SINGLE RECEIVE ANTENNA AND FIXED ENCODING ORDER We consider the case when each receiver has only one an- tenna (n r =1) and the encoding order in (1) is fixed. With- out loss of generality, we assume that the encoding order is K,..., 1. We consider the problem min Si K  i=1 tr(S i ) (2) s.t. 1 2 log  1+ H i S i H H i 1+ ∑ i-1 j=1 H i S j H H i  = ˜ R i (3) Since each receiver has only one antenna, H i is a row matrix and S i is rank 1. To solve this problem, we utilize the BC- MAC duality [2] to solve the problem as a MAC. The dual MAC channels are H T i and the decoding order is 1,...,K. Thus, the MAC problem is min Qi K  i=1 tr(Q i ) s.t. 1 2 log |I + ∑ K m=i H H m Q m H m | |I + ∑ K m=i+1 H H m Q m H m | = ˜ R i where Q i ,i =1,...,K are the MAC transmit covariance matrices to be found. Since H i ’s are row matrices, Q i ’s are positive scalars. Drawing on this fact, we can solve for Q i analytically: Q i =       H i  I + K  m=i+1 H H m Q m H m  - 1 2       -2 (2 2 ˜ Ri - 1) 45 0-7803-8549-7/04/$20.00 c  2004 IEEE