Opportunistic Beamforming with Precoding for Spatially Correlated Channels Tareq Y. Al-Naffouri and Mohamed E. El-Tayeb Electrical Engineering Dept., King Fahd University of Petroleum and Minerals, Saudi Arabia Email: {naffouri, melgaily}@kfupm.edu.sa Abstract—Random beamforming (RBF) exploits multiuser di- versity to increase the sum-rate capacity of MIMO broadcast channels. However, in the presence of spatial correlation between the downlink channels, multiuser diversity can not be exploited and the sum-rate suffers a signal to noise (SNR) hit. In this paper, we explore precoding techniques that minimize this hit. Basically, we derive an optimum and an approximate precoding matrix that minimizes the sum-rate hit of RBF. As a by product, we introduce a technique that evaluates the cumulative distribution function (CDF) of weighted norms of Gaussian random variables. I. I NTRODUCTION Multiple antennas in multiuser systems have been intro- duced as an effective means to boost wireless system capacity. While dirty paper coding (DPC) is known for achieving the ca- pacity region in a broadcast scenario, it requires full feedback and it is computationally expensive [1]. Other less expensive techniques like random beamforming were able to capture most of the DPC capacity with less feedback requirements [2]. In a large user regime, the sumrate of RBF and DPC coincide at R = M log log n + log P M + o(1) where P is the transmitted power, M is the number of transmit antennas and n is the number of users. In the presence of spatial correlation between the users’ channels, the sum-rate capacity experiences a hit and becomes, R = M log log n + log P M + log c + o(1) where log c represents the hit and c ≤ 1. In this paper, we investigate different precoding techniques that minimize the sum-rate hit in the presence of spatial correlation. The paper is organized as follows. After the introduction in section I, we introduce the channel model in section II. Random beamforming with precoding techniques are reviewed in section III and in section IV we show our simulations results followed by our conclusions in section V. II. CHANNEL MODEL We consider a multi-antenna Gaussian broadcast channel with n receivers equipped with one antenna and a transmitter with M antennas. The received signal at the kth user is expressed as Y k (t)= √ PH k S(t)+ W k , k =1,...,n, (1) where k denotes a user index. S(t) denotes transmitted symbol and satisfies the power constraint E{S ∗ S}≤ P . The channel matrix H i consists of complex Gaussian random variables CN (0,R) and W k is the additive complex Gaussian noise with CN (0, 1). The covariance matrix R is a measure of the spatial correlation and is assumed to be non-singular with tr(R)= M . III. RANDOM BEAMFORMING WITH PRECODING In the presence of spatial correlation, we can precode the transmitted symbol with a general matrix A before beamform- ing, i.e.transmit αAS(t). The parameter α satisfies the power constraint (α = M tr(AA ∗ ) ) and the sum-rate eventually becomes R P rec = M log log n + M log P M − h P rec with a hit of h Prec = M log tr(AA ∗ ) M + ME log ( ‖φ m ‖ 2 ˜ Λ −1 ) . (2) It should be noted that ˜ Λ is constructed from the eigenvalues of ˜ R and the effective channel gain is ˜ H k = AH k . Finding the optimum precoding matrix A opt is challenging, but one can show that the optimum precoding matrix takes the following form A opt = Q Aopt D Aopt where Q Aopt is orthonormal and D Aopt is a diagonal matrix with positive entries 1 . The proof of the above expression is straight forward and for brevity we omit it here. Finding Q Aopt and D Aopt is not easy. An intuitive choice would be to set Q Aopt = Q R and optimize over D Aopt . In the following sections, we examine various choices of the diagonal matrix. A. Random Beamforming with Zero Forcing A natural choice of the precoding matrix is one which cancels the effect of the correlation, i.e. A ZF = Q R Λ − 1 2 R From (2), this choice would results in the following hit h ZF = M log tr(R −1 ) M 1 It is shown in [6]that this intuitive choice is actually optimum 978-1-4244-3401-5/09/$25.00 ©2009 IEEE 63