Asymptotic capacity of beamforming with limited feedback

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Abstract
We study the channel capacity of a point-to-point communication system with multiple antennas and limited feedback. The receiver with per- fect channel knowledge can relay B bits, which specify a beamforming vector, to the transmitter. We show that a Random Vector Quantization scheme is asymp- totically optimal and give a simple expression for the associated capacity. I. Summary We consider a flat Rayleigh fading channel with M trans- mit and N receive antennas. The received vector is given by y = Hvx+w where H is an N ◊M channel matrix whose el- ement is a complex Gaussian random variable with zero mean and unit variance, v is an M ◊ 1 beamforming vector, x is a transmitted symbol with zero mean and unit variance. y is an N ◊ 1 received symbol vector, and w is an AWGN vector with covariance matrix 1 I where I is an identity matrix. The mutual information I(x,y) can be maximized over the beamforming vector v subject to a power constraint kvk 1. With B feedback bits, we can construct a vector quantizer with a codebook V = {v1,··· ,v2B}. Assuming that H is known at the receiver, it chooses vj from V that maximizes I(x,y), and relays the corresponding index back to the trans- mitter. Optimizing the codebook V for finite M and N is quite dicult; however, as ( M,N) ! 1, the eigenvectors of H†H

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