Error Rate of Transmit Beamforming with Delayed and Limited Feedback

In this paper, we study the effect of delayed and limited feedback of beamformer codebook index on the error rate of transmit beamforming (BF) over Rayleigh fading channels. Closed-form expressions for the moment generating function (MGF) and the probability density function (PDF) of the receiver output signal-to-noise ratio (SNR) including the effects of outdated and finite-rate feedback are derived. Based on these, we further provide analytical expressions for error rate of BF systems. Our results show that for channels with non-bandlimited fading spectra, the error rate performance monotonically degrades when the feedback delay increases. On the other hand, for channels with bandlimited fading spectra the error rate may not monotonically degrades as the delay goes to infinity. However, delayed feedback may cause the transmit diversity gain being reduced to one for high SNRs, that is, no diversity gain but only an array gain is attained. The performance loss caused by delayed feedback may be much more detrimental than the limited-rate feedback for all SNRs.