Asymptotic behavior of the pairwise error probability with applications to transmit beamforming and rake receivers

We derive the precise asymptote of the pairwise error probability for high signal-to-noise ratio (SNR) and apply it to obtain new results concerning transmit beamforming and selective Rake receivers. For downlink beamforming (with N transmit antennas and independently identically distributed (i.i.d.) Rayleigh fading) based on quantized feedback from the mobile, we show that at least ┌log₂(N)┐ bits of feedback (per coherence time) is required to obtain full diversity, and among all beamforming schemes using ┌log₂(N)┐ bits of feedback, selection diversity is optimal. We give the exact expression for the SNR loss of selection diversity with respect to ideal beamforming based on perfect knowledge of fading coefficients. Further, we study selective Rake receivers for independent arbitrary fading distribution and arbitrary power delay profile (PDP). In particular, we show that the SNR loss of the SRake receiver with respect to the all-Rake receiver does not depend on the PDP, and we also propose a transformation to adapt the expressions known for the symbol error probability for the case of i.i.d. Rayleigh fading to the general case.