Capacity of RAKE reception with energy randomization

We consider a RAKE receiver for coherent binary orthogonal signaling over a frequency selective multipath Rayleigh fading channel. The receiver uses maximal-ratio combining such that the weight estimation errors are not independent of the additive noise. We find the capacity-achieving energy randomization scheme with two energy levels for the cases of imperfect and perfect channel estimates. We observe that the capacity-achieving probability gets closer to 1/2 with increase in the number of paths. We also show that the capacity is higher if channel estimates are perfect and that the channel estimation errors have more pronounced effect on the capacity at low signal-to-noise ratios.