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The uniqueness of the capacity-achieving distribution over noncoherent flat-fading channels

We prove the uniqueness of the capacity-achieving input distribution for a noncoherent flat-fading channel (Rayleigh or Rician), subject to a second moment constraint. For this purpose, we first show the uniqueness and symmetry of the capacity-achieving output distribution. Then, we establish the assertion by proving that every fading channel is a bijective mapping from the space of symmetric, input distributions to the induced space of symmetric output distributions. This proof is constructive in the sense that for a given symmetric output distribution, one can uniquely determine the input distribution that induced such an output distribution.