Capacity of the AWGN channel in bits per channel use (bpcu) is C(SNR)=12log2(1+SNR) where SNR is the Signal-to-Noise-Ratio.
We consider equidistant
The input-output-relation is:
Y=Δ⋅X+Z.A good choice for PX is the Maxwell-Boltzmann (MB) distribution with parameter ν, which resembles a sampled Gaussian distribution: PXν(x)=e−νx2∑˜x∈Xe−ν˜x2.
In order to find the optimal ν and Δ, we solve the optimization problem:
RMB(SNR)=maxν,Δ:E[(ΔXν)2]≤PI(Xν;ΔXν+Z), where I(Xν;ΔXν+Z) is the mutual information of input Xν and output Y=ΔXν+Z.