Processing math: 100%
Logo der Uni Stuttgart
ASK Input distributions

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 bi-polar amplitude shift keying (ASK) constellations X with 2m signal points:X={±1,±3,,±(2m1)}.

The input-output-relation is:

Y=ΔX+Z.
  • Input X with distribution PX on X.
  • Δ scales the constellation.
  • Average signal power P=E[(ΔX)2].
  • Noise term Z is zero mean Gaussian with power one: ZN(0,1).
  • Signal-to-Noise-Ratio is SNR=P/1=E[(ΔX)2].

A good choice for PX is the Maxwell-Boltzmann (MB) distribution with parameter ν, which resembles a sampled Gaussian distribution: PXν(x)=eνx2˜xXeν˜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.