On the capacity of some deterministic non-linear channels subject to additive white Gaussian noise

We consider a variety of memoryless discrete-time noisy communication channels where the noise is modeled as an additive white Gaussian noise process, and where the input of the channel is distorted according to a deterministic function f(X) of the form i) f(X) = αX
ⁿ
ii) f(X) = α|X|
ⁿ
iii) f(X) = α|X|
^1/q
iv) f(X) = αsgn(X)|X|
^1/q
, for all α ϵ R*, n ϵ N*\{1}. Subject to an average power constraint, we study the capacity-achieving input distributions of these classes of channels and prove them to be discrete except for the linear case f(X) = αX, where the optimal input is Gaussian distributed. Furthermore, we prove that these optimal input distributions have a finite number of mass points for classes iii) and iv). The results are reached through the development of a methodology and tools that are based on standard decompositions in a Hilbert space with the Hermite Polynomials as a basis, and we conjecture that these bases are natural candidates for general information-theoretic studies of the capacity of channels affected by additive white Gaussian noise.