On the detrimental effect of assuming a linear model for non-linear AWGN channels

In communication theory, one of the best understood and commonly adopted channel model is the average-power con strained linear AWGN channel, the capacity of which is given by the expression 1/2 log (1 + SNR). But what if the channel is not linear? How bad is it to adopt a linear model for a non-linear channel? In this paper, we answer these questions by considering generic deterministic memoryless non-linear channel models. We study these models under an even-moment, a peak or a mixture input constraint. We prove that for the majority of the studied channels, the capacity achieving input distributions are of a discrete nature with a finite number of mass points. The linear model under the average power constraint and other "equivalent channels " being the only exceptions. We establish our results using Hermite bases in a standard Hilbert space decomposition of some relevant information quantities in channels affected by AWGN. We present numerical results for two sample sub-linear channels. We determine their optimal inputs and plot their capacity curves showing that adopting a linear model for slightly non-linear channels will have serious implications on achievable rates and their achieving distributions.