Statistics of companding noise for QAM transmission

In modern voiceband data communication, the received signal is subject to nonlinear quantization noise due to companding. Under certain conditions, this quantization noise may become dominant and cause serious degradation in performance. In order to design better signal constellations for this environment, it is of interest to characterize this noise. We study the conditional probability density function (PDF) for the quantization noise found on a PCM companding channel. We obtain an analytical expression for the conditional PDF of the nonlinear quantization noise and an algorithm to compute the conditional PDF of the filtered noise, which is the actual noise at the slicer input of the linear receiver. We compare the results to a Gaussian PDF (with second-order moments that depend on the signal point) and conclude that a Gaussian approximation, although not entirely accurate, is not unreasonable