The exact impact of amplitude quantization on multi-dimensional, high-order moments estimation

In this paper, we study the exact impact of amplitude quantization on the process of estimating a multi-dimensional (m-D) higher-order moment using quantized signal processing schemes. First, we derive the relationship between an m-D, finite-order unquantized moment and its quantized counterpart and express it as the sum of two key terms, one of which is wanted as it embodies the desired information, namely the m-D unquantized moment to be estimated, in addition to some other quantity that gives an exact measure of the unavoidable component of the impact of amplitude quantization. As for the second term, it wholly represents the other unwanted component of the impact under consideration that can, at least in theory, be exactly computed given the input statistics or avoided if certain specific conditions are satisfied by the input statistics. The issue of avoidance of this unwanted component will not be discussed here as it is reported elsewhere as referenced in the text. The m-D results relevant to the impact of amplitude quantization in the 1-D and 2-D cases are then derived and a complete statistical characterization of the quantization noise in these 2 cases is then obtained using simple and intuitive arguments. The theoretical results obtained in these 2 cases are applied to the important class of Gaussian signals, leading to a new interpretation of a Gaussian-driven quantized channel. Finally, numerous simulation tests have been carried out and have resulted in a very good support of the theoretical predictions.