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An Almost Linear Relation Between the Step Size Behavior and the Input Signal Intensity in Robust Adaptive Quantization

A considerable amount of information regarding the behavior and performance of robust adaptive quantizers is contained in the function C(\sigma) which denotes the dependence of the central (or typical) step size, C , on the intensity, σ, of the input random variables. The graphical representation of C(\sigma) is called the load curve. It is quite remarkable that plots of log C(\sigma) are found to be almost linearly dependent on log σ. One of our contributions is to show that this linearity follows from an unusual, but quite effective, approximation to the Gaussian and Laplacian distributions of the input random variables. The central result is a quite explicit expression for the function C(\sigma) , and thus for the load curve, derived from the aforementioned approximation: \log C(\sigma) = S \log \sigma + T . The quantities S and T are explicitly given in terms of the various fixed parameters of the system. The expression for S provides insight into the tradeoffs between the dynamic range of the step size, the robustness in the presence of channel errors and the ability to track changes in the signal intensity σ.