Analysis of adaptive differential PCM of a stationary Gauss - Markov input

Abstract

An adaptive matched differential pulse-code modulator (AMDPCM) is analyzed. The adaptation of the symmetric uniform quantizer parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Delta_{n}</tex> is performed by fixed multipliers assigned to the quantizer output levels. The input is stationary first-order Gauss-Markov. The correlation of the samples is used as the leakage parameter in the matched integrator, with the predictive reconstruction similarly matched. For a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</tex> -level quantizer and multipliers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\gamma^{-1}, \gamma)</tex> the limiting joint distribution of the prediction error and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Delta_{n}</tex> is derived and the asymptotic sample-point and time-averaged mean-square error (rose) and mean and variance of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Delta_{n}</tex> as functions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\gamma \in (1,2]</tex> are computed and plotted. It is found that the asymptotic performance of AMDPCM does not depend on the choice of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Delta_{0}</tex> , that the increase in mse incurred by using A(M)DPCM instead of (M)DPCM with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Delta_{opt}</tex> is small, with mse(A(M)DPCM) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\downarrow \min_{\Delta}</tex> mse ((M)DPCM) as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\gamma \downarrow 1</tex> , and that the signal-to-noise ratio of AMDPCM does not depend on the input power.