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

An adaptive matched differential pulse-code modulator (AMDPCM) is analyzed. The adaptation of the symmetric uniform quantizer parameter
\Delta_{n}
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
4
-level quantizer and multipliers
(\gamma^{-1}, \gamma)
the limiting joint distribution of the prediction error and
\Delta_{n}
is derived and the asymptotic sample-point and time-averaged mean-square error (rose) and mean and variance of
\Delta_{n}
as functions of
\gamma \in (1,2]
are computed and plotted. It is found that the asymptotic performance of AMDPCM does not depend on the choice of
\Delta_{0}
, that the increase in mse incurred by using A(M)DPCM instead of (M)DPCM with
\Delta_{opt}
is small, with mse(A(M)DPCM)
\downarrow \min_{\Delta}
mse ((M)DPCM) as
\gamma \downarrow 1
, and that the signal-to-noise ratio of AMDPCM does not depend on the input power.