Stochastic stability for feedback quantization schemes

Abstract

Feedback quantization schemes (such as delta modulation. adaptive quantization, differential pulse code modulation (DPCM), and adaptive differential pulse code modulation (ADPCM) encode an information source by quantizing the source letter at each time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> using a quantizer, which is uniquely determined by examining some function of the past outputs and inputs called the state of the encoder at time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> . The quantized output letter at time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> is fed back to the encoder, which then moves to a new state at time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i+1</tex> which is a function of the state at time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> and the encoder output at time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> . In an earlier paper a stochastic stability result was obtained for a class of feedback quantization schemes which includes delta modulation and some adaptive quantization schemes. In this paper a similar result is obtained for a class of feedback quantization schemes which includes linear DPCM and some ADPCM encoding schemes. The type of stochastic stability obtained gives almost-sure convergence of time averages of functions of the joint input-state-output process. This is stronger than the type of stochastic stability obtained previously by Gersho, Goodman, Goldstein, and Liu, who showed convergence in distribution of the time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> input-state-output as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i \rightarrow \infty</tex> .