Theory of an Adaptive Quantizer

In an adaptive quantizer that has been used for speech encoding, the entire amplitude range expands or contracts by a multiplicative constant after each input sample. The constant M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> depends only on the magnitude of the current quantizer output. Assuming independent identically distributed input samples, we show that the sequence of quantizer ranges is a stochastically stable process. Furthermore, we derive a key design equation, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Σ pᵢx</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">logMᵢ = O</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pᵢ(x)</tex> is the probability that an input sample is in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> th magnitude interval when the ratio of quantizer range to rms signal level is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> . A designer may specify <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> and solve this equation for multipliers that provide the desired steady-state performance. There are many such sets of multipliers and we show that the adaptation time constant associated with each set decreases as the ratio of the largest multiplier to the smallest multiplier is increased. On the other hand, the spread of the steady-state range distribution about the operating point can be made as small as desired by making this ratio sufficiently small. A bound is obtained for the tradeoff between responsiveness to changing input level and steady-state range accuracy.