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, Mi, 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, ??pi(X) log Mi=0, where Pi(X) is the steady-state probability that a quantized sample lies in the ith magnitude level when the ratio of quantizer range scale to rms signal level is X. A designer may specify X 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 depends on MN/Ml, the ratio of largest to smallest multiplier.