Adaptive quantization with a one-word memory

• Published in: Bell System Technical Journal Vol. 52; no. 7; pp. 1119 - 1144
• Main Author: Jayant, N. S.
• Format: Journal Article
• Language: English
• Published: Oxford, UK American Telephone and Telegraph Company 01.09.1973; Blackwell Publishing Ltd
• ISSN: 0005-8580; 1538-7305
• DOI: 10.1002/j.1538-7305.1973.tb02008.x

We discuss a quantizer which, for every new input sample, adapts its step-size by a factor depending only on the knowledge of which quantizer slot was occupied by the previous signal sample. 1 Specifically, if the outputs of a uniform B-bit quantizer (B > 1) are of the form the step-size Δ r , is given by the previous step-size multiplied by a time-invariant function of the code-word magnitude: The adaptations are motivated by the assumption that the input signal variance is unknown, so that the quantizer is started off, in general, with a suboptimal step-size Δ START . Multiplier functions that maximize the signal-to-quantization-error ratio (SNR) depend, in general, on Δ START and the input sequence length N. For example, if the signal is stationary and N → ∞ best multipliers, irrespective of Δ START , have values arbitrarily close to unity. On the other hand, small values of N and suboptimal values of Δ START necessitate M values further away from unity. By including an adequate range of values for N and Δ START in a generalized SNR definition, we show how one can determine stable multiplier functions M OPT that are optimal for a given signal. In computer simulations of 2- and 3-bit quantizers with first-order Gauss-Markovian inputs, we note that, except when the magnitude of the correlation C between adjacent samples is very high, M OPT has the property of calling for fast increases and slow decreases of step-size. We derive optimum multipliers theoretically for two simple cases: