Adaptive Quantization With a One-Word Memory

Abstract

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. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> Specifically, if the outputs of a uniform B-bit quantizer (B > 1) are of the form the step-size Δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> , 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 <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">START</inf> . Multiplier functions that maximize the signal-to-quantization-error ratio (SNR) depend, in general, on Δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">START</inf> and the input sequence length N. For example, if the signal is stationary and N → ∞ best multipliers, irrespective of Δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">START</inf> , have values arbitrarily close to unity. On the other hand, small values of N and suboptimal values of Δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">START</inf> necessitate M values further away from unity. By including an adequate range of values for N and Δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">START</inf> in a generalized SNR definition, we show how one can determine stable multiplier functions M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">OPT</inf> 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 <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">OPT</inf> has the property of calling for fast increases and slow decreases of step-size. We derive optimum multipliers theoretically for two simple cases: