STABILITY OF THE DPCM TRANSMISSION-SYSTEM

Bounded input-bounded output stability of the DPCM transmission system is investigated. For the input signal s, the output s is calculated by a nonlinear feedback loop. In the feedforward path the quantizer characteristic can be taken continuous. It exhibits a threshold and has a linear part with a variable slope p greater than 1. In the feedback path the linear filter R is recursive. Stability of R is sufficient but not necessary to ensure stability of the DPCM system. This paper considers stability of an order 1 predictor. It is ensured if its coefficient a lies in ] - 1, 1[. The major result is that, subject to suitable initialization, for a constant input s(n) = epsilons(s > 0), the system is stable if and only if, a lies in an interval ]a(lim)'(s, p), a(lim)(s, p) [ which is strictly larger than ] -1, 1 [. For a time-varying input with maximum amplitude s, the ''if'' part remains but the ''only if'' part does not. Comparison with the classical concept of stability for linear systems displays three specific features of the DPCM case: i) the conditions on the prediction coefficient are less stringent; ii) the kind of stability is weaker because there is, for any a, a bound S(a,p) over the input amplitude s; iii) for the same value of Absolute value of a, the negative case has a better stability than the positive case. The reason for the third dissimilarity is that the analysis uses a stationary point technique in the former case, and a second-order cycle technique in the latter case. As a function of the quantizer slope, the maximum prediction coefficient a(lim) exhibits a maximum. The corresponding value P(opt) delimits two ranges. For smaller slopes the quantizer threshold effect degrades the stability. At the optimum, the range of stability for a is the same with and without the quantizer threshold. In practical applications of DPCM systems, especially in adaptive systems, advantage can be taken from all the above properties.