Step Response of an Adaptive Delta Modulator

Abstract

N. S. Jayant has proposed a simple but effective form of adaptive delta modulation which uses two positive parameters, P and Q, to adjust the step size. The values P = Q = 1 describe linear delta modulation (LDM), and Jayant has recommended using Q = 1/P and 1 < P < 2. In this paper, we study the step response of this scheme for arbitrary P and Q. For each P and Q, we are able to define an integer n, the stability exponent for P and Q, such that the step response is unstable when P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> Q > 1, it converges to the new level when P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> Q < 1, and when P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> Q = 1, it eventually settles into a periodic (2n + 2)-step cycle, for almost all initial conditions. For P ≧ 2, and for some combinations of P and Q with P between 1.6 and 2, it is possible to have both PQ < 1 and P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> Q ≧ 1, so that PQ < 1 is not sufficient for convergence. When a system is convergent, but a minimum step size δ is imposed, the eventual periodic hunting will not necessarily resemble that of LDM, but will be bounded by δP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> .