Dynamical study of a second-order DPCM transmission system modeled by a piecewise-linear function

Abstract

This paper analyzes the behavior of a second-order differential pulse code modulation (DPCM) transmission system when the nonlinear characteristic of the quantizer is taken into consideration. In this way, qualitatively new properties of the DPCM system have been unraveled, which cannot be observed and explained if the nonlinearity of the quantizer is neglected. For the purpose of this study, a piecewise-linear nondifferentiable quantizer characteristic is considered. The resulting model of the DPCM is of the form of iteration equations (i.e., map), where the inverse iterate is not unique (i.e., noninvertible map). Therefore, the mathematical theory of noninvertible maps is particularly suitable for this analysis, together with the more classic tools of nonlinear dynamics. This study allowed us, in addition, to show, from a theoretical point of view, some new properties of nondifferentiable maps, in comparison with differentiable ones. After a short review of noninvertible maps, the presented methods and tools for noninvertible maps are applied to the DPCM system. An original algorithm for calculation of bifurcation curves for the DPCM map is proposed. Via the studies in the parameter and phase plane, different nonlinear phenomena such as the overlapping of bifurcation curves causing multistability, chaotic behavior, or multiple basins with fractal boundary are pointed out. All observed phenomena show a very complex dynamical behavior even in the constant input signal case, discussed here.