Short-time signal representation by nonlinear difference equations
April 6, 1990
The solution of a nonlinear difference equation can take on complicated deterministic behavior which appears to be random for certain values of the equation's coefficients. Due to the sensitivities to initial conditions of the output of such "chaotic" systems, it is difficult to duplicate the waveform structure by parameter analysis and waveform synthesis techniques. In this paper, methods are investigated for short-time analysis and synthesis of signals from a class of second-order difference equations with a cubic nonlinearity. In analysis, two methods are explored for estimating equation coefficients: (1) prediction error minimization (a linear estimation problem) and (2) waveform error minimization (a nonlinear estimation problem). In the latter case, which improves on the prediction error solution, an iterative analysis-by-synthesis method is derived which allows as free variables initial conditions, as well as equation coefficients. Parameter estimates from these techniques are used in sequential short-time synthesis procedures. Possible application to modeling "quasi-periodic" behavior in speech waveforms is discussed.