Nonlinear oscillations in quantized linear discrete-time systems
Burrus, C. Sidney
Doctor of Philosophy
This research presents the analysis of limit cycles occurring in the fixed-point implementations of digital filters due to multiplication rounding. The implemented digital filters are referred to as quantized linear discrete-time systems. Existence of such limit cycles in autonomous second order systems is proved and then the results are extended to higher order and non-autonomous systems. Upper bounds on magnitudes and periods of the limit cycles are easily derived from the existence theorems. Exact solutions for limit cycles can be obtained by using either an integer programming method or a direct search algorithm. Since both are of exhaustive type method, they are capable of providing the complete set of all limit cycles in a given system. Approximate solutions are desirable when exact solutions are not needed. Two classic approximation methods, namely, harmonic balance and describing function methods, are extended to quantized discrete systems for predicting frequencies and amplitudes of near-harmonic limit cycles. A comparative study of limit cycles in different configurations of a digital filter is also presented. It is found that a digital filter is more likely to oscillate if it is realized as a combination of first and second order difference equations. A simple technique for quenching such limit cycles is also included.