9 - Quantization and two's complement arithmetic in digital filters

Abstract

This chapter analyzes the quantization and two's complement arithmetic in digital filters. The second-order digital filter associated with both quantization and two's complement arithmetic is described. It is found that if the number of states of the machine is large enough and the period is very long, then the system may exhibit near-chaotic behaviors and the phase portrait may visually exhibit a near-fractal pattern if the chaotic behavior is exhibited in the corresponding infinite state machine for the same filter parameters and initial conditions. On the other hand, if linear or limit cycle behaviors are exhibited in the corresponding infinite state machine, one would intuitively expect that for the same filter parameters and initial conditions the finite state machine would also exhibit linear or limit cycle behaviors. However, this intuitive expectation is not true and a counter intuitive phenomenon is stated in a later observation. A finite state machine may exhibit a near-chaotic behavior even when its corresponding infinite state machine does not exhibit any chaotic behavior. The nonlinear behavior of unstable second-order digital filters is also elaborated in the chapter.