Some Effects of Quantization and Adder Overflow on the Forced Response of Digital Filters

In this paper ive derive a criterion for the satisfactory behavior of second- The effects of quantization (i.e., roundoff, truncation, etc.) and adder overflow are present in any special-purpose computer type realization of a digital filter. When taken into account, these effects cause an otherwise linear system to become quite nonlinear. To date, the analysis of limit cycle phenomena in such nonlinear digital filters has been concerned with the study of the zero-input response of second-order filters. 1-3 A more fundamental problem is that of determining whether or not a filter's response to a nonzero input (the forced response) is in some meaningful sense close to the ideal response. This problem seems to have been ignored. If we consider input sequences, the levels of which are sufficiently small (in the sense that when the input sequence is applied to the linear 863 864 T H E BELL SYSTEM TECHNICAL JOURNAL, APRIL 1972 model of the filter, the response eventually lies within the open interval determined by the most positive and the most negative machine numbers), then it is tempting to conjecture, as if the system were linear, that when the filter's zero-input response can be made to admit only limit cycles of small amplitude by using sufficiently many bits in the representation of the data so that the quantization errors are made sufficiently small, then the deviation of the filter's forced response from the ideal can also be made small in the same manner. As will be shown by counterexamples, however, this conjecture is false.