Continuous Time Analysis of the Response of the Pseudomedian and Related Filters to Periodic Signals

We introduce a continuous time method to analyze the response of median, pseudomedian, average (mean), and midrange filters to certain periodic signals. The filter definitions are generalized to continuous time, and these definitions are applied to periodic signals such as triangle, square, and sinusoidal waves of varying frequencies. These operations yield “amplitude response” measures which are analytic functions of the frequency of the input signal. In addition, a “correlation” measure is defined to indicate the level of distortion introduced by each filter. Examples of this analysis for the median, pseudomedian, average, and midrange filters show similarities and differences among them. Although these theoretical measures do not perfectly demonstrate the performance of the discrete time filters, continuous time analysis does provide valuable insights into the filter behavior. The response of the continuous time median filter shows its susceptibility to high frequency periodic noise and proves, again, the existence of infinite-length bi-valued fast-fluctuating roots of this filter. The pseudomedian filter, in contrast, completely attenuates amplitude-symmetric periodic signals above a certain frequency, and has no infinite-length fast-fluctuating roots. Continuous time filter analogues are therefore an important theoretical tool for understanding the behavior of both linear and nonlinear filters.