Impulsive Noise, Fuzzy Uncertainty, and the Analog Median Filter

Additive fuzzy systems combined with supervised or unsupervised learning have been proposed to deal with impulsive noise in discrete-time signals, but suffer from the curse of dimensionality. Faster methods, possibly based on new paradigms, would be welcome. Order statistics filters and their fuzzy counterparts provide alternative and robust ways of dealing with discrete-time data, but the idea of ranking (or fuzzy ranking) does not appear at first sight to be meaningful in the continuous-time case. This chapter investigates to which extent the concept of “sorting” is meaningful for continuous-time signals. It presents a tutorial on the basic concepts behind sorting, and applies the results to the study of the single-input single-output analog median filter. Interestingly, the concept of sorting or rearrangement, which plays a fundamental role in the development, appears naturally when trying to define the uncertainty associated with a general set membership function.