FIR Filtering of Discontinuous Signals: A Random-Stratified Sampling Approach

This paper presents a novel approach, based on random stratified sampling (StSa) technique, to estimate the output of a finite impulse response (FIR) filter when the input signal is either a piecewise-continuous function having first-derivative discontinuities (FDDs), or a piecewise-discontinuous function, i.e. having zero-derivative discontinuities (ZDDs). The proposed approach investigates the implications of such discontinuities on the output signal and its statistical properties. Mainly, we devise mathematical expressions for the variance of the StSa estimator in the two cases above, along with other minor special cases. It is found that the uniform convergence rate of the estimator, in the FDDs case, is N
^-3
, where N is the number of random samples. However, the variance in the ZDDs case is adversely affected by the existence of discontinuities. We prove that it converges more slowly with a uniform rate of N
^-2
.