Approximating the frequency response of a median filter

Question

A median filter is a non-linear and lossy process, so it doesn't have a closed form frequency response as would a FIR filter (say a box filter of the same length) in an LTI system.

• But how closely can something similar to a frequency response of a median filter be approximated?
• How would this scale with the length of a median filter?
• Under what conditions or for what class of signals might this approximation be ballpark "close"?
• For what class of signals might this approximation be very inaccurate?
• What kinds of frequency domain distortion or additive noise does a median filter produce?

Answer 1

For a start, any non-linear system will not have an easily-identifiable frequency response. So, it's really a nonsensical question. I intend no offense; nonsensical questions are often the most enlightening!

However one way to try to answer your question is to assume that the LTI filter involved is the mean (rather than the median) of the windowed data.

Then your question:

Under what conditions or for what class of signals might this approximation be ballpark "close"?

becomes:

Under what conditions or for what class of signals might the mean be ballpark "close" to the median.

In that case, for a purely stochastic signal, the mean and median are similar when the probability density function (PDF) of the signal is symmetric about the mean.

For what class of signals might this approximation be very inaccurate?

When the signal's PDF is "very" asymmetric.