Some properties of order-Statistics filters

Abstract

LetX1,X2,... be a stationary sequence of random variables with Pr{Xt, ≤x}=F(x),t=1, 2,... Also leti∶n,(t),i=1,...,n, denote the ith order statistic (OS) in the moving sample (Xt−N,...,Xt,...,Xt+N) of odd sizen=2N+1. ThenYt=∑aiXi∶n(t) with ∑ai=1 is an order-statistics filter. In practiceai≥0,i=1,...,n. Fort>N, the sequence {Yt} is also stationary. IfX1X2, ... are independent, the autocorrelation function ρ(r)=corr(Yt,Yt+r) is zero forr >n − 1 and forr ≤n − 1 can be evaluated directly in terms of the means, variances, and covariances of the OS in random samples of sizen +r fromF(x).