SOME PROPERTIES OF ORDER-STATISTICS FILTERS

Let X1, X2,... be a stationary sequence of random variables with Pr{X(t) less-than-or-equal-to x} = F(x), t = 1, 2,.... Also let X(i:n)(t), i = 1,..., n, denote the ith order statistic (OS) in the moving sample (X(t-N),..., X(t),..., X(t+N)) of odd size n = 2N + 1. Then Y(t) = SIGMA-a(i)X(i:n)(t) with SIGMA-a(i) = 1 is an order-statistics filter. In practice a(i) greater-than-or-equal-to 0, i = 1,..., n. For t > N, the sequence {Y(t)} is also stationary. If X1, X2,... are independent, the autocorrelation function rho(r) = corr(Y(t), Y(t+r) is zero for r > n-1 and for r less-than-or-equal-to n-1 can be evaluated directly in terms of the means, variances, and covariances of the OS in random samples of size n + r from F(x). In special cases several authors have observed that the spectral density function f(omega) of {Y(t)} is initially decreasing for omega > 0. This result is made more precise and shown to hold generally under white noise. The effect of outliers (impulses) is also discussed.