Robust Target Detection Using Order Statistic Filters

In radar, sonar, and ultrasonic detection system, interference due to clutter can severely deteriorate the quality of the received signal to the point of concealing the target. This paper presents theoretical analysis of order statistic processors for improved target detection. The sort function is used to provide in s i i t into the optimal rank for detection of various targets and clutter environments where observations are independent and identically distributed. When observations do not contain identical statistical information, the analysis becomes more complex. Both simulation and experimental results are used to illustrate the extent of robustness that can be obtained from a particular rank in the presence of observations containing insignificant target information (i.e. null observations). Order statistics (OS) have been studied in many areas of statistical signal processing such as speech, image, and sonar. The OS filter is a nonlinear processor that can be expressed by: + n = O S r : n { ~ 1 , ~ I , ~ 3 * z,,} for l s r s n (1) where q is the unordered observed input value from the sequence of size n (window size) and r is the rank of the input from the ordered sequence that becomes the output +,,. This filter is the median detector when r=(n+1)/2 (for n odd), the maximum detector when r=n, and the minium detector when r=l . This paper develops a statistical analysis of the OS filter to obtain general input-output relationships so that predictions concerning the performance of the OS filter in various targetclutter situations can be made. The relation between the optimal rank and properties of the target and clutter distributions are discussed. The expected value of the output of the OS filter is examined to establish the OS filter as an estimator of the quantilm of the input signal distribution. Then the performance of the OS filter is analyzed through simulation for various target-clutter distributions ushg sort function analysis for cases in which the observations are and are not independent and identically diatributed. Finally, results from ultrasonic experimental measurements using split-spectrum processing (i.e. a method of obtaining frequency diverse observations) are examined for supporting theoretical predictions. Detection Properties of OS Filters In order to determine the expected value of the output of the OS filter, the general expression for the output probability density function, fx,:.(-), of the OS filter is derived in terms of r and n as defined in Equation (1) along with the input probability density function. The input signals, q, are assumed to be independent and identically distributed with the distribution function, Fx(.), and the density function, fx(*). In the following derivation let X represent the random variable for the input of the OS filter and Xr:n represent the random variable for the output of the OS filter with rank r. The distribution function for Xr:n is defined as Fx,,, = Pr {X(r:nl<z}. The probability that at least r of the n values are less than z can be found by applying the binomial distribution [4], Fx,:.(4 = c [ ;) F$(~)(I -F~(Z))"' for 1 l r l n . (2) The density function can be found by taking the derivative fx,,.(z) = r I:] ~ ~ ' ( z ) ( l ~ x ( z ) ) " ' f x ( z ) for 1 ~ r l n . (3) The density function for Xrtn is the product of the probability density function of a single input, fx(z), and another function given by: