Detection threshold calculations for Weibull interference

Abstract

The Weibull probability density function (pdf) has been used to represent the noise-only output of an envelope detected matched filter when the input noise contains non-Gaussian, high amplitude components. This situation may occur in active sonar when the total interference is dominated by reverberation. The familiar Rayleigh pdf associated with Gaussian input noise is a member of the Weibull family with a shape factor of 2. If the threshold for detection is based on an assumed Rayleigh density and the noise changes to Weibull with a shape factor less than 2, the false alarm rate will increase dramatically even if the average noise power is unchanged. Therefore, in order to maintain a constant false alarm rate (CFAR), the threshold must be raised. This will result in an increase in the signal to noise ratio required to achieve a specific probability of detection (Pd). It is necessary to determine the signal plus noise pdf to quantitatively evaluate this effect on detectability. An integral formulation of this function has been derived by Eckstrom, and it has been numerically integrated for a few cases by Nilsson and Glisson. The present authors have developed a much simpler approximate procedure for computing receiver operating characteristic (ROC) curves for a linear detector in Weibull noise that does not require numerical integration. This method is based on the premise that the signal plus noise pdf is not strongly affected by the Weibull shape parameter, at least over the 10% < Pd < 90% range. In this paper, the details of this approximate technique are presented, and some comparisons with more exact calculations are provided.