Mean and variance CFAR performance via fiber integration

A closed-form analysis of the performance of two-parameter (mean and variance) CFAR normalizers is described. Typically, CFAR normalizers remove the background mean, then display and/or detect using a fixed threshold. If the background variance is varying and unknown, as is oftentimes the case across frequency or beamformer bins, a different threshold must be set for each bin independently, and human operators must view displays with large dynamic ranges. To avoid these problems, the variance may also be normalized. However, in contrast to the case of mean-level normalizers, the output distributions of mean and variance normalizers was previously unknown due to the complicated dependence of their outputs on the mutually dependent sample mean and variance. This paper describes the approach of fiber integration to obtain the distributions of these normalizers. This approach is very general, and can be used to obtain the output statistics of a very wide class of CFAR normalizers and input statistics. An analysis is presented for the case of chi-squared background statistics input to three different two-parameter normalizers: the so-called deflection ratio (z - /spl mu/)//spl sigma/, the log deflection ratio (log z - /spl mu//sub log/)//spl sigma//sub log/, and a new power domain method, called the log-gamma CFAR normalizer, -log/spl Gamma/(v,vz//spl mu/)//spl mu///spl Gamma/(v), where z is the input data, /spl mu/ and /spl sigma//sup 2/ are the sample estimates of the background mean and variance, and v = /spl mu//sup 2///spl sigma//sup 2/. Compared to mean level CFAR, it is shown that the deflection ratios has the least amount of CFAR loss (typically less than 1 dB) of these three, the log-gamma method has slightly more (typically 1 to 2 dB), and that the log-deflection ratio has by far the greatest (typically 4 dB or more).