Statistics of the scattering cross-section of a small number of random scatterers

Complex radar targets are often modeled as a number of individual scattering elements randomly distributed throughout the spatial region containing the target. While it is known that as the number of scatterers grows large the distribution of the scattered signal power or intensity is asymptotically exponential, this is not true for a small number of scatterers. The authors study the statistics of measured power or intensity, and hence scattering cross section, resulting from a small number of constant amplitude scatterers each having a random phase. They derive closed-form expressions for the probability density function (pdf) of the scattered signal intensity for one, two, and three scatterers having arbitrary amplitudes. For n>3 scatterers, they derive expressions for the pdf when the individual scatterers have identical constant amplitudes and independent random phases; these expressions are Gram-Charlier type expansions with weighting functions determined by the asymptotic form of the intensity pdf for a large number of scatterers n. The Kolmogorov-Smirnov goodness-of-fit test is used to show that the series expansions are a good fit to empirical pdfs computed using Monte-Carlo simulation of targets made up of a small number of constant amplitude scatterers with random phase