A common basis for analytical clutter representations

In order to understand the problems intractable problems nature presents us, we are forced to make simplifications till we arrive at simple analytical models that we are capable of understanding. Such canonical models are rare, but useful as tools for constructing more realistic models that we can use analyze nature. The class of analytical models for clutter analysis limited to those that consist of various amplitude models with the phase noise assumed to have a probability density function that is uniformly distributed. These analytical models can be extended by relaxing the assumption of uniform phase noise to phase noise with non-uniform distributions. It is shown how to determine the probability density function for these non-uniform distributions in general and then the method is illustrated with such distributions as Gaussian, Laplacian, and Chi-squared. This enables one to determine the probability density functions of the individual components of clutter model such as x_t = v_t cos(Φ_c). Using the rule for determining the products of distributions, we show that the functional form for x_t is reduced to evaluating integrals that reduce to elliptical functions. Once the functional form has been determined, it is easy to determine the moments of the PDF and hence completely characterize its statistics.