Scattering function estimation

The estimation of the scattering function of a random, zero-mean, homogeneous, time-variant, linear filter is considered. The sum of the random filter output and independent noise is the input to an estimator. The estimator structure is equivalent to a bank of linear filters followed by squared-envelope detectors; the envelope detector outputs are the input to a final linear filter. The estimator output is shown to be an unconstrained linear operation on the ambiguity function of the estimator input. Except for a bias term due to the additive noise, the mean of the estimator output is an unconstrained linear operation on the scattering function of the random filter. The integral variance of the output is found for a Gaussian channel. The mean and variance clearly indicate the tradeoff between resolution and variance reduction obtained by varying the estimator structure. For any well-behaved channel it is shown that an effectively unbiased estimate of the scattering function can be obtained if the input signal has both sufficient energy and enough time and frequency spread to resolve the random filter; the random filter is not required to be underspread. The variance of an estimate can be further reduced by increasing the time or frequency spread of the transmitted signal.