3.9 Probability density estimation

Probability density estimation

Many signal processing algorithms, implicitly or explicitly, assume that the signal and the observation noise are each welldescribed as Gaussian random sequences. Virtually all linear estimation and prediction filters minimize the mean-squarederror while not explicitly assuming any form for the amplitude distribution of the signal or noise. In many formal waveformestimation theories where probability density is, for better or worse, specified, the mean-squared error arises fromGaussian assumptions. A similar situation occurs explicitly in detection theory. The matched filter is probably the optimumdetection rule only when the observation noise is Gaussian. When the noise is non-Gaussian, thedetector assumes some other form. Much of what has been presented in this chapter is based implicitly on a Gaussian model for both the signal and the noise. When non-Gaussian distributions areassumed, the quantities upon which optimal linear filtering theory are based, covariance functions, no longer suffice tocharacterize the observations. While the joint amplitude distribution of any zero-mean, stationary Gaussian stochasticprocess is entirely characterized by its covariance function; non-Gaussian processes require more. Optimal linear filteringresults can be applied in non-Gaussian problems, but we should realize that other informative aspects of the process arebeing ignored.

This discussion would seem to be leading to a formulation of optimal filtering in a non-Gaussian setting. Would that suchtheories were easy to use; virtually all of them require knowledge of process characteristics that are difficult tomeasure and the resulting filters are typically nonlinear [Lipster and Shiryayev: Chapter 8] Rather than present preliminary results, we take the tack that knowledge is better than ignorance: At least thefirst-order amplitude distribution of the observed signals should be considered during the signal processing design. Ifthe signal is found to be Gaussian, then linear filtering results can be applied with the knowledge than no otherfiltering strategy will yield better results. If non-Gaussian, the linear filtering can still be used and theengineer must be aware that future systems might yield "better" results. Note that linear filtering optimizes the mean-squared error whether the signalsinvolved are Gaussian or not. Other error criteria might better capture unexpected changes in signal characteristicsand non-Gaussian processes contain internal statistical structure beyond that described by the covariancefunction.

Types

When the observations are discrete-valued or made so by digital-to-analog converters, estimating the probability massfunction is straightforward: Count the relative number of times each value occurs. Let $$

r 0 … r L 1