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Parameter Estimation

Module by: Don Johnson. E-mail the author

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Determining signal parameter values or a probability distribution's parameters are the simplest estimation problems. Their fundamental utility in signal processing, much less array processing, is unquestioned. How do we estimate noise power? What is the best estimator of signal amplitude? How should array outputs be effectively combined to estimate propagation delay? Examination of useful estimators, and evaluation of their properties and performances constitute a case study of estimation problems. As expected, many of these issues are interrelated and serve to highlight the intricacies that arise in estimation theory.

All parameters of concern here have unknown values; we classify parameter estimation problems according to whether the parameter is stochastic or not. If so, then the parameter has a probability density and choosing the density, as we have said so often, narrows the problem considerably, suggesting that measurement of the parameter's density would yield something like what was assumed! If the density is not known, the parameter is termed nonrandom, and its values range unrestricted over some interval. The resulting nonrandom-parameter estimation problem differs greatly from the random-parameter problem. We consider first the latter problem, letting θθ be a scalar parameter having the a priori (before any data are available) density pθ p θ . The impact of the a priori density becomes evident as various error criteria are established, and an "optimum" estimator is derived.

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