Simultaneous optimum detection and estimation of signals in noise

Abstract

The problem of simultaneous detection and estimation of signals in noise is formulated in the language of statistical decision theory. Optimum structures and corresponding general measures of system performance are derived under the Bayes criterion of minimum average risk for the detectors and estimators appropriate to this type of joint operation: It is shown that whereas the structures of the resulting optimum detectors require a class of modified likelihood ratios, the structures of the optimum estimators, which act on the data when there is uncertainty as to the presence of a signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[p(H_{1}) &lt; 1 ]</tex> , have a common canonical form for a wide variety of operating strategies. This form is identical with that obtained for estimation alone <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[p(H_{1}) &lt; 1 ]</tex> , even though there is generally mutual coupling between detector and extractor. A simple structure is obtained for the estimation of amplitude and waveform in the case of a quadratic cost function (least mean-square error), where it is found that the estimator which is optimum here <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[p(H_{1}) &lt; 1 ]</tex> is the product of the corresponding Bayes estimator in the "classical" case [P(H1) = 1 ] and a simple algebraic function of the generalized likelihood ratio. In this case, one can also show that estimators that are unbiased in the classical sense remain unbiased. In parallel with the classical theory, a generalized version of unconditional maximum likelihood estimation is obtained for the "simple" cost function when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(H_{1} &lt; 1</tex> . It is found that estimators that are linear in the classical case <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[p(H_{1}) = 1 ]</tex> are nonlinear in the more general situation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[p(H_{1})&lt;1]</tex> , where increased structural complexity is always the rule for both detectors and estimators. A specific example involving the coherent estimation of signal amplitude illustrates the approach. Analogous extensions to prediction and filtering are formulated, making it evident that a broad area for further generalizations of classical Bayes detection and extraction theory is available for systematic investigation.