New Results in the Theory of Simultaneous Optimum Detection and Estimation of Signals in Noise

Earlier work of the authors [IEEE Trans. Inf. Theory, 1968, vol. 14, no. 3, pp. 434–444] is extended to the sequential or adaptive decision-making processes involving optimum signal detection and extraction, when exact knowledge as to the signal’s presence is unavailable at any given stage. In particular, the uncoupled cases are developed, and extension to the coupled cases is considered. These problems are typical of many “multidecision” situations occurring in adaptive processing, unsupervised learning, and pattern recognition. Explicit development of the theory here is restricted to time-invariant parameters and deterministic signal waveforms. An improved approach to the single-decision problem of joint detection and estimation is also given, and illustrated by an example of coherent signal detection, and amplitude estimation, in which optimum and sub-optimum performance is compared. Various distinctions between different classes of estimators that are possible in the adaptive case are explicitly discussed, including their convergence properties. The paper concludes with a summary of the results obtained and suggestions for future work.