Estimation of a Signal Parameter in Gaussian White Noise

The asymptotic properties of the maximum-likelihood estimator (MLE), truncated MLE, and Bayes estimators of a one-dimensional parameter are investigated for the transmission of a continuous signal in a channel with Gaussian white noise. The conditions are determined under which the consistency and asymptotic efficiency of these estimators (i.e., in the terminology of Kotel'nikov [Theory of Potential Noise Immunity, Moscow–Leningrad, Gosenergoizdat, 1956 (in Russian)], the absence of anomaly) are guaranteed. The results obtained in this connection may be regarded as a mathematically rigorous expression of the assertion in the book cited that anomaly is absent if the noise is sufficiently small and different branches of the signal curve do not pass too close to one another. Frequency modulation is studied in closer detail. In particular, an exponentially exact bound is found for the width of the band in which a consistent estimate of the parameter is obtainable, i.e., anomaly is absent, for a given transmission time.