The report presents two methods for the estimation of a periodic signal in additive noise. Both methods assume that only a finite time sample of the signal plus noise is available for processing. The estimate of the signal is chosen to minimize the mean square error between the estimate and the sample of signal plus noise. This is also a maximum likelihood estimate if the noise is white and Gaussian. The first method is frequency domain analysis. Estimates for the period of the signal and the complex amplitudes of its harmonics are derived. The second method is time domain analysis. Estimates for the period of the signal and for the waveform of one period are derived. Under the assumption of white noise and large signal-to-noise ratio, formulas for the expected values and variances of the period estimates are derived. The estimates for the period are found to be the same by both methods. The estimate is unbiased and has a variance inversely proportional to the signal-to-noise ratio, and inversely proportional to the cube of the number of periods in the given sample. The expected values of the estimates of the waveform itself are derived, and the estimates are found to be biased.

The report proposes a design of an adaptive receiver for the detection and estimation of nearly periodic signals in additive Gaussian noise. A nearly periodic signal is defined to be a sample function of a Gaussian random process which can be divided into equal length intervals, called periods, in such a manner that the correlation between periods decreases exponentially with their separation. The receiver computes a low signal-to-noise ratio conditional likelihood ratio from which the observer must make decisions. The likelihood ratio is conditional because the receiver estimates any unknown parameters necessary for the computation of the true likelihood ratio. Thus the receiver can only compute a likelihood ratio conditioned upon these estimates being the true values of the unknown parameters. The receiver consists of pre-emphasis filters followed by a comb filter, energy detector, and weighted summation. A theoretical evaluation of the receiver, in terms of ROC curves, is made for the special case of nearly periodic signals with statistically independent equal-strength harmonics in white noise of known power. (Author).

Describes the leading techniques for analyzing noise. Discusses methods that are applicable to periodic signals,aperiodic signals, or random processes over finite or infiniteintervals. Provides readers with a useful reference when designing ormodeling communications systems.

Detection of Signals in Noise serves as an introduction to the principles and applications of the statistical theory of signal detection. The book discusses probability and random processes; narrowband signals, their complex representation, and their properties described with the aid of the Hilbert transform; and Gaussian-derived processes. The text also describes the application of hypothesis testing for the detection of signals and the fundamentals required for statistical detection of signals in noise. Problem exercises, references, and a supplementary bibliography are included after each chapter. Students taking a graduate course in signal detection theory.