Time Delay Estimation
NAVAL UNDERWATER SYSTEMS CENTER NEW LONDON CT
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This study investigates methodologies for passive estimation of the bearing to a slowly moving acoustically radiating source. The mathematics for the solution to such a problem is analogous to estimating the time delay or group delay between two time series. The estimation of time delay is intimately related to the coherence between two time series. New results on using coherence to provide information about linear and nonlinear systems are presented. The maximum likelihood ML estimate of time delay is derived the explicit dependence of the estimate on coherence is evident in the realization in which the two time series are prefiltered to accentuate frequency bands of high coherence and subsequently crosscorrelated. Also included are statistics of the estimates of the magnitude-squared coherence MSC, including the probability density function, the cumulative distribution function, and the m-th moment of the MSC estimate. A complete discussion of the bias and the variance of the MSC estimates is presented. The receiver operating characteristics of a linearly thresholded coherence estimation detector are also presented. A general FORTRAN 4 computer program using the fast Fourier transform to estimate time delay is given.
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