The present paper derives the statistical measures on the bearing estimates (bias, variance, Cramer-Rao Bound) for unequally separated clusters of hydrophones and also unequal variances on the time delay estimates. The present results reduce to those found in the literature for direction finding systems that presume equally separated clusters of hydrophones and equal time delay variances. The derivation takes into account those differences in spatial and temporal characteristics in the direction finding system to design an optimum weighting filter. When applied, the filter is shown to lead to improved direction finding estimates through reduction of both bias and variance. The relative influence of the various approaches on the quality of the bearing estimates is identified. Finally, it is shown that merging without weighting of the available time delay data may deteriorate the bearing estimate instead of improving it.

Simulation results are presented to compare the effectiveness of various windowing functions for the generalized correlator when a strong spectral peak or sinusoid is present. In this case, the time delay of the desired source and of the interfering sinusoid are allowed to be different. The results show that all windows except HBII and SCOT deteriorate markedly. These two windows remain robust to the ambiguity problems presented by the sinusoid, independent of whether the spectral peak is included in the signal spectrum or in the noise spectrum. Also, window HBII performs better than SCOT in terms of probability of correct detection, variance of time delay, and SNR threshold.

A structure is presented for passive estimation of range, bearing as well as velocity of a source from a linear array. It uses a quasi-optimal post-processor of the time delays, which are obtained from a generalized correlator with finite observation time. The post-processor ultimately maps the sequential time delay observations onto invariant source trajectory parameters over which smoothing is performed to reduce, jointly, the variance and the bias in the estimate of the source kinematics. The present approach remains viable for moving sources at long ranges, off-broadside source directions and high time delay variances. Analysis and simulation results are presented to justify its usefulness under the stated stringent conditions. Review of and comparison to existing approaches are made to highlight the viability of present approach in the estimation of source trajectory.

Various optimum and sub-optimum filters have been added to the basic cross-correlator in order to enhance its time delay estimation capability in the presence of noise. These pre-detection filters include the Hannan-Thomson, the Eckart, the Smoothed Coherence Transform, some least-squares, and two optimum filters recently derived by the authors. Here, a quantitative comparison of all six filters has been conducted by extensive simulation for various input signal and noise ratio and signal and noise bandwidths. The results define the actual enhancement in time delay detection when using the optimum filters and the amount of performance degradation from the optimum for the other filters. The rank ordering of the filters is carried out. Histograms of the time delay estimates, their mean and variance are presented. Confirmation of analytical predictions is noted.

Expanded statistical measures in the cepstrum are obtained when estimating the time delay\taubetween the wavelets of a composite signal embedded in noise. Previous considerations have been limited to physical instances where the modulusaand the ratio of signal to noise spectra\Phi_{y}/ \Phi_{n}yield values forx = (1 +a^{2}) \Phi_{y}/ \Phi_{n}and/or\gamma = 2|a| / (1 + a^{2})much less than one. The physical parameters,a, \Phi_{y}, and\Phi_{n}are allowed to vary throughout the spectrum, thus introducing new elements to the analysis. The required statistical measures are derived for the complete series of cross-terms, for a higher order perturbation in the modulation term, and for its logarithmically generated harmonics. Under these open conditions, the explicit behavior of the additional statistical measures are delineated as a function of the modulusaand the ratio of signal to noise spectra. The effect of the interfering harmonics with the modulation term is assessed for signal and noise with exponential autocorrelation functions.

Often, the cepstrum has been used when estimating the time delay r between elements of a composite signal embedded in noise,n(t). Here a probabilistic analysis is conducted dealing with the effect of stationary Gaussian noise on the characteristics of such a nonlinear processor. The input noise,n(t)is reflected at the cepstrum's output as: 1) modulationM_{1}(\omega)of sinusoidal information carrier with subsequent reduction in the height of echo peak at\tau; and 2) generation of background noises\hat{phi}(\omega)and\hat{psi}(\omega)with tendency to obscure that echo peak. The resulting probabilities of these terms are derived. The expected mean and standard deviation of reduction in the peak level at\taudue to noise are analytically described. Similar statistical measures on the background noise are also obtained. The results point out the dependence of the-statistical measures upon the pointwise variation of input signal to noise spectra. Thus the total signal power to total noise power(S/N)alone is an insufficient measure for setting the cepstrum performance, and relative bandwidths of signal and noise are also needed. The detection probabilityP_{D}of a peak corresponding to\tauis calculated for exponential spectra with equal bandwidths in signal and noise and with echo strengtha = 0.4; P_{D}is found to be close to one for anSINgreater than -4 dB; deterioration in detection is found to be rapid with thresholding near anS/N \sim -6db whereP_{D}falls to 0.3.

Characteristic polynomials are here generated for a class of eigenvalue problems using the singularity condition previously encountered in the solution of corresponding forced system. The process is efficient and requires n2/2 operations in contrast to n3 operations required by techniques such as the Leverrier-Faddeev method.

It is shown that in cepstrum analysis of multipath signals the stringent conditions imposed on the strength of the delayed echo a can be relaxed to require at most only that a be different from one.

A solution is presented for a class of a very large system of linear algebraic equations. The given set of simultaneous equations is initially transformed to yield an equivalent system of equations in triangular form whose solution is straightforward. With the present technique, the number of operations on the computer is minimized while the convergence to a unique solution is guaranteed.