No abstract

Higher-order correlations and spectra may be used for detection, time delay estimation, classification, and discrimination of signals. For these applications, a detailed knowledge of their attributes can be highly useful. In this paper, the properties of the bicorrelation and tricorrelation of bandlimited deterministic transients, i.e., energy signals, and their spectra, the bispectrum and trispectrum are studied. Bandlimited transients that contain frequencies down to and including zero and those that have a nonzero lower cutoff frequency are both considered. Using symmetries inherent in the bispectrum of a signal, the entire bispectrum can be mapped from bispectral elements defined in two polygons, one for the unaliased and one for the aliased domain, each of which is one-twelfth the area of its total domain. The nonredundant unaliased region of the trispectrum is contained in two principal unaliased polyhedra, each replicated 48 times to reproduce the full trispectrum. If there is aliasing in the trispectrum, then the total aliased domain can similarly be mapped from two principal aliased polyhedra. It is shown that the critical sampling interval for the bicorrelation, i.e., the sampling interval sufficient to avoid aliasing, is 2/3 the critical sampling interval for the ordinary correlation, and the critical sampling interval for the tricorrelation is 1/2 the critical sampling interval for the ordinary correlation. If the lowest frequency of the bandlimited transient fb is greater than half the highest frequency ft, then there is no unaliased bispectrum. The unaliased trispectrum is made up of two different replicated domains, one of which disappears for 3fb>ft.

Higher order statistical blind deconvolution methods are implemented for use in removing multipath distortion from passively received underwater acoustic transient signals. Using single channel data and simulations, it is demonstrated that a fourth order method based on cumulant maximization can work well if the associated multipath Green's function is sufficiently "sparse." The iterative method is parameterized by filter length, and while there is a range of values at which the best solutions are obtained with conventional convergence criteria, useful solutions exist across a much broader range of filter lengths if the iterations are not always allowed to proceed to convergence. The fourth order objective functional is generalized to arbitrary order, and the method is shown to also produce good results for the third order objective functional.

A comparison of the spareseness (simplicity) norm criterion blind deconvolution methods of Cabrelli and Wiggins is made in order to ascertain relative performance for underwater acoustic transient source signal estimation, especially in the presence of noise. Both methods perform well at high signal-to-noise ratios, producing source estimates that are significant improvements over the original received signal for classification purposes. At moderate and lower SNRs, the Cabrelli method tends to generate results that are superior to the Wiggins method. This is especially true for a damped sinusoid transient source, for which the Wiggins method fails completely at lower SNRs, while the Cabrelli method can still produce good source estimates.