Properties of higher-order correlations and spectra for bandlimited, deterministic transients

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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 f b is greater than half the highest frequency f t , then there is no unaliased bispectrum. The unaliased trispectrum is made up of two different replicated domains, one of which disappears for 3f b >f t .
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