SAMPLING REQUIREMENTS FOR NTH-ORDER CORRELATIONS

As has been derived previously [Pflug, MS thesis, Univ. of New Orleans (1990); Pflug et al., J. Acoust. Soc. Am. 91, 975-988 (1992); Pflug et al., J. Acoust. Soc. Am. 94, 2159-2172 (1993)], the sampling intervals required to prevent nonremovable and removable aliasing in higher-order correlations calculated from discrete-time data are Delta t(3) less than or equal to<1/(3f(t)) for the bicorrelation and Delta t(4) less than or equal to 1/(4f(t)) for the tricorrelation, where f(t) is the highest or top frequency present in a signal. It is shown here that a general expression for the sampling required to prevent aliasing for each order n of correlation calculated from discrete-time data is Delta t(n) less than or equal to 1/(nf(t)). Time-domain computer calculations of correlation central ordinate values for orders two through eight are consistent with this result. Removable aliasing can be eliminated for data which are sampled such that Delta t less than or equal to 1/(2f(t)) by (1) the application of n-dimensional transform or time-domain masking filters, (2) one-dimensional filtering of the transform domain diagonal factor, or (3) interpolation of the data before correlating.