Sampling requirements for nth-order correlations

Abstract

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 Δt3≤1/(3ft) for the bicorrelation and Δt4≤1/(4ft) for the tricorrelation, where ft 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 Δtn≤1/(nft). 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 Δt≤1/(2 ft) 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.