Bispectral analysis of continuous-time processes under random sampling schemes

In signal processing, a continuous-time process is often sampled at discrete intervals. The discretized sampling intervals are usually equally spaced. In general these discretized observations will not be able to give full information about the original signal. Aliasing is one such problem. Non-equally spaced sampling intervals occur naturally also. For irregularly sampled data, most existing literature are concerned with second-order properties. This paper is concerned with the estimation of the bispectral density function of a continuous-time process which is sampled at irregular intervals. The sampling time interval is modeled by a point process. Conditions under which consistent estimate of the bispectral density function are given. Asymptotic bias and covariances are obtained. Poisson sampling scheme is used as a benchmark for comparisons. Contrast with second-order spectral estimates is presented. It is shown that in order to have a computationally effective consistent bispectral estimator, the sampling point process has to be second-order orthogonal increment. In such case the variance of the estimate is the same as that of Poisson sampling process for all sampling schemes. To have a bispectral estimator which is more efficient than Poisson, the sampling process cannot be a delayed-renewal process. (C) 1998 Elsevier Science B.V. All rights reserved.