The Bispectral Aliasing Test: A Clarification and Some Key Examples

Abstract

Controversy regarding the correctness of a test for aliasing proposed by Hinich and Wolinsky has been surprisingly long-lived. Two factors have prolonged this controversy. One factor is the presence of deep-seated intuitions that such a test is fundamentally incoherent. Perhaps the most compelling objection is that, given a set of discrete-time samples, one can construct an unaliased continuous-time series which exactly fits those samples. Therefore, the samples alone can not show that the original time series was aliased. The second factor prolonging the debate has been an inability of its proponents to unseat those objections. In fact, as is shown here, all objections can be met and the test as stated is correct. In particular, the role of stationarity as knowledge in addition to the sample values turns out to be crucial. Under certain conditions, including those addressed by the bispectral aliasing test, the continuous-time signals reconstructed from aliased samples are non-stationary. Therefore detecting aliasing in (at least some) stationary continuous-time processes both makes sense and can be done. The merits of the bispectral test for practical use are briefly addressed, but our primary concern here is its theoretical soundness.