The bispectral aliasing test: a clarification and some key examples

Controversy regarding the correctness of a test for aliasing proposed by Hinich and Wolinsky [3] 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 discretetime samples, one can construct an unaliased continuoustime 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 continuoustime signals reconstructed from aliased samples are nonstationary. 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. 1. THE BISPECTRAL ALIASING TEST The domain of the discrete-time bispectrum is the two dimensional bifrequency {ω1, ω2} plane. Assuming a realvalued discrete time series, the usual replication phenomenon dictates that all non-redundant information is confined to the square 0 ≤ ω1, ω2 ≤ π. When one fully accounts for symmetries, the non-redundant information in the bispectrum is confined to a particular triangle inside this square [1, 9]. This triangle naturally divides into two pieces. One piece is an isosceles triangle and is unproblematic. The other piece, somewhat unusual in shape, is the source of the controversy under discussion. Naive consideration of this triangle shows that it involves frequencies higher than the Nyquist This work was supported by Los Alamos National Laboratory LDRD 97028. frequency and therefore must have something to do with aliasing. Hinich and Wolinsky considered this more carefully and showed that the naive intuition is correct: if the discrete time series arises from sampling a stationary, bandlimited, continuous-time process, and if the sampling rate is sufficiently rapid to avoid aliasing, then the discrete bispectrum is non-zero only in the isosceles triangular subset of the fundamental domain. Conversely, if the bispectrum of a sampled stationary continuous-time process is non-zero in the outer triangle, then the sampling rate was too slow to avoid aliasing. It should be clearly understood that there is no assertion that aliasing in general can be detected. The statement is not “if a signal is aliased, then the outer triangle will have a non-zero bispectrum.” Rather, the assertion is the converse, “if the outer-triangle shows a non-zero bispectrum, the (underlying) continuous-time signal must have been aliased.” At one level, this result is obvious and, in fact, the result was initially so-regarded [6]. However, doubt soon arose. Perhaps the most important source for suspicion is the argument based on reconstruction alluded to above. In light of this objection, one is led to reconsider the association of the outer triangle with aliasing. One can take the position that there is no relation, as in [2]. One can decide that something is aliased, but that it is the bispectral estimator rather than the signal. There is some plausibility to this claim, for the frequencies that are involved in the outer triangle are ω1, ω2, and ω1 + ω2 − 2π. This seems to be the position of Pflug et al. [5]. Or, one can try to delineate the conditions, if any, under which the test makes sense. This was done by Hinich and Messer in 1995[4]. They confirmed the validity of the original argument and stated its conclusions more carefully. In particular they conclude that a non-zero bispectrum in the outer triangle indicates a non-random signal or one of the following: • a random, but non-stationary signal ; • a random, stationary, but aliased signal, or; • a random, stationary, properly-sampled signal which violates the mixing condition. We believe that the analysis of Hinich and Messer, while entirely correct, did little to persuade the detractors of the test. In particular their analysis did not address the reconstruction objection and may have left the impression that the circumstances for which the test applies are unlikely to be met in practice. In this paper, we show that the reconstruction objection is far from fatal. We further establish that stationarity is the only property which is crucial to the test. Since this property is required in order to define the bispectrum, one can legitimately apply the aliasing test whenever one is entitled to compute a bispectrum. Therefore the bispectral aliasing test is as theoretically sound as the bispectrum itself. 2. THE SELECTION RULE AND BRILLINGER’S FORMULA The bispectrum, defined to be the triple Fourier transform of the third-order autocorrelation, reduces to a function of two frequencies since stationarity confines the spectrum to the plane through the origin of the frequency domain perpendicular to the vector (1,1,1). F123(c3(t1, t2, t3)) = b(ω1, ω2)δ(ω1 + ω2 + ω3) (1) Another way of computing the bispectrum is to switch the order in which one does the Fourier transforming and the ensemble averaging. This leads to the following result. b(ω1, ω2) = 〈X(ω1)X(ω2)X(ω3 = ω1 + ω2)〉 (2) If the process is bandlimited and X(ω) = 0 for |ω| > π, then the bispectrum is confined to the intersection of the (1, 1, 1) plane and the π-cube (i.e. (ω1, ω2, ω3) ∈ (−π, π)⊗ (−π, π) ⊗ (−π, π) ). The plane and its projection onto the (ω1, ω2) plane is shown in Figure 1. Upon sampling with unit time step, one obtains the usual replication in three dimensions. (Doing everything in 3-dimensions and projecting at the end keeps things simpler and makes it easier to avoid errors.) In particular, one gets that if the process is sampled at a frequency greater than twice the highest frequency component, then the bispectrum is confined to the replications of the tilted hexagon shown. The replication gives the discrete-time bispectrum bd: