2130 IEEE TRANSAcnONS ON SIGNAL PROCESSING. VOL 43. NO.9. SEP1'EMBER 1995 On the Principal Domain of the Discrete Bispectrum of a Stationary Signal Melvin J. Hinich and Hagit Messer, Senior Member, IEEE Abstract- This paper presents a simplifyiag, yet gmenI ap- proach to detennbling the symmetry structure of a bispec:tnun. The prindpaJ domain (PD) of the bispednun lIDd its rePm of positive support are derived in a way that illuminates the cootroveny surrouncIinI the t:rianaIe in the PD, which is called the outer triangle (OT) by Hinich IlDd WoIinlIIr.y,wheft the blspectnun is zero for. stationary random sampled pl"OCeSS that is DOt aIlasecL The basic statistlcal issues of testing for DODZe1'O bispectnII structure are reviewed. 1. INTRODUCTION H IGHER order spectral analysis is considerably more complicated than spectral analysis. Going from one to two or more frequencies introduces many complicating tech- nical issues.The symmetriesof the bispectrum are notobvious, and those of the trispectrum are even more complicated.This paper introduces a fresh approach to bispectral analysis. The statistical properties of higher order spectral estimates are more complex than spectral estimates. For example, the variances of the bispectrum depend on the trispectrum and the sixth-order cumulant spectrum, whereas the variances of spectral estimates depend on the power spectrum and the trispectrum. These variance parameters play a crucial role in computing the asymptotic properties (both bispectrum and trispectrum) based on the tests for nonlinear and nonGaussian structure that have received considerable interest and appli- cation in a number of fields, including engineering signal processing [10], [11]. This paper presents a window to the technical issues inher- ent in the successful application of higher order spectra for signal analysis. One of the major benefits in using a higher than second- order cumulant is the possibility to discriminate between Gaussian and nonGaussian random signals. Since the bis- pectrum of a Gaussian sequence is zero over the entire principaldomain, a test for Gaussianity based on the estimated bispectrum has been suggested [4], [14]. This test is well accepted by the signal processing community and has been used for signal detection problems (e.g., [6], [9]). Hinich and Wolinsky [5J have studied the discrete bispec- trum of a sequence created by sampling a stationary,random Manuscript received May 13, 1994; revised February I, t995. This work was supported UDderOffice of Naval Research Contract NOOOI4-91-J-1276. TIle associate editor coordinating the review of this paper and approving it for publication was Dr. Athina Petropulu. . M. J. Minich is with the Applied Resean:h Laboratories. The University of Texas at Austin, Austin, TX 78713-8029 USA. H. Messer is with the Department of Electrical Engineering-Systems. Tel Aviv Univenity, Tel-Aviv Israel. IEEE Log Number 9413322. process. They have shown that the principal domain can be divided into two triangles:the innertriangle (IT) and the outer triangle (OT). H the sampling rate agrees with the Nyquist rule, the discrete bispectrum over the OT is identically zero. Based on this observation, they suggested a test for aliasing [5J. Sharfer and Messer [13] used it for testing for jitter in the sampling clock. This test, as well as a bispectral-based test for a transient coherent signal in stationary noise [7] and other advancedtestsbased on zero bispectrum over subregions of the principal domain, has been criticized lately on several occasions (e.g., [3J, [12], [15]). In this paper, we prove, in a way different from [5J, that the bispectrum of a discrete sequence is zero over the OT under certain conditions, We study these conditions, and we explain how each of them can be related to properties of the tested signai.Thus, its violation can be tested using estimates of the discrete bispectrum over the OT. II. THEoRETICAL BACKGROUND Assume firstthat x(t) is a real, zero-mean l , continuous AI) random process. Define C(tl,t2,t3) = E{X(tl)X(t2)X(t3)}, and assume it is finite. The 3-D Fourier transform of C(tl, t2, t3) (which is the third-order simple cumulant of x(t) is the third-order cumulant spectrum of x(t) CS(Wl,W2,W3) = L:L:L: C(tl,t2,t3) x e-jw,t'e-jw.t'e-jw3t3 dtl dt2 dt3' (I) H x(t) is at least third-order A2) stationary, its third-order cumulant spectrum is only a function of two variables C(tI,t2,t3) = e(tl - t2, t2 - t 3 ) = e(TllT2). A 2-D Fourier transform of this bicovariance function e( TI, T2) = E{x(t)x(t - TI)X(t - T2)}' can then be applied.The resultant 2-D function of WI and W2 is the blspectrum of the stationary, random signal x(t) B(WI,W2) = L:L: C(TI, T2)e- jw ,T, e- jw2 1) dTldT2' (2) For a stationary x(t), the third-order cumulant spectrum must satisfy IThe assumptions of zero mean and real signal can be relaxed. We use them only for simplicity. I. I I 1053-587XJ95$04.00 @ 1995 IEEE