On the computation and interpretation of auto- and cross-trispectra

This paper discusses the principal domains of auto-and cross trispectra. It is shown that the cumulant and moment based trispectra are identical except on certain planes in trifrequency space. If these planes are avoided, their principal domains can be derived by considering the regions of symmetry of the fourth order spectral moment. The fourth order averaged periodogram will then serve as an estimate for both cumulant and moment trispectra. Statistics of estimates of normalised trispectra or tricoherence are also discussed. The auto-trispectrum, Cxxxx(f1; f2; f3), of a stationary and ergodic (at least to the fourth order) random process, x(t), is usually deened 1] as the Fourier transform of its fourth cumulant, cxxxx(1; 2; 3). (1) cxxxx(1; 2; 3) = C x(t)x(t + 1)x(t + 2)x(t + 3)] (2) where F represents Fourier transformation, C represents the cumulant operation on random variables, f's and 's represent frequencies and lags, respectively. The cumulant based trispectrum is guaranteed to exist provided the cu-mulant function decreases with increasing lag and belongs to the class L1 of absolutely integrable functions. This is a suucient but not necessary condition for the existence of the Fourier transform of the cumulant function. Alternatively , a moment based auto-trispectrum, Mxxxx(f1; f2; f3), can be deened as the Fourier transform of the fourth order (3) mxxxx(1; 2; 3) = E x(t)x(t + 1)x(t + 2)x(t + 3)] (4) where E represents the expectation operation. The fourth order cumulant can be expressed in terms of moments as (5) Taking Fourier transforms on both sides of equation 5, invoking separability and performing changes of variables such as u = 1 ? 2 2 and v = 1 + 2 where is the Dirac delta function and Mxx represents the second order moment spectrum of the process. The second order moment and cumulant spectra are both identical to the power spectrum for a zero-mean process. Equation 6 shows that the cumulant based auto-trispectrum and the moment based auto-trispectrum are identical except along the planes f1 = ?f2, f2 = ?f3, and f3 = ?f1 in trifrequency space. Using the Fourier-Stieltjes representation of the random process x(t) = Z 1 ?1 e ?j2f dX(f) (7) where dX(f) is a stochastic function, and invoking the sta-tionarity property, it has been shown 1] that E dX(f1) dX(f2) dX(f3) dX(f4)] = Mxxxx(f1; f2; f3)(f1 + f2 + f3 + f4) df1 df2 df3 df4 (8) The region …