MEAN AND VARIANCE OF ESTIMATES OF THE BISPECTRUM OF A HARMONIC RANDOM PROCESS - AN ANALYSIS INCLUDING LEAKAGE EFFECTS

Analytical expressions are derived for the mean and variance of estimates of the bispectrum of a real-valued time series assuming a cosinusoidal model. The effects of spectral leakage, inherent in the discrete Fourier transform operation when the modes present in the signal have nonintegral number of wavelengths in the record, are included in the analysis. A single phase-coupled triad of modes can cause the bispectrum to have a nonzero mean value over the entire region of computation owing to leakage. The variance of bispectral estimates in the presence of leakage has contributions from individual modes and from triads of phase-coupled modes. Time-domain windowing reduces the leakage. The theoretical expressions for the mean and variance of bispectral estimates are derived in terms of a function dependent on an arbitrary symmetric time-domain window applied to the record, the number of data, and the statistics of the phase coupling among triads of modes. The theoretical results are verified by numerical simulations for simple test cases, and applied to laboratory data to examine phase coupling in a hypothesis testing framework.