Robust coherence analysis in the frequency domain

In this paper, a new cross periodogram, called Laplace cross periodogram, is introduced for robust coherence analysis of multiple time series in the frequency domain. It is derived by replacing the ordinary Fourier transform that defines the ordinary cross periodogram with what we call the Laplace-Fourier transform obtained from trigonometric least-absolute-deviations (LAD) regression. Under certain stationarity assumptions, the Laplace cross periodogram is found through an asymptotic analysis to be associated with what we call the Laplace cross spectrum, a function proportional to the Fourier transform of cross zero-crossing rates. Robustness of the Laplace cross periodogram and the corresponding coherency estimator is demonstrated by numerical examples.