Spectral Analysis of Stationary Random Bivariate Signals

A novel approach toward the spectral analysis of stationary random bivariate signals is proposed. Unlike existing approaches, the proposed framework exhibits a natural link between well-defined statistical objects and physical parameters for bivariate signals. Using the quaternion Fourier transform, we introduce a quaternion-valued spectral representation of random bivariate signals seen as complex-valued sequences. This makes possible the definition of a scalar quaternion-valued spectral density and the corresponding autocovariance for bivariate signals. This spectral density can be meaningfully interpreted in terms of frequency-dependent polarization attributes. A natural decomposition of the spectral density of any random bivariate signal in terms of unpolarized and polarized components is introduced. Nonparametric spectral density estimation is investigated, and we introduce the polarization periodogram of a random bivariate signal. Numerical experiments support our theoretical analysis, illustrating the relevance of the approach on synthetic data.