Estimation of cross-power and auto-power spectral densities in frequency domain by subspace methods

In this paper, frequency-domain subspace-based algorithms are proposed to estimate discrete-time cross-power spectral density (cross-PSD) and auto-power spectral density (auto-PSD) matrices of vector auto-regressive moving-average and moving-average (ARMAMA) models from sampled values of the Welch cross-PSD and auto-PSD estimators on uniform grids of frequencies. The proposed algorithms are shown to be strongly consistent. A link between the well-known time-domain covariance-based spectrum estimation methods and the frequency-domain realization-based algorithms of this paper is also established. The consistency of the proposed identification algorithms is somewhat unexpected since they use the averaged periodograms as the data, which are known to be only asymptotically unbiased spectrum estimates with a constant variance independent from the size of the data record.