Estimation of cross-power and auto-power spectral densities in frequency domain by subspace methods H¤ useyin Akc ‚ay Abstract 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 identication algorithms is somewhat unexpected since they use the averaged periodograms as the data, which are known to be only asymp- totically unbiased spectrum estimates with a constant variance independent from the size of the data record. Index Terms Cross-power spectrum, auto-power spectrum, identication, subspace method, periodogram, Welch estimator, strong consistency, ARMAMA. I. I NTRODUCTION The cross-spectral analysis is a fundamental and powerful technique to investigate an unknown relationship between two time series in frequency-domain. It is widely used in many engineering problems; e.g., time delay estimation of spatial sensors [1], blind equalization in communications [2], speech enhancement [3], analysis of feedback systems [4], system identication of mechanical vibration systems [5], atmospheric problems [6], and vibration testing of structures against earthquake and wind loads [7]. The cross-PSD function has mostly been computed in a nonparametric way by using the fast Fourier transform (FFT). A parametric approach was recently proposed in [8] for the cross-spectral analysis. In this approach, each of the two scalar observable outputs: x[k] = s x [k]+ n x [k], (1) y[k] = s y [k]+ n y [k] (2) is modeled by an auto-regressive (AR) term and two moving- average (MA) terms where the signals s x [k] and s y [k] are the outputs of two innite-impulse response lters driven by a common scalar input and n x [k] and n y [k] are the noise terms which are the outputs of two innite-impulse response lters driven by two independent scalar inputs. The latter inputs are also independent from s x [k] and s y [k]. The model dened by Eqs. (1) and (2) was called the ARMAMA model in [8]. In [8], the parameters of the scalar ARMAMA model in Eqs. (1) and (2) were found by estimating the cross- correlation and the auto-correlation functions of x[k] and Department of Electrical and Electronics Engineering, Anadolu Univer- sity, Eskisehir 26470, Turkey. E-mail address: huakcay@anadolu.edu.tr. y[k]. Both estimation problems amount to solving some generalized Yule-Walker equations in a least-squares sense, and moreover the solutions are obtained by inverting certain Toeplitz matrices. The latter operation is carried out itera- tively by the numerically efcient Levinson method. In this paper, we propose estimating the cross-PSD and the auto-PSD matrices by subspace algorithms using the averaged periodograms, or more generally, the Welch auto- PSD and the cross-PSD estimates [9] computed at uniform grids of frequencies from the observed outputs in Eqs. (1) and (2). The proposed algorithms are realization based and the AR parameters are computed by Hankel matrix factor- ization followed by extraction of the so-called observability range space, and the MA parameters are estimated from the data by a least-squares procedure. An additional step, which is implemented as solution to a convex semidenite programming problem, comes into action when the lack of positivity of the estimated auto-PSD has been detected. The identication algorithms developed in this paper di- rectly utilize numerically efcient and reliable FFT and data obtained from different experiments can be merged easily. Furthermore, the cross-PSD and the auto-PSD function esti- mates delivered by these algorithms are strongly consistent. The consistency of the proposed algorithms is somewhat unexpected since the periodogram estimator is known to be only asymptotically unbiased and its variance does not vanish even if the amount of data increases unboundedly. A relation between the frequency-domain subspace algo- rithms of this paper and the covariance approaches in the spectral estimation literature is also derived. This relation is based on the interpolation properties of the discrete Fourier transform when restricted to nite-dimensional systems, which was recently exploited in [10], and plays a central role in showing the consistency of the proposed algorithms. The contents of this paper are as follows. In Section 2, the denition of the ARMAMA model introduced in [8] is extended to vector stochastic processes. Section 3 is devoted to review of a particular nonparametric approach to the cross-PSD and the auto-PSD estimation, namely the averaged periodogram and the Welch estimators. In Sec- tion 4, subspace-based algorithms for the estimation of the cross-PSD and the auto-PSD matrices from spectrum mea- surements at equidistantly spaced frequencies are developed and their convergence properties are studied. Due to space limitations, numerical examples and real life applications are omitted. They will be reported elsewhere. 51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA 978-1-4673-2064-1/12/$31.00 ©2012 IEEE 3445 978-1-4673-2066-5/12/$31.00 ©2012 IEEE