Estimation of multiple sinusoidal frequencies using truncated least squares methods

and then p(S) = 1. Now suppose that p (S) 2 p > 1 and that the boundary values of Gp-I are infinite. From (12) a boundary value of Gp can be finite only if /BPI = 1, then from (1 1) we obtain P k = O c,(t;p) = hkXj-(t-k) = 0 , andp(S) = p. Finally, we must show that forp 2 p (S) the MLE does not exist. reflection coefficients with IBkI < 1 fork < p and IPpI = 1 leading through (5) to the condition (6) in the definition ofp(S). From the expression (10) of Gp we obtain, for p, in a neighbourhood of BD 9 where C stands for some constant, so G, tends to zero in this boundary point. V. CONCLUSION We have proved that for almost all sets of n records of length m of complex data, the MLE in AR (p) models exists if and only if the n records can not be exactly fitted by complex undamped si-nusoids using the same set o f p distinct frequencies. So in estimating AR (p) models with increasing order p , the maximum likelihood method can be applied until p = m-1 or stops with p just smaller than the minimal number of frequencies p (S) used by the sinusoids fitting the data exactly. The method in [4] working on the reflection coefficients, its use withp = p (S) gives, in the limit, the borderline values B1,. * * , with IBp0,/ = 1 describing the average of the discrete spectrum of these sinusoids. Maximum likelihood estimation for complex autoregressive model and Toeplitz interspectral matrices, " in The role of likelihood and entropy in incomplete-data problems: Applications to estimating point process intensities and Toeplitz constrained covariances, " S. Degerine, " Comportement au bord et caracterisation d'un maximum pour la vraisemblance d'un vecteur altatoire gaussien centre avec contrainte sur sa structure de covariance, " C. Absfracf-Various SVD-based methods have been shown effective for resolving closely spaced frequencies. However, the massive computations required by SVD makes it unsuitable for real-time applications. To reduce the computational complexity, three truncated QR methods are proposed: I) truncated QR without column pivoting (TQR); 2) truncated QR with reordered columns (TQRR); and 3) truncated QR with column pivoting (TQRP). I t is demonstrated that many of the benefits of the SVD-based methods a r e achievable under the truncated QR …