Unitary matrix pencils in signal processing

Abstract

The problem of retrieving harmonics of a measured signal can be solved via solving an eigenvalue problem for a unitary Hessenberg matrix H built from the first Schurparameters of the Toeplitz matrix of the signal's autocorrelation coefficients. From the eigenvalues eιθk· we get the approximated frequencies θ k and from the first components of the corresponding eigenvectors we get the amplitudes. The eigenvalue problem for H is equivalent to the eigenvalue problem for the unitary matrix pencil G o − λG e , called the Schurparameter pencil, the eigenvalues being the same for both problems. The advantage of working with this generalized eigenvalue problem instead of solving the standard eigenvalue problem for H lies in the sparsity of the matrix pencil G o − λG e , the only nonzero entries being in fact the Schurparameters. Solving the retrieval of harmonics problem as a unitary eigenvalue problem also has the advantage that we can make use of the mathematically rich eigenvalue structure of unitary matrices. This allows in particular to give bounds on the distance of the computed approximations to the frequencies to the actual” frequencies. We present such results and numerical examples showing the accuracy of the method and the effectiveness of the perturbation results.