Best Least Squares Paraunitary Approximation: Analytic Procrustes Problem

This paper addresses the analytic Procrustes problem, which aims to find the best least-squares paraunitary approximation of a square matrix of analytic transfer functions, or the best paraunitary transformation between two rectangular analytic matrices. This is accomplished by generalising the Procrustes solution from ordinary matrices to the case of matrices of analytic functions via their analytic singular value decomposition (SVD). Different from the ordinary matrix case, the analytic SVD is not restricted to singular values being nonnegative. In the case that singular values do not possess any zero crossings, we can find an analytic paraunitary matrix analogously to the standard Procrustes approach. In the case that singular values exhibit any zero crossings, the solution does not only depend on the left- and right-singular vectors, but also on a discontinuous and hence non-analytic switching function that forces those analytic singular values to become nonnegative real. We show that a close approximation of this switching function can be achieved via a complex-valued allpass filter, for which we suggest a new suitable design to minimise the overall least squares error of the fit. In addition, we propose a DFT domain algorithm to approximate this polynomial Procrustes solution, which avoids ambiguities in the analytic SVD, and possesses proven convergence. Generally, this solution requires a delay for causality, and this delay grows with the approximation order. Examples and simulations demonstrate our proposed method.