Analytic Gram-Schmidt Orthogonalisation and QR Decomposition for Polynomial Matrices

We extend the concept of Gram-Schmidt orthogonalisation from a set of ordinary vectors to vectors of analytic functions. Firstly, we address the normalisation of such a vector. Even if its Euclidean norm possesses spectral zeros, we show that it is always possible to find an analytic normalised vector that has unit norm everywhere on the unit circle. Secondly, a sequence of such normalisation steps can be utilised for a Gram-Schmidt procedure applied to analytic matrices of full spatial rank. Analogous to the case of ordinary matrices, where the Gram-Schmidt procedure leads to a QR factorisation, we prove that an analytic QR decomposition exists with an analytic paraunitary matrix and an upper-right triangular matrix of analytic functions as factors. Thirdly, we present algorithms with proven convergence for both the analytic vector normalisation and the analytic QR decomposition. This type of decomposition can find applications in broadband multiple-input multiple-output systems, and we compare our algorithmic realisation to existing solutions in the literature, which do not necessarily converge towards an analytic decomposition.