Computing the Minimum-Phase Filter Using the QL-Factorization

We investigate the QL-factorization of a time-invariant convolutive filtering matrix and show that this factorization not only provides the finite length equivalent to the minimum-phase filter, but also gives the associated all-pass filter. The convergence properties are analyzed and we derive the exact convergence rate and an upper bound for a simple single-input single-output system with filter length L=2. Finally, this upper bound is used to derive an approximation of the convergence rate for systems of arbitrary length. Implementation-wise, the method has the advantage of being numerically stable and straight forward to extend to the multiple-input multiple-output case. Furthermore,due to the existence of fast QL-factorization methods, it is possible to compute the filters efficiently.