Computing the Minimum-Phase Filter Using the QL-Factorization

Abstract

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.

Matrix spectral factorization for SA4 multiwavelet

“…Matrix Spectral Factorization (MSF) plays a crucial role in the solution of various applied problems for MIMO systems in communications and control engineering [46]. It has been applied to designing minimum phase FIR filters and the associated all-pass filter [19], quadrature-mirror filter banks [3], MIMO systems for optimum transmission and reception filter matrices for precoding and equalization [15], precoders [12], and many other applications.…”

Design of a Simple Orthogonal Multiwavelet Filter by Matrix Spectral Factorization

“…A well-known method for MSF is Bauer's method [6]. This method has been successfully applied in [14,30,35,39]. Details of the algorithm are given in section 4.1 below.…”

Efficient minimum-phase prefilter computation using fast QL-factorization

“…Furthermore, it can be seen that each of these columns will correspond to the all-pass filter associated with the minimum-phase filter. For a detailed description of this, see [5], [6]. In the finite length case, each row of L (column of Q) will not be exactly the same, but as can be seen in [6], the values in each row of L will converge toward the true minimum-phase filter as a function of the row number 2 , likewise the columns of Q will converge toward the associated all-pass filter.…”

“…For a detailed description of this, see [5], [6]. In the finite length case, each row of L (column of Q) will not be exactly the same, but as can be seen in [6], the values in each row of L will converge toward the true minimum-phase filter as a function of the row number 2 , likewise the columns of Q will converge toward the associated all-pass filter. Thus, the accuracy of the estimated filter coefficients (compared to the true filters) depends on where in L and Q we take out the filter coefficients.…”

“…Recently, it has been discovered that the minimum-phase filter and its associated all-pass filter can be obtained by performing a QL-factorization of the channel matrix, H, [5], [6]. When we perform the factorization,…”