Fast and stable least-squares approach for the design of linear phase FIR filters

In this paper, we present a new numerical algorithm for solving the normal equations associated with the least-squares design of linear phase FIR filters. The usual solution methods have a computational complexity of O(N3). Moreover, solving the normal equations with Gaussian elimination commonly yields numerical errors, especially when the filter is long. Here, we convert a least-squares method into the problem of constructing a system of orthonormal functions. The proposed design algorithm needs only O(N2) computations, and numerical errors can be reduced. Some examples are given to show the excellent performance of the algorithm.