A fast algorithm for computing inverse cosine transforms for designing zero-phase FIR filters in frequency domain

A new algorithm for computing inverse cosine transforms or for designing zero-phase FIR filters from nonuniform frequency samples is presented. The algorithm is simple, fast, recursive and can be used in 1-D or 2-D applications. Based on the three-term recursive relation of the Chebyshev polynomials, the cosine matrix is decomposed into LU products using parallel computations. Two alternative approaches-a direct and a progressive-suitable for serial computations are also derived. Given N samples, the direct version requires 2.5N/sup 2/+O(N) flops for computing the inverse cosine transforms or for calculating the filter coefficients, whereas the progressive version needs only O(5N) flops when the next N+1th sample appears. The algorithm guarantees real results and produces accurate solutions even in cases of designing high-order 1-D or 2-D FIR filters or when the interpolation matrix is ill conditioned. It can be also used in LU-factorization and can be extended to m-D filter design.<>