Iterative DFT-Domain Inverse Filter Optimization Using a Weighted Least-Squares Criterion

For many inverse filtering problems, finite impulse response filters are designed according to least-squares criteria, where time-domain and frequency-domain weights are often applied to achieve optimal results for the considered application. While least-squares-optimal filter coefficients are given by an explicit formula, the computation cost to compute its solution is proportional up to the third power of the number of jointly optimized filter coefficients. A joint optimization of all filter coefficients is necessary whenever a time-domain or a frequency-domain weight is introduced. This imposes limits for filter lengths and numbers of channels in many real-world scenarios. In this contribution, an algorithm is presented that yields time-domain filter coefficients optimized to meet such a weighted least-squares criterion, while performing the most expensive computation steps efficiently in the discrete Fourier transform domain. As a consequence, the demands on computational power and memory are kept on a moderate level, even for large-scale problems. A rigorous mathematical derivation is provided that identifies all approximations used in the algorithm. Additionally, an effective regularization method is proposed that does not depend much on the regularization parameters. Furthermore, the proposed approach is experimentally evaluated considering a sound-zones scenario, which is one of many possible application areas. In that way, the applicability of the proposed approach is verified.