A closed-form solution for the design of complex FIR filters and its application to filter banks

In this paper, the least-square design problem of complex FIR filters with arbitrary frequency response characteristics is investigated. A closed-form solution is obtained by solving the approximation problem analytically, enabling a very fast calculation of the impulse response of the filter to be designed. An application of the derived solution to QMF bank design is explored. By using the Butterworth approximation to obtain an IIR filter for the analysis part and by employing the closed-form solution to approximate a nonlinear-phase FIR filter for the synthesis part, a class of nearly perfect-reconstruction QMF bank is developed. It is shown that the mixed IIR/FIR filter bank not only has a very good reconstruction performance but also enjoys an extremely low design complexity.