A Direct Design of Oversampled Perfect Reconstruction FIR Filter Banks

We address a problem to find optimal synthesis filters of oversampled uniform finite-impulse-response (FIR) filter banks (FBs) yielding perfect reconstruction (PR), when we are given an analysis FB, in the case where all the filters have the same length that is twice a factor of downsampling. We show that in this class of FBs, a synthesis FB that achieves PR can be found in closed form with elementary matrix operations, unlike conventional design methods with numerical optimization. This framework allows filter coefficients to be complex as well as real. Due to the extra degrees of freedom in a synthesis FB provided by oversampling, we can determine optimal coefficients of synthesis filters that meet certain criteria. We introduce in this paper two criteria: variance of additive noise and stopband attenuation. We show theoretical results of optimal synthesis filters that minimize these criteria and design examples of oversampled linear-phase FIR FBs and DFT-modulated FBs. Moreover, we discuss applications to signal reconstruction from incomplete channel data in transmission and inverse transform of windowed discrete Fourier transform with 50% overlapping