Optimal design for synthesis filters of oversampled uniform perfect reconstruction filter banks with 50% overlapping

We address a problem to find optimal synthesis filters of oversampled uniform FIR filter banks (FBs) yielding perfect reconstruction (PR), in which the all filters have the same length that is twice a factor of downsampling. In this framework, filter coefficients are allowed to be complex as well as real. Optimality is specified by two criteria: variance of additive noise and stopband attenuation. We first clarify that a synthesis FB yielding PR, which has a degree of freedom in design, can be found in closed form with simple matrix operations. Owing to a degree of freedom, we can specify optimal synthesis FBs which meet those criteria. Theoretical results for finding optimal synthesis filters are shown. Moreover, an example of optimal inverses of a windowed discrete Fourier transform with 50% overlapping is provided.