A closed form expression for an efficient class of quadrature mirror filters and its FIR approximation

We find a simple closed form expression for an efficient class of Quadrature Mirror Filters (QMF's) by exploiting the inherent symmetry of the QMF property. We derive a simple rule of thumb to calculate the maximum feasible frequency selectivity of the filter for a given number N of filter taps. We show that, for even n, the frequency selectivity of a 2n+1 or 2n tap filter can be increased if and only if the number of taps is increased by at least 4. Most existing QMF's closely match the derived analytical expression as well as verify the results on frequency selectivity. We obtain FIR implementations of the aforementioned analytical expression by using the Remez allocation algorithm. We choose weighting functions that confine the error to the intersection of the transition band and the stop band of the filter, as well as force the magnitude of the passband ripple to be much lower than that of the stopband ripple. We make such a choice in order to optimally satisfy the power complementarity condition as well as to attain high stop band attenuation. Our implementations match existing designs in performance.