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 Quad rature Mirror Filters (QMF's) by exploiting the inherent symmetry of t he QMF property. We derive a simple rule of thumb to calculate the max imum feasible frequency selectivity of the filter for a given number N of filter taps. We show that, for even n, the frequency selectivity o f 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 fr equency selectivity. We obtain FIR implementations of the aforemention ed analytical expression by using the Remez allocation algorithm. We c hoose weighting functions that confine the error to the intersection o f the transition band and the stop band of the filter, as well as forc e the magnitude of the passband ripple to be much lower than that of t he stopband ripple. We make such a choice in order to optimally satisf y the power complementarity condition as well as to attain high stop b and attenuation. Our implementations match existing designs in perform ance.