Two-channel perfect-reconstruction QMF banks are not very useful because the analysis filters have poor frequency selectivity. The selectivity characteristics can be improved, however, if we allow the system response Tⅇⅈω T ω to have magnitude-response ripples while keeping its linear phase.

Say that H 0 z H 0 z is causal, linear-phase, and has impulse response length NN. Then it is possible to write H 0 ⅇⅈω H 0 ω in terms of a real-valued zero-phase response H ~ 0 ⅇⅈω H ~ 0 ω , so that Note that if NN is odd, ⅇⅈωN-1=1 ω N 1 1 , A null in the system response would be very undesirable, and so we restrict NN to be an even number. In that case, Johnston's idea was to assign a cost function that penalizes deviation from perfect reconstruction as well as deviation from an ideal lowpass filter with cutoff ω 0 ω 0 . Specifically, real symmetric coefficients h 0 n h 0 n are chosen to minimize where 0<λ<1 0 λ 1 balances between the two conflicting objectives. Numerical optimization techniques can be used to determine the coefficients, and a number of popular coefficient sets have been tabulated. (See Crochiere and Rabiner, Johnston, and Ansari and Liu)