Johnston's QMF Banks 2.4 2002/11/04 2005/10/01 15:50:10.141 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Michael Haag mjhaag@rice.edu filter banks filterbanks johnston qmf quadrature mirror filterbank reconstruction This module examines a type of Quadrature Mirror Filterbank (QMF) in regards to its reconstruction properties and the ideas presented by Johnston. 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 T ω to have magnitude-response ripples while keeping its linear phase. Say that H 0 z is causal, linear-phase, and has impulse response length N. Then it is possible to write H 0 ω in terms of a real-valued zero-phase response H ~ 0 ω , so that H 0 ω ω N 1 2 H ~ 0 ω T ω H 0 ω 2 H 0 ω 2 ω N 1 H ~ 0 ω 2 ω N 1 H ~ 0 ω 2 ω N 1 H ~ 0 ω 2 N 1 H ~ 0 ω 2 Note that if N is odd, ω N 1 1 , ω 2 T ω 0 A null in the system response would be very undesirable, and so we restrict N to be an even number. In that case, T ω ω N 1 H ~ 0 ω 2 H ~ 0 ω 2 ω N 1 H 0 ω 2 H 0 ω 2 The system response is linear phase, but will have amplitude distortion if H 0 ω 2 H 0 ω 2 is not equal to a constant. 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 . Specifically, real symmetric coefficients h 0 n are chosen to minimize J λ ω ω 0 H 0 ω 2 1 λ ω 0 1 H 0 ω 2 H 0 ω 2 where 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) "12B" Filter As an example, consider the "12B" filter from Johnston: h 0 0 -0.006443977 h 0 11 h 0 1 0.02745539 h 0 10 h 0 2 -0.00758164 h 0 9 h 0 3 -0.0913825 h 0 8 h 0 4 0.09808522 h 0 7 h 0 5 0.4807962 h 0 6 which gives the following DTFT magnitudes ().

The top plot shows the analysis filters and the bottom one shows the system response.

R.E.Crochiere and L.B. Rabiner Multirate Digital Signal Processing Prentice Hall 1983 Englewood Cliffs, NJ J.D. Johnston A filter family designed for use in Quadrature mirror filterbanks Proc IEEE Internat. Conf. on Acoustics, Speech, and Signal Processing 1980 291-294 April R. Ansari and B. Liu Multirate Signal Processing Handbook for Digital Signal Processing Wiley Interscience S.K. Mitra and J.F. Kaiser New York chp. 14 pp. 981-1084