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Johnston's QMF Banks

Module by: Phil Schniter

Summary: This module examines a type of Quadrature Mirror Filterbank (QMF) in regards to its reconstruction properties and the ideas presented by Johnston.

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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

H 0 ⅇⅈω=ⅇ-ⅈωN−12 H ~ 0 ⅇⅈω H 0 ω ω N 1 2 H ~ 0 ω (1)
Tⅇⅈω= H 0 2ⅇⅈω− H 0 2ⅇⅈω−π=ⅇ-ⅈωN−1 H ~ 0 2ⅇⅈω−ⅇ-ⅈω−πN−1 H ~ 0 2ⅇⅈω−π=ⅇ-ⅈωN−1 H ~ 0 2ⅇⅈω−ⅇⅈπN−1 H ~ 0 2ⅇⅈω−π 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 (2)
Note that if NN is odd, ⅇⅈωN−1=1 ω N 1 1 ,
Tⅇⅈω|ω=π2=0 ω 2 T ω 0 (3)
A null in the system response would be very undesirable, and so we restrict NN to be an even number. In that case,
Tⅇⅈω=ⅇ-ⅈωN−1 H ~ 0 2ⅇⅈω+ H ~ 0 2ⅇⅈω−π=ⅇ-ⅈωN−1| H 0 ⅇⅈω|2+| H 0 ⅇⅈω−π|2 T ω ω N 1 H ~ 0 ω 2 H ~ 0 ω 2 ω N 1 H 0 ω 2 H 0 ω 2 (4)

note:

The system response is linear phase, but will have amplitude distortion if | H 0 ⅇⅈω|2+| H 0 ⅇⅈω−π|2 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 ω 0 . Specifically, real symmetric coefficients h 0 n h 0 n are chosen to minimize
J=λ∫ ω 0 π| H 0 ⅇⅈω|2dω+1−λ∫0∞1−| H 0 ⅇⅈω|2−| H 0 ⅇⅈπ−ω|2dω J λ ω ω 0 H 0 ω 2 1 λ ω 0 1 H 0 ω 2 H 0 ω 2 (5)
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)

Example 1: "12B" Filter

As an example, consider the "12B" filter from Johnston: h 0 0=-0.006443977= h 0 11 h 0 0 -0.006443977 h 0 11 h 0 1=0.02745539= h 0 10 h 0 1 0.02745539 h 0 10 h 0 2=-0.00758164= h 0 9 h 0 2 -0.00758164 h 0 9 h 0 3=-0.0913825= h 0 8 h 0 3 -0.0913825 h 0 8 h 0 4=0.09808522= h 0 7 h 0 4 0.09808522 h 0 7 h 0 5=0.4807962= h 0 6 h 0 5 0.4807962 h 0 6 which gives the following DTFT magnitudes (Figure 1).

Figure 1: The top plot shows the analysis filters and the bottom one shows the system response.
Figure 1 (john_qmf.png)

References

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

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