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Spectral Audio Signal Processing
    Wavelet Filter Banks
       Wavelets
          Discrete Wavelet Filterbank

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Discrete Wavelet Filterbank


$\displaystyle h_k(t)$ $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{a^k}} h\left(\frac{t}{a^k}\right), \quad a>1$  
$\displaystyle \longleftrightarrow\quad H_k(\omega )$ $\displaystyle =$ $\displaystyle \sqrt{a^k} H(a^k\omega )$  

  • Wavelet channel-filter $ H_k(\omega)$ is a scaling of channel-filter $ H_0(\omega)$ (scaling in time domain also)
  • In STFT, channel filter $ H_k(\omega)$ is a shift of channel-filter $ H_0(\omega)$ (modulation in time domain)
  • As $ k$ increases, $ h_k$ lengthens, $ H_k$ narrows
  • Dyadic filter bank ($ a=2$):
    Figure 12.6: Dyadic filter bank.
    \includegraphics[scale=0.8]{eps/dyadicFilters}
  • $ H_0(\omega ) = $ top-octave bandpass (BP) filter
  • $ H_1(w) = \sqrt{2} H_0(2\omega ) = $ BP for next octave down
  • $ H_2(w) = 2H_0(4\omega ) = $ octave bandpass below that, etc.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

 

hamid_kh_56 wrote:

9/26/2008
 
thx.
how can i computed filter coefficients to use in image Processing?
 

JOS wrote:

9/26/2008
 
I typically use the window method of FIR filter design whenever it is good enough (since it's so simple). There are some examples in Rabiner and Schafer 1978 (Digital Processing of Speech Signals - cited in the bibliography). See, for example, figures 6.31 and 6.32.

A "very easy" method described by Vaidyanathan (1993 - also in bibliography) is to design a two-channel paraunitary QMF bank, and repeat recursively to keep splitting the lower-half of the spectrum down to some depth.

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