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Spectral Audio Signal Processing
    Multirate Filter Banks
       Wavelet Filter Banks
          Geometric Signal Theory
             Discrete Wavelet Transform

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Discrete Wavelet Transform

The discrete wavelet transform is a discrete-time, discrete-frequency counterpart of the continuous wavelet transform of the previous section:

$\displaystyle X(k,n)$ $\displaystyle =$ $\displaystyle s^{-k/2} \int_{-\infty}^\infty x(t) h\left(nT-a^{-k}t\right) dt$  
  $\displaystyle =$ $\displaystyle \int_{-\infty}^\infty x(t) h\left(na^k T-t\right) dt$  

where $ n$ and $ k$ range over the integers, and $ h$ is the mother wavelet, interpreted here as a (continuous) filter impulse response.

The inverse transform is, as always, the signal expansion in terms of the orthonormal basis set:

$\displaystyle x(t) = \sum_k \sum_n X(k,n) \underbrace{\varphi _{kn}(t)}_{\hbox{basis}} $

We can show that discrete wavelet transforms are constant-Q by defining the center frequency of the $ k$th basis signal as the geometric mean of its bandlimits $ \omega_1$ and $ \omega _2$, i.e.,

$\displaystyle \omega _c(k) \isdef \sqrt{\omega _1(k)\,\omega _2(k)} \eqsp \sqrt{a^k\omega _1(0)\,a^k\omega _2(0)} \eqsp a^k\omega _c(0). $

Then

$\displaystyle Q(k) \isdefs \frac{\omega _c(k)}{\omega _2(k) - \omega _1(k)} \eqsp \frac{a^k\omega _c(0)}{a^k\omega _2(0) - a^k\omega _1(0)} \eqsp Q(0) $

which does not depend on $ k$.

Previous: Continuous Wavelet Transform
Next: Discrete Wavelet Filterbank

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About the Author: 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.


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