Discrete Wavelet Transform

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Discrete Wavelet Transform

Transform (filterbank form):

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

		$\displaystyle n,k\;\hbox{integers}$

Inverse:

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

Wavelet filter banks always Constant-:
- $Q \isdef$ center-frequency / bandwidth
- Center-frequency $\isdef$ geometric mean of bandlimits $\omega_1$ and $\omega_2$ :

: $\omega _c \isdef \sqrt{\omega _1\omega _2}$
: $\omega _c(k) \isdef \sqrt{\omega _1(k)\omega _2(k)} = \sqrt{a^k\omega _1(0)a^k\omega _2(0)} = a^k\omega _c(0)$
: $Q(k) \isdef \frac{\omega _c(k)}{\omega _2(k) - \omega _1(k)} = \frac{a^k\omega _c(0)}{a^k\omega _2(0) - a^k\omega _1(0)} = Q(0)$

$&bull#bullet;$: In dyadic filter bank, $Q=\sqrt{2}$ :
: $\;$ center-frequency $\isdef \sqrt{\omega _0(\omega _0+\hbox{bandwidth})} = \sqrt{2}\omega _0$
: $\; \implies Q = \frac{\sqrt{2}\omega _0}{2\omega _0 - \omega _0} = \sqrt{2}$

Figure 12.3 shows a block diagram of the discrete dyadic wavelet filterbank, and Fig.12.4 shows its time-frequency tiling as compared to that of the STFT.

**Figure 12.3:** Dyadic Wavelet Filterbank
$\includegraphics[width=\textwidth]{eps/DyadicFilterbank}$

**Figure 12.4:** Time-frequency tiling for the Short-Time Fourier Transform (left) and dyadic wavelet filterbank (right).
$\includegraphics[width=\textwidth]{eps/DyadicTiling}$

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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.

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Spectral Audio Signal Processing
    Wavelet Filter Banks
       Wavelets
          Discrete Wavelet Transform