Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Sponsor

Evaluate multicore with Texas Instruments:
Low-cost evaluation module & free software development kit available NOW.

Chapters

See Also

Embedded SystemsFPGAElectronics

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

Chapter Contents:

Search Spectral Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Discrete Wavelet Filterbank

In a discrete wavelet filterbank, each basis signal is interpreted as the impulse response of a bandpass filter in a constant-Q filter bank:

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

Thus, the $ k$th channel-filter $ H_k(\omega )$ is obtained by frequency-scaling (and normalizing for unit energy) the zeroth channel filter $ H_0(\omega )$. The frequency scale-factor is of course equal to the inverse of the time-scale factor.

Recall that in the STFT, channel filter $ H_k(\omega )$ is a shift of the zeroth channel-filter $ H_0(\omega )$ (which corresponds to ``cosine modulation'' in the time domain).

As the channel-number $ k$ increases, the channel impulse response $ h_k$ lengthens by the factor $ a^k$., while the pass-band of its frequency-response $ H_k$ narrows by the inverse factor $ a^{-k}$.

Figure 10.33 shows a block diagram of the discrete wavelet filter bank for $ a=2$ (the ``dyadic'' or ``octave filter-bank'' case), and Fig.10.34 shows its time-frequency tiling as compared to that of the STFT. The synthesis filters $ f_k$ may be used to make a biorthogonal filter bank. If the $ h_k$ are orthonormal, then $ f_k=h_k$.

Figure 10.33: Dyadic Biorthogonal Wavelet Filterbank
\includegraphics[width=\twidth]{eps/DyadicFilterbank}

Figure 10.34: Time-frequency tiling for the Short-Time Fourier Transform (left) and dyadic wavelet filter bank (right).
\includegraphics[width=\twidth]{eps/DyadicTiling}

Previous: Discrete Wavelet Transform
Next: Dyadic Filter Banks

Order a Hardcopy of Spectral Audio Signal Processing

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.


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.

Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )