Discrete Wavelet Filterbank

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Geometric Signal Theory

Discrete Wavelet Transform

Dyadic Filter Banks

The Short Time Fourier Transform (STFT) computes the spectrum (DFT) of successive time frames of a signal. — Click for https://ccrma.stanford.edu/~jos/sasp/Short_Time_Fourier_Transform.html

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

The impulse signal is the shortest pulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). The impulse is typically denoted by the Greek delta symbol δ and is often called the 'Dirac delta function', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

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A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

The impulse response of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

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A set of filters that decompose a signal into a set of components — Click for http://en.wikipedia.org/wiki/Filter_bank

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Geometric Signal Theory

Discrete Wavelet Transform

Dyadic Filter Banks

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

Figure 11.32 shows a block diagram of the discrete wavelet filter bank for (the ``dyadic'' or ``octave filter-bank'' case), and Fig.11.33 shows its time-frequency tiling as compared to that of the STFT. The synthesis filters may be used to make a biorthogonal filter bank. If the are orthonormal, then .

**Figure 11.32:** Dyadic Biorthogonal Wavelet Filterbank
$\includegraphics[width=\twidth]{eps/DyadicFilterbank}$

**Figure 11.33:** Time-frequency tiling for the Short-Time Fourier Transform (left) and dyadic wavelet filter bank (right).
$\includegraphics[width=0.8\twidth]{eps/DyadicTiling}$

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Discrete Wavelet Filterbank

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University