Dyadic Filter Banks

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Geometric Signal Theory

Discrete Wavelet Filterbank

Dyadic Filter Bank Design

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The midpoint of the passband. — Click for http://en.wikipedia.org/wiki/Bandpass

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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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

A bandpass filter accepts input signal energy in a certain spectral band, and rejects the energy outside the band. — Click for http://en.wikipedia.org/wiki/Band-pass_filter

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

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

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

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Geometric Signal Theory

Discrete Wavelet Filterbank

Dyadic Filter Bank Design

Dyadic Filter Banks

**Figure 11.34:** Frequency response magnitudes for a dyadic filter bank (amplitude scalings optional).
$\includegraphics[width=0.7\twidth]{eps/dyadicFilters}$

A dyadic filter bank is any octave filter bank,^12.6 as illustrated qualitatively in Figure 11.34. Note that $H_0(\omega )$ is the top-octave bandpass filter, $H_1(w) = \sqrt{2} H_0(2\omega )$ is the bandpass filter for next octave down, $H_2(w) = 2H_0(4\omega )$ is the octave bandpass below that, and so on. The optional scale factors result in the same sum-of-squares for each channel-filter impulse response.

A dyadic filter bank may be derived from the discrete wavelet filter bank by setting and relaxing the exact orthonormality requirement on the channel-filter impulse responses. If they do happen to be orthonormal, we may call it a dyadic wavelet filter bank.

For a dyadic filter bank, the center-frequency of the th channel-filter impulse response can be defined as

$\displaystyle \omega _c(k) \eqsp \sqrt{\omega _0(\omega _0+\hbox{bandwidth})} \eqsp \sqrt{2}\omega _0$

(12.123)

so that

$\displaystyle Q \eqsp \frac{\sqrt{2}\omega _0}{2\omega _0 - \omega _0} \eqsp \sqrt{2}.$

(12.124)

Thus, a dyadic filter bank is a special case of a constant-Q filter bank for which the is $\sqrt{2}$ .

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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

Dyadic Filter Banks

``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