PSF Dual and Graphical Equalizers

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Spectral Audio Signal Processing

Dual of Constant Overlap-Add

Strong COLA

PSF and Weighted Overlap Add

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

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

dc stands for direct current in electrical engineering and signal processing (as opposed to 'alternating current'). The mean (average value) of a signal may be called its dc component. Similarly, frequency zero is called the dc frequency, because a sinusoid with zero frequency is a constant signal. — Click for https://ccrma.stanford.edu/~jos/filters/Mathematical_Sine_Wave_Analysis.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

Fast Fourier Transforms (FFT) are fast algorithms for computing the Discrete Fourier Transform (DFT) — Click for https://ccrma.stanford.edu/~jos/mdft/Fast_Fourier_Transform_FFT.html

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

An audio equalizer is a filter used to adjust the spectrum of an audio signal, usually in some number of frequency bands. High quality audio equalizers provide one or more bands of adjustment per critical band of human hearing. — Click for https://github.com/bdejong/musicdsp/blob/master/source/Filters/197-rbj-audio-eq-cookbook.rst

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 Fourier transform of a signal is a frequency-domain function — Click for https://ccrma.stanford.edu/~jos/mdft/

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Every periodic signal has a fundamental frequency given by the inverse of the period. If the period is in units of seconds, then the fundamental frequency is in units of Hertz (cycles per second). According to Fourier theory, every periodic signal can be expressed as a sum of sinusoids at frequencies given by integer multiples of the fundamental frequency. For integers greater than 1, these frequencies are called harmonic frequencies, and the sinusoids at harmonic frequencies are typically called harmonics. — Click for https://www.physicsclassroom.com/class/sound/Lesson-4/Fundamental-Frequency-and-Harmonics

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Dual of Constant Overlap-Add

Strong COLA

PSF and Weighted Overlap Add

PSF Dual and Graphical Equalizers

Above, we used the Poisson Summation Formula to show that the constant-overlap-add of a window in the time domain is equivalent to the condition that the window transform have zero-crossings at all harmonics of the frame rate. In this section, we look briefly at the dual case: If the window transform is COLA in the frequency domain, what is the corresponding property of the window in the time domain? As one should expect, being COLA in the frequency domain corresponds to having specific uniform zero-crossings in the time domain.

Bandpass filters that sum to a constant provides an ideal basis for a graphic equalizer. In such a filter bank, when all the ``sliders'' of the equalizer are set to the same level, the filter bank reduces to no filtering at all, as desired.

Let denote the number of (complex) filters in our filter bank, with pass-bands uniformly distributed around the unit circle. (We will be using an FFT to implement such a filter bank.) Denote the frequency response of the ``dc channel'' by $W(e^{j\omega T})$ . Then the constant overlap-add property of the -channel filter bank can be expressed as

$\displaystyle W \in \hbox{\sc Cola}(2\pi/N)$

(9.35)

which means

$\displaystyle S(\omega) = \sum_{k=0}^{N-1} W\left(e^{j(\omega-\omega_k)T}\right) = \hbox{constant}$

(9.36)

where $\omega_k T\isdef k\cdot 2\pi/N$ as usual. By the dual of the Poisson summation formula, we have

$\displaystyle \zbox {W \in \hbox{\sc Cola}(2\pi/N) \quad \Leftrightarrow \quad w \in \hbox{\sc Nyquist}(N)} \protect$

(9.37)

where $w\in \hbox{\sc Nyquist}(N)$ means that is zero at all nonzero integer multiples of , i.e.,

$\displaystyle w(n)=0, \quad n=\pm N,\pm 2N, \pm 3N, \ldots\,.$

(9.38)

Thus, using the dual of the PSF, we have found that a good -channel equalizer filter bank can be made using bandpass filters which have zero-crossings at multiples of samples, because that property guarantees that the filter bank sums to a constant frequency response when all channel gains are equal.

The duality introduced in this section is the basis of the Filter-Bank Summation (FBS) interpretation of the short-time Fourier transform, and it is precisely the Fourier dual of the OverLap-Add (OLA) interpretation [9]. The FBS interpretation of the STFT is the subject of Chapter 9.

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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

PSF Dual and Graphical Equalizers

``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