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I’ve been reading up on approaches to pitch shifting, and have found some common approaches use fourier transforms to achieve a change of pitch.

I’m curious, is fourier transform technology used in the “guts” of digital equalizers?

Any suggestions for references to gain an understanding on how digital equalizers are implemented?

Something at the level of the content of the web pages initiated at this link about pitch shifting would be great:

Guitar Pitch Shifter

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  • $\begingroup$ If your pitch shifter is being applied to a single monotone instrument and if the application is real time, you don't wanna use any fourier transform to do the pitch shifting. You wanna do it all in the time domain. You're gonna need a decent pitch detector. $\endgroup$ Sep 28, 2023 at 16:25

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Fourier transform technology

Well, Fourier theory is behind every kind of equalizer. We wouldn't know what "equal" means without; the Fourier transform is how we know how to describe a signal in the frequency domain, and that's what ends up being "of equal power everywhere" after an equalizer.

If we interpret "technology used in the guts" as "does it employ a Fourier transform during operation":

That fully depends on the type of equalizer! The whole class of Frequency-Domain Equalizers are based on it. But even if you have something like a 12-band EQ made of biquad sections, somehow, somewhere you need to find the "is" state of the observed spectrum to make an equalizer actually equalize and not just arbitrarily shape the spectrum. And the Fourier transform is the standard way of doing that.

Any suggestions for references to gain an understanding on how digital equalizers are implemented?

I'd say: start with about any good DSP textbook; mine was a German one co-authored by one of my favourite lecturers, so that got me hooked on DSP. then, I read Oppenheim/Schafer's Discrete-Time Signal Processing, which I bought for a handful of Euros used; having had but a short glimpse into Richard G. Lyon's (he's on here, nice person) Understanding Digital Signal Processing seems quite nice, as well. I think that's the book I'd recommend to you, or maybe his more introductory The Essential Guide to Digital Signal Processing, which I think is more for management-level, less for people who are/have been already studying electrical engineering, math or similar topics.

Sooner than you'd like you'll appreciate the Fourier transform, its discrete-time variant (DFT, as usually implemented by an FFT), and what you can do. It will also tell you where all the formulas that just happened to appear in the website you link to come from!

Now, I'm from a communications engineering background, so the kind of equalizers we build have a different application than those that people in the audio signal processing trade build. Their mathematical construction, however, is the same, as the job of an equalizer is always to remove the effect of echos that constructively and destructively overlay with the original signal, or, identically, since the impulse response (i.e., when do the echoes come, and at which phase and amplitude) and the frequency response (i.e., how does the room==channel==system attenuate and phase-shift different frequencies) are inherently the same information – linked through the Fourier transform!

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    $\begingroup$ "Their mathematical construction, however, is the same, as the job of an equalizer is always to remove the effect of echos that constructively and destructively overlay with the original signal,"... $$ $$ That's not always the case. In audio, equalizers are used to aesthetically "equalize" something. In audio, equalizers are normally linear filters with controls that humans or automation manipulate. Often those controls are in two classes: "Parametric equalizers" or "Graphic equalizers". Different knobs but they're both filters. $\endgroup$ Sep 27, 2023 at 19:19
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    $\begingroup$ But it's certainly possible, and sometimes it's done this way, that the DFT (using the FFT algorithm) is used to perform the filtering of equalization. To do that (and be linear), you must use ether the Overlap-add (OLA) or Overlap-save (OLS, sometimes called Overlap-scrap) to perform the filtering. One of these two techniques must be used because the DFT naturally performs circular convolution when the filter needs to perform linear convolution. $\endgroup$ Sep 27, 2023 at 19:21
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Unlikely. An equalizer (unless it's offline) should not have large delays, and Fourier transforms introduce latency. Equalizers are more likely to use IIR filters and/or possibly quadrature mirror filter banks.

Continuous Fourier transforms/integrals are a tool for analysis/design. Discrete Fourier transforms are not directly applicable to filtering. They can be used as computational shortcut with overlap/add and overlap/save techniques. There is also some applicability using overlapping windowing and short-time transforms, but those make more sense in the context of dynamically adapting transfer functions.

At any rate, FFTs are an efficient Swiss army knife in the context of convolutions, and they have relations to the continuous Fourier transform that is a good tool for describing things in the frequency domain, so latency or not, they are likely to end up in experimental stuff. I just doubt that you'll find them a lot in commercial stuff.

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