The author describes a process called "ontological remodeling", which is when a change in viewpoint radically simplifies a previously intractable problem. This is the story, not just of Sudoku, but all of mathematics.
It's also common in audio manipulation, e.g. change to the frequency domain in order to modify pitch, then change back to the time domain.
Funnily enough, yes but actually no. For understanding and mathematical proofs the Fourier Transform is obviously essential. But when you first get into audio DSP programming it might seem that the FFT is crucial as well. But virtually all digital audio filters directly operate on the on the time domain.
The problem is that we always work with a sampled signal. And while the Nyquist theorem tells us that as long as our sample frequency is at least twice as high as the highest frequency in our input that our sampling doesn't lose any information, we still have to be aware of it.
When you convert a fixed frequency sampled signal into the frequency domain, you get a sum of a fixed set (co)sine functions, because we're still sampled. Now some operations you can do in this frequency domain exactly and simply. E.g. if 100Hz is part of our fixed set of functions and we wish to subtract a 100Hz signal, we can directly do that on our coefficient. A pitch shift by an exact multiple of the frequency sample delta is possible exactly too, assuming the lowest/highest frequencies shifted off are inaudible.
But you almost never want to do these operations. A classic example of something you might want to do is a low-pass filter. Simple right? FFT to frequency domain, zero out the coefficients above the cutoff frequency, and convert back. No! Zeroing out coefficients is equivalent to subtracting those specific sine waves. But that is not a low-pass. As an example, suppose our sample frequency is such that the FFT's sine wave coefficients are 10Hz apart, and we wish to do a 30Hz low-pass filter. This means that a 25Hz signal should be completely unaffected, but a 25Hz signal can't be represented just using the 10Hz and 20Hz coefficients - you need higher order terms! Thus if we zero those out, we distort our 25Hz signal.
So to solve this in the real-world you get into the difficult topic of filter design, or other audio DSP algorithms that are vastly more involved than a simple FFT.
Specifically regarding the latter part of "extrapolating sampled data", I would highly recommend watching this video: https://xiph.org/video/vid2.shtml. As long as your input signal is low-passed to below 22kHz, the 44.1kHz sampling is perfect. No information is lost, no distortion.
I however am not qualified enough to tell you how the naive FFT filter approach changes in distortion as you raise the sampling frequency.
(edit) e.g. bandwidth limited signal with only a single non-zero sample does not represent a rectangular function, but a sinc.
So, FFT is a lie. But very useful one.
Other complicated DSP algorithms I've read about (but haven't come close to fully grokking) are:
- Band-limited oscillators. Ask yourself, how do you generate a square wave signal? It's "just" 1 for t seconds and -1 for t seconds, repeating, right? But what if t isn't a multiple of our sampling frequency? Also, a square wave can be interpreted as an infinite sum of ever increasing frequency sine waves. But Nyquist tells us we can't have those higher frequencies. So even if t was a multiple of our sampling frequency, it still wouldn't be right to have t/f samples of 1 followed by t/f samples of -1. But just taking the first N terms of the infinite sum (those below Nyquist) is slow, so you get things like polyblep: https://www.kvraudio.com/forum/viewtopic.php?t=375517.
- Resampling. So a sampled bandlimited signal is perfectly represented by the samples through the Whittaker–Shannon interpolation formula. To resample we simply have to pick equidistant samples using the interpolation formula. However this is slow (and global, so not real-time), so there's a bunch of techniques to speed this up with minimal distortion: http://ldesoras.free.fr/doc/articles/resampler-en.pdf
- Pitch shift. This seems innocuous enough, right? But suppose it were easy to shift pitch. Then we could resample our audio to 2x the frequency, shift the pitch up by an octave and play back at the original speed. Now we've made our audio twice as long without changing the pitch! Pitch shifting and timescaling are two sides of the same coin. And I don't understand either side. Fundamentally I don't really understand what it means to make an audio signal "longer", without changing the pitch, at the waveform level. But people do it anyway: https://en.wikipedia.org/wiki/Audio_time_stretching_and_pitc...
- The rabbit hole goes on...
it's such a powerful tool that the only problem it leaves is, what benefit is there to calling it ontological remodeling instead of "a change in viewpoint"?