I want to get into audio signal processing. I know how to generate a tone, but I have no idea how filters work. What do I do to a set of samples to boost/cut a frequency with adjustable bandwidth? What is FFT? What's the code behind a parametric equalizer?
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you need to familiarize with concepts such us:
FIR filters and their frequency response: https://en.wikipedia.org/wiki/Finite_impulse_response
Nyquist frequency and sample frequency:https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem
And then, have a look to the filters used for equalization: https://en.wikipedia.org/wiki/Equalization_(audio)
Hope these basic links help you to do a bit more of research :)