Which (Fourier transform?) processing method to use for time variant audio processing? (newbie) Edit:
I've since been able to understand the how the relvant parts of Fourier transform work and how they relate to my problem / what I want to do.
And have come to the conclusion that Fourier transform is not the best suited to my problem and that I have an alternative way to do what I want that is much simpler to control / behave in the way I want in practice.

Since I'm new here I don't know what's the right thing to do with my question on this site. Just in case anybody was following this / was preparing an answer, I'm leaving this question up for one more day with this edit and after that I'll delete the question.



I've devloped a basic algorithm for processing audio. I have some programming experience but none with Fourier transforms (but will dedicate all of 2020 to learn this full time if need be).
My question relates to how best to program my algorithm in practice.

What I need to do is the following:


Analyze the frequency content at exact locations of a 44.1kHz PCM
audio stream / file.
Detect and modify the amplitude of specific frequencies at specific
time locations (I do not need to modify phase or have knowledge of
phase to do this).
Then turn this modified version back into audio without audible artifacts from processing errors/limitations.


As I said, I currenty do not know much about Fourier transform. So far my understanding is that an FFT (with windowing) will allow me to analyze the frequency spectrum of a block, detect and modify the amplitude of specific frequencies. But not modify the amplitude of specific frequencies at a specific time in that block and not the same frequency at a different time in that same block? For instance if for 44.1kHz sample rate and block size 8192 as far as I understand it normal FFT will not allow me to detect for instance two seperate 10kHz transients in that block as seperate and modify only one of them?

If my above understanding is correct. Can anybody suggest proper (Fourier transform like?) processing method(s) to use / investigate to achieve my goal?

I've throught of a rough way to do what I want with FFT/IFFT by dividing the frequency spectrum (5Hz to Nyquist for instance) into seperate octave bands all processed seperately, for instance first band 11.025kHz to 22.05kHz under 44.1kHz sample rate and block size 4 for instance not sure of the right block size just an estimate (so I get the transients in one block and still sufficient frequency resolution for that frequency band), then 5.5125kHz to 11.025kHz band under 22.05kHz sample rate (downsampled) also with block size 4 as an estimate, etc with a total of 12 bands down to about 5Hz for instance.
But this will be a lot of work and will take me a lot of time to develop and surely someone must have done something similar and smarter a long time ago to do the type of processing I wish to do :) If anybody can point me in the right direction (and perhaps if I got some assumptions wrong correct me) I will be very thankful!
I believe the answer to your question is Short-time Fourier Transform or Short-term Fourier Transform.  There is the wikipedia article.  I tried to show the essential math in this answer.

Both the Phase-Vocoder and Sinusoidal Modeling are done with the STFT of some form or another.  The two methods sorta merge in concept at the STFT level.  Remember that the analysis window for STFT need not be the same as the synthesis window in constructing the processed audio for output.