Reddit mentions of Mathematics of the Discrete Fourier Transform (DFT): with Audio Applications ---- Second Edition

It will take a sharper mind than mine to give this topic the full, thorough ELI5 treatment, but basically:

• Take the audio file and cut it up into chunks of n samples. Picking a larger n lets the DFT output lower frequencies, but requires more computational power and can create artifacts in complex signals like music, so it's a trade off. You usually pick n to be a power of two, because the most used FFT algorithm (Cooley-Tukey) requires it. If the last chunk falls short of the full n samples, you just pad the end with zeros.
  
  
• Apply whatever FFT algorithm you're using to each successive chunk. (I'm not gonna go into the math here.)
  
  
• For each chunk, you now have a list of wave lengths (in samples) and a complex number representing their respective amplitudes. So if you wanted to find the peak frequency in a file you'd take the magnitudes of all the complex numbers across all the chunks, find the largest one, and pick the corresponding wave length in the corresponding chunk. Let's say the wave length is 100 samples. So now you divide the audio file's sample rate (say, 44.1KHz) by the wave-length of your peak frequency and you get 44100Hz/100 = 441Hz. That's your peak frequency.
  
  There are a ton of little details and traps and quirks along the way I didn't mention, but that's the general idea. One thing that I will point out, since it's an important part of any audio application of the DFT is that the transform averages out all the frequencies that happen within each chunk, and there is some quantization error. Abrupt jumps at the beginning and end of each chunk (remember, these are just random chunks - you didn't pick them to begin and end nicely on a zero-level sample) also cause the DFT to output high frequencies that aren't really there. To deal with these two issues, we do something called windowing. Essentially, instead of picking contiguous chunks of samples, we pick overlapping chunks. For example, the first chunk goes from sample 0 to sample 255; the second from 128 to 383, the third from 256 to 511 - each chunk is 256 samples long, but the first 128 samples of each chunk are the same as the last 128 samples from the previous chunk. Then we apply a window function to each chunk which basically fades the beginning and end of each one down to zero. Window functions are symmetrical, though, so when you reconstruct the chunks and overlap them back over each other, the whole thing adds up to the original signal!
  
  The whole thing with the overlapping chunks and windowing is sometimes referred to as the Short-Time Fourier Transform (STFT) and, in the case of digital audio, the Discrete-time Short-Time Fourier Transform (Discrete-time STFT.) Whenever digital audio is being taken to the frequency-domain (i.e., when the DFT is applied to audio) this is what they're talking about.
  
  If you're serious about understanding and implementing the DFT for audio, get yourself Julius Smith's Mathematics of the Discrete Fourier Transform with Audio Applications. It's the best book I've found on the subject by far. The whole book is available for free on Dr. Smith's website too; I'm sure Google can find it for you.
  
  ELI5 FTW!

read these pages from the beginning. this is the best introduction I've read. https://ccrma.stanford.edu/~jos/mdft/mdft.html

or buy the book which is better formatted https://www.amazon.com/dp/097456074X/ref=cm_sw_r_cp_api_E0JNzbCFC34Q1