Best way to get the bin amplitudes of an audio DFT "normalized" 0-1

Question

I'm working on a spectral processing plugin that's predicated on allowing the user to manipulate per-bin values in an STFT algorithm. I normalize the bin amplitudes by diving them by $N$ (number of samples used in the DFT), but the resulting numbers often inhabit a range that is difficult to work with directly, e.g. it is very difficulty to precisely gate/limit an amplitude value that is often well below 0.1.

Obviously, once normalized, each DFT bin has an absolute amplitude range of 0-1, but what I'd like to do is scale them somehow so they tend to span that entire range, factoring in things specific to audio signals like the bass content often being much greater than the treble. My first instinct is to either apply some logarithmic curve, or maybe even use a table of values derived from the Fletcher-Munson curve, but I wanted to ask here and see if I couldn't go about it in a smarter way. Thank you!

EDIT: I believe I might not have explained my intentions well enough! The normalization I'm planning on doing is going to be temporary, so that the user can easily set values without needing to work with tiny decimal places. E.G., the user can set a "gate" level of 0.3, and then all bins with a "normalized" amplitude under 0.3 would be muted. What I'd like is for a single gate value to work reasonably well for all frequencies, even though lower-frequency bins tend to have a much amplitude in most musical signals.

Answer 1

I'm working on a spectral processing plugin that's predicated on allowing the user to manipulate per-bin values in an STFT algorithm

This begs the question "why" ? That's not particularly useful and very difficult to implement without creating significant artifacts (due to time domain aliasing). It would help to understand what your specific application and requirements are.

Best way to get the bin amplitudes of an audio DFT "normalized" 0-1

There is no best way that I'm aware of, because that's typically not done. Normalizing each bin individually makes the spectrum white. Audio is decidedly non-white and by normalizing you probably also get a lot of noise amplification in the low-energy bins.

Human perception of audio is typically log/log, i.e. using a logarithmic frequency axis and logarithmic level axis (in dB). The STFT isn't a great fit for that since the bin spacing is linear. The bins at low frequencies tend to be too wide and the bins at high frequencies are way too narrow.

The common approach is to process the date in log frequency bands (3rd octaves for example). This can be done with a weighted STFT or a bandpass filter bank. The STFT needs to be large enough to accommodate the lowest frequency resolution you need. This may cause some non-trivial latency which may or may not be a problem.

Then convert the energy in dB. If your original audio signal is calibrated in real sound pressure (units of Pascal), than you can normalize to dBSPL (using $20 \mu Pa$ as a reference). If not, people typically use dBFS (Full Scale), where $0dB$ is your clipping point.