How can we extract the frequency content of an audio signal given the histogram of beat energy versus beats per minutes?
Fourier Transform of an audio signal
signal processing
Related Solutions
Electret microphone capsules are very easy to use. Just give them a bias voltage and amplify the output signal. They have a built-in JFET that does the hard work. Here is an example schematic (from Sparkfun).
Many electret mic capsules are available, with slightly different size, pedigree and frequency response. The 2.2K RL is more-or-less standard.
I am not sure about intuition in general, but regarding the step-function FT being a sync function:
Note that the shape will remain the same, but the frequencies over which the FT of a particular step-function resides is a function of the pulse-width of the original signal. Namely, expanding a function in the time-domain actually shrinks the corresponding frequency-domain function (think slowing down voice recordings, the sound gets very low i.e. lower frequency).
That being said, as you decrease the pulse-width of a particular step-function the frequency components of that signal increase because now there is more change happening (to use a loose descriptor) in a shorter amount of time.
In contrast, if we expand the step-function in the time-domain to have a longer pulse-width then there is less change and the corresponding frequency components must be much lower.
In general, I look at a function and try and get a feel for how quickly it might be changing to get a rough idea. But as I said, I don't know of any general rule of thumb here.
Best Answer
You can't. The histograms you have shown are meaningless in the context of the fourier analysis of the full musical signal. In fact the histograms you have shown don't actually represent any part of the frequency spectrum of an audio signal.
To extract beat information from an audio signal requires demodulation of that signal via a frequency selective envelope detector - this largely loses any other spectral content that you might be wishing to extract.