How to extract semi-precise frequencies from a WAV file using Fourier Transforms

Question

Let us say that I have a WAV file. In this file, is a series of sine tones at precise 1 second intervals. I want to use the FFTW library to extract these tones in sequence. Is this particularly hard to do? How would I go about this?

Also, what is the best way to write tones of this kind into a WAV file? I assume I would only need a simple audio library for the output.

My language of choice is C

Answer 1

To get the power spectrum of a section of your file:

• collect N samples, where N is a power of 2 - if your sample rate is 44.1 kHz for example and you want to sample approx every second then go for say N = 32768 samples.

• apply a suitable window function to the samples, e.g. Hanning

• pass the windowed samples to an FFT routine - ideally you want a real-to-complex FFT but if all you have a is complex-to-complex FFT then pass 0 for all the imaginary input parts

• calculate the squared magnitude of your FFT output bins (re * re + im * im)

• (optional) calculate 10 * log10 of each magnitude squared output bin to get a magnitude value in dB

Now that you have your power spectrum you just need to identify the peak(s), which should be pretty straightforward if you have a reasonable S/N ratio. Note that frequency resolution improves with larger N. For the above example of 44.1 kHz sample rate and N = 32768 the frequency resolution of each bin is 44100 / 32768 = 1.35 Hz.

Answer 2

You are basically interested in estimating a Spectrum -assuming you've already gone past the stage of reading the WAV and converting it into a discrete time signal.

Among the various methods, the most basic is the Periodogram, which amounts to taking a windowed Discrete Fourier Transform (with a FFT) and keeping its squared magnitude. This correspond to Paul's answer. You need a window which spans over several periods of the lowest frequency you want to detect. Example: if your sinusoids can be as low as 10 Hz (period = 100ms), you should take a window of 200ms o 300ms or so (or more). However, the periodogram has some disadvantages, though it's simple to compute and it's more than enough if high precision is not required:

The raw periodogram is not a good spectral estimate because of spectral bias and the fact that the variance at a given frequency does not decrease as the number of samples used in the computation increases.

The periodogram can perform better by averaging several windows, with a judious choosing of the widths (Bartlet method). And there are many other methods for estimating the spectrum (AR modelling).

Actually, you are not exactly interested in estimating a full spectrum, but only the location of a single frequency. This can be done seeking a peak of an estimated spectrum (done as explained), but also by more specific and powerful (and complicated) methods (Pisarenko, MUSIC algorithm). They would probably be overkill in your case.