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
WAV files contain linear pulse code modulated (LPCM) data. That just means that it is a sequence of amplitude values at a fixed sample rate. A RIFF header is contained at the beginning of the file to convey information like sampling rate and bits per sample (e.g. 8 kHz signed 16-bit).
The format is very simple and you could easily roll your own. However, there are several libraries available to speed the process such as libsndfile. Simple Direct-media Layer (SDL)/SDL_mixer and PortAudio are two nice libraries for playback.
As for feeding the data into FFTW, you would need to buffer 1 second chunks (determine size by the sample rate and bits per sample). Then convert all of the samples to IEEE floating-point (i.e. float or double depending on the FFTW configuration--libsndfile can do this for you). Next create another array to hold the frequency domain output. Finally, create and execute an FFTW plan by passing both buffers to fftw_plan_dft_r2c_1d and calling fftw_execute with the returned fftw_plan handle.
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.
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.