Thursday, October 10, 2013

Epilogue: Fourier analysis and testing


It's time for the final update of my GSoC project. The last part of my GSoC was all about improving the resampler test cases for PulseAudio. Since a lot of DSP testing depends on the Fourier transformation, I'll try to explain it briefly here.

The Fourier transform converts a periodic signal into a sum of sines and cosines of varying frequencies and phases. This way we can see the amplitude and frequency of the fundamental sine waves that make out our signal.

Lets take a look at a 440Hz sine wave like the one below.

~100 samples of a 440Hz sine wave

This is a simple sine wave without any higher harmonics so we don't expect to see much in our transformed signal. The sample format of our sine wave is 16-bit PCM. We can see that our amplitude is at about 50% (32768 would be loudest for 16-bit PCM and we are at about 16000) or -6dB.

Below we can see the Fourier transform of our sine wave.

We can clearly see a nice spike at 440Hz with an amplitude of -6dB. If our sine wave would consist of a fundamental sine wave at 440Hz and a higher harmonic at 880Hz we would see two spikes.

Now there are some other spikes, albeit rather small ones, which means that our original sine wave did not only contain a fundamental wave at 440Hz. The other spikes are considered to be noise. This way we can measure the signal to noise ratio (SNR). We take a look at the amplitude of our signal and the highest amplitude of an unwanted spike and divide them. Easy, isn't it?

Lets take a final look at a transformed signal. Below is the Fourier transform of a logarithmic chirp (logarithmically swept sine wave) from 440Hz to 48kHz which was run through a resampler.

We can see that the start frequency is indeed around 440Hz but at about 20kHz our amplitude starts to fall off. This happened because our resampler was smart enough to filter out frequencies above the audible level (or frequencies that are higher than half of the sampling frequency) to avoid aliasing.

These test and graphs are now easily reproducible using the new and shiny 'resampler-quality-test' which generates arbitrary sine or chirp signals and runs them through the resampler (the result can be saved) and a script which plots the Fourier transform of a WAVE file.

Some more improvements were made to the already existing resampler test case but these are not so interesting.

That's all for this year's GSoC. It was a fun and productive experience. I want to take a moment and thank my mentor for this year, Peter Meerwald, for all the help and friendly exchanges during the summer, and my last year's mentor Tanu Kaskinen for the same nice treatment last year.


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