posted 26 May 2001 01:49            

windowing.kym

The attached sound file contains three example Sounds demonstrating an effect I ran into today during my work...

Whenever one does FFT based filtering, it is usually a good idea to "window" your data, especially when you are trying to get a spectral estimate. But I had assumed all along that windowing of data for the purposes of convolution -- i.e., FFT based filtering, would be a bad idea since the result would have the window shape imposed on it.

Back a while ago I discovered that the SSC technique of using a Bristow-Johnson window on the output of the FFT filtering was a pretty clever idea. The Bristow-Johnson window shares the property along with the triangular Bartlett window, and the Hann window, of being a perfect mixer. A perfect mixer is a window that always sums to one for any sample of the window and a sample taken a half-period away.

I knew a long time ago that windowing both input and outputs was not such a good idea if you want spectral purity.

But I found out today, that it is actually better, for streamed data, i.e., sounds, to put the windowing ahead of the FFT blocks. The attached sounds try to demonstrate these effects. You can get a better demonstration by recording a sample of the outputs and examining with a better spectrum analyzer.

The reason that pre-windowing works better can be seen by considering that the default window for chopped data sections is a rectangular window. Discontinuities at the edges of these blocks excites high-frequency energy, in the FFT analysis, that really isn't in the data.

By pre-windowing we can prevent this spurious excitation. Using a Bristow-Johnson window is desirable here, since the window is a perfect mixer, and so the results when combined by addition at the output of the inverse FFT, using 50% overlap between successive blocks, reconstructs the signal correctly.

The net result is significantly reduced spurious sidebands at the output.

[On my system here, a recording made of the windowing after the FFT with a 1 KHz signal shows spurious sidebands at around 1.5 KHz with a level about 48 dB down from the signal peak amplitude.

Meanwhile the same signal recorded using windowing ahead of the FFT's shows that the next sidebands, just below the carrier frequency, have levels more than 78 dB down from the signal peak.]

FWIW!

- DM