Saturday, November 2, 2013

Little bit on FFT based filtering and leakage

* There are no brick wall filters.  Basically all digital filters have ripple in the stopband due to the basis in the sinc (sin(x)/x) function.

*  FFT's also have aliased ripple out past their 'stop' band; this is controlled with the windowing function used.

*  The less the stopband ripple, the wider (in terms  of bins) the passband is (for  any given bin)

*  The reason this is so is due to the way discrete Fourier transforms (including the FFT) assume that the data being transformed is periodic; the problem occurs when the signal period doesn't  end exactly on the DFT's boundaries.

*  The response of the default rectangular window for all frequencies except multiples of the FFT frequency (sampling rate divided by number of points) is the sinc function, and the ripple out of band (out of each bin's band) is 12dB down for the first sidelobe, and the sidelobes roll off at the leisurely rate of 6dB per octave. This is the 'tail' you see on narrow spectral features in FFT's.

* Adding points on an FFT is basically the same thing as adding taps on a filter, both computationally (in terms of CPU load) and spectrally.

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