Why is it a bad idea to filter by zeroing out FFT bins?

Question

It's very easy to filter a signal by performing an FFT on it, zeroing out some of the bins, and then performing an IFFT. For instance:

t = linspace(0, 1, 256, endpoint=False)
x = sin(2 * pi * 3 * t) + cos(2 * pi * 100 * t)
X = fft(x)
X[64:192] = 0
y = ifft(X)

The high frequency component is completely removed by this "brickwall" FFT filter.

But I've heard this is not a good method to use. Why is it generally a bad idea? Are there circumstances in which it's an ok or good choice?

[as suggested by pichenettes]

Answer 1

Zeroing bins in the frequency domain is the same as multiplying by a rectangular window in the frequency domain. Multiplying by a window in the frequency domain is the same as circular convolution by the transform of that window in the time domain. The transform of a rectangular window in the Sinc function (sin(wt)/wt). Note that the Sync function has lots of large ripples and ripples that extend the full width of time domain aperture. If a time-domain filter that can output all those ripples (ringing) is a "bad idea", then so is zeroing bins.

These ripples will be largest for any spectral content that is "between bins" or non-periodic in the FFT aperture width. So if your original FFT input data is a window on any data that is somewhat non-periodic in that window (e.g. most non-synchronously sampled "real world" signals), then those particular artifacts will be produced by zero-ing bins.

Another way to look at it is that each FFT result bin represents a certain frequency of sine wave in the time domain. Thus zeroing a bin will produce the same result as subtracting that sine wave, or, equivalently, adding a sine wave of an exact FFT bin center frequency but with the opposite phase. Note that if the frequency of some content in the time domain is not purely periodic in the FFT width, then trying to cancel it by adding the inverse of an exactly periodic sine wave will produce, not silence, but something that looks more like a "beat" note (AM modulated sine wave of a different frequency). Again, probably not what is wanted.

Conversely, if your original time domain signal is just a few pure unmodulated sinusoids that are exactly periodic in the FFT aperture width, then zero-ing FFT bins will remove the designated ones without artifacts.