Why not use Fourier transforms for filtering?

Question

Are there any resources for constructing filters using discrete Fourier transforms (DFTs)? Or is it as easy as it seems. For example, for lowpass filtering I obtain the DFT of my signal, zero out components above the selected cutoff frequency and then transform back to get my signal minus the high frequency (noise) components. It seems to behave correctly for the signal I've applied it to, but am I missing something. Upsides would appear to be (1) no time delay between the input and output and (2) the filter is sharp with no transition band. Downside is that this method is slower than using impulse filters, but are there other upsides, downsides, or considerations I'm missing?

Answer 1

The technique of filtering using frequency domain truncation and IFFT is used in specific areas where computation time is essential (to avoid the time domain convolution with a long impulse response), like image processing. However the technique also has some downsides you should be aware of.
The frequency domain filtering performed on a data record is totally equivalent to a circular convolution made on your time signal with the impulse response of your ideal low-pass filter (typically a sin(x)/x function). The fact what we are doing a circular convolution will result in unwanted artifacts and you'll see some "parts" of the beginning of your signal at the end of the filtered signal and vice-versa.
Try for example to create a short signal (like a sharp pul
se) and extend it with zeros. After filtering you will see a sort of "pre-shoot" of your original signal at the very end of your signal. If you then rotate your time signal you'll discover that the signal is actually continuous (in a circular way).
You can reduce the problem by, for example, adding "some" zeros at the beginning and at the end of your signal (zero-pad) before your filtering operation (before the FFT) and then remove these additional samples again from you filtered signal (after the IFFT).