# What's a “Fourier filter”?

E.g. the constant Q-transform is built by adding so called "Fourier filters".

What's a "Fourier filter"?

People (usually from fields outside signal processing) sometimes use the term Fourier filter for a filtering operation in the FFT domain, which simply works by multiplying the FFT bins of a signal with a given filter function (often just ones and zeros, corresponding to pass bands and stop bands, respectively). Why this is generally not such a good idea is explained here.

Also in Computer Vision, the term Fourier filter is used as explained above.

In the document you linked to in a comment, the term is used to describe the computation of the Discrete-Time Fourier Transform (DTFT) at a given frequency from a finite length portion of a signal. This computation can be interpreted as a filtering operation, because it is a sum of products. The corresponding filter is a band pass filter with center frequency equal to the given DTFT frequency. More more information on the filter interpretation of the D(T)FT have a look at this page.

• See my comment on the question. – mavavilj Dec 7 '15 at 12:04
• Should those filter functions need to be transformed (presumably into Frequency domain) as well before multiplying with the FFT bins of a signal? I am beginner in applying theory into practice. Please bare with trivial things. – jomegaA Feb 10 '20 at 12:22
• @jomegaA: Normally, they are very simple, so they can be "designed" directly in the frequency domain (e.g., they might only consist of 1's and 0's, as mentioned in my answer). – Matt L. Feb 10 '20 at 12:24
• @MattL. Ok. If I choose Hann window for example, I must transform the window into frequency domain before multiplying with FFT bins of signal, presume I work in Frequency domain. In time-domain I convolve the signal with window to get the output. – jomegaA Feb 10 '20 at 12:27
• @jomegaA: A window is not meant to be a filter; usually you multiply your time domain signal with a window, but you don't filter it with a window. – Matt L. Feb 10 '20 at 12:43

Each bin of a DFT or FFT result can be considered to be the result of a filter operation. Each filter's frequency response is either Sinc shaped, or if a non-rectangular window is applied before the FFT, the response will be shaped like the transform of that window, with a passband width roughly proportional to from Fs/N to twice that.

Computing a single unwindowed DFT bin is nearly identical to a Goertzel filter. So that's yet another potential synonym for the same thing.