E.g. the constant Q-transform is built by adding so called "Fourier filters".
What's a "Fourier filter"?
$\begingroup$
$\endgroup$
2
-
$\begingroup$ Here (doc.ml.tu-berlin.de/bbci/material/publications/Bla_constQ.pdf) the Fourier filter is claimed to be $\sum_{n<N}x[n]e^{-2\pi i nz /N}$. $\endgroup$ – mavavilj Dec 7 '15 at 12:03
-
$\begingroup$ I've updated my answer, see below. $\endgroup$ – Matt L. Dec 7 '15 at 12:34
Add a comment
|
$\begingroup$
$\endgroup$
5
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.
-
-
$\begingroup$ 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. $\endgroup$ – jomegaA Feb 10 '20 at 12:22
-
$\begingroup$ @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). $\endgroup$ – Matt L. Feb 10 '20 at 12:24
-
$\begingroup$ @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. $\endgroup$ – jomegaA Feb 10 '20 at 12:27
-
$\begingroup$ @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. $\endgroup$ – Matt L. Feb 10 '20 at 12:43
$\begingroup$
$\endgroup$
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.
