FFT windowing time and frequency resolution

According to you, some windowing functions have less frequency bleed, true, because they are closer to either Gaussian or Sinc (both infinite, so only approximations; only ‘flat top’ approximates Sinc).

But you also said that windowing reduces the frequency resolution. Indeed, because that’s a side effect. Window size determines the width of FFT, not the area. Windowing functions concentrate time near the middle, similarly to a narrow window size except that it’s smoother. This both increases time resolution and decreases frequency resolution.

If you are asking a question, it is not clear what your question is.

I’m talking about the fact that you biasingly said that windows decrease frequency resolution but didn’t say about the compensated time resolution increase!

Are you referring to something in the Audacity manual? If so, please add a link to the relevant page of the on-line manual and quote the exact text so that I can find it on the page.



“Changing to the Blackman-Harris Window Type gets rid of much of the spectral leakage at the expense of lower frequency resolution (note that the red band near the 2.0k mark is wider).”

“Changing to a rectangular window causes the track to be redrawn a little faster at the expense of very bad spectral leakage. However, the frequency resolution is better (the red band near the 2.0k mark is narrower).”

Note that it talks about lower frequency resolution, but nothing about the higher time resolution.

See higher up the same page: Time Smearing and Frequency Smearing

Clearly such an issue will be reduced with a window that concentrates itself near the middle more!

Let’s say that height there is the volume. With a rectangle window, there is smear:

_ + _ = ––

With a triangle window, it’s reduced:

–^–– + ––^– = –^^–

It forms 2 peaks because triangle concentrates itself near the middle more than rectangle. Triangle will also have worse frequency resolution because of that concentration. So aside from “spectral leakage”, the other factor seems to be just another time–frequency tradeoff. Am I missing anything?

OK, I think I see what you are saying. Are you asking if window types that give less smearing in the frequency domain, give more smearing in the time domain, and vice verse?

The answer is that the window type has little effect on the time resolution. It’s the size of the window that has the major effect on the time resolution. Here are some comparisons:

Your time observations make sense, because Blackman Harris doesn’t have 2 times less area than a Triangle. 512 does have 2 times less area than 1024. The smoother windowings produce time leakage to a small extent, with a theoretical infinite sinc window producing bad time leakage but no frequency leakage? Rectangle 2048 would have no “time leakage” and Triangle 4096 would have some “time leakage”. Both have the same area, but Triangle produces less frequency spectral leakage because it doesn’t cut abruptly.