methematica logic of windowing to reduce spectral leakage

[yefj]

Hello,I am trying to understand mathematicky why convolving with a window function makes spectral leakage smaller?
its not a magic thing, our data tone can be anywhere why mathematckly the convolution helps wit spectral leakage?
Thanks.

 

[D.A.(Tony)Stewart]

https://www.edn.com/windowing-functions-improve-fft-results-part-i/

https://www.edn.com/windowing-functions-improve-fft-results-part-ii/

 

[FvM]

Windowing is multiplication of time domain signal with window function of same length, not convolution.

 

[D.A.(Tony)Stewart]

By applying a window function to a signal before taking its Fourier transform, you can reduce the amount of spectral leakage and obtain a more accurate representation of the signal's frequency content. Your choice of window depends on the signal content for differences in depth and spacing of signals to enhance the weak signal above the leakage baseline and not compromise the accuracy of the result.

 

[kaz1]

By applying a window function to a signal before taking its Fourier transform, you can reduce the amount of spectral leakage and obtain a more accurate representation of the signal's frequency content. Your choice of window depends on the signal content for differences in depth and spacing of signals to enhance the weak signal above the leakage baseline and not compromise the accuracy of the result.

Some fun statements on windows:
-A window is just a scaling vector, to kill/mute the sides of another vector but keeps its central section alive.
-Any scaling vector that goes up then down is a window.
-Rectangular (boxcar) window is the only window that is not a window, yet has two names.
-Windowing is popular but there are cases when it should not be applied though you can apply boxcar window.

 

[D.A.(Tony)Stewart]

-Window shopping is when you browse for things you can't afford, while window programming is when you code without debugging.
-When you close a window, you're basically telling the computer to forget everything you just did.
-Window cleaners must have a pane-staking job.
-When it comes to computer programming, sometimes it's better to open a new window rather than trying to fix the old one.