# Is there a closed form expression for main-lobe width increase given a window?

We know that when we window a signal, we increase the main-lobe width. Let 'main-lobe-width' here be the null-to-null bandwidth of the main lobe. Let us further more say that the main-lobe width of a square window is '1'.

What I would like to know, is given a particular symmetric window, is there a closed form solution for how much the main-lobe will increase by, relative to that of a square window?

I can look up this percentage increase for various windows just fine by the way. I am asking if there is a closed form solution for what the percentage increase is, given an arbitrary symmetric window.

• Do you want a closed form solution for the first null in the spectrum of an arbitrary window function? I think there is insufficient information to determine that. If all we know is that the window symmetric (even function) we can say for sure that the Fourier transform is symmetric. In theory, one could derive formulas for each window, case-by-case. – Atul Ingle Mar 4 '14 at 17:29
• @AtulIngle Yes, closed form for position of first null would do as well. The only constraint we have is that the window is symmetric. (Like many of the classical windows we see, hamming, hanning, etc). – TheGrapeBeyond Mar 4 '14 at 17:37
• Isn't this similar to asking if there is a closed form solution to the FT of a arbitrary symmetric function, other than the FT? – hotpaw2 Mar 4 '14 at 18:35
• @hotpaw2 Why 'other than the FT'? Closed form soln to FT of an arbitrary FT for symmetric function, yes. – TheGrapeBeyond Mar 4 '14 at 18:40