# 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

## 1 Answer

If I understand you correctly, you are in fact asking for a closed-form solution for the position of the first zero of a given window function. Such a solution cannot exist, because what you would need is an analytic solution to a polynomial equation, which does not exist for polynomials of degree 5 or higher (that's the Abel-Ruffini Theorem).

• I had never heard of this theorem before, thanks! Let me then re-phrase and make it simpler: Let us say you are given the classical hamming window. How then, did/do people determine the location of its first zero? Tables exist that give the location of the first zero, but how did they get it? Are you saying that they must have found it numerically? Thanks again! – TheGrapeBeyond Mar 5 '14 at 13:07
• It depends on the window. For rectangular and triangular windows, you can easily compute their Fourier transform and determine their first zero-crossing exactly. For the raised-cosine family of windows (Hamming, Hann, Blackman, and a few others), it is also possible to find an analytic expression for their Fourier transforms. I do not know if it's possible to get an exact value for the first zero-crossing from these Fourier transforms. Probably not, because tables usually give "approximate main lobe widths". – Matt L. Mar 5 '14 at 13:36
• MattL, I then suppose that for more complicated windows, we have to literally just 'compute and see'? That surprises me to some extent, but it would seem from all this that this is indeed the case. – TheGrapeBeyond Mar 5 '14 at 13:48
• I'm afraid that's the way it seems to be. – Matt L. Mar 5 '14 at 13:54