# How are windowing functions' coefficients extracted?

Started by March 31, 2004
```Hi all,

I just wonder why for example Hamming has chosen 'that' special set of
coefficient for his windowing function, or any other guy such as
Kaiser, Van Haan etc. Is there any mathematical reasoning behind these
coefficients or are they just 'discovered' by performing several
tests?

Vahid
```
```interesting question that i had at another time in my life.

In article 961ee3fc.0403310805.4d8992d9@posting.google.com, Vahid Raissi at
vraissi@hotmail.com wrote on 03/31/2004 11:05:

> I just wonder why for example Hamming has chosen 'that' special set of
> coefficient for his windowing function, or any other guy such as
> Kaiser, Van Haan etc. Is there any mathematical reasoning behind these
> coefficients or are they just 'discovered' by performing several
> tests?

in a sense, by the latter method.  Hamming wrote a book called "Digital
Filters" that is useful, to me, only for interesting historical context.  in
it, he describes his thought process a little, noticing that the sidelobes
of a rectangular window of double width where mostly opposite polarity of
the sidelobes of the Hann window.  so he programs a computer (they had those
back in the 50s and 60s) to plot the height of the biggest sidelobe vs. some
parameter i'll call "a" (the deviation from the Hann coefs).

W(f) = (1-a)*Hann(f) + a*rect(f/2)

= 0.5*( (1+a) + (1-a)*cos(pi*x) )

i think i did that right.  the a comes out to be about 0.08 but not exactly
(it was also a function of the window length, N, but settled down as N got
very large).

```
```Vahid,

Fredric J. Harris, "On the Use of Windows for Harmonic Analysis with the
Discrete Fourier Transform," Proceedings of the IEEE, Vol. 66, No. 1,
January 1978].

--Randy

vraissi@hotmail.com (Vahid Raissi) writes:

> Hi all,
>
> I just wonder why for example Hamming has chosen 'that' special set of
> coefficient for his windowing function, or any other guy such as
> Kaiser, Van Haan etc. Is there any mathematical reasoning behind these
> coefficients or are they just 'discovered' by performing several
> tests?
>
> Vahid

--
%  Randy Yates                  % "And all that I can do
%% Fuquay-Varina, NC            %  is say I'm sorry,
%%% 919-577-9882                %  that's the way it goes..."
%%%% <yates@ieee.org>           % Getting To The Point', *Balance of Power*, ELO
```
```On 31 Mar 2004 08:05:17 -0800, vraissi@hotmail.com (Vahid Raissi)
wrote:

>Hi all,
>
>I just wonder why for example Hamming has chosen 'that' special set of
>coefficient for his windowing function, or any other guy such as
>Kaiser, Van Haan etc. Is there any mathematical reasoning behind these
>coefficients or are they just 'discovered' by performing several
>tests?
>
>Vahid

Hello Vahid,

the Hamming and Kaiser windows were developed
using some fairly complex mathematics.
I don't remember the history anymore, but the
Kaiser window was developed with with *much*
mathematical reasoning.  kaiser did *not* accidently
come up with a window function that
approximates prolate spheroid functions using
Bessel functions.

[-Rick-]

```