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How are windowing functions' coefficients extracted?

Started by Vahid Raissi 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,

Read the old (ancient?) text:

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 http://home.earthlink.net/~yatescr
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-]