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From: dbd <dbd@ieee.org>
Newsgroups: comp.soft-sys.matlab
Subject: Re: how to reverse windowing effect?
Date: Mon, 19 May 2008 23:02:47 -0700 (PDT)
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On May 19, 4:59 pm, "ggk giust" <ggkm...@comcast.net> wrote:
> Greetings,
>
> I'd like to apply the blackman-harris window to my real-
> time data array, before taking its FFT and applying a
> frequency-domain filter to it.
>
> My question is, when I then do the inverse FFT, how do I
> reverse the effect of windowing (in this case using
> blackman-harris 4 term, -92db) so that the resulting real-
> time waveform is as accurate as possible?
>
> Is there some equation I can simply apply? I see a lot of
> references to windowing in the literature, but no
> information about how to reverse the effect of windowing
> when going back from the frequency to the time domain.
> Looking for practical advice how to compute this.
>
> Best regards, -GGK

A few thoughts:

1) There is no need to apply a window just to implement a frequency
domain filter.

2) If you want to generate a windowed power spectrum from the data you
are filtering, apply the window in the frequency domain after the FFT
and after pulling off the unwindowed frequency domain data for the
filter processing. The window you suggest belongs to the family of
cosine-summed filters that can be applied as small convolutions in the
frequency domain.

3) Why are you considering the 4-term -92 db window instead of the
correctly optimized 4-term -98 dB version Nuttall published in a
correction to the harris paper?

4) Windows can be applied as a matrix multiply in the frequency domain
after the FFT. For the cosine-summed windows, this matrix has only
near diagonal elements. For a discussion of the conditions under which
the matrix is invertible and can be used to remove the windowing see
section V of the article:
Practical methods for rapid and accurate computation of
interferometric spectra for remote sensing
applications
Barnet, C.D.   Blaisdell, J.M.   Susskind, J.
Gen. Sci. Corp., Beltsville, MD, USA ;
Geoscience and Remote Sensing, IEEE Transactions on
Jan. 2000   Volume: 38 , Issue: 1   page(s): 169 - 183

5) For examples of the use of windows in a transform-modify-
inverse_transform context, look at pre and post processing used in
audio compression schemes like MP3. Also look in the music analysis/
synthesis literature.

I hope that will give you a few ideas to consider.

Dale B. Dalrymple
http://dbdimages.com