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Discussion Groups | Audio Signal Processing | Question about cross-spectrum and convolution

Technical discussions related to Audio Signal Processing (digital effects, acoustics, noise reduction, musical signal processing, etc).

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Question about cross-spectrum and convolution - Ze Ji - Dec 19 11:11:00 2005

Hi, I have a question regarding the concept of cross-spectrum, which is defined as the Fourier transform of convolution between two signals X and Y:

FT(X*Y)

Isn't it the same as:

FT(X) FT(Y)

My simple clue is:

X*Y = IFT( FT(X) FT(Y) )

So, FT(X*Y) = FT(X) FT(Y).

Am I right?

Thanks

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Re: Question about cross-spectrum and convolution - Jeff Brower - Dec 31 5:51:00 2005

Ze Ji-

> Hi, I have a question regarding the concept of cross-spectrum, which is
> defined as the Fourier transform of convolution between two signals X and
> Y:
>
> FT(X*Y)
>
>
> Isn't it the same as:
>
> FT(X) FT(Y)
>
> My simple clue is:
>
> X*Y = IFT( FT(X) FT(Y) )
>
> So, FT(X*Y) = FT(X) FT(Y).
>
> Am I right?

Yes right, and correlation amounts to division in the frequency domain;
i.e. transfer function calculation.  Easy to conceive, but there are lots
of details to actual implemention in a real-time system that operates on
continuous data.

For use of FFT to accomplish frequency domain processing in real-time
systems, here is a good book that takes out the guesswork:

  The Fast Fourier Transform and its Applications
  E Oran Brigham
  ISBN 0-13-307505-2

-Jeff

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