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Re: I Need Help..

From: Mike Miller
Subject: Re: I Need Help..
Date: Mon, 25 Nov 2013 12:17:20 -0500

On Mon, 25 Nov 2013 10:35:14 +0000, Damian Harty wrote:
>>> 1)      Why only use the real part of the results?
>> Assuming the original signal is real, taking the real part of the inverse 
>> FFT and multiplying by 2
>> is essentially restoring the spectral symmetry of the FFT. Basically because 
>> the low-pass
>> filtering should have preserved the 8 conjugate-symmetric bins at the end of 
>> the array.
> But a "real" signal (I'm unaware of any real world processes that generate 
> imaginary numbers in the time domain, if I'm honest) generates imaginary 
> components of the FFT depending on the phasing of the sine component? So 
> sin(wt) gives an imaginary coefficient and cost (wt) gives a real 
> coefficient, sin (wt+phi) gives a complex coefficient depending on phi.
> So it's still not really clear to me why slice off the imaginary part of the 
> signal. Multiplying by 2 satisfies Parseval's theorem and is all good, from 
> my point of view - but taking only the real component (as distinct from the 
> magnitude of the complex number) manifestly doesn't satisfy Parseval and 
> represents some kind of "polarizing filter" on the data, letting through 
> components with a phase angle close to pi/2 unhindered but completely 
> blocking components with a phase angle of zero Without more context it's 
> difficult to understand if it's deliberate and very clever or just a bit 
> muddled.

Because the type of low-pass filtering being done in this script is
discarding the conjugate symmetric frequency components, so it is
effectively converting the real input into an analytic output (with
half the power). And the way to recover the real signal from an
analytic signal is to simply take the real part. Multiplying by 2
accounts for the fact that half the power was lost when discarding the
components from Fs/2 to Fs. But since the DC component doesn't have a
symmetric component, it has now been doubled.

And as the OP followed up that what looked like a second signal was
really the same as the original signal, taking away the mean of the
original is actually correcting for the DC component being doubled, so
that clears that up.


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