sqrt in time - effect in freq?

This relates to an earlier post - "filter of amplitude versus filter
of (I,Q)?".

I would like to learn what is the effect in the frequency domain of
taking the square root of a quantity on the time domain. Also, of
taking the square (actually magnitude of a complex number) in the time
domain?

I think I know the second - the magnitude is number*complex_conjugate
which is autocorrelation of the frequency spectrum - but I dont know
the first.

Thanks,

Chris
==================
Chris Bore
BORES Signal processing
www.bores.com
chris@bores.com

Chris Bore wrote:
> This relates to an earlier post - "filter of amplitude versus filter
> of (I,Q)?".
> 
> I would like to learn what is the effect in the frequency domain of
> taking the square root of a quantity on the time domain. Also, of
> taking the square (actually magnitude of a complex number) in the time
> domain?
> 
> I think I know the second - the magnitude is number*complex_conjugate
> which is autocorrelation of the frequency spectrum - but I dont know
> the first.

I don't have my notes from '55 and I won't do it over, but I can quote a 
result that might be relevant.

If you remove one sideband from an ordinary double-sideband signal 
(leaving the other sideband and the carrier intact) the demodulated 
result will be increasingly distorted as the modulation percentage 
increases. One way to achieve low-distortion is increasing the carrier 
amplitude ("exalted carrier operation") to reduce the modulation 
percentage. A more interesting way is using a square-law detector, 
effectively squaring the signal.

To a first order of approximation, removing one of the sidebands 
effectively creates the square root of the modulation. (That's not to 
imply that using the square root of the modulation will eliminate a 
sideband! :-) )

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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Chris Bore schrieb:
> This relates to an earlier post - "filter of amplitude versus filter
> of (I,Q)?".
> 
> I would like to learn what is the effect in the frequency domain of
> taking the square root of a quantity on the time domain. Also, of
> taking the square (actually magnitude of a complex number) in the time
> domain?
> 
> I think I know the second - the magnitude is number*complex_conjugate
> which is autocorrelation of the frequency spectrum - but I dont know
> the first.
> 
> Thanks,
> 
> Chris
> ==================
> Chris Bore
> BORES Signal processing
> www.bores.com
> chris@bores.com


Hello,

your right on the part with the multiplication. This is basically a 
convolution of the signal in the frequency domain with itself. When 
using sinoids this has the same effect like a mixer which doubles the 
carrier frequency.
However taking the square root is something much more complicated, since 
the considered signal neither represents a linear time invariant nor a 
linear frequency invariant system and therefore I am afraid there does 
not exist a closed form solution for the Fourier Transform. So you have 
to do try it numerically or use a linearization operation (taylor 
series) for the specific signal.

Greetz,

Sebastian Doht