Basic Power Spectral Density question

Hi,

I'm looking for a more fundamental answer here. I'm starting to learn
about PSD and autocorrelation etc., and while time domain makes sense to
me, I have trouble imagining what Fourier tranform does to the
autocorrelation function except tranforming it the frequency domain. Or
put differently, if PSD shows how is the power (mean square value) spreads
through different frequencies,  then what frequencies are we talking about
here? Frequencies of what? Is there a less abstract way of seeing where
these freqs. are coming from? In addition, what do negative freqs.
represent? 

Thanks a lot!
		
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sergio wrote:
> Hi,
> 
> I'm looking for a more fundamental answer here. I'm starting to learn
> about PSD and autocorrelation etc., and while time domain makes sense to
> me, I have trouble imagining what Fourier tranform does to the
> autocorrelation function except tranforming it the frequency domain.

That is correct.  Taking the Fourier transform of an autocorrelation 
function takes the time-domain information and transforms it into the 
frequency domain.  You aren't adding or subtracting information, you're 
only looking at it from a different point of view.

> Or
> put differently, if PSD shows how is the power (mean square value) spreads
> through different frequencies,  then what frequencies are we talking about
> here? Frequencies of what? Is there a less abstract way of seeing where
> these freqs. are coming from? 

Usually you're taking an autocorrelation in time, so the frequencies 
you're talking about are the frequencies of the component sinusoids in 
the autocorrelation function.  Without a concrete problem to deal with 
there isn't a less abstract way of seeing where the frequencies are 
coming _from_, but one concrete way of thinking where the frequencies 
are going _to_ is to imagine your signal impinging on a receiver with a 
bandpass filter -- only those frequencies in the PSD that can get 
through the filter do, the rest are absorbed or reflected by the filter.

> In addition, what do negative freqs.
> represent? 

A sine wave just bounces around, up and down in real-number space, so 
the sign of the frequency is more or less meaningless.  When you take 
the fourier transform using e^{jw}, however, e^{jw} has a constant 
absolute value, and the number inscribes a circle on the complex plane 
at a certain speed (frequency) and direction (sign).

If you're working with purely real, baseband data, then all of the 
information about the signal is contained in the positive-frequency half 
of the transform -- the negative-frequency half is constrained by the 
"purely real" requirement to have a certain relationship to the positive 
frequency half.

When you modulate data for transmission, however, it is handy to be able 
to throw away one of these halves and only transmit the other -- see 
"single sideband" for information.  In DSP it becomes very convenient to 
generate a complex signal from a real one internally in the software -- 
then keeping track of negative frequencies and other esoterica makes 
life more convenient, overall.
> 
> Thanks a lot!
> 		
> 		This message was sent using the Comp.DSP web interface on DSPRelated.com


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Tim Wescott
Wescott Design Services
http://www.wescottdesign.com