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I am new in signal processing field, and actually I just need some basic tools for what I am working on. I've learnt that the autocorrelation function is the inverse Fourier transform of the signal's power spectrum, and that if the power spectrum is flat then the autocorrelation function is the Dirac delta, so that the signal is uncorrelated in time. But what if the power spectrum is constant only within the range of the cutoff frequencies? In particular, I am given the following power spectrum for a signal $x(t)$:

$ S_{xx}(f)= \frac{1}{2(f_u-f_l)}$, for $\vert f\vert \in [f_l,f_u]$,

where $f_u$ and $f_l$ are the upper and the lower cutoff frequency respectively.

Does the uncorrelation still hold? If not, does the signal satisfy any property anyways?

Thank you in advance.

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  • $\begingroup$ well, you already know what to do to get the autocorrelation function to that power spectrum. So, apply that Fourier transform to that rectangular spectrum. What do you get? $\endgroup$ – Marcus Müller May 21 '19 at 15:45
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    $\begingroup$ I guess you may have to distinguish continuous (analog) and discrete (digital) signals. For analog signals, the autocorrelation function is a Dirac if and only if the spectrum is flat for all frequencies. Typically though what we care about is whether or not our digital samples are uncorrelated. It is easier to safisfy this. If your low-pass filter before your ADC is, say an ideal one, your ACF will be a (squared) sinc. If you get things right, your samples are in the zero crossings of the sinc and the discrete ACF looks like a discrete delta... $\endgroup$ – Florian May 21 '19 at 15:53
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    $\begingroup$ Thank you very much! Yes, when applying the Fourier transform I get the sinc function but I didn't think about to sample my signal only in the zero crossings of the sinc function! This way I can look at the signal as it is uncorrelated :-) Thank you @Florian $\endgroup$ – user43292 May 21 '19 at 16:18
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    $\begingroup$ @Florian care to post that as answer to this question? I think user43292 would in fact accept that as answer :) $\endgroup$ – Marcus Müller May 21 '19 at 16:49
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Since this does seem to answer the question, I'm reposting my comment as an answer:

I guess you may have to distinguish continuous (analog) and discrete (digital) signals. For analog signals, the autocorrelation function is a Dirac if and only if the spectrum is flat for all frequencies. Typically though what we care about is whether or not our digital samples are uncorrelated. It is easier to safisfy this. If your low-pass filter before your ADC is, say an ideal one, your ACF will be a sinc. If you get things right, your samples are in the zero crossings of the sinc and the discrete ACF looks like a discrete delta...

As the OP mentioned, the trick is to sample the signals in the zero crossings of the sinc function. For an ideal (rectangular) lowpass filter with (two-sided) bandwidth $B$ this means sampling exactly at a rate $1/B$.

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  • $\begingroup$ I would have another little question, taking the inverse Fourier transform of the rect function I find the sinc function, not a squared sinc. Am I missing anything? $\endgroup$ – user43292 May 22 '19 at 16:15
  • $\begingroup$ You're right, it's a sinc, not a squared sinc. I was thinking of another related problem. $\endgroup$ – Florian May 23 '19 at 7:21
  • $\begingroup$ Ok, thank you very much again for your help! $\endgroup$ – user43292 May 24 '19 at 14:07
  • $\begingroup$ You're welcome! If the answer solved your problem, please consider accepting it. :) $\endgroup$ – Florian May 24 '19 at 15:03

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