# What kind of power spectrum implies uncorrelation in time?

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?

• 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? May 21, 2019 at 15:45
• 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... May 21, 2019 at 15:53
• 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 May 21, 2019 at 16:18
• @Florian care to post that as answer to this question? I think user43292 would in fact accept that as answer :) May 21, 2019 at 16:49

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 $$1/B$$.