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Let's say I have a noise power-spectral-density (PSD) which is not flat and ranges from 0 to $f_1$ Hz in frequency. As we know, the total area under the PSD is equal to the total average power of the signal (according to Parseval's theorem). Let's say this value is $\sigma^2$. This means the value of the auto-correlation function at zero-time (which is the total power in the signal) will also be equal to $\sigma^2$.

Now assume we want to approximate the above colored noise with a band-limited (from 0 to $f_1$) white Gaussian noise with the same noise power (i.e. $\sigma^2$). That means the variance of the AWGN is $\sigma^2$, the auto-correlation function would be a delta function (i.e. $\sigma^2 * \delta(t)$) or more precisely a sinc() function, and the PSD would be flat (equal to $\sigma^2$) from 0 to $f_1$. The total area under the PSD should be equal to the total average power of the noise. The total area under the PSD in this case simply would be $\sigma^2 * f_1$ which is not equal to the initial assumption of the total power $\sigma^2$. What am I missing here? Should I consider the flat PSD value is $\sigma^2/ f_1$ instead of $\sigma^2$ (however this does not satisfy the auto-correlation vs PSD relation)?

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    $\begingroup$ Note that true white Gaussian noise (with delta autocorrelation) has no variance. What you're looking at is filtered white noise, whose autocorrelation is indeed a sinc, and has a well-defined variance. The PSD should be equal to $\sigma^2/f_1$. $\endgroup$ – MBaz Sep 18 '18 at 22:24
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I think what your missing is that the Power Spectral Density, as a density is the power per unit frequency, so assuming units of Hz as is typically done, $\sigma^2/f_1$ in your case is the power per Hz; and therefore to get the total power, $\sigma^2$, you would multiply that by the total frequency $f_1$ resulting is the expected $\sigma^2$.

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