Question about calculation of Autocorrelation for 1/f noise I obtained time records of 1/f noise by different methods (filtered white noise, Voss-Mcartney method):
Flicker Noise
I plot the PSD and it does look like 1/f (the slope is right). I know that, if 1/f noise wasnt "real" but generated from filtered white noise with a pole at very low frequency (-10 dB/dec), the autocorrelation should be a decreasing exponential:
$$
f(t)=\frac1{2\pi}\int_{-\infty}^\infty\frac{B} {a+\mathrm{j}w}\mathrm{e}^{\mathrm{j}wt}\,\mathrm{d}w\,=B\mathrm{e}^{\mathrm-at}h(t)
$$
The closer the pole is to zero (tends to 1/f) the longer the decay time of the exponential.
To my astonishment, when I calculate the autocorrelation from the time records that are used to to plot the PSD I dont obtain an exponential but something that has the shape of and inverted logarithm:
Autocorrelation (The calculated autocorrelation value is in blue). The
equation of the orange curve is: $$-0.115log(\tau) + 1.05 $$
To calculate the autocorrelation I use the simple formula of doing the average of the product of points separated tau for the complete record, I also used plot_acf function in python to check my result and it matched perfectly.
I dont understand why the autocorrelation is not a decaying exponential, I tried with different methods of generating 1/f noise and the result is the same, any ideas?
 A: Tricky subject. I have done quite some biblio search in the past on this subject. I obtained similar numerical results as well, and I can share some clues.
In short:

*

*I think your derivation of the exponentially decaying autocorrelation is at least partially incorrect/inappropriate;

*The autocorrelation you get numerically, with a logarithmic dependence on time... is GOOD, I guess?! I can give you a reference.

Problem with exponential decay. I am not sure you can just "low pass" a PSD like that. I see your general idea... you have a low-pass filter with response function $1/(1+j\omega\tau)$, etc. This is something you can do with the Fourier transform of a signal, not with the PSD. Also, remember the PSD is by definition a real function, so that denominator with an imaginary part seems just wrong to me. Maybe this makes sense in some limit? I am not sure, I am doubtful that the PSD you wrote is a legitimate one, in any sense.
Log time dependence. Some time ago I found a very interesting reference reporting exactly that trend. Look here => "Metrology and 1/f noise arXiv:1407.7760v6", equation (27). The paper has been later published in the journal "Metrologia" (2015), have a look. In my view, it contains an unusually clear discussion about flicker noise. Simplifying a bit, a better derivation for the autocorrelation is
$$R(t) = \int_0^\infty \frac{A\cos(\omega t)}{\omega}d\omega,$$
which is however divergent. This is connected to the fact that the variance of flicker noise will diverge over an infinite time span. One way to remove this is to take a cut-off $\omega_c \approx 1/T$, where $T$ is the total measurement time. Indeed, any flicker component at a frequency below $\omega_c$ will look like a constant over the duration of the measurement. If you now consider
$$R(t) = \int_{\omega_c}^\infty \frac{A\cos(\omega t)}{\omega}d\omega \approx const-A\log(t)$$
you indeed obtain that log dependence. In principle, you should get minus the integral of $\cos(x)/x$ evaluated in $\omega_ct$, which is different from a log. However, since $\omega_ct<1$ you can show that the function is well approximated by the log function.
Tricky subject.
PS if anyone has further good references/books discussing these details... I would be really glad to know. I have found it extremely hard to find good references. Thanks in advance!
