I understand that the time domain representation of white noise looks like impulses.
What do their autocorrelation functions look like? (for color noise)
Actually the time domain representation of white noise doesn't necessarily look like impulses. For all signals, the autocorrelation and the power spectral density form a Fourier transform pair (Wiener–Khinchin theorem). This implies that a perfectly white signal has an autocorrelation equal to an impulse at zero. The corresponsing signal in the time domain is therefore either the impulse function itself, or any other function with no correlation whatsoever between the different samples. Whether they are Gaussian distributed, 1 and -1, or whatever: as long as they are "random" the signal will be white noise. Typically the distribution is assumed to be Gaussian, though.
Any coloured noise can be thought of as filtered white noise, so this will still look like a somewhat random signal, but with less "sharp" transitions between the samples because the high frequency components have been filtered out.
The most defined Noise after White Noise is Red (Brownian) Noise. For Which you can easily find out its statistical properties by some predefined process such as "Ornstein-Uhlenbeck process" that is also called "OU" Process .
For the Detail Description of OU process i have attached one snapshot which will hopefully let u to visualize the colored noise in time domain.
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