I know the white noise is uncorrelated. So a perfectly white signal has an autocorrelation equal to an impulse at zero. This implies that the corresponding signal in the time domain is any function with no correlation whatsoever between the different samples.
And I know that any colored 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. This means that in colored noise, the low frequency noise will have higher variance than the higher frequency noise.
But other than that can you give more intuition of the difference between the white and colored noise in time domain?
Lets say I have two exact same signals and I add white noise to one and colored noise with the same total noise power to the other. Does that mean in the signal which I added colored noise the DC part of the signal experiences a noise with higher average variance than the AC part of the signal?
If I use a white noise of the same power instead of colored noise, then what do I lose and how bad this approximation would be in a system simulation?