# How the white and colored noise differ in time domain

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?

• "Colored" doesn't necessarily mean that the "high frequency components have been filtered out". It just means that there's some correlation between the samples, which might as well have been obtained from filtering white noise with a high pass filter (i.e., filtering out the low frequency components) or any other filter. – Matt L. Sep 19 '18 at 19:37

White noise looks like the plot below on the left in the time domain. A key point is no matter how much you zoom into it, it will still generally look like this. Why? Because by definition every sample (let's assume a sampled system although this is not limited to that) is uncorrelated from the next in a white noise signal. The auto correlation is therefore an impulse.

If it happens to be a Gaussian White Noise process, then it will also have a magnitude distributed as shown in the histogram on the right, but these are two different characteristics, it can be White and not Gaussian, or Gaussian and not White.

Now for comparison, below is the same result for a "Colored Noise", and in this case, Gaussian distributed process. Interestingly here the histogram of the distribution, given sufficient samples is identical, but the key point is as you zoom into the time domain plot on the left, we see the result of "memory" due to the bandwidth limiting that was done. Because of low pass filtering (in this case), the signal cannot transition from one sample to any other arbitrary location on the next sample; the location of the next sample depends on previous samples and is therefore constrained. (The more we low pass filter, the longer the effective memory depth). So there is more predictability on the next sample based on previous samples; samples are correlated to adjacent samples, and the auto-correlation function will no longer be an impulse but will be spread out with a width depending on how narrow the filtering is in frequency.

In the frequency domain, the power spectral density for the "Colored Noise" could look like the plot below; in particular this plot is the low pass filtered signal discussed above (but colored noise could be any selective band of frequencies, or even multiple bands). In contrast the white noise would have the same level of power spectral density for all frequencies. So in case this point wasn't obvious to all readers; white similar to white light that contains all frequencies vs colored with again the analogy to light in that individual colors are given by selective frequencies of light.

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?

The phase at DC is a random variable. The power spectrum represents the magnitude square power. Adding noise can add constructively or destructively. The levels can have a greater range of values or equivalently a greater variance. The sum of two zero mean Gaussian variables is a zero mean random variable. A particular sum will not be zero but as you repeat the average will tend toward zero.

Usually when simulating, one could add noise with the same expected power or the same measured power. The likelihood that two random finite waveforms would have exactly the same measured power in nature is essentially zero.

If I use a white noise of the same power instead of colored noise, then what do I loose and how bad this approximation would be in a system simulation?

What do you mean by lose and bad? It would depend on details of the particular simulation. Why would you use white noise instead of colored noise in a simulation? What do you lose when using loose instead of lose? Clarity. (I had to look it up to be sure, thank Google) Fundamentally, What level of fidelity do you need?

A lot of natural noise has multiple sources where some predominate over others. White noise is an idealization. Nothing in nature have a flat spectrum over $-\infty < f < \infty$. Engineering White noise is flat over a band of interest. If one samples too fast (or upsamples) over this band the samples will show correlation. There will be a colored band and the Nyquist band is the full band.

• Sorry for the typo. Yes, I meant lose and not loose. – shampar Sep 20 '18 at 21:02
• The reason I am using white instead of colored noise is because I have the PSD of the noise (which is not flat over the frequency). For some reason I don't want to apply the noise directly from its PSD, instead I get the equivalent rms of this noise by the calculating the area under it (let's say $\sigma$) and by using randn() function in MATLAB I apply white noise with the same power (i.e. $\sigma^2$). Now, by bad and lose, I mean how far this approximation would be if I want to determine system performance in terms of BER or eye opening, or any other metric of this kind. – shampar Sep 20 '18 at 21:02
• I didn't quite get my answer from your comments on the first part. $S_1 = S + N_1$ and $S_2 = S + N_2$, where $N_1$ is colored (LP filtered noise) and $N_2$ is white noise with the same power. Is it right to say that in $S_2$ the variance of noise is the same throughout all time snapshots of it, but in $S_1$ the noise on average has more variance in lower frequency parts of the signal – shampar Sep 20 '18 at 21:14
• as i said it would depend – user28715 Sep 20 '18 at 22:01