# Conceptual Questions on Colored Noise Process

I am having a tough time finding answers to some specific questions and finding references where there is information regarding Brownian noise or Red Noise. I'm referring to white and colored noises where white noise can be Gaussian or Uniform Noise where the power spectrum is flat. Along the same line we have colored noise whose frequency is $$1/f^{\alpha}$$ where $$\alpha =1$$ for pink noise and $$\alpha =2$$ for brown or red noise. Matlab Code to generate colored noise presents the code to generate colored noise by filtering white noise simulated using randn.

This question is inspired from a previous question asked here: Stationary vs non-stationary signals?

Shall be obliged for the answers to the following specific questions. Please correct me wherever I am wrong.

1. Is brown noise stationary or non-stationary? If stationary then, the signal does not change with time. Then how come it appears to be random?

2. If a signal is corrupted with brown noise and pink noise respectively, how would the noisy signal's characteristics differ? I think since brown and pink noise samples are correlated, the resulting noisy signal's samples if it was uncorrelated when clean, would now become correlated? Not sure about this. It's important for me to know so that I can apply specific noise filtering techniques for correlated and uncorrelated signals.

3. An example of non-stationary process?

4. Do the terms "noise" and "process" mean the same?

• Is brown noise the same as Brownian noise or the same as a Brownian process? A Brownian process is not stationary in any sense of word. – Dilip Sarwate Jul 29 '20 at 19:25
• I thought noise and process is interchangeable....I'm referring to white and colored noises where white noise can be Gaussian or Uniform Noise where the power spectrum is flat. Along the same line we have colored noise whose frrequency is $1/f^{\alpha}$ where $\alpha =1$ for pink noise and $\alpha =2$ for brown or red noise. – Sm1 Jul 29 '20 at 19:32

• Thank you very much for explaining it so lucidly. A bit of a clarification is requested:(1) why the term randomness is associated with stationary -- where the randomness comes? If the initial seed generator of randn and rand functions are fixed, then we get the same signal generated everytime I run the code i..e., deterministic in that sense. (2) consider a signal of interest is $iid$ uncorrelated which is corrupted with additive white gaussian noise (AWGN), pink and brownian (red) noise. Will the noisy signal with the AWGN still be uncorrelated? However, for pink and brown correlated? – Sm1 Jul 30 '20 at 12:39