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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?

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  • $\begingroup$ 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. $\endgroup$
    – Dilip Sarwate
    Commented Jul 29, 2020 at 19:25
  • $\begingroup$ 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. $\endgroup$
    – Sm1
    Commented Jul 29, 2020 at 19:32

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  1. Once you talk about the Spectrum of noise / process you implicitly says it is stationary in the wide sense.
  2. What does it mean to have a signal with uncorrelated samples? Do you understand it means you can't using linear predictor, to have any information from all past samples ion the current one? Usually this is not how signals behave. This is exactly why correlated noise is hard. It behaves like a signal hence it is harder to separate them apart. One way to deal with the model of Signal + Colored Noise is to apply whitening transform on the data, apply White Noise methods and then apply the inverse transform.
  3. Any process where the probability for a specific value depends on the index value. Here are some specific examples (From Investopedia - An Introduction to Stationary and Non-Stationary Processes):

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  1. Noise is used when dealing with signals. We call noise to the non deterministic added (Usually, we have multiplicative noise as well) term of the signal. Since we have some index (Time / Spatial, etc...) for the signal we have it for the noise as well which means it is a process.
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  • $\begingroup$ 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? $\endgroup$
    – Sm1
    Commented Jul 30, 2020 at 12:39
  • $\begingroup$ 1. Stationary is a property of some random processes. A process being stationary allows analyzing it using well known tools as Fourier Analysis. Hence it is very popular. Because we can deal with it effectively which means people are using this model. 2. I am not sure I understand. But if you process is addition of uncorreletad and correlated processes than the end result is still correlated. $\endgroup$
    – Royi
    Commented Jul 31, 2020 at 6:09

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