Hey Luis,

1. When you say it's white, then it means that it's first of all wide sense stationary.

2. There exists such noise. For example if you take a white Gaussian noise and filter it through some LTI system, and remains Gaussian but becomes colored.

3. There is not necessarily a connection, besides the fourier transform connection to the second order moments.

Best,

Shahar

Mr. Shahar,

First of all, thanks for contributing.

1. I am not so sure about the veracity of your affirmation. I have seen in many scientific papers that authors explicitly qualify some white Gaussian noise as stationary. However, it wouldn't be necessary to specify that it is stationary if it was always true for AWGN. For example, you can find a noise which variance changes with time. It would still be Gaussian, and even it's spectrum could still be flat, but it wouldn't be stationary anymore.

2. How do you know that after filtering AWGN you will still get a Gaussian process? Any filter would apply this property? It's not so obvious.

3. If you are referring to the characteristic function when you consider Fourier transform, I not so sure that it has an spectrum meaning.

Please, provide me scientific references where I can confirm your statements.

Thank you very much.

In regard to question number one, I propose to run this simple MatLab script that shows that not always white Gaussian noise is stationary. There is a stationary AWGN and a Gaussian noise with a time-varying variance. Both have a flat amplitude spectrum, however, one is stationary and the other is not.

Can anybody tell if there is something wrong in this code?

Dear  Luis,

                     I hope the below lines may solve your doubt 

Normally in classical communication theory it is often assumed that the noise is stationary AWGN Process.Although for some problems it is a valid assumption , which might me mathematically convenient for  us but in real scenario noise is often time varying , correlated & non gaussian.

For example: impulsive noise and acoustic( these two can't be modelled as AWGN)

Non stationary and non gaussian can be moddeled by Markovian chain of stationary sub process.  

Answer as per my Literature survey about the same...All the best...!

Pls see file attached. regards.

Dear Luis,

Regarding my first answer:

1. You are correct, I'm too used to white noise being stationary, but it doesn't have to be so.

However, for a WSS AWGN my answers to questions 2 and 3 are correct. To validate that, you may use any undergraduate course book in stochastic processes, for example [Papoulis]_Probability,_Random_Variables_and_Stochastic_processes.

The Fourier transform I talked about is of the autocorrelation function of the stochastic process and not the Fourier transform of the stochastic process itself, as it does not necessarily exists. It has nothing to do with characteristic function.

Best regards,

Shahar

1. No, not always.

2. If you filter an AWGN noise, it is still Gaussian but not White.

3. Spectral and statistical properties are independent. That is the reason to specify "White" and "Gaussian" in AWGN

1) Yes- this means Wide Sense Stationary since the notion of "whiteness" is related to the Power Spectral Density and this has no meaning unless it is Wide Sense Stationary.

2) Strict White Noise does not exist in nature. It would imply an infinite power- since it has to be flat at a non zero value to infinity. We live in a finite universe. This is strictly a mathematical fiction to allow performance bounds to be easily computed. However, given an understanding of the constraints the existing performance bounds are generally good approximations.

3) The spectral properties only give you information about the first and second order moments of the underlying Stochastic Process. To have a complete characterization of the Stochastic Process you need to have all moments- you can see this from the Characteristic Function.

Thanks to everybody for contributing to this debate. I appreciate your time.

As long as I see, there is so much confussion regarding to this topic. There is a great contrast among your different answers.

I specially thank to Professor Mahmoud for the Lecture slides he provided, although there are no references to Stationary Noise, only White Noise is considered, and no links between them are stated.

I insist when I say that in many scientific papers, authors explicitly qualify some white Gaussian noise as stationary, like if it wasn´t always the case.

Thank you all again. I think I finally agree to Mr. Aguilera. His three (concise) answers look very logical to me.

Hi Luis,

         Generally we consider white Gaussian noise for the theoritical point of view. Generally in real channel there will not be White Gaussian noise, it will be coloured Gaussian noise depends.

Hi Luis,

1) White implies WSS.

2)As Kenneth said, white noise is a fictitious assumption, real noise is in fact colored. For instance, in an array system, the noise at the output of each array (when close together) couldn't be assumed to be white.

3)Spectral properties give mean and variance information. Stochastic properties require higher order moments.

After some research, I can finally close these questions with the following answers:

"Whiteness" means: flat power spectral density, thus zero autocorrelation for lag different than zero (uncorrelated samples), thus wide sense stationary (time-independent autocorrelation and mean).

1st. Does always white Gaussian noise means wide-sense stationary noise (WSS)? Answer: Yes, due to whiteness, not due to Gaussian. Noise of any statistical distribution is WSS if it is said to be white.

2nd. Does even exist Gaussian noise without been white? Answer: White Gaussian noise passing through a linear filter always produces Gaussian noise, only changing its parameters (mean and variance), since any linear combination of uncorrelated Gaussian random variables produces a Gaussian random variable. However, the whiteness is lost in general, and thus samples become correlated after passing a linear filter.

3rd. How are related the spectral content of a stochastic process with it's statistical distribution? Answer: The spectral content on a random process is only dependent on the correlation of samples (and thus on joint distribution), but not on the statistical distribution itself. It can be Gaussian, or uniform or whatever the distribution. If every pair of different samples of the process are uncorrelated, the power spectral density is flat, otherwise, it is not flat (which is always the case in practical scenarios).

Thanks everybody for the contribution.

difference between AWGN and Gaussian Random noise?

The difference consists on the "whiteness" property of AWGN. It implies a flat power spectral density, thus zero autocorrelation for lag different than zero (uncorrelated samples), thus wide sense stationary (time-independent autocorrelation and mean).

Gaussian noise does not have spectral restrictions, it only has the Gaussian pdf restriction. Its samples may be correlated, since it is not white.

Additive White Gaussian Noise (AWGN) is the highly statistical in nature with random radio noise overriding signal characterized by a wide frequency range depending upon the communication channel with proportionate to a signal strength .