Gaussian noise with different SNR levels are usually used in research works to simulate a realistic environment. How can researchers guarantee that Gaussian noise can simulate the reality of a System?
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3$\begingroup$ Why Gaussianity? $\endgroup$– EmreCommented Aug 27, 2014 at 19:52
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5 Answers
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Gaussian is a very good assumption for every process or system that's subject to the Central Limit Theorem. See http://en.wikipedia.org/wiki/Central_limit_theorem
What this means is that when gaussian random variables are added, the result is gaussian (so you can apply similar statistics after the addition as were before), and besides that, when any random variables (that have finite variance, so Cauchy r.v. does not apply) are added, they tend to become more gaussian in their p.d.f. as you add 'em up.
What's also very cool about the "normalized" gaussian function, $e^{-\pi t^2}$ is that its Fourier transform is exactly the same: $$\mathcal{F}\left\{e^{-\pi t^2}\right\}=e^{-\pi f^2}$$ that sometimes makes the math fun and easy. Regarding the gaussian p.d.f., that means the corresponding characteristic function is also gaussian. And when you add random variables, you convolve their p.d.f.'s and that means you multiply their characteristic functions. When you multiply two gaussians, what then do you get?
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$\begingroup$ that is interesting. so I can say Guassian noise cannot guarantee a realistic environment. but due to Central Limit Theorem, Gaussian is a very good assumption. actually a reviewer commented on my paper and asked how do you guarantee that Guassian noise can simulate a realistic environment. The signal was active power of a tie line. The noise has been added to simulate the measurement noise. so I should convince him lol :P SOS @robertbristow-johnson $\endgroup$– SAHCommented Aug 27, 2014 at 19:34
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$\begingroup$ Thanks for the answer @hilmar please read above comment. $\endgroup$– SAHCommented Aug 27, 2014 at 19:35
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$\begingroup$ would you provide more information about that theory please? @hilmar $\endgroup$– SAHCommented Aug 27, 2014 at 20:52
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As pointed out in other answers, the Central Limit Theorem is one reason why Gaussian noise is so important as a model. One other reason that hasn't been explicitly mentioned and that I would like to point out is the fact that Gaussian noise is completely described by second order statistics, which are relatively easy to measure. E.g., if you have a univariate distribution and if it is Gaussian, then you know everything you can ever know about it by knowing (measuring) its mean (first order) and its variance (second order). There are no higher order statistics, which is good because they are much harder to measure reliably. This fact is of course no justification for using a Gaussian model, but it is a very handy property if we think that a Gaussian model is justified.
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$\begingroup$ Thanks for the answer @matt-l would you give more information about your reason, I didnt get it :( $\endgroup$– SAHCommented Aug 27, 2014 at 20:45
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$\begingroup$ @Electricman: A Gaussian random variable (RV) is completely characterized by its mean and its variance. This is normally not the case for RVs with other distributions. I.e. after measuring the mean and the variance you know everything about the RV if it is Gaussian. This is a big advantage because measuring the mean and the variance is relatively easy compared to measuring higher order moments (which may be necessary if the RV is not Gaussian). $\endgroup$– Matt L.Commented Aug 27, 2014 at 20:53
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$\begingroup$ what is Gaussian white noise? is it same as zero mean Gaussian noise? @matt-l $\endgroup$– SAHCommented Aug 28, 2014 at 6:08
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$\begingroup$ well I used zero mean Gaussian noise to simulate measurement noises. Now a reviewer asked why did you used Gaussian , how it guarantee a realistic environment. I thought its better to cite some other papers that used the same noise model. but as I saw, they used white Gaussian noise. can I still refer to them? Thanks :) @matt-l $\endgroup$– SAHCommented Aug 28, 2014 at 7:00
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A realistic environment is created by characterizing the noise, and then using a distribution or simulation with that characterization (or as close an approximation as can be used.). If an accurate characterization of the system's noise turns out to be Gaussian, then the researcher is good to go.
In the event that one does not have a good characterization (nor even know the number of possible noise and error sources), a Gaussian distribution is the limit of the sum of a large number of unknown (but bounded?) noise sources, so might be a reasonable guess. But that's only a guess, so beware trusting the simulation as being the same as reality. (An unbounded noise source, "fat tail" or "Black Swan" event, can easily have reality do something never seen in a simulation that uses only Gaussian noise sources. Thus, no guarantee for the research work.)
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$\begingroup$ what is Gaussian white noise? is it same as zero mean Gaussian noise? @hotpaw2 $\endgroup$– SAHCommented Aug 28, 2014 at 6:09
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2$\begingroup$ Gaussian white noise? That sounds like a new different question. Why don't you ask it? (this is just the comment section.) $\endgroup$– hotpaw2Commented Aug 28, 2014 at 8:12
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How can researchers guarantee that Gaussian noise can simulate the reality of a System?
They cannot guarantee that it is an accurate reflection of all systems. In fact, though it is a decent approximation of many systems we know that there are many systems that it doesn't accurately reflect at all.
So why do we use gaussian noise? Two reasons. First, because it does accurately reflect many systems. Second, because it is very easy to deal with mathematically, making it an attractive model to use.
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$\begingroup$ Thanks for the answer @jim-clay do you know any reference that has said Gaussian noise is a decent approximation of many systems? $\endgroup$– SAHCommented Aug 27, 2014 at 18:27
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$\begingroup$ No, I don't. But take a look at the binomial distribution (en.wikipedia.org/wiki/Binomial_distribution). It looks very gaussian. In fact, if you convolve lots of independent distributions together the result tends to look pretty gaussian. Thus, when you have lots of different statistical "events" (like thermal atomic noise) happening the sum of all those millions/trillions/whatever events often looks pretty gaussian. $\endgroup$– Jim ClayCommented Aug 27, 2014 at 18:49
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$\begingroup$ what is Gaussian white noise? is it same as zero mean Gaussian noise? @jim-clay $\endgroup$– SAHCommented Aug 28, 2014 at 6:09
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In my experience in audio, I find that Gaussian noise is an excellent model of the time domain values of environmental background noise. In audio, the noise is most often colored, but the density is still well described by a Gaussian model.
When you get into specific noises, like keyboard clicks, or engine noises, often there are non-Gaussian components, and so other models need to be used. But for high level general background noise, Gaussian is a great model.
As the other people have answered, the central limit theorem plays a role here.
