# Is there a mathematical method to determine if noise is Gaussian?

Is there a mathematical method to determine if a signal's noise is Gaussian?

The only way I know so far is to analyze the histogram and layover a Gaussian distribution to visually determine if the distribution is Gaussian. I would like to know if there is a mathematical way to determine if the noise is Gaussian and how accurate the result is.

• en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence if you want to compare two PDFs; if you want to compare a discrete observation to a PDF, it's probably worth reading en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test – Marcus Müller Nov 1 '17 at 22:21
• note that these two methods give you a sort of a distance; whether or not this is relevant to you really depends on why you need to know whether the noise follows a Gaussian distribution. – Marcus Müller Nov 1 '17 at 22:32
• If this noise is additive and dominant, the histogram approach would be okay. In my opinion, we'll have to use a noise model assumption anyway, such as additive noise or noise characteristics (mean, variance, ..). For example, let's assume the the model y = x + n, where x is the signal and n is Gaussian noise. If you average the corrupted signal y and analyze the residuals, may be you can get some idea about noise characteristics. – dhanushka Nov 2 '17 at 1:58
• I am not specialist but the keyword (other than what suggested by Marcus Müller) may be density estimation en.wikipedia.org/wiki/Density_estimation – AlexTP Nov 2 '17 at 10:21

There are several statistical tests if a time series is Gaussian, although in statistics, the term "tests for normality" is usually how you search for them.

The Nist EDA site is a good place to look and the probability plot is better for shorter data sets than the sample histogram.

http://www.itl.nist.gov/div898/handbook/eda/section3/probplot.htm

Near the bottom of the page, there are references to q-q plots, KS, Chi squared, and other goodness of fit tests. You can find ample information about them on the web and replicating here isn't going to add anything.

Matlab has qqplot and prob plot in the Statistics toolbox, and the qqplot with a single argument is specific to Gaussian distributions. SAS has all these tests. R has the tests.

I recommend this book, written by 2 Engineers, and they cover several tests including for things like independence, and stationarity. The book is oriented towards the practical, minimum of mathematics.

Bendat, Julius S., and Allan G. Piersol. Random data: analysis and measurement procedures. Vol. 729. John Wiley & Sons, 2011.

The wrinkle of these tests is that they don't conform to a Signal plus Noise scenario. The tests generally assume that the time series is all Gaussian or not. A constant mean isn't a problem. Signals are not usually Gaussian and a simple test can't tell the difference.

Signal processing operations such as a DFT, tend to manifest central limit theorem effects on data, so you need to be aware that even linear transformations will not preserve a non Gaussian pdf.

It should be also noted that from a practical perspective, Gaussianity isn't black and white. Algorithms that have Gaussian assumptions usually work well even if the Gaussianity assumption is not strictly valid. Things like bi-modality and non-symmetry are more important to know about. Cauchy (heavy tails) like noise and multiplicative noise are also important to know about.

• Could you please explain more about "Signal processing operations such as a DFT, tend to manifest central limit theorem effects on data, so you need to be aware that even linear transformations will not preserve a non Gaussian pdf." ? I mean why the fact that linear transformations do not preserve non Gaussian pdf is surprised. Thanks. – AlexTP Nov 3 '17 at 10:34
• Surprised? Are you surprised? – user28715 Nov 3 '17 at 18:22
• I just dont understand what you want to say. Due to my humble knowledge, I do know only the statement that linear transformations do preserve Gaussian pdf. – AlexTP Nov 3 '17 at 19:56
• and how does a linear transformation not preserve a non Gaussian pdf conflict with your humble knowledge. Fundamentally, since I said the tests were not suitable to determine normality in the presence of a signal, it would occur to many that filtering out the signal and then applying the test to what remains would be an approach, so a linear transformation, i.e. FFT based convolution would alter any non Gaussian properties of the noise and bias the test toward normality. Your humble knowledge would include the CLT, wouldn't it? – user28715 Nov 3 '17 at 20:44
• No, what you are saying does not conflict with my humble knowledge which fortunately does include CLT. Just dont know that "linear transforms (or even non-linear transforms) preserve non Gaussian pdf" is a common mistake that people wrongfully expect. I mean if a pdf is non Gaussian, I usually dont expect anything about its transformed counterpart. Thanks for explaining. – AlexTP Nov 4 '17 at 7:15