# Determining the “whiteness” of noise

How does one quantify how "white" some noise is? Are there any statistical measures, or any other measures (FFTs for example) that can quantify how close to white noise a particular sample is?

• Are you interested in suggestions on how to compare different noise sources/signals or are you looking for a "industry standard" metric that applies to the amount of "color" in a noise source? I am not aware of a general metric that applies, but you can compare the amount of coloration by looking at the noise power distribution in an FFT or PSD (flatter = whiter) or you can compare autocorrelation fucitons (narrower = flatter). – user2718 Jan 29 '13 at 13:44
• If I understand you correctly you are looking for an automatic black box calculator of 'whiteness', correct? – Spacey Jan 29 '13 at 15:31
• +1 for computing the Power Spectral Density of the source. For the record, I would like to add that white noise can't be sampled in practice, as its PSD is flat in -∞ < f < ∞. – Serge Jan 29 '13 at 21:04
• @Mohammad - Not necessarily a black box to calculate. I'm just curious if there's a mathematical estimator of whiteness. – Kitchi Jan 30 '13 at 6:21
• @BruceZenone - For a real sample of data, as Serge pointed out, the PSD will never be completely flat, no? But I'm still guessing that the flatter it is, the closer it comes to being "true" white noise. – Kitchi Jan 30 '13 at 6:22

You could form a statistical test, based on the autocorrelation of the potentially-white sequence. The Digital Signal Processing Handbook suggests the following.

This may be implemented in scilab as below.

Running this function over two noise sequences: a white noise one, and a lightly filtered white noise one, then the following plot results. Script for generation of each realization of the noise sequences is at the end.

The mean of the statistic for the white noise is 9.79; the mean of the statistic for the filtered noise is 343.3.

Looking at a chi-squared table for 10 degrees of freedom, we get:

and we see that there is no significance level at which 9.79 (in the table) that the white noise isn't white. We also see that the value of 343.3 is very likely to be non-white (comparing it to the 23.2093 value in the $p=0.01$ significance column).

function R = whiteness_test(x,m)
N = length(x);
XC = xcorr(x);
len = length(XC);
lags = len/2+1 + [1:m];
R = N*sum(XC(lags).^2)/XC(len/2+1).^2;
endfunction


X = rand(1,1000,'normal');
Y = filter(1,[1 -0.5],X)
R = [R; whiteness_test(X,10)];
R2 = [R2; whiteness_test(Y,10)];

• im not a big statistican... But i have a concern regarding the general validity of the above metioned test for non gaussian white noise processes: As far as i understand white noise only means that there is no correlation in time and thus the autocorrelation is an impulse at 0 lag. White does not necessarily mean that the amplitudes are normally distributed, which is what the test assumes... Thus as far as i understand the test is valid for white gaussian noise (because the sum of squared gaussian distributions is Chi-squared) and not for general white noise? Am i right or is there anything wr – Fabian Jun 12 '17 at 9:21
• @Fabian : Yes and no. You are correct in that the test assumes that the autocorrelation values are Gaussian. If the original noise is just about any distribution, then the central limit theorem means that the distribution of the autocorrelation estimates will be Gaussian. There are some pathological cases where the autocorrelation coefficients will not be Gaussian, but these are generally few and far between (and perhaps autocorrelation analysis isn't the best thing to be doing in those cases). – Peter K. Jun 12 '17 at 12:04
• @PeterK.Wouldn't a "harder" test be to determine the flatness of the PSD? This way, no assumptions are made and the distribution of the noise samples is irrelevant. – Envidia Jun 12 '17 at 19:01
• @Envidia : The two are equivalent, are they not? The PSD is just the DFT of the autocorrelation sequence. – Peter K. Jun 12 '17 at 19:10
• @PeterK. In your example yes, they are essentially equivalent. However the procedure does assume i.i.d. where as generally, the samples can be distributed in any fashion. I do understand that the Central Limit theorem does come into play and is valid, hence why I use the term "harder". Maybe a better term would be "general". – Envidia Jun 12 '17 at 19:19

I would use the signal's autocorrelation properties or flatness of PSD to determine this. The autocorrelation of theoretical white noise is an impulse at lag 0. Furthermore, the PSD of the fourier transform of the autocorrelation function, the PSD of theoretical white noise is constant.

Either of these should give you a good idea of the whiteness of your noise.

Whiteness is equivalent to independence.

You can look at the diehard https://en.m.wikipedia.org/wiki/Diehard_tests

Volume 2 of Knuth's Seminumerical Algorithms has a section on random number genentators and testing.

The problem with DFT based tests is that there is a little bit of spectral leakage the technique introduces some correlation, which if you make your transforms "long" can typically be neglected.

There are tests for random bit streams as well at NIST

• Forgot to say, Stan: +1 for those diehard tests! I hadn't seen that list. :-) – Peter K. Jun 27 '17 at 11:01