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how to generate gaussian noise with certain covariance and zero mean ?

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i have a signal and i want to add gaussian noise to it with zero mean and 0.1 covariance.
variance = 0.1
W = sqrt(variance).*randn(1,size(Xmodt,2)); %Gaussian white noise W
mysignal = mysignal + W; %Add the noise
this code lets me define variance. but i need an algorithm or code to generate gaussian noise with specific covariance and zero mean.
thanks in advance

Accepted Answer

David Ding
David Ding on 27 Sep 2017
Hi Tejas,
If I understand your question correctly, you wish to generate AWGN with certain co-variance. In this case, you would have a vector of zero-mean Gaussian noises that are statistically dependent. In order to model this in MATLAB, your workflow would be to generate an n x 1 noise vector and then pre-multiply that by the co-variance matrix.
For example:
% Generate a 2 x 1 Gaussian noise vector with covariance
noiseVec = randn(2, 1);
var1 = 0.1;
var2 = 0.2;
covar = 0.05;
cMatrix = [var1, covar; covar, var2];
noiseVec = cMatrix * noiseVec;
  4 Comments
Shah Mahdi Hasan
Shah Mahdi Hasan on 14 Aug 2020
I agree with JUNHO. There should have been sqrt(). Think about the scalar analogy. If you have a standard normal random variable x~N(0,1) and want to have a certain variance sigma, then you would multiply the following:
y ~ N(0,sigma^2) = sigma*x
Brendan Nichols
Brendan Nichols on 26 Nov 2020
I agree with Junho as well. Test out a variation of David's answer:
noiseVec = randn(2, 1e6);
var1 = 0.1;
var2 = 0.2;
covar = 0.05;
cMatrix = [var1, covar; covar, var2];
noiseVec = cMatrix * noiseVec;
cov(noiseVec(1, :), noiseVec(2, :))
Note the covariance is not equal to cMatrix. Try the following instead:
noiseVec = randn(2, 1e6);
var1 = 0.1;
var2 = 0.2;
covar = 0.05;
cMatrix = [var1, covar; covar, var2];
noiseVec = chol(cMatrix, 'lower') * noiseVec;
cov(noiseVec(1, :), noiseVec(2, :))
Note the addition of the Cholesky decomposition to get the covariance of the noise to match.

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More Answers (1)

getahun kibret
getahun kibret on 22 Feb 2023
i want to get a matlab code that adds AWGN with zero mean and variance 0.85

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