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

Question

Tejas Appaji 2017-9-25

1
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https://ww2.mathworks.cn/matlabcentral/answers/358204-how-to-generate-gaussian-noise-with-certain-covariance-and-zero-mean

回答： getahun kibret 2023-2-22

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

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Answer 1

David Ding 2017-9-27

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https://ww2.mathworks.cn/matlabcentral/answers/358204-how-to-generate-gaussian-noise-with-certain-covariance-and-zero-mean#answer_283344

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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;

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Shah Mahdi Hasan 2020-8-14

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 2020-11-26

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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, :))