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

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

Answer 1

[David Ding]

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;

[Tejas Appaji]

thank you. this is exactly what i was looking for.

[JUNHO KWEON]

Hi, thank you for your kind answer. By the way, as I know, randn has variance of 1. In this case, if covar =0, var1 = 4, var2 = 9, why don't we put 'sqrt' such as

% Generate a 2 x 1 Gaussian noise vector with each variance. (orthogonal)

noiseVec = rand(2,1);
var1 = 4;
var2 = 9;

cMatrix = [sqrt(4) 0; 0 sqrt(9)];

noiseVec = cMatrix*noiseVec;

Here, var1 = E[x1*x1.']; var2 = E[x2*x2.']; where E is expectation. x1 and x2 are zero-mean noise with each variance values.

Or did you mean var1 and var2 are standard deviation values?

[Shah Mahdi Hasan]

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]

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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