How to gaussian white noise with mean, variance and autocorrelation function?

I want to input a white gaussian noise (ζ_z )in my system with mean = 0, variance, σ= 3,6 and 12*10^-4 and exponential autocorrelation function with correlation time 𝜏 =0.1. Can anyone help me with this?

The equations are given as-

Answer 1

[Chaitanya Mallela]

The White Gaussian Noise should contain a multivariate random numbers with three different realizations, each realization with mean 0 and variance 3,6,0.0012 respectively.

N=1000;                               % Input sample length
M = [0 0 0];                          % Mean vector 
var = [3 6 12*10^-4];                 % Variance vector
Cov = var.*eye(3,3);                  % Covariance matrix
x = mvnrnd(M,Cov,N);                  % MultiVariate random vector added to the input
                                      
x1 = autocorr(x(:,1));                % Space Correlation of three realizations
x2 = autocorr(x(:,2));
x3 = autocorr(x(:,3));

Since time is a separate dimension, we get the Gaussian Random Variables by fixing a particular time instant(say t) in the Gaussian Random Process to obtain the above three realizations. So, we consider the time instants in such a way it satisfies the time step auto correlation function of the Gaussian Random Process.

[Musanna Galib]

Thank you very much for the reply. I am not expert in this field. I didn't understand the explanation related to implement auto-correlation time. Can you kindly elaborate how to implement the autocorrelation time. I am actually trying to reproduce some result. They used these parameters mentioned in the question.

This noise will be used to give a force. I am quoting the line - "The stochastic force is an exponentially correlated noise with fixed standard deviation and correlation time 0.1 s with exponential autocorrelation function."