Let x ~ CN (0,a) and y ~ CN(0,b), with a, b positive constants. Let z = x + y. I need to find the distribution of the vector t=[x; z]. My basic understanding is that vector t is also complex Gaussian distributed and it has mean [0;0]. I do not know how to compute the covariance matrix of t. Can anybody help me? Thank you.
Because x and y are independent random variables, Var(z) = Var(x) + Var(y) = a + b;
One way to compute the covariance Cov(x,z) = Cov(z,x) is to simply apply the definition of covariance.
Cov(x,z) = E[xz*]-E[x]E[z*], and here we have E[x]=E[z*]=0.
Cov(x,z) = E[x(x+y)* ]=E[xx*+xy*]=E[xx*]+E[xy*]. Note that E[xy*]=E[x]E[y*]=0.
Cov(x,z) = E[xx*].
Observe that Var(x)= E[xx*]-E[x]E[x*] = 1, and E[x]=E[x*]=0 in this exercise.
Finally, Cov(x,y) = a.
Having computed the variances and co variances it is then straightforward to write the correlation matrix.