There are two approaches to ensuring a sample mean of exactly 0, and a sample variance of exactly 1.
x = 1/2*(randn(N, 1) +1i*randn(N,1))
First approach: modify the values as a group
x2 = x2 ./ sqrt(xVariance);
The number of degrees of freedom of the original x is 10 -- N*2 independent random values.
The number of degrees of freedom of the modified group, x2, is only 8: although each of the values is still random, together as a group they are correlated in ways that reduces the freedom.
Second approach: take only initial entries and generate the remaining values as whatever is necessary to make the group fit. This is proving to be more difficult than I thought it would be.