Fourier transform and the multivariate normal I am wondering about how to specify multivariate normal distributions for vectors that have undergone a Fourier transform.  For instance:
Say we have the mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$ of a multivariate normal given in the not-yet Fourier transformed domain.  I can draw a vector sample $\mathbf{x}$ from this distribution,
$$ \mathbf{x} \sim N(\boldsymbol{\mu}, \boldsymbol{\Sigma})$$
I can compute the PDF of my sample $\mathbf{x}$ using $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$.  Then, this sample $\mathbf{x}$ is Fourier transformed, $FT(\mathbf{x}) = \mathbf{\tilde x}$.  
What steps do I take to transform $N(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ so that I can compute the PDF of the Fourier transformed sample $\mathbf{\tilde x}$ directly?
 A: The multivariate normal has some nice properties.  In particular, if $x\sim N(\mu,\Sigma)$, then, for any matrix $A$, $Ax \sim N(A\mu, A\Sigma A^T)$. 
Noting that a (discrete) Fourier transform can an be written in matrix form as $FT(x) = Fx$, we see that $FT(x) \sim N(F\mu, F\Sigma F^T)$.  
You can prove this by checking the first and second moments.  $E(Fx) = FE(x)$ and similar for $E((x- F\mu)(x-F\mu)^T)$).
This idea is particularly important when $\Sigma$ is a circulant matrix, in which case $F\Sigma F^T$ is diagonal (and hence easier to work with numerically and theoretically). 
A: Unfortunately, the definition and conventions for complex Multivariate Normal is not completely standardized. Perhaps "yours" is consistent with equatoion 2 of http://cran.r-project.org/web/packages/cmvnorm/vignettes/complicator.pdf , which is different than https://en.wikipedia.org/wiki/Complex_normal_distribution . 
You can invert the Fourier transform of a real Multivariate Normal per 0.7 equation 9b of https://www.cs.nyu.edu/~roweis/notes/gaussid.pdf . That will leave you with a real Multivariate normal, plus your existing one; then simulate to your heart's content. Once you have the exact definition and convention used in your complex Multivariate Normal, equation 9b needs to be adapted accordingly. You ought to be able to handle it at that point.
The other way of going is to use 0.7 equation 9a of https://www.cs.nyu.edu/~roweis/notes/gaussid.pdf on x. But no matter which way you go, you're living dangerously if you don't understand the conventions used in your Fourier transform of μ. You seem to be playing fast and loose, as evidenced by your terminology of μ as the mean of a Multivariate Normal, and also as something on which you are applying Fourier transform to get μ~. It doesn't mean anything is wrong, but you can easily mix up things which use different conventions, and wind up with the wrong answer.
