Let $\mathbf{x}$ be a random column vector in $\Bbb{R}^n$, given as follows $\mathbf{x}=(x_1,\ldots,x_n)^\top$. We require $\mathbf{x}$ to follow a multivariate Gaussian distribution with mean vector $\mu$ and covariance matrix $\Sigma$, i.e. $\mathbf{x}\sim N(\mu,\Sigma)$. We assume that the $t$-th entry of the mean vector $\mu$ is given by a linear law, as follows $$ \mu_t = \mathbf{w}_t\cdot\mathbf{u}+b_t, $$ where $\mathbf{w}_t,\mathbf{u}\in\Bbb{R}^m$ are not random variables and $b_t\in\Bbb{R}$, $t=1,\ldots,n$.
With what type of noise should we contaminate $\mathbf{x}$, such that the mean vector is given as above and the covariance matrix depends on $\mathbf{u}$?
For instance, if we contaminate $\mathbf{x}$ with additive $\mathbf{e}\sim N(0,\sigma I)$, that is, $x_t = \mathbf{w}_t\cdot(\mathbf{u}+\mathbf{e})+b_t$, then indeed the mean vector is given by $$ \mu_t = E\big[x_t\big] = \mathbf{w}_t\cdot\mathbf{u}+b_t, $$ since $E\big[\mathbf{e}\big]=0$, but the covariance matrix $\Sigma = \big(\sigma_{ij}\big)_{ij=1}^n$ is given by $$ \sigma_{ij} = \sigma\mathbf{w}_i\cdot\mathbf{w}_j, $$ as follows after some manipulations, which unfortunately does not depend on $\mathbf{u}$.
So! How and with what type of noise should we contaminate $\mathbf{x}$ such that the mean vector is given as above and the covariance matrix depends on $\mathbf{u}$?
What would be the noise if the resulting covariance matrix needs to be anisotropic, i.e., not in the form $\Sigma=aI$?