How does uncertainty propagate through an equation with complex variables? I am trying to understand how uncertainty propagates through systems with complex variables. 
Given the general error propagation formula
$$
\sigma^2_u = \left(\frac{\partial u}{\partial x}\right)^2\sigma_x^2 + \left(\frac{\partial u}{\partial y}\right)^2 \sigma_y^2 + \ldots
$$
so that if x is uncertain then in the case of multiplication by some constant, if
$$
u = Ax 
$$
then simply
$$
\sigma_u = A\sigma_x.
$$
I understand that variance can never be complex so what would happens in the case that A is complex? so for example:
$$
u = xe^{(-2\pi i f)}.
$$
I am assuming that x is drawn from a normal distribution with known parameters. 
 A: When variable is complex, you use the product by conjugate complex instead of the square of the value (remember that for a complex number, $\vert z \vert^2 = zz^\dagger$).
\begin{eqnarray}
\sigma_u & = & \sqrt{E \left[(u-E[u])(u-E[u])^\dagger \right]} \\
& = & \sqrt{E \left[uu^\dagger - uE[u]^\dagger - E[u]u^\dagger + E[u]E[u]^\dagger\right]}\\
& = & \sqrt{E[uu^\dagger] - E\left[uE[u]^\dagger\right] - E\left[E[u]u^\dagger\right] + E\left[E[u]E[u]^\dagger \right]}\\
& = & \sqrt{ E[\vert u \vert ^2] - E\left[uE[u]^\dagger\right] - E\left[E[u]u^\dagger\right] + E\left[ \left\vert  E[u] \right\vert ^2 \right]}
\end{eqnarray} 
Now, if $u = Ax$ where $A$ is a complex value and we know that $x = x^\dagger$ because $x$ is a real variable.
\begin{eqnarray}
\require{cancel}
\sigma_u & = & \sqrt{ E[\vert Ax \vert ^2] - E\left[AxE[Ax]^\dagger\right] - E\left[E[Ax](Ax)^\dagger\right] + E\left[ \left\vert  E[Ax] \right\vert ^2 \right]}\\
& = & \sqrt{ \vert A \vert^2 E[\vert x \vert ^2] \cancel{- \vert A \vert^2 E\left[xE[x]\right] - \vert A \vert^2 E\left[E[x]x\right] }+ \vert A \vert^2 E\left[ \left\vert  E[x] \right\vert ^2 \right]}\\
& = & \sqrt{ \vert A \vert^2 E[ x ^2] + \vert A \vert^2 E\left[   E[x]  ^2 \right]}\\
& = & \sqrt{ \vert A \vert^2 \left( E[ x ^2] + E\left[   E[x]  ^2 \right]\right)}\\
& = & \vert A \vert \sqrt{  \left( E[ x ^2] + E\left[   E[x]  ^2 \right]\right)}\\
\sigma_u & = & \vert A \vert \sigma_x
\end{eqnarray}
