How does uncertainty propagate through an equation with complex variables?

Question

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.

Answer 1

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}