I have the following problem: $$b = \Re\{Ax\} + w,$$ where $A$ and $x$ are complex, and $b$ and $w$ are real. I'd like to compute the least squares solution, i.e., $$\hat{x} = \arg\,\min_x\|b-\Re\{Ax\}\|_2^2.$$
I know if everything were real or complex, the solution would be $(A^H A)^{-1}A^H b$. Is there a similar form that works for real measurements of a complex system?
The only thing I can think of is splitting $A$ and $x$ into real/imaginary components, as in $$A_r = \Re\{A\}, ~A_i = \Im\{A\}, ~A_c = [A_r, A_i] $$ $$x_r = \Re\{x\}, ~x_i = \Im\{x\}, ~x_c = [x_r^T,x_i^T]^T$$ so that $$b = A_c x_c + w.$$ Then the solution would be $(A_c^T A_c)^{-1}A_c^T b$. However, the matrix is bigger now and takes longer to compute. Also, if I wanted to apply some sort of regularization to $x$, I would be dealing with the real/imaginary components separately.