linear least squares -- complex observations, real estimate constraint Consider the following least squares optimization problem:
$$
\hat{x} = \arg\min_x \| y - A x\|^2
$$
where the observations are complex $y\in{\cal C}^{N\times 1}$, and the complex design matrix $A\in {\cal C}^{N\times K}$ is full rank ($K$).  Is there a simple closed-form solution if $x$ is constrained to be real (i.e., $x\in{\cal R}^{K \times 1}$)?

If you write $y=y_1+iy_2$ and $A=A_1+iA_2$, then your problem is equivalent to finding:
$$\min_{x\in\mathbf{R}^K}f(x)=\lVert y_1-A_1x\rVert^2+\lVert y_2-A_2 x\rVert^2$$
Now we are left with minimizing an unconstrained convex function:
$$\nabla f(x)=-2y_1^T A_1+2x^TA_1^T A_1-2y_2^T A_2+2x^TA_2^T A_2$$
Note that $A_1^TA_1+A_2^T A_2$ is still positive definite, so the minimizer is:
$$x_0=(A_1^TA_1+A_2^TA_2)^{-1}(A_1^T y_1+A_2^T y_2)$$
