Widely linear system of equations, revisited

It seems I just cannot stop talking about widely linear equations. Sorry for being to monotonous. My colleague Dominik recently made me aware of the fact that it is possible to solve the widely linear system of equations

$$ \ma{A} \cdot \ma{x} + \ma{B} \cdot \ma{x}^* = \ma{c}$$

that I already visited in an earlier blog post in a much more convenient form. I was suggesting to stack the real and the imaginary parts of both $\ma{x}$ and $\ma{c}$ into long vectors which then transforms the entire system of $N$ complex-valued equations into a system of $2N$ real-valued equations, using real- and imaginary parts of $\ma{A}$ and $\ma{B}$ to construct the coefficients.

It is possible to completely avoid that. Simply conjugate the original equation, this gives you

$$ \ma{A}^* \cdot \ma{x}^* + \ma{B}^* \cdot \ma{x} = \ma{c}^*.$$

Combining both equations in one we can write

$$ \begin{bmatrix} \ma{A} & \ma{B} \\ \ma{B}^* & \ma{A}^* \end{bmatrix} \cdot \begin{bmatrix} \ma{x}  \\ \ma{x}^*  \end{bmatrix} = \begin{bmatrix} \ma{c}  \\ \ma{c}^*  \end{bmatrix}. $$

Therefore, provided the inverse exists, we have

$$ \begin{bmatrix} \ma{x}  \\ \ma{x}^*  \end{bmatrix} = \begin{bmatrix} \ma{A} & \ma{B} \\ \ma{B}^* & \ma{A}^* \end{bmatrix}^{-1} \cdot  \begin{bmatrix} \ma{c}  \\ \ma{c}^*  \end{bmatrix}.$$

This is of course redundant, since we only need to compute $\ma{x}$ (once) and not $\ma{x}$ and $\ma{x}^*$. Instead of simply ditching the last $N$ rows, we can avoid computing them altogether, by applying the rules for inverting two by two block matrices. The final solution then becomes

$$ \ma{x} = \Big(\ma{A} – \ma{B} (\ma{A}^*)^{-1} \ma{B}^*\Big)^{-1} \cdot \ma{c} \; – \ma{A}^{-1} \ma{B} \Big(\ma{A}^* – \ma{B}^* \ma{A}^{-1} \ma{B}\Big)^{-1} \cdot \ma{c}^*.$$

So simple. Why didn’t I see it earlier?

2 thoughts on “Widely linear system of equations, revisited”

  1. You might probably also mention that if the inverse of the
    augmented matrix does not exist, then its pseudo inverse
    can be multiplied to augmented vector c to get the WLLS
    solution.

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