I am doing an implementation of the Poly-Reference Least Squares Complex Frequency Domain algorithm for modal analysis as described in various papers like: "A poly-reference implementation of the least-squares complex frequencydomain estimator" by Guillaume et. al.
I have the following equation:
$$H(w) = \sum_{i=1}^n \left(\frac{ v_i \ l_i^T}{jw-\lambda_i} + \frac{ v_i^* \ l_i^H}{jw-\lambda_i^*} \right)- \frac{LR}{w^2} + UR$$
$$v_i \in \mathbb{C}^l, l_i \in \mathbb{C}^m$$ $$LR, UR \in \mathbb{R}^{l \times m}$$
This needs to be solved in a linear least squares fashion, I need the values for $v_i$, $LR$ and $UR$. Assuming at first that $l,m =1$, I would solve this the classical way via a parameter vector
$$\theta = [v_1, ..., v_n,v_1^*, ..., v_n^*, LR, UR]$$
and the pseudo-inverse. But since I don't know how to enforce that $v_i$ and $v_i^*$ are actually complex conjugated, I believe that approach can not be correct. I don't see a way how to solve for $v_i$ only or likewise transform the first term to not contain the complex conjugates of $v_i$.
Any help or reference would be much appreciated.
EDIT: After splitting $v_i$ into real and imaginary parts and rewriting and solving the problem, I actually do not get only real coefficients for x and y, which leads me to believe, that I have done something wrong. Is there any way how I could avoid splitting $v_i$ into real and imaginary or trying to solve for both complex and complex conjugate $v_i$ and $v_i^*$?