How to calculate the least squares estimator

Question

We have a complex samples y¯^(1) that are modeled by a known real vector a = [a1,..., an]^T multiplied by an unknown complex scalar α: y¯(1) ≈ a¯ . α

What would be a least squares estimator for α?

Answer 1

Let $y_j = y'_j + i y''_j$ be the complex samples. The least squares optimization problem is minimizing $$f(u, v) := \sum_{j=1}^n |(y'_j + iy''_j) - \underbrace{(u+iv)}_{=:\alpha}a_j|^2 = \sum_{j=1}^n (y'_j-ua_j)^2 + \sum_{j=1}^n (y''_j - va_j)^2.$$

Taking the partial derivatives with respect to $u$ and $v$ and setting each to zero will give you an expression for the optimal $u$ and $v$.