What is the relation between the following two complex functions:
$$g(\theta)=\sum_n x[n]\ y[n]\ e^{in\theta}$$
and
$$f(\theta)=\sum_n \left(x[n]\pm i\sqrt{1-x[n]^2}\right)\ y[n]\ e^{in\theta}$$
where $\theta$ ranges from $-\pi$ to $\pi$ and $x[n]$ is real and smaller than 1 and $y[n]$ is in general complex ? Can we find a single equation that relates the two functions ?