I have a problem which I initially thought was simple but am not so sure anymore. I'd like to solve the following system of equations
$$\alpha_1 = e^{j\mathbf{x}\theta_1} + e^{j\mathbf{x}\theta_2} + e^{j\mathbf{x}\theta_3}$$
$$\alpha_2 = e^{j\mathbf{y}\theta_1} + e^{j\mathbf{y}\theta_2} + e^{j\mathbf{y}\theta_3}$$
$$\alpha_3 = e^{j\mathbf{z}\theta_1} + e^{j\mathbf{z}\theta_2} + e^{j\mathbf{z}\theta_3}$$
where $\mathbf{x},\mathbf{y},\mathbf{z}$ and each $\alpha \in \mathbb{C}$ are given and $\theta_{i}, i \in [1,2,3]$ are the variables to be solved for. Is there some trick to this I'm not seeing?