Let's say we have the following :
$$\quad x_1=Ae^{j\phi_1}+Be^{j\phi_2} $$
$$\quad x_2=Ce^{j\phi_1}+De^{j\phi_2} $$
where:
$\phi_1\ , \phi_2$
,$\,A\,$,$\,B\,$,$\,C\,$,$\,D\,$ are different and unknown
$A, B, C, D \in \mathbb{R}$
All what we know:
$A+B\,$ is known
$C+D\,$ is known
Also what's common between $x_1$ and $x_2$ is that they have the same $\phi_1$ and the same $\phi_2$ Another thing it's easy to have other candidates $x_3\,,$ $x_4\,,$ $x_5\,...$ with the same form above.
My question is :
Is there any way to make $x_2=x_1$ ? Using the fact that we know $C+D$ in $x_2$ and $A+B$ in $x_1$ ?
Or at least, can we suppress $C$ and $D$ in $x_2$ by using the value $C+D$ that we know ? where by suppress I mean to make their value as small as possible.
Motivation
Of course if we only have one complex number, the problem will be easy, like:
if: $$\quad x_1=Ae^{j\phi_1} $$
$$\quad x_2=Ce^{j\phi_1} $$
1) Divide $x_2$ by $C$ $$\frac{x_2}{C}=\frac{Ce^{j\phi}}{C} =e^{j\phi} $$
2) Multiply the result by $A$ : $$A*e^{j\phi} = Ae^{j\phi} = x_1$$
** Note that in this case we Know $A$ and $C$ but above we only know the sum $A+B$ and $C+D$
Idea: you have a linear system with unknowns $A,B,C,D$ (see also the comment by Alex Francisco): $$A + e^{j(\phi_2 - \phi_1)}B = C + e^{j(\phi_2 - \phi_1)}D,$$ $$A + B = k;$$ $$C + D = k.$$ Apply the general theory of linear systems...