Assume $x_1=r_1e^{j\theta_1}，x_2=r_2e^{j\theta_2},x_2=r_3e^{j\theta_3}$, $r_1,r_2,r_{3}$, where $r_1,r_2,r_3$ are magnitudes of the three complex values.
Let $$x_{12}= x_1+x_2$$ $$x_{w12} = e^{jw_1} x_1+e^{jw_2}x_2$$ $$x_{123} = x_1+x_2+x_3$$ $$x_{w123} = e^{jw_1} x_1+e^{jw_2}x_2+e^{jw_3} x_3$$
And what I have known are all the magnitudes, including$r_1,r_2,r_3,|x_{12}\;|,|x_{w12}\;|,|x_{123}\;|,|x_{w123}\;|$, and $w_1,w_2,w_3$, how can I determine the phase difference between the three complex values? Namely, $\Delta_{2,1} = \theta_2-\theta_1$,$\Delta_{3,1} = \theta_3-\theta_1$.
Actually, I tried to use 'acos' to determine the phase difference $\theta = acos\left(\cfrac{|x_{12}\;|^2-r_1^2-r_2^2}{2r_1r_2}\right)$, In that way, the angle may be postive or negative, so there is ambiguity. And then I tried to rotate, Namely, $$ \arg \min_{\theta,-\theta} \left(\vert x_{w12}\; \vert- |e^{j\theta}e^{jw_1} r_1+ e^{jw_2} r_2|\right)^2$$ but it didn't work for three values.
Any comments would appreciated! Thanks!
We have the situation depicted in the sketch
In doing the addition of $x_1,x_2$, known in polar form, the sign of $\alpha = \theta_{12}- \theta_1$ is determined by the sign of $\theta_2 -\theta_1$.
In fact we can write $$ \eqalign{ & r_{\,12} e^{\,i\,\theta _{\,12} } = r_{\,1} e^{\,i\,\theta _{\,1} } + r_{\,2} e^{\,i\,\theta _{\,2} } = r_{\,1} e^{\,i\,\theta _{\,1} } \left( {1 + {{r_{\,2} } \over {r_{\,1} }}e^{\,i\,\left( {\theta _{\,2} - \theta _{\,1} } \right)} } \right) \cr & e^{\,i\,\left( {\theta _{\,12} - \,\theta _{\,1} } \right)} = {{r_{\,1} } \over {r_{\,12} }}\left( {1 + {{r_{\,2} } \over {r_{\,1} }}e^{\,i\,\left( {\theta _{\,2} - \theta _{\,1} } \right)} } \right) = {{r_{\,1} } \over {r_{\,12} }} + {{r_{\,2} } \over {r_{\,12} }}e^{\,i\,\left( {\theta _{\,2} - \theta _{\,1} } \right)} \cr & sign\left( {\theta _{\,12} - \,\theta _{\,1} } \right) = sign\left( {\theta _{\,2} - \theta _{\,1} } \right) \cr} $$
Knowing the length of the triangle sides $r_1, \, r_2, \, r_{12}$, then by the sines law you can determiine the absolute values of the angles $\alpha, \, \beta, \, \gamma$.
If $\theta_{12}$ is known, then you can place the diagonal $x_{1,2}$, however it remains undetermined the reflection of the triangle around it.