Is it possible to solve a system of equations for the phase of complex exponentials
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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
$$alpha_1 = e^{jmathbf{x}theta_1} + e^{jmathbf{x}theta_2} + e^{jmathbf{x}theta_3}$$
$$alpha_2 = e^{jmathbf{y}theta_1} + e^{jmathbf{y}theta_2} + e^{jmathbf{y}theta_3}$$
$$alpha_3 = e^{jmathbf{z}theta_1} + e^{jmathbf{z}theta_2} + e^{jmathbf{z}theta_3}$$
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?
systems-of-equations non-convex-optimization
edited 2 days ago
Torsten Schoeneberg
3,3761832
asked Nov 12 at 20:48
Mark Wagner
62
add a comment |
up vote
1
down vote
favorite
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
$$alpha_1 = e^{jmathbf{x}theta_1} + e^{jmathbf{x}theta_2} + e^{jmathbf{x}theta_3}$$
$$alpha_2 = e^{jmathbf{y}theta_1} + e^{jmathbf{y}theta_2} + e^{jmathbf{y}theta_3}$$
$$alpha_3 = e^{jmathbf{z}theta_1} + e^{jmathbf{z}theta_2} + e^{jmathbf{z}theta_3}$$
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?
systems-of-equations non-convex-optimization
edited 2 days ago
Torsten Schoeneberg
3,3761832
asked Nov 12 at 20:48
Mark Wagner
62
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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
$$alpha_1 = e^{jmathbf{x}theta_1} + e^{jmathbf{x}theta_2} + e^{jmathbf{x}theta_3}$$
$$alpha_2 = e^{jmathbf{y}theta_1} + e^{jmathbf{y}theta_2} + e^{jmathbf{y}theta_3}$$
$$alpha_3 = e^{jmathbf{z}theta_1} + e^{jmathbf{z}theta_2} + e^{jmathbf{z}theta_3}$$
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?
systems-of-equations non-convex-optimization
edited 2 days ago
Torsten Schoeneberg
3,3761832
asked Nov 12 at 20:48
Mark Wagner
62
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
$$alpha_1 = e^{jmathbf{x}theta_1} + e^{jmathbf{x}theta_2} + e^{jmathbf{x}theta_3}$$
$$alpha_2 = e^{jmathbf{y}theta_1} + e^{jmathbf{y}theta_2} + e^{jmathbf{y}theta_3}$$
$$alpha_3 = e^{jmathbf{z}theta_1} + e^{jmathbf{z}theta_2} + e^{jmathbf{z}theta_3}$$
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?
systems-of-equations non-convex-optimization
systems-of-equations non-convex-optimization
edited 2 days ago
Torsten Schoeneberg
3,3761832
asked Nov 12 at 20:48
Mark Wagner
62
edited 2 days ago
Torsten Schoeneberg
3,3761832
asked Nov 12 at 20:48
Mark Wagner
62
edited 2 days ago
Torsten Schoeneberg
3,3761832
edited 2 days ago
Torsten Schoeneberg
3,3761832
edited 2 days ago
Torsten Schoeneberg
3,3761832
3,3761832
asked Nov 12 at 20:48
Mark Wagner
62
asked Nov 12 at 20:48
Mark Wagner
62
asked Nov 12 at 20:48
Mark Wagner
62
62
add a comment |
add a comment |
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