Solving complex exponential polynomials

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SUMMARY

The discussion focuses on solving the complex exponential polynomial equation e^(j*m*θ1) + e^(j*m*θ2) + e^(j*m*θ3) + e^(j*m*θ4) + e^(j*m*θ5) = 0, where θ1 < θ2 < θ3 < θ4 < θ5 and m is an integer. Participants suggest that the integer m can be absorbed into the thetas, simplifying the problem. It is concluded that the sum of three terms must be smaller than 2 in magnitude, allowing for the determination of the remaining two exponentials. The reference to the Pentagon theorem indicates a geometric approach to finding solutions.

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beechner224
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Are there any general methods to solve the following complex exponential polynomial without relying on numerical methods? I want to find all possible solutions, not just a single solution.

e^(j*m*[tex]\theta1[/tex]) + e^(j*m*[tex]\theta2[/tex])+e^(j*m*[tex]\theta3[/tex]) + e^(j*m*[tex]\theta4[/tex]) + e^(j*m*[tex]\theta5[/tex]) = 0

where

[tex]\theta1[/tex]<[tex]\theta2[/tex]<[tex]\theta3[/tex]<[tex]\theta4[/tex]<[tex]\theta5[/tex]

and

m is a integer
 
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At first a question:
Why do you include m? You could absorb it into the thetas?
The ordering of the thetas probably doesn't play a role either.

With that in mind you could probably solve these equations this way:
The sum of three terms should be smaller than 2 in magnitude. Then it is always possible to find the final two exponentials.
 
Last edited:

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