Consider the following system of signals: \(S = \{ a, b, \mu, \nu \}\). We assume that we know the following \(\{m_k\}\):
\[\label{eq:TProny2D-mk} m_k = U(k\delta) = a e^{-2\pi i \mu k \delta} + b e^{-2\pi i \nu k \delta} = a (e^{-2\pi i \mu \delta})^k + b (e^{-2\pi i \nu \delta})^k, \quad k = 0, 1, 2.\]
Introduce
\[x = e^{-2\pi i \mu \delta}, \quad y = e^{-2\pi i \nu \delta}.\]
Then the source system can be rewritten in the following form:
\[m_k = ax^k + by^k.\]
The solution can be obtained from the following formulas:
\[M_k = |m_k|, \quad \phi = -2\pi \mu \delta, \quad \theta = -2\pi \nu \delta, \quad \Delta = \phi - \theta,\]
\[\Delta = \arccos \Bigg( \dfrac{2 M_1^2 - M_0^2 - M_2^2}{2(M_0^2 - M_1^2)} \Bigg)\]
\[D = M_0^2 - 4\dfrac{(M_1^2 - M_0^2)^2}{M_2^2 + 3M_0^2 - 4M_1^2}\]
\[a = \dfrac{M_0 \pm \sqrt{D}}{2}, \quad b = \dfrac{M_0 \mp \sqrt{D}}{2}\]
\[\theta = -\arccos \Big( \dfrac {\Re m_1 (a\cos\Delta + b) + \Im m_1 (a\sin\Delta)} {(a\cos\Delta + b)^2 + (a\sin\Delta)^2} \Big), \quad \phi = \theta + \Delta.\]