General solving approach for general complex Prony problems doesn’t work well for noisy system. Consider the following numerical example: we have two-dimensional trigonometry Prony system (\(S_1\)). Let’s add some noise to the moments and try to reconstruct original system by the general approach (\(S_2\)) and the trigonometric approach (\(S_3\)). The numerical results are in the table \ref{tbl:noisy}.
\label{tbl:noisy}
| \(x_1\) | \(x_2\) | \(a_1\) | \(a_2\) | \(m_0\) | \(m_1\) | \(m_2\) | \(m_3\) | |
|---|---|---|---|---|---|---|---|---|
| \(S_1\) | 0.891-0.454i | 0.951-0.309i | 1+0i | 1+0i | 2+0i | 1.84-0.76i | 1.4-1.4i | 0.74-1.8i |
| \(S_2\) | 0.919-0.385i | 1.93-6.15i | 2.04+0.09i | 0.0005-0.0002i | 2.04+0.09i | 1.91-0.71i | 1.46-1.39i | 0.76-1.7i |
| \(S_3\) | 0.953-0.304i | 0.916-0.402i | 1.02+0i | 1.02+0i | 2.04+0.09i | 1.91-0.71i | 1.46-1.39i | 0.76-1.7i |
| \(S_2-S_1\) | 0.0277+0.0691i | 0.98-5.84i | 1.04+0.09i | -0.999-0i | 0.0425+0.0877i | 0.067+0.0542i | 0.0624+0.0105i | 0.019+0.0951i |
| \(S_3-S_1\) | 0.062+0.15i | -0.0355-0.0932i | 0.0222+0i | 0.0222+0i | 0.0425+0.0877i | 0.067+0.0542i | 0.0624+0.0105i | 0.019+0.0951i |
We can see that the general approach gives a big error for \(x_2\) and \(a_2\). The trigonometric approach solves the noisy problem well.