where is the amplitude, is the sampling period, is the damping factor, is the angular frequency, is the initial phase. If we let , we then derive the concise form below:

The approximation problem above can be solved based on the error minimization:

This turns to be a nonlinear problem. It can be solved using Prony method that utilizes linear equation solutions. If there are as many data samples as parameters of the approximation problem, the above M equations 18 can be expressed:

20 can be written in a matrix form as below:

Prony proposed to define the polynomial that has the above as its roots (Prony, 1795):

Equation 22 can be rewritten in the form below:

Shifting the index on equation 20 from to , and multiplying by parameter , then we derive:

Notice are roots of equation 23, then equation 24 be written as:

Solving equation 25 for the polynomial coefficients. In subsequent steps we compute the frequencies, damping factors and the phases according to Algorithm 1. After all the parameters are computed, we then compute the components of the input signal. For details see Algorithm 1 as follows:

2020-07-18