The complex exponential algorithm
The complex exponential- or Prony-algorithm
represents one possibility to
determine the external
(amplitude, phase) and internal parameters
(damping beta ,
frequency w ) of solutions
of a
linear dynamic
system .
Starting from the general solution of a linear
dynamic system, a
z-transform is carried
out:
The complex exponential algorithm takes a linear combination
on both sides of the equation
with yet unknown coefficients
a . Interchanging the summation indices
leads to:
Setting the resulting complex polynominal
on the right hand side to zero, relates it to
the characteristic polynominal of the linear dynamic
system and assigns
a finite difference equation
to the linear dynamic
system.
The originally non-linear problem is now split into two
linear system of equations 1) and 3) and the
non-linearity is put into the determination of the
roots of the complex
polynominal 2).
The first step 1) of the complex exponential algorithm represents
an autoregressive model
with the autoregressive coefficients a .
The finally derived parameters are
very sensitive to the
accuracy of these autoregressive coefficients in the
overdetermined linear system of equations
1).