Talk:Prony's method

How is it that there are only N samples for F(n) whereas the matrix equation makes use of F from 0 to 2N-1? Bsofly 16:47, 4 December 2006 (UTC)

Formulation is clearly wrong. —Preceding unsigned comment added by 130.203.193.252 (talk) 19:05, 10 October 2007 (UTC)

The formulation is correct. See Eq. 11 of the citation [1]. --Chassin (talk) 19:01, 10 October 2010 (UTC)

Error regarding minus signs

In the article I read:

"Because the summation of complex exponentials is the homogeneous solution to a linear difference equation, the following difference equation will exist:

${\hat {f}}(\Delta _{t}n)=-\sum _{m=1}^{M}{\hat {f}}[\Delta _{t}(n-m)]P_{m},\quad n=M,\dots ,N-1.$

The key to Prony's Method is that the coefficients in the difference equation are related to the following polynomial:

$z^{M}-P_{1}z^{M-1}-\dots -P_{M}=\prod _{m=1}^{M}\left(z-e^{\lambda _{m}}\right).$"

Considering the single-mode situation, where:

${\hat {f}}(\Delta _{t}n)={\hat {f}}(\Delta _{t}0)(e^{\lambda _{1}})^{n}$

we clearly have:

${\hat {f}}(\Delta _{t}n)-e^{\lambda _{1}}{\hat {f}}(\Delta _{t}(n-1))=0$.

Transforming this into a difference equation (whether or not this is the same as Z-transforming, I am not completely sure, but I am more or less following from the article), we get:

$z-e^{\lambda _{1}}=0$.

But this difference equation, according to the notation of the second equation quoted from the article, corresponds to:

$z-P_{1}=0$,

with a minus sign in front of $P_{1}$, so we can identify:

$P_{1}=e^{\lambda _{1}}$.

Substituting this into the first equation quoted from the article, as it is until now, with $M=1$, yields:

${\hat {f}}(\Delta _{t}n)=-{\hat {f}}[\Delta _{t}(n-1)]e^{\lambda _{1}}$,

which clearly contradicts the second equation derived by me.

I conclude that the minus sign on the right-hand side of the first equation quoted above is erroneous, and that a similar flaw is present in the first matrix-vector equation in the article, as it is until now. I will therefore correct these errors shortly.Redav (talk) 17:30, 28 June 2020 (UTC)