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)