Appendix B: non-stationary Prony method and shaping regularization

The equation 25 can be rewritten as

$\displaystyle \sum_{m=1}^{M}\hat{a}_m x[n-m] = x[n].$ (26)

If the $\hat{a}_m, m=1,2,\cdots,M$ in equation 26 are time dependent, then we have

$\displaystyle \sum_{m=1}^{M}\hat{a}_m[n] x[n-m] \approx x[n],$ (27)

which is under-determined. There many methods for the solving under-determined linear system. For example, Tikhonov (1963) used the regularization method for making the under-determined problem well-posed by adding constraints on the estimated model.