# Sum-of-exponentials modeling via Hankel low-rank approximation with palindromic kernel structure

Estimation of a sum-of-damped-exponentials signal from noisy samples of the signal is a classic signal processing problem. It can be solved by maximum likelihood as well as suboptimal subspace methods. In this paper, we consider the related problem of sum-of-exponentials modeling, in which the model is constrained to have no damping. This constraint is difficult to impose in the subspace methods. We develop solution methods using an equivalent Hankel matrix low-rank approximation formulation. A necessary condition for the model to have no damping is that a vector in the kernel of the Hankel matrix has palindromic structure. Imposing this necessary condition in solution methods is trivial. Simulation results show that even for a relatively high noise-to-signal ratios the necessary condition is in fact also sufficient, \ie, the identified model has no dumping. Another contribution of the paper is a method for sum-of-exponentials modeling based on circulant embedding: low-rank approximation of a circulant matrix constructed from the given data. This method imposes the constraint that the model has no damping plus an addition constraint that the model frequencies are on the discrete-Fourier transform's grid.

## Reference

- I. Markovsky and D. Toon Verbeke. Sum-of-exponentials modeling via Hankel low-rank approximation with palindromic kernel structure. Technical report, Dept. ELEC, Vrije Universiteit Brussel, 2018.