  We introduce a new method for Estimation of Signal Parameters based on
Iterative Rational Approximation (ESPIRA) for sparse exponential sums. Our
algorithm uses the AAA algorithm for rational approximation of the discrete
Fourier transform of the given equidistant signal values. We show that ESPIRA
can be interpreted as a matrix pencil method applied to Loewner matrices. These
Loewner matrices are closely connected with the Hankel matrices which are
usually employed for signal recovery. Due to the construction of the Loewner
matrices via an adaptive selection of index sets, the matrix pencil method is
stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT
and the matrix pencil method but with less computational effort. Moreover,
ESPIRA strongly outperforms ESPRIT and the matrix pencil method for noisy data
and for signal approximation by short exponential sums.
