In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form $$f(t) = \sum _{j=1}^{K} \gamma _{j} \, \cos (2\pi a_{j} t + b_{j})$$ f ( t ) = ∑ j = 1 K γ j cos ( 2 π a j t + b j ) , where the frequency parameters $$a_{j} \in {\mathbb {R}}$$ a j ∈ R (or $$a_{j} \in {\mathrm i} {\mathbb {R}}$$ a j ∈ i R ) are pairwise different. Our method is based on the recently proposed numerically stable iterative rational approximation algorithm in Nakatsukasa et al. (SIAM J Sci Comput 40(3):A1494–A1522, 2018). For signal reconstruction we use a set of classical Fourier coefficients of f with regard to a fixed interval (0, P) with $$P>0$$ P > 0 . Even though all terms of f may be non-P-periodic, our reconstruction method requires at most $$2K+2$$ 2 K + 2 Fourier coefficients $$c_{n}(f)$$ c n ( f ) to recover all parameters of f. We show that in the case of exact data, the proposed iterative algorithm terminates after at most $$K+1$$ K + 1 steps. The algorithm can also detect the number K of terms of f, if K is a priori unknown and $$L \ge 2K+2$$ L ≥ 2 K + 2 Fourier coefficients are available. Therefore our method provides a new alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.

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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 (MPM) applied to Loewner matrices. These Loewner matrices are closely connected with the Hankel matrices that are usually employed for signal recovery. Due to the construction of the Loewner matrices via an adaptive selection of index sets, the MPM is stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT and the MPM, but with less computational effort. Moreover, ESPIRA strongly outperforms ESPRIT and the MPM for noisy data and for signal approximation by short exponential sums.

In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form [Formula: see text], where the frequency parameters [Formula: see text] are pairwise distinct. In order to reconstruct [Formula: see text] we employ a finite set of classical Fourier coefficients of [Formula: see text] with regard to a finite interval [Formula: see text] with [Formula: see text]. For our method, [Formula: see text] Fourier coefficients [Formula: see text] are sufficient to recover all parameters of [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The recovery is based on the observation that for [Formula: see text] the terms of [Formula: see text] possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA Algorithm for rational approximation, SIAM J. Sci. Comput. 40(3) (2018) A1494A1522]. If a sufficiently large set of [Formula: see text] Fourier coefficients of [Formula: see text] is available (i.e. [Formula: see text]), then our recovery method automatically detects the number [Formula: see text] of terms of [Formula: see text], the multiplicities [Formula: see text] for [Formula: see text], as well as all parameters [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], [Formula: see text], determining [Formula: see text]. Therefore, our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.