Exact reconstruction of sparse non-harmonic signals from their Fourier coefficients

Abstract

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.

From ESPRIT to ESPIRA: Estimation of Signal Parameters by Iterative Rational Approximation

“…(As we will show in Section 3.5, the type of the resulting function r M (z) will indeed be (M − 1, M ) for exact data.) At the iteration step J ≥ 1, we proceed as follows to compute a rational function r J−1 of type (J − 1, J − 1), see also [24,13,27]. From the previous iteration step we have already a given partition S J ∪ Γ J = I of index sets which have been found by an adaptive (greedy) procedure.…”

“…The zeros of the denominator qM (z) are the poles z j of r M (z) and can be computed by solving an (M + 2) × (M + 2) generalized eigenvalue problem (see [24] or [27]), that has for…”

“…Two eigenvalues of this generalized eigenvalue problem are infinite and the other M eigenvalues are the wanted zeros z j of qM (z) (see [21,24,27] for more detailed explanation). We apply Algorithm 5 to the output of Algorithm 4.…”

“…In our recent work [13,27], we have investigated a different approach for parameter reconstruction for exponential sums, where we employed a finite set of Fourier coefficients of the signal f that occur in a Fourier expansion of f on a finite interval [0, P ]. We have shown in [13,27] that the signal recovery problem can then be rephrased as a rational interpolation problem, which in turn is solvable in a stable way using the AAA algorithm [24]. The AAA algorithm is available in Chebfun, see [14].…”

Modifications of Prony's Method for the Reconstruction of Structured Functions

“…. , M and j = k, are also analysed in [PPD21]. The authors show that such functions can be reconstructed using Fourier coefficients and rational approximation.…”