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 (2pi a_{j} t + b_{j}), where the frequency parameters a_{j} in {mathbb {R}} (or a_{j} in {mathrm i} {mathbb {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. Even though all terms of f may be non-P-periodic, our reconstruction method requires at most 2K+2 Fourier coefficients 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 steps. The algorithm can also detect the number K of terms of f, if K is a priori unknown and L ge 2K+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.
Highlights
Classical Fourier analysis methods provide for any real square integrable signal f (t) a Fourier series representation on a given interval (0, P), P > 0, of the form f (t) = ∞cn ( f ) e2πint/P n=−∞ α0 cos(β0) 2 + αn n=1 cos 2π nt P − βn (1.1)with Fourier coefficients
For f as given in (1.3) the parameter K ∈ N, as well as the parameters γ j ∈ (0, ∞), b j ∈ [0, 2π ), j = 1, . . . , K, and 0 < a1 < a2 < · · · < aK < ∞ are uniquely defined from the function values f (t) for t ∈ T, where T ⊂ R is an interval of positive length
(1.3) from their Fourier coefficients. This method exploits the special structure of the Fourier coefficients of f on some interval [0, P)
Summary
Classical Fourier analysis methods provide for any real square integrable signal f (t) a Fourier series representation on a given interval (0, P), P > 0, of the form f (t). Compared to other rational interpolation algorithms, a further important advantage of the employed modified AAA algorithm is that we do not need a priori knowledge on the number K of terms in (1.3) but can determine K in the iteration process, supposed that L ≥ 2K + 1 Fourier coefficients are available. That a signal f with K non-P-periodic terms f j as in (1.3) can theoretically be determined from 2K Fourier coefficients cn( f ) with n ∈ ⊂ N if K is known beforehand. It turns out that there is no a priori information needed about possibly occurring P-periodic terms of f In this case, we first compute the rational function r (z) that determines the non-P-. I.e., our method determines the number K of terms as well as all
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