Reconstruction of Stationary and Non-stationary Signals by the Generalized Prony Method

Transcription

1 Reconstruction of Stationary and Non-stationary Signals by the Generalized Prony Method Gerlind Plonka Kilian Stampfer Ingeborg Keller March 7, 018 Abstract. We employ the generalized Prony method to derive new reconstruction schemes for a variety of sparse signal models using only a small number of signal measurements. Introducing generalized shift operators, we study the recovery of sparse trigonometric and hyperbolic functions as well as sparse expansions of shifted Gaussians and Gabor functions with Gaussian windows. Furthermore, we show how to reconstruct sparse polynomial expansions and sparse non-stationary signals with structured phase functions. Key words: generalized Prony method, exponential sums, trigonometric sums, Gabor expansions, shifted Gaussians, non-stationary signals, signal reconstruction, empirical mode decomposition. Mathematics Subect Classification: 15A18, 39A10, 41A30, 45Q05, 65F15. 1 Introduction The recovery of signals possessing a given structure is an important problem in various applications as e.g. wireless telecommunication [1], image super-resolution [3], nondestructive testing [5] and phase retrieval [3]. We assume that we can employ certain a priori knowledge about the underlying signal model, where a small number of parameters needs to be determined from given signal measurements in order to recover the structure of the signal. A prototype of such a signal is an exponential sum of the form fx c e xα 1.1 with unknown complex parameters c and α, 1,..., M, which need to be recovered from measurement values of f. Here and in all other signal models we always assume that the parameters α are pairwise different and that all coefficients c are Institute for Numerical and Applied Mathematics, Göttingen University, Lotzestr , Göttingen, Germany. {plonka,i.keller}@math.uni-goettingen.de Institute for Mathematical Stochastics, Göttingen University, Goldschmidtstr. 7, Göttingen, Germany. k.stampfer@math.uni-goettingen.de

2 1 Introduction nonzero, otherwise, the exponential sum can be suitably simplified to a sum with less terms. It is well-known, that this recovery problem can be solved by Prony s method using equidistant point evaluations fx 0 + kh, k 0,..., M 1, where x 0 R can be arbitrarily chosen, and h 0 is a suitable step size. Indeed, the model 1.1 already covers many important applications. For example, taking α it with t R, 1.1 is closely related to the model gt c δt t, 1. a finite stream of Diracs, since the Fourier transform ĝx : gt e itx dt of g is equal to f in 1.1. Therefore, g can be recovered from equidistant Fourier samples. Signals as given in 1. are said to have finite rate if innovation, []. More generally, considering a finite linear combination of arbitrary shifts of a given function φ, gt c φt t 1.3 with t R leads by Fourier transform to a product of an exponential sum M c e it x of the form 1.1 with the Fourier transform φx, and can therefore also be recovered from suitable equidistant Fourier samples of g, see [9,17,19,]. Applying the Laplace transform to 1.1 with the restriction α R, we find hs Lfs c s α. Therefore, the rational function hs can be reconstructed from equidistant samples of its inverse Laplace transform using Prony s method. Also, sparse expansions of shifted Lorentzian functions with the Fourier transform e it x α x /, where t and α denote the shifts and the function width, can be recovered using the model 1.1, see []. However, many applications require more general signal models. In this paper we regard a signal as stationary if it can be e.g. written in the form fx c x e iφ x or c x cosφ x where the amplitudes c x are constants and the phase functions φ x are linear polynomials. Thus the exponential sum in 1.1 is stationary for α it, t R. We say that the signal is non-stationary, if these assumptions are not longer satisfied. In Section 6, we will consider expansions with phase functions being quadratic polynomials or of the form φ x α x p + β with p R +. Non-stationary signals play an important role in many applications in signal processing, since signals naturally change their behavior in time. Stationary signal analysis methods are usually not well suited for decomposing these signals, and some effort has been put into deriving new techniques for non-stationary signal decomposition. In [11] the empirical mode decomposition EMD has been proposed, a

3 1 Introduction 3 greedy non-parametric method to decompose non-stationary signals into so-called intrinsic mode functions. For applications and further investigation of EMD we refer to [1] and the references therein. Mathematically improved techniques include the synchrosqueezed wavelet transform [10] and the signal decomposition method using a signal separation operator [6]. Other recent attempts for signal reconstruction are based on hybrid methods employing wavelet analysis and the finite rate of innovation approach, [3]. Having more a priori information about the signal structure in a sparse model, we aim at a direct identification of the important signal components. In [4], the reconstruction of piecewise sinusoidal signals has been studied with finite rate of innovation methods based on the model 1.1, but this model is still restricted to linear phase functions. Recently, the generalized Prony method has been proposed in [15]. This approach allows the recovery of sparse sums of eigenfunctions of linear operators. It covers the reconstruction of exponential sums in 1.1 as a special case, where the exponentials are interpreted as eigenfunctions of the shift operator. In this paper we want to exploit the generalized Prony method introduced in [15] and derive reconstruction procedures for different signal models that go essentially beyond the exponential sum in 1.1. Here, we particularly restrict ourselves to models that can be recovered ust from direct measurement values, i.e., point evaluations of the signal. In Section.3 we will indicate how other sampling schemes can be also used instead. Employing generalized shift operators we will present new recovery methods for a variety of signal models as e.g. linear combinations of Gaussians, Gabor expansions with Gaussian windows and non-stationary trigonometric expansions. For each of these new models, we will present a reconstruction method and show, which signal measurements are needed for the recovery. The paper is organized as follows. In Section we recall the Prony method and present its interpretation as a recovery method for sparse sums of eigenfunctions of the shift operator. We also describe some sampling schemes to reconstruct the exponential sum which we are allowed to use by exploiting the generalized Prony method. In Subsection.3, we introduce generalizations of the shift operator and show their basic properties. Sections 3 6 are devoted to the investigation of various signal models that are based on the generalized shift operators. In Section 3 we consider sparse expansions of trigonometric and hyperbolic functions that can be reconstructed using the symmetric shift operator. In Section 4 we study the recovery of expansions of shifted Gaussians and Gabor expansions with shifted Gaussian windows. In Section 5 we reconstruct sparse expansions of monomials and complex Gaussians with different scaling. Moreover, sparse expansions of Chebyshev polynomials and linear combinations of non-stationary exponentials as e.g. e α cos can be recovered. Section 6 is devoted to the recovery of further non-stationary signals whose components possess non-linear phase functions of the form α x p + β with known real parameter p > 0 and unknown parameters α, β R, or with phase functions of the form x + α x + β. Finally, we illustrate the signal models by some numerical examples in Section 7. Stability issues of the numerical methods will be more closely considered in a forthcoming paper.

4 Prony s Method for Exponential Sums Revisited 4 Prony s Method for Exponential Sums Revisited.1 The Prony method Let us first recall Prony s method to reconstruct fx in 1.1 using equidistant samples of f. The function fx in 1.1 can be interpreted as the solution of a linear difference equation with constant coefficients. This observation is the key of this recovering method. The main idea is to reconstruct the parameters e α in a first step, and the coefficients c in a second step. If α C, as we have assumed here, we need to put attention to the fact that the α may not be uniquely determined by e α since e ix is π-periodic. We assume therefore that we have an a priori known bound Im α < T. We choose a sampling size h < π T such that α h < π and will reconstruct the values e αh. Then α can be uniquely extracted. We define the characteristic polynomial Prony polynomial M M P z : z e αh z λ.1 with λ : e α h. Assuming that P z has the monomial representation P z k0 M 1 p k z k z M + p k z k, we consider the homogeneous linear difference equation of order M for f in 1.1, k0 p k fhk + m k0 k0 p k M c λ k+m c λ m P λ 0, c λ m M p k λ k k0 which is satisfied for all m Z. Exploiting p M 1 we derive the linear system M 1 k0 p k fhk + m fhm + m, m Z. from the given function values fhl, l 0,..., M 1. The coefficient matrix H fhk + m M 1 k,m0 in. has Hankel structure, and the linear system in. is uniquely solvable, provided that the values λ e αh in 1.1 are pairwise different and c 0 for 1,..., M. This can be easily seen from the factorization H V diag c 1,..., c M V T, where V denotes the Vandermonde matrix V λ k M 1 k,0. Having found the coefficients p k of the Prony polynomial P z by solving., we can extract the zeros λ e α h, 1,..., M and finally determine the parameters c in 1.1 by solving the overdetermined linear system fl c e lα c λ l, l 0,..., M 1.

5 Prony s Method for Exponential Sums Revisited 5 The described method has used the function values fhl, l 0,..., M 1. But a careful inspection of this approach shows that there is no reason to start with f0. We can equivalently start with an arbitrary value x 0 R and apply the samples fx 0 + lh, l 0,..., M 1 to reconstruct 1.1. As before, we get p k fx 0 + hk + m k0 k0 p k M c e x 0α λ k+m M c e x 0α λ m p k λ k k0 c e x 0α λ m P λ 0,.3 which leads to a similar Hankel system as in.. For a recent survey on Prony s method and its applications we refer to [18].. Revisiting Prony s Method Using the Shift Operator Following the ideas in [15], we now want to look at the exponential sum in 1.1 from a different point of view, i.e. as an expansion of eigenfunctions of a suitably chosen operator. We consider the usual shift operator S h : CR CR acting on the vector space CR of continuous functions on R, given by Then, for each α C we have S h f : f + h, h R \ {0}..1 S h e α x e αh+x e αh e αx, i.e., e αx is an eigenfunction of S h with the eigenvalue e αh. Therefore, we can interpret the signal fx in 1.1 as a sparse linear combination of eigenfunctions of the shift operator S h and obtain S h fx c e αx+h c e αh e αx c λ e α x with λ : e α h, and the eigenvalues λ of the active eigenfunctions in the exponential sum are the zeros of the Prony polynomial P z in.1. As we have seen before, the eigenfunction e α x is only determined uniquely by its eigenvalue e α h, if we have the a priori information that Im α is contained in a fixed interval of length π h, since the eigenspace of eα h is very large. For α C the relation S h e α x e α h e α x also implies S h e α + πik h x e x+hα + πik h e αh e α + πik h x. Therefore, for each eigenvalue e αh we find the eigenspace spanned by the eigenfunctions {e xα + πik h : k Z}. We can now reinterpret Prony s method for the reconstruction of the exponential sum in 1.1 as follows. Assume that all wanted parameters α satisfy Im α < T and h < π T.

6 Prony s Method for Exponential Sums Revisited 6 Then, a given finite linear combination of M eigenfunctions of the shift operator S h with the corresponding M pairwise different eigenvalues e α h, 1,..., M, and with nonzero complex coefficients c can be completely recovered from the values S l h fx 0 S hl fx 0, l 0,..., M 1, with x 0 R. Using the shift operator S h in.1 the homogeneous difference equation in.3 can be simply rewritten as k0 p k S k+m h fx 0 M p k S hk+m fx 0 p k S hk+m c e α x 0 k0 k0 k0 M p k c S hk+m e α x 0 M c λ m p k λ k e α x 0 0. k0 c M k0 p k λ m+k e α x 0 As before, the coefficients p k of the Prony polynomial can be computed by the Hankel M 1 system as in. with the coefficient matrix H S hk+l fx 0, and the procedure to evaluate all parameters in 1.1 is the same as before. k, To make this procedure work, we have essentially used two properties of the shift operator, namely, S h is a linear operator, e αx is the unique eigenfunction of S h to the eigenvalue e αh for each α C with Im α π/h, π/h..3 Sampling Schemes for Recovering Exponential Sums As we have seen, the exponential sum in 1.1 can be completely reconstructed using the equidistant function values fx 0 + hl, l 0,..., M 1. These values can be understood as the application of a point evaluation functional F x0 with F x0 S l h f F x 0 S hl f : fhl + x 0. However, the generalized Prony method in [15] allows a higher flexibility of sampling schemes. In [15], it has been shown that instead of using a point evaluation functional we can also employ another linear functional F to S hl f satisfying the assumption that all eigenfunctions of the shift operator which may play an active role in the exponential sum to be recovered do not vanish under the action of F. Using this generalized approach, we replace our measurements S l h fx 0 fx 0 + hl by F S l h fx. Then the recovery of the parameters α can be still achieved by evaluating the coefficients of the Prony polynomial P z in the first step, k0 p k F S k+m h f k0 p k F c S k+m h M k0 M M c e α p k Therefore, we obtain the Hankel system M 1 k0 M 1 p k F S k+m h f k0 p k λ m+k F e α k0 c F S k+m h e α M c λ m p k λ k k0 F e α 0. p k F f + hk + m F f + hm + m,.

7 Prony s Method for Exponential Sums Revisited 7 for m Z, where the Hankel matrix H have M 1 F S k+m h f is invertible, since we k,m0 H V diag c 1,..., c M diag F e α 1,..., F e α M V T with the Vandermonde matrix V λ k M 1 k,0 eαhk M 1 k,0. To illustrate the variety of possible sampling schemes, we give two examples. 1. Assume that we know a priori that the parameters in 1.1 satisfy Im α 0, T and choose 0 < h < π T. Then we can consider F f : x0 +h x 0 fx dx fx χ [0,h] x x 0 dx f, χ [0,h] x 0, where χ [0,h] denotes the characteristic function on [0, h], and the condition F e α x0 +h x 0 e αx dx 0 is obviously satisfied for all α 0, T since e αh 1. With this sampling functional, it is sufficient to take the values F S l h f x0 +h x 0 fx + hl dx x0 +hl+1 x 0 +hl fx dx, l 0,..., M 1, to reconstruct f.. With the assumption α it and t 1, 1, we consider the functional F f : fx Φx dx f, Φ with Φx 1 π sinc x 1 sinx π x. The Fourier transform of the sinc kernel is the box function Φt χ [ 1,1] t and we obtain by Parseval identity with ĝx fx, F f + hl f + hl, Φ ĝ + hl, Φ g, Φ e ilh. In particular, since g in 1. is a stream of diracs, it follows that F f + hl c δ t, Φ e ilh c χ [ 1,1] t e ilht c e ilht. Therefore, also the samples f + hl, Φ, l 0,..., M 1, are sufficient to recover f in Generalized Shift Operators We want to generalize the shift operator in order to be able to recover many more signal models beyond exponential sums. In particular, we consider the following linear operators. A Let S h, h : CR CR denote the symmetric shift operator for given h > 0 by S h, h fx : 1 fx h + fx + h 1 S h + S h fx..3

8 Prony s Method for Exponential Sums Revisited 8 B Let K : R C be a given continuous function satisfying the property Kx, h 1 + h Kx, h Kx + h, h 1 Kx, h 1 Kx + h 1, h..4 We define for h 0 the shift operator S K,h : CR CR by S K,h fx : Kx, h fx + h..5 C Let the function G: [a, b] R be continuous and strictly monotonous in the sampling domain [a, b] R and let G 1 denote its inverse function. We introduce the shift operator S G,h : C[a, b] CR for h 0 by These operators have the following properties. S G,h fx : fg 1 Gx + h..6 Theorem.1. Let the operators S h, h, S G,h and S K,h be defined as above. Then the following holds, S h, h S h1, h 1 f S h1, h 1 S h, h f Sh1 +h, h 1 +h f + S h1 h, h 1 h f,.7 In particular, we have S k h, h f 1 k 1 1 S G,h1 S G,h f S G,h S G,h1 f S G,h1 +h f,.8 S K,h1 S K,h f S K,h S K,h1 f S K,h1 +h f..9 k 1/ S k G,h f S G,khf, S k K,h f S K,khf, k 1 k S l k lh, k lh f + δ k/, k/ k f, k/ where δ k/, k/ 1 if k is even and vanishes otherwise. Proof. We find for the symmetric shift 1 S h, h S h1, h 1 fx S h, h fx + h 1 + fx h fx + h 1 + h + fx h 1 + h + fx + h 1 h + fx h 1 h 1 Sh1 +h, h 1 +h fx + S h1 h, h 1 h fx. Repeated application of the operator S h, h yields S k h, h f 1 k S h + S h k f 1 k k 1 k 1/ k 1 k 1 k 1/ k S h l l Sk l h f k 1 S l k lh + S k lh f + δ k/, k/ k k f k/ k 1 k S l k lh, k lh f + δ k/, k/ k f. k/

9 3 Reconstruction of Expansions of Trigonometric and Hyperbolic Functions 9 Let the continuous function G be strictly monotonous in [a, b] R, and let x, x + h 1, x + h, x + h 1 + h [a, b]. Then S G,h S G,h1 fx S G,h fg 1 G + h 1 x fg 1 GG 1 Gx + h 1 + h fg 1 Gx + h 1 + h S G,h1 +h fx S G,h1 S G,h fx. Finally, using the special properties.4 of the function K, it follows that S K,h S K,h1 fx S K,h K, h 1 f + h 1 x Kx, h Kx + h, h 1 fx + h 1 + h Kx, h 1 Kx + h 1, h fx + h 1 + h S K,h1 S K,h fx. Remark.. 1. Operators of the form.6 are well established as generalized operators in the field of quantum mechanics and are used to solve evolution operator equations in quantum field theory, see [7, 8].. Besides using the generalized shift operators introduced in.3,.4 and.5, we can also combine these shift operators to generate further operators. For example, we will consider S G,h, h fx : 1 fg 1 Gx h + fg 1 Gx + h in Sections 5.3 and 6.1. Similarly, a combination of S K,h and S h, h can be applied. 3. By Theorem.1, the iterated symmetric shift Sh, h k can be presented as a linear combination of shifts S hl, hl for l 0,..., k. Applying the symmetric shift operator we will therefore always use these shifts instead of Sh, h l, l 0,..., k. Since the Chebyshev polynomials T l z : cosl arccos z satisfy the relation x k 1 k 1 k 1/ k 1 k T k l x + δ l k/, k/ k T k/ k/, it will be advantageous to write the Prony polynomial P z as an expansion of Chebyshev polynomials. 3 Reconstruction of Expansions of Trigonometric and Hyperbolic Functions First we employ the symmetric shift operator in order to derive a new method to reconstruct expansions of trigonometric functions. We observe that for each h R, we have and S h, h cosαx 1 [cosαx + h + cosαx h] cosαh cosαx, S h, h sinαx 1 [sinαx + h + sinαx h] cosαh sinαx, i.e., the symmetric shift operator S h, h possesses the eigenfunctions cosαx and sinαx for all α R.

10 3 Reconstruction of Expansions of Trigonometric and Hyperbolic Functions Reconstruction of Cosine Expansions We want to reconstruct an expansion of the form fx c cosα x, 3.10 where we need to recover the unknown coefficients c R \ {0} and frequency parameters α R, 1,..., M. Theorem 3.1. Assume that all parameters α are in the range [0, K R and let h π K. Then, f in 3.10 can be uniquely reconstructed using the M samples fkh, k 0,..., M 1. More generally, for x 0 R satisfying α x 0 k + 1π/ for k Z the 4M 1 sample values fx 0 + hk, k M + 1,..., M 1, are sufficient to reconstruct f in Proof. We define the Prony polynomial which can be written as P z : M z coshα P z p l T l z, where T l z : cosl arccos z denotes the Chebyshev polynomial of first kind of degree l. In particular, p M M+1 since for l 1 the leading coefficient of T l is l 1. As the first step we compute the coefficients p l of the polynomial P z using the sample values. By definition of the Prony polynomial and Theorem.1 we have for f in 3.10, 1 1 p l S lh, lh S mh fx 0 p l fx 0 + m + lh + fx 0 + m lh M p l c [cosα x 0 + m + lh + cosα x 0 + m lh] c cosα x 0 + mh p l cosα lh c cosα x 0 + mh p l T l cosα h 0 for all m 0,..., M 1. Similarly, it follows that p l S lh, lh S mh fx 0 c cosα x 0 mh for all m 0,..., M 1. For x 0 0, we obtain the linear system M 1 p l T l cosα h 0 p l fm + lh + fm lh fm + Mh + fm Mh 3.11 M

11 3 Reconstruction of Expansions of Trigonometric and Hyperbolic Functions 11 for m 0,..., M 1. Since f is an even function, it suffices to know the signal values fkh, k 0,..., M 1 to build this system. The quadratic coefficient matrix has Toeplitz-plus-Hankel structure, M 1 H fm + lh + fm lh m, M M 1 c cosα mh cosα lh V diagc M V T, 3.1 m, with the generalized Vandermonde matrix M 1,M V T k cosα h k0, The matrix V is always invertible, since the terms cosα h are nonzero and pairwise distinct by assumption. Therefore, H is invertible if c 0 for 1,..., M. For x 0 0, we need to take all values S lh, lh S mh, mh fx 0 into account. Here, we apply M 1 p l fx 0 + m + lh+fx 0 m + lh + fx 0 + m lh + fx 0 m lh M fx 0+m+Mh + fx 0 m+mh + fx 0 +m Mh + fx 0 m Mh, and, similarly as in 3.1, the coefficient matrix factorizes in the form H fx 0 +m+lh+fx 0 m+lh+fx 0 +m lh+fx 0 m lh 3.14 M 1 l,m0 M M 1 c cosα x 0 cosα mh cosα lh Vdiagc cosα x 0 M V T. l,m0 The diagonal matrix diagc cosα x 0 M is invertible if we have c 0 and α x 0 k + 1π/ for all k Z and all 1,..., M. Having found the coefficients of the Prony polynomial, we can extract the zeros coshα, 1,..., M. In the second step, we solve the linear system fx 0 + hk c cosα x 0 + hk, k 0,... M 1 in order to compute c, 1,... M. 3. Reconstruction of Sine Expansions The symmetric shift operator can also be applied for the reconstruction of sparse linear combination of sines of the form fx c sinα x, 3.15 with unknown coefficients c R \ {0} and α R \ {0}. This recovery problem is closely related to the problem in 3.10, but we need to pay attention at some details. For example, f0 does not give us any information here, since the function f in 3.15 is odd, and a frequency α 0 does not occur.

12 3 Reconstruction of Expansions of Trigonometric and Hyperbolic Functions 1 Theorem 3.. Assume that all parameter α are in the range 0, K and let h π K. Then, f in 3.15 can be reconstructed using the 4M 1 sample values fx 0 + hk, k M + 1,..., M 1, where x 0 R satisfies sinα x 0 0 for 1,..., M. In particular, the M samples fkh, k 1,..., M are sufficient to reconstruct f in Proof. We proceed similarly as in the last proof. We define the Prony polynomial M P z : z coshα p l T l z, with coefficients p l in the Chebyshev expansion and p M M+1. To compute the coefficients of P z, we obtain a linear system as in 3.14, this time with the coefficient matrix M 1 H fx 0 + m+lh + fx 0 m+lh + fx 0 + m lh + fx 0 m lh M M 1 4 c sinα x 0 cosα mh cosα lh l,m0 4 V diagc sinα x 0 M V T l,m0 with V as in Invertibility follows if sinα x 0 0. This is satisfied for x 0 π K h. Thus, the function values fx 0 + hl fhl + 1, l 0,..., M 1, are already sufficient for the reconstruction, since we have { 0 l 1, fx 0 hl fh1 l fhl 1 l. Having found P z we obtain the zeros coshα and can compute the coefficients c in 3.15 by solving a linear system using the sample values. Remark The symmetric shift operator possesses also the eigenfunctions sinhαx and coshαx with α R. Therefore, sparse expansions of the form and fx fx c coshα x, 3.16 c sinhα x, 3.17 can be reconstructed using at most 4M 1 consecutive sample values fx 0 + kh. Taking the samples fhl, l 0,..., M 1 for 3.16 or fhl, l 1,..., M for 3.17 is also sufficient for reconstructing these expansions.. Obviously, the considered expansions can also be studied using the well-known exponential sums by expanding the trigonometric and hyperbolic functions into sums of exponentials. But in this case, the number of terms in the sparse sums is doubled from M to M. 3. Using the Laplace transform with Lcos α s s and Lsin α s s +α α, we can also reconstruct signals of the form s +α fs c s s + α or fs c α s + α.

13 4 Reconstruction of Expansions Using the Operator S K,h 13 Analogously, models arising for the Laplace transform of cosh and sinh can be recovered. 4 Reconstruction of Expansions Using the Operator S K,h In this section, we study signal models that arise using the generalized shift operator in.5. S K,h fx Kx, h fx + h 4.1 Reconstruction of Expansions of shifted Gaussians Let gx : e βx for some given β C \ {0}. We want to reconstruct an expansion of shifted Gaussians of the form fx c gx α c e βx α, 4.18 and need to recover the M coefficients c C and α R, 1,..., M. Let K 1 x, h : e βhx+h such that for all h 1, h R the relation K 1 x, h 1 + h K 1 x, h 1 K 1 x + h 1, h K 1 x, h K 1 x + h, h 1 is satisfied. Then, the functions e β α are eigenfunctions of S K1,h for all α R, since S K1,h e β α x e βhx+h e βx+h α e βα h e βx α. Theorem 4.1. If Re β 0, the stepsize h R \ {0} can be taken arbitrarily. If Re β 0, we assume that α T, T for 1,..., M for some given T and π choose 0 < h Im β T. Then, f in 4.18 can be reconstructed using the M sample values fx 0 + hk, k 0,..., M 1, where x 0 R is an arbitrary real number. Proof. We define the Prony polynomial M P z : z e hβα p l z l, where the parameters p l denote the coefficients of P z in monomial representation with p M 1. Then we find p l S K1,l+mhfx 0 p l e βhl+mx0+hl+m fx 0 + hl + m p l e βhl+mx 0+hl+m c e βx 0+hl+m α c e βx 0+hm α e βhmx 0+hm p l e βh l +hlx 0 +hm α e βhlx 0+hl+m

14 4 Reconstruction of Expansions Using the Operator S K,h 14 c e βx 0+hm α e βhmx 0+hm p l e β hlα 0 for m 0,..., M 1. The coefficients p 0,..., p M 1 of P z can therefore be computed by the linear system M 1 p l e βhl+mx 0+hl+m fx 0 + hl + m 4.19 e βhm+mx 0+hM+m fx 0 + hm + m, m 0,..., M 1. This system matrix has Hankel structure and its invertibility follows from the factorization H : K 1 x 0, hl + mfx 0 + hl + m M 1 l,m0 e βhl+mx 0+hl+m c e βx 0+hl+m α c e βx 0 α e βhl+mα V h diag c e βα x 0 V T h M 1 l,m0 M 1 l,m0 with the Vandermonde matrix e βα 1h e βα h... e βα M h V h :... e M 1βα 1h e M 1βα h... e M 1βα M h. Having solved 4.19, we can find the zeros e hβα of the Prony polynomial and extract the parameters α, 1,..., M. This is always possible using the supposed restrictions on h. Finally, we solve the linear system fx 0 + hk c e βx 0+hk α in order to compute the coefficients c in Remark The recovery of sums of Gaussians has also been considered in in [14] and in a short note in the multivariate case, see [16], but without using the property, that the Gaussian is an eigenfunction of a suitable generalized shift operator.. The model 4.18 is also of the form 1.3 and can therefore be reconstructed using equidistant Fourier values, as it has been done e.g. in [19].

15 4 Reconstruction of Expansions Using the Operator S K,h Reconstruction of Gabor Expansions with the Gaussian Window Function Similarly as in the previous subsection, we can even consider modulated Gaussians. We now want to recover a Gabor expansion of the form fx c e πixα gx s, 4.0 where gx : e βx is the Gaussian window with known β R \ {0}, and where we have to recover the parameters c, α R, and the shifts s R, 1,..., M. Using again the shift operator S K1,h in.5 with K 1 x, h e βhx+h we observe that indeed e πixα gx s e πixα e βx s are eigenfunctions of S K1,h, S K1,h e πiα β s x 0 e βhx 0+h e πix 0+hα e βx 0+h s e hβs +πiα e πix 0α βx 0 s. Theorem 4.3. Assume that all parameters α are in the range K, K for 1,..., M and let 0 < h 1 K. Then, f in 4.0 can be reconstructed using the M sample values fx 0 + hk, k 0,..., M 1, where x 0 R is an arbitrary real number. Proof. This time, we define the Prony polynomial in the form M P z : z e hπiα +βs p l z l, where p l denote the coefficients of P z in monomial representation with p M 1. We observe that the zeros of the Prony polynomial are complex, where the imaginary part covers the modulation parameters α and the real part the shift parameters s. Then we find p l S K1,l+mhfx 0 p l e βhl+mx0+hl+m fx 0 + hl + m p l e βhl+mx 0+hl+m c e πix 0+hm+lα e βx 0+hl+m s c e βx 0+hm s e βhmx0+hm e πix 0+hmα M p l e lhπiα +βs 0 for m 0,..., M 1. The coefficients p 0,..., p M 1 of P z can therefore be computed by the same system as in 4.19, and we can extract the parameters α and s from the zeros of the Prony polynomial. Finally, the coefficients c are determined by inserting the function values fx 0 + hk into the model 4.0 and by solving the obtained linear system. 4.3 Reconstruction of Generalized Exponential Sums We want to reconstruct expansions of the form fx c xα r e xα, 4.1

16 4 Reconstruction of Expansions Using the Operator S K,h 16 where r R is known and where we have to recover c, α C. Here we employ the shift operator S K,h with K x, h x h+x r satisfying x r x r x + h 1 r. h 1 + h + x h 1 + x h 1 + h + x Then xα r e xα are eigenfunctions of S K,h for each α C, since S K,hα r e α x0 r x 0 α x 0 + h r e α x 0 +h e αh α x 0 r e α x 0. h + x 0 For pairwise different α it follows that the eigenvalues e α h are pairwise different, if the imaginary part of α h is in a fixed interval of length π. Therefore, we will assume that h is chosen such that Im α [ π/h, π/h holds. Theorem 4.4. Let h R \ {0} be such that Im α [ π/h, π/h. Then, f in 4.1 can be reconstructed using the M sample values fx 0 + hk, k 0,..., M 1, where x 0 R \ {0} is an arbitrary real number. Proof. We employ the Prony polynomial P z M z eαh M p lz l with p M 1. We simply observe that p l S K,l+mhfx 0 x r 0 M p l c x 0 + hl + mα r e x 0+hl+mα x 0 + hl + m M c x 0 α r e x 0+hmα p l e hα l 0, and the coefficients p l of P z can be computed by the system M 1 p l x 0 x 0 + hl + m fx 0 + hl + m x 0 x 0 + hm + m fx 0 + hm + m for m 0,..., M 1. The system matrix has Hankel structure with the factorization x 0 M 1 fx 0 + hl + m x 0 + hl + m V diag c x 0 α r e x 0α M V T l,m0 where the Vandermonde matrix V e hαk M 1,M k0, is generated by the knots ehα, 1,..., M. Invertibility is ensured since e hα are pairwise different, c 0 and x 0 0. Using p l to construct the Prony polynomial we firstly compute its roots, recover e αh and afterwards the coefficients c in 4.1 by solving a linear system. Remark 4.5. The expansion in 4.1 is equivalent to fx c x r e xα if we take c c α r. We observe that also more general expansions of the form fx c Hx e xα

17 5 Reconstruction of Expansions Using the Operator S G,h 17 can be recovered, if we assume that the function Hx is known and that K x, h Hx/Hx + h is well-defined. Then, obviously S K,hH e α Hx0 x 0 Hx 0 + h e α x 0 +h e αh Hx 0 e α x 0. Hh + x 0 Alternatively, we can also consider fx fx/hx which is again a simple exponential sum. 5 Reconstruction of Expansions Using the Operator S G,h Now we consider the generalized shift operator S G,h in.6 with S G,h fx fg 1 Gx + h. This operator gives us a lot of freedom to generate generalized shifts. 5.1 Reconstruction of Expansions of Monomials If we consider for example Gx ln x and G 1 x e x, we obtain S ln,h fx fe ln x+h fxe h fxa with a : e h > 0. With other words, S ln,h is the dilation operator with the dilation factor a e h. In particular, all functions of the form x p k with p k C are eigenfunctions of this operator, S ln,h p k x e ln x+h p k x p k e hp k x p k a p k. If the p k are pairwise distinct and if Im p k [ π/h, π/h, then the eigenvalues e hp k are pairwise distinct. We now want to recover an expansion of the form fx c x p, 5. where c C and p C with Im p [ π/h, π/h for all 1,..., M. Theorem 5.1. Let h C \ {0} and a : e h / {e πik/n : k Z} for all N < M be such that a k, k 0,..., M 1 are pairwise distinct values. Then f in 5. can be reconstructed by the M sample values fa k x 0, where x 0 C \ {0} can be chosen arbitrarily. Proof. We define and observe that M P z : z a p k1 p l z l p l fa l+m x 0 M p l c a l+mp x p M 0 c a m x 0 p p l a pl 0

18 5 Reconstruction of Expansions Using the Operator S G,h 18 for m 0,..., M 1, leading to the system M 1 fa l+m x 0 p l fa M+m x 0, m 0,..., M 1. The invertibility of the Hankel matrix H fa l+m x 0 M 1 l,m0 follows from H V diag c x p 0 M V T with the Vandermonde matrix V a pk M 1,M k0, generated by the N pairwise different knots a p, 1,..., M. Once the coefficients of the Prony polynomial P z are found, we compute a p as the zeros of P z, extract p and finally compute c in 5. by solving the system fx 0 a k c x 0 a k p, k 0,..., M 1. Remark 5.. This example was already discussed in [15] using the dilation operator D a with D a fx : fax. Moreover, using the substitution x e x, the model 5. can be transferred to the original model 1.1. If the parameters p are positive integers, then f in 5. is a sparse polynomial, and its reconstruction can be performed using the Ben-Or and Tiwari Algorithm, see e.g. [1, 13]. 5. Expansions of Complex Gaussians with Different Scaling Let now Gx x and G 1 x x for x 0. We consider the corresponding generalized shift operator S x,hfx f x + h and observe that the complex Gaussians e αx with α C are eigenfunctions of this operator, S x,h e α x e α x +h e αh e αx. The parameter α can be uniquely recovered from the eigenvalues e αh if Im αh [ π, π, i.e. if Im α [ π/h, π/h. We now consider the reconstruction of expansions of the form fx c e α x 5.3 where we need to recover c C and α C. Theorem 5.3. Let Im α K, K for some K > 0 for all 1,..., M. We choose h : 1/K. Then the expansion in 5.3 can be uniquely recovered from the samples f x 0 + kh, k 0,..., M 1, where x 0 R can be chosen arbitrarily.

19 5 Reconstruction of Expansions Using the Operator S G,h 19 Proof. We define M P z : z e αh p l z l and observe for the monomial coefficients p l of P z, p l f x 0 + m + lh M p l c e α x 0 +m+lh M c e α x 0 +mh p l e αhl 0 for all m 0,..., M 1. Therefore this system can be used to compute the coefficients p l using p M 1, since the corresponding matrix f x M m + lh can be factorized in the form M 1 f x 0 + m + lh l,m0 V diagc e α x 0 M V T l,m0 with the Vandermonde matrix V e hαk M 1,M k0, generated by the pairwise different knots e hα, 1,..., M. Having found P z, we can extract its zeros, recover p and finally also c by solving a linear system. Remark 5.4. Similarly, using Gx x p and G 1 x p x with given p > 0 we can recover expansions of the form fx c e α x p p by reconstructing c, α C from the samples f x p 0, + hk k 0,..., M 1. Taking e.g. Gx cosx for x [0, π] and G 1 x arccosx we obtain the operator S cos,h fx farccoscosx + h which has the eigenfunctions e α k cos x with S cos,h e α k cos x e α k cosarccoscos x+h e α kcos x+h. In this way, we can also recover expansions of the form fx c e α cos x with parameters c, α C using the samples farccoscos x+kh, k 0,... M 1 with suitably chosen h.

20 5 Reconstruction of Expansions Using the Operator S G,h Sparse Expansions of Chebyshev Polynomials Let now x [ 1, 1] and Gx arccos x. Then G is monotonous in [ 1, 1] and G 1 y cos y for y [0, π]. We apply a combination of the symmetric shift operator S h, h and S G,h, S G,h, h fx : 1 fcosarccosx + h + fcosarccosx h, which is also called Chebyshev shift operator, see also [15, 0]. Then the Chebyshev polynomials T k x cosk arccos x of degree k 0 are eigenfunctions of this operator, S G,h, h T k x 1 T kcosarccosx + h + T k cosarccosx h 1 cos karccosx + h + cos karccosx h coskh cosk arccos x coskh T k x. We want to reconstruct a sparse Chebyshev expansion of the form fx c T n x, 5.4 where we need to recover the unknown indices 0 n 1 < n <... < n M and the coefficients c R. We assume that an upper bound K of the degree n M of the polynomial in 5.4 is a priori known. Theorem 5.5. Let K be a bound of the degree of the polynomial f in 5.4 and let 0 < h π K. Then the Chebyshev expansion in 5.4 can be uniquely recovered from the samples fcoskh, k 0,..., M 1. Proof. Let P z M z cosn h M p lt l z, where p l are the coefficients of P z in the Chebyshev expansion, and p M M+1. Then, since the cosine is even, we obtain p l SG,m+lh, m+lh fcos0 + S G,m lh, m lh fcos0 p l fcosm + lh + fcosm lh M p l c Tn cosm + lh + T n cosm lh c M p l cosn ml cosn lh c cosn mh p l T l cosn h 0. This observation leads to the linear system fcosm + lh + fcosm lh M 1 l,m0 p M+1 fcosm + Mh M 1 m0

21 6 Reconstruction of Non-stationary Signals 1 to evaluate the vector p p 0,..., p M 1 T of Prony polynomial coefficients. It can be simply shown that the coefficient matrix of this system is invertible, provided that cosn h are pairwise distinct. Having P z it is simple to recover the indices n and afterwards the coefficients c. Remark Similarly as in Sections 3.1 and 3., Theorem 5.5 can be generalized by taking samples fcosx 0 + kh, k M + 1,..., M 1, and the choice of x 0 governs the number of needed function values taking into account that cosine is even.. A numerical algorithm for the reconstruction of sparse expansions of Chebyshev polynomials can be already found in [0]. The approach can also be transferred to Chebyshev polynomials of second, third and fourth kind, see [0]. However, in [0] the connection to shift operators with Chebyshev polynomials as eigenfunctions has not been explicitly used. 6 Reconstruction of Non-stationary Signals Within the last years, more efforts have been made to reconstruct non-stationary signals of the form fx c x cosφ x. The empirical mode decomposition method described in [11,1] is a heuristic iterative method to decompose a given non-stationary signal into certain signal components. However, this algorithm does not always provide the wanted signal components in a suitable way. If there is more a priori information on the envelope functions c x and the phase functions φ x available we may be able to exploit it in a direct way. Using the Prony method with generalized shifts we will consider some models of non-stationary signals. In particular, we will study constant envelope functions and polynomial phase functions with a priori known polynomial structure. 6.1 Non-stationary Signals with Phase Functions φ x α x p + β We want to recover signals of the form fx c cosα x p + β, 6.5 where the odd integer p > 0 is a priori known, and where the coefficients c, α R and β [ π, π need to be recovered. We assume here that the α are pairwise different and nonnegative. This last assumption is not a restriction since cos x is an even function. First, we construct a generalized shift operator that possesses the eigenfunctions cosα x p + β. We employ the operator S x p,h, h : CR CR with h > 0, which is a combination of the symmetric shift operator S h, h and the operator S G,h with Gx x p sgnx x p and G 1 x sgnx p x, given by S x p,h, hfx : 1 f sgnx p + h p x p + h + f sgnx p h p x p h. Here sgnx denotes the sign of x and is 1 for x > 0, 1 for x < 0, and 0 for x 0. We find S x p,h, h cos α x p + β 1 cos α sgnx p + h p p x p + h + β

22 6 Reconstruction of Non-stationary Signals + 1 cos α sgnx p h p p x p h + β 1 cos α x p + h + β + cos α x p h + β cos α x p + β cosα h. The eigenvalues cosα h and cosα k h are pairwise different for α α k if α, α k [0, π/h]. We conclude Theorem 6.1. Let f be of the form 6.5 with known odd integer p > 0, and with unknown parameters c R, β [0, π and pairwise different α [0, K for some K > 0 for all 1,..., M. Let h : π/k. 1. If the parameters β do not appear in the model 6.5, then f can be uniquely recovered from its signal values f sgnx 0 + hk p x 0 + hk, k M + 1,..., M 1, where x 0 0 only needs to satisfy cosα x 0 0 for 1,..., M. Taking x 0 0, the function values f p hk, k 0,..., M 1 are already sufficient for the reconstruction of the parameters α, c, 1,..., M.. If the nonzero parameters β appear in 6.5, then the parameters α, 1,..., M, can be recovered from signal values f sgnx 0 + hk p x 0 + hk, k M + 1,..., M 1 in a first step. Using in a second step additionally the signal values f sgnx 0 + hk π/α p x 0 + hk π/α for k M + 1,..., M 1, the parameters c and β can be reconstructed. The value x 0 needs to be chosen such that cosα x 0 + β 0 for 1,..., M. Proof. We consider the Prony polynomial of the form M P z : z cosα h p l T l z, where p l denote the coefficients in the representation of P z using the Chebyshev polynomials T l z cosl arccos z with p M M+1. Then we observe that M 1 p l S x p,hm+l, hm+lf p x 0 + S x p,hm l, hm lf p x 0 1 p l f p x 0 + hm + l + fsgnx 0 hm + l p x 0 hm + l +fsgnx 0 + hm l p x 0 + hm l +fsgnx 0 hm l p x 0 hm l M 1 p l c cos α x 0 + hm + l + β + cosα x 0 hm + l + β + cos α x 0 + hm l + β + cosα x 0 hm l + β c cosα x 0 + β cosα hm p l cosα hl c cosα x 0 + β cosα hm p l T l cosα h 0

23 6 Reconstruction of Non-stationary Signals 3 for all m 0,..., M 1. We obtain the linear system with p p 0,..., p M 1 T and with H f p x0 +hm+l +f Hp M+1 f M +f sgnx 0 hm+l p x 0 hm+l +f +f sgnx 0 hm+m p x 0 hm+m +f sgnx 0 +hm l p x 0 +hm l f M f p x0 +hm+m +f sgnx 0 +hm M p x 0 +hm M sgn x 0 hm l p x 0 hm l M 1, m, sgnx 0 hm M p x 0 hm M M 1, m0 to compute the coefficients p l of the Prony polynomial in Chebyshev representation. The coefficient matrix of this system has the form H V diagc cosα x 0 + β M V T 6.6 with the generalized Vandermonde matrix V cosα hl M 1 l,0 T lcosα h M 1 l,0 as in These Vandermonde matrices are invertible if the values cosα h are pairwise different, which is ensured by the choice of h. The invertibility of the diagonal matrix is ensured if c 0 and if α x 0 + β is not of the form πk + 1/ for some k 0. In particular, for vanishing β we can simply use x 0 0 and need only the function values flh, l 0,... M 1 to recover α, since f in 6.5 is an even function. Having found the parameters α by the described procedure, in the case of vanishing β we can simply obtain the values c cosα x 0 by solving the linear system c cosα x 0 + lh + cosα x 0 lh c cosα x 0 cosα lh f p x 0 + lh + fsgnx 0 lh p x 0 lh for l 0,..., M 1, where the coefficient matrix is the same generalized Vandermonde matrix V as in 6.6. If the model contains nonvanishing parameters β, then we have to solve the system c cosα x 0 + lh + β + cosα x 0 lh + β c cosα x 0 + β cosα lh f p x 0 + lh + fsgnx 0 lh p x 0 lh to obtain d : c cosα x 0 + β. In addition, we have to solve c cosα x 0 + lh π + β + cosα x 0 lh π + β c cosα x 0 π + β cosα lh

24 6 Reconstruction of Non-stationary Signals 4 f sgnx 0 π α +lh p x0 π α +lh +f sgnx 0 π α lh p x0 π α lh for l 0,..., M 1, to find d : c cosα x 0 π + β sinα x 0 + β for 1,..., M. Thus, we conclude c d + d, β argd + i d α x 0 mod π. 6. Non-stationary Signals with Quadratic Phase Functions Finally, we consider signals of the form fx c cosx + α x + β 6.7 with parameters c R, α T, T for some T > 0, and β [ π, π ], 1,..., M. This model can be rewritten as fx c cos β cosx + α x c sin β sinx + α x c,1 cosx + α x c, sinx + α x c,1 + i c, e ix +α x + c,1 i c, b e ix+α / α /4 + b e ix+α / α /4 Re Re b e iα /4 e ix+α / e ix +α x d e ix+α /, 6.8 where we have used the substitutions c,1 : c cos β, c, : c sin β, b : and d : b e iα /4. Similarly, we observe that fx c sinx + α x + β c cosx + α x + β π Im d e ix+α / c,1 +ic,, with d b e iα /4. The model is therefore closely related to the model in 4.18 with β i. We conclude

25 7 Numerical examples 5 Theorem 6.. Assume that β [ π, π and that α T, T, 1,..., M, for some T > 0 and let 0 < h π T. Then, f in 6.7 can be reconstructed using the M sample values fx 0 + hk, k 0,..., M 1 and the M sample values fx 0 + hk, where x 0 R is an arbitrary real number. Proof. Considering the function hx fx + i fx, we can apply Theorem 4.1 to recover the parameters d and α for 1,..., M. The original parameters c and β, 1,..., M, are then obtained using the relations b d e iα /4, c,1 Re b, c, Im b, c b, β arg b, sgn c sgn c,1. 7 Numerical examples In this section we want to illustrate the recovery method for non-stationary signals with some examples. Example 7.1. We start with considering the recovery of an expansion of complex shifted Gaussians, fx c gx α c e βx α, with M 5, gx e ix, i.e., β i, and with complex coefficients c and real shifts α given in Table 1. The coefficients have been obtained by applying a uniform random choice from the intervals 5, 5 + i, for c and from π, π for α. For reconstruction, we have used the 10 signal values f, 1,..., 8, indicated by in Figure 1 left. The maximal error for recovering the parameters is given by max c c , max α α , where c and α denote the computed parameters Re c Im c α Table 1 Coefficients c C and α R for the expansion of shifted Gaussians in Example 7.1. Example 7.. Next, we consider the recovery of a Gabor expansion of the form fx c e πixα gx s, with gx e x /, M 6, real coefficients c, α and s as given in Table. The coefficients have been obtained by applying a uniform random sampling from

26 7 Numerical examples 6 15 real part 5 real part imaginary part 4 imaginary part Figure 1 Left: Real and imaginary part of the signal fx consisting of shifted Gaussians given in Example 7.1. Right: Real and imaginary part of the Gabor expansion considered in Example 7.. Stars indicate the used signal values. the intervals 10, 10 for c, from 5, 5 for s and from 0, 1 for α. For the reconstruction we have used the 1 signal values fl, l 0,..., 11 indicated by in Figure 1 right. For the errors we obtain in this example max c c , max α α < , max s s < , where c, α, s denote the parameters computed by the numerical procedure c s α Table Coefficients c, α, s R for the Gabor expansion in Example 7.. Example 7.3. Finally, we consider two examples for the model with quadratic phase function fx c cosx + α x + β in 6.7. In Figure left, we display a signal with M 3 components with corresponding parameters given in Table 3. For reconstruction, we have used the signal values fl, l 0,..., 5. In Figure right, we give a second example with coefficients given in Table 4. Here, M 6, and we have used the signal values f 1 + 5l 1, l 0,..., 11 for reconstruction. The coefficients have been obtained by applying a uniform random sampling from the intervals 1, 5 for c in the first and from 0, 5 in the second example, from π, π for α and from π/, π/ for β for both examples. The reconstruction errors in the first example with M 3 terms are max c c , max α α , max β β

27 8 Acknowledgement 7 For the second example with M 6 we obtain max c c , max α α , max β β Figure Left: non-stationary signal fx with quadratic phase function with parameters given in Table 3. Right: non-stationary signal fx with quadratic phase function with parameters given in Table 4. Stars indicate the used signal values. 1 3 c α β Table 3 Coefficients c, α, β R for the non-stationary signal in Figure left c α β Table 4 Coefficients c, α, s R for the non-stationary signal in Figure right. 8 Acknowledgement The authors gratefully acknowledge partial support by the German Research Foundation in the proect PL 170/16-1 and in the framework of the RTG 088. Further, the authors thank Thomas Peter for valuable suggestions to improve the manuscript. References [1] M. Ben-Or and P. Tiwari, A deterministic algorithm for sparse multivariate polynomial interpolation, in: Proc. Twentieth Annual ACM Symp. Theory Comput., ACM Press New York, 1988, pp

28 References 8 [] G. Baechler, A. Scholefield, L. Baboulaz, and M. Vetterli, Sampling and exact reconstruction of pulses with variable width, IEEE Trans. Signal Process. 6510, 017. [3] R. Beinert and G. Plonka, Sparse phase retrieval of one-dimensional signals by Prony s method, Frontiers of Applied Mathematics and Statistics 3:5 017, open access, doi: /fams [4] J. Berent, P.L. Dragotti, and T. Blu, Sampling piecewise sinusoidal signals with finite rate of innovation methods, IEEE Trans. Signal Process , [5] F. Boßmann, G. Plonka, T. Peter, O. Nemitz, and T. Schmitte, Sparse deconvolution methods for ultrasonic NDT, Journal of Nondestructive Evaluation , [6] C.K. Chui and H.N. Mhaskar, Signal decomposition and analysis via extraction of frequencies, Appl. Comput. Harmon. Anal , [7] G. Dattoli, P.L. Ottaviani, A. Torre, and L. Vázquez, Evolution operator equations: integration with algebraic and finite- difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory, Rev. Nuovo Cim , [8] G. Dattoli, P.E. Ricci, and D. Sacchetti, Generalized shift operators and pseudopolynomials of fractional order, Appl. Math. Comput , [9] P.L. Dragotti,, M. Vetterli, and T. Blu, Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets Strang-Fix, IEEE Trans. Signal Process [10] I. Daubechies, J. Lu, and H.-T. Wu, Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool, Appl. Comput. Harmon. Anal , [11] N.E. Huang, Z.Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.-C. Yen, C.C. Tung, and H.H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. Lond. A , [1] N.E. Huang and S.S.P. Shen eds., Hilbert-Huang transform and its applications, World Scientific Publishing, Singapore, second edition, 014. [13] E. Kaltofen and W.-s. Lee, Early termination in sparse interpolation algorithms, J. Symb. Comput , [14] T. Peter, Generalized Prony Method, PhD thesis, University of Göttingen, Der Andere Verlag 014. [15] T. Peter and G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, Inverse Problems 9 013, [16] T. Peter, G. Plonka, and R. Schaback, Prony s Method for Multivariate Signals, Proc. Appl. Math. Mech , [17] T. Peter, D. Potts, and M. Tasche, Nonlinear approximation by sums of exponentials and translates, SIAM J. Sci. Comput. 334, [18] G. Plonka and M. Tasche, Prony methods for recovery of structured functions, GAMM- Mitt , [19] G. Plonka and M. Wischerhoff, How many Fourier samples are needed for real function reconstruction? J. Appl. Math. Comput , [0] D. Potts and M. Tasche, Sparse polynomial interpolation in Chebyshev bases, Linear Algebra Appl , [1] R. C. Qiu and I-T. Lu, Multipath resolving with frequency dependence for wide-band wireless channel modeling, IEEE Trans. Vehicular Technology , [] M. Vetterli, P. Marziliano, and T. Blu, Sampling signals with finite rate of innovation, IEEE Trans. Signal Process , [3] X. Wei and P.L. Dragotti, FRESH FRI based single-image super-resolution algorithm, IEEE Trans. Image Process ,