Nonlinear Approximation by Sums of Exponentials and Translates

Abstract

Abstract. In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem. Let h be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands if finitely many perturbed, uniformly sampled data of h are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too. The second approximation problem is related to the first one and reads as follows: Let ϕ be a given 1-periodic window function as defined in section 4. Further let f be a linear combination of translates of ϕ. Determine all shift parameters, all coefficients, and the number of translates if finitely many perturbed, uniformly sampled data of f are given. Using Fourier technique, this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by APM. The stability of the solution is discussed in the square and uniform norm too. Numerical experiments show the performance of our approximation methods.

What’s the Frequency, Kenneth?: Sublinear Fourier Sampling Off the Grid

“…Given bounds on both the amplitudes |a j − a j | and the frequencies |ω j − ω j |, it is possible to compute the two terms in the error. This is standard in the literature of polynomial-time algorithms to recover real frequencies (e.g., [9], with which our result is comparable).…”

Deterministic sparse FFT algorithms

“…This idea has been used e.g. in [14], [36], [44]. Bounds for the condition number of the corresponding Vandermonde matrix can be found e.g.…”

Modal Analysis Using Co-Prime Arrays

“…Other methods include approximate least squares or maximum likelihood estimation [2], [3], reduced rank linear prediction [4], [5], MUSIC [6], and ESPRIT [7], [8]. While there are extensions of MUSIC and ESPRIT for direction of arrival estimation from non-uniformly sampled data (see, e.g., [9]- [13]), Prony-like methods have mainly been developed for uniformly sampled data, and extending such methods to non-uniformly sampled data has not received much attention (exceptions being [14] and [15]). …”

“…In [14], the authors approach the modal estimation problem by fitting a polynomial to the non-uniform samples and estimating the parameters of the exponentials using linear regression. For the case that the modes are on the unit circle, in [15] a truncated window function is fitted to the non-uniform measurements in the least squares sense, and then an approximate Prony method is proposed to estimate the frequencies of the exponentials.…”