Parameter estimation for exponential sums by approximate Prony method

Abstract

The recovery of signal parameters from noisy sampled data is a fundamental problem in digital signal processing. In this paper, we consider the following spectral analysis problem: Let f be a real-valued sum of complex exponentials. Determine all parameters of f , i.e., all different frequencies, all coefficients, and the number of exponentials from finitely many equispaced sampled data of f . This is a nonlinear inverse problem. In this paper, we present new results on an approximate Prony method (APM) which is based on [1]. In contrast to [1], we apply matrix perturbation theory such that we can describe the properties and the numerical behavior of the APM in detail. The number of sampled data acts as regularization parameter. The first part of APM estimates the frequencies and the second part solves an overdetermined linear Vandermonde-type system in a stable way. We compare the first part of APM also with the known ESPRIT method. The second part is related to the nonequispaced fast Fourier transform (NFFT). Numerical experiments show the performance of our method.

Nonlinear Approximation by Sums of Exponentials and Translates

“…See also the impressive results in [5,6]. Recently, the two last named authors of this paper have investigated the properties and the numerical behavior of APM in [26], where only real-valued exponential sums (1.1) were considered. Further, the APM is generalized to the parameter estimation for a sum of nonincreasing exponentials in [27].…”

“…In contrast to [3,4], we prefer an approach to the APM by the perturbation theory for a singular value decomposition ofH (see [26]). In this paper, we investigate the stability of the approximation of (1.1) in the square and uniform norm for the first time.…”

Deterministic sparse FFT algorithms

“…Unfortunately, Prony's method is in general numerically unstable, see [44]. Therefore, there has been some effort to develop stabilized versions of Prony's method.…”

“…Therefore, there has been some effort to develop stabilized versions of Prony's method. In [44] and [45], the approximate Prony method is proposed which is based on [5]. Further approaches to methods for parameter identification include MUSIC [49], ESPRIT [48] or the matrix pencil method [24].…”

“…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.…”