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Beyond Classical Exponential Analysis: Generalizations, Connections and Applications
Annie Cuyt

Last modified: 2018-07-30


In classical exponential analysis the objective is to recover the values of the linear coefficients and the mutually distinct nonlinear parameters in the argument of each exponential term through sparse interpolation. When the exponential model contains n terms, then the above inverse problem can be solved from 2n samples. If the sparsity n is not known, then theoretically at least one more sample is required. Of course in a noisy setting, both for the recovery of the unknowns and to determine the correct n, a larger number N > 2n + 1 of samples is used.
The problem dates back to the real-valued exponential fitting problem of de Prony from the 18-th century. In the past decades several popular numerical methods for its solution were formulated, mostly based on some generalized eigenvalue reformulation. When dealing with complex-valued input data, all algorithms collect their samples following the Shannon-Nyquist rate which guarantees that the solution for the nonlinear parameters is unique and does not suffer from periodicity issues.
We present a technique [2] that allows to break the traditional Shannon-Nyquist sampling condition. While solving the aliasing effect that is a result from not adhering to it, we are also able to recondition the numerical problem statement. The latter proofs to be very useful in the case of clustered frequencies.
We also reformulate the exponential analysis as a tensor decomposition and a rational approximation problem, thereby offering the possibility to use facts and techniques from other domains, such as the coupling of tensor decompositions and the Froissart doublet theory.
Last but not least, we point out that the toolbox of methods put together with these new generalizations and connections, has great potential in several engineering applications, such as DOA or direction of arrival problems, ISAR or inverse synthetic aperture radar imaging, automotive radar, peak fitting in analytical chemistry, seismic data processing, and other application problems.

In cooperation with Prof Wen-shin Lee, wen-shin.lee@stir.ac.uk


exponential analysis, sparse interpolation, inverse problems