The fit of a sum of exponentials to noisy data

摘要

Given a set of noisy data y0,…, ym, measured at equidistant points on the time axis, find n (n <12m) exponents bi and coefficients ci such that ∑i ci exp(bit) yields a good approximation of the data. We study two algorithms for the reconstruction of the exponents and coefficients. We show that the error in the reconstructed exponents is of the order of the gap σn + 1/(σn − σn + 1) between the smallest accepted singular value σn and the largest rejected singular value σn + 1 of a Hankel matrix constructed from the data. Moreover, we provide a statistical analysis, which gives some insight in the optimal choice of the dimensions of the Hankel matrix to be used, given a fixed number of datapoints, and we illuminate this by several clarifying examples.