L. Daniel Abreu
We obtain estimates for the Mean Squared Error (MSE) for the multitaper spectral estimator and certain compressive acquisition methods
for multi-band signals. We confirm a fact discovered by Thomson [Spectrum estimation and harmonic analysis, Proc. IEEE, 1982]: assuming bandwidth W and N time domain observations, the average of the square of the first K=2NW Slepian functions approaches, as K grows, an ideal band-pass kernel
for the interval [-W,W]. We provide an analytic proof of this fact and measure the corresponding rate of convergence in the L1 norm. This
validates a heuristic approximation used to control the MSE of the multitaper estimator. The estimates have also consequences for the method
of compressive acquisition of multi-band signals introduced by Davenport and Wakin, giving MSE approximation bounds for the dictionary formed by
modulation of the critical number of prolates. We will also discuss extensions of the methods to other settings, in particular for the analysis of acoustic signals, which are in general non-stationary. This is a joint work with Jos√© Luis Romero.