MSE Estimates for Multitaper Spectral Estimation and Off-Grid Compressive Sensing

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 <inline-formula> <tex-math notation="LaTeX">$W$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> time domain observations, the average of the square of the first <inline-formula> <tex-math notation="LaTeX">$K=\lfloor 2NW\rfloor$ </tex-math></inline-formula> Slepian functions approaches, as <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> grows, an ideal bandpass kernel for the interval <inline-formula> <tex-math notation="LaTeX">$[ -W,W]$ </tex-math></inline-formula>. We provide an analytic proof of this fact and measure the corresponding rate of convergence in <inline-formula> <tex-math notation="LaTeX">$L^{1}$ </tex-math></inline-formula> 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.