Performance of super-resolution methods in parameter estimation and system identification

Extracting information from low-resolution data is a challenging task in many applications ranging from astronomy and radar to medical imaging, microscopy, and spectroscopy. This work examines the performance of super-resolution methods that enable the recovery of the high-frequency information of a signal from low-frequency, i. e., low-resolution, measurements by exploiting a priori knowledge on the signal structure. First, we develop a mathematical theory of super-resolution from short-time Fourier transform (STFT) measurements. Then, we study the problem of identifying a linear time-varying system characterized by a discrete set of delays and Doppler shifts, which is closely related to the super-resolution problem from STFT measurements. Finally, we analyze the performance of so-called high-resolution parametric subspace methods for the estimation of the spectrum of a signal from a finite number of samples. Our results are enabled by new bounds on the extremal singular values of complex-valued Vandermonde matrices, with entries in the unit disk.