  Estimating the spectral characteristics of a nonstationary random process is
an important but challenging task, which can be facilitated by exploiting
structural properties of the process. In certain applications, the observed
processes are underspread, i.e., their time and frequency correlations exhibit
a reasonably fast decay, and approximately time-frequency sparse, i.e., a
reasonably large percentage of the spectral values is small. For this class of
processes, we propose a compressive estimator of the discrete Rihaczek spectrum
(RS). This estimator combines a minimum variance unbiased estimator of the RS
(which is a smoothed Rihaczek distribution using an appropriately designed
smoothing kernel) with a compressed sensing technique that exploits the
approximate time-frequency sparsity. As a result of the compression stage, the
number of measurements required for good estimation performance can be
significantly reduced. The measurements are values of the discrete ambiguity
function of the observed signal at randomly chosen time and frequency lag
positions. We provide bounds on the mean-square estimation error of both the
minimum variance unbiased RS estimator and the compressive RS estimator, and we
demonstrate the performance of the compressive estimator by means of simulation
results. The proposed compressive RS estimator can also be used for estimating
other time-dependent spectra (e.g., the Wigner-Ville spectrum) since for an
underspread process most spectra are almost equal.
