  In this paper, we propose a convex optimization-based estimation of sparse
and smooth power spectral densities (PSDs) of complex-valued random processes
from mixtures of realizations. While the PSDs are related to the magnitude of
the frequency components of the realizations, it has been a major challenge to
exploit the smoothness of the PSDs, because penalizing the difference of the
magnitude of the frequency components results in a nonconvex optimization
problem that is difficult to solve. To address this challenge, we design the
proposed model that jointly estimates the complex-valued frequency components
and the nonnegative PSDs, which are respectively regularized to be sparse and
sparse-smooth. By penalizing the difference of the nonnegative variable that
estimates the PSDs, the proposed model can enhance the smoothness of the PSDs
via convex optimization. Numerical experiments on the phased array weather
radar, an advanced weather radar system, demonstrate that the proposed model
achieves superior estimation accuracy compared to existing sparse estimation
models, regardless of whether they are combined with a smoothing technique as a
post-processing step or not.
