Convex Estimation of Sparse-Smooth Power Spectral Densities From Mixtures of Realizations With Application to Weather Radar

Abstract

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.

Sharpening Minimization Induced Penalties

“…While this paper constructs the novel GME-MI model from the MI penalty ψ defined by the minimization in (3), the condition ( 7) is common with the existing studies [20], [21], [23], [25] for linearly involved prox-friendly penalties. Thus, we can set B satisfying (7) for any A and L by, e.g., [25].…”

“…Although the expression (6) represents the cost function J of the GME-MI model (5) as the sum of convex functions under the condition (7), it involves non-prox-friendly functions difficult to handle: ψ defined by the minimization in (3) and the conjugate of the sum of ψ and 1 2 ∥B • ∥ 2 . Due to this situation, the existing solvers [18]- [25] for the GME models for (linearly involved) prox-friendly penalties are inapplicable to the GME-MI model.…”

“…We refer to this class of penalties as minimization induced (MI) penalties. For example, to resolve the problem of unknown block partitions of blocksparse signals, which arises in many applications, e.g., [1]- [7], the latent optimally partitioned (LOP)-ℓ 2 /ℓ 1 penalty [8] is designed to minimize the mixed ℓ 2 /ℓ 1 norm [9] over the set of candidate block partitions. Another important example of the MI penalties is the total generalized variation (TGV) penalty [10], which is derived as a sound high-order extension of the total variation penalty [11], and optimally balances highorder derivatives.…”