Nonlinear Beamforming Based on Group-Sparsities of Periodograms for Phased Array Weather Radar

We propose
nonlinear
beamforming for phased array weather radars (PAWRs). Conventional beamforming is
linear
in the sense that a backscattered signal arriving from each elevation is reconstructed by a weighted sum of received signals, which can be seen as a linear transform for the received signals. For
distributed targets
such as raindrops, however, the number of scatterers is significantly large, differently from the case of
point targets
that are standard targets in array signal processing. Thus, the spatial resolution of the conventional linear beamforming is limited. To improve the spatial resolution, we exploit two characteristics of a
periodogram
of each backscattered signal from the distributed targets. The periodogram is a series of the powers of the discrete Fourier transform (DFT) coefficients of each backscattered signal and utilized as a nonparametric estimate of the
power spectral density
. Since each power spectral density is proportional to the Doppler frequency distribution, 1) major components of the periodogram are concentrated in the vicinity of the
mean Doppler frequency
and 2) frequency indices of the major components are similar between adjacent elevations. These are expressed as
group-sparsities
of the DFT coefficient matrix of the backscattered signals, and we propose to reconstruct the signals through convex optimization exploiting the group-sparsities. We consider two optimization problems. One problem roughly evaluates the group-sparsities and is relatively easy to solve. The other evaluates the group-sparsities more accurately but requires more time to solve. Both problems are solved with the
alternating direction method of multipliers
(ADMM) including
nonlinear
mappings. Simulations using synthetic and real-world PAWR data show that the proposed method dramatically improves the spatial resolution.