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, (i) major components of the periodogram are concentrated in the vicinity of the mean Doppler frequency, and (ii) 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- parsities. We consider two optimization problems which evaluate the group-sparsities in slightly different ways, and both problems are solved with the alternating direction method of multipliers including nonlinear mappings. Simulations using real PAWR data show that the proposed method dramatically improves the spatial resolution.
