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

Abstract

We propose <i>nonlinear</i> beamforming for phased array weather radars (PAWRs). Conventional beamforming is <i>linear</i> 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 <i>distributed targets</i> such as raindrops, however, the number of scatterers is significantly large, differently from the case of <i>point targets</i> 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 <i>periodogram</i> 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 <i>power spectral density</i>. 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 <i>mean Doppler frequency</i> and 2) frequency indices of the major components are similar between adjacent elevations. These are expressed as <i>group-sparsities</i> 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 <i>alternating direction method of multipliers</i> (ADMM) including <i>nonlinear</i> mappings. Simulations using synthetic and real-world PAWR data show that the proposed method dramatically improves the spatial resolution.