Noise Shaping for Phased Array Beamforming

Quantization of phase and/or amplitude has far-reaching effects on the radiation characteristics of the phased array (PA), including gain, minor lobe level, and point deviation. Traditionally, one common method to address such a nonlinear distortion is using random-phasing to interrupt the error periodicity. Here, we show that the distortion due to the quantization can be better remediated by spectrally shaping the error compared to the random-phasing (dithering) approaches. We adapted the method for phase-only and amplitude-phase synthesis of planar array designed based on analog beamforming (ABF). To do that, for the first time, 2-D real- and complex-coefficient minimum-phase digital finite impulse response (FIR) filters are designed based on the discrete Hilbert transform (DHT) method. In particular, the digital filter design for phase-only synthesis is comprehensively investigated, respecting the error spectra in the beamspace domain. It is shown that by pushing the error out of the so-called visible region, the decrease of antenna directivity due to the quantization can be compensated to some extent, which provides a quite advantage over the uniform distribution of error. For some cases, pushing the error out of the visible region might be impossible. For such cases, we proposed using the spaced-notches filter. It is also shown that the method is on maximum efficacy when both phase and amplitude of the excitation signal are controllable. Thus, complex-valued noise shaping (CV-NS) can be exploited for the phase-amplitude synthesis of the PA, showing quite promising performance.