Equivalence of aperture reduction in element space and constrained combination of DFT beams in beamspace

In this paper, we present an analytical proof of equivalence of the signal processing in the reduced aperture element space and in beamspace produced by the combination of multiple adjacent DFT beams with a subsequent constraining of the resulting magnitudes. This link finds applications in millimeter wave (mmWave) communications and radars that are typically equipped with a small number of RF chains and employ hybrid beamforming with analog phase shifters. This result unifies the transceiver designs, reduces complexity, and proves the applicability of state-of-the-art beamspace-based methods. It has a special implication for channel estimation at the initial stage when terminals acquire coarse estimates of the Sectors-of-Interest (SoIs). We show that the constrained groups of beams are equivalent to DFT beamformers of a smaller size aperture and present a closed-form expression of the corresponding effective aperture length as a function of the number of beams. We also derive an approximation of this expression to find the indices of active array elements in a closed form. Finally, we verify this theory and analyze the accuracy of the proposed approximation using numerical simulations.