New Insights into Channel vs Subspace Codes for Large-Scale Beamspace MIMO Channel Sensing

This paper provides novel insights into channel and subspace codes in nonadaptive channel sensing with a single RF chain. Observing that this problem naturally maps to a noncoherent decoding problem, we show that the sensing performance of the maximum likelihood (ML) angle estimator, which does not require knowledge of the typically unknown channel coefficient, is governed by two key terms: the minimum subspace distance and beam gain of the used beamformers. We derive an exact expression for the subspace distance of binary linear channel codes mapped to BPSK, which illuminates the relationship between subspace and Hamming distance, used to design subspace and channel codes, respectively. Our result also reveals why good Hamming distance alone is insufficient for sensing, and shows that well-known families of channel codes such as Reed-Muller codes, yield zero subspace distance and thereby poor sensing performance when used naively without proper codebook pruning. Finally, we introduce so-called beamspace subspace codes based on sparse antenna selection patterns (Golomb rulers), which we show provide near-optimal subspace distance. We demonstrate that this property of judiciously designed sparse arrays can be leveraged together with beamforming gain via convolutional beamspaces, enabling hardware- and sample-efficient channel sensing with theoretical guarantees in large-scale multiantenna communications.