Abstract
We investigate the age-limited capacity of the Gaussian many channel with total <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> users, out of which a random subset of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K_{a}$ </tex-math></inline-formula> users are active in any transmission period, and a large-scale antenna array at the base station (BS). In an uplink scenario where the transmission power is fixed among the users, we consider the setting in which both the number of users, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> , and the number of antennas at the BS, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> , are allowed to grow large at a fixed ratio <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\zeta = {M}/{N}$ </tex-math></inline-formula> . Assuming perfect channel state information (CSI) at the receiver, we derive the achievability bound under maximal ratio combining. As the number of active users, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K_{a}$ </tex-math></inline-formula> , increases, the achievable spectral efficiency is found to increase monotonically to a limit <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log _{2}\left ({1+\frac {M}{K_{a}}}\right)$ </tex-math></inline-formula> . Further extensions of the analysis to the zero-forcing receiver as well as imperfect CSI are provided, demonstrating the channel estimation penalty in terms of the mean squared error in estimation. Using the age of information (AoI) metric, first coined by Kaul et al., as our measure of data timeliness or freshness, we investigate the trade-offs between the AoI and spectral efficiency in the context massive connectivity with large-scale receiving antenna arrays. As an extension of Liu and Yu, based on our large system analysis, we provide an accurate characterization of the asymptotic (finite system size) spectral efficiency as a function of the number of antennas and the number of users, the attempt probability, and the AoI. It is found that while the spectral efficiency can be made large, the penalty is an increase in the minimum AoI obtainable. The proposed achievability bound is further compared against recent massive MIMO-based massive unsourced random access (URA) schemes.
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