Energy Efficiency of Massive Random Access in MIMO Quasi-Static Rayleigh Fading Channels With Finite Blocklength

This paper considers the massive random access problem in multiple-input multiple-output (MIMO) quasi-static Rayleigh fading channels. Specifically, we derive achievability and converse bounds on the minimum energy-per-bit required for each active user to transmit <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$J$ </tex-math></inline-formula> bits with blocklength <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> , power <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> receive antennas under a per-user probability of error (PUPE) constraint, in the cases with and without a priori channel state information at the receiver (CSIR and no-CSI). In the case of no-CSI, we consider both the settings with and without the knowledge of the number <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> of active users at the receiver. Numerical evaluation shows that the gap between achievability and converse bounds is less than 2.5 dB for the CSIR case and less than 4 dB for the no-CSI case in most considered regimes. Under the condition that the distribution 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> is known in advance, the uncertainty of the exact value 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> entails only a small penalty in terms of energy efficiency. Our results show the significance of MIMO for the massive random access problem. As an example, we show that the spectral efficiency grows approximately linearly with the number of receive antennas in the case of CSIR, whereas the growth rate decreases in the case of no-CSI. Moreover, in the case of no-CSI, we demonstrate the suboptimality of the pilot-assisted scheme, especially when the number of active users is large. Building on non-asymptotic results, assuming all users are active and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$J=\Theta (1)$ </tex-math></inline-formula> , we obtain scaling laws of the number of supported users as follows: when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L = \Theta \left ({n^{2}}\right)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P=\Theta \left ({\frac {1}{n^{2}}}\right)$ </tex-math></inline-formula> , one can reliably serve <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K = \mathcal {O}(n^{2})$ </tex-math></inline-formula> users in the case of no-CSI; under mild conditions in the case of CSIR, the PUPE requirement is satisfied if and only if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {nL\ln KP}{K}=\Omega \left ({1}\right)$ </tex-math></inline-formula> .