Non-Bayesian Activity Detection, Large-Scale Fading Coefficient Estimation, and Unsourced Random Access With a Massive MIMO Receiver

Abstract

In this paper, we study the problem of user <italic>activity detection</italic> and large-scale fading coefficient estimation in a random access wireless uplink with a massive MIMO base station with a large number <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> of antennas and a large number of wireless single-antenna devices (users). We consider a block fading channel model where the <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-dimensional channel vector of each user remains constant over a <italic>coherence block</italic> containing <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> signal dimensions in time-frequency. In the considered setting, the number of potential users <inline-formula> <tex-math notation="LaTeX">$K_{\text {tot}}$ </tex-math></inline-formula> is much larger than <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> but at each time slot only <inline-formula> <tex-math notation="LaTeX">$K_{a} \ll K_{\text {tot}}$ </tex-math></inline-formula> of them are active. Previous results, based on compressed sensing, require that <inline-formula> <tex-math notation="LaTeX">$K_{a}\le L $ </tex-math></inline-formula>, which is a bottleneck in massive deployment scenarios. In this work, we show that such limitation can be overcome when the number of base station antennas <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> is sufficiently large. More specifically, we prove that with a coherence block of dimension <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> and a number of antennas <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$K_{a}/M = o(1)$ </tex-math></inline-formula>, one can identify <inline-formula> <tex-math notation="LaTeX">$K_{a} = O\left({L^{2}/\log ^{2}\left({\frac {K_{\text {tot}}}{K_{a}}}\right)}\right)$ </tex-math></inline-formula> active users, which is much larger than the previously known bounds. We also provide two algorithms. One is based on Non-Negative Least-Squares, for which the above scaling result can be rigorously proved. The other consists of a low-complexity iterative componentwise minimization of the likelihood function of the underlying problem. While for this algorithm a rigorous proof cannot be given, we analyze a constrained version of the Maximum Likelihood (ML) problem (a combinatorial optimization with exponential complexity) and find the same fundamental scaling law for the number of identifiable users. Therefore, we conjecture that the low-complexity (approximated) ML algorithm also achieves the same scaling law and we demonstrate its performance by simulation. We also compare the discussed methods with the (Bayesian) MMV-AMP algorithm, recently proposed for the same setting, and show superior performance and better numerical stability. Finally, we use the discussed approximated ML algorithm as the inner decoder in a concatenated coding scheme for <italic>unsourced random access</italic>, a grant-free uncoordinated multiple access scheme where all users make use of the same codebook, and the receiver must produce the list of transmitted messages, irrespectively of the identity of the transmitters. We show that reliable communication is possible at any <inline-formula> <tex-math notation="LaTeX">$E_{b}/N_{0}$ </tex-math></inline-formula> provided that a sufficiently large number of base station antennas is used, and that a sum spectral efficiency in the order of <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(L\log (L))$ </tex-math></inline-formula> is achievable.