Improved Scaling Law for Activity Detection in Massive MIMO Systems

In this paper, we study the problem of activity detection (AD) in a massive MIMO setup, where the Base Station (BS) has M 1 antennas. We consider a flat fading channel model where the M -dim channel vector of each user remains almost constant over a coherence block (CB) containing Dc signal dimensions. We study a setting in which the number of potential users Kc assigned to a specific CB is much larger than the dimension of the CB Dc (Kc Dc) but at each time slot only Ac Kc of them are active. Most of the previous results, based on compressed sensing, require that Ac ≤ Dc, which is a bottleneck in massive deployment scenarios such as Internetof-Things (IoT) and Device-to-Device (D2D) communication. In this paper, we propose a novel scheme for AD and show that it overcomes this limitation when the number of antennas M is sufficiently large. We also derive a scaling law on the parameters (M,Dc,Kc, Ac) and also Signal-to-Noise Ratio (SNR) under which our proposed AD scheme succeeds. Our analysis indicates that with a CB of dimension Dc, and a sufficient number of BS antennas M = O(Ac), one can identify the activity of Ac = O(D 2 c/ log( Kc Ac )) active users, which is much larger than the previous bound Ac = O(Dc) obtained via traditional compressed sensing techniques. In particular, in our proposed scheme one needs to pay only a negligible logarithmic penalty O(log(c Ac )) for increasing the number of potential users Kc, which makes it perfect for AD in IoT setups. We propose very low-complexity algorithms for AD and provide numerical simulations to illustrate the validity of our results.