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 $M \gg 1$ 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 $D_c$ signal dimensions. We study a setting in which the number of potential users $K_c$ assigned to a specific CB is much larger than the dimension of the CB $D_c$ ($K_c \gg D_c$) but at each time slot only $A_c \ll K_c$ of them are active. Most of the previous results, based on compressed sensing, require that $A_c\le D_c$, which is a bottleneck in massive deployment scenarios such as Internet-of-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 $(M, D_c, K_c, A_c)$ and also Signal-to-Noise Ratio (SNR) under which our proposed AD scheme succeeds. Our analysis indicates that with a CB of dimension $D_c$, and a sufficient number of BS antennas $M=O(A_c)$, one can identify the activity of $A_c=O(D_c^2/\log (\frac{K_c}{A_c}))$ active users, which is much larger than the previous bound $A_c=O(D_c)$ obtained via traditional compressed sensing techniques. In particular, in our proposed scheme one needs to pay only a negligible logarithmic penalty $O(\log (\frac{K_c}{A_c}))$ for increasing the number of potential users $K_c$, 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.