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 block 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 $A_{c}\leq D_{c}$, which is a bottleneck in massive deployment scenarios such as Internet-of-Things (IoT) and Device-to-Device (D2D) communication. In this paper, we show that one can overcome this fundamental limitation when the number of BS antennas $M$ is sufficiently large. More specifically, we 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 $M$ with $A_{c}/M=o(1)$, one can identify the activity of $A_{c}=O(D_{c}^{2}/\log^{2}(\frac{K_{c}}{A_{c}}))$ 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 poly-logarithmic penalty $O(\log^{2}(\frac{K_{c}}{A_{c}}))$ for increasing the number of potential users $K_{c}$, which makes it ideally suited for AD in IoT setups. We propose low-complexity algorithms for AD and provide numerical simulations to illustrate our results.