# A New Scaling Law for Activity Detection in Massive MIMO Systems

## TL;DR

This paper introduces a new scaling law for activity detection in massive MIMO systems, demonstrating that with enough antennas, the number of detectable active users can significantly surpass traditional compressed sensing limits, especially for IoT applications.

## Contribution

The paper derives a novel scaling law showing activity detection success with more active users than previous bounds, leveraging large antenna arrays in massive MIMO systems.

## Key findings

- Active user detection scales with the square of the coherence block dimension.
- Detection success is achievable with a poly-logarithmic penalty for increasing potential users.
- Proposed algorithms outperform existing methods in simulations.

## Abstract

In this paper, we study the problem of \textit{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 \textit{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 show that one can overcome this fundamental limitation when the number of BS antennas $M$ is sufficiently large. More specifically, we derive a \textit{scaling law} on the parameters $(M, D_c, K_c, A_c)$ and also \textit{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. We also compare the performance of our proposed AD algorithms with that of other competitive algorithms in the literature.