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 block 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 K
_c
assigned to a specific CB is much larger than the dimension of the CB D
_c
(K
_c
≫ D
_c
) but at each time slot only A
_c
≪ K
_c
of them are active. Most of the previous results, based on compressed sensing, require that A
_c
≤ 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 Dc, 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
((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
((K
_c
)/(A
_c
))) for increasing the number of potential users Kc, 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.