Non-Bayesian Activity Detection, Large-Scale Fading Coefficient Estimation, and Unsourced Random Access With a Massive MIMO Receiver
Alexander Fengler, Saeid Haghighatshoar, Peter Jung, Giuseppe Caire
Abstract
In this paper, we study the problem of user activity detection and large-scale fading coefficient estimation in a random access wireless uplink with a massive MIMO base station with a large number M of antennas and a large number of wireless single-antenna devices (users). We consider a block fading channel model where the M-dimensional channel vector of each user remains constant over a coherence block containing L signal dimensions in time-frequency. In the considered setting, the number of potential users K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tot</sub> is much larger than L but at each time slot only K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> <; <; K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tot</sub> of them are active. Previous results, based on compressed sensing, require that K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> ≤ L, which is a bottleneck in massive deployment scenarios. In this work, we show that such limitation can be overcome when the number of base station antennas M is sufficiently large. More specifically, we prove that with a coherence block of dimension L and a number of antennas M such that K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> /M = o(1), one can identify K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> = O(L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tot</sub> / K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> )) active users, which is much larger than the previously known bounds. We also provide two algorithms. One is based on Non-Negative Least-Squares, for which the above scaling result can be rigorously proved. The other consists of a low-complexity iterative componentwise minimization of the likelihood function of the underlying problem. While for this algorithm a rigorous proof cannot be given, we analyze a constrained version of the Maximum Likelihood (ML) problem (a combinatorial optimization with exponential complexity) and find the same fundamental scaling law for the number of identifiable users. Therefore, we conjecture that the low-complexity (approximated) ML algorithm also achieves the same scaling law and we demonstrate its performance by simulation. We also compare the discussed methods with the (Bayesian) MMV-AMP algorithm, recently proposed for the same setting, and show superior performance and better numerical stability. Finally, we use the discussed approximated ML algorithm as the inner decoder in a concatenated coding scheme for unsourced random access, a grant-free uncoordinated multiple access scheme where all users make use of the same codebook, and the receiver must produce the list of transmitted messages, irrespectively of the identity of the transmitters. We show that reliable communication is possible at any E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> provided that a sufficiently large number of base station antennas is used, and that a sum spectral efficiency in the order of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (Llog(L)) is achievable.