Joint Message Detection and Channel Estimation for Unsourced Random Access in Cell-Free User-Centric Wireless Networks

We consider unsourced random access (uRA) in a cell-free (CF) user-centric wireless network, where a large number of potential users compete for a random access slot, while only a finite subset is active. The random access users transmit codewords of length $L$ symbols from a shared codebook, which are received by $B$ geographically distributed radio units (RUs) equipped with $M$ antennas each. Our goal is to devise and analyze a \emph{centralized} decoder to detect the transmitted messages (without prior knowledge of the active users) and estimate the corresponding channel state information. A specific challenge lies in the fact that, due to the geographically distributed nature of the CF network, there is no fixed correspondence between codewords and large-scale fading coefficients (LSFCs). This makes current activity detection approaches which make use of this fixed LSFC-codeword association not directly applicable. To overcome this problem, we propose a scheme where the access codebook is partitioned in location-based subcodes, such that users in a particular location make use of the corresponding subcode. The joint message detection and channel estimation is obtained via a novel {\em Approximated Message Passing} (AMP) algorithm for a linear superposition of matrix-valued sources corrupted by noise. The statistical asymmetry in the fading profile and message activity leads to \emph{different statistics} for the matrix sources, which distinguishes the AMP formulation from previous cases. In the regime where the codebook size scales linearly with $L$, while $B$ and $M$ are fixed, we present a rigorous high-dimensional (but finite-sample) analysis of the proposed AMP algorithm. Exploiting this, we then present a precise (and rigorous) large-system analysis of the message missed-detection and false-alarm rates, as well as the channel estimation mean-square error.