Unsourced Random Access in MIMO Quasi-Static Rayleigh Fading Channels: Finite Blocklength and Scaling Law Analyses

This paper considers the unsourced random access (URA) problem with a random and unknown number of active users in multiple-input multiple-output (MIMO) quasi-static Rayleigh fading channels. We derive non-asymptotic achievability bounds on the probability of incorrectly estimating the number of active users, and provide scaling laws on the gap between the estimated and true numbers of active users. We prove that the error probability reaches a plateau as the power <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> and blocklength <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> increase, whereas it decays exponentially with the number <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> of receive antennas and eventually vanishes. Then, we explore the fundamental limits of URA by deriving non-asymptotic achievability bounds and converse bounds (including two single-user converse bounds and one multi-user ensemble converse bound) on the minimum energy-per-bit required by each active user to transmit <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> bits with blocklength <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> under misdetection and false-alarm constraints. Numerical results show that the extra required energy-per-bit due to the uncertainty in the number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {K}}_{a}$ </tex-math></inline-formula> of active users decreases as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {E}[{\mathrm {K}}_{a}]$ </tex-math></inline-formula> increase and the error requirement becomes milder. In the non-asymptotic regime, using codewords distributed on a sphere outperforms Gaussian random coding. Existing schemes are shown to exhibit a large gap to our bounds when the number of active users is large, calling for more advanced schemes that perform energy-efficiently in this case. In the asymptotic regime with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\to \infty $ </tex-math></inline-formula>, we establish scaling laws on the minimum required <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> to reliably support <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {K}}_{a}$ </tex-math></inline-formula> active users as functions of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>, which highlight the potential of MIMO in enabling low-cost communication and indicate that it is possible for the minimum required <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> to remain on the same order when the number of active users increases but stays below a threshold.