  The wireless channel is undergoing continuous changes, and the block-fading
assumption, despite its popularity in theoretical contexts, never holds true in
practical scenarios. This discrepancy is particularly critical for user
activity detection in grant-free random access, where joint processing across
multiple resource blocks is usually undesirable. In this paper, we propose
employing a low-dimensional approximation of the channel to capture variations
over time and frequency and robustify activity detection algorithms. This
approximation entails projecting channel fading vectors onto their principal
directions to minimize the approximation order. Through numerical examples, we
demonstrate a substantial performance improvement achieved by the resulting
activity detection algorithm.
