The Cost of Learning the Channel under Partial Visibility: Exact Noncoherent Degrees of Freedom of the XL-MIMO Uplink

In this paper, we determine the cost of learning the channel, in signal dimensions lost per coherence block of length T, in the uplink of an extremely large array whose K single-antenna users see only known parts of the N antennas, their visible parts. With full visibility, N ≥ K and T ≥ 2K, this cost is the number of users squared. We prove that, for every T ≥ 2K−1 and every fixed N, it equals the number of ordered pairs of users, not necessarily distinct, with nested or equal visible parts, even under cooperation. This law holds whenever two antenna assignments exist. First, the users can be given distinct antennas of their visible parts. Second, for every user j, those whose visible parts are not within that of j can be given distinct antennas in their parts outside that of j. Then, when no visible part contains another, every user attains its single-user degrees of freedom (DoF) for every T ≥ K. For example, 64 users on staggered windows reach a sum DoF of 63 at T = 64, whereas full visibility allows at most 16. When only the second assignment fails, each unit of shortfall costs one more dimension for every T ≥ 2K−1. In particular, with K antennas, each nonzero channel entry costs one dimension. Within this model, the overlap of the visible parts, not the array size, determines the cost. To the best of our knowledge, the converse is new for nested visible parts.