In antenna engineering, two questions seem independent. First: how many independent signals can I send through a spatial channel? This is the question of degrees of freedom — the number of distinguishable communication modes supported by the geometry of the transmitter, receiver, and the space between them. Second: how strong is each signal? This is the question of channel strength — how much power each mode carries from transmitter to receiver.
Gustafsson and Brick (arXiv:2603.28749, March 2026) develop a geometric framework showing that both questions are governed by the same operator-theoretic structure. The degrees of freedom and the channel strengths are both encoded in the singular value spectrum of a single propagation operator. The number of significant singular values gives the degrees of freedom. The magnitudes of those singular values give the channel strengths. What appeared to be two separate engineering problems is one mathematical object viewed from two angles.
The unification matters in the near field, where the standard far-field approximation breaks down. In the far field, the relationship between degrees of freedom and channel strength is well understood — the singular values decay predictably, and the transition from significant to negligible is sharp. In the near field, the decay is more gradual, the transition zone is broader, and the interplay between how many modes exist and how strong each one is becomes the central design problem. The single-operator framework provides exact answers where approximations fail.
The structural observation: the apparent independence of “how many” and “how much” was an artifact of the far-field regime, where the sharp decay made the two questions effectively separable. Move closer — into the near field — and the separation dissolves. The number and the strength are always coupled; the far field merely hid the coupling by making one question trivial. Two engineering specifications that are designed, budgeted, and optimized separately turn out to be constrained by a single underlying object, and any design that optimizes one while ignoring the other is working with an incomplete model.