An antenna array estimates the direction of an incoming signal by measuring phase differences across its elements. For narrowband signals, this works cleanly — the phase at each antenna is a linear function of direction, and solving for direction is a linear algebra problem. For ultra-wideband signals, three problems compound: the array may be spatially undersampled (elements too far apart for the highest frequency), the absolute phase at each element may be unknown (asynchronous hardware), and the beam direction shifts with frequency (beam squint). Each problem has been solved independently. Tian and colleagues solve all three simultaneously by reframing the problem geometrically.
Instead of treating the received signal as a vector of complex amplitudes, they represent it as a curve in a high-dimensional space parameterized by frequency. Different directions of arrival produce curves with different geometric shapes — different curvatures, torsions, and orientations in this space. The direction of arrival is encoded not in the absolute position of the curve (which depends on unknown phase offsets) but in its shape (which doesn't).
This geometric representation absorbs all three problems at once. Spatial undersampling creates ambiguity in phase, but not in the shape of the curve. Unknown absolute phase shifts the curve without changing its shape. Beam squint changes the parameterization of the curve but preserves its geometric invariants. The direction of arrival is recovered from shape features — curvature ratios, projection angles — that are invariant to all three sources of corruption.
The physics didn't change. The representation did. By lifting the signal from amplitudes to geometry, the three problems that were hard separately become one problem that is naturally solved.