Finite-SNR Closed Form for the Square Marčenko-Pastur MIMO Capacity
For the canonical independent and identically distributed (IID) single-user multiple-input multiple-output (MIMO) channel with full channel state information at the transmitter (CSIT), the asymptotic capacity per receive antenna is a waterfilling integral over the Marčenko-Pastur law, whose shape depends only on the ratio of transmit to receive antennas. Closed forms are known for every ratio other than the square one, and there only above a finite signal-to-noise ratio (SNR). The square ratio behaves differently because the Marčenko-Pastur support reaches the origin. The waterfilling cutoff then stays strictly inside the support at every finite SNR, and the water level has been available only numerically. This letter closes that gap with a classical trigonometric parametrization of the square law, which removes the edge singularity and replaces the moving cutoff by a single angle. Both the SNR and the capacity become explicit functions of that angle, so the capacity curve is traced by sweeping the angle rather than by solving a scalar constraint at each operating point. The same parametrization delivers the low-SNR behavior in closed form, where the slope of capacity in SNR equals the upper edge of the limiting spectrum.