Uniform Arrays Pay One More Dimension: Exact Noncoherent Degrees of Freedom of Sparse MIMO Channels

In this paper, we study the degrees of freedom (DoF) of multiple-antenna channels with K specular paths, N_t transmit antennas on a half-wavelength grid and a uniform array of N_r >= 2K receive antennas, with path gains and angles unknown at both ends and redrawn in every block of T channel uses. We define the learning cost c as the number of signal dimensions per block lost to learning when T >= N_t. We prove that c >= 3/2 for every array once K >= N_t, with equality for non-uniform arrays once K >= 3N_t - 2, and already from K = N_t + 2 for every non-uniform array with N_t <= 4 and for the first explicit infinite family at that threshold, that c = 5/2 for uniform arrays with K >= N_t + 2 except at N_t = T = 2, and that c grows linearly in N_t at K = N_t + 1. For T >= N_t, these values need only K + 2 receive antennas. Here, c is half the real dimension of a space of symmetries of the transmit array, and the converses rest on genie and covering arguments adapted to these symmetries. For N_t >= 3, we also obtain the DoF of uniform arrays with K >= N_t + 2 at every T >= 2, and of every array at T = 2. When K >= N_t + 2 paths renew at staggered times known to both ends, the learning cost vanishes.