  Modern wireless channels are increasingly dense and mobile making the channel
highly non-stationary. The time-varying distribution and the existence of joint
interference across multiple degrees of freedom (e.g., users, antennas,
frequency and symbols) in such channels render conventional precoding
sub-optimal in practice, and have led to historically poor characterization of
their statistics. The core of our work is the derivation of a high-order
generalization of Mercer's Theorem to decompose the non-stationary channel into
constituent fading sub-channels (2-D eigenfunctions) that are jointly
orthogonal across its degrees of freedom. Consequently, transmitting these
eigenfunctions with optimally derived coefficients eventually mitigates any
interference across these dimensions and forms the foundation of the proposed
joint spatio-temporal precoding. The precoded symbols directly reconstruct the
data symbols at the receiver upon demodulation, thereby significantly reducing
its computational burden, by alleviating the need for any complementary
decoding. These eigenfunctions are paramount to extracting the second-order
channel statistics, and therefore completely characterize the underlying
channel. Theory and simulations show that such precoding leads to
${>}10^4{\times}$ BER improvement (at 20dB) over existing methods for
non-stationary channels.
