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The high mobility, density and multi-path evident in modern wireless systemsmakes the channel highly non-stationary. This causes temporal variation in thechannel distribution that leads to the existence of time-varying jointinterference across multiple degrees of freedom (DoF, e.g., users, antennas,frequency and symbols), which renders conventional precoding sub-optimal inpractice. In this work, we derive a High-Order Generalization of Mercer'sTheorem (HOGMT), which decomposes the multi-user non-stationary channel intotwo (dual) sets of jointly orthogonal subchannels (eigenfunctions), that resultin the other set when one set is transmitted through the channel. This dualityand joint orthogonality of eigenfuntions ensure transmission over independentlyflat-fading subchannels. Consequently, transmitting these eigenfunctions withoptimally derived coefficients eventually mitigates any interference across itsdegrees of freedoms and forms the foundation of the proposed jointspatio-temporal precoding. The transferred dual eigenfuntions and coefficientsdirectly reconstruct the data symbols at the receiver upon demodulation,thereby significantly reducing its computational burden, by alleviating theneed for any complementary post-coding. Additionally, the eigenfunctionsdecomposed from the time-frequency delay-Doppler channel kernel are paramountto extracting the second-order channel statistics, and therefore completelycharacterize the underlying channel. We evaluate this using a realisticnon-stationary channel framework built in Matlab and show that our precodingachieves ${\geqslant}$4 orders of reduction in BER at SNR${\geqslant}15$dB inOFDM systems for higher-order modulations and less complexity compared to thestate-of-the-art precoding.

Joint Spatio-Temporal Precoding for Practical Non-Stationary Wireless Channels