4D Fresnel Space-Time Modulation for Near-Field ELAA: Kinematic Multiplexing and O(N log N) Precoding at Sub-THz Frequencies

Extremely Large Antenna Arrays (ELAA) operating at sub-terahertz frequencies introduce a regime where near-field Fresnel propagation and high-mobility carrier Doppler interact simultaneously, creating a four-dimensional signal space that existing schemes exploit only partially. This paper proposes \textbf{4D Fresnel Space-Time Modulation (4D-FSM)}, a unified framework encoding information jointly across angle, depth, synthetic velocity, and QAM amplitude through a structured symbol manifold $\mathcal{S}$. Synthetic velocity is introduced via Space-Time Modulation (STM): a linear phase ramp $u(ξ,t) = \exp(j[Ωt + g_kξ])$ induces a Doppler-equivalent shift without physical motion, creating velocity-orthogonal bubbles that resolve co-located users. We derive the joint orthogonality surface governing simultaneous user separability in depth and velocity, revealing that users separated in depth require strictly less velocity separation to remain orthogonal -- a multiplexing gain with no counterpart in OTFS or LDMA. The Discrete Fresnel Transform (DFnT) factorization $\mathbf{H} = \mathbf{F}_D \mathbf{C}(z) \mathbf{P}$ reduces precoder complexity from $\mathcal{O}(N^3)$ to $\mathcal{O}(N\log N)$, completing within \SI{500}{\nano\second} against a \SI{5.4}{\micro\second} coherence window. Monte Carlo evaluation at $f_c = \SI{140}{\giga\hertz}$, $N = 4096$ confirms $ρ\approx 0.998$ across the full velocity range, \SI{6.16}{\bit\per\second\per\hertz} spectral efficiency where all baselines collapse, and $K_{\max} = 64$ orthogonal users -- a $248\times$ sum-rate advantage over TTD at $K = 50$.