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 $\mathbf{\text{4D}}\mathbf{\text{Fresnel}}\mathbf{\text{Space-Time}}\mathbf{\text{Modulation}}\mathbf{\text{(4D-FSM)}}$, 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 $u(\xi ,t)=\exp (j[\mathrm{\Omega }t+g_k\xi ])$ 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 $f_c=\text{SI}140\text{giga}\text{hertz}$, $N=4096$ confirms $\rho \approx 0.998$ 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 $K=50$.