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

Abstract

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 $u(\xi ,t)=\exp (j[\Omega t+g_k\xi ])$ 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 $f_c = \SI{140}{\giga\hertz}$, $N=4096$ 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 $K=50$.

The paper contains inconsistencies in the proposed 4D-FSM communication framework that affect the formulation and interpretation of the numerical results. These issues require further technical investigation and revision to ensure the consistency and correctness of the presented analysis