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 $u(ξ,t) = \exp(j[Ωt + g_kξ])$ 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 $K = 50$.

Figures (10)

• Figure 1: Illustrative system architecture for 4D-FSM showing the end-to-end pipeline: bit mapping, 4D symbol manifold, DFnT precoding, STM-enabled array, near-field propagation, and receiver detection.
• Figure 2: Illustration of kinematic separation via synthetic velocity: users sharing the same spatial coordinates $(x,z)$ are resolved along the velocity dimension, creating "velocity-orthogonal bubbles."
• Figure 3: STM phase-ramp-to-Doppler chain for three velocity channels $v_1 < v_2 < v_3$. (a) Linear phase ramps $\phi_n = g_k\xi_n$. (b) Element phasors showing progressive wavefront tilt. (c) Resolved sinc$^2$ Doppler peaks in velocity-orthogonal bubbles (green shading).
• Figure 4: Normalised DFnT beam intensity $|AF(x,z)|^2$ for $v_1 = 50m\per s$ (heatmap) and $v_2 = 100m\per s$ (cyan contours) at identical spatial coordinates. The spatially coincident contours and the inset lateral cut confirm that the STM gradient $g_k$ shifts only the velocity coordinate of $\mathcal{S}$, leaving the Fresnel focal spot unchanged.
• Figure 5: Normalised manifold correlation $\rho(v)$ vs. user velocity. 4D-FSM maintains $\rho \approx 1$ across the full velocity range. TTD and LDMA collapse to $|\mathrm{sinc}(f_D T_{\rm int})| \approx 0.003$ at 200ms, matching the analytical prediction (grey dotted curve). BTSM oscillates around the quantisation floor $|\mathrm{sinc}(\Delta f_D T_{\rm int})| \approx 0.028$.

Theorems & Definitions (6)

• Proposition 1: Joint Orthogonality Surface
• Remark 1
• Corollary 1: Depth--Velocity Multiplexing Gain
• Remark 2
• Remark 3
• Remark 4