Extended wave propagators as pulsed-beam communication channels

TL;DR

This paper introduces an analytic extension of the wave propagator to model extended pulsed-beam antennas as complex spacetime entities, forming a new framework for wave communication channels.

Contribution

It proposes a novel analytic continuation of the wave propagator to represent extended antennas in complex spacetime, enabling a more realistic modeling of wave communication channels.

Findings

The analytic propagator models extended emitters and receivers as complex spacetime entities.

Causality constraints restrict antenna orientations to the future cone in complex spacetime.

Invariance under imaginary spacetime translations has significant implications for wave transmission.

Abstract

Let P_0(x_r-x_e) be the causal propagator for the wave equation, representing the signal received at the spacetime point x_r due to an impulse emitted at the spacetime point x_e. Such processes are highly idealized since no signal can be emitted or received at a precise point in space and at a precise time. We propose a simple and compact model for extended emitters and receivers by continuing P_0 to an analytic function P(z_r - z_e), where z_e=x_e+iy_e represents a circular pulsed-beam emitting antenna centered at x_e and radiating in the spatial direction of y_e while z_r=x_r-iy_r represents a circular pulsed-beam receiving antenna centered at x_r and receiving from the spatial direction of y_r. The space components of y_e and y_r give the spatial orientations and radii of the antennas, while their time components s_e, s_r represent the time a signal takes to propagate along the antennas between the center and the boundary. The analytic propagator P(z_r - z_e) represents the transmission amplitude, forming a communication channel. Causality requires that the extension/orientation 4-vectors y_e and y_r belong to the future cone V, so that z_e and z_r belong to the future tube and the past tube in complex spacetime, respectively. The invariance of P(z_r - z_e) under imaginary spacetime translations has nontrivial consequences.