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28 March 2024
 
  » arxiv » math-ph/0108006

 Article overview


Communication via Holomorphic Green Functions
Gerald Kaiser ;
Date 15 Aug 2001
Subject Mathematical Physics; Complex Variables MSC-class: 31-XX, 32-XX, 35-XX, 78-XX | math-ph math.CV math.MP
AbstractLet G(x_r-x_e) be the causal Green function for the wave equation in four spacetime dimensions, representing the signal received at the spacetime point x_r due to an impulse emitted at the spacetime point x_e. Such emission and reception processes are highly idealized, since no signal can be emitted or received at a single (mathematical) point in space and time. We present a simple model for sl extended m emitters and receivers by extending G analytically to a function ilde G(z_r- z_e), where z_e=x_e+iy_e is a complex spacetime point representing a circular sl pulsed-beam emitting antenna dish m centered at x_e and emitting in the direction of y_e, and z_r=x_r-iy_r represents a circular sl pulsed-beam receiving antenna dish m centered at x_r and receiving from the direction of y_r. The holomorphic Green function ilde G(z_r-z_e) represents the sl coupling m between the emission from z_e and the reception at z_r. To preserve causality and give nonsingular coupling, the orientation vectors y_e and y_r must belong to the sl future cone m V_+ in spacetime. Equivalently, z_e and z_r belong to the sl future and past tubes m in complex spacetime, respectively. The space coordinates of y_e and y_r give the spatial orientations and radii of the dishes, while their time coordinates determine the sl duration and focus m of the emission and reception processes. The sl directivity m D(y) of the communication process is a convex function on V_+, i.e., D(y_r+y_e)le D(y_r)+D(y_r). This shows that the efficiency of the communication can be no better than the sum of its emission and reception components.
Source arXiv, math-ph/0108006
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