Unified field theory for space-time systems

The communication efficiency associated with the problems of transmitting information in space and time is known to be dependent upon the characteristics of the transmission medium. Notably Turin (1977), Kailath (1961), and Bello (1963) have considered the temporal channel response as a function of the channel time and delay spread processes. The physics of the medium have not been considered. Middleton and Groginsky (1965) and Wittwer (1980) have attempted to incorporate the physics of the medium in a strictly temporal channel model. This paper presents a theory that generalizes the earlier work for characterizing medium-induced effects upon deterministic and random space-time fields, and the theory is applicable to any frequency band in the electromagnetic spectrum. For a linear medium, a space-time system field function (S-TSFF) is introduced as a system model for an arbitrary (turbulent or material) medium (Lo 1988). The unified system field theory states that the output space-time field is related to the input space-time field and the S-TSFF through a set of space-time superposition integrals. The geometry between the energy source, the medium, and the observed world-point area or volume is dependent on the chosen coordinate system, and is implicitly expressed by the space-time superposition integrals. (The term world-point was used by Minkowski (1908) to denote a point in a four-dimensional space-time coordinate system. To distinguish our approach from the analysis of arbitrary events in a space-time manifold, the terms world-area and world-volume are chosen to denote the ‘simultaneity in space’ as opposed to the notation of world-line and world-space used in the theory of relativity.) For a deterministic input space-time field, the statistics of the output field are solely dependent on the S-TSFF. For a random input space-time field, the statistics of the output field are related to the statistics of the input field and the S-TSFF. In either case, the statistical characterization can be expressed in terms of a space-time correlation function, or, equivalentIy, the space-time power spectral density of the input-output fields and of the S-TSFF. The space-time correlation function of the S-TSFF generalizes the concept of the mutual coherence function (MCF) used in statistical optics (Born and Wolf 1970). A dual property for the S-TSFF is observed and it is shown to be consistent with the duality principle in physics and the linear system theory in engineering. This space-time duality concept leads to the conclusion that fundamental properties of matter are imbedded in all physically realizable systems. This conclusion has far reaching implications in many detection, instrumentation, and measurement systems—for example, the uncertainty principle can be applied to analyse the stability of atomic clocks and trap ion frequency standards. The medium's response to an applied field is dependent upon whether the observed field is due to a space-time point source, a space-time plane wave, or something in-between. The space-time system field theory has numerous applications in system engineering, e.g. communications, radar, sonar, optics, and various imaging systems.