This paper presents a unified analysis of the three-parameter aperture distributions for both sum and difference antenna patterns, suitable for communications or telemetry applications with either a stationary or tracking antenna, and with the parameters automatically determined by Particle-Swarm Optimization (PSO). These distributions can be created, for example, by reflector, phased array, or other antenna systems. The optimizations involve multiple objectives, for which Pareto efficiency concepts apply, and are accelerated by compact, analytical closed-form equations for key metrics of the distributions, including the far-field radiation pattern and detection slope of the difference pattern. The limiting cases of the threeparameter distributions are discussed and shown to generalize other distributions in the literature. A derivation of the generalized vector far fields provides the background for the distribution study and helps clarify the definition of cross-polarization in the far-field. Examples are given to show that the three-parameter (3P) distributions meet a range of system-level constraints for various applications, including a sidelobe mask for satellite ground stations and maximizing pointing error detection sensitivity while minimizing clutter from sidelobes for tracking applications. The equations for the relative angle sensitivity for the difference pattern are derived. A study of the sensitivity of the 3P parameter values is presented.
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