Derivation of Reactance Time Function and Diversity Analysis for Two-Element ESPAR Antenna

This paper presents a theoretical and simulation-based study of a two-element Electronically Steerable Parasitic Array Radiator (ESPAR) antenna designed for antenna pattern multiplexing (APM) in single-RF-chain receiver systems. We derive a closed-form analytical expression for the time-varying reactance required at the parasitic element to generate a continuously evolving radiation pattern, modeling the array factor as a rotating phasor in the complex plane. Two reactance profiles are analyzed: a practically realizable bounded case and an idealized unbounded case as its special form. The bounded formulation constrains the phase variation to a finite range, resulting in multiple spectral components, with the two strongest components exhibiting low correlation (≈ 0.20), suitable for diversity reception. The unbounded formulation produces exactly two dominant spectral components at DC and + fs, but yields a higher correlation coefficient (≈ 0.48), reducing its diversity advantage despite its spectral compactness. A Fourier-based analysis quantifies the harmonic content, and correlation coefficients between dominant components are used to assess diversity potential. The estimated diversity gains are 7.6 dB for the bounded case and 9.0 dB for the unbounded case. The results demonstrate that bounded reactance control offers a viable and effective implementation for compact, low-complexity diversity antennas, while the unbounded case serves as a useful theoretical reference.