Achievable Rate with Antenna Size Constraint: Shannon meets Chu and Bode

Using ideas from Chu and Bode/Fano theories, we characterize the maximum achievable rate over the single-input single-output wireless communication channels under a restriction on the antenna size at the receiver. By employing circuit-theoretic multiport models for radio communication systems, we derive the information-theoretic limits of compact antennas. We first describe an equivalent Chu's antenna circuit under the physical realizability conditions of its reflection coefficient. Such a design allows us to subsequently compute the achievable rate for a given receive antenna size thereby providing a physical bound on the system performance that we compare to the standard size-unconstrained Shannon capacity. We also determine the effective signal-to-noise ratio (SNR) which strongly depends on the antenna size and experiences an apparent finite-size performance degradation where only a fraction of Shannon capacity can be achieved. We further determine the optimal signaling bandwidth which shows that impedance matching is essential in both narrowband and broadband scenarios. We also examine the achievable rate in presence of interference showing that the size constraint is immaterial in interference-limited scenarios. Finally, our numerical results of the derived achievable rate as function of the antenna size and the SNR reveal new insights for the physically consistent design of radio systems.