Achievable Rate with Antenna Size Constraint: Shannon meets Chu and Bode
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The achievable rate of existing wireless systems is commonly studied based on mathematical models that disregard the physical limitations of antennas in terms of size, bandwidth, and efficiency. Antenna models are indeed essential to properly analyze the achievable rate as a key performance metric of wireless communication systems. In this work, we use ideas from Chu and Bode/Fano theories to establish the achievable rate under antenna-size constraint. Specifically, we characterize the maximum achievable rate over single-input single-output (SISO) 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 size of the receive antenna thereby providing a realistic upper-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 small fraction of Shannon capacity can be achieved. We further determine the optimal signaling bandwidth which shows that impedance matching is essential in bandwidth-limited 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 a function of the antenna size and the SNR reveal new insights into concepts for the physical design of radio systems.
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