Convexity of the Capacity of One-Bit Quantized Additive White Gaussian Noise Channels

Abstract

In this study, the maximum error-free transmission rate of an additive white Gaussian noise channel with a symmetric analog-to-digital converter (ADC) was derived as a composite function of the binary entropy function, Gaussian Q-function, and the square root function, assuming that the composite function was convex on the set of all non-negative real numbers. However, because mathematically proving this convexity near zero is difficult, studies in this field have only presented numerical results for small values in the domain. Because the low-signal-to-noise (SNR) regime is considered to be a major application area for one-bit ADCs in wireless communication, deriving a concrete proof of the convexity of the composite function on small SNR values (non-negative values near zero) is important. Therefore, this study proposes a novel proof for convexity, which is satisfied for all non-negative values, based on the continuity of the involved functions.