Bounds for joint source-channel coding at high SNR

Shannon's capacity and rate-distortion function, combined with the separation principle, provide tight bounds for the minimum possible distortion in joint source-channel coding. These bounds, however, are usually achievable only in the limit of large block length. In their 1973 paper, Ziv and Zakai provide a family of alternative capacity and rate-distortion functions, based on functionals satisfying the data-processing inequality, which potentially give tighter bounds for systems with a small block length, e.g., for scalar modulation. We examine a recently proposed approximation for the Ziv-Zakai bounds based on the Rényi-divergence functional. For the specific case of a uniform source, we derive explicit bounds on the Ziv-Zakai-Rényi rate- distortion function, which prove this approximation in the limit of small distortion. Our results can be extended, using the same technique, to more general sources.