Capacity and Normalized Optimal Detection Error in Gaussian Channels

For vector Gaussian channels, a precise differential connection between channel capacity and a quantity termed normalized optimal detection error (NODE) is presented. Then, this C–NODE relationship is extended to continuous-time Gaussian channels drawing on a waterfilling characterization recently found for the capacity of continuous-time linear time-varying channels. In the latter case, the C–NODE relationship becomes asymptotic in nature. In either case, the C–NODE relationship is compared with the I–MMSE relationship due to Guo et al. connecting mutual information in Gaussian channels with the minimum mean-square error (MMSE) of estimation theory.