# Capacity and Normalized Optimal Detection Error in Gaussian Channels

## TL;DR

This paper establishes a precise differential relationship between channel capacity and normalized optimal detection error (NODE) in Gaussian channels, extending it to continuous-time channels and comparing it with the I-MMSE relationship.

## Contribution

It introduces a new differential connection between capacity and NODE for Gaussian channels and extends this to continuous-time channels using waterfilling, comparing it with existing I-MMSE results.

## Key findings

- Derived a differential relationship between capacity and NODE in vector Gaussian channels.
- Extended the C-NODE relationship to continuous-time Gaussian channels with asymptotic accuracy.
- Compared the C-NODE relationship with the I-MMSE relationship, highlighting similarities and differences.

## Abstract

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.