Designing capacity achieving coding schemes for the band-limited additive Gaussian channel with colored noise has been and is still a challenge. In this paper, the capacity of the band-limited additive Gaussian channel with colored noise is studied from a fundamental algorithmic point of view by addressing the question of whether or not the capacity can be algorithmically computed. To this aim, the concept of Turing machines is used, which provides fundamental performance limits of digital computers. It has been shown that there exist Gaussian colored noise with a computable continuous noise spectral density whose capacity is a non-computable number. Moreover, it has been demonstrated that for these channels, it is not possible to find a computable sequence of asymptotically sharp upper bounds for their capacity.

翻译：针对带限有色噪声加性高斯信道设计逼近容量的编码方案始终是一个挑战。本文从基础算法视角研究带限有色噪声加性高斯信道的容量，通过探讨容量是否可被算法计算这一核心问题展开分析。为此，本文采用图灵机概念——该概念为数字计算机提供了基础性能极限。研究表明，存在可计算连续噪声谱密度的有色高斯噪声，其容量为不可计算数。进一步证明，对于此类信道，无法找到其容量的渐近紧上界的可计算序列。