Designing capacity-achieving coding schemes for the band-limited additive colored Gaussian noise (ACGN) channel has been and is still a challenge. In this paper, the capacity of the band-limited ACGN channel 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. t is shown that there are band-limited ACGN channels having computable continuous spectral densities whose capacity are non-computable numbers. Moreover, it is demonstrated that for those channels, it is impossible to find computable sequences of asymptotically sharp upper bounds for their capacities.

翻译：设计针对带限加性有色高斯噪声（ACGN）信道的容量逼近编码方案始终是一个挑战。本文从基本算法视角研究带限ACGN信道的容量，探讨其容量是否可被算法计算的问题。为此，采用图灵机概念——这一数字计算机的根本性能极限——作为工具。研究表明：存在具有可计算连续谱密度的带限ACGN信道，其容量为不可计算数。进一步证明，对于此类信道，不可能找到可计算的渐近紧致上界序列来逼近其容量。