This paper explores the computational complexity involved in determining the capacity of the band-limited additive colored Gaussian noise (ACGN) channel and its capacity-achieving power spectral density (p.s.d.). The study reveals that when the noise p.s.d. is a strictly positive computable continuous function, computing the capacity of the band-limited ACGN channel becomes a $\#\mathrm{P}_1$-complete problem within the set of polynomial time computable noise p.s.d.s. Meaning that it is even more complex than problems that are $\mathrm{NP}_1$-complete. Additionally, it is shown that the capacity-achieving distribution is also $\#\mathrm{P}_1$-complete. Furthermore, under the widely accepted assumption that $\mathrm{FP}_1 \neq \#\mathrm{P}_1$, it has two significant implications for the ACGN channel. The first implication is the existence of a polynomial time computable noise p.s.d. for which the computation of its capacity cannot be performed in polynomial time, i.e., the number of computational steps on a Turing Machine grows faster than all polynomials. The second one is the existence of a polynomial time computable noise p.s.d. for which determining its capacity-achieving p.s.d. cannot be done within polynomial time.

翻译：本文探讨了带限加性有色高斯噪声（ACGN）信道容量及其可达容量的功率谱密度（p.s.d.）计算所涉及的复杂度。研究表明，当噪声功率谱密度为严格正的可计算连续函数时，在多项式时间可计算的噪声功率谱密度集合中，计算带限ACGN信道的容量属于#P1-完全问题，这意味着其复杂度甚至高于NP1-完全问题。此外，研究还表明，可达容量的分布同样属于#P1-完全问题。在广泛接受的FP1 ≠ #P1假设下，这对ACGN信道有两个重要影响：其一，存在一个多项式时间可计算的噪声功率谱密度，其容量无法在多项式时间内计算，即图灵机上的计算步数增长速度超过所有多项式函数；其二，存在一个多项式时间可计算的噪声功率谱密度，其可达容量的功率谱密度无法在多项式时间内确定。