Characterization of the Complexity of Computing the Capacity of Colored Noise Gaussian Channels

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.