Characterization of the Complexity of Computing the Capacity of Colored Gaussian Noise 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 NP
₁
-complete. Additionally, it is shown that computing the capacity-achieving distribution is also
$\#\mathrm {P}_{1}$
-complete. Furthermore, under the widely accepted assumption that
$\mathrm {FP}_{1} \neq \#\mathrm {P}_{1}$
, this 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 the corresponding ACGN 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 is the existence of a polynomial time computable noise p.s.d. for which determining the corresponding ACGN’s capacity-achieving p.s.d. cannot be done within polynomial time. This implies that either the sequence of achievable rates with guaranteed distance to capacity is not polynomial time computable, or the corresponding blocklength sequence is not polynomial time computable.