  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.
