Zero-Error Feedback Capacity for Bounded Stabilization and Finite-State Additive Noise Channels

This article studies the zero-error feedback capacity of
causal
discrete channels with memory. First, by extending the classical zero-error feedback capacity concept, a new notion of
uniform zero-error feedback capacity

$C_{0f} $
for such channels is introduced. Using this notion a tight condition for bounded stabilization of unstable noisy linear systems via causal channels is obtained, assuming no channel state information at either end of the channel. Furthermore, the zero-error feedback capacity of a class of additive noise channels is investigated. It is known that for a discrete channel with correlated additive noise, the ordinary capacity with or without feedback is equal
$\log q-\mathcal {H}_{ch} $
, where
$\mathcal {H}_{ch} $
is the entropy rate of the noise process and
$q $
is the input alphabet size. In this paper, for a class of finite-state additive noise channels (FSANCs), it is shown that the zero-error feedback capacity is either zero or
$C_{0f} =\log q -h_{ch} $
, where
$h_{ch} $
is the
topological entropy
of the noise process. A condition is given to determine when the zero-error capacity with or without feedback is zero. This, in conjunction with the stabilization result, leads to a “Small-Entropy Theorem”, stating that stabilization over FSANCs can be achieved if the sum of the topological entropies of the linear system and the channel is smaller than
$\log q$
.