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We consider finite-state channels (FSCs) where the channel state isstochastically dependent on the previous channel output. We refer to these asNoisy Output is the STate (NOST) channels. We derive the feedback capacity ofNOST channels in two scenarios: with and without causal state information (CSI)available at the encoder. If CSI is unavailable, the feedback capacity is$C_{\text{FB}}= \max_{P(x|y')} I(X;Y|Y')$, while if it is available at theencoder, the feedback capacity is $C_{\text{FB-CSI}}= \max_{P(u|y'),x(u,s')}I(U;Y|Y')$, where $U$ is an auxiliary RV with finite cardinality. In bothformulas, the output process is a Markov process with stationary distribution.The derived formulas generalize special known instances from the literature,such as where the state is i.i.d. and where it is a deterministic function ofthe output. $C_{\text{FB}}$ and $C_{\text{FB-CSI}}$ are also shown to becomputable via convex optimization problem formulations. Finally, we present anexample of an interesting NOST channel for which CSI available at the encoderdoes not increase the feedback capacity.

The Feedback Capacity of Noisy Output is the STate (NOST) Channels