Bounded State Estimation over Finite-State Channels: Relating Topological Entropy and Zero-Error Capacity

Abstract

We investigate bounded state estimation of linear systems over finitestate erasure and additive noise channels in which the noise is governed by a finite-state machine without any statistical structure. Upper and lower bounds on their zero-error capacities are derived, revealing a connection with the topological entropy of the channel dynamics. Some examples are introduced and separate capacity bounds based on their specific features are derived and compared with bounds from topological entropy. Necessary and sufficient conditions for linear state estimation with bounded errors via such channels are then obtained, by extending previous results for nonstochastic memoryless channels to finite-state channels. These estimation conditions bring together the topological entropies of the linear system and the discrete channel.

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

“…However, we do not need to use their random coding technique. This result is also analogous to our previous work on linear state estimation over channels with memory, [37], [39], where we showed that bounded estimation errors could be achieved iff C 0 > h lin . However, the feedback interconnection here between the channel and the LTI system requires different proof techniques.…”

“…The results presented in this paper go well beyond our preliminary work reported in [1] as well as our work on state estimation in [37], [39]. This paper includes new results on general causal channels and stabilization in addition to the proof of Theorem 1 and new properties discussed in Section III-B.…”

“…This result is a zero-error analogue of the formula C f = log q − H ch [36] for the ordinary feedback capacity of stochastic additive noise channels with Shannon entropy rate H ch . In previous work on state estimation, we have shown that log q − h ch upper bounds C 0f [37]. Here we show that this upper bound is, in fact, achievable.…”

“…Moreover, in [37], we have shown that the zero-error capacity (without feedback) is lower bounded by…”

“…Example 5. A binary (w, d) sliding-window channel, has at most d errors in each sliding window of length w [37]. We define the state as a binary word of length w, in which 0 indicates no error and 1 label the erroneous symbol swaps that can occur.…”