Capacity of Finite-State Channels with Delayed Feedback

TL;DR

This paper shows that the capacity of a finite-state channel (FSC) with delayed feedback of delay d is equal to the instantaneous-feedback capacity of a reformulated FSC with expanded state $\tilde{S}_{t-1}=(S_{t-d},X_{t-d+1}^{t-1})$ and $\tilde{Y}_t=Y_{t-d+1}$, enabling the application of established instantaneous-feedback techniques. It develops two computable bound families—$Q$-graph bounds and duality-based bounds—via a transformed unifilar FSC, and extends these to the delayed-feedback setting. The paper derives exact capacity for the trapdoor channel with two-step delayed feedback, $\mathcal{C}^{\mathrm{fb}}_2 = \log_2(\tfrac{3}{2})$, and provides tight upper bounds on the feedforward capacity through related upper bounds on delayed feedback for the input-constrained BSC and the dicode erasure channel, showing that feedback can increase capacity in these channels. Overall, the delay-to-instantaneous reduction yields tractable, computable bounds and concrete capacity results for classic FSCs, clarifying how feedback delay reshapes information rates in channels with memory.

Abstract

In this paper, we investigate the capacity of finite-state channels (FSCs) in presence of delayed feedback. We show that the capacity of a FSC with delayed feedback can be computed as that of a new FSC with instantaneous feedback and an extended state. Consequently, graph-based methods to obtain computable upper and lower bounds on the delayed feedback capacity of unifilar FSCs are proposed. Based on these methods, we establish that the capacity of the trapdoor channel with delayed feedback of two time instances is given by $\log_2(3/2)$. In addition, we derive an analytical upper bound on the delayed feedback capacity of the binary symmetric channel with a no consecutive ones input constraint. This bound also serves as a novel upper bound on its non-feedback capacity, which outperforms all previously known bounds. Lastly, we demonstrate that feedback does improve the capacity of the dicode erasure channel.

Figures (7)

• Figure 1: Finite-state channel with delayed feedback of $d$ time instances.
• Figure 2: An example of a $Q$-graph with $\mathcal{Q}=2$ and $\mathcal{Y}=\{0,1\}$.
• Figure 3: The trapdoor channel. The channel can be viewed as a box in which at time $t$ a labelled ball $s_{t-1}$ (channel state) lies. Then, a new ball $x_t$ (channel input) is inserted into the box, and the channel output $y_t$ is chosen with equal probability as either $s_{t-1}$ or $x_t$. The remaining ball in the box (either $s_{t-1}$ or $x_t$) is now called $s_t$ and serves as the channel state for the next time-instance.
• Figure 4: The DEC. The inputs take values from the binary alphabet while the outputs take values in $\mathcal{Y}=\{-1,0,1,?\}$. Given an input $x_t$, the output of the DEC is $y_t=x_t-x_{t-1}$ with probability $1-p$, or $y_t=?$ with probability $p$, where $p \in [0,1]$ is the channel parameter. The channel state is the previous input, i.e. $s_{t-1}=x_{t-1}$.
• Figure 5: The trapdoor channel with delayed feedback of two time instances as a new unifilar FSC with instantaneous feedback.

Theorems & Definitions (27)

• Theorem 1: PermuterCuffVanRoyWeissman08, Th. 3
• Theorem 2: Bellman equation, Arapos93_average_cose_survey
• Remark 1
• Theorem 3
• Theorem 4
• Remark 2
• Remark 3
• Theorem 5: Computable upper bounds
• Theorem 6: Bellman equation
• Remark 4