Capacity of the Trapdoor Channel with Delayed Feedback

We show that the trapdoor channel's capacity with delayed feedback of two time-instances is given by C₂ᶠᵇ = log₂(3/2). This demonstrates that the feedback capacity degrades sharply even with a single time-instance delay of the channel outputs. The capacity result is established by showing that the delayed feedback capacity can be formulated as a capacity problem with instantaneous feedback and an extended state. Consequently, graph-based methods can be applied to obtain new computable upper and lower bounds on the capacity, which are shown to coincide for the trapdoor channel.