Conditions for tightness of capacity–distortion bounds under noisy feedback

Establish conditions under which the lower and upper bounds on the capacity–distortion functions C_DIF(D) and C_RIF(D) for joint identification and sensing over state-dependent discrete memoryless channels with noisy strictly causal feedback coincide.

Background

The paper derives lower and upper bounds on the capacity–distortion function for both deterministic and randomized identification with noisy strictly causal feedback over state-dependent discrete memoryless channels (Theorems 1 and 2). These bounds generally do not coincide, leaving the exact capacity–distortion characterization unresolved in the noisy-feedback setting.

Identifying regimes (e.g., channel/feedback/noise/state distributions or distortion measures) where the achievability and converse bounds meet would yield a complete capacity–distortion characterization and clarify the impact of feedback noise on joint identification and sensing.