Joint Identification, Transmission, and Sensing over State-Dependent Channels with Noiseless Feedback

I. INTRODUCTION

• The joint interplay of identification, transmission, and sensing over channels with feedback remains largely unexplored.Source: §I · p. 1
• The model uses randomized encoding to send transmission and identification messages while estimating the unknown state sequence under an average distortion constraint.Source: §I · p. 1
• The randomized ITS capacity-distortion region has a single-letter separation structure: rates are governed by the averaged channel subject to the sensing-induced input constraint.For a given transmission rate, the identification rate is the largest output entropy compatible with the distortion and transmission-rate constraints.Source: §I · p. 1
• Achievability combines constant-composition transmission coding, equiprobable output partitioning, and combinatorial array constructions; the converse uses averaged-channel reduction and feedback-identification arguments.The converse also uses distortion-constrained input characterization.Source: §I · p. 1

II. SYSTEM MODEL AND MAIN RESULT

• The channel is specified by finite alphabets X, S, and Y and a conditional distribution W_S(y|x,s).Source: §II · p. 1
• The state sequence is i.i.d. according to P_S, known in distribution but not in realization to the transmitter or receiver.Source: §II · p. 2
• For an input distribution P, the induced output distribution is denoted P overbar W.Source: §II · p. 2

A. Feedback

• The encoder can use all previous outputs without noise when selecting the next input.Source: A. Feedback · p. 2
• A deterministic feedback strategy is a sequence of functions mapping past output histories to channel inputs.Source: A. Feedback · p. 2
• A randomized feedback strategy is a probability distribution over deterministic feedback strategies.Source: A. Feedback · p. 2

B. Sensing

• The distortion function is a nonnegative function on state-estimate pairs, and expected distortion is taken under the joint distribution induced by coding, feedback, the channel, and estimation.Source: B. Sensing · p. 2
• An optimal estimator is defined, with an arbitrary choice permitted when multiple minimizers exist.Source: B. Sensing · p. 2
• The paper defines minimum distortion for each input symbol and the corresponding minimum distortion for an input distribution.Source: B. Sensing · p. 3

• For a fixed transmission rate, the identification rate is the largest output entropy achievable under the distortion and transmission-rate constraints.Source: C. Identification, Transmission, and… · p. 4
• Theorem 1 characterizes the randomized ITS capacity-distortion region for distortion levels D ≥ Dmin, assuming positive capacity of the averaged channel.Source: C. Identification, Transmission, and… · p. 4
• The region has a single-letter separation structure: communication rates are determined by the averaged channel, with input distributions restricted by the sensing constraint.Source: C. Identification, Transmission, and… · p. 4
• If distortion is unconstrained, the region reduces to the classical randomized identification-transmission region with feedback for the averaged channel.Source: C. Identification, Transmission, and… · p. 4

• The symbolwise estimator h⋆ lower-bounds distortion for any estimator, and the i.i.d.-state, causal-feedback setting makes the posterior of St depend only on (Xt, Yt).Source: §III · p. 4
• For input types approaching P, the constant-composition construction supplies codewords and disjoint decoding regions when rates satisfy R′ < I(P; W̄) and R′′ < H(W̄|P).Source: §III · p. 4
• The output-partition lemma provides partitions whose probability masses are nearly equal, with deviation from uniformity vanishing exponentially.Source: §III · p. 5
• The identification construction uses an array whose distinct rows have Hamming distance greater than (1 − ε)Sn when log Nn ≤ cεSn.Source: §III · p. 5
• The counting argument bounds output-sequence counts using the largest output entropy over distortion-admissible inputs, while atypicality and distortion violations have vanishing probability.Source: §III · p. 6

• For I(P⋆; W̄) = 0 and H(P⋆W̄) > 0, only RT = 0 is possible, while any RID < H(P⋆W̄) is constructed.Source: A. Code Construction · p. 6
• For a fixed P⋆ in the distortion-feasible set, a constant-composition codeword and conditionally typical outputs are partitioned into approximately equiprobable sets.Source: A. Code Construction · pp. 6–7
• The first-kind error vanishes for rates below H(P⋆W̄), and the asymptotic distortion is at most d⋆(P⋆) ≤ D.Source: A. Code Construction · p. 7
• Combinatorial array rows assign tags to identities and partition indices, with distinct identification rows separated by Hamming distance greater than (1 − ε)Bn.Source: A. Code Construction · p. 7
• The feedback index selects an identification tag sent using a length-⌈√n⌉ transmission code on the averaged channel.Source: A. Code Construction · pp. 7–8

B. Achievability

• The construction produces an (n + t0)-length randomized ITS code with the claimed message sizes, error probabilities, and distortion guarantee.The proof verifies the first-kind and second-kind error criteria, then checks average distortion and rates.Source: B. Achievability · pp. 8–9
• Distinct identification messages are protected by controlling false acceptance after transmission and tag decoding, including collisions in the identification arrays.For two different identification messages, false acceptance requires a first-phase decoding error, a tag error, or an array collision; the proof bounds collision counts and the resulting probability.Source: B. Achievability · p. 8
• The symbolwise estimator uses h⋆(Xt, Yt), and constant-composition Phase I codewords ensure their induced input type approaches a distortion-feasible distribution.The Phase I distortion converges to at most D; because t0 = o(n), Phase II contributes vanishing normalized distortion.Source: B. Achievability · p. 9
• The rate construction reaches the target identification rate, covers positive transmission-information cases, and handles zero-rate boundaries separately.The proof also treats I(P⋆; W̄) = 0 with H(P⋆W̄) > 0 separately, while zero output entropy makes both rates zero.Source: B. Achievability · p. 9

C. Converse Proof

• The transmission converse gives RT ≤ max over P in P_D of I(P; W̄).The bound follows from the channel converse and compactness and continuity.Source: C. Converse Proof · p. 10
• For RT > 0, trimming feedback-strategy regions bounds RID by the maximum output entropy over distortion-feasible inputs meeting the transmission-rate constraint.The proof takes limits in the trimming parameters to obtain the feasible-set bound.Source: C. Converse Proof · p. 12
• A feasible input P⋆ exists with RT ≤ I(P⋆; W̄) and RID ≤ H(P⋆W̄), giving both rate bounds under the same distortion constraint.Compactness and continuity ensure the maximum is attained.Source: C. Converse Proof · p. 12

V. CONCLUSION AND FUTURE WORK

• For each distortion-feasible input P, transmission and identification rates are bounded by I(P; W̄) and H(PW̄), respectively.When P_D = P(X), the characterization recovers the classical randomized identification-transmission region with feedback; otherwise, sensing may restrict achievable rates.Source: §V · p. 12
• Future directions include noisy or rate-limited feedback, encoder state information, secrecy constraints, multiuser and Gaussian models, and finite-blocklength analysis.Source: §V · p. 12