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An Information-Theoretic Approach to Joint Sensing and Communication

Mehrasa Ahmadipour, Mari Kobayashi, Miche`le Wigger, Giuseppe Caire

arXiv:2107.14264v2cs.IT

TL;DR

The paper studies how a transmitter can communicate and estimate a receiver’s state through one waveform with generalized feedback. It characterizes point-to-point and physically degraded broadcast-channel tradeoffs, develops numerical evaluation, and provides bounds for general broadcast channels. Illustrative examples show gains from optimal co-design over time-sharing baselines.

  • Problem

    The paper asks for the fundamental rate-distortion limits of joint communication and transmitter-side state sensing, beyond resource-splitting and task-specific waveform designs.

  • Method

    The paper characterizes capacity-distortion tradeoffs for point-to-point and physically degraded broadcast channels, gives general broadcast-channel bounds, and develops a Blahut-Arimoto-type numerical method.

  • Results

    Optimal co-design schemes outperform basic and improved time-sharing schemes in illustrative examples, while general broadcast channels admit inner and outer bounds on the capacity-distortion region.

  • Takeaways & Limitations

    Communication and sensing can be jointly optimized through the common waveform, with some matched cases achieving capacity without compromising sensing performance.

Abstract

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A communication setup is considered where a transmitter wishes to convey a message to a receiver and simultaneously estimates the state of that receiver through a common waveform. The state is estimated at the transmitter by means of generalized feedback, i.e., a strictly causal channel output, and the known waveform. The scenario at hand is motivated by joint radar and communication, which aims to co-design radar sensing and communication over a shared spectrum and hardware. For the case of memoryless single receiver channels with i.i.d. time-varying state sequences, we fully characterize the capacity-distortion tradeoff, defined as the largest achievable rate below which a message can be conveyed reliably while satisfying some distortion constraints on state sensing. We propose a numerical method to compute the optimal input that achieves the capacity-distortion tradeoff. Then, we address memoryless state-dependent broadcast channels (BCs). For physically degraded BCs with i.i.d. time-varying state sequences, we characterize the capacity-distortion tradeoff region as a rather straightforward extension of single receiver channels. For general BCs, we provide inner and outer bounds on the capacity-distortion region, as well as a sufficient condition when this capacity-distortion region is equal to the product of the capacity region and the set of achievable distortions. A number of illustrative examples demonstrate that the optimal co-design schemes outperform conventional schemes that split the resources between sensing and communication.

I. INTRODUCTION

The paper develops an information-theoretic framework for jointly communicating and sensing a receiver’s state through a shared waveform and generalized feedback. It characterizes key capacity-distortion tradeoffs, gives numerical evaluation methods, and establishes bounds and special-case results for broadcast channels.

  • Motivation: Joint sensing and communication is motivated by shared-spectrum applications requiring continuous state estimation while exchanging information.The model integrates sensing and communication through one waveform and hardware rather than separately allocating resources.
  • Research gap: The paper addresses fundamental performance limits that prior system-specific waveform studies did not characterize.Its central performance measure is the capacity-distortion tradeoff, balancing communication rate against state-estimation distortion.
  • Point-to-point results: For point-to-point channels, the capacity-distortion-cost tradeoff is exactly characterized, with sensing and communication coupled only through the common input waveform.The communication scheme can ignore generalized feedback, which is used for state sensing; matched cases can achieve capacity and minimum distortion simultaneously.
  • Broadcast-channel results: For broadcast channels, physically degraded cases are characterized, while general cases receive inner and outer bounds and a sufficient product-region condition.General broadcast-channel feedback can improve communication and sensing, but optimal sensing depends on the waveform rather than the feedback-code construction.
  • Point-to-point results: A Blahut-Arimoto-type algorithm and alternating optimization method evaluate the point-to-point tradeoff numerically.The paper also characterizes a deterministic symbol-by-symbol estimator that depends on the current input and feedback signal.
  • Tradeoff properties: The tradeoff is nondecreasing and concave in distortion and cost, and can saturate at the no-estimation channel capacity.When a capacity-achieving input also minimizes distortion, the tradeoff remains equal to channel capacity for all allowed distortions above the minimum.

C. Examples

The examples compare optimal co-design with time-sharing baselines for binary and fading Gaussian channels. In both settings, the tradeoff is computed from input distributions, with figures illustrating the resulting performance comparisons.

  • Baseline Schemes: The Basic TS scheme separates sensing and communication, while Improved TS uses one waveform but time-shares between sensing- and communication-prioritized modes.Basic TS cannot perform both tasks simultaneously; Improved TS can, but still interpolates between two operating points.
  • Baseline Schemes: The optimal co-design tradeoff contains the Improved TS endpoints, whereas its other operating points are typically suboptimal.The endpoints (Rmin(B), Dmin(B)) and (CNoEst(B), Dmax(B)) lie on C(D, B).
  • Example 1: Binary Channel: For the binary multiplicative Bernoulli channel, the tradeoff is parameterized by C = qHb(p) and D = p min{q, 1 −q}, for p ∈ [0, 1/2].This specialization uses Hamming distortion and perfect output feedback, with no cost constraint.
  • Example 1: Binary Channel: For q = 0.4, the binary-channel figure compares the tradeoff with both TS baselines and reports a significant gain for optimal co-design.The capacity-achieving input is uniform, while minimum distortion Dmin = 0 is achieved by always sending X = 1.
  • Example 2: Real Gaussian Channel: At B = 10 dB and σ2_fb = 1, the Gaussian baseline includes Rmin(B) = 0.733 and Dmin(B) = (1 + σ2_fb)/(1 + P + σ2_fb).Communication without sensing achieves distortion Dtrivial(B) = 1, while sensing without communication achieves (0, Dmin(B)).
  • Example 2: Real Gaussian Channel: For the fading AWGN example, the numerical tradeoff is approximated using a modified Blahut–Arimoto procedure after quantizing the input, noise, and state.The approximation uses a 16-point PAM input, a 50-point noise alphabet, and an 8000-point quantizer for the state variable S^2.

III. MULTIPLE RECEIVERS

The broadcast-channel section formulates joint communication and state estimation for two receivers, then characterizes exact results for physically degraded channels and bounds for general channels. It also compares these regions with time-sharing baselines.

  • System Model: The two-receiver model sends a common message and two private messages while estimating both receivers’ state sequences from generalized feedback.Receiver k observes its own state sequence, and the transmitter produces state estimates using the waveform and feedback.
  • System Model: A broadcast-channel code contains three message sets, feedback-dependent encoders, receiver decoders, and transmitter state estimators.The estimators map the full input and feedback sequences to reconstruction sequences for each receiver.
  • Capacity-Distortion Region: The capacity-distortion region is the closure of all achievable tuples (R0, R1, R2, D1, D2) satisfying reliable decoding and average distortion constraints.The model uses bounded per-symbol distortion functions and joint probability of error.
  • Capacity-Distortion Region: Optimal estimator functions are symbol-by-symbol and independent of the encoding and decoding functions.This parallels the single-receiver case and allows the estimation functions to be determined separately from the communication code.
  • Results and Baselines: Exact characterization is available for physically degraded state-dependent broadcast channels, while general channels receive inner and outer bounds.A full general characterization is difficult because even the corresponding broadcast-channel capacity regions with and without feedback are unknown.
  • Results and Baselines: Broadcast-channel baselines use either separate sensing and communication or a common waveform that prioritizes one function, with operating points depending on the chosen input distribution.Because there are two distortions and three rates, the optimal input distribution for each mode need not be unique.

B. Capacity-Distortion Region for Physically Degraded SDMBCs

For physically degraded state-dependent broadcast channels, the paper characterizes the capacity-distortion region exactly and evaluates it on binary examples. Feedback supports sensing but does not improve communication, while waveform choice couples the achievable rates and distortions.

  • Characterization: Theorem 2 gives the exact capacity-distortion region as the closure of tuples induced by a joint law P_UX and two rate constraints.The result applies to physically degraded SDMBCs with generalized feedback.
  • Characterization: Superposition coding can achieve the region while ignoring feedback for data communication and using optimal estimators for sensing.The converse follows standard steps, whereas achievability combines superposition coding with the optimal estimators.
  • Binary examples: In Example 3, zero distortions D1 = D2 = 0 require deterministic X = 1 and yield zero communication rates.The communication-optimal input is Xmax ∼B(1/2), which simultaneously maximizes both communication rates.
  • Binary examples: For γ = 0.5 and q = 0.6, the optimal co-design boundary shows a significant gain over basic and improved time-sharing baselines.The plotted boundary is the projection onto (R1, R2, D1); D2 is omitted because it is a scaled version of D1.
  • Characterization: The region expression captures rate-rate, rate-distortion, and distortion-distortion tradeoffs through the parameters r and p.The parameter r controls the tradeoff between the two rates, while p controls the rate-distortion and distortion-distortion tradeoffs.
  • Binary examples: With flipping inputs, rate constraints remain identical but distortion constraints are relaxed because receiver-2 outputs provide additional state information.When X = 0 and Y2 = 1, the transmitter infers S2 = 1 and therefore S1 = 1.

C. Capacity-Distortion Region for General SDMBCs

For general state-dependent broadcast channels, the paper provides inner and outer bounds because the capacity region is unknown in general. It also identifies a sufficient condition under which rates and distortions separate as a Cartesian product.

  • Outer bound: Theorem 3 gives an outer bound on the capacity-distortion region using auxiliary variables U1 and U2 and constraints on individual and sum rates.The bound also includes average distortion constraints.
  • Inner bound: Proposition 1 gives an inner bound based on auxiliary variables U0, U1, U2 and compression variables V0, V1, V2.Achievability uses block-Markov coding, Marton coding for fresh data, and compression information describing the previous block's inputs and outputs.
  • Scope: The inner and outer bounds coincide only in special cases because even the feedback and no-feedback capacity regions are generally unknown.This uncertainty makes complete characterization of the general SDMBC capacity-distortion region challenging.
  • No rate-distortion tradeoff: Under Proposition 2's condition, the capacity-distortion region equals the Cartesian product of the SDMBC capacity region and distortion region.The condition requires functions ψ1 and ψ2 satisfying the stated relations irrespective of the input distribution PX.

D. Example 5: Erasure BC with Noisy Feedback

The state-dependent erasure broadcast channel with noisy feedback admits a Cartesian-product capacity-distortion region under the stated conditions. The example then contrasts sensing and communication modes and characterizes inner and outer bounds for the state-dependent Dueck broadcast channel.

  • Erasure BC with noisy feedback: The state-dependent erasure BC uses binary states and erasure variables at both receivers, with Hamming distortion measures for state estimation.The channel and distortion setup is specified in (76)–(77).
  • Erasure BC with noisy feedback: The erasure BC’s capacity-distortion region equals the Cartesian product of its capacity region and distortion region.This conclusion is stated as Corollary 5 under the specified conditions.
  • State-dependent Dueck BC: For the state-dependent Dueck BC, the capacity-distortion region is a product of capacity and distortion regions when q ∈[0, 1/2].This is the no-rate-distortion-tradeoff case identified in Corollary 6.
  • State-dependent Dueck BC: For the general Dueck case, the paper provides an outer bound and an inner bound, with the inner bound including their convex hull.The bounds characterize both the distortion and capacity regions, while Fig. 7 compares them with time-sharing baselines for q = 3/4.
  • Time-sharing comparisons: In the Dueck example, sensing with communication achieves R1 + R2 ≤1 and Dk ≥5/32, whereas sensing without communication has zero rate.The communication-mode baseline instead uses an i.i.d. Bernoulli-1/2 input with sum rate R1 + R2 = 25/16 and distortions Dmax,1 = Dmax,2 = 11/64.

IV. CONCLUSION

The paper characterizes sensing–communication tradeoffs for single-user and physically degraded broadcast channels, and bounds them for general broadcast channels. Its examples show gains from co-design and cases without a rate-distortion tradeoff, while channels with memory remain future work.

  • Main conclusions: The paper fully characterizes the capacity-distortion tradeoff for single-user and physically degraded broadcast channels.For general broadcast channels, it presents inner and outer bounds.
  • Main conclusions: Optimal co-design yields non-negligible gains over basic and improved time-sharing schemes in the illustrative examples.The comparison includes schemes that either separate sensing and communication or prioritize one task through the common waveform.
  • Main conclusions: In some cases, capacity is achieved without compromising sensing performance.The paper also states that optimal sensing depends on the waveform rather than the underlying coding scheme in the studied single-transmitter systems.
  • Future work: Characterizing the capacity-distortion tradeoff for channels with memory is identified as future research.The discussion notes that feedback increases point-to-point capacity for channels with memory.

APPENDIX A PROOF OF LEMMA 1

The appendix develops the coding and optimization arguments underlying the single-user capacity-distortion tradeoff. It uses symbol-by-symbol estimation, random coding, and an alternating Blahut–Arimoto-type optimization over input distributions.

  • Achievability construction: The encoder reconstructs the state symbol by symbol from each transmitted input and observed feedback signal.The reconstruction sequence applies the optimal estimator ˆs∗ to every pair (xi, zi).
  • Achievability construction: Random coding achieves reliable communication whenever R < I(X; Y |S), while the expected distortion is evaluated under the induced state-input-feedback distribution.The proof combines a packing argument for error probability with an averaged distortion bound.
  • Tradeoff optimization: Time-sharing makes the achievable rate-distortion set convex, so its boundary is obtained through a parameterized optimization for each µ ≥0.The optimization incorporates communication and distortion objectives under the input-cost constraint.
  • Optimization proof: The proof establishes the optimization updates using conditional mutual-information identities, concavity in the input distribution, and KKT conditions for the cost constraint.The conditional distribution update is optimized by matching the relevant conditional marginal, while the input update follows from KKT conditions.
  • Tradeoff optimization: Alternating maximization over the auxiliary conditional distribution and input distribution yields an optimal convergent input distribution for each parameter choice.Varying µ traces the tradeoff at fixed cost B, and varying B gives the full capacity-distortion-cost boundary.

APPENDIX D PROOF OF COROLLARY 1

The appendix proves that under a structural condition involving a function of the input and feedback, the expected distortion is independent of the input distribution. Consequently, the rate-distortion function remains at channel capacity above the minimum distortion.

  • Distortion independence: The proof introduces T = ψ(X, Z) and rewrites expected distortion using the conditional state distribution given the input and feedback.The Markov relation S–T–(X, Z) supports the reduction.
  • Distortion independence: Because (T, S) is independent of X under the stated condition, the expected distortion does not depend on the input distribution PX.This is the key step establishing the absence of a rate-distortion tradeoff in the covered cases.
  • Distortion independence: For any input cost B, the rate-distortion function equals channel capacity for all allowed distortions D ≥ Dmin.The appendix connects distortion independence to the constant rate-distortion tradeoff.

APPENDIX E PROOF OF REMARK 1

The achievability proof constructs codes and estimators, then shows that decoding errors and sensing distortions satisfy the required constraints under mutual-information conditions.

  • Code construction: The proof fixes an input distribution and estimator achieving the target capacity-distortion-cost tuple, then adapts codebook generation, encoding, decoding, and estimation to the receiver’s observed state.The construction replaces the state used in decoding with the receiver state while retaining the general coding and estimation procedure.
  • Decoding: The decoder declares the unique message index whose codeword satisfies the prescribed joint typicality condition; otherwise, it declares an error.This is the decoding rule used after codebook generation.
  • Error analysis: The average error probability is bounded by the union of two error events, each vanishing asymptotically by the weak law of large numbers and the packing lemma.The analysis averages over channel noise, states, and the random code construction.
  • Rate condition: R < I(X; Y |S_R) is the decoding condition ensuring reliable communication for the receiver-state formulation.The condition appears as the central rate constraint in the achievability argument.
  • Constraints: The construction satisfies the cost constraint by design, while the distortion constraint follows as the approximation parameter ϵ approaches zero.The proof then establishes the existence of deterministic codebooks satisfying the required constraints.

APPENDIX G PROOF OF THEOREM 3

The converse proof enhances the physically degraded broadcast channel in two ways, derives sum-rate bounds for both, and combines them to establish the desired outer bound.

  • Channel enhancements: The first enhanced channel gives Receiver 1 both state sequences and both outputs, preserving physical degradedness for every input distribution.This enhancement yields a channel whose total rate is bounded using the joint outputs and states.
  • Sum-rate bounds: Both enhancements imply the sum-rate bound R0 + R1 + R2 ≤ I(X; Y1, Y2 | S1, S2) + ϵn.The bound is obtained separately for the two enhanced broadcast channels.
  • Channel enhancements: The reversely enhanced channel gives Receiver 2 both state sequences and both outputs while Receiver 1 retains only its original observation.The same converse strategy is applied with the receiver indices exchanged.
  • Conclusion: Combining the enhanced-channel inequalities and taking n → ∞ followed by ϵn ↓ 0 establishes the converse result.The limiting operations remove the finite-blocklength slack from the outer bound.
  • Estimator analysis: For the example’s estimator analysis, the optimal state estimate depends on the observed outputs and input relation, including cases where the state is deterministic or uninformative.The proof distinguishes equal and unequal input cases and derives corresponding estimators.

B. Minimum distortion

The minimum-distortion analysis derives optimal estimators from conditional state distributions and evaluates the resulting distortion as a function of the input disagreement probability.

  • Distortion evaluation: The expected distortion is evaluated for an input distribution summarized by t := Pr[X1 ≠ X2].The calculation proceeds separately for the distortion of state S2 and uses the optimal estimators.
  • Estimator derivation: The optimal estimators are obtained by calculating conditional state probabilities given the inputs and observed outputs.The analysis uses these conditional distributions to select the minimizing state estimate.
  • Estimator derivation: When the observations determine a state, the minimum conditional distortion is zero; when they reveal nothing, the prior state distribution remains relevant.The proof identifies both deterministic and uninformative observation cases.
  • Distortion evaluation: The resulting distortion expression contains the term 2 min{P_S(1), P_S(0)(1 + P_S(0))}.This term arises in the evaluated minimum-distortion expression for the example.
  • Capacity consequence: The converse sum-rate constraint is maximized at t = 1/2, completing the converse to the capacity region in the example.The maximizing disagreement probability is combined with the preceding distortion analysis.

D. Proof of Achievability Results

The achievability analysis selects auxiliary variables and input distributions, evaluates the resulting constraints, and varies t to characterize achievable rate-distortion tuples.

  • Input choice: The construction uses Bernoulli-1/2 variables X0, X1, and X2, with X0 independent of (X1, X2) and Pr[X1 ≠ X2] controlled by t.This parameterization provides the input family used to evaluate the achievable region.
  • Auxiliary variables: The auxiliary variables are chosen as V1 = (X0, X1), V2 = (X0, X2), and V0 = X1 ⊕ Y′1.These variables are substituted into the general achievable-region constraints.
  • Achievable region: For any t ∈ [0, 1], rate-distortion tuples satisfying the derived distortion and rate constraints are achievable.The achievable set is parameterized by the input disagreement probability.
  • Time-sharing: For q ≤ 1/2, the distortion constraints do not depend on t, so t = 1/2 can be chosen without loss of optimality to achieve the capacity-region sum-rate bound.This choice, together with the other rate constraints, establishes achievability of the stated capacity region.
  • Time-sharing: For q > 1/2, varying t over an appropriate interval produces the achievable set, whose convex hull may require combinations involving t = 0 or t = 1.The required endpoint depends on whether q lies in [2/3, 1] or [1/2, 2/3].
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