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Networked Embodied Communication: From Collective Distinguishability to Communication Reliability

Yewen Cao, Yulin Shao

arXiv:2609.06969v2eess.SPcs.ITmath.IT

TL;DR

When an agent’s active communication interface is unavailable or inactive, it can convey messages through deliberately selected physical states sensed by infrastructure. This paper develops a distributed-receiver framework and analyzes distinguishability and reliability under repeated sensing. It shows how receiver cooperation can resolve ambiguities that repetition at an individual AP cannot, and uses the analysis to guide joint network and alphabet design.

  • Problem

    A single sensing viewpoint can leave physically different message-bearing states indistinguishable, and repetition cannot resolve distinctions absent from that observation.

    Source: §I · p. 2

  • Method

    The paper analyzes collective distinguishability and repeated-sensing reliability, then uses the results to jointly design the receive AP set and embodied alphabet.

    Source: §I · p. 2, A. Cell-Free Sensing Infrastructure · p. 3

  • Results

    For a finite alphabet, the largest indistinguishable message group determines the asymptotic maximum-error floor, while the minimum pairwise Chernoff information determines the error exponent once all messages are distinguishable.

    Source: §I · p. 2, Proof of Theorem 4 · p. 15

  • Takeaways & Limitations

    Additional receive viewpoints can reveal distinctions absent from individual observations, whereas repetition only accumulates evidence for distinctions already present.

    Source: §VII · p. 13

Abstract

from arXiv · show

Embodied agents need to convey information to surrounding infrastructure, but their active communication interfaces may be unavailable, constrained, or intentionally inactive. Their ability to manipulate physical states offers a complementary path: messages can be encoded in deliberately selected configurations and recovered through infrastructure sensing. This principle underlies embodied communication. Yet physical differences do not guarantee distinguishable messages: a single sensing viewpoint may leave ambiguities that repeated sensing cannot resolve. This paper develops networked embodied communication, where distributed access points (APs) jointly observe message-bearing scatterer positions under fixed illumination. Under a correlated Gaussian sensing model, we characterize the additional distinguishability supplied by receive APs, establish exact redundancy conditions, and reveal how distinctions absent from individual observations can emerge through cross-AP statistical relationships. We then establish the exact asymptotic optimal maximum-error behavior of a finite alphabet under repeated independent sensing. The largest group of indistinguishable messages determines the error floor; once all messages are distinguishable, the minimum pairwise Chernoff information determines the error exponent. For a given alphabet, receiver cooperation can therefore eliminate an error floor that repetition at any individual AP cannot overcome. Building on these results, we derive finite-budget reliability conditions and jointly design the receive AP set and message-bearing positions. Numerical results show that the proposed search closely approaches exact benchmarks on reduced instances with substantially fewer candidate evaluations than exhaustive enumeration, while receiver cooperation reduces the sensing intervals needed to guarantee reliable decoding.

I. INTRODUCTION

Embodied communication encodes messages in deliberately selected physical states, but physical differences alone do not ensure distinguishable sensing observations. This section motivates distributed receive APs and models how they observe a controllable scatterer under fixed illumination.

I. INTRODUCTION

  • Physical states can carry messages recovered through infrastructure sensing, without requiring an agent to transmit a data-bearing waveform.Embodied communication may use configurations, motion, or interactions and RF, visual, infrared, acoustic, or multimodal sensing.Source: §I · p. 1
  • A single sensing viewpoint may leave physically different scatterer positions statistically indistinguishable, which repetition under that same configuration cannot resolve.Distributed receivers can jointly reveal distinctions missing from individual viewpoints.Source: §I · p. 1
  • Networked embodied communication requires collective distinguishability and reliability analysis because an AP’s value depends on the distinctions already observed by the network.The receive network and embodied alphabet are coupled: the alphabet sets the states to distinguish, while the APs determine which distinctions are available.Source: §I · p. 2

II. SYSTEM MODEL

  • The system maps each equiprobable message to an admissible scatterer position, which the agent maintains while the message is sensed.The infrastructure illuminates the scatterer; selected receive APs observe its bistatic echoes and forward observations for CPU decoding.Source: §II · p. 3

A. Cell-Free Sensing Infrastructure

  • The design fixes the transmit AP configuration and probing signals while selecting receive APs from a distributed candidate set.This isolates how receive-side sensing diversity changes message distinguishability and reliability.Source: §I · p. 2, A. Cell-Free Sensing Infrastructure · p. 3

B. Multistatic Observation Model

C. Network Observation Statistics

For each message and receive AP set, the CPU observes a Gaussian sensing distribution whose covariance depends on the scatterer position. Selecting more APs adds observations by marginal extension, making network-level statistical separation the basis for distinguishability.

C. Network Observation Statistics

  • Each embodied symbol induces a random CPU observation, with its covariance depending on the scatterer position and the common residual covariance shared across symbols.The CPU is assumed to know the covariance for each candidate position, for example from a statistical model, calibration, or measurements collected over time.Source: C. Network Observation Statistics · p. 4
  • Repeated sensing intervals provide conditionally independent, identically distributed observations for each embodied symbol.The induced distribution family therefore depends on the alphabet and the selected receive AP set.Source: C. Network Observation Statistics · p. 4
  • For nested receive sets, the smaller-set observation is a marginal of the larger-set observation, enabling analysis of the distinguishability added by new APs.Changing the receive set can alter the statistical distinctions available to the CPU without changing the alphabet positions.Source: C. Network Observation Statistics · p. 4

III. COLLECTIVE DISTINGUISHABILITY UNDER DISTRIBUTED SENSING

  • Chernoff information measures pairwise separation between the joint Gaussian distributions produced by the selected receive network.It generalizes the Bhattacharyya-distance characterization used for the RF model in [9].Source: §III · p. 4

A. Pairwise Chernoff Information

Pairwise Chernoff information quantifies separation between the sensing distributions for two embodied symbols and characterizes their binary error behavior. Its Gaussian form exposes which covariance modes contribute, while the general model can require optimizing its parameter away from the Bhattacharyya midpoint.

A. Pairwise Chernoff Information

  • Under independent repetitions, Chernoff information gives the exact asymptotic binary error exponent and bounds binary error exponentially.The optimizing parameter also tightens the exponential upper bound within the Chernoff family.Source: A. Pairwise Chernoff Information · p. 5
  • For distinct Gaussian hypotheses, only generalized eigenmodes with values different from one contribute to pairwise distinguishability.The Chernoff parameter can be optimized by a one-dimensional root search because the objective is strictly concave for distinct distributions.Source: A. Pairwise Chernoff Information · p. 5
  • Figure 2 illustrates complementary APs: each leaves a different symbol pair indistinguishable, but their joint observations identify the alphabet.The example has AP r1 unable to distinguish q2 from q3 and AP r2 unable to distinguish q1 from q2.Source: Figure 2 · p. 5
  • In the general distributed-sensing model, optimizing the Chernoff parameter gives a bound no looser than the Bhattacharyya bound, with strict improvement when its unique maximizer is not 1/2.Bistatic gains, array responses, illumination, and scattering correlations can break the symmetry that fixes the midpoint in the common-SNR rank-one case.Source: A. Pairwise Chernoff Information · p. 5
  • Zero pairwise Chernoff information is equivalent to identical observation distributions under the selected receive AP set.Positive information marks a statistical distinction present at the CPU; additional APs may resolve an unresolved pair or strengthen an existing distinction.Source: A. Pairwise Chernoff Information · p. 5

B. Receive AP Complementarity

Receive APs can supply complementary distinctions, including distinctions that are absent from individual observations. The section characterizes their incremental value through conditional observation statistics, exact redundancy conditions, and identifiability criteria.

B. Receive AP Complementarity

  • Neither AP identifies the alphabet alone, but their observations identify it collectively.An added AP’s value depends on the distinctions it contributes beyond those already available at the CPU.Source: B. Receive AP Complementarity · p. 5
  • Adding receive observations cannot reduce pairwise Chernoff distinguishability, and adds none exactly when both conditional means and innovation covariances match.For a pair, the equality condition is A_i = A_j and V_i = V_j.Source: B. Receive AP Complementarity · p. 6
  • The larger AP set refines the equivalence partition, and identifies the alphabet exactly when every pair unresolved under the smaller set gains a conditional distinction.For each such pair, a difference in conditional means or innovation covariances is sufficient.Source: B. Receive AP Complementarity · p. 6
  • Added observations contribute through changed conditional innovation covariance or conditional means, the latter arising from cross-AP correlations.The conditional mean contribution can distinguish symbols even when the added observations’ marginal statistics alone do not.Source: B. Receive AP Complementarity · p. 6
  • With independent receive AP observations, the alphabet is identifiable exactly when the selected APs collectively distinguish every symbol pair.This pair-coverage characterization generally does not hold under correlated observations, where an AP’s contribution can depend on its conditional relationship to existing observations.Source: B. Receive AP Complementarity · p. 7

IV. RELIABILITY OF THE EMBODIED ALPHABET

This section formulates reliability for a fixed receive AP set and an embodied symbol held across repeated sensing intervals. It preserves the distinction between unresolved symbols and weak but distinguishable ones in its maximum-error framework.

IV. RELIABILITY OF THE EMBODIED ALPHABET

  • Repeated independent sensing preserves the equivalence partition, while pairwise Chernoff functions and information scale with the number of intervals.The observations are modeled as independent and identically distributed conditioned on the transmitted message.Source: §IV · p. 7
  • It defines the message observation density for L intervals and the maximum error probability of the maximum-likelihood decoder.The supplied passages introduce these quantities but do not state their resulting asymptotic behavior.Source: §IV · p. 7
  • The section uses optimal maximum error probability so reliability cannot be improved by sacrificing a particularly difficult embodied symbol.Randomized decoders are allowed to balance decision probabilities among statistically indistinguishable symbols.Source: §IV · p. 7

A. Asymptotic Reliability

Repeated sensing cannot resolve messages that induce identical observation distributions, while receive AP cooperation can change that ambiguity structure. For finite sensing budgets, the paper gives sufficient and necessary reliability bounds whose gap leaves some designs unresolved.

A. Asymptotic Reliability

  • The exact asymptotic error floor strictly decreases between two nonidentifiable designs if and only if the larger equivalence class becomes smaller.If the added observations make the alphabet identifiable, the error floor is eliminated and the error decays exponentially.Source: A. Asymptotic Reliability · p. 7
  • Once the alphabet is identifiable, the error exponent improves if and only if every pair attaining the current minimum pairwise Chernoff information gains a distinction.A newly distinguishing observation must make the pair’s A or V differ.Source: A. Asymptotic Reliability · pp. 7–8
  • Receive AP cooperation can identify an alphabet that remains nonidentifiable at every individual AP, removing an error floor that temporal repetition cannot overcome.Repetition accumulates evidence for distinctions already present under a fixed receive AP set; cooperation changes the available observations.Source: A. Asymptotic Reliability · pp. 7–8

B. Reliability with a Finite Sensing Budget

  • For a finite sensing budget, the necessary and sufficient bounds leave an interval in which these bounds do not resolve whether the target reliability is attainable.Below C⋆min < Cnec it is impossible for any decoder; at C⋆min ≥ Creq it is guaranteed under ML decoding.Source: B. Reliability with a Finite Sensing… · p. 8
  • When C⋆min(C, S) ≥ Creq, joint ML decoding over S is guaranteed to meet the target, even when no decoder using any individual AP can meet it.This is a finite-budget benefit of joint processing for the same alphabet and sensing budget.Source: B. Reliability with a Finite Sensing… · p. 8

C. Communication Rate

Communication rate is evaluated over the sensing duration, while the receive AP set and embodied alphabet must be designed together because each changes the distinctions relevant to reliability. The joint criterion first resolves ambiguity and then strengthens the weakest pairwise distinctions.

C. Communication Rate

  • A design with J symbols conveys log2 J bits over L sensing intervals when its minimum pairwise Chernoff information meets Creq(J, ϵ, L).The rate normalization counts sensing duration but excludes actuation, settling, and waiting time.Source: C. Communication Rate · p. 8

V. JOINT RECEIVE AP AND EMBODIED ALPHABET DESIGN

  • The receive AP set and embodied alphabet are coupled: an AP useful for one alphabet may be redundant for another, so both must be designed jointly.The alphabet determines required distinctions, while the AP set determines the observations available at the CPU.Source: §V · p. 8

A. Joint Design Criterion

  • The lexicographic design criterion prefers smaller ambiguity-class-size vectors first, then larger sorted pairwise Chernoff-information vectors.For identifiable alphabets, the first entry of the Chernoff vector determines the asymptotic error exponent; later entries break ties.Source: A. Joint Design Criterion · p. 9
  • Adding receive AP observations can only refine the equivalence classes and cannot decrease pairwise Chernoff information, so using the full AP budget does not worsen the criterion.The equality constraint on the AP budget does not restrict the optimum relative to an at-most budget.Source: A. Joint Design Criterion · p. 9

B. Receive AP Selection Structure

A receive AP’s value depends on the current alphabet and AP set: complementary observations can resolve ambiguities, but the resulting gains need not have diminishing returns. The paper also reduces repeated Chernoff evaluations and uses a joint search that revisits AP and alphabet choices.

B. Receive AP Selection Structure

  • An AP can split an unresolved equivalence class, while in an identifiable alphabet it improves the minimum Chernoff information only if every currently limiting pair gains distinguishability.Standalone sensing strength therefore does not determine an AP’s value in the existing network.Source: B. Receive AP Selection Structure · p. 9
  • For a fixed alphabet, minimum pairwise Chernoff information is monotone in the receive AP set but is not generally submodular.In the counterexample, either AP alone gives zero gain, while their joint observations identify the alphabet and yield a positive minimum.Source: B. Receive AP Selection Structure · p. 9

C. Efficient Chernoff Evaluation

  • The reduced Chernoff representation limits determinant order to rank(Ui) + rank(Uj), rather than the full CPU observation dimension DS.In whitened coordinates, only the span of the two hypotheses’ echo factors can contribute to pairwise distinguishability.Source: C. Efficient Chernoff Evaluation · p. 10

D. Joint Search Algorithm

  • The search initializes alphabets from the position pair with greatest Chernoff information, then inserts and refines positions under the lexicographic criterion.It starts from multiple AP singleton sets, refines the alphabet after AP additions, and tests AP exchanges to revisit earlier choices.Source: D. Joint Search Algorithm · p. 10
  • The finite search terminates because each accepted update strictly improves the lexicographic criterion over a finite set of feasible designs.This applies to position refinement and AP exchange as well as the insertion stages.Source: D. Joint Search Algorithm · p. 10

E. Alphabet Size Selection

The paper selects the largest tested alphabet size whose returned joint design meets the finite-budget reliability condition. Numerical results show that AP value depends on the existing observations, and that geometric imbalance can make Chernoff optimization more consequential for some symbol pairs.

E. Alphabet Size Selection

  • For each tested alphabet size J, the search returns a joint design and guarantees the target reliability when its minimum pairwise Chernoff information meets Creq(J, ϵ, L).The selected size is the largest tested J satisfying that condition; if none does, the specified fallback size is 1.Source: E. Alphabet Size Selection · p. 11

A. Pairwise Distinguishability

  • The median relative gain over the Bhattacharyya distance is 0.03%, 0.37%, and 1.98% across balanced, asymmetric, and nearby-AP layouts, respectively.The 95th percentile reaches 8.63% in the nearby-AP case; geometric imbalance can shift the Chernoff maximizer from the midpoint and create an appreciable gap for some pairs.Source: A. Pairwise Distinguishability · p. 11, Figure 4 · p. 11
  • At −36°, the candidate AP’s optimized increment is 5.323, while its standalone value is 6.112; at −4°, the standalone value is 6.979 and the increment is 2.343.The strongest standalone AP is not the most useful addition to the existing set.Source: A. Pairwise Distinguishability · p. 11, Figure 5 · p. 11

VI. NUMERICAL RESULTS

  • Figure 6 plots the estimated maximum ML decoding error versus Nrx for different numbers L of sensing intervals, with a fixed embodied alphabet.Source: Figure 6 · p. 12

B. Reliability with Networked Sensing

Across these results, networked sensing supports reliable decoding through complementary AP observations, while joint design approaches the exact benchmark and reduces the sensing intervals needed for guaranteed rates. The conclusion emphasizes that receiver value depends on the distinctions added beyond the current network.

B. Reliability with Networked Sensing

  • The target error is first reached at Nrx = 5 for L = 2, at Nrx = 3 for L = 3 and 4, and at Nrx = 2 for L = 5 and 6; it is not reached for L = 1.This is for the tested fixed alphabet, AP candidates, SNRref, and error target.Source: B. Reliability with Networked Sensing · p. 12

C. Joint Design Validation

  • 1.0 median objective ratio is achieved by Algorithm 1 for every tested J, with mean relative gap no greater than 0.33%.At J = 10, the median ratios are 0.942 for the sequential design and 0.667 for the insertion only design.Source: Figure 7 · p. 12, C. Joint Design Validation · p. 12
  • Approximately 2.8× to 1.10 × 103 fewer candidate evaluations are needed relative to exhaustive enumeration as J grows from 3 to 10.Source: C. Joint Design Validation · p. 12

D. Guaranteed Communication Rate

  • The guaranteed rate reaches its one-interval ceiling at fewer APs with higher SNR, while larger J raises that ceiling but adds pairwise reliability constraints.For example, at 16 dB the ceilings for J = 6 and J = 8 are reached at Nrx = 2 and 3, respectively.Source: D. Guaranteed Communication Rate · p. 12, Figure 8 · p. 13

VII. CONCLUSION

  • An AP’s network value depends on the new message distinction it contributes, including distinctions arising from cross-AP statistical relationships invisible in marginal observations.Source: §VII · p. 13
  • Repeated sensing accumulates evidence for existing distinctions, whereas additional viewpoints can remove an error floor; after identifiability, exponent gains require improving every currently limiting symbol pair.Source: §VII · p. 13

SUPPLEMENTARY MATERIAL

The supplementary proofs connect Gaussian Chernoff information and AP-set equivalence to reliability behavior. They show how indistinguishable classes determine the asymptotic error floor, while the weakest pair controls decay once the alphabet is identifiable.

SUPPLEMENTARY MATERIAL

  • The correlated-Gaussian Chernoff calculation separates mean and covariance contributions, each nonnegative and zero exactly when its corresponding parameters match.Source: Proof of Theorem 2 · p. 15
  • Adding AP observations refines the equivalence partition, and two symbols remain equivalent exactly when their A and V matrices match.Source: Proof of Theorem 2 · p. 15

Proof of Corollary 3

  • With independent AP observations, two symbols are jointly distinguishable exactly when at least one selected AP distinguishes them locally.Source: Proof of Corollary 3 · p. 15

Proof of Theorem 4

  • The asymptotic maximum-error floor is determined by the largest indistinguishable class, with floor 1 − 1/κmax(C, S).The proof decodes among distinct classes and then randomizes within the selected class.Source: Proof of Theorem 4 · p. 15
  • Adding APs removes a nonzero error floor exactly when every pair still ambiguous under the original set becomes distinguishable under the expanded set.For each such pair, the condition is Ai ≠ Aj or Vi ≠ Vj.Source: Proof of Theorem 4 · p. 16
  • Once identifiable, the minimum pairwise Chernoff information increases only if every pair attaining the current minimum gains distinguishability.Source: Proof of Theorem 4 · p. 16

Proof of Proposition 5

The proof bounds maximum-likelihood error by considering pairwise confusions over repeated independent observations. A union bound over competing symbols then yields a sufficient finite-budget reliability condition.

Proof of Proposition 5

  • For each transmitted symbol, an ML error requires at least one competing likelihood to match or exceed its likelihood.Source: Proof of Proposition 5 · p. 16
  • The Chernoff bound controls each pairwise confusion over L independent sensing intervals, and a union bound sums over the J − 1 competitors.Source: Proof of Proposition 5 · p. 16
  • Requiring the resulting maximum-error upper bound to be no greater than ϵ gives the stated sufficient reliability condition.Source: Proof of Proposition 5 · p. 16

Proof of Proposition 6

These proofs establish a necessary pairwise reliability threshold, show that the AP-set objective is monotone but need not have diminishing returns, and simplify a correlated-Gaussian determinant expression.

Proof of Proposition 6

  • A J-ary decoder meeting the target error induces binary decoders, yielding a necessary lower bound on the minimum pairwise Chernoff information.Source: Proof of Proposition 6 · p. 16

Proof of Proposition 7

  • The AP-set objective is monotone, but a two-AP example violates diminishing returns because one AP’s gain is zero alone and positive after the other is selected.Source: Proof of Proposition 7 · p. 16

Proof of Proposition 8

  • For the correlated-Gaussian expression, the proof cancels the log det K terms and applies Sylvester’s determinant identity to the remaining determinants.Source: Proof of Proposition 8 · p. 17
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