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Fundamental Limits of Joint Target Detection and Parameter Estimation - Characterizing Mixed-State Sensing Limits via Posterior Entropy Volume

Dazhuan Xu, Nan Wang, Han Zhang

arXiv:2609.09667v1cs.ITeess.SPmath.IT

TL;DR

Joint target sensing must resolve both target presence and continuous parameters, yet decision procedures typically depend on specified losses. The paper uses mixed typical-set and volume arguments to show that posterior mixed entropy volume contracts from its prior scale by 2^-I(Ξ;Y).

Abstract

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The development of integrated sensing and communication calls for a unified theoretical foundation for sensing. This paper models target-presence patterns and continuous physical parameters as a mixed discrete-continuous state $\Xi$ on a branched reference measure, and treats posterior entropy volume and joint mutual information as two complementary representations of the same limit. Entropy volume carries physical units, can be compared with engineering scales such as resolution cells, and remains meaningful when the number of active targets varies; mutual information is dimensionless, invariant to coordinates and units, and additive through the chain rule. We define entropy number, entropy volume, and mixed entropy volume for discrete, continuous, and mixed states, respectively. The maximum-entropy principle for the uniform distribution on a support of fixed measure explains the measure-theoretic meaning of the exponential entropy scale, while the main limit follows from a mixed asymptotic equipartition property and a posterior probability-volume inequality through conditional typical sets. For asymptotically reliable high-probability sensing regions, the minimum achievable first-order posterior mixed entropy volume equals the prior mixed entropy volume multiplied by $2^{-I(\Xi;Y)}$, where $I(\Xi;Y)=I(V;Y)+I(X_V;Y\mid V)$. Thus, detection and estimation contributions multiply in the volume domain and add in the bit domain, and one sensing bit halves the posterior effective measure. We further prove that posterior-preserving cascades attain the direct-inference limit, while arbitrary intermediate compression incurs the exact information loss $I(\Xi;Y\mid Z)$. Numerical results for a single-target presence-range model illustrate the information composition, posterior entropy-volume contraction, and cascade-interface loss.

I. INTRODUCTION

The paper addresses the lack of a posterior-volume limit for sensing states that combine target presence with continuous parameters. It models those states on branched reference measures and characterizes their posterior uncertainty independently of any particular estimator or detector.

I. INTRODUCTION

  • The paper seeks a posterior-volume limit for joint sensing of discrete target presence and continuous parameters, beyond fixed-alphabet or distortion-based formulations.The target is the complete mixed posterior, with both achievability and a converse.Source: §I · pp. 1–2

A. General Sensing Channel

  • Posterior uncertainty is evaluated before task-dependent outputs such as MAP, MMSE, or hard decisions are applied.The sensing channel is specified by the physical-state prior and observation kernel, rather than by optimizing the prior as a channel input.Source: A. General Sensing Channel · pp. 2–3

B. Finite-Order Multi-Target Branched State

  • A finite set of presence patterns determines which continuous target parameters exist, so absent targets contribute no physical degree of freedom.The reference measure combines counting measure over branches with Lebesgue measure over active coordinates; arbitrary random target counts, unlabeled permutations, and data association are outside the stated model.Source: B. Finite-Order Multi-Target Branched… · p. 3

C. General Matrix Sensing-System Model

  • A general matrix sensing model links active-target physical modes and responses to observations, while the theorems require only the conditional observation kernel.The model allows noise, interference, or unmodeled disturbances; the results do not depend on linearity or Gaussian noise.Source: C. General Matrix Sensing-System Model · p. 3

D. Two Special Cases: Ranging and Array Direction Finding

The paper uses exponential entropy to represent uncertainty as an effective physical measure relative to a specified reference. It applies this framework to ranging and array direction finding, and combines discrete presence uncertainty with continuous parameter uncertainty.

D. Two Special Cases: Ranging and Array Direction Finding

A. Basic Principle of the Unified Measure

  • Exponential entropy gives an effective measure scale relative to the reference measure; entropy volume depends on that physical scale, whereas mutual information is coordinate- and unit-invariant.The reference measure is specified by the physical measurement system, and the two representations serve complementary roles.Source: A. Basic Principle of the Unified… · p. 3

B. Entropy Number

  • The entropy number is the effective number of posterior candidates, not the literal cardinality of the posterior support.For binary target presence it lies between one and two, approaching one when the posterior concentrates on a single hypothesis.Source: B. Entropy Number · p. 4

C. Entropy Volume

  • For a continuous state, entropy volume is an exponential volume scale in the chosen reference coordinates, with units inherited from the physical state.For range–Doppler, the reference scale is m·Hz; Gaussian calibration is confined to a fixed continuous dimension.Source: C. Entropy Volume · p. 4

D. Mixed Entropy Volume

  • Mixed entropy volume is the product of the presence entropy number and the conditional continuous entropy volume, combining variable-dimensional branches on a branched reference measure.Uncertainty therefore multiplies in the physical-measure domain while information adds in the logarithmic domain.Source: D. Mixed Entropy Volume · p. 4

E. Maximum-Entropy Characterization of Entropy Volume

The uniform maximum-entropy principle explains why an exponential entropy scale is a minimum support-measure scale. The paper combines that result with mixed typical-set arguments to obtain posterior-volume limits and a decomposition of joint sensing information.

E. Maximum-Entropy Characterization of Entropy Volume

  • For a distribution supported on a finite-measure set, exponential entropy is the minimum support-measure scale compatible with its entropy, with equality for the uniform distribution.The conclusion applies relative to the reference measure in discrete, continuous, and mixed spaces.Source: E. Maximum-Entropy Characterization of… · pp. 4–5
  • A single-observation maximum-entropy bound concerns full posterior support; high-probability regions require the mixed AEP and a probability–volume inequality, and the single-observation high-probability bound does not hold.The asymptotic region measure is considered over an m-fold extension, not as a one-scene geometric region.Source: E. Maximum-Entropy Characterization of… · p. 5

F. Joint Sensing Information and the Entropy-Volume Contraction Law

  • The joint sensing information decomposes into target-presence information and conditional parameter information, and contracts prior mixed entropy volume by a factor of 2^-I(Ξ;Y).One sensing bit halves the posterior effective measure under the same reference measure.Source: F. Joint Sensing Information and the… · p. 5

A. Probability Space, Reference Measure, and Regularity Conditions

  • The analysis assumes an i.i.d. extension and finite absolute first moments of relevant log-densities, permitting finite entropies and weak-law arguments.For finite-order branched states, the integrability condition can be checked branch by branch.Source: A. Probability Space, Reference… · pp. 5–6

B. Mixed Asymptotic Equipartition

The mixed AEP supplies typical-set density and measure bounds under the stated integrability condition. Those bounds also control conditional atypical probability on a set of observations whose probability tends to one.

B. Mixed Asymptotic Equipartition

  • Under the integrability assumption, the weak law establishes mixed AEP bounds for state sequences and their typical-set measures, without requiring Lebesgue or counting measure specifically.The proof uses a density relative to the reference measure and integrability of its logarithm.Source: B. Mixed Asymptotic Equipartition · p. 6

C. Conditional Atypical Probability

  • Although conditional atypical probability need not vanish pointwise, it is uniformly small for observations in a set whose probability tends to one.The argument applies Markov's inequality to the average atypical probability obtained from the joint typical-set result.Source: C. Conditional Atypical Probability · p. 6

D. Conditional Mixed Asymptotic Equipartition

The conditional mixed AEP characterizes posterior density and typical-set measure at their first exponential order for high-probability observations. Sensing operators represent this scale through the per-state geometric mean of a region’s mixed measure.

D. Conditional Mixed Asymptotic Equipartition

  • The posterior-volume lower bound from the conditional typical set applies only on a high-probability observation set; the density and measure bounds hold for their corresponding objects more generally.The restriction on the lower bound is essential to the typical-set measure comparison.Source: D. Conditional Mixed Asymptotic… · p. 7
  • For high-probability observation sequences, the conditional posterior is approximately uniform over its typical set to first exponential order, with conditional entropy determining the density and measure scales.This is the asymptotic posterior counterpart of the maximum-entropy principle.Source: D. Conditional Mixed Asymptotic… · p. 7

E. Sensing Operators and Empirical Entropy Volume

  • A sensing operator outputs a measurable region from the observations and may also provide a representative state sequence.A typical-set sensor uses posterior sampling for that representative sequence.Source: E. Sensing Operators and Empirical… · p. 7
  • Taking the mth root of the extended system’s mixed measure expresses it as an equivalent posterior volume per physical state.The empirical quantities are random variables.Source: E. Sensing Operators and Empirical… · p. 7

V. POSTERIOR ENTROPY-VOLUME LIMIT OF JOINT TARGET DETECTION AND PARAMETER ESTIMATION

The section establishes an achievability–converse route to the posterior entropy-volume limit using conditional typical sets and a posterior probability–volume bound. A typical-set sensor achieves the asymptotic posterior scale.

V. POSTERIOR ENTROPY-VOLUME LIMIT OF JOINT TARGET DETECTION AND PARAMETER ESTIMATION

  • The proof combines a high-probability posterior-volume inequality with typical-set achievability and a region converse to obtain the joint detection-and-estimation limit.The static maximum-entropy relation motivates entropy volume but is not a premise of the main proof chain.Source: §V · pp. 7–8

A. Achievability

  • The typical-set sensor’s failure probability vanishes, while its empirical entropy, sensing information, and entropy volume converge in probability to their conditional-entropy and mutual-information limits.Its representative sequence has the same distribution as the true state sequence.Source: A. Achievability · p. 8

B. Posterior Probability–Entropy-Volume Bound

  • The posterior probability–volume bound limits a region’s conditional success probability by its mixed measure scaled by the conditional-entropy exponent, plus the atypical-posterior probability.The bound also applies to sensing operators with internal randomness after conditioning and averaging.Source: B. Posterior… · p. 8

C. Region Converse

For asymptotically reliable sensing regions, the paper gives a first-order posterior entropy-volume limit and its logarithmic information form. The typical-set sensor attains the limit.

C. Region Converse

  • The region converse constrains the random sensing region itself for every asymptotically reliable sensing-operator sequence.A lower bound on expected region measure alone does not constrain empirical entropy volume in the needed direction.Source: C. Region Converse · pp. 8–9

E. Information Limit: Logarithmic Form of Entropy-Volume Contraction

  • The information limit is the logarithmic representation of the same posterior-volume limit, not an independent limit.The typical-set sensor attains the mutual-information limit as well as the entropy-volume bound.Source: E. Information Limit: Logarithmic Form… · p. 9

D. Posterior Entropy-Volume Limit: Primary Form

F. Entropy Bound for Nonvanishing Failure Probability

The paper extends the entropy argument to nonvanishing failure probability and shows when a detection–estimation cascade preserves the direct-inference limit. The cascade result requires preservation of the complete posterior information.

F. Entropy Bound for Nonvanishing Failure Probability

  • When failure probability does not vanish, the direct posterior-volume converse no longer follows, so an entropy-domain bound adds a failure-probability penalty.The result assumes finite total state-space measure, which holds automatically for bounded location domains with finitely many branches.Source: F. Entropy Bound for Nonvanishing… · p. 10

A. Posterior Factorization and Lossless Cascades

  • A posterior-preserving intermediate representation retains the direct mutual-information limit, with the joint information splitting into presence-pattern and conditional-parameter terms.The cascade’s detection and estimation regions combine to attain the same limit as direct inference.Source: A. Posterior Factorization and… · pp. 10–11
  • The cascade’s mixed posterior volume combines branches of differing continuous dimensions through the first-order exponential rate of their summed measures.The branch measures should not be compared term by term as if they shared a dimension.Source: A. Posterior Factorization and… · p. 11

B. Information Loss under Arbitrary Intermediate Compression

The cascade’s information loss is exactly the observation information that remains after the intermediate representation is known. A cascade preserves the direct-inference limit when its representation preserves the posterior of the complete mixed state; hard decisions are generally insufficient.

B. Information Loss under Arbitrary Intermediate Compression

  • The loss is exactly I(Ξ;Y|Z), and it is zero iff p(Ξ|Y)=p(Ξ|Z) almost surely.Thus losslessness is determined by whether the intermediate representation is sufficient for the complete mixed state.Source: B. Information Loss under Arbitrary… · p. 11

C. Hard-Decision Cascades

  • Hard decisions or coarse quantization are generally not sufficient for Ξ; transmitting only the presence decision usually fails to preserve observation information needed for continuous parameters.This identifies a possible performance gap in two-step joint detection–estimation without invalidating that framework.Source: C. Hard-Decision Cascades · p. 11

VII. DETECTION AND ESTIMATION AS TWO BOUNDARY COROLLARIES

The paper presents pure parameter estimation and pure target detection as boundary cases of its joint theorem. In a single-target presence–range simulation, the detection, estimation, and joint information terms compose as predicted across SNR.

VII. DETECTION AND ESTIMATION AS TWO BOUNDARY COROLLARIES

  • With fixed presence or no continuous uncertainty, the joint theorem reduces to continuous estimation or detection, respectively, with posterior entropy volume contracting by 2^-I(X;Y) or 2^-I(V;Y).In the pure-detection case, the branched reference measure reduces to counting measure.Source: §VII · p. 12

A. Numerical Model and Implementation

  • The benchmark uses V∈{0,1} with P(V=1)=1/2 and X uniform on [0,32); it evaluates the posterior on 1024 grid points using 2×10^4 Monte Carlo realizations per SNR.The prior location uncertainty is 5 bits, or 32 resolution cells.Source: A. Numerical Model and Implementation · p. 12

B. Decomposition of Joint Sensing Information

  • At 12 dB, I_D=0.882 bits, I_E=3.119 bits, and I_J=4.001 bits; at higher SNR, detection information approaches 1 bit while continuous information continues to grow.The figure plots detection, conditional estimation, and joint sensing information versus SNR; the maximum discrepancy between direct I_J and I_D+I_E calculations is approximately 4.2 × 10^-2 bits, about 1% of I_J.Source: Figure 1 · p. 13, B. Decomposition of Joint Sensing… · p. 13

C. Posterior Entropy-Volume Contraction

The section relates joint sensing information to posterior entropy-volume contraction, then examines how cascade interfaces preserve or lose that information. It also clarifies the physical reference-measure interpretation and the asymptotic and finite-order scope.

C. Posterior Entropy-Volume Contraction

  • Detection and estimation information add in bits, while their corresponding uncertainty-contraction factors multiply in the entropy-volume domain.The normalized relation in (52) equates the mixed posterior contraction to 2^-I_J.Source: C. Posterior Entropy-Volume Contraction · p. 13

D. Physical Meaning of One Sensing Bit

  • One, two, three, and four sensing bits reduce normalized posterior entropy volume to 1/2, 1/4, 1/8, and 1/16 of its prior value.At 12 dB, the location posterior contracts to less than one resolution cell, while the effective target-presence candidate count is about 1.09.Source: D. Physical Meaning of One Sensing Bit · p. 13

E. Interface Information Loss in Cascaded Receivers

  • At 12 dB, interface information losses are approximately 3.20 bits (bV only), 1.31 bits (Q = 16), 0.52 bits (Q = 64), and 0.29 bits (Q = 256); posterior-preserving loss is zero.As interface resolution increases, the loss approaches the zero-loss limit.Source: E. Interface Information Loss in… · p. 14, Figure 4 · p. 15

F. Numerical Convergence

  • At 12 and 16 dB, the difference between G = 1024 and G = 2048 is on the order of 10^-3.Grid results for G = 512, 1024, and 2048 agree closely over the main operating range; grid-spacing correction prevents artificial entropy-volume shifts.Source: F. Numerical Convergence · p. 14

IX. DISCUSSION: MEASURES, DIMENSIONS, AND SCOPE

  • Entropy volume depends on the physical reference measure and can be compared with engineering scales, while mutual information is coordinate-invariant and additive.The unified theorem does not require a common Euclidean dimension across branches.Source: §IX · p. 15
  • Achievability is asymptotic under an m-fold independent extension, asymptotic equipartition, and a sensing-region formulation.Finite blocklength, non-i.i.d. extensions, model mismatch, and computational complexity are left for future work; the finite-order model also excludes arbitrary random target counts, unlabeled states, permutation equivalence classes, and data association.Source: §IX · p. 15

X. CONCLUSION

  • The fundamental sensing limit is the achievable posterior volume under the physical reference measure, with mutual information giving its logarithmic contraction rate.A posterior-preserving cascade attains the direct-inference limit; arbitrary intermediate compression incurs information loss I(Ξ;Y|Z).Source: §X · p. 15

APPENDIX A BIOGRAPHY SECTION

The appendix biographies summarize the authors’ affiliations, career stages, and research interests.

Biography

  • Dazhuan Xu is a professor jointly affiliated with Nanjing University of Aeronautics and Astronautics and Purple Mountain Laboratories, with research interests including information theory for sensing and statistical signal processing.He has authored or coauthored more than 300 journal and conference papers.Source: Appendix A · p. 16
  • Nan Wang received her Ph.D. in 2025 and is a lecturer at Shenyang Aerospace University, with research interests including information theory for sensing and radar signal processing.Source: Appendix A · p. 17
  • Han Zhang received his Ph.D. in 2023 and is a lecturer at Nanjing University of Posts and Telecommunications, with research interests in radar signal processing and information theory.Source: Appendix A · p. 17
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