联合目标检测与参数估计的基本极限——通过后验熵体积刻画混合状态感知极限

中文摘要 AI 辅助

集成感知与通信的发展需要统一的感知理论基础。本文将目标存在模式与连续物理参数建模为分支参考测度上的混合离散-连续状态$\Xi$，并将后验熵体积与联合互信息视为同一极限的两种互补表示。熵体积具有物理单位，可与分辨率单元等工程尺度比较，且在活动目标数量变化时仍有意义；互信息无量纲，对坐标和单位不变，并通过链式法则可加。我们分别对离散、连续和混合状态定义了熵数、熵体积和混合熵体积。固定测度支撑上均匀分布的最大熵原理解释了指数熵尺度的测度论意义，而主要极限则通过条件典型集的混合渐近等分性质和后验概率-体积不等式得出。对于渐近可靠的高概率感知区域，最小可达一阶后验混合熵体积等于先验混合熵体积乘以$2^{-I(\Xi;Y)}$，其中$I(\Xi;Y)=I(V;Y)+I(X_V;Y\mid V)$。因此，检测与估计的贡献在体积域中相乘，在比特域中相加，一个感知比特将后验有效测度减半。我们进一步证明，保持后验的级联可达到直接推断极限，而任意中间压缩会造成精确的信息损失$I(\Xi;Y\mid Z)$。针对单目标存在-距离模型的数值结果说明了信息组成、后验熵体积收缩和级联接口损失。

英文摘要

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 $Ξ$ 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 $2^{-I(Ξ;Y)}$, where $I(Ξ;Y)=I(V;Y)+I(X_V;Y\mid V)$. 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 $I(Ξ;Y\mid Z)$. Numerical results for a single-target presence-range model illustrate the information composition, posterior entropy-volume contraction, and cascade-interface loss.

发表机构

• Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
• Purple Mountain Laboratories（紫金山实验室）
• Shenyang Aerospace University（沈阳航空航天大学）
• Nanjing University of Posts and Telecommunications（南京邮电大学）

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