Heuristic Sensing Schemes for Four-Target Detection in Time-Constrained Vector Poisson and Gaussian Channels

TL;DR

This work addresses sensing scheme design for detecting four targets under a fixed sensing time in time-constrained vector Poisson and Gaussian channels. It adopts four time-sharing configurations (singlets, pairs, triplets, quadruplet) and analyzes both unconstrained and constrained objectives using mutual information $I(X;Y)$ and Bayes detection metrics, employing Monte Carlo to manage the high-dimensional problem. Key findings show that $I(X;Y)$ is concave in the allocated times $T$ for both channels, but optimality under Bayes risk differs from information-theoretic optimality: for Poisson, individual sensing often wins when the prior $p\ge0.5$, whereas for Gaussian channels, triplet-sensing frequently performs best; the Bayes-detection results do not always align with the $I$ results. Under fixed-time constraints with hybrid configurations parameterized by $\alpha$, $I$ remains concave in $\alpha$, and the Poisson case again trends toward individual sensing for larger $p$, while Gaussian performance favors triplets. These insights inform sensor scheduling strategies under time limits and high-dimensional observation models, though the optimal scheme remains problem-dependent and further theoretical guarantees are open questions.

Abstract

In this work, we investigate the different sensing schemes for the detection of four targets as observed through a vector Poisson and Gaussian channels when the sensing time resource is limited and the source signals can be observed through a variety of sum combinations during that fixed time. For this purpose, we can maximize the mutual information or the detection probability with respect to the time allocated to different sum combinations, for a given total fixed time. It is observed that for both Poisson and Gaussian channels; mutual information and Bayes risk with $0-1$ cost are not necessarily consistent with each other. Concavity of mutual information between input and output, for certain sensing schemes, in Poisson channel and Gaussian channel is shown to be concave w.r.t given times as linear time constraint is imposed. No optimal sensing scheme for any of the two channels is investigated in this work.

Figures (6)

• Figure 1: Illustration of sensing paradigm for detection of $4-$long hidden random vector $X$ from $15-$long observable random vector $Y$ through a vector Poisson channel under a total time constraint of $T=\sum_{i=1}^{15} T_i.$
• Figure 2: Illustration of sensing paradigm for detection of $4-$long hidden random vector $X$ from $15-$long observable random vector $Y$ through a vector Gaussian channel under a total time constraint. Where $w_i(t)$ are independent white noise processes. Only one of the integrators becomes active for a time $T_i$ such that total time constraint $T=\sum_{i=1}^{15} T_i$ is satisfied by all the integrators.
• Figure 3: Poisson channel: (Left) $I(X;Y)$ vs. $T$, (right) $P_d$ vs. $T$ for varying prior probabilities $p$.
• Figure 4: Poisson channel : $\rm{Config-1:} (\frac{T-\alpha}{4},\frac{T-\alpha}{4},\frac{T-\alpha}{4},\frac{T-\alpha}{4},0,0,0,0,0,0,0,0,0,0,\alpha )$; $\rm{Config-2:} (0,0,0,0,\frac{T-\alpha}{6},\frac{T-\alpha}{6},\frac{T-\alpha}{6},\frac{T-\alpha}{6},\frac{T-\alpha}{6},\frac{T-\alpha}{6},0,0,0,0,\alpha )$ and $\rm{Config-3:} (0,0,0,0,0,0,0,0,0,0, \frac{T-\alpha}{4},\frac{T-\alpha}{4},\frac{T-\alpha}{4},\frac{T-\alpha}{4},\alpha )$ where $0 \le \alpha \le T$ and time constraint $\sum_{i=1}^{15} T_i=T$ for $\lambda_0=2$, $\lambda_1=20$, and varying prior probability $p$.
• Figure 5: Gaussian channel: (Left) $I(X;Y)$ vs. $T$, (right) $P_d$ vs. $T$ for varying prior probabilities $p$.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 4
• Corollary 2
• Theorem 5
• Corollary 3
• Conjecture 1