Sensing Method for Two-Target Detection in Time-Constrained Vector Gaussian Channel

TL;DR

The paper investigates time-constrained sensing design for a three-sensor vector Gaussian channel aiming to detect a two-component hidden vector. It develops both information-theoretic and detection-theoretic formulations: the former uses mutual information $I(X;Y)$ with $Y$ as a Gaussian mixture and estimates $H(Y)$ via Monte Carlo, revealing concavity along an affine time-split line; the latter formulates Bayes risk and maximizes $P_d=1-r$ under the same time constraint. Through extensive Monte Carlo simulations, it shows that the time allocations optimal for maximizing $I$ can differ from those maximizing $P_d$, with hybrid sensing ($T_1=T_2$ and nonzero $T_3$) frequently optimal for detection. The work highlights a fundamental trade-off between information-theoretic and detection-theoretic objectives in sensor scheduling and suggests directions for exploring broader concavity properties and parameter regimes.

Abstract

This paper considers a vector Gaussian channel of fixed identity covariance matrix and binary input signalling as the mean of it. A linear transformation is performed on the vector input signal. The objective is to find the optimal scaling matrix, under the total time constraint, that would: i) maximize the mutual information between the input and output random vectors, ii) maximize the MAP detection. It was found that the two metrics lead to different optimal solutions for our experimental design problem. We have used the Monte Carlo method for our computational work.

Figures (7)

• Figure 1: Illustration of sensing paradigm for detection of $2-$long hidden random vector $X$ from $3-$long observable random vector $Y$ through a vector Gaussian channel under a total time constraint $T=\sum_{i=1}^{3} T_i$. $w_i(t)$ are independent white Gaussian noise processes. Only one of the integrators becomes active for a time $T_i$ such that time constraint is always satisfied after the total sensing time $T$ is consumed. Objective is to maximize the mutual information between input and output, $I(X_1, X_2; Y_1, Y_2, Y_3)$, and Bayes probability of total detections, $P_d$, by satisfying the time constraint.
• Figure 2: Mutual information $I(X;Y)$ vs. $T_3$ and probability of total correct detections $Pd$ vs. time $T_3$ in a time constraint $T_1+T_2+T_3=1$ such that $(T_1,T_2,T_3) := (\frac{1-T_3 }{2},\frac{1-T_3 }{2},T_3 )$ where $0 \le T_3 \le 1$.
• Figure 3: $I(X;Y)$ vs. $(T_1,T_2,T_3)$ under time constraint $T_1+T_2+T_3=10$ for $\lambda_0=0$, $\lambda_1=2$, and varying prior probability $p$.
• Figure 4: Mutual information $I(X;Y)$ vs. $T_3$ and probability of total correct detections $P_d$ vs. $T_3$ for prior probabilities of $0.125,0.5,0.75$ and $0.99.$
• Figure 5: Mutual information $I(X;Y)$ vs. $T_3$ and probability of total correct detections $P_d$ vs. $T_3$ for prior probabilities of $0.125,0.5,0.75$ and $0.99.$

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Theorem 3