The Theoretical Limit of Radar Target Detection

TL;DR

This work develops an information-theoretic framework for radar target detection by introducing Detection Information ($I(\boldsymbol{y};v)$) as a bit-based performance metric and showing it to be the theoretical limit of detection. It derives an equivalent detection channel, the a posteriori target-state distribution, and analytical expressions for false alarm and detection probabilities, revealing that, for large observation intervals, $P_{FA}=\pi(1)$. A stochastic sampling-a-posteriori (SAP) detector is proposed, with empirical DI approaching the theoretical DI as snapshots grow, establishing the DI as the fundamental limit and demonstrating that MAP and NP detectors cannot exceed it. Numerical results confirm the theoretical claims, illustrating the DI upper bound and the convergence behavior of SAP versus traditional detectors. Overall, the paper provides a rigorous information-theoretic justification for a target-detection limit and a detector that can attain it in the large-sample regime, with implications for hypothesis testing and related fields.$

Abstract

In this paper, we solve the optimal target detection problem employing the thoughts and methodologies of Shannon's information theory. Introducing a target state variable into a general radar system model, an equivalent detection channel is derived, and the a posteriori probability distribution is given accordingly. Detection information (DI) is proposed for measuring system performance, which holds for any specific detection method. Moreover, we provide an analytic expression for the false alarm probability concerning the a priori probability. In particular, for a sufficiently large observation interval, the false alarm probability equals the a priori probability of the existing state. A stochastic detection method, the sampling a posteriori probability, is also proposed. The target detection theorem is proved mathematically, which indicates that DI is an achievable theoretical limit of target detection. Specifically, when empirical DI is gained from the sampling a posteriori detection method approaches the DI, the probability of failed decisions tends to be zero. Conversely, there is no detector whose empirical DI is more than DI. Numerical simulations are performed to verify the correctness of the theorems. The results demonstrate that the maximum a posteriori and the Neyman-Pearson detection methods are upper bounded by the theoretical limit.

Figures (9)

• Figure 1: Equivalent communication system model for radar detection systems
• Figure 2: Target detection system for $m$ snapshots
• Figure 3: The typical sets and the conditional typical set for the target detection system.
• Figure 4: $P_\mathrm{FA}$ and $P_\mathrm{D}$ versus $\rm{SNR}$ for the observation interval is $N=32$ and different $\pi(1)$.
• Figure 5: $P_\mathrm{FA}$ versus $\pi(1)$ for the observation interval $\left( N=64, 512, 2048 \right)$ and $\mathrm{SNR} =0 \mathrm{dB}$.

Theorems & Definitions (2)

• Lemma 1
• proof