# Exploring ISAC: Information-Theoretic Insights
## 7 ISAC with Detection-Error Exponents  
Radar is not only used to estimate parameters such as vehicle velocities, arrival angles, etc., but is also extensively employed to detect obstacles, or more generally, other terminals. In this context, the sensing problem must be framed as a detection or hypothesis testing problem. This framework allows us to incorporate memory into the channel transition law, analogous to slow-fading channels, thereby making the model more reflective of realistic scenarios. This involves considering multiple hypotheses (e.g., the presence or absence of one or more obstacles), and to ensure the model is fully general, we allow the communication channel to depend on the chosen hypothesis. Such a scenario arises, for example, when an obstacle obstructs the line of sight between the Tx and Rx, thereby significantly altering the communication channel characteristics. Of course, the general model also accommodates simpler cases where the communication channel remains independent of the hypothesis.  
In this section, the sensing performance is quantified by detection-error probabilities under the different hypotheses. Since these error probabilities can be made to vanish asymptotically with infinite observation lengths, the focus here will be on the exponential decay rate of these error probabilities. Much of the existing literature on the fundamental limits of ISAC with detection-error exponents has focused on mono-static radar, where sensing (detection) is performed at the communication Tx [57; 71; 72; 73; 74; 114]. A notable exception is [114], where detection is carried out at the communication receiver.  
From an information-theoretic perspective, the detection-error ISAC problem is considerably more challenging than the previously presented distortion-based ISAC setup. The primary difficulty stems from the sensing aspect, as the optimal performance of hypothesis testing systems is not well understood, even without the additional communication component present in ISAC systems. In particular, the simple estimation and communication strategies employed in the distortion-ISAC setup are suboptimal. Instead, the Tx can improve its detection performance by generating inputs according to a smart, sequential strategy that adapts based on previously observed outputs. Specifically, in a multi-hypothesis testing problem, the Tx might initially produce inputs based on a particular distribution, and once the observed signals provide sufficient evidence to discriminate one hypothesis with high confidence, it can switch to a different input distribution that better discriminates among the remaining hypotheses.  
Adaptive systems are complex to implement, which motivates the practical interest in excluding them from certain considerations. When restricting attention to non-adaptive strategies, similar closed-form expressions for the fundamental ISAC performance limits and trade-offs can be derived, as in the distortion ISAC setup.  
Figure 18: Comparison of the achievable rate–distortion tradeoffs under different secrecy constraints.  
The next subsection explains the model both for the adaptive and non-adaptive coding scenarios, and is followed by a section presenting the existing information-theoretic results on ISAC with detection exponents. The last subsection has a slightly different flavor for the sensing task and the goal is to detect a change point.