# Exploring ISAC: Information-Theoretic Insights
## 7 ISAC with Detection-Error Exponents
### The Memoryless Block Model  
Consider the model in Figure 19, where the Tx wishes to communicate a message to a Rx over a channel that depends on a single parameter \(\theta\in\Theta\), for \(\Theta\) a finite set, and at the same time aims to determine this parameter based on the backscattered (generalized-feedback) signals. The parameter is assumed to take value in a discrete and finite set, transforming the sensing problem into a hypothesis testing/detection problem. For a given parameter \(\theta\) in a finite set \(\Theta\), the communication channel to the Rx as well as the generalized feedback to the Tx are assumed to be stationary and memoryless, and are described by a joint transition law \(P_{YZ|X}^{\theta}\). In the information-theory literature, such a communication channel with a fixed but a prior unknown parameter \(\theta\) is known as a _compound channel_[20; 115]. The switch indicates whether the Tx can employ adaptive/closed-loop coding or non-adaptive/open-loop coding. More precisely, if the switch is closed, the \(i\)-th channel input \(X_{i}\) can depend on the previous generalized feedback signals \(Z_{1},\ldots,Z_{i-1}\), and if the switch is open then all inputs only depend on the message \(W\). Formally, the Tx produces the channel inputs either as (in the case of non-adaptive coding)  
\[X^{n}=\phi^{(n)}(W) \tag{74}\]  
or as (in the case of adaptive coding)  
\[X_{i}=\phi_{i}(W,Z_{1},\ldots,Z_{i-1}),\quad i=1,\ldots,n, \tag{75}\]  
where \(\phi^{(n)}\) and \(\phi_{1},\ldots,\phi_{n}\) are encoding functions on appropriate domains. The Tx further guesses the parameter \(\theta\) as  
\[\hat{\theta}=h(X_{1},\ldots,X_{n},Z_{1},\ldots,Z_{n}), \tag{76}\]  
using some appropriate detection function \(h(\cdot)\). As before, the Rx decodes the transmitted message using an appropriate decoding function: \(\hat{W}=g(Y_{1},\ldots,Y_{n})\). Communication performance is measured as before in terms of rate \(R\) of message \(W\), where the Tx and Rx have to be designed in a way that the decoding error probability  
Figure 19: ISAC with a detection sensing problem. \(\Pr[\hat{W}\
eq W|\theta]\) vanishes asymptotically when the blocklength \(n\) increases under _any of the hypothesis \(\theta\)_. In this sense, the communication rate is defined in the same way as for the compound channel [20; 115]. Sensing performance is measured in terms of the asymptotic detection-error exponents  
\[E_{\theta}\triangleq-\frac{1}{n}\log\Pr[h(Z^{n},X^{n})\
eq\theta\mid\theta], \quad\theta\in\mathbf{\Theta}, \tag{77}\]  
where the conditioning on \(\theta\) simply indicates that the \(Z^{n}\) sequence is generated from \(X^{n}\) according to the memoryless law \(P^{\theta}_{Z|X}\). Different requirements on the detection-error exponents have been considered in the literature [71; 72; 73; 74; 57]. We summarize the requirements in the following definition. **Definition 7**. : _Let \(\Theta=\{0,1\}\). Then we say that a rate-detection-error exponent \((\mathsf{R},\mathsf{D})\) is achievable in the Stein setup if there exists a sequence (in the blocklength \(n\)) of encoding, decoding and detection functions so that the following three conditions are satisfied simultaneously:_  
\[\lim_{n\to\infty}\Pr[\hat{W}\
eq W]\ =\ 0 \tag{78}\]  
\[\lim_{n\to\infty}\Pr[h(Z^{n},X^{n})\
eq 0\mid\theta=0]\ =\ 0 \tag{79}\]  
\[E_{1}\ \geq\ \mathsf{D}. \tag{80}\]  
_Similarly, the triple \((\mathsf{R},\mathsf{D}_{0},\mathsf{D}_{1})\) is achievable in a exponent-region sense, if above sequences exist so that (78) holds, as well as_  
\[E_{\theta}\ \geq\ \mathsf{D}_{\theta},\quad\theta\in\Theta. \tag{81}\]  
_Let now \(\Theta\) be arbitrary.