# Exploring ISAC: Information-Theoretic Insights
## 7 ISAC with Detection-Error Exponents
### Sequential (Variable-Length) ISAC with Change-Point Detection  
A related model has also been considered in [75]. Communication again takes place over a fixed block of \(n\) channel uses. However, the channel starts in the state \(\theta=0\) and at a random time \(\
u\), it will change to a state \(\theta=1\). The goal of the state estimator is to detect this change point \(\
u\) with smallest delay. So, detection is variable-length as in the previous subsection, however data communication is fixed-length over \(n\) channel uses. Inputs have to be generated in an non-adaptive fashion.  
Formally, the Tx generates its inputs as \(X^{n}=f^{(n)}(W)\), where \(W\) is a uniform message of rate \(R\) and \(\phi\) an encoding function of appropriate domain. The Rx guesses the message \(W\) as \(\hat{W}=g(Y^{n})\) using a guessing function \(g\). We again assume perfect feedback \(Z=Y\) and the Tx thus estimates the change point using a stopping rule based on all inputs \(X^{n}\) and the past outputs \(Y_{1},\ldots,Y_{t}\). We denote the estimate of the change point by the random variable \(N\in\{1,\ldots,n+1\}\), where \(n+1\) indicates that the channel did not change state. Notice that \(N\) being a stopping rule based on the inputs and previous outputs, formally means that each event \(N=i\) is measurable with respect to \(Y^{i}\) and \(X^{n}\).  
As usual, decoding error probability \(\Pr[\hat{W}\
eq W]\) is required to tend to 0 as the blocklength \(n\rightarrow\infty\). Typical performance measures for the change-point detection problem are the _false alarm rate (FAR)_, which in the present setup, should be defined as  
\[\text{FAR}=\varlimsup_{n\rightarrow\infty}\max_{w}\frac{1}{\mathbb{E}_{ \infty}[N|\mathbf{x}^{n}(w)]}. \tag{98}\]  
and the _worst-case average detection delay (WADD)_, which here is defined as:  
\[\text{WADD}=\sup_{v\geq 1}\varlimsup_{n\rightarrow\infty}\ \max_{w}\ \text{ess sup}_{\mathbf{Y}^{n-1}}\mathbb{E}_{v}\Big{[}(N-v+1)^{+}| \mathbf{x}^{n}(w),\mathbf{Y}^{v-1}\Big{]}. \tag{99}\]  
Here, we defined \(\mathbb{E}_{v}[\cdot]\) as the expectation operator assuming that the change point is at time \(v\). Similarly, \(\mathbb{E}_{\infty}[\cdot]\) denotes the expectation when the state of the channel never changes and remains \(\theta=0\) throughout.  
Definition 9: The pair \((R,\Delta)\) is called achievable in this quickest change-point detection problem, if for arbitrary small \(\alpha>0\), there exists a sequence (in the blocklength \(n\)) of stopping rules \(N\), encoding functions \(f^{(n)}\) and decoding functions \(g\), such that the following conditions are satisfied:  
\[\text{FAR}\leq\alpha \tag{100}\]  
\[\text{WADD}\leq\frac{\alpha}{\Delta} \tag{101}\]  
\[\lim_{n\rightarrow\infty}\Pr[\hat{W}\
eq W]=0. \tag{102}\]  
In above definition, \(\Delta\) describes the ratio between the FAR and the WADD.  
The work in [75] establishes a set of achievable \((R,\Delta)\) pairs by using sub-block composition codes [117], which not only ensure a given empirical statistics (type) across any given codeword \(x^{n}(w)\) but also within each sub-block.  
Theorem 17: _For any choice of \(P_{X}\), all pairs of \((R,\Delta)\) are achievable that satisfy_  
\[R\ \leq\ I(X;Y) \tag{103}\]  
\[\Delta\ \leq\ \mathbb{E}_{P_{X}}\Big{[}D\Big{(}P_{Y|X}^{1}\|P_{Y|X}^{0}\Big{)} \Big{]}. \tag{104}\]