# Exploring ISAC: Information-Theoretic Insights
## 7 ISAC with Detection-Error Exponents
### Sequential (Variable-Length) ISAC with Detection-Exponents  
In [116], a variable-length version of the ISAC setup with detection-error exponents in the symmetric setup is considered. In this variable-length version, the transmission duration is not fixed from the beginning, but varies as a function of the generalized feedback signals, which in this case, has to coincide with the Rx's channel outputs to ensure synchronization of the communication. After each time \(t\), the Tx will decide based on the past channel outputs \(Y_{1},\ldots,Y_{t}\) whether to stop or continue communication. Let \(T\) be the random time where transmission stops. The model in [116] imposes that \(T\) be smaller than a given threshold \(n\) with high probability.  
Given that the communication duration is random, the number of transmitted message bits, and thus the rate of communication, are also allowed to be random. In fact, the message bits are supposed to consist of a stream of i.i.d. Bernoulli-\(1/2\) bits \(\{U_{i}\}_{i=1}^{\infty}\), and an increasing sequence of numbers \(\{M_{t}\}_{t=1}^{T}\), which indicates at each time \(t\), how many information bits have been transmitted until then and have to be decoded at the Rx if transmission stops at time \(t\). Since transmission stops at time \(T\), the Rx has to decode \(W_{T}\) information bits. The rate is defined as \[R_{T}=\frac{W_{T}}{n},\] (91) where recall that \(T\) is the stopping time of the communication and \(n\) is the given constraint (upper bound) on this stopping time. Formally, the encoder, decoder, and state detector are described as follows: * At each time \(t=1,2,\ldots\), the Tx forms the channel input as \(X_{t}=f_{t}(U_{1},\ldots,U_{W_{t}},Z^{t-1})\), for an appropriate encoding function \(f_{t}\); * At the end of the transmission, the Tx guesses the state as \(\hat{\theta}=h(X^{T},Z^{T})\), for an appropriate guessing function \(h\); * At the end of transmission, the Rx decodes the transmitted message bits as \(\hat{U}^{W_{T}}=g(Y_{1},\ldots,Y_{T})\) for an appropriate decoding function \(g\).  
**Definition 8**.: _A rate-detection exponent \((R,D_{\mathrm{VL}})\) is achievable in this variable-length setup if there exists a sequence (in the blocklength constraint \(n\)) of stopping rules, encoding functions, decoding functions, and state guessing functions, as defined above, such that:_  
\[\lim_{n\to\infty}\max_{\theta\in\Theta}\max_{\mathbf{u}}\Pr[T>n]=0 \tag{92}\]  
\[\lim_{n\to\infty}\min_{\theta\in\Theta}\min_{\mathbf{u}}\Pr\Big{[}R^{(n)}\geq R \Big{]}\ =\ 1 \tag{93}\]  
\[\lim_{n\to\infty}\max_{\theta\in\Theta}\max_{\mathbf{u}}\Pr\Big{[}\hat{U}^{W_{T}} \
eq U^{W_{T}}\big{|}\theta,U^{W_{T}}=\mathbf{u}\Big{]}\ =\ 0, \tag{94}\]  
\[\lim_{n\to\infty}-\frac{1}{n}\log\max_{\theta\in\Theta}\max_{\mathbf{u}}P\Big{[} \hat{\theta}\
eq\theta\,\big{|}\,\theta,U^{W_{T}}=\mathbf{u}\Big{]}\ \geq\ D_{\mathrm{VL}}. \tag{95}\]  
A set of achievable rate-detection-exponent pairs \((R,E_{\mathrm{VL}})\) for the described setup was presented in [116]:  
**Theorem 16**.: _All rate-detection exponents \((R,D_{\mathrm{VL}})\) that for each \(\theta\in\Theta\) satisfy the following two conditions for some choice of \(P_{X}\) (which can depend on \(\theta\))_  
\[R \leq I(P_{X},P_{Y|X}^{\theta}) \tag{96}\] \[D_{\mathrm{VL}} \leq\min_{\theta\
eq\theta}\mathbb{E}_{P_{X}}\Big{[}D\Big{(}P_{Z |X}^{\theta^{\prime}}\|P_{Z|X}^{\theta}\Big{)}\Big{]} \tag{97}\]  
_are achievable._  
This result looks similar to the achievability result in Theorem 15, where however the variable-length coding allows an improvement in the detection-error exponent from the Chernoff information in (90) to the Kullback-Leibler divergence in (97). An example in [116] illustrates well this benefit of variable-length coding by means of a numerical plot.