# Exploring ISAC: Information-Theoretic Insights
## 3 Mono-Static ISAC with Sensing Distortion
### Channels with Memory  
The previous sections assumed a memoryless stationary model both for the channel and the distribution of the target/state that the transmitter wishes to estimate. The work [60] relaxed both assumptions and considered a general model, where the state process \(\{S_{t}\}\) follows an arbitrary joint distribution and the channel is characterized by a general sequence of transition laws \(P_{Y_{i}Z_{i}|X|^{3}\hat{S}^{n}-1\gamma^{i-1}}\), for \(i=1,2,\ldots\). The distortion constraint is also generalized beyond average block distortion constraints by requiring that  
\[p-\varlimsup_{n\to\infty}\frac{1}{n}d(S^{n},\hat{S}^{n})\leq\mathsf{D}, \tag{24}\]  
for a general non-negative distortion function \(d(\cdot,\cdot)\).  
Using [PERSON]'s information spectrum method, [60] characterized the capacity-distortion tradeoff ( Additional cost constraints can be included in the model and the results in a standard way) for this setup with memory. It is given by [60]  
\[\mathsf{C}(\mathsf{D}):=\sup_{\{P_{\mathcal{X}^{n}}\}_{n}}p-\varliminf_{n\to \infty}\frac{1}{n}i(X^{n};Y^{n}) \tag{25}\]  
where the supremum is over all input distributions \(\{P_{\mathcal{X}^{n}}\}\) and estimators \(\{\hat{S}^{n}(Z^{n},X^{n})\}\) satisfying \(\varlimsup_{n\to\infty}\frac{1}{n}d(S^{n},\hat{S}^{n}(X^{n},Z^{n}))\leq\mathsf{D}\). Here, \(i(X^{n};Y^{n})\) is the previously defined information density between sequences \(X^{n}\) and \(Y^{n}\).  
A slightly different model was considered in [86], where the transmitter has to estimate the state sequence in an online manner, i.e., a state estimate \(\hat{S}_{i}\) has to be produced after having produced the time-\(i\) channel input \(X_{i}\) and having observed the time-\(i\) feedback signal \(Z_{i}\). The capacity-distortion tradeoff was derived for this related model, but limited to the class of ergodic channels where the sequences of information densities are sure to converge.  
While the generality of the presented \"arbitrary/ergodic non-i.i.d\". models is appealing, the complexity of the expressions (both from an analytical perspective as well as in view of numerical evaluations) limits the utility of the results. An interesting approach is  
Figure 4: Achievability and converse bounds on the rate–distortion–error trade-off for \(\epsilon=10^{-3}\), \(q=0.4\), \(\mathsf{K}=0.5\) and different values of the blocklength \(n\).  
to consider larger (not only i.i.d.) classes of channels and source sequences for which the capacity-distortion tradeoff still has a relatively simple form.  
In this spirit, the work in [86] characterized the capacity-distortion tradeoff of a class of channels that have previously been introduced and studied in the context of pure capacity calculations. This class of channels is also particularly interesting because the numerical calculation of capacity [87] as well as of the capacity-distortion tradeoff can be cast into the framework of Markov decision processes and thus solved using reinforcement learning (RL) as well as its many more advanced alternatives that have been introduced in recent years such as Q-learning, etc.  
An RL approach was followed in [86] to evaluate the capacity-distortion tradeoff for a specific class of binary channels. Interestingly, the authors of [86] also analyzed the influence of the size of the state space considered in the RL approach, which corresponds to the memory in the coding strategy employed at the transmitter. Figure 5 plots a weighted sum between the information rate and the distortion in function of the weight factor \(\beta\). It shows the performances achieved by four versions of the RL approach allowing for different sizes of the state spaces: a full state space; a highly-reduced state space that only allows to implement memoryless policies; and intermediate state spaces with sizes equal to 10% or 40% of the full state space and thus allowing the implementation of coding strategies with a limited amount of memory.