# Exploring ISAC: Information-Theoretic Insights
## 3 Mono-Static ISAC with Sensing Distortion
### The Memoryless Model  
The first information-theoretic model for a single-Tx single-Rx ISAC system was introduced by [PERSON], [PERSON], and [PERSON] in [56] and is depicted in Figure 2.  
In this model, a single Tx wishes to communicate a message \(W\) of \(nR\) independent and uniform data bits to the single Rx by communicating over \(n\) uses of a state-dependent discrete memoryless channel. The state sequence \(S_{1},\ldots,S_{n}\) models the parameters one wishes to estimate (e.g., the accelerations of a given target) and in this model, are assumed i.i.d. according to a given and known distribution \(P_{S}\). At the same time, the Tx also aims to estimate the state sequence \(S_{1}^{n}=(S_{1},\ldots,S_{n})\) from generalized feedback signals, which here model the backscatters observed at the Tx. The communication channel and the radar channel (i.e., the generation of the backscatters) are jointly modeled by a DMSC with a stationary channel law \(P_{YZ|XS}(y,z|x,s)\). That means, if at a given time \(i\in\{1,\ldots,n\}\) (more precisely for a given channel use \(i\)), the Tx feeds input \(X_{i}=x_{i}\) to the channel and the state realization is \(S_{i}=s_{i}\), then the Rx's time-\(i\) observed channel output \(Y_{i}\) and the generalized  
Figure 2: The first information-theoretic ISAC model.  
feedback signal \(Z_{i}\) backscattered to the Tx are generated according to the conditional pmf \(P_{YZ|XS}(\cdot,\cdot|x_{i},s_{i})\), irrespective of the past inputs, outputs, and state realizations.  
The transmitter produces its (potentially random) channel inputs \(X_{1},\ldots,X_{n}\) as a function of the message \(W\) and the backscattered signals. So, it produces the time-\(i\) input as \(X_{i}=\phi_{i}(W,Z_{1},\ldots,Z_{i-1})\), for \(i=1,\ldots,n\). The receiver observes the channel outputs \(Y_{1},\ldots,Y_{n}\) corresponding to these inputs and based on the entire sequence produces a guess of the message \(\hat{W}=g(Y_{1},\ldots,Y_{n})\).  
Based on the backscattered sequence \(Z_{1},\ldots,Z_{n}\) and its produced inputs \(X_{1},\ldots,X_{n}\), the transmitter also produces the state estimates \((\hat{S}_{1},\ldots,\hat{S}_{n})=h(X_{1},\ldots,X_{n},Z_{1},\ldots,Z_{n})\). The quality of these state estimates is measured by the expected average per-block distortion  
\[\Delta^{(n)}:=\mathbb{E}[d(S^{n},\hat{S}^{n})]=\frac{1}{n}\sum_{i=1}^{n} \mathbb{E}[d(S_{i},\hat{S}_{i})] \tag{1}\]  
where \(d:\mathcal{S}\times\hat{\mathcal{S}}\mapsto\mathbb{R}_{0}^{+}\) is a given bounded _distortion function_:  
\[\max_{(s,s)\in\mathcal{S}\times\hat{\mathcal{S}}}d(s,\hat{s})<\infty. \tag{2}\]  
Examples of commonly used distortion functions are the Hamming distortion \(d(s,\hat{s})=1\left\{s\
eq\hat{s}\right\}\), which measures the fraction of wrongly reconstructed symbols, or the mean-squared error distortion \(d(s,\hat{s})=(s-\hat{s})^{2}\), which measures the average squared distance between the reconstruction and source sequences.  
A last feature of this model is a cost constraint (such as a power constraint) on the input sequence, as imposed by many practical communication systems. These cost constraints can often be expressed as  
\[\mathbb{E}[b(X^{n})]=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[b(X_{i})] \tag{3}\]  
for some given cost functions \(b\colon\mathcal{X}\mapsto\mathbb{R}_{0}^{+}\). In the case of an input power constraint, the cost function \(b(x)=x^{2}\) is implied for radio channels where the power is proportional to the square of the emitted signal (which represents the electromagnetic field).