# Exploring ISAC: Information-Theoretic Insights
## 3 Mono-Static ISAC with Sensing Distortion
### The Capacity-Distortion-Cost Tradeoff
: _Note that the above result also remains valid if the expected distortion constraint (4b) is replaced by an excess distortion constraint where the probability that the sequences exceed average distortion \(\mathsf{D}\) is required to vanish asymptotically in the blocklength \(n\). The work in [57] considered such an excess distortion criterion but imposed weaker constraints where both the excess distortion probability and the decoding error probability do not necessarily need to vanish asymptotically but can simply be bounded by given positive constants \(\delta,\epsilon\in(0,1)\). The analysis in [57] showedthat the fundamental limits remain unchanged when the sum of both allowed error probabilities \(\delta+\epsilon<1\). If maximum error probabilities are considered instead of average error probabilities (over messages), then the results remain for all \(\epsilon,\delta\in(0,1)\), and the so-called strong converse holds. Note that similar non-zero error probability behaviors apply also to only communication but no sensing, because communication takes place over a compound channel, see [20]._  
It has been shown in [64] that the rate-distortion tradeoff function \(\mathsf{C}(\mathsf{D},\mathsf{B})\) is non-decreasing and concave in \(\mathsf{D}\geq\mathsf{D}_{\min}\) and \(\mathsf{B}\geq 0\), and for any \(\mathsf{B}\geq 0\) saturates at the channel capacity without distortion constraints \(\mathsf{C}_{\text{NoDist}}(\mathsf{B})\). For many channels, given \(\mathsf{B}\geq 0\), the tradeoff \(\mathsf{C}(\mathsf{D},\mathsf{B})\) is strictly increasing in \(\mathsf{D}\) until it reaches \(\mathsf{C}_{\text{NoDist}}(\mathsf{B})\). However, for SDMBCs and costs \(\mathsf{B}\geq 0\) where the capacity-achieving input distribution \(P_{X_{\max}}:=\operatorname*{argmax}_{P_{X}\in\mathcal{P}_{\mathsf{B}}}I(X;Y \mid S)\) also achieves minimum distortion \(\mathsf{D}_{\min}(\mathsf{B})\) in (9), the capacity-distortion tradeoff is constant, irrespective of the allowed distortion \(\mathsf{D}\). This is in particular the case when the expected distortion \(\mathsf{E}d(S,s^{*}(X,Z))\) does not depend on the input distribution \(P_{X}\). To understand the result in Theorem 3.1, consider the example of a real Gaussian channel with Rayleigh fading and noisy feedback. (For more examples, see [56; 64].) The channel output is thus given by:  
\[\overline{\lim}_{n\to\infty}\;\Delta^{(n)} \leq \mathsf{D},\]0  
where \(X_{i}\) is the channel input satisfying \(\overline{\lim}_{n\to\infty}\frac{1}{n}\sum_{i}\mathsf{E}|X_{i}|^{2}\leq \mathsf{B}=10\) dB, and both sequences \(\{N_{i}\}\) and \(\{S_{i}\}\) are independent of each other and i.i.d. Gaussian with zero mean and unit variance. The Tx observes the noisy feedback  
\[\overline{\lim}_{n\to\infty}\;\Delta^{(n)} \leq \mathsf{D},\]1  
where \(\{N_{\text{fb},i}\}\) are i.i.d. zero-mean Gaussian of variance \(\sigma_{\text{fb}}^{2}\geq 0\). We consider the quadratic distortion measure \(d(s,\hat{s})=(s-\hat{s})^{2}\). The capacity of this channel is achieved with a Gaussian input \(X_{\max}\sim\mathcal{N}(0,\mathsf{B})\), and thus the communication mode with sensing achieves the rate-distortion pair  
\[\overline{\lim}_{n\to\infty}\;\Delta^{(n)} \leq \mathsf{D},\]2 \[\overline{\lim}_{n\to\infty}\;\Delta^{(n)} \leq \mathsf{D},\]3  
where the numerical values correspond to \(\sigma_{\text{fb}}^{2}=1\) and \(P=10\) dB and the logarithm here is with respect to the natural unit and thus, measured in _nats_. Minimum distortion \(\mathsf{D}_{\min}\) is achieved by 2-ary pulse amplitude modulation (PAM), and thus, the sensing mode with communication achieves the rate-distortion pair \((R_{\min}(\mathsf{B}),\)\(\mathsf{D}_{\min}(\mathsf{B}))=(0.733,\frac{1+\sigma_{\text{fb}}^{2}}{1+P+\sigma_{ \text{fb}}^{2}}=0.166)\) where the numerical value again corresponds to \(\sigma_{\text{fb}}=1\) and \(B=10\) dB. Next, they characterize the performance of the basic TS baseline scheme.