# Exploring ISAC: Information-Theoretic Insights
## 4 Sensing at the Rx (Rx-ISAC) with Sensing Distortion
### Capacity-Distortion Tradeoffs
: _The capacity-distortion function for strictly causal state communication is_  
\[rClC_{\text{Strc-caus.}}(\text{D}) = \max_{P_{X}P_{U|XS}}\bigg{(}I(U,X;Y)-I(U,X;S)\bigg{)}, \tag{28}\]  
_where the maximum is over all laws \(P_{X}P_{U|XS}\) such that \(\mathbb{E}[d(S,\hat{s}^{*}(U,X,Y))]\leq\text{D}\), where \((U,S,X,Y)\) are distributed according to \(P_{S}P_{X}P_{U|XS}P_{Y|XS}\) and_  
\[\hat{s}^{*}(u,x,y)=\arg\min_{s^{\prime}\in\mathcal{S}}\sum_{s\in\mathcal{S}}P_ {S|UXY}(s|u,x,y)d(s,s^{\prime}). \tag{29}\]  
_The capacity-distortion function for causal state communication is_  
\[rClC_{\text{Caus.}}(\text{D}) = \max\big{(}I(U,V;Y)-I(U,V;S)\big{)} \tag{30}\]_where the maximum is over all laws \(P_{V}P_{U|V,S}\) and functions \(x(v,s)\) such that \(\mathbb{E}[d(S,s^{*}(U,V,Y))]\leq\mathrm{D}\), for \((U,V,S,X,Y)\) distributed according to \(P_{V}P_{S}P_{U|VS}\mathbb{1}\left\{X=x(V,S)\right\}P_{V|XS}\) and here,_  
\[\hat{s}^{*}(u,v,y)=\arg\min_{s^{\prime}\in\mathcal{S}}\sum_{s\in\mathcal{S}}P_{ S|UVY}(s|u,x,y)d(s,s^{\prime}). \tag{31}\]  
In above expressions, the \(U\)- and \(V\)-auxiliaries stand for the auxiliary codewords. The subtracted mutual information terms arise because the Tx transmits compression information together with the data, and thus the rate of the compression information needs to be deduced from the total rate of communication that can be sustained from the Tx to the Rx. When the Tx observes the state sequence non-causally, no block Markov strategies are necessary. Gel'fand-Pinsker (GP) coding [91], which achieves capacity for channels with non-causal state information at the Tx, is used to transmit the data and the compression information to the Rx. In GP coding, the channel inputs are again obtained as a function of auxiliary codewords and the state sequence. The Rx thus again cannot reconstruct the sequence of channel inputs, even after decoding the messages correctly. However, again, a symbol-by-symbol estimator based on the decoded codewords and the observed sequences achieves the optimal Rx sensing performance. For this setup, with non-causal state information at the Tx, the exact capacity-distortion tradeoff is generally still an open problem, as only upper and lower bounds are known [89]. The work in [62] has characterized the exact capacity-distortion tradeoff in the case of a Gaussian model with mean squared error (MSE) distortion. This is, for a scenario where the time-\(t\) channel output is given by \(Y_{t}=X_{t}+S_{t}+N_{t}\) for \(X_{t}\) the channel input, \(S_{t}\) a Gaussian state of variance \(Q\); and \(N_{t}\) a Gaussian noise of variance \(N\), and \(d(s,\hat{s})=(s-\hat{s})^{2}\). For this case, the capacity-distortion tradeoff was derived in [62]. Theorem 4.2: _The capacity-distortion tradeoff with non-causal state information at the Tx in the Gaussian case is given by:_  
\[r\mathrm{CIC_{\mathrm{Caus}}}(\mathrm{D}) = \max_{r\in[0,1]}\frac{1}{2}\log\left(1+\frac{rP}{N}\right), \tag{32}\]  
_where the maximum is over all values of \(r\) satisfying_  
\[r\mathrm{CID}\geq Q\frac{rP+N}{(\sqrt{Q}+\sqrt{(1-r)P})^{2}+rP+N}. \tag{33}\]  
In the above theorem, the parameter \(r\in[0,1]\) indicates the fraction of the transmit power that the Tx uses for data transmission, i.e., to encode the message. The rest of the power, i.e., a fraction \(1-r\) of the total power, is used to send channel state information (in an uncoded manner) to the Rx.