Fundamental Limits of Communication-Assisted Sensing in ISAC Systems

Integrated sensing and communication (ISAC) system is widely acknowledged for its potential to enhance sensing and communications (S&C) performance by sharing the use of hardware, spectrum, and signaling strategies [1]. Recently, the ISAC system has been officially approved as one of the six critical usage scenarios of 6G by the International Telecommunications Union (ITU). Over the past few decades, substantial research efforts have been directed toward signal processing and waveform design (cf. [2, 3], and the reference therein). However, the fundamental limits and the resulting performance tradeoff between S&C have remained longstanding challenges within the research community [4].

Several groundbreaking studies have recently delved into exploring the fundamental limits within various ISAC system configurations. The authors of [5, 6] consider a scenario where the transmitter (Tx) communicates with a user through a memoryless state-dependent channel, simultaneously estimating the state from generalized feedback. The capacity-distortion-cost tradeoff of this channel is characterized to illustrate the optimal achievable rate for reliable communication while adhering to a preset state estimation distortion. Concurrently, another study in [7, 8] focuses on a general scenario where the Tx senses arbitrary targets rather than the specified communication state, namely, the separated S&C channels. The deterministic-random tradeoff between S&C within the ISAC signaling strategy is unveiled by characterizing the Cramér-Rao bound (CRB)-communication rate region [7]. Subsequently, the work in [8] extended this tradeoff to any well-defined sensing distortion metrics.

Unfortunately, there is limited research on exploring the coordination gains offered by mutual support of S&C functionalities in ISAC systems. Notably, within intelligent 6G applications requiring high-quality sensing service, a natural question arises regarding how sensing performance can be enhanced by incorporating communication functionality, and what the fundamental limits are for such a system. To fill this research gap, we introduce a novel system setup in this paper, which is referred to as communication-assisted sensing (CAS). As illustrated in Fig. 1, such the CAS system can endow users with beyond-line-of-sight (BLoS) sensing capability without the need of additional sensors. Specifically, the base station (BS) with favorable visibility illuminates the targets and captures observations containing the interested parameters through device-free wireless sensing abilities, then conveys the relevant information to the users. Thus, the users can acquire the parameters of interest to the BLoS targets.

The main contributions of this paper are summarized below. First, we establish a novel CAS system framework, delving deeper into the coordination gain facilitated by the ISAC technique. Second, we analyze the information-theoretic aspects of the CAS framework, characterizing the fundamental limits of the achievable sensing distortion at the end-user. Finally, we illustrate a practical transmission scheme, designing an ISAC waveform to achieve the minimum sensing distortion.

The information-theoretic framework of the proposed CAS system is depicted in Fig. 2. The Tx employs ISAC signal to simultaneously implement S&C tasks. The state random variables $S$ of the target, the S&C channel input $X$, the sensing channel output $Z$, and the communication channel output $Y$ take values in the sets $\mathcal{S}$, $\mathcal{X}$, $\mathcal{Z}$, $\mathcal{Y}$, respectively. Here, the state sequence $\{S_{i}\}_{i\geq 1}$ follows independently and identically distributed (i.i.d.) with a prior distribution $P_{S}$. The CAS process mainly contains the following two parts.

$\bullet$ Sensing Process: The sensing channel output at sensing receiver (Rx) $Z_{i}$ at a given time $i$ is generated based on the sensing channel law $Q_{Z|SX}(\cdot|x_{i},s_{i})$ given the $i$th channel input $X_{i}=x_{i}$ and the state realization $S_{i}=s_{i}$. We assume that the sensing channel output $Z_{i}$ is independent to the past inputs, outputs and state signals. Let $\tilde{\mathcal{S}}$ denote the estimate alphabet, the state estimator is a map from the acquired the observations $\mathcal{Z}^{n}$ to $\tilde{\mathcal{S}}^{n}$. Thus, the expected average per-block estimation distortion in the sensing process can be defined by

where $d(\cdot,\cdot)$ is a distortion function bounded by $d_{\text{max}}$.

$\bullet$ Communication Process: The Tx encodes the estimate $\tilde{S}_{i-1}$ at the last epoch to communication channel input $X_{i}$. The communication Rx receives the channel output $Y_{i}$ according to channel law $Q_{Y|X}$. The decoder is a map from $\mathcal{Y}^{n}$ to $\hat{\mathcal{S}}^{n}$, where $\hat{\mathcal{S}}$ denotes the reconstruction alphabet. Thus, the expected average per-block estimation distortion in the communication process can be defined by

The overall sensing quality of service (QoS) may be measured by the distortion between the source and its reconstruction at the user-end, i.e.,

In the CAS process, a $(2^{n\mathsf{R}},n)$ coding scheme consists of

1) a state parameter estimator $h$: $\mathcal{X}^{n}\times\mathcal{Z}^{n}\to\tilde{\mathcal{S}}^{n}$;

2) a message set (also the estimate set) $\tilde{\mathcal{S}}=[1:2^{n\mathsf{R}}]$;

3) an encoder $\phi$: $\tilde{\mathcal{S}}^{n}\to\mathcal{X}^{n}$;

4) a decoder $\psi$: $\mathcal{Y}^{n}\to\hat{\mathcal{S}}^{n}$.

For a practical system, the S&C channel input $X$ may be restricted by the limited system resources. Define the cost-function $b(x):\mathcal{X}\to\mathbb{R}^{+}$ to characterize the S&C channel cost, then a rate-distortion-cost tuple $(\mathsf{R},D,B)$ is said achievable if there exists a sequence of $(2^{n\mathsf{R}},n)$ codes that satisfy

Remark: We emphasize that while our CAS system setup shares some similarities with conventional remote estimation [9, 10, 11] and remote source coding [12, 13, 14] problems, they are essentially distinct. In remote estimation, a sensor measures the target’s state and transmits its observations to a remote estimator via a wireless channel. In remote source coding, the encoder cannot access the original source information but only its noisy observations. Note that in these schemes, the state observations (which may be acquired by additional sensors) are independent of the communication channel input. In contrast, the S&C channels are coupling due to the employment of the ISAC signaling strategy. For instance, $S\leftrightarrow Z\leftrightarrow X\leftrightarrow Y\leftrightarrow\hat{S}$ forms a Markov chain in the remote source coding, which is, however, no longer a Markov chain due to the dependency of observation $Z$ on the dual-functional channel input $X$ in the CAS framework. As a result, the overall distortion may not be well-defined by applying the remote source coding theory.

In this subsection, we analyze the optimal estimator in the sensing process and the communication channel capacity constrained by the sensing distortion and resource limitation.

Lemma 1 [6]: By recalling that $S\leftrightarrow XZ\leftrightarrow\tilde{S}$ forms a Markov chain, the sensing distortion $\Delta_{s}^{(n)}$ is minimized by the deterministic estimator

where

is independent to the choice of encoder and decoder.

The detailed proof of Lemma 1 can be found in [6]. By applying the estimator (6), we may define the estimate-cost function as $e(x)=\mathbb{E}[d(S,\tilde{s}^{\star}(X,Z))|X=x]$. Then, a give estimation distortion $D_{s}$ satisfies

Within the ISAC signaling strategy, the S&C channel input $X$ may be constrained by both the estimate-cost and resource-cost functions. Namely, the feasible set of the probability distributions of $X$ can be described by the intersection of the two sets as follows.

Definition 1: In communication process, the information-theoretic capacity constrained by estimation and resource cost is defined by

where $I(X;Y)$ denotes the mutual information (MI) between the communication channel input and output.

From (9), we can observe the coupling relationship between S&C processes, namely, the achieved sensing distortion concurrently impacts the communication channel capacity. Moreover, the achieved communication distortion $D_{c}$ is determined by the channel capacity in terms of the rate-distortion theory in lossy data transmission strategy.

Definition 2: The information-theoretic rate distortion function can be defined by

According to the optimal sensing estimator in Lemma 1, we have $I(\tilde{S};\hat{S})=I(X,Z;\hat{S})=I(X;\hat{S})$ due to the fact that $\hat{S}$ is conditional independent to $Z$ with given $X$.

We start with a converse to show that any achievable coding scheme must satisfy (11). Consider any $(2^{n\mathsf{R}},n)$ coding scheme defined by the encoding and decoding functions $\phi$ and $\psi$. Let $\tilde{S}^{n}=h^{\star}(X^{n},Z^{n})$ be the estimate sequence as given in (5) and $\hat{S}^{n}=\psi(\phi(\tilde{S}^{n}))$ be the reconstruction sequence corresponding to $\tilde{S}^{n}$.

By recalling from the proof of the converse in lossy source coding, we have

where $(a)$ follows the Definition 2, $(b)$ and $(c)$ are due to the convexity and non-increasing properties of the rate distortion function.

On the other hand, by recalling from the proof of the converse in channel coding, we have

where $(d)$ follows the Definition 1, $(e)$ and $(f)$ are due to the concavity and non-decreasing properties of the capacity constrained by estimation and resource costs [6]. Additionally, we have the data processing inequality $I(\tilde{S};\hat{S})=I(X;\hat{S})\leq I(X;Y)$ benefit from the Markov chain $X\leftrightarrow Y\leftrightarrow\hat{S}$. By combing (12), (13), we complete the proof of the converse.

Let $S^{n}$ be drawn i.i.d. $\sim P_{S}$, we will show that there exists a coding scheme for a sufficiently large $n$ and rate $\mathsf{R}$, the distortion $\Delta^{n}$ can be achieved by $D$ if (11) holds. The core idea follows the famous random coding argument and source channel separation theorem with distortion.

1) Codebook Generation: In source coding with rate distortion code, randomly generate a codebook $\mathcal{C}_{s}$ consisting of $2^{n\mathsf{R}}$ sequences $\hat{S}^{n}$ which is drawn i.i.d. $\sim P_{\hat{S}}$. The probability distribution is calculated by $P_{\hat{S}}=\sum_{\tilde{S}}P_{\tilde{S}}P_{\hat{S}|\tilde{S}}$, where $P_{\hat{S}|\tilde{S}}$ achieves the equality in (10). In channel coding, randomly generate a codebook $\mathcal{C}_{c}$ consisting of $2^{(n\mathsf{R})}$ sequences $X^{n}$ which is drawn i.i.d. $\sim P_{X}$. The $P_{X}$ is chosen by satisfying the capacity with estimation- and resource-cost in (9). Index the codeword $\tilde{S}^{n}$ and $X^{n}$ by $w\in\{1,2,\cdots,2^{n\mathsf{R}}\}$.

2) Encoding: Encode the $\tilde{S}^{n}$ by $w$ such that

where $\mathcal{T}^{(n)}_{d,\epsilon_{s}}(P_{\tilde{S},\hat{S}})$ represents the distortion typical set [15] with joint probability distribution $P_{\tilde{S},\hat{S}}=P_{\tilde{S}}P_{\hat{S}|\tilde{S}}$. To send the message $w$, the encoder transmits $x^{n}(w)$.

3) Decoding: The decoder observes the communication channel output $Y^{n}=y^{n}$ and look for the index $\hat{w}$ such that

where $\mathcal{T}^{(n)}_{\epsilon_{c}}(P_{X,Y})$ represents the typical set with joint probability distribution $P_{X,Y}=P_{X}P_{Y|X}$. If there exists such $\hat{w}$, it declares $\hat{S}^{n}=\hat{s}^{n}(\hat{w})$. Otherwise, it declares an error.

4) Estimation: The encoder observes the channel output $Z^{n}=z^{n}$, and computes the estimate sequence with the knowledge of channel input $x^{n}$ by using the estimator $\tilde{s}^{n}=h^{\star}(x^{n},z^{n})$ given in (5).

5) Distortion Analysis: We start by analyzing the expected communication distortion (averaged over the random codebooks, state and channel noise). In lossy source coding, for a fixed codebook $\mathcal{C}_{s}$ and choice of $\epsilon_{s}>0$, the sequence $\tilde{s}^{n}\in\tilde{\mathcal{S}}^{n}$ can be divided into two categories:

$\bullet$ $(\tilde{s}^{n},\hat{s}^{n}(w))\in\mathcal{T}^{(n)}_{d,\epsilon_{s}}$, we have $d(\tilde{s}^{n},\hat{s}^{n}(w))<D_{c}+\epsilon_{s}$;

$\bullet$ $(\tilde{s}^{n},\hat{s}^{n}(w))\notin\mathcal{T}^{(n)}_{d,\epsilon_{s}}$, we denote $P_{e_{s}}$ as the total probability of these sequences. Thus, these sequence contribute at most $P_{e_{s}}d_{\max}$ to the expected distortion since the distortion for any individual sequence is bounded by $d_{\max}$.

According to the achievability of lossy source coding [15, Theorem 10.2.1], we have $P_{e_{s}}$ tends to zero for sufficiently large $n$ whenever $\mathsf{R}\geq R^{\text{IT}}(D_{c})$.

In channel coding, the decoder declares an error when the following events occur:

$\bullet$ $(x^{n}(w),y^{n})\notin\mathcal{T}^{(n)}_{\epsilon_{c}}$;

$\bullet$ $(x^{n}(w^{\prime}),y^{n})\in\mathcal{T}^{(n)}_{\epsilon_{c}}$, for some $w^{\prime}\neq w$, we denote $P_{e_{c}}$ as the probability of the error occurred in decoder. The error decoding contribute at most $P_{e_{c}}d_{\max}$ to the expected distortion. Similarly, we have $P_{e_{c}}\to 0$ for $n\to\infty$ whenever $\mathsf{R}\leq C^{\text{IT}}(D_{s},B)$ according to channel coding theorem [15].

On the other hand, the expected estimation distortion can be upper bounded by

Note that $(s^{n},x^{n}(w),\tilde{s}^{n})\in\mathcal{T}^{(n)}_{\epsilon_{e}}(P_{S,X,\tilde{S}})$ where $P_{S,X,\tilde{S}}$ denotes the joint marginal distribution of $P_{S,X,Z,\tilde{S}}=P_{S}P_{X}Q_{Z|SX}\mathbb{I}\{\tilde{s}=\tilde{s}^{\star}(x,z)\}$ with $\mathbb{I}(\cdot)$ being the indicator function, we have

according to the typical average lemma [6]. In summary, the overall sensing QoS can be calculated by

where $(g)$ follows the definition of the separable distortion metric in Theorem 1 and in $(h)$ we omit the terms containing the product of $P_{e_{s}}$ and $P_{e_{c}}$. Consequently, taking $n\to\infty$ and $P_{e_{s}},P_{e_{c}},\epsilon_{c},\epsilon_{e},\epsilon_{s}\to 0$, we can conclude that the expected sensing distortion (averaged over the random codebooks, state and channel noise) tends to be $D\to D_{c}+D_{s}$ whenever $R^{\text{IT}}(D_{c})\leq\mathsf{R}\leq C^{\text{IT}}(D_{s},B)$.

By applying the estimator (6), the mean squared error (MSE) between $\mathbf{s}$ and its estimate $\tilde{\mathbf{s}}$ may be obtained by [17]

Fortunately, when we adopt MSE, a widely used metric in parameter estimation, as the distortion metric, the condition of separable distortion in the CAS process holds. Namely, the overall distortion $D$ can be equivalently decomposed into the sum of the estimation and communication distortions,

where $(i)$ holds from the properties of the conditional expectation

Here, $\tilde{\mathbf{s}}=\mathbb{E}\left[\mathbf{s}|\mathbf{z},\mathbf{x}\right]$ is the nimimum mean squared error (MMSE) estimator, and $f(\mathbf{z},\mathbf{x})$ represents an arbitrary function with respect to $\mathbf{z},\mathbf{x}$. Furthermore, the MMSE estimate $\tilde{\mathbf{s}}$ can be expressed by [17]

where $\mathbf{I}_{M_{s}}\otimes\mathbf{R}_{z}$ is the covariance matrix of the observation $\mathbf{z}$ with the expression of $\mathbf{R}_{z}=\mathbf{X}^{H}\bm{\Sigma}_{s}\mathbf{X}+\sigma^{2}_{s}\mathbf{I}_{T}$.

The estimate $\tilde{\mathbf{s}}$ also follows complex Gaussian distribution $\mathcal{CN}(\mathbf{0},\mathbf{R}_{\tilde{\mathbf{s}}})$, with the covariance matrix of

Based on the rate distortion function for the Gaussian distribution source, we have

which is in a reverse water-filling form as

where $\lambda_{i}(\mathbf{R}_{\tilde{\mathbf{s}}})$ represents the $i$th eigenvalue of $\mathbf{R}_{\tilde{\mathbf{s}}}$, and $\xi$ is the reverse water-filling factor.

Our aim is to optimize the probability distribution $P_{X}$. However, there may be no explicit expression of the communication channel capacity for an arbitrary distribution of channel input. To circumvent numerical computing, we temporarily restrict the waveform to the Gaussian distribution with $\mathcal{CN}(\mathbf{0},\mathbf{R}_{x})$ which is widely considered in communication systems. It should be noted that although Gaussian distribution is optimal for communication process, but not necessarily achieving optimal estimation performance, leading to an uncertain overall sensing QoS in the CAS system. Evidently, this also exhibits a deterministic-random tradeoff discussed in [7, 8].

The statistical covariance matrix $\mathbf{R}_{x}$ can be approximated by sample covariance $1/T\mathbf{X}\mathbf{X}^{H}$. Thus, the MI between the received and transmitted signals conditioned on communication channel $\mathbf{H}_{c}$ can be approximately characterized by

The original problem (19) can be reformulated by

where $P_{T}$ is the power budget. We derive the explicit expressions of MI and rate distortion function in the scenario of TRM estimation. However, it is still challenging to solve the non-convex problem (29) since the imposed reverse water-filling constraint introduces an unknown nuisance parameter (i.e., the factor $\xi$), which may only be determined through a numerical search algorithm in general. The solution of (29) is omitted in this paper due to the limited space. We refer the interested reader to [18] for more details.

We conduct a comparative analysis of the performance of two signaling strategies: the separated S&C waveforms (SW) scheme and the ISAC scheme. In the SW scheme, optimal S&C waveforms are independently selected for the S&C processes. The SW scheme is more similar to the remote estimation due to the decoupling of the sensing observations from the communication channel input. Unlike remote estimation, which typically employs additional sensors for observation collection, the SW scheme detect the target through wireless sensing. Consequently, the S&C resource competition exists, defined by $\text{Tr}(\mathbf{X}_{s}\mathbf{X}_{s}^{H})+\text{Tr}(\mathbf{X}_{c}\mathbf{X}_{c}^{H})\leq TP_{T}$ in this paper, due to the shared use of hardware platforms.

We can observe from Fig. 3 that the power competition between the S&C processes dominates the overall sensing QoS in the regimes of low SNRs. As anticipated, the ISAC scheme outperforms the SW scheme due to the benefits derived from the resource multiplexing gains. Conversely, in high SNR regimes, the SW scheme exhibits superior performance compared to the ISAC scheme, implying favorable quality of the S&C channels. As resource multiplexing gains approach saturation levels, the optimal waveform structures take precedence in influencing the sensing QoS. Specifically, achieving a balance between S&C channel for the eigenspace of the waveform becomes crucial.