Waveform Design for Communication-Assisted Sensing in 6G Perceptive Networks

The typical use cases of CAS are shown in Fig. 1, where a user wishes to sense a target within its Non-LoS (NLoS) paths, and hence requests sensing service from the nearby BS, who observes the target of interest from a LoS path. Let us denote $\bm{\upeta}\in\mathbb{R}^{K}$ as the parameter vector to be sensed, such as range, azimuth, TRM, etc., which takes value on a set $\mathcal{A}$ with a prior distribution $p_{\bm{\upeta}}(\bm{\eta})$. The procedure that the BS senses the target and transmits the estimated results to the user, can be equivalently considered as that the target information successively “passes through” the sensing and communication channels before its arrival at the user. In such a case, the original $\bm{\upeta}$, the estimated $\hat{\bm{\upeta}}_{s}$ at the BS, and the recovered $\hat{\bm{\upeta}}_{sc}$ at the user forms a Markov chain $\bm{\upeta}\to\hat{\bm{\upeta}}_{s}\to\hat{\bm{\upeta}}_{sc}$. The detailed signal models are as follows.

$\bullet$ Sensing side (S-side): The BS transmits sensing waveform to the target and yields the estimated parameter, namely, $\hat{\bm{\upeta}}_{s}$ through the noisy received echo signals with the expression of

where $\textbf{X}_{s}\in\mathbb{C}^{N\times T}$ is the transmitted sensing waveform. $\textbf{H}_{s}(\bm{\upeta})\in\mathbb{C}^{M_{s}\times N}$ represents the sensing channel matrix with respect to (w.r.t.) the latent parameters $\bm{\upeta}$. $\textbf{Z}_{s}\in\mathbb{C}^{M_{s}\times T}$ is the additive noise, whose entries follow an independently and identically circularly symmetric complex Gaussian distribution with $\mathcal{CN}(0,\sigma_{s}^{2}\textbf{I}_{TM_{s}})$. $T$, $N$ and $M_{s}$ are the numbers of symbols, transmitting and receiving antennas at the BS, respectively.

$\bullet$ Communication side (C-side): The BS transmits the estimated information $\hat{\bm{\upeta}}_{s}$ to the user through source-channel coding, and the user reconstruct an estimate $\hat{\bm{\upeta}}_{sc}$ from the following received signal

where $\textbf{X}_{c}(\hat{\bm{\upeta}}_{s})\in\mathbb{C}^{N\times T}$ is the transmitted communication waveform carrying the information of $\hat{\bm{\upeta}}_{s}$. $\textbf{H}_{c}\in\mathbb{C}^{M_{c}\times N}$ represents the communication channel matrix. $\textbf{Z}_{c}\in\mathbb{C}^{M_{c}\times T}$ follows similar distribution as to $\textbf{Z}_{s}$ with variance $\sigma_{c}^{2}$. $M_{c}$ is the number of the equipped antennas at the user.

A reasonable metric for sensing QoS is the mean square error (MSE) between the recovery $\hat{\bm{\upeta}}_{sc}$ at the user and the ground truth $\bm{\upeta}$. Then the average distortion can be defined by

We adopt the minimum MSE (MMSE) estimator at S-side, i.e., $\hat{\bm{\upeta}}_{s}=\mathbb{E}_{\bm{\upeta}}\left[\bm{\upeta}|\textbf{Y}_{s}\right]$, the distortion $D_{sc}$ can be recast by

where $(a)$ holds from the properties of the conditional expectation $\mathbb{E}\left[(\bm{\upeta}-\mathbb{E}\left[\bm{\upeta}|\textbf{Y}_{s}\right])^{T}f(\textbf{Y}_{s})\right]=0$. In (4), we demonstrate that the sensing QoS depends on both the estimation distortion $D_{s}$ at S-side and the recovery distortion $D_{c}$ at C-side. Moreover, it is worth highlighting that the above CAS process is to some extent similar to the remote source coding (also known as the CEO problem) in information theory, where the encoder can only access a noisy observation rather than the original source of interest [8].

Evidently, the distortion $D_{s}$ and $D_{c}$ depends on the transmitting waveform $\textbf{X}_{s}$ and $\textbf{X}_{c}$, respectively, which motivates us to design the optimal waveform such that the distortion $D_{sc}$ achieved at the user is minimized. Before presenting the formulation, we briefly introduce the rate-distortion (R-D) theory to characterize the relationship between the distortion $D_{c}$ and the channel capacity at C-side. For a random variable $\bm{\upeta}$ and its estimate $\hat{\bm{\upeta}}$, the R-D function is given by [9]

where $I(\bm{\upeta};\hat{\bm{\upeta}})$ and $d(\bm{\upeta},\hat{\bm{\upeta}})$ represent the mutual information (MI) and distance metric between $\bm{\upeta}$ and $\hat{\bm{\upeta}}$, respectively. $R(D)$ characterizes the minimal information rate required to achieve a preset distortion in lossy transmission. Based on the source-channel separation theorem with distortion [9, Th. 10.4.1], the recovery distortion $D_{c}$ at C-side is achievable if and only if $R(D_{c})\leq C$, where $C$ represents the capacity of communication channel. Consequently, the general optimization problem for CAS can be formulated as

where $P_{T}$ is the total power budget at BS. The first constraint is referred as to the separation theorem constraint and the second as power constraint. Although the CAS formulation presents a quite concise form, it is difficult to characterize the distortion $D_{sc}$ with the explicit expressions for arbitrary estimation tasks and prior distributions $p_{\bm{\upeta}}(\bm{\eta})$ in general. In what follows, we will focus on the specified TRM estimation as similarly considered in [10], to reveal the insights into CAS and leave the more general cases for future research.

In this subsection, we will derive the explicit expressions for sensing distortion $D_{s}$, R-D function $R(D)$ and communication channel capacity $C$ in the case of CAS-based TRM estimation task, where the parameter to be estimated is the target channel matrix itself, i.e., $\bm{\upeta}=\text{vec}(\textbf{H}_{s}^{T})$. The vector form of sensing model (1) can be rewritten as

where $\textbf{y}_{s}=\text{vec}(\textbf{Y}_{s}^{T})$, $\tilde{\textbf{X}}_{s}=\textbf{I}_{M_{s}}\otimes\textbf{X}_{s}^{T}$, $\textbf{h}_{s}=\text{vec}(\textbf{H}_{s}^{T})$, $\textbf{z}_{s}=\text{vec}(\textbf{Z}_{s}^{T})$. Here, we make the following assumptions on the parameter’s prior distribution and the random waveform

A1. The vector $\textbf{h}_{s}$ is a Gaussian random vector with the distribution $\textbf{h}_{s}\sim\mathcal{CN}(0,\sigma_{\eta}^{2}\textbf{I}_{NM_{s}})$.

A2. The Gaussian signal is used to transmit information at the C-side in order to achieve the capacity, namely, $\textbf{X}_{c}\sim\mathcal{CN}(0,\textbf{R}_{c_{x}})$, where $\textbf{R}_{c_{x}}$ is the covariance matrix.

(1) Sensing Distortion: Under the assumption A1, the sensing model (7) is exactly a Gaussian linear model (GLM), where the observations $\textbf{y}_{s}$ follows complex Gaussian distribution with zero mean and the covariance matrix

Thus, by adopting the MMSE estimator at S-side, the estimated $\hat{\bm{\upeta}}_{s}$ can be obtained at the BS as [11]

In this case, the sensing distortion is the MMSE, expressed by

(2) R-D Function: From (9), we note that the estimated parameter also follows complex Gaussian distribution with zero mean and the covariance matrix

At the C-side, the BS transmits this Gaussian source $\hat{\bm{\upeta}}_{s}$ to the user and adopts the MSE as the distortion metric. This model is also referred to as the quadratic Gaussian problem[8]. In such case, the R-D function (5) has the following expression according to the result in [9, Th. 10.3.3] ¹¹1The coefficient $\frac{1}{2}$ is vanished since the complex variable is considered.

where $\lambda_{\upeta_{i}}$ represents the $i$-th eigenvalue of the matrix $\textbf{R}_{\upeta_{s}}$, the distortion $D_{i}$ satisfies the inverse water-filling constraint as follows [9]

with $\xi$ being the inverse water-filling factor.

(3) Channel Capacity: Under the assumption A2, the communication channel capacity is given by

In this scenario, the BS transmits individual waveform for sensing and communication tasks, which can achieve optimal S$\&$C performance at S- and C- sides, respectively. Therefore, problem (21) can be divided into two sub-problems that are coupled by the total transmit power budget at the BS. Let us denote $P_{s}$ and $P_{c}$ as the power allocated to S- and C- sides satisfying $P_{s}+P_{c}=P_{T}$. Thus, for the given power $P_{s}$ at S-side, the MMSE minimization problem can be written by

Problem (22) has the classical water-filling solution as [12]

At the C-side, since the $D(R)$ is the monotonically decreasing function with respect to $R$, minimizing the distortion $D_{c}$ is equivalent to maximizing the channel capacity with the following form

Again, the closed-form water-filling solution and the resulting channel capacity are given by [12]

where $\xi_{c}$ is the water-filling factor such that $\sum_{i=1}^{N}\lambda_{c_{i}}=P_{c}$. Finally, we can calculate $\lambda_{\upeta_{i}}$ by substituting (23) into (19), and obtain the distortion $D_{c}$ for a given $C^{\star}$ through the reverse water-filling procedure (13), thereby the total distortion $D_{sc}$.

Based on the above analysis, one can readily reconstruct the optimal S$\&$C waveform when the water-filling factors $\xi_{s}$ and $\xi_{c}$ are determined. Meanwhile, these two factors are coupled in the power constraint. Accordingly, problem (21) can be transformed into the following form

where $(\cdot)^{+}=\text{max}\{\cdot,0\}$. Problem (26) is nothing but to find how much power allocated to S- and C- sides such that the minimum distortion can be achieved at the user. Therefore, the optimal solution of (26) can be found through the grid search over the power interval $[0,P_{T}]$, whose detailed flow is summarized in Algorithm 1.

Remark 1 (Power Allocation Trade-off): The separated waveform design demonstrates a clear power allocation trade-off between S- and C- sides. Evidently, excessively allocating resources to either side will cause the sensing QoS significant degradation at the user. Although only the power allocation is considered in this paper, the system resources including bandwidth, multi-beams, dwell time, etc., may also affect the CAS performance, which is left for our future research.

In this subsection, we focus on the dual-functional waveform applied for both S- and C-sides, i.e., $\textbf{X}_{s}=\textbf{X}_{c}=\textbf{X}$. This corresponds to that the BS transmits the parameter data of the last epoch at C-side and uses the same waveform for sensing at the current epoch. We should highlight the following essential difference compared to the above separated waveform design.

Remark 2 (Water-filling Trade-off): We can observe that the sensing-optimal waveform prefers to uniformly allocate the power in (23), whereas the communication-optimal waveform needs to allocate power in terms of the channel $\textbf{H}_{c}$ through water-filling method in (25). One cannot find a dual-functional waveform that simultaneously optimizes both S- and C- sides with high probability, unless the sensing and communication channels are sufficiently similar. This leads to a water-filling trade-off.

By transforming the reverse water-filling (13) into closed-form, the original problem for dual-functional waveform design can be recast by

where the variables are transformed into the shared use power allocation $\tilde{\lambda}_{i}$ and inverse water-filling factor $\xi$. This non-convex problem is difficult to cope with in general due to the inverse water-filling procedure and the fact that the variables are coupled in the separation theorem constraint. To tackle this problem, we present a heuristic sub-optimal search algorithm. The core idea is setting the sensing (or communication) optimal solution as the initial point, gradually re-allocating the power to improve the communication (or sensing) performance, until the objective function is minimized.

Initially, we set the sensing-optimal scheme $\bm{\lambda}^{\text{s-opt}}$ or communication-optimal scheme $\bm{\lambda}^{\text{c-opt}}$ as the initial point. The $\bm{\lambda}^{\text{s-opt}}$ and $\bm{\lambda}^{\text{c-opt}}$ are the solutions of problem (22) and (24) where the power budget is replaced by the total power $P_{T}$, respectively. Let us take $\bm{\lambda}^{\text{(0)}}=\bm{\lambda}^{\text{s-opt}}=[\tilde{\lambda}^{(0)}_{i},\cdots,\tilde{\lambda}^{(0)}_{N}]^{T}$ as an example. The distortion $D_{sc}^{(0)}$ can be readily obtained by following the similar procedure in the separated scheme. Then, the search direction is chosen to improve the communication performance, i.e.,

Thus, the solution is updated by

where the second formula in (29) is to meet the power constraint. For a given solution $\bm{\lambda}^{\text{(1)}}$, we can update the distortion $D_{sc}^{(1)}$ in the new iteration. Furthermore, the step size $\beta$ is chosen by the classical backtracking line search, ensuring the value of objective $D_{sc}$ non-increasing at each iteration. Therefore, the iterative procedure is stopped until $D_{sc}$ remains constant. In summary, the proposed dual-functional waveform design algorithm can be found in Algorithm 2.