Electromagnetic Property Sensing: A New Paradigm of Integrated Sensing and Communication

The electromagnetic (EM) property serves as a crucial indicator to identify the material of targets, playing a pivotal role in various industries and enabling automation processes to operate efficiently and accurately [1]. In conventional material identification techniques, the infrared emissivity as a material specific property is investigated to increase the material classification reliability [1]. However, the conventional infrared material identification techniques typically demand specialized devices that are costly and challenging to manufacture and maintain [2]. Besides, infrared light faces challenges in penetrating the target, particularly when the target surface is coated with camouflage material, rendering conventional infrared sensing methods ineffective. In contrast, the capability of EM waves to penetrate objects makes EM property sensing a viable and promising solution for accurate material identification. Besides, EM property sensing technology can be used in inspection and imaging of a human body, which meets the increasing demands from social security, environmental monitoring, and medical applications.

Recently, integrated sensing and communication (ISAC) has opened up numerous game-changing opportunities for future wireless sensing systems. ISAC allows communications systems and sensing systems to share the scarce radio spectrum, which saves a large amount of resource cost [3, 4, 5, 6]. In contrast to the dedicated sensing or communication functionality, the ISAC design methodology exhibits two types of gains. Firstly, the shared use of limited resources, namely, spectrum, energy, and hardware platforms, offers improved efficiency for both sensing and communications (S&C), and thus provides integration gain. Secondly, mutual assistance between S&C may offer coordination gain and further boost the dual performance. Hence, ISAC is expected to benefit both communication and sensing functionality in the near future [7, 8, 9, 10]. Due to its numerous advantages, ISAC is envisioned to be a key enabler for many future applications including intelligent connected vehicles, Internet of Things (IoT), and smart homes and cities [11, 12, 13, 14].

While ISAC has achieved notable success in localization [3], tracking [10], imaging [15], and various other applications [7, 9, 16, 11], the realm of EM property sensing within ISAC remains unexplored to the best knowledge of the authors. Since the EM property of the target material is implicitly encoded into the channel state information (CSI) from the transmitter to the receiver, there are promising prospects to incorporate EM property sensing capabilities into the ISAC paradigm. In the realm of ISAC, the primary function of sensing is to create an accurate representation that bridges the real physical world with its digital twin counterpart [17], [18]. In this context, the sensing of EM properties becomes essential. Unlike digital twins in image-based applications that may primarily rely on shape and location, digital twins in communications would need to reconstruct the communications channel, necessitating detailed material information. Thus, accurate EM property data is crucial to construct digital twin scenarios precisely, as it provides the material characteristics essential for channel reconstruction. This ensures that the digital representation not only mirrors the physical world’s form but also its interactive characteristics.

In this paper, we develop a novel scheme that utilizes OFDM pilot signals in ISAC systems to sense the EM property and identify the material of the target. The main contributions are summarized as follows:

• $\bullet$

  ISAC EM property sensing formulation: We establish an end-to-end EM propagation model from the perspective of Maxwell equations. A closed-form expression for a vector related to material EM property is derived, incorporating the Lippmann-Schwinger equation and the method of moments for discretization. The objective is to reconstruct the relative permittivity and conductivity distribution (RPCD) within a specified detection region that contains the target.

• $\bullet$

  Compressive sensing-based RPCD reconstruction: We propose a multi-frequency-based EM property sensing and material identification method using compressive sensing techniques, where the joint sparsity of the EM property vector is exploited to reconstruct RPCD.

• $\bullet$

  Beamforming design for enhanced sensing: To further improve the sensing accuracy, we optimize the sensing matrix by designing the transmitter beamforming matrix based on the Born approximation. The design involves minimizing the difference between the Gram matrix of the designed beamforming matrix and a target matrix. The optimization problem is cast in terms of the Gram matrix and is solved iteratively to obtain the optimal beamforming matrix.

Simulation results demonstrate that the proposed method can achieve both high-quality RPCD reconstruction and accurate material classification. Moreover, the RPCD reconstruction quality and material classification accuracy can be improved by increasing the signal-to-noise ratio (SNR) at the receiver, or by decreasing the distance between the target and the transmitter.

The rest of this paper is organized as follows. Section II presents the system model and formulates the EM property sensing problem. Section III derives the end-to-end electromagnetic propagation formula. Section IV elaborates the strategy of sensing target material with multiple measurements. Section V describes the transmitter beamforming design criterion. Section VI provides the extensive numerical simulation results, and Section VII draws the conclusion.

Notations: Boldface denotes vector or matrix; $j$ corresponds to the imaginary unit; $(\cdot)^{H}$, $(\cdot)^{T}$, and $(\cdot)^{*}$ represent the Hermitian, transpose, and conjugate, respectively; $\circ$ denotes the Khatri-Rao product; $\odot$ denotes the Hadamard product; $\otimes$ denotes the Kronecker product; $\mathrm{vec}(\cdot)$ denotes the vectorization operation; $\mathbf{A}[:,i]$ denotes the submatrix composed of all elements in column $i$ of matrix $\mathbf{A}$; $\mathbf{I}$, $\mathbf{1}$, and $\mathbf{0}$ denote the identity matrix, the all-ones vector, and the all-zeros vector with compatible dimensions; $\left\|\mathbf{a}\right\|_{2}$ denotes $\ell 2$-norm of the vector $\mathbf{a}$; $\left|a\right|$ denotes the absolute value of the complex number $a$; $\Re(\cdot)$ and $\Im(\cdot)$ denote the real and imaginary part of complex vectors or matrices, respectively; $\left\|\mathbf{A}\right\|_{F}$ denotes Frobenius-norm of the matrix $\mathbf{A}$; $\mathbf{A}\succeq\textbf{0}$ indicates that the matrix $\mathbf{A}$ is positive semi-definite. The distribution of a circularly symmetric complex Gaussian (CSCG) random vector with zero mean and covariance matrix $\mathbf{A}$ is denoted as $\mathcal{CN}(\mathbf{0},\mathbf{A})$.

Consider a downlink communications scenario, in which an $N_{t}$-antenna base station (BS) communicates with an $N_{r}$-antenna mobile user with orthogonal frequency division multiplexing (OFDM) signaling. The transmitter adopts fully digital precoding structure where the number of RF chains $N_{RF}$ is equal to the number of antennas $N_{t}$. To enable EM property sensing, we resort to the pilot transmission, in which a total of $K$ subcarriers are used to transmit pilots. The central frequency and the subcarrier interval of OFDM signals are $f_{c}$ and $\Delta f$, respectively. Then the transmission bandwidth is $W=(K-1)\Delta f$, and the frequency of the $k$-th subcarrier is $f_{k}=f_{c}+(2k-K-1)\Delta f/2$, where $k=1,2,\cdots,K$. The corresponding wavelength of the $k$-th subcarrier is $\lambda_{k}=\frac{c}{f_{c}+(2k-K-1)\Delta f/2}$. Furthermore, we consider that a total of $I$ pilot symbols are transmitted in each subcarrier.

The comprehensive sensing of a target in ISAC is accomplished through the following steps. Firstly, the existence of the target is ascertained through target detection. This initial phase is focused on discovering the presence of the target, which is a fundamental prerequisite for any further analysis. Secondly, once the target is detected, the task of ISAC is pinpointing the precise location and velocity of the target. These parameters are crucial as they provide the spatial context needed to accurately engage with the target. The methods to acquire these parameters have been comprehensively discussed in conventional ISAC literature [7, 9, 10, 11, 16, 12, 13, 5]. Thirdly, we proceed to sense the EM property of the target, acquiring the material-specific information. Therefore, when sensing the EM property of the target, we may assume that other properties like the position of the target have already been obtained and are thus known values.

Suppose the target scatters the pilot signals from the transmitter to the receiver as shown in Fig. 1. Since only the signals scattered by the target carry the information of its EM property, we may send the pilot signals towards the target by beamforming properly at the transmitter. The overall ISAC channel from the transmitter to the receiver of the $k$-th subcarrier hence only consists of the scattering path channel $\mathbf{H}_{k}\in\mathbb{C}^{N_{r}\times N_{t}}$. Thus, the received signals can be formulated as:

where $\tilde{\mathbf{w}}_{k}\in\mathbb{C}^{N_{t}\times 1}$ is the digital beamforming weights on all transmitting antennas; $x_{k}$ is the normalized pilot symbol; $\tilde{\mathbf{n}}_{k}\sim\mathcal{CN}\left(\mathbf{0},\sigma_{k}^{2}\mathbf{I}_{N_{r}}\right)$ is the complex Gaussian noise at the receiver of the $k$-th subcarrier. For notational brevity, we denote $\mathbf{w}_{k}=\tilde{\mathbf{w}}_{k}x_{k}$ that is perfectly known by the receiver during the pilot transmission process.

Assume prior knowledge has determined that the target is contained within a certain sensing domain $D$, which can be discretized into a total of $M$ sampling points. Let the vectors $\mathbf{E}_{k}^{i,D}\in\mathbb{C}^{M\times 1}$ and $\mathbf{J}_{k}^{D}\in\mathbb{C}^{M\times 1}$ collect the incident electric field and the equivalent contrast source at all $M$ sampling points in $D$, respectively [19, 20, 21]. Define $\mathbf{H}_{1,k}\in\mathbb{C}^{M\times N_{t}}$ as the channel matrix of the transmitter-to-target path that maps $\mathbf{w}_{k}$ to $\mathbf{E}_{k}^{i,D}$, and define $\mathbf{H}_{2,k}\in\mathbb{C}^{N_{r}\times M}$ as the channel matrix of the target-to-receiver path that maps $\mathbf{J}_{k}^{D}$ to $\tilde{\mathbf{y}}_{k}$. The ISAC channel from the transmitter through the domain $D$ to the receiver can then be described as:

where $\mathbf{X}_{k}\in\mathbb{C}^{M\times M}$ maps $\mathbf{E}_{k}^{i,D}$ to $\mathbf{J}_{k}^{D}$ and represents the influence on the signal transmission caused by the existence of target. The material-related EM property of the target is implicitly incorporated in the formulation of $\mathbf{X}_{k}$, which can be leveraged to estimate the material of the target. The matrices $\mathbf{H}_{1,k}$ and $\mathbf{H}_{2,k}$ are also known as the radiation operator matrices and are solely related to the physical laws of EM propagation [22]. Since we assume that the positions of the receiver, the sensing domain $D$, and the transmitter are known, $\mathbf{H}_{1,k}$ and $\mathbf{H}_{2,k}$ can be calculated by the computational EM methods such as method of moments (MOM) [23], [24], finite element method (FEM) [25], and finite-difference time-domain (FDTD) [26]. In this paper, the objective is to retrieve the EM property of the target using the received signals $\tilde{y}_{k}$ and to further identify the material of the target from $\mathbf{X}_{k}$.

In this section, a closed-form expression of $\mathbf{X}_{k}$ is derived to unfold the influence on the received signals caused by the target scattering process. Sensing the target EM property is equivalent to reconstructing the difference between the complex relative permittivity of the target and the complex relative permittivity of air at each point $\mathbf{r}^{\prime}$ in domain $D$, denoted as $\chi_{k}(\mathbf{r}^{\prime})$. Since the complex relative permittivity of air is approximately equal to $1$, we can formulate the contrast function as:

where $\epsilon_{r}(\mathbf{r}^{\prime})$ is the real relative permittivity at point $\mathbf{r}^{\prime}$, $\sigma(\mathbf{r}^{\prime})$ is the conductivity at point $\mathbf{r}^{\prime}$, $\epsilon_{0}$ is the vacuum permittivity, and $\omega_{k}=2\pi f_{k}$ is the angular frequency of electromagnetic waves of the $k$-th subcarrier. The distributions of $\epsilon_{r}(\mathbf{r}^{\prime})$ and $\sigma(\mathbf{r}^{\prime})$ are the targets that need to be recovered in the EM property sensing task. The total electric field corresponding to the $k$-th subcarrier inside the domain $D$ can be expressed using the Lippmann-Schwinger (LS) equation as [19], [27]

where $k_{k}=\frac{2\pi}{\lambda_{k}}$ is the wavenumber of the $k$-th subcarrier, while $E_{k}^{t}({\mathbf{r}})$, $E_{k}^{i}({\mathbf{r}})$, and ${G_{k}({\mathbf{r},\mathbf{r}}^{\prime})}$ denote the total electric field, the incident electric field, and the Green’s function, respectively. Consider a 2D transverse magnetic (TM) scenario, and ${G_{k}({\mathbf{r},\mathbf{r}}^{\prime})}$ can then be formulated as [19], [27]

where $H_{0}^{(2)}$ is the zero-order Hankel function of the second kind.

Let the vectors $\mathbf{E}_{k}^{s,D}\in\mathbb{C}^{M\times 1}$, $\mathbf{E}_{k}^{t,D}\in\mathbb{C}^{M\times 1}$, and $\boldsymbol{\chi}_{k}\in\mathbb{C}^{M\times 1}$ collect the scattering electric field, the total electric field, and $\chi_{k}(\mathbf{r})$ at all $M$ sampling points in $D$, respectively. Denote $\mathbf{G}_{k}\in\mathbb{C}^{M\times M}$ as the discretized matrix of the integral kernel $k_{k}^{2}G_{k}\left(\mathbf{r}-\mathbf{r}^{\prime}\right)$ in (4) using MOM, where both $\mathbf{r}$ and $\mathbf{r}^{\prime}$ are on the discretized sampling points within the domain $D$. The total electric field in $D$ can be obtained by adding the incident field and the scattering field, which means $\mathbf{E}_{k}^{t,D}$ can be written in the discretized form of (4) as:

In order to yield a closed-form expression of $\mathbf{E}_{k}^{t,D}$, we reformulate equation (6) in the non-linear form with respect to $\textrm{diag}\left({\boldsymbol{\chi}_{k}}\right)$ as:

After obtaining $\mathbf{E}_{k}^{t,D}$, the equivalent contrast source in $D$ is defined as [19], [27]

which can be regarded as the source of scattering EM waves. Considering the radiation from the equivalent contrast source $\textrm{diag}\left({\boldsymbol{\chi}_{k}}\right)\mathbf{E}_{k}^{t,D}$ transmitted through the channel $\mathbf{H}_{2,k}$, we can formulate $\tilde{\mathbf{y}}_{k}$ as [20], [21]

Taking into account $\mathbf{E}_{k}^{i,D}=\mathbf{H}_{1,k}\mathbf{w}_{k}$, the received signal transmitted through the target scattering path can be written as:

Therefore, the closed-form expression of $\mathbf{X}_{k}$ in (2) can be written as:

For those sampling points occupied by the air within $D$, the corresponding rows of $\mathbf{X}_{k}$ are all zeros, and those points are not regarded as EM sources of scattering waves. Note that (11) is a general formulation regardless of the positions of the transmitter and the receiver. Although we consider a bistatic sensing scenario in this paper, (11) can also be used in a monostatic sensing scenario where the transmitter is in close proximity to the receiver.

Denote $\tilde{\mathbf{W}}_{k}=\left[\mathbf{w}_{k,1},\mathbf{w}_{k,2},\cdots,\mathbf{w}_{k,I}\right]\in\mathbb{C}^{N_{t}\times I}$ as the digital transmitter beamformer stacked by time of the $k$-th subcarrier. Then the overall received pilot signals $\mathbf{Y}_{k}\in\mathbb{C}^{N_{r}\times I}$ can be formulated in a compact form as:

Denote the vectorized noise as $\mathbf{n}_{k}=\left[\tilde{\mathbf{n}}_{k,1}^{\top},\tilde{\mathbf{n}}_{k,2}^{\top},\cdots,\tilde{\mathbf{n}}_{k,I}^{\top}\right]^{\top}$. Then we can vectorize $\mathbf{Y}_{k}$ to $\mathbf{y}_{k}\in\mathbb{C}^{IN_{r}\times 1}$ as:

where $\overset{(a)}{=}$ in (13) comes from the equality: $\mathrm{vec}\left(\mathbf{A}\textrm{diag}(\mathbf{b})\mathbf{C}\right)=\left(\mathbf{C}^{T}\circ\mathbf{A}\right)\mathbf{b}$. In order to derive a quasi-linear sensing model, we further define the sensing matrix $\mathbf{D}_{k}\in\mathbb{C}^{IN_{r}\times M}$ as:

Then the sensing equation with respect to $\boldsymbol{\chi}_{k}$ can be formulated as:

Note that although we call (15) a quasi-linear sensing model, $\mathbf{y}_{k}$ is actually nonlinear with respect to $\boldsymbol{\chi}_{k}$ because $\mathbf{D}_{k}$ also depends on $\boldsymbol{\chi}_{k}$ and is therefore unknown.

In this section, the beamforming matrix is designed to optimize the initial sensing matrix $\tilde{\mathbf{E}}^{0}$ based on BA. The beamforming matrix design is carried out before the sensing iteration, and is unrelated to the target RPCD distribution. For notational brevity, we will omit the superscript $0$ in the remainder of this section.

In this section, we use the EM simulation software Ansys HFSS to simulate the transmission process in the ISAC system and to generate the received signals, which are used in reconstructing the EM property and identifying the material of the target. Suppose the center of the target is located at the origin. We consider a uniform linear array (ULA) transmitter equipped with $N_{t}=1024$ antennas. The ULA transmitter is parallel to the $y$-axis, and its center is located at $(x_{t},y_{t})=(d,0)$ m, where $d$ denotes the distance from the transmitter to the target. Suppose a ULA receiver equipped with $N_{r}=16$ antennas is located at $(x_{r},y_{r})=(-20,-20)$ m and is parallel to the ULA transmitter. The inter-antenna spacing for both the transmitter and the receiver is set as $\lambda_{c}/2$. The central frequency is $f_{c}=30$ GHz and the frequency step of OFDM subcarriers is $\Delta f=200$ KHz. In each subcarrier, $I=32$ pilot OFDM symbols are transmitted. Suppose prior knowledge determines that the target is located within the region $D=[-1,1]\times[-1,1]$ $\mathrm{m}^{2}$. We choose the number of sampling points in the domain $D$ as $M=32\times 32=1024$. Each sampling point in $D$ can be regarded as a pixel in the image, i.e., the RPCD image is composed of $32\times 32=1024$ pixels. More pixels could hardly improve the RPCD reconstruction quality due to the resolution limit.

Besides, we introduce the normalized mean square error (NMSE) of RPCD reconstruction as the criterion to quantitatively describe the performance of EM property sensing