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ISAC system assisted by RIS with sparse active elements
EURASIP Journal on Advances in Signal Processing volume 2023, Article number: 20 (2023)
Abstract
In this paper, we consider an integrated sensing and communications system assisted by a reconfigurable intelligent surface (RIS). A small number of sparsely placed active sensors are applied in the RIS to perform effective channels and, thereby, enable optimized beamforming for both communications and sensing objectives, namely, establishing reliable communication links with communication users (CUs) and effectively localize targets. The timevarying multipath channels between the RIS and the CU as well as the timevarying channel between the RIS and the targets are estimated by exploiting an interpolated Hermitian and Toeplitz covariance matrix followed by directionofarrival estimation using the MUSIC algorithm. Based on such results, we jointly optimize the transmit beamformer at the base station and the unitmodulus RIS passive beamformer. The RIS beamformer is optimized to maximize its minimum beampattern gain towards the desired sensing angles subject to the minimum signaltonoise ratio requirement at the CU. Simulation results verify the effectiveness of the proposed approach, and the performance of different sparse array configurations is compared.
1 Introduction
Nextgeneration wireless communication systems are required to deliver information in an extremely fast, more trustworthy, lowlatency, and secure manner. Recent communications networks have emerged to use the millimeterwave (mmWave) frequency band to provide innovative features and enable high data rate applications, such as highquality video transmission, vehicletoinfrastructure communications, intra, and intervehicle messaging, devicetodevice communications, and Internet of Things frameworks, and tactile Internet [1,2,3,4]. Similarly, modern radar systems use a wide frequency spectrum for a variety of applications such as air surveillance, longrange meteorological, and automobile radars.
The demand for higher data rates and higher sensing resolution in recent communication and sensing systems has rendered spectral congestion a critical problem. On the other hand, wireless sensing and communication systems are evolving to achieve wider frequency bands, larger antenna arrays, and reduced size. As a result, their hardware architectures, channel properties, and signal processing are becoming increasingly complicated. An effective solution to reduce the hardware cost and save radio spectrum is to incorporate sensing capability into the wireless communication infrastructures, thereby allowing future wireless networks to provide ubiquitous sensing services in addition to the regular communication functions [5]. There has been a great interest in developing integrated sensing and communication (ISAC) techniques to enable shared spectrum access [6, 7]. ISAC can be implemented in several forms. For example, dualfunction radarcommunication (DRFC) systems share the same spectrum and hardware for both wireless communications and radar sensing, thereby achieving high spectral and cost efficiency [8,9,10,11].
Recently, reconfigurable intelligent surface (RIS) has emerged as a promising means for the nextgeneration wireless communication systems to boost the channel capacity and increase service coverage [12]. RIS is a metasurface made up of a large number of passive reflecting elements, typically in a rectangular shape, in which each element can be digitally controlled to adjust the amplitude and/or phase of the incident signal, thereby allowing optimized control of the propagation channels [13,14,15]. The capability of RIS to transform the wireless propagation environments which are traditionally considered unmanageable into controllable ones opens a new direction to change the paradigm of wireless communications and sensing with much higher capability and flexibility.
Naturally, RIS is found to be attractive in ISAC applications. For example, RIS can assist the base station (BS) to better communicate and detect communication users (CUs) and targets that are outside of the lineofsight (LOS) region of the BS [16,17,18,19,20,21,22,23]. In [21], the authors describe a simplified RISassisted DFRC system to detect targets in crowded areas, and the output signaltonoise ratio (SNR) of the radar signal is maximized to enhance the target Localization performance while ensuring the output SNR at the CU. It also considers the scenario that the targets are located in the nonlineofsight (NLOS) region of the BS and, thereby, target sensing and localization become much more challenging. It is shown in [19] that utilization of RIS transforms the NLOS targets from the BS into LOS ones from the RIS, thus enabling estimation of the targets’ directionofarrival (DOA) with respect to the RIS. In [16], the authors jointly optimize the transmit information and sensing beamforming at the BS and the reflective beamforming at the RIS to maximize the minimum beampattern gain of the RIS towards the desired sensing angles, subject to the minimum SNR requirement at the CU and the maximum transmit power constraint at the BS.
A conventional RIS system in which all elements are passive does not have the capability of estimating the individual channel state information (CSI) involved with the RIS. Instead, only the cascade CSI is considered, thus bringing inaccuracies and higher complexity [24, 25]. An attractive alternative effective solution is to use a hybrid analog/digital architecture [26,27,28]. By utilizing a small number of active elements in a RIS, it can achieve highprecision CSI estimation and, as a result, significantly facilitates simpler and more effective optimization of the adaptive beamformers and system resources. Compare with the random deployment of RIS active elements, their deployment arranged in welldesigned sparse array configurations provides better performance.
In this paper, we consider a wireless communication system that, assisted by a RIS, implements ISAC functions to deliver information to a CU with multipath propagation channels and to localize targets that have LOS only with the RIS. To provide effective beamforming optimization at both the BS and the RIS and achieve accurate target localization, we exploit a small number of active sensors in the RIS, whose majority of elements are passive, to enable estimation of the timevarying multipath channels between the RIS and the CU as well as that between the RIS and the targets. Modeling the elements in the RIS as a uniform rectangular array (URA), the active sensors are designed to form an Lshaped sparse array with each of the two subarrays in the horizontal (denoted as the xaxis) and elevation (denoted as the zaxis) directions. In order to effectively utilize the small number of active RIS sensors, these sensors should be sparsely placed. Sparse array configurations are known to offer many desirable advantages, such as extending the array aperture and providing a higher number of degreesoffreedom [29,30,31,32]. In particular, we choose the recently developed hybrid optimized nonredundant array (ONRA) structure [33, 34] to deploy the active sparse RIS elements in each subarray because this structure achieves the highest number of unique difference lags and the flexible design capability of the array aperture. By performing sparse array interpolation and pairing between the azimuth and elevation angle estimates, we achieve CSI estimation associated with the CU multipath and targets in the twodimensional (2D) coordinates. The estimated CSI is then used for the optimization of the BS and RIS beamformers for both communication information delivery and target localization objectives. The effectiveness of the proposed technique in terms of the beampattern gain and target localization accuracy is verified using simulation results.
The remainder of this paper is structured as follows. The models of the RISassisted ISAC system and signals are introduced in Sect. 2. In Sect. 3, we provide an interpolation technique for the channel estimation algorithm, an iterative method for maximizing active beamforming at BS and optimizing passive beamforming at RIS while simultaneously maintaining a minimum SNR for the communication user and detecting the targets at RIS from the RIS reflection beams. Section 4 assesses how well the suggested active RIS designs function across various array structures. Finally, conclusions are drawn in Sect. 5.
Notations: We use lowercase (uppercase) bold characters to denote vectors (matrices). In particular, \({{{\textbf {I}}}}_N\) denotes the \(N \times N\) identity matrix. \((\cdot )^*\) denotes complex conjugate, \((.)^{\textrm{T}}\) and \((.)^{\textrm{H}}\) respectively represent the transpose and the conjugate transpose of a matrix or a vector, and \((\cdot )^{\dagger }\) represents MoorePenrose inverse of a matrix. In addition, \(\Vert \cdot \Vert _{\textrm{F}}\) and \(\Vert \cdot \Vert _{{*}}\) respectively denote the Frobenius norm and the nuclear norm of a matrix, \(\otimes\) and \(\odot\) respectively denote the Kronecker product and the Hadamard product, and \({{\mathcal {T}}}({\varvec{x}})\) denotes the Hermitian and Toeplitz matrix with \({\varvec{x}}\) as its first column. We use \(\textrm{diag}({{\textbf {a}}})\) to represent a diagonal matrix that uses the elements of vector \({{\textbf {a}}}\) as its diagonal elements and \(\textrm{tr}(\cdot )\) represents the trace operator. Furthermore, \([{{\textbf {A}}}]_{u,v}\) denotes the (u, v)th element of matrix \({{\textbf {A}}}\), \([{{\textbf {A}}}]_{:,v}\) denotes the vth column of matrix \({{\textbf {A}}}\), and \({{\textbf {A}}} \succcurlyeq 0\) stands for matrix \({{\textbf {A}}}\) to be positive semidefinite. \({\mathbb {E}}[\cdot ]\) is the statistical expectation operator.
2 System and signal models
2.1 System model
Consider a RISassisted ISAC system as shown in Fig. 1a, which consists of an Msensor BS in a vertical uniform linear array (ULA) structure, a CU with a single antenna, an Nelement RIS in a URA structure, and multiple potential target locations at the NLOS areas of the BS. As shown in Fig. 1b, the rectangular RIS has \(N_x\) elements and \(N_z\) elements in the azimuth and elevation dimensions, and the total number of RIS elements is \(N=N_x \times N_z\). The RIS elements are separated by \(d=\lambda /2\) in both dimensions, where \(\lambda\) denotes the signal wavelength.
Among the N RIS elements, \({\bar{N}}\) elements (\({\bar{N}}\ll N\)) are active and are arranged in an Lshaped sparse array structure. We denote \({\bar{N}}_x\) and \({\bar{N}}_z\) as the numbers of active elements in the x and zaxis subarrays, respectively. Because the element in the corner of the Lshape is shared by both subarrays, the total number of active elements in the RIS is \({\bar{N}}={\bar{N}}_x+{\bar{N}}_z1\). The positions of the active elements along the x and the zaxes are represented by \({{\mathbb {X}}} =\{p_{0}, p_{1}, \cdots , p_{{\bar{N}}_x1}\}\lambda /2\) and \({{\mathbb {Z}}}= \{q_{0}, q_{1}, \cdots , q_{{\bar{N}}_z1}\}\lambda /2\), respectively, where \(p_i\) and \(q_i\) are integers for all i, and \(p_0 = q_0 = 0\) is assumed. We also denote \(W_x= p_{{\bar{N}}_x1}+1\) and \(W_z= p_{{\bar{N}}_z1}+1\) as the numbers of active and passive elements included within the respective apertures of the x and zaxis subarrays. In this paper, we consider the hybrid ONRA structure [33, 34] in both subarrays.
The operation of the ISAC system consists of CSI estimation and optimized beamforming for data transmission and target sensing. When performing CSI estimation, the active RIS elements work to sense the signals and perform channel estimation [28, 35]. The DOAs and channel gains associated with the channels are estimated at the BS and the RIS which collectively allow the BS to jointly optimize the BS beamformer and RIS reflecting coefficients under the maximum power constraints. On the other hand, when performing data transmission, all the active and passive elements work as passive reflectors.
2.2 Signal model
In this subsection, we describe the signal models for both communication and target sensing subsystems. The communication subsystem involves the CURIS channel \({{\textbf {h}}}_r \in {\mathbb {C}}^{N \times 1}\), the CUBS channel \({{\textbf {h}}}_d \in {\mathbb {C}}^{M \times 1}\), and the BSRIS channel \({{\textbf {G}}} \in {\mathbb {C}}^{N \times M}\). It is assumed that the channel between the BS and the RIS is fixed and the CSI is known at the BS. On the other hand, the CURIS and CUBS channels are timevarying and need to be periodically estimated. The sensing subsystem involves the same BSRIS channel \({{\textbf {G}}} \in {\mathbb {C}}^{N \times M}\) and the timevarying roundtrip RIStargetsRIS channel \({{\textbf {H}}}_{ t}\in {\mathbb {C}}^{N \times N}\). We assume the timedivision duplexing (TDD) protocol and, by taking advantage of channel reciprocity between the uplink and downlink transmissions, channel estimation is considered only in the uplinks.
We separately describe the channels in four separate parts given below.
2.2.1 CURIS channel
Consider that the signal from a farfield CU impinges on the RIS with \(L_u\) uncorrelated multipath. Denote the 2D DOA of the \(l_u\)th path from the CU and observed at the RIS as \(\{\theta _{l_u}, \phi _{l_u}\}\), \(l_u=1, \cdots , L_u\), where \(\theta _{l_u}\in [{\pi }/2,{\pi }/2]\) and \(\phi _{l_u}\in [{\pi }/2,{\pi }/2]\) are respectively the elevation and the azimuth angles. The 2D steering vector \({{\textbf {a}}}(\theta _{l_u},\phi _{l_u})\) at the RIS is expressed as [36]
where
Since the CU contains a single antenna, the channel of the CURIS link is expressed as
where \(\beta _{l_u}\) is the path gain. The signal vector received at the RIS corresponding to the x and the zaxes are respectively given as [35, 36]
where \({s}_u(t)\) denotes the signal transmitted by the CU, \({{\textbf {s}}}(t) = [\beta _1, \beta _2, \cdots , \beta _{L_u}]^{\textrm{T}} s_u(t)\), \({{\textbf {A}}}_x= [{{\textbf {a}}}_x(\phi _1), {{\textbf {a}}}_x(\phi _2),\cdots ,{{\textbf {a}}}_x(\phi _{L_u})]\), and \({{\textbf {A}}}_z= [{{\textbf {a}}}_z(\theta _1), {{\textbf {a}}}_z(\theta _2),\cdots ,{{\textbf {a}}}_z(\theta _{L_u})]\). In addition, \({{\textbf {n}}}_x(t) \sim {\mathcal{C}\mathcal{N}}(0, \sigma _{\textrm{n}}^2 {{\textbf {I}}}_{N_x})\) and \({{\textbf {n}}}_z(t) \sim {\mathcal{C}\mathcal{N}}(0, \sigma _{\textrm{n}}^2 {{\textbf {I}}}_{N_z})\) are the additive white Gaussian noise (AWGN) vectors observed at the x and zaxis direction elements.
In the sensing mode, the signals are only observed at the two sparse subarrays respectively consisting of \({{{\bar{N}}}}_x\) and \({{{\bar{N}}}}_z\) elements. We define binary mask matrices for the two subarrays, \({{\textbf {U}}}_x \in {\mathbb {C}}^{W_x\times N_x}\) and \({{\textbf {U}}}_z \in {\mathbb {C}}^{W_x\times N_z}\), given as
Then, the observed signal vectors corresponding to the two subarrays become
Note that only \({{{\bar{N}}}}_x\) and \({{{\bar{N}}}}_z\) elements respectively in \({\tilde{{{\textbf {y}}}}_x}(t)\) and \({\tilde{{{\textbf {y}}}}_z}(t)\) corresponding to the active element positions are nonzero.
In Sect. 3, we will use the masked signals observed in the sparse subarrays to interpolate the missing elements within the respective subarray aperture and estimate the signal channels.
2.2.2 CUBS channel
Similar to the CURIS link, we consider that the signal transmitted from the CU arriving at the BS through \(L_d\) multipath. The timevarying channel between the BS and the CU is formulated as
where \(\gamma _{l_d}\) is the \(l_d\)th channel path gain and \({{\textbf {f}}}(\theta _{l_d})\in {\mathbb {C}}^{M \times 1}\) is the array steering vector at the BS for \(l_d=1, \cdots , L_d\). Note that we assume that the M antennas in the BS array are vertically placed so only the elevation components are considered.
2.2.3 BSRIS channel
The channel between the BS, in which the M antennas form a vertically aligned ULA, and the RIS is assumed to be fixed and thus is known at the BS. This channel between the BS and the RIS can be decomposed into \(L_b \le \textrm{min}(M,N)\) independent paths, given as
where, for the \(l_b\)th path, \(\alpha _b\) denotes the gain, \({{\textbf {a}}}(\theta _{l_r},\phi _{l_r}) = {{\textbf {a}}}_z(\theta _{l_r})\otimes {{\textbf {a}}}_x(\phi _{l_r})\in {\mathbb {C}}^{N\times 1}\) is the corresponding steering vector at the RIS, \({{\textbf {f}}}^{\textrm{H}}(\theta _{l_b})\in {\mathbb {C}}^{M\times 1}\) is the steering vector at the BS, and \({\varvec{\alpha }}_{{\textrm{BI}}} = [\alpha _1,\alpha _2,\cdots ,\alpha _{L_b}]^{\textrm{T}}\). The channel is decomposed into \(L_b\) independent paths with \(\alpha _b\) denoting the gain of the \(l_b\)th path.
2.2.4 RIStargetRIS channel
To illustrate the offerings of RIS in providing localization of targets in the NLOS region of the BS, we consider potential target locations which have LOS to the RIS but not the BS. As such, the target locations are estimated based on signals observed at the RIS.
The RIS reflects the impinging signals from the BS to the targets and receives the signal reflected from the target. We assume \(L_t\) targets in the scene. The direction of the \(l_t\)th target observed at the RIS is denoted as \((\theta _{l_t},\phi _{l_t})\), which is shared for both the outgoing RIStarget path and the returning targetRIS path. The roundtrip channel of the RIStargetRIS link is modeled as
where \(\delta _{l_t}\) combines the roundtrip path gain and the target radar cross section (RCS), \({{\textbf {b}}}(\theta _{l_t},\phi _{l_t})= {{\textbf {b}}}_z(\theta _{l_t})\otimes {{\textbf {b}}}_x(\phi _{l_t}) \in {\mathbb {C}}^{N\times 1}\) denotes the array steering vector of the RIS from the target, and \({\varvec{\delta }}_{{\textrm{TI}}} = [\delta _1,\delta _2,\cdots ,\delta _{L_t}]^{\textrm{T}}\) corresponding to the \(l_t\)th target for \(l_t = 1, \cdots , L_t\).
Similar to the CURIS channel considered in Sect. 2.2.1, the RIS only observes masked signals at the sparsely placed active element positions. The observed channel matrix corresponding to the signal received on the zaxis subarray denoted as the RIStargetsRIS(z) link, is described as
where only \({{{\bar{N}}}}_z\) rows are nonzero. Similarly, the RIStargetsRIS(x) link is expressed as
where only \({{{\bar{N}}}}_x\) rows are nonzero.
3 Channel estimation, joint beamforming, and target localization
In this section, we consider the channel estimation, joint beamforming at the BS and the RIS, and the target localization in terms of 2D DOA estimation. These objectives are addressed in the following three phases. The first phase estimates the timevarying channel between the CU and the RIS using the Lshaped sparse subarrays at the RIS. In the second phase, we maximize the minimum beampattern gain of the RIS towards the desired sensing directions while ensuring the minimum SNR requirement at the CU under the maximum transmit power constraint at the BS. The objective of the third phase is to determine the target 2D DOAs by the sparse active elements at the RIS.
3.1 Phase I: CURIS channel estimation
In the first phase, the signal vectors \({\tilde{{{\textbf {y}}}}}_x(t)\) and \({\tilde{{{\textbf {y}}}}}_z(t)\) observed at the two active subarrays of the RIS are used to estimate the uplink multipath channels between the CU and the RIS. The interpolation technique is applied to obtain the full covariance matrices of vectors \({{{\textbf {y}}}}_x\) and \({{{\textbf {y}}}}_z\) corresponding to all elements spanned by the active subarray aperture, namely, for elements located at positions \(p_0, p_0+1, \cdots , W_x 1\) and \(q_0, q_0+1, \cdots , W_z 1\).
3.1.1 Covariance matrix interpolation
Assuming that the noise is uncorrelated to the signals, the covariance matrices of \({\tilde{{{\textbf {y}}}}}_x(t)\) is expressed as:
where \({{\textbf {R}}}_s= \textrm{diag}(\sigma _1^2, \sigma _2^2, \cdots , \sigma _{L_u}^2)\) is the source covariance matrix, \(\sigma _{l_u}^2 = \beta _{l_u}^2 \sigma _u^2\) represents the power of the \(l_u\)th path signal, \(\sigma _u^2 = {{\mathbb {E}}}({s_u(t)^2})\) is the source signal power. Because of the sparse placement of the active RIS elements, the covariance matrices \({\tilde{{{\textbf {R}}}}}_{x}\) contain missing holes. We exploit the matrix interpolation of \({\tilde{{{\textbf {R}}}}}_{x}\) to obtain an estimate of the interpolated covariance matrices \({{{\textbf {R}}}}_{x} \in {{\mathbb {C}}}^{W_x \times W_x}\).
The matrix interpolation for the xaxis subarray is formulated as the following nuclear norm minimization problem [37]:
where \({{\mathcal {T}}}({\varvec{w}}) \in {{\mathbb {C}}}^{W_x \times W_x}\) denotes the Hermitian and Toeplitz matrix with \({\varvec{w}} \in {{\mathbb {C}}}^{W_x \times 1}\) as its first column, \(\Vert {{\mathcal {T}}}({\varvec{w}}) \Vert _* = \text {tr}(\sqrt{{{\mathcal {T}}}^{\textrm{H}}({\varvec{w}}){{\mathcal {T}}}({\varvec{w}})})\) is the nuclear norm of \({{\mathcal {T}}}({\varvec{w}})\), and \(\zeta\) is a tunable regularization parameter. In addition, \({{\textbf {Q}}}_x = {{\textbf {U}}}_x{{\textbf {U}}}_x^{\textrm{T}}\) is the binary mask of the sparse covariance matrix. The obtained \({{\mathcal {T}}}({\varvec{w}})\) becomes the estimate of \({{{\textbf {R}}}}_{x}\), denotes as \({\hat{{{\textbf {R}}}}}_{x}\). We can similarly perform matrix interpolation at the z axis and obtain the estimate of the interpolated covariance matrix \({{\textbf {R}}}_{z}\), denoted as \({\hat{{{\textbf {R}}}}}_{z}\).
As the interpolated covariance matrices are full rank, subspacebased methods, such as multiple signal classification (MUSIC), can be applied to \({\hat{{{\textbf {R}}}}}_{x}\) to obtain the azimuth DOAs of the multipath signals. The elevation angles must be paired with the estimated azimuth angles and their estimation is discussed below.
3.1.2 Pairmatched 2D DOA estimation
When there are multiple paths between the CU and the RIS, it is important to determine the correct pairing between the estimated azimuth and elevation angles. The array manifold matrix corresponding to the estimated azimuth angles \({{\hat{\phi }}}_1, {{\hat{\phi }}}_2, \cdots , {{\hat{\phi }}}_{L_u}\) can be constructed as \({\hat{{{\textbf {A}}}}}_x= [{{\textbf {a}}}_x({\hat{\phi }}_1), {{\textbf {a}}}_x({\hat{\phi }}_2),\cdots ,{{\textbf {a}}}_x({\hat{\phi }}_{L_u})]\). We will estimate the manifold matrix \({{\textbf {A}}}_z\) of the zaxis subarray from the following crosscovariance matrix between \({\tilde{{{\textbf {y}}}}}_x(t)\) and \({\tilde{{{\textbf {y}}}}}_z(t)\):
Performing the eigendecomposition of the covariance matrices \({\hat{{{\textbf {R}}}}}_{x}\) and \({\hat{{{\textbf {R}}}}}_{z}\) yields in
and
where \({\hat{{{\textbf {V}}}}}_{xs}\) and \({\hat{{{\textbf {V}}}}}_{zs}\) denote the estimated signal subspaces for the two linear arrays, whereas \({\hat{{\varvec{\Lambda }}}}_{xs}\) and \({\hat{{\varvec{\Lambda }}}}_{zs}\) are the diagonal matrices containing the eigenvalues corresponding to the signal subspaces. As the position of the active elements for x and zaxis are identical, we can write \(W_x = W_z\) and \({{\textbf {I}}}\) \(= {{\textbf {I}}}_{W_x} = {{\textbf {I}}}_{W_z} \in {\mathbb {C}}^{W_x \times W_x}\). Exploiting the relationship between the components spanning the signal subspace in the above formulations, \({{\textbf {R}}}_{{{\textbf {s}}}}\) can be estimated as [38]
Thus, an estimate of the array manifold matrix, \({\hat{{{\textbf {A}}}}}_{z}= [{{\textbf {a}}}_{z}{({\hat{\theta }}_1)}, {{\textbf {a}}}_{z}{({\hat{\theta }}_2)}, \cdots , {{\textbf {a}}}_{z}{({{\hat{\theta }}}_{L_u})}] \in {\mathbb {C}}^{W_z \times L_u}\), can be obtained from (16) as
Note that matrix \({\tilde{{{\textbf {R}}}}}_{xz}\) is not a Hermitian and Toeplitz matrix and thus cannot be directly interpolated using interpolation techniques utilizing such properties. Instead, we indirectly obtain the interpolated \({{\textbf {R}}}_{xz}\) from the following operations:
As a result, we can rewrite (20) as
For the \(l_u\)th path, \(l_u = 1,2, \cdots , L_u\), the elevation angle can be estimated as
From the paired azimuth and elevation angle estimation, we can generate the array manifold matrix for the CURIS channel as \({\hat{{{\textbf {A}}}}} = \left[ {{\textbf {a}}}( {{\hat{\theta }}}_{1},{{\hat{\phi }}}_{1}), \cdots , {{\textbf {a}}}({{\hat{\theta }}}_{L_u}, {{\hat{\phi }}}_{L_u}) \right] \in {\mathbb {C}}^{N \times L_u}\).
3.1.3 Path gain estimation
Because the path gains are identical for the x and zaxis subarrays, computation in one of these two subarrays will suffice. To estimate the path gain of the CURIS channel, the CU transmits pilot signal \(s_u(t)\) to the RIS. At the RIS, the received signal at the xaxis elements is given as
where \({{\textbf {g}}} = [\beta _{1}, \beta _{2}, \cdots , \beta _{L_u}]^{\textrm{T}}\) represents the path gains and can be estimated from
where \(\bar{{{\textbf {y}}}}_{x} ={\mathbb {E}}\{{{{\textbf {y}}}}_{x}(t) s_u^*(t)\}\). From the above DOAs and path gain estimation, we can reconstruct the CURIS multipath channel \({{\textbf {h}}}_r\).
Similarly, we can estimate the CUBS channel at the BS which, however, does not require interpolation because all BS antennas are active. The normalized rootmeansquare error (RMSE) of the CURIS channel at the RIS is estimated as
where \(K_\mathrm{{cu}}\) is the number of independent trials.
3.2 Phase II: joint BS and RIS beamforming optimization
In phase II, our objective is to maximize the minimum beampattern gain of the RIS towards the desired sensing angles, while ensuring the minimum SNR requirement at the CU under the maximum transmit power constraint at the BS. Let \({{\textbf {v}}} = [e^{j\psi _1},\cdots , e^{j\psi _N}]^{\textrm{T}} \in {\mathbb {C}}^{N\times 1}\) denote the reflective phase shift vector at the RIS where \(\varvec{\Phi }= \text {diag}({{\textbf {v}}})\). By combining the signals transmitted through the direct BSCU link and the reflected BSRISCU link, the received signal at the CU is expressed as
where \({{\textbf {w}}}\in {\mathbb {C}}^{M\times 1}\) denotes the transmit beamforming vector at the BS, \(\textrm{s}(t) \sim \mathcal{C}\mathcal{N}(0,\sigma _{{\textrm{s,BS}}}^2)\) is the transmitted random symbol, and \(n_c(t) \sim \mathcal{C}\mathcal{N}(0,\sigma _{{\textrm{n,CU}}}^2)\) represents the AWGN at the CU. The received SNR at the CU is computed as
Next, we consider the radar sensing towards the potential target locations which are assumed to be at the NLOS areas of the BS. In this case, we use the virtual LOS links created by the RIS reflection to sense the targets. The beampattern gain of the RIS towards the desired sensing angles are used as the sensing performance metric. The beampattern gain from the RIS towards the target angel \(({\theta _{l_t}, \phi _{l_t}})\) is given as
We are interested in sensing the prospective targets at \(L_t\) directions observed at the RIS. To achieve the aforementioned objective, i.e., maximizing the minimum beampattern gain at these \(L_t\) angles while ensuring the minimum SNR requirement at the CU under the maximum transmit power constraint at the BS, we formulate the following SNRconstrained minimum beampattern gain maximization problem,
where \({\mathcal {L}} =\{1, 2, \cdots , L_t\}\), \(P_0\) is the maximum power allowed at the BS, and \(\Gamma\) is the required SNR by the CU. \(v_n\) is the nth element of \({{\textbf {v}}}\) and the constraint \(v_n =1\) ensures that the RIS weights are unit modulus, thereby achieving phaseonly beamforming which is desired for convenience RIS implementations [39]. This problem is highly nonconvex and thus is difficult to be directly solved. It can be solved, however, by using the techniques of alternating optimization and semidefinite relaxation (SDR) [16]. The BS beamforming weight vector \({{\textbf {w}}}\) is first optimized with an initial value of the \(\varvec{\Phi }\) and then optimize the RIS reflecting beamforming matrix \(\varvec{\Phi }\) with optimized \({{\textbf {w}}}\). These procedures are described in the following two subsections.
3.2.1 Transmit beamforming optimization at BS
First, we optimize the transmit beamformer \({{\textbf {w}}}\) in the above problem under a presumed reflective beamformer \({\varvec{\Phi }}\). This problem is formulated as
To perform SDR, we introduce \({{\textbf {W}}} = {{\textbf {w}}}{{\textbf {w}}}^{\textrm{H}}\) with \({{\textbf {W}}}\succeq 0\) and \({\text {rank}{({{\textbf {W}}})} = 1}\). Let \({{\textbf {h}}} = {{\textbf {G}}}^{\textrm{H}}{\varvec{\Phi }}^{\textrm{H}} {{\textbf {h}}}_{r} + {{\textbf {h}}}_d\) denote the combined channel vector from the BS to the CU accounting for both BSCU and BSIRACU channels. Then, the transmit beamforming optimization in problem (31) is reformulated as,
However, this problem is still nonconvex due to the rankone constraint on \({{\textbf {W}}}\) in (32e). Relaxing this rankone constraint renders the SDR version of the problem (32), expressed as
This problem is a semidefinite programming (SDP) that can be efficiently solved using convex solvers, such as CVX. Once the optimal solution of \({{\textbf {W}}}\) is solved from (32) and denoted as \({{\textbf {W}}}^{{\textrm{opt}}}\), the optimized transit beamforming vector at the BS, \({{{\textbf {w}}}}\), is obtained as [16]
3.2.2 Reflective beamforming optimization at RIS
Next, we optimize the reflective beamformer \({\varvec{\Phi }}\) in problem (30) when the transmit beamformer of the BS is obtained from (33) and (34). Then, the sensing beampattern gain from the RIS towards angle \(\{ \theta _{l_t}, \phi _{l_t}\}\) is given as
where
Define
Then, by substituting (37) into (35), we have \(\rho (\theta _{l_t},\phi _{l_t})= {\bar{{\textbf {v}}}}^{\textrm{H}}{{\textbf {R}}}_2(\theta _{l_t},\phi _{l_t}){\bar{{\textbf {v}}}}\).
Let \({{\textbf {H}}} = \text {diag}({{\textbf {h}}}_r^{\textrm{H}}){{\textbf {G}}} \in {\mathbb {C}}^{N\times M}\). The received signal power at the CU can be written as \(({{\textbf {h}}}_r^{\textrm{H}} {\varvec{\Phi }} {{\textbf {G}}}+ {{\textbf {h}}}_d^{\textrm{H}}){{\textbf {w}}}^2= ({{\textbf {v}}}^{\textrm{H}}{{\textbf {H}}}+{{\textbf {h}}}_d^{\textrm{H}}){{\textbf {w}}}^2\). Then, the output SNR constraint in (30b) is reformulated as,
which is equivalent to
with
As a result, the optimization of \(\varvec{\Phi }\) in problem (30) becomes the optimization of \(\bar{{\textbf {v}}}\) in the following problem:
To solve \({\bar{{\textbf {v}}}}\) using SDR, we define \(\bar{{\textbf {V}}}= {\bar{{\textbf {v}}}\bar{{\textbf {v}}}^{\textrm{H}}}\) with \({\bar{{\textbf {V}}}}\succeq 0\) and \(\text {diag}{(\bar{{\textbf {V}}})}=1\). Noting \({\bar{{\textbf {v}}}}^{\textrm{H}}{{\textbf {R}}}_2(\theta _{l_t},\phi _{l_t}){\bar{{\textbf {v}}}}= \text {tr}({{\textbf {R}}}_2(\theta _{l_t},\phi _{l_t}){\bar{{\textbf {V}}}})\) and \({\bar{{\textbf {v}}}}^{\textrm{H}}{{\textbf {R}}}_3{\bar{{\textbf {v}}}}= \text {tr}({{\textbf {R}}}_3{\bar{{\textbf {V}}}) }\), the reflective beamforming optimization in problem (41) is reformulated as
Similarly, we relax the rankone constraint and accordingly obtain the SDR version of the problem (42) as
Let \(\bar{{\textbf {V}}}^{\textrm{opt}}\) denote the obtained optimal solution to \({\bar{{\textbf {V}}}}\) from problem (43). When it has a high rank, Gaussian randomization can be used to construct an approximate rankone solution. At any iteration, if the updated gain is higher than the prior gain, the targets’ beampattern gain and \({{\textbf {v}}}\) are updated at the same time. If not, \({{\textbf {v}}}\) is again randomly generated.
It is noted that, with a sufficient number of randomizations, the objective value after solving the problem (42) will be monotonically nondecreasing [15]. As a result, the convergence of the proposed alternating optimizationbased algorithm for solving the problem (30) is ensured.
3.3 Phase III: NLOS target localization
In this section, using the reflected RIS signal, we estimate the target 2D DOA. The RIS has a sparse placement of the Lshaped active elements to allow for independent determination of the targets’ azimuth and elevation angles. Here, we demonstrate the estimation of the elevation angles of the targets using the RIS zaxis subarray. The signals received at the azimuth subarray can be similarly formulated and the azimuth angles of the targets can be similarly computed using the matrix interpolation and pairmatched MUSIC processes.
The echo channel reflected from the target over the RISreflected link, denoted as BSRIStargetsRIS(z), is given by
As such, the received signals at the vertical RIS subarray sensors are given by
For the estimation of the target directions, the received signal from the BS, \({{\textbf {Gw}}}s(t)\) in the above expression, act as interference. Because \({{\textbf {G}}}\) is fixed and known, we can remove this component and obtain the interferencefree target echo signal as
As the active elements on the RIS are in a sparse structure, similar to (15), we can perform matrix completion to fill in the missing elements and generate the interpolated covariance matrix \({{\hat{{{\textbf {R}}}}}_Z}\). \({{\textbf {B}}}_t\) at RIS can be obtained by following the same procedures of the pairmatched MUSIC algorithm and gain estimation as described in Sect. 3.1.1.
The normalized RMSE of the target response at the RIS is estimated as
where \(K_t\) is the number of independent trials.
4 Simulation results
In this section, we provide simulation results to demonstrate the effectiveness of the proposed active RISassisted ISAC scheme and the superiority of the Lshaped hybrid ONRAbased sparse array structure over other Lshaped structures. The RIS contains \(N_x = N_z = 23\) elements in each dimension, rendering a total number of \(N = 529\) elements.
We use a small number of \({{\bar{N}}}=11\) active elements in all Lshaped sparse arrays and examine the channel estimation and target DOA estimation performance. Figure 2 shows the structures of different sparse active subarrays in a single axis. Each array uses 6 antenna elements. Both the x and zaxis directions use the same subarrays and they together form Lshape sparse active arrays. Note that the two subarrays share the element located at the 0th position. For the hybrid ONRA configuration, the positions of the active elements along the x and the z axes are \({{\mathbb {X}}} = {{\mathbb {Z}}} = \{0,3,7,12,20,22\}\lambda /2\). It yields 16 nonnegative lags located as \({{\mathbb {D}}}_{\textrm{self}}^{X} = {{\mathbb {D}}}_{\textrm{self}}^{Z} = \{0,2,3,4,5,7,8,9,10,12,13,15,17,19,20,22\}\lambda /2\). It achieves the largest array aperture which coincides with the size of the RIS, rendering \(W_x = N_x = W_y = N_y = 23\) in this case.
For comparison, we compare the performance of the Lshaped hybrid ONRAbased sparse array structure with Lshaped arrays consisting of orthogonally placed ULA structures and popularly used sparse array structures, including the coprime array [30], nested array [29], coprime array with minimum lag redundancies (CAMLR) [40], and super nested array (SNA) [41]. These five sparse array configurations being compared respectively achieve 5, 9, 12, 12, and 12 unique nonnegative coarray lags and have smaller array apertures as shown in Fig. 2 [34]. The transmit array of the BS is equipped with a vertically placed ULA with \(M = 4\) antennas.
The largescale path loss for a CURIS path with distance r is given as \(\textrm{PL}(r)[\text {dB}]= 10\ \text {log}_{10}(4\pi f_c/c)^2+10 \, \alpha \, \text {log}_{10}(r/r_0)\), where \(f_c\), \(\alpha\), and \(r_0\) are the carrier frequency, the path loss exponent, and the reference distance, respectively [42]. In this paper, \(f_c=28\) GHz and \(r_0=1\ \text {m}\) are assumed.
The BS and the RIS have a fixed distance of \(d_{\text {BI}}=10.2\) m, and the distance between the RIS and the CU is \(d_{\text {IC}}=120\) m. There are two targets and their 2D angels with respect to the RIS are respectively [\(10^\circ , 30^\circ\)] and [\(30^\circ , 10^\circ\)]. The twoway complexvalued path gain is \(\delta _{l_t} = \sqrt{{\lambda ^2k}/({64\pi ^3d_{\textrm{IT}}^4})}\) denotes the signal attenuation caused by the propagation from RIS to the target and then from the target to RIS. We assumed that both targets are equally distant from the RIS, where \(d_{\text {IT}}=10.2\) m is the distance between the RIS and target, and \(k = 7\) dBm denotes the radar crosssection.
For the CU, we assume \(L_u= 2\) paths for the CURIS channel \({{\textbf {h}}}_r\).
We consider two distinct scenarios. In the first scenario, we intend to evaluate how well the different array structures would function, and the multipath is chosen to be closely spaced with the LOS and their DOA separation observed at the RIS is only \(2^{\circ }\) in both azimuth and elevation. In the second scenario, we consider well separated multipath whose azimuth and elevation angles both deviate from the LOS by \(20^{\circ }\). The near and far spacing between this multipath can be used to determine the performance advantages of the hybrid ONRA array construction.
Figure 3 considers the first case with closely spaced multipath and compares the normalized RMSE performance of the estimated channels with respect to the number of snapshots for different array configurations. In this case, the CU’s transmit power is 10 dBm, whereas the noise power is \(80\) dBm. The outcomes clearly demonstrate that the RIS with Lshaped hybrid ONRA active elements achieves noticeably higher channel estimation accuracy than other sparse array structures utilizing the same number of active elements (\({\bar{N}}=11\)). In Fig. 4, we increase the transmit power from the CU to 20 dBm, and it is observed that the RMSE performance of all array configurations improves due to the increased transmit power.
In Fig. 5, we consider the second scenario with well separated multipath signals. With a \(20^{\circ }\) separation in both azimuth and elevation, the normalized RMSE of all sparse structures is much lower and the differences between different array configurations are smaller.
Figure 6 depicts the CURIS channel estimation results for different angular separations between the CUIRS multipath, where the transmit power from the CU is 10 dBm and the number of snapshots is 5000. When the difference between the two azimuth angles and the two elevation angles was only \(0.1^\circ\), the channel estimation accuracy was very poor for all array configurations in this figure. For all array designs, channel estimation performance improves with increasing multipath separation, but for ONRA structures, channel estimation performance is exceptionally accurate even with a \(1^\circ\) separation when compared to other structures. This figure further highlights the superior estimate performance of the hybrid ONRA over the other array architectures.
Figure 7 depicts the normalized RMSE performance of the estimated CURIS channels as the transmit power varies between 0 dBm and 30 dBm, where the number of snapshots is fixed to 5000, and the two paths are closely spaced with \(2^{\circ }\) separation in both azimuth and elevation. It is observed that the normalized RMSE generally reduces as the transmit power increases and, for this closely spaced multipath scenario, the hybrid ONRAbased active RIS configuration performs much better than other sparse array structures.
In Fig. 8, we compare the normalized RMSE of the estimated CURIS channel versus the separation between the CU and the IRS. The number of snapshots is 5000, the transmit power is 10 dBm, and the angular separation is \(2^{\circ }\) in both azimuth and elevation. It is observed that, as the distance increases, the signal is more attenuated, yielding higher normalized RMSE results. The ONRA structure consistently outperforms the other array configurations.
Figure 9 shows the convergence performance of the minimum beampattern gain obtained in phase II based on the alternating optimizationbased algorithm. Here, the transmit power from BS is 40 dBm, the noise power is \(80\) dBm, and \(\Gamma = 10\) dB is assumed. Because the convergence performance depends on randomization, the plotted results are obtained by averaging over 5 independent trials. For comparison, we also plotted the result with true CSI of the BSRIS channel, BSCU channel, and CURIS channel to be perfectly known at the BS during optimization. We can observe that most of the array configurations achieve convergence in 6–8 iterations. Sparse subarrays using the ONRA structure achieve the highest minimum beampattern gain which is very close to that obtained under the assumption that accurate CSI is known.
In phase III, we determine the target reflected signals. Figure 10 demonstrates the normalized RMSE performance versus the number of snapshots, where the transmit power from the BS is 40 dBm and the noise power is \(109\) dBm. It is again verified that the hybrid ONRA structure offers better performance than other sparse array configurations.
5 Conclusion
In this paper, we presented an ISAC system assisted by a RIS with partial active elements. Lshaped sparse arrays are used for channel estimation and target DOA estimation. A nuclear normbased interpolation technique was used to fill in the gaps in the covariance matrix and achieve improved channel estimation and target DOA estimation performance. The array and RIS gains are optimized to concurrently maintain a specific SNR requirement toward the communication user and beampattern gains toward the targets which are assumed to have LOS only with the RIS. Simulation findings demonstrate that our proposed hybrid ONRA RIS structure performs better than other contemporary RIS structures.
Availability of data and materials
Data sharing is not applicable to this study.
Abbreviations
 2D:

Twodimensional
 AWGN:

Additive white Gaussian noise
 BS:

Base station
 CAMLR:

Coprime array with minimum lag redundancy
 CSI:

Channel state information
 CU:

Communication user
 DOA:

Directionofarrival
 DOD:

Directionofdeparture
 DRFC:

Dualfunction radarcommunications
 RIS:

Reconfigurable intelligent surface
 ISAC:

Integrated sensing and communication
 LOS:

Lineofsight
 MUSIC:

Multiple signal classification
 NLOS:

Nonlineofsight
 ONRA:

Optimized nonredundant array
 RMSE:

Rootmeansquare error
 SDP:

Semidefinite programming
 SDR:

Semidefinite relaxation
 SNA:

Super nested array
 SNR:

Signaltonoise ratio
 ULA:

Uniform linear array
 URA:

Uniform rectangular array
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Mirza Asif Haider conceived of the presented idea, carried out the simulations, and prepared the original draft; Yimin D. Zhang verified the methods and edited the manuscript; Elias Aboutanios verified the methods and edited the manuscript. All authors discussed the results, contributed to the manuscript, and approved the final manuscript.
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Asif Haider, M., Zhang, Y.D. & Aboutanios, E. ISAC system assisted by RIS with sparse active elements. EURASIP J. Adv. Signal Process. 2023, 20 (2023). https://doi.org/10.1186/s13634023009775
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DOI: https://doi.org/10.1186/s13634023009775
Keywords
 Intelligent reflecting surface
 Integrated sensing and communications
 Sparse array
 Structured matrix completion
 Directionofarrival estimation