Atomic Norm Minimization-based DoA Estimation for IRS-assisted Sensing Systems

The future sixth-generation (6G) wireless networks are anticipated to require high-resolution and high-accuracy sensing to support environment-aware applications such as autonomous driving [1]. However, current cellular base station (BS)-based sensing encounters challenges, including dead zones for sparsely deployed BSs, blockage issue for obstructed obstacles, and severe path loss for round-trip propagation in millimeter-wave systems.

To address these challenges, intelligent reflecting surface (IRS) is regarded as a promising technology[2]. Generally, IRS is composed of many passive reflecting elements (REs) which can independently impose phase shifts on the incident signals. It can create a non-line-of-sight (NLoS) link for dead zones or blocked areas, thus ensuring uninterrupted sensing. Additionally, IRS can provide high beamforming gain due to its large aperture, thus extending the sensing coverage. Consequently, IRS-aided sensing systems are widely studied [3, 4, 5]. For instance, the authors in [5] design the measurement matrix of IRS and adopt an efficient method for target angle estimation.

However, the severe path loss of the IRS-aided sensing systems limits their application dramatically. The path loss of the BS-IRS-target-IRS-BS link is much higher than that of the conventional BS-target-BS link. Therefore, semi-passive IRS, composed of passive REs and active sensing elements (SEs), is proposed to address the path loss issue [6]. IRS SEs are employed to collect echo signals reflected from targets and directly sense them, thus reducing path loss by avoiding the IRS-BS link. Some researchers have explored semi-passive IRS-aided sensing systems [7, 8, 9]. The authors in [7] compare the detection signal-to-noise ratio and estimation Cramér-Rao bound (CRB) performance of fully-passive and semi-passive IRS-enabled sensing systems, and reveal the outstanding performance of semi-passive IRS except for scenarios with an extremely large number of REs. The authors in [8] propose a new beam scanning-based protocol for simultaneously target sensing and user communication. The authors in [9] consider a joint time-of-arrival and direction-of-arrival (DoA) estimation problem, and adopt the multiple signal classification (MUSIC) method for DoA estimation. However, IRS not only establishes a NLoS link for obstructed targets, but also measures targets with IRS phase offsets. The MUSIC method relies solely on the receive antenna structure, and fails to fully exploit the antenna resources in the semi-passive IRS-aided sensing system. More specifically, in addition to SEs, the target angle information is also embedded in IRS REs, a factor not considered in [9]. Therefore, the DoA estimation performance can be significantly enhanced by further leveraging the DoA information from IRS REs.

Based upon the background above, we address a multi-target DoA estimation problem in a semi-passive IRS-assisted sensing system as illustrated in Fig. 1. First, in order to fully exploit the DoA information in both IRS REs and SEs matrices, a reconstruction problem is proposed by utilizing the atomic norm minimization (ANM) method to reconstruct a sparse signal composed of finite atoms from the received echo signal. The original ANM problem is intractable and is thus transformed into a convex form according to the Lagrangian dual theorem. Next, the CRB for DoA estimation is derived to characterize the estimation performance. When the covariance matrix of the IRS measurement matrix is an identity matrix, the CRB is inversely proportional to $MN^{3}+NM^{3}$ for the single-target case, where $M$ and $N$ are the numbers of IRS SEs and REs, respectively. Finally, extensive numerical results validate the superior accuracy and resolution performance of the proposed ANM-based DoA estimation method over the MUSIC method. Furthermore, when the measurement matrix of IRS REs is configured as a discrete Fourier transform (DFT) matrix, the mean square error (MSE) of the proposed method closely approximates the CRB.

We consider a localization system with the assistance of a semi-passive IRS as illustrated in Fig. 1, where an $N_{b}$-antenna BS attempts to detect the angles of $K$ targets. The direct links between the BS and targets are assumed to be blocked due to an unfavorable propagation environment. Thus, an IRS is deployed to establish NLoS links for target angle estimation. Here, a semi-passive IRS consisting of $N$ passive REs and $M$ active SEs is adopted, which offers better sensing performance compared to the fully-passive IRS [6]. We consider a quasi-static flat-fading channel within the considered time slots.

In order to estimate the target angles, the MUSIC method is employed in [9]. However, the MUSIC method is solely dependent on the receive antenna structure, which means that it only utilizes the information of $\mathbf{B}(\bm{\theta})$ for DoA estimation. In the semi-passive IRS-assisted sensing model, DoA information is embedded in both the IRS SEs matrix $\mathbf{B}(\bm{\theta})$ and the REs matrix $\mathbf{Q}(\bm{\theta})$, marking a significant departure from the conventional sensing model where DoA information is typically confined to the receive antenna matrix $\mathbf{B}(\bm{\theta})$. Therefore, an ANM-based DoA estimation method is proposed in this work to fully exploit $\mathbf{B}(\bm{\theta})$ and $\mathbf{Q}(\bm{\theta})$.

ANM is a well-known infinite-size dictionary based compressive sensing approach, which is widely used in the DoA estimation problem [4, 5]. For a matrix $\mathbf{X}\in\mathbb{C}^{M\times N}$, the atomic norm of $\mathbf{X}$ is defined as

where $c_{i}\in\mathbb{R}$ describes the atomic decomposition and $\varphi_{i}$ denotes the corresponding phase. The received signal $\mathbf{Y}$ is composed of $K$ atoms of $\mathbf{b}(\theta_{k})\mathbf{a}_{r}^{H}(\overline{\theta}_{k})$, and is thus sparse in the domain constructed by the atoms $\mathbf{b}(\theta_{i})\mathbf{a}_{r}^{H}(\overline{\theta}_{i})$. Consequently, we aim to reconstruct a sparse signal $\mathbf{X}$ from the received echo signal $\mathbf{Y}$, yielding the following reconstruction problem

where $\beta$ is a parameter balancing the sparsity and reconstruction performance, and $\|\mathbf{X}\|_{\mathcal{A}}$ quantifies the sparsity of $\mathbf{X}$. The problem (8) is intractable and challenging to solve. Therefore, we transform it into a tractable problem below.

First, the original problem (8) can be rewritten as

By introducing a Lagrangian parameter $\mathbf{G}\in\mathbb{C}^{M\times N}$, the corresponding Lagrangian function is given by

where $\langle\mathbf{X}-\mathbf{Z},\mathbf{G}\rangle\triangleq\Re\left\{\operatorname{Tr}\left(\mathbf{G}^{H}(\mathbf{X}-\mathbf{Z})\right)\right\}$. Thereby, the dual function can be obtained as

where $\|\mathbf{G}\|_{\tilde{\mathcal{A}}}$ is the dual norm of the atomic norm, defined as $\|\mathbf{G}\|_{\tilde{\mathcal{A}}}\triangleq\sup_{\|\mathbf{U}\|_{\mathcal{A}}\leq 1}\langle\mathbf{U},\mathbf{G}\rangle$, and $I_{S}(x)$ is an indicator function that equals zero if $x\in S$ and infinity otherwise. Hence, the optimization problem (9) can be transformed into

The dual norm $\|\mathbf{G}\|_{\tilde{\mathcal{A}}}$ can be simplified to

Next, we construct a Hermitian matrix $\begin{bmatrix}\mathbf{W}&\mathbf{G}\\ \mathbf{G}^{H}&\varrho\mathbf{I}_{N}\end{bmatrix}$, where $\varrho\in\mathbb{R}$ is a hyperparameter and $\mathbf{W}\in\mathbb{C}^{M\times M}$ is a Hermitian matrix. This matrix is semi-definite positive, i.e.,

if and only if

Therefore, for any given matrix $\bm{\Psi}\in\mathbb{C}^{M\times N}$, we have

yielding

Let $\bm{\Psi}=\mathbf{b}(\theta)\mathbf{a}_{r}^{H}(\overline{\theta})$, we have

For the matrix $\mathbf{W}$, if we have

the expression (III) indicates that

Finally, the original problem (8) can be transformed into

The problem above is convex and can be efficiently solved by convex optimization toolbox, such as the CVX toolbox. The computational complexity primarily depends on the size of the semi-definite positive matrix in (14), resulting in an approximated complexity of $\mathcal{O}\left((N+M)^{3.5}\right)$ [12]. Denoting the optimal solution in (23) as $\hat{\mathbf{G}}$, the DoAs can be obtained by identifying the $K$ largest peak values of $f(\theta)$ defined below according to the dual constraint in (III),

The CRB serves as a lower bound on the variance of any unbiased estimator. In this section, we would like to explore the performance limitation of DoA estimation based on the IRS-assisted sensing model in (II-B).

Let $\bm{\xi}\triangleq[\theta_{1},\ldots,\theta_{K},\overline{\alpha}_{1},\ldots,\overline{\alpha}_{K}]^{T}\in\mathbb{R}^{3K\times 1}$ denote the vector of the unknown targets’ parameters, where $\overline{\bm{\alpha}}_{k}\triangleq[\Re\{\alpha_{k}\},\Im\{\alpha_{k}\}]$. By vectorizing (II-B), we have

where $\bm{\Upsilon}=\mathbf{B}(\bm{\theta})\bm{\Lambda}\mathbf{Q}^{H}(\bm{\theta})\mathbf{D}$. Let $\mathbf{F}\in\mathbb{R}^{3K\times 3K}$ denote the Fisher information matrix for estimating $\bm{\xi}$. Since $\operatorname{vec}(\mathbf{Y})\sim\mathcal{CN}(\operatorname{vec}(\bm{\Upsilon}),\sigma^{2}\mathbf{I}_{ML})$, each entry of $\mathbf{F}$ is given by [13]

where $(a)$ holds due to $\operatorname{Tr}(\bm{\Upsilon}^{H}\bm{\Upsilon})=\left(\operatorname{vec}(\bm{\Upsilon})\right)^{H}\operatorname{vec}(\bm{\Upsilon})$. Let $\mathbf{e}_{i}$ denote the $i$-th column of the identity matrix, we have

where $\dot{\mathbf{B}}\triangleq\left[\frac{\partial\mathbf{b}(\theta_{1})}{\partial\theta_{1}},\ldots,\frac{\partial\mathbf{b}(\theta_{K})}{\partial\theta_{K}}\right]$, $\dot{\mathbf{Q}}\triangleq\left[\frac{\partial\mathbf{a}_{r}(\overline{\theta}_{1})}{\partial\theta_{1}},\ldots,\frac{\partial\mathbf{a}_{r}(\overline{\theta}_{K})}{\partial\theta_{K}}\right]$. Note that

Consequently, by letting $\mathbf{R}_{D}\triangleq\mathbf{D}\mathbf{D}^{H}$ denote the covariance matrix of $\mathbf{D}$, we have

Similarly, by letting $\bm{\alpha}\triangleq[\overline{\bm{\alpha}}_{1},\ldots,\overline{\bm{\alpha}}_{K}]^{T}\in\mathbb{R}^{2K\times 1}$, we have

Thus, the Fisher information matrix can be expressed as

Therefore, the CRB of the $k$-th DoA estimation is given by

where $\left[\mathbf{F}^{-1}\right]_{k,k}$ denotes the $k$-th diagonal entry of the matrix $\mathbf{F}^{-1}$. In the case of a single target and $\mathbf{R}_{D}=L\mathbf{I}_{N}$, the CRB for DoA estimation can be simplified as [8]

It is observed from (35) that the CRB is inversely proportional to $MN^{3}+NM^{3}$. This relationship can be elucidated as follows. When $\mathbf{R}_{D}=L\mathbf{I}_{N}$, the IRS REs sense the entire space, analogous to the IRS SEs that can receive the signals from the entire space. It is also noteworthy that $\mathbf{B}$ and $\mathbf{Q}$ in (IV), (31) and (32) are interchangeable with $\mathbf{R}_{D}=L\mathbf{I}_{N}$, indicating the equivalent roles of IRS REs and SEs. Let us take a closer look at $\mathcal{O}(NM^{3})$. Note that the IRS SEs inherently provide a spatial direction gain of $\mathcal{O}(M^{3})$ [14]. If the IRS REs are consistently directed towards a specific angle during the entire $L$ time slots, a beamforming gain of $\mathcal{O}(N^{2})$ can be achieved [2, 9]. However, in order to sense the targets, the IRS REs should adjust the reflection coefficients in each time slot, thus contributing only a gain of $\mathcal{O}(N)$. Therefore, the combined gain totals $\mathcal{O}(NM^{3})$, and the CRB exhibits an inverse proportionality to $MN^{3}+NM^{3}$. When there are multiple targets, the interference among them implies that the CRB for one target serves as a lower bound for that of multiple targets, i.e., $\mathrm{CRB}(\theta_{0})\leq\mathrm{CRB}(\theta_{k})$.

In this section, numerical experiments are conducted to evaluate the performance of the proposed ANM-based DoA estimation method. The carrier frequency is set to $f_{c}=28$ GHz, and the transmit signal is $s(t)=1,\forall t$. Other system parameters are configured as follows unless specified otherwise: $N_{b}=64$, $N=64$, $M=8$, $L=N$, $\vartheta_{BI}=-60^{\circ}$, $d_{BI}=30$ m, $K=3$, $\bm{\theta}=[-10^{\circ},10^{\circ},30^{\circ}]^{T}$, $d_{k}=5$ m, $\kappa_{k}=10$ dBsm, $\sigma^{2}=-120$ dBm, $\varrho=\beta^{2}/N=1000$, and $\mathbf{D}$ is set as the DFT matrix. The root-MSE (RMSE) is defined as $\mathrm{RMSE}\triangleq\sqrt{\frac{1}{KN_{\text{NS}}}\sum_{i=1}^{N_{\text{NS}}}\|\bm{\theta}-\hat{\bm{\theta}}_{i}\|^{2}}$, where $N_{\text{NS}}$ is the number of Monte Carlo trials, and $\hat{\bm{\theta}}_{i}$ denotes the estimated DoA in the $i$-th trial. The root-CRB (RCRB) is defined as $\mathrm{RCRB}\triangleq\sqrt{\frac{1}{K}\sum_{k=1}^{K}\mathrm{CRB}(\theta_{k})}$. The benchmark is the MUSIC-based method [9] labeled “MUSIC”.

First, Fig. 2 illustrates the RMSE versus the transmit power $P_{t}$ for three cases: 1) Case 1: $\bm{\theta}=[-10^{\circ},10^{\circ},30^{\circ}]^{T}$ and $\mathbf{D}=\mathrm{DFT}$; 2) Case 2: $\bm{\theta}=[-10^{\circ},10^{\circ},30^{\circ}]^{T}$, and the phase shifts of $\mathbf{D}$ follow a uniform distribution over $[0,2\pi)$; 3) Case 3: $\bm{\theta}=[-10^{\circ},10^{\circ},20^{\circ}]^{T}$, and the phase shifts of $\mathbf{D}$ follow a uniform distribution over $[0,2\pi)$. Several interesting findings emerge. First, the proposed ANM-based method outperforms the MUSIC-based method in all cases, as it fully utilizes the active and passive element resources (i.e., $\mathbf{B}(\bm{\theta})$ and $\mathbf{Q}(\bm{\theta})$), whereas the MUSIC method only utilizes the receive SEs (i.e., $\mathbf{B}(\bm{\theta})$). Second, the proposed method nearly reaches the performance limit when employing the DFT measurement matrix. This phenomenon may be attributed to the following reason. The DFT matrix results in an identity covariance matrix, implying that each target receives equivalent power from the CRB perspective. As a result, the proposed method effectively estimates DoAs. However, with a random measurement matrix, certain targets receive more power while others receive less, intensifying their interference and diminishing estimation performance. Consequently, the results deviate significantly from the CRB. Third, as for case 3 where two targets are close to each other, the performance of MUSIC degrades dramatically, while the performance of the proposed method varies little. This is because the resolution of the proposed method, which mainly depends on the number of IRS REs $N$, outperforms the MUSIC method, which mainly depends on the number of IRS SEs $M$.

We then investigate the RMSE versus the number of targets $K$ in Fig. 5 considering various transmit power levels. The target angles are taken from the set $\{\pm 60^{\circ},\pm 45^{\circ},\pm 30^{\circ},\pm 15^{\circ},0^{\circ}\}$. The MUSIC method performs well in the case of one target, but experiences a dramatic performance drop as $K$ increases due to its limited resolution. It fails when $K\geq M=8$. In contrast, the proposed method performs well even when $K=9$, since $N>K$ and it utilizes both $\mathbf{B}(\bm{\theta})$ and $\mathbf{Q}(\bm{\theta})$ for DoA estimation.

Next, Fig. 5 displays the RMSE versus the number of IRS SEs $M$ when $K=3$ and $N+M=72$ under different transmit power levels. The DoA estimation accuracy increases as $M$ grows since more echo power can be obtained. The proposed method can even work when $M=1$ since it can also utilize $\mathbf{Q}(\bm{\theta})$ to sense the targets. However, its performance is limited when the transmit power is low. Therefore, a moderate number of IRS SEs should be chosen to balance the hardware cost and sensing performance.

Finally, we explore the RMSE with respect to the measurement length $L$ in Fig. 5, where $P_{t}=20$ dBm and the phase shifts of $\mathbf{D}$ follow a uniform distribution over $[0,2\pi)$. The proposed method performs slightly worse than MUSIC when $L<23$, likely due to limited measurements. However, when $L>30$, the performance of the proposed method tends to stabilize and significantly outperforms MUSIC. Therefore, the proposed method is likely to operate with moderate measurements.

This work investigates a multi-target DoA estimation problem in a semi-passive IRS-assisted localization system. The ANM method is adopted to fully extract the DoA information embedded in the IRS REs and SEs matrices. The CRB for DoA estimation is then derived to evaluate the sensing performance. Simulation results have validated the superior accuracy and resolution performance of the proposed ANM-based DoA estimation method over MUSIC. Future work may consider efficient DoA estimation approaches under few IRS measurements.