Robust Design for IRS-Aided Communication Systems with User Location Uncertainty

Intelligent reflecting surface (IRS) has recently emerged as a promising method to tackle the coverage problem of dead-zone users [1, 2]. The IRS is a meta-surface consisting of a large number of low-cost, passive reflecting elements, each of which can independently reflect the incident signal with adjustable phase shifts. Through proper design of the phase shifts at the IRS [3, 4], the signal received at the user can be significantly enhanced.

To facilitate the design of phase shifts, channel state information (CSI) is compulsory. As such, plenty of works have been devoted to the estimation of IRS-aided channels, including the channel between the BS and IRS, as well as the reflection channel between the IRS and user. In general, the proposed methods can be categorized into two types. The first type is to estimate the cascaded channels[5, 6]. However, with a large number of reflecting elements, the training overhead would become prohibitive. The second type is to directly estimate two channels separately [7], assuming that the IRS has both reflection mode and receive mode. However, this requires a large number of receive radio frequency (RF) chains (equal to the number of reflecting elements), which would significantly increase the hardware cost as well as the power consumption of the IRS.

To tackle the above issues, one possible way is to only estimate the angles of arrivals (AOAs) and angles of departures (DOAs)[8, 9, 10]. Because the locations of the IRS and BS remain fixed, the channel between the BS and IRS varies very slowly and can be accurately estimated by computing the AOAs and AODs [11]. Then, exploiting the user location information provided, for example, by global positioning system (GPS), the angular information of the reflection channel from the IRS to the user can be obtained. However, due to user mobility and the precision of GPS, the user location information may not be accurate, resulting in imperfect angular information.

Motivated by this, this letter proposes a robust design framework in the presence of user location uncertainty.¹¹1 Unlike [11] and [12], which adopt conventional error model, the current work first proposes a location-based channel estimation scheme, and then presents a practical channel error model related to location uncertainty. In addition, the method to tackle the optimization problem is also different from [11] and [12]. Specifically, considering a bounded spherical error model, the transmit beamforming vector and phase shifts are jointly designed to minimize the transmit power, subject to the minimum achievable rate constraint for all possible user location errors. To tackle the non-convex worse-case rate constraint, the second-order Taylor expansion is used to approximate the achievable rate, and the S-Procedure is used to convert the resultant semi-definite constraint into a matrix inequality. Then, applying the semi-definite relaxation (SDR) method, an alternating algorithm is designed, which optimizes the transmit beamforming vector and phase shifts iteratively by solving a sequence of semi-definite programming (SDP) sub-problems. Simulation results show that the proposed robust algorithm can guarantee the target rate regardless of the user location errors, which substantially outperforms the non-robust algorithm proposed in [3].

We consider an IRS-aided system as illustrated in Fig.1, where one BS with $N$ antennas communicates with a single-antenna user, which is assisted by an IRS with $M$ reflecting elements. Like most works on IRS, e.g.,[3, 5], we assume that the IRS operates in the far-field regime. Furthermore, we assume that direct link between the BS and the user does not exist, due to blockage or unfavorable propagation environments. Both the BS and the IRS are equipped with uniform rectangular arrays (URAs) with the size of $N_{y}\times N_{z}$ and $M_{y}\times M_{z}$ respectively, where $N_{y}$ ($N_{z}$) and $M_{y}$ ($M_{z}$) denote the numbers of BS antennas and reflecting elements along the $y$ ($z$) axis, respectively.

In the IRS-aided communication system, due to the fixed location of the IRS, the channel from the BS to the IRS, i.e., ${\bf G}_{\text{B2I}}$, usually remains constant over a long period, and thus can be accurately estimated. Therefore, we assume that ${\bf G}_{\text{B2I}}$ is perfectly known. In contrast, due to user mobility, the reflection channel ${\bf g}_{\text{I2U}}$ varies over the time, hence should be estimated. As such, we propose to exploit user location information to estimate the reflection channel ${\bf g}_{\text{I2U}}$.

The elevation and azimuth AODs ($\theta_{\text{I2U}}$ and $\phi_{\text{I2U}}$) have the following relationship with the locations of the IRS and user:

where $d_{\text{I2U}}$ is the distance between the IRS and the user, $(x_{\text{I}},y_{\text{I}},z_{\text{I}})$ and $(x_{\text{U}},y_{\text{U}},z_{\text{U}})$ are locations of the IRS and the user respectively. Substituting (11) into (8) and (9), we obtain the relationship between the effective AODs ($\vartheta_{\text{z-I2U}}$ and $\vartheta_{\text{y-I2U}}$) and locations of the IRS and the user:

In general, user location information obtained from the GPS is imperfect. Let $(\hat{x}_{\text{U}},\hat{y}_{\text{U}},\hat{z}_{\text{U}})$ denote the estimated location of the user and ${\bf\Delta}\triangleq[\Delta x_{\text{U}},\Delta y_{\text{U}},\Delta z_{\text{U}}]^{T}$ denote the estimation error. The estimated effective AODs from the IRS to the user are given by

Invoking the results given by Proposition 1, the reflection channel can be expressed as

where $\odot$ stands for Hadamard product, $\hat{\bf g}_{\text{I2U}}=\alpha_{\text{I2U}}{\bf b}(\hat{\vartheta}_{\text{z-I2U}},\hat{\vartheta}_{\text{y-I2U}})$ is the estimated channel, and ${\bf e}_{\text{I2U}}$ is the estimation error with $\left[{\bf e}_{\text{I2U}}\right]_{i}=e^{j\pi{\bf f}_{i}^{T}{\mbox{\boldmath$\Delta$}}}$, where ${\bf f}_{i}=[a_{i_{m}i_{n}},b_{i_{m}i_{n}},c_{i_{m}i_{n}}]^{T}$ with

Assuming that the location error is bounded by a sphere with the radius $\Upsilon$, i.e.,$\|{\bf\Delta}\|^{2}\leq\Upsilon^{2}$, we aim to minimize the total transmit power through joint design of beamforming vector ${\bf w}$ and phase shift vector $\xi$ under the worst-case QoS constraint, namely, the achievable rate should be above a threshold $r$. Mathematically, the worst-case robust design problem can be formulated as

Substituting (10) and (15) into (20), we have

In this section, we provide numerical results to evaluate the performance of the proposed algorithm. The considered system is assumed to operate at $28$ GHz with bandwidth of 100 MHz and noise spectral power density of -169 dBm/Hz. The IRS is at the origin $(0,0,0)$ of a Cartesian coordinate system. The locations of the BS and the user are $(100,-100,0)$ and $(20,20,-20)$, respectively. Unless otherwise specified, the following setup is used: $N=16$, $M=100$ and $\Upsilon=4$.²²2As pointed out in [16], synchronously operating a large number of phase shifters is a non-trivial task. The IRS may suffer from phase errors. It have been shown in [17] that the average received SNR is attenuated by $\phi^{2}$ where $\phi\triangleq\mathbb{E}\left\{e^{j\theta}\right\}$ with $\theta$ being the phase error. One possible way to reduce the phase error is to use phase shifters with adaptive tuning capability.

Fig. 2 compares the proposed robust scheme with a “Non-robust” optimization scheme [3]. It is worth noting that the target rate $r$ is a predefined threshold, while the x-axis in Fig.2 ($R$) is the rate achieved under different location errors of each channel realization. The proposed robust beamforming scheme requires that the achievable rate under all possible location errors should exceed the threshold $r$, while the non-robust beamforming scheme regards the estimated channels as perfect channels, and aims to maximize the achievable rate without location uncertainty, using the same power as the proposed beamforming scheme. As can be readily seen, the variation of the user rate with robust optimization is much smaller than that with non-robust optimization. Moreover, the target rate can be always guaranteed regardless of user location errors. By contrast, with non-robust optimization, the user rate varies over a wide range. For instance, with $r=6$, the user rate ranges from near 0 bits/s/Hz to about 7.2 bits/s/Hz. Moreover, there is a high probability (over $80\%$) that the target rate can not be achieved.

Fig. 3 presents the transmit power versus the target rate with different location uncertainty and numbers of reflecting elements. Obviously, to achieve a higher target rate, more power is required. Also, as location uncertainty (measured by $\Upsilon$) increases, the required transmit power becomes larger so that the target rate can be achieved for all possible location errors. Besides, since both the beamforming gain and the aperture gain of the IRS grow with the number of reflecting elements, the transmit power drops significantly with the number of reflecting elements, implying the benefit of the IRS in power saving.

In this paper, considering user location uncertainty, we study the robust beamforming design for an IRS-aided communication system. We first handle the location uncertainty by exploiting techniques of Taylor expansion and S-Procedure. Then SDR is used to transform the non-convex problem into a sequence of SDP sub-problems, which can be efficiently solved via some optimization tools, for example, CVX. Simulation results demonstrate that with our proposed robust algorithm, the QoS requirement is always met regardless of location uncertainty, while with a non-robust optimization algorithm [3] which has the same transmit power as the proposed algorithm, the QoS requirement is met with a probability below $20\%$.