Location Information Aided Multiple Intelligent Reflecting Surface Systems

With the desirable feature of low energy consumption and low hardware complexity, the intelligent reflecting surface (IRS) has recently emerged as a key candidate technology for future wireless communication systems [1]. Specifically, an IRS is a meta-surface comprising a large number of low-cost, nearly passive, reflecting elements, each of which can independently reflect the incident signal with adjustable phase shifts. By properly adjusting its phase shifts, the IRS can enhance the desired signal power at the receiver, thereby helping combat the unfavorable wireless propagation environment.

Due to the aforementioned benefits, IRS-aided wireless communications have attracted considerable research interests. In [2, 3, 5, 6, 4], the joint transmit beamforming and IRS beamforming problem was studied, while in [7], transmit power allocation and phase shift beamforming were jointly optimized. Later on, the works [8] and [9] investigated a more realistic case where the IRS only has a finite number of phase shifts at each element. The multi-user multi-IRS scenario has been considered in [10], where joint active and passive beamformers are designed based on the assumption that global channel state information (CSI) is available. Moreover, the integration of IRS with other promising technologies has been studied, including cognitive radio [11], non-orthogonal multiple access (NOMA) [12, 13, 14, 15], two-way communications [16], millimeter wave (mmWave)[17, 18], terahertz communications [19], physical-layer security [20, 21, 22] and simultaneous wireless information and power transfer (SWIPT)[23, 24].

However, there are two main challenges when performing IRS beamforming in practice. The first one is the requirement of instantaneous CSI, as adopted by all the aforementioned works. Due to the passive architecture of the IRS and large number of reflecting elements, CSI acquisition is highly non-trivial. The conventional CSI estimation techniques are not applicable, and the typical method is to estimate the cascaded channel instead of individual channels [25, 26, 27, 28, 29]. However, with a large number of reflecting elements, the channel training overhead becomes prohibitively high. To reduce the training overhead, the works [12, 26] proposed to divide the IRS elements into groups where only the effective channel for all elements in each group are estimated. But this method comes at the cost of degraded IRS passive beamforming performance, since the phase shifts of reflecting elements in each group needs to be set identical.

Another challenge is that a separate link is required for information exchange between the base station (BS) and the IRS. In general, phase shifts are designed at the BS according to the instantaneous CSI, and then shared with the IRS via the separate link [30, 26]. Due to the time varying nature of the wireless channel, phase shifts designed with instantaneous CSI need to be updated frequently. In addition, as the number of reflecting elements becomes larger, the required capacity of the separate link also increases. To reduce the frequency of information exchange between the BS and IRS, the work [31, 32] proposed to use statistical CSI for phase shift design.

To address the aforementioned two challenges, we propose a novel location information aided multi-IRS design framework, where the design of the transmit beam and phase shifts only relies on statistical CSI obtained from location information. Compared with the traditional design framework which usually requires full CSI and involves large training overhead, our proposed design framework has the following three major advantages. First of all, the location information can be easily obtained by global position system (GPS), hence the training overhead can be substantially reduced. Secondly, the location information varies much slower compared with the instantaneous CSI, hence no frequent update is required. Thirdly, only very small amount of location information needs to be shared between the BS and IRS, hence only low-capacity connection is required, which further reduces the hardware implementation cost. The main contributions of this paper are summarized as follows:

• •

  Utilizing the imperfect location information, the angle information of the line-of-sight (LOS) path is estimated, and the statistical properties of the estimation error are characterized.

• •

  Based on the estimated angle information, a low-complexity beamforming scheme is proposed, where closed-form expressions of the transmit beam and phase shift beam are obtained. Simulation results show that when the individual user rate requirement is low, the proposed beamforming scheme is superior to the joint optimization scheme in [3].

• •

  The closed-form expression for the achievable rate of the proposed scheme is presented, which facilitates the study of the impact of key system parameters such as location accuracy, number of reflecting elements, number of antennas and Rician K-factor.

• •

  Finally, under the proposed beamforming scheme, an optimal power control scheme is proposed to minimize the total transmit power, subject to individual user rate constraints.

The remainder of the paper is organized as follows. In Section II, we introduce the multi-IRS system, while in Section III, we propose an angle estimation scheme based on location information. According to the estimated effective angles, a low-complexity scheme for BS beamforming and IRS beamforming is presented in Section IV. Then, we give a detailed analysis of the achievable rate in Section V. Also, an optimal power control scheme is proposed in VI. Numerical results and discussions are provided in Section VII, and finally Section VIII concludes the paper.

Notation: Boldface lower case and upper case letters are used for column vectors and matrices, respectively. The superscripts ${\left(\cdot\right)}^{*}$, ${\left(\cdot\right)}^{T}$, ${\left(\cdot\right)}^{H}$, and ${\left(\cdot\right)}^{-1}$ stand for the conjugate, transpose, conjugate-transpose, and matrix inverse, respectively. Also, the Euclidean norm, absolute value, Hadamard product are denoted by $\left\|\cdot\right\|$, $\left|\cdot\right|$ and $\odot$ respectively. In addition, $\mathbb{E}\left\{\cdot\right\}$ is the expectation operator, and $\text{tr}\left(\cdot\right)$ represents the trace. For a matrix ${\bf A}$, ${[\bf A]}_{mn}$ denotes its entry in the $m$-th row and $n$-th column, while for a vector ${\bf a}$, ${[\bf a]}_{m}$ denotes the $m$-th entry of it. Besides, $j$ in $e^{j\theta}$ denotes the imaginary unit. Finally, $z\sim\mathcal{CN}(0,{\sigma}^{2})$ denotes a circularly symmetric complex Gaussian random variable (RV) $z$ with zero mean and variance $\sigma^{2}$, and $z\sim\mathcal{N}(0,{\sigma}^{2})$ denotes a real valued Gaussian RV.

We consider a multi-IRS system, as illustrated in Fig. 1, where one BS with $N$ antennas communicates with $K$ single antenna users, each of which is assisted by an IRS with $M$ reflecting elements. ¹¹1 When the number of users is large, it is infeasible to associate each user with a unique IRS. In such scenario, the IRSs can be assigned to users with poor coverage. Alternatively, effective user scheduling protocol can be designed to exploit the benefit of IRSs. We further assume that the direct links between the BS and users do not exist due to blockage or unfavorable propagation environments [7]. The BS is connected with IRSs via low-capacity hardware links so that they can exchange information (e.g. CSI and phase shifts). As in [11, 33], we assume that both the BS and IRSs use uniform linear arrays (ULA). ²²2 The main reason to use ULA instead of uniform planar array (UPA) is that the simple structure of the ULA helps to get some useful and intuitive insights. In addition, the UPA can be regarded as multiple ULAs. As such, the analytical results obtained for ULA can be readily extended to the UPA structure. Without loss of generality, we assume that the BS is located at the origin of a Cartesian coordinate system, and the ULAs of both the BS and IRSs are along the $y$ axis.

To fully exploit the potential of IRSs, they are usually deployed in desirable locations with LOS paths to both the BS and users. As such, we use angle-domain Rician fading to model the channels between IRSs and the BS or users[8]. Specifically, the channel from the BS to the $m$-th IRS can be expressed as

$\displaystyle{\bf G}_{\text{B2I},m}\!=\!\sqrt{\alpha_{\text{B2I},m}\frac{v_{\text{B2I},m}}{v_{\text{B2I},m}\!+\!1}}\bar{\bf G}_{\text{B2I},m}\!+\!\sqrt{\alpha_{\text{B2I},m}\frac{1}{v_{\text{B2I},m}\!+\!1}}\widetilde{\bf G}_{\text{B2I},m},$

(1)

where $\widetilde{\bf G}_{\text{B2I},m}{\in\mathbb{C}^{M\times N}}$ is the non-line-of-sight (NLOS) component, whose elements follow the $\mathcal{CN}\left(0,1\right)$ distribution, $\alpha_{\text{B2I},m}$ models large-scale fading, and $v_{\text{B2I},m}$ is the Rician K-factor. The LOS component $\bar{\bf G}_{\text{B2I},m}{\in\mathbb{C}^{M\times N}}$ is given by

$\displaystyle\bar{\bf G}_{\text{B2I},m}={\bf b}\left(\vartheta_{\text{y-B2Ia},m}\right){\bf a}^{T}\left(\vartheta_{\text{y-B2I},m}\right),$

(2)

where ${\bf a}\left(\vartheta_{\text{y-B2I},m}\right){\in\mathbb{C}^{N\times 1}}$ is the array response vector at the BS, with the effective angle of departure (AOD), i.e., the phase difference between two adjacent antennas along the $y$ axis, given by

$\displaystyle\vartheta_{\text{y-B2I},m}=-\frac{2\pi d_{\text{BS}}}{\lambda}\cos\theta_{\text{B2I},m}\sin\phi_{\text{B2I},m},$

(3)

where $d_{\text{BS}}$ is the distance between two adjacent BS antennas, $\lambda$ is the carrier wavelength, $\theta_{\text{B2I},m}$ and $\phi_{\text{B2I},m}$ are the elevation and azimuth AODs from the BS to the $m$-th IRS, respectively.

Similarly, ${\bf b}\left(\vartheta_{\text{y-B2Ia},m}\right){\in\mathbb{C}^{M\times 1}}$ is the array response vector at the $m$-th IRS, where the effective angle of arrival (AOA) is given by

$\displaystyle\vartheta_{\text{y-B2Ia},m}=\frac{2\pi d_{\text{IRS}}}{\lambda}\cos\theta_{\text{B2Ia},m}\sin\phi_{\text{B2Ia},m},$

(4)

where $d_{\text{IRS}}$ is the distance between two adjacent reflecting elements, $\theta_{\text{B2Ia},m}$ and $\phi_{\text{B2Ia},m}$ are the elevation and azimuth AOAs at the $m$-th IRS, respectively. Furthermore, without loss of generality, we assume $d_{\text{BS}}=d_{\text{IRS}}=\frac{\lambda}{2}$.

Specifically, the $n$-th element of ${\bf a}\left(\vartheta\right)$ and the $l$-th element of ${\bf b}\left(\vartheta\right)$ are given by

	$\displaystyle a_{n}=e^{j\pi\left(n-1\right)\vartheta},n=1,...,N,$		(5)
	$\displaystyle b_{l}=e^{j\pi\left(l-1\right)\vartheta},l=1,...,M.$		(6)

Similarly, the channel from the $m$-th IRS to the $k$-th user is given by ${\bf g}_{\text{I2U},mk}^{T}{\in\mathbb{C}^{1\times M}}$:

$\displaystyle{\bf g}_{\text{I2U},mk}^{T}=\sqrt{\alpha_{\text{I2U},mk}\frac{v_{\text{I2U},mk}}{v_{\text{I2U},mk}+1}}\bar{\bf g}_{\text{I2U},mk}^{T}+\sqrt{\alpha_{\text{I2U},mk}\frac{1}{v_{\text{I2U},mk}+1}}\widetilde{\bf g}_{\text{I2U},mk}^{T},$

(7)

with the LOS component $\bar{\bf g}_{\text{I2U},mk}^{T}{\in\mathbb{C}^{1\times M}}$ given by

$\displaystyle\bar{\bf g}_{\text{I2U},mk}^{T}={\bf b}^{T}\left(\vartheta_{\text{y-I2U},mk}\right).$

(8)

During the downlink data transmission phase, the BS broadcasts the signal

$\displaystyle{\bf x}=\sum\limits_{i=1}^{K}{\bf w}_{i}s_{i},$

(9)

where ${\bf w}_{i}{\in\mathbb{C}^{N\times 1}}$ is the transmit beamforming vector and $s_{i}$ is the symbol for the $i$-th user, satisfying $\mathbb{E}\left\{|s_{i}|^{2}\right\}=1$.

Then, the signal received at the $k$-th user can be expressed as

$\displaystyle y_{k}=\sum\limits_{m=1}^{K}\sum\limits_{i=1}^{K}{\bf g}_{\text{I2U},mk}^{T}{\mbox{\boldmath$\Theta$}_{m}}{\bf G}_{\text{B2I},m}{\bf w}_{i}s_{i}+n_{k},$

(10)

where $n_{k}\sim\mathcal{CN}\left(0,\sigma_{0}^{2}\right)$ is the noise at the $k$-th user. The phase shift matrix of the $m$-th IRS is given by ${\mbox{\boldmath$\Theta$}}_{m}={\text{diag}}\left({\mbox{\boldmath$\xi$}}_{m}\right){\in\mathbb{C}^{M\times M}}$ with the phase shift beam ${\mbox{\boldmath$\xi$}}_{m}={[e^{j\vartheta_{m,1}},...,e^{j\vartheta_{m,n}},...,e^{j\vartheta_{m,M}}]}^{T}{\in\mathbb{C}^{M\times 1}}$.

To facilitate the design of the transmit beam ${\bf w}_{i}$ and phase shift beam ${\mbox{\boldmath$\xi$}}_{m}$, CSI of individual channels is necessary. In this section, we exploit the user location information provided by the GPS to obtain angle information of the LOS path. Furthermore, due to the fact that IRS locations are fixed, we assume that they are perfectly known by the BS.

Without loss of generality, we assume the BS is located at the origin $(0,0,0)$, and the $m$-th IRS is located at $(x_{\text{I},m},y_{\text{I},m},z_{\text{I},m})$, then the effective AOD from the BS to the $m$-th IRS can be computed as

$\displaystyle\vartheta_{\text{y-B2I},m}$

$\displaystyle=-\frac{y_{\text{I},m}}{d_{\text{B2I},m}},$

(11)

where $d_{\text{B2I},m}=\sqrt{x_{\text{I},m}^{2}+y_{\text{I},m}^{2}+z_{\text{I},m}^{2}}$ is the distance between the BS and the $m$-th IRS.

Similarly, the effective AOA at the $m$-th IRS can be calculated as

$\displaystyle\vartheta_{\text{y-B2Ia},m}$

$\displaystyle=\frac{y_{\text{I},m}}{d_{\text{B2I},m}}.$

(12)

Due to user mobility or other unfavorable conditions, the user location information provided by the GPS is imperfect. Specifically, the accurate location of user $k$, i.e., $(x_{\text{U},k},y_{\text{U},k},z_{\text{U},k})$, is uniformly distributed within a sphere with the radius $\Upsilon$ and center point $(\hat{x}_{\text{U},k},\hat{y}_{\text{U},k},\hat{z}_{\text{U},k})$, where $(\hat{x}_{\text{U},k},\hat{y}_{\text{U},k},\hat{z}_{\text{U},k})$ is the estimated location of user $k$, provided by the GPS.

The $m$-th IRS calculates its effective AOD from itself to the $k$-th user as

$\displaystyle\hat{\vartheta}_{\text{y-I2U},mk}$

$\displaystyle=\frac{y_{\text{I},m}-\hat{y}_{\text{U},k}}{\hat{d}_{\text{I2U},mk}},$

(13)

where $\hat{d}_{\text{I2U},mk}=\sqrt{(x_{\text{I},m}-\hat{x}_{\text{U},k})^{2}+(y_{\text{I},m}-\hat{y}_{\text{U},k})^{2}+(z_{\text{I},m}-\hat{z}_{\text{U},k})^{2}}$ is the distance between the $m$-th IRS and the $k$-th user.

Furthermore, the estimation error $\epsilon_{\text{y-I2U},mk}$ has the following distribution.

After obtaining the angle information, the BS uses the estimated angles to design transmit and phase shift beams. Since transmit and phase shift beams are coupled, the resultant optimization problem is non-convex, hence the global optimal solution is in general intractable. As such, many sub-optimal algorithms have been proposed such as alternating optimization. However, due to the large number of reflecting elements, the computational complexity of these algorithms is very high. Motivated by this, instead of aiming for the global optimal solution, we perform low-complexity local optimization at the BS and each IRS, separately, where closed-form solutions can be obtained.

Specifically, the BS utilizes the angle information of the BS-IRS link to design the transmit beam. Without loss of generality, we assume the $k$-th user is assisted by the $k$-th IRS. Thus, the transmit beam for the $k$-th user should be aligned to the $k$-th IRS. As such, the transmit beam is designed as

$\displaystyle{\bf w}_{k}=\sqrt{\frac{\eta_{k}\rho_{d}}{N}}{\bf{a}}_{\text{B2I},k}^{*},$

(20)

where ${\bf{a}}_{\text{B2I},k}\triangleq{\bf a}\left(\vartheta_{\text{y-B2I},k}\right){\in\mathbb{C}^{N\times 1}}$, $\rho_{d}$ is the transmit power of the BS and $0<\eta_{k}<1$ is the power control coefficient.

The estimated angle information of the IRS-user link is used to design the phase shift beam. For the $k$-th user, we aim to maximize its received signal via the $k$-th IRS by optimizing the phase shift beam $\mbox{\boldmath$\xi$}_{k}$. Specifically, the optimization problem is formulated as

$\displaystyle\begin{array}[]{ll}\max\limits_{{\mbox{\boldmath$\Theta$}}_{k}}&\left|{\bf g}_{\text{I2U},kk}^{T}{\mbox{\boldmath$\Theta$}_{k}}{\bf G}_{\text{B2I},k}{\bf w}_{k}\right|^{2},\\ \operatorname{s.t.}&\begin{array}[t]{lll}\left|{\left[{\bf\Theta}_{k}\right]}_{ii}\right|=1,i=1,...,M.\end{array}\end{array}$

(24)

According to the rule that ${\bf E}^{T}{\bf X}{\bf F}={\bf x}^{T}\left({\bf E}\odot{\bf F}\right)$ with ${\bf X}={\text{diag}}\left({\bf x}\right)$, we have

$\displaystyle{\bf g}_{\text{I2U},kk}^{T}{\mbox{\boldmath$\Theta$}_{k}}{\bf G}_{\text{B2I},k}{\bf w}_{k}={\mbox{\boldmath$\xi$}_{k}^{T}}\left({\bf g}_{\text{I2U},kk}\odot{\bf G}_{\text{B2I},k}{\bf w}_{k}\right)={\mbox{\boldmath$\xi$}_{k}^{T}}\left({\bf g}_{\text{I2U},kk}\odot{\bf b}_{\text{B2I},k}\right){\bf a}^{T}_{\text{B2I},k}{\bf w}_{k},$

(25)

where ${\bf b}_{\text{B2I},k}\triangleq{\bf b}\left(\vartheta_{\text{y-B2Ia},k}\right){\in\mathbb{C}^{M\times 1}}$.

Based on the above equation, the objective function in (24) becomes

$\displaystyle\left|{\mbox{\boldmath$\xi$}_{k}^{T}}\left({\bf g}_{\text{I2U},kk}\odot{\bf b}_{\text{B2I},k}\right)\right|^{2}\left|{\bf a}^{T}_{\text{B2I},k}{\bf w}_{k}\right|^{2}.$

(26)

Noticing that $\left|{\bf a}^{T}_{\text{B2I},k}{\bf w}_{k}\right|^{2}$ is a constant independent of ${\mbox{\boldmath$\xi$}}_{k}$ , the optimization problem is equivalent to

$\displaystyle\begin{array}[]{ll}\max\limits_{{\mbox{\boldmath$\xi$}}_{k}^{T}}&\left|{\mbox{\boldmath$\xi$}_{k}^{T}}\left({\bf g}_{\text{I2U},kk}\odot{\bf b}_{\text{B2I},k}\right)\right|^{2},\\ \operatorname{s.t.}&\begin{array}[t]{lll}\left|\left[{\mbox{\boldmath$\xi$}}_{k}\right]_{i}\right|=1,i=1,...,M.\end{array}\end{array}$

(30)

Obviously, the solution for the above optimization problem is

$\displaystyle{\mbox{\boldmath$\xi$}}_{k}=\left({\bf g}_{\text{I2U},kk}\odot{\bf b}_{\text{B2I},k}\right)^{*}.$

(31)

Using the estimated angles, we design the phase shift beam as

$\displaystyle{\mbox{\boldmath$\xi$}}_{k}=\left({\hat{\bar{\bf g}}}_{\text{I2U},kk}\odot{\bf b}_{\text{B2I},k}\right)^{*},$

(32)

where ${\hat{\bar{\bf g}}}_{\text{I2U},mk}={\bf b}\left(\hat{\vartheta}_{\text{y-I2U},mk}\right){\in\mathbb{C}^{M\times 1}}$.

In this section, we present a detailed investigation on the achievable rate of the multi-IRS system. Without loss of generality, let us focus on the achievable rate of the $k$-th user. We consider the realistic case where the $k$-th user does not have access to the instantaneous CSI of the effective channel gain. As such, we can rewrite ${{y}}_{k}$ as

$\displaystyle y_{k}=\underbrace{\mathbb{E}\left\{{\bf g}_{k}^{T}{\bf w}_{k}\right\}s_{k}}_{\text{desired signal}}+\underbrace{{\left({\bf g}_{k}^{T}{\bf w}_{k}-\mathbb{E}\left\{{\bf g}_{k}^{T}{\bf w}_{k}\right\}\right)}s_{k}}_{\text{leakage signal}}+\underbrace{{\bf g}_{k}^{T}\sum\limits_{i\neq k}^{K}{\bf w}_{i}s_{i}}_{\text{inter-user interference}}+\underbrace{n_{k}}_{\text{noise}},$

(33)

where we define the equivalent channel of the $k$-the user as

$\displaystyle{\bf g}_{k}^{T}=\sum\limits_{m=1}^{K}{\bf g}_{\text{I2U},mk}^{T}{\mbox{\boldmath$\Theta$}_{m}}{\bf G}_{\text{B2I},m}.$

(34)

Invoking the result in [34], the achievable rate of the $k$-th user is given by

$\displaystyle R_{k}=\log_{2}\left(1+\frac{A_{k}}{B_{k}+\sum\limits_{i\neq k}^{K}C_{k,i}+\sigma_{0}^{2}}\right),$

(35)

where

	$\displaystyle A_{k}\triangleq{\left|\mathbb{E}\left\{{\bf g}_{k}^{T}{\bf w}_{k}\right\}\right|}^{2},$		(36)
	$\displaystyle B_{k}\triangleq\mathbb{E}\left\{{\left|{{{\bf g}_{k}^{T}{\bf w}_{k}-\mathbb{E}\left\{{\bf g}_{k}^{T}{\bf w}_{k}\right\}}}\right|}^{2}\right\},$		(37)
	$\displaystyle C_{k,i}\triangleq\mathbb{E}\left\{{\left|{{\bf g}_{k}^{T}{\bf w}_{i}}\right|}^{2}\right\},$		(38)

denote the desired signal power, leakage power and interference, respectively.

To derive the expression of $R_{k}$, we first give the following Lemma:

Theorem 2 presents a closed-form expression for the achievable rate, which quantifies the impacts of some key parameters, such as antenna number, user number, the number of reflecting elements, Rician K-factor and user location uncertainty. For instance, $R_{k}$ is an increasing function with respect to the Rician K-factor, because a small Rician K-factor implies more interference caused by NLOS paths. Also, increasing the number of users would degrade the individual rate, due to stronger inter-user interference. In addition, as $\Upsilon$ becomes larger, $\zeta_{\text{y-I2U},mk,s}\to 0$ and thus $R_{k}$ gradually goes to zero, indicating that the achievable rate would significantly degrade as user location uncertainty becomes larger. Moreover, we can see that $C_{k,i}$ increases with the correlation coefficient between ${\bf a}_{\text{B2I},i}$ and ${\bf a}_{\text{B2I},m},m\neq i$, implying that IRSs should be deployed at distinct directions relative to the BS to reduce the interference from multiple IRSs. Ideally, it is desired that ${\bf a}_{\text{B2I},m}^{T}{\bf a}_{\text{B2I},i}^{*}\to 0,m\neq i$.

We now consider some special cases, where simplified expressions are available.