On outage performance of IRS-assisted wireless communication with user mobility following RWP model

An intelligent reﬂecting surface (IRS)-assisted wireless communication system with user mobility following random waypoint (RWP) model is investigated, where the user receives the reﬂection signal from the IRS controlled by the access point (AP) via a programmable controller, and the IRS is mounted on an unmanned aerial vehicle which is only an auxiliary device to enable IRS reﬂection communication at a certain altitude. As user moves, the AP makes the reﬂected beamforming of the IRS following with the user through coding the unit-element of the IRS. To investigate the effect of joint IRS feature and user mobility on the outage performance, the cumulative distribution function of the received signal-to-noise ratio of the mobile user is ﬁrst derived and then the close-form expression for the outage probability is obtained. Numerical simulations are performed to validate the accuracy of the an- alytical results. The solutions are also compared with that for conventional amplify-and-forward relay system. Insights are drawn related to the number of meta-surface elements, and the maximum activity radius of mobile user.

✉ E-mail: sunjy@guet.edu.cn An intelligent reflecting surface (IRS)-assisted wireless communication system with user mobility following random waypoint (RWP) model is investigated, where the user receives the reflection signal from the IRS controlled by the access point (AP) via a programmable controller, and the IRS is mounted on an unmanned aerial vehicle which is only an auxiliary device to enable IRS reflection communication at a certain altitude. As user moves, the AP makes the reflected beamforming of the IRS following with the user through coding the unit-element of the IRS. To investigate the effect of joint IRS feature and user mobility on the outage performance, the cumulative distribution function of the received signal-to-noise ratio of the mobile user is first derived and then the close-form expression for the outage probability is obtained. Numerical simulations are performed to validate the accuracy of the analytical results. The solutions are also compared with that for conventional amplify-and-forward relay system. Insights are drawn related to the number of meta-surface elements, and the maximum activity radius of mobile user.
Introduction: Over the last several years, intelligent reflecting surface (IRS)-assisted wireless communication [1][2][3] has attracted tremendous research interests from both industry and academia. A large amount of meta-surface elements are combined into a IRS, which is able to passively reflect the incident electromagnetic wave to it and change the phase of the wave under the control of a programmable controller. Due to its high efficiency in energy and cost, IRS can be deployed on arbitrarily shaped surfaces to meet various requirements. Through the reasonable deployment of the IRS, the coverage [4], reliability [5] and security [6] of IRS-assisted wireless communication networks have been greatly improved.
On the other hand, the mobility of the nodes in wireless networks can bring time-varying feature into the received signal. To make a comprehensive assessment on the performance of IRS-assisted wireless communication networks, it is necessary to take the mobility of the nodes into account. The uplink transmission between a ground transmitter via a IRS and a low earth orbit satellite is analysed in [7], where the orbit is known a priori and the position of the satellite can be totally predictable that is utilized by the IRS to optimize the link reliability of the satellite communication. An integrated IRS and unmanned aerial vehicle (UAV) system is proposed in [8] where the IRS is deployed onto the UAV and the effect of changing the altitude of the UAV on the system outage performance is considered. However, none of the works for IRS-assisted wireless communication has studied the impact of the nodes mobility utilizing mobility model. Various mobility models have been adopted in mobile wireless network [9,10], visible light communication and LiFi network [11,12], and UAV communication system [13], where the random waypoint (RWP) model appropriate for the user moving in a specific region has been widely studied.
Motivated by the above, we consider an IRS-assisted communication system with mobile user moving as a RWP model, and the user receives the reflection signal from the IRS which is mounted on a UAV and controlled by the access point (AP) via a smart controller. As user moves, the AP makes the reflected beamforming of the IRS following with the user through coding the unit-element of the IRS. To investigate the effect of joint IRS feature and user mobility on the outage performance, we first analyse the cumulative distribution function (CDF) of the received signal-to-noise ratio (SNR) of the mobile user, and then obtain the close-form expression for the outage probability (OP). Finally, numeri- System and channel model: The proposed IRS-assisted wireless communication system considering user mobility following RWP model is shown in Figure 1, which includes an AP, an IRS (mounted on a UAV) and a mobile user (called Bob) moving as a typical RWP model in a circular region. The line of sight (LoS) link between the AP and Bob is assumed blocked and the same assumption has been adopted in [2,8]. Thus Bob can only receive the reflected signal from the IRS, which is controlled by the AP via a programmable controller. Note that the UAV adopted here is only an auxiliary device to enable IRS reflection communication at a certain altitude and it is stationary during the IRS communication. And it also should be noted that as Bob moves, by coding the unit-element of the IRS to appropriately adjust the phases of the IRS, the AP makes the reflected beamforming of the IRS following with Bob.
The proposed system is practical for various scenarios. For example, in the rebuilding of the emergency network after an earthquake disaster, the users in the affected area cannot communicate directly with the AP in the outside world since the collapsed buildings hinders the information transmission, thus an IRS attached to UAVs communication system can be deployed to help reestablish connection conveniently and quickly; on the other hand, for indoor application, if the direct link between the AP and users is hampered, an IRS can be deployed in the ceiling or attached to the ceiling lamp to assist communication.
For the proposed system, the vertical distance between the IRS plane and the user plane is H , which is assumed to be far larger than the scale of the IRS, hence the projection of the IRS in the user plane is located at the center of a circle. Bob moves as a RWP model in the circle. The spatial separating distance between the mobile user and the projection center in the user plane is denoted as r, 0 ≤ r ≤ R, where R is the maximum activity radius of mobile user.
Signal x is transmitted from the AP via IRS to Bob, then the received signal y by Bob is y = √ P a hx + n, where h is the gain of the reflection channel from the AP via IRS to Bob, P a is the transmit power of the AP, and n is the additive white Gaussian noise (AWGN) with zero mean and variance N 0 .
To make the reflected beamforming of the IRS follow with the mobile user, the phase of the IRS is reconfigured under a smart programmer controlled by the AP. Let the reconfigurable phase of the IRS be ϕ i (i = 1, . . . , N), where N is the total number of meta-surface of the IRS, then the gain of the reflection channel can be expressed as where h 1,i and h 2,i is the channel gain from the AP to the i-th (i = 1, . . . , N) element of the IRS, and from the ith element of the IRS to Bob, given respectively by h 1,i = l −ε/2 μ i exp(− jφ i ) and h 2,i = d −ε/2 ν i exp(− jθ i ), where l and d are the propagation distances from the AP to the IRS, and from the IRS to Bob, respectively; The path-loss coefficient is denoted as ε, being a constant in this paper; μ i and φ i represent the amplitude and phase of the Rayleigh fading channel h 1,i , respectively; and ν i and θ i denote the amplitude and phase of the Rayleigh fading channel h 2,i , respectively. In the proposed IRS-assisted wireless system, φ i and θ i is known and the reconfigurable phase ϕ i of the IRS can be deployed as ϕ i = φ i + θ i to make the received SNR at Bob maximum [14]. The estimation of phase φ i and θ i is not covered in this paper, which will be investigated in the future. Therefore, the reflection channel gain h can be simplified as Then the received SNR at Bob is given by It is obvious from (3) that variable X is relevant to the IRS feature and the channel amplitude fading, while variable Y is relevant to the propagation distance d from the IRS to Bob, and d = √ r 2 + H 2 depends on the spatial distance r of the mobile user deviating from the central point of the circular region. Thus Y is relevant to user mobility. The generation mechanism of X and Y is totally different, thus X and Y can be considered as independently distributed random variable.
To investigate the performance of the proposed IRS-assisted wireless communication system with mobile user, it is necessary to derive first the statistics characteristic of the received SNR Q at Bob.

Statistics characteristic of SNR:
In the following section, on the basis of obtaining the statistics characteristic of X and Y respectively, we finally get the statistics characteristic of the received SNR Q at Bob.
First, the statistics characteristic of variable X is analysed. We can know from [15] that if channel amplitude μ i and ν i are independently and identically Rayleigh distributed random variable, then X obeys squared K G distribution. The probability density function (PDF) and CDF of X can be respectively expressed as [15] f X (x) = 2 g+s (g) (s) = gs/EX (2) and EX (2) is the 2nd-order moment ofX . The derivation of g, s and EX (2) can be referred to Equations (14)-(18) of [17].
Then, we analyse the statistics characteristic of Y . Since Bob moves as a RWP model in the circular region with a spatial separation distance r from the central point, the PDF of r can be given by [9] f r (r) = 12 73 27r Due to the propagation distance d from the IRS to Bob is d = √ r 2 + H 2 and the random variable Y = Cd ε , employing the distribution of the function of the random variable, we can get the PDF of Y given by for Y ∈ [Y min , Y max ], where Y max = C(R 2 + H 2 ) ε/2 and Y min = CH ε , and D 0 = 12 73 1 εC , D 1 = 27 R 2 + 35H 2 R 4 + 8H 4 R 6 , D 2 = 35 R 4 + 16H 2 R 6 , and D 3 = 8 R 6 . and the CDF of Y can be expressed as 6 ( Ymin C ) 6/ε . Finally, we analyse the statistics characteristic of Q. Since Q = X Y , and X and Y are independently distributed, the CDF of Q can be given by where Ymin . In the following, we will calculate F 1 (Q) and F 2 (Q), respectively.
With (5) and (8), F 1 (Q) can be written as On the other hand, F 2 (Q) can be given by where   Outage performance evaluation: The OP performance of the IRSassisted wireless communications system with user mobility following RWP model is analysed in this section. When the received SNR Q of a wireless link is below the threshold Q th , an outage occurs. Hence the OP at Bob is expressed by

Fig 3 OP comparison for the IRS-assisted system and the conventional AF relay system versus various maximum activity radius
Numerical results and discussions: In this section, we present the analytical results for the OP of the proposed IRS-assisted wireless communication system. For the comparison, we also get the OP of a conventional AF relay system where the IRS in the IRS-assisted system is substituted by an AF relay with a fixed amplifying coefficient. Figure 2 plots the OP versus the transmission SNR for a range of values of the number of meta-surface N. We can know that changing the number of meta-surface N from 8, 9 to 10 will cause the OP drops, that is to say, a larger N makes the outage performance better; increasing the transmission power can make the outage performance improved; we can also know that the outage performance of the IRS-assisted wireless system is always superior to that of the conventional AF relay system, which demonstrates that precisely controlling the propagation direction of the signal through the IRS can greatly improve the outage performance. In addition, we can know the simulation results are always consistent with the analytical results. Figure 3 plots the OP versus the transmission SNR for various values of maximum activity radius R of the mobile user. We can see that given maximum activity radius, the outage performance of the IRS-assisted system is always superior to that of the AF relay system. However, given transmission SNR, increasing the maximum activity radius will make the outage performance deteriorated for both the IRS-assisted system and the conventional AF relay system. This is because enlarging R will make the user's received SNR decreased and further make the outage performance degraded.

Conclusion:
The OP of the IRS-assisted wireless communication system with a mobile user moving as a RWP model is investigated. Due to the mobility of the user, the received SNR is affected by both the IRS characteristics and user mobility. Thus on the basis of deriving the PDF and CDF of the independently distributed random variable of X (relevant to IRS feature and channel amplitude fading) and Y (relevant to user mobility), the statistics characteristics of the received SNR Q = X/Y of the mobile user is obtained. Then, the close-form expression of OP for the proposed IRS-assisted system is derived. Numerical results validate the accuracy of the analytical results. The superiority of the proposed IRSassisted system over the conventional AF relay system has been demon-strated. Insights are drawn related to the number of meta-surface elements, and the maximum activity radius of mobile user. The obtained results from this work are helpful for system designer to evaluate the effect of joint IRS feature and user mobility on the system performance.