Recently, reconfigurable intelligent surface (RIS) has been proposed as a novel and cost-effective solution to achieve high spectral and energy efficiency for wireless communications via only low-cost reflecting elements . With a large number of elements whose electromagnetic response (e.g. phase shifts) can be controlled by simple programmable PIN diodes , the RIS can reflect the incident signal and generate a directional beam, and thus enhancing the link quality and coverage.
In literature, most existing works focus on the phase shifts optimizations in the RIS assisted wireless communications. In , the authors proposed a hybrid beamforming scheme for a multi-user RIS assisted multi-input multi-output (MIMO) system together with a limited phase shifts optimization algorithm to maximize the user data rate. However, the performance limits of the RIS assisted communications lack investigations.
In this letter, we consider an uplink cellular network where the direct link between the base station (BS) and the user suffer from deep fading. To improve the quality-of-services (QoS) at the BS, we utilize a practical RIS with limited phase shifts to reflect signal from the user to the BS. To evaluate the performance limits of the RIS assisted communications, we provide an analysis on the achievable data rate with continuous phase shifts of the RIS, and then discuss how the limited phase shifts influence the data rate based on the derived achievable data rate.
The rest of this paper is organized as follows. In Section II, we introduce the system model for the RIS assisted communications. In Section III, the achievable data rate is derived. The impact of limited phase shifts is discussed in Section IV. Numerical results in Section V validate our analysis. Finally, conclusions are drawn in Section VI.
Ii System Model
Consider a narrow-band uplink cellular network consisting of one BS and one cellular user. Due to the dynamic wireless environment involving unexpected fading and potential obstacles, the Light-of-Sight (LoS) link between the cellular user and the BS may not be stable or even falls into a complete outage. To tackle this problem, we adopt an RIS to reflect the signal from the cellular user towards the BS to enhance the QoS. In the following, we first introduce the RIS assisted communication model, and then present the reflection-dominant channel model.
Ii-a RIS assisted Communication Model
The RIS is composed of electrically controlled RIS elements. Each element can adjust the phase shift by leveraging positive-intrinsic-negative (PIN) diode. In Fig. 1, we give an example of the element’s structure. The PIN diode can be switched between “ON” and “OFF” states by controlling its biasing voltage, based on which the metal plate can add a different phase shift to the reflected signal. It is worthwhile to point out that the phase shifts are limited rather than continuous in practical systems .
In this letter, we assume that the RIS is bit coded, that is, we can control the PIN diodes to generate patterns of phase shifts with a uniform interval . Therefore, the possible phase shift value can be given by , where is an integer satisfying . Without loss of generality, the reflection factor of RIS element at the -th row and the -th column is denoted by , where
where the reflection amplitude is a constant.
Ii-B Reflection Dominant Channel Model
In this subsection, we introduce the channel modeling between the BS and the user. As shown in Fig. 2, let and be the distance between the BS and RIS element , and that between element and the user, respectively. Define the transmission distance through element as , where . According to , the reflected LoS component of the channel between the BS and the user via RIS element can then be given by
where is the path loss parameter, is the antenna gain, and is the wave length of the signal.
Benefited from the directional reflections of the RIS, the BS-RIS-user link actually dominates the multipath effect between the BS and the user. Therefore, the channel model between the BS and the user via RIS element can be written by
where is the path loss model for the NLOS link, is the small-scale NLOS component, and is the Rician factor indicating the ratio of the reflected component to the scattered one. Using the geometry information, we can rewritten the channel gain by the following proposition, and the proof can be found in Appendix A.
When the distance between RIS element and the BS and the distance between element and the user are much larger than the horizontal and vertical distances between two adjacent elements, and , i.e., , we have
where and are constants.
Iii Achievable Data Rate Analysis
After traveling through the reflection dominated channel, the received signal at the user can be expressed by
where is the additive white Gaussian noise, is the transmit power, and is the transmitted signal with
. Therefore, the received Signal-to-Noise Ratio (SNR) can be expressed by
where is the conjugate of a complex number .
The received SNR can be maximized by optimizing the response of each RIS element, and thus the achievable data rate can be expressed by
Here, , , and . Derivations are given in Appendix B.
To maximize the data rate, we need to let for any and . Thus, we have the following proposition.
The optimal phase shifts with continuous value should satisfy the following equation:
where is an arbitrary constant. The achievable data rate is
Based on the expression of the achievable data rate, we can have the following remarks to show the upper and lower bounds of the data rate.
The upper bound of the data rate is achieved when we consider the pure LoS channel, i.e., , where an asymptotic received power gain of can be obtained.
The lower bound of the data rate is achieved when we consider the Rayleigh channel, i.e., , where an asymptotic squared received power gain of can be obtained.
These two Remarks show that the data rate grows with , since the received SNR increases from the order of to that of .
Iv Analysis on the Number of Phase Shifts
In this section, we will discuss the influence of limited phase shifts on the data rate. Since the number of phase shifts is finite in practice, we will select the one which is the closest to the optimal one as given in (9), and denote it by . Define the phase shift errors caused by limited phase shifts as
With coding bits, we have .
The SNR expectation with limited phase shifts can be written by
Since , we have . Thus, the following inequality holds:
To quantify the data rate degradation, we define the error brought by limited phase shifts as the ratio of the data rate with limited phase shifts to that with continuous ones. To guarantee the system performance, should be larger than with , i.e.,
Recalling (12), we can obtain the requirements on the coding bits as
Therefore, the required number of coding bits can be written as in (16).
In the following, we will provide a proposition to discuss impact of the RIS size on the required number of coding bits, i.e., the number of phase shifts.
Given the performance threshold , the required coding bits is a decreasing function of the RIS size. Specially, when the RIS size is sufficiently large, i.e., , 1 bit is sufficient to satisfy the performance threshold.
We first discuss how the number of phase shifts influences the required coding bits. Define the RIS size , and
According to (16), the required coding bits has the same monotonicity as . Therefore, we only need to investigate how changes when increases.
Let and . We have
Note that and , we have
This implies that decreases as grows, i.e., the required number of coding bits decreases as the RIS size grows.
When the size of the RIS is sufficiently large, i.e., , since , we have . Therefore, the required number of coding bits should be . ∎
V Simulation Results
In this section, we verify the derivation of the achievable data rate and evaluate the optimal phase shift design for the RIS-assisted cellular network, the layout of which is given in Fig. 3. The parameters are selected according to 3GPP standard  and existing works [2, 3]. The distances between the BS and the user to the center of the RIS are given by m and m, and the heights of the BS and the RIS are 25 m and 10 m, respectively. The RIS separation is set as m and the reflection amplitude is assumed to be ideal, i.e., . The transmit power is dBm, the noise power is dBm, and the carrier frequency is 5.9 GHz. The UMa model in  is utilized to describe the path loss for both LoS and NLoS components. For simplicity, we assume that in this simulation. All numeral results are obtained by 3000 Monte Carlo simulations.
In Fig. 4, we plot the achievable data rate vs. the RIS size with continuous phase shifts. From this figure, we can observe that our theoretic results are tightly close to the simulated ones. We can also observe that the data rate increases with the RIS size , as more energy is reflected. In addition, when the RIS size is sufficiently large, the slope of the curve with the pure LoS channel is 4, which implies that the received SNR is proportional to the squared number of RIS elements111From (10), when is sufficiently large in the pure LoS scenario, the data rate can be written by , where is a constant. Since we use the logarithmic coordinates for the RIS size , the slope of the curve being 4 is equal to the received SNR being an order of .. This result is consistent with Remark 1. Similarly, when the RIS size is sufficiently large, the slope of the curve with the Rayleigh channel is 2, which corresponds to Remark 2. In addition, we can observe that the data rate increases with the Rician factor, i.e., .
In Fig. 5, we plot the performance degradation vs. coding bit with in the Rician channel model  and threshold . From this figure, we can find that the required number of phase shifts with the Rician channel model is: 1) 3 bit when the RIS size is small, e.g., ; 2) 2 bit when the RIS size is moderate, e.g., ; 3) 1 bit when the RIS size approaches to infinity, e.g., . These observations imply that the required coding bits decrease as the number of RIS elements grows, and 1 bit is enough when the RIS size goes to infinity, which verify Proposition 3.
In this letter, we have derived the achievable data rate of the RIS assisted uplink cellular network and have discussed the impact of limited phase shifts design based on this expression. Particularly, we have proposed an optimal phase shift design scheme to maximize the data rate and have obtained the requirement on the coding bits to ensure that the data rate degradation is lower than the predefined threshold.
From the analysis and simulation, given the location of the RIS, we can have the following conclusions: 1) We can achieve an asymptotic SNR of the squared number of RIS elements with a pure LoS channel model, and an asymptotic SNR of the number of RIS elements can be obtained when the channel is Rayleigh faded; 2) The required number of phase shifts will decrease as the RIS size grows given the data rate degradation threshold; 3) A number of phase shifts are necessary when the RIS size is small, while 2 phase shifts are enough when the RIS size is infinite.
Appendix A Proof of Proposition 1
Assume that the BS is located at the local origin and the angle between the direction of the RIS and the - plane as . Define the principle directions of the RIS as and , we have and , where , , and are directions of , , and axis.
Denote as the position of RIS element , where
Here, is the projected distance on -asix between the BS and the RIS. Therefore, we have
which can be achieved by when . We can obtain the expression of the distance between the RIS and the user using the similar method. Since the distance between two RIS elements is much smaller than that between the BS and user, the pathloss of the BS and user via different RIS element can be regarded a constant.
Appendix B Derivations of Equation (8)
Due to the property of the logarithmic function, we have
Since is constant, we will derive in the following.
Since has a zero mean, the final term in (23) equals to 0. Moreover, since is independent for different elements and , the following equation holds:
Therefore, we have
This ends the proof.
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