Is RIS-Aided Massive MIMO Promising with ZF Detectors and Imperfect CSI?

11/02/2021
by   Kangda Zhi, et al.
FAU
Queen Mary University of London
0

This paper provides a theoretical framework for understanding the performance of reconfigurable intelligent surface (RIS)-aided massive multiple-input multiple-output (MIMO) with zero-forcing (ZF) detectors under imperfect channel state information (CSI). We first propose a low-overhead minimum mean square error (MMSE) channel estimator, and then derive and analyze closed-form expressions for the uplink achievable rate. Our analytical results demonstrate that: 1) regardless of the RIS phase shift design, the rate of all users scales at least on the order of 𝒪(log_2(MN)), where M and N are the numbers of antennas and reflecting elements, respectively; 2) by aligning the RIS phase shifts to one user, the rate of this user can at most scale on the order of 𝒪(log_2(MN^2)); 3) either M or the transmit power can be reduced inversely proportional to N, while maintaining a given rate. Furthermore, we propose two low-complexity majorization-minimization (MM)-based algorithms to optimize the sum user rate and the minimum user rate, respectively, where closed-form solutions are obtained in each iteration. Finally, simulation results validate all derived analytical results. Our simulation results also show that the maximum sum rate can be closely approached by simply aligning the RIS phase shifts to an arbitrary user.

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I Introduction

Massive multiple-input multiple-output (MIMO) has been widely recognized as a cornerstone technology for fifth-generation (5G) and beyond wireless communications[lu2014overview, bjornson2017massive, ngo2013energy, zhang2014power, xiao2020uav, ozdogan2019massive, liu2021cellfree]. Thanks to their spatial multiplexing gains, massive MIMO systems can simultaneously provide high quality of service for multiple users on the same time-frequency resource. Massive MIMO also has some other appealing properties, e.g., the transmit power can be reduced inversely proportional to the number of antennas without sacrificing the achievable rate.

However, conventional massive MIMO still has some drawbacks. The first one is the blockage problem. Due to the complex environment and user mobility, communication links may be blocked, in which case the channel strength could be severely degraded. Another problem is the high cost and energy consumption of the active radio-frequency (RF) chains. Massive MIMO commonly employs hundreds of antennas, each of which will be connected to a RF chain. Hence, this system incurs high hardware cost and energy consumption.

The recently developed technology of reconfigurable intelligent surfaces (RISs)[di2020smart, pan2020intelligent, wu2019intelligent, pan2020multicell, huang2019reconfigu, yuxianghao2020robost], also referred to as intelligent reflecting surfaces (IRSs), is a promising solution for tackling the above two issues in massive MIMO systems. On the one hand, since the RIS is a small, thin and light surface, it can be flexibly deployed at a carefully selected location with a favorable propagation environment. Therefore, RISs enable additional high-quality communication paths to overcome the blockage problem. On the other hand, RISs are comprised of low-cost passive reflecting elements, which are much cheaper than active RF chains. Therefore, it is envisioned that RISs are beneficial for improving the energy efficiency of conventional massive MIMO systems.

Due to these appealing features, RIS-aided massive MIMO has gained growing research interests with many activities, focusing on various applications and different perspectives, such as channel estimation [he2019cascaded], dual-polarized transmission[de2021polarize], millimeter wave (mmWave) communications[wang2021massiveMIMO], hardware impairments[papazafeiropoulos2021intelligent], multi-RISs co-design[mei2021multibeam], cell-free systems[van2021reconfigurable], antenna selection[he2020reconfigurable], and power scaling law analysis [bijoson2020nearField, zhi2020power, zhi2021twotimescale].

To fully understand the potential of RISs, it is essential to draw theoretical insights from information-theoretical expressions, which rigorously demonstrate the impact of the various system parameters. Fundamental information-theoretical expressions for conventional massive MIMO systems have been provided in, e.g., [bjornson2017massive, ngo2013energy, zhang2014power]. It was shown that the achievable rate of conventional massive MIMO systems with antennas scales on the order of . This naturally raises the question what the corresponding scaling law for massive MIMO systems after the integration of RISs is. To answer this question, explicitly analytical rate expressions are required. It has already been shown that in RIS-aided single-user systems with reflecting elements, the achievable rate could scale as [wu2019intelligent, han2019large], or even [9060923] if two RISs cooperate. Similar scaling orders were also reported for some other RIS-aided communication scenarios, such as the RIS-aided relay[kang2021irs], RIS with scattering parameter analysis [shen2021parameter], and RISs with hardware impairments[qian2021scaling, xing2021hw]. However, these works focused on the simple single-user case, and cannot be easily generalized to multi-user systems.

In fact, it is challenging to provide an insightful analysis for the rate scaling order of RIS-aided multi-user systems. This is because the resulting signal-to-interference-plus-noise ratio (SINR) expressions are more complicated and more involved than the interference-free signal-to-noise ratio (SNR) expressions for single-user systems, and also because the optimal RISs passive beamforming vectors cannot be given in closed form in case of multiple users. Some initial results were provided in

[zhi2020power] and [zhi2021twotimescale] by considering RIS-aided massive MIMO with simple maximal ratio combining (MRC). For uncorrelated Rayleigh fading channels, it was proved that the achievable rate scales only as with respect to . This is due to the severe multi-user interference, since the common RIS-base station (BS) channel is used by all users. To tackle this issue, most recently, the authors in [zhi2021ergodic] firstly revealed that a rate scaling order is achievable with zero-forcing (ZF), which demonstrates the huge potential of ZF detectors in RIS-aided massive MIMO systems.

However, there are two main limitations in [zhi2021ergodic]. Firstly, ideal channel state information (CSI) of the aggregated channel including the superimposition of the direct channel and the reflected channel, was assumed. Secondly, the authors in [zhi2021ergodic] only considered some initial performance analysis and RIS phase shift optimization, which lacks further insightful analysis. By contrast, this work aims to provide an analytical framework to gain an in-depth analysis for the performance of RIS-aided massive MIMO systems with ZF detectors under the realistic assumption of imperfect CSI.

Specifically, in this work, we first propose a low-overhead channel estimation scheme, in which the required pilot length is independent of . We next perform a comprehensive theoretical analysis to reveal the explicit rate scaling order and answer the fundamental question whether the RIS-aided massive MIMO with ZF detectors is promising or not. Finally, based on majorization-minimization (MM) algorithms, we respectively optimize the RIS phase shifts to maximize the sum user rate and the minimum user rate. The detailed contributions are summarized as follows.

1) Low-overhead channel estimation: We first propose a minimum mean square error (MMSE)-based method to estimate the aggregated channel in the systems, which is a superimposition of cascaded RIS channels and the direct channels. The length of pilots only needs to be no smaller than the number of users. We also analyze the impacts of various system parameters on the mean square error (MSE).

2) Reveal rate scaling orders: We derive the closed-form ergodic rate expression and its insightful lower and upper bounds. The lower bound shows that the data rates of all users are guaranteed to be on the order of , regardless of the RIS phase shift design. The upper bound shows that the data rate of a specific user can be on the order of , if the RIS phase shift is designed to align its beamforming to that user. We also demonstrate that these two analytical results are robust to RIS phase shift quantization errors.

3) Answer the question whether the considered system is promising or not: Based on the analytical results, we prove that RIS-aided massive MIMO systems with ZF detector are promising for three applications. It can provide ultra-high network throughput according to the high data rate scaling order for all users; it can help reduce inversely proportional to without sacrificing the data rate, which helps avoid the power hungry RF chains and is promising for green communications; it can help all users communicate with small transmit power, inversely proportional to , which is promising for IoT applications.

4) Low-complexity RIS optimization:

We design the RIS phase shifts to maximize the sum user rate and minimum user rate, based on the MM algorithm with closed-form solution in each iteration. We also show that aligning RIS phase shifts to an arbitrary user is an effective heuristic approach for maximizing the sum user rate. In addition, we demonstrate that maximizing the sum rate can also ensure a high minimum user date.

The rest of this paper is organized as follows. Section II describes the system and channel model. Section III proposes the MMSE channel estimation scheme. Section IV theoretically proves that RIS-aided massive MIMO is promising with ZF detectors. Section V proposes the MM algorithm for solving the sum rate and minimum user rate maximization problems. Section VI provides extensive simulations to verify the the correctness of analytical results and the effectiveness of proposed optimization algorithms. Finally, Section VII concludes this work.

Notations: Boldface lower case and upper case letters denote the vectors and matrices, respectively. The inverse, conjugate transpose, conjugate and transpose of matrix are denoted by , , , , respectively. The -th and -th elements of the matrix are represented by and . and respectively denote that is definite positive and semi-positive. denotes the standard big-O notation. and

denote the maximal eigenvalue and the phase of matrix

. and denote the mean and covariance operators.

Ii System and Channel Model

Fig. 1: Massive MIMO systems assisted by an RIS.

As shown in Fig. 1, the uplink transmission of an RIS-assisted massive MIMO system is considered. The considered system consists of users with a single antenna, a BS with antennas, and an RIS with reflecting elements. Besides, we assume a quasi-static channel model with each channel coherence interval (CCI) spanning time slots. In each CCI, we denote the instantaneous channel between the users and the RIS, and that between the RIS and the BS as and , respectively. Then, the cascaded user-RIS-BS channel is , where is the RIS phase shift matrix. Meanwhile, the direct channels between the users and the BS are denoted as . Finally, in each CCI, the instantaneous aggregated channels from the users to the BS are given by .

It has been shown in [zhi2021twotimescale] that it is better to place an RIS close to the users rather than close to the BS in the massive MIMO systems. Therefore, in this paper, we assume that the RIS is deployed on the facade of a tall building in the proximity of the users, as illustrated in Fig. 1. Since the RIS has a certain height and is close to the users, the user-RIS channels would be line-of-sight (LoS) dominant. For analytical tractability, we assume that the user-RIS channels are purely LoS as follows

(1)

where is the large-scale path loss factor for user , and is the deterministic LoS channel between user and the RIS.

Since the RIS is installed close to the users, it may be located far away from the BS. Therefore, both LoS and non-LoS (NLoS) transmission paths would exist in . As a result, we characterize the RIS-BS channel by Rician fading, which is expressed as

(2)

where is the path loss factor, and is the Rician factor which represents the ratio between the power of LoS component and the power of NLoS component . The elements of

are independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance. For a rich-scattering environment, we can assume

and then the RIS-BS channel reduces to a Rayleigh fading channel containing only NLoS paths. For a scattering-free environment, we have and then the RIS-BS channel is purely LoS.

Finally, since the users might be located far away from the BS, and rich scatterers (trees, cars, buildings and so on) are distributed on the ground, we assume that the channels between the users and the BS are Rayleigh fading [han2019large]. Thus, we have

(3)

where is the channel between user and the BS with large-scale fading coefficient and small-scale fading vector comprised of i.i.d. complex Gaussian random variables with zero mean and unit variance. Here, and

We adopt the two-dimensional uniform rectangular array (URA) to model the LoS channels[wu2019intelligent]. For an LoS channel , we first decompose into two closest integers and , where . Then, the -th element of is given by

(4)

where and denote the azimuth and elevation angles of arrival (AoA) or corresponding angles of departure (AoD). Based on (4), it can be shown that . Then, we can express that

(5)
(6)

Iii Channel Estimation

To design the ZF detector, the channels are estimated by the BS using a pilot-based method. For conventional massive MIMO systems, only the direct channel needs to be estimated, and the minimum pilot sequence length is . In RIS-aided massive MIMO systems, the required pilot overhead can be prohibitive due to the extremely large channel dimension of in the RIS-BS link. To reduce the pilot overhead, we only estimate the aggregated channel , for which the minimum pilot sequence length is still , which is the same as for conventional massive MIMO systems.

Specifically, in each CCI, the users are assigned mutually orthogonal pilot sequences with length . The pilot sequence of user is denoted by . Let , where due to the orthogonality. Then, at the beginning of each CCI, time slots are used for the users to transmit the pilot signal to the BS. The received pilot signal at the BS can be given by , where is the transmitted pilot power of each user, and is the noise matrix whose elements are i.i.d. Gaussian variables following . Then, we can obtain the observation vector for the channel of user by multiplying the term to , as follows

(7)

where , the -th column of , denotes the aggregated channel of user .

Lemma

Channel and noise in (7

) are complex Gaussian distributed, where

, and .

Proof: Please refer to Appendix A.

From Lemma III, it is seen that the considered channel is still Gaussian distributed as conventional massive MIMO systems[ozdogan2019massive, Eq. (1)], but with the different mean and variance. Therefore, we can still apply the well-used MMSE estimator to obtain the channel estimate of .

Theorem

The MMSE estimate of channel is given by

(8)

where . Denote the estimation error as , where the error is independent of the estimate . Then, the MSE matrix for the channel estimation is

(9)

Proof: Please refer to Appendix B.

Based on (9), the MSE can be calculated as . Clearly, the MSE is a decreasing function of , , and , but an increasing function of , , , , , and . This is because represents the pilot SNR, and increasing its value improves the estimation quality. is the Rician factor, and increasing its value makes the RIS-aided channels more deterministic and therefore decreases the estimation error. Also, the increase of introduces more communication paths between the users and the BS, which also increases the estimation error.

Note that in the absent of the RIS (i.e., ) or for a purely LoS RIS-BS channel (), the MSE matrix in (9) reduces to , which is the same as for conventional massive MIMO systems [ngo2013energy]. Let denote the estimated aggregated channel of the users. Then, based on (8), we have

(10)

where .

Iv Ergodic Rate Analysis

In the transmission phase, the users transmit symbols where , and the received signal at the BS can be expressed as

(11)

where and . To eliminate the multi-user interference, the BS adopts the linear ZF detectors , which leads to . Then, in each CCI, the BS detects the received signal as follows

(12)

whose -th entry can be further expressed as

(13)

Iv-a Derivatives of the Achievable Rate

Based on (13), the accurate ergodic rate of user can now be given by

(14)

where a factor captures the rate loss caused by pilot overhead, and the expectation is taken over random channel components in . It is difficult to derive an exact expression of (14) due to the expectation operator before the logarithm symbol. Since the function is convex of , we utilize the Jensen’s inequality to obtain the following lower bound

(15)
(16)

where is defined in (9), utilizes the independence between the channel estimate and the estimation errors, and is due to the result in (9) and .

Theorem

The achievable rate of user is lower bounded by

(17)

where .

Proof: Please refer to Appendix C.

The rate expression in Theorem IV-A depends only on the slowly varying statistical CSI. Therefore, when designing the phase shifts to maximize the rate in (17), we only need to update the RIS’s phase shifts over a much large time scale, which could effectively reduce overhead and computational complexity. Before the design of the phase shifts, we first analyze (17) to shed some light on the benefits of the RIS, and to answer the question whether RIS-aided massive MIMO is promising or not.

Iv-B Conventional Systems without RIS

Corollary

When the RIS is switched off (i.e., ), the data rate (17) reduces to

(18)

When the RIS is switched off, the RIS-aided massive MIMO systems degrade to the conventional massive MIMO systems with Rayleigh fading channels (), which has been studied in [ngo2013energy]. As expected, the obtained rate (18) is the same as [ngo2013energy, Eq. (42)]. Based on (18), it can be seen that the rate is on the order of , and the rate can maintain a non-zero value when the power is scaled down proportionally to , as the number of antennas , where is a constant. Specifically, we have

(19)

Note that the achievable rate in (18) and power scaling law in (19) will serve as baselines and help us identify the benefits enabled by introducing an RIS.

Iv-C What’s New After Integrating An RIS?

The order of magnitude of in (17) with respect to is , since and are independent of . However, it is challenging to determine how scales with , due to the unknown value of and the inverse operator. For tractability, we propose an insightful lower bound for in the following.

Corollary

A -independent lower bound is given by

(20)

where equality holds when , and the gap enlarges after optimizing . Besides, (20) can be approximated as

(21)

which scales on the order of .

Proof: Please refer to Appendix D.

Interestingly, if we treat as a new path-loss factor, (21) possesses the same form as (18). This reveals two fundamental impacts of the RIS: Positive effect: RIS enhances the channel strength by a factor ; Negative effect: RIS results in larger channel estimation errors . However, the channel strength always increases with since is an increasing function of , but the estimation error saturates to as . Therefore, for large , the benefits of the RIS outweigh its drawbacks in massive MIMO systems.

Corollary IV-C proves that even with imperfect CSI, RIS-aided massive MIMO systems can achieve an ergodic rate at least on the order of . This promising gain comes from the additional paths contributed by the RIS for each user, such that more signals can be collected by the BS. Compared with in conventional systems, Corollary IV-C proves that much higher capacity can be achieved after integrating an RIS. More importantly, the scaling law indicates that if we want to maintain a fixed rate, the number of antennas can be reduced inversely proportional to the number of RIS elements. For better understanding, we provide a quantitative relationship for a special case.

Corollary

When and for large , to achieve for a given , the required number of antennas is approximately given by

(22)

Proof: When , we have . Then, using (21), for large , we have , and . Solving the equation completes the proof.

Corollary IV-C corresponds to the scenarios with rich scattering. Eq. (22) clearly exhibits the inverse proportional relationship between and . Meanwhile, intuitively, increases with , , and , but decreases with the link strengths and . Since the RIS’s reflecting elements consume much less energy than RF chains, Corollary IV-C states that the energy efficiency can be remarkably improved by integrating an RIS.

Corollary

If the RIS-BS channel is purely LoS (), RIS-aided massive MIMO systems perform no worse than conventional massive MIMO systems, i.e., .

Proof: Substituting into (20), , and , it can be shown that . Then, we have .

Corollary IV-C corresponds to the scenario where the RIS is carefully deployed to reduce the scatters and obstacles between the BS and the RIS. In this case, the additional channel estimation error in , caused by the RIS, vanishes. Therefore, the RIS only has the positive effect of enhancing the channel strength, which improves the achievable rate. We emphasize that even though we can only prove that RIS-aided systems are no worse than conventional systems when , in general, it could perform much better because the second lower bound is not as tight as the first lower bound if is carefully designed.

Iv-D Power Scaling Law

In conventional massive MIMO systems, an attractive feature is that the transmit power can be scaled down proportionally by increasing [zhang2014power, ngo2013energy, bjornson2017massive]. After introducing an RIS, we reveal a new power scaling law with respect to , and compare it to (19).

Corollary

As , when the power is scaled proportionally to , the achievable rate in (17) can maintain a non-zero value , where

(23)

with

Proof: Substitute into (17). As , we have , , , , and , which help us arrive at the first equation in (23). Then, using the inequality in (D), we can obtain the lower bound.

Comparing (23) with (19), it can be seen that this new scaling law has a high order of magnitude with respect to . Besides, by comparing (23) with (18), it is interesting to find that (23) can be interpreted as the SNR achieved by a conventional massive MIMO system with transmit power and path-loss . To sum up, for large and , transmit power can be significantly reduced while achieving high data rates.

Iv-E Comparison with MRC-based Systems

Corollary

When or or is large, ZF-based RIS-aided massive MIMO outperforms its MRC-based counterpart. Besides, the severe fairness problem in MRC-based RIS-aided massive MIMO system [zhi2021twotimescale, Remark 2] does not exist in the considered ZF-based systems.

Proof: According to Corollary IV-C, when or grows without bound, it is found that . Thus, all users can have infinite data rates. However, as proved in [zhi2021twotimescale, Remark 2], when using MRC detectors, due to the mutual interference, the rate is still bounded when or is large. Meanwhile, the rates of all users in the considered system are at least on the order of . However, when using MRC, the rate of only one user can be on the order of , while the rates of all other users degrade to zero when is large, which results in a serious fairness problem.

ZF-based RIS systems perform better since RIS-aided systems suffer from severe multi-user interference. This is because multiple users share the common RIS-BS channel, and thus the users’ channels are highly correlated. The highly correlated channels result in severe interference and low data rate. However, by using ZF, the severe multi-user interference issue can be addressed, which leads to promising performance for various aspects.

Iv-F The Upper Bound

The analysis based on the lower bound is rigorous but conservative, since it ignores the performance gain achieved by optimizing . We next provide an upper bound to unveil the maximum gain achieved by optimizing .

Corollary

The rate is upper bounded by , where

(24)
(25)

Based on (24), is at least on the order of . Based on (25), is on the order of .

Proof: Please refer to Appendix E.

We emphasize that (24) holds for all users but (25) does not. This is because (25) is achieved by aligning the RIS phase shifts to a specific user , i.e., . However, when , it is known that , , is bounded even for [zhi2021twotimescale]. Thus, the additional -fold gain in (25) comes from the concentration of passive beamforming on user . Combining the lower bound in Corollary IV-C and this upper bound, we highlight the following conclusion:

Remark

If we align the RIS phase shifts for one user, the rate of this user will scale at most on the order of , while the rates of the other users scale at least on the order of , which is high as well.

Based on these two achievable rate scaling laws, the sum user rate will be high for large and , if we simply align the RIS phase shifts for an arbitrary user, which constitutes a low-complexity heuristic approach for the sum-rate maximization problem.

Corollary

The quantization error caused by RIS discrete phase shifts does not impact the derived achievable rate scaling orders.

Proof: First, the lower bound does not depend on , and hence, is not affected by quantization errors. Secondly, holds for an RIS with -bit quantization[han2019large]. Therefore, scaling order still holds for .

Iv-G Summary

We summarize that RIS-aided massive MIMO with ZF detectors is promising for

  • Green communications (Corollary IV-C) : The number of BS antennas can be reduced inversely proportional to the number of RIS elements, while maintaining a constant rate.

  • Enhanced mobile broadband (Corollary IV-C, IV-F, IV-F, Remark IV-F) : According to the rate scaling orders, ultra-high throughput requirement can be achieved for large and .

  • Internet of things (Corollary IV-D) : For large and , all users can significantly reduce their transmit powers while maintaining high data rates.

V RIS Phase Shift Design

In this section, based on the derived rate expression in (17) and the low-complexity MM technique[sun2017MM], we aim to solve the sum user rate maximization (Max-Sum) and the minimum user rate maximization (Max-Min) problems, respectively. The Max-Sum problem maximizes the utility but may sacrifice fairness. On the contrary, the Max-Min problem guarantees fairness but may sacrifice utility. Thus, simultaneously investigating both problems can help us understand which optimization criterion is more suitable for the considered systems. For tractability, variable is rewritten as , where . Then, we can transform the design of to the design of vector .

Lemma

The rate in (17) can be rewritten as , where

(26)

and . Besides, we have and .

Proof: We can complete the proof by substituting the last equality in (D) into (17), and using and . Besides, we have due to , which results in . Since the rate must be non-negative due to its definition in (15), we obtain , which means that .

Define for brevity. Since the same factor is included in , we can ignore it and formulate the following two optimization problems

(27)
(28)

To successfully solve the above two problems under the MM algorithm framework, tractable lower-bound surrogate functions need to be constructed for objective functions in (27) and (28), and then closed-form optimal solutions are expected to be derived via the surrogate functions.

V-a Max-Sum Problem

Lemma

For a fixed point , a lower bound of is given by

(29)

where

(30)

Proof: Please refer to Appendix F.

Then, the Max-Sum problem (27) can be directly solved based on the proposed surrogate function in Lemma V-A. Denoted by the solution in the -th iteration, the closed-form optimal solution in the -th iteration is given by

(31)

V-B Max-Min Problem

Next, we focus on the Max-Min problem (28), which is more challenging since the objective function is non-differential. Therefore, we first adopt the log-sum-exp approximation in [xingsi1992entropy] to obtain a lower-bounded smooth objective function, as follows

(32)

where is a constant for controlling the approximation accuracy, and the last inequality can be proved similar as [xingsi1992entropy, (15)].

Lemma

For a fixed point , in (32) is lower bounded by

(33)

where

(34)
(35)

Proof: Please refer to Appendix G.

Based on the MM algorithm, the Max-Min problem (28) can be solved by maximizing the lower bound