Maximizing Spectral and Energy Efficiency in Multi-user MIMO OFDM Systems with RIS and Hardware Impairment

1 Maximizing Spectral and Ener gy Ef ficienc y in Multi-user MIMO OFDM Systems with RIS and Hardware Impairment Mohammad Soleymani, Member , IEEE , Ignacio Santamaria, Senior Member , IEEE , A ydin Sezgin, Senior Member , IEEE , and Eduard Jorswieck, F ellow , IEEE Abstract —An emerging technology to enhance the spectral efficiency (SE) and energy efficiency (EE) of wireless communica- tion systems is reconfigurable intelligent surface (RIS), which is shown to be v ery powerful in single-carrier systems. However , in multi-user orthogonal frequency division multiplexing (OFDM) systems, RIS may not be as promising as in single-carrier systems since an independent optimization of RIS elements at each sub- carrier is impossible in multi-carrier systems. Thus, this paper in vestigates the performance of various RIS technologies like reg- ular (reflective and passive), simultaneously transmit and r eflect (ST AR), and multi-sector beyond diagonal (BD) RIS in multi- user multiple-input multiple-output (MIMO) OFDM broadcast channels (BC). This requires to f ormulate and solve a joint MIMO precoding and RIS optimization problem. The obtained solution reveals that RIS can significantly improve the system performance ev en when the number of RIS elements is relati vely low . Moreover , we develop resource allocation schemes for ST AR- RIS and multi-sector BD-RIS in MIMO OFDM BCs, and show that these RIS technologies can outperform a r egular RIS, especially when the r egular RIS cannot assist the communications for all the users. Index T erms —Improper Gaussian signaling, I/Q imbalance, MIMO OFDM systems, reconfigurable intelligent surface, spec- tral and energy efficiency . I . I N T R O D U C T I O N Smart radio en vironments can be realized by employing modern intelligent metasurface technologies [2]. Indeed, wire- less channels in systems aided by reconfigurable intelligent surfaces (RIS) are not determined by nature only and can be optimized [3]. This can yield additional degrees of freedom A preliminary version of a portion of this work was presented in Pr oc. of IEEE CAMSAP 2023 [1]. Mohammad Soleymani is with the Signal and System Theory Group, Uni- versit ¨ at Paderborn, Germany , (email: mohammad.soleymani@uni- paderborn. de). Ignacio Santamaria is with the Department of Communications Engineer- ing, Universidad de Cantabria (email: i.santamaria@unican.es). A ydin Sezgin is with the Ruhr University Bochum, Germany (email: aydin. sezgin@rub .de). Eduard Jorswieck is with the Institute for Communications T echnology , T echnische Universit ¨ at Braunschweig, 38106 Braunschweig, Germany (e- mail: jorswieck@ifn.ing.tu- bs.de) The work of Ignacio Santamaria was funded by MCIN/ AEI /10.13039/501100011033, under Grant PID2022-137099NB-C43 (MADDIE). This work of A ydin Sezgin is funded by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM Research Hub under Grant 16KISK037. The work of Eduard Jorswieck was supported by the Federal Ministry of Education and Research (BMBF , Germany) through the Program of “Souver ¨ an. Digital. V ernetzt. ” joint Project 6G-RIC, under Grant 16KISK031. (DoF) in designing systems, which can be utilized to substan- tially improv e the performance of wireless systems, especially when the system is single carrier [4], [5]. Howe ver , in multi- carrier systems, the sub-carriers undergo dif ferent channel states. Thus, the relativ e improv ement per carrier might be lower since RIS elements cannot be optimized independently at each sub-carrier . Hence, it should be in vestigated how well RIS can perform in multi-carrier systems in which one is unable to optimize each carrier independently of other carriers. In this paper, we address this issue and propose a framework to enhance the spectral efficiency (SE) and energy ef ficiency (EE) of multi-user multiple-input, multiple-output (MIMO) orthogonal frequenc y division multiplexing (OFDM) systems by jointly optimizing the power/transmit co variance matrices and the RIS elements. Note that SE and EE improv ements are always among the main concerns of modern wireless communication systems, where a goal in 6G is to achiev e SE 10 times and EE 100 times higher than 5G systems [6], [7]. This work shows that the RIS benefits in multi-user MIMO OFDM systems decrease with the number of sub-carrier/sub- bands, but the benefits are still significant e ven when the number of RIS elements per user/sub-carrier is very lo w . A. Literatur e re view Note that there are various technologies for intelligent metasurfaces. The simplest architecture is the passi ve reflecti ve RIS, which is referred to as a regular RIS in this paper . It has been shown that RIS can be a po werful technology to impro ve the SE and EE of sev eral single-carrier systems [8]–[24]. For e xample, the papers in [9], [10] illustrated that RIS can improv e the performance of a single-cell multiple- input single-output (MISO) broadcast channel (BC). In [11], [14], the superiority of RIS was examined in multi-cell MIMO BCs by considering different performance metrics such as the minimum and sum rate, global EE and minimum EE of users when transceivers suf fer from hardware impairment (HWI). The authors in [17] sho wed that RIS can increase the weighted sum rate of the secondary users in a MIMO cognitive radio system. In [12], it was demonstrated that RIS can enlarge the achiev able rate region of a single-cell MIMO BC. In [18], [19], RIS was employed as a tool to reduce the interference leakage of the K -user interference channels. The paper in [20] exhibited that RIS can enhance the resilience of cell-free MIMO systems. 2 A drawback of a regular RIS is that it cannot provide an omni-directional coverage. In other words, the transmitter and receiv er ha ve to be in the reflection space of a regular RIS so that the regular RIS can bring any benefit to the system. Thus, a regular RIS supports only a half-space cov erage. T o address this issue, another technology for intelligent metasurfaces is proposed that can allo w transmission and reflection at the same time, which in kno wn as simultaneously transmit and reflect (ST AR) RIS [25]. Note that ST AR-RIS is also kno wn as intelligent omni-surfaces since it can provide an omni- directional coverage [25], [26]. The superiority of ST AR- RIS o ver regular RIS has been inv estigated in [16], [27]– [31] by taking into account dif ferent scenarios. For instance, in [27]–[29], it was sho wn that ST AR-RIS can improv e the performance of a two-user BC with single-antenna receiv ers. In [16], we proposed schemes to increase the EE of a single- cell MIMO ST AR-RIS-aided BC. In [30], it was demonstrated that ST AR-RIS can increase the minimum rate of users in a multi-cell BC with I/Q imbalance (IQI). Moreov er , in [31], it was shown that ST AR-RIS can enhance the performance of MISO ultra-reliable low-latency communications (URLLC) BCs. The concept of ST AR-RIS can further be generalized to multi-sector beyond diagonal (BD) RIS in which the cov erage area is divided into multiple sectors (possibly more than two) [32], [33]. In this case, the multi-sector BD-RIS can receiv e signal in a sector and partially reflect it in all the sectors, which enables a full-space coverage similar to ST AR-RIS. In this sense, multi-sector BD-RIS can be also cate gorized as omni-directional surfaces. Note that in BD-RIS, the matrix for the RIS coefficients can be non-diagonal since the BD-RIS elements can be connected via a circuit design. In this paper , we consider the single-connected multi-sector BD-RIS in which the RIS elements in each sector are not connected to the other RIS elements in the sector . It should be emphasized that the main advantage of multi-sector BD-RIS over ST AR-RIS is that the multi-sector BD-RIS can provide more directional beams in each sector , which can enhance the channel gain [33]. Additionally , the channel of a user in a sector can be optimized independently of the channels of the users in the other sectors. This simplifies optimization of the channels of the users and enhance the performance of RIS. Y et, one of the main factors limiting the performance of wireless communication systems is HWI. In practice, de vices nev er perform ideally , and if we do not take the device imperfections into consideration when designing a system, the system performance can drop significantly [34]–[37]. A common source of HWI is I/Q imbalance (IQI), which happens because of an imbalance in the in-phase and quadrature signals [35], [36]. IQI is modeled as a widely linear transformation (WL T), which makes the output signal improper when the input signal is proper [35], [36]. T o compensate for IQI, one can resort to improper Gaussian signaling (IGS), which is also an effecti ve interference-management tool, especially in single-carrier systems [34], [38], [39]. Ho wev er , the benefits of IGS as an interference-management technique may disappear in multi-carrier systems as the number of resources per users increases, which in turn may help to easier manage interfer- T ABLE I: Summary of the most related papers on OFDM RIS-aided systems. This paper [41]–[50] [51], [52] [53] MIMO √ √ EE metrics √ Fairness rate √ Multi-user √ √ HWI √ √ IGS √ ST AR-RIS √ Multi-sector BD-RIS √ ence by a simpler transmission strategy [40]. Now , single-carrier systems are very efficient in frequency- flat channels. Ho wev er , this might not necessarily be the case when the channels are frequency selective since a demand- ing process is needed for equalizing the channels in single- carrier systems. T o easier cope with frequency selectivity of wireless channels, multi-carrier techniques such as OFDM can be employed, which are able to more efficiently utilize the spectrum. Multi-carrier systems divide the frequency band into sev eral sub-bands, which permits a more intelligent power allocation, based on the channel response on each frequency sub-band. Ho wev er, RIS cannot exploit this feature since it is impossible to independently optimize RIS at each sub-band [45]. In a more realistic scenario, the RIS elements remain approximately constant across all frequency subbands, which is highly suboptimal when the number of subbands gro ws. Thus, one might e xpect that the benefits of RIS may disappear when the bandwidth is large, and consequently , the number of subbands is high. Although limited, yet a few studies exist on RIS-aided multi-user OFDM systems [41]–[66]. Many studies on RIS- aided OFDM systems considered a single-user system [51]– [66]. For instance, in [55], the authors proposed a joint channel estimation and passi ve beamforming scheme for a RIS-aided OFDM system and sho wed that RIS can substantially improve the SE of the system. A lo w-cost passi ve beam forming was proposed in [56] for point-to-point SISO RIS-aided OFDM systems, and it was sho wn that RIS can increase the achie vable rate of the system. The authors in [58] considered a point- to-point single-input single-output (SISO) RIS-aided OFDM system and proposed schemes to estimate channels as well as to optimize RIS elements. In [59], the authors dev eloped a practical transmission protocol for point-to-point SISO RIS- aided OFDM systems. The authors in [51] proposed schemes to optimize a MIMO OFDM simultaneous wireless infor- mation and power transfer (SWIPT) RIS-aided system with a non-linear energy harvesting model and showed that RIS can enhance the system performance. In [52], the authors considered a point-to-point MIMO RIS-aided OFDM system and showed that RIS can improv e the SE of the system. The performance of RIS in a multi-user OFDM system has been studied in [41]–[49]. In [43], [45], it was sho wn that RIS can increase the sum-rate of multi-user multiple-input single-output (MISO) RIS-aided OFDM systems. In [44], the authors studied a multi-user SISO RIS-aided OFDM BC and optimized the RIS elements to maximize the total power receiv ed by users for a given transmit power . The authors in 3 [42] considered an uplink of a multi-user OFDM system, and showed that RIS can decrease the bit error rate of the system. B. Motivation W e provide a brief summary for the papers on RIS in OFDM systems in T able I, based on the system model, the considered performance metric, assumptions regarding the devices, and the considered RIS technology . Even though the studies in [41]–[49], [51], [52], [54]–[66] ha ve provided a valuable insight on the performance of RIS in OFDM systems, there are still some open problems in RIS-aided OFDM systems, which are the focus of this paper . For instance, the performance of RIS has not been studied in multi-user MIMO OFDM systems. Most of the modern wireless communication systems such as 4G and 5G employ MIMO and OFDM, which suggests developing resource management techniques for RIS-aided MIMO OFDM systems. Additionally , there is no w ork on RIS-aided OFDM systems that considers the EE metrics, even in single-antenna, single-user OFDM systems. As indicated before, a main goal of 6G is to increase EE by an order of approximately 100 times compared to 5G systems, which makes energy-ef ficient schemes vital for 6G. Furthermore, there is only one paper (i.e., [53]) on RIS-aided OFDM systems, which considered a single-user user system with additiv e hardware distortion noise. Indeed, IQI has not been studied in RIS-aided OFDM systems, while in practice, devices are ne ver ideal and may suf fer from IQI and/or other sources of imperfections. Moreov er , there is no work on RIS- aided OFDM systems with more recent RIS technologies such as ST AR-RIS or multi-sector BD-RIS. As a summary , there are still some open research lines in MU-MIMO RIS-aided OFDM systems, which require further in vestigations. These include de veloping HWI-aware designs, energy-ef ficient techniques and fairness-aw are schemes. More- ov er , there are some ne w technologies and/or concepts for RIS that have not been studied in MU-MIMO OFDM systems. C. Contribution In this paper, we maximize the SE and the EE of multi- user MIMO RIS-aided OFDM systems by considering various technologies for RIS like regular , ST AR and multi-sector BD- RIS. T o the best of our knowledge, this is the first work on RIS-aided OFDM systems that considers EE metrics and pro- poses IQI-aw are schemes for RIS-aided OFDM systems. In the conference version of this work [1], we proposed optimization schemes for maximizing the minimum rate of users in a single- cell MIMO RIS-aided BC with perfect devices. This paper extends the results in [1] to multi-cell MIMO RIS-aided BC with IQI, and considers other technologies for RIS, including ST AR-RIS and multi-sector BD-RIS. The proposed algorithms in this work can be applied to a wide range of optimization problems in which the objective function and/or constraints are linear functions of the rates and/or EE of users. Such optimization problems include many practical objecti ve functions such as minimum and/or sum rate, global EE, and minimum EE. Furthermore, we can apply the algorithms to an y multi-user MIMO RIS-aided OFDM system with IQI at transceivers. In this work, we consider a multi- cell MIMO RIS-aided OFDM BC as an example of such a system and sho w that RIS (either regular , ST AR, or multi- sector) can substantially increase the SE and EE even with a relativ ely low number of RIS components. In more detail, our numerical results show that optimizing the RIS elements can provide a high gain in OFDM systems ev en when we cannot independently optimize the RIS elements at each subband. These results show that RIS can be a promising technology to enhance the performance of OFDM systems. W e, additionally , propose IQI-aware schemes for MU MIMO RIS-aided OFDM systems. T o this end, we employ IGS since IQI makes the signals improper if proper signaling is emplo yed, which reduces the achie vable rate [35]. W e can also employ IGS as an interference-management technique, especially in single-carrier systems. Howe ver , as sho wn in [40], the IGS benefits as an interference-management tool vanish when the number of subbands increases. Indeed, the higher the number of subbands is, the more resources per user are av ailable, which reduces the interference le vel. Hence, the IGS gains in this scenario are mainly because of the ability to compensate for IQI. Note that IQI can significantly decrease the system performance, especially when we neglect it in designing signaling schemes. Finally , we dev elop optimization schemes for ST AR-RIS as well as multi-sector BD-RIS in multi-user MIMO OFDM BCs. W e sho w that ST AR-RIS ev en with random coefficients can outperform a regular RIS with optimized elements. Addi- tionally , it is sho wn that optimizing ST AR-RIS elements can result in substantial impro vements, and ST AR-RIS can highly outperform a regular RIS when all the users are not in the cov erage area of the regular RIS. Moreover , our results sho w that multi-sector BD-RIS can significantly outperform ST AR- RIS. W e consider different schemes for operating ST AR- RIS and multi-sector BD-RIS. Firstly , we assume that all the multi-sector BD-RIS elements can operate in all sectors simultaneously , which we refer to as the ener gy splitting (ES) scheme. Secondly , we assume that each multi-sector BD-RIS element can operate in only one sector at a time, which is called the mode switching (MS) scheme. W e sho w that the performance of the MS scheme is comparable with ES, while it has much lower computational and implementation complexities. I I . S Y S T E M M O D E L W e consider a multi-cell BC consisting of L multiple- antenna BS with N B antennas each. In each cell, there are K users with N U antennas each. Moreover , there are N RISs (either multi-sector , ST AR or regular) with N R elements to assists the BS, and there is at least one RIS in each cell. W e assume that each BS transmits a linear superposition of multicarrier OFDM signals with N i sub-carriers. A. RIS model This paper studies different RIS technologies such as reg- ular , ST AR and multi-sector in a multi-user MIMO OFDM system. Belo w , we first describe the concept of ST AR-RIS 4 S T A R RI S R e f l e c ti on S pa c e T X S pa c e T r a nsm it R e f l e c t 1 BS 12 U 11 U 13 U 14 U (a) ST AR-RIS ( N s = 2 ). 1 BS 11 U 12 U 15 U Sector 1 Sector 2 Sector 3 Sector 4 B D - R I S 13 U 14 U 16 U 17 U 18 U (b) Four -sector BD-RIS ( N s = 2 ). Fig. 1: A broadcast channel aided by a multi-sector BD-RIS. and multi-sector BD-RIS briefly . Then we state the effecti ve channels in MIMO OFDM systems, aided by these RIS tech- nologies. Moreover , we explain dif ferent operational modes for a multi-sector BD-RIS and/or ST AR-RIS. 1) Concept of multi-sector BD-RIS: Multi-sector BD-RIS is a generalization of ST AR-RIS and can be categorized as intelligent omni-surfaces, which can provide a full 360 ◦ cov- erage. In a ST AR-RIS, each element can transmit and reflect simultaneously , which realizes an omni-directional cov erage [26]. Thus, there are two spaces for each ST AR-RIS, which are referred to as the reflection space and the transmission space (see Fig. 1a). The concept/technology of ST AR-RIS can be extended to multi-sector BD-RIS in which each RIS can hav e multiple sectors, each covering a set of users and/or a sub-space, as shown in Fig. 1b . In the most general case, a multi-sector BD-RIS can ha ve N s sectors, and when N s = 2 , the multi-sector BD-RIS is equiv alent to a ST AR-RIS. Thus, in a multi-sector-BD-RIS-aided system, each user belongs to a region/space, cov ered by a sector of the multi-sector BD-RIS. For instance, in Fig. 1b, users u 11 , u 12 and u 13 are cov ered by the first sector of the multi-sector BD-RIS, while users u 17 and u 18 are cov ered by the fourth sector of the multi-sector BD-RIS. 2) Effective channels in fr equency domain: W e consider the channels in frequency domain. T o this end, we employ the channel model in [45, Eq. (12)]. Thus, the channel between the j -th BS and the k -th user associated to the l -th BS, denoted by u lk , at subband i is H lk,j,i ( { Φ } ) = N X n =1 G lk,n,i Φ n s n G nj,i + F lk,j,i (1) where G lk,n,i is the channel between u lk and the n -th RIS in subband i , G nj,i is the channel between the n -th multi-sector BD-RIS and the j -th BS in subband i , F lk,j,i is the direct link between u lk and the j -th BS in subband i . Moreover , the matrix Φ n s n contains the coefficients for the n s -th sector of the multi-sector BD-RIS in which u lk is located. The matrices Φ n s n s for n s = 1 , · · · , N s are diagonal as Φ n s n = diag  ϕ n s n 1 , ϕ n s n 2 , · · · , ϕ n s nN R  ∀ n s , m. (2) Note that the multi-sector BD-RIS, which is considered in this paper , is referred to as the cell-wise single-connected multi- sector BD-RIS in [33]. T o operate in a passi ve mode, the amplitudes of the coefficients for each multi-sector BD-RIS element should satisfy N s X n s =1 | ϕ n s nm | 2 ≤ 1 , ∀ n, m, (3) which is a conv ex constraint. W e represent the feasibility set, corresponding to this constraint as T U = ( ϕ n s nm : N s X n s =1 | ϕ n s nm | 2 ≤ 1 ∀ n, m ) . (4) Assuming passiv e and lossless operation for each multi-sector BD-RIS, we ha ve the following constraint [67, Eq. (3)] N s X n s =1 | ϕ n s nm | 2 = 1 , ∀ n, m. (5) W e represent the feasibility set for the constraint (5) as T I = ( ϕ n s nm : N s X n s =1 | ϕ n s nm | 2 = 1 ∀ n, m ) . (6) In these two models, it is assumed that each multi-sector BD- RIS operates in a passi ve mode, which yields a constraint only on the amplitude of the multi-sector BD-RIS elements. How- ev er , there might be some additional constraints on the phases of the coef ficients for each multi-sector BD-RIS component in practice. Unfortunately , multi-sector BD-RIS has not been implemented yet, and we ha ve to consider only T U and T I . Nev ertheless, according to the model in [27], the phases of the reflection and transmission coefficients for each ST AR- RIS element are highly dependent as ∠ ϕ 1 nm = ∠ ϕ 2 nm ± π 2 , which results in T S N =  ϕ 1 nm , ϕ 2 nm : | ϕ 1 nm | 2 + | ϕ 2 nm | 2 = 1 , R n ϕ 1 ∗ nm ϕ 2 nm o = 0 ∀ n, m o . (7) According to [31, Lemma 1], the constraints for each ST AR- RIS component in the set T S N can be re written as | ϕ 1 nm + ϕ 2 nm | 2 ≤ 1 , (8) | ϕ 1 nm − ϕ 2 nm | 2 ≤ 1 , (9) 5 | ϕ 1 nm | 2 + | ϕ 2 nm | 2 = 1 . (10) Indeed, T S N is only valid for ST AR-RIS ( N s = 2 ) and is more stringent than the other feasibility sets since it has additional constraints on the phases of the ST AR-RIS components. Moreov er , it can be easily verified that T S N ⊂ T I ⊂ T U . Hereafter , we represent the feasibility set of RIS coefficients by T , unless we explicitly refer to one of these feasibility sets. Note that according to the model in [45], the reflecting co- efficients cannot be independently optimized at each subband. Thus, we assume that the coef ficients remain constant across all the frequency subbands. Additionally , it should be noted that as can be verified through (1), the channels are linear functions of { Φ } = { Φ n s n } ∀ n s ,n . Howe ver , for notational simplicity , we do not state this explicitly hereafter . 3) Operational modes for multi-sector BD-RIS: There are different possibilities to operate a multi-sector BD-RIS. For instance, all the multi-sector BD-RIS components can acti vely operate in all sectors, which is referred to as the ES scheme. This scheme is the most general case to operate a multi-sector BD-RIS, and is expected to outperform the other operational schemes. Howe ver , the ES scheme has also higher computa- tional and implementation comple xities. In this mode, there are N s complex-v alued optimization parameters per multi-sector- BD-RIS element, which can provide more design flexibility at the cost of higher computational complexities. Additionally , operating each component of the multi-sector BD-RIS in all sectors may require more adv anced circuit designs. As indicated, the coefficients corresponding to each sector might be also highly dependent not only through the amplitudes, but also through the phases, which may make ES schemes inefficient in practice. T o cope with the challenges of ES schemes, we can consider different operational modes for multi-sector BD-RIS with lower comple xities. For instance, each component of the multi- sector BD-RIS can operate only in a sector at a time. T o realize such schemes, one possibility can be to divide the components of the multi-sector BD-RIS into N s groups. In the n s -th group, all the components of the multi-sector BD-RIS operate only in the n s -th sector , which is called the MS scheme. T o further clarify MS, we provide an example for N s = 2 , which is equiv alent to a ST AR-RIS. In this case, the components are divided into tw o groups. In the first group, all the ST AR-RIS components operate only in the reflection mode, while the ST AR-RIS elements in the other group operate only in the transmission mode. Another possibility could be to operate all the elements of a multi-sector BD-RIS in one sector in a time slot, and switch the operating sector in ne xt time slots in a round robin to cover all the sectors, which we call the time switching (TS) scheme. In a ST AR-RIS, it means that all the ST AR-RIS elements operate in a reflection mode in a time slot, while the y all operate in a transmission mode in the next time slot. Of course, there can be also hybrid schemes, which are a combination of the ES, MS and TS schemes, but we do not consider such schemes in this w ork. The ES and MS schemes can provide a full cov erage at a time, while the TS scheme covers only a subspace and is unable to provide a full coverage in a single time slot. As a result, we consider only the ES and MS schemes, but our proposed schemes can be easily applied to TS and/or hybrid schemes. B. I/Q imbalance model W e consider the IQI model in [35]. Note, that the model in [35] considers single-carrier systems, howe ver we can be extend it to multi-carrier systems. If the system at hand is wide-band, it is expected that the IQI parameters differ at each subband, based on [68, T able I]. Howe ver , in narrow-band systems, the IQI parameters can be the same in all subbands. In this paper , we in vestigate the most general case in which each subband can have different IQI parameters. W e briefly restate the model in [35] for the sake of completeness below . When IQI occurs at a device, the output signal is a WL T of the input signal. Hence, if we denote the input signal of a MIMO OFDM system at subband i as x i , the actual transmitted signal is x t,i = Γ t 1 ,i x i + Γ t 2 ,i x ∗ i , (11) where x ∗ i is the conjugate of x i , and the coef ficients Γ t 1 ,i and Γ t 2 ,i are, respectiv ely , gi ven by Γ t 1 ,i = I + A T ,i e j ψ T ,i 2 , Γ t 2 ,i = I − Γ ∗ t 1 ,i , (12) where A T ,i = a t,i I N U and ψ T ,i = ψ t,i I N U are, respectiv ely , the coefficients corresponding to the amplitude and phase mismatches at the transmitter side. Note that a t,i and ψ t,i are real-valued scalar parameters, and the device is ideal if a t,i = 1 and ψ t,i = 0 . At the receiv er side, we ha ve a similar model in which the output signal is a WL T of the receiv ed signal as y i = Γ r 1 ,i y r,i + Γ r 2 ,i y ∗ r,i , (13) where y r,i = H i x t,i + n i is the receiv ed signal, where H i and n i are, respectively , the MIMO channel, and additi ve noise at the i -th subband. Moreover , the coefficients Γ r 1 ,i and Γ r 2 ,i are Γ r 1 ,i = I + A R,i e j ϕ R,i 2 , Γ r 2 ,i = I − Γ ∗ r 1 ,i , (14) respectiv ely , where A R,i = a r,i I N B and ϕ R,i = ϕ r,i I N B are, respectiv e, the coefficients corresponding to the amplitude and phase mismatches at the receiver side. Similarly , a r,i and ϕ r,i are real-valued scalar parameters, and the receiver is perfect if a r,i = 1 and ϕ r,i = 0 . Finally , the signal at the output of the receiv er is y i =  Γ r 1 ,i H i Γ t 1 ,i + Γ r 2 ,i H ∗ i Γ ∗ t 2 ,i  x t,i +  Γ r 1 ,i H i Γ t 2 ,i + Γ r 2 ,i H ∗ i Γ ∗ t 1 ,i  x ∗ t,i + Γ r 1 ,i n i + Γ r 2 ,i n ∗ i . (15) In the follo wing lemma, we restate (15) in real vector -valued variables by employing the real-decomposition method. Lemma 1 ( [35]) . Equation (15) can be re written in a r eal domain as y i = H i x i + n i , 6 wher e y i , x i , and n i ar e, r espectively , the r eal decomposition of y i , x i , and Γ r 1 ,i n i + Γ r 2 ,i n ∗ i . Note that the effective noise at the output of the r eceiver is n i , which is zer o-mean Gaussian with covariance matrix C i = σ 2 Γ i Γ T i , where σ 2 is the noise variance at each received antenna, and Γ i can be obtained as in [14, Eq. (13)]. Mor eover , the equivalent channel H i is given by [14, Eq. (11)]. C. Signal model BS l intends to transmit x l,i = K X k =1 x lk,i , (16) at subband i , where x lk,i is the transmit signal of BS l intended for u lk at subband i . As described in Section II-B, the signal that BS l transmits is a WL T of x l,i , based on (11). Employing Lemma 1, the receiv ed signal receiv ed by u lk in subband i is y lk,i = L X j =1 H lk,j,i x j,i + n lk,i , where n lk,i is the real-decomposition of the ef fectiv e addi- tiv e zero-mean Gaussian noise at user k in subband i with cov ariance matrix C lk,i , and H lk,j,i is the equi valent channel between BS j and u lk at subband i , giv en by Lemma 1. Note that each channel is a linear function of { Φ } , according to (1). Hence, the effecti ve channel is also a linear function of { Φ } . Moreov er , note that all the signals x lk,i s, are independent zero- mean and possibly improper Gaussian random vectors, where P lk,i = E { x lk,i x T lk,i } . The achiev able rate of u lk is equal to r lk = N i X i =1 r lk,i , (17) where r lk,i is the rate of decoding x lk,i at u lk treating intra-cell and inter-cell interference as noise r lk,i = 1 2 log 2    I + D − 1 lk,i S lk,i    (18a) = 1 2 log 2 | D lk,i + S lk,i | | {z } r lk,i 1 − 1 2 log 2 | D lk,i | | {z } r lk,i 2 , (18b) where S lk,i = H lk,l ,i P lk,i H T lk,l ,i is the cov ariance matrix of the useful signal at u lk in subband i , and D lk,i is the cov ariance matrix of noise plus interference at u lk in subband i , giv en by D lk,i = L X n =1 ,n  = l H lk,n,i P n,i H T lk,n,i + K X m =1 ,m  = k H lk,l ,i P lm,i H T lk,l ,i + C lk,i . (19) The EE of u lk can be written as [69] e lk = r lk p c + η P i T r ( P lk,i ) , (20) where η and p c are defined as in [14, Eqs. (26)-(28)]. Addi- tionally , the global EE (GEE) can be obtained as [69] GE E = P ∀ l,k r lk LK p c + η P l T r ( P l ) , (21) where P l = P k,i P lk,i . D. Pr oblem statement W e aim at maximizing the SE and EE, which can be formulated as max { P } , { Φ }∈T f 0 ( { P } , { Φ } ) (22a) s.t. f g ( { P } , { Φ } ) ≥ 0 , ∀ g , (22b) X ∀ k,i T r ( P lk,i ) ≤ P l , ∀ l, (22c) P lk,i ≽ 0 , ∀ l, k , i (22d) where { P } = { P lk,i } ∀ l,k,i and { Φ } = { Φ m } ∀ m are the optimization variables, and P l is the power budget of BS l . The functions f 0 and f i include EE and SE metrics, and thus, can be considered as a linear function of r lk , and/or e lk , and/or GE E . The optimization problems that can be cast as (22) hav e been discussed in [11, Sec. II.D]. Such problems include, but not limited to, the maximization of the minimum/sum rate, minimum EE and global EE. Note that these objectiv e functions and/or optimization problems are among the most practical performance metrics in wireless communication systems. I I I . O P T I M I Z AT I O N A L G O R I T H M S T O S O L V E (22) T o solve the non-con vex problem (22), we emplo y majoriza- tion minimization (MM) and alternating optimization (A O). T o this end, we first fix the RIS coefficients to { Φ ( t − 1) } , and solve (22) to obtain  P ( t )  . W e then fix the transmit cov ariance matrices to  P ( t )  and update { Φ } by solving (22). In the following, we describe our proposed algorithm to update { P } and Φ in separate subsections. A. Updating transmit covariance matrices When RIS components are fixed to Φ ( t − 1) , (22) is max { P } f 0  { P } , { Φ ( t − 1) }  (23a) s.t. f g  { P } , { Φ ( t − 1) }  ≥ 0 , ∀ g , (23b) (22c) , (22d) , (23c) which is non-conv ex since r lk  { P } , Φ ( t − 1)  for all l , k is not concav e in { P } . T o provide a suboptimal solution for (23), we obtain suitable conca ve lower bounds for r lk  { P } , { Φ ( t − 1) }  . T o this end, we employ the lower bound in [11, Lemma 3], which results in the concav e lower bounds for the rates that are provided in Lemma 2. Lemma 2. A concave lower bound for r lk is ¯ r lk = P i ¯ r lk,i , wher e ˜ r lk,i is r lk,i ≥ ¯ r lk,i = r lk,i 1 ( { P } ) − r ( t − 1) lk,i 2 7 − K X j =1 ,  = k R ( T r H H lk,l ,i ( D ( t − 1) lk,i ) − 1 H lk,l ,i ln 2  P lj,i − P ( t − 1) lj,i  ! ) − L X n =1 ,  = l R ( T r H H lk,n,i ( D ( t − 1) lk,i ) − 1 H lk,n,i ln 2  P n,i − P ( t − 1) n,i  ! ) , wher e r ( t − 1) lk,i 2 = r lk,i 2  { P ( t − 1) }  , and D ( t − 1) lk,i = D lk,i  { P ( t − 1) }  . Substituting r lk with ¯ r lk , we ha ve max { P } ¯ f 0  { P } , { Φ ( t − 1) }  (24a) s.t. ¯ f g  { P } , { Φ ( t − 1) }  ≥ 0 , ∀ g , (24b) (22c) , (22d) , (24c) which is con vex in { P } when SE metrics such as minimum and/or sum rate are considered. F or EE metrics such as GEE and/or minimum weighted EE, we can employ Dinkelbach- based algorithms to obtain a global optimal solution of (24). Note that the solution of (24) forms the new set of the transmit cov ariance matrices, i.e.,  P ( t )  , which is utilized in the next step. B. Updating RIS elements In this subsection, we first propose optimization algorithms to update { Φ } for the ES scheme since it is more general than the MS scheme. Indeed, the MS scheme can be considered as a special case of the ES scheme as discussed in Section II-A, which means that the solutions for the ES scheme can be applied to the MS scheme as well. At the end of this subsection, we explain ho w the ES solutions can be modified to include the MS scheme. When transmit cov ariance matrices are fixed to  P ( t )  , (22) is equiv alent to max { Φ }∈T f 0 n P ( t ) o , { Φ }  (25a) s.t. f g n P ( t ) o , { Φ }  ≥ 0 , ∀ g , (25b) which is non-conv ex since r lk  P ( t )  , { Φ }  is not concave in { Φ } , and moreov er , T is not a conv ex set for T I and T S N . T o provide a suboptimal solution for (25), we first obtain a conca ve lo wer bound for r lk  P ( t )  , { Φ }  , and then con ve xify T I , and/or T S N . T o this end, we employ the bound in [14, Lemma 2], which gi ves the surrogate functions for the rates as in the lemma below . Lemma 3. A concave lower-bound for the rate of users is ˆ r lk = P i ˆ r lk,i , where ˆ r lk,i is r lk,i ≥ ˆ r lk,i = r ( t − 1) lk,i − 1 ln 2  T r  ¯ S lk,i ¯ D − 1 lk,i  − T r  ( ¯ D − 1 lk,i − ( ¯ S lk,i + ¯ D lk,i ) − 1 ) H ( S lk,i + D lk,i )  +2 R n T r  ¯ V H lk,i ¯ D − 1 lk,i V lk,i o , (26) wher e r ( t − 1) lk,i = r lk,i  { Φ ( t − 1) }  , V lk,i = H lk,l ,i ( Φ ) P ( t ) 1 / 2 lk,i , ¯ D lk,i = D lk,i  { Φ ( t − 1) }  , ¯ S lk,i = S lk,i  { Φ ( t − 1) }  , and ¯ V lk,i = H lk,l ,i  { Φ ( t − 1) }  P ( t ) 1 / 2 lk,i . Substituting r lk with ˆ r lk , (25) becomes max { Φ }∈T ˆ f 0 n P ( t ) o , { Φ }  (27a) s.t. ˆ f g n P ( t ) o , { Φ }  ≥ 0 , ∀ g , (27b) which is con vex only for T U in which the con ver gence to a stationary point of (22) is ensured. Howe ver , (27) is non- con ve x for T I (or T S N ) because of the constraint in (5) (or (10)). W e can rewrite P N s n s =1 | ϕ n s mn | 2 = 1 as the two constraints N s X n s =1 | ϕ n s mn | 2 ≤ 1 ∀ m, n, (28) N s X n s =1 | ϕ n s mn | 2 ≥ 1 ∀ m, n, (29) The constraint (28) is con vex. Ho wev er , (29) is a non-conv ex constraint, which makes (27) non-conv ex for T I (or T S N ). T o approximate (29) with a conv ex constraint, we can employ the con ve x-concave procedure (CCP) and relax the constraint for a faster conv ergence as [14] N s X n s =1  2 R  ϕ n ( t − 1) ∗ s mn ϕ n s mn  − | ϕ n ( t − 1) s mn | 2  ≥ 1 − ϵ, ∀ m, n, (30) where ϵ > 0 . Substituting the constraints (28) and (30) in (27) yields max { Φ } ˆ f 0 n P ( t ) o , { Φ }  (31a) s.t. ˆ f g n P ( t ) o , { Φ }  ≥ 0 , ∀ g (31b) (28) , (30) , (31c) which can be ef ficiently solv ed since (31) is con vex. W e denote the solution of (31) as { Φ ( ⋆ ) } . It might happen that { Φ ( ⋆ ) } does not meet the constraint in (5) due to the relaxation in (30). T o generate a feasible solution, we normalize { Φ ( ⋆ ) } as ˆ ϕ n s mn = ϕ n ( ⋆ ) s mn r P N s n s =1    ϕ n ( ⋆ ) s mn    2 , ∀ m, n. (32) Finally , we update { Φ } as { Φ ( t ) } =      { ˆ Φ } if ˆ f 0   P ( t )  , { ˆ Φ }  ≥ ˆ f 0  P ( t )  , { Φ ( t − 1) }  { Φ ( t − 1) } otherwise , (33) where { ˆ Φ } = { ˆ Φ n s m } ∀ n s ,m , where ˆ Φ n s m = diag ( ˆ ϕ n s m 1 , ˆ ϕ n s m 2 , · · · , ˆ ϕ n s m N RIS ) . (34) The updating policy in (33) ensures conv ergence because of generating a non-decreasing sequence of ˆ f 0 ( · ) . Note that for the set T S N , we can employ a similar approach since the constraints (9) and (8) are con vex, and we can handle (10) similar to P N s n s =1 | ϕ n s mn | 2 = 1 for N s = 2 . Hence, to update { ϕ } , we ha ve to solv e the con vex problem max { ϕ } ˆ f 0 n P ( t ) o , { Φ }  (35a) 8 s.t. ˆ f g n P ( t ) o , { Φ }  ≥ 0 , ∀ g (35b) (9) , (8) , (28) , (30) , (35c) Then we update { Φ } according to the rule in (33). Now we present our solution for the MS scheme. That is, we randomly di vide the multi-sector BD-RIS elements into N s groups such that there are at least ⌊ N RI S N s ⌋ elements per group. Then the RIS elements in the n s -th group, indicated by G mn s , operate only in the sector n s , and thus, ϕ n s mn = 0 if n / ∈ G mn s . Inserting this rule in (27) and (31) giv es the MS solution for T U and T I , respectiv ely . For T I , we have to update { Φ } based on the rule in (33). Note that the sets T I and T S N are equi valent for MS schemes since the phase dependency constraints are automatically satisfied when each ST AR-RIS element is acti ve in only one sector . I V . N U M E R I C A L R E S U LT S In this section, we employ Monte Carlo simulations to provide numerical results. W e assume that the small-scale fading for the channels G i and G ki for all i, k is Rician similar to [11], [13] since there is a line of sight (LoS) link between the BS and RIS as well as between the RIS and the users. Howe ver , as the links between the BS and the users are assumed to be non-LoS (NLoS), the small-scale fading for F ki for all k , i is assumed to be Rayleigh distributed. Note that it is also assumed that there is no correlation between the channels at different subbands, which is an extreme case, and it can be expected that the performance of RIS is improved with correlated channels. The simulation scenario is based on [11, Fig. 2], unless explicitly mentioned otherwise. For more descriptions on the simulation parameters and setup, we refer the reader to [11], [15]. A. Maximization of the minimum rate Here, we inv estigate the effecti veness of the various RIS technologies from a SE point of view by considering the maximization of the minimum rate. T o this end, we consider the impact of dif ferent system parameters, including the power budget at the BSs, the number of subbands, the number of RIS elements, and IQI parameters. 1) Impact of power budg et: Fig. 2 shows the av erage minimum rate versus P for N B = N U = 2 , K = 3 , N R = 100 , L = 2 , N = 2 , and N i = 16 . In this figure, we can observe that the regular RIS can substantially increase the minimum rate e ven when the RIS elements are not optimized (RIS- Rand). Additionally , there is almost a constant gap between the proposed scheme for RIS-aided systems and the scheme with random RIS coef ficients (RIS-Rand). Indeed, e ven though the RIS elements cannot be independently optimized at each subband, we can get a significant gain by optimizing RIS coefficients, which shows the effecti veness of RIS in multi- user MIMO OFDM systems. 2) Impact of the number of subbands: Fig. 3 sho ws the av- erage performance improv ements versus P for N B = N U = 2 , K = 3 , N R = 100 , L = 1 , N = 1 , different N i and a t,i = a r,i = 1 for all i . The relati ve performance curves in Fig. 3a and Fig. 3b are obtained by comparing the av erage 0 4 8 12 17 10 20 30 35 P (dB) Minimum Rate (b/s/Hz) RIS RIS-Rand No-RIS Fig. 2: A verage minimum rate v ersus P for the case without RIS (No- RIS), randomly configured RIS (RIS-Rand), and optimized regular RIS (RIS) with N B = N U = 2 , K = 3 , N R = 100 , L = 2 , N = 2 , and N i = 16 . 0 4 8 12 17 100 200 300 400 500 600 700 P (dB) Relative Performance Improvement (%) N i = 2 N i = 16 N i = 32 (a) RIS compared to No-RIS. 0 4 8 12 17 20 40 60 80 100 110 P (dB) Relative Performance Improvement (%) N i = 2 N i = 16 N i = 32 (b) RIS compared to RIS-Rand. Fig. 3: Improv ements by RIS versus P for N B = 2 , N U = 2 , K = 3 , L = 1 , N = 1 , and different N i . minimum rate achie ved by our scheme for RIS-aided OFDM systems with the a verage minimum rate of OFDM systems without RIS and with the a verage minimum rate of OFDM systems with random RIS coefficients, respectiv ely . According to Fig. 3, RIS can provide a huge gain. Howe ver , the benefits of optimizing RIS components highly decrease with N i since RIS elements cannot be independently optimized at each subband, and as the number of subbands for a fixed N R increases, the ef fectiv eness of optimizing Φ decreases. Interestingly , the benefits of optimizing RIS elements are still significant e ven when there are slightly higher than 1 RIS elements per user per subband ( N R / ( K N i ) ≃ 1 . 04 when N i =32). Furthermore, the gains of optimizing RIS elements are much higher in low SNR regimes. Since we also consider a po wer/covariance matrix optimization, it may happen that the signals for a user are transmitted o ver a few subbands when the BS power b udget is low . Thus, the effecti ve number of the utilized subbands is lower than N i , especially at low SNR regimes, which enhances the benefits of optimizing RIS elements. Additionally , as shown in our pre vious studies [11], [14], the benefits of employing RIS are higher in low SNR regimes, which may enhance the gain of a proper optimization of RIS elements. 3) Impact of N R : Fig. 4 shows the a verage performance improv ements versus N R for N B = N U = 1 , K = 3 , L = 1 , 9 20 40 60 80 100 0 200 400 600 800 N RI S Relative Performance Improvement (%) SNR=0dB SNR=10dB SNR=17dB (a) RIS compared to No-RIS. 20 40 60 80 100 0 20 40 60 80 100 120 N RI S Relative Performance Improvement (%) SNR=0dB SNR=10dB SNR=17dB (b) RIS compared to RIS-Rand. Fig. 4: Improv ements by RIS versus P for N B = 2 , N U = 2 , K = 3 , L = 2 , N = 2 , and N i = 16 . 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 10 15 20 25 a t Minimum Rate (b/s/Hz) IQI-aware (RIS) IQI-Unaware (RIS) IQI-aware (No-RIS) IQI-Unaware (No-RIS) Fig. 5: Impact of IQI on the av erage minimum rate of users for N B = 2 , N U = 2 , K = 2 , N R = 100 , L = 2 , N = 2 , and N i = 16 . N i = 16 , N = 1 , and different SNRs. In this figure, the benefits of employing RIS almost linearly increases with N R , especially at lo w SNR. Additionally , we can observe that the slope of the RIS benefits is higher in lo wer SNR, which is in line with the results in the Figs. 2 and 3. Moreo ver , optimizing RIS elements can yield substantial gains e ven when N R = 10 , which is lo wer than the number of sub-bands (in this case N R / ( K N i ) ≃ 0 . 21 ). 4) Impact of IQI: Fig. 5 shows the a verage minimum rate of users v ersus a t for N B = 2 , N U = 2 , K = 2 , N R = 100 , L = 2 , N = 2 , and N i = 16 . In this figure, we assume that both transcei vers suf fer from IQI with the same IQI parameters at all subbands, which means that a r,i = a t,i = a t for all i . This figure shows that the average minimum rate decreases with IQI level at transcei vers ev en if it is considered in the system designed and compensated by IGS. Furthermore, we can observe that IQI can significantly decrease the a verage minimum rate especially when we do not take it into account in optimizing the parameters. Indeed, IQI may even vanish the benefits of RIS in systems with highly imbalanced de vices if the employed schemes are not robust against IQI. Additionally , we observe that IQI has a similar impact on the systems with and without RIS; howe ver , the benefits of IQI-aware schemes are a bit higher in RIS-aided systems. 0 4 8 12 17 0 20 40 60 80 100 120 P (dB) Minimum Rate (b/s/Hz) S-RIS-ES S-RIS-ES N S-RIS-MS S-RIS-Rand RIS No-RIS Fig. 6: A verage minimum rate versus P for the case without RIS (No-RIS), randomly configured ST AR-RIS (S-RIS-Rand), optimized ST AR-RIS (S-RIS) with different operational mode and feasibility sets, and optimized regular RIS (RIS) for N B = N U = 2 , K = 2 , N R = 100 , L = 1 , N = 1 , and N i = 16 . 5) Comparison of differ ent technologies for RIS: Here, we assume L = 1 and N = 1 , which means that we consider only one RIS (either multi-sector BD-RIS, ST AR or regular). T o show the impact of an omin-directional co verage, we consider a scenario in which the users are located such that the regular RIS can cover only half of the users, while the ST AR-RIS and/or multi-sector BD-RIS can cover all the users. In Fig. 6, we show the av erage minimum rate versus P for N B = N U = 2 , K = 2 , N R = 100 , L = 1 , N = 1 , and N i = 16 . This figure compares the performance of a regular RIS and a ST AR-RIS by considering different feasibility sets as well as different operational modes for the ST AR-RIS. In this example, RIS (either regular or ST AR) can highly increase the av erage minimum rate. Additionally , a ST AR-RIS with ev en random coefficients can outperform the proposed scheme for the regular RIS. Moreov er , optimizing the ST AR- RIS coefficients can significantly increase the a verage mini- mum rate. Furthermore, the ES scheme outperforms the MS scheme; howe ver , considering the higher computational and implementation complexities of ES schemes, the MS scheme performance is comparable with the ES scheme, especially for T S N in which the phases of transmit and reflection coefficients cannot be independently optimized. Fig. 7 compares the performance of v arious RIS technolo- gies from an av erage minimum rate point of vie w for K = 4 , N R = 32 , L = 1 , N = 1 , N i = 16 , and different N B , N U . T o this end, we consider a four-sector BD-RIS ( N s = 4 ), a ST AR-RIS and a regular RIS. W e assume that there is one user in each sector of the multi-sector BD-RIS. Moreover , to make the comparison fair , we assume that the total number of the RIS elements is the the same and is equal to 32 in these different technologies. It means that there are only N R / N s elements per each sector of the multi-sector BD-RIS, which is equal to 8 in this example. Note that we assume that ST AR- RIS has N R (32 in this example) elements since ST AR-RIS 10 0 4 8 12 17 0 5 10 15 20 25 30 P (dB) Minimum Rate (b/s/Hz) MSBD-RIS RIS S-RIS-MS No-RIS S-RIS-Rand (a) N B = N U = 1 . 0 4 8 12 17 0 10 20 30 40 50 P (dB) Minimum Rate (b/s/Hz) MSBD-RIS RIS S-RIS-MS No-RIS S-RIS-Rand (b) N B = N U = 2 . Fig. 7: The average minimum rate versus P for the case without RIS (No-RIS), randomly configured ST AR-RIS (S-RIS-Rand), optimized ST AR-RIS with mode switching scheme (S-RIS-MS), optimized regular RIS (RIS), and optimized multi-sector BD-RIS with mode switching scheme (MSBD-RIS) for K = 4 , N R = 32 , L = 1 , N = 1 , and N i = 16 . 0 4 8 12 17 0 200 400 600 800 1 , 000 P (dB) Relativ e Performance Improvement (%) MSBD-RIS over ST AR-RIS ( N RIS = 32 ) MSBD-RIS over No-RIS ( N RIS = 32 ) MSBD-RIS over ST AR-RIS ( N RIS = 64 ) MSBD-RIS over No-RIS ( N RIS = 64 ) Fig. 8: The average improvements by multi-sector BD-RIS versus P for N B = N U = 1 , K = 4 , L = 1 , N = 1 , N i = 16 , and N R . can employ a different technology comparing to multi-sector BD-RIS. Furthermore, we assume that the antenna gain of each multi-sector BD-RIS is computed based on the idealized model in [33, Eq. (17)]. Here, we consider only the MS scheme since it has lower computational and implementation complexities while pro viding a competitiv e performance, as sho wn in Fig. 6. In Fig. 7, all the considered RIS technologies can substan- tially increase the average minimum rate with ev en a very low N R . In this example, the number of RIS elements per user per subband is 0 . 5 , which is e ven lower than 1. W e can also observ e that ST AR-RIS can highly outperform a regular RIS. Additionally , the multi-sector BD-RIS can provide a significant gain e ven though the number of RIS elements per sector is much lo wer than the number of ST AR-RIS elements. Fig. 8 shows the average improv ements by multi-sector BD- RIS ov er ST AR-RIS and/or No-RIS versus P for N B = 1 , N U = 1 , K = 4 , L = 1 , N = 1 , N i = 16 , and N R . As can be observed, multi-sector BD-RIS can substantially improv e the SE, especially in low SNR regimes. Additionally , the benefits of employing RIS increase with N R . 0 4 8 12 17 0 50 100 150 200 P (dB) Sum Rate (b/s/Hz) RIS No-RIS RIS-Rand (a) A verage sum rate. 0 4 8 12 17 0 50 100 150 200 250 300 350 P (dB) Relative Performance Improvement (%) RIS over No-RIS RIS over RIS-Rand (b) Relativ e performance im- prov ement. Fig. 9: The average sum rate versus P for N B = 2 , N U = 2 , K = 3 , L = 2 , N = 2 , N i = 16 , and N R = 50 . 3 4 5 6 7 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 P c (W) Minimum EE (b/J/Hz) RIS No-RIS RIS-Rand (a) A verage minimum EE. 3 4 5 6 7 20 40 60 80 100 120 140 P c (W) Relative Performance Improvement (%) RIS over No-RIS RIS over RIS-Rand (b) Relativ e performance im- prov ement. Fig. 10: The a verage minimum EE versus P c for N B = N U = 2 , K = 2 , N R = 50 , L = 2 , N = 2 , and N i = 16 . B. Maximization of the sum rate Fig. 9 shows the average sum rate versus P for N B = N U = 2 , K = 3 , L = 2 , N = 2 , N i = 16 , and N R = 50 . In this example, RIS can substantially increase the av erage sum rate. Moreover , the benefits of RIS decrease with the power budget, which supports the results in Fig. 2. Indeed, in low SNR regimes, it is likely that the BS transmits over a fe w number of subbands, which can increase the benefits of optimizing RIS elements since the RIS coef ficients cannot be independently optimized at each subband. Additionally , as shown in, e.g., [14], RIS can provide higher gains in lo w SNR regimes in single-carrier systems since the recei ved power of cell-edge users is v ery lo w in systems without RIS when the BS transmission power is lo w . Hence, we can expect that this result hold in multi-carrier case, which highly increases the benefits of RIS in low SNR regimes. C. Maximization of the minimum EE Fig. 10 shows the average minimum EE versus P c for N B = N U = 2 , K = 2 , N R = 50 , L = 2 , N = 2 , and N i = 16 . In this example, RIS can significantly improv e the average minimum EE of OFDM systems with different P c . Ho wev er , the EE benefits of RIS are lo wer than its SE benefits. Additionally , the RIS benefits slightly increase with P c . The reason is that, when P c asymptotically increases, the solutions of the maximization of the minimum rate and 11 3 4 5 6 7 0 . 2 0 . 4 0 . 6 P c (W) Global EE (b/J/Hz) RIS No-RIS RIS-Rand Fig. 11: The average global EE v ersus P c for N B = 2 , N U = 2 , K = 2 , N R = 40 , L = 2 , N = 2 , and N i = 16 . 2 8 16 24 32 20 40 60 80 100 120 140 N i Relativ e Performance Improvement (%) RIS over No-RIS RIS over RIS-Rand Fig. 12: The average benefits of RIS from a global EE point of view versus N i for N B = 2 , N U = 2 , K = 2 , N R = 40 , L = 2 , N = 2 , and P c = 5 W . the maximization of the minimum EE are identical since the EE maximization for a large P c is equi valent to the rate maximization. Hence, as P c increases, the EE benefits of RIS become closer to the SE benefits, which makes the EE benefits of RIS an increasing function of P c . D. Maximization of the global EE Fig. 11 sho ws the av erage global EE v ersus P c for N B = 2 , N U = 2 , K = 2 , N R = 40 , L = 2 , N = 2 , and N i = 16 . In this figure, RIS can significantly enhance the global EE of the OFDM system for different P c . Moreover , optimizing RIS elements provides more that 32% improvements in the particular example with a relativ ely low number of RIS elements per user per subband ( N R / ( K N i ) ≃ 1 . 25 ). Fig. 12 sho ws the av erage benefits of RIS from a global EE point of view versus N i for N B = 2 , N U = 2 , K = 2 , N R = 40 , L = 2 , N = 2 , and P c = 5 W . As can be observed, the benefits of employing RIS (compared to No- RIS) increase with N i , while the benefits of optimizing RIS elements decrease with N i . It should be noted that the benefit of optimizing RIS components in more than 25% for N i = 32 in this particular example, which is still significant. V . C O N C L U S I O N This paper proposed resource allocation algorithms with both precoding and RIS element optimizations for a multi- user MIMO RIS-aided OFDM BC with HWI to maximize the SE and EE. W e showed that RIS can substantially enhance the SE and EE of the OFDM BC ev en when the number of RIS elements is low . Moreover , we showed that the benefits of optimizing RIS elements in low SNR regimes are much higher than in high SNR regimes. Additionally , we showed that the benefits of optimizing RIS elements are still significant in OFDM systems ev en though the RIS coef ficients cannot be optimized at each subband independently . Furthermore, we showed that IQI can significantly reduce the average minimum rate e ven if it is compensated by emplo ying IGS. 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