I Introduction
The wireless propagation environment between the base station (BS) and the user equipments (UEs) is generally regarded as uncontrollable. Thanks to the recent advances of metamaterials, the emerging reconfigurable intelligent surface (RIS) is able to control the propagation of incident signals via the interaction between electromagnetic waves and materials coated on the surface [basar2019wireless]. Thus, the development of RIS makes it possible to manipulate the wireless propagation environment by tuning the reflection coefficients of a large number of RIS elements, which effectively improves the system capacity and reduces the communication outage [Ntontin'19, jung2019reliability]. To this end, RIS has been considered as a promising technology for 6G wireless communication systems [liang2019large].
Ia Prior works
As an emerging technology, RIS has recently attracted extensive attention including performance analysis [basar2019wireless], precoding design [wang2020intelligent, huang2019reconfigurable, pan2020multicell, wu2019beamforming, liu2021compact], channel state information acquisition [mishra2019channel, alwazani2020intelligent], and hardware implementation [Dai2020reconfigurable], etc.
It has been revealed in [basar2019wireless] that the received signal power in RISaided wireless communication systems is quadratically proportional to the number of RIS elements. To achieve this exciting result, precoding design is essential for RISaided communication systems to jointly optimize the precoder at the BS and the reflection coefficients at the RIS. Different optimization metrics have been studied in recent works. Specifically, the suboptimal analytical solution of precoding design for multiple RISs was derived in [wang2020intelligent], which aimed to maximize the received signal power of a singleantenna user. The energy efficiency of an RISaided communication system was investigated by joint power allocation and precoding optimization for multiuser cases in [huang2019reconfigurable], which employed gradient descent search and sequential fractional programming. Moreover, the weighted sumrate of all users in an RISaided communication system was maximized by utilizing the block coordinate descent algorithm, which alternately optimized the active precoding at the BS and the phase shifts at the RIS [pan2020multicell]. In addition, the transmit power at the BS was minimized by applying the successive refinement method [wu2019beamforming].
Although the precoding design of RISaided communication systems has been widely studied in the literature, prior works usually considered the classical regular RIS, where the RIS elements are regularly arranged on a grid with a constant interelement spacing. Since the performance bound based on regular RIS highly relies on the number of RIS elements, the array gain provided by RIS can only be improved by increasing the number of RIS elements. However, the number of RIS elements is usually limited in practical systems, which is mainly caused by two reasons. The first one is the high overhead for the acquisition of RISaided channels. Specifically, the required pilot overhead of existing cascaded channel estimation schemes is usually proportional to the product of the number of RIS elements and the number of UEs, such as the ON/OFF protocol based channel estimation
[mishra2019channel]and the discrete Fourier transform protocol based channel estimation
[alwazani2020intelligent]. Consequently, with a large number of RIS elements, the overhead for channel state information acquisition will be very high [hu2019two, basar2019wireless]. The second one is the limitation of the RIS power consumption. Most existing works have assumed that the power consumption of RIS elements is negligible. However, when the number of RIS elements is large, the RIS power consumption could be high. For example, the dissipated power of an RIS element is about 10 mW [huang2019reconfigurable], so the total power consumption of RIS is up to 1 W with 100 RIS elements. Due to these two reasons, a not very large number of RIS elements is preferred in practice, which however limits the capacity of RISaided communication systems. Therefore, it is important to improve the capacity of RISaided communication systems with a limited number of RIS elements.IB Contributions
In this paper, we focus on improving the capacity of RISaided communication systems with a limited number of RIS elements. Inspired by the advantages of sparse phased arrays [wang2017analysis], unlike the existing regular RIS, we propose an irregular RIS structure to improve the system capacity. The contributions of this paper are summarized as follows.

We propose a new topological structure of RIS called irregular RIS, where a given number of elements are irregularly deployed on an enlarged surface. Considering that RIS is easy to be integrated into the wireless environment (such as the facades of buildings or ceilings) [basar2019wireless], space available for the deployment of an enlarged surface is usually sufficient. To the best of our knowledge, the irregular RIS has not been investigated in the literature, which is studied in this paper for the first time. Since the selection of feasible locations for RIS elements can exploit spatial degrees of freedom (DoFs), the proposed irregular RIS with an enlarged surface can enhance the system capacity.

We develop the system model for the irregular RISaided communication systems, and formulate a joint topology and precoding optimization problem to improve the weighted sumrate. Specifically, the topology design is formulated as an integer programming problem, while the precoding design is formulated to jointly optimize the precoder at the BS and the reflection coefficients at the irregular RIS.

To solve the formulated optimization problem, we decouple the decision variables to be optimized and propose a joint optimization algorithm to alternately optimize the RIS topology and the precoding design. Particularly, an adaptive topology design method based on the tabu search algorithm [glover1989tabu] (ATS) is used to design the irregular RIS topology, and a neighbor extractionbased crossentropy (NECE) precoding method is introduced for precoding design. We also analyze the complexity of the proposed joint optimization algorithm, which is much lower than that of the exhaustive search method. Finally, simulation results show that the proposed irregular RIS can significantly improve the capacity of RISaided communication systems.
IC Organization and notations
Organization: The rest of the paper is organized as follows. The model of the proposed irregular RISaided communication system is introduced in Section II. We formulate the weighted sumrate optimization problem for the proposed system in Section III. A joint optimization algorithm is proposed to solve the formulated optimization problem in Section IV. Simulation results are shown in Section V, and the conclusions are finally drawn in Section VI.
Notations:Vectors and matrices are denoted by lowercase and uppercase boldface letters, respectively. , , , and denote the conjugate transpose, transpose, inverse, and th entry of matrix , respectively. and denote the norm and Euclidean norm of vector , respectively. denotes a diagonal matrix whose diagonal elements consist of corresponding entries in vector . denotes the cardinality of set .
denotes the identity matrix of size
. Finally, denotes the allone matrix of size .Ii Irregular RISAided Communication System
In this section, we first introduce the concept of the proposed irregular RIS. Then, the model of the irregular RISaided communication system is illustrated.
We consider an RISaided multiuser downlink multipleinput multipleoutput (MIMO) wireless communication system in this paper. The existing regular RISaided communication system is shown in Fig. 1 (a), where a BS equipped with antennas and a regular RIS composed of elements simultaneously serve singleantenna users. As shown in Fig. 1 (b), different from the regular RIS whose elements are arranged on a regular surface with a constant interelement spacing, we propose a new concept of irregular RIS, where RIS elements are irregularly distributed over grid points of an enlarged surface (). To simplify the description of the proposed concept, the grid constraint with a fixed grid spacing is assumed in this paper^{1}^{1}1The more complex scenario where elements can be arbitrarily distributed within the surface aperture is left for future works.. Without loss of generality, the grid spacing between adjacent grid points is assumed to be half wavelength of the carrier frequency [wang2017analysis]. The proposed irregular RIS can be equivalently implemented by selecting feasible locations for RIS elements from all grid points. It is worth noting that, the proposed irregular RIS is different from the sparse phased arrays, which aims to shape the beam pattern in the free space and suppress the sidelobe [rocca2016unconventional]. Actually, the proposed irregular RIS works as a reflection array to assist wireless communication systems, which takes the influence of channel conditions and multiple UEs into account.
Let denote the topology matrix representing the topology of RIS, where . We define to indicate whether an RIS element is deployed at the th grid point or not (), i.e., means that the th grid point is selected for an RIS element, while denotes that the th grid point is not selected.
The received signal for all users can be expressed as
(1) 
where represents the transmitted signal at the BS, each component of
donates the additive white Gaussian noise (AWGN) with zero mean and variance
, the reflection coefficients of grid points of the irregular RIS can be represented as(2) 
and denotes the BSRIS channel. For simplifying the expression, we define , , where and represent the channel from the BS to user , and that from the RIS to user , respectively.
In this paper, we consider the fully digital precoder at the BS, i.e.,
(3) 
where denotes the precoding vector for user , and denotes the transmitted symbol vector satisfying for users. Considering the practical hardware implementation of RIS, we assume the constant reflection amplitude constraint and finite discrete phase shift constraint at the RIS [wu2019beamforming, Dai2020reconfigurable, tan2021hybrid, yang2016A]. To this end, for , the reflection amplitude satisfies , and the phase shift takes discrete values from the quantized phase shift set given by
(4) 
where is the number of quantized bits of finite discrete phase shifts.
Based on the system model (1), the signaltointerferenceplusnoise (SINR) of user in the irregular RISaided communication system can be derived as
(5) 
Based on (5), a joint topology and precoding optimization problem will be formulated for the proposed irregular RISaided communication system in the next section.
Iii Problem Formulation of the Proposed Irregular RISAided Communication System
To fully improve the system performance of the proposed irregular RISaided communication system, the topology design and the corresponding precoding design should be carefully designed. In this section, the joint topology and precoding design problem of the irregular RIS is firstly formulated to maximize the weighted sumrate (WSR). Then, we convert the formulated problem to a precoding optimization problem for a specific topology of RIS by decoupling the decision variables.
Let denote the fully digital precoder at the BS, denote the weight of user . The transmit power at the BS is given by
(6) 
which should be lower than the maximum allowable transmit power . We focus on maximizing the WSR by jointly designing the irregular RIS topology and the corresponding precoding design, i.e., , , and . The WSR maximization problem can be formulated as
(7a)  
(7b)  
(7c)  
(7d)  
(7e) 
where denotes the transmit power constraint, denotes the discrete phase shift constraint, and denote the irregular RIS constraints. Note that and restrict the irregular topology design of RIS, where diagonal elements of the topology matrix are assigned with the value of , and the rest diagonal elements are assigned with .
Note that the objective function (7a), the phase shift constraint , and the irregular constraints and are nonconvex. Thus, it is difficult to directly solve the WSR maximization problem . To tackle this challenge, we can decouple the RIS topology design and the corresponding precoding design. Specifically, for a given topology , can be reduced to
(8a)  
(8b)  
(8c)  
(8d) 
The subproblem can be efficiently solved by alternating optimization, which will be discussed later in Section IV.
Remark 1
The system model (1) of the proposed irregular RISaided communication systems is reduced to the system model of the existing regular RISaided communication systems by simply assuming that
(9) 
Thus, the formulated problem can be seen as a general formulation for both regular RIS and irregular RIS. Similarly, if , the problem is reduced to as
(10a)  
(10b)  
(10c) 
We can find that the simplified problem is just the precoding design problem in the existing regular RISaided communication systems [pan2020multicell].
Remark 2
Note that, the complete channel state information (CSI) relevant to the number of grid points of RIS is required to solve , which results in high pilot overhead. Fortunately, since the covariance matrix of the BSRIS channel remains unchanged over a long period of time [hu2019two], the designed topology of RIS can be temporarily fixed. That is, the designed topology of RIS only needs to be changed adaptively in a large timescale. Then, the precoding design with a given specific RIS topology, i.e., the solution to , is only optimized in a small timescale to track the timevarying BSUE channel and RISUE channel due to the user mobility. Since the dimension of the equivalent CSI for a given RIS topology is relevant to the number of selected RIS elements, the resultant CSI acquisition overhead is the same as that for a regular RIS with the same number of RIS elements.
Iv The Proposed Joint Optimization Algorithm
In this section, we propose a joint optimization algorithm to solve . First, an exhaustive search method is introduced in Subsection IVA. Next, the overview of the proposed joint optimization algorithm is provided in Subsection IVB. Specifically, we use an adaptive topology design method based on the tabu search algorithm (ATS) to optimize the irregular RIS topology in Subsection IVC. Then, with a given specific RIS topology, the coupled variables and in are decoupled and solved by a neighbor extractionbased crossentropy (NECE) precoding method in Subsection IVD.
Iva Exhaustive search method
In this subsection, we first introduce an exhaustive search method to solve , which can be seen as a benchmark for all other possible solutions. Since the feasible sets of and are discrete sets with finite elements, the optimal solution can be obtained by the exhaustive search method.
The exhaustive search solution to can be illustrated as follows. Firstly, all the irregular RIS topologies from possible options are traversed. Then, for each specific topology , the phase shift for each RIS element is successively chosen to generate the reflection coefficients from possible candidates. Subsequently, given and , the precoder can be obtained by the zeroforcing (ZF) based linear precoder. The equivalent channel between the BS and UEs is given by
(11) 
Thus, the ZF precoder can be expressed as [wu2019beamforming]
(12) 
where represents the power allocation at the BS. Through the exhaustive search method, we calculate the WSR for all different combinations of and , from which the optimal solution will be finally found. However, the possible number of searches of the exhaustive search method is up to , which is prohibitively high for a large RIS size.
IvB Overview of the proposed joint optimization algorithm
Due to the intolerable complexity of the exhaustive search method, we propose a suboptimal joint optimization algorithm with low complexity to solve . As shown in Fig. 2, we decouple the decision variables and alternately optimize the RIS topology and the precoding design.
Specifically, instead of traversing all possible topological structures of RIS, we search for the suboptimal design of RIS iteratively by an ATS method. Specifically, in each iteration, the problem is reduced to , where the RIS topology is fixed. We generate the neighbors of and obtain corresponding precoding designs by a NECE method with ZF precoder at the BS. By comparing the WSR of these neighbors, the optimal neighbor is selected for the next iteration. When the maximum number of iterations is reached or the termination condition is satisfied, we obtain the suboptimal solution to , which is denoted by , , and .
The details of topology design of the irregular RIS will be provided in the following Subsection IVC. The precoding design of the irregular RISaided communication system will be provided in Subsection IVD. The generalization of our proposed joint optimization algorithm will be discussed in Subsection IVE. The complexity analysis of the proposed joint optimization algorithm will be analyzed in Subsection IVF.
IvC ATSbased irregular topology design
Inspired by the tabu search algorithm in [glover1989tabu], we propose an ATS method to optimize the irregular topology of RIS, which obtains a suboptimal RIS topoloy by iteratively searching the possible topologies according to an adaptive move criterion.
The proposed ATS method is summarized in Algorithm 1, and its explanations are detailed as follows. Firstly we initialize the RIS topology by randomly selecting ones and zeros in step 1. For a given , the WSR can be calculated based on ZF precoder at the BS and discrete phase shift adjustment at the RIS in step 2, which will be discussed in the next subsection. In step 4, we focus on the neighbors of generated by a redefined move criterion in the th iteration, where we randomly swap ones for zeros among the diagonal elements of . Here is defined as the neighbor distance, which will be dynamically adjusted according to the value of . We can choose a large at the beginning of iterations for a wide searching range. As the iteration progresses, we decrease the neighbor distance for fine tuning. Next, the obtained neighbors should be checked in the tabu list, which avoids one topology being searched more than once. Those tabu solutions will be discarded in step 5. In this way, we obtain neighbors and separately calculate the corresponding WSR for all candidates in step 6. Then, we select the best candidate with the maximum WSR, which is saved as the new topology for the next iteration. The optimal and are updated accordingly in step 7. The tabu list is updated in step 811. Specifically, the feasible solution is added to the tabu list to avoid cyclic search and the earliest tabu solution in the current list is removed. After the iteration threshold is reached, the suboptimal topology of RIS can be obtained.
Different from the tabu search algorithm developed in [glover1989tabu], the neighborhood, move criterion, and objective function of the proposed ATS method are redefined accordingly in terms of the irregular topology design of RIS. Besides, the neighbor distance in the move criterion is adaptively adjusted based on the iteration process. A large value of corresponds to a wide searching range, while a smaller one aims at fine tuning and better convergence.
IvD NECEbased precoding design
The use of the ATS method requires joint precoding at the BS and the RIS for a specific RIS topology, which corresponds to solving . We propose a NECE method to solve
, which iteratively optimizes the precoder at the BS and the phase shifts at the RIS with probability distribution updated during the iterative process.
The proposed NECE precoding method is summarized in Algorithm 2, and further explanations are detailed as follows. Let
denote the probability matrix of
for a given , where is the probability vector of satisfying . Each component of denotes the probability of taking different values from as shown in (4). Let denote the th element of . We assign an indicator function expressed as(13) 
Firstly, we initialize the probability matrix in step 1 given by
(14) 
which represents that the value of is selected from the elements in with the same probability. In step 3, we randomly generate candidates based on the probability distribution function (PDF) in the th iteration, which can be expressed as
(15) 
Note that is the generated th candidate. Then, we are able to calculate the effective channel for all candidates according to (11) in step 4. The precoders and WSR can be calculated in step 56 according to (12) and (7a), respectively. Next, we sort in a descending order in step 7 as
(16) 
where represents the th candidate in the descending sequence, from which best candidates are selected as primary elites in step 8.
Then, we propose a neighbor extraction method to expand the searching range by changing the phase shift of each diagonal element of . Notice that only diagonal elements of are effective since the trace of is according to (7e), whose values can be selected from . In this way, we can obtain extra candidates in step 9 and calculate the corresponding precoders at the BS as well as the WSR. The candidate whose WSR is larger than is selected as a supplementary elite in step 10. The total number of elites is updated as , where is the number of selected supplementary elites. The weight of the th elite is calculated in step 11 as follows:
(17) 
which represents the ratio of the WSR of elite to the average WSR of all elites. Note that the larger WSR corresponds to the larger weight. After that, in step 12, we update according to the probability transfer criterion, i.e.,
(18) 
The updated probability matrix is employed in the next loop until the iteration threshold is reached. Finally, the suboptimal precoding design can be obtained.
The contributions of the NECE method proposed in this paper are summarized as follows. The probability vector and objective function are redefined in terms of the specific problem. In addition, different from the crossentropy algorithm developed in [rubinstein2013cross], neighbor extraction and individual weights of elites are introduced for expanded searching range and improved accuracy.
So far we have introduced the details of the proposed joint optimization algorithm. The generalization and the complexity will be discussed in the following subsections.
IvE Generalization of the proposed joint optimization algorithm
The proposed joint optimization algorithm can be generalized in various optimization scenarios. For example, the transmit power minimization problem is discussed in this subsection, which provides another perspective to evaluate our proposed schemes.
Let denote the practical SINR of user as expressed in (5), and denote the target SINR of user . The transmit power minimization problem can be described as
(19a)  
(19b)  
(19c)  
(19d)  
(19e) 
can also be solved by the proposed joint optimization algorithm through sightly modifying the optimization target. Similar to the procedure to solve , we search for the suboptimal design of RIS topology iteratively by the ATS method. In each iteration, with a fixed RIS topology , we generate the neighbors of and obtain the corresponding precoding designs by the NECE method. By comparing the required transmit power, the optimal neighbor is selected for the next iteration. Finally, we can obtain the suboptimal solution to .
IvF Complexity analysis
In this subsection, we quantitatively compare the searching complexity, i.e., the maximum number of searches, of the exhaustive search method and the proposed joint optimization algorithm.
In the exhaustive search method, possible RIS topologies are traversed with elements deployed over grid points. For a specific topology, the possible combinations of phase shifts at the RIS with elements is up to , where is the quantized bit number. Thus, the searching complexity of the optimal solution is
(20) 
By contrast, we only search possible RIS topologies in the ATS method, where is the required number of iterations for convergence, and represents the neighborhood size. As for the NECE precoding method, given a specific , combinations are tested, where is the number of iterations of the NECE method, and denote the number of primary candidates and extra candidates in each iteration, respectively. Therefore, the searching complexity of the proposed joint optimization algorithm is
(21) 
Notice that and of the proposed algorithms are usually small, which will be confirmed by simulations in Section V.
The comparison of the searching complexity between our proposed algorithm and the exhaustive search method is shown in Table I. The typical parameters are set as , , , , and for a largescale system with and , which can guarantee the convergence of our proposed algorithm in simulations. One can observe that the searching complexity of the proposed algorithm is much lower than that of the exhaustive search method. Actually, as the system size increases, the searching complexity of the exhaustive search method becomes unacceptable. However, the searching complexity of the proposed algorithm is just linear with , which is much more computationally efficient.
Exhaustive search method  Proposed algorithm  

3294720  1872000  
2088000 
V Simulation Results
In this section, we provide simulation results to evaluate the performance of the proposed irregular RISaided communication systems by employing the proposed joint optimization algorithm.
Va Simulation setup
As shown in Fig. 3, we consider an irregular RISaided multiuser MIMO communication system for simulations. The BS equipped with antennas is located at . The irregular RIS with elements distributed over a rectangular surface with grid points is located at . singleantenna users are randomly scattered in a circular area, whose center is located at .
The uncorrelated Rayleigh fading channel model is adopted to account for the smallscale fading. The largescale fading, i.e., the distancedependent path loss, is considered as well. Specifically, the path loss of the BSRISUE channel can be expressed as [Ozdogan'19, zhang2020joint]
(22) 
where and are the distance between the BS and RIS, and that between the RIS and UEs, respectively. denotes the effect of channel fading and antenna gain. and denote the path loss exponent. Similarly, the path loss of the BSUE channel is expressed as . The parameters are set as dB, dB, , , and [zhang2020joint].
Considering that the 1bit phase shift for metasurfaces is relatively easy to implement with diodes [yang2016A], we set the quantized bit number as . Besides, the noise power is set as dBm, the weights of users are set as . The parameters of proposed iteration algorithms are set as , , , , , if not particularly indicated. The neighbor distance depends on the irregular ratio of RIS, i.e., , and is dynamically changing during the iteration. For instance, if and , is set as 3 at the beginning and is reduced to 2 when the iteration reaches half of the maximum number of iterations. Due to the excessive possible cases of RIS topologies, the size of the tabu list can be simply set as to expand the searching range, with a very low probability of falling into a loop. To validate the superiority of the proposed irregular RIS and joint optimization algorithm, we assume that the CSI is perfectly known [wang2020intelligent, huang2019reconfigurable, pan2020multicell, wu2019beamforming, liu2021compact].
VB WSR performance
In this subsection, we illustrate the WSR performance of the proposed irregular RISaided communication system. The optimal solution to based on the exhaustive search method serves as the upper bound for comparison. As for the proposed joint optimization algorithm, the ATS method is utilized in the irregular topology design of RIS. Besides, the successive refinement (SR) method proposed in [wu2019beamforming] is adopted to optimize the phase shifts of RIS for comparison with the NECE precoding method proposed in this paper. The solution to , i.e., the WSR optimization problem of the classical regular RISaided communication systems[pan2020multicell], is also used as a benchmark.
We first consider a smallscale system with , , , and . The WSR versus the transmit power for the small size system is shown in Fig. 4. By comparing the upper bounds, one can observe that the proposed irregular RISbased scheme with a limited number of RIS elements can enhance the system capacity compared with the existing solution. For example, the WSR of the proposed scheme increases by at the transmit power of dBm. Besides, note that both the NECE method and the SR method are effective, we can conclude that the capacity enhancement of the proposed irregular RISaided communication system does not depend on a specific precoding algorithm. The reason is that, the selection of feasible locations for RIS element deployment leads to additional spatial DoFs, which enables us to select a element subset with the optimal channel conditions. Therefore, the irregular RIS with elements can achieve the spatial diversity benefits of grid points, which enhances the received signal power and thus improves the system capacity via the welldesigned topology and precoding design. It also implies that the proposed joint optimization algorithm based on ATS method and NECE method is able to achieve the suboptimal performance with much lower complexity, which verifies the reliability and convergence of our proposed joint optimization algorithm.
Then, we consider a largescale system with , , , and . Since the complexity of the exhaustive search method is unacceptable for such a large system size, we only provide the suboptimal solution by the proposed joint optimization algorithm. Here we provide an optimized RIS topology in Fig. 5, where the colored squares represent the selected locations for RIS elements. The precoding optimization for the classical regular RISaided communication systems based on the NECE method is utilized as a benchmark scheme. Besides, the SR method for both cases of the classical regular RIS and the proposed irregular RIS is realized for comparison. The WSR versus the transmit power is shown in Fig. 6, where similar conclusions can be drawn. The WSR of the proposed irregular RIS is improved compared with that of the regular RIS. For instance, provided the NECE method is adopted, the WSR of the proposed irregular scheme increases by at the transmit power of dBm. In addition, the proposed NECE method outperforms the SR method for both cases of the existing regular RIS and the proposed irregular RIS, which confirms the effectiveness of the proposed NECE method.
VC Irregular ratio of RIS
The effect of different irregular ratios of RIS is analyzed in this subsection. First we consider an irregular RIS with a fixed number of elements, where , , and . The size of the irregular RIS is variable, which is represented by , i.e., the number of grid points of the surface. The simulation results are provided based on the proposed joint optimization algorithm with the transmit power set as dBm. The existing regular RISbased scheme with , , and serves as a benchmark, whose surface size is fixed. The WSR versus the number of grid points of the irregular RIS is shown in Fig. 7. It is observed that altering the irregular ratio of the irregular RIS via enlarging the surface size effectively improves the WSR performance, where the number of RIS elements is assumed to be a constant. Moreover, we consider several regular RISbased scenarios where more antennas are deployed at the BS, namely, , , and . The number of RIS elements and the number of users remain unchanged. It implies that the irregular RISs with enlarged size, namely, , , and , outperform the classical regular RISs with , , and , respectively. Note that when the irregular ratio is , i.e., and , the irregular scheme saves half of the number of antennas at the BS. Thus, the irregular RIS can provide a feasible solution to improve the system performance without increasing antennas and RF chains at the BS, whose hardware cost and power consumption are usually high. Nevertheless, with the increased number of grid points of the irregular RIS, the growth of performance slows down and the system complexity increases. Thus, we should make a tradeoff between the cost and the performance by carefully designing the irregular ratio of RIS in practice.
Then, we consider an RIS with variable elements and a fixed size. The transmit power is set as dBm. Other parameters are set as , , and . The WSR versus the number of RIS elements is shown in Fig. 8, where the classical regular RISbased schemes with , , and different numbers of RIS elements serve as the benchmark schemes. One can observe that, the proposed irregular RISbased schemes with achieve higher WSR than classical regular schemes. Besides, the red curve illustrates the scalability of our proposed irregular RISbased scheme. Specifically, the system capacity increases with the increased number of RIS elements. Therefore, it implies that we can design an RIS with a large aperture and the number of elements selected for communication depends on the capacity requirements. It is worth noting that the performance gap between the proposed irregular RIS and the classical regular RIS becomes smaller as the irregular ratio of RIS increases, hence the benefit of further increasing RIS elements with a very high irregular ratio is negligible. The tradeoff between the cost and performance can be a possible future research topic.
VD Transmit power consumption
The solution of the transmit power minimization problem is analyzed in this subsection. We show the transmit power versus the target SINR of UEs in Fig. 9 with parameters set as , , and . It is revealed that the proposed irregular scheme requires much lower transmit power than the classical one for a certain target SINR. In addition, the proposed NECE method outperforms the SR method for both schemes. Thus, we can draw the conclusion that the irregular scheme can significantly decrease the transmit power consumption, which further verifies the effectiveness of the proposed irregular RIS and the corresponding joint optimization algorithm.
Vi Conclusions
The capacity of the existing regular RISaided wireless communication systems with a limited number of RIS elements is restricted. To tackle this challenge, we investigated the design of irregular RIS in this paper. Firstly, we proposed an irregular RIS structure with a given number of elements distributed over an enlarged surface. Then, for the proposed irregular RISaided communication system, we formulated a WSR maximization problem to optimize the system capacity. Finally, a joint optimization algorithm with low complexity was proposed to iteratively solve the optimization problem. Specifically, an ATS method was used to design the irregular RIS topology, and a NECE method was introduced to optimize the precoding design. Simulation results validated that, with a limited number of RIS elements, the proposed irregular RIS can enhance the system capacity. Several open problems are left for future works. For example, how to deploy the RIS elements arbitrarily within the surface aperture of other possible shapes [rocca2016unconventional] remains to be investigated. Moreover, other metrics such as the energy efficiency of irregular RISaided communication systems are worth further investigations.
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