Capacity Improvement for Intelligent Reflecting Surface-Assisted Wireless Systems With a Small Portion of Active Elements

In this study, the use of intelligent reflecting surfaces (IRSs) to improve the capacity of a multi-user multiple-input-single-output downlink system is investigated. Unlike existing IRS-assisted wireless systems equipped with entirely passive IRS elements or purely active IRS elements, we propose a new IRS architecture in which each IRS element can operate either in active mode or in passive mode. To reduce power consumption, only a small portion of the IRS elements operate in active mode. We seek to maximize the capacity of the proposed IRS-assisted wireless system by jointly optimizing the phase shifts of all IRS elements, the transmit beamforming of the base station, and the IRS active-passive mode vector, which leads to a non-convex problem. In this study, the non-convex problem is addressed by using a probability learning-based algorithm from the cross-entropy optimization framework with an explicit expression for learning the targeted sampling distribution’s tilting parameters. The simulation results reveal that a significant capacity improvement over traditional IRS-assisted wireless systems with entirely passive IRS elements can be achieved using the proposed scheme, which employs only a small portion of active IRS elements.

the base station (BS) towards the desired directions. Unfor- 23 tunately, the use of purely passive IRS elements introduces 24 the double-fading problem, which originates from the product 25 of path losses on the BS-IRS and IRS-user links. Although 26 this problem can be alleviated by increasing the number of 27 passive IRS elements [10], the resulting physical size and 28 power consumption of the IRS module is impractical for some 29 The associate editor coordinating the review of this manuscript and approving it for publication was Xujie Li . scenarios, and the complexity of IRS optimization increases 30 considerably. 31 Recent studies propose the use of active elements in 32 IRS-assisted wireless systems [11], [12], [13], [14], [15], 33 [16] to tackle double-fading via the use of low-power 34 reflection-type power amplifiers (PAs) such that the phases 35 and amplitudes of the incident signals from the BS can 36 be simultaneously adjusted. For example, Dong et al. [15] 37 investigate the physical layer security enhancement of active 38 IRS-assisted wireless systems. Moreover, resource allocation 39 algorithm design for active IRS-assisted wireless systems has 40 also been studied [16]. The results indicate that IRS-assisted 41 systems with active elements achieve significantly higher 42 transmission rates than IRS-assisted systems with passive ele-43 ments. However, although the reflection-type PAs consume 44 low power (e.g., 6-20 mW [17]), their power consumption 45 still exceeds than that of a conventional passive IRS element 46 without PA (e.g., 5 mW [18]). 47 1 A hybrid active and passive IRS architecture was introduced in [19] to improve performance and reduce power consumption. However, the system model and problem formulation considered in [19] are explicitly intended for an IRS-assisted single-user downlink system. Moreover, [19] assumes that the phase shifts of the IRS element have an infinite resolution rather than a practical finite resolution. Note that according to recent research on the implementation of phase shifters [20], high-resolution phase shifters are not feasible. Additionally, phase shifters consume more power at higher resolution. For instance, the power consumption for 1-, 2-, and 5-bit resolution phase shifters are 10, 20, and 50 mW, respectively [21], [22], [23]. It would therefore be impractical to assume infinite-resolution phase shifters because of power consumption and hardware costs. This study, in contrast to [19], focuses on a multiuser scenario, which is more complex because of the intricately coupled variables that result from multiuser interference. Furthermore, each phase shift of the IRS element in this study has a finite resolution due to practical considerations. In this case, the algorithm proposed in [19], which is tailored for an IRS-assisted single-user system with infinite-resolution phase shifters, may not be applicable to the scenario of this study. Most importantly, the proposed algorithm simultaneously determines the phase shifts of all IRS elements and their corresponding operation modes without alternating procedures such as the algorithm developed in [19]. a complex Gaussian distribution with zero-mean and 88 variance σ 2 . ∼ denotes ''distributed as.'' Lastly, j √ −1 89 is an imaginary unit.

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As shown in Fig. 1, we consider an IRS-assisted multiuser 93 downlink wireless system that comprises an M -antenna BS, 94 U single-antenna users, and an N -element IRS. In contrast 95 to traditional IRS-assisted wireless systems, each IRS ele-96 ment has two modes of operation: (i) passive mode, where 97 the IRS element is used to reflect the incident signals from 98 the BS, and (ii) active mode, where the IRS element with 99 a low-power reflection-type PA serves as an active relay. 2 100 In passive mode, an IRS element reflects signals without 101 amplification. Therefore, generally, the reflection amplitude 102 of the IRS element in the passive mode is assumed to be 103 one. However, in active mode, the IRS elements amplify the 104 reflected signals, indicating that the reflection amplitude of 105 the IRS element with the active mode is no smaller than one. 106 In this study, we assume that the reflection amplitude of the 107 IRS element in active mode is fixed (and denoted by η > 1) 108 to simplify the IRS circuitry [13]. Although the reflection-109 type PA consumes low power (e.g., 6-20 mW [17]), its power 110 consumption is still higher than that of a conventional passive 111 IRS element without a PA (e.g., 5 mW [18]). Consequently, 112 only the K < N IRS elements operate in active mode when 113 considering power consumption, while the remaining IRS 114 elements operate in passive mode.

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For ease of notation, we introduce a binary diagonal mode 116 selection matrix, denoted by B diag(b) with b = {b n } N n=1 , 117 to indicate the mode in which the n-th IRS element operates 118 in, i.e., active or passive mode. Specifically, b n = 1 if the 119 n-th IRS element operates in active mode, and b n = 0 if 120 the n-th IRS element operates in passive mode. In this case, 121 the reflection matrix at the IRS can be written as the product 122 of the reflection amplitude matrix, denoted by A diag (a) 123 with a = {a n } N n=1 , and the phase shift matrix, denoted by Here, a n ∈ {1, η} 125 denotes the reflection amplitude of the n-th IRS element, 126 where a n = 1 if b n = 0, and a n = η if b n = 1. Alternatively, 127 we can express a n = η b n , which indicates that the correspond-128 ing reflection amplitude matrix A is also determined once the 129 mode selection matrix B is determined. θ n ∈ [0, 2π) is the 130 phase shift of the n-th IRS element. In practice, each phase 131 shift θ n has only a finite resolution. Therefore, θ n is restricted 132 to the values of F λ { , 2 , . . . , 2 λ } with = 2π 2 λ for 133 the phase shift using λ bits of resolution.

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At the BS, the data intended for the u-th user s u with 135 E{|s u | 2 } = 1 is multiplied by the associated transmit beam-136 forming vector w u ∈ C M ×1 . Thus, the transmit signal vector 137 at the BS is constructed by U u=1 w u s u , and the total transmit 138 power constraint is expressed as U u=1 w u 2 2 ≤ ℘, where 139 2 For cost-effective consideration, only the IRS element with the active mode is connected to a low-power reflection-type PA, similar to the antenna selection technique [25]. Therefore, only the K < N reflection-type PAs are required in the proposed scheme.   of (1) and the definition above, the signal-to-interference-150 plus-noise ratio at the u-th user can be expressed as

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Our objective is to maximize the sum-rate of an IRS-assisted The optimization problem in (3) is non-convex because 165 of the coupled optimization variables and the non-convex 166 constraints in (3c) and (3d). We adopt a classical ZF-based 167 at 168 the BS to simplify the optimization problem in (3), where 169 . . , f U ] is the effective channel matrix from the BS to all 171 users. In addition, the obtained transmit beamforming matrix 172 W must be normalized by a factor of fulfill the power constraint in (3c). Accordingly, the objective 174 function in (3a) can be represented solely in terms of B (or b) 175 and (or x), and P1 is recast as which can be solved using an exhaustive search. However, 180 the search size of P2 is up to N ! (N −K )!K ! × 2 λN , which is pro-181 hibitively high when N is large.

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CEO is an effective probability learning-based frame-184 work for solving complex optimization problems [24].
where E (t) denotes the elite set constituted by the L elite 206 where 0 < ϕ ≤ 1 is a fixed smoothing parameter. 207 We must first design a conveniently parameterized sam-208 pling distribution to generate samples to use the CEO frame- mode. Accordingly, the sampling distribution for generating 218 a random mode selection vector, b, can be expressed as

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Once the parameterized sampling distribution is estab-240 lished, in each iteration t, a collection of random sam-241 ples {ϒ } L =1 is sampled from the pre-242 designed parametrized sampling distribution P( · ; π (t−1) ) = 243 P( · ; v (t−1) , P (t−1) ) defined in (9). However, the generated 244 samples {b we propose a restricted operator to ensure that each generated 247 sample b (t) is feasible without changing the distribution of 248 P( · ; v (t−1) ) as follows:  [ ] } L elite =1 thus obtained, the parameterized 264 sampling distribution π (t−1) is updated to π (t) by solving 265 the problem in (5). Proposition 1 provides an optimal update 266 formula.     is considerably lower than that of an exhaustive search.

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The parameter settings for the CEO-based framework are 284 problem-dependent. However, the desired performance can 285 be obtained in conformity with the recommendation of [24]. 286 It is suggested to set L = εN with 1 < ε < 10 and ς = 0.1. of the proposed algorithm is also provided in Fig. 2(b) when 318 ℘ = 10 dBm. Accordingly, the CEO with a fixed mode 319 Algorithm 1 CEO for Solving Problem (4) Input: Sample size L, parameter ς , and smoothing parameter ϕ Employ proposed restricted operator to ensure that each generated sample b (t) is selection and K = 8 (K = 16) achieves performance compa-351 rable with the SA with N entirely active reflecting elements 352 while reducing power consumption. Notably, the CEO with 353 dynamic mode selection and K = 16 even performs bet-354 ter than the SA with purely active reflecting elements when 355 η = 2. This phenomenon is due to the fact that SA may lead 356 to premature convergence when an inappropriate initial guess 357 is applied. Additionally, as plotted in Fig. 5, the performance 358 of the CEO with dynamic mode selection is further improved 359 by increasing λ.
360 Figure 6 compares the energy efficiency of the test algo-361 rithms with transmission power, where the energy efficiency 362 is defined as [16], [21], [22], [23] Here, P total is the total power consumption and is given by 365 (12), as shown at the bottom of the page, where P BB , P RFchain , 366 P λ-PS , and P SW are the power consumptions of the baseband 367 processor, a single RF chain, a λ-bit phase shifter, and a 368 switch, respectively. P I is the circuit power required to sup-369 port one IRS element. In the simulation, the power consump-370 tion values of these components are set to P BB = 200 mW, 371 P RFchain = 300 mW, P 1-PS = 10 mW, P SW = 5 mW, and 372 P I = 2 mW [16], [21], [22], [23], [27], [28]. In addition, 373   In contrast to (5), an additional constraint is required to ensure 403 that the sum total probability of an element in set F λ being 404 chosen as x n is one, that is, |F λ | k=1 p nk = 1. Therefore, the 405 optimal update problem in our case can be expressed as To proceed with this optimization, we first calculate the par-415 tial derivatives of ln P(ϒ (t) ; π ) with respect to v n and p nk , 416 yielding 417 ∂ ∂v n ln P(ϒ (t) Next, differentiating (A.2) with respect to v n and applying the 420 result of (A.3a) yields Subsequently, solving for v n of (A.4) yields the following 423 analytic expression for v n :