Analysis of Hierarchical Rate Splitting for Intelligent Reflecting Surfaces-Aided Downlink Multiuser MISO Communications

An intelligent reflecting surface (IRS) is an emerging technology in the next-generation (B5G and 6G) wireless communications with the aim of improving the spectral/energy efficiency of the wireless networks. In this paper, we focus on multiple IRS-aided multiuser multiple-input single-output (MISO) downlink network, where each IRS is deployed near to the cell boundary of the cellular network in order to help the downlink transmission to the cell-edge users. To achieve the significant performance benefits provided by the rate-splitting (RS) transmission in multiuser scenario, two-layer hierarchical rate splitting (2L-HRS) technique is deployed at the base station (BS) to serve all the users. Furthermore, we propose an On-Off scheme for controlling the IRSs with practical phase shifts. In order to analyze the performance of the users, we derive the closed-form expressions of outage probability for both cell-edge users and near users. By extensive Monte Carlo simulations, we demonstrate that the proposed design framework outperforms the corresponding one-layer RS (1L-RS), the multiuser linear precoding (MU-LP) and 2L-HRS without IRS. In addition, we reveal the advantages of introducing RS-based IRS in improving the cell-users’ performance. The impact of channel estimation errors due to imperfect channel state information, and the various network’s parameters, such as the number of reflecting elements and the number of cell-users, on the network performance is demonstrated.

activities in the areas of IRS-assisted wireless networks and highlighted the open research problems, while in [12], the large intelligent surfaces (LIS)-space shift keying and LIS-spatial modulation schemes have been investigated to enhance the network spectral efficiency along with signal quality at the end user. While in [13], an energy-efficient resource allocation scheme have been investigated for downlink multi-user communication systems, the authors in [14] have studied the emerging research field of reconfigurable intelligent surface (RIS)-empowered smart radio environments (SREs) and highlighted the most suitable applications of RISs in wireless networks along with presenting an electromagnetic-based communication-theoretic framework for analyzing and optimizing metamaterial-based RISs. The work [16] has proposed an enexact-alternative-optimization method to design the optimal transmit beamforming vectors at the base stations and the reflecting beamforming vectors at the IRS. Further, the resource allocation scheme has been studied in [18] for IRS-aided full duplex cognitive radio networks. While a non-orthogonal multiple access (NOMA) transmission scheme has been proposed in [21], [22], the beamforming design algorithms have been studied in [23]- [29]. For example: an IRS-aided single-cell wireless system have been studied in [23] with discrete phase shifters where a single IRS is deployed to help in the communications between a multi-antenna access point (AP) and multiple single-antenna users, while by jointly optimizing the active transmit precoders at the AP and the passive reflect phase shifts at all IRSs, the transmit power at the AP has been minimized in [24]. The authors in [25] have proposed a robust beamforming algorithm for IRS-assisted multi-user multiple-input single-output (MISO) network, whereas [26] has considered an IRS-assisted multi-cell multiple-input multiple-output (MIMO) network and focused on maximizing the weighted sum rate by jointly optimizing the active transmit precoder matrix at the base stations (BSs) and the phase-shift matrix at the IRS. Furthermore, the passive beamforming and information transfer strategy has been developed in [29] for IRS-assisted multiuser MIMO networks. However, none of the authors in [23]- [29] have analyzed the performance of rate-splitting multiple access (RSMA) based IRS-assisted communication network and thus, it motivates us to focus on this problem.
In the last few years, rate splitting (RS) has emerged as a remedy for improving the spectrum efficiency [30]- [36]. For instant, the authors in [30] have highlighted recent advancements in RS specially for MIMO networks, while the advantages of RSMA over NOMA technology have been illustrated in [32]. One of the most important characteristic of RSMA is that it partially decodes interference and partially treats interference as noise through the split into common and privates messages, whereas the NOMA completely relies on SIC and fully decodes interference. In addition, The NOMA is more suited to overloaded networks whereas the RSMA is suitable for all type of network loads. The techniques for stream combining and linear precoder design have been investigated in [34] for RS in multiuser MIMO systems. Moreover, to optimize the precoders, message splits, time slot allocation along with relaying user scheduling, the authors in [36] have formulated a max-min fairness problem of a K-user cooperative RS in MISO network. Similar to [36], the authors in [37] have adopted a RS strategy and developed a precoder design algorithm for maximizing minimum of all clusters rates in multi-group multicasting. However, to the best of the authors knowledge, the performance analysis of RS-based IRSs-aided multiuser MISO networks has not yet been investigated in the literature.
Inspire by the aforementioned discussions, in this paper we consider a multiuser MISO downlink network, where multiple IRS are deployed near to the cell boundary of the cellular network in order to assist the downlink transmission to cell-edge users and deploy two-layer hierarchical rate splitting (2L-HRS) at the BS. In the light of existing work [10], [12] [29], the major contributions of this work are outlined as follows.
• We primarily emphasis on multiple IRS-aided downlink MISO hierarchical RS (HRS) network wherein numerous IRS are deployed near to the cell boundary of the cellular network for serving the cell-edge users. To achieve the significant performance benefits provided by RS transmission specially in multiuser scenario, we deploy 2L-HRS transmission strategy at the BS. • We propose an On-Off scheme for controlling the IRSs with practical phase shifts. Next, to analyze the performance of the users, we derive the closed-form expressions of outage probability for both cell-edge users and near users by considering both common as well as private data signal-to-interference-plus-noise ratio (SINR). • By extensive Monte Carlo simulations, we validate the effectiveness of the proposed scheme and demonstrate that the proposed design framework outperforms the corresponding one-layer RS (1L-RS), the multiuser linear precoding (MU-LP) and 2L-HRS without IRS. In addition, we show the advantages of introducing RS-based IRS in improving the cell-users' performance. • Moreover, we demonstrate the impact of channel estimation errors because of imperfect channel state information and the various network's parameters, such as the number of reflecting elements and the number of cell-users, on the network performance.
Organization: The rest of the paper is structured as follows. The system model is described in Section II. Section III presents the SINRs expressions for the considered HRS framework. While the proposed on-off scheme for controlling the IRS with practical phase shifts is illustrated in Section IV, outage analysis for IRS-aided HRS is depicted in Section V. In Section VI, numerical results are presented. At last, conclusions are drawn in Section VII.
Notations: The following notations are used throughout the paper. The lowercase and uppercase boldface letters (e.g., a and A) are used to denote a vector and a matrix, respectively. (·) H is defined for complex conjugate. The matrix I M denotes an M × M identity matrix. Tr[A] is the trace of a matrix A and ||·|| 2 used to denote the square-norm. Besides, the list of main notations utilized in this paper are depicted in Table 1.

II. SYSTEM MODEL
We consider a downlink multi-user scenario as illustrated in Fig. 1 where a multi-antenna BS (with N s antennas) utilizes rate splitting to serve multiple single antenna users. Let us consider a total of K single antenna users indexed by a set K = {1, 2, . . . , K}. We divide these K users into M + 1 groups, 1 denoted as G m , m ∈ M, where M = {0, 1, . . . , M}. The users having a direct link with BS, i.e., near users (NUs), are grouped into a single group G 0 , whereas M nonoverlapping groups are formed to divide all the cell-edge users (CEUs), i.e., the users with no direct connection from the BS. Each group consists a subset of users, K m ⊂ K, such that K m ∩ K n = ∅, m = n, m, n ∈ M. If we assume that the cardinality of m-th subset is K m , i.e., |K m | = K m , them the groups satisfy m∈M |K m | = K.
The BS communicates with a cell-edge group (CEG) G m m ∈ M , where M = {1, 2, . . . , M} through an IRS consisting of N r reflecting elements. It is assumed that only the signals reflected by the IRS once are considered while those reflected by IRS for two and more times are ignored. The 1. User grouping is a well known technique for the reduction of latency and interference [38], [39]. In this work, we assume that there exists a method [38], [39] to group the users based on the location in order to control the interference and we are provided with grouped users. BS utilizes two-layer hierarchical rate splitting (2L-HRS) technique to serve all the users as explained below.

A. TWO-LAYER HIERARCHICAL RATE SPLITTING (2L-HRS)
In this technique, the BS splits the message of each user into three part-messages known as private, common and commonto-group part-messages. Similar to 1L-RS, the common partmessages of all the users are jointly encoded to obtain a common data stream of order K intended to all the users, and the private part-messages of each user is separately encoded to obtain K different 1-order private data streams intended to individual user. In addition to this, common-to-group partmessages of the users in a group G m , m ∈ M are also jointly encoded to get a K m -order data stream intended for the users in G m only.
Let the combined data vector to be sent by the BS using 2L-HRS is where x K is the common data stream of order K, x K m , m ∈ M represents the K m -order common-to-group data stream to be decoded by the users in G m , and x k , k ∈ K is the unit-order private data stream corresponding to k-th user. We also assume unity average power constraint such that Tr[xx H ] = 1. The data vector x is linearly precoded with the help of an N s × (K + M + 2) precoder matrix P = [p K , p K 0 , p K 1 , . . . , p K M , p 1 , p 2 , . . . , p K ], where the column vector p K precodes the common data stream x K , the column vector p K m is used as a precoding vector for common-to-group data stream x K m for group G m , and vector p k is the precoding vector for private data stream x k . The data vector broadcasted by the BS can therefore be given as We assume a total transmit power constraint at the BS as

B. DATA TRANSMISSION
The signal received at the q-th NU, q ∈ K 0 can be given as where g q denotes the 1 × N s channel vector between the BS and the q-th NU and e q is the zero mean AWGN at the q-th NU in group G 0 with variance σ 2 . Let all the elements of vector g q are independent and identically distributed (i.i.d.) as zero mean complex Gaussian (ZMCG) random variables with variance g . Assuming that IRS m is placed close to m-th group G m , m ∈ M so that the only the CEUs in group G m can hear from it. The received signal at the k-th CEU, k ∈ K, k ∈ K m , m ∈ M obtained through the m-th IRS can be written as where H m denotes the N r × N s channel matrix between the BS and IRS m , f m,k denotes the channel vector between IRS m and the k-th CEU in group G m . All the channel coefficients in matrix H m and vector f m,k are assumed to be i.i.d. as ZMCG random variables with variances h and f , respectively. The variable e k denotes the zero mean AWGN at the receiving antenna of the CEU having variance σ 2 . The transmit SNR can therefore be defined as ρ B = P B σ 2 . Matrix m is a N r × N r diagonal matrix containing the reflection coefficients of each reflecting element at IRS m and can be defined as where ϕ m,n and β m,n (ϕ m,n ), n = 1, 2, . . . , N r , denote the phase shift and the phase dependent reflection amplitude at n-th reflecting element of IRS m . If we write the elements of diagonal matrix m in a column vector, denoted by φ m , and define a diagonal matrix F m,k containing the elements of vector f m,k , then we can rewrite (3) as can be utilized to initialize private precoders of RS, while singular value decomposition (SVD) can be utilized to initialize the common precoder [32], [33]. However, for tractability of the analysis and in order to satisfy the transmit power constraint, we consider that each precoding vector p i = √ α i P Bp withp being an arbitrary complex-valued unit-norm vector and α K + m∈M α K m + k∈K α k = 1.

III. SINR FOR HRS
As per the 2L-HRS scheme, three different data streams (i.e., x K , x K m , and x k ) need to be decoded by each NU and CEU. The decoding order is based on the order of the data streams, i.e., a higher order data stream is decoded prior to a lower order data stream.

A. CHARACTERIZATION OF SINR
Each user first decodes the K-order common data stream by considering the interference of all other data streams as noise. Therefore, the SINR for common data stream at q-th NU and k-th CEU can be obtained from (2) and (5), respectively, as whereh = H mp . On successful decoding of common data stream x K , its contribution is removed from the received signals y q and y k , respectively. Next, the m-th common-to-group data stream x K m , m ∈ M, is decoded by the all the users in group G m by treating all other common-to-group data streams and all private data streams as noise. Therefore, the SINR for decoding K 0 -order common-to-group data stream at q-th NU can be given as Similarly, the SINR for decoding the K m -order common-togroup data stream at k-th CEU, k ∈ K m , can be given as In the end, each NU removes the contribution of the decoded common-to-group data stream from the received signal y q and then decodes its own private data stream x q . Thus, the SINR for decoding 1-order data stream x q can be written as Similarly, the k-th CEU removes the contribution of the decoded common-to-group data stream from the received signal y k and decodes its own private data stream x k . The SINR for decoding the private stream x k can be written as It can be seen from (7), (9), and (11) that the choice of vector φ m is crucial for analyzing the system performance of CEUs in IRS-aided downlink network. There are different solutions available in the literature such as ideal or Zero Forcing (ZF) design, finite resolution-based design using discrete fourier transform (DFT), and cost-effective on-off control [22]. Due to the simplicity and scalability of the design (mainly for large reflecting elements) to configure and control the IRS, we consider on-off scheme in this work. Note that the cell-edge users' effective channel vectors are determined by the choice of m (for m = 1, 2, . . . , M), where ϕ m,n and β m,n (ϕ m,n ), n = 1, 2, . . . , N r are usually arbitrary. In literature [10], [12], [18], the choices for ϕ m,n and β m,n (ϕ m,n ), n = 1, 2, . . . , N r are usually arbitrary. However, due to the limitations on hardware of the IRS this assumption may not always hold true. Hence, in this paper we assume finite resolution beamforming through the ON-OFF control approach, whereby each diagonal element of m is either 1 (on) or 0 (off), like the one proposed in [22].

IV. PROPOSED ON-OFF SCHEME TO CONTROL IRS WITH PRACTICAL PHASE SHIFTS
In this section, we describe the proposed On-Off scheme to control IRSs with practical phase shifts. In this scheme, each reflecting element of IRS can either be OFF (i.e., inactive) or ON (i.e., active). For inactive elements, the reflection amplitude will be zero and for active elements, we consider a practical reflection amplitude 2 β n (ϕ n ) defined as [40] where β min ≥ 0 denotes the minimum reflection amplitude, θ ≥ 0 is the difference between −π/2 and the phase corresponding to β min , and is a positive constant related to the circuitry of the reflecting element. In order to have a finite resolution based design, we consider that the continuous phase ϕ n is first quantized uniformly by utilizing a ν-bit quantizer to give a discrete phase shift value from L = 2 ν discrete levels. Therefore, if ϕ n ∼ U(−π, π), then the quantized phase shift valuẽ L . Let the number of reflecting elements at IRS is factored as N r = L r D r , such that L r and D r are positive integers, 2. For simplicity, we skip the subscript 'm' corresponding to m-th IRS. The discussion in this section is valid for any arbitrary IRS among the considered IRSs. then we define a L r × N r matrix A as . . , L r , represents a 1×D r vector containing the practical reflection amplitudes of D r active reflecting elements and 0 is the 1 × D r vector of all zeros. Let a , = 1, 2, . . . , L r , denotes the -th row vector of matrix A, then we choose φ H = a , such that a maximizes the SINRs in (15). Therefore, the optimum SINRs at the k-th user can be given as where Remark 1: It is worth noting that for a given , random SINRs γ k,K (a ), γ k,K m (a ), and γ k (a ) (defined in (15)) are dependent on each other. However, the set of SINRs Further, the Monte Carlo simulations verify that maximum values of three SINRs, i.e.,γ k,K ,γ k,K m , andγ k are obtained at same a . In other words, if a particular choice of φ m (i.e., a particular active reflecting element pattern) maximizes the K-order common message SINR, then the same φ m also maximizes both the K m -order common-to-group SINR and unity order private message SINR.

A. OUTAGE PROBABILITY FOR CEU
Under 2L-HRS scheme, a CEU will said to be in outage if and only if the three optimum SINRs (defined in (14)) corresponding to common-to-all, common-to-group, and private messages simultaneously fall below the respective threshold SINRs. Thus, the outage probability at k-th user can be defined as, (16) where τ K , τ K m , and τ 0 are threshold SINRs for commonto-all, common-to-group, and private messages, respectively. Pr{·, ·, ·} represents the joint probability. Following [41], we can write (16) as, where F X 1 (·) denotes the cumulative distribution function (CDF) of a RV X 1 , F X 1 ,X 2 (·, ·) and F X 1 ,X 2 ,X 3 (·, ·, ·) are the joint CDFs of two and three RVs, respectively. If we define a set X ≡ {γ k,K , γ k,K m , γ k }, then for all X 1 , X 2 , X 3 ∈ X and X j max({X j (a )} L r =1 ), j = 1, 2, 3, following can be written using the discussion in Remark 1, where τ 1 , τ 2 , τ 3 ∈ T, with T ≡ {τ K , τ K m , τ 0 }. It is clear from Remark 1 that the set of random variables {X 1 (a ), X 2 (a ), X 3 (a )} is independent of another set {X 1 (a κ ), X 2 (a κ ), X 3 (a κ )} for all κ = , κ, = 1, 2, . . . , L r . This also implies that random variable X 1 (a ) is independent of X 1 (a κ ), X 2 (a ) is independent of X 2 (a κ ), and X 3 (a ) is independent of X 3 (a κ ) for all κ = . Hence, the CDF of the maximum of L r independent random variables (or set of random variables) can be written as the product of their marginal (or joint) CDFs. CN (0, δ 2 b ), for p = 1, 2, . . . , P, then for any arbitrary real-valued vector c = [c 1 , c 2 , . . . , c P ] T with c p ∈ (0, 1), the CDF of W = |c H Db| 2 can be given as (19) where K v (·) is the v-th order modified Bessel's function of second kind, λ j = 1 , andc j , j = 1, 2, . . . , J, is a non-zero element of vector c.
Proof: Let vector c contains J (J ≤ P) non-zero elements c j , j ∈ {1, 2, . . . , J}, then we can define a RV U as Assuming that random vector b is known, we can write that The Exp(λ) represents the exponential distribution with rate parameter λ.
. Since the sum of J i.n.i.d. exponential RVs follows Hypoexponential distribution [42], we can get the probability density function (PDF) of V as Now, we can obtain the PDF of W by using (22) in and by substituting (23) in F W (w) = w 0 f W (x) dx along with the use of [43, eq. (6.561.12)], we get (19).
Corollary 1: For J=1 (special case), i.e., vector c contains only one non-zero element (sayc), the CDF of W = |c H Db| 2 can be given as
Observation 1: For real and positive threshold SINRs for common-to-all, common-to-group, and private messages (i.e., τ K , τ K m , τ k > 0), it can be observed by using (28) in (27) that the three power allocation coefficients must satisfy the following conditions as and such that η K , η K m , η k > 0. The optimum values of these power allocation coefficients are chosen so as to minimize the outage probability. In Fig. 2, Fig. 3, and Fig. 4, we have shown the outage probability variations of a CEU for varying α K , α K m , and α k , respectively, under 4-user scenario with equal NUs and CEUs. For Fig. 2, we have considered τ K = 0.2, 0.3 with τ K m = τ k = 0.2 and ρ B = 20 dB. Furthermore, we consider α K 1 = 2α K 0 = 0.2 and remaining power is divided in four private messages as (i) each CEU and NU getting 35% and 15% of the power available for private messages, i.e., α k = 0.35(1 − α K − α K 0 − α K 1 ), and (ii) each CEU and NU getting 30% and 20% of the power available for private messages, i.e., . Under these settings, it can be observed from Fig. 2 that the valid range for α K with N r = 4, 8 both is α K > 0.17 for τ K = 0.2 and α K > 0.24 for τ K = 0.3, which is inline with (31). Moreover, the outage probability is minimized at α K ≈ 0.24, 0.22 for scenario (i) and (ii) given above, respectively with τ K = 0.2. However, if we have increase the threshold SINR of common-to-all message to 0.3, we observe an increase in the optimum value of α K under both power allocation scenarios. In Fig. 3, we have used the optimum values of α K to observe the variations of outage probability with power allocation coefficient of common-to-group data. We have also fixed ρ B = 20 dB, τ K = τ K m = τ k = 0.2, and α K 0 = 0.5α K 1 . It can be noticed from Fig. 3 that for a group of 2 CEUs having 30% of total private messages power each, 0.13 < α K 1 < 0.35, and CEU outage probability is minimized at α K 1 ≈ 0.18. Similarly, if we allocate 35% of total private messages power to each CEU, we get 0.13 < α K 1 < 0.38, and the minimum occurs at In the end, we have optimized the outage probability for the power allocation coefficient for private message of the CEU (i.e., α k , k ∈ K 1 ) in Fig. 4. We have taken α K = 0.25, α K 0 = 0.1 and α K 1 = 0.2 in Fig. 4. Furthermore, we have assumed that all the users in a group are allocated with equal power. It can be seen from Fig. 4 that for all values of ρ B and N r considered in the figure, the outage probability is minimum for α k > 0.16, which is approximately 35% of (1 − α K − α K 0 − α K 1 ), i.e., fraction of available power for all private messages.
In the numerical results section, the optimum values of power allocation coefficients (for given threshold SINRs of common-to-all, common-to-group, and private data) are chosen in accordance to Observation 2.

B. ASYMPTOTIC OUTAGE OF CEU FOR LARGE N r
In order to find the further insights, we investigate the outage behavior of a CEU for sufficiently large number of reflecting elements, i.e., N r → ∞. For this, we assume the ideal phase shift scenario (i.e., β min = 1) at each reflecting element of IRS. Therefore, β n (ϕ n ) = 1 for all n and λ j = 1 h f = λ, for all j. Following the results in Corollary 2, we can write the CDF in (27) as where (τ 1 , η 1 ) ∈ T m . Under ideal reflection conditions, X 1 (a ) will have equal average statistics due to the fact that u = 1 1×D r for = 1, 2, . . . L r . Thus, the CDF ofX 1 (given in (18)) can be written as For sufficiently large number of reflecting elements (i.e., N r → ∞), we have L r → ∞ for a fixed D r . Using the relation lim n→∞ (1 − x) n ≈ e −nx with some algebra, we can write the asymptotic CDF ofX 1 as In a similar fashion, we obtain the asymptotic expressions of joint CDFs (defined in (18)) as and Utilizing these asymptotic expressions of marginal and joint CDFs from (34)- (36) in (17), we get the asymptotic outage probability of k-th CEU for N r → ∞ as Further, we define a metric called Log-Asymptotic Outage Order (O N r ), which is the parameter associated with large values of reflecting elements (N r ), as the negative of the ratio of natural logarithm of asymptotic outage probability (for large N r ) and number of reflecting elements, i.e., where P OUT k is given in (37). It is analogus to the diversity order which is obtained at high SNR values. This depicts the rate of variations of outage probability at high values of N r . It explains about how quickly or slowly the outage probability is varying with respect to transmit power (or transmit SNR) under the assumptions of large reflecting surfaces. The insights obtained from the Log-Asymptotic Outage Order are discussed in the numerical results section.

C. DIVERSITY ORDER ANALYSIS
Further, we analyze the outage behavior of a CEU for high transmit power conditions, i.e., ρ B → ∞. Assuming the ideal phase shift scenario at IRS as considered in previous Subsection and following Corollary 2, we can write the joint CDF in (30) as Lemma 2: For high transmit power conditions, i.e., ρ B → ∞, and a given reflection coefficient vector φ H = a , the jont CDF in (39) can be approximated as Proof: As ρ B → ∞, the argument of the modified Bessel's function in (39) tends to zero. Following [43], we utilize two different approximations of K D (x) for Using these asymptotic values in (39), we get (40).
Corollary 3: By applying the L'Hôpital's rule in (40), we can show that for all values of D r with ρ B → ∞, Following Lemma 2 along with the assumptions of ideal reflecting conditions, we can also write the following: and Remark 2: Utilizing (42) and (43) along with the result of Corollary 1 in (17), it can be deduced that for high transmit SNR conditions, the outage probability of the considered IRS-assisted downlink network with 2L-HRS using one-bit control achieves a diversity order of L r , i.e.,

D. OUTAGE PROBABILITY FOR NU
The outage probability for q-th NU, q ∈ K 0 , can be defined as the probability of all the three SINRs at q-th NU (i.e., γ q,K , γ q,K 0 , and γ q ) falling below the corresponding threshold SINRs and can be evaluated as which can further be written (similar to (17)) as Since g q ∼ CN (0, g I), we may write |g H qp | 2 ∼ Exp( 1 g ). Utilizing (6), (8), and (10) with few algebraic manipulations, we can write the CDFs of instantaneous SINRs as where Similarly, the joint CDFs of instantaneous SINRs at NU can be obtained as Utilizing (47) and (48) in (46), we get the outage probability for the NU.

VI. NUMERICAL RESULTS
In this section, we present the analytical and simulated results for the outage performance of IRS-assisted downlink multi-user multi-group MISO communication system with 2L-HRS. Throughout the numerical results, we have assumed θ = 0.43π and = 1.6. The continuous phase ϕ n is uniformly quantized using a 4-bit quantizer to give L = 16 discrete levels. The continuous phase ϕ n is randomly generated using Uniform distribution between −π and π and the n-th reflection amplitude is evaluated by numerically averaging 10 5 iterations of ϕ n . Unless mentioned explicitly, we take N s = 4 and K = 4 (i.e., four user downlink scenario) with equal number of NUs and CEUs (i.e., K 0 = 2). Both the CEUs are considered to be in a single group making M = 1 and K 1 = 2. It is assumed that 25% of the total transmit power (P B ) is assigned to common data precoder p K , (i.e., α K = 0.25). However, the common-to-group data precoders p K 0 and p K 1 for groups G 0 and G 1 , respectively are assigned a power of 0.1P B and 0.2P B , respectively. The remaining 45% of the power is divided among all private precoding vectors, such that K k=1 α k = 0.45. For all the results presented in this section (except Fig. 8), we consider D r = 1 (or, L r = N r ). Fig. 5 shows the outage performance of a CEU in the IRS-assisted downlink four-user scenario with the BS utilizing 2L-HRS. The impact of practical reflection amplitude and phase shift at IRS element is shown on the outage performance of the CEU by fixing h = f = 1. We also assume equal threshold SINRs for all three SINRs, i.e., τ K = τ K 1 = τ 0 = 0.2. It can be observed from Fig. 5 that outage performance deteriorates with larger deviation of practical reflection amplitude from ideal conditions (i.e., β min = 1). Further, it can be noticed that the performance loss (due to practical reflection amplitude) is more for N r = 8 as compared to that for N r = 4.  with the degradation of average channel powers for both the BS-IRS and IRS-CEU channels. Again, the degradation is higher for larger number of reflecting elements at IRS. It is clear from Fig. 5 that the all the analytical curves are matching with the simulated performance. It clearly depicts that the analytical expression of the CEU outage probability obtained through (17) is applicable to arbitrary values of h , f and β min . Fig. 6 shows a comparison of outage performances for IRS-assisted 2L-HRS scheme with IRS-assisted 1L-RS and multi-user linear precoding (MU-LP) schemes. In 1L-RS scheme, the data of each user is divided in two parts, i,e, common and private parts. However, in MU-LP scheme, no common message is generated. Thus, we set α K 0 = α K 1 = 0 for 1L-RS and α K = α K 0 = α K 1 = 0 for MU-LP in Fig. 6. We have considered h = f = 1 and β min = 0.8. It is clear from Fig. 6 that the 2L-HRS with IRS outperforms all the other schemes considered in the figure. Furthermore, we compare the IRS-assisted 2L-HRS scheme with no-IRS situation, where the BS directly communicates to the CEUs by utilizing 2L-HRS. It can be noticed from Fig. 6 that the use of IRS significantly enhances the performance of MISO downlink scenario. In addition, we have shown the outage performance of an NU in a 4-user based MISO system assuming that the average channel power ( g ) of BS-NU channel is 15 dB. It can be observed from Fig. 6 that the use of IRS enables a CEU to outperform the NU significantly.
The impact of channel estimation errors (for both BS-IRS and IRS-CEU channels) on the outage performance of the considered network is shown in Fig. 7. We have considered two different settings for average channel gains of BS-IRS m and IRS m -CEU k as (i) h = f = 0 dB, and (ii) h = −2 dB, f = −1 dB. The error variance of channel estimation error in estimating the channel between the BS and m-th IRS is defined as , σ 2 f ,e = P −0.1 B , i.e., imperfect CSI of both BS-IRS and IRS-CEU channels. It can be observed from Fig. 7 that the performance degradation due to condition (b) is small as compared to that due to condition (c) for both the identical and non-identical average channel power scenarios. For example, to attain an outage probability of 0.01 with h = f = 0 dB, an additional transmit SNR (with respect to the perfect CSI condition (a)) of approximately 1 dB and 3.5 dB is required under conditions (b) and (c), respectively. However, if we consider h = −2 dB, f = −1 dB, then the same outage probability is achieved at a transmit SNR expense of approximately 1.5 dB and 5 dB under conditions (b) and (c), respectively. Further, it should be noted that the transmit SNR requirement further increases as we set the lower outage probability constraints. Fig. 8 illustrates the impact of D r on the outage performance of IRS-assisted 2L-HRS framework with K = 4 and N r = 12. It can be observed that for all the transmit SNR (ρ B ) values, the outage probability improves with decreasing D r and the best outage performance is achieved for D r = 1. In addition, it can be noticed from Fig. 8 that the diversity gain improves significantly as L r increases.
In Fig. 9, we have shown the outage performance of a CEU with respect to the number of reflecting elements N r at IRS for four-user (K = 4) and five-user (K = 5) scenarios with different values of ρ B . We have assumed two NUs in both the scenarios and only one CEU group (i.e., M = 1). It can be observed from Fig. 9 that for all the parameters considered in the figure, the outage performance of a CEU enhances with increasing N r . This is due to the fact that higher value of N r provides larger degrees of freedom for reflecting the data received by the BS. Furthermore, it can be noticed from Fig. 9 that the performance of a CEU deteriorates if we put an additional CEU in the same group. The behavior is intuitive as more CEUs in a group will result in large interference power. It can be deduced from Fig. 9 that outage probability versus reflecting element performance of a CEU depends significantly on two parameters, (i) transmit SNR ρ B and (ii) fraction of the power allocated to the concerned CEU. In Fig. 9, we have assumed that users with indices 1 and 2 are NUs and remaining users are CEUs. We have set α K = 0.25, α K 0 = 0.1, and α 1 = α 2 = 0.075 under both four-user and five-user scenarios. In order to evaluate the performance of a CEU (say user 3) we have considered two conditions, (a) uniform power distribution among CEUs and (b) non-uniform power distribution with higher power to user 3, with fixed α K 1 . It can be seen from Fig. 9 that the performance of user 3 enhances by allocating more power to the private data of user 3 as well as with high transmit SNR. Moreover, the performance gain is high under two-CEU case as compared to three-CEU scenario. Fig. 10 shows the variations of Log-Asymptotic Outage Order for large values of N r with respect to transmit SNR at the BS. For this figure, we consider a dense network with 20 users out of which 16 are close to the cell boundary. These 16 CEUs are equally grouped into 2 groups G 1 and G 2 . It is assumed that all the users within a group have equal power allocation coefficients. We also assume τ K = τ K 1 = 0.1, τ 0 = 0.01, and L r = N r . In Fig. 10, the Log-Asymptotic Outage Order of k-th CEU, k ∈ K 1 is examined for different values of common-to-group and common-to-all power allocation coefficients. It can be noticed from Fig. 10 that for all the parameter settings considered in the figure, O k N r increases with ρ B upto certain limit and then saturates at 0.5, which means that O k N r ≤ 0.5 under all parameter settings. Furthermore, it can be observed from Fig. 10 that increasing the common-to-group power allocation coefficient of G 1 (i.e., the group in which the user under consideration exists) provides improved O k N r values for a fixed commonto-all power. However, if we reduce the common-to-group power by say 25% (for each group G 1 and G 2 ), and increase the common-to-all power by the same amount (i.e., 50%), then the performance degrades significantly. Thus, it can be deduced from Fig. 10 that the impact of common-to-group power is higher on the overall performance as compared to the impact of common-to-all power, specially under the conditions of large reflecting elements.

VII. CONCLUSION
In this paper, we have considered multiple IRS-assisted downlink MISO HRS communication network and deployed a RS transmission strategy at the BS. Next, an On-Off scheme for controlling the IRSs with practical phase shifts has been proposed. In addition, we have derived the closedform expressions of outage probability for both cell-edge users and near users by considering both common as well as private data SINR. Through Monte Carlo simulations, we have shown the effectiveness of the proposed scheme and demonstrated that the proposed design framework outperforms the corresponding 1L-RS, MU-LP and 2L-HRS without IRS. Moreover, the impact of channel estimation errors and increasing the number of reflecting elements N r and number of cell-edge users K m on the outage probability performance has been unvieled.