Intelligent reflecting surface (IRS) is a new and disruptive technology to achieve spectrum- and energy-efficient as well as cost-efficient wireless networks, thus has drawn fast-growing interests from both academia  and industry . IRS consists of a large number of low-cost reflecting elements and each element can passively reflect a phase-shifted version of the incident electromagnetic field . Thus, the reflected signal propagation can be smartly configured by adjusting the phase shifts to achieve certain communication objectives such as received-signal power boosting, interference mitigation and secure transmission . For an IRS-assisted multiuser multiple-input-single-output (MISO) wireless system, the transmit beamforming at the base station (BS) and the passive beamforming at the IRS are jointly optimized to minimize the total transmission power in , while the power allocation at the BS and the phase shifts at the IRS are jointly optimized to maximize the system’s sum rate in  and energy efficiency in . The minimum-secrecy-rate maximization problem was studied in  for an IRS-assisted downlink multiuser MISO system with multiple eavesdroppers. Notice that IRS differs significantly from existing backscatter communication technology. IRS operates in full-duplex mode without introducing self-interference, and provides additional path for the traditional wireless signal without conveying its own information. In contrast, a backscatter device transmits its own information by modulating the incident signal from either dedicated source  or ambient source [10, 11, 12].
On the other hand, non-orthogonal multiple access (NOMA) which can serve multiple users in the same resource block, has been recognized as a promising technology for future wireless communication systems, due to its appealing advantages such as enhanced spectrum efficiency, massive wireless connectivity and low latency 
. Specifically, power-domain NOMA exploits the difference in the channel gain among multiple users for multiplexing and relies on successive-interference-cancellation (SIC) for decoding multiple data flows. For a downlink NOMA system, the outage probability and achievable sum rate were analyzed in, and the optimal transmission power allocation with user fairness assurance was investigated in  . Recently, a backscatter-NOMA system which integrates NOMA and a novel symbiotic radio paradigm (also termed as cooperative ambient backscatter communication  ) was proposed in , and the outage as well as ergotic rate performances were analyzed therein.
It’s well-known that downlink NOMA achieves obvious spectrum-efficiency gain than traditional orthogonal multiple access (OMA) only when the channel strengthes of multiple users are significantly different . The beamforming technology at the BS and/or the users is applicable to enlarge the channel-strength differences to some extent for some cases. However, for the special case when the users are in the same direction of the BS, the bemaforming only at the BS and the users will enhance the channel strengthes of all users simultaneously, resulting into small spectrum-efficiency gain of NOMA compared to OMA. Notice that IRS can digitally manipulate the reflected electromagnetic waves by intelligently adjusting the phase shifts of its passive elements. Hence, we expect to explore the use of IRS to provide additional channel pathes to construct stronger combined channels with significant strength difference in an artificial manner, thus enhancing the NOMA performance gain.
Motivated by the above reasons, in this paper, we consider an IRS-assisted NOMA system in which a BS transmits downlink superposed signals to multiple users. To our best of knowledge, there is no existing work focusing on the performance optimization for such a system. In order to optimize the rate performance and ensure user fairness, we maximize the minimum decoding signal-to-interference-plus-noise-ratio (SINR) (i.e., equivalently the rate) of all users, by jointly optimizing the power allocation at the BS and the phase shifts at the passive IRS, subject to the BS’s power-allocation constraints, the IRS’s phase-shift constraints, and the users’ SINR constraints for NOMA decoding. However, the formulated problem is non-convex and difficult to be solved optimally. To tackle the coupled variables and the non-convex constraints, we propose an efficient iterative algorithm based on the block coordinated decent (BCD), successive convex optimization (SCO) and semidefinite relaxation (SDR) techniques, in which the randomization method is used to obtain an approximate phase-shift solution for the IRS. The convergence is proved and the complexity is analyzed for the proposed algorithm. A low-complexity solving scheme is further proposed. Simulation results show that the IRS can enhance the rate performance of downlink NOMA transmission significantly, even for the scenario in which the users have the same or comparable channel strength(es). Moreover, practical IRS structure with low phase resolution can approximate the best-achievable rate performance achieved by continuous phase shifts, e.g., a 3-bit phase quantizer is sufficient to ensure rate degradation of less than 3.4%. The proposed low-complexity solving scheme is numerically shown to suffer from negligible rate performance degradation.
Ii System Model
As illustrated in Fig. 1, we consider an IRS-assisted downlink NOMA communication system, in which a single-antenna BS transmits superposed signals to single-antenna users in the same time and frequency block with the assistance of an IRS. The IRS consists of passive reflecting elements, and each element can reflect a phase-shifted version of the incident signal. A smart controller connected to the IRS can intelligently adjust the phase shifts to assist the NOMA transmission.
Consider the special case when the users have comparable channel strengthes. For this case, NOMA achieves small spectrum-efficiency gain than traditional OMA. We expect to explore the use of IRS to provide additional channel pathes to construct stronger combined channels with significant strength difference in an artificial manner, thus enhancing the NOMA performance gain.
The channel between the BS and user , is denoted as . All BS-to-User channels ’s are assumed to be mutually independent and Rayleigh fading, i.e., , where denotes the circularly symmetric complex Gaussian (CSCG) distribution with mean
, since the line-of-sight (LoS) path may be blocked. Notice that the IRS is typically pre-deployed such that it can exploit LoS path with the fixed BS. Hence, we use Rician fading to model the channel vectorbetween the BS and the reflecting elements of the IRS, i.e.,
where is the Rician factor of , and are the LoS component and non-LoS (NLoS) component, respectively. The elements of are independent and each element follows the distribution . Since the IRS is typically deployed close to the users to enhance their performance, the channel vector between the IRS and each user is modeled as
where is the Rician factor of , and are the LoS component and the NLoS component, respectively. We assume that all channel state information (CSI) is perfectly known.
The BS transmits a linear superposition of data flows by allocating a fraction of the total transmission power to the -th data flow. That is, the transmitted complex baseband signal is
where is the data flow intended to user .
The signal received at user is then given by
where the IRS’s diagonal phase-shift matrix with denoting the phase shift of the -th reflecting element, for , and denotes the additive white Gaussian noise (AWGN) at user .
The users in downlink NOMA systems employ SIC technique to decode signals, and the decoding order is from the weakest user to the strongest user  . However, the decoding order for IRS-assisted NOMA may be any one of all the different orders, since the combined channel (i.e., ) also depends on , and the phase-shift values . Denote the set of all orders as , where with the index referring to the user with -th weakest (combined) channel. Each user is always able to sequentially decode the signal of the -th user for , and then extract the -th user’s interference from the received signal. The corresponding SINR for user decoding the -th data flow is given by
After cancelling the interference signals from all weaker users with indexes , user decodes the -th data flow by treating the signals from the rest users as interference. The SINR for user decoding its own signal is expressed as
From (6), the corresponding rate for user is thus
Iii Problem Formulation
To maximize the system’s rate performance while ensuring the fairness among users, as in  , we maximize the minimum SINR given in (6) (equivalently the achievable rate given in (7)) of users by jointly optimizing the power allocation (i.e., ) at the BS and the phase shifts (i.e., ) at the IRS. The optimal max-min SINR can be obtained as , where is the optimal max-min SINR for given decoding order . The can be obtained by solving the following problem. For notational simplicity, we omit the decoding order index in the sequel.
Note that (8b) ensures that the rate of each user exceeds , where is a slack variable signifying the minimum SINR to be maximized, (8c) ensures that SIC can be performed correctly, e.g., the SINR for user decoding the -th data flow needs to be no smaller than certain threshold , (8e) and (8d) is the normalization constraint and non-negative constraints of the BS’s power allocation coefficients, (8f) is the phase-shift constraints of the IRS’s reflecting elements.
Iv Optimal Solution
It is challenging to solve problem (P1) due to the non-convex constraints (8b) and (8c). Since there are two blocks of variables (i.e., and ) coupled in (8), we exploit the BCD (i.e., blocked coordinate decent), SCO (i.e., successive convex optimization), and SDR (i.e., semidefinite-relaxation) techniques to solve it approximately. In each iteration , we optimize different blocks of variables alternatively. In the sequel, the and with superscript indicate their values in the -th algorithmic iteration. Therefore, the original problem is decoupled into two sub-problems, which are described as follows.
Iv-a Power Allocation Optimization
In each iteration , for given phase shifts , the power allocation can be optimized by solving the problem
As problem (P1.1) is non-convex due to the non-convex constraint (8b), we adopt the SCO technique to obtain an efficient approximate solution which is guaranteed to converge to at least a locally optimal solution. The basic idea is to successively maximize a lower bound of (9). Notice that the left-hand-side (LHS) (i.e., ) of (8b) is continuously differentiable and jointly convex with respect to , thus it can be globally lower-bounded by its first-order Taylor expansion at any point. We derive a concave lower bound on the LHS of (8b) as follows
By introducing a slack variable , sub-problem (P1.1) in (9) is approximated as
Problem (P1.2) is convex and thus can be solved by existing tools like CVX .
Iv-B Phase Shift Optimization
In each iteration , for given power allocation coefficients , the phase shifts can be optimized by solving the following problem
Recall . We denote , and . Let . Then the term can be rewritten as . We further reduce this term to , where
Note that . We define the matrix , which needs to satisfy and . Since the rank-one constraint is non-convex, we exploit the SDR technique to relax problem (P1.3) as follows
However, problem (P1.4) is still non-convex due to the non-convex constraint of (13b). To tackle the coupled variables and , we use the bisection method to solve problem (P1.4). Specifically, with certain and , we replace the in problem (P1.4) by , and solve the resulting feasibility problem reduced from (P1.4). The update of and depends on whether a feasible can be found. It can be checked that for sufficiently large and small , the above bisection search over can give a globally optimal phase-shift-related matrix in the -th alterative iteration.
However, the obtained by solving problem (P1.4) generally doesn’t satisfy the rank-one constraint, and thus the optimal objective value of problem (P1.4) only serves an upper bound of problem (P1.3). To obtain a rank-one solution, we apply a Gaussian randomization scheme which is described as followed. Firstly, from
, we obtain the singular-value-decomposition (SVD) ofas . Define . A random vector is generated as follows
where the random vector . Then we generate as , where denotes the vector contains the first elements in . The objective value of problem (P1.3) is approximated as the maximal one achieved by the best among all ’s. It has been verified that SDR technique followed by such randomization scheme can guarantee at least an -approximation of the optimal objective value of problem (P1.3) .
Iv-C Overall Algorithm
The overall algorithm is summarized in Algorithm 1. As shown, the algorithm optimizes and alternatively in the out-layer iteration, and the bisection search together with the randomization method are adopted to obtain an approximate phase-shift solution. The algorithm terminates when the the increase of the objective value is smaller than a smaller .
We prove the convergence of Algorithm 1 as follows.
Algorithm 1 is guaranteed to converge.
First, in step 3 of Algorithm 1, since the optimal solution is obtained for given , we have the following inequality on the minimum rate
where (a) and (c) hold since the Taylor expansion in (10) is tight at given local point and , respectively, and () comes from the fact that is the optimal solution to problem (P1.2).
Second, in steps 4-24 of Algorithm 1, since is the optimal solution to problem (P1.3), the following inequality holds
The inequality in (17) indicates that the objective value of problem (P1) is always non-decreasing after each iteration, although an approximated optimization problem (P1.2) is solved to obtain the optimal power allocation in each iteration. On the other hand, since the objective is continuous over the compact feasible set of problem (P1), it is upper-bounded by some finite positive number . Hence, the proposed Algorithm 1 is guaranteed to converge, which completes the proof. ∎
Notice that no global optimality can be assured for Algorithm 1. The reasons are two fold. First, the problem (P1) is not jointly convex with respect to , and . Second, only an approximated problem (P1.2) is solved for sub-problem (P1.1), and the adopted method of SDR followed by Gaussian randomization for solving sub-problem (P1.3) does not guarantee the global optimality of solution.
The complexity of Algorithm 1 is affordable, since it needs to solve only one approximated convex problem (P1.2) to update in each iteration, and solve a convex SDR problem followed by Gaussian randomization to update in each iteration.
Iv-D A Low-Complexity Solving Scheme
Notice that the solving scheme described at the beginning of Section III is of high complexity, since it needs to solve optimization problems (P1)’s, by exhaustively searching over different decoding orders. We propose a low-complexity solving scheme. That is, we choose the decoding order according to all users’ combined channel strengthes each of which is obtained by optimizing the IRS phase shifts. With the chosen decoding order, the optimization problem (P1) needs to be solved once. Hence, the above solving scheme needs to solve optimization problems, being less than the original scheme in Section III. Fortunately, numerical results show that the above low-complexity scheme suffers from slight rate performance degradation.
Specifically, the maximally achievable strength of the combined channel for each user can be obtained by solving the following problem
Similar to Section IV-B, by introducing the matrix , the term can be rewritten as , which needs to satisfy and . Since the rank-one constraint is non-convex, we exploit the SDR technique to relax problem (P2) as follows
The optimal obtained by solving (P2.1) generally doesn’t satisfy the rank-one constraint, and a Gaussian randomization scheme can be applied again to obtain a rank-one solution, which is similar to that described in Section IV-B and thus omitted herein.
V Numerical Results
Numerical results are provided in this section. As in , we assume that the BS-to-User channels are Rayleigh fading and the large-scale pathloss is , where is the distance with unit of meter (m). Both the BS-to-IRS channel and the IRS-to-User channels are assumed to be Rician fading, and their pathloss are and respectively. We consider the case of two users, i.e., . The BS-to-IRS distance is set as 50 m, both BS-to-User distances are set as 60 m, and both IRS-to-User distances are set as 15 m. We set the Rician factors . As in  and , we set dB, dBm. Let and . For communication performance comparison, we consider two benchmarks, i.e., traditional NOMA (without IRS but with optimized power allocation at the BS) and the traditional OMA (without IRS but with optimized time allocation for both users), and plot the rate (computed from (7)) based on 10000 random channel realizations..
Fig. 2 compares the per-user rate performance for the proposed IRS-assisted NOMA and the two benchmarks, for the number of IRS reflecting elements , respectively. The IRS-assisted NOMA achieves significant rate gain compared to the traditional NOMA, which in general comes from the enhanced combined-channel strength and larger channel-strength differences introduced by the IRS. We further observe that larger rate-gain is achieved for larger number of reflecting elements . For instance, given dBm, a rate increase of 30.2%, 55.5% and 77.7% for , respectively. Moreover, the user 1 and user 2 achieves almost the same throughput, achieving good fairness. As shown, the traditional NOMA achieves almost the same rate performance as the OMA. This is because that the BS has the same (average) channel strength with user 1 and user 2. The practical significance of this proposed IRS-assisted NOMA lies in that it enables the NOMA system to achieve large rate gain than traditional NOMA system, for the scenario of comparable or even the same channel strength for multiple users.
In practical systems, the IRS structure has finite phase resolution and the implemented phase shifts depend on the number of quantization bits denoted as . We numerically verify the effect of IRS’s finite phase resolution on the rate performance. Each optimized continuous phase shift is quantized to its nearest discrete value in the set . Fig. 3 plots the max-min rate of both users versus the transmit power for different phase-quantization bits ’s and fixed . We observe that the IRS’s finite phase resolution in general degrades the max-min rate compared to IRS with infinite phase resolution, but the rate performance degradation fast becomes negligible as increases. For instance, given dBm, the max-min rate degrades by 22.2%, 10.3%, 3.4%, 3.2% and 2.7%, for and 5, respectively. Even for the case of 1-bit phase quantizer, the proposed IRS-assisted NOMA improves the max-min rate by and compared to the benchmarks of traditional OMA and NOMA, respectively.
Fig. LABEL:fig:FigSim3 compares the rate performance of the original solving scheme and the low-complexity scheme. We fix and consider continuous phase shifts. It is observed that the low-complexity scheme with a chosen suboptimal decoding order suffers from slight rate performance degradation less than 6% for the transmit power =0 dBm, compared to the original scheme which exhaustively searches all possible decoding orders. Moreover, the rate performance degradation becomes negligible as increases.
This paper has investigated the problem of rate optimization for an IRS-assisted downlink NOMA system. The minimum SINR (i.e., equivalently the rate) of all users are maximized by jointly optimizing the BS’s power allocation and the IRS’s phase shifts. An efficient algorithm is proposed to find a suboptimal solution to the formulated non-convex problem, by leveraging the block coordinated decent, successive convex optimization and semidefinite relaxation techniques. Numerical results show that the IRS can enhance the performance of downlink NOMA significantly even for the scenario when the users have the same or comparable channel strength(es), and practical IRS with low phase resolution can approximate the best-achievable rate performance achieved by continuous phase shifts. The IRS-assisted NOMA with transmit beamforming at the BS will be further studied. Other interesting future work for IRS-assisted NOMA includes the outage performance analysis, rate performance under imperfect CSI, etc.
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