I Introduction
The explosion of mobile data and the everincreasing demand for higher data rates have continuously driven the advancement of wireless communication technologies in the past decade, such as polar code, massive multipleinput multipleoutput (MIMO) and millimeter wave (mmWave) communications, among others. Moreover, a 1000fold increment in network capacity with ubiquitous connectivity and low latency is envisioned for the forthcoming fifthgeneration (5G) wireless network [1]. Meanwhile, the energy efficiency of future wireless networks is also aimed to be improved by several orders of magnitude so as to maintain the power consumption at increasingly higher data rates.
Recently, intelligent reflecting surface (IRS) has been proposed as a promising solution to achieve the above goals in a costeffective way [2, 3, 4, 5, 6]. Specifically, IRS is a reconfigurable planar array comprising a vast number of passive reflecting elements, which are able to independently induce a phase shift to the incident signal and thus collaboratively alter the reflected signal propagation to achieve desired channel responses in wireless communications. By properly adjusting the phase shifts of IRS’s elements, their reflected signals can combine with those from other paths coherently at the receiver to maximize the link achievable rate. Moreover, an amplitude reflection coefficient between zero and one can be designed for the incident signal at each element, so as to further enhance the performance. Hence, different from the conventional halfduplex amplifyandforward (AF) relay, IRS achieves high beamforming gains by intelligent reflection in a fullduplex manner, thus without consuming any energy or requiring additional time/frequency resource for signal regeneration and retransmission. It is worth noting that passive reflectarray antennas have been applied in radar and satellite communication systems before. However, their uses in mobile wireless communications are rather limited, since traditional passive arrays only allow for fixed phaseshift patterns once fabricated, and are thus unable to adapt to the dynamic wireless channel due to user mobility. Fortunately, the advances in radio frequency (RF) micro electromechanical systems (MEMS) and metamaterial (e.g., metasurface) have made it feasible to reconfigure the phase shifts in real time [7], thus greatly enhancing the functionality and applicability of IRS for wireless communications.
Prior works on IRSaided wireless systems [2, 3, 4, 5] have considered frequencyflat (nonselective) fading channels for narrowband communication. However, the design of IRS passive array coefficients for the more general frequencyselective fading channels for broadband communication still remains open, to the authors’ best knowledge. Motivated by this, we consider in this paper an IRSaided orthogonal frequency division multiplexing (OFDM)based wireless system over frequencyselective channels, as shown in Fig. 1. For the purpose of exposition, we consider the case of downlink communication from a base station (BS) to one single user in the vicinity of an IRS, where the IRS and the user are both far away from the BS (e.g., in a celledge user scenario). We will extend our results to the uplink communication as well as the general case with multiple users served by the BS/IRS in the journal version of this work. As IRS has a large number of reflecting elements, how to jointly design their reflection coefficients (i.e., phase shifts and amplitude attenuations) so as to achieve the optimal constructive superposition of the reflected signals by the IRS and those from other paths at the user receiver is crucial to maximizing the link achievable rate. However, this is a nontrivial problem to solve under our considered setup due to two main reasons. Firstly, under the frequencyselective channel with multiple paths, the reflection coefficients of the IRS need to cater to the channel gains and delays of all paths to the user, including both the reflected paths by the IRS and the remaining nonreflected paths directly from the BS. Secondly, the achievable rate for the user is determined by both the IRS reflection coefficients and the transmit power allocation at the BS over OFDM subcarriers (SCs), which are intricately coupled and thus need to be jointly optimized. To tackle the above challenges, we formulate a new optimization problem aiming to maximize the downlink achievable rate by jointly optimizing the transmit power allocation at the BS and the passive array coefficients at the IRS, which is nonconvex and thus difficult to solve. This motivates us to propose an iterative algorithm to alternately optimize the power allocation and passive array coefficients with a customized initialization scheme, which is guaranteed to converge to at least a locally optimal solution. It is shown by numerical results that the proposed algorithm achieves significantly improved rate performance compared to systems without using the IRS or with heuristic IRS reflection coefficient designs. In addition, it is shown that the proposed initialization scheme is able to achieve a good performance very close to that by the iterative algorithm, and is thus suitable for lowcomplexity implementation.
Ii System Model
We consider a singleuser downlink OFDMbased wireless system, wherein an IRS is employed to enhance the communication between a BS and a user, as illustrated in Fig. 1. For the purpose of exposition, we assume that the BS and the user are both equipped with a single antenna. The IRS is assumed to comprise passive reflecting units, denoted by the set , and is connected to a controller, which adjusts the IRS pattern for desired signal reflection. A separate wireless control link serves for information exchange between the IRS controller and the BS on the channel state information (CSI) and other information needed for implementing the joint design of the BS transmission and IRS reflection. It is further assumed that the signals that are reflected by the IRS more than once have negligible power due to severe path loss and are thus ignored. We consider a quasistatic block fading channel model for all channels involved and focus on one particular fading block where the channels remain approximately constant. To obtain the optimal design and the corresponding performance upper bound, we assume that perfect CSI of all channels is available at the BS via channel training and/or feedback from the IRS/user, and thus it can compute the optimal solution and inform it to the IRS for implementation.
Similar to conventional OFDMbased systems, the total bandwidth of the system is equally divided into orthogonal SCs, denoted by the set . Moreover, let , where each denotes the transmit power allocated to SC at the BS. Assume the total transmission power at the BS is . Thus, the power allocation at the BS should satisfy .
Let
denote the zeropadded
tap baseband equivalent multipath channel of the BSuser direct link, wheredenotes an allzero matrix of size
. Moreover, there exists an tap baseband equivalent multipath channel of the BSIRSuser link, through which the signal transmitted by the BS is reflected by the IRS before arriving at the receiver of the user. Let denote the zeropadded tap baseband equivalent BSIRS channel, where each corresponds to the BSIRS channel at the th tap, . Similarly, let denote the baseband equivalent channel of IRSuser link, where each corresponds to the IRSuser channel at the th tap, . At the IRS, each element rescatters the received signal with an independent reflection coefficient. Specifically, let denote the IRS reflection coefficients, where each comprises an amplitude coefficient and a phase shift , i.e., . Let denote the reflection coefficient matrix of the IRS, where is a square diagonal matrix with the elements of on the main diagonal. The composite BSIRSuser channel, denoted by , is thus the concatenation of the BSIRS channel, IRS reflection, and IRSuser channel, which is given by(1) 
where each corresponds to the effective BSIRSuser channel at the th tap, . Hence, the superposed channel impulse response (CIR) from the BS to the user by combining the BSuser (direct) channel and the BSIRSuser (IRSreflected) channel is given by
(2) 
Assume OFDM modulation at the BS with a cyclic prefix (CP) of length , which is no smaller than . Then the channel frequency response (CFR) of the CIR is given by
(3) 
where
denotes the discrete Fourier transform (DFT) matrix. Define
, where . We then have and . Hence, the overall CFR can be rewritten as , and the CFR at each th SC is given by(4) 
where denotes the th row of the DFT matrix . The achievable rate in bits per second per Hertz (bps/Hz) is thus obtained as (by accounting for the CP overhead)
(5) 
where
is the gap from channel capacity owing to a practical modulation and coding scheme (MCS); the receiver noise is assumed to be independent over all SCs, and is modelled as a circularly symmetric complex Gaussian (CSCG) random variable with mean zero and variance
.Iii Problem Formulation
In this paper, we aim to maximize the achievable rate by jointly optimizing the BS transmit power allocation and the IRS reflection coefficients. Therefore, we formulate the following optimization problem
(6)  
(7)  
(8) 
Note that Problem (P1) is a nonconvex optimization problem. Particularly, it can be shown that the objective function of (P1) is nonconcave over ; moreover, the variables and are coupled in the objective function, which makes their joint optimization difficult. To overcome the above challenges, in the following section, we propose an alternating optimization algorithm to find an approximate solution to (P1), by iteratively optimizing one of and with the other fixed at each time. In addition, we devise a customized method to obtain an initial solution of , denoted by , as the starting point of the proposed alternating optimization algorithm.
Iv Proposed Solution
Iva Power Allocation Optimization Given IRS Coefficients
Note that given a set of IRS coefficients , the CFR is fixed. The optimal BS transmit power allocation is thus given by the wellknown waterfilling (WF) solution [8], i.e.,
(9) 
where , is the effective channeltonoise power ratio (CNR) for SC , and is the cutoff CNR that satisfies
(10) 
IvB IRS Coefficient Optimization Given Power Allocation
With given power allocation, Problem (P1) is simplified as
(11) 
It can be shown that (P1.1) is not a convex optimization problem. In the following, we adopt the successive convex approximation (SCA) technique to obtain a locally optimal solution to (P1.1). First, by introducing a set of auxiliary variables ’s, ’s, and ’s, we transform (P1.1) into the following equivalent problem
(12)  
(13)  
(14)  
(15) 
where and denote the real and imaginary part of a complex number, respectively. Define , which is a convex and differentiable function over and . Thus, given any and , the firstorder approximation of at the point serves as a lower bound to it, i.e.,
(16) 
where equality holds if and only if and . Note that is an affine function over and , which also has the same gradient over and as at the point .
Next, we consider the following optimization problem
(17) 
Problem (P1.2) is a convex optimization problem, which can be solved efficiently via existing software in polynomial time with respect to and , e.g., CVX [9]. Therefore, an approximate solution to (P1.1’) and thus (P1.1) can be obtained by successively updating and based on the optimal solution to (P1.2), which is summarized in Algorithm 1. It can be shown that monotonic convergence of Algorithm 1 is guaranteed, and the obtained solution is a locally optimal solution to (P1.1) [10].
To summarize, the overall iterative algorithm to solve (P1) is given in Algorithm 2. It is worth noting that starting from an initial point denoted by , the initial value for Algorithm 1 in each iteration of Algorithm 2 is set as the obtained in the previous iteration. It can be shown that the objective value of (P1) is nondecreasing over each iteration of Algorithm 2, which is also upperbounded by a finite value. Therefore, Algorithm 2 is guaranteed to converge. Moreover, the obtained solution to (P1) can be shown to be at least a locally optimal solution based on [11]. Note that the performance of Algorithm 2 is critically dependent on the choice of the initial IRS reflection coefficients . In the following subsection, we propose a customized method for finding efficiently.
IvC Initialization Method
Note that the IRS is able to increase the link rate mainly due to the increased effective channel power between the BS and the user, by creating an additional strong CIR via the BSIRSuser channel that can constructively combine with that of the BSuser direct channel. Motivated by this, we propose to design the initial value of , i.e., , by maximizing the effective channel power from the BS to the user, which is given by , with denoting the norm. Therefore, we formulate the following optimization problem
(18) 
Note that Problem (P2) is a nonconvex quadratically constrained quadratic problem (QCQP), for which we apply the semidefinite relaxation (SDR) [12] technique to obtain an approximate solution for it, as follows. Define and , Problem (P2) is thus equivalent to
(19)  
(20) 
Note that ; similarly, and hold, where denotes the matrix trace. By defining and , we transform (P2) into the following problem
(21)  
(22)  
(23) 
where ,
denotes the identity matrix of size
, and the constraint in (23) ensures is positive semidefinite. Note that (P2) can be shown to be equivalent to (P2SDR) with the additional constraint of .Problem (P2SDR) is a convex semidefinite program (SDP), which can be solved efficiently via existing software, e.g., CVX [9], with polynomial complexity in [12]. Let denote the optimal solution to (P2SDR). If , the relaxation from (P2) to (P2SDR) is tight and the optimal to Problem (P2) can be obtained as , where
is the eigenvalue decomposition (EVD) of the upper left
submatrix of , denoted by , anddenotes the column vector formed by the main diagonals of
. On the other hand, if , the optimal objective value of Problem (P2SDR) serves as an upper bound to that of Problem (P2) and additional processing is required to construct a rankone solution according to . In particular, we consider a customized Gaussian randomization method to find an approximate solution to Problem (P2). To enhance the performance of the proposed algorithm, a number (denoted by ) of ’s are generated based on , from which the one that yields the largest objective value of Problem (P2) is selected. The overall algorithm for solving (P2) is summarized in Algorithm 3, where the output of Algorithm 3 is then set as the initial for Algorithm 2.V Numerical Results
In this section, we examine the performance of our proposed algorithm via numerical results. We consider and for both the BSuser link (i.e., the direct link) and the BSIRSuser link (i.e., the reflected link), among which taps at random delays are nonzero for each link and are modeled as CSCG random variables with an exponential power decay profile. The CP length is set as . The total average channel power of the reflected link over all taps is given by , and that of the direct link is given by . The average signaltonoise ratio (SNR) is thus given by , while we normalize the total average channel power of the two links as for convenience, unless stated otherwise. Let denote the average power ratio of the reflected link to the direct link. Hence, indicates that the user is located far away from the IRS, thus its channel with the BS is dominated by the BSuser direct link; while on the other hand, indicates that the user is located in close vicinity of the IRS. The SNR gap is set as dB, and the number of randomizations in Algorithm 3 is chosen as . All the results are averaged over 100 independent channel realizations.
For comparison, we consider the following benchmark schemes:

Channel Power Maximization (CPM): In this scheme, we adopt the IRS coefficients as obtained via the initialization method based on CPM proposed in Section IVC, and the WFbased power allocation based on .

Random Phase: We assume the IRS coefficients have random phase and maximum amplitude. As the channel coefficients are randomly generated, this is equivalent to setting , based on which we obtain the WFbased power allocation. Note that the IRS behaves like a lossless reflective mirror in this case.

Without IRS: We consider the WF power allocation and achievable rate based on the BSuser direct link only.
First, we evaluate the convergence behavior of Algorithm 2. The number of reflecting elements is set as , and the power ratio of the reflected link to the direct link is set as . For comparison with the CPMbased initialization method proposed in Section IVC, we consider a benchmark initialization method with random phase (or ). Fig. 2 shows the achievable rate over iterations at SNR dB for a random channel realization. Monotonic convergence is observed for both initialization methods, which is consistent with our discussions in Section IV. Moreover, it is observed that the proposed CPMbased method converges much faster compared to the random phase method (i.e., versus iterations), while both methods achieve the same converged achievable rate (i.e., bps/Hz). This thus validates the efficiency of the proposed CPMbased initialization method.
Fig. 3 shows the performance of the iterative algorithm and the benchmark schemes at different SNR values, with and . It is observed that all the three schemes with IRS outperform the scheme without IRS, due to the IRSenhanced average channel power from the BS to the user. Moreover, the proposed iterative algorithm and CPMbased initialization scheme both achieve significantly improved achievable rates over the random phase scheme, since the direct channel and the reflected channel are superposed more constructively via designing the IRS reflection coefficients. Furthermore, it is observed that the performance of the proposed CPMbased initialization scheme is very close to that of the iterative algorithm, and the performance gap decreases as the SNR increases. Therefore, this scheme is suitable for practical implementation with lower complexity.
Fig. 4 compares the performance of the iterative algorithm and the benchmark schemes versus , where we set the reference SNR with as dB and the reference power ratio with as . It is observed that the achievable rates for both the proposed iterative algorithm and initialization scheme increase with , owing to the passive beamforming gain harvested by properly designing the IRS reflection coefficients according to the CSI; while on the other hand, the achievable rate for the random phase scheme is nearly invariant with . Moreover, it is observed that the performance gain of the proposed schemes over the scheme without IRS becomes more pronounced as increases.
Fig. 5 shows the performance of the iterative algorithm and the benchmark schemes versus the reflected to direct link power ratio with . For ease of illustration, we set the reference SNR with only the direct link as dB, thus the overall SNR is . Note that as increases, this corresponds to a practical scenario where the user maintains a fixed distance with the BS (e.g., on a circle centered at the BS), and gradually moves towards the IRS. It is observed that when is very small, all the schemes yield similar performance since the user is far away from the IRS, whose effect is thus negligible. On the other hand, as increases, the performance gain of the proposed iterative algorithm and CPMbased initialization method over the benchmark schemes increase drastically, due to the increased dominance of the IRS reflected link over the direct link. Moreover, it is observed that all the schemes with IRS achieve significantly improved performance compared to that without IRS when is large. This indicates that even though the user is located far away from the BS in the celledge scenario, a nearby IRS is effective in enhancing the link rate.
Vi Conclusion
In this paper, we proposed a novel approach to enhance the achievable rate of an OFDM system by utilizing the IRS, and jointly designed the IRS reflection coefficients and BS transmit power allocation to maximize the link achievable rate. By leveraging optimization techniques, we proposed computationally efficient methods to find highquality suboptimal solutions for the formulated problem. Numerical results showed the effectiveness of IRS in boosting the achievable rate of a celledge user aided by the IRS. The proposed initialization method was also shown to achieve very close rate performance compared to the proposed iterative method, but with much lower complexity for implementation.
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