Reduced-Complexity Intelligent Reflecting Surface Optimization for Single-Carrier Transmission in Frequency-Selective Fading Channel

In this letter, we propose a novel reduced-complexity intelligent reflecting surface (IRS) optimization for cyclic-prefixed single-carrier transmission, which optimizes IRS reflection coefficients by increasing the achievable information rate. More specifically, our algorithm is designed based on coherently combining a single dominant channel tap for IRS beamforming. We demonstrate that significant improvement of the achievable information rate of the system is accomplished by employing the IRS with appropriate optimization schemes. Furthermore, our proposed algorithm achieves nearly identical performance to solving the full optimization problem in line-of-sight (LoS) and non-LoS scenarios while maintaining a practical RIS optimization and detection complexity.


I. INTRODUCTION
T HE DEMAND for high-rate and reliable wireless com- munication systems has grown stronger over the years, and the next-generation wireless standard is expected further to use millimeter-wave (mmWave) from its predecessor while opening up the Terahertz (THz) band.However, it is a challenging task to exploit the full potential of such high frequency bands in wireless communications.Typically, the mmWave and THz signals are highly susceptible to severely high propagation loss due to the short wavelength.Also, it is crucial to ensure a line-of-sight (LoS) connection between the transmitter and the receiver.To combat this limitation, intelligent reflecting surface (IRS) has been proposed as a technology to approach the above-mentioned problems [1], [2], [3], [4].Most IRSs consist of a planar array of reflecting elements, each having the capability of controlling phase rotation and amplitude variation from the incoming incident wave to the reflected one, hence allowing passive beamforming.Since each IRS element simply reflects impinging waves, it is more power-efficient than its active counterparts, which inevitably consume power by transmitting signals.
In high-rate broadband communication systems, the signal typically experiences frequency-selective fading [5].Most existing studies in a broadband IRS scenario are constituted The authors are with the Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan (e-mail: sugiura@iis.u-tokyo.ac.jp).
Digital Object Identifier 10.1109/LWC.2024.3401154 under the assumption of employing orthogonal frequencydivision multiplexing (OFDM) [6], [7], [8], [9].In [6], the grouping method of IRS elements was proposed, which reduces the channel estimation overhead, while presenting a sub-optimal iterative solution for IRS optimization.In [7], Zheng and Zhang proposed the low-complexity IRS optimization for OFDM, which is referred to as the strongest channel impulse response (CIR) maximization (SCM) method, where the IRS reflection coefficients are optimized so as to maximize the received signal corresponding to the strongest tap.In [8], Xu et al. presented the analytical model of the time-varying frequency-selective Rician fading channel was constructed to investigate the performance of IRS-enhanced OFDM under the LoS-dominated high-mobility unmanned aerial vehicle scenario.In [9], deep learning is utilized for constructing a direct mapping from the sampled channels and the optimal IRS reflection coefficients in the IRS-aided OFDM system with a small number of active IRS elements.In contrast to its benefits, OFDM has a well-known drawback associated with a high peak-to-average power ratio, which imposes the power amplifier on a large linear dynamic range [10], [11].Hence, single-carrier transmission is typically preferable in broadband uplink transmission.However, to the best of our knowledge, there has not been well-investigated broadband IRS-assisted single-carrier transmission due to the presence of the challenging issue related to inter-symbol interference, unlike OFDM.Only the exceptions are constituted by [12], [13].More specifically, in [12], the cyclic-prefixed single-carrier (CPSC) transmission was investigated for IRS-aided uplink, where the IRS is used to change the reflected signal to configure a cyclically delayed version of the transmitted symbol block in order to achieve the cyclic delay diversity (CDD), rather than the beamforming gain.Most recently, in [13], the progressive channel estimation method was proposed to alternately transmit pilots and data in the context of the IRS-assisted CDD scheme [12].Note that the beneficial IRS-assisted beamforming gain necessary for mmWave communication cannot be attained by CDD.
Apart from the full IRS optimization based on estimated channel coefficients, training-based IRS approaches [14], [15] attain the passive beamforming gain with low optimization complexity and pilot overhead.In [14], Fathnan et al. proposed the IRS whose reflection pattern is controlled by the pulse width of a transmitted signal, while in [15], the combination of pattern training and differential modulation allows us to dispense with channel estimation in IRS optimization and data detection.Note that these benefits are attained at the sacrifice of reduced beamforming gain due to the use of pre-determined reflection patterns.Against this background, the novel contributions of this letter are as follows.We propose a low-complexity IRS optimization scheme for CPSC transmission to attain the beneficial beamforming gain.This allows us to increase the maximum information rate in the mmWave communication scenario rather than attain the cyclic diversity gain of the previous schemes [12], [13].We formulate an optimization problem to maximize the channel capacity of IRS-empowered CPSC transmission, and the formulated non-convex problem is sub-optimally solved by the successive convex approximation (SCA) technique [16].Additionally, to reduce the computational complexity imposed by the optimization, we propose a reduced-complexity algorithm by focusing on a single dominant out of delay-spread taps inspired by the SCM [7].It is demonstrated that the proposed algorithm achieves performance nearly identical to that of solving the high-complexity optimization problem by SCA in terms of the achievable information rate while maintaining significantly low computational complexity.Also, we demonstrate that the proposed IRS-aided CPSC scheme is capable of achieving a beneficial frequency diversity gain, unlike the conventional OFDM counterpart, especially in an NLoS scenario. 1  II.SYSTEM MODEL Fig. 1 shows the system model of our uplink communication scenario, where a base station (BS), user equipment (UE), and an IRS are placed on the three-dimensional rectangular coordinate system.Each of the BS and the UE is equipped with a single antenna, and the IRS is composed of M reflection elements.We assume that a link between the BS and the UE is blocked, and hence, the assistance of the IRS is necessary.

A. Channel Model
In this letter, we consider the frequency-selective fading channel.More specifically, the link between the UE and the IRS (UE-IRS) is represented by M baseband equivalent multipath channel coefficients, i.e., g m ∈ C L 1 (m = 1, . . ., M ), where L 1 denotes the number of delay taps.Similarly, the link between the IRS and the BS (IRS-BS) is given by h m ∈ C L 2 with L 2 the number of the associated delay taps.Each first tap of g m and h m corresponds to the channel coefficient associated with the LoS path, while other taps represent the channel coefficients for non-LoS (NLoS) components.The equivalent baseband channel between the UE and the BS via the m-th IRS element (UE-IRS-BS) is defined as g m * h m . Let where and φ m denotes the phase rotation of the m − th IRS element.

B. Transmitter Model
Each transmission frame is divided into two subframes.The first subframe corresponds to the data frame, which consists of N information symbols x = [x 1 , x 2 , . . ., x N ] T ∈ C N , each having the transmit power of σ 2 s .The second subframe is the cyclic prefix, which corresponds to the last N CP symbols of the data frame, which is used for preventing inter-block interference under the assumption that N CP is sufficiently higher than L 0 .

C. Receiver Model
At the receiver, after discarding the cyclic prefix N CP from the received signal, the received frame is expressed as follows: where n ∈ C N is the associated additive white Gaussian noise (AWGN) components, which follow the complex-valued Gaussian distribution of N c (0, σ 2 n I N ).Here, σ 2 n represents the noise variance.Furthermore, Ξ ∈ C N ×N is the circulant matrix, whose first column is given by the zero-padded overall CIR ξ .For simplicity, we assume that the perfect CIR is acquired at the receiver.
The received frame y is demodulated with the aid of the frequency-domain minimum mean-square error (MMSE) equalizer.The estimated symbols are given by multiplying the weights T ∈ C N ×N by the received frame as follows: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where while Ω is the diagonal matrix constructed by the eigenvalues of Ξ.Note that Ω is calculated by exploiting the fast Fourier transform with the complexity order as low as O(N log N ).
To be specific, the circulant matrices can be decomposed by the unitary discrete Fourier transform (DFT) matrix D N as follows:

D. Theoretical Information Rate Bound
In this section, we introduce the theoretical information rate bound.From (3), (4), and (5), the demodulated symbol is rewritten by Hence, the mean-square error (MSE) of the estimated frame is formulated by where Note that the term |f H n Vφ| 2 in (8) represents the channel frequency response (CFR) of the n-th subcarrier.Moreover, in (8), since the coefficient matrices multiplied by n and x in the third and second terms, respectively, are circulant, the instantaneous signal-to-interference plus noise ratio (SINR) is identical for each of N symbols [17].Therefore, the MSE of each estimated symbol is given by E/N .Furthermore, it is known that symbols xn estimated by the MMSE weights of (5) are biased, i.e., the original symbols x n , multiplied by a constant q, which is expressed by [18] Removal of this bias by multiplying the equalizer output by the inverse of the constant provides the unbiased output, which allows us to achieve a lower error probability compared to the biased counterpart [19].More specifically, the unbiased SINR ρ un is given by [18] where E un is the unbiased MSE, which is represented by [18]: Finally, the information rate of the frequency-domain-MMSE-assisted CPSC scheme is given by

III. PROPOSED IRS OPTIMIZATION
To maximize the information rate ( 14), we formulate the following optimization problem: noting that constant terms are omitted for brevity.Similar to [6], the non-convex problem (15a) is solvable with the SCA to acquire a stationary point.However, the associated complexity is as high as O(M 4.5 N 3.5 ) [6], which is unrealistic since the optimization has to be updated within the channel's coherence time.
To reduce the above-mentioned complexity, we propose an algorithm that maximizes the received signal power associated with the strongest tap, inspired by the SCM scheme proposed for the OFDM arrangement [7].The problem of channel gain maximization is formulated by Let us introduce a matrix of to express the CFR of the cascaded UE-IRS-BS channel.Then, (16a) is rewritten by where ω n,m is the n-th entry of ω m .
Based on Parseval's theorem, the maximization of the total channel power in the time domain is equivalent to that of the frequency domain.Hence, instead of (19), we can consider the following problem: Note that the number of CIR taps in the time domain is sufficiently lower than the data frame length, i.e., L 0 N .This implies that the channel gains in the time domain are more localized than in the frequency domain.
To efficiently increase the beneficial channel gain, the IRS phase shifts are optimized to coherently combine the channel gains associated with a single specific tap.More specifically, the tap index with the strongest CIR gain, noted as l , is decided as follows: Then, to coherently combine the channels associated with the l -th CIR tap, the IRS phase shift of each element is set to Note that the complexity of our algorithm, locating l in ( 21) is as low as O(L 0 M ), which is significantly lower than the complexity of the SCA-based algorithm in (15a).

IV. PERFORMANCE RESULTS
We carried out Monte Carlo simulations to evaluate the achievable performance of the proposed reflection optimization algorithm.The UE, IRS, and BS are located at (x, y, z) = (5,150,1), (0,150,2), (150,0,2), respectively, as shown in Fig. 1.We consider the single-slope model P r /P t = K (d /d r ) γ to calculate the path loss P r /P t at a distance d, where P t and P r denote the transmit power of the UE and the received power at the BS, respectively.Here, we set a reference distance d r = 1 m and a path loss at the reference distance K = −30 dB.Also, γ is the path loss exponent.
For each channel, we assume the frequency-selective Rician fading and the total power of CIR taps is fixed to the value calculated by the above-mentioned path-loss model.A uniform power delay profile is considered for the NLoS taps, where each tap coefficient is generated as zero-mean complex-valued Gaussian random variables.The numbers of delay taps for the UE-IRS link and the IRS-BS link are set as L 1 = 13 and L 2 = 4, and the associated path-loss exponents are given by γ = 2.2 and 2.8, respectively.The Rician K factor for the IRS-BS link is maintained to be 3 dB, while that for the UE-IRS link was given by 0 dB and −20 dB.
We consider the uniform planar array (UPA) [20] for the IRS with M = 100 elements, which is located in the yzplane with each row consisting of 10 elements.The spacing between each element is set as the half wavelength.Since the distance between the IRS and the BS/UE is longer than the size of the array, the LoS paths are assumed to be the same in amplitude and correlated in phase among each element.For the UE-IRS link, the angle of arrival (AoA) is expressed by an elevation angle θ and an azimuth angle ψ, and the phase offset of the LoS path ω for an element is calculated as ω(y, z ) = 2π λ d ((y − 1) sin θ sin ψ + (z − 1) cos θ).For the IRS-BS link, the phase offset is calculated in the same manner by changing the AoA to the angle of departure.The transmission frame consists of the cyclic prefix with N CP = 16 symbols and the data frame with N = 64 information symbols.
In our simulations, we plotted the ideal bound based on the SCA algorithm, where the optimization parameters of our SCM scheme are used as the initial condition.Additionally, we considered two benchmarks, i.e., the scheme with no phase shift and that with the random phase shift at the IRS elements.Note that in each scheme, the reflection coefficients are assumed to have a unit amplitude.

A. Optimized CIR Results
We evaluated the effects of the proposed optimization scheme on the resultant CIRs.Fig. 2(a) shows the CIR of the cascaded channel for the UE-IRS-BS link when the K-factor for the UE-IRS link is given by 0 dB.It is observed that the ideal SCA bound, and the proposed SCM-based scheme exhibited nearly identical CIRs, which enhances the first (LoS) CIR tap by approximately 20 dB, compared with the CIRs of the random phase shift benchmark.Also, the 20 dB power enhancement of the LoS tap corresponds to the number of IRS elements M = 100.
Next, Fig. 2(b) compares the cascaded CIRs for the UE-IRS-BS link, where the K factor of the UE-IRS link was changed to −20 dB.Compared with Fig. 2(a), the resultant channel power is dispersed in the entire taps because of the NLoS-dominant scenario.Note that each value calculated in Fig. 2 is the one averaged over random channel realizations.In the NLoS-dominant scenario of Fig. 2(b), the SCM enhances channel powers of tap numbers 5-13 by approximately 10.45 dB.Despite the channel power dispersion, the beneficial CIR power enhancement of the proposed SCM scheme remained observable.

B. Information Rate Analysis
Fig. 3(a) shows the achievable information rate for the K factor of the UE-IRS link K = 0 dB.The frequencydomain MMSE equalizer was employed for the proposed and benchmark schemes.Observe in Fig. 3(a) that the proposed SCM scheme significantly outperforms the two benchmarks while exhibiting performance nearly identical to the ideal SCA bound as expected from the CIR optimization results of Fig. 2(a).
Finally, Fig. 3(b) shows the achievable information rate for the K factor of the UE-IRS link −20 dB, while other system parameters were the same as those used in Fig. 3(a).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.It was found that the proposed SCM scheme achieved a higher information rate than the two benchmarks, even in the NLoS-dominant channel, similar to Fig. 2(b).By contrast, the proposed SCM scheme exhibited a marginal performance loss from the ideal SCA bound, where the gap is as low as 1 dBm.

C. BER Performance Analysis
Finally, in Fig. 4, we carried out Monte Carlo simulations to evaluate the bit error rates (BERs) of the proposed IRSaided CPSC scheme and the conventional IRS-aided OFDM counterpart, where the K factor of the UE-IRS link was given by K = 0 dB and K= −20 dB in Figs.4(a) and 4(b), respectively.In each scheme, the BER with SCA algorithm and that of the SCM bound are calculated.Observe in Fig. 4(b) that decreasing the K factor, the proposed CPSC scheme and its bound exhibited a better BER performance than the OFDM counterparts, owing to the beneficial frequency diversity gain.Note that the frequency diversity is more apparent in Fig. 4(b) than in Fig. 4(a) since the of frequency-selective fading are dominant in the NLoS V. CONCLUSION This letter proposed the novel low-complexity IRS-assisted transmission scheme.Our algorithm focuses on the maximization of the channel gain associated with a single strongest CIR tap to significantly reduce the complexity while achieving the near-optimal information rate.In our simulations, we demonstrated that the proposed scheme achieves performance nearly identical that of solving the full optimization problem while attaining the complexity order as low as O(L 0 M ), which remains unchanged by the block length N.

Manuscript received 9
April 2024; accepted 12 May 2024.Date of publication 15 May 2024; date of current version 11 July 2024.This work was supported in part by Japan Science and Technology Agency (JST) ASPIRE under Grant JPMJAP2345, and in part by Japan Society for the Promotion of Science (JSPS) KAKENHI under Grant 23H00470.The associate editor coordinating the review of this article and approving it for publication was G. Zheng.(Corresponding author: Shinya Sugiura.)

1
Notation: Scalars, vectors, and matrices are lowercase Italic, bold lowercase Italic, and bold uppercase letters, respectively.(•) T , (•) H , and (•) −1 denote transpose, Hermitian transpose, and inversion matrix.D N and D H N denote the unitary discrete Fourier transform (DFT) matrix and the inverse DFT (IDFT) matrix of shape N × N. I N denotes the identity matrix of shape N × N. 0 N ×M denotes the zero matrix of shape N × M. ∠(•) denotes the phase of a complex number.(x * y ) denotes the convolution operation on the two vectors.

Fig. 2 .
Fig. 2. The optimized CIR results of the cascaded UE-IRS-BS with the K factor of the UE-IRS link: (a) 0 dB and (b) −20 dB.

Fig. 3 .
Fig. 3. Achievable information rate of the proposed SCM and benchmark schemes with the K factor of the UE-IRS link: (a) 0 dB and (b) −20 dB.

Fig. 4 .
Fig. 4. BER of the proposed SCM and benchmark schemes with the K factor of the UE-IRS link: (a) 0 dB and (b) −20 dB.