Journal of information and communication convergence engineering 2024; 22(4): 267-272
Published online December 31, 2024
https://doi.org/10.56977/jicce.2024.22.4.267
© Korea Institute of Information and Communication Engineering
Correspondence to : Jingon Joung (E-mail: jgjoung@cau.ac.kr)
Department of Electrical and Electronics Engineering, Chung-Ang University, Seoul 06974, Republic of Korea
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This study proposes a novel power control method for amplify-and-forward (AF) relay systems operating in time-varying channels. Transmit power control between the source and relay nodes significantly enhances the performance of the AF relay system, and the improvements are proportional to the number of antennas. However, these enhancements are restricted by the presence of time-varying channels, and this limitation increases in severity with increasing number of antennas. To address the challenges posed by these channel variations, the proposed method adaptively optimizes the power-scaling factors to minimize the mean-squared errors under a power-inequality constraint. Numerical results demonstrate that the proposed method effectively mitigates the bit-error-rate performance degradation caused by channel variations, maintaining robust performance even in systems with a large number of antennas. This approach offers a promising solution for improving the reliability and efficiency of AF relay systems under dynamic channel conditions.
Keywords Amplify-and-forward relay, Minimum mean-squared error (MMSE), Power control, Time-varying channels
The number of mobile devices has been increasing over the last few decades, and this trend will continue due to the proliferation of mobile broadband services. Consequently, mobile networks now require higher data rates and wider coverage areas than were previously required [1]. Traditionally, relays have effectively enhanced the data rate and extended the coverage of mobile networks without requiring an additional wired backhaul [2]. Accordingly, mobile network standards such as IEEE 802.16, long-term evolution, and new radio have adopted relay systems.
Relaying between a source node (SN) and destination node (DN) can be accomplished via several schemes, which are distinguished by the signal processing of a relay node (RN). The most straightforward relaying scheme is amplify-andforward (AF), in which the signals received at the RN are linearly processed and retransmitted to the DN [3]. Owing to the absence of a decoding procedure at the RN, the AF scheme is less complex than other strategies such as the decode-and-forward and compressed-and-forward schemes. However, the noise amplification induced by the RN limits the AF scheme and causes performance degradation. To circumvent this problem, various power control methods based on channel information in AF relay systems have been adopted [3]. For example, a power control framework was developed to maximize the signal-to-noise ratio (SNR) of the DN [4]. In [5], the optimal power-scaling factors of the RN were represented using two objective functions: the minimum mean-squared error (MSE) and zero forcing. Further, power control methods based on the minimum MSE (MMSE) criterion under power-inequality constraints were designed [6]. Channel information is uncertain in AF relay systems because it is obtained from channel estimation at an SN or DN (and feedback from them). Thus, channel uncertainty should be carefully considered when designing power controls, particularly when the channel varies over time. However, all the previously mentioned studies assumed timeinvariant channels throughout the data transmission. In timevarying channel environments, the power control performance degrades owing to outdated information. Therefore, channel variations should be carefully considered to avoid significant performance degradation.
In this study, we propose a power control method for AF relay systems in time-varying channels. Closed-form expressions of the optimal power-scaling factors are derived from the MMSE problem under a local power-inequality constraint. However, obtaining a solution is challenging because the optimal power-scaling factors are heavily entangled. To obtain a feasible solution, we employ an alternating-optimization-based iterative algorithm that sequentially optimizes the power-scaling factors while keeping all others fixed. The simulation results confirm that the proposed power control method effectively mitigates bit-error-rate (BER) performance degradation caused by channel variations when each node has four or more antennas.
The remainder of this paper is organized as follows: The system and signal models for the considered AF relay system and time-varying channels are introduced in Section II, while the proposed power control method is presented in Section III. The simulation results and conclusions are presented in Section IV and Section V, respectively. The following notation is used throughout the paper: boldface lower-case and capital letters denote vectors and matrices, respectively, while lower-case letters denote scalars; superscript H represents a conjugate transpose; |x| and |x| denote the absolute value of x and the 2-norm of x, respectively; In represents an n × n identity matrix;
In this study, we consider downlink communication in a two-hop half-duplex AF relay system with scaled eigen beamforming [6]. The SN, RN, and DN are equipped with NS, NR, and ND antennas, respectively. For beamforming purposes, NS and ND are greater than or equal to two, while NR has no constraints. The direct path between the SN and DN is not considered. Denoting T as the number of transmitted symbols per frame, Fig. 1 illustrates the frame structure, which consists of one pilot and T−1 data symbols. The pilot symbol is transmitted at the first transmission time. The channels are estimated separately at the RN and DN and then fed back to every node through a broadcasting channel. In this study, perfect channel feedback is assumed. After the channel estimation and feedback completed, the remaining transmission time is used to transmit T−1 data symbols.
Let
Here, the elements of
For the beamforming, all nodes should know the perfect channel information (i.e., H(t) and F(t) for
where the elements of
Here,
The SN transmits the data over the first-hop link. The tth transmitted symbol vector of the SN is expressed as
where the t th data symbol is denoted by
Here,
Then,
where
In this section, we design the power-scaling factors {α(t), β(t), γ(t)} to minimize the MSE between d(t) and
The transmit power of the SN and RN are independently bounded by PS and PR (i.e.,
where αL, βL, and γL are the power-scaling factors under the local power-inequality constraint. Using the Lagrange multiplier method and substituting (8), (10), and (12) into (13), we obtain
where λS and λR are the non-negative Lagrangian multipliers, and JL is the cost function under the local power-inequality constraint, which is given by
Here, the corrupted singular values
where δh and δf are defined as
Direct computation of the optimal solution for (18)-(22) is challenging because αL(t), βL(t), γL(t), λS(t), and λR(t) are functions of other optimization variables. Therefore, we adopt the alternating optimization presented in Algorithm 1. Here, the subscript k denotes the iteration index. The maximum number of iterations and threshold for the stopping criterion are denoted by kmax and ε, respectively. In line 7, {αL,k (t), βL,k (t), γL,k (t)} is updated by inputting the power-scaling factors and Lagrangian multipliers obtained at the (k−1)th iteration into (18)-(20). When implementing Algorithm 1, an infeasible solution
Algorithm 1: Power control algorithm under local power-inequality constraint.
1 | Input: Correlation coefficients |
2 | Output: Relay processing matrix and source-destination beamforming vectors |
3 | for t = 1 to T - 1 do |
4 | |
5 | Initialize |
6 | for k = 1 to Kmax do |
7 | Update |
8 | Obtain |
9 | Compute |
10 | if |
11 | Break. |
12 | end if |
13 | end for |
14 | |
15 | end for |
The performance of the proposed power control algorithm is evaluated through numerical simulations. The simulation parameters are listed in Table 1. The local power-inequality constraint is set to PS = PR = 1. For the simulation, the received SNRs at the RN and DN are defined as
Table 1 . Simulation parameters for verifying the performance of the proposed algorithm
Parameters | Values |
---|---|
Local power-inequality constraint | |
SNR at RN | |
Data symbol modulation | 16-QAM |
Channel correlation coefficients | |
Maximum number of iterations | Kmax =1000 |
Threshold for stopping criterion | ε = 10−4 |
Variance of channel estimation error |
where the coefficients
Table 2 . Numerical values of
a2 | a1 | b2 | b1 | |
---|---|---|---|---|
{2, 2, 2} | 0.7961 | −1.8052 | 2.1372 | 1.0164 |
{4, 4, 4} | 1.3696 | −3.0867 | 7.4859 | 1.0584 |
{8, 8, 8} | 2.1750 | −4.8452 | 19.3744 | 1.1544 |
In Fig. 2, the BER performance is evaluated over
In this study, we proposed a power control method under the local power-inequality constraint for reliable communication in multi-input multi-output amplify-and-forward relay systems in time-varying channels. In the proposed method, the power-scaling factors of each node were obtained from an alternating optimization-based iterative algorithm. The simulation results demonstrated that the proposed method effectively mitigated the BER performance degradation caused by time-varying channels when each node had more than four antennas.
This work was supported in part by the Institute of Information and Communications Technology Planning and Evaluation (IITP) Grant funded by the Korea Government (MSIT) (No.2021-0-00874, Development of Next Generation Wireless Access Technology Based on Space Time Line Code, 30%; No. 2022-0-00635, Development of 5G Industrial Terminal Technology Supporting 28 GHz Band/Private 5G Band/NR-U Band, 25%); in part by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIT) under Grant 2022R1A2C1003750 & RS-2024-00405510; and in part by the Chung-Ang University research grant in 2024.
Han-Gyeol Lee
received the B.S. and M.S. degrees in electrical and electronics engineering from Chung-Ang University, Seoul, South Korea, in 2021 and 2023, respectively, where he currently is pursuing the Ph.D. degree with the Department of Electrical and Electronical Engineering. His areas of interest include multicarrier systems and signal processing techniques for nextgeneration wireless communication.
Duckdong Hwang
received the B.S. and M.S. degrees in electronics engineering from Yonsei University, South Korea, and the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, CA, USA, in May 2005. From 1993 to 1998, he worked as a Research Engineer with Daewoo Electronics, South Korea. In 2005, he joined the Digital Research Center, Samsung Advanced Institute of Technology, as a Research Staff Member. Since 2012, he has been a Research Associate Professor with the School of Information and Communication Engineering, Sungkyunkwan University, the Department of Electrical Engineering, Konkuk University, and the Department of Electronics, Information and Communication Engineering, Sejong University, South Korea. His research interests include physical layer aspect of the next generation wireless communication systems, including multiple antenna techniques, interference alignment and management, and cooperative relays and their applications in the heterogeneous small cell networks and wireless security issues.
Jingon Joung
received the B.S. degree in radio communication engineering from Yonsei University, Seoul, South Korea, in 2001, and the M.S. and Ph.D. degrees in electrical engineering and computer science from KAIST, Daejeon, South Korea, in 2003 and 2007, respectively. He was a Postdoctoral Fellow with KAIST, and UCLA, CA, USA, in 2007 and 2008, respectively. He was a Scientist with the Institute for Infocomm Research, Singapore, from 2009 to 2015, and joined Chung-Ang University (CAU), Seoul, in 2016, as a Faculty Member. He is currently a Professor with the School of Electrical and Electronics Engineering, CAU, where he is also the Principal Investigator of the Intelligent Wireless Systems Laboratory. His research interests include signal processing, numerical analysis, algorithms, and machine learning. Dr. Joung is an inventor of a Space-Time Line Code (STLC), that is a fully symmetric scheme to a Space-Time Block Code.
Journal of information and communication convergence engineering 2024; 22(4): 267-272
Published online December 31, 2024 https://doi.org/10.56977/jicce.2024.22.4.267
Copyright © Korea Institute of Information and Communication Engineering.
Han-Gyeol Lee 1, Duckdong Hwang
2, and Jingon Joung1*
1Department of Electrical and Electronics Engineering, Chung-Ang University, Seoul 06974, Republic of Korea
2Department of Information and Communications Engineering, Sejong University, Seoul 143-747, Republic of Korea
Correspondence to:Jingon Joung (E-mail: jgjoung@cau.ac.kr)
Department of Electrical and Electronics Engineering, Chung-Ang University, Seoul 06974, Republic of Korea
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This study proposes a novel power control method for amplify-and-forward (AF) relay systems operating in time-varying channels. Transmit power control between the source and relay nodes significantly enhances the performance of the AF relay system, and the improvements are proportional to the number of antennas. However, these enhancements are restricted by the presence of time-varying channels, and this limitation increases in severity with increasing number of antennas. To address the challenges posed by these channel variations, the proposed method adaptively optimizes the power-scaling factors to minimize the mean-squared errors under a power-inequality constraint. Numerical results demonstrate that the proposed method effectively mitigates the bit-error-rate performance degradation caused by channel variations, maintaining robust performance even in systems with a large number of antennas. This approach offers a promising solution for improving the reliability and efficiency of AF relay systems under dynamic channel conditions.
Keywords: Amplify-and-forward relay, Minimum mean-squared error (MMSE), Power control, Time-varying channels
The number of mobile devices has been increasing over the last few decades, and this trend will continue due to the proliferation of mobile broadband services. Consequently, mobile networks now require higher data rates and wider coverage areas than were previously required [1]. Traditionally, relays have effectively enhanced the data rate and extended the coverage of mobile networks without requiring an additional wired backhaul [2]. Accordingly, mobile network standards such as IEEE 802.16, long-term evolution, and new radio have adopted relay systems.
Relaying between a source node (SN) and destination node (DN) can be accomplished via several schemes, which are distinguished by the signal processing of a relay node (RN). The most straightforward relaying scheme is amplify-andforward (AF), in which the signals received at the RN are linearly processed and retransmitted to the DN [3]. Owing to the absence of a decoding procedure at the RN, the AF scheme is less complex than other strategies such as the decode-and-forward and compressed-and-forward schemes. However, the noise amplification induced by the RN limits the AF scheme and causes performance degradation. To circumvent this problem, various power control methods based on channel information in AF relay systems have been adopted [3]. For example, a power control framework was developed to maximize the signal-to-noise ratio (SNR) of the DN [4]. In [5], the optimal power-scaling factors of the RN were represented using two objective functions: the minimum mean-squared error (MSE) and zero forcing. Further, power control methods based on the minimum MSE (MMSE) criterion under power-inequality constraints were designed [6]. Channel information is uncertain in AF relay systems because it is obtained from channel estimation at an SN or DN (and feedback from them). Thus, channel uncertainty should be carefully considered when designing power controls, particularly when the channel varies over time. However, all the previously mentioned studies assumed timeinvariant channels throughout the data transmission. In timevarying channel environments, the power control performance degrades owing to outdated information. Therefore, channel variations should be carefully considered to avoid significant performance degradation.
In this study, we propose a power control method for AF relay systems in time-varying channels. Closed-form expressions of the optimal power-scaling factors are derived from the MMSE problem under a local power-inequality constraint. However, obtaining a solution is challenging because the optimal power-scaling factors are heavily entangled. To obtain a feasible solution, we employ an alternating-optimization-based iterative algorithm that sequentially optimizes the power-scaling factors while keeping all others fixed. The simulation results confirm that the proposed power control method effectively mitigates bit-error-rate (BER) performance degradation caused by channel variations when each node has four or more antennas.
The remainder of this paper is organized as follows: The system and signal models for the considered AF relay system and time-varying channels are introduced in Section II, while the proposed power control method is presented in Section III. The simulation results and conclusions are presented in Section IV and Section V, respectively. The following notation is used throughout the paper: boldface lower-case and capital letters denote vectors and matrices, respectively, while lower-case letters denote scalars; superscript H represents a conjugate transpose; |x| and |x| denote the absolute value of x and the 2-norm of x, respectively; In represents an n × n identity matrix;
In this study, we consider downlink communication in a two-hop half-duplex AF relay system with scaled eigen beamforming [6]. The SN, RN, and DN are equipped with NS, NR, and ND antennas, respectively. For beamforming purposes, NS and ND are greater than or equal to two, while NR has no constraints. The direct path between the SN and DN is not considered. Denoting T as the number of transmitted symbols per frame, Fig. 1 illustrates the frame structure, which consists of one pilot and T−1 data symbols. The pilot symbol is transmitted at the first transmission time. The channels are estimated separately at the RN and DN and then fed back to every node through a broadcasting channel. In this study, perfect channel feedback is assumed. After the channel estimation and feedback completed, the remaining transmission time is used to transmit T−1 data symbols.
Let
Here, the elements of
For the beamforming, all nodes should know the perfect channel information (i.e., H(t) and F(t) for
where the elements of
Here,
The SN transmits the data over the first-hop link. The tth transmitted symbol vector of the SN is expressed as
where the t th data symbol is denoted by
Here,
Then,
where
In this section, we design the power-scaling factors {α(t), β(t), γ(t)} to minimize the MSE between d(t) and
The transmit power of the SN and RN are independently bounded by PS and PR (i.e.,
where αL, βL, and γL are the power-scaling factors under the local power-inequality constraint. Using the Lagrange multiplier method and substituting (8), (10), and (12) into (13), we obtain
where λS and λR are the non-negative Lagrangian multipliers, and JL is the cost function under the local power-inequality constraint, which is given by
Here, the corrupted singular values
where δh and δf are defined as
Direct computation of the optimal solution for (18)-(22) is challenging because αL(t), βL(t), γL(t), λS(t), and λR(t) are functions of other optimization variables. Therefore, we adopt the alternating optimization presented in Algorithm 1. Here, the subscript k denotes the iteration index. The maximum number of iterations and threshold for the stopping criterion are denoted by kmax and ε, respectively. In line 7, {αL,k (t), βL,k (t), γL,k (t)} is updated by inputting the power-scaling factors and Lagrangian multipliers obtained at the (k−1)th iteration into (18)-(20). When implementing Algorithm 1, an infeasible solution
Algorithm 1: Power control algorithm under local power-inequality constraint.
1 | Input: Correlation coefficients |
2 | Output: Relay processing matrix and source-destination beamforming vectors |
3 | for t = 1 to T - 1 do |
4 | |
5 | Initialize |
6 | for k = 1 to Kmax do |
7 | Update |
8 | Obtain |
9 | Compute |
10 | if |
11 | Break. |
12 | end if |
13 | end for |
14 | |
15 | end for |
The performance of the proposed power control algorithm is evaluated through numerical simulations. The simulation parameters are listed in Table 1. The local power-inequality constraint is set to PS = PR = 1. For the simulation, the received SNRs at the RN and DN are defined as
Table 1 . Simulation parameters for verifying the performance of the proposed algorithm.
Parameters | Values |
---|---|
Local power-inequality constraint | |
SNR at RN | |
Data symbol modulation | 16-QAM |
Channel correlation coefficients | |
Maximum number of iterations | Kmax =1000 |
Threshold for stopping criterion | ε = 10−4 |
Variance of channel estimation error |
where the coefficients
Table 2 . Numerical values of
a2 | a1 | b2 | b1 | |
---|---|---|---|---|
{2, 2, 2} | 0.7961 | −1.8052 | 2.1372 | 1.0164 |
{4, 4, 4} | 1.3696 | −3.0867 | 7.4859 | 1.0584 |
{8, 8, 8} | 2.1750 | −4.8452 | 19.3744 | 1.1544 |
In Fig. 2, the BER performance is evaluated over
In this study, we proposed a power control method under the local power-inequality constraint for reliable communication in multi-input multi-output amplify-and-forward relay systems in time-varying channels. In the proposed method, the power-scaling factors of each node were obtained from an alternating optimization-based iterative algorithm. The simulation results demonstrated that the proposed method effectively mitigated the BER performance degradation caused by time-varying channels when each node had more than four antennas.
This work was supported in part by the Institute of Information and Communications Technology Planning and Evaluation (IITP) Grant funded by the Korea Government (MSIT) (No.2021-0-00874, Development of Next Generation Wireless Access Technology Based on Space Time Line Code, 30%; No. 2022-0-00635, Development of 5G Industrial Terminal Technology Supporting 28 GHz Band/Private 5G Band/NR-U Band, 25%); in part by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIT) under Grant 2022R1A2C1003750 & RS-2024-00405510; and in part by the Chung-Ang University research grant in 2024.
Table 1 . Simulation parameters for verifying the performance of the proposed algorithm.
Parameters | Values |
---|---|
Local power-inequality constraint | |
SNR at RN | |
Data symbol modulation | 16-QAM |
Channel correlation coefficients | |
Maximum number of iterations | Kmax =1000 |
Threshold for stopping criterion | ε = 10−4 |
Variance of channel estimation error |
In-Ho Lee
Journal of information and communication convergence engineering 2017; 15(1): 21-27 https://doi.org/10.6109/jicce.2017.15.1.21