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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

MMSE-Based Power Control for Amplify-and-Forward Relay Systems in Time-Varying Channels

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

Received: June 12, 2024; Revised: July 13, 2024; Accepted: July 16, 2024

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; E is the expectation of a random variable or matrix; CN0,σ2 represents a zero-mean complex Gaussian distribution with variance σ2; (x)+ denotes (x + |x|)/2 for an arbitrary real number x; and x is the real part of x.

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.

Fig. 1. Frame structure with one pilot and T-1 data symbols

Let HtNR×NS and FtND×NR denote the firstand second-hop channel matrices at the t th symbol time, respectively, for symbol time index t0,  ,  T1. The channel model conforms to Rayleigh fading, where the elements of H(t) and F(t) are independent random variables following CN0, 1. The first-order Gauss-Markov process is assumed to model the time variation of the first- and secondhop channels as follows [7]:

Ht=ρhHt1+Zht=ρhtH0+Σi=1tρhtiZhi,
Ft=ρfFt1+Zft=ρftF0+Σi=1tρftiZfi.

Here, the elements of ZhtNR×NS and ZftND×NR are independent random variables following CN0,1ρh2 and CN0,1ρf2, respectively, where ρh0,1 and ρf0,1 are the first- and the second-hop correlation coefficients, respectively. In this study, we assume that ρh and ρf are known at each node because the channel correlation coefficient can be accurately estimated [7].

For the beamforming, all nodes should know the perfect channel information (i.e., H(t) and F(t) for t0,,T1). However, only the estimates of H(0) and F(0), denoted by H^0NR×NS and F^0ND×NR, respectively, are available. The estimated channel matrices are modeled as:

H^0=H0+Δh,
F^0=F0+Δf,

where the elements of ΔhNR×NS and ΔfND×NR are independent random variables following CN0,σh2 and CN0,σf2, respectively. Consequently, the relay processing matrix and source-destination beamforming vectors are obtained from H^0 and F^0, as follows [6]:

Wt=αtvfuhHNR×NR,
at=βtvhNS×1,
bt=γtufND×1.

Here, uhNR×1 and vhNS×1 are the left and right singular vectors, respectively, corresponding to the largest singular value ηh of H^0; ufND×1 and vfNR×1 are the left and right singular vectors, respectively, corresponding to the largest singular value ηf of F^0. The power-scaling factors αt, βt, and γt are designed in Section III.

The SN transmits the data over the first-hop link. The tth transmitted symbol vector of the SN is expressed as

xst=atdtNS×1,

where the t th data symbol is denoted by dt with Edt2=1. At the RN, the t th received symbol vector is given by

yRt=HtxSt+nRtNR×1.

Here, nRtNR×1 is an additive white Gaussian noise (AWGN) vector with EnRtnRHt=σR2INR. The RN performs linear processing of yRt with W(t) as follows:

xRt=WtyRtNR×1.

Then, xRt is transmitted to the DN over the second-hop link. The t th received symbol vector at the DN is given by

yDt=FtxRt+nDtND×1,

where nDtND×1 is an AWGN vector with EnDtnDHt=σD2IND. Substituting (8)-(10) into (11), the DN combines yDt to obtain the t th estimated data symbol d^t=bHtyDt, which is expressed as

d^t=bHtFtWtHtatdt+bHtFtWtnRt+bHtnDt.

In this section, we design the power-scaling factors {α(t), β(t), γ(t)} to minimize the MSE between d(t) and d^t under the local inequality constraint. Henceforth, the time index t is omitted whenever convenient.

The transmit power of the SN and RN are independently bounded by PS and PR (i.e., ExS2PS and ExR2PR), respectively, under the local power-inequality constraint. The MMSE of the data symbol is expressed as follows:

minαL,βL,γLE{|dd^2},s.tExs2PS and Exs2PS

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

minαL,βL,γL,λS,λRJL,

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

JL =E{|dd^2}+λSExS 2PS+λRExR 2PR=12αLβLγLρhtηhρftηf+αL2βL2γL2ρh2t η˜ h2ρf2t η˜ f2+αL2γL2ρf2t η˜ f2σR2+γL2σD2+λSβL2PS+λRαL2βL2ρh2t η˜ h2+αL2σR2PR

Here, the corrupted singular values η˜h and η˜f satisfy

η˜h2=E{|ηhuhHΔhvh+Σi=1tρhiuhHZhivh|2}η˜h2=E{|ηhuhHΔhvh+Σi=1tρhiuhHZhivh|2},
η˜f2=E{|ηfufHΔfvf+Σi=1tρfiufHZfivf|2}=ηf2+2ηfEδf+E{|δf2}+ρf2t1,

where δh and δf are defined as uhHΔhvh and ufHΔfvf, respectively. By equating the derivatives of JL with respect to the optimization variables to zero, the optimal power-scaling factors and Lagrangian multipliers under the local power-inequality constraint are expressed as

αL=βLγLρhtηhρftηfγL2ρL2t η˜f2+λRβL2ρh2t η˜h2+σR2 ,
βL=αLγLρhtηhρftηfαL2γL2ρh2t η˜h2ρf2t η˜f2+λRαL2ρh2t η˜h2+λS ,
γL=αLβLρhtηhρftηfαL2βL2ρh2t η˜h2ρf2t η˜f2+αL2σR2ρf2 η˜f2+σD2 ,
λR=βLγLρhtηhρhtηfPR βL2ρh2t η˜h2+σR2γLρf2tη˜f2+,
λS=αLγLρhtηhρftηfPSαL2ρh2tη˜h2γL2ρf2t η˜f2+λR+.

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 αL,kt=βL,kt=γL,k= is obtained if βL,0 (t) and γL,0 (t) are initialized by zero. To resolve this issue, {βL,0 (t), γL,0 (t), λS,0 (t), λR,0 (t)} are set to {1, 1, 0, 0} at t = 1.

Algorithm 1: Power control algorithm under local power-inequality constraint.

1 Input: Correlation coefficients ρh,ρf, estimated singular values ηh,ηf corrupted singular values η˜ht,η˜ft and local power constraints PS,PR.
2 Output: Relay processing matrix and source-destination beamforming vectors Wt,at,bt.
3 for t = 1 to T - 1 do
4 βL,0t=γL,0t=1 and λS,0t=λR,0t=0.
5 Initialize JL,0t=0.
6 for k = 1 to Kmax do
7 Update αL,kt, βL,kt, and γL,kt as (18)–(20).
8 Obtain λR,kt and λS,kt from (21) and (22).
9 Compute JL,kt from (15).
10 if 0JL,k1tJL,ktϵ then
11 Break.
12 end if
13 end for
14 Wt=αL,ktvfuhH, at=βL,ktvh, bt, bt=γL,ktuf.
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 PS/σR2 and PR/σD2, respectively [6]. Because the radio link between the SN and RN is usually in good condition during downlink communication [8], PS/σR2 is fixed at 25 dB. The data symbols are modulated using 16-quadrature amplitude modulation (QAM). The channel correlation coefficient typically ranges between 0.9 and 0.99 [9]. Accordingly, the correlation coefficients ρh and ρf are set to 0.99 and 0.97, respectively. The maximum number of iterations kmax and threshold value ε are 1000 and 10−4, respectively. The variance of channel estimation errors σh2 and σf2 is fixed at 0.1. From (18)-(22), the square of the corrupted singular values η˜h2t and η˜f2t are required to obtain the power-scaling factors. Thus, assuming channel uncertainty of up to 10% (i.e., σh20.1 and σf20.1), the following model is employed [6]:

Table 1 . Simulation parameters for verifying the performance of the proposed algorithm

ParametersValues
Local power-inequality constraintPS=PR=1
SNR at RNPS/σR2= 25 dB
Data symbol modulation16-QAM
Channel correlation coefficientsρh,ρf=0.99,0.97
Maximum number of iterationsKmax =1000
Threshold for stopping criterionε = 10−4
Variance of channel estimation errorσh2=σf2=0.1


Eδa2σ2+a1σ2,
Eδ2b2σ2+b1σ2,

where the coefficients a2,a1,b2,b1 are numerically computed using the polynomial fitting method. Table 2 lists the values of a2,a1,b2,b1 used in the simulations. By substituting (23) into (16) and (17), η˜h2t and η˜f2t are obtained.

Table 2 . Numerical values of a2,a1,b2,b1 in (23)

NS,NR,NDa2a1b2b1
{2, 2, 2}0.7961−1.80522.13721.0164
{4, 4, 4}1.3696−3.08677.48591.0584
{8, 8, 8}2.1750−4.845219.37441.1544


In Fig. 2, the BER performance is evaluated over PR/σD2 and T to demonstrate the robustness of the proposed power control method against time-varying channels. For comparison, the power control method in [6] is adopted as a baseline method, in which the power-scaling factors obtained at t = 0 are applied for t1. The simulation results show that the performance gap between the proposed and baseline methods is negligible when the number of antennas at each node is two. In contrast, the proposed method effectively mitigates BER performance degradation caused by the time-varying channels when the number of antennas at each node is four or eight. Particularly, the proposed method with {NS, NR, ND}={4, 4, 4} outperforms the baseline method with {NS, NR, ND}={8, 8, 8} when PR/σD210 or T12. The baseline method exhibits limited BER performance because the power-scaling factors are not adjusted to the channel variations. In contrast, the proposed method adaptively controls the power-scaling factors at each symbol time to minimize the MMSE of the data symbols at the DN.

Fig. 2. 16-QAM BER performance comparison with PS/σR2==25 dB. (a) BER over PR/σD2 for T=16. (b) BER over T for PR/σD2=20 dB.

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.

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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.


Article

Regular paper

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.

MMSE-Based Power Control for Amplify-and-Forward Relay Systems in Time-Varying Channels

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

Received: June 12, 2024; Revised: July 13, 2024; Accepted: July 16, 2024

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.

Abstract

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

I. INTRODUCTION

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; E is the expectation of a random variable or matrix; CN0,σ2 represents a zero-mean complex Gaussian distribution with variance σ2; (x)+ denotes (x + |x|)/2 for an arbitrary real number x; and x is the real part of x.

II. SYSTEM AND SIGNAL MODELS

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.

Figure 1. Frame structure with one pilot and T-1 data symbols

Let HtNR×NS and FtND×NR denote the firstand second-hop channel matrices at the t th symbol time, respectively, for symbol time index t0,  ,  T1. The channel model conforms to Rayleigh fading, where the elements of H(t) and F(t) are independent random variables following CN0, 1. The first-order Gauss-Markov process is assumed to model the time variation of the first- and secondhop channels as follows [7]:

Ht=ρhHt1+Zht=ρhtH0+Σi=1tρhtiZhi,
Ft=ρfFt1+Zft=ρftF0+Σi=1tρftiZfi.

Here, the elements of ZhtNR×NS and ZftND×NR are independent random variables following CN0,1ρh2 and CN0,1ρf2, respectively, where ρh0,1 and ρf0,1 are the first- and the second-hop correlation coefficients, respectively. In this study, we assume that ρh and ρf are known at each node because the channel correlation coefficient can be accurately estimated [7].

For the beamforming, all nodes should know the perfect channel information (i.e., H(t) and F(t) for t0,,T1). However, only the estimates of H(0) and F(0), denoted by H^0NR×NS and F^0ND×NR, respectively, are available. The estimated channel matrices are modeled as:

H^0=H0+Δh,
F^0=F0+Δf,

where the elements of ΔhNR×NS and ΔfND×NR are independent random variables following CN0,σh2 and CN0,σf2, respectively. Consequently, the relay processing matrix and source-destination beamforming vectors are obtained from H^0 and F^0, as follows [6]:

Wt=αtvfuhHNR×NR,
at=βtvhNS×1,
bt=γtufND×1.

Here, uhNR×1 and vhNS×1 are the left and right singular vectors, respectively, corresponding to the largest singular value ηh of H^0; ufND×1 and vfNR×1 are the left and right singular vectors, respectively, corresponding to the largest singular value ηf of F^0. The power-scaling factors αt, βt, and γt are designed in Section III.

The SN transmits the data over the first-hop link. The tth transmitted symbol vector of the SN is expressed as

xst=atdtNS×1,

where the t th data symbol is denoted by dt with Edt2=1. At the RN, the t th received symbol vector is given by

yRt=HtxSt+nRtNR×1.

Here, nRtNR×1 is an additive white Gaussian noise (AWGN) vector with EnRtnRHt=σR2INR. The RN performs linear processing of yRt with W(t) as follows:

xRt=WtyRtNR×1.

Then, xRt is transmitted to the DN over the second-hop link. The t th received symbol vector at the DN is given by

yDt=FtxRt+nDtND×1,

where nDtND×1 is an AWGN vector with EnDtnDHt=σD2IND. Substituting (8)-(10) into (11), the DN combines yDt to obtain the t th estimated data symbol d^t=bHtyDt, which is expressed as

d^t=bHtFtWtHtatdt+bHtFtWtnRt+bHtnDt.

III. POWER CONTROL METHOD IN TIME-VARYING CHANNELS

In this section, we design the power-scaling factors {α(t), β(t), γ(t)} to minimize the MSE between d(t) and d^t under the local inequality constraint. Henceforth, the time index t is omitted whenever convenient.

The transmit power of the SN and RN are independently bounded by PS and PR (i.e., ExS2PS and ExR2PR), respectively, under the local power-inequality constraint. The MMSE of the data symbol is expressed as follows:

minαL,βL,γLE{|dd^2},s.tExs2PS and Exs2PS

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

minαL,βL,γL,λS,λRJL,

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

JL =E{|dd^2}+λSExS 2PS+λRExR 2PR=12αLβLγLρhtηhρftηf+αL2βL2γL2ρh2t η˜ h2ρf2t η˜ f2+αL2γL2ρf2t η˜ f2σR2+γL2σD2+λSβL2PS+λRαL2βL2ρh2t η˜ h2+αL2σR2PR

Here, the corrupted singular values η˜h and η˜f satisfy

η˜h2=E{|ηhuhHΔhvh+Σi=1tρhiuhHZhivh|2}η˜h2=E{|ηhuhHΔhvh+Σi=1tρhiuhHZhivh|2},
η˜f2=E{|ηfufHΔfvf+Σi=1tρfiufHZfivf|2}=ηf2+2ηfEδf+E{|δf2}+ρf2t1,

where δh and δf are defined as uhHΔhvh and ufHΔfvf, respectively. By equating the derivatives of JL with respect to the optimization variables to zero, the optimal power-scaling factors and Lagrangian multipliers under the local power-inequality constraint are expressed as

αL=βLγLρhtηhρftηfγL2ρL2t η˜f2+λRβL2ρh2t η˜h2+σR2 ,
βL=αLγLρhtηhρftηfαL2γL2ρh2t η˜h2ρf2t η˜f2+λRαL2ρh2t η˜h2+λS ,
γL=αLβLρhtηhρftηfαL2βL2ρh2t η˜h2ρf2t η˜f2+αL2σR2ρf2 η˜f2+σD2 ,
λR=βLγLρhtηhρhtηfPR βL2ρh2t η˜h2+σR2γLρf2tη˜f2+,
λS=αLγLρhtηhρftηfPSαL2ρh2tη˜h2γL2ρf2t η˜f2+λR+.

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 αL,kt=βL,kt=γL,k= is obtained if βL,0 (t) and γL,0 (t) are initialized by zero. To resolve this issue, {βL,0 (t), γL,0 (t), λS,0 (t), λR,0 (t)} are set to {1, 1, 0, 0} at t = 1.

Algorithm 1: Power control algorithm under local power-inequality constraint.

1 Input: Correlation coefficients ρh,ρf, estimated singular values ηh,ηf corrupted singular values η˜ht,η˜ft and local power constraints PS,PR.
2 Output: Relay processing matrix and source-destination beamforming vectors Wt,at,bt.
3 for t = 1 to T - 1 do
4 βL,0t=γL,0t=1 and λS,0t=λR,0t=0.
5 Initialize JL,0t=0.
6 for k = 1 to Kmax do
7 Update αL,kt, βL,kt, and γL,kt as (18)–(20).
8 Obtain λR,kt and λS,kt from (21) and (22).
9 Compute JL,kt from (15).
10 if 0JL,k1tJL,ktϵ then
11 Break.
12 end if
13 end for
14 Wt=αL,ktvfuhH, at=βL,ktvh, bt, bt=γL,ktuf.
15 end for

IV. SIMULATION RESULTS

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 PS/σR2 and PR/σD2, respectively [6]. Because the radio link between the SN and RN is usually in good condition during downlink communication [8], PS/σR2 is fixed at 25 dB. The data symbols are modulated using 16-quadrature amplitude modulation (QAM). The channel correlation coefficient typically ranges between 0.9 and 0.99 [9]. Accordingly, the correlation coefficients ρh and ρf are set to 0.99 and 0.97, respectively. The maximum number of iterations kmax and threshold value ε are 1000 and 10−4, respectively. The variance of channel estimation errors σh2 and σf2 is fixed at 0.1. From (18)-(22), the square of the corrupted singular values η˜h2t and η˜f2t are required to obtain the power-scaling factors. Thus, assuming channel uncertainty of up to 10% (i.e., σh20.1 and σf20.1), the following model is employed [6]:

Table 1 . Simulation parameters for verifying the performance of the proposed algorithm.

ParametersValues
Local power-inequality constraintPS=PR=1
SNR at RNPS/σR2= 25 dB
Data symbol modulation16-QAM
Channel correlation coefficientsρh,ρf=0.99,0.97
Maximum number of iterationsKmax =1000
Threshold for stopping criterionε = 10−4
Variance of channel estimation errorσh2=σf2=0.1


Eδa2σ2+a1σ2,
Eδ2b2σ2+b1σ2,

where the coefficients a2,a1,b2,b1 are numerically computed using the polynomial fitting method. Table 2 lists the values of a2,a1,b2,b1 used in the simulations. By substituting (23) into (16) and (17), η˜h2t and η˜f2t are obtained.

Table 2 . Numerical values of a2,a1,b2,b1 in (23).

NS,NR,NDa2a1b2b1
{2, 2, 2}0.7961−1.80522.13721.0164
{4, 4, 4}1.3696−3.08677.48591.0584
{8, 8, 8}2.1750−4.845219.37441.1544


In Fig. 2, the BER performance is evaluated over PR/σD2 and T to demonstrate the robustness of the proposed power control method against time-varying channels. For comparison, the power control method in [6] is adopted as a baseline method, in which the power-scaling factors obtained at t = 0 are applied for t1. The simulation results show that the performance gap between the proposed and baseline methods is negligible when the number of antennas at each node is two. In contrast, the proposed method effectively mitigates BER performance degradation caused by the time-varying channels when the number of antennas at each node is four or eight. Particularly, the proposed method with {NS, NR, ND}={4, 4, 4} outperforms the baseline method with {NS, NR, ND}={8, 8, 8} when PR/σD210 or T12. The baseline method exhibits limited BER performance because the power-scaling factors are not adjusted to the channel variations. In contrast, the proposed method adaptively controls the power-scaling factors at each symbol time to minimize the MMSE of the data symbols at the DN.

Figure 2. 16-QAM BER performance comparison with PS/σR2==25 dB. (a) BER over PR/σD2 for T=16. (b) BER over T for PR/σD2=20 dB.

V. CONCLUSION

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.

ACKNOWLEDGEMENTS

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.

Fig 1.

Figure 1.Frame structure with one pilot and T-1 data symbols
Journal of Information and Communication Convergence Engineering 2024; 22: 267-272https://doi.org/10.56977/jicce.2024.22.4.267

Fig 2.

Figure 2.16-QAM BER performance comparison with PS/σR2==25 dB. (a) BER over PR/σD2 for T=16. (b) BER over T for PR/σD2=20 dB.
Journal of Information and Communication Convergence Engineering 2024; 22: 267-272https://doi.org/10.56977/jicce.2024.22.4.267

Table 1 . Simulation parameters for verifying the performance of the proposed algorithm.

ParametersValues
Local power-inequality constraintPS=PR=1
SNR at RNPS/σR2= 25 dB
Data symbol modulation16-QAM
Channel correlation coefficientsρh,ρf=0.99,0.97
Maximum number of iterationsKmax =1000
Threshold for stopping criterionε = 10−4
Variance of channel estimation errorσh2=σf2=0.1

Table 2 . Numerical values of a2,a1,b2,b1 in (23).

NS,NR,NDa2a1b2b1
{2, 2, 2}0.7961−1.80522.13721.0164
{4, 4, 4}1.3696−3.08677.48591.0584
{8, 8, 8}2.1750−4.845219.37441.1544

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JICCE
Dec 31, 2024 Vol.22 No.4, pp. 267~343

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