Full length articleA novel user pairing scheme for functional decode-and-forward multi-way relay network
Introduction
Multi-way relay networks (MWRNs), where a single relay facilitates all users in the network to exchange information with every other user, have important potential applications in teleconferencing, data exchange in a sensor network or file sharing in a social network [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. A MWRN is a generalization of two-way relay networks (TWRNs), which enable bidirectional information exchange between two users and are widely recognized in the literature for their improved spectral efficiency, compared to conventional relaying [13], [14], [15], [16], [17], [18], [19]. Note that multi-user TWRNs [20], [21], [22], [23], [24], [25], where each user exchanges information with a pre-assigned user only, can be considered as a special case of MWRNs.
The users in a MWRN can adopt either pairwise transmission [1], [5], [9] or non-pairwise transmission [4], [6], [8], [26] strategy for message exchange. Though non-pairwise transmission can offer larger spectral efficiency, its benefits come at the expense of additional signal processing complexity at the relay [6]. Hence, in this paper, we focus on pairwise transmission strategy. Recently, pairwise transmission based MWRNs have been studied for different relaying protocols, e.g., functional decode and forward (FDF) [1], decode and forward [4], amplify and forward [5] and compute and forward [7] protocols. It was shown in [1] that pairwise FDF with binary linear codes for MWRN, where the relay decodes a function of the users’ messages rather than the individual messages from a user pair, is theoretically the optimal strategy since it achieves the common rate. Also it was shown in [2] that for a MWRN with lattice codes in an Additive White Gaussian Noise (AWGN) channel, the pairwise FDF achieves the common rate. Hence, in this paper, we consider FDF MWRN.
In a pairwise transmission based FDF MWRN, user pair formation is a critical issue. In this regard, two different pairing schemes have been proposed in the literature. In the pairing scheme in [1], two consecutive users are paired with each other (i.e., user 1 with user 2, user 2 with user 3, user with user , etc. where is the number of users in the MWRN). Thus, in general, the th and the th users form a pair at the th time slot, where . In the pairing scheme in [9], instead of consecutive users as in the pairing scheme in [1], two users in a pair are chosen from the two ends of a sequence such that the second user in one pair becomes the first user in the next pair (i.e., the pairs would be , , , etc.). Thus, in general, the th and the th user form a pair at the th time slot when and the th and th user form a pair at the th time slot when , where denotes the floor operation. The achievable rates for these two existing pairing schemes were analyzed in [1], [2], [9], while the average bit error rate (BER) for the first pairing scheme was analyzed in [27].
A major drawback of the pairing scheme in [1] is that they do not take the users’ channel information into account when pairing the users. In [9], the authors have considered only one case of asymmetric channel conditions, which is . However, they have not utilized the channel gain information for intelligent choice of the user pairs. Rather, both in [1] and [9], the users with good and bad channel conditions transmit the same amount of time. The only difference between the pairing schemes in [1] and [9] is the ordering of the user pair. Thus, if the performance metrics of [1] and [9] are being averaged over a number of settings for average channel gain, [1] and [9] would give the same results. Though in [9], the authors have shown that their pairing scheme is optimal in terms of the common rate for DF protocol and the considered channel conditions, the pairing schemes in [1] and [9], are not optimal in terms of the sum rate and error performance. This is because both in [1] and [9], the users with good and bad channel conditions transmit the same amount of time and the overall throughput will be less than that if the user with good channel conditions are allowed to transmit more times. Moreover, in a MWRN, the decision about each user depends on the decisions about all other users transmitting before it. Thus, in the above pairing schemes, if any user experiences poor channel conditions, it can lead to incorrect detection of another user’s message, which can adversely impact the system performance due to error propagation. We also note that a recent paper on opportunistic pairing [11] also suffers from the error propagation problem similar to [1].
In this paper, we propose a novel pairing scheme for user pair formation in a FDF MWRN. In this scheme, each user is paired with a common user, which is chosen by the relay as the user with the best average channel gain. This allows the user with the best channel conditions to contribute to improving the overall system performance by reducing the error propagation in the network. In our prior work in [28], we considered a pairing scheme to reduce error propagation in an amplify and forward (AF) MWRN. However, we considered simple binary phase shift keying (BPSK) modulation in [28], which is not suitable for practical high data rate systems. Also, our prior work in [28] reduces error propagation for a specific channel gain scenario but may not be optimal in terms of the common rate and sum rate. These major limitations of our prior work have motivated us to generalize and extend the prior work. The major contributions of this paper are as follows:
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Considering an -user FDF MWRN employing sufficiently large dimension lattice codes, we derive upper bounds for the common rate and sum rate with the proposed pairing scheme (cf. Theorems 1–2).
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Considering an -user FDF MWRN with -ary quadrature amplitude modulation (QAM) based transmission, which is a special case of lattice code based transmission, we derive the asymptotic average SER with the proposed pairing scheme (cf. Theorem 3).
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We present important insights, obtained from a careful analysis of the results in Theorem 1, Theorem 2, Theorem 3, in the form of Propositions 1–9. Analyzing the results in Theorem 1, Theorem 2, Theorem 3, we compare the performance of the proposed pairing scheme with the existing pairing schemes and show that:
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For the equal average channel gain scenario, the average common rate and the average sum rate are the same for the proposed and existing pairing schemes, but the average SER improves with the proposed pairing scheme (cf. Propositions 1, Proposition 4, Proposition 7).
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For the unequal average channel gain scenario, the average common rate, the average sum rate and the average SER all improve for the proposed pairing scheme (cf. Propositions 2, Proposition 5, Proposition 8).
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For the variable average channel gain scenario, the average common rate for the proposed pairing scheme is practically the same as the existing schemes, whereas, the average sum rate and the average SER improve for the proposed pairing scheme (cf. Propositions 3, Proposition 6, Proposition 9).
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The rest of the paper is organized as follows. The generalized system model is presented in Section 2. The proposed pairing scheme is discussed in Section 3 and the general lattice code based transmissions with the proposed pairing scheme are presented in Section 4. The common rate and the sum rate for a FDF MWRN with the proposed scheme is derived in Section 5. The average SER is derived in Section 6. The numerical and simulation results for verification of the analytical solutions are provided in Section 7. Finally, conclusions are provided in Section 8.
Throughout this paper, we use the following notations: denotes the estimate of a message, denotes that the message is estimated for the second time, denotes absolute value of a complex variable, denotes Euclidean norm, denotes the argument, denotes the maximum value, denotes the minimum value, denotes the expected value with respect to random channel coefficients, denotes the floor operation, denotes logarithm to the base two and is the Gaussian Q-function.
Section snippets
Generalized system model
We consider a generalized -user MWRN, where all the users exchange their information with each other through a single relay. In this setup, a pair of users communicates with each other at a time, while, the remaining users are silent. We assume that the users transmit in a half-duplex manner and they do not have any direct link in between them. The information exchange takes place in two phases–multiple access and broadcast phase–each comprising time slots for an -user MWRN [1]. In the
Proposed pairing scheme for MWRN
In this section, we propose a new pairing scheme for user pair formation in the multiple access phase. Before that we briefly discuss the pairing processes of the two existing pairing schemes in the following subsection.
Signal transmissions with the proposed pairing scheme
In this section, we discuss the general lattice code based transmissions with the proposed pairing scheme in a MWRN. The signal transmission protocols presented in the following section are directly applicable to equal and variable average channel gain scenarios. For the unequal average channel gain scenario, they are applicable with replaced by . We denote the th user as the common user and the th user as the other users, where, and . For the rest of this paper, we
Common rate and sum rate analysis
In this section, we investigate common rate and sum rate of the MWRN with the proposed pairing scheme. We first analyze the SNR of each user pair in a MWRN and use these results to obtain expressions for the achievable rates. For the rest of this paper, we simplify the notations by omitting the time slot superscript .
Error performance analysis
In this section, we characterize the error performance of a FDF MWRN with the new pairing scheme. We provide the analytical derivations for -QAM modulation, which is a 2 dimensional lattice code and is widely used in practical wireless communication systems.
Results
In this section, we provide numerical results to verify the insights provided in Propositions 1–6. We also provide simulation results to verify Propositions 7–9. We consider an user FDF MWRN where each user transmits a packet of bits and uses 16-QAM modulation. The power at the users, and the power at the relay, are assumed to be equal and normalized to unity. The SNR per bit per user is defined as . Following [6], the channel variance for the th user is modeled by
Conclusions
In this paper, we have proposed a novel user pairing scheme in a FDF MWRN. We have derived the upper bound on the average common rate (Theorem 1) and the average sum rate (Theorem 2) and the asymptotic average SER (Theorem 3) for the proposed pairing scheme. We have analyzed the results in Theorems 1–3 to compare the performance of the proposed scheme with existing pairing schemes under different channel scenarios. The main insights are summarized in Propositions 1–9. Our analysis shows that
Shama Naz Islam received the B.Sc. (1st class honours) degree in Electrical and Electronic Engineering from Bangladesh University of Engineering & Technology (BUET), Dhaka, Bangladesh in October, 2009. From 2010 to 2011, she had been a lecturer in the department of Electrical and Electronic Engineering in Bangladesh University of Engineering and Technology. She is currently studying towards Ph.D. in Research School of Engineering, The Australian National University, Canberra, Australia. In
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Shama Naz Islam received the B.Sc. (1st class honours) degree in Electrical and Electronic Engineering from Bangladesh University of Engineering & Technology (BUET), Dhaka, Bangladesh in October, 2009. From 2010 to 2011, she had been a lecturer in the department of Electrical and Electronic Engineering in Bangladesh University of Engineering and Technology. She is currently studying towards Ph.D. in Research School of Engineering, The Australian National University, Canberra, Australia. In 2012, she received the best student paper award in Women in Engineering category in IEEE Australia Council student paper contest. She is an associate fellow of the Higher Education Academy, UK. Her research interests are mainly in the areas of cooperative communication, multi-way relay network, wireless network coding, LTE network and information theory.
Salman Durrani (S’00–M’05–SM’10) received the B.Sc. (1st class honours) degree in Electrical Engineering from the University of Engineering & Technology, Lahore, Pakistan in 2000. He received the Ph.D. degree in Electrical Engineering from the University of Queensland, Brisbane, Australia in December 2004. He has been with the Australian National University, Canberra, Australia, since 2005, where he is currently Senior Lecturer in the Research School of Engineering, ANU College of Engineering & Computer Science. His current research interests are in wireless communications and signal processing, including synchronization in communication systems, outage and connectivity of wireless energy harvesting systems and ad-hoc networks and signal processing on the unit sphere. He has co-authored more than 70 publications to date in refereed international journals and conferences. He is a Member of Engineers Australia and a Senior Fellow of The Higher Education Academy, UK.
Parastoo Sadeghi (S02-M06-SM07) received the B.E. and M.E. degrees in Electrical Engineering from Sharif University of Technology, Tehran, Iran, in 1995 and 1997, respectively, and the Ph.D. degree in Electrical Engineering from The University of New South Wales, Sydney, Australia, in 2006. From 1997 to 2002, she worked as a Research Engineer and then as a Senior Research Engineer at Iran Communication Industries (ICI) in Tehran, Iran and at Deqx (formerly known as Clarity Eq) in Sydney, Australia. She is currently a Fellow at the Research School of Engineering, The Australian National University, Canberra, Australia. She has visited various research institutes, including the Institute for Communications Engineering, Technical University of Munich in 2008 and MIT in 2009 and 2013. She has co-authored more than 100 refereed journal or conference papers and a book on Hilbert Space Methods in Signal Processing, published by Cambridge University Press in 2013. She is a Chief Investigator in a number of Australian Research Council Discovery and Linkage Projects. In 2003 and 2005, she received two IEEE Region 10 student paper awards for her research in the information theory of time-varying fading channels. Her research interests are mainly in the areas of network coding, wireless communications systems and signal processing.