Abstract
Considering a symmetric Gaussian multiway relay channel (MWRC) with K users, this work compares two transmission strategies, namely oneway relaying (OWR) and multiway relaying (MWR), in terms of their achievable rates. While in OWR, only one user acts as data source at each time and transmits in the uplink channel access, users can make simultaneous transmissions in MWR. First, we prove that for MWR, latticebased relaying ensures a gap less than bit from the capacity upper bound while MWR based on decodeandforward (DF) or amplifyandforward (AF) is unable to guarantees this rate gap. For DF and AF, we identify situations where they also have a rate gap less than bit. Later, we show that although MWR has higher relaying complexity, surprisingly, it can be outperformed by OWR depending on K and the system SNR. Summarily speaking, for large K and small users’ transmit power, OWR usually provides higher rates than MWR.
Keywords:
Multiway relay channel; Multiway relaying; Oneway relaying; Capacity gap; Achievable rate1 Introduction
As an extension of twoway relay channels (TWRCs) [13], multiway relay channels (MWRCs) have been introduced by Gunduz et al. [4] to improve the spectral efficiency in multicast communication networks [5,6]. In an MWRC, several users want to (fully) share their information with the help of one or more relays. Some practical examples of MWRCs are conference calls in a cellular network, file sharing between several wireless devices, and devicetodevice communications.
Different relaying schemes are applicable to MWRCs. One approach is to divide the data transmission time into several oneway relaying (OWR) phases. Conventional relaying strategies, i.e. amplifyandforward (AF) and decodeandforward (DF), can be accommodated by OWR. A more recent approach is to employ multiway relaying (MWR) where several users are allowed to simultaneously transmit to the relay. For MWR, several schemes have been proposed including AF, DF, and compressandforward (CF) [7]. Further, an MWR approach based on lattice codes has been proposed [810] which is called functionaldecodeforward (FDF). In the following, we generally use OWR and MWR to refer to the discussed relaying schemes for MWRCs. Note that MWR generally has a higher relaying complexity than OWR.
There exist several works studying the performance of MWRCs in terms of their achievable rate. In [4], it is shown that MWR with CF can achieve to within bit of the common rate capacity where K is the number of users. Also, for TWRCs with FDF, it is shown that the capacity gap is less than bit [11] while FDF achieves the capacity for binary MWRCs [9]. Ong et al. [10] show that under some conditions, FDF achieves the common rate capacity of Gaussian MWRCs. Furthermore, they briefly discuss the capacity gap of FDF when all users and the relay have equal power.
In this work, a detailed performance comparison between MWR and OWR is provided. More specifically, we focus on the common rate of these relaying schemes over symmetric Gaussian MWRCs. The Gaussian symmetric model can be practically associated with situations where dynamic power adjustment mechanism at the users is applied to compensate for the slow fading effect. For instance, in a cellular CDMA system, dynamic power adjustment is used to equalize the received power of users at the basestation (relay) resulting in a higher achievable rates in the system [12].
For MWR, we prove that similar to CF, FDF assures a gap less than bit with the common rate capacity of symmetric Gaussian MWRCs. For AF and DF, we first show that they may have a larger than bit capacity gap and then we find the SNR regions where a gap smaller than bit is guaranteed. In the next step, we study the achievable rate of MWRCs using OWR. For this purpose, we consider OWR with AF and DF and show that for the considered MWRC setup, DF always outperforms AF when OWR is used. Then, the achievable rate of MWR with CF and FDF (guaranteeing a less than bit gap) is compared with the rate of DF OWR. Surprisingly, in spite of a higher relaying complexity, MWR is not always superior to OWR and we find the SNR regions where OWR indeed outperforms MWR. According to our study, by decreasing SNR or increasing K, we may see OWR surpassing MWR.
The article is organized as follows: Section 2 provides the system model and some definitions. The capacity gap analysis for MWR is discussed in Sections 3 and 4 focuses on the rate study for OWR. Rate comparison between MWR and OWR is presented in Sections 5 and 6 concludes the article. Further, all proofs are provided in Appendix.
2 Preliminaries
Consider an MWRC where K ≥ 2 users want to share their data without having direct usertouser links. It means that each user aims to receive all other users data as well as to transmit its data to all other users. We name users by u_{1},u_{2},…,u_{K} and their data by X_{1},X_{2},…,X_{K}. Each user has a limited average power P, thus, for all i, . To enable data communication between users, a relay, , with average transmit power P_{r }is employed.
Data communication consists of uplink and downlink phases. In the uplink phase, users transmit their data to while in the downlink phase broadcasts its message. We assume that the received signals at the relay and users are contaminated by a zeromean Gaussian noise with unit variance. Due to considering AWGN channel, we refer to this MWRC by Gaussian MWRC.
In this article, we consider the common rate capacity of Gaussian MWRCs. The common rate capacity is the maximum data rate at which all users can reliably transmit and receive data. In other words, if we denote the achievable data rates at all user by a Ktuple (R_{1},R_{2},…,R_{K}), where R_{i} is achievable at u_{i}, then
For more details on common rate definition and its applications in MWRCs, the reader is referred to [4,10]. Note that for a general Gaussian MWRC, the common rate capacity is yet to be known. Thus, in the following, we use the capacity upper bound for our capacity gap analysis instead of the capacity itself. For this purpose, we borrow the following lemma from [4].
Lemma 1
An upper bound on the common rate capacity of a symmetric Gaussian MWRC is
Please notice that in this article, log(·) represents the logarithm in base 2.
3 Rate analysis for MWR
Here, we focus on the achievable rate of MWR and study the capacity gap for FDF, DF and AF. We prove that similar to CF, FDF guarantees a capacity gap less than bit.
3.1 Capacity gap of FDF
As suggested in [10], for MWR with FDF, the uplink transmission is divided into K−1 multipleaccess (MAC) slots. In each MAC slot, a pair of users transmit their data to the relay. Each user encodes its data using nested lattice codes [13]. This enables to directly decode the modulosum of the users data, instead of decoding their data separately, after receiving the superimposed users’ signals. Then, encodes the sum value and broadcasts it to the users. This pairwise transmission continues for K − 1 times until u_{K−1 }and u_{K }transmit their data. Now, in addition to own data, each user has received K − 1 independent linear combinations of other users data. By forming a system of linear equations, consisting of these K − 1 independent equations, each user can find the data of any other user. For more information on this pairwise transmission strategy, the interested reader is referred to [911].
The achievable rate of latticebased relaying was first studied in [8] for TWRC. Later, the following lemma was proposed [10] for the achievable common rate of FDF.
Lemma 2
The maximum achievable rate of FDF over a symmetric Gaussian MWRC is
Proof
Please see [10]. □
The following theorem states the performance of FDF in comparison with the capacity upper bound.
Theorem 1
The gap between the achievable rate of FDF and the capacity of a Kuser symmetric Gaussian MWRC is less than bit.
Proof
See Appendix. □
For numerical illustrations, the achievable rate of FDF and the capacity upper bound for several cases are depicted in Figures 1, 2 and 3. In Figure 1, users’ SNR effect on the capacity gap is studied while the effect of the relay SNR and K are presented in Figures 2 and 3, respectively. As seen, the achievable rate of FDF always sits above the bit gap. Further, when downlink limits the rate, FDF achieves the capacity.
Figure 1. Achievable rates of relaying schemes when K = 3 and P_{r }= 15 dB.
Figure 2. Achievable rates of relaying schemes when K = 3 and P = 10 dB.
Figure 3. Achievable rates of relaying schemes when P = 10 dB and P_{r }= 15 dB.
3.2 Capacity gap of DF
For DF MWR, all users share the same uplink transmission time and simultaneously send their data to the relay. Then, decodes the data of all users and broadcasts them over the downlink as described in [4].
Lemma 3
The maximum achievable common rate of DF MWR is
Proof
See [4]. □
Our analysis reveals that depending on SNR and K, DF may not be able to guarantee a bit gap to . The following theorem summarizes the result.
Theorem 2
The gap between and is less than bit if either or (K − 1)P < P_{r }and .
Proof
Please see Appendix. □
As the numerical results in Figures 1, 2 and 3 indicate, in some SNR regions and depending on the number of users, the capacity gap might be larger than bit for DF.
3.3 Capacity gap of AF
When AF is used for MWR, similar to DF, all users simultaneously transmit their data to the relay. Unlike DF, however, only amplifies the received signal, while meeting the relay power constraint, and transmits it back to the users [4]. Then, each user cancels out its own signal from the broadcast signal and decodes the other users data. In this case, it is easy to prove the following lemma for the achievable rate of AF.
Lemma 4
In a Kuser symmetric Gaussian MWRC,
is the maximum common rate that AF can achieve.
Now, the following theorem is presented on the capacity gap of AF.
Theorem 3
The gap between and is less than if P_{r }≤ (K − 1)P and or (K − 1)P < P_{r }and K(K − 1)P^{2 }− P − 1 < P_{r} + (K − 1)PP_{r}.
Proof
Please see Appendix. □
Depending on the SNR and K, the achievable rate of AF may fall under the bit gap from the capacity upper bound (Figures 1, 2 and 3).
4 Rate analysis for OWR
In this section, we study the achievable rate of OWR. In a MWRC with OWR, transmission time in both uplink and downlink phases is divided into K slots. In each slot, one user serves as the source and the rest are the data destinations. First, the source user transmits in the uplink slot and then broadcasts the data back to the users in the downlink slot. Since each user transmits in only one uplink slot and stays silent in the rest, it can upscale its power to KP during its transmission turn without violating the power constraint.
When DF is employed for OWR, first decodes the received data from the source in the uplink and then broadcasts it to the users. Then, destination users decode the received signal from the relay. It is easy to show that the achievable rate of DF OWR is
For AF, amplifies and forwards the received signal in the uplink without further processing. The decoding is done at the destination users. The achievable rate of this scheme is
It can be shown that OW (with DF or AF) does not guarantee a bit gap.
Now, we like to compare the performance of AF and DF for OW. Using the achievable rates in (6) and (7), we can derive the following theorem.
Theorem 4
In a symmetric Gaussian MWRC with OWR, DF always outperforms AF in terms of the achievable rate.
Proof
See Appendix. □
5 Comparison between the rate of OWR and MWR
In this section, we compare the performance of OWR and MWR. For OWR, we consider DF which has the superior performance over AF. Also, FDF and CF are considered for MWR since they provide a guaranteed rate performance (capacity gap).
5.1 Comparison of DF OWR and FDF MWR
In this region, it is clear that MWR outperforms OWR due to its smaller prelog factor. However, increasing K decreases the gap between MWR and OWR. Consider the second SNR region where and
In this SNR region, FDF MWR surpasses DF OWR if
Since the right hand side of (10) is an increasing function of P, it can be concluded that for a fixed P_{r}, decreasing P reduces the chance of holding the inequality (10). It means that when the relay’s received SNR decreases, OWR may start performing better than MWR.
Now, we consider a third region where KP ≤ P_{r}. Here,
Thus, MWR performs better if
From (12) and noticing that is a decreasing function and , it can be concluded that decreasing P or increasing K (without violating KP ≤ P_{r}) is in favor of OWR. Numerical results for the comparison between the achievable rate of DF OWR and FDF MWR are presented in Figure 4 and 5. As seen, when K = 2, for small P (low receive SNR at the relay), OWR performs close to MWR and even outperforms FDF. Increasing SNR causes the gap between OWR and MWR to largen. By setting K = 8, we see that for a significant SNR region OWR surpasses FDF.
Figure 4. Comparison between the achievable rates of OWR and MWR when P_{r }= 15 dB and K = 2.
Figure 5. Comparison between the achievable rates of OWR and MWR when P_{r }= 15 dB and K = 8.
5.2 Comparison of DF OWR and CF MWR
To compare the performance of DF OWR and CF MWR, we use two SNR regions. First, assume P_{r }< KP. Thus,
From (13), we can conclude that MWR outperforms OWR in this SNR region when
In (14), if P_{r }≥ 1, using the derivative of the right hand side of (14), it can be shown that when P decreases, MWR may lose its advantage over OWR. Now, we consider the second SNR region where KP ≤ P_{r}. Thus
MWR with CF performs better than DF OWR if
It can be concluded that for low SNRs, (16) does not hold and OWR outperforms MWR. Further, the left side of (16) is an increasing function of K. Thus, by increasing K, we may start seeing higher rates from OWR than MWR. Figures 4 and 5 depict the comparison between the achievable rate of DF OWR and CF MWR.
6 Conclusion
In this article, we compared the performance of OWR and MWR in a symmetric Gaussian MWRC where several users want to share their data through a relay. To this end, we first proved that FDF always have a capacity gap less than bit while depending on the users’ and relay SNR, AF and DF may have a capacity gap larger than . Furthermore, for OWR, we showed that DF is always superior to AF. By comparing the achievable rate of DF OWR with CF and FDF MWR, we concluded that MWR is likely to outperform OWR in high SNR regions or for small K. Conducting a similar study for asymmetric Gaussian channels is considered as a direction for future research.
Appendix
Before presenting proofs, we state the following propositions based on Lemma 1, 2 and 3.
Proposition 1
If P_{r }≤ (K − 1)P, i.e. downlink is the rate bottleneck, we have
Otherwise
Proposition 2
In a Gaussian MWRC with FDF MWR, if , then downlink is the bottleneck resulting in
Proposition 3
When , downlink constrains the rate of DF MWR and
Further, when , uplink is the rate bottleneck and
Proof of Theorem 1
We start the proof by partitioning the range of P_{r }and P using Proposition 1 and 2. Then, the achievable rate of FDF and the rate upper bound are compared in each region in order to complete the proof. The partitions specify which constraints in (2) and (3) are active. Since K ≥ 2, we have . To this end, the regions of interest are specified as , , and (K − 1) P < P_{r}. These partitions are denoted by , and , respectively.
Capacity gap on: The achievable rate of FDF as well as the upper bound is determined by downlink on this region. Using Proposition 1 and 2, we conclude that . In other words, FDF achieves the capacity upper bound and the gap between and , , is 0.
Capacity Gap on: For this region, the achievable rate of FDF is bounded by uplink while the rate upper bound is forced by downlink. Thus,
Since log(·) is an increasing function, the maximum of G_{U }happens when P_{r} has its maximum value on A_{2}. Since P_{r }< (K − 1)P, it is easy to show that
Capacity Gap on: Both and are limited by the uplink in this case. Thus, using Proposition 1 and 2
Now, it is inferred from (25) that .
Proof of Theorem 2
Similar to FDF, we partition the SNR region and study the capacity gap for DF over different partitions. First, we point out that . To this end, we define three SNR regions namely , , and denoting , , and (K − 1) P < P_{r}, respectively.
Capacity gap on: and are limited by downlink. Using propositions 1 and 3, it is concluded that .
Capacity gap on: For this partition
Now, the capacity gap is less than bit if
Considering that , it is easy to show that (27) does not necessarily hold for all values of P_{r} and P in this region.
Capacity gap on: Here, uplink is the rate bottleneck for the upper bound as well as DF. Thus,
and if which does not necessarily hold for all P and P_{r }values within .
Proof of Theorem 3
We again define SNR regions, called and based on Proposition 1. The first region is where P_{r }≤ (K − 1) P and the second region includes (K − 1)P < P_{r}.
Capacity Gap on: In this region, we have
Now, from (29), one can show that if
Capacity Gap on: On this partition,
Using (30), it is easy to conclude that if K(K − 1)P^{2 }− (K − 1)PP_{r }< 1 + P_{r} + P then AF has a capacity gap smaller than .
Proof of Theorem 4
First assume P_{r }< KP. Since
is always correct. As a consequence, for this SNR region still holds.
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Alberta Innovates Technology Futures (AITF) for supporting our research.
References

Y Wu, PA Chou, S Kung, Information exchange in wireless networks with network coding and physicallayer broadcast. Tech. rep., Microsoft Research

B Rankov, A Wittneben, Spectral efficient signaling for halfduplex relay channels. Asilomar Conference on Signals, Systems, and Computers (Pacific Grove, USA, 2005), pp. 1066–1071

P Popovski, H Yomo, The antipackets can increase the achievable throughput of a wireless multihop network. in IEEE Intl, ed. by . Conf. on Communications (ICC) (Istanbul, Turkey, 2006), pp. 3885–3890

D Gunduz, A Yener, A Goldsmith, H Poor, The multiway relay channel. in IEEE Intl, ed. by . Symp. on Inf. Theory (ISIT) (IEEE, Seoul, South Korea, 2009), pp. 339–343

A Amah, A Klein, Regenerative multigroup multiway relaying. IEEE Trans. Veh. Technol 60(7), 3017–3029 (2011)

C Hausl, O Iscan, F Rossetto, Resource allocation for asymmetric multiway relay communication over orthogonal channels. EURASIP J. Wirel. Commun. Netw 2012(20), 1–12 (2012)

T Cover, A Gamal, Capacity theorems for the relay channel. IEEE Trans. Inf. Theory 25(5), 572–584 (1979). Publisher Full Text

W Nam, SY Chung, Y Lee, Capacity bounds for twoway relay channels. IEEE International Zurich Seminar on Communications (Zurich, Switzerland, 2008), pp. 144–147

L Ong, S Johnson, C Kellett, An optimal coding strategy for the binary multiway relay channel. IEEE Commun. Lett 14(4), 330–332 (2010)

L Ong, C Kellett, S Johnson, Capacity theorems for the AWGN multiway relay channel. in IEEE Intl, ed. by . Symp. on Inf. Theory (ISIT) (USA, Austin, 2010), pp. 664–668

W Nam, SY Chung, Y Lee, Capacity of the Gaussian twoway relay channel to within 1/2 bit. IEEE Trans. Inf. Theory 56(11), 5488–5494 (2010)

S Ariyavisitakul, LF Chang, Signal and interference statistics of a CDMA system with feedback power control. IEEE Trans. Commun 41(11), 1626–1634 (1993). Publisher Full Text

U Erez, R Zamir, Achieving 1/2 log (1+SNR) on the AWGN channel with lattice encoding and decoding. IEEE Trans. Inf. Theory 50(10), 2293–2314 (2004). Publisher Full Text