On the achievable rates of symmetric Gaussian multi-way relay channels
© Noori and Ardakani; licensee Springer. 2013
Received: 28 September 2012
Accepted: 3 December 2012
Published: 22 January 2013
Considering a symmetric Gaussian multi-way relay channel (MWRC) with K users, this work compares two transmission strategies, namely one-way relaying (OWR) and multi-way 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, lattice-based relaying ensures a gap less than bit from the capacity upper bound while MWR based on decode-and-forward (DF) or amplify-and-forward (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.
As an extension of two-way relay channels (TWRCs)[1–3], multi-way relay channels (MWRCs) have been introduced by Gunduz et al. 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 device-to-device communications.
Different relaying schemes are applicable to MWRCs. One approach is to divide the data transmission time into several one-way relaying (OWR) phases. Conventional relaying strategies, i.e. amplify-and-forward (AF) and decode-and-forward (DF), can be accommodated by OWR. A more recent approach is to employ multi-way relaying (MWR) where several users are allowed to simultaneously transmit to the relay. For MWR, several schemes have been proposed including AF, DF, and compress-and-forward (CF). Further, an MWR approach based on lattice codes has been proposed[8–10] which is called functional-decode-forward (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, 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 while FDF achieves the capacity for binary MWRCs. Ong et al. 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 base-station (relay) resulting in a higher achievable rates in the system.
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.
Consider an MWRC where K ≥ 2 users want to share their data without having direct user-to-user 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 u1,u2,…,u K and their data by X1,X2,…,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 zero-mean Gaussian noise with unit variance. Due to considering AWGN channel, we refer to this MWRC by Gaussian MWRC.
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.
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, for MWR with FDF, the uplink transmission is divided into K−1 multiple-access (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. This enables to directly decode the modulo-sum 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 pair-wise transmission continues for K − 1 times until uK−1and 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[9–11].
Please see. □
The following theorem states the performance of FDF in comparison with the capacity upper bound.
The gap between the achievable rate of FDF and the capacity of a K-user symmetric Gaussian MWRC is less than bit.
See Appendix. □
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.
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.
The gap between and is less than bit if either or (K − 1)P < P r and.
Please see Appendix. □
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. 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.
is the maximum common rate that AF can achieve.
Now, the following theorem is presented on the capacity gap of AF.
The gap between and is less than if P r ≤ (K − 1)P and or (K − 1)P < P r and K(K − 1)P2 − P − 1 < P r + (K − 1)P P r .
Please see Appendix. □
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.
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.
In a symmetric Gaussian MWRC with OWR, DF always outperforms AF in terms of the achievable rate.
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
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.
5.2 Comparison of DF OWR and CF MWR
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. Figures4 and5 depict the comparison between the achievable rate of DF OWR and CF MWR.
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.
Before presenting proofs, we state the following propositions based on Lemma 1, 2 and 3.
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.
As a consequence,.
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.
Considering that, it is easy to show that (27) does not necessarily hold for all values of P r and P in this region.
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 .
Now, from (29), one can show that if
Using (30), it is easy to conclude that if K(K − 1)P2 − (K − 1)P P r < 1 + P r + P then AF has a capacity gap smaller than.
Proof of Theorem 4
is always correct. As a consequence, for this SNR region still holds.
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.
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