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
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
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
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,
Data communication consists of uplink and downlink phases. In the uplink phase, users
transmit their data to
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
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
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
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
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,
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
Theorem 2
The gap between
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
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,
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
Proof
Please see Appendix. □
Depending on the SNR and K, the achievable rate of AF may fall under the
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
When DF is employed for OWR,
For AF,
It can be shown that OW (with DF or AF) does not guarantee a
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
First, assume
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
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
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
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
If
Proposition 3
When
Further, when
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
Capacity gap on
Capacity Gap on
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
As a consequence,
Capacity Gap on
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
Capacity gap on
Capacity gap on
Now, the capacity gap is less than
Considering that
Capacity gap on
and
Proof of Theorem 3
We again define SNR regions, called
Capacity Gap on
Now, from (29), one can show that
Capacity Gap on
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
holds, then
is always correct. As a consequence, for this SNR region
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.
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