 Research Article
 Open Access
Joint ChannelNetwork Coding for the Gaussian TwoWay TwoRelay Network
 Ping Hu^{1},
 ChiWan Sung^{1} and
 Kenneth W. Shum^{2}Email author
https://doi.org/10.1155/2010/708416
© Ping Hu et al. 2010
 Received: 1 October 2009
 Accepted: 13 March 2010
 Published: 31 May 2010
Abstract
New aspects arise when generalizing twoway relay network with one relay to twoway relay network with multiple relays. To study the essential features of the twoway multiplerelay network, we focus on the case of two relays in our work. The problem of how two terminals, equipped with multiple antennas, exchange messages with the help of two relays is studied. Five transmission strategies, namely, amplifyforward (AF), hybrid decode amplify forward (HLC), hybrid decode amplify forward (HMC), decode forward (DF), and partial decode forward (PDF), are proposed. Their designs are based on a variety of techniques including network coding, multiplexed coding, multiinput multioutput transmission, and multiple access with common information. Their performance is compared with the cutset outer bound. It is shown that there is no dominating strategy and the best strategy depends on the channel conditions. However, by studying their multiplexing gains at high signaltonoise (SNR) ratio, it is shown that the AF scheme dominates the others in high SNR regime.
Keywords
 Relay Node
 Network Code
 Transmission Strategy
 MIMO Channel
 Rate Pair
1. Introduction
Relay channel, which considers the communication between a source node and a destination with the help of a relay node, was introduced by van der Meulen in [1]. Based on this channel model, Cover and El Gamal developed coding strategies known as decodeforward (DF) and compressforward (CF) in [2]. These techniques now become standard building blocks for cooperative and relaying networks, which have been extensively studied in the literature (e.g., [3, 4]).
For many applications, communication is inherently twoway. A typical example is the telephone service. In fact, the study of twoway channel is not new and can be traced back to Shannon's work in 1961 [5]. However, the model of twoway relay channel, though natural, did not attract much attention. Recently, probably due to the advent of network coding [6] in the last decade, there is a growing interest in this model. The application of DF and CF to twoway relay channel was considered in [7]. The halfduplex case was studied in [8, 9]. The results in [10] showed that feedback is beneficial only in a twoway transmission. Network coding for the twoway relay channel was studied in [11, 12]. Physical layer network coding based on lattices is considered recently [13], and shown to be within 0.5 bit from the capacity in some special cases [14].
All the aforementioned works are for one relaying node. It is easy to envisage that in real systems, more than one relay can be used. Schein in [15] started the investigation of the network with one sourcedestination pair and two parallel relays in between. This model was further studied in [16] under the assumption of halfduplex relay operations. For oneway multiplerelay networks in general, cooperative strategies were proposed and studied in [17]. We remark that a notable feature that does not exist in the singlerelay case is that the multiple relays can act as a virtual antenna array so that beamforming gain can be reaped at the receiver. In this paper, we follow this line of research and consider twoway communications. Tworelays are assumed, for this simple model already captures the essential features of the more general multiplerelay case. We are interested in knowing how different techniques can be used to construct transmission strategies for the twoway tworelay network and how they perform under different channel conditions. In particular, we apply the idea of network coding to both the physical layer and the network layer. Besides, channel coding techniques for multiple access channel (MAC) and multiinput multioutput (MIMO) channel are also employed. Several transmission strategies are thus constructed and their achievable rate regions are derived.
We remark that the channel model that we consider in this paper is also called the restricted twoway tworelay channel [7]. This means that the signal from a source node depends only on the message to be transmitted, but not on the received signal at the source. Besides, our results are obtained under the halfduplex assumption, which is realistic for practical systems. Each node is assumed to transmit one half of the time and receive during the other half of the time. The performance of our proposed strategies can be further improved if the ratio of transmission time and receiving time is optimized. We do not consider this more general case, since it complicates the analysis but provides no new insights.
This paper is organized as follows. Our network model is described in Section 2. Some basic coding techniques are reviewed in Section 3. Based on these coding techniques, several transmission strategies are devised in Section 4. Their performance at high signaltonoise ratio regime is analyzed in Section 5. The rate regions of these strategies are compared under some typical channel realizations in Section 6. The conclusion is drawn in Section 7.
2. Channel Model and Notations
for , where , , , and denote the power constraints on terminals and and relays 1 and 2, respectively.
A rate pair is said to be achievable if there exists a sequence of codes, satisfying the power constraints in (5) and (6), with as .
for .
Notations
be the signal to noise ratio of the signal received at relay from terminal . Shannon's capacity formula is denoted by . Also, for matrices, we let , where denote the identity matrix. The reason for the factor of 0.25 before the log function, instead of a factor of 0.5 in the original capacity formula, is due to the fact that the total transmission time is divided into two stages of equal length. All logarithms in this paper are in base 2. The set of nonnegative real numbers is denoted by . Gaussian distribution with mean zero and covariance matrix is denoted by .
3. Review of Coding Techniques and Capacity Regions from Information Theory
The proposed transmission strategies are based on a host of existing coding techniques and capacity results. A review of them is given in this section.
3.1. PhysicalLayer Network Coding
where is an additive noise. For simplicity, it is assumed that both link gains from the sources to the relay are equal to one. In the second phase, the relay amplifies the received signal , and transmits a scaled version of , where is a scalar chosen so that the power requirement is met. Since source 1 knows , the component within the received signal at source 1 can be treated as known interference, and hence be subtracted. Similarly, source 2 can subtract from the received signal. Decoding is then based on the signal after interference subtraction.
3.2. Multiplexed Coding
Multiplexed coding [22] is a useful coding technique for multiuser scenarios in which some user knows the message of another user a priori. Consider the twoway relay channel as in the previous paragraph. Node 1 wants to send message to node 2 via the relay node, and node 2 wants to send message to node 1 via the relay node. For , let be the number of bits used to represent message . The transmission of the nodes is divided into two phases. In the first phase, the two source nodes transmit. Suppose that the relay node is able to decode and . For the encoder at the relay, we generate a array of codewords. Each codeword is independently drawn according to the Gaussian distribution such that the total power of each codeword is less than or equal to . In the second phase, the relay node sends the codeword in the entry in this array. Suppose that the received signal at source node is corrupted by additive white Gaussian noise with variance , for . At source 1, since is known, the decoder knows that one of the codewords in the row corresponding to had been transmitted. Out of these codewords, it then declares the one based on the maximal likelihood criterion. By the channel coding theorem for the pointtopoint Gaussian channel, source 1 can decode reliably at a rate of . Likewise, by considering the columns in the array of codewords, source 2 can decode at a rate of .
Multiplexed coding can be implemented using concepts from network coding. We assume, without loss of generality, that . We identify the possible messages from source node with the vectors in the dimensional vector space over the finite field of size 2, , and identify the messages from source node with a subspace of of dimension , say . We generate Gaussian codewords independently, one for each vector in . To send messages and in the second phase, the relay node transmits the codeword corresponding to , where the addition is performed using arithmetics in . The output of the decoder at node 1 is a vector in . We subtract from it the vector in corresponding to . If there is no decoding error, this gives the codeword corresponding to , and the value of is recovered.
Now let us consider node 2. Since is known a priori, node 2 is certain that the signal transmitted from the relay is associated with one of the vectors in the affine space . The message can be estimated by comparing the likelihood function of the codewords associated with . It can be seen that the maximal data rate is the same as in the array approach mentioned in the previous paragraph, but the size of the codebook at the relay reduces from to .
3.3. Capacity Region for MIMO Channel
3.4. Capacity Region for MultipleAccess Channel (MAC)
We refer the reader to [24] for more details on the optimal coding scheme for MAC.
4. ChannelNetwork Coding Strategies
We develop five transmission schemes for TWTR network. In the first scheme (AF), the received signals at both relay nodes are amplified and forwarded back to terminals and . In the second and third scheme (HLC, HMC), one of the relays employs the amplify forward strategy, while the other decodes the messages from terminals and . In the fourth scheme (DF), both relays decode the messages from terminals and . In the last strategy (PDF), another mixture of decodeforward and amplifyforward strategy is described.
4.1. Amplify Forward (AF)
is then transmitted in the second stage. At the end of the second stage, each terminal, who has the information of itself, subtracts the corresponding term and obtains the desired message from the residual signal.
The message from terminal can then be decoded using a decoding algorithm for pointtopoint MIMO channel. The received signal at terminal is treated similarly.
Theorem 1.
are satisfied.
Proof.
The residual signal (27) at terminal can be written as plus a noise vector with covariance matrix . The residual signal at terminal equals plus a noise vector with covariance matrix . Therefore, after selfsignal subtraction, the resultant channels can be considered MIMO channels with two transmit antennas and receive antennas. From (19), we obtain the rate constraints in (28). The inequalities in (30) are the power constraints for terminals and , and those in (31) are the power constraints for relays 1 and 2.
4.2. Hybrid DecodeAmplify Forward with Linear Combination (HLC)
In this strategy, relay 1 decodes the messages from terminals and , and meanwhile, relay 2 employs the amplifyforward strategy. In order to obtain beamforming gain, after decoding the two messages, relay 1 reconstructs the codewords corresponding to the decoded messages and sends a linear combination of them in the second stage.
for some and . Relay 2 amplifies by a scalar factor and transmits .
The decoding is done by using decoding method for MIMO channel.
Theorem 2.
are satisfied.
In (35), the product of a real number and a set is defined as .
Proof.
From the rate constraints for MAC channel in (22)–(24), we have the rate constraints for relay 1 in (35). We multiply by a factor of one half because the first phase only occupies half of the total transmission time.
The conditions in (36) and (37) are derived from the capacity formula for MIMO channel with colored noise in (19). The inequalities in (39) are the power constraints for sources and . The inequalities in (40) and (41) are the power constraints for relays 1 and 2, respectively.
The parameters , , , and can be obtained by running an optimization algorithm. For example, we can aim at maximizing a weighted sum . The values of , , and which maximize the weighted sum are chosen.
4.3. Hybrid DecodeAmplify Forward with Multiplexed Coding (HMC)
As in the previous strategy, relay 1 decodes and forwards the messages from and , and relay 2 amplifies and transmits the received signal. However, in this strategy, relay 1 reencodes the messages into a new codeword to be sent out in the second stage. Terminals and decode the desired messages based on multiplexed coding.
Theorem 3.
, are covariance matrices satisfying (39), and satisfies (41).
Proof.
The proof is by random coding argument and we will sketch the proof below. More details can be found in [25].
Our objective is to show that any rate pair that satisfies the condition in the theorem is achievable. For , terminal randomly generates a Gaussian codebook with codewords with length , satisfying the power constraint in (5). Label the codewords by , for . For relay 1, we generate a array of Gaussian codewords of length and power . The codeword in row and column is denoted by , and satisfies the power constraint in (6).
After the first stage, relay 1 is required to decode both messages from terminals and . This can be accomplished with arbitrarily small probability of error if the rate constraints for MAC in (22) to (24) are satisfied. This corresponds to the rate constraint in (42). Let the estimated messages from and be and .
In the second stage, relay 1 transmits . Relay 2 amplifies its received signal and transmits . From (41), the amplified signal has average power no more than .
Note that terminal A knows its message , and with probability arbitrarily close to one if (42) is satisfied. The idea of multiplexed coding can then be used. In (48), the covariance matrix of the signal in square bracket is given by in (46), and the covariance of the noise term is given by . Applying the capacity expression, we obtain the rate constraint in (44). In a similar manner, we obtain (43).
4.4. Decode Forward (DF)
In the DF strategy, terminal node , ( ) splits the message into two parts: the common part and the private part . The two common messages are transmitted via both relay nodes. The private message is decoded by relay 1 only, and can be interpreted as going through the path from terminal to relay 1 to terminal . Symmetrically, the private part of message is decoded by relay 2 only, and can be interpreted as going through the path from terminal to relay 2 to terminal . After the first stage, relay 1 decodes the common messages of both terminals and the private message of terminal . Relay 2 decodes the common messages of both terminals and the private message of terminal . The encoding and decoding schemes in the first stage is similar to those developed by Han and Kobayashi for the interference channel (IC) in [26]. Since both relays have access to the common messages, the channel in the second stage can be considered a multiple access channel with common information. Furthermore, since terminals and have information of themselves, we can further improve the rate region by the idea of multiplexed coding.
We have the following characterization of the rate region for the DF strategy:
Theorem 4.
for some nonnegative and .
Details of the DF coding scheme and the proof of Theorem 4 are given in the Appendix.
4.5. Partial Decode Forward (PDF)
In the PDF strategy, both relays decode the message of terminal . Each relay then subtracts the reconstructed signal of terminal from the received signal. Call the resulting signal the residual signal. The message of terminal is reencoded into a new codeword, and linearly combined with the residual signal. This linear combination is then transmitted in the second stage. Since both relays know the message of terminal , the tworelays can jointly reencode the message of terminal using some encoding scheme for a MIMO channel with two transmit antennas and receive antennas.
Theorem 5.
for . (Here, denotes the th diagonal entry in .)
Proof.
The tworelays treat the signal originated from terminal as noise, and decode the message of terminal . The rate requirement in (58) guarantees that the message of terminal can be decoded with arbitrarily small probability of error at both relays. Let the decoded message of terminal be denoted by .
For , the reconstructed signal is then subtracted from . The residual signal at relay is .
for some amplifying factor . The inequality in (63) ensures that the power constraint is satisfied at the relays.
From the capacity formula for MIMO channel (19), terminal can recover the message from terminal reliably if (60) is satisfied.
This is equivalent to a MIMO channel with link gain matrix and colored noise. Recall that is the covariance matrix of the encoded signal. By the capacity formula of MIMO channel (19), we obtain the rate constraint in (59).
Remark 1.
We note that the matrices , , and , for , are invertible. Indeed, by checking that is strictly positive for all nonzero , we see that the matrix is positive definite, and hence invertible.
5. Performance in High SNR Regime
In this section, we compare the performance of the five strategies described in the previous section in the high SignaltoNoise Ratio (SNR) regime.
as the performance measure at high SNR. At high SNR, that is, when is very small, we can approximate the sum rate by if the multiplexing gain is equal to .
We first suppose that the covariance matrices and , and the amplifying constants and , are fixed. Note that if the power constraint in (31) holds, then it continues to hold if becomes smaller. Therefore, when , the power constraints in (30) and (31) are satisfied.
depends only on the rank of the matrix , and equals 0, 0.5, or 1, if the rank of is 0, 1, or 2, respectively. The problem of determining the multiplexing gain now reduces to determining the rank of the matrices in (71) and (72).
Similarly, we can show that the rank of the matrix in (71) is equal to two.
Since the above argument holds for all invertible and , and positive and , we conclude that the multiplexing gain of the AF strategy is equal to 2.
Similarly, the multiplexing gain of DF is also limited by the decoding of messages at the relays. The rate constraints (50) and (51) imply that it is no more than 0.5.
provided that the has full rank. Therefore, its maximal multiplexing gain is 1.5.
Multiplexing gains of the transmission schemes in the high SNR regime.
Scheme  AF  HMC, HLC, DF  

Multiplexing gain  2  0.5  1.5 
6. Numerical Examples
Theorem 6 (Outer bound).
for some real number between 0 and 1, and covariance matrices and such that holds for .
In Figure 3, we plot the rate regions when all link gains are large (the link gain is 10 for all links). As mentioned in the previous section, the AF strategy has the largest multiplexing gain in the high SNR regime. We can see in Figure 3 that the AF strategy achieves the largest sum rate.
Performance guideline for the twoway tworelay network in the medium SNR regime.



 Scheme 

G  G  G  G  AF 
G  B  B  G  DF 
G  G  B  B  HMC, HLC 
G  B  G  B 
7. Conclusion
We have devised several transmission strategies for the TWTR network, each of which is derived from a mixandmatch of several basic building blocks, namely, amplifyforward strategy, decodeforward strategy, and physicallayer network coding, and so forth. We can see from the numerical examples that there is no single transmission strategy that can dominate all other strategies under all channel realizations. In other words, transmission strategy should be tailormade for a given environment. In this paper, we have investigated the pros and cons of different building blocks and demonstrated how they can be used to construct transmission strategies for the TWTR network. We believe that the idea can be applied to other relay networks as well.
While in this paper we only consider the case where there are only tworelays, the ideas of our proposed schemes can be applied to the case with more than tworelays. In particular, AF and PDF can be directly implemented without any change. As for DF, HMC, and HLC, the design may be more complicated, since we have to determine which relay to decode which source's message. On the other hand, the idea behind remains the same.
In our work, we have assumed that the channels are static. When link gains are time varying, our result reveals that a static strategy can only be suboptimal. To fully exploit the available capacity of the network, adaptive strategies that can switch between several modes are needed. How to determine a good strategy based on channel state information is an open problem. It is especially difficult if the switching is based on local information only, and we leave it for future work.
Appendix
Proof of Theorem 4
The following informationtheoretic argument shows that any rate pair satisfying the conditions in Theorem 4 is achievable.
Codebook Generation
By (56) and (57), with very high probability the power constraints on node and node are satisfied.
There is a common codebook for relay 1 and relay 2. We generate an array of codewords with rows and columns. The codewords have length and each component is drawn independently from . Label the codewords by , for and .
Since is strictly less than 1, satisfies the power constraint of node 1 with very high probability.
The codeword satisfies the power constraint of node 2 by the hypothesis that .
Encoding
For source node , to send the message , it sends to the relays.
In the second stage, relay 1 and relay 2 transmit and . The messages indicated by is the estimated version of the original message.
Decoding
The receiver at relay 1 treats the signal component as noise, and tries to decode , and . It reduces to a MAC with two users, but three independent messages; two messages from node and one message from node . In order to decode these three messages reliably, we need the requirement in (50). Likewise, we have the requirement in (51) for correct decoding at node 2.
Relay 2 treats the signal component as noise, and tries to decode , and . This can be done with arbitrarily small error if the condition in (51) holds.
where is the mutual information function. This gives the conditions in (54) and (55).
Similarly, we have the conditions in (52) and (53) for successful decoding in terminal . This completes the proof of Theorem 4.
Declarations
Acknowledgment
This work is supported by a grant from the City University of Hong Kong (Project no. SRG 7002386).
Authors’ Affiliations
References
 van der Meulen EC: Transmission of information in a Tterminals discrete memoryless channel, Ph.D. dissertation. University of California, Berkeley, Calif, USA; June 1968.Google Scholar
 Cover TM, El Gamal AA: Capacity theorems for the relay channel. IEEE Transactions on Information Theory 1979, 25(5):572584. 10.1109/TIT.1979.1056084MathSciNetView ArticleMATHGoogle Scholar
 Sendonaris A, Erkip E, Aazhang B: User cooperation diversity—part I: system description. IEEE Transactions on Communications 2003, 51(11):19271938. 10.1109/TCOMM.2003.818096View ArticleGoogle Scholar
 Laneman JN, Tse DNC, Wornell GW: Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Transactions on Information Theory 2004, 50(12):30623080. 10.1109/TIT.2004.838089MathSciNetView ArticleMATHGoogle Scholar
 Shannon CE: Twoway communications channels. Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, June 1961 611644.Google Scholar
 Ahlswede R, Cai N, Li SYR, Yeung RW: Network information flow. IEEE Transactions on Information Theory 2000, 46(4):12041216. 10.1109/18.850663MathSciNetView ArticleMATHGoogle Scholar
 Rankov B, Wittneben A: Achievable rate regions for the twoway relay channel. Proceedings of IEEE International Symposium on Information Theory (ISIT '06), July 2006, Seattle, Wash, USA 16681672.Google Scholar
 Larsson P, Johansson N, Sunell KE: Coded bidirectional relaying. Proceedings of the 63rd IEEE Vehicular Technology Conference (VTC '06), MayJuly 2006, Melbourne, Australia 2: 851855.Google Scholar
 Kim SJ, Mitran P, Tarokh V: Performance bounds for bidirectional coded cooperation protocols. IEEE Transactions on Information Theory 2008, 54(11):52355241.MathSciNetView ArticleMATHGoogle Scholar
 Dash D, Khoshnevis A, Sabharwal A: An achievable rate region for a multiuser halfduplex twoway channel. Proceedings of the 40th Asilomar Conference on Signals, Systems, and Computers (ACSSC '06), OctoberNovember 2006, Pacific Grove, Calif, USA 707711.Google Scholar
 Liu CH, Xue F: Network coding for twoway relaying: rate region, sum rate and opportunistic scheduling. Proceedings of IEEE International Conference on Communications (ICC '08), May 2008, Beijing, China 10441049.Google Scholar
 Baik IJ, Chung SY: Network coding for twoway relay channels using lattices. Proceedings of IEEE International Conference on Communications (ICC '08), May 2008, Beijing, China 38983902.Google Scholar
 Narayanan K, Wilson MP, Sprintson A: Joint physical layer coding and network coding for bidirectional relaying. In Proceedings of the 45th Annual Allerton Conference on Communication, Control, and Computing, June 2007. University of Illinois;Google Scholar
 Nam W, Chung SY, Lee YH: Capacity bounds for twoway relay channels. Proceedings of International Zurich Seminar on Communications (IZS '08), March 2008, Zurich, Germany 144147.Google Scholar
 Schein B: Distributed coordination in network information theory, Ph.D dissertation. MIT, Cambridge, Mass, USA; 2001.Google Scholar
 Xue F, Sandhu S: Cooperation in a halfduplex Gaussian diamond relay channel. IEEE Transactions on Information Theory 2007, 53(10):38063814.MathSciNetView ArticleMATHGoogle Scholar
 Kramer G, Gastpar M, Gupta P: Cooperative strategies and capacity theorems for relay networks. IEEE Transactions on Information Theory 2005, 51(9):30373063. 10.1109/TIT.2005.853304MathSciNetView ArticleMATHGoogle Scholar
 Zhang S, Liew SC, Lam PP: Hot topic: physicallayer network coding. Proceedings of the 12th Annual International Conference on Mobile Computing and Networking (MOBICOM '06), September 2006, Los Angeles, Calif, USA 358365.View ArticleGoogle Scholar
 Katti S, Gollakota S, Katabi D: Embracing wireless interference: analog network coding. Proceedings of the ACM SIGCOMM Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications (ACM SIGCOMM '07), August 2007, Kyoto, Japan 397408.Google Scholar
 Zhang S, Liew SC, Lu L: Physical layer network coding schemes over finite and infinite fields. Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM '08), NovemberDecember 2008, New Orleans, La, USA 37843789.Google Scholar
 Dey BK, Katti S, Jaggi S, Katabi D, Médard M, Shintre S: "Real" and "complex" network codes: promises and challenges. Proceedings of the 4th Workshop on Network Coding, Theory, and Applications (NetCod '08), January 2008, Hong Kong 16.Google Scholar
 HøstMadsen A: Capacity bounds for cooperative diversity. IEEE Transactions on Information Theory 2006, 52(4):15221544.View ArticleMathSciNetMATHGoogle Scholar
 Telatar E: Capacity of multiantenna Gaussian channels. European Transactions on Telecommunications 1999, 10(6):585595. 10.1002/ett.4460100604View ArticleGoogle Scholar
 Cover TM, Thomas JA: Elements of Information Theory. WileyInterscience, New York, NY, USA; 1991.View ArticleMATHGoogle Scholar
 Hu P: Cooperative strategies for Gaussian parallel relay networks, M.S. thesis. City University of Hong Kong, Hong Kong; September 2009.Google Scholar
 Han TS, Kobayashi K: A new achievable rate region for the interference channel. IEEE Transactions on Information Theory 1981, 27(1):4960. 10.1109/TIT.1981.1056307MathSciNetView ArticleMATHGoogle Scholar
 Zheng L, Tse DNC: Diversity and multiplexing: a fundamental tradeoff in multipleantenna channels. IEEE Transactions on Information Theory 2003, 49(5):10731096. 10.1109/TIT.2003.810646View ArticleMATHGoogle Scholar
 Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge, UK; 1985.View ArticleMATHGoogle Scholar
 Khojastepour MA, Sabharwal A, Aazhang B: Bounds on achievable rates for general multiterminal networks with practical constraints. Proceedings of the 2nd International Conference on Information Processing in Sensor Networks (IPSN '03), 2003, Palo Alto, Calif, USA, Lecture Notes in Computer Science 2634: 146161.View ArticleMATHGoogle Scholar
 Kramer G: Topics in MultiUser Information Theory, Foundations and Trends in Communications and Information Theory. Volume 4. NOW Publishers; 2007.MATHGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.