# Joint Channel-Network Coding for the Gaussian Two-Way Two-Relay Network

- Ping Hu
^{1}, - ChiWan Sung
^{1}and - Kenneth W. Shum
^{2}Email author

**2010**:708416

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 two-way relay network with one relay to two-way relay network with multiple relays. To study the essential features of the two-way multiple-relay 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, amplify-forward (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, multi-input multi-output transmission, and multiple access with common information. Their performance is compared with the cut-set 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 signal-tonoise (SNR) ratio, it is shown that the AF scheme dominates the others in high SNR regime.

## Keywords

## 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 decode-forward (DF) and compress-forward (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 two-way. A typical example is the telephone service. In fact, the study of two-way channel is not new and can be traced back to Shannon's work in 1961 [5]. However, the model of two-way 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 two-way relay channel was considered in [7]. The half-duplex case was studied in [8, 9]. The results in [10] showed that feedback is beneficial only in a two-way transmission. Network coding for the two-way 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 source-destination pair and two parallel relays in between. This model was further studied in [16] under the assumption of half-duplex relay operations. For one-way multiple-relay networks in general, cooperative strategies were proposed and studied in [17]. We remark that a notable feature that does not exist in the single-relay 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 two-way communications. Two-relays are assumed, for this simple model already captures the essential features of the more general multiple-relay case. We are interested in knowing how different techniques can be used to construct transmission strategies for the two-way two-relay 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 multi-input multi-output (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* two-way two-relay 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 half-duplex 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 signal-to-noise 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 .

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 non-negative 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. Physical-Layer 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 multi-user scenarios in which some user knows the message of another user *a priori*. Consider the two-way 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 point-to-point 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 Multiple-Access Channel (MAC)

We refer the reader to [24] for more details on the optimal coding scheme for MAC.

## 4. Channel-Network 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 decode-forward and amplify-forward 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 point-to-point 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 self-signal 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 Decode-Amplify Forward with Linear Combination (HLC)

In this strategy, relay 1 decodes the messages from terminals and , and meanwhile, relay 2 employs the amplify-forward 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 Decode-Amplify 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 re-encodes 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.

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 re-encoded 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 two-relays can jointly re-encode 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 two-relays 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 non-zero , 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 Signal-to-Noise Ratio (SNR) regime.

*multiplexing gain*(also called

*degree of freedom*) [27], defined by

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).

**,**and are invertible matrices, the rank of the matrix in (76) is equal to the rank of . Then we get

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.

## 7. Conclusion

We have devised several transmission strategies for the TWTR network, each of which is derived from a mix-and-match of several basic building blocks, namely, amplify-forward strategy, decode-forward strategy, and physical-layer 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 tailor-made 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 two-relays, the ideas of our proposed schemes can be applied to the case with more than two-relays. 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 information-theoretic 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 T-terminals 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):572-584. 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):1927-1938. 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):3062-3080. 10.1109/TIT.2004.838089MathSciNetView ArticleMATHGoogle Scholar - Shannon CE: Two-way communications channels.
*Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, June 1961*611-644.Google Scholar - Ahlswede R, Cai N, Li S-YR, Yeung RW: Network information flow.
*IEEE Transactions on Information Theory*2000, 46(4):1204-1216. 10.1109/18.850663MathSciNetView ArticleMATHGoogle Scholar - Rankov B, Wittneben A: Achievable rate regions for the two-way relay channel.
*Proceedings of IEEE International Symposium on Information Theory (ISIT '06), July 2006, Seattle, Wash, USA*1668-1672.Google Scholar - Larsson P, Johansson N, Sunell K-E: Coded bi-directional relaying.
*Proceedings of the 63rd IEEE Vehicular Technology Conference (VTC '06), May-July 2006, Melbourne, Australia*2: 851-855.Google Scholar - Kim SJ, Mitran P, Tarokh V: Performance bounds for bidirectional coded cooperation protocols.
*IEEE Transactions on Information Theory*2008, 54(11):5235-5241.MathSciNetView ArticleMATHGoogle Scholar - Dash D, Khoshnevis A, Sabharwal A: An achievable rate region for a multiuser half-duplex two-way channel.
*Proceedings of the 40th Asilomar Conference on Signals, Systems, and Computers (ACSSC '06), October-November 2006, Pacific Grove, Calif, USA*707-711.Google Scholar - Liu C-H, Xue F: Network coding for two-way relaying: rate region, sum rate and opportunistic scheduling.
*Proceedings of IEEE International Conference on Communications (ICC '08), May 2008, Beijing, China*1044-1049.Google Scholar - Baik I-J, Chung S-Y: Network coding for two-way relay channels using lattices.
*Proceedings of IEEE International Conference on Communications (ICC '08), May 2008, Beijing, China*3898-3902.Google Scholar - Narayanan K, Wilson MP, Sprintson A: Joint physical layer coding and network coding for bi-directional relaying. In
*Proceedings of the 45th Annual Allerton Conference on Communication, Control, and Computing, June 2007*. University of Illinois;Google Scholar - Nam W, Chung S-Y, Lee YH: Capacity bounds for two-way relay channels.
*Proceedings of International Zurich Seminar on Communications (IZS '08), March 2008, Zurich, Germany*144-147.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 half-duplex Gaussian diamond relay channel.
*IEEE Transactions on Information Theory*2007, 53(10):3806-3814.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):3037-3063. 10.1109/TIT.2005.853304MathSciNetView ArticleMATHGoogle Scholar - Zhang S, Liew SC, Lam PP: Hot topic: physical-layer network coding.
*Proceedings of the 12th Annual International Conference on Mobile Computing and Networking (MOBICOM '06), September 2006, Los Angeles, Calif, USA*358-365.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*397-408.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), November-December 2008, New Orleans, La, USA*3784-3789.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*1-6.Google Scholar - Høst-Madsen A: Capacity bounds for cooperative diversity.
*IEEE Transactions on Information Theory*2006, 52(4):1522-1544.View ArticleMathSciNetMATHGoogle Scholar - Telatar E: Capacity of multi-antenna Gaussian channels.
*European Transactions on Telecommunications*1999, 10(6):585-595. 10.1002/ett.4460100604View ArticleGoogle Scholar - Cover TM, Thomas JA:
*Elements of Information Theory*. Wiley-Interscience, 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):49-60. 10.1109/TIT.1981.1056307MathSciNetView ArticleMATHGoogle Scholar - Zheng L, Tse DNC: Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels.
*IEEE Transactions on Information Theory*2003, 49(5):1073-1096. 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: 146-161.View ArticleMATHGoogle Scholar - Kramer G:
*Topics in Multi-User Information Theory, Foundations and Trends in Communications and Information Theory*.*Volume 4*. NOW Publishers; 2007.MATHGoogle Scholar

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