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A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages



Bin Duo, Zhenyong Wang and Xuemai Gu
 
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ABSTRACT

The traditional relay channel as an important component of wireless networks has received considerable attention in recent years. In order to fully utilizing the wireless resources, a novel relay channel model, the Relay Channel with Forwarding Own Messages (RC-FOM), is proposed. In such study, the relay has an own message for the destination in addition to the traditional communication from the source to the destination with the help of the relay. Upper bounds as well as an achievable rate region are derived based on the Decode-and-Forward (DF) strategy for the discrete memoryless RC-FOM. For the Additive White Gaussian Noise (AWGN) channels, the corresponding achievable rate region is proved theoretically and validated numerically. This new achievable scheme can be flexibly reduced to the classic relay channel and thus seen as a generalization of the relay channel.

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  How to cite this article:

Bin Duo, Zhenyong Wang and Xuemai Gu, 2014. A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages. Information Technology Journal, 13: 359-364.

DOI: 10.3923/itj.2014.359.364

URL: https://scialert.net/abstract/?doi=itj.2014.359.364
 
Received: June 08, 2013; Accepted: September 18, 2013; Published: February 07, 2014



INTRODUCTION

User cooperation in wireless networks is an effective technique that enables users to cooperate with each other in their transmissions, thus increasing the transmission efficiency. Relay channels are a special class of cooperative transmission networks, where a source wishes to transmit a message to a destination with the help of relays. The information theoretic study of relay channels have been broadly investigated (Cover and Gamal, 1979; Kramer et al., 2005; Host-Madsen and Zhang, 2005; Reznik et al., 2004; Paulraj et al., 2003; Wang et al., 2005) and practical relaying protocols have also been developed (Laneman et al., 2004).

Van Der Meulen (1971) was the first one who introduced the relay channel. This channel model was then studied deeply by Cover and Gamal (1979). Up to now, the studies on relay channels have been expanded to large-scale hybrid networks, for example (1) The wireless sensor networks (Youn and Kang, 2008; Shakir and Wang, 2008; Arivubrakan and Dhulipala, 2013), (2) The multi-way relay networks (Ong et al., 2011, 2012; Timo et al., 2013) and (3) The LTE-advanced networks (Sasikala and Srivatsa, 2006; Abed et al., 2011). In these relay modes, the relay nodes only assist the source with relaying messages to the destination without sending new messages of their own. When the traditional relay channel is applied to a practical large network, in order to make full use of the channel capacity, one has to consider the capacity limits if a relay node must play a role both in relaying the source messages, as well as its own messages. A representative channel model is proposed as illustrated in Fig. 1 which is referred to as Relay Channel with Forwarding Own Messages (RC-FOM). This RC-FOM consists of a point-to-point communication channel between the relay and the destination and relayed channels in comparison with the traditional relay channel.

In this study, the new channel model is established and the impact of the relay with forwarding own messages on capacity limits on upper bounds is studied. By proposing the random coding method with jointly typical sequences, achievable rates for both the discrete memoryless and Gaussian RC-FOM are derived. Finally, the achievable rate regions under different channel conditions are compared and validated by simulations.

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
Fig. 1: Information flow of relay channel with forwarding own messages

MATERIALS AND METHODS

Experimental system model: The relay channel is a three-terminal discrete memoryless channel consisting of finite sets denoted by X1, X2, Y1, Y2 (i.e., cardinalities of random variables X1, X2, Y1 and Y2) and a transition probability distribution p(y1,y2|x1,x2) where x1εX1, x2εX2, y1εY1 and y2εY2. In this channel, X1 and X2 are the channel inputs from the source and the relay, respectively while Y1 and Y2 are the channel outputs at the relay and destination, respectively. This channel was described by Van Der Meulen (1971) as the first work on the achievable rate regions for the relay channel. Different cooperation schemes such as Decode-and-Forwarding (DF) and Compress-and-Forwarding (CF) were introduced. However, the relay can also send independent and new messages in addition to relaying messages that help the destination in decoding the source's message. This novel system model is named as the relay channel with forwarding own messages in this study. Correspondingly, the relay uses DF scheme as the transmission strategy.

A Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages code for the RC-FOM consists of the following:

Two sets of integers, the source message Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages and the relay message Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
An encoder at the transmitter, X1: W1→X1n where, XijImage for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages(Xi,1, Xi,2,... Xi,j)
A set of relay functions {fi}ni = 1 at the relay such that:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(1)

  where, x2,i is the i-th component of the codeword x2 = (x2,1,... x2,n) that is only dependent of the past received information (y1,1,... y1,i-1) and the information wo
A decoder at the destination, φ: Yn→W1xW0

The average error probability of decoding W1 and W0 is defined as:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(2)

Figure 2 illustrates the encoding and decoding structure of the two messages. Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages denote the estimates of W1 and Wo. The source and the relay generate their messages (W1, Wo) independently and uniformly over W1xW0.

A rate pair (R1, Ro) is said to be achievable for the RC-FOM if there exist a sequence of codes Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages such that for any ò>0, the average probability of error:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(3)

as n→∞.

A relay channel with forwarding own messages is said to be degraded if its transition probability satisfies:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(4)

That is to say, Y2 is independent of X1 on the condition of having Y1 and X2. Note that the definition of degradation precisely describes the notation that one channel is worse than other one in the RC-FOM.

General method: In this subsection, the result of the max-flow-min-cut theorem by Cover and Thomas (1991) is used as a method to establish upper rate bounds on the capacity region of the RC-FOM.

Given any Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages code for RC-FOM, the joint probability distribution satisfies:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(5)

According to Fano’s inequality, the information entropy H(W1, W0|Y2) can be bounded as:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(6)

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
Fig. 2: The encoding and decoding process for the relay channel with forwarding own messages

As the message can be decoded from Y2, Pne→0 and δn defined as (R1+Ro)Pne+1/n, also goes to zero as n→∞.

In the following, R1 is upper bounded as:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(7)

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(8)

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(9)

where, Eq. 8-9 follow as conditioning reduces entropy. Now, let Q be a random variable uniformly distributed over {1,...,n}, set V1 = QYi1,i+1W1, V2 = QYi-12, Y1 = Y1,Q, Y2 = Y2,Q.

Hence:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(10)

Also:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(11)

Hence:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(12)

Then, for the relay-destination channel, Ro can be bounded easily as below:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(13)

Based on the derivation of the upper bounds in Eq. 10, 12 and 13 by taking n→∞, this subsection draws a conclusion. For any rate pair (R1, Ro) with Pne→0 of the RC-FOM, there exist some random variables U→(V1, V2)→(X1, X2)→(Y1, Y2), such that (R1, Ro) satisfies the following conditions:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(14)

PROPOSED APPROACH

Here, proposes a random coding technique with jointly typical sequences to derive the achievable rate region for the RC-FOM based on the well-known DF strategy (Cover and Gamal, 1979). Here, the relay is assumed to be able to fully recover the source’s message forwarded to the destination. Then, it re-encodes the decoded source message W1 together with its own message Wo for the destination.

The coding schemes combine ideas from the relay channel (Cover and Gamal, 1979), the broadcast channel (Liang and Veeravalli, 2007) and the multiple access channel (Willems, 1982).

Random codebook generation:

Randomly generated independently and identically distributed (i.i.d) n-sequence u with probability:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

Label them as u(w’1). Assume that all the nodes know u

For each u(w’1) generate Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages conditionally independent n-sequence x1 at the source each drawn randomly according to:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

index them as Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

For each u(w’1) generate at random Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages i.i.d n-sequence x2 at the relay, each with distribution:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

index them as Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

Assume a transmission period of B blocks, each consisting of n transmissions. A sequence of B-1 i.i.d source messages Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages iε[1:B-1] and a sequence of B-1 i.i.d relay messages Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages iε[1:B-1], are to be sent over the channel in nB transmissions, respectively. Note that the average rate pairs over B blocks are:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

which can be made as close to (R1, Ro) as desired when B→∞. The encoding and decoding are explained with the help of Table 1.

Encoding: At first consider block i, where i ≠ 1, B which means it is not the first or the last block:

The encoder at the source sends x1(w1(i), w1(i-1)), where w1(i-1) was denoted above as w’1
Assuming that the relay already has a correct estimation of w1(i-1) from the previous block, then it sends x2(w0(i), w1(i-1))

When i = 1, the source sends x1(w1(1), 1) and the relay sends x2(w0(1)1), where w1(0) = 1 by convention. When i = B, the source sends x1(1, w1(B-1)) and the relay sends x2(1, w1(B-1)), where w1(B) = w0(B) = 1 by convention.

Decoding:

Assuming the relay has decoded w1(i-1) which was sent at block i-1, it can decode w1(i) by looking for a unique Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages(i) such that:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

  are jointly typical. If:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(15)

  then based on joint Asymptotic Equipartition Property (AEP), one has Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages(i) = w1(i) with probability goes to 1
The destination decodes from the last block until all blocks are received. Assuming that it has decode w1(i) in block (i+1), then in block i, it declare that Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages(i-1) is received, if (u(Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages(i-1)), x1(w1(i), Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages(i-1)), y2(i)) are jointly typical. It is easy to see that if:R1<I(U, X1; Y2)

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(16)

  Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages(i-1) = w1(i-1) with probability goes to 1, as n increases
Having w1(i-1), the destination decodes w0(i) by looking for a unique Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages(i) such that (u(w1(i-1)), x1(w1(i), w1(i-1)), x2(Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages(i), w1(i-1)), y2(i)) are jointly typical. If there does not exist such unique sequences, the destination occurs an error. Then based on AEP, the error probability will go to zero if:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(17)

By combining Eq. 15-17, the rate pairs in the closure of the convex hull of all (R1, Ro) for the RC-FOM satisfying: R1<min{I(X1; Y1|X2, U), I(U, X1, Y2)}

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(18)

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(19)

for some joint distribution:

p(u, v1, v2, x1, x2, y1, y2) = p(u) p(v1|u) p(v2|u)
p(x1, x2|v1, v2) p(y1, y2|x1, x2)

are achievable using DF scheme.

Note that if U = X2, Eq. 18 reduces to one of the main results by Cover and Gamal (1979).

NUMERICAL RESULTS AND ANALYSIS

Here, the achievability results in Eq. 18-19 are extended to the Gaussian RC-FOM because it approximately presents realistic wireless networks. The signals received, respectively at the relay and the destination node are given by:

Table 1: The encoding and decoding process for the RC-FOM
Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

Y1 = X1+Z1
(20)

Y2 = X1+X2+Z2
(21)

where, Z1~N(0, N1) and Z2~N(0, N2) are i.i.d gaussian noise. The source and the relay have average power constraints:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

and:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

If an AWGN RC-FOM is said to be degraded, then the relay has all the knowledge that the destination knows, i.e., the channel input-output relations in Eq. 20 and 21 are changed to:

Y1 = X1+Z1
(22)

Y2 = X1+X2+Z1+Z’
(23)

where, Z1~N(0, N1) and Z’~N(0, N2-N1) are i.i.d gaussian noise for N1<N2.

Let:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

denote the capacity function and let:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages

in the rest of this section. All achievable rate values are expressed in bps Hz-1.

Let U~N (0, P), X’1~N(0, αP1) and X’2~N(0, βP2). Also, let:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(24)

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(25)

In the above expressions the parameter β shows the relay’s participation level in the help of relaying source’s message which controls the trade-off between the relayed messages and relay’s own messages. Although, for the traditional degraded relay channel it is always optimal to relay source’s message with full power, this may not be the case when the relay sends its own messages.

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
Fig. 3: The achievable rate regions for the degraded gaussian relay channel with forwarding own messages

Therefore, the rate pairs in the closure of the convex hull of all (R1, Ro) for the Gaussian relay channel with forwarding own messages satisfying:

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(26)

Image for - A New Achievable Rate Region for the Relay Channel with Forwarding Own Messages
(27)

are achievable using DF scheme for some α, βε[0, 1].

Note that, set β = 0 and Eq. 26 will reduce to the capacity of the degraded Gaussian relay channel as another main result by Cover and Gamal (1979).

Figure 3 illustrates the achievable rate regions to demonstrate the relationship between the source rate (R1) and the rate of relaying own messages (Ro) for two scenarios for comparison:

Case 1: The degraded gaussian RC-FOM with P1 = P2 = 10 dB, N1 = 2 dB and N1 = 4 dB
Case 2: The non-degraded gaussian RC-FOM with P1 = P2 = 10 dB, N1 = 5 dB and N1 = 4 dB

In the case 2, using the relay performs worse than the case 1 because the system no longer benefits from using the relay sufficiently to achieve the best achievable rate when the relay has more noise. Therefore, it is profitable to use the relay whenever it is more helpful in forwarding the source message to the destination, i.e., the source-destination channel is the degraded version of the source-relay channel.

CONCLUSION

In this study, the relay channel with forwarding own messages was studied. In particular, the cooperative DF scheme was proposed and the coding strategy was developed for this channel. Furthermore, upper bounds on the capacity region and the corresponding achievable performance bounds for this channel were obtained in discrete memoryless channel and Gaussian channel, respectively. Finally, numerical results established the critical role of relay’s own message in the capacity region and shed light on the generalization of this channel.

This study can be further extended. First, achievable rate regions by using CF and partial decode-and-forward schemes can also be derived in RC-FOM. Then, RC-FOM in fading channels is a practical application in wireless networks. Finally, in order to get greater flexibility of power allocation, global power constraint instead of a per-node power constraint is worth studying.

ACKNOWLEDGMENTS

This research study is supported by the National Natural Science Foundation of China (Grant No. 61101125 and Grant No. 61201143).

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