Sum-Rate Analysis of the Two-Way Relay Channelin Spectrum-Sharing Environments

Najmeh MadaniKousha Communication Industries, Tehran, Iran

Email:njmadani@gmail.com

Abstract—This paper examines the two-way relay channel(TWRC) in a spectrum-sharing environment. Two secondaryusers, a primary user and a relay are the involved nodes inour model. The transmit power of the secondary users and therelay are adapted optimally to approach the maximum achievablesum-rate while keeping the interference level at the primaryuser below a threshold. Numerical simulations are conductedto demonstrate the performance of the proposed scenario.

I. INTRODUCTION

Relay-based networks have been widely perceived as greatassets to wireless communications in terms of coverage exten-sion, throughput improvement and saving network resources.In this context, two-way relaying which provides a bi-directioncommunication is of great importance. The two-way relaychannel (TWRC) is privileged as it helps in redeeming thespectral loss one-way relaying suffers from [1]. In a sim-ple TWRC model, two terminals exchange their messagesvia a relay terminal. The terminals operate in half-duplexmode. Protocols such as amplify-and-forward (AF), decode-and-forward (DF), and compress-and-forward (CF) have beenutilized for information transmission in TWRC [2].

The merits of TWRC in power saving and spectral efficiencymake it to be best adopted in spectrum-sharing networks. Inthese networks, secondary users use the spectrum of primaryusers in a way that the links of the primary users are not af-fected [3]. To protect the QoS of primary users, the secondaryusers have to be cautious about the transmit power. Applicationof the TWRC in spectrum-sharing networks has been recentlystudied in [4]. This work has concentrated on sum-rate opti-mization by relay beamforming and power allocation at thesecondary users while maintaining the received power at theprimary user below a certain level. The amplify-and-forwardprotocol has been considered in this research.

In this paper, we aim at the maximum achievable sum-rate of the TWRC using decode-and-forward protocol in aspectrum-sharing environment. The decode-and-forward sup-ports different coding schemes at the relay node, e.g., superpo-sition coding [1], [5], XOR coding [6], [5], and optimal coding[7], [8]. We follow the superposition scheme. However, theproposed solution is flexible and can accommodate other cod-ing strategies. It is assumed that the channel state information(CSI) is provided at the nodes. The sum-rate maximizationis performed with the help of CSI while complying powerconstraints. The constraints are introduced by the spectrum-sharing system and the transmit power limitations of the nodes.

We consider a Rayleigh flat fading environment.The rest of the paper is organized as follows. Section II

describes the system model and the problem formulation. Sec-tion III provides the sum-rate optimization strategy. Simulationresults are presented in section IV and finally conclusions aremade in section V.

II. SYSTEM MODEL

Two source nodes 𝑆1 and 𝑆2 exchange information withthe help of a relay node 𝑅. All nodes operate in half-duplexmode and information exchange is performed within two time-slots. The source nodes use the same spectrum band and shareit with a primary user. All the channels are assumed to beconstant during the two time-slots. In the first time-slot, 𝑆1 and𝑆2 transmit to 𝑅 simultaneously, so 𝑅 receives the followingsignal:

𝑦𝑟 [𝑛] =√

ℎ1𝑠𝑥1 [𝑛] +√

ℎ2𝑠𝑥2 [𝑛] + 𝑧𝑟 [𝑛] , (1)

where 𝑛 represents the time index, ℎ1𝑠 and ℎ2𝑠 are the channelpower gains from 𝑆1 to 𝑅 and from 𝑆2 to 𝑅, respectively, and𝑧𝑟[𝑛] indicates the AWGN with the power spectral density of𝑁0. The transmit power of 𝑆1 and 𝑆2 are denoted by 𝑝1 and𝑝2 and are constrained to the predefined value of 𝑄𝑠:

𝑝1 ≤ 𝑄𝑠,

𝑝2 ≤ 𝑄𝑠. (2)

As spectrum-sharing entails, signal transmission betweensource nodes and relay should not harm the performance of theprimary user. Let 𝑄𝑝 indicates the peak power that the primaryuser can tolerate without QoS violation, thus the receivedinterference at the primary user in the first time-slot shouldbe limited as:

𝑝1ℎ1𝑝 + 𝑝2ℎ2𝑝 ≤ 𝑄𝑝, (3)

where ℎ1𝑝 and ℎ2𝑝 indicate the channel power gains from 𝑆1 tothe primary user, and from 𝑆2 to the primary user, respectively.The relay employs a two-phase decode-and-forward strategy[9]. It decodes the received signals, re-encode them andbroadcasts the signals to the source nodes in the second time-slot. The received signals at 𝑆1 and 𝑆2 are expressed as:

𝑦1 [𝑛] =√

ℎ1𝑠��1 [𝑛] +√

ℎ1𝑠��2 [𝑛] + 𝑧1 [𝑛] , (4)

𝑦2 [𝑛] =√

ℎ2𝑠��1 [𝑛] +√

ℎ2𝑠��2 [𝑘] + 𝑧2 [𝑛] , (5)

2011 8th International Symposium on Wireless Communication Systems, Aachen

978-1-61284-402-2/11/$26.00 ©2011 IEEE 101

𝑄𝑝ℎ1𝑝

𝑄𝑝ℎ2𝑝

𝑃1

𝑃2

(a)

𝑄𝑠

𝑄𝑝ℎ1𝑝

𝑃1

𝑃2

(b)

𝑄𝑠

𝑄𝑝ℎ2𝑝

𝑃1

𝑃2

(c)

𝑄𝑠

𝑄𝑠

𝑃1

𝑃2

(d)

𝑄𝑠

𝑄𝑠

𝑃1

𝑃2

(e)

Fig. 1. Possible power regions for source nodes.

where 𝑧1[𝑛] and 𝑧2[𝑛] refer to the AWGN with the powerdensity of 𝑁0. ��1[𝑛] and ��2[𝑛] are the re-encoded signalscorresponding to the signal streams of 𝑆1 and 𝑆2, respectively.𝑅 divides the power between ��1[𝑛] and ��2[𝑛] according to thesuperposition coding. Let 𝑄𝑟 defines the maximum power that𝑅 can transmit, 𝑝𝑟1 denotes the assigned power to ��1[𝑛] and𝑝𝑟2 indicates the allocated power to ��2[𝑛]. It is obvious that:

𝑝𝑟1 + 𝑝𝑟2 ≤ 𝑄𝑟. (6)

To protect the primary user from harmful interference, thesignal transmission in the second time-slot should satisfy thefollowing constraint:

𝑝𝑟1ℎ𝑝 + 𝑝𝑟2ℎ𝑝 ≤ 𝑄𝑝, (7)

where ℎ𝑝 is the channel power gain between 𝑅 and theprimary user. It is assumed that the channel power gainsare independent and identically distributed (i.i.d). It is furtherassumed that the source nodes and the relay are fully awareof codebooks and channel power gains. We continue with theassumption of 𝑁0𝐵 = 1, where 𝐵 is the available bandwidth.

In the first time-slot, when the source nodes transmit andthe relay receives, we have a multiple access channel (MAC).Using sequential decoding at 𝑅, the achievable rate region ischaracterized by the following relations [10]:

𝑅1𝑅 ≤ log (1 + 𝑝1ℎ1𝑠) , (8)

𝑅2𝑅 ≤ log (1 + 𝑝2ℎ2𝑠) , (9)

𝑅1𝑅 +𝑅2𝑅 ≤ log (1 + 𝑝1ℎ1𝑠 + 𝑝2ℎ2𝑠) . (10)

The second time-slot represents a broadcast channel wherethe achievable rates are derived by the fact that each nodeknows about it’s transmitted signal. Each source node can sub-tract it’s self-interference from the received signal, describedby (4) or (5). In this time-slot the rates satisfy:

𝑅𝑅1 ≤ log (1 + 𝑝𝑟1ℎ1𝑠) , (11)

𝑅𝑅2 ≤ log (1 + 𝑝𝑟2ℎ2𝑠) . (12)

The rates are expressed in terms of bits/sec/Hz. The achievablerate from 𝑆1 to 𝑆2 via 𝑅 is expressed as the minimum of therates during the two time-slots.

𝑅12 ≤ 1

2min {log (1 + 𝑝1ℎ1𝑠) , log (1 + 𝑝𝑟2ℎ2𝑠)} . (13)

Similarly, the achievable rate from 𝑆2 to 𝑆1 satisfies:

𝑅21 ≤ 1

2min {log (1 + 𝑝2ℎ2𝑠) , log (1 + 𝑝𝑟1ℎ1𝑠)} . (14)

The factor 1/2 follows because the information transmissionis performed in two time-slots. Our goal in this paper is tomaximize the achievable sum-rate of the modeled TWRCby power allocation at the source nodes and the relay. Thisoptimization problem which is subjected to the constraints (2),(3), (6) and (7) can be formulated as follows:

𝑅𝑠𝑢𝑚𝑚𝑎𝑥 = max

𝑝1,𝑝2,𝑝𝑟1,𝑝𝑟2

min

{𝑅12 +𝑅21,

1

2log(1 + 𝑝1ℎ1𝑠 + 𝑝2ℎ2𝑠)

}.

(15)

III. SUM-RATE ANALYSIS

The constraints introduced in the previous section definetransmit power regions for the nodes. As Fig.1 demonstrates,(2) and (3) shape the power regions of the source nodes. Weproceed with the Fig.1a. The solution will be extended nextto cover all the cases. The selected region corresponds to thefollowing conditions:

ℎ1𝑝 >𝑄𝑝

𝑄𝑠,

ℎ2𝑝 >𝑄𝑝

𝑄𝑠. (16)

For the case of 𝑅, the transmit power should satisfy (6) and(7). We can brief these two conditions as:

𝑝𝑟1 + 𝑝𝑟2 ≤ 𝐴, 𝐴 =

{𝑄𝑟 ℎ𝑝 <

𝑄𝑝

𝑄𝑟𝑄𝑝

ℎ𝑝ℎ𝑝 >

𝑄𝑝

𝑄𝑟

. (17)

In the broadcast phase, the sum-rate is:

𝑅12 +𝑅21 =1

2{log(1 + 𝑝𝑟1ℎ1𝑠) + log(1 + 𝑝𝑟2ℎ2𝑠)} . (18)

The maximum value it can reach while satisfying (17) can beeasily calculated and is as follows:

𝑅𝐵𝐶𝑚𝑎𝑥 =

1

2log(1+

𝐴2

4ℎ1𝑠ℎ2𝑠+

𝐴

2(ℎ1𝑠+ℎ2𝑠)+

(ℎ2𝑠 − ℎ1𝑠)2

4ℎ1𝑠ℎ2𝑠).

(19)This maximum occurs when 𝑝∗𝑟1 = 𝐴

2 + 12

(1

ℎ2𝑠− 1

ℎ1𝑠

)and

𝑝∗𝑟2 = 𝐴2 +

12

(1

ℎ1𝑠− 1

ℎ2𝑠

)and is valid as long as −𝐴 < 1

ℎ2𝑠−

1ℎ1𝑠

< 𝐴. To have 𝑅𝐵𝐶𝑚𝑎𝑥 as the maximum achievable sum-rate

of the modeled TWRC, the source nodes should transmit withthe power levels that ensure:

𝑝1ℎ1𝑠 ≥ 𝑝∗𝑟2ℎ2𝑠 equivalently 𝑝1 ≥ 𝐴ℎ2𝑠

2ℎ1𝑠+

(ℎ2𝑠 − ℎ1𝑠)

2ℎ21𝑠︸ ︷︷ ︸

𝑝∗1

,

(20)

102

P1

P2

𝛼

𝛽

𝑥1

𝑥2

(𝑝∗1, 𝑝2)

(𝑝1, 𝑝∗2)

(a)

𝛼

𝛽

𝑥1

𝑥2

(𝑝∗1, 𝑝2)

(𝑝1, 𝑝∗2)

𝑃1

𝑃2

(b)

𝛼

𝛽

𝑥1

𝑥2

(𝑝∗1, 𝑝2)

(𝑝1, 𝑝∗2)

𝑃1

𝑃2

(c)

Fig. 2. Shaded areas indicate the power levels of the source nodes which result in a maximized broadcast sum-rate.

and

𝑝2ℎ2𝑠 ≥ 𝑝∗𝑟1ℎ1𝑠 equivalently 𝑝2 ≥ 𝐴ℎ1𝑠

2ℎ2𝑠+

(ℎ1𝑠 − ℎ2𝑠)

2ℎ22𝑠︸ ︷︷ ︸

𝑝∗2

.

(21)Meanwhile, the transmit power of the source nodes shouldresult in a MAC sum-rate greater than 𝑅𝐵𝐶

𝑚𝑎𝑥:

𝑝1ℎ1𝑠 + 𝑝2ℎ2𝑠 ≥ 𝐴2

4ℎ1𝑠ℎ2𝑠 +

𝐴

2(ℎ1𝑠 + ℎ2𝑠) +

(ℎ2𝑠 − ℎ1𝑠)2

4ℎ1𝑠ℎ2𝑠.

(22)Fig.2 illustrates (20), (21), and (22) with the power region

of the source nodes in the 𝑃1 − 𝑃2 plane. Line 1 and Line2 which represent boundary points of (3) and (22) have thefollowing equations, respectively:

Line 1:𝑝1𝑥1

+𝑝2𝑥2

= 1, (23)

where 𝑥1 =𝑄𝑝

ℎ1𝑝and 𝑥2 =

𝑄𝑝

ℎ2𝑝.

Line 2:𝑝1𝛼

+𝑝2𝛽

= 1, (24)

where 𝛼 = 𝐴ℎ2𝑠

ℎ1𝑠+ (ℎ2𝑠−ℎ1𝑠−𝐴ℎ1𝑠ℎ2𝑠)

2

4ℎ21𝑠ℎ2𝑠

and 𝛽 = 𝐴 +(ℎ2𝑠−ℎ1𝑠−𝐴ℎ1𝑠ℎ2𝑠)

2

4ℎ1𝑠ℎ22𝑠

. The shaded parts in Fig.2a, Fig.2b, andFig.2c indicate the regions where (20), (21), and (22) aresatisfied. Here, the broadcast sum-rate is the controlling rateand 𝑅𝐵𝐶

𝑚𝑎𝑥 is the response of our optimization problem definedin (15).

Line 1 and Line 2 might be positioned in a way thatno points in the power region validate (20), (21), and (22)concurrently. In this case, 𝑅𝐵𝐶

𝑚𝑎𝑥 is not supported anymore.Fig.3a demonstrates a possible scenario. In Fig.3a, the area ofthe power region which fulfills (22) and (21) dose not qualify(20). Hear, 𝑅12+𝑅21 = 1

2{log(1+𝑝1ℎ1𝑠)+log(1+𝑝𝑟1ℎℎ1𝑠)}is the controlling sum-rate. To improve the sum-rate, 𝑝𝑟1 and𝑝𝑟2 are selected as follows:

𝑝𝑟1 = 𝐴− 𝑝𝑟2, 𝑝𝑟2 = 𝑝1𝑠ℎ1𝑠

ℎ2𝑠. (25)

It can be verified that the controlling sum-rate is ascendingin terms of 𝑝1 as long as 𝑝1 ≤ 𝑝∗1. This is on the contrary

to the MAC sum-rate which is descending as 𝑝1 increases.The downward sum-rate of the MAC phase is concluded fromthe fact that 𝑥2

𝑥1> ℎ1𝑠

ℎ2𝑠and 𝑝2 = 𝑥2(1 − 𝑝1

𝑥1) is adopted. The

optimum value for the controlling sum-rate is achieved whenit equals the MAC sum-rate:

1+𝑝1ℎ1𝑠+𝑥2(1− 𝑝1𝑥1

)ℎ2𝑠 = (1+𝑝1ℎ1𝑠)(1+(𝐴−𝑝1 ℎ1𝑠

ℎ2𝑠))ℎ1𝑠).

(26)The power level of 𝑆1 which comes from the above equationis:

𝑝1 =𝐴ℎ2𝑠

2ℎ1𝑠+

𝑥2ℎ22𝑠

2𝑥1ℎ31𝑠

− 1

2ℎ1𝑠−√

(ℎ21𝑠 − 𝑥2ℎ2

2𝑠

𝑥1−𝐴ℎ2

1𝑠ℎ2𝑠)2 − 4ℎ31𝑠(𝑥2ℎ2

2𝑠 −𝐴ℎ1𝑠ℎ2𝑠)

2ℎ31𝑠

,

(27)

and the maximum achievable sum rate is:

𝑅𝑠𝑢𝑚𝑚𝑎𝑥 =

1

2log(1 + (

𝐴ℎ21𝑠ℎ2𝑠 +

𝑥2ℎ22𝑠

𝑥1− ℎ2

1𝑠

2𝑥1ℎ31𝑠

−√(ℎ2

1𝑠 − 𝑥2ℎ22𝑠

𝑥1−𝐴ℎ2

1𝑠ℎ2𝑠)2 − 4ℎ31𝑠(𝑥2ℎ2

2𝑠 −𝐴ℎ1𝑠ℎ2𝑠)

2𝑥1ℎ31𝑠

)

(𝑥1ℎ1𝑠 − 𝑥2ℎ2𝑠) + 𝑥2ℎ2𝑠). (28)

Fig.3b presents another case where 𝑅𝐵𝐶𝑚𝑎𝑥 is not supported.

The points of the power region which fulfill (22) and (20)do not satisfy (21). The controlling sum-rate is 𝑅12 +𝑅21 =12{log(1+𝑝2ℎ2𝑠)+log(1+(𝐴− 𝑝2ℎ2𝑠

ℎ1𝑠)ℎ2𝑠)} and it’s maximum

is obtained similar to the previous case:

𝑅𝑠𝑢𝑚𝑚𝑎𝑥 =

1

2log(1 + (

𝐴ℎ22𝑠ℎ1𝑠 +

𝑥1ℎ21𝑠

𝑥2− ℎ2

2𝑠

2𝑥2ℎ32𝑠

−√(ℎ2

2𝑠 − 𝑥1ℎ21𝑠

𝑥2−𝐴ℎ2

2𝑠ℎ1𝑠)2 − 4ℎ32𝑠(𝑥1ℎ2

1𝑠 −𝐴ℎ2𝑠ℎ1𝑠)

2𝑥2ℎ32𝑠

)

(𝑥2ℎ2𝑠 − 𝑥1ℎ1𝑠) + 𝑥1ℎ1𝑠). (29)

103

𝛼

𝛽

𝑥1

𝑥2

(𝑝∗1, 𝑝2)(𝑝∗

1, 𝑝2)(𝑝∗

1, 𝑝2)

(𝑝1, 𝑝∗2)(𝑝1, 𝑝∗

2)(𝑝1, 𝑝∗

2)

𝑃1

𝑃2

(a)

𝛼

𝛽

𝑥1

𝑥2

(𝑝∗1, 𝑝2)

(𝑝1, 𝑝∗2)

𝑃1

𝑃2

(b)

𝛼

𝛽

𝑥1

𝑥2(𝑝∗

1, 𝑝2)

(𝑝1, 𝑝∗2)

𝑃1

𝑃2

(c)

Fig. 3. No points in the power region support the maximized broadcast sum-rate.

To reach this rate, the power level of 𝑆2 should be:

𝑝2 =𝐴ℎ1𝑠

2ℎ2𝑠+

𝑥1ℎ21𝑠

2𝑥2ℎ32𝑠

− 1

2ℎ2𝑠−√

(ℎ22𝑠 − 𝑥1ℎ2

1𝑠

𝑥2−𝐴ℎ2

2𝑠ℎ1𝑠)2 − 4ℎ32𝑠(𝑥1ℎ2

1𝑠 −𝐴ℎ2𝑠ℎ1𝑠)

2ℎ32𝑠

.

(30)

Fig.3c refers to a new case where 𝑥1 < 𝛼 and 𝑥2 < 𝛽.No points in the power region validate (22) and consequentlythe MAC sum-rate is always lower than 𝑅𝐵𝐶

𝑚𝑎𝑥. For this casethe maximum achievable sum-rate occurs in the MAC phase.Adopting 𝑝1 = 𝑥1(1− 𝑝2

𝑥2), the sum-rate is:

𝑅12 +𝑅21 =1

2log(1 + 𝑥1ℎ1𝑠 + 𝑝2(ℎ2𝑠 − 𝑥1ℎ1𝑠

𝑥2)). (31)

As long as 𝑥2

𝑥1> ℎ1𝑠

ℎ2𝑠, the rate is increasing in terms of 𝑝2.

Therefore, the maximum sum-rate is as follows:

𝑅𝑀𝐴𝐶𝑚𝑎𝑥 =

1

2log(1 + 𝑥2ℎ2𝑠). (32)

This is obtained when 𝑝2 = 𝑥2. To have 𝑅𝑀𝐴𝐶𝑚𝑎𝑥 as the

optimum sum-rate, it is necessary that 𝐴 ≥ 𝑥2ℎ2𝑠

ℎ1𝑠. This leads

to a one-way relaying system where 𝑆1 transmits and 𝑆2

receives. When 𝐴 < 𝑥2ℎ2𝑠

ℎ1𝑠, the relay can not support 𝑅𝑀𝐴𝐶

𝑚𝑎𝑥 .Similar to our reasoning in the broadcast phase, the controllingsum-rate, it’s maximum and the power allocation strategy atthe nodes are according to (25), (27), and (28). For the casewhere 𝑥2

𝑥1< ℎ1𝑠

ℎ2𝑠and 𝐴 ≥ 𝑥1ℎ1𝑠

ℎ2𝑠, the maximum achievable

sum-rate is:𝑅𝑀𝐴𝐶

𝑚𝑎𝑥 =1

2log(1 + 𝑥1ℎ1𝑠). (33)

if 𝐴 < 𝑥1ℎ1𝑠

ℎ2𝑠, (29) and (30) determine the result.

In summery, to obtain the maximum sum-rate of the consid-ered TWRC, 𝑅𝐵𝐶

𝑚𝑎𝑥 and 𝑅𝑀𝐴𝐶𝑚𝑎𝑥 are determined. The one with

the lower value limits the sum-rate and is a possible answerfor (15). Based on the channel power gains and the powerconstraints of the nodes, it is then decided that the sum-rateis supported or a lower rate according to (28) or (30) is thetrue rate. The power level of the source nodes and the relayare assigned correspondingly. To have a general solution, some

modifications are necessary. In evaluating 𝑅𝐵𝐶𝑠𝑢𝑚, the following

extension is applicable:⎧⎨⎩

𝑝∗𝑟1 = 𝐴, 𝑝∗𝑟2 = 0,

𝛼 = 𝐴, 𝛽 = 𝐴ℎ1𝑠

ℎ2𝑠,

if 1ℎ2𝑠− 1

ℎ1𝑠≥ 𝐴

𝑝∗𝑟1 = 0, 𝑝∗𝑟2 = 𝐴,

𝛼 = 𝐴ℎ2𝑠

ℎ1𝑠, 𝛽 = 𝐴,

if 1ℎ2𝑠− 1

ℎ1𝑠≤ −𝐴

as stated before, otherwise.

(34)

We reconsider 𝑅𝐵𝐶𝑠𝑢𝑚 = 1

2{log(1+𝑝∗𝑟1ℎ1𝑠)+log(1+𝑝∗𝑟2ℎ2𝑠)},𝑝∗1 =

𝑝∗𝑟2ℎ2𝑠

ℎ1𝑠and 𝑝∗2 =

𝑝∗𝑟1ℎ1𝑠

ℎ2𝑠. 𝑄𝑠 is another important factor

that should be included in sum-rate optimization. It has a rolein shaping the power region of the source nodes, and thereforethe MAC sum-rate is directly dependent in. We assume that𝑥2

𝑥1> ℎ1𝑠

ℎ2𝑠. The maximum Mac sum-rate is:

𝑅𝑀𝐴𝐶𝑚𝑎𝑥 =

1

2log(1 +𝑚𝑖𝑛1ℎ1𝑠 +𝑚𝑖𝑛2ℎ2𝑠) (35)

Where 𝑚𝑖𝑛2 = min(𝑄𝑠, 𝑥2), ��𝑠 = 𝑥1(1 − 𝑚𝑖𝑛2

𝑥2), and

𝑚𝑖𝑛1 = min(𝑄𝑠, ��𝑠). Since 𝑆2 has a better link condition,ℎ2𝑝

ℎ2𝑠<

ℎ1𝑝

ℎ1𝑠, the maximum MAC sum-rate is derived by 𝑝2. If

𝑅𝑀𝐴𝐶𝑚𝑎𝑥 < 𝑅𝐵𝐶

𝑚𝑎𝑥 then:

𝑅𝑠𝑢𝑚𝑚𝑎𝑥 =

⎧⎨⎩

12 (log(1 +𝑄𝑠ℎ2𝑠) + log(1 + (𝐴−𝑄𝑠

ℎ2𝑠

ℎ1𝑠)ℎ2𝑠)),

if 𝑄𝑠 < 𝑥2, 𝑄𝑠 < 𝑝∗2 and𝑄𝑠ℎ2𝑠

ℎ1𝑠+ 𝑚𝑖𝑛1ℎ1𝑠

ℎ2𝑠(1+𝑄𝑠ℎ2𝑠)> 𝐴

12 (log(1 +𝑄𝑠ℎ1𝑠) + log(1 + (𝐴−𝑄𝑠

ℎ1𝑠

ℎ2𝑠)ℎ1𝑠)),

if 𝑄𝑠 < 𝑥2, 𝑄𝑠 < ��𝑠, 𝑄𝑠 < 𝑝∗1 and𝑄𝑠ℎ1𝑠

ℎ2𝑠+ 𝑄𝑠ℎ2𝑠

ℎ1𝑠(1+𝑄𝑠ℎ1𝑠)> 𝐴

12 (log(1 + 𝑝𝑜ℎ1𝑠) + log(1 + (𝐴− 𝑝𝑜

ℎ1𝑠

ℎ2𝑠)ℎ1𝑠)),

if 𝑥2 < 𝑄𝑠 and 𝑥2 > 𝐴ℎ1𝑠

ℎ2𝑠

or,𝑄𝑠 < 𝑥2, ��𝑠 < 𝑄𝑠, ��𝑠 < 𝑝∗1 and��𝑠ℎ1𝑠

ℎ2𝑠+ 𝑄𝑠ℎ2𝑠

ℎ1𝑠(1+��𝑠ℎ1𝑠)> 𝐴

𝑅𝑀𝐴𝐶𝑚𝑎𝑥 ,otherwise

(36)where 𝑝𝑜 = min((27), 𝑄𝑠). As (36) shows, 𝑄𝑠 may limit the

104

−5 0 5 10 150.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Qp (dB)

Ave

rage

sum

−ra

te (

bits

/sec

/Hz)

Fig. 4. Maximum achievable sum-rate in terms of the primary user’sconstraint.

maximum sum-rate. For the case 𝑅𝑀𝐴𝐶𝑚𝑎𝑥 > 𝑅𝐵𝐶

𝑚𝑎𝑥 we have:

𝑅𝑠𝑢𝑚𝑚𝑎𝑥 =

⎧⎨⎩

𝑅𝐵𝐶𝑚𝑎𝑥,if 𝑝∗1 < min(𝑄𝑠, 𝑥𝑝) and 𝑝∗2 ≤ 𝑄𝑠

12 (log(1 +𝑄𝑠ℎ2𝑠) + log(1 + (𝐴−𝑄𝑠

ℎ2𝑠

ℎ1𝑠)ℎ2𝑠)),

if 𝑝∗1 < min(𝑄𝑠, 𝑥𝑝) and 𝑝∗2 > 𝑄𝑠12 (log(1 + 𝑝𝑜ℎ1𝑠) + log(1 + (𝐴− 𝑝𝑜

ℎ1𝑠

ℎ2𝑠)ℎ1𝑠)),

if 𝑥𝑝 < 𝑄𝑠 and 𝑝∗1 > 𝑥𝑝12 (log(1 +𝑄𝑠ℎ1𝑠) + log(1 + (𝐴−𝑄𝑠

ℎ1𝑠

ℎ2𝑠)ℎ1𝑠)),

if 𝑄𝑠 < 𝑥𝑝 and 𝑝∗1 > 𝑄𝑠

(37)where 𝑥𝑝 is the abscissa of the intersection of Line 1 andLine2. The maximum achievable sum-rate for 𝑥2

𝑥1< ℎ1𝑠

ℎ2𝑠can

be obtained similarly.

IV. SIMULATION RESULT

In this section we provide numerical simulations to illustratethe maximum achievable sum-rate of the considered TWRCsystem. All the channel gains have exponential distributionwith unit mean. The average sum-rate is obtained over1000000 channel realization. Fig.4 shows how the achievablesum-rate varies in terms of the allowed interference at theprimary user. The transmit power constraints at the sourcenodes and the relay are set to 10 dB. As the power constraintat the primary user gets more relaxed, the system sum-rateapproaches the sum-rate of a TWRC system in a non-sharingenvironment.

Fig.5 depicts the achievable sum-rate of the system versus𝑄𝑠 for the optimal power allocation (OPA) and equal power al-location (EPA). In EPA scheme, the source nodes are assignedequal power based on the available power region. The relayemploys the same strategy. The power constraints of the relayand the primary user are 10 dB and 0 dB, respectively. As itwas expected OPA leads to a superior sum-rate performance.The simulations confirm that relaxing a constraint dose notresult in a considerable sum-rate increase. It can be seen inthe Fig.5 that the upward trend of the sum-rate slows down

0 5 10 15 200.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

Qs (dB)

Ave

rage

sum

−ra

te (

bits

/sec

/Hz)

OPAEPA

Fig. 5. Comparison of achievable sum-rate for OPA and EPA.

after a while. In this situation, 𝑄𝑝 is the dominant constraintand drives the sum-rate.

V. CONCLUSION

In this research, the sum-rate performance of the TWRC ina spectrum-sharing environment has been studied. It has beenshown that the proposed power allocation improves the sum-rate comparing to the equal power allocation. Furthermore,the impact of the model constraints in achievable sum-ratehas been investigated.

REFERENCES

[1] B. Rankov and A. Wittneben, “Spectral efficient protocols for half-duplex fading relay channels,” IEEE J. Sel. Areas Commun., vol. 25,no. 2, pp. 379–389, Feb. 2007.

[2] ——, “Achievable rate regions for the two-way relay channel,” in Proc.IEEE Int. Symp. on Information Theory (ISIT), Jul. 2006, pp. 1668 –1672.

[3] S. Haykin, “Cognitive radio: brain-empowered wireless communica-tions,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb.2005.

[4] K. Jitvanichphaibool, Y.-C. Liang, and R. Zhang, “Beamforming andpower control for multi-antenna cognitive two-way relaying,” in Proc.IEEE Wireless Communications and Networking Conference (WCNC),Apr. 2009, pp. 515–520.

[5] I. Hammerstrom, M. Kuhn, C. Esli, J. Zhao, A. Wittneben, and G. Bauch,“MIMO two-way relaying with transmit CSI at the relay,” in Proc. IEEESignal Processing Advances in Wireless Communications, (SPAWC), Jun.2007, pp. 1–5.

[6] Y. Wu, P. A. Chou, and S. Y. Kung, “Information exchange in wirelessnetworks with network coding and physical-layer broadcast,” in Proc.39th Annual Conf. on Information Sciences and Systems (CISS), Mar.2005.

[7] T. J. Oechtering, C. Schnurr, I. Bjelakovic, and H. Boche, “Broadcastcapacity region of two-phase bidirectional relaying,” IEEE Trans. Inf.Theory, vol. 54, no. 1, pp. 454–458, Jan. 2008.

[8] R. F. Wyrembelski, T. J. Oechtering, and H. Boche, “Decode-and-Forward Strategies for Bidirectional Relaying,” in Proc. 19th AnnualIEEE Int. Symp. on Personal, Indoor and Mobile Radio Communications(PIMRC ’08), Sep. 2008, pp. 1–6.

[9] S. J. Kim, P. Mitran, and V. Tarokh, “Performance bounds for bidirec-tional coded cooperation protocols,” IEEE Trans. Inf. Theory, vol. 54,no. 11, pp. 5235–5241, Nov. 2008.

[10] T. Cover and J. Thomas, Elements of Information Theory. New York:Wiley, 1991.

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