On the Capacity of Pairwise Collaborative Networks

1 On the Capacity of P airwise Collaborati v e Networks Saeed A. Astaneh † , Saeed Gazor † , Hamid Behroozi † † † Department of Electrical and Computer Engineering, Queen’ s Uni versity , Kingston, ON, Canada † † Department of Mathematics and Statistics, Queen’ s Uni versity , Kingston, ON, Canada Email: astaneh, s.gazor , behroozi@queensu.ca Abstract —W e derive expressions for the achievable rate r egion of a collaborative coding scheme in a tw o-transmitter , tw o- recei ver Pairwise Collaborative Network (PCN) where one trans- mitter and recei ver pair , namely relay pair , assists the other pair , namely the source pair , by partially decoding and f orwarding the transmitted message to the intended receiv er . The relay pair pro vides such assistance while handling a private message. W e assume that users can use the past channel outputs and can transmit and receiv e at the same time and in the same frequency band. In this collaborative scheme, the transmitter of the sour ce pair splits its inf ormation into tw o independent parts. Ir onically , the r elay pair employs the decode and f orward coding to assist the source pair in delivering a part of its message and re-encodes the decoded message along with private message, which is intended to the receiver of the relay pair , and broadcasts the results. The recei ver of the relay pair decodes both messages, retrieves the private message, re-encodes and transmits the decoded massage to the intended destination. W e also characterize the achiev able rate r egion f or Gaussian PCN. Finally , we provide numerical results to study the rate trade off for the in volved pairs. Numerical result shows that the collaboration offers gain when the channel gain between the users of the relay pair are strong. It also shows that if the channel conditions between transmitters or between the recei vers of the relay and source pairs are poor , such a collaboration is not beneficial. Index T erms —Pairwise collaborativ e network, rate splitting, decode and forward. I . I N T RO D U C T I O N In a multi user network users may collaborate to jointly con vey the information. V an der Meulen [1] introduced the relay channel where a relay forwards the data from a source to the destination. Cov er and El Gamal [2] proved some capacity theorems for a single relay channel. In a collaborative network, users may collaborate to transmit message of other users while handling their o wn priv ate messages; this can be regarded as a generalization of the traditional relay channel. W e present a pairwise relaying collaboration model where a pair of transmitter and receiv er collaborates with the source pair in deliv ering the message of the source pair along with its own priv ate message. Our proposed model dif fers from previous research in that we consider collaboration schemes that the transmitter and recei ver of the relay pair handles a priv ate message, which, to the best of our knowledge, no previous work has considered in this setting. Figure 1 represents such a network where the 1st user, intends to send a message to the 4th user, and the 2nd user to the 3rd user . W e propose two collaboration schemes where in the first scheme, the transmitter of the relay pair, the 1st user, splits its message into two independent parts. The relay pair collaborates with Fig. 1. Collaboration schemes in the PCN: the partial decode and forward scheme; the source pair transmitter splits its information into two independent parts. The relay pair employs decode and forward scheme to assist the source pair in delivering only one part of its message. the source pair via decode and forward coding to transmit a part of the message of the source pair and the priv ate message of the relay pair . In the second scheme the relay pair partially cancels the interference of other users and sends the compressed observed signal to the intended receiver of source pair , the 4th user . Collaboration between wireless users has been in vestigated recently by several authors. Liang and V eera valli [3] studied a cooperative relay broadcast channel with three users where relay links are incorporated into standard two-user broadcast channels to support user cooperation. Liang and Kramer [4] hav e found improved bounds for the relay broadcast channel. T annious and Nosratinia in [5] dev eloped decode and forward and compress and forward strategies for a network of one relay channel with pri v ate messages where in addition to the traditional communication from source to destination (assisted by relay), the source has a priv ate message for the relay , and the relay has a priv ate message for the destination (see [6] for a survey on decode and forward and compress and forward strategies). Akhavan and Gazor [7] in vestigated multi-hopping strategies and resource allocation in such networks. Reznik, Kulkarni and V erdu in [8] further studied the relay broadcast collaborativ e model for the case of more than two destinations. Sendonaris, Erkip and Aazhang in [9], [10] showed that collaboration enlarges the achie vable rate region in a channel with two collaborative transmitters and a single receiver . Laneman, Tse and W ornell considered a fading channel with two cooperati ve transmitters and two non-cooperati ve recei vers [11]. Host-Madsen in [12], [13] presented the achiev able rate regions for channels with transmitter and/or receiv er collabo- 2 ration. Ng, Jindal, Goldsmith and Mitra [14] inv estigated ca- pacity improv ement from transmitter and recei ver cooperation in a two-transmitter , two-recei ver network. In this paper , we extend the results of [5], [7] and study the achiev able rate region of the decode and forward coding schemes in the PCN. W e present in Section II the network model. In Section III we dev elop the rate splitting in con- junction with decode and forward coding scheme for the PCN and determine its capacity . W e in vestigate the additi ve white Gaussian noise PCN rate region in Section IV. Finally in Section V we give the concluding remarks. I I . S Y S T E M M O D E L The PCN consists of inputs x i where i ∈ { 1 , 2 , 3 } , out- puts y j where j ∈ { 2 , 3 , 4 } and the transition probability p ( y 2 , y 3 , y 4 | x 1 , x 2 , x 3 ) (see Figure 1). The 1st user wishes to send the message w 1 to the 4th user , while the 2nd user wishes to send the message w 2 to the 3rd user . W e define an additive white Gaussian noise (A WGN) PCN with the input output relation: Y i = Z i + P j 6 = i p h ij X j where X i , Y i and Z i denote input, output and channel noise with normal distributions, i.e. Z i ∼ N (0 , N i ) , respectively . Let denote the power gain of the communication channel between the i th and j th user by h ij . W e impose the po wer constraints E  X 2 i  6 P i for all channel inputs. W e assume that the users can transmit and receiv e at the same time and in the same frequency band. In this paper , let X , x and x denote a random variable, a scalar and a vector , respectively . W e define ¯ x = 1 − x and C ( x ) = 1 2 log(1 + x ) . I I I . C O L L A B O R A T I O N V I A P A RT I A L D E C O D E - A N D - F O RW A R D In this section we consider a collaborative scheme, partial decode and forward, which includes rate splitting technique at the source pair transmitter and decode and forward relaying at the relay pair transmitter and receiv er . The source pair transmitter splits the message w 1 into two independent parts, w 11 and w 12 . The source pair transmitter, 1st user , encodes w 11 and w 12 to the code word x 1 . The relay pair transmitter, 2nd user, decodes w 12 and re-encodes both its message, i.e. w 2 , and w 12 , to x 2 . The relay pair receiver , 3rd user , retrie ves w 12 and w 2 from y 3 and re-encodes w 12 to x 3 . Finally , the source pair recei ver , 4th user , by using y 4 estimates its intended message w 12 and w 11 . In the following we prov e that the rates ( R 1 , R 2 ) , giv en by (1) sho wn at the top of the next page, are achie vable for the PCN for some joint distrib ution p ( x 3 ) p ( u 1 | x 3 ) p ( u 2 | u 1 x 3 ) p ( x 2 | u 1 x 3 ) p ( x 1 | u 1 u 2 x 3 ) . For this achie vable re gion, we apply Fourier-Motzkin elim- ination to eliminate R 11 and R 12 from the bounds and then obtain the region (2) which provides a simpler form. W e use the coding strategies dev eloped in [2], [5], [6], [15] for relay and multiple access channels (MACs). The 1st user uses a three-le vel superposition block Marko v encoding, while the 2nd user uses a two-lev el and the 3rd user a single-lev el superposition coding. Furthermore, we use the regular encoding/backward decoding techniques. W e divide the messages w 1 and w 2 into B blocks for b = 1 , 2 , ..., B and send these message blocks in B + 2 transmission blocks. In the follo wing we construct the codebooks and discuss the decoding in each block. Random Codebook Construction: 1) W e generate 2 nR 12 i.i.d. x 3 = ( x 31 , x 32 , ..., x 3 n ) se- quences, each with distribution p ( x 3 ) = n Q i =1 p ( x 3 i ) and label them x 3 ( w 00 12 ) . 2) F or each x 3 ( w 00 12 ) , we generate 2 nR 12 i.i.d. u 1 se- quences, each with distribution p ( u 1 ) = n Q i =1 p ( u 1 i | x 3 i ) and label them u 1 ( w 0 12 , w 00 12 ) . 3) F or each pair u 1 ( w 0 12 , w 00 12 ) and x 3 ( w 00 12 ) , we gen- erate 2 nR 12 i.i.d. u 2 sequences, each with distrib u- tion p ( u 2 ) = n Q i =1 p ( u 2 i | x 3 i , u 1 i ) and label them u 2 ( w 12 , w 0 12 , w 00 12 ) . 4) F or each pair u 1 ( w 0 12 , w 00 12 ) and x 3 ( w 00 12 ) , we generate 2 nR 2 i.i.d. x 2 sequences, each with distribution p ( x 2 ) = n Q i =1 p ( x 2 i | x 3 i , u 1 i ) and label them x 2 ( w 2 , w 0 12 , w 00 12 ) . 5) F or each triplet u 1 ( w 0 12 , w 00 12 ) , x 3 ( w 00 12 ) and u 2 ( w 12 , w 0 12 , w 00 12 ) , we generate 2 nR 11 i.i.d. x 1 sequences, each with distribution p ( x 1 ) = n Q i =1 p ( x 1 i | x 3 i , u 1 i , u 2 i ) and label them x 1 ( w 11 , w 12 , w 0 12 , w 00 12 ) . Encoding: For each time b = 1 , 2 , ..., B + 2 the users send the follo wing sequences: 1) x 1 ( w 11 , b , w 12 , b , 1 , 1) , x 2 ( w 2 , b , 1 , 1) , x 3 (1) b = 1 2) x 1 ( w 11 , b , w 12 , b , w 12 , b − 1 , 1) , x 2 ( w 2 , b , w 12 , b − 1 , 1) , x 3 (1) b = 2 3) x 1 ( w 11 , b , w 12 , b , w 12 , b − 1 , w 12 , b − 2 ) , x 2 ( w 2 , b , w 12 , b − 1 , w 12 , b − 2 ) , x 3 ( w 12 , b − 2 ) b = 3 , ..., B 4) x 1 (1 , 1 , w 12 , b − 1 , w 12 , b − 2 ) , x 2 (1 , w 12 , b − 1 , w 12 , b − 2 ) , x 3 ( w 12 , b − 2 ) , b = B + 1 5) x 1 (1 , 1 , 1 , w 12 , b − 2 ) , x 2 (1 , 1 , w 12 , b − 2 ) , x 3 ( w 12 , b − 2 ) b = B + 2 Decoding: 1) The 2nd user decodes w 12 , b by looking for ˆ w 12 , b such that y 2 , b , x 2 ( w 2 , b , w 12 , b − 1 , w 12 , b − 2 ) , x 3 ( w 12 , b − 2 ) , u 1 ( w 12 , b − 1 , w 12 , b − 2 ) and u 2 ( w 12 , b , w 12 , b − 1 , w 12 , b − 2 ) are jointly typical. The decoding is reliable if R 12 < I ( Y 2 ; U 2 | X 2 , X 3 , U 1 ) . 2) The 3rd user decodes w 12 , b , w 12 , b − 1 , w 11 , b and w 2 , b by looking for ˆ w 12 , b , ˆ w 12 , b − 1 , ˆ w 11 , b and ˆ w 2 , b such that y 3 , b , x 1 ( w 11 , b , w 12 , b , w 12 , b − 1 , w 12 , b − 2 ) , x 2 ( w 2 , b , w 12 , b − 1 , w 12 , b − 2 ) , x 3 ( w 12 , b − 2 ) , u 1 ( w 12 , b − 1 , w 12 , b − 2 ) and u 2 ( w 12 , b , w 12 , b − 1 , w 12 , b − 2 ) are jointly typical. Here, the 1st and 2nd users attempt to transmit a common message w 12 along with their priv ate messages, i.e. w 11 and w 2 , respectiv ely . It is shown in [15], [16] that this step can be made reliably 3                    R 11 < min { I ( Y 4 ; X 1 | U 1 , U 2 , X 2 , X 3 ) , I ( Y 3 ; X 1 | X 2 , X 3 , U 1 , U 2 ) } R 12 < I ( Y 2 ; U 2 | U 1 , X 2 , X 3 ) R 2 < min { I ( Y 3 ; X 2 | X 1 , X 3 , U 1 , U 2 ) , I ( Y 4 ; X 2 | U 1 , U 2 , X 1 , X 3 ) } R 2 + R 11 < min { I ( Y 3 ; X 1 , X 2 | X 3 , U 1 , U 2 ) , I ( Y 4 ; X 1 , X 2 | U 1 , U 2 , X 3 ) } R 2 + R 11 + R 12 < min { I ( Y 3 ; X 1 , X 2 , U 1 , U 2 | X 3 ) , I ( Y 4 ; U 1 , U 2 , X 1 , X 2 , X 3 ) } R 1 = R 11 + R 12 (1)                    R 1 < I ( Y 2 ; U 2 | U 1 , X 2 , X 3 ) + φ 1 R 2 < min { I ( Y 3 ; X 2 | U 1 , U 2 , X 1 , X 3 ) , I ( Y 4 ; X 2 | U 1 , U 2 , X 1 , X 3 ) } R 1 + R 2 < min { φ 2 , I ( Y 2 ; U 2 | U 1 , X 2 , X 3 ) + φ 3 } φ 1 = min { I ( Y 3 ; X 1 | U 1 , U 2 , X 2 , X 3 ) , I ( Y 4 ; X 1 | U 1 , U 2 , X 2 , X 3 ) } φ 2 = min { I ( Y 4 ; U 1 , U 2 , X 1 , X 2 , X 3 ) , I ( Y 3 ; U 1 , U 2 , X 1 , X 2 | X 3 ) } φ 3 = min { I ( Y 3 ; X 1 , X 2 | U 1 , U 2 , X 3 ) , I ( Y 4 ; X 1 , X 2 | U 1 , U 2 , X 3 ) } (2) if          R 2 < I ( Y 3 ; X 2 | X 1 , X 3 , U 1 , U 2 ) R 11 < I ( Y 3 ; X 1 | X 2 , X 3 , U 1 , U 2 ) R 2 + R 11 < I ( Y 3 ; X 1 , X 2 | X 3 , U 1 , U 2 ) R 2 + R 11 + R 12 < I ( Y 3 ; X 1 , X 2 , U 1 , U 2 | X 3 ) 3) The 4th user decodes w 2 , b , w 12 , b and w 12 , b − 1 and w 12 , b − 2 . W e hav e the problem of sending correlated sources over a MA C as treated in [15], [16]. The decoding can be done reliably if          R 11 < I ( Y 4 ; X 1 | U 1 , U 2 , X 2 , X 3 ) R 2 < I ( Y 4 ; X 2 | U 1 , U 2 , X 1 , X 3 ) R 11 + R 2 < I ( Y 4 ; X 1 , X 2 | U 1 , U 2 , X 3 ) R 11 + R 12 + R 2 < I ( Y 4 ; U 1 , U 2 , X 1 , X 2 , X 3 ) The rate region in (1) follows from combining the achiev able regions derived above. I V . C A PAC I T I E S O F AW G N P C N S In this section, we in vestigate the achie vable rate region of in volved pairs when two pairs of source and relay collaborate in sending information to the intended receivers. W e compare the achie vable rate of the proposed collaboration scheme with the scenario where pairs do not collaborate. In the absence of collaboration between pairs, we model the channel by interference channel in which transmission of information of a pair interferes with the communication between the other pair [17], [18]. The capacity of the interference channel (IFC) is an open problem, howe ver , in the case where h 14 ≤ h 24 and h 23 ≤ h 13 the capacity re gion of interference channel is completely characterized as:          R 1 < C  h 14 P 1 N 4  R 2 < C  h 23 P 2 N 3  R 1 + R 2 < min n C  h 14 P 1 + h 24 P 2 N 4  , C  h 23 P 2 + h 13 P 1 N 3 o (4) Now , we concentrate on the A WGN PCN. Employing partial decode and forward scheme, the rates ( R 1 , R 2 ) , are achiev able for the A WGN PCN:                  R 1 < C  α ¯ β h 12 P 1 ¯ αh 12 P 1 + N 2  + φ 1 R 2 < min  C  ¯ δ h 23 P 2 N 3  , C  ¯ δ h 24 P 2 N 4  R 1 + R 2 < min  φ 2 , C  α ¯ β h 12 P 1 ¯ αh 12 P 1 + N 2  + φ 3  (5) where φ 1 , φ 2 and φ 3 are given by (3) shown at the top of the following page. W e use the following independent normal distributions to find the rate region for the partial decode and forward coding scheme (2): A ∼ N (0 , ¯ αP 1 ) , B ∼ N  0 , α ¯ β P 1  , C ∼ N (0 , αβ ¯ γ P 1 ) , D ∼ N (0 , αβ γ P 1 ) , E ∼ N  0 , ¯ δ P 3  , where α, β , γ , δ ∈ [0 , 1] . Furthermore, we let X 1 = A + B + C + D , X 2 = E + C + D , X 3 = D , U 1 = C + D and U 2 = B + C + D . Here, we mov e on to study the condition under which collaboration improves the achie vable rate of pairs. Numerical result sho ws that the collaboration offers capacity gain when the channel gain between the source pair is week and the corre- sponding channel between relay users is strong. It also shows that if the channels condition between the transmitters, h 12 and between the receivers, h 34 , are poor, such a collaboration is not beneficial. W e consider the A WGN PCN with P 1 = P 2 = P 3 = 1 and N 2 = N 3 = N 4 = 1 and we examine the proposed collaboration scheme under different channel conditions. W e compare the achiev able rate region of the proposed scheme with the scheme where pairs do not collaborate, i.e. interfer - ence channel. W e also consider the scenario where both pairs acquire the same rate as R 1 = R 2 and study the capacity gain of the schemes. Figure 2 demonstrates the trade off between the achieved rate region of the source and relay pairs. The achie vable rate region of the source pair expands as the transmitter increases its transmit power . Similar to the source pair , increasing the transmit power of the relay pair increases the achiev able rate of the relay pair . 4                                        φ 1 = min  C  ¯ αh 13 P 1 N 3  , C  ¯ αh 14 P 1 N 4  φ 2 = min                    C    h 14 P 1 ¯ α + α ¯ β + αβ ¯ γ  1 + q h 24 h 14 δ P 2 αβ P 1  2 + αβ γ  1 + q h 24 h 14 δ P 2 αβ P 1 + q h 34 h 14 P 3 αβ γ P 1  2 N 4    , C    h 13 P 1 ¯ α + α ¯ β + αβ ¯ γ  1 + q h 23 h 13 δ P 2 αβ P 1  2 + h 23 h 13 ¯ δ P 2 P 1 N 3                       φ 3 = min  C  ¯ αh 13 P 1 + ¯ δ h 23 P 2 N 3  , C  ¯ αh 14 P 1 + ¯ δ h 24 P 2 N 4  (3) First we in vestigate the case where the communication channel between the source pair is poor . The channel condition h 12 = 1 , h 13 = 10 , h 14 = 1 , h 23 = 10 , ; h 24 = 10 and h 34 = 1 ex emplifies such a condition. Figure 2(a) shows the rate region of the in volved pairs employing the proposed collaboration scheme in conjunction with rate region of interference channel. W e observe that under this condition, collaboration offers small capacity gain. W e also observe that the relay pair has incentiv e to collaborate and obtain more rate than interference channel, only if the source pair demands for more rate. W e plotted the line X = Y to in vestigate the achie vable rate on condition that both pairs demands equal rates, i.e. R 1 = R 2 . W e observe that under this condition collaboration is not beneficial. Exploiting strong communication links between the 1st and the 2nd users and between the 3rd and the 4th users, pairs ob- tain a considerable capacity g ain which is sho wn in Figure 2(b) with h 12 = 10 , h 13 = 10 , h 14 = 1 , h 23 = 10 , ; h 24 = 10 and h 34 = 10 . In this scenario, the source pair is suffering from poor direct channel gain, between the 1st and the 4th users, howe ver , the communication channel between pair users to relay users is strong. In that case collaboration enhances the achiev able rate of both pairs. It also offers dramatic gain if both pairs are interested in equal rates. Collaboration only improves the achiev able rate if the direct link between relay pair, i.e. the 2nd and the 3rd user , is strong. Otherwise as shown in Figure 2(c), with h 12 = 10 , h 13 = 10 , h 14 = 10 , h 23 = 1 , ; h 24 = 10 and h 34 = 10 , collaboration does not enlarge the rate region. Howe ver , increasing the channel gain between the 3rd and 4th users and between the 1st and the 2nd, the pairs gain as much as the non collaborativ e scheme (see Figure 2(d)), with h 12 = 1 , h 13 = 10 , h 14 = 10 , h 23 = 10 , ; h 24 = 10 and h 34 = 1 . This emphasizes that the ef ficiency of proposed scheme significantly depends on the channel gain between the relay users. Lastly , in equal channel condition for both pairs, i.e. h 12 = 10 , h 13 = 10 , h 14 = 10 , h 23 = 10 , ; h 24 = 10 and h 34 = 10 (Figure 2(e)), the channel gain between the 1st and the 2nd users and between the 3rd and the 4th users, increases the capacity gain for inv olved pairs. V . C O N C L U S I O N W e hav e considered a network of collaborati ve transmitter- receiv er pairs in which one pair (relay pair) acts as relay to assist the source pair in deliv ering the message of the source pair as well as its own priv ate message. W e have studied partial decode and forward collaborative schemes and established the capacity of this coding schemes for the PCN. In the proposed scheme we let the transmitter of the source pair to split its message into two independent parts. The relay pair decodes and forwards only one part of the message of the source pair and re-encodes and transmits the decoded message along with the relay pair pri vate message. Having decoded both messages, the receiv er of the relay pair decodes and transmits the message of the source pair to the intended destination. For A WGN PCNs, we have characterized the achiev able rate regions. W e hav e also provided numerical results and compared the proposed collaboration scheme with achiev able rate of a non collaborati ve scheme, i.e. interference channel. W e have examined the channel conditions under which such a collaboration is beneficial. W e have shown that when the channel gain between the source pair is week collaboration offers capacity gain to both pairs. Howe ver , if the channels condition between the in volv ed pairs are poor such a collaboration is not beneficial. R E F E R E N C E S [1] E. C. v an der Meulen, “Three-terminal communication channels, ” Adv . Appl. Prob , vol. 3, no. 1, pp. 120–154, 1971. [2] T . Cov er and A. Gamal, “Capacity theorems for the relay channel, ” IEEE T rans. Inf. Theory , v ol. 25, no. 5, pp. 572–584, 1979. [3] Y . Liang and V . V . V eerav alli, “Cooperati ve relay broadcast channels, ” IEEE T rans. Inf. Theory , vol. 53, no. 3, pp. 900–928, March 2007. [4] Y . Liang and G. Kramer , “Capacity theorems for cooperative relay broadcast channels, ” in Pr oc. 40th Annual Conference on Information Sciences and Systems , pp. 1719–1724, 22-24 March 2006. [5] R. T annious and A. Nosratinia, “Relay Channel With Priv ate Messages, ” IEEE T rans. Inf. Theory , vol. 53, no. 10, pp. 3777–3785, 2007. [6] G. Kramer , M. Gastpar , and P . Gupta, “Cooperativ e Strategies and Capacity Theorems for Relay Networks, ” IEEE T rans. Inf. Theory , vol. 51, no. 9, pp. 3037–3063, 2005. [7] S. A. Astaneh and S. Gazor, “Joint Relay and Node Selection in Collaborativ e Networks, ” 24th Biennial Symposium on Communications QBSC’08 , June 2008. [8] A. Reznik, S. Kulkarni, and S. V erdu, “Broadcast-relay channel: capacity region bounds, ” Information Theory , 2005. ISIT 2005. Pr oceedings. International Symposium on , pp. 820–824, 4-9 Sept. 2005. 5 (a) (b) (c) (d) (e) Fig. 2. The achiev able rate region for collaborativ e, partial decode and forward (PDF) and non collaborati ve, interference channel (IFC) in a pairwise collaborativ e network with different scenarios for channel conditions: a) h 12 = 1 , h 13 = 10 , h 14 = 1 , h 23 = 10 , ; h 24 = 10 and h 34 = 1 b) h 12 = 10 , h 13 = 10 , h 14 = 1 , h 23 = 10 , ; h 24 = 10 and h 34 = 10 c) h 12 = 10 , h 13 = 10 , h 14 = 10 , h 23 = 1 , ; h 24 = 10 and h 34 = 10 d) h 12 = 1 , h 13 = 10 , h 14 = 10 , h 23 = 10 , ; h 24 = 10 and h 34 = 1 e) h 12 = 10 , h 13 = 10 , h 14 = 10 , h 23 = 10 , ; h 24 = 10 and h 34 = 10 . [9] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity . Part I. System description, ” IEEE T rans. Commun. , vol. 51, no. 11, pp. 1927–1938, 2003. [10] ——, “User cooperation diversity . Part II. Implementation aspects and performance analysis, ” IEEE T rans. Commun. , vol. 51, no. 11, pp. 1939– 1948, 2003. [11] J. Laneman, D. Tse, and G. W ornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior, ” IEEE T rans. Inf. Theory , vol. 50, no. 12, pp. 3062–3080, 2004. [12] A. Host-Madsen, “On the achiev able rate for receiver cooperation in ad-hoc networks, ” ISIT 2004. Pr oceedings. International Symposium on Information Theory . [13] ——, “Capacity Bounds for Cooperati ve Div ersity, ” IEEE T rans. Inf. Theory , vol. 52, no. 4, pp. 1522–1544, 2006. [14] C. Ng, N. Jindal, A. Goldsmith, and U. Mitra, “Capacity Gain From T wo-Transmitter and T wo-Receiv er Cooperation, ” IEEE T rans. Inf . The- ory , vol. 53, no. 10, pp. 3822–3827, 2007. [15] T . Cover , A. Gamal, and M. Salehi, “Multiple access channels with arbitrarily correlated sources, ” IEEE T rans. Inf. Theory , vol. 26, no. 6, pp. 648–657, 1980. [16] Y . Liang and H. V . Poor, “Multiple-access channels with confidential messages, ” IEEE Tr ans. Inf. Theory , vol. 54, no. 3, pp. 976–1002, March 2008. [17] T . Cover and J. Thomas, Elements of Information Theory . W iley- Interscience New Y ork, 2006. [18] I. Sason, “On achiev able rate regions for the Gaussian interference channel, ” Information Theory , IEEE Tr ansactions on , vol. 50, no. 6, pp. 1345–1356, 2004.