Convolutional Network-Coded Cooperation in Multi-Source Networks with a Multi-Antenna Relay

# Convolutional Network-Coded Cooperation in Multi-Source Networks with a Multi-Antenna Relay

Alireza Karbalay-Ghareh, Masoumeh Nasiri-Kenari, , and
Mohsen Hejazi
The authors are with the Wireless Research Laboratory (WRL), Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran (email: karbalayghareh, mhejazi @ee.sharif.edu, mnasiri@sharif.edu).
###### Abstract

We propose a novel cooperative transmission scheme called Convolutional Network-Coded Cooperation (CNCC) for a network including sources, one -antenna relay, and one common destination. The source-relay (S-R) channels are assumed to be Nakagami- fading, while the source-destination (S-D) and the relay-destination (R-D) channels are considered Rayleigh fading. The CNCC scheme exploits the generator matrix of a good systematic convolutional code, with the free distance of designed over , as the network coding matrix which is run by the network’s nodes, such that the systematic symbols are directly transmitted from the sources, and the parity symbols are sent by the best antenna of the relay. An upper bound on the BER of the sources, and consequently, the achieved diversity orders are obtained. The numerical results indicate that the CNCC scheme outperforms the other cooperative schemes considered, in terms of the diversity order and the network throughput. The simulation results confirm the accuracy of the theoretical analysis.

Cooperative networks, linear network coding, convolutional network-coded cooperation, diversity order, network throughput.

## I Introduction

one of the most important and intrinsic features of the wireless networks is fading. This phenomenon induces many adverse effects in the networks, and considerably reduces the performance. Diversity is a well-known technique to deal with fading, which is used in the various domains such as time, frequency, and space. Cooperative relay-based networks have been proposed to combat fading, by benefiting from the spatial diversity through the relays or the antennas of a multi-antenna relay [1, 2, 3, 4]. The traditional cooperative transmission schemes, such as Amplify-and-Forward (AF) and Decode-and-Forward (DF), have been introduced and evaluated in the the numerous papers like [4, 5, 6, 7, 8]. The basic and common shortcoming of these schemes is the reduction of the network throughput in the multi-sources networks [9, 10]. To eliminate this problem, the idea of using network coding in the cooperative networks has been suggested in the recent years.

Network Coding was first introduced by Ahlswede et al. in the seminal work [11], and then proposed as Linear Network Coding (LNC) in [12]. The main idea of the LNC is that in a multi-hop network, the intermediate nodes combine linearly the received data from the source nodes, and transmit them to the next hops, instead of transmitting each received data separately. This approach can dramatically reduce the delay, and consequently, increase the network throughput. The initial researches on network coding were related to the wired networks, but it was gradually generalized to the wireless networks [13, 14]. The network codes were first considered in the network layer with the assumption of an ideal and error-free physical layer, but afterwards were extended to the physical layer by considering the effects of fading and noise. The most spectrally efficient Physical Layer Network Coding (PLNC) was proposed in [15, 16, 17], in which the nodes simultaneously transmit their own data, and the other nodes receive the linear combinations of the data transmitted plus the noise during just one time slot; nonetheless, a high level of synchronization must be hold in order this scheme to be viable.

Recently, the network coding has been exploited in the cooperative relay networks due to its capability to increase the network throughput. Quite a few of these works have somehow demonstrated that utilizing the linear network coding in the multi-source cooperative networks instead of the traditional schemes leads to the same diversity order, while improves the network throughput. [18] has investigated the network coding in a double-source cooperative network, where each node creates a linear combination of its own symbol and its partner’s correctly decoded symbol on . In [20], a cooperative transmission scheme based on the network coding along with the multi-user detection has been proposed to improve the users’ Bit Error Rate (BER). [21] has introduced a cooperative scheme based on the network coding and the best relay selection technique in a network with source-destination pairs and intermediate relays. In [21], each destination must correctly decode the other sources’ symbols in order to recover its own symbol, and without this assumption the diversity order decreases from to . The aforementioned papers have used the binary field to build the linear combinations. Nevertheless, some papers considered using the higher fields . Specifically, [23] showed that in the multi-source networks with the error-prone source-relay channels, the network coding on can not lead to the full diversity order. Furthermore, [25] considered a network with sources where each source acts as a relay for the other sources, and a maximum distance separable (MDS) linear block code is utilized by the relays. The scheme of [25] leads to the network throughput equal to symbol per channel use (spcu) and diversity order of . Although based on the Singleton bound [32], the diversity order of was expected in [25], but due to the error possibility in the inter-source channels this expectation had not been realized. [27] has generalized the idea of [25], in which a () MDS linear block code is used as the network coding matrix, which leads to the improved Singleton bound with the same throughput of spcu. Although these schemes have boosted the diversity order, but they are still suffering from low network throughput. To address this problem, [28] has introduced Complex Field Network Coding (CFNC) scheme in a network including sources, which reaches to the diversity order of , as well as the network throughput of approximately spcu. In [29], the CFNC has been also utilized in a network including sources, relays, and one common destination, which reaches to the full diversity order of along with the network throughput of symbol per source per channel use (spspcu) (or equivalently spcu), while the network throughput of the traditional (AF and DF) and the finite field linear network coding schemes in such a network are respectively equal to spcu and spcu. The main reason for the higher throughput of the CFNC schemes in [28] and [29] is the utilization of the complex fields instead of the finite fields, in which the symbols of different nodes, similar to the PLNC, are simultaneously transmitted and their linear combinations in the complex field are received in the other nodes; as a result, this scheme possesses highly complicated synchronization concerns of the PLNC. Furthermore, in [30], a cooperative transmission scheme based on the linear network codes designed over has been introduced for improving the Diversity-Multiplexing Trade-off (DMT) in a network consisting of source-destination pairs, and intermediate relays, which reaches the full diversity order of as well as the network throughput of spcu. In [30], the parity check matrix of a MDS linear block code is used as the network coding matrix. However, the field size required for such MDS codes increases exponentially with the values of and , which can significantly enhance the system complexity.

In this paper, to increase the diversity order without the degradation of the network throughput, we propose a new cooperative transmission scheme called “Convolutional Network-Coded Cooperation” (CNCC) in a network including sources, one relay with antenna, and one common destination. The CNCC scheme uses the generator matrix of a systematic convolutional code designed on throughout the network as the network coding matrix. In this scheme, the systematic packets of the sources are directly transmitted from the sources to the destination, which are simultaneously received and decoded by the relay. If the relay correctly decodes all the packets, it will generate the parity packets by using the underlying convolutional code, and finally will transmit the parity packets by its best antenna (the strongest R-D channel) to the destination. In contrast to the linear network coding scheme, in our proposed CNCC scheme, the diversity order can be enhanced without the reduction of the network throughput, by the increase of the underlying convolutional code’s constraint length. It will be shown that the proposed scheme improves the network throughput, and its diversity order at the worst scenarios (weak S-R channels) is equal to , and at the best scenarios (strong enough S-R channels) can be much greater than . In fact, when the relay is located close to the sources, the diversity order can reach to , where , and is the free distance of the exploited convolutional code. It should be noted that the free distance of the convolutional code is increased by the constraint length at the expense of the decoder’s complexity. In contrast to the linear network coding scheme, in our proposed scheme, the network throughput does not depend on the number of antennas, but on the number of parity bits of the convolutional codes.

The rest of this paper is organized as follows. In Section II, the system model is described. In Section III, the proposed CNCC scheme is introduced. In Section IV, the performance of the CNCC scheme is analyzed in terms of the sources’ BER and the diversity order. Four examples for the CNCC scheme are presented in Section V. The numerical results are provided in Section VI. Ultimately, Section VII concludes the paper, and suggests some future works.

## Ii System Model

We consider a cooperative network consisting of single-antenna sources, one -antenna relay, and one common single-antenna destination, as shown in Fig. 1. The relay’s antennas are omnidirectional which can be used for both the reception and the transmission. All the S-D, R-D, and S-R channels are assumed to be block fading channels with the depth of bit transmission intervals, which change independently and identically from one block to another block. Each source has information bits for transmission which is divided to the packets of length bits, where . We assume a perfect interleaving process throughout the network. That is, the packets in the sources and the relay are interleaved before transmissions, and the received packets at the relay and the destination are deinterleaved before decoding. The interleavers must have a sufficient depth (usually greater than ) such that the successive bits of a packet experience almost independent fading coefficients. Furthermore, the Binary Phase Shift Keying (BPSK) modulation is used for the transmissions.

The S-D and R-D channels are assumed to be Rayleigh fading channels. Due to the fact that the relay is close to the sources, the S-R channels are assumed to be Nakagami- fading channels which for are reduced to the Rayleigh, and for act stronger than Rayleigh channels, such that for Nakagami- channel converts to the AWGN channel.

All the transmissions are accomplished through orthogonal time slots like the Time Division Multiple Access (TDMA) protocol. The duration of each time slot for transmission of a packet (with the length of bits) is second. By using a proper Cyclic Redundancy Check (CRC) code, we assume that the relay is able to detect the erroneous decoded packets. The relay receives the packet of each source, during the source’s dedicated time slots, via its antenna, and then decodes the packet using the Maximum Ratio Combining (MRC) of the received signals.

## Iii The Proposed CNCC Scheme

The proposed CNCC scheme exploits a good systematic convolutional code designed over with the generator matrix of

 G(D)=[ IN×N ∣∣ PN×M′(D) ] (1)

as the network coding matrix implemented in the network level, where is the identity matrix, and is an matrix whose entries are either a polynomial or a rational function of . Furthermore, and are respectively the number of the parity bits and the constraint length of the convolutional code.

The encoder of the convolutional code in (1) is minimally realized in the relay. That is, the relay contains memories (shift registers). The systematic packets, related to the first section of the , i.e., , are directly transmitted from the sources to the destination within the first time slots. The parity packets are generated from the correctly decoded sources’ packets in the relay, pertaining to the second section of the , i.e., , and then are transmitted from the best antennas of the relay to the destination during the consequent time slots. The best antenna of the relay at each time slot is defined as the antenna that possesses the strongest R-D channel, which is recognized by the destination.

Specifically, the transmission strategy in the CNCC scheme is as follows. The sources transmit their own interleaved packets of the length during their dedicated time slots to the destination, where the relay simultaneously receives them through its antenna, and after deinterleaving the packets, decodes each packet by the MRC method. If the relay correctly decodes all the -th sources’ packets (success situation), it will produce the corresponding parity packets by using the convolutional code in (1), and after interleaving, will transmit them from its best antennas to the destination during the dedicated time slots. However, if the relay fails to correctly decode all the packets (failure situation), it will not generate any parity packets, and will inform the destination. In the failure situations, the sources’ packets are merely decoded based on the signals received through the direct S-D paths without the help of the relay. But, in the success situations, the destination uses both the sources’ systematic packets and the parity packets received from the relay, and after deinterleaving, runs the Viterbi algorithm to decode all the packets of the sources.

Due to the proximity of the relay to the sources, the number of the failure situations is negligible compared with the number of the success situations. Hence, the network throughput in the proposed CNCC scheme is tightly lower bounded as spcu (symbol per channel use). We are interested in the lower value for to increase the network throughput. Accordingly, by selecting the number of parity outputs of convolutional codes less than (or equal to) the number of the relay’s antennas , the network throughput of the CNCC scheme ( spcu) will be greater than (or equal to) that of the LNC scheme with single-antenna relays ( spcu). That is, the network throughput of the CNCC scheme is not a function of , and consequently, it remains constant and does not decrease with the increase of the number of antennas.

## Iv Performance Analysis

In this section, we first analyze the the BER of the proposed scheme, and then determine the achieved diversity order. As mentioned previously, there is two situations ( and ) in the relay that must be considered in the BER analysis. Hence, the end-to-end BER of the network’s sources can be written as

 Pb=Pb|sPs+Pb|fPf=Pb|s(1−Pf)+Pb|fPf (2)

where is the BER of the sources. is the probability of the failure situation in which the relay fails to correctly decode all the packets of the sources. is the probability of the success situation in which the relay correctly decodes all the sources’ packets, where . and are respectively the BER of the sources in the failure and the success situations.

### Iv-a Computation of Pf

The -th received signal from the -th source () at the -th antenna of the relay () is as

 ysi,rj(t)=√Eb hsi,rj(t) xsi(t)+nsi,rj(t) (3)

where , , and . s are Nakagami- fading coefficients from the to the . is the BPSK signal transmitted from the . Moreover, is the additive white Gaussian noise with the zero mean and the variance of . is the transmitted energy per bit. We assume that all the S-R channels have the same average energy; that is, . Hence, the probability density functions (pdf) of the coefficients are as

 fhsi,rj(hsi,rj)=2⎛⎝m¯¯¯¯¯¯¯h2sr⎞⎠mhsi,rj2m−1Γ(m) e−mhsi,rj2¯¯¯¯¯¯h2sr, (4)

where is the Gamma function, and for integer values of is equal to .

First, we calculate the bit error probability at the relay. By using the MRC, the conditional bit error probability of a BPSK signal is as

 Pe|γsi,r=Q(√2γsi,r), (5)
 γsi,r=EbN0M∑j=1h2si,rj, (6)

where is the received SNR of the transmitted signal from the source at the relay. By defining , and due to the fact that is the Nakagami- random variable, given in (4), will be a Chi Square random variable with degrees of freedom as

 fh′si,rj(h′si,rj)=⎛⎝m¯¯¯¯¯¯¯h2sr⎞⎠mh′si,rjm−1Γ(m) e−m h′si,rj¯¯¯¯¯¯h2sr (7)

. Hence, in (6), which is the sum of independent Chi Square random variables each of which with degrees of freedom, has the Chi Square pdf with degrees of freedom. By defining as the average received SNR of the S-R channels, we have

 fγsi,r(γsi,r)=(m¯¯¯γsr)Mmγsi,rMm−1Γ(Mm) e−mγsi,r¯γsr. (8)

As a result, the unconditional bit error probability, , can be easily obtained from (5) and (8) as

 Pe=∫∞γsi,r=0Pe|γsi,rfγsi,r(γsi,r)dγsi,r, (9)
 Pe=[12(1−μsr)]MmMm−1∑w=0(Mm−1+ww)[12(1+μsr)]w, (10)

where .

Now, we compute the failure probability, . Because of assuming a perfect interleaving, the successive bits within each sources’ packets experience independent fadings. As a result, the probability that one packet of a specific source can be correctly decoded in the relay is equal to . In (2), is the probability that all the packets of the sources corresponding to the successive slots can be correctly decoded in the relay. Therefore, the success probability at the relay is as follows

 Ps=(1−Pe)Nn. (11)

As a result, in (2) is computed as

 Pf=1−Ps=1−(1−Pe)Nn, (12)

where is given by (10).

At the high SNRs (), and can be respectively approximated as

 Pe≈(2Mm−1Mm)(m4¯¯¯γsr)Mm, (13)
 Pf≈nNPe≈K(1¯¯¯γsr)Mm, (14)

where is a constant coefficient.

### Iv-B Computation of Pb|f

When the relay fails to correctly decode the packets of the sources, it does not participate in the cooperation phase, and consequently, these packets are decoded only based on the received signals through the direct S-D channels. Hence, the bit error probability of the sources in the failure situation is simply obtained similar to (10) by setting and , and substituting instead of . As a result, we have

 Pb|f=12⎛⎜⎝1− ⎷¯¯¯γsd1+¯¯¯γsd⎞⎟⎠, (15)

where is defined as the average received SNR from the S-D channels at the destination. Similar to (13), at the high SNRs (), can be approximated as

 Pb|f≈14¯¯¯γsd. (16)

### Iv-C Computation of Pb|s

In the situation, based on the packets of the sources, the relay produces the corresponding parity packets using the systematic convolutional code given in (1). Finally, the parity packets are transmitted through the best antenna of the relay during their dedicated time slots. The destination runs the Viterbi algorithm to decode the sources’ packets. Hence, the BER of the sources in this situation is equal to the BER of the exploited convolutional code described by in (1) whose systematic and parity packets are respectively transmitted through the Rayleigh fading S-D channels, and the best of available Rayleigh fading R-D channels.

Due to interleaving with sufficient depth, the successive bits of each packet sent by the sources and the relay are well assumed to experience independent fadings. The received signals from the sources and the best selected antennas of the relay in the destination are respectively as follows

 ysi,d(t)=√Ebhsi,d(t)xsi,d(t)+zsi,d(t),\vspace−.7cm (17)
 yrsel,d(t′)=√Ebhrsel,d(t′)xrsel,d(t′)+zrsel,d(t′), (18)

where , , and . The parameters in (17) and (18) are as follows. : the transmitted energy per bit. : the -th transmitted bit from the -th source. and : respectively, the Rayleigh fading coefficient, and the Gaussian noise with zero mean and the variance of . : the -th parity bit transmitted from the best selected antenna of the relay. and : respectively, the Rayleigh fading coefficient, and the Gaussian noise with zero mean and the variance of .

The BER of a convolutional code with the free distance of is upper bounded as

 Pbconv.<1N∞∑d=dfreeBdPd, (19)

where s are the coefficients of the Bit Weight Enumerating Function (BWEF) of the convolutional code as

 B(X)=∞∑d=dfreeBdXd=∂A(W,X)∂W∣∣W=1. (20)

is the Input-Output Weight Enumeration Function (IOWEF) of the code, which can be easily computed from the state diagram of the convolutional code. Furthermore, in (19) is the Pairwise Error Probability (PEP) with the Hamming weight of . It must be noticed that only depends on the S-D and R-D channels.

#### Iv-C1 Computation of PEP

The destination uses the Maximum Likelihood criterion to decode the sequence transmitted by the sources as well as the best antennas of the relay. Hence, according to (17) and (18), the conditional PEP can be easily obtained as [34]

 Pd|{γsd,γrsel,d}=Q⎛⎜⎝ ⎷2d1∑k=1γsd(tk)+2d2∑k=1γrsel,d(t′k)⎞⎟⎠, (21)

where . and are the error positions related to the transmitted bits respectively from the sources and the relay. Moreover, and are, respectively, the instantaneous received SNRs from the sources and the best antennas of the relay. The s are independent for different , and have the exponential pdf as

 fγsd(tk)(γsd(tk))=1¯¯¯γsdexp−γsd(tk)¯¯¯γsd, (22)

where is the average received SNR from the S-D channels. Furthermore, is as

 γrsel,d(t′k)=maxj=1,...,Mγrj,d(t′k), (23)

where is the received SNR from the -th antenna of the relay at the destination, and has the exponential pdf and cdf, respectively, as

 fγrj,d(t′k)(γrj,d(t′k))=1¯¯¯γrdexp−γrj,d(t′k)¯¯¯γrd\vspace−.5cm (24)
 Fγrj,d(t′k)(γrj,d(t′k))=1−exp−γrj,d(t′k)¯¯¯γrd (25)

, where is the average received SNR from the R-D channels. Then, from (23)-(25), the pdf of is easily obtained as

 fγrsel,d(t′k)(γrsel,d(t′k))=M[1−exp−γrsel,d(t′k)¯¯¯γrd]M−11¯¯¯γrdexp−γrsel,d(t′k)¯¯¯γrd (26)

Now, we can compute by averaging in (21) over the distributions of and given respectively in (22) and (26) as

 Pd=∫∞0...∫∞0Q⎛⎜⎝ ⎷2d1∑k=1γsd(tk)+2d2∑k=1γrsel,d(t′k)⎞⎟⎠(d1∏k=11¯¯¯γsdexp−γsd(tk)¯¯¯γsd)× ⎛⎜⎝d2∏k=1M[1−exp−γrsel,d(t′k)¯¯¯γrd]M−11¯¯¯γrdexp−γrsel,d(t′k)¯¯¯γrd⎞⎟⎠d1∏k=1dγsd(tk)d2∏k=1dγrsel,d(t′k). (27)

By using the upper bound and the Binomial expansion, , and after some straightforward simplifications, the upper bound for is obtained as

 Pd≤12(11+¯¯¯γsd)d1(MM−1∑w=0(M−1w)(−1)w11+w+¯¯¯γrd)d2, (28)

where . It can be easily demonstrated that the following inequality holds

 MM−1∑w=0(M−1w)(−1)w11+w+¯¯¯γrd

Hence, from (28) and (29), we also have

 Pd≤12(11+¯¯¯γsd)d1⎛⎝M(M−1)!¯¯¯γ   Mrd⎞⎠d2, (30)

where .

From (28) and (30), is not an explicit function of , but a function of and such that . Hence, we can not directly use the equations (19) and (20) to compute the BER of the sources in the success situations, because the given BWEF is an explicit function of . For the problem on hand, we define a modified IOWEF, , which is the function of and as follows

 AMod(W,Y,Z)=∑w,d1,d2Aw,d1,d2WwYd1Zd2, (31)

where and . In this equation, the exponents of , , and denote respectively the Hamming weights of the input bits, the systematic output bits, and the parity output bits of the convolutional code. Furthermore, is the number of paths in the state diagram of the code, which originate from the zero state, and finally return to the zero state, such that their numbers of nonzero input bits, nonzero systematic output bits, and nonzero parity output bits are respectively equal to , , and . Since the exploited convolutional code in the CNCC scheme is systematic, we always have . To derive , in the state diagram of the exploited code in (1), we assign a gain to each branch , such that , , and are respectively the Hamming weights of the input block (among input bits), the systematic output block (among the first output bits), and the parity output block (among the last output bits) of the branch . Again, we have . With this dedicated gain to each branch, the transfer function from the initial zero state to the final zero state in the modified state diagram yields the . Similar to (20), the modified BWEF is obtained from the modified IOWEF as

 BMod(Y,Z)=∑d1,d2Bd1,d2Yd1Zd2=∂AMod(W,Y,Z)∂W∣∣W=1, (32)

where and . is the total number of the nonzero input bits in the paths of the modified state diagram with the number of the nonzero systematic output bits equal to , and the number of the nonzero parity output bits equal to .

Now, similar to (19), the BER of the sources in the success situations can be expressed as

 Pb|s<1N∑d1,d2Bd1,d2Pd. (33)

Ultimately, from (28) and (30)-(33), the closed form upper bounds for the are obtained as

 Pb|s<12NBMod(Y=11+¯¯¯γsd, Z=MM−1∑w=0(M−1w)(−1)w11+w+¯¯¯γrd), (34)

and

 Pb|s<12NBMod⎛⎝Y=11+¯¯¯γsd, Z=M(M−1)!¯¯¯γ   Mrd⎞⎠. (35)

### Iv-D End-to-end BER and the achieved diversity order

Taking the path loss effect into account leads to the following relationships: and , where , , and denote respectively the S-R, S-D, and R-D distances, and is the path loss exponent. Without loss of generality, we assume that the relation approximately holds. Therefore, we have and , where . We define as the average received SNR in the destination. Hence, from (2), (10), (12), (15), and (34), the upper bound of the end-to-end BER of the network’s sources is obtained as

 Pb<12NBMod(Y=11+¯¯¯γ, Z=MM−1∑w=0(M−1w)(−1)w11+w+¯¯¯γrd)× (1−[12(1−μsr)]MmMm−1∑w=0(Mm−1+ww)[12(1+μsr)]w)nN +12(1−√¯¯¯γ1+¯¯¯γ)×⎛⎝1−(1−[12(1−μsr)]MmMm−1∑w=0(Mm−1+ww)[12(1+μsr)]w)nN⎞⎠ (36)

where , , , and .

Now, we aim to compute the diversity order of the CNCC scheme as

 D≜−lim¯¯¯γ→∞logPb(¯¯¯γ)log¯¯¯γ. (37)

To this end, we first analyze the achieved diversity order in the success situation from (35) and (32). Let’s introduce the set as

 F={ (d1,d2) ∣∣ Yd1Zd2 exists in the series expression of BMod(Y,Z) }. (38)

We then define

 (d⋆1,d⋆2)=argmin(d1,d2)d1+Md2   s.t.   (d1,d2)∈F. (39)

It would be worthy to mention that the pair is not necessarily unique. Then, from (32), (35), and (37)-(39), it can be easily observed that the diversity order in the success situation is equal to

 D⋆=d⋆1+Md⋆2, (40)

which can be rewritten as

 D⋆=d⋆1+d⋆2+(M−1)d⋆2. (41)

Due to the fact that and are respectively related to the error patterns of the sources and the relay, and . Moreover, we have . Hence, from (41), can be lower bounded as

 D⋆≥dfree+M−1. (42)

Finally, from (1), (14), (16), (35), (38), and (39), at high SNRs (), is approximated as

 Pb≈K′(1¯¯¯γ)d⋆1(M(M−1)!¯¯¯γM)d⋆2+K′′(1¯¯¯γ)Mm+1 (43)

where and are the constant terms. According to (37), (40), and (43), the achieved diversity order of the sources in the CNCC scheme is equal to

 DCNCC=min (D⋆,Mm+1). (44)

By a proper value of in the exploited convolutional code, we can always have , and consequently, . We consider two special cases.

#### Iv-D1 Weak S-R channels

In this case, we consider the Rayleigh fading channels (Nakagami- with ), the same as the S-D and the R-D channels. Hence, the diversity order will be .

#### Iv-D2 Strong S-R channels

For this case which is more likely to happen, according to the assumption that the relay is close to the sources, the Nakagami- S-R channels with are well assumed. As a result, the diversity order up to can be achieved, which is much more than , as will be shown in the examples of the next section.

## V Illustrative Examples For The CNCC Scheme

In this section, we consider four networks with two sources, and different number of the relay’s antennas and . For all the examples considered, the constraint lengths of the exploited convolutional codes have been set to .

### V-a First (N=2,M=1,M′=1) and second (N=2,M=2,M′=1) networks

We select in these networks. Hence, the CNCC scheme exploits the good systematic convolutional code as

 G1(D)=⎡⎢⎣101+D+D2+D31+D+D3011+D2+D31+D+D3⎤⎥⎦ (45)

with . The encoder of this code can be minimally realized in the relay by three memories, as shown in Fig. 2, where and are the input bits, and are the systematic output bits, and is the parity output bit generated and transmitted by the relay. The modified BWEF of has been computed by Matlab. Its first several terms which play the significant roles in the performance of the CNCC scheme is as follows

 BMod1(Y,Z)=(3Y3Z)+(4Y4Z+9Y3Z2+