FiniteSNR analysis for partial relaying cooperation with channel coding and opportunistic relay selection
 Thang X. Vu^{1}Email authorView ORCID ID profile,
 Pierre Duhamel^{2},
 Symeon Chatzinotas^{1} and
 Bjorn Ottersten^{1}
https://doi.org/10.1186/s1363401704650
© The Author(s) 2017
Received: 15 December 2016
Accepted: 18 April 2017
Published: 2 May 2017
Abstract
This work studies the performance of a cooperative network which consists of two channelcoded sources, multiple relays, and one destination. To achieve high spectral efficiency, we assume that a single time slot is dedicated to relaying. Conventional networkcodedbased cooperation (NCC) selects the best relay which uses network coding to serve the two sources simultaneously. The bit error rate (BER) performance of NCC with channel coding, however, is still unknown. In this paper, we firstly study the BER of NCC via a closedform expression and analytically show that NCC only achieves diversity of order two regardless of the number of available relays and the channel code. Secondly, we propose a novel partial relayingbased cooperation (PARC) scheme to improve the system diversity in the finite signaltonoise ratio (SNR) regime. In particular, closedform expressions for the system BER and diversity order of PARC are derived as a function of the operating SNR value and the minimum distance of the channel code. We analytically show that the proposed PARC achieves full (instantaneous) diversity order in the finite SNR regime, given that an appropriate channel code is used. Finally, numerical results verify our analysis and demonstrate a large SNR gain of PARC over NCC in the SNR region of interest.
Keywords
1 Introduction
Cooperation among nodes is an effective technique to widen the coverage and to improve the performance of wireless networks both in terms of signaltonoise ratio (SNR) and diversity gain [1, 2]. Such improvements, however, usually comes at the price of an additional orthogonal channel, resulting in a reduced spectral efficiency, which can be significant in multiplerelay networks. In order to reduce this loss, opportunistic relay selection (RS) has been proposed to select the best relay for cooperation [3]. It has been shown that RS achieves full diversity order for singlesource multiplerelay networks and outperforms other relaying schemes in terms of SNR gain and effective capacity [4].
Network coding (NC) has gained tremendous attention because of its potential improvement in diversity gain and throughput over classical routing techniques [5]. The principle of NC is to allow intermediate nodes to combine multiple input packets into a single output. Recently, there has been much interest on combining NC together with RS to further improve the spectral efficiency. It is shown via outage probability (OP) analysis that the use of RS in a twoway relay channel (TWRC) could achieve full diversity order and a significant SNR gain [6–9]. The authors in [6] propose a joint design of NC with RS for decodeandforward (DF) TWRC based on the maxmin criterion in order to maximize the worst relay channel. In [7], an SNRbased suboptimal relay ordering is proposed for twoway amplifyandforward (AF) relay networks. A similar method is studied in [8] to derive the BER, OP, and diversity order. While research on RS in TWRC is readily available in the literature, research on RS in unidirectional relay networks is still limited. This problem was first considered in [10], which analyzes diversity multiplexing tradeoff (DMT) and shows that full diversity order is achieved. However, [10] relies on an unrealistic assumption that unintended packets are available at all destinations. A generalized DMT analysis is presented in [11]. Likewise, [11] relies on an optimistic assumption that the selected channels are independent, which is infeasible because these channels belong to an ordered SNR sequence, and thus are highly correlated [7]. By removing this unrealistic assumption, it was shown in our previous work that NCC fails to achieve full underlying diversity gain [12] regardless of the number of available relays. The analysis of the counterpart AF in interuser interference channels was studied in [13–16]. In [17, 18], the impacts of outdated and imperfect channel state information (CSI) were analyzed via closedform expression for system OP and pairwise error probability. It is worth noting that the abovementioned works study the system diversity via the upperbound limit of the BER or OP in the absence of channel coding, which is not the case in many practical scenarios where nodes are usually protected by some forward error correction codes.
In this paper, we investigate the performance of cooperative networks under practical conditions, i.e., the transmitted signals are protected by convolutional codes (CC). In the considered system, two sources communicate with a common destination with the aid of multiple relays. Such a scenario can find applications in the uplink cellular mobile systems where two mobile users send data to the base station and some friendly, idle users act as relays. Due to the constraints on spectral efficiency and processing delay, it is assumed that only one time slot is dedicated to cooperation. The best RS is employed to effectively exploit the spatial diversity [3]. At the destination, cooperative maximal ratio combining (CMRC) detector [19] is employed prior to channel decoding to mitigate error propagation. We note that CMRC is a suboptimal detector but provides full diversity gain and performance close to the maximum likelihood (ML) receiver [20].

Firstly, we analyze the BER in closedform expressions for the NCC, in which one selected relay helps the two sources by applying network coding on the estimated codewords. From the analyzed BER, we analytically show that NCC always achieves a diversity of order two regardless of the channel code and the total number of relays. This result coincides with the diversity order derived from OP analysis in [11, 12]. It would be noted that we analyze NCC in the presence of RS and channel coding, whereas [21] considered a singlerelay network and [9] studied symbolbased NCC without channel coding.

Secondly, we propose a partial relaying based cooperation (PARC). The key difference between PARC and NCC is that the former selects two relays for cooperation, each one helping one source independently. Compared to [22, 23], our proposed scheme has two main differences: (i) we analyze the system via BER, whereas these papers study the system OP, which is fundamentally different from our setting (we can obtain the actual BER for arbitrary SNR value); and (ii) we investigate RS to improve the spectral efficiency, while these papers consider singlerelay networks.

Thirdly, insightful theoretical analysis is provided for PARC in the finiteSNR regime. In particular, closedform expressions for the BER and instantaneous diversity order^{1} are derived, which reveal the dependency of the instantaneous diversity order on the operating SNR value and the minimum distance of the channel code.

Finally, numerical results demonstrate the effectiveness of our proposed scheme. It is shown via both analytical and simulation results that PARC can achieve full (instantaneous) diversity order in the low and medium SNR regime when a suitable CC is used. This result is important since the practical systems usually operate in the finite SNR regime.
The rest of the paper is organized as follows: Section 2 describes the system model. Section 3 provides details for the relay selection process. Section 4 analyzes the BER and diversity order of NCC. Section 5 analyses the performance analysis of PARC. Section 6 shows numerical results. Finally, Conclusions and Discussions are given in Section 7.
2 System model
2.1 Network codingbased cooperation (NCC)
where P _{ XY } with X∈{S _{1},S _{2}},Y∈{R _{ NC },D} is the average received power at node Y from node X, including the path loss; h _{ XY } is the channel fading coefficient between X and Y, which is independent and identically distributed (i.i.d.) complex Gaussian random variable with zero mean and unit variance, i.e., \(\mathbb {E}\left \{h_{XY}^{2}\right \}=1\), and is mutually independent among X→Y channels; n _{(.)} is a noise vector whose components are i.i.d. complex Gaussian random variables with zero mean and variance σ ^{2}.
At the end of the first phase, R_{ NC } decodes the estimate \(\hat {\mathbf {x}}_{iR}\) of x _{ i },i=1,2, using the ML detector as \(\hat {c}_{iR,k} = \arg \min _{c_{i,k}\in \{0,1\}} \{y_{S_{i}R_{NC},k}  \sqrt {P_{S_{i}R_{NC}}}h_{S_{i}R_{NC}} x_{i,k}^{2}\}, i \in \{1,2\},~ 1 \leq k \leq N\), where x _{ i,k }, the kth symbol of x _{ i }, is the modulated symbol of c _{ i,k }. Then R_{ NC } performs network encoding to get \(\hat {\mathbf {c}}_{NC}\), where \(\hat {c}_{NC,k} = \hat {c}_{1R,k} \oplus \hat {c}_{2R,k},\ \forall k\), and ⊕ denote the binary XOR operation.
where 0 is a zero matrix with the same size as g. See [21] for full details of the joint decoding at the destination.
2.2 Partial relayingbased cooperation (PARC)
Motivated by our previous work which shows that full diversity order can be achieved for the threenode relay network in the low and medium SNR regimes even when parts of the codeword is forwarded [21], we propose to combine RS with partial relaying in PARC to select two relays, each one is the best relay (maximizing the endtoend SNR) for one source. Since there are two active relays in the relaying phase, each relay only occupies half of the relaying time slot, as shown in Fig. 2 b. Consequently, the selected relays can only forward half of the estimated codeword to the destination.
At the end of the first phase, the selected relay estimates the source coded symbols and forwards them to the destination. In the proposed scheme, the selected relay \(\mathrm {R}^{\star }_{i}, i = 1,2,\) uses half of the relaying time slot to forward half of the codeword c _{ i } to the destination. More specifically, the selected relay \(\mathrm {R}^{\star }_{i}\) first estimates L=N/2 (without loss of generality, assuming N is even) source coded symbols to form an estimated punctured codeword \(\hat {\mathbf {c}}_{R^{\star }_{i}} = \{\hat {c}_{R^{\star }_{i},l}\}_{l\in \Theta }\), where
∀k _{ l }∈Θ, where \(x_{i,k_{l}}\) being the corresponding modulated symbol of \(c_{i,k_{l}}\). Next, \(\mathrm {R}^{\star }_{i}\) modulates \(\hat {\mathbf {c}}_{R^{\star }_{i}}\) into the modulated signal \(\hat {\mathbf {x}}_{R^{\star }_{i}}\) and then forwards it along with the index set Θ to the destination. The cost for conveying the index set is negligible since it can send, e.g., the seed of the random interleaver, to the destination.
where \(h_{R^{\star }_{i}D}\) is the channel coefficient from \(\mathrm {R}^{\star }_{i} \to \mathrm {D}\), and \(\mathbf {n}_{R^{\star }_{i}D}\) is a noise vector whose components are Gaussian random variable with zero mean and variance σ ^{2}.
In (7), \(\lambda _{R^{\star }_{i}}\) is the parameter of the CMRC detector which is computed as \(\lambda _{R^{\star }_{i}} \triangleq \frac {\min (\gamma _{S_{i}R^{\star }_{i}}, \gamma _{R^{\star }_{i}D})}{\gamma _{R^{\star }_{i}D}}\), where γ _{ XY }=P _{ XY }h _{ XY }^{2}/σ ^{2} being the instantaneous SNR of the channel X→Y. The CMRC detector then computes loglikelihood ratio values of the coded bits and sends them to the channel decoder. Finally, the channel decoder applies the BCJR algorithm [26] to decode the transmitted data.
Remark 1
In our protocol, the selected relay always forwards the estimated symbols to the destination. Fortunately, possible decoding error in \(\hat {c}_{R^{\star }_{i},l}\), hence error propagation, is effectively mitigated by \(\lambda _{R^{\star }_{i}}\) in CMRC. For example, if the sourcerelay channel is too noisy, i.e., \(\gamma _{S_{i}R^{\star }_{i}}\) is too small, it is highly probable that \(\mathrm {R}^{\star }_{i}\) decodes with errors. In this case, however, \(\lambda _{R^{\star }_{i}}\) is small and the contribution of the relayed signal is negligible.
3 Relay selection for NCC and PARC
In this section, we describe in details the relay selection process in NCC and PARC. Furthermore, essential properties of the selected relay channels are analyzed.
3.1 Relay selection in NCC
3.2 Relay selection in PARC
The selection process is executed at the beginning of each block in a distributed manner based on the maxmin criterion that maximizes the worst endtoend SNR and reduces computational complexity [6]. After the channel estimation, the relays set a timer that is inversely proportional to their channel gain. The first relay whose timer is zero will send a pulse to the destination. Upon receiving the pulse, the destination declares the chosen relay [3]. It is observed that the endtoend performance of relayed symbols is determined by the weaker between sourcerelay and relaydestination connections. We thus model a twohop sourcerelaydestination link by an equivalent singlehop channel, which is highly accurate for DMF relaying protocol [3].
where \( \frac {1}{\overline {\gamma }_{R^{\star }_{i},j}} = \sum \limits _{k=n_{1}}^{n_{j}} \left (\frac {1}{\overline \gamma _{S_{i}R_{k}}} + \frac {1}{\overline \gamma _{R_{k}D}}\right). \)
Remark 2
The relay selection process in PARC is performed for each source separately, which is different from NCC. Also, the decoding process at the destination is performed separately for each source.
4 Performance analysis for network codingbased cooperation
In this section, we analyze the BER and diversity order of NCC. Using the equivalent channel, the twohop networkcoded signal can be modeled as if it was conveyed by a single channel whose instantaneous SNR is γ _{NC} [20].
4.1 Bit error rate analysis
where F is the minimum distance of the compound code G, w _{ i }(d) denotes input weights (number of nonzero information bits) corresponding to source S_{ i } in the compound codeword, and \(\overline {\text {PEP}}^{\text {NC}}(d)\) is the unconditioned pairwise error probability (UPEP)^{3} of receiving a compound codeword with the output weight d (number of nonzero coded bits), assuming that the allzero codeword, e.g., c _{1}=c _{2}=0, has been transmitted. To derive (10), it requires the knowledge of the minimum distance F of the compound code, the input weight w _{ i }(d), and how d output wrights in the compound codeword X are distributed among the three channels S_{1}→D, S_{2}→D and R_{ NC }→D. Denote W _{ d }={d _{1},d _{2},d _{ R }} as the weight pattern that specifies how d weights are distributed among these channels, where d _{ i } is the output weight of the individual codeword transmitted via the channel S_{ i }→D or channel R_{ NC }→D. By definition, d=d _{1}+d _{2}+d _{ R }. The input weight and the pattern can be computed via heuristic searching of the trellis of G. The following result is important for further analysis:
Lemma 1
The minimum distance F of the compound code G is equal to twice the minimum distance f of the single code g, F=2f, and the weight pattern W _{ F } has one of the following values {f,f,0}, {f,0,f}, {0,f,f}.
Lemma 2
For any pattern W _{ d }={d _{1},d _{2},d _{ R }} of the compound codeword X with output weight d>F, there are at least two nonzero elements in W _{ d }.
The proof of Lemma 1 and Lemma 2 are given in [21]. Lemma 1 and Lemma 2 provide an important information about the output weights of the compound code: d weights of the compound code always experience at least two independent channels. Furthermore, the number of patterns is finite and strictly defined by G.
where \(\gamma _{\Sigma _{NC}} = d_{1}\gamma _{S_{1}D} + d_{2}\gamma _{S_{2}D} + d_{R}\gamma _{NC}\) is defined as the total SNR at the destination in NCC, and \(Q(x) = \frac {1}{\sqrt {2\pi }} \int _{x}^{+\infty } e^{t^{2}/2}dt\) denotes the Qfunction.
Applying the MGF method [27] we can derive the UPEP \(\overline {\text {PEP}}^{\text {NC}}\left (d\mathbf {W}_{d}\right)\) in NCC as in Theorem 1.
Theorem 1
Given the weight pattern W _{ d }={d _{1},d _{2},d _{ R }}, d=d _{1}+d _{2}+d _{ R }, the UPEP \(\overline {\text {PEP}}^{\text {NC}}\left (d\mathbf {W}_{d}\right)\) of the compound code in NCC has a form given by:
Proof
See Appendix A. □
Input weight and output weight distribution at d=F=24 of compound code G in (4), g= [ 23, 35, 37]
w _{1}  w _{2}  d _{1}  d _{2}  d _{ R } 

0  12  0  12  12 
12  0  12  0  12 
12  12  12  12  0 
4.2 Diversity analysis
Since the BER is linearly proportional to the UPEP \(\overline {\text {PEP}}^{\text {NC}}\left (d\mathbf {W}_{d}\right)\) via corresponding input weights, the diversity order of NCC is equal to diversity order of the UPEP. Let \(x \propto \overline {\gamma }^{\eta }\) denote the exponential equivalence, i.e., x achieves diversity of order η, where \(\overline {\gamma }\) stands for the general average SNR. The diversity order of the \(\overline {\text {PEP}}^{\text {NC}}\left (d\mathbf {W}_{d}\right)\) is given by the following theorem.
Theorem 2
Proof
See Appendix B. □
It is shown from (11) and Theorem 2 that the BER in NCC is a combination of three factors whose respective diversity orders are 2, N _{ r }+1 and N _{ r }+2. As the contribution of these factors are comparable and equal to the input weights of the compound code (shown in Table 1 as an example), the diversity order of NCC is dominated by the diversity order 2 factor. Consequently, NCC achieves diversity order 2 regardless of the channel code and the total number of available relays. This result is in line with the diversity order of NCC obtained via outage analysis [11, 12].
5 Performance analysis for partial relayingbased cooperation
In this section, we analyze the BER and diversity order of PARC by using the equivalent channel model. Since PARC is symmetric, the analysis for two sources is similar. For ease of presentation, we drop the source subscript in this section. After two phases, the destination receives two signals from the source S and the selected relay R^{⋆}. The combined signal at the CMRC detector’s output can be classified into two groups: the first group consists of symbols which are helped by the selected relay, and the second group includes the rest of the symbols which are not relayed. In other words, the received signal at the destination can be seen as an output of a block fading channel with two blocks: one block consisting of the N/2 symbols which only see channel γ _{ SD }, and the other one containing the other N/2 symbols which see both channel γ _{ SD } and \(\phantom {\dot {i}\!}\gamma _{R^{\star }}\).
5.1 Bit error rate analysis
where \(\mathcal {C}^{n}_{k} = \frac {n!}{(nk)!\times k!}\).
where \(\phantom {\dot {i}\!}\gamma _{\Sigma } = d_{1}\gamma _{SD} + d_{2}(\gamma _{SD}+\gamma _{R^{\star }}) = d\gamma _{SD} + d_{2} \gamma _{R^{\star }}\).
Taking into account the independence between γ _{ SD } and \(\phantom {\dot {i}\!}\gamma _{R^{\star }}\), we have \(\phantom {\dot {i}\!}\Psi _{\gamma _{\Sigma }}(s) = \Psi _{\gamma _{SD}}(ds) \times \Psi _{\gamma _{R^{\star }}}(d_{2}s)\).
Theorem 3
where \(\mathcal {I}_{1}\left (a,b\right)\) has been defined in Theorem 1.
Proof
See Appendix C. □
Substituting \(\overline {\text {PEP}}^{\text {PA}}\left (d\mathbf {D}_{d}\right)\) from Theorem 3 into (16) and (13), we obtain the upper bound for the BER. Note that even though d in (13) can be as large as the codeword’s length, i.e., N, the BER usually depends on a few first values in fading channels. To give insightful understanding of PARC, we analyze the system diversity order.
5.2 Diversity analysis for \(\overline {\text {PEP}}^{\text {PA}}(d\mathbf {D}_{d})\)
We first analyze the diversity order of the UPEP \(\overline {\text {PEP}}^{\text {PA}}(d\mathbf {D}_{d})\) for a given weight pattern D _{ d }, which determines how the selected relay contributes to the overall system performance.
Theorem 4
where ∝ denotes the proportional relation.
Proof
See Appendix D. □
Theorem 4 states that \(\overline {\text {PEP}}^{\text {PA}}(d\mathbf {D}_{d})\) can have either diversity order one or diversity order of N _{ r }+1 depending on the weight pattern D _{ d }.
5.3 Diversity analysis of PARC
where K is the normalized constant that depends on the channel code and network topology.
which obviously matches the classical definition of diversity when the SNR tends to infinity. Because the diversity order depends on the SNR, we refer to \(\zeta (\overline {\gamma })\) as instantaneous diversity order. The key idea behind the definition is that it allows the system behavior to be studied at any SNR value.
An important observation from (21) is that the instantaneous diversity order of PARC depends on the operating SNR value and the channel code, which provides a criterion design to achieve desirable diversity order in the finiteSNR regime. By choosing a proper channel code whose minimum distance f, such as \(K2^{f} \overline {\gamma }_{*}^{N_{r}} \ll 1\), then the PARC achieves full (instantaneous) diversity order of N _{ r }+1 in the SNR region \([0, \overline {\gamma }_{*}]\). This result is crucial because the operating SNR regime is usually finite in practice.
6 Numerical results
This section presents simulation results to confirm the effectiveness of the proposed PARC. All channels are subject to quasistatic block Rayleigh fading plus AWGN. Because we focus on the diversity order, which is not affected by modulation order, BPSK modulation and binary network coding are carried out in simulations. The data packet length is equal to 1024 bits. We consider symmetric network, i.e., \(\overline {\gamma }_{S_{i}R_{j}} = \overline {\gamma }_{SR}, \overline {\gamma }_{R_{j}D} = \overline {\gamma }_{RD}, \overline {\gamma }_{S_{i}D} = \overline {\gamma }_{SD}, \forall i,j\). Unless otherwise stated, the relays are located in the middle of the sources and the destination. The path loss exponent is equal to 3.5. As a result, the average SNR in sourcerelay channels and relaydestination channels are 10.5 dB better than sourcedestination channels. Note that our analysis holds for arbitrary locations of the relays. The channel code is chosen as the one that optimizes both the minimum distance and distance spectrum in block Rayleigh fading channels [31]. Different channel codes g are compared.
We also present the performance of two reference schemes. The first reference scheme (named Reference 1 in the figures) is based on fractional repetition coding cooperation [22, 23]. The second reference scheme employs fractional repetition coding together with network coding (named Reference 2 in the figures). All relays are active and share the relaying phase in two reference schemes. In Reference 1, since the relays help the sources separately, each relay forwards 1/(2N _{ r }) of the estimated codeword. In Reference 2, all relays use NC to help the sources, and each relay randomly forwards 1/N _{ r } of the networkcoded codeword. Note that no relay selection is used in the reference schemes.
In conclusion, the most effective of the proposed PARC is the capability of achieving full (instantaneous) diversity order in the low and medium SNR regime, which in turn results in a large SNR gain in the finiteSNR regime. Such gain is crucial for practical systems because their operating SNRs is usually finite.
7 Conclusions
We have proposed a novel cooperative scheme for a twosource multiplerelay network that combines the best relay selection and partial relaying cooperation to effectively exploit the spatial diversity. We have shown that the instantaneous diversity order is a function of the minimum distance of the channel code and the operating SNR. It has been shown by both analytical and simulation results that our proposed scheme can gain full diversity order in the finiteSNR regime when a suitable channel code is used.
The proposed scheme can easily be extended to general multisource multirelay networks. In this case, the selected relays might forward a number of symbols which is less than half of the codeword length. The major challenge in this case is how to select best (multiple) relays for network coding and partial relaying. A promising application of PARC is to design a cooperation scheme to support multiple sources with different error correction capacities to achieve a given target BER. This problem can be solved by carefully designing how many symbols of each source should be relayed depending on the corresponding channel code’s strength.
8 Endnotes
^{1} Instantaneous diversity order is measured as the slope of the BER curve in loglog scale, which allows to study the system behavior at arbitrary SNR value. This definition coincides with the conventional diversity definition in the high SNR regime [32]
^{2} Other selection of Θ, e.g., optimal index set, can be employed, but are beyond the scope of this paper.
^{3} The UPEP does not depend on the fading channels.
9 Appendix A: proof of Theorem 1

Case 1: d _{ R } = 0, there is not any weight on the channel γ _{ NC }. The total SNR is equal to \(\gamma _{\Sigma _{NC}} = d_{1}\gamma _{S_{1}D} + d_{2} \gamma _{S_{2}D}\), and its MGF is given as follows:$$\begin{array}{*{20}l} \Psi_{\gamma_{\Sigma_{NC}}}(s) &= \Psi_{\gamma_{S_{1}D}}(d_{1}s) \times \Psi_{\gamma_{S_{2}D}}(d_{2}s) \\ &= \frac{1}{1 + d_{1}\overline{\gamma}_{S_{1}D} s}\frac{1}{1 + d_{2}\overline{\gamma}_{S_{2}D} s}. \end{array} $$(22)The UPEP \(\overline {\text {PEP}}^{\text {NC}}\left (d\mathbf {W}_{d}\right)\) can be computed using the MGF method [27] as follows:$$\begin{array}{*{20}l} \overline{\text{PEP}}^{\text{NC}}\left(d\mathbf{W}_{d}\right)&=\frac{1}{\pi}\int \limits_{0}^{\pi/2} \Psi_{\gamma_{\Sigma_{NC}}}\left(\frac{1}{\sin^{2}\theta}\right)d\theta \\ & = \mathcal{I}_{1}\left(d_{1}\overline{\gamma}_{S_{1}D},d_{2}\overline{\gamma}_{S_{2}D}\right), \end{array} $$(23)where$$\mathcal{I}_{1}(a,b) = \frac{1}{2}\left(1  \frac{a}{ab}\sqrt{\frac{a}{1+a}}  \frac{b}{ba}\sqrt{\frac{b}{1+b}}\right). $$

Case 2: d _{1}=0. In this case, the total SNR equals \(\gamma _{\Sigma _{NC}} = d_{2}\gamma _{S_{2}D} + d_{R} \gamma _{NC}\). Given the MGF of γ _{ NC } in (8), the MGF of the total SNR is given as follows:$$ {\begin{aligned} \Psi_{\gamma_{\Sigma_{NC}}}(s) &= \Psi_{\gamma_{S_{2}D}}(d_{2}s) \times \Psi_{\gamma_{{NC}}}(d_{R}s) \hfill \\ &= \sum \limits_{j=1}^{N_{r}} \!\left((1)^{j1} \sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots \neq {n_{j}} \end{subarray}}^{N_{r}}\! \frac{1}{1 +\! d_{2} \overline{\gamma}_{S_{2}D}s}\frac{1}{1+\! d_{R} \overline{\gamma}_{NC,j}s}\right). \end{aligned}} $$(24)The UPEP \(\overline {\text {PEP}}^{\text {NC}}\left (d\mathbf {W}_{d}\right)\) is computed using the MGF method as follows:$$ {\begin{aligned} {}\overline{\text{PEP}}^{\text{NC}}\!\left(d\mathbf{W}_{d}\right) &= { \sum \limits_{j=1}^{N_{r}}} \left((1)^{j1}\!\! \sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots\neq {n_{j}} \end{subarray}}^{N_{r}}\right. \\ &\quad\left.{\vphantom{\sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots\neq {n_{j}} \end{subarray}}^{N_{r}}}} \frac{1}{\pi} \int \limits_{0}^{\pi/2} \frac{\sin^{4}\theta}{\left(\sin^{2}\theta + d_{2} \overline{\gamma}_{S_{2}D}\right)\left(\sin^{2}\theta + d_{R} \overline{\gamma}_{{NC}, j}\right)}d\theta \right)\\ &= { \sum \limits_{j=1}^{N_{r}}} \left((1)^{j1} \sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots \neq {n_{j}} \end{subarray}}^{N_{r}} \mathcal{I}_{1}\left(d_{2}\overline{\gamma}_{S_{2}D},d_{R}\overline{\gamma}_{NC,j}\right) \right). \end{aligned}} $$(25)

Case 3: d _{2}=0. Similar to Case 2 we have \( \overline {\text {PEP}}^{\text {NC}}\left (d\mathbf {W}_{d}\right)\) equals$$\begin{array}{*{20}l} { \sum \limits_{j=1}^{N_{r}}} \left((1)^{j1} \sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots \neq {n_{j}} \end{subarray}}^{N_{r}} \mathcal{I}_{1}\left(d_{1}\overline{\gamma}_{S_{1}D},d_{R}\overline{\gamma}_{NC,j}\right) \right). \end{array} $$

Case 4: d _{1} d _{2} d _{ R }≠0. In this case, \(\gamma _{\Sigma _{NC}} = d_{1}\gamma _{S_{1}D} + d_{2}\gamma _{S_{2}D} + d_{R} \gamma _{NC}\). The MGF of \(\gamma _{\Sigma _{NC}}\) is given as follows:$$ {\begin{aligned} {}\Psi_{\gamma_{\Sigma_{NC}}}(s) &= \Psi_{\gamma_{S_{1}D}}(d_{1}s) \times \Psi_{\gamma_{S_{2}D}}(d_{2}s) \times \Psi_{\gamma_{{NC}}}(d_{R}s) \hfill \\ &=\! { \sum \limits_{j=1}^{N_{r}}} \!\left(\!\!(1)^{j1}\!\!\! \sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots \neq {n_{j}} \end{subarray}}^{N_{r}} \!\! \frac{1}{1\,+\,d_{1} \overline{\gamma}_{S_{1}D}s} \frac{1}{1\,+\, d_{2} \overline{\gamma}_{S_{2}D}s}\frac{1}{1\,+\, d_{R} \overline{\gamma}_{NC,j}s}\!\!\right)\!. \end{aligned}} $$(26)Applying the MGF method to compute the UPEP, we have:where \(\mathcal {I}_{2}(a,b,c)\) has been defined in Theorem 1.$$ {\begin{aligned} &\overline{\text{PEP}}^{\text{NC}}\left(d\mathbf{W}_{d}\right) = { \sum \limits_{j=1}^{N_{r}}} \left((1)^{j1} \sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots \neq {n_{j}} \end{subarray}}^{N_{r}} \frac{1}{\pi}\int \limits_{0}^{\pi/2} \right.\\ &\left.\frac{\sin^{6}\theta}{(\sin^{2}\theta\,+\, d_{1} \overline{\gamma}_{S_{1}D}) (\sin^{2}\theta\,+\, d_{2} \overline{\gamma}_{S_{2}D})(\sin^{2}\theta\! +\! d_{R} \overline{\gamma}_{NC,j})}d\theta {\vphantom{\sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots \neq {n_{j}} \end{subarray}}}}\right) \\ &= { \sum \limits_{j=1}^{N_{r}}} \left((1)^{j1}\! \sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots \neq {n_{j}} \end{subarray}}^{N_{r}}\! \mathcal{I}_{2}(d_{1}\overline{\gamma}_{S_{1}D}, d_{2}\overline{\gamma}_{S_{2}D},d_{R}\overline{\gamma}_{NC,j}) \right), \end{aligned}} $$
Combining these four cases gives the proof of Theorem 1.
10 Appendix B: proof of Theorem 2

Case 1: d _{ R }=0. In this case, all weights are located in the sourcedestination channels, resulting in \(\Psi _{\gamma _{\Sigma _{NC}}}\left (1/2\right) = \Psi _{\gamma _{S_{1}D}}\left (d_{1}/2\right)\times \Psi _{\gamma _{S_{2}D}}\left (d_{2}/2\right)\). The diversity order in this case is given by:$$ {\begin{aligned} \tau &\geq {\lim}_{\overline{\gamma} \rightarrow \infty} \frac{\log \Psi_{\gamma_{S_{1}D}}\left(d_{1}/2\right)}{\log \overline{\gamma}} {\lim}_{\overline{\gamma} \rightarrow \infty} \frac{\log \Psi_{\gamma_{S_{2}D}}\left(d_{2}/2\right)}{\log \overline{\gamma}} \\ & \geq \!{\lim}_{\overline{\gamma} \rightarrow \infty}\!\left(\!\frac{\left(1 + d_{1} G_{S_{1}D}\overline{\gamma}/2\right)^{1}} {\log \overline{\gamma}}\!\right) \,\, {\lim}_{\overline{\gamma} \rightarrow \infty}\!\left(\frac{\left(1 + d_{2} G_{S_{2}D}\overline{\gamma}/2\right)^{1}} {\log \overline{\gamma}}\right) \\ &= 1 + 1 = 2. \end{aligned}} $$(29)
This is enough to say the UPEP has diversity order of 2 when d _{ R }=0, and we can write \(\overline {\text {PEP}}^{\text {NC}}\left (dd_{R} = 0\right) \propto \overline {\gamma }^{2}\).

Case 2: d _{1}=0. The MGF of the total SNR in this case has a form of \(\Psi _{\gamma _{\Sigma _{NC}}}\left (1/2\right) = \Psi _{\gamma _{S_{2}D}}\left (d_{2}/2\right)\times \Psi _{\gamma _{NC}}\left (d_{R}/2\right)\). Consequently, the diversity order is given as follows:$$\begin{array}{*{20}l} {}\tau &\geq \,\,{\lim}_{\overline{\gamma} \rightarrow \infty} \!\frac{\log \Psi_{\gamma_{S_{2}D}}\!\left(d_{2}/2\right)}{\log \overline{\gamma}} \underbrace {\,\,{\lim}_{\overline{\gamma} \rightarrow \infty}\! \frac{\log \Psi_{\gamma_{NC}}\left(d_{R}/2\right)}{\log \overline{\gamma}}}_{\mathcal{J}} \\ &= 1 + \mathcal{J}, \end{array} $$(30)
where \(\mathcal {J}\) is the diversity order of the best relayed signal without the direct link, which is equal to diversity order of the best relay selection for twoway relay channels. It has been shown in [6] that this diversity order is equal to N _{ r }. Therefore, the system diversity order when d _{1}=0 is equal to N _{ r }+1. In order words, \(\overline {\text {PEP}}^{\text {NC}}(dd_{1} = 0) \propto \overline {\gamma }^{(N_{r}+1)}\).

Case 3: d _{2}=0. Similar to case 2, the diversity order is equal to N _{ r }+1.

Case 4: d _{1} d _{2} d _{ R }≠0. In this case the MGF of \(\gamma _{\Sigma _{NC}}\) is a product of three terms:
$$\begin{array}{*{20}l} \Psi_{\gamma_{\Sigma_{NC}}}\left(1/2\right) =& \Psi_{\gamma_{S_{1}D}}\left(d_{1}/2\right)\times \Psi_{\gamma_{S_{2}D}}\left(d_{2}/2\right) \\ &\times \Psi_{\gamma_{NC}}\left(d_{R}/2\right). \end{array} $$(31)Substituting (31) into (28) we have
$$\begin{array}{*{20}l} {}\tau &\geq \,\,{\lim}_{\overline{\gamma} \rightarrow \infty} \!\frac{\log \Psi_{\gamma_{S_{1}D}}\!\left(d_{1}/2\right)}{\log \overline{\gamma}} \,\,{\lim}_{\overline{\gamma} \rightarrow \infty}\!\! \frac{\log \Psi_{\gamma_{S_{2}D}}\left(d_{2}/2\right)}{\log \overline{\gamma}}~ \\ &\quad\underbrace{{\lim}_{\overline{\gamma} \rightarrow \infty} \frac{\log \Psi_{\gamma_{NC}}\left(d_{R}/2\right)}{\log \overline{\gamma}}}_{\mathcal{J}} \\ &= 1 + 1 + \mathcal{J} = N_{r} + 2. \end{array} $$(32)We can write \(\overline {\text {PEP}}^{\text {NC}}\left (dd_{1}d_{2}d_{R} \neq 0\right) \propto \overline {\gamma }^{(N_{r}+2)}\).
These four cases prove Theorem 2.
11 Appendix C: proof of Theorem 3

Case 1: D _{ d }=D _{1}. In this case, all d weights are located in the sourcedestination block, resulting in γ _{ Σ }=d γ _{ SD } and \(\Psi _{\gamma _{\Sigma }}(s) = \Psi _{\gamma _{SD}}(ds)\). In this case, we have$$\begin{array}{*{20}l} \overline{\text{PEP}}(d\mathrm{D}_{1}) &= \frac{1}{\pi}\int_{0}^{\pi/2} \frac{\sin^{2}\theta}{\sin^{2}\theta + d\overline{\gamma}_{SD}}d\theta \\ &= \frac{1}{2}\left(1  \sqrt{\frac{d\overline{\gamma}_{SD}}{1 + d\overline{\gamma}_{SD}}}\right). \end{array} $$(33)

Case 2: D _{ d }≠D_{1}. There is always d _{2} weights are relayed, resulting in \(\Psi _{\gamma _{\Sigma }}(s) = \Psi _{\gamma _{SD}}(ds)\times \Psi _{\gamma _{R^{\star }}}\left (d_{2}s\right)\phantom {\dot {i}\!}\). From (9) we have$$ {\begin{aligned} {}\overline{\text{PEP}}\left(d\mathbf{D}_{d}\right) &=\! { \sum \limits_{j=1}^{N_{r}}}\! \left(\!(1)^{j1}\!\!\! \sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots \neq {n_{j}} \end{subarray}}^{N_{r}}\! \frac{1}{\pi}\! \int \limits_{0}^{\pi/2}\!\!\frac{\sin^{4}\theta}{\left(\sin^{2}\!\theta \,+\, d\overline{\gamma}_{SD}\right)\!\left(\sin^{2}\!\theta \,+\, d_{2}\overline{\gamma}_{R^{\star}, j}\right)}d\theta\! \right) \\ &=\! { \sum \limits_{j=1}^{N_{r}}} \!\left(\!(1)^{j1} \sum \limits_{\begin{subarray}{c} {n_{1}}=1,\dots,{n_{j}} = 1 \\ {n_{1}} \neq \dots \neq {n_{j}} \end{subarray}}^{N_{r}} \mathcal{I}_{1}\left(d\overline{\gamma}_{SD}, d_{2}\overline{\gamma}_{R^{\star}_{i},j}\right)\right), \end{aligned}} $$(34)
where \(\mathcal {I}_{1}\left (a,b\right)\) is defined in Theorem 1, and \(\bar {\gamma }_{R^{\star },j}\) is defined in Section 3.2.
12 Appendix D: proof of Theorem 4

Case 1: D _{ d }=D_{1}={d,0}. There is no symbol helped by the relay and thus \(\Psi _{\gamma _{\Sigma }}\left (1/2\right) = \Psi _{\gamma _{SD}}\left (d/2\right)\). The diversity order in this case is given as
$$\begin{array}{*{20}l} \tau &\geq {\lim}_{\overline{\gamma} \rightarrow \infty} \frac{\log \Psi_{\gamma_{SD}}\left(d/2\right)}{\log \overline{\gamma}} \\ & \geq {\lim}_{\overline{\gamma} \rightarrow \infty}\left(\frac{\left(1 + d G_{SD}\overline{\gamma}/2\right)^{1}} {\log \overline{\gamma}}\right) = 1, \end{array} $$(36)which states that the UPEP has diversity order of one when d _{2}=0. We may write \(\overline {\text {PEP}}\left (d\mathrm {D}_{1}\right) \propto \overline {\gamma }^{1}\).

Case 2: D _{ d }≠D_{1}, then \(\Psi _{\gamma _{\Sigma }}\left (1/2\right) = \Psi _{\gamma _{SD}}\left (d/2\right)\times \Psi _{\gamma _{R^{\star }}}\left (d_{2}/2\right)\phantom {\dot {i}\!}\). Consequently, the diversity order is given as
$$\begin{array}{*{20}l} {}\tau &\geq {\lim}_{\overline{\gamma} \rightarrow \infty} \frac{\log \Psi_{\gamma_{SD}}\left(d/2\right)}{\log \overline{\gamma}} \underbrace {{\lim}_{\overline{\gamma} \rightarrow \infty} \frac{\log \Psi_{\gamma_{R^{\star}}}\left(d_{2}/2\right)}{\log \overline{\gamma}}}_{\tau_{Sel}} \\ &= 1 + \tau_{Sel}, \end{array} $$(37)where τ _{ Sel } is the diversity order of the best relay signal (without the direct link). It is shown in [3] that the best relay selection achieves full diversity order of N _{ r }. Therefore, we can write \(\overline {\text {PEP}}\left (d\mathbf {D}_{d} \neq \mathrm {D}_{1}\right) \propto \overline {\gamma }^{(N_{r}+1)}\).
Combining these two cases, we complete the proof of Theorem 4. [9, 17, 18].
Declarations
Acknowledgements
The research work is supported in part by the Luxembourg FNR Core 2013 SeMiGod under the grant code I2RSIGPFN13SEMI.
Authors’ contributions
TXV substantially contributes to this paper. PD, SC, and BO equally contribute to the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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