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Improved twoway doublerelay selection technique for cooperative wireless communications
EURASIP Journal on Wireless Communications and Networking volume 2021, Article number: 57 (2021)
Abstract
In this article, a novel twoway doublerelay selection strategy with its bit error rate (BER) performance analysis is proposed. In this novel strategy, as a first step we choose two relays out of a set of relaynodes in a way to maximize the system performance in terms of BER and complexity. In the second step, the selected relays apply orthogonal spacetime coding scheme using the threephase protocol to establish a twoway communication between the transceivers, which will lead to a significant improvement in the achievable diversity and coding gain with a very low decoding complexity by using a symbolwise decoder. Furthermore, the performance of the overall system is further enhanced through the use of a network coding method at the selected relaynodes. Moreover, this paper proposes the analytical approximation of the BER performance. As well, we show that the analytical results match perfectly the simulated ones. In addition, we prove that our strategy outperforms the current stateoftheart ones by proposing a better cooperative communication system in terms of BER.
Introduction
Multiuser interference and channel impairments such as multipath propagation and timevarying fading can affect the performance of wireless communications in terms of achievable data rate and bit error rate (BER) [1,2,3,4,5,6,7,8,9,10,11,12,13]. The use of distributed spacetime coding techniques [2, 3, 8], distributed beamforming techniques [4,5,6] and relay selection techniques [9,10,11,12,13,14,15,16,17,18,19,20,21,22] helps in reducing the impact of channel impairments and as a result enhancing the performance of wireless communication systems. The performance of these systems in terms of achievable data rate and BER can be enhanced through the use of these techniques by allowing the receiver side to receive different versions with different phases and amplitudes of the transmitted information symbols which will be combined efficiently together in order to achieve the former.
In relaying techniques, the relaynodes receive and process the transmitted data before retransmitting the resulting signals in an orthogonal channels or using a specific technique to maximize the received signaltonoise ratio or achievable data rate or to minimize the overall BER. By increasing the number of relays, the spatial diversity gain will be improved. These latter techniques, named as diversity techniques, use oneway or twoway relaying schemes [1,2,3,4,5,6,7]. Some of the diversity techniques consider that the availability of channel state information (CSI) is essential at all nodes of the network with slow fading channels [23, 24]. Other techniques consider that the availability of CSI is essential only at the receiving antennas [2, 13]. Recently, noncoherent and differential diversity techniques are proposed as new approaches with no need of CSI at both the transmitter and receiver nodes [2,3,4,5,6]. Due to the nonnecessity of CSI, these techniques suffer from a low BER, a low spectral efficiency and high latency as well as high decoding complexity.
Bidirectional communication techniques are proposed in order to enhance system performance where they consider two entities send and receive their data through different relaynodes [2,3,4,5,6,7]. Bidirectional relaying schemes are classified into three groups, twophase [2, 3, 7, 8], threephase [2, 3] and fourphase [4,5,6] protocols based on the number of time slots required for message exchanging between all the nodes. Decreasing the number of transmission periods from four phase to three phase or to two phase will lead to a significant improvement in the spectrum efficiency of these systems. This will make the performance of the two and three phase better than the performance of the fourphase system [2,3,4, 14]. Furthermore, through the use of orthogonal channels the relaynodes can transmit the received signals, as well, they can encode them using orthogonal or nonorthogonal coding techniques. Orthogonal coding techniques such as spacetime coding (STC) enjoy a full diversity gain with low decoding complexity [13]. Nonorthogonal coding techniques improve the overall system performance and achievable data rate as well as enjoy full diversity gain and high coding gain; however, they suffer from high decoding complexity. Full spatial diversity gain with low decoding complexity in nonorthogonal coding can be obtained through the use of relay selection techniques [8,9,10,11,12,13,14]. For this purpose, many relaying techniques with single and dualrelay selection strategies have been investigated and proposed [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. The authors in [25] introduced a general overview of channelcoded physicallayer network coding in bidirectional relay communications. In [26], the authors discussed a general overview of a protograph lowdensity parity check code called rootprotograph code over fading channels from the codedesign perspective. Note that both decodeandforward physical network coding twoway relaying scheme [25] and the codedaided relaying, i.e., coded cooperation scheme [26], can achieve capacityapproaching performance.
The authors in [10,11,12] proposed a max–min method that selects the optimal relaynode from a set of relaynodes to maximize either the signaltonoise ratio (SNR) or the overall achievable data rate. Several relay selection methods are recently suggested specifically to select the best one or two relaynodes from a group of relays. For example, the authors in [11] proposed several techniques to select the best relaynode using max–min method and the best two relaynodes using doublemax method in certain scenarios to achieve the best performance. In [13, 14], the authors proposed doublerelay selection strategy and proved that dualrelay selection strategies outperform singlerelay ones.
The stateoftheart articles investigating the singlerelay selection techniques consider only the weakest or strongest channels to select the best one or two relaynodes [10,11,12,13], e.g., they choose the best relay among all available relaynodes that owns the strongest forward or backward channel or they select two relays where one of them has the strongest forward channel and the other owns the strongest backward channel. Therefore, the motivation of this work is that the current techniques are not considering the differences among the close value channels, while this can significantly improve the performance of relaying techniques. Therefore, in this paper, we propose a novel doublerelay selection technique based on STC using the threephase protocols. Furthermore, in the proposed technique, network coding is applied at the relay sides in order to decrease power consumption through the combination of symbols of the communicating transceivers in one symbol with the same constellation. From simulations and analytical results, we show that our novel technique delivers a better performance compared to the already existing relaying techniques. This article is summarized as follows: Sects. 2 and 3 explain the system model and the proposed technique, respectively. Section 4 demonstrates the BER analysis, while Sects. 5 and 6 show the experimental methods and the results and discussions, respectively. The conclusion is introduced in Sect. 7.
System model
In our system model shown in Fig. 1, we consider a halfduplex wireless relay network with R + 2 singleantenna nodes, i.e., two peripheralnodes (\({\text{PN}}_{1}\) and \({\text{PN}}_{2}\)), that intend to communicate with each other, but they cannot communicate directly due to transmission range limitation. Therefore, a group of R intermediate singleantenna relaynodes that are located between them need to facilitate this communication. More specifically, two intermediate relaynodes will be chosen to facilitate the required communication. Also, we assume having independent Rayleigh flatfading channels with zero mean and unit variance where the channel from \({\text{PN}}_{1}\) to relaynode \(i\) is denoted as \(f_{i }\) and from \({\text{PN}}_{2}\) to relaynode \(i\) as \(b_{i}\). Furthermore, all the communication channels between peripheral and relaynodes are assumed to be reciprocal for the transmission from \({\text{PN}}_{1}\) to \({\text{PN}}_{2}\) and vice versa and perfect CSI is assumed to be available at all receiving nodes. The channels are assumed to stay constant during two time slots and change to an independent realization afterward. In addition, perfect synchronization and timing are considered. Finally, we assume here that all the relaynodes and peripheralnodes have limited average transmit powers denoted as \(P_{{{\text{PN}}_{j} }} , j = 1,2\) for peripheralnodes and \(P_{{{\text{RN}}_{i} }} , i = 1,2, \ldots ,R\) for relaynodes. In this article, we use the following notations .\(,.,.\), (.)* and \(E\left( . \right)\) to denote the absolute value, the floor function which rounds toward zero, the Frobenius norm, the complex conjugate and the statistical expectation, respectively.
Now, let us consider the example shown in Fig. 2 which illustrates the concept of this novel technique. We assume that all channel values between the intermediate relaynodes and the two peripheralnodes are given. Based on the max–min selection criterion [10,11,12], \({\mathcal{R}}{\text{N}}_{1}\) and \({\mathcal{R}}{\text{N}}_{3}\) will be chosen as the best two available relaynodes since they have the maximum value links of (7 and 6.95) among the group of minimum value links (7, 6.9, 6.95, 6.85, 4, and 3). However, we can clearly see that \({\mathcal{R}}{\text{N}}_{2}\) and \({\mathcal{R}}{\text{N}}_{4}\) will probably have a much better performance as compared to \({\mathcal{R}}{\text{N}}_{1}\) and \({\mathcal{R}}{\text{N}}_{3}\), as the maximum difference between their minimum value links is less than 3% ((7–6.85)/7) × 100% in the best scenario, while the maximum value links for \({\mathcal{R}}{\text{N}}_{2}\) and \({\mathcal{R}}{\text{N}}_{4}\) are (20 and 25), respectively, compared to (8 and 7.5) for \({\mathcal{R}}{\text{N}}_{1}\) and \({\mathcal{R}}{\text{N}}_{3}\), which is at least 60% improvement in favor of \({\mathcal{R}}{\text{N}}_{2}\) and \({\mathcal{R}}{\text{N}}_{4}\). Therefore, our proposed tworelaynode selection technique, whose flowchart is shown in Fig. 3, will first find the minimum value link for each relaynode (w = 7, 6.9, 6.95, 6.85, 4, and 3). Then, it sorts those minimum value links in descending order, so (w_{n} = 7, 6.95, 6.9, 6.85, 4, and 3). After that, it will check the successive difference between the minimum value links, to find a subgroup of relaynodes that have close values of their minimum value links (i.e., successive difference less than threshold). Once the successive difference becomes above the specified threshold (e.g., Threshold = 10%), this comparison will terminate, and the technique will move for further processing of the selected subgroup of relaynodes (w_{g} = 7, 6.95, 6.9, 6.85). At this stage, the proposed technique will estimate the expected performance of each node in the selected subgroup by multiplying the values of their both channels (\(f_{i} \times b_{i}\)) to produce (7 × 8 = 56, 6.95 × 7.5 = 52, 6.9 × 20 = 138, and 6.85 × 25 = 171). After that it selects the two relaynodes (\({\mathcal{R}}{\text{N}}_{2}\) and \({\mathcal{R}}{\text{N}}_{4}\)) with the highest possible performance (138, and 171). Section 3 explains the proposed dualrelay selection technique in more details.
The proposed bidirectional dualrelay selection technique
In this proposed technique, the threephase decodeandforward (DF) communication protocol will be executed [2, 3]. The first peripheralnode \({\text{ PN}}_{1}\) broadcasts its message vector \({\varvec{z}}_{{P_{1} }}\) in the first available time slot; then, the second peripheralnode \({\text{PN}}_{2}\) broadcasts its message vector \({\varvec{z}}_{{P_{2} }}\) in the second available time slot, so that the rth intermediate relaynode \({\text{RN}}_{r}\) receives the following during the first and second time slots, respectively:
Considering that \(\left[ {{\varvec{z}}_{{P_{1} }} } \right]_{i} \in Z_{{P_{1} }}\), \(\left[ {{\varvec{z}}_{{P_{2} }} } \right]_{i} \in Z_{{P_{2} }}\), \(E\left\{ {\left {\left[ {{\varvec{z}}_{{P_{1} }} } \right]_{i} } \right^{2} } \right\} = 1\), \(E\left\{ {\left {\left[ {{\varvec{z}}_{{P_{2} }} } \right]_{i} } \right^{2} } \right\} = 1\), \(Z_{{P_{1} }}\) and \(Z_{{P_{2} }}\) are possibly two different constellations, and \({\varvec{n}}_{{{\mathcal{R}}{\text{N}}_{1} ,r}}\) and \({\varvec{n}}_{{{\mathcal{R}}{\text{N}}_{2} ,r}}\) represent the noise signal vectors at the rth relaynode \({\mathcal{R}}{\text{N}}\)_{r} in the first and the second time slots. After that, the rth intermediate relaynode \({\text{RN}}\)_{r} decodes the two received message vectors using the maximum likelihood (ML) decoder as:
It is worth mentioning here that the encountered decoding complexity by the intermediate relaynodes is very low, as they apply a symbolwise decoding capable of providing a linear decoding complexity to detect the received messages. Later, the rth relaynode \({\mathcal{R}}{\text{N}}\)_{r} will combine the two received message vectors \(\tilde{\user2{z}}_{{{\text{PN}}_{1,r} }}\) and \(\tilde{\user2{z}}_{{{\text{PN}}_{2,r} }}\) to form a single message vector as follows:
where \({\mathcal{F}}\left( {., .} \right)\) is a combination function of the two received message vectors used at the intermediate relaynodes. Such combination functions have been recently discussed by many articles where modular arithmetic (MA) proposed in [2], XOR function discussed in [1, 2] and the combination function suggested in [2] are just some examples of the available combination functions. In the relay decoding described by Eqs. (3) and (4), the relaynodes simply round the real and imaginary parts of the received messages to the nearest constellation point in case of an integer constellation. Furthermore, the use of the combination function \({\mathcal{F}}\left( {.,.} \right)\) in the proposed scheme has a negligible complexity.
The two selected intermediate relaynodes (\({\text{RN}}_{i} \;{\text{and}}\;{\text{RN}}_{j}\)) will be chosen based on the proposed relaynodes selection technique, discussed in Sect. 2, as follows:
Step 1 Select the first relaynode based on max–min selection criterion, such that:
Step 2 Select the next relaynode based on max–min selection criterion when \(k = \left\{ {2, 3} \right\}\), such that:
Step 3 The third step of the proposed technique consists of sorting the relaynodes in a descending order based on their minimum value links, then calculating the difference in the quality of links between the second relay \({\text{RN}}_{{a_{2} }}\) and the third relay \({\text{RN}}_{{a_{3} }}\) using the following equation when \(k = 2\)
If Diff > Threshold, then the best selected relaynodes are \({\text{RN}}_{{a_{k} }} , \;k = \left\{ {1,2} \right\}\). Since they have the best links between the two communicating peripheralnodes that offer the best performance in the network.
Step 4 For Diff < Threshold, repeat Step 2 and 3 when \(k = 4\). After that, if Diff > Threshold, then \(M_{k} = 3\) else \(M_{k} = 4\). This step aims to shortlist all the intermediate relaynodes that have very close minimum value links, and will be repeated once again as long as the Diff value is below the targeted threshold.
Step 5 Select the best two relaynodes among the previous shortlisted relaynodes based on their expected performance, which will be calculated as the multiplication of their forward and backward links as below:
From (6d), it is clearly observed that \(i = a_{k}\) and \(j = a_{l}\). An example for the proposed selection technique is available in Sect. 2 as well as more details related to the previous steps are explained in the flowchart shown in Fig. 3. Let us now discuss the complexity of the proposed relayselection technique. First of all, steps 1 and 2 have the same complexity of the bestknown relay selection technique [10,11,12]. Therefore, we have added steps 3, 4, and 5 to modify the bestknown technique. Note that the sorting of the forward and backward channels is already applied in steps 1 and 2. In general, the proposed algorithm terminates after checking two to four relaynodes causing a negligible complexity in this case. However, let us discuss the worstcase scenario when the proposed algorithm checks all the available relaynodes. In step 3, the algorithm performs (R − 1) subtractions and (R − 1) divisions, and in step 4, it compares the results of step 3 with the threshold value, resulting in (R − 1) comparisons. In step 5, it multiplies the forward channel value with the backward channel value for the selected relaynodes and selects the ones with the highest two values resulting in (R − 1) multiplications and (2R − 3) comparisons. This means that the proposed algorithm performs (3R − 4) comparisons, (R − 1) subtractions, (R − 1) divisions and (R − 1) multiplications. Therefore, in the worstcase scenario, the complexity of these three added steps is linearly increasing with the increase of the number of available relaynodes. However, in almost all scenarios, the algorithm terminates after 2–4 iterations only, without the need to go through R − 1 iterations. Note that the complexity of the proposed algorithm can be fixed by setting the number of iterations to K. This means that in steps 1 and 2, the algorithm selects the best K relays, e.g., K = 4. In step 3, the algorithm performs only (K − 1 = 3) subtractions and (K − 1 = 3) divisions. In step 4, it performs (K − 1 = 3) comparisons. In step 5, it performs (2K − 3 = 5) comparisons and (K − 1 = 3) multiplications. In the aforementioned scenario, the complexity of the proposed algorithm becomes negligible. After that, those two selected intermediate relaynodes (\({\text{RN}}_{i} \;{\text{and}}\;{\text{RN}}_{j}\)) will perform orthogonal STC technique using Alamouti scheme to enhance the overall system’s diversity and gain without adding extra decoding complexity. In the following step, the two selected intermediate relaynodes (\({\text{RN}}_{i} \;{\text{and}}\;{\text{RN}}_{j}\)) broadcast their combined and decoded message vectors in the third time slot toward the peripheralnodes, so that, the second peripheralnode will receive:
where \(A_{i} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right], A_{j} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ {  \,1} & 0 \\ \end{array} } \right]\), and \({\varvec{n}}_{{{\text{PN}}_{2} }}\) represents the noise signal vector at the second peripheralnode PN_{2} received in the third time slot. Similarly, ML decoding scheme will also be performed at the peripheralnodes to recover the received message vectors. Note here that the concerned peripheralnode can retrieve the data message vectors using a symbolbysymbol detector, instead of applying the ML decoder, which has a linear decoding complexity based on the knowledge of its own data message \({\varvec{z}}_{{P_{2} }} ,\user2{ }\) and by applying the inverse of the combination function used at the rth relaynode \({\mathcal{R}}{\text{N}}\)_{r} similarly as in [2, 3]. Note that if only one relay (\(R = 1\)) is available between the communicating terminals, the selection technique does not have any option other than the available relay to be selected to facilitate the needed communication between them. In this case, the relay receives the information symbols from the communicating terminals, combine them using (5) and forward the resulting symbol to the terminals without applying any STC technique.
BER performance analysis
In this section, we derive the mathematical model of the average BER performance of the proposed relay selection technique using binary phase shift key (BPSK) modulation according to the assumptions proposed in Sect. 2. In this analysis, we assume that all relaynodes are ideal as in [13, 14, 27] and all noise signals are drawn from an independently distributed Gaussian random variables with zero mean and covariance \(\sigma^{2} {\varvec{I}}_{T}\). Now, the proposed relaynode selection technique, defined in Eqs. (6a)–(6d) above, selects the ith and jth relaynodes with \({P}_{{RN_{i} }} = {P}_{{{\text{RN}}_{j} }} = {P}_{{{\text{RN}} }} ,\) and considering that the SNR of the link between \({\mathcal{R}}{\text{N}}_{r}\) and \({\text{PN}}_{1}\) is denoted by \({\upgamma }_{r}^{{{\text{PN}}_{1} }} = \gamma \left {f_{r} } \right^{2}\), the SNR of the link between \({\mathcal{R}}{\text{N}}_{r}\) and \({\text{PN}}_{2}\) is denoted by \({\upgamma }_{r}^{{{\text{PN}}_{2} }} = \gamma \left {b_{r} } \right^{2}\), and \(\gamma = P_{{{\mathcal{R}}{\text{N}}}} /\sigma^{2}\) is the mean SNR at the \(r{{\rm th}}\) relay \({\mathcal{R}}{\text{N}}_{r}\). Let us rearrange \({\upgamma }_{r}^{{{\text{PN}}_{t} }} , r = 1, \ldots ,{\mathcal{R}},{{t}} = 1,{ }2\) increasingly such that \({\upgamma }_{1}^{{{\text{PN}}_{t} }} \le {\upgamma }_{2}^{{{\text{PN}}_{t} }} \le \ldots \le {\upgamma }_{R}^{{{\text{PN}}_{t} }}\) and denote \(w_{1}^{{{\text{PN}}_{t} }} = {\upgamma }_{1}^{{{\text{PN}}_{t} }}\) and \(w_{l}^{{{\text{PN}}_{t} }} = {\upgamma }_{l}^{{{\text{PN}}_{t} }}  {\upgamma }_{{\left( {l  1} \right) }}^{{{\text{PN}}_{t} }}\) for \(l = 2, \ldots , R.\) Note here that the independent factors, \(w_{l}\) for \(l = 1, \ldots ,R\), are following a probability distribution function as depicted in the below equation [13, 14, 27]:
In this relay selection technique, as a first step the ith relaynode \({\mathcal{R}}{\text{N}}_{i}\) is selected among the \(R\) available relaynodes based on the selection criterion, explained in Eqs. (6a)–(6d), considering that \(b_{i}\) is the link between \({\mathcal{R}}{\text{N}}_{i}\) and \({\text{PN}}_{2}\) with \({\upgamma }_{{\alpha_{1} }}^{{{\text{PN}}_{2} }}\) and \(f_{i}\) is the link between \({\mathcal{R}}{\text{N}}_{i}\) and \({\text{PN}}_{1}\) with \({\upgamma }_{{u_{1} }}^{{{\text{PN}}_{1} }}\) where \({\upgamma }_{{u_{1} }}^{{{\text{PN}}_{1} }} \ge \gamma_{{{\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{{\text{PN}}_{2} }}\), and \(u_{1}\) could be any uth greater than \(\alpha_{1}\). For the second step, the jth relaynode \({\mathcal{R}}{\text{N}}_{j}\) is selected similarly from the remaining \(\left( {{\text{R}}  1} \right)\) relaynodes with \(b_{j}\) is the link between \({\mathcal{R}}{\text{N}}_{j }\) and \({\text{PN}}_{2}\) with \({\upgamma }_{{\alpha_{2} }}^{{{\text{PN}}_{2} }}\), and \(f_{j}\) is the link between \({\mathcal{R}}{\text{N}}_{j }\) and \({\text{PN}}_{1}\) with \({\upgamma }_{{u_{1} }}^{{{\text{PN}}_{1} }} \ge {\upgamma }_{{\alpha_{2} }}^{{{\text{PN}}_{2} }}\), and \(u_{2}\) could be any uth greater than \(\alpha_{2}\) and assuming that \({\upgamma }_{{\alpha_{1} }}^{{{\text{PN}}_{2} }} \ge {\upgamma }_{{\alpha_{2} }}^{{{\text{PN}}_{2} }} .\) As a result, the average BER of the proposed doublerelay selection technique can be expressed as:
It can be clearly seen that this equation is a combination of four terms where the first term (P_{1}) can be calculated using the momentgeneration function (MGF) as:
Then, the MGF of \(Q_{1}\) can be also calculated as [13, 14, 27]:
Now, we can use the partial fraction method to simplify the given equation above to:
Therefore, the term \(P_{1}\) can now be expressed as [13, 14, 27]
Finally, combining Eqs. (11), (12) and (13), then \(P_{1}\) can be rewritten as:
Similarly, the terms \(P_{2} ,\) \(P_{3}\), and \(P_{4}\) presented in Eq. (9) can be found by following the same procedure that leads to the calculation of \(P_{1 }\) presented in Eq. (14). For \(P_{2}\), considering that \(u_{1} \ge u_{2}\) and using the MGF of \(Q_{2} = {\upgamma }_{{u_{1} }}^{{{\text{PN}}_{1} }} + {\upgamma }_{{u_{2} }}^{{{\text{PN}}_{1} }}\), for \(P_{3}\) given that \(u_{2} \ge \alpha_{1}\), and using MGF of \(Q_{3} = {\upgamma }_{{u_{2} }}^{{{\text{PN}}_{1} }} + {\upgamma }_{{\alpha_{1} }}^{{{\text{PN}}_{2} }}\), and for \(P_{4}\) given that \(u_{1} \ge \alpha_{2}\) and using MGF of \(Q_{4} = {\upgamma }_{{u_{1} }}^{{{\text{PN}}_{1} }} + {\upgamma }_{{\alpha_{2} }}^{{{\text{PN}}_{2} }} , P_{2}\), \(P_{3}\) and \(P_{4}\) can be calculated as follows:
Again, using partial fraction expansion we get:
Then, the term of P_{2}, P_{3} and P_{4} can be rewritten as:
Finally, using the final equations of \(P_{1}\), \(P_{2} ,\) \(P_{3} ,\) and \(P_{4}\) then the average BER of the suggested relaynode selection technique can be now calculated using:
Now, assuming that the threshold value is equal to zero, then the ith relaynode \({\mathcal{R}}{\text{N}}_{i}\) is selected among the \(R\) available relaynodes based on the selection criterion explained in Sect. 2, given that \(b_{i}\) is the link between \({\mathcal{R}}{\text{N}}_{i}\) and \({\text{PN}}_{2}\) with \({\upgamma }_{{{\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{{\text{PN}}_{2} }}\) and \(f_{i}\) is the link between \({\mathcal{R}}{\text{N}}_{i}\) and \({\text{PN}}_{1}\) with \({\upgamma }_{{u_{1} }}^{{{\text{PN}}_{1} }}\) where \({\upgamma }_{{u_{1} }}^{{{\text{PN}}_{1} }} \ge \gamma_{{{\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{{\text{PN}}_{2} }}\), and \(u_{1}\) could be any uth greater than \({\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}\). After that, the jth relaynode \({\mathcal{R}}{\text{N}}_{j}\) is selected similarly from the remaining \(\left( {{{R}}  1} \right)\) relaynodes with \(b_{j}\) which is the link between \({\mathcal{R}}{\text{N}}_{j }\) and \({\text{PN}}_{2}\) with \({\upgamma }_{{{\raise0.7ex\hbox{${\left( {R  1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {R  1} \right)} 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{{\text{PN}}_{2} }}\), and \(f_{j}\) is the link between \({\mathcal{R}}{\text{N}}_{j }\) and \({\text{PN}}_{1}\) with \({\upgamma }_{{u_{2} }}^{{{\text{PN}}_{1} }}\) where \({\upgamma }_{{u_{1} }}^{{{\text{PN}}_{1} }} \ge {\upgamma }_{{{\raise0.7ex\hbox{${\left( {R  1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {R  1} \right)} 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{{\text{PN}}_{2} }}\) and \(u_{2}\) could be any \(u{{\rm th}}\) greater than \({\raise0.7ex\hbox{${\left( {R  1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {R  1} \right)} 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}.\) Then, the average BER of the proposed doublerelaynode selection technique can be expressed as:
Again, this equation is composed of four terms, where the first term \(P_{1}\) can be calculated using the MGF, such that:
Then, the MGF of \(Q_{1}\) can be also calculated as [13, 14, 27]:
Now, we can use the partial fraction method to simplify the given equation above to:
Therefore, the term \(P_{1}\) can now be expressed as [13, 14, 27]
Finally, combining Eqs. (33), (34) and (35), then \(P_{1}\) can be rewritten as:
Similarly, the terms \(P_{2} ,\) \(P_{3}\) and \(P_{4}\) in Eq. (31) can be found by following the same procedure. For \(P_{2}\), considering that \(u_{1} \ge u_{2}\), and using the MGF of \(Q_{2} = {\upgamma }_{{u_{1} }}^{{{\text{PN}}_{1} }} + {\upgamma }_{{u_{2} }}^{{{\text{PN}}_{1} }}\), for \(P_{3}\) given that \(u_{2} \ge {\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}\), and using MGF of \(Q_{3} = {\upgamma }_{{u_{2} }}^{{{\text{PN}}_{1} }} + {\upgamma }_{{{\raise0.7ex\hbox{$R$} \!\mathord{\left/ {\vphantom {R 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{{\text{PN}}_{2} }}\), and for \(P_{4}\) given that \(u_{1} \ge {\raise0.7ex\hbox{${\left( {R  1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {R  1} \right)} 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}\) and using MGF of \(Q_{4} = {\upgamma }_{{u_{1} }}^{{{\text{PN}}_{1} }} + {\upgamma }_{{{\raise0.7ex\hbox{${\left( {R  1} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {R  1} \right)} 2}}\right.\kern\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{{\text{PN}}_{2} }}\), then:
Again, using partial fraction expansion we get:
Then, the terms of P_{2}, P_{3} and P_{4} can be rewritten as:
Finally, using the final equations of \(P_{1}\),\(P_{2} ,\) \(P_{3} ,\) and \(P_{4}\) then the average BER of the suggested relaynode selection technique can be now calculated using:
Methods/experimental
In this section, we present the results obtained from both our conducted simulations and mathematical model presented in Sect. 4. In the simulation results, Monte Carlo runs have been used to generate the simulated performance of the proposed technique to be compared with the best known techniques and the theoretical performance attained from the mathematical equation given by (30) in Sect. 4. Note that in our system model shown in Fig. 1, we consider a halfduplex relay network with R + 2 singleantenna nodes consisting of two communicating terminals, i.e., (PN_{1} and PN_{2}), that intend to communicate with each other through R intermediate singleantenna relaynodes. We assume independent slow Rayleigh flatfading channels with zero mean and unit variance. Furthermore, all CSI is perfectly known at all receiving nodes and all communication channels between the terminals and relays are assumed to be reciprocal. The channels are assumed to stay constant during two time slots and change to an independent realization afterward. Moreover, perfect synchronization and timing are considered. In Fig. 4, we show the simulated performance of the proposed relayselection technique in terms of BER with the theoretical performance attained from the mathematical equation given by (30) in Sect. 4 under the scenario of a wireless cooperative network with various number of available intermediate relaynodes R = {2; 4; 6} using BPSK constellation and assuming no direct link available between the first peripheralnode \({\text{PN}}_{1}\) and the second peripheralnode \({\text{PN}}_{2}\). This figure shows a very close matching between our simulations and mathematical model. Figure 5 shows a performance comparison in terms of the BER between our proposed doublerelay selection technique with the doublemax technique proposed in [11], the max–min technique proposed in [12], and the dualrelay selection technique proposed in [14] under the scenario of R = {2, 4} intermediate relaynodes using 4QAM constellation. Figure 6 also shows a performance comparison between the previously mentioned relaynode selection techniques but under the scenario of R = {2, 4, 6} intermediate relaynodes using 8PSK constellation. Both figures clearly show that our proposed technique outperforms the best available stateoftheart relaynode selection techniques. Figure 7 shows the BER performance of our proposed dualrelay selection technique using the scenario of four available intermediate relaynodes (R = 4) and 4QAM constellation under different threshold values, i.e., threshold varies between 0 and 5. Note here that our proposed technique with threshold value set to zero is equivalent to the selection strategy proposed in [14]. The figure clearly shows that the best performance is achieved when the threshold value is two. To confirm this result, Fig. 8 shows the BER results of our proposed relay selection technique using 4QAM modulation with four available intermediate relaynodes and SNR of 20 dB under different threshold values. This figure clearly supports our findings in Fig. 7.
Similarly, Fig. 9 shows the BER performance of our proposed dualrelay selection technique using the scenario of six available intermediate relaynodes (R = 6) using 4QAM constellation under different threshold values, i.e., threshold varies between 0 and 5. Again, this figure clearly shows also that the best performance is achieved when the threshold value is two. Figure 10 shows the BER results of our proposed relay selection technique using 4QAM modulation with six available intermediate relaynodes and SNR of 15 dB under different threshold values. This figure clearly supports our findings in Figs. 7, 8 and 9.
Furthermore, Fig. 11 shows the BER performance of our proposed dualrelay selection technique using the scenario of four available intermediate relaynodes (R = 4) using 8PSK constellation under different threshold values, i.e., threshold varies between 0 and 5. Again, this figure clearly shows also that the best performance is achieved when the threshold value is two. Figure 12 shows the BER results of our proposed relay selection technique using 8PSK modulation with four available intermediate relaynodes and SNR of 25 dB under different threshold values. This figure clearly supports our findings in Figs. 7, 8 and 9.
Similarly, Fig. 13 shows the BER performance of our proposed dualrelay selection technique using the scenario of six available intermediate relaynodes (R = 6) using 8PSK constellation under different threshold values, i.e., threshold varies between 0 and 5. Again, this figure clearly shows also that the best performance is achieved when the threshold value is two. Figure 14 shows the BER results of our proposed relay selection technique using 8PSK modulation with six available intermediate relaynodes and SNR of 20 dB under different threshold values. This figure clearly supports our findings in Figs. 7, 8 and 9.
Results and discussion
From Fig. 4, we observe that the simulated performance of the suggested strategy in terms of BER is very close to the theoretical BER performance attained from Eq. (30) in Sect. 4. In Figs. 5 and 6, it can be observed that the suggested strategy that uses the threephase relaying protocol outperforms the current stateoftheart strategies that perform the same relaying protocol.
From Fig. 7, it is clearly shown that our proposed threephase dualrelaynode selection technique under different threshold values outperforms the threephase doublerelaynode selection technique proposed in [14]. Noting that our proposed technique using a threshold value of zero is similar to the threephase doublerelaynode selection technique proposed in [14]. It is worth mentioning that the proposed technique which implements a threephase relaying protocol outperforms the current stateoftheart strategies that perform the same relaying protocol. From Fig. 8, it is observed that the BER improves while increasing the threshold value until the best performance is achieved at threshold value of two (threshold = 2); then, the performance will degrade for threshold values greater than three.
Figure 9 clearly shows that our proposed threephase dualrelaynode selection technique under different threshold values outperforms the threephase doublerelaynode selection technique proposed in [14]. Figure 10 shows the relation between the BER performance of the relaynode network scenario discussed for Fig. 9 and the threshold values. It is very obvious that this relation is similar to that noticed in Fig. 8, the BER will improve while increasing the threshold value until the best performance is achieved at threshold value of two (Threshold = 2), then the performance will degrade for threshold values greater than three. Furthermore, we note here that the BER has been reduced by almost 3 dB when we change the threshold value in our proposed technique from zero (standard max–min selection criteria) to two.
Finally, observed results from Figs. 11 and 13 are similar to those found in Figs. 7 and 9. In addition, observed results from Figs. 12 and 14 are similar to those found in Figs. 8 and 10.
Conclusion
In this work, we have proposed a novel dualrelay selection strategy using the threephase protocol based on STC. Furthermore, in this technique we have applied a network coding scheme at the selected relays in order to combine the symbols of the communicating symbols into one symbol with the same constellation. As a result, additional coding gain will be achieved. Furthermore, the analytical BER of this novel technique is investigated and proposed. The BER expression is compared to the simulation results in order to validate the analytical results. We proved as well that our novel method outperforms the most recent work and the current techniques.
Availability of data and materials
A part of the simulation data used or analyzed during the current study is available from the corresponding author on reasonable request.
Abbreviations
 BER:

Bit error rate
 bpcu:

Bit per channel use
 BPSK:

Binary phase shift key
 CSI:

Channel state information
 MGF:

Moment generation function
 QAM:

Quadrature amplitude modulation
 SNR:

Signaltonoise ratio
 STC:

Spacetime coding
 ML:

Maximum likelihood
 QAM:

Quadrature amplitude modulation
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This work was supported by College of Engineering and Technology, the American University of the Middle East, Kuwait.
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All contributions are jointly done by all authors starting from proposing the system model analyzing the system performance, optimizing the system model, verifying the correctness of the derivations, till writing and modifying the paper writing and smoothing the words in the paper. All authors read and approved the final manuscript.
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Alabed, S., Maaz, I. & AlRabayah, M. Improved twoway doublerelay selection technique for cooperative wireless communications. J Wireless Com Network 2021, 57 (2021). https://doi.org/10.1186/s13638020018467
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Keywords
 Twoway relay networks
 Cooperative diversity techniques
 Distributed spacetime coding
 Relay selection
 Network coding