Priority-Based Max Link Selection for Double Differential Modulation-Based Buffer-Aided Relaying With Offsets

In this paper, we consider a decode-and-forward (DF) buffer-aided (BA) multi-relay system using double differential (DD) encoding and decoding for the transmission and reception of data packets. The proposed system does not require carrier frequency offset (CFO) information and channel state information (CSI) at any communicating node. A priority-based max link selection protocol has been adopted to select links based on the buffer status and channel quality. The Markov-chain approach is used for developing the state transition matrix, using which the steady state probability of the system is obtained. The outage probability and average bit error rate (ABER) expressions are derived using the steady-state probability. The proposed setup performance is then compared with the conventional coherent BA system. It is established that the considered setup is affected by a signal-to-noise ratio (SNR) penalty of $\approx 3$ dB, which is considerably lower than the well-known 6 dB SNR penalty existing for DD modulation compared to coherent modulation. Also, the proposed setup outcompetes the coherent max-link approach in the presence of a CFO.

SECTION I. Introduction

Spatial diversity techniques can overcome the channel fading phenomenon’s ill effects in wireless communication systems. Cooperative relaying helps us achieve spatial diversity, improving the throughput and reliability of wireless communication by forming a virtual antenna array [1]. In an elementary cooperative system, a source node communicates with the destination $(D)$ node via a relay $(R)$ node with the help of various relaying protocols such as decode-and-forward (DF), amplify-and-forward (AF) and incremental relaying [2], [3], [4]. In a typical cooperative system, $S$ -$R$ and $R$ -$D$ links keep on transmitting irrespective of the channel quality, which leads to inferior performance as few links may be in an outage at a given time interval, which causes a loss of data packets. This limitation has led to a surge of interest in buffer-aided (BA) relaying, which helps us manage the transmission or reception of data packets based on the link quality and buffer status [5]. It also helps us achieve higher diversity and better performance [6]. However, all these gains come at the cost of an additional packet delay [7].

A. Literature Review

Several BA cooperative relaying scenarios have been covered in the literature [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. Authors in [11] compared fixed and selective scheduling for the BA system. Authors in [12], propose a max-max link selection strategy that helps achieve overall spatial diversity. A max-link selection strategy is described in [13], where the link with the highest channel gain is chosen to transmit or receive a data packet at every time slot; this protocol helps us to achieve a diversity order of twice the number of relaying nodes but adds an additional packet delay. Authors in [14] discuss the link selection strategy based on the buffer state; in this protocol, authors prioritized link selection based on the buffer status. A hybrid link selection approach is presented in [15], which combines max-link and max-max link selection strategies. In [16], authors discussed a link selection strategy in which the $S$ -$R$ link is only selected when the $R$ -$D$ link is unavailable for transmission. This strategy reduces the overall packet delay at the expense of the system’s diversity order. A priority-based max-link selection approach has been proposed in [17], where the data packet is transmitted and received depending on the channel quality and the relay buffer status.

A few more BA setups with imperfect configurations have been discussed in [18], [19], [20], [21]. In [18], authors explore a BA relaying setup with outdated channel state information (CSI). In contrast, in [19], authors discussed a single relay setup with a BA relay suffering from self-interference. Authors in [20], [21] discussed the security aspect of the BA setup. In [20], authors analyzed a security aspect by considering the energy harvesting relays, while in [21], authors analyzed a secure communication system by taking out the ratios of all the available links and then selecting the link with the max ratio for the transmission or reception of data. An NBA system using the Markov chain (MC) approach for the formation of energy states is analyzed in [22]; the only relay in the system is the energy harvesting relay, and it uses energy queues instead of data packets for the formation of MC states.

Moreover, most practical cooperative relaying scenarios consider that perfect knowledge of the channel state of all the communicating links is available to different nodes; this assumption is practically non-feasible, so for such cases, different single differential modulation (SDM) schemes for cooperative communication have been discussed in the literature [23], [24]. The SDM assumes that the channel remains constant for at least two consecutive intervals, and it helps with communication, even without perfect knowledge of the CSI. It is noteworthy, too, that, in wireless cooperative systems, transceiver nodes may be mobile, due to which the problem of carrier frequency offset (CFO) may arise. Hence, the channel state does not remain constant over two consecutive time intervals. Thus, SDM suffers huge performance loss [25] because of the imperfect CFO compensation. Double differential modulation (DDM) is useful in such a scenario, as DDM avoids the need for CSI and CFO estimation; therefore, DD-encoded data can be decoded without their knowledge. The fundamental difference between SDM and DDM is that DDM uses three consecutively received data samples to decode the transmitted symbol. In other words, two levels of a single differential scheme are employed [26].

Implementation of DDM for different relaying protocols has been carried out in [27], [28], [29]. A low complexity piecewise linear (PL) decoder for the coherent system is explained in [30], and its differential counterpart is given in [31]; in both cases, PL decoder significantly improves the BER performance of the system. In [32], DDM is used in which relays employ cyclic redundancy bits and forward the data only when there is no decoding error; this helps reduce decoding error; however, the overall data rate is reduced. Authors in [33], introduced a DF cooperative system with DDM employs a PL detector and established analytically that the PL decoder achieves full diversity. Authors in [34], discussed the DD transmission for AF two-way relaying nodes. Also, in [35], authors examined the DDM approach for the mobile wireless sensor networks with multiple CFOs, while in [36], authors used DDM for Leo-based land mobile satellite communication.

B. Novelty and Contributions

In wireless communication networks, obtaining the perfect CSI is a challenging task, and it further becomes more difficult in a multi-node system as the overall training requirement grows with increasing nodes. Furthermore, in communication systems where nodes are mobile, the problem of CFO exists due to the relative movement of nodes. The channel estimation may lead to a significant error floor for such a system due to the imperfect CFO compensation [25]. In such scenarios, the DDM scheme is useful as it allows the receiver to bypass the CFO estimation along with CSI; hence, DDM does not encounter any residual CFO problems due to a well-designed DD receiver. Moreover, it is well known that BA coherent relaying systems can improve the performance of wireless systems by offering us the liberty to select the max-link at any given instant. Buffers at the relay nodes help us store the data whenever the link faces an outage, thereby avoiding packet drop. Hence, we concur that by combining BA relaying with the DDM-based scheme, we can inherit the benefits of both of them. Specifically, the main advantages we can obtain by combining the DDM-based scheme with the BA-relaying are as follows:

• Narrow down the 6 dB SNR gap, which was earlier observed for the NBA DDM-based scheme.

• With the help of a DDM-based scheme, we can communicate even without the knowledge of CSI and CFO.

• In the presence of a CFO, the considered system might outperform the existing coherent BA relaying scheme.

The key idea of this work is to study wireless BA cooperative relaying, where the transceiver nodes may be mobile, leading to time-varying channels because of the presence of a CFO. Due to the time-varying nature of the wireless channels, the existing coherent BA protocols are either not applicable or offer poor performance for the considered CFO-perturbed cooperative networks. Since DDM offers a satisfactory performance in the presence of a CFO in a cooperative network, it is envisaged that the multiple relay DDM-based BA system can overcome the problem of CFO and CSI estimation. Nevertheless, the necessary performance improvement cannot be merely achieved by adding a buffer to relays; hence, a modified link selection mechanism that complies with the requirements of DDM is needed in a DDM-based BA relaying. Specifically, the important contributions of the presented work can be summed up as follows:

• A $K$ relay BA cooperative setup using DDM with a buffer size $L$ is proposed.

• A comprehensive link selection technique based on the buffer status and link quality is proposed; also, a novel index-based approach is proposed that satisfies the prerequisites of DDM and helps in encoding and decoding the received signals.

• The closed-form expressions of ABER and outage probability are derived, and the derived results are then compared with the coherent counterparts.

• The system’s ABER performance is further examined for a different set of values of $K$ , power allocation factor, and relay location. In the presence of the CFO, the proposed multiple relay DDM-based BA system outperforms the coherent BA setup.

• Additionally, the proposed scheme closes down the 6dB signal-to-noise ratio (SNR) gap between the DDM and coherent systems that have been previously reported in the literature.

The remainder of the article is structured as follows: A discussion of DDM-based BA relaying and the proposed protocol are both carried out in Section II. The performance analysis of the proposed setup is covered in Section III. Section IV covers numerical results and discusses the proposed DDM-based BA relaying. Section V serves as the paper’s conclusion and provides the future directions of the proposed work.

SECTION II. Buffer-Aided Double Differential Relaying

This Section proposes a DDM-based relaying setup with the BA relay nodes. Firstly, we discussed the system model used, then we discussed the DDM, and finally, we proposed the novel index-based transmission and reception protocol for DDM-based BA relaying.

A. System Model

A dual-hop cooperative relaying DDM-based system is depicted in Fig. 1; it consists of a source node $(S)$ , destination node $(D)$ , and $K$ DF-based relay nodes represented as $R_{k}, k\in 1, 2,\ldots, K$ . It is assumed that every node is mobile with a single antenna and operates in half-duplex (HD) mode. Also, no direct communication takes place between $S$ and $D$ as a direct link is absent between them. At any given time instant, assuming total transmitted power to be $P_{T}$ , which is distributed in between $S$ and the $k$ -th relay chosen for transmission, respectively, as $P_{S} = \delta P_{T}$ and $P_{R} = (1 - \delta)P_{T}$ , where $\delta \in \{0,1\}$ is power allocation factor, $P_{R}$ is the power of $k$ -th relay node and $P_{S}$ is the source power.

Fig. 1.

Double differential-based buffer-aided cooperative system.

It is assumed that all the relays exist in a cluster, which is placed exactly in between the $S$ and $D$ , such that $S$ -$R_{k}$ distance is given by $d_{S{R_{k}}}$ and $R_{k}$ -$D$ distance is denoted by $d_{R_{k}D}$ . In addition, it is assumed that relays are far enough from $S$ and $D$ such that inter-relay distance is insignificant when compared with $S$ -$R_{_{k}}$ and $R_{k}$ -$D$ distance; therefore, all the relays are assumed to be at equal distance from $S$ and $D$ , i.e., $d_{SR_{k}}=d_{SR}$ and $d_{R_{k}D}=d_{RD}$ for all values of $k$ . Hence, we can say that $d_{SR}+d_{RD}=d_{SD}$ , where $d_{SD}$ is the distance between $S$ to $D$ . Each relay node has a data buffer of $L$ , represented by $Q_{k}$ . The total number of data packets in the buffer $Q_{k}$ at any instance is denoted by $\psi (Q_{k})$ , where $0\leq \psi (Q_{k}) \leq L$ . At the beginning, the buffer of each relay node is presumed to be empty, i.e., $\psi (Q_{k})=0$ , and the buffer status increases whenever the relay node receives the data. In contrast, during the transmission of the data from the relay, the buffer status is decremented by one. Also, $S$ is always assumed to have data for transmission.

The system experiences an independent and identically distributed Rayleigh-block fading, i.e., the channel remains unchanged for a fixed time frame of size $T$ , and the above Rayleigh distribution will have a mean zero and variance $\sigma _{h}^{2}$ ; with the channel parameters between any $ij$ link $i \in \{S,R_{k}\}$ , $j \in \{R_{k}, D\}$ , $i \neq j$ is denoted by $h_{ij}$ . The system is also perturbed by a zero mean and $N_{0}$ variance additive white Gaussian noise (AWGN). The average SNRs for S-R and R-D links can be evaluated as $\overline {\gamma }_{SR}=P_{S} \times {}{}\frac { d_{SR}^{-{\alpha }}} {N_{0}}$ and $\overline {\gamma }_{RD}=P_{R} \times {}{}\frac { d_{RD}^{-{\alpha }}} {N_{0}}$ , respectively, where $\alpha $ represents the path loss exponent and can be defined as attenuation in the power of the transmitted signal over a distance. As the nodes are mobile, the Doppler shift frequency experienced by the receiving node when the transmitting node is moving towards it will be $f_{_{1}}=f_{_{c}}(1+v/c)$ , where $v$ is relative velocity of two nodes, $c$ denotes speed of light, and $f_{_{c}}$ represents carrier frequency. Similarly, when the transmitting node is moving away from the receiving node, then the observed frequency shift will be $f_{_{2}}=f_{_{c}}(1-v/c)$ . The Doppler effect introduces a time-varying phase shift in the received signal, which makes the channel time-selective, leading to the problem of CFO.

B. Double Differential Modulation

Fig. 2a shows the DD encoding of the data at the source node. The equation for the DDM signal at $S$ is given as \begin{align*} v(t)=v(t-1)p(t);p(t)=p(t-1)x(t),\, t=0,\,1,\,\ldots, \,T, \tag{1}\end{align*} where $v(t)$ is DD encoded symbol with an assumptions $v(0)=1$ , $|v(t)|^{2}=1$ , also $p(t)$ is single differential (SD) encoded symbol with $p(0)=1$ and $|p(t)|^{2}=1$ . The symbol $x$ is the original symbol of $S$ in the $t$ -th time interval of a block, and it belongs to the unit norm constellation $\Xi $ . Now, if the total of $0\leq m\leq K S$ -$R$ links are eligible to communicate (based on the buffer status of the respective relay nodes), then for such a scenario, the signal received at $R_{k}$ through the best link is denoted by \begin{equation*} y_{_{R_{k}}}(t)=h_{_{SR_{k}}}e^{jtw_{_{R_{k}}}}v(t)+e_{_{R_{k}}}, \tag{2}\end{equation*} where $h_{_{SR_{k}}}$ is the channel coefficient for $S$ -$R_{k}$ link, $e_{_{R_{k}}}$ is the AWGN at $R_{k}$ , and $w_{R_{k}}$ is the CFO at $R_{k}$ with $w_{R_{k}} \in [-\pi,\,\pi]$ , and it is as per [39, eq. (9.7.26)],. Assuming that the symbol is received at $R_{k}$ at any time interval, it is then DD decoded as shown in Fig. 2b, the MLD in the f1gure 1 s the maximum likelihood (ML) detector, and it is represented as \begin{equation*} x_{_{R_{k}}}(l)=\text {arg}\,\underset {x\in \Xi }{\text {max}}\,\text {Re}\,\left \{{Y_{_{R_{k}}}^{\ast}(l)Y_{_{R_{k}}}^{\ast}(l-1)x(l)}\right \}, \tag{3}\end{equation*} here $\{\cdot \}^{\ast}$ denotes conjugate of a symbol and $\text {Re}\{\cdot \}$ represents the real part of the symbol. The simplified MLD of the DD encoded symbol at $R_{k}$ is given by \begin{equation*} x_{_{R_{k}}}(l)=\text {arg}\,\underset {x(l)\in \Xi }{\text {max}}\,\text {Re}\,\left \{{y_{_{R_{k}}}^{\ast}(l)y_{_{R_{k}}}^{\ast}(l-2)y_{_{R_{k}}}^{2}(l-1)x(l)}\right \}. \tag{4}\end{equation*} The decoded symbol is then DD encoded at $R_{k}$ in the $l$ -th time interval as \begin{align*} v_{_{R_{k}}}(l)=&v_{_{R_{k}}}(l-1)p(l);p_{_{R_{k}}}(l) \\=&p_{_{R_{k}}}(l-1)x_{_{R_{k}}}(l);\,\,l=0,\,1,\,\ldots, \,T. \tag{5}\end{align*} This encoded symbol at $R_{k}$ is then transmitted to $D$ , the symbol received at $D$ is given as \begin{equation*} y_{D}(l)=h_{_{R_{k}D}}e^{jlw_{_{D}}}v_{R}(l)+e_{_{D}}(l), \tag{6}\end{equation*} where $h_{_{R_{k}D}}$ is the channel coefficient for $R_{K}$ -$D$ link, $w_{_{D}}$ is the CFO at $D$ with $w_{_{D}} \in \{-\pi,\,\pi \}$ it is as per [39, eq. (9.7.26)], and $e_{_{D}}$ is the AWGN at $D$ . The conditional probability density function (PDF) for the symbol $y_{_{D}}=y_{_{D}}(l)y_{_{D}}(l-2)$ is given by \begin{align*} p_{y_{_{D}}}\left [{y_{_{D}}|y_{_{D}}(l-1)}\right]=&\mathbf {\epsilon }p_{y_{_{D}}}\left ({y_{{_{D}}}|y_{{_{D}}}(l-1)}\right),x_{_{R}}(l)\neq x(l) \\&+ \left ({1-\mathbf {\epsilon }}\right)p_{y_{_{D}}}\left ({y_{{_{D}}}|y_{{_{D}}}(l-1)}\right),x_{_{R}}(l)= x(l), \tag{7}\end{align*} where $\mathbf {\epsilon }$ is the ABER for single $S$ -$R$ link. Now, since our system can have $m S$ -$R$ links, from here onwards, we will denote the BER of $S$ -$R$ link as $\epsilon _{m}$ , a detailed discussion about which will be covered in Section III. For the above-discussed DD model, the log-likelihood ratio (LLR) detector is derived using [33, eq. (9)], as \begin{equation*} f\left ({t_{_{1}}(m)}\right)=\text {ln}\left [{\frac {\left ({1-\mathbf {\epsilon }_{m}}\right) e^{t_{{_{1}}}(m)}+\mathbf {\epsilon }_{m}}{\mathbf {\epsilon }_{m}e^{t_{{_{1}}}(m)}+\left ({1-\mathbf {\epsilon }_{m}}\right)}}\right], \tag{8}\end{equation*} where $\text {ln}(\cdot)$ represents the natural logarithm and \begin{equation*} t_{{_{1}}}(m)=\frac {2\text {Re}\left \{{y_{_{D}}^{\ast}(l)y_{_{D}}^{2}(l-1)y_{_{D}}^{\ast}\left ({l-2}\right.}\right \}}{3|y_{_{D}}(l-1)|^{2} N_{0}}. \tag{9}\end{equation*} The LLR provided in (8) can be approximated with the help of a piece-wise linear (PL) detector as \begin{align*} f_{_{PL}}\left ({t_{_{1}}(m)}\right)= \begin{cases} -T_{_{1}}(m),& \text {for }\, t_{_{1}}(m)< -T_{_{1}}(m), \\ T_{_{1}}(m),& \text {for }\, t_{_{1}}(m)>T_{_{1}}(m), \\ t_{_{1}}(m),& \text {for}\, -T_{_{1}}(m)< t_{_{1}}(m)< T_{_{1}}(m), \end{cases} \tag{10}\end{align*} where, $T_{1}(m)=\text {ln}\left[{{}{}\frac {1-\mathbf {\epsilon }_{_{m}} }{\mathbf {\epsilon }_{_{m}}} }\right]$ . Now, $y_{_{D}}(l) \sim \mathcal {CN} (y_{_{D}}(l-1)e^{jw_{_{D}}}p(l), 2N_{0})$ and $y_{_{D}}(l-2)\sim \mathcal {CN} (y_{_{D}}(l-1)e^{jw_{_{D}}}p(l-1), 2N_{0})$ , using pdfs of $y_{_{D}}(l)$ and $y_{_{D}}(l-2)$ the PDF of $t_{_{1}}(m)$ can be obtained using [33, eq. (15)], as \begin{align*}&p_{_{t_{1}}}\left ({w|y_{_{D}}(l-1),x_{_{R}}(l)}\right)(m) \\&=\frac {3}{4}e^{-\left ({\frac {3|w|}{2}+\left ({\frac {3|w|+x_{_{R}}(r)w}{4|w|N_{2}}}\right)}\right)} \\&{}\times \sum _{k=0}^{\infty }\frac {|y_{_{D}}(l-1)|^{2k}\left ({|w|+x_{_{R}}w}\right)^{k}}{N_{2}^{k}k!4^{k}|w|^{k}} \\&{}\times \sum _{l=0}^{k}\frac {|3w|^{k-l}}{\Gamma (k-l+1)}L_{r}\left ({-\frac {|w|-x_{_{l}}(r)w}{4|w|N_{2}}|y_{_{D}}(l-1)|^{2}}\right), \tag{11}\end{align*} where $L_{r}(\cdot)$ is the Laguerre polynomial and $\Gamma (\cdot,\cdot)$ represents upper incomplete gamma function.

Fig. 2.

a) encoder b) decoder.

C. Proposed Transmission-Reception Protocol

In the given system, we have divided the relay nodes into three groups, namely:

• $\mathcal {I}_{1}$ : Set with the cardinality of $|\mathcal {I}_{1}|$ and it contains relays with buffer status full.

• $\mathcal {I}_{2}$ : Set with the cardinality of $|\mathcal {I}_{2}|$ and it contains relays with buffer status empty.

• $\mathcal {I}3$ : Set with the cardinality of $|\mathcal {I}_{3}|$ and it contains relays with buffer status partially full.

After dividing relays into groups, the transmission/reception of data packets from/at R is carried out as follows:

1. For buffer status full, if the best $R$ -$D$ link’s SNR is higher than the threshold $\gamma _{\text {th}}$ , $R$ will use the best $R$ -$D$ link for transmitting the data packet to $D$ , for a fixed time frame $T$ . The bits inside the data packets are DD encoded and are indexed also so that received data can be decoded by $D$ .

2. For buffer status empty, if the best $S$ -$R$ link’s SNR is higher than the $\gamma _{\text {th}}$ , $S$ will use the best $S$ -$R$ link for transmitting the data packet to $R$ , for a fixed time frame $T$ . The bits inside the data packets are DD encoded and are also indexed so that received data can be decoded by the selected relay node.

3. Whenever the buffer is partially filled

   •Step 1 shall be replicated in case the max $R$ -$D$ link’s SNR is higher than $\gamma _{\text {th}}$ .

   •When the max $R$ -$D$ link does not have SNR greater than $\gamma _{\text {th}}$ , step 2 shall be replicated if max $S$ -$R$ link’s SNR is bigger than $\gamma _{\text {th}}$ .

We assume all channels are static for the proposed DDM-based system over intervals of at least three symbols. SNR is evaluated in a non-data-aided manner [40], [41]. Additionally, the suggested protocol solely relies on the relay’s buffer status to determine which relay is selected at any given time.

Remark 1:

The proposed link selection strategy uses packet-based data scheduling. However, the given approach differs from the available max-link strategies in that it adds an index to every transmitted data bit to keep track of the bits that have already been sent/received, which helps perform DD encoding/decoding at various communicating nodes. Also, by giving higher priority to $R$ -$D$ link, we had taken care of the fact that the BER of the $R$ -$D$ link depends upon the BER of $S$ -$R$ link (see eq. (11)). Hence, it will lead to a lower BER.

SECTION III. Performance Analysis

In this section, the proposed system’s outage and ABER performances are analyzed. The evolution of the buffer status is analyzed using the state transition matrix, which is modeled by utilizing the MC approach, whose state at any instance is determined by the total number of data packets available in the relay buffers. The total number of states in a system with $K$ relay nodes and $L$ -sized buffers on each relay node is $\mathcal {Z}=(L+1)^{K}$ , with any arbitrary state denoted as $s_{v}$ , where $s_{v}\triangleq (\psi (Q_{1}),\psi (Q_{2}),\,\ldots, \,\psi (Q_{K})),\, 1\leq v \leq \mathcal {Z}$ .

A. Steady State Probability

For evaluating the steady state probability of the given system, we should develop a state transition matrix A of $\mathcal {Z}\times \mathcal {Z}$ dimension. Let $A_{uv}$ denotes the $uv$ -th entry of A, where $u,\,v \in \{1,\,2,\,\ldots, \, \mathcal {Z}\}$ , also $A_{uv}$ is the probability of transition from state $s_{v}$ to $s_{u}$ or mathematically we can say $A_{uv}=\text {Pr}(X_{t+1}=s_{u}|X_{t}=s_{v})$ . When no change happens in the state because of an outage, we denoted that element by $A_{vv}$ . The entries in A are a result of the events when (i) reception of data takes place at $D$ through the best $R$ -$D$ link (ii) transmission of data takes place from the source node to the selected relay node via best $S$ -$R$ link, and (iii) no transmission or reception takes place, i.e., outage event. For the given system, at any time interval for an arbitrary state $s_{v}$ , the probability of successfully transmitting the data by a relay when buffer status is full is given by \begin{equation*} P_{_{|\mathcal {I}_{1}|}}^{F}= \frac {1}{|\mathcal {I}_{1}|}\left [{1-\left [{1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{_{RD}}}}}\right]^{|\mathcal {I}_{1}|}}\right]\,. \tag{12}\end{equation*} Consequently, the probability for unsuccessful transmission from relay when buffer status is full is given by \begin{equation*} \overline {P}_{_{|\mathcal {I}_{1}|}}^{F}=\left [{1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{_{RD}}}}}\right]^{|\mathcal {I}_{1}|}\,\,. \tag{13}\end{equation*} Similarly, for the case of buffer status empty, the probability of successful and unsuccessful reception of the data packet through the selected $S$ -$R$ link is depicted, accordingly, by \begin{align*} P_{_{|\mathcal {I}_{2}|}}^{E}=&\frac {1}{|\mathcal {I}_{2}|}\left [{1-\left [{1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{_{SR}}}}}\right]^{|\mathcal {I}_{2}|}}\right]\, \text {and} \tag{14a}\\ \overline {P}_{|\mathcal {I}_{2}|}^{E}=&\left [{1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{_{SR}}}}}\right]^{|\mathcal {I}_{2}|}. \tag{14b}\end{align*} For the partially filled relay buffer status, there are the same number of $R$ -$D$ and $S$ -$R$ links in the system and they are equal to the cardinality of set $\mathcal {I}_{3}$ , i.e., $|\mathcal {I}_{3}|$ ; hence the probability for successful and unsuccessful transmission, respectively, through the best $R$ -$D$ link will be \begin{align*} P_{_{|\mathcal {I}_{3}|,RD}}^{\mathcal {P}}=&\frac {1}{|\mathcal {I}_{3}|}\left [{1-\left [{1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{_{RD}}}}}\right]^{|\mathcal {I}_{3}|}}\right]\,\,\,\,\,\,\text {and} \tag{15a}\\ \overline {P}_{_{|\mathcal {I}_{3}|,RD}}^{\mathcal {P}}=&\left [{1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{_{RD}}}}}\right]^{|\mathcal {I}_{3}|}. \tag{15b}\end{align*} Similarly, for the $S$ -$R$ link, probabilities for successful and unsuccessful transmission of data packets can be, respectively, given as \begin{align*} P_{_{|\mathcal {I}_{3}|,SR}}^{\mathcal {P}}=&\frac {1}{|\mathcal {I}_{3}|}\left [{1-\left [{1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{_{SR}}}}}\right]^{|\mathcal {I}_{3}|}}\right]\,\text {and} \tag{16a}\\ \overline {P}_{_{|\mathcal {I}_{3}|,SR}}^{\mathcal {P}}=&\left [{1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{_{SR}}}}}\right]^{|\mathcal {I}_{3}|}. \tag{16b}\end{align*} Next, we have to formulate the state transition matrix of the system for the proposed priority-based max-link scheme; the $uv$ -th element of $\boldsymbol {A}$ is represented as \begin{align*} \boldsymbol {A}_{_{uv}}=\begin{cases} P_{|\mathcal {I}_{1}|}^{F}\,\text {if } s_{u} \in \mathcal {I}_{1} \\ P_{|\mathcal {I}_{2}|}^{E}\, \text {if } s_{u} \in \mathcal {I}_{2},\,\mathcal {I}_{1} = \phi \\ P_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}}\, \text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{1}, \,\mathcal {I}_{2}=\phi \\ \overline {P}_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}} P_{|\mathcal {I}_{3}|,\,SR}^{\mathcal {P}},\,\text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{1}, \,\mathcal {I}_{2}=\phi \\ \overline {P}_{|\mathcal {I}_{1}|}^{F}P_{|\mathcal {I}_{2}|}^{E},\text {if } s_{u} \in \mathcal {I}_{2},\,\mathcal {I}_{1}\neq \phi \\ \overline {P}_{|\mathcal {I}_{1}|}^{F}P_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}},\,\text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{1}\neq \phi \\ \overline {P}_{|\mathcal {I}_{1}|}^{F}\overline {P}_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}} P_{|\mathcal {I}_{3}|,\,SR}^{\mathcal {P}},\,\text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{1}\neq \phi. \\ \overline {P}_{|\mathcal {I}_{2}|}^{E}P_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}},\,\text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{2}\neq \phi \\ \overline {P}_{|\mathcal {I}_{2}|}^{E}\overline {P}_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}}P_{|\mathcal {I}_{3}|,\,SR}^{\mathcal {P}},\,\text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{2}\neq \phi \\ \overline {P}_{|\mathcal {I}_{1}|}^{F} \overline {P}_{|\mathcal {I}_{2}|}^{E} P_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}},\,\text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{1},\mathcal {I}_{2}\neq \phi \\ \overline {P}_{|\mathcal {I}_{1}|}^{F} \overline {P}_{|\mathcal {I}_{2}|}^{E}\overline {P}_{|\mathcal {I}_{3}|,RD}^{\mathcal {P}} P_{|\mathcal {I}_{3}|,SR}^{\mathcal {P}},\,\text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{1},\,\mathcal {I}_{2}\neq \phi \end{cases} \tag{17}\end{align*} When all the available links go into the outage, then for such case, the whole setup goes into the outage, and no transmission takes place, the term $\boldsymbol {A}_{_{vv}}$ is denoted by\begin{align*}&\boldsymbol {A}_{_{vv}} \\&=\begin{cases} \overline {P}_{|\mathcal {I}_{1}|}^{F},\,\text {if } s_{u}=s_{v},\,\mathcal {I}_{1}\neq \phi \\ \overline {P}_{|\mathcal {I}_{2}|}^{E},\,\text {if } s_{u}=s_{v},\,\mathcal {I}_{2} \neq \phi \\ \overline {P}_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}}\overline {P}_{|\mathcal {I}_{3}|,\,SR}^{\mathcal {P}},\,\text {if } s_{u}=s_{v},\,\mathcal {I}_{3}\neq \phi \\ \overline {P}_{|\mathcal {I}_{1}|}^{F}\overline {P}_{|\mathcal {I}_{2}|}^{E},\,\text {if } s_{u}=s_{v},\,\mathcal {I}_{1}\neq \phi,\,\,\mathcal {I}_{2} \neq \phi \,\,\,\,\,\,. \\ \overline {P}_{|\mathcal {I}_{1}|}^{F},\overline {P}_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}}\overline {P}_{|\mathcal {I}_{3}|,\,SR}^{\mathcal {P}},\,\text {if } s_{u}=s_{v},\,\mathcal {I}_{3}\neq \phi,\,\mathcal {I}_{1}\neq \phi \\ \overline {P}_{|\mathcal {I}2|}^{E},\overline {P}_{|\mathcal {I}_{3}|,\,RD}^{P}\overline {P}_{|\mathcal {I}_{3}|,\,SR}^{\mathcal {P}},\,\text {if } s_{u}=s_{v},\,\mathcal {I}_{3}\neq \phi,\,\mathcal {I}_{2}\neq \phi \\ \overline {P}_{|\mathcal {I}_{1}|}^{F},\overline {P}_{|\mathcal {I}2|}^{E}\overline {P}_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}}\overline {P}_{|\mathcal {I}_{3}|,\,SR}^{\mathcal {P}},\,\text {if } s_{u}=s_{v},\,\mathcal {I}_{1},\mathcal {I}_{2},\mathcal {I}_{1}\neq \phi \end{cases} \tag{18}\end{align*} In the above-developed state transition matrix, when we cannot go from one state to another in a single step, we keep that element as zero, i.e., $A_{uv}\,\,{=}$ 0. The developed matrix $\boldsymbol {A}$ inherits the following characteristics:

• Irreducible: Every state of the MC is accessible regardless of from where we started.

• Column stochastic: Evey column in the matrix $\boldsymbol {A}$ will have a sum of unity.

• Aperiodic: Each state has a finite probability of an outage; hence each state has a finite probability of remaining in any state at any moment.

Because of these characteristics, we can express the steady-state probability for the considered setup as \begin{equation*} \boldsymbol {\pi } =\left ({\mathbf {A}-\mathbf {I}+\mathbf {B}}\right)^{-1}\mathbf {b}, \tag{19}\end{equation*} where $\mathbf {B}$ is a matrix of dimension $\mathcal {Z}\times \mathcal {Z}$ with all the elements equal to one, $\mathbf {I}$ is the identity matrix having dimension $\mathcal {Z}\times \mathcal {Z}$ , $\boldsymbol {\pi }=[\pi _{1},\pi _{2},\,\ldots, \,\pi _{\mathcal {Z}}]$ with $\pi _{v}=\text {Pr}(s_{v})$ , and $\mathbf {b}$ is the column matrix with dimension $\mathcal {Z}$ and all its entries are equal to unity. The outage probability of the proposed system can be evaluated with the help of (19) as \begin{equation*} P_{\text {out}}= \sum _{v=1}^{\mathcal {Z}} \pi _{v} A_{vv}=\text {diag}\left ({\mathbf {A}}\right)\boldsymbol {\pi }, \tag{20}\end{equation*} where the row vector $\text {diag}(\mathbf {A})$ has the dimension $\mathcal {Z}$ and it contains diagonal entries of $\mathbf {A}$ .

B. ABER Analysis

While investigating the ABER of the proposed DDM configuration, it is assumed that data transmission across any link happens only when we have instantaneous SNR greater than $\gamma _{\text {th}}$ . Assume that the given link’s instantaneous SNR pdf is represented as \begin{equation*} f_{\gamma _{\varphi }}(y)=\frac {1}{\overline {\gamma }_{\varphi }}e^{-\frac {y}{\overline {\gamma }_{\varphi }}}, \tag{21}\end{equation*} where $\varphi \in \{SR_{k},R_{k}D\}$ and $\overline {\gamma }_{\varphi }$ is the average SNR for a given link $\varphi $ . Let us assume that there are $\mathcal {L}$ total links involved in the communication; then, the pdf of the chosen max-link is provided by \begin{align*} f_{\Gamma _{\varphi }}(x)=\frac {\text {d}}{\text {d}x}\left [{\int _{0}^{x}\frac {1}{\overline {\gamma }_{\varphi }}e^{-\frac {y}{\overline {\gamma }_{\varphi }}} \text {d}y }\right]^{\mathcal {L}} =\frac {\mathcal {L}}{\overline {\gamma }_{\varphi }}e^{\frac {x}{\overline {\gamma }_{\varphi }}}\left [{1-e^{\frac {x}{\overline {\gamma }_{\varphi }}}}\right]^{\mathcal {L}-1}. \tag{22}\end{align*} With the help of the above expressions, the ABER of the chosen max-link can be evaluated as \begin{equation*} \overline {P}_{e_{\varphi }}\left ({\mathcal {L}}\right)= \frac {\int _{\gamma _{\text {th}}}^{\infty } P_{e_{\varphi }}(x)f_{\Gamma _{\varphi }}(x) \text {d}x}{\int _{\gamma _{\text {th}}}^{\infty } f_{_{\Gamma _{\varphi }}}(x)\text {d}x}. \tag{23}\end{equation*}

1) ABER analysis of $S$ -$R_{k}$

The expressions of the ABER of the $S$ -$R_{k}$ links can be evaluated as given below \begin{align*} \overline {P}_{e_{SR}}\left ({\mathcal {M}}\right)=\frac {\int _{\gamma _{\text {th}}}^{\infty }Q\left ({\sqrt {2y}}\right)\frac {\mathcal {M}}{\overline {\gamma }_{SR}}e^{-\frac {y}{\overline {\gamma }_{SR}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{SR}}}}\right]^{\mathcal {M}-1}\text {d}y}{\int _{\gamma _{\text {th}}}^{\infty }\frac {\mathcal {M}}{\overline {\gamma }_{SR}}e^{-\frac {y}{\overline {\gamma }_{SR}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{SR}}}}\right]^{\mathcal {M}-1}\text {d}y}, \tag{24}\end{align*} where $Q(\cdot)$ represents Gaussian q-function and $\mathcal {M}$ is the number of available $S$ -$R$ links. The expression of ABER of $S$ -$R_{k}$ link can be estimated approximately by using the Chernov bound as \begin{equation*} \overline {P}_{e_{SR}}\left ({\mathcal {M}}\right)=\frac {\int _{\gamma _{\text {th}}}^{\infty }\frac {\mathcal {M}}{\overline {\gamma }_{SR}}e^{-y}e^{-\frac {y}{\overline {\gamma }_{SR}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{SR}}}}\right]^{\mathcal {M}-1}\text {d}y}{2\int _{\gamma _{\text {th}}}^{\infty }\frac {\mathcal {M}}{\overline {\gamma }_{SR}}e^{-\frac {y}{\overline {\gamma }_{SR}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{SR}}}}\right]^{\mathcal {M}-1}\text {d}y}. \tag{25}\end{equation*} After binomially expanding the above expression and then evaluating it further, we derived the expression of $\overline {P}_{e_{SR}}(\mathcal {M})$ as \begin{equation*} \overline {P}_{e_{SR}}\left ({\mathcal {M}}\right)=\frac {\mathcal {M}\times {\sum _{k=0}^{\mathcal {M}-1}}\binom {\mathcal {M}-1}{k}\frac {(-1)^{k}e^{-\gamma _{\text {th}}\left [{\frac {\overline {\gamma }_{SR}+k+1}{\overline {\gamma }_{SR}}}\right]} }{\left ({\overline {\gamma }_{SR}+k+1}\right)}}{2\left [{1-\left ({1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{SR}}}}\right)^{\mathcal {M}}}\right]}. \tag{26}\end{equation*} With the help of (26), we can obtain the ABER of $S$ -$R$ link for different values of $\mathcal {M}$ .

2) ABER analysis of $R_{k}$ -D

For the case of a binary phase shift keying (BPSK) signal, with the total of $\mathcal {M} S$ -$R$ links available, the decision rule at $D$ will be of the following form:\begin{equation*} f_{_{PL}}\left ({t_{_{1}}\left ({\mathcal {M}}\right)}\right)\underset {-1}{\overset {1}{\gtrless }} 0. \tag{27}\end{equation*} Now, using (10), the following error events can occur at D:

1. $\,t_{_{1}}(\mathcal {M})< 0$ if $-T_{_{1}}(\mathcal {M})< t_{_{1}}(\mathcal {M})< T_{_{1}}(\mathcal {M})$

2. $\,T_{_{1}}(\mathcal {M})< 0$ if $t_{_{1}}(\mathcal {M})>T_{_{1}}(\mathcal {M})$

3. $\,-T_{_{1}}(\mathcal {M})< 0$ if $t_{_{1}}(\mathcal {M})>-T_{_{1}}(\mathcal {M})$

As $\epsilon _{_{\mathcal {M}}}$ will always take a value less than 0.5, hence $T_{1}(\mathcal {M})$ will always be greater than 1; then the probability of error at $D$ can be approximated to \begin{align*}&P_{e}\left ({y_{_{D}}\left ({l-1}\right)}\right) \\&= \text {Pr}\left [{t_{1}\left ({\mathcal {M}}\right)< -T_{1}\left ({\mathcal {M}}\right)|x(l)=1;y_{_{D}}(l-1)=-1}\right] \\&+ \text {Pr}\left [{-T_{1}\left ({\mathcal {M}}\right)\leq t_{1}\left ({\mathcal {M}}\right)< 0|x(l)=1;y_{_{D}}(l-1)=-1}\right]. \tag{28}\end{align*} Assuming $\mathcal {N}$ represents the total number of $R$ -$D$ links available for data packet transmission, then we have \begin{align*} \overline {P}_{_{e_{RD}}}\left ({\mathcal {M},\mathcal {N}}\right)=&\left ({1-\mathbf {\epsilon }_{\mathcal {_{M}}}}\right)\left [{\overline {P}_{_{e_{RD}}}^{1}\left ({\mathcal {M},\mathcal {N}}\right)+\overline {P}_{_{e_{RD}}}^{2}\left ({\mathcal {M},\mathcal {N}}\right)}\right] \\&+ \mathbf {\epsilon }_{\mathcal {_{M}}}\left [{\overline {P}_{_{e_{RD}}}^{3}\left ({\mathcal {M},\mathcal {N}}\right)+\overline {P}_{_{e_{RD}}}^{4}\left ({\mathcal {M},\mathcal {N}}\right)}\right]. \tag{29}\end{align*} The expressions for $\overline {P}_{_{e_{RD}}}^{1}(\mathcal {M},\mathcal {N})$ , $\overline {P}_{_{e_{RD}}}^{2}(\mathcal {M},\mathcal {N})$ , $\overline {P}_{_{e_{RD}}}^{3 }(\mathcal {M},\mathcal {N})$ , and $\overline {P}_{_{e_{RD}}}^{4}(\mathcal {M},\mathcal {N})$ are derived in the Appendix. We represented the error matrix of the setup as $\mathbf {E}$ , and in order to evaluate the overall ABER, we first have to obtain each entry of $\mathbf {E}$ . The matrix $\mathbf {E}$ will have a dimension of $\mathcal {Z}\times \mathcal {Z}$ , and at any given time instant, its $uv$ -th entry is represented by \begin{align*}&\boldsymbol {E}_{_{uv}} \\&=\begin{cases} \frac {\overline {P}_{e_{RD}}\left ({\left ({L-\mathcal {I}_{1}+1}\right),\mathcal {I}_{1}}\right)}{|\mathcal {I}_{1}|}\,\, \text {if } s_{u} \in \mathcal {I}_{1},\,\mathcal {I}_{2}=\phi,\mathcal {I}_{3}=\phi \\ \frac {\overline {P}_{e_{SR}}\left ({\mathcal {I}_{2}}\right)}{|\mathcal {I}_{2}|}\,\, \text {if } s_{u} \in \mathcal {I}_{2},\,\mathcal {I}_{1}=\phi,\mathcal {I}_{3}=\phi \\ P_{|\mathcal {I}_{3}|,\,RD}^{\mathcal {P}}\overline {P}_{e_{RD}}\left ({L,\mathcal {I}_{3}}\right)\,\, \text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{1}=\phi,\mathcal {I}_{2}=\phi \\ \overline {P}_{_{\mathcal {I}_{3},\,RD}}^{\mathcal {P}} \frac {\overline {P}_{e_{SR}}\left ({\mathcal {I}_{3}}\right)}{|\mathcal {I}_{3}|}\,\, \text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{1}=\phi,\mathcal {I}_{2}=\phi \\ \overline {P}_{_{\mathcal {I}_{1},\,RD}}^{F} \frac {\overline {P}_{e_{SR}}\left ({\mathcal {I}_{2}}\right)}{|\mathcal {I}_{2}|}\,\, \text {if } s_{u} \in \mathcal {I}_{2},\,\mathcal {I}_{1}\neq \phi,\mathcal {I}_{3}=\phi \\ P_{_{\mathcal {I}_{1},\,RD}}^{F} \overline {P}_{e_{RD}}\left ({L-\mathcal {I}_{1}+1,\mathcal {I}_{1}}\right)\,\, \text {if } s_{u} \in \mathcal {I}_{1},\,\exists \,\mathcal {I}_{2} \text {or } \exists \,\mathcal {I}_{3} \\ \overline {P}_{_{\mathcal {I}_{1},\,RD}}^{F} P_{_{\mathcal {I}_{2},\,SR}}^{E}\overline {P}_{e_{SR}}\left ({\mathcal {I}_{2}}\right)\,\, \text {if } s_{u} \in \mathcal {I}_{2},\,\mathcal {I}_{1}\neq \phi,\mathcal {I}_{3}\neq \phi \\ \overline {P}_{_{\mathcal {I}_{1},\,RD}}^{F} \frac {\overline {P}_{e_{RD}}\left ({\left ({L-\mathcal {I}_{1}}\right),\mathcal {I}_{3}}\right)}{|\mathcal {I}_{3}|}\,\, \text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{1}\neq \phi,\mathcal {I}_{2}=\phi. \\ \overline {P}_{_{\mathcal {I}_{3},\,RD}}^{\mathcal {P}}\overline {P}_{_{\mathcal {I}_{1},\,RD}}^{F}\frac {\overline {P}_{e_{SR}}\left ({\mathcal {I}_{3}}\right)}{|\mathcal {I}_{3}|}\,\, \text {if } s_{u} \in \mathcal {I}_{3},\,\mathcal {I}_{1}\neq \phi,\mathcal {I}_{2}=\phi \\ P_{_{\mathcal {I}_{2},\,SR}}^{E} \overline {P}_{e_{SR}}\left ({\mathcal {I}_{2}}\right)\,\, \text {if } s_{u} \in \mathcal {I}_{2},\,\exists \,\mathcal {I}_{3}, \,\mathcal {I}_{1} =\phi \\ \overline {P}_{_{\mathcal {I}_{2},\,SR}}^{E}\frac { \overline {P}_{e_{RD}}\left ({L,\mathcal {I}_{3}}\right)}{|\mathcal {I}_{3}|}\,\, \text {if } s_{u} \in \mathcal {I}_{3},\,\,\mathcal {I}_{2}\neq \phi, \,\mathcal {I}_{1} =\phi \\ \overline {P}_{_{\mathcal {I}_{3},\,RD}}^{\mathcal {P}}\overline {P}_{_{\mathcal {I}_{2},\,SR}}^{E}\frac { \overline {P}_{e_{SR}}\left ({\mathcal {I}_{3}}\right)}{|\mathcal {I}_{3}|}\,\, \text {if } s_{u} \in \mathcal {I}_{3},\, \,\mathcal {I}_{2}\neq \phi, \,\mathcal {I}_{1} =\phi \\ \overline {P}_{_{\mathcal {I}_{1},\,RD}}^{F}\overline {P}_{_{\mathcal {I}_{2},\,SR}}^{E}\frac { \overline {P}_{e_{RD}}\left ({\left ({L-\mathcal {L}_{1}}\right),\mathcal {I}_{3}}\right)}{|\mathcal {I}_{3}|}\,\, \text {if } s_{u} \in \mathcal {I}_{3},\,\,\mathcal {I}_{2}\neq \phi, \,\mathcal {I}_{1} \neq \phi \\ \overline {P}_{_{\mathcal {I}_{3},\,RD}}^{\mathcal {P}}\overline {P}_{_{\mathcal {I}_{1},\,RD}}^{F}\overline {P}_{_{\mathcal {I}_{2},\,SR}}^{E}\frac { \overline {P}_{e_{SR}}\left ({\mathcal {I}_{3}}\right)}{|\mathcal {I}_{3}|}\,\, \text {if } s_{u} \in \mathcal {I}_{3},\, \,\mathcal {I}_{2}\neq \phi, \,\mathcal {I}_{1} \neq \phi \\ \end{cases} \tag{30}\end{align*} It is important to notice from (30) that we set $\mathbf {E}_{vv}=0$ when a movement between states is not possible in a single step and when there is an outage. For the considered DDM-based BA system overall ABER is expressed as \begin{equation*} \overline {P}_{e}=\mathbf {b}^{T}\mathbf {E}\boldsymbol {\pi },\, \tag{31}\end{equation*} where $\mathbf {E}$ is given in (30), and $\boldsymbol {\pi }$ in (19).

Remark 2:

For deriving the ABER expressions of the $R$ -$D$ and $S$ -$R$ link, we have taken into account the threshold requirement for non-incremental DDM relaying. Also, the number of available $S$ -$R$ and $R$ -$D$ links has been considered. Hence, the proposed PL detector differs significantly from the existing PL detector used for the NBA system based on DDM.

C. Development of ${A}$ and ${E}$ matrices

In this subsection, to better comprehend the development of $\mathbf {A}$ and $\mathbf {E}$ matrices, we will try to construct it for the simplest case of $L=2$ and $K=2$ . For the considered case of $L=2$ and $K=2$ , we will have a total of $(2+1)^{2}$ states, i.e., 9 states, represented by $s_{v}=\{00,01,02,10,11,12,20,21,22\}$ . Assuming symmetric channels, i.e., $\overline {\gamma }_{SR}=\overline {\gamma }_{RD}$ and then using (12), (14a), (15a), and (16a) we get \begin{equation*} P_{m,\,RD}^{F}=P_{m,\,SR}^{E}=P_{m,\,RD}^{P}=P_{m,\,SR}^{P}=\chi _{m}. \tag{32}\end{equation*} where $m$ represents the number of available links. Similarly using (13), (14b), (15b), and (16b) we have \begin{equation*} \overline {P}_{m,\,RD}^{F}=\overline {P}_{m,\,SR}^{E}=\overline {P}_{m,\,RD}^{P}=\overline {P}_{m,\,SR}^{P}=\overline {\chi }_{m}. \tag{33}\end{equation*} With the aforementioned simplified assumptions and using (17) and (18) the matrix $\mathbf {A}$ will be given by \begin{align*} \mathbf {A}= \begin{bmatrix} \overline {\chi }_{_{2}} & \overline {\chi }_{_{1}} \chi _{_{1}} & 0 & \overline {\chi }_{_{1}}\chi _{_{1}} & 0 & 0 & 0 & 0 & 0 \\ \chi _{_{2}} & \overline {\chi }_{_{1}}^{3} & \chi _{_{1}} & 0 & \chi _{_{2}} & 0 & 0 & 0 & 0 \\ 0 & \overline {\chi }_{_{1}}^{2} \chi _{_{1}} & \overline {\chi }_{_{1}}^{2} & 0 & 0 & \overline {\chi }_{_{1}} \chi _{_{1}} & 0 & 0 & 0 \\ \chi _{_{2}} & 0 & 0 & \overline {\chi }_{_{1}}^{3} & \chi _{_{2}} & 0 & \chi _{_{1}} & 0 & 0 \\ 0 & \chi _{_{1}} & 0 & \chi _{_{1}} & \overline {\chi }_{_{2}}^{2} & \chi _{_{1}} & 0 & \chi _{_{1}} & 0 \\ 0 & 0 & \overline {\chi }_{_{1}}\chi _{_{1}} & 0 & \overline {\chi }_{_{2}}\chi _{_{2}} & \overline {\chi }_{_{1}}^{3} & 0 & 0 & \chi _{_{2}} \\ 0 & 0 & 0 & \overline {\chi }_{_{1}}^{2}\chi _{_{1}} & 0 & 0 & \overline {\chi }_{_{1}}^{2} & \overline {\chi }_{_{1}} \chi _{_{1}} & 0 \\ 0 & 0 & 0 & 0 & \overline {\chi }_{_{2}}\chi _{_{2}} & 0 & \overline {\chi }_{_{1}} \chi _{_{1}} & \overline {\chi }_{_{1}}^{3} & \chi _{_{2}} \\ 0 & 0 & 0 & 0 & 0 & \overline {\chi }_{_{1}}^{2} \chi _{_{1}} & 0 & \overline {\chi }_{_{1}}^{2} \chi _{_{1}} & \overline {\chi }_{_{2}} \end{bmatrix}. \tag{34}\end{align*} Similarly, let us designate the ABER for communicating $S$ -$R$ and $R$ -$D$ links as $\Upsilon _{m}$ and $\Upsilon _{m,n}$ , respectively, i.e., \begin{align*}&\overline {P}_{e_{SR}}(m)=\Upsilon _{m} \tag{35a}\\&\overline {P}_{e_{RD}}\left ({m,n}\right)=\Upsilon _{m,n} \tag{35b}\end{align*} Using the above expressions of ABERs the error matrix of the proposed DDM-based setup for the case of $L=2$ and

$K=2$ will be \begin{align*}&\mathbf {E} \\&=\begin{bmatrix} 0 & \overline {\chi }_{_{1}} \Upsilon _{_{2,1}} & 0 & \overline {\chi }_{_{1}}\Upsilon _{_{1}} & 0 & 0 & 0 & 0 & 0 \\ \frac { \Upsilon _{_{2}}}{2} & 0 & \chi _{_{1}}\Upsilon _{_{2,}} & 0 & \chi _{_{2}}\frac {\Upsilon _{_{2,2}} }{2} & 0 & 0 & 0 & 0 \\ 0 & \overline {\chi }_{_{1}}^{2} \Upsilon _{_{1}} & 0 & 0 & 0 & \overline {\chi }_{_{1}} \Upsilon _{_{1,1}} & 0 & 0 & 0 \\ \frac {\Upsilon _{_{2}}}{2} & 0 & 0 & 0 & \Upsilon _{_{1,2}}\frac {\Upsilon _{_{2,2}} }{2} & 0 & \chi _{_{1}}\Upsilon _{_{1,1}} & 0 & 0 \\ 0 & \Upsilon _{_{1}} & 0 & \chi _{_{1}} \Upsilon _{_{1}} & 0 & \chi _{_{1}}\Upsilon _{_{2,1}} & 0 & \chi _{_{1}}\Upsilon _{_{1,1}} & 0 \\ 0 & 0 & \overline {\chi }_{_{1}}\Upsilon _{_{1}} & 0 & \overline {\chi }_{_{2}}\frac {\Upsilon _{_{2}} }{2} & 0 & 0 & 0 & \Upsilon _{_{1,2}} \\ 0 & 0 & 0 & \overline {\chi }_{_{1}}^{2}\Upsilon _{_{1}} & 0 & 0 & 0 & \overline {\chi }_{_{1}} \Upsilon _{_{1}} & 0 \\ 0 & 0 & 0 & 0 & \overline {\chi }_{_{2}}\frac {\Upsilon _{_{2}} }{2} & 0 & \overline {\chi }_{_{1}} \Upsilon _{_{1,1}} & 0 & \Upsilon _{_{1,2}} \\ 0 & 0 & 0 & 0 & 0 & \overline {\chi }_{_{1}}^{2} \chi _{_{1}} & 0 & \overline {\chi }_{_{1}}^{2} \Upsilon _{_{1}} & 0 \end{bmatrix} \tag{36}\end{align*}

D. Asymptotic ABER Analysis

The coding gain $(\mathcal {CG})$ and diversity order $(\mathcal {DO})$ of the considered DDM-based setup will be analyzed in this subsection. The analysis for both SR and RD links will be carried out separately, and then, with the help of that, the overall $\mathcal {DO}$ and $\mathcal {CG}$ will be evaluated. For this, let us consider a high SNR regime, such that $\overline {\gamma }_{SR}\rightarrow \infty $ and $\overline {\gamma }_{RD}\rightarrow \infty $ .

1) $\mathcal {CG}$ calculations for $S$ -$R$ link

Using a Maclaurin series, we can approximate the expression of $e^{-x}$ as $e^{-x} \mathop{\approx }\limits_{x\rightarrow 0} 1-x $ ; using the given approximation in (26), we obtain the asymptotic expression of $\overline {P}_{e_{SR}}(\mathcal {M})$ as \begin{equation*} \overline {P}_{e_{SR}}\left ({\mathcal {M}}\right)\approx \left ({\mathcal {M}/2\overline {\gamma }_{SR}}\right)\times {\sum _{k=0}^{\mathcal {M}-1}}\binom {\mathcal {M}-1}{k}(-1)^{k}e^{-\gamma _{\text {th}}}. \tag{37}\end{equation*} Now, for any arbitrary link, the conventional relation for the $\mathcal {DO}$ and $\mathcal {CG}$ is given by \begin{equation*} P_{e}=\left ({\mathcal {CG}/\overline {\gamma }}\right)^{\mathcal {DO}}. \tag{38}\end{equation*} Using (37) and (38), we will get the $\mathcal {DO}$ and $\mathcal {CG}$ of the $S$ -$R$ link respectively as \begin{align*}&\mathcal {CG}_{_{SR}}=\left ({\mathcal {M}/2}\right)\times {\sum _{k=0}^{\mathcal {M}-1}}\binom {\mathcal {M}-1}{k}(-1)^{k}e^{-\gamma _{\text {th}}}. \qquad \tag{39a}\\&\mathcal {DO}_{_{SR}}=1. \tag{39b}\end{align*}

2) $\mathcal {CG}$ calculation of $R$ -$D$ link

For evaluating the $\mathcal {CG}$ of the RD link, we must first obtain the asymptotic expression of different $\overline {P}_{e}$ terms provided in (52), (56), (61) and (66). Now, using the approximation $e^{-x} \mathop{\approx }\limits_{x\rightarrow 0} 1-x $ in (52), the final asymptotic expression of $\overline {P}_{_{e_{RD}}}^{1}(\mathcal {M},\mathcal {N})$ is denoted by \begin{align*} \overline {P}_{_{e_{RD}}}^{1}\left ({\mathcal {M},\mathcal {N}}\right)\approx&e^{-T_{_{1}}\left ({\mathcal {M}}\right)} \times \frac {\mathcal {N}\times {\sum _{k=0}^{\mathcal {N}-1}}\binom {\mathcal {N}-1}{k}(-1)^{k}e^{-\gamma _{\text {th}}}}{\left ({\overline {\gamma }_{RD}}\right)\left [{1-\left ({\frac {\gamma _{\text {th}}}{\overline {\gamma }_{SR}}}\right)^{\mathcal {N}}}\right]}, \\\approx&\frac {1}{T_{_{1}}\left ({\mathcal {M}}\right)}\times \frac {\mathcal {N}\times {\sum _{k=0}^{\mathcal {N}-1}}\binom {\mathcal {N}-1}{k}(-1)^{k}e^{-\gamma _{\text {th}}}}{\overline {\gamma }_{RD}}. \tag{40}\end{align*} Proceeding similarly, we will get the approximated expression of the term $\overline {P}_{_{e_{RD}}}^{2}(\mathcal {M},\mathcal {N})$ at the high SNR region as \begin{align*} \overline {P}_{_{e_{RD}}}^{2}\left ({\mathcal {M},\mathcal {N}}\right)\approx \left [{1-\frac {1}{T_{_{1}}\left ({\mathcal {M}}\right)}}\right]\frac {\mathcal {N}\times {\sum _{k=0}^{\mathcal {N}-1}}\binom {\mathcal {N}-1}{k}(-1)^{k}e^{-\gamma _{\text {th}}}}{\overline {\gamma }_{RD}} \tag{41}\end{align*} In order to obtain the expression of $\overline {P}_{_{e_{RD}}}^{3}(\mathcal {M},\mathcal {N})$ at a high SNR regime we have to first approximate the expression of $\beta _{1}$ , which is used in (61); using the approximation $e^{-x} \mathop{\approx }\limits_{x\rightarrow 0} 1-x $ and further simplifying it we will get the asymptotic expression of $\beta _{1}$ as \begin{equation*} \mathcal {B}_{1}\approx \frac {\mathcal {B} {\sum _{p=0}^{\mathcal {N}-1}}(-1)^{p} \binom {\mathcal {N}-1}{p}}{2\overline {\gamma }_{RD}\left [{1-\left ({1-1+\frac {\gamma _{\text {th}}}{\overline {\gamma }_{RD}}}\right)^{\mathcal {N}}}\right]}. \tag{42}\end{equation*} Substituting (42) into (61) and after few simplifications, we obtain \begin{equation*} \overline {P}_{_{e_{RD}}}^{3}\left ({\mathcal {M},\mathcal {N}}\right)\approx \frac {\mathcal {B} {\sum _{p=0}^{\mathcal {N}-1}}(-1)^{p} \binom {\mathcal {N}-1}{p}\Gamma \left [{k+1,\gamma _{\text {th}}}\right]}{2\overline {\gamma }_{RD}\left [{1-\left ({1-1+\frac {\gamma _{\text {th}}}{\overline {\gamma }_{RD}}}\right)^{\mathcal {N}}}\right]}. \tag{43}\end{equation*} For obtaining the asymptotic expression of $\overline {P}_{_{e_{RD}}}^{4}(\mathcal {M},\mathcal {N})$ we first have to simplify the expression of $\mathcal {C}_{1}$ shown in (66) at high SNR regime as \begin{equation*} \mathcal {C}_{1}\approx \frac {\mathcal {C} {\sum _{p=0}^{\mathcal {N}-1}}(-1)^{p} \binom {\mathcal {N}-1}{p}}{2\overline {\gamma }_{RD}\left [{1-\left ({1-1+\frac {\gamma _{\text {th}}}{\overline {\gamma }_{RD}}}\right)^{\mathcal {N}}}\right]}. \tag{44}\end{equation*} Employing (44) in (66) and then further simplifying, we get \begin{align*} \overline {P}_{_{e_{RD}}}^{4}\left ({\mathcal {M},\mathcal {N}}\right)\approx \frac {\mathcal {C} {\sum _{p=0}^{\mathcal {N}-1}}(-1)^{p} \binom {\mathcal {N}-1}{p}}{2\overline {\gamma }_{RD}\left [{1-\left ({1-1+\frac {\gamma _{\text {th}}}{\overline {\gamma }_{RD}}}\right)^{\mathcal {N}}}\right]} \Gamma \left [{k+1,\gamma _{\text {th}}}\right]. \tag{45}\end{align*} Now, using (40), (41), (43), (45), and then applying the approximation $1-\mathbf {\epsilon }_{\mathcal {M}}\approx 1$ in (29) and further excluding the higher order SNR terms, we will get the overall asymptotic expression of $\overline {P}_{_{e_{RD}}}(\mathcal {M},\mathcal {N})$ as \begin{equation*} \overline {P}_{_{e_{RD}}}\left ({\mathcal {M},\mathcal {N}}\right)\approx \frac {\mathcal {N}\times {\sum _{k=0}^{\mathcal {N}-1}}\binom {\mathcal {N}-1}{k}(-1)^{k}e^{-\gamma _{\text {th}}}}{\overline {\gamma }_{RD}} \tag{46}\end{equation*} Using (38) and (46) the $\mathcal {CG}$ and $\mathcal {DO}$ of the $R$ -$D$ link can be respectively given as \begin{align*}&\mathcal {CG}_{RD}=\mathcal {N}\times {\sum _{k=0}^{\mathcal {N}-1}}\binom {\mathcal {N}-1}{k}(-1)^{k}e^{-\gamma _{\text {th}}}. \tag{47a}\\&\mathcal {DO}_{RD}=1. \tag{47b}\end{align*}

Remark 3:

In order to obtain the overall $\mathcal {CG}$ of the considered setup we have to substitute the expressions of $\mathcal {CG_{SR}}$ and $\mathcal {CG_{RD}}$ into the error matrix E, after substitution we have to evaluate the ABER expressions as per (31) for different values of $L$ and $K$ .

SECTION IV. Numerical Results

In this section, we presented the numerical results of the considered BA DDM-based setup; the obtained results are further verified using Monte-Carlo simulations of $10^{6}$ BPSK signals. The target rate of the system is denoted by $\mathcal {R}$ and using it, we obtain the pre-determined threshold of the system as $\gamma _{\text {th}}=2^{2\mathcal {R}}-1\,\,{=}$ 1, for $\mathcal {R}=1$ . The $S$ -$D$ distance has been normalized to one with the assumption that relays are placed between $S$ and $D$ , and they all exist in the cluster such that $d_{SR}+d_{RD}=1$ . Also, the path loss coefficient $\alpha $ is fixed at 3.

Fig. 3 depicts the plots for ABER versus the total available power of the considered setup; channels are assumed to be symmetric $\overline {\gamma }_{SR}=\overline {\gamma }_{RD}$ , with the value of $K$ being varied from 1 to 4. For the given setup, it has been assumed that there is no CFO in the system. The performance of the considered DDM-based setup is compared with the coherent counterpart at the different values of $K$ , keeping the buffer size fixed at 2, i.e., $L=2$ . The plots show that the system’s performance keeps improving as we increase the value of $K$ . To get more insights into the performance enhancements obtained through setup, we compared its ABER performance with a coherent BA relaying system. We observed that the considered DDM system suffers an SNR penalty of $\sim 2$ -2.5 dB at the BER level of $10^{-5}$ for different values of $K$ . This comparison indicates that in the given setup, we had overcome an SNR penalty of 6 dB observed between NBA differential and coherent relaying for $K=1$ , in a practical BER regime of $10^{-5}$ . Also, no such multiple relay DDM system is available in the literature, so the performance gap for such a DDM-based BA system with the coherent BA system is given for the first time.

Fig. 3.

ABER versus total power at fixed $L$ and varying $K$ .

Remark 4:

The usage of the relay buffer and the proposed priority-based max-link link selection technique are primarily responsible for the enhancement in the ABER performance, which results in the narrowing of the SNR gap between coherent and DDM-based systems from 6 dB to 2-2.5 dB.

Plots for ABER versus total available power are shown in Fig. 4; for the plots demonstrated, we have assumed that the system encounters a CFO, which is uniformly distributed over $[-\pi,\pi]$ . We have also altered the value of $K$ from 1 to 3 while keeping $L$ fixed at 2. Plots indicate that the proposed DDM configuration performs remarkably well throughout the entire SNR range, proving that the proposed DDM technique completely eliminates the CFO effect. One of the most significant observations from the plot is the degradation in the performance of a coherent BA setup when the CFO is present in the system; hence, it can be inferred that the presented DDM-based setup is much more capable of countering the effects of CFO as compared to coherent DDM-based setup.

Fig. 4.

ABER versus total power for varying $K$ in presence of CFO.

The graphs for outage probability versus total available power for symmetric channels, i.e., $\overline {\gamma }_{_{SR}}=\overline {\gamma }_{_{RD}}$ , are shown in Fig. 5. Also, we set the value of $L$ at 2 and then varied $K$ from 2 to 4. After observing the performance, we can establish that it is improving as $K$ increases, which validates our analytical findings. Additionally, we achieve the same outage performance as the coherent BA relaying configuration. This finding is based on the observation that the outage performance of any relaying configuration solely depends on the average SNR of all communicating links, which in turn depends on the path loss exponents, the placements of the relays, and the transmit power of the source and relays.

Fig. 5.

Outage probability versus total power at varying $K$ .

Fig. 6 shows the system’s ABER performance with respect to the power allocation factor $(\delta)$ for various values of $K$ while keeping $L$ constant at 3. Plots were produced for SNR values of 10 dB and 15 dB. From the figure, we can see that the minima of the system occur at $\delta =0.4$ for $K=1,2$ and it is located at $\delta =0.3$ for $K=3$ ; therefore, it can be deduced that the source node requires lesser power for better ABER performance compared to $R$ ; hence we need to allocate more power to $R$ -$D$ link. It must be noted down over here that the overall system power is distributed among the source node and selected relay nodes, so delta plays a significant role while analyzing the system.

Fig. 6.

ABER versus power allocation factor ($\delta $ ).

Fig. 7 displays the ABER versus relay location plots for different $K$ and total transmit power levels. It is noteworthy over here that we have normalized the distance, i.e., we have divided both $d_{SR}$ and $d_{RD} $ with $d_{SD}$ while keeping the buffer size fixed at 3 for every relay node. Transmit power is assumed to be uniformly allocated to the $S$ and $R$ . It can be seen from the figure that for $K=1,\,2$ , the minima exists at $d_{SR}=0.6$ . In contrast, for the case of $K=3$ , the minima get shifted to $d_{SR}=0.7$ ; so it can be deduced that we need to keep the relay closer to $D$ for better performance.

Fig. 7.

ABER versus relay location ($d_{SR}$ ) for different $K$ and fixed $L$ .

SECTION V. Conclusion and Future Works

A novel system employing multiple relays for a DDM-based BA relaying system has been proposed. A new protocol for transmitting symbols from relay nodes is given that utilizes different time intervals for the reception and transmission of data symbols. Also, a priority-based link selection protocol has been proposed, To make it suitable for DDM-based relaying, we have used indexing of the bits, which helps in decoding and encoding by keeping track of the last two bits. The outage probability for the considered DDM-based setup is obtained by using the MC model. Also, by using the PL detector and MC model, the ABER of the considered setup was evaluated. The ABER for $S$ -$R$ and $R$ -$D$ links have been obtained separately, and it has been observed that the obtained ABER expressions depend upon the number of communicating links.

Next, we compared the performance of the proposed DDM-based setup with its coherent counterpart. Based on the numerical findings, we can deduce that the considered DDM setup encountered an SNR penalty of roughly 2.5 dB at the BER range of $10^{-4}$ . However, the system under consideration does not need CSI or CFO information at any of the communicating links. Also, under the assumption of the CFO, the proposed setup completely outperforms the coherent max-link schemes. Moreover, the proposed DDM setup minimizes the 6 dB SNR penalty between differential and coherent NBA systems.

The above-developed system can be used to communicate a signal in a remote location where the channel quality is not good enough, and the estimation of CSI and CFO is difficult. Also, it can be used in hybrid links where one link has a high data rate, so a relay node will store the data packet and transmit it to a link with a lower data rate later. Furthermore, in future work, with the help of obtained ABER and outage probability expression, the optimal buffer size can be obtained for the proposed BA DDM-based relaying system. An analysis of the average packet delay can also be done with the trade-off between system performance and average packet delay can be obtained.

Appendix Derivation of $\overline{P}_{_{{e}_{RD}}}^{1}{(}\mathcal{M},\mathcal{N}{)}$ , $\overline{P}_{_{{e}_{RD}}}^{2}{(}\mathcal{M},\mathcal{N}{)}$ , $\overline{P}_{_{{e}_{RD}}}^{3 }{(}\mathcal{M},\mathcal{N}{)}$ , and $\overline{P}_{_{{e}_{RD}}}^{4}{(}\mathcal{M},\mathcal{N}{)}$

In (32) the term $\overline {P}_{_{e_{RD}}}^{1}(\mathcal {M},\mathcal {N})$ is expressed as \begin{equation*} \overline {P}_{_{e_{RD}}}^{1}\left ({\mathcal {M},\mathcal {N}}\right)=\frac {\int _{\gamma _{\text {th}}}^{\infty } P_{_{e_{RD}}}^{1}\left ({\mathcal {M},\mathcal {N}}\right)(y)f_{\Gamma _{RD}}(y) \text {d}y}{\int _{\gamma _{\text {th}}}^{\infty } f_{_{\Gamma _{RD}}}(y)\text {d}y}. \tag{48}\end{equation*} The term $P_{_{e_{RD}}}^{1}(\mathcal {M},\mathcal {N})$ can be mathematically given by \begin{align*} P_{_{e_{RD}}}^{1}\left ({m,n}\right)=&\text {Pr}\left [{t_{_{1}}(m)< -T_{_{1}}(m)|x(l)=1;x_{R}(l)=1}\right] \\=&\int _{-\infty }^{-T_{1}\left ({\mathcal {M}}\right)}f_{t_{1}}\left [{w|y_{D}(l-1),x_{R}(l)}\right]dw. \tag{49}\end{align*} Using the cumulative distribution function (CDF) given in [33, eq. (16)], the above expression can be expressed as \begin{equation*} P_{_{e_{RD}}}^{1}\left ({\mathcal {M},\mathcal {N}}\right)=\frac {e^{-y/2}\times e^{-3T_{_{1}}\left ({\mathcal {M}}\right)/2}}{2}. \tag{50}\end{equation*} Substituting expression of $P_{_{e_{RD}}}^{1}(\mathcal {M},\mathcal {N})$ from (50) into (48) and then substituting the expression of $f_{\Gamma _{RD}}(x)$ from (22), (32) can be represented by \begin{align*}&\overline {P}_{_{e_{RD}}}^{1}\left ({\mathcal {M},\mathcal {N}}\right) \\&=\frac {\int _{\gamma _{\text {th}}}^{\infty }e^{-y/2}e^{-3T_{_{1}}\left ({\mathcal {M}}\right)/2}\frac {\mathcal {N}}{\overline {\gamma }_{RD}}e^{-\frac {y}{\overline {\gamma }_{RD}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{RD}}}}\right]^{\mathcal {N}-1}}{2\int _{\gamma _{\text {th}}}^{\infty }\frac {\mathcal {N}}{\overline {\gamma }_{RD}}e^{-\frac {y}{\overline {\gamma }_{RD}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{RD}}}}\right]^{\mathcal {N}-1}}.\qquad \tag{51}\end{align*} Now, expanding binomially and then simplifying (51) we get \begin{align*}&\overline {P}_{_{e_{RD}}}^{1}\left ({\mathcal {M},\mathcal {N}}\right) \\&=\frac {\mathcal {N}e^{-3T_{_{1}}\left ({\mathcal {M}}\right)/2} {\sum _{p=0}^{\mathcal {N}-1}}\binom {\mathcal {N}-1}{p}\frac {(-1)^{p}e^{-\overline {\gamma }_{\text {th}}\left [{\frac {\overline {\gamma }_{RD}+2p+2}{\overline {\gamma }_{RD}}}\right]}}{\left ({\overline {\gamma }_{RD}+2p+2}\right)} }{\left [{1-\left ({1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{RD}}}}\right)^{\mathcal {N}}}\right]}.\,\, \tag{52}\end{align*} The expression for $\overline {P}_{_{e_{RD}}}^{2}(\mathcal {M},\mathcal {N})$ is given as \begin{equation*} \overline {P}_{_{e_{RD}}}^{2}\left ({\mathcal {M},\mathcal {N}}\right)=\frac {\int _{\gamma _{\text {th}}}^{\infty } P_{_{e_{RD}}}^{2}\left ({\mathcal {M},\mathcal {N}}\right)(y)f_{\Gamma _{RD}}(y) \text {d}y}{\int _{\gamma _{\text {th}}}^{\infty } f_{_{\Gamma _{RD}}}(y)\text {d}y}. \tag{53}\end{equation*} The term $P_{_{e_{RD}}}^{2}(\mathcal {M},\mathcal {N})$ in the above equation can be mathematically given by \begin{align*} P_{_{e_{RD}}}^{2}\left ({\mathcal {M},\mathcal {N}}\right)=&\text {Pr}\left ({t_{_{1}}\left ({\mathcal {M}}\right)< T_{_{1}}\left ({\mathcal {M}}\right)}\right)|\left.{d(l)=1;\tilde {d}(l)=-1}\right) \\=&\int _{-T_{_{1}}\left ({\mathcal {M}}\right)}^{0} f_{t_{_{1}}\left ({\mathcal {M}}\right)}\left [{w|y_{D}(l-1),x_{R}(l)}\right]dw \\=&\frac {1}{2}e^{-y/2}\left [{1- e^{-\frac {3T_{_{1}}\left ({\mathcal {M}}\right)}{2}}}\right]. \tag{54}\end{align*}

Substituting the expressions of $f_{\Gamma _{RD}}(x)$ and $P_{_{e_{RD}}}^{2}(\mathcal {M},\mathcal {N})$ from (22) and (54), respectively, in (53), we get \begin{align*}&\overline {P}_{_{e_{RD}}}^{2}\left ({\mathcal {M},\mathcal {N}}\right) \\&=\frac {\int _{\gamma _{\text {th}}}^{\infty }e^{-y/2}\left [{1- e^{-\frac {3T_{_{1}}\left ({\mathcal {M}}\right)}{2}}}\right]\frac {\mathcal {N}}{\overline {\gamma }_{RD}}e^{-\frac {y}{\overline {\gamma }_{RD}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{RD}}}}\right]^{\mathcal {N}-1}\text {d}y}{2\int _{\gamma _{\text {th}}}^{\infty }\frac {\mathcal {N}}{\overline {\gamma }_{RD}}e^{-\frac {y}{\overline {\gamma }_{RD}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{RD}}}}\right]^{\mathcal {N}-1}\text {d}y}. \tag{55}\end{align*} Now, applying binomial expansion and then solving (55) further, we have \begin{align*}&\overline {P}_{_{e_{RD}}}^{2}\left ({\mathcal {M},\mathcal {N}}\right) \\&=\frac {\mathcal {N}\left ({1-e^{-3T_{_{1}}\left ({\mathcal {M}}\right)/2}}\right) {\sum _{p=0}^{\mathcal {N}-1}}\binom {\mathcal {N}-1}{p}\frac {(-1)^{p}e^{-\gamma _{\text {th}}\left [{\frac {\overline {\gamma }_{RD}+2p+2}{\overline {\gamma }_{RD}}}\right]}}{\left ({\overline {\gamma }_{RD}+2p+2}\right)} }{\left [{1-\left ({1-e^{-\frac {\gamma _{\text {th}}}{\overline {\gamma }_{RD}}}}\right)^{\mathcal {N}}}\right]}. \tag{56}\end{align*} The expression of $\overline {P}_{_{e_{RD}}}^{3}(\mathcal {M},\mathcal {N})$ is evaluated as \begin{equation*} \overline {P}_{_{e_{RD}}}^{3}\left ({\mathcal {M},\mathcal {N}}\right)=\frac {\int _{\gamma _{\text {th}}}^{\infty } P_{_{e_{RD}}}^{3}\left ({\mathcal {M},\mathcal {N}}\right)(y)f_{\Gamma _{RD}}(y) \text {d}y}{\int _{\gamma _{\text {th}}}^{\infty } f_{_{\Gamma _{RD}}}(y)\text {d}y}. \tag{57}\end{equation*} The term $P_{_{e_{RD}}}^{3}(\mathcal {M},\mathcal {N})$ is evaluated as \begin{align*} P^{(3) }_{e_{RD}}\left ({\mathcal {M},\mathcal {N}}\right)=&\text {Pr}\,\left [{t_{_{1}}\left ({\mathcal {M}}\right)< -T_{_{1}}\left ({\mathcal {M}}\right)|x(l)=1;x_{R}(l)=-1}\right]\, \\=&\int _{-\infty }^{-T_{_{1}}\left ({\mathcal {M}}\right)} f_{t_{_{1}}}\left ({\mathcal {M}\left [{w|y_{D}(l-1),x_{R}(l)}\right]dw. }\right. \tag{58}\end{align*} Now, substituting the value of $x_{R}(l)=-1$ in the probability density function (PDF) given in [33, eq. (15)], and then substituting the pdf in the (58) and later evaluating the integration in (58), we have \begin{equation*} P^{(3) }_{e_{RD}}\left ({\mathcal {M},\mathcal {N}}\right)=\left ({1/2}\right)e^{-y} \mathcal {B}y^{k} \, \tag{59}\end{equation*} where $\mathcal {B}= \sum _{k=0}^{\infty } \sum _{l=0}^{k}{}{}\frac {\Gamma (k-l+1,3T_{1}(\mathcal {M})/2)}{2^{l}k!(k-l)!}$ . Substituting $f_{\Gamma _{RD}}(x)$ from (22) and $P^{(3) }_{e_{RD}}(\mathcal {M},\mathcal {N})$ from (59) into (57), the expression for $\overline {P}_{_{e_{RD}}}^{3}(\mathcal {M},\mathcal {N})$ can be rewritten as\begin{align*} \overline {P}_{_{e_{RD}}}^{3}\left ({\mathcal {M},\mathcal {N}}\right)=\frac {\int _{\gamma _{\text {th}}}^{\infty } e^{-y} \mathcal {B}y^{k}\frac {\mathcal {N}}{\overline {\gamma }_{RD}}e^{-\frac {y}{\overline {\gamma }_{RD}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{RD}}}}\right]^{\mathcal {N}-1}\text {d}y}{2\int _{\gamma _{\text {th}}}^{\infty }\frac {\mathcal {N}}{\overline {\gamma }_{RD}}e^{-\frac {y}{\overline {\gamma }_{RD}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{RD}}}}\right]^{\mathcal {N}-1}\text {d}y}. \tag{60}\end{align*} Now, applying binomial expansion and then further solving the above equation, we get \begin{equation*} \overline {P}_{_{e_{RD}}}^{3}\left ({\mathcal {M},\mathcal {N}}\right)=\mathcal {B}_{1} \Gamma \left [{k+1,\gamma _{\text {th}}\left [{\frac {\overline {\gamma }_{RD}+p+1}{\overline {\gamma }_{RD}}}\right]}\right], \tag{61}\end{equation*} where, $\mathcal {B}_{1}={}{}\frac {\mathcal {B} {\sum _{p=0}^{\mathcal {N}-1}}(-1)^{p} \binom {\mathcal {N}-1}{p}{}{}\frac {\overline {\gamma }_{RD}^{k}}{(\overline {\gamma }_{RD}+p+1)^{k+1}}}{2\left [{1-\left({1-e^{-{}{}\frac {\gamma _{\text {th}}}{\overline {\gamma }_{RD}}}}\right)^{\mathcal {N}}}\right]}$ .

The expression of $\overline {P}_{_{e_{RD}}}^{4}(\mathcal {M},\mathcal {N})$ is evaluated by \begin{equation*} \overline {P}_{_{e_{RD}}}^{4}\left ({\mathcal {M},\mathcal {N}}\right)=\frac {\int _{\gamma _{\text {th}}}^{\infty } P_{_{e_{RD}}}^{4}\left ({\mathcal {M},\mathcal {N}}\right)(y)f_{\Gamma _{RD}}(y) \text {d}y}{\int _{\gamma _{\text {th}}}^{\infty } f_{_{\Gamma _{RD}}}(y)\text {d}y}. \tag{62}\end{equation*} In the above equation the term $P_{_{e_{RD}}}^{4}(\mathcal {M},\mathcal {N})$ is expressed as \begin{align*} P^{(4) }_{e_{RD}}\left ({\mathcal {M},\mathcal {N}}\right)=&\text {Pr}\,\left [{t_{_{1}}\left ({\mathcal {M}}\right)< 0|-T_{_{1}}\left ({\mathcal {M}}\right)\leq t_{_{1}}\left ({\mathcal {M}}\right)}\right. \\&\qquad \left.{\leq T_{_{1}}\left ({\mathcal {M}}\right);x(l)=1;x_{R}=-1}\right]\, \\=&\int _{-T_{_{1}}\left ({\mathcal {M}}\right)}^{0} f_{t_{_{1}}\left ({\mathcal {M},\mathcal {N}}\right)}\left [{w|y_{D}(l-1),x_{R}(l)}\right]dw. \tag{63}\end{align*} Now, with the help of pdf of $f_{t_{_{1}}(\mathcal {M})}[w|y_{D}(l-1),x_{R}(l)]$ as given in (12), the above equation can be solved, the final expression for $P^{(4) }_{e_{RD}}(\mathcal {M},\mathcal {N})$ is given by \begin{equation*} P^{(4) }_{e_{RD}}\left ({\mathcal {M},\mathcal {N}}\right)=\left ({1/2}\right)e^{-y}\times \,\mathcal {C} y^{k}/k!, \tag{64}\end{equation*} where, $\mathcal {C}= \sum _{k=0}^{\infty } \sum _{l=0}^{k}{}{}\frac {\gamma (k-l+1,3T_{1}(\mathcal {M}/2))}{2^{l}(k-l)!k!}$ . Now, putting expressions of $P^{(4) }_{e_{RD}}(\mathcal {M},\mathcal {N})$ and $f_{\Gamma _{RD}}(x)$ into (62), we get \begin{align*}&\overline {P}_{_{e_{RD}}}^{4}\left ({\mathcal {M},\mathcal {N}}\right) \\&=\frac {\int _{\gamma _{\text {th}}}^{\infty } e^{-y}\times \,\mathcal {C} \frac {y^{k}}{k!}\frac {\mathcal {N}}{\overline {\gamma }_{RD}}e^{-\frac {y}{\overline {\gamma }_{RD}}}\left [{1-e^{-\frac {y}{\gamma _{RD}}}}\right]^{\mathcal {N}-1}}{2\int _{\gamma _{\text {th}}}^{\infty }\frac {\mathcal {N}}{\overline {\gamma }_{RD}}e^{-\frac {y}{\overline {\gamma }_{RD}}}\left [{1-e^{-\frac {y}{\overline {\gamma }_{RD}}}}\right]^{\mathcal {N}-1}}. \tag{65}\end{align*} Applying binomial expansion and further solving the above equation can be expressed as \begin{equation*} \overline {P}_{_{e_{RD}}}^{4}\left ({\mathcal {M},\mathcal {N}}\right)=\mathcal {C}_{1} \Gamma \left [{k+1,\gamma _{\text {th}}\left [{\frac {\overline {\gamma }_{RD}+p+1}{\overline {\gamma }_{RD}}}\right]}\right], \tag{66}\end{equation*} where, $\mathcal {C}_{1}={}{}\frac {\mathcal {C} {\sum _{p=0}^{\mathcal {N}-1}}(-1)^{p} \binom {\mathcal {N}-1}{p}{}{}\frac {\overline {\gamma }_{RD}^{k}}{(\overline {\gamma }_{RD}+p+1)^{k+1}}}{2\left [{1-\left({1-e^{-{}{}\frac {\gamma _{\text {th}}}{\overline {\gamma }_{RD}}}}\right)^{\mathcal {N}}}\right]}$ .