In this section, the SOPs of \(U_i\) to detect \(s_i\) are respectively derived for the proposed relay–antenna selection schemes. For each relay, each user, and the eavesdropper, two practical signal processing techniques, i.e., SC and MRC, are considered. As such, the proposed relay–antenna selection schemes should be designed in four cases^{Footnote 5}: (1) *SC-SC w/ CSI*: SC at the legitimate nodes (relays and users), and SC at the eavesdropper with the CSI of the eavesdropping channel; (2) *SC-MRC w/ CSI*: SC at the legitimate nodes, and MRC at the eavesdropper with the CSI of the eavesdropping channel; (3) *SC-SC w/o CSI*: SC at the legitimate nodes, and SC at the eavesdropper without the CSI of the eavesdropping channel; (4) *SC-MRC w/o CSI*: SC at the legitimate nodes, and MRC at the eavesdropper without the CSI of the eavesdropping channel.

### 3.1 The SOP derivation for ORAS w/ CSI

#### 3.1.1 SC at the eavesdropper

In the case of SC at the eavesdropper, each antenna wiretaps the transmission date independently. Hence, the individual secrecy performance is limited by the antenna with the best channel condition. In such case, the probability density function (PDF) of the equivalent eavesdropping channel gain is given by [22]

$$\begin{aligned} f_{Z}(z)=\sum _{k=1}^{K}{\left( {\begin{array}{c}K\\ k\end{array}}\right) }(-1)^{k+1}\frac{k}{\Omega _Z}e^{-\frac{kz}{\Omega _Z}}, \end{aligned}$$

(19)

where *Z* stands for eavesdropping channel gain, \(\Omega _{Z}\) represents the reciprocal of the expected gain of the eavesdropping channel.

As a result, the SOP of \(U_1\) can be expressed as

$$\begin{aligned} P_{U_{1},w/}^{SC-SC}={\prod \limits _{n{\mathrm {=1}}}^{N}} \underbrace{\Pr \left\{ {{\left| {\Phi _{U_{1},w/}}\right| =0}} \right\} }_{\Psi _{1,w/}^{{SC-SC}}}, \end{aligned}$$

(20)

and the SOP of \(U_2\) can be expressed as

$$\begin{aligned} P_{U_{2},w/}^{SC-SC}&=\prod _{n=1}^{N} \bigg [ {\underbrace{\Pr \left( {{\left| {\Phi _{U_2,w/}}\right| =0}}\right) }_{\Psi _{2,w/}^{SC-SC}}}+\sum _{m=1}^{M} {\underbrace{\Pr \left( {{\left| {\Phi _{U_2,w/}}\right| =m}}\right) }_{\Psi _{3,w/}^{SC-SC}}} \nonumber \\&\quad \times {\underbrace{\Pr \left( {{\left. { {C_{_{sec,{U_{2}}}} < {R_{2}^{s}}} }\right| \left| {\Phi _{U_2,w/}}\right| = m} }\right) }_{\Psi _{4,w/}^{SC-SC}}}\bigg ], \end{aligned}$$

(21)

To further derive the expression of the SOPs in (20) and (21), Lemma 1 is developed in the following.

### Lemma 1

By using the statistics of channel gains, the analytical expressions of \(\Psi _{1,w/}^{SC-SC}\), \(\Psi _{2,w/}^{SC-SC}\), \(\Psi _{3,w/}^{SC-SC}\) and \(\Psi _{4,w/}^{SC-SC}\) can be derived as

$$\begin{aligned} {\left\{ \begin{array}{ll} \Psi _{i,w/}^{SC-SC} = \prod _{m=1}^{M} \left( 1-\Delta _{i,w/}^{SC-SC} \right) ,i\in (1,2), {~{}} \\ \Psi _{3,w/}^{SC-SC} = {\left( {\begin{array}{c}M\\ m\end{array}}\right) } (\Delta _{2,w/}^{SC-SC})^{m}(1-\Delta _{2,w/}^{SC-SC})^{M-m}, {~{}} \\ \Psi _{4,w/}^{SC-SC} = ( 1-\Delta _{3,w/}^{SC-SC}/\Delta _{2,w/}^{SC-SC})^{m} , \end{array}\right. } \end{aligned}$$

(22)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta _{1,w/}^{SC-SC} = Q_1 Q_2 J_{1} \exp {(- W_1)}, {~{}} \\ \Delta _{2,w/}^{SC-SC} = Q_1 Q_3 J_{1} \exp {(- W_2)}, {~{}} \\ \Delta _{3,w/}^{SC-SC} = Q_1 Q_3 G_{1} O_1 \mu _1^{-1} , \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\left\{ \begin{array}{ll} Q_1 = \sum _{k=1}^{K}{\left( {\begin{array}{c}K\\ k\end{array}}\right) }(-1)^{k+1}, {~{}} \\ Q_2 = \sum _{l_1=1}^{L_1}{\left( {\begin{array}{c}L_1\\ l_1\end{array}}\right) }(-1)^{l_1+1}, {~{}} \\ Q_3 = \sum _{l_2=1}^{L_2}{\left( {\begin{array}{c}L_2\\ l_2\end{array}}\right) }(-1)^{l_2+1}, {~{}} \\ \mu _1 = { \frac{B_{1}}{{\Omega _{SR_{n,m}}}}+ \frac{l_{2}B_{1}}{{\Omega _{R_{n,m}U_{2}}}}}+ \frac{k}{\Omega _{R_{n,m}E}} , {~{}} \\ G_1 = \frac{k}{\Omega _{R_{n,m}E}}\exp \left( -\frac{ A_{1}}{\Omega _{SR_{n,m}}}-\frac{l_{2}A_{1}}{\Omega _{R_{n,m}U_{2}}}\right) , {~{}} \\ W_1 = \frac{k E (t +1)}{2\Omega _{R_{n,m}E}}+\left( { \frac{{1}}{{\Omega _{SR_{n,m}}}}+ \frac{{l_1}}{{\Omega _{R_{n,m}U_{1}}}}}\right) F(t), {~{}} \\ W_2 = \frac{k E (t +1)}{2\Omega _{R_{n,m}E}}+\left( { \frac{{1}}{{\Omega _{SR_{n,m}}}}+ \frac{{l_2}}{{\Omega _{R_{n,m}U_{2}}}}}\right) F(t), {~{}} \\ J_1 =\sum _{l_{0}=1}^{N_0} \frac{k\pi E\sqrt{1-t^2}}{2 N_o \Omega _{R_{n,m}E}} , O_1 = 1-\exp \left( - \mu _1 E \right) , \end{array}\right. } \end{aligned}$$

\(t=\cos \left( \frac{2l_0-1}{2N_0}\pi \right)\), \(A_0=2^{2R_{1}^{s}}-1\), \(B_0=\rho \left( A_0+\alpha \right)\), \(C_0=\rho \left( \alpha +\alpha A_0 -A_0 \right)\), \(D_0=\rho ^{2} A_0 \left( 1-\alpha \right)\), \(E= \frac{C_0}{D_0}\), \(A_1=\frac{2^{2R_{2}^{s}}-1}{\rho \left( 1-\alpha \right) }\), \(B_1=2^{2R_{2}^{s}}\), \(F\left( t\right) =\frac{ 2A_0+B_0E\left( t+1\right) }{2C_0-D_0E\left( t+1\right) }\), and \(N_0\) denotes the number of terms for the quadrature approximation.

###
*Proof*

The proof of Lemma 1 is shown in Appendix 1.

By substituting (22) into (20) and (21), we can get the closed-form expressions of the SOPs for \(P_{U_{1},w/}^{SC-SC}\) and \(P_{U_{2},w/}^{SC-SC}\). Thus, in the presence of SC at the eavesdropper, the total SOP can be expressed as

$$\begin{aligned} P_{out,w/}^{SC-SC}= \frac{\left( P_{U_{1},w/}^{SC-SC}+P_{U_{2},w/}^{SC-SC} \right) }{2}. \end{aligned}$$

(23)

Accordingly, the SOP for NOMA system with SC at the eavesdropper in ORAS scheme w/ CSI can be approximated as

$$\begin{aligned} P_{out,w/}^{SC-SC} =\frac{1}{2} \prod _{n=1}^{N}\prod _{m=1}^{M} \left( 1-\Delta _{1,w/}^{SC-SC} \right) +\frac{1}{2} \prod _{n=1}^{N}\prod _{m=1}^{M}\left( 1-\Delta _{3,w/}^{SC-SC} \right) . \end{aligned}$$

(24)

#### 3.1.2 MRC at the eavesdropper

In the case of MRC at the eavesdropper, the antenna cooperates with each other to wiretap the transmission date. Hence, the PDF of the equivalent eavesdropping channel gain is given by [22]

$$\begin{aligned} f_{Z}(z)=\frac{Z^{K-1}}{{\Omega _Z}^{K} (K-1)!}e^{-\frac{z}{\Omega _Z}}. \end{aligned}$$

(25)

In this scheme, the SOPs of \(U_1\) and \(U_2\) can be gotten by changing the superscript “SC-SC” in (20) and (21) to “SC-MRC”. However, the derivation of each term is different. Similar to Lemma 1, Lemma 2 is given as follows.

### Lemma 2

The analytical expressions of \(\Psi _{1,w/}^{SC-MRC}\), \(\Psi _{2,w/}^{SC-MRC}\), \(\Psi _{3,w/}^{SC-MRC}\) and \(\Psi _{4,w/}^{SC-MRC}\) can be expressed as

$$\begin{aligned} {\left\{ \begin{array}{ll} \Psi _{i,w/}^{SC-MRC} = \prod _{m=1}^{M} \left( 1-\Delta _{i,w/}^{SC-MRC} \right) ,i\in (1,2), {~{}} \\ \Psi _{3,w/}^{SC-MRC} = {\left( {\begin{array}{c}M\\ m\end{array}}\right) } (\Delta _{2,w/}^{SC-MRC})^{m}(1-\Delta _{2,w/}^{SC-MRC})^{M-m}, {~{}} \\ \Psi _{4,w/}^{SC-MRC} = ( 1-\Delta _{3,w/}^{SC-MRC}/\Delta _{2,w/}^{SC-MRC})^{m} , \end{array}\right. } \end{aligned}$$

(26)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta _{1,w/}^{SC-MRC} = Q_2 J_{2} \exp {(-W_3 )}, {~{}} \\ \Delta _{2,w/}^{SC-MRC} = Q_3 J_{2} \exp {(-W_4 )}, {~{}} \\ \Delta _{3,w/}^{SC-MRC} =Q_3 G_{2} O_2, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\left\{ \begin{array}{ll} G_2 = \frac{\exp \left( -\frac{ A_{1}}{\Omega _{SR_{n,m}}}-\frac{l_{2}A_{1}}{\Omega _{R_{n,m}U_{2}}}\right) }{(\Omega _{R_{n,m}E})^K (K-1)!}, {~{}} \\ \mu _2 = { \frac{B_{1}}{{\Omega _{SR_{n,m}}}}+ \frac{l_{2}B_{1}}{{\Omega _{R_{n,m}U_{2}}}}}+ \frac{1}{\Omega _{R_{n,m}E}} , {~{}} \\ O_2 = (\mu _2)^{-K} ( \Gamma (K)-\Gamma (K,\mu _2 E)),{~{}}\\ W_{3} = \frac{ E (t+1)}{2\Omega _{R_{n,m}E}}+\left( { \frac{{1}}{{\Omega _{SR_{n,m}}}}+ \frac{{l_1}}{{\Omega _{R_{n,m}U_{1}}}}}\right) F(t), {~{}} \\ W_{4} = \frac{ E (t+1)}{2\Omega _{R_{n,m}E}}+\left( { \frac{{1}}{{\Omega _{SR_{n,m}}}}+ \frac{{l_2}}{{\Omega _{R_{n,m}U_{2}}}}}\right) F(t), {~{}} \\ J_2 =\sum _{l_{0}=1}^{N_0} \frac{\pi E \sqrt{1-t^2}}{2 N_o (\Omega _{R_{n,m}E})^K (K-1)!} \bigg [\frac{{E}{(t+1)}}{2}\bigg ]^{K-1}, \end{array}\right. } \end{aligned}$$

where \(\Gamma (x)\) is the Gamma function, and \(\Gamma (x,y)\) is the upper incomplete Gamma function [23].

Similar to that of the “SC-SC” case in the ORAS scheme w/ CSI, the total SOP of the “SC-MRC” case can be computed by (23). The detailed proof is omitted here to avoid redundancy.

By substituting (26) into (20) and (21), the closed-form expression of the SOP with MRC at the eavesdropper can be obtained accordingly. Thus, for the ORAS scheme w/ CSI, the SOP with MRC at the eavesdropper can be approximated as

$$\begin{aligned} P_{out,w/}^{SC-MRC} =\frac{1}{2} \prod _{n=1}^{N}\prod _{m=1}^{M} \left( 1-\Delta _{1,w/}^{SC-MRC} \right) +\frac{1}{2} \prod _{n=1}^{N}\prod _{m=1}^{M}\left( 1-\Delta _{3,w/}^{SC-MRC} \right) . \end{aligned}$$

(27)

### 3.2 The SOP derivation for ORAS w/o CSI

#### 3.2.1 SC at the eavesdropper

In this scenario, the SOPs of the users \(U_1\) and \(U_2\) can be expressed as (28) and (29), respectively,

$$\begin{aligned} P_{U_{1},w/o}^{SC-SC}&= {\underbrace{\Pr \left( {{\left| {\Phi _{U_1,sub.}}\right| =(0,0)}}\right) }_{\Psi _{1,w/o}^{SC-SC}}}+\sum _{n=1}^{N} {\underbrace{\Pr \left( {{\left| {\Phi _{U_1,w/o}}\right| =(n,m)}}\right) }_{\Psi _{2,w/o}^{SC-SC}}} \nonumber \\&\quad\times {\underbrace{\Pr \left( {{\left. { {C_{_{sec,{U_{1}\leftarrow s_1}}} < {R_{1}^{s}}} }\right| \left| {\Phi _{U_1,w/o}}\right| = (n,m)} }\right) }_{\Psi _{3,w/o}^{SC-SC}}}, \end{aligned}$$

(28)

and

$$\begin{aligned} P_{U_{2},w/o}^{SC-SC}&= {\underbrace{\Pr \left( {{\left| {\Phi _{U_2,w/o}}\right| =(0,0)}}\right) }_{\Psi _{4,w/o}^{SC-SC}}}+\sum _{n_{+}=1}^{N} {\underbrace{\Pr \left( {{\left| {\Phi _{U_2,w/o}}\right| =(n_{+},m_{+})}}\right) }_{\Psi _{5,w/o}^{SC-SC}}}\nonumber \\&\quad\times {\underbrace{\Pr \left( {{\left. { {C_{_{sec,{U_{2} }}} < {R_{2}^{s}}} }\right| \left| {\Phi _{U_2,w/o}}\right| = (n_{+},m_{+})} }\right) }_{\Psi _{6,w/o}^{SC-SC}}}, \end{aligned}$$

(29)

To further derive the expressions of the above SOPs, Lemma 3 is developed in the following.

### Lemma 3

The analytical expressions of \(\Psi _{1,w/o}^{SC-SC}\), \(\Psi _{2,w/o}^{SC-SC}\), \(\Psi _{3,w/o}^{SC-SC}\), \(\Psi _{4,w/o}^{SC-SC}\),\(\Psi _{5,w/o}^{SC-SC}\) and \(\Psi _{6,w/o}^{SC-SC}\) can be expressed as

$$\begin{aligned} {\left\{ \begin{array}{ll} \Psi _{1,w/o}^{SC-SC} = \prod _{m=1}^{M} \prod _{n=1}^{N} \left( 1-\Delta _{1,w/o}^{SC-SC} \right) , {~{}} \\ \Psi _{2,w/o}^{SC-SC} =\beta _1 \left( \Delta _{1,w/o}^{SC-SC} \right) ^{\zeta _1}\left( 1-\Delta _{1,w/o}^{SC-SC} \right) ^{MN-\zeta _1}, {~{}} \\ \Psi _{3,w/o}^{SC-SC} = 1-\Delta _{2,w/o}^{SC-SC} /\left( \Delta _{1,w/o}^{SC-SC} \right) ^{\zeta _1}, {~{}} \\ \Psi _{4,w/o}^{SC-SC} = \prod _{m=1}^{M} \prod _{n=1}^{N} \left( 1-\Delta _{3,w/o}^{SC-SC} \right) , {~{}} \\ \Psi _{5,w/o}^{SC-SC} =\beta _2 \left( \Delta _{3,w/o}^{SC-SC} \right) ^{\zeta _2}\left( 1-\Delta _{3,w/o}^{SC-SC} \right) ^{MN-\zeta _2}, {~{}} \\ \Psi _{6,w/o}^{SC-SC} = 1-\Delta _{4,w/o}^{SC-SC} /\left( \Delta _{3,w/o}^{SC-SC} \right) ^{\zeta _2}, \end{array}\right. } \end{aligned}$$

(30)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta _{1,w/o}^{SC-SC} =Q_2 \exp \left( -\frac{ C_{1}}{\Omega _{SR_{n,m}}}-\frac{l_{1}C_{1}}{\Omega _{R_{n,m}U_{1}}}\right) , {~{}} \\ \zeta _{1}=m_1+\ldots +m_n, \zeta _{2}=m_{+}^{1}+\ldots +m_{+}^{n_+}, {~{}} \\ \beta _{1}={\left( {\begin{array}{c}N\\ n\end{array}}\right) }\sum _{m_{1}=1}^{M}\ldots \sum _{m_{n}=1}^{M}{\left( {\begin{array}{c}M\\ m_1\end{array}}\right) }\ldots {\left( {\begin{array}{c}M\\ m_n\end{array}}\right) }, {~{}} \\ \Delta _{2,w/o}^{SC-SC} = Q_1Q_4 G_3 J_1 \exp (-W_5) ,{~{}} \\ \Delta _{3,w/o}^{SC-SC} =Q_{3} \exp \left( -\frac{ D_{1}}{\Omega _{SR_{n,m}}}-\frac{l_{2}D_{1}}{\Omega _{R_{n,m}U_{2}}}\right) ,{~{}} \\ \beta _{2}={\left( {\begin{array}{c}N\\ n_+\end{array}}\right) }\sum _{m_{+}^{1}=1}^{M}\ldots \sum _{m_{+}^{n_+}=1}^{M}{\left( {\begin{array}{c}M\\ m_{+}^1\end{array}}\right) }\ldots {\left( {\begin{array}{c}M\\ m_{+}^{n_+}\end{array}}\right) }, {~{}} \\ \Delta _{4,w/o}^{SC-SC} = Q_1 Q_5 G_4 \mu _3^{-1} k/\Omega _{R_{n,m}E}, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\left\{ \begin{array}{ll} Q_{4} = \sum _{j=0}^{\zeta _1} {\left( {\begin{array}{c}\zeta _1\\ j\end{array}}\right) }(-1)^{j} (Q_2)^{j}, {~{}} \\ Q_{5} = \sum _{j=0}^{\zeta _2} {\left( {\begin{array}{c}\zeta _2\\ j\end{array}}\right) }(-1)^{j} (Q_3)^{j}, {~{}} \\ \mu _3 =(\frac{jB_{1}}{{\Omega _{SR_{n,m}}}}+ \frac{j l_{2}B_{1}}{{\Omega _{R_{n,m}U_{2}}}}+ \frac{k}{\Omega _{R_{n,m}E}}), {~{}} \\ G_3 = \exp \left( -\frac{ C_{1}(\zeta _1-j)}{\Omega _{SR_{n,m}}}-\frac{ l_1 C_{1}(\zeta _1-j)}{\Omega _{R_{n,m}U_{1}}}\right) , {~{}} \\ C_1 = \frac{2^{2R_{1}^{th}}-1}{\rho \left( 1-(1-\alpha ) 2^{2R_{1}^{th}}\right) }, D_1 = \frac{2^{2R_{2}^{th}}-1}{(1-\alpha ) \rho },{~{}} \\ W_{5} = \frac{k E (t+1)}{2\Omega _{R_{n,m}E}}+\left( { \frac{{j}}{{\Omega _{SR_{n,m}}}}+ \frac{{j l_1}}{{\Omega _{R_{n,m}U_{1}}}}}\right) F(t), {~{}} \\ G_4 =\exp \left( -\frac{ D_{1}(\zeta _2-ii)+jA_1}{\Omega _{SR_{n,m}}}-\frac{l_{2}D_{1}(\zeta _2-j)+j l_2 A_1}{\Omega _{R_{n,m}U_{2}}}\right) . \end{array}\right. } \end{aligned}$$

###
*Proof*

The proof of Lemma 3 is shown in Appendix 2.

By substituting (30) into (28) and (29), the closed-form expression of the SOP can be obtained by

$$\begin{aligned} P_{out,w/o}^{SC-SC} =\frac{1}{2} \bigg (1- Q_1 Q_6 G_5 J_1\exp (-W_6)\bigg )+\frac{1}{2} \left( 1- Q_1 Q_7 G_6 \mu _4^{-1} k/\Omega _{R_{n,m}E}\right) , \end{aligned}$$

(31)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} Q_{6} = \sum _{i_{o}=0}^{MN} {\left( {\begin{array}{c}MN\\ i_o\end{array}}\right) }(-1)^{i_o} (Q_2)^{i_o}, {~{}} \\ Q_{7} = \sum _{i_{oo}=0}^{MN} {\left( {\begin{array}{c}MN\\ i_{oo}\end{array}}\right) }(-1)^{i_{oo}} (Q_3)^{i_{oo}}, {~{}} \\ \mu _4 =(\frac{i_{oo}B_{1}}{{\Omega _{SR_{n,m}}}}+ \frac{i_{oo} l_{2}B_{1}}{{\Omega _{R_{n,m}U_{2}}}}+ \frac{k}{\Omega _{R_{n,m}E}}), {~{}} \\ G_5 = \exp \left( -\frac{ C_{1}(MN-i_{o})}{\Omega _{SR_{n,m}}}-\frac{ l_1 C_{1}(MN-i_{o})}{\Omega _{R_{n,m}U_{1}}}\right) , {~{}} \\ W_{6} = \frac{k E (t+1)}{2\Omega _{R_{n,m}E}}+\left( { \frac{{i_{o}}}{{\Omega _{SR_{n,m}}}}+ \frac{{i_{o} l_1}}{{\Omega _{R_{n,m}U_{1}}}}}\right) F(t), {~{}} \\ G_6 =\exp \left( -\frac{ D_{1}(MN-i_{oo})+i_{oo}A_1}{\Omega _{SR_{n,m}}}-\frac{l_{2}D_{1}(MN-i_{oo})+i_{oo} l_2 A_1}{\Omega _{R_{n,m}U_{2}}}\right) . \end{array}\right. } \end{aligned}$$

#### 3.2.2 MRC at the eavesdropper

In this case, the SOPs of \(U_1\), \(U_2\) have the similar forms with (28) and (29). Following Lemma 3, Lemma 4 is given as follows.

### Lemma 4

The analytical expressions of \(\Psi _{1,w/o}^{SC-MRC}\), \(\Psi _{2,w/o}^{SC-MRC}\), \(\Psi _{3,w/o}^{SC-MRC}\), \(\Psi _{4,w/o}^{SC-MRC}\),\(\Psi _{5,w/o}^{SC-MRC}\) and \(\Psi _{6,w/o}^{SC-MRC}\) can be expressed as

$$\begin{aligned} {\left\{ \begin{array}{ll} \Psi _{1,w/o}^{SC-MRC} = \Psi _{1,w/o}^{SC-SC}, {~{}} \\ \Psi _{2,w/o}^{SC-MRC} =\Psi _{2,w/o}^{SC-SC}, {~{}} \\ \Psi _{3,w/o}^{SC-MRC} = 1-\Delta _{2,w/o}^{SC-MRC} /\left( \Delta _{1,w/o}^{SC-SC} \right) ^{\zeta _1}, {~{}} \\ \Psi _{4,w/o}^{SC-MRC} = \Psi _{4,w/o}^{SC-SC}, {~{}} \\ \Psi _{5,w/o}^{SC-MRC} =\Psi _{5,w/o}^{SC-SC}, {~{}} \\ \Psi _{6,w/o}^{SC-MRC} = 1-\Delta _{4,w/o}^{SC-MRC} /\left( \Delta _{3,w/o}^{SC-SC} \right) ^{\zeta _2}, \end{array}\right. } \end{aligned}$$

(32)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta _{2,w/o}^{SC-MRC} = Q_4 G_3 J_2 \exp (-W_7),{~{}} \\ \Delta _{4,w/o}^{SC-MRC} = Q_5 G_7 (\mu _5)^{-K} \Gamma (K), \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu _5 = { \frac{j B_{1}}{{\Omega _{SR_{n,m}}}}+ \frac{j l_{2}B_{1}}{{\Omega _{R_{n,m}U_{2}}}}}+ \frac{1}{\Omega _{R_{n,m}E}} , {~{}} \\ G_7 = \frac{\exp \left( -\frac{ (\zeta _2-j) A_{1}}{\Omega _{SR_{n,m}}}-\frac{(\zeta _2-j) l_{2}A_{1}}{\Omega _{R_{n,m}U_{2}}}\right) }{(\Omega _{R_{n,m}E})^K (K-1)!} , {~{}} \\ W_7 = \frac{ E (t+1)}{2\Omega _{R_{n,m}E}}+\left( { \frac{{j}}{{\Omega _{SR_{n,m}}}}+ \frac{{j l_1}}{{\Omega _{R_{n,m}U_{1}}}}}\right) F(t). \end{array}\right. } \end{aligned}$$

The derivation is similar to that for the “SC-SC” case in the ORAS scheme w/o CSI. The detailed proof is omitted here to avoid redundancy.

By substituting (32) into (28) and (29), the closed-form of the SOP can be expressed as

$$\begin{aligned} P_{out,w/o}^{SC-MRC} =\frac{1}{2} \bigg (1- Q_6 G_5 J_2\exp (-W_8)\bigg )+\frac{1}{2} \left( 1- Q_7 G_{6} O_3\right) , \end{aligned}$$

(33)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu _6 =(\frac{i_{oo}B_{1}}{{\Omega _{SR_{n,m}}}}+ \frac{i_{oo} l_{2}B_{1}}{{\Omega _{R_{n,m}U_{2}}}}+ \frac{1}{\Omega _{R_{n,m}E}}), {~{}} \\ W_{8} = \frac{ E (t+1)}{2\Omega _{R_{n,m}E}}+\left( { \frac{{i_{o}}}{{\Omega _{SR_{n,m}}}}+ \frac{{i_{o} l_1}}{{\Omega _{R_{n,m}U_{1}}}}}\right) F(t), {~{}} \\ O_3 =(\mu _6)^{-K} \Gamma (K)\bigg {/}\bigg ((\Omega _{R_{n,m}E})^K (K-1)!\bigg ). \end{array}\right. } \end{aligned}$$