Outage Performance and Diversity Analysis of Cognitive Triple-Hop Cluster-Based Networks Under Interference Constraint

Abstract

In this paper, a full diversity for a triple-hop underlay cognitive network is investigated. In secondary network, multiple decode-and-forward (DF) relays located in two clusters help a source forwarding data to a destination under interference constraint with a primary user. We analyze and compare the outage performance and diversity order of two proposed protocols of the three hops model: the hop between a source node and multiple relays in the first cluster, the hop between the first and second clusters, and the hop between the second cluster and a destination node. One optimal relay (or a best relay) in each cluster is chosen by the selection strategies of two proposed protocols, which take the role of decoding and forwarding the message from the previous node to the next node. To be more specific, the optimal relay in the first cluster decodes the source signal and forward the re-encoded one toward the best relay in second cluster; next, the best relay in this cluster DFs the received signal to the destination node. Finally, we determine the approximate expressions of the end-to-end outage probability and obtain the diversity order for each protocol when the value of the Gaussian noise is small.

Appendices

Appendix 1: Asymptotic Expression for ${\bf{\Pr _1^{out}}}$

We note that the ${{\textit{SNR}}_{SR_1^i}}(i \in \{1,2,\ldots ,{N_1}\}$ are not mutually independent because they belong to the same RVs $|{h_{SP}}{|^2}$. From (2) and (3), the probability of the decoding set ${C_1}$ is formulated as follows:

$$\begin{aligned} \Pr \left[ {{C_1}\left| {|{h_{SP}}{|^2}} \right. } \right] &= \prod \limits _{p = 1}^{{m_1}} {\Pr }\, \left[ {\min \, \left( {{P_{th}}|{h_{SR_1^p}}{|^2},{I_P}{\frac{{|{h_{SR_1^p}}{|^2}}}{{|{h_{SP}}{|^2}}}}} \right) \ge {{\textit{SNR}}_{th}}{N_0}} \right] \\ &\quad \times \, \prod \limits _{q = {m_1} + 1}^{{M_1}} {\Pr }\,\left[ {\min \, \left( {{P_{th}}|{h_{SR_1^qS}}{|^2},{I_P}{\frac{{|{h_{SR_1^q}}{|^2}}}{{|{h_{SP}}{|^2}}}}} \right) < {{\textit{SNR}}_{th}}{N_0}} \right] \end{aligned}$$

(20)

We divide the Eq. (20) into two cases as $|{h_{SP}}{|^2} < {\frac{{{I_P}}}{{{P_{th}}}}}$ and $|{h_{SP}}{|^2} > {\frac{{{I_P}}}{{{P_{th}}}}}$:

$$\begin{aligned} \Pr \left[ {{C_1}\left| {|{h_{SP}}{|^2}} \right. } \right] = \left[ \begin{array}{ll} \left\{ \begin{array}{ll} \prod \limits _{p = 1}^{{m_1}} {\Pr } \left[ {|{h_{SR_1^p}}{|^2} \ge {\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{P_{th}}}}}} \right] \\ \quad \times \prod \limits _{q = {m_1} + 1}^{{M_1}} {\Pr } \left[ {|{h_{SR_1^q}}{|^2} < {\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{P_{th}}}}}} \right] \end{array} \right. \quad if \, |{h_{SP}}{|^2} < {\frac{{{I_P}}}{{{P_{th}}}}}\\ \left\{ \begin{array}{ll} \prod \limits _{p = 1}^{{m_1}} {\Pr }\, \left[ {\frac{{|{h_{SR_1^p}}{|^2}}}{{|{h_{SP}}{|^2}}} \ge {\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{I_P}}}}} \right] \\ \quad \times \prod \limits _{j = {k_1} + 1}^{{N_1}} {\Pr }\, \left[ {\frac{{|{h_{SR_1^q}}{|^2}}}{{|{h_{SP}}{|^2}}} < {\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{I_P}}}}} \right] \end{array} \right. \quad if \,|{h_{SP}}{|^2} > {\frac{{{I_P}}}{{{P_{th}}}}} \end{array} \right. \end{aligned}$$

(21)

The probability density function (PDF) and the cumulative distribution function (CDF) of RVs $|{h_{SR_1^i}}{|^2}$ are respectively given as:

$${f_{_{\left|{h_{SR_1^i}}\right|^2}}}(x)= {\lambda _{S{R_1}}}\exp \left( {- {\lambda _{S{R_1}}}x} \right)$$

(22)

$${F_{_{\left|{h_{SR_1^i}}\right|^2}}}(x)= 1 - \exp \left( {- {\lambda _{S{R_1}}}x} \right)$$

(23)

Then, we have the following probabilities:

$$\Pr \left[ {\left|{h_{SR_1^i}}\right|^2 \le x} \right]= {F_{_{|{h_{SR_1^i}}{|^2}}}}(x)$$

(24)

$$\Pr \left[ {\left|{h_{SR_1^i}}\right|^2 > x} \right]= 1 - {F_{{\left|{h_{SR_1^i}}\right|^2}}}(x)$$

(25)

Substituting (24) and (25) into (21), we obtain as

$$\begin{aligned} \Pr \left[ {{C_1}\left| {|{h_{SP}}{|^2}} \right. } \right] = \left[ \begin{array}{ll} \left\{ \begin{array}{ll} \prod \limits _{p = 1}^{{m_1}} {\left[ {1 - {F_{\left|{h_{SR_1^p}}\right|^2}}\left( {\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{P_{th}}}}} \right) } \right] } \\ \quad \times \prod \limits _{q = {m_1} + 1}^{{M_1}} {{F_{\left|{h_{SR_1^q}}\right|^2}}\left( {\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{P_{th}}}}} \right) } \end{array} \right. \quad if\,|{h_{SP}}{|^2} < {\frac{{{I_P}}}{{{P_{th}}}}}\\ \left\{ \begin{array}{ll} \prod \limits _{p = 1}^{{m_1}} {\left[ {1 - {F_{\left|{h_{SR_1^p}}\right|^2}}\left( {\frac{{{{\textit{SNR}}_{th}}{N_0}|{h_{SP}}{|^2}}}{{{I_P}}}} \right) } \right] } \\ \quad \times \prod \limits _{q = {m_1} + 1}^{{M_1}} {{F_{\left|{h_{SR_1^q}}\right|^2}}\left( {\frac{{{{\textit{SNR}}_{th}}{N_0}|{h_{SP}}{|^2}}}{{{I_P}}}} \right) } \end{array} \right. \quad if\,|{h_{SP}}{|^2} > {\frac{{{I_P}}}{{{P_{th}}}}} \end{array} \right. \ \end{aligned}$$

(26)

Substituting (22) and (23) into (26), we obtain as

$$\begin{aligned} \Pr \, \left[ {{C_1} \left| {|{h_{SP}}{|^2}} \right. } \right] = \left[ \begin{array}{ll} \left\{ \begin{array}{ll} \exp \left( { - {m_1}{\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{P_{th}}}}}} \right) \\ \quad \times {\left( {1 - \exp \left( {- {\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{P_{th}}}}}} \right) } \right) ^{{M_1} - {m_1}}} \end{array} \right. \quad if\,|{h_{SP}}{|^2} < {\frac{{{I_P}}}{{{P_{th}}}}}\\ \left\{ \begin{array}{ll} \exp \left( { - {m_1}{\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}|{h_{SP}}{|^2}}}{{{I_P}}}}} \right) \\ \quad \times {\left( {1 - \exp \left( {- {\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}|{h_{SP}}{|^2}}}{{{I_P}}}}} \right) } \right) ^{{M_1} - {m_1}}} \end{array} \right. \quad if\,|{h_{SP}}{|^2} > {\frac{{{I_P}}}{{{P_{th}}}}} \end{array} \right. \end{aligned}$$

(27)

Let X be an i.i.d. exponential RV with parameter ${\lambda _{SP}}$, the probability for choosing decoding set ${C_1}$ is obtained as

$$\Pr \left[ {{C_1}} \right] = {\int _0^{+ \infty } {\Pr \left[ {{C_1}\left| x \right. } \right] \times f} _X}(x)dx$$

(28)

From (27) and (28), we obtain:

$$\begin{aligned} \Pr \left[ {{C_1}} \right] &= \exp \left( {- {m_1}{\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{P_{th}}}}}} \right) \times {\left( {1 - \exp \left( { - {\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{P_{th}}}}}} \right) } \right) ^{{M_1} - {m_1}}}\\ &\quad \times \, \int _0^{\frac{{{I_P}}}{{{P_{th}}}}} {{\lambda _{SP}}} \exp ( - {\lambda _{SP}}x)dx + \int _{\frac{{{I_P}}}{{{P_{th}}}}}^{+ \infty } {\left\{ {\exp \left( { - {m_1}{\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}x}}{{{I_P}}}}} \right) } \right. } \\ &\quad \times \, \left. {{{\left( {1 - \exp \left( {- {\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}x}}{{{I_P}}}}} \right) } \right) }^{{M_1} - {m_1}}} \times {\lambda _{SP}}\exp ( - {\lambda _{SP}}x)dx} \right\}\end{aligned}$$

(29)

From [11, chapter 4], we have the asymptotic expansion of the exponential function as follows

$${e^x}\mathop = \limits _{x \rightarrow 0} 1 + x + {\frac{{{x^2}}}{2}} + {\frac{{{x^3}}}{6}} + O({x_4})$$

(30)

When $\theta = {\frac{1}{{{N_0}}}} \rightarrow \infty$ or ${{\textit{SNR}}_{th}}{N_0} \rightarrow 0$, from (30) we have the asymptotic expression as:

$$\exp \left( {- {\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{P_{th}}}}}} \right) = 1 - {\frac{{{\lambda _{S{R_1}}}}}{{{P_{th}}}}}{{\textit{SNR}}_{th}}{N_0} + O\left( {{{\left[ {- \frac{{{\lambda _{S{R_1}}}}}{{{P_{th}}}}{{\textit{SNR}}_{th}}{N_0}} \right] }^2}} \right)$$

(31)

From (31), we have the approximate expression as follows

$$1 - \exp \left( {- {\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}}}{{{P_{th}}}}}} \right) \approx {\frac{{{\lambda _{S{R_1}}}}}{{{P_{th}}}}}{{\textit{SNR}}_{th}}{N_0}$$

(32)

Similar to (32), we also obtained as

$$1 - \exp \left( {- {\lambda _{S{R_1}}}{\frac{{{{\textit{SNR}}_{th}}{N_0}x}}{{{I_P}}}}} \right) \approx {\frac{{{\lambda _{SR}}x}}{{{I_P}}}}{{\textit{SNR}}_{th}}{N_0}$$

(33)

Substituting (32) and (33) into (29), we obtain the asymptotic expression of $\Pr [{C_1}]$ when $\theta \rightarrow \infty$ or ${{\textit{SNR}}_{th}}{N_0} \rightarrow 0$ as follows

$$\begin{aligned} \Pr _{S{R_1}}^{out} &= {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{P_{th}}}}{{\textit{SNR}}_{th}}{N_0}} \right) ^{{M_1} - {m_1}}} \times \int _0^{\frac{{{I_P}}}{{{P_{th}}}}} {{\lambda _{SP}}} \exp ( - {\lambda _{SP}}x)dx\\ &\quad +\, \int _{\frac{{{I_P}}}{{{P_{th}}}}}^{+ \infty } {{{\left( {\frac{{{\lambda _{S{R_1}}}x}}{{{I_P}}}{{\textit{SNR}}_{th}}{N_0}} \right) }^{{M_1} - {m_1}}} \times {\lambda _{SP}}} \exp ( - {\lambda _{SP}}x)dx \end{aligned}$$

(34)

By using the Eq. 3.381.3 in Gradshteyn and Tadzhik [12]: $\int _u^\infty {{x^{v - 1}}\exp ( - \mu x)dx} = {\mu ^{- v}}\varGamma (v,\mu u)$, we obtain the followings result:

$$\begin{aligned} & \int _{\frac{{{I_P}}}{{{P_{th}}}}}^{+ \infty } {{{\left( {\frac{{{\lambda _{S{R_1}}}x}}{{{I_P}}}{{\textit{SNR}}_{th}}{N_0}} \right) }^{{M_1} - {m_1}}}{\lambda _{SP}}} \exp ( - {\lambda _{SP}}x)dx\\ &\quad = {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{I_P}}}{{\textit{SNR}}_{th}}{N_0}} \right) ^{{M_1} - {m_1}}}{\lambda _{SP}}{\left( {{\lambda _{SP}}} \right) ^{- \left( {{M_1} - {m_1} + 1} \right) }}\varGamma \left( {{M_1} - {m_1} + 1,{\frac{{{I_P}{\lambda _{SP}}}}{{{P_{th}}}}}} \right) \\ &\quad = {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{I_P}{\lambda _{SP}}}}} \right) ^{{M_1} - {m_1}}}\varGamma \left( {{M_1} - {m_1} + 1,{\frac{{{I_P}{\lambda _{SP}}}}{{{P_{th}}}}}} \right) {\left( {{{\textit{SNR}}_{th}}{N_0}} \right) ^{{M_1} - {m_1}}} \end{aligned}$$

(35)

Substituting (35) into (34), we obtain

$$\begin{aligned} \Pr \left[ {{C_1}} \right] \ &= {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{P_{th}}}}} \right) ^{{M_1-m_1}}}\left( {1 - \exp \left( {- \frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}} \right) } \right) {\left( {{{\textit{SNR}}_{th}}{N_0}} \right) ^{{M_1-m_1}}}\\ &\quad +\, {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{I_P}{\lambda _{SP}}}}} \right) ^{{M_1-m_1}}}\varGamma \left( {{M_1-m_1} + 1,{\frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}}} \right) {\left( {{{\textit{SNR}}_{th}}{N_0}} \right) ^{{M_1-m_1}}}\\ &= \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{P_{th}}}}} \right) ^{{M_1-m_1}}}\left( {1 - \exp \left( {- \frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{I_P}{\lambda _{SP}}}}} \right) ^{{M_1-m_1}}}\varGamma \left( {{M_1-m_1} + 1,{\frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {\left( {\frac{{{{\textit{SNR}}_{th}}}}{\theta }} \right) ^{{M_1}-m_1}} \end{aligned}$$

(36)

where $\varGamma \left( {.,.} \right)$ is the incomplete upper gamma function [12].

From (36), we have the probability of decoding set ${C_1}$ when ${m_1} = 0$:

$$\begin{aligned} \Pr _{S{R_1}}^{out} &= \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{P_{th}}}}} \right) ^{{M_1}}}\left( {1 - \exp \left( {- \frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{I_P}{\lambda _{SP}}}}} \right) ^{{M_1}}}\varGamma \left( {{M_1} + 1,{\frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {\left( {\frac{{{{\textit{SNR}}_{th}}}}{\theta }} \right) ^{{M_1}}}\\ &= {X_1} \times {\theta ^{- {M_1}}} \end{aligned}$$

(37)

which $X_1$ is denoted as

$$\begin{aligned} {X_1} = \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{P_{th}}}}} \right) ^{{M_1}}}\left( {1 - \exp \left( { - \frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{I_P}{\lambda _{SP}}}}} \right) ^{{M_1}}}\varGamma \left( {{M_1} + 1,{\frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {\left( {{{\textit{SNR}}_{th}}} \right) ^{{M_1}}} \end{aligned}$$

From (37), the outage probability of the links ${{\textit{CR}}_1} - {{\textit{CR}}_2}$ and ${{\textit{CR}}_2} - D$ can be formulated similar to that of the $S - {{\textit{CR}}_1}$ link due to the independence of these three links:

$$\begin{aligned} Pr _{{R_1}{R_2}}^{out}&= \left[ \begin{array}{ll} \left( {\frac{{{\lambda _{{R_1}{R_2}}}}}{{{P_{th}}}}} \right) ^{{M_2}}\left( {1 - \exp \left( { - \frac{{{\lambda _{{R_1}P}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{{R_1}{R_2}}}}}{{{I_P}{\lambda _{{R_1}P}}}}} \right) ^{{M_2}}}\varGamma \left( {{M_2} + 1,{\frac{{{\lambda _{{R_1}P}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {\left( {\frac{{{{\textit{SNR}}_{th}}}}{\theta }} \right) ^{{M_2}}} \\ &= {X_2} \times {\theta ^{ - {M_2}}} \end{aligned}$$

(38)

$$\begin{aligned} \Pr _{{R_2}D}^{out}= & \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{{R_2}D}}}}{{{P_{th}}}}} \right) ^1}\left( {1 - \exp \left( { - \frac{{{\lambda _{{R_2}P}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{{R_2}D}}}}{{{I_P}{\lambda _{{R_2}P}}}}} \right) ^1}\varGamma \left( {1 + 1,{\frac{{{\lambda _{{R_2}P}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {\left( {\frac{{{{\textit{SNR}}_{th}}}}{\theta }} \right) ^1} \\=\, & {X_3} \times {\theta ^{ - 1}} \end{aligned}$$

(39)

where $X_2$ and $X_3$ are given as

$$\begin{aligned} {X_2}= & \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{{R_1}{R_2}}}}}{{{P_{th}}}}} \right) ^{{M_2}}}\left( {1 - \exp \left( { - \frac{{{\lambda _{{R_1}P}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{{R_1}{R_2}}}}}{{{I_P}{\lambda _{{R_1}P}}}}} \right) ^{{M_2}}}\varGamma \left( {{M_2} + 1,{\frac{{{\lambda _{{R_1}P}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {\left( {{{\textit{SNR}}_{th}}} \right) ^{{M_2}}}\\ {X_3}= & \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{{R_2}D}}}}{{{P_{th}}}}} \right) ^1}\left( {1 - \exp \left( { - \frac{{{\lambda _{{R_2}P}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{{R_2}D}}}}{{{I_P}{\lambda _{{R_2}P}}}}} \right) ^1}\varGamma \left( {1 + 1,{\frac{{{\lambda _{{R_2}P}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {{\textit{SNR}}_{th}} \end{aligned}$$

In addition, at high $\theta$ value $\left( {{{\textit{SNR}}_{th}}{N_0} \rightarrow 0} \right)$, we can approximate an expression for $\Pr _1^{out}$ as

$$Pr_1^{out} = Pr _{S{R_1}}^{out} + Pr _{{R_1}{R_2}}^{out} + Pr _{{R_2}D}^{out} = {X_1}{\theta ^{ - {M_1}}} + {X_2}{\theta ^{ - {M_2}}} + {X_3}{\theta ^{ - 1}}$$

(40)

From [13, Eq. (3)], the diversity gain is calculated as follows

$${D_1} = - \mathop {\lim }\limits _{\theta \rightarrow + \infty } {\frac{{\log \left( {\Pr _1^{out}} \right) }}{{\log \theta }}}$$

(41)

From (40) and (41), the diversity gain of the NCT protocol is obtained as

$$\begin{aligned} {D_1} = - \mathop {\lim }\limits _{\theta \rightarrow + \infty } {\frac{{\log \left( {{X_1} \times {\theta ^{ - {M_1}}} + {X_2} \times {\theta ^{ - {M_2}}} + {X_3} \times {\theta ^{ - 1}}} \right) }}{{\log \theta }}}\\ \,\,\,\,\,\,\,\,\,\,= - \max \left( { - {M_1}, - {M_2}, - 1} \right) = 1 \end{aligned}$$

(42)

Appendix 2: Asymptotic Expression for ${\bf{\Pr _2^{out}}}$

We have the end-to-end outage probability for the CT3H protocol as in (16):

$$P_2^{out} = \sum \limits _{{m_1} = 0}^{{M_1}} {C_{{M_1}}^{{m_1}}} \Pr \left[ {{C_1}} \right] \sum \limits _{{m_2} = 0}^{{M_2} \times {m_1}} {C_{{M_2} \times {m_1}}^{{m_2}}} \Pr \left[ {{C_2}} \right] \ (Pr _{{R_2}D}^{out})^{{m_2}}$$

(43)

From (33), we obtain the result of $\sum \nolimits _{{m_1} = 0}^{{M_1}} {C_{{M_1}}^{{m_1}}} \Pr \left[ {{C_1}} \right]$ as follows:

$$\begin{aligned} \sum \limits _{{m_1} = 0}^{{M_1}} {C_{{M_1}}^{{m_1}}} \Pr \left[ {{C_1}} \right] &= \sum \limits _{{m_1} = 0}^{{M_1}} {C_{{M_1}}^{{m_1}}} \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{P_{th}}}}} \right) ^{{M_1} - {m_1}}}\left( {1 - \exp \left( { - \frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{I_P}{\lambda _{SP}}}}} \right) ^{{M_1} - {m_1}}}\varGamma \left( {{M_1} - {m_1} + 1,{\frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] \\ &\quad \times \, {\left( {\frac{{{{\textit{SNR}}_{th}}}}{\theta }} \right) ^{{M_1} - {m_1}}}\\ &= \sum \limits _{{m_1} = 0}^{{M_1}} {C_{{M_1}}^{{m_1}}} {X_4}{\theta ^{ - ({M_1} - {m_1})}} \end{aligned}$$

(44)

where $X_4$ is denoted as

$$\begin{aligned} {X_4} = \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{P_{th}}}}} \right) ^{{M_1} - {m_1}}}\left( {1 - \exp \left( { - \frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{S{R_1}}}}}{{{I_P}{\lambda _{SP}}}}} \right) ^{{M_1} - {m_1}}}\varGamma \left( {{M_1} - {m_1} + 1,{\frac{{{\lambda _{SP}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {\left( {{{\textit{SNR}}_{th}}} \right) ^{{M_1} - {m_1}}} \end{aligned}$$

Also, we can obtain

$$\begin{aligned} &\sum \limits _{{m_2} = 0}^{{M_2} \times {m_1}} {C_{{M_2} \times {m_1}}^{{m_2}}} \Pr \left[ {{C_2}} \right] = \sum \limits _{{m_2} = 0}^{{M_2} \times {m_1}} {C_{{M_2} \times {m_1}}^{{m_2}}} \\ &\quad \times \, \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{{R_1}\,{R_2}}}}}{{{P_{th}}}}} \right) ^{{M_2} \times {m_1} \,-\, {m_2}}}\left( {1 - \exp \left( { - \frac{{{\lambda _{{R_1}\,P}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{{R_1}\,{R_2}}}}}{{{I_P}{\lambda _{{R_1}\,P}}}}} \right) ^{{M_2} \times {m_1}\, -\, {m_2}}}\varGamma \,\left( {{M_2}\, - \,{m_2}\, +\, 1,{\frac{{{\lambda _{{R_1}\,P}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] \\ &\quad \times \, {\left( {\frac{{{{\textit{SNR}}_{th}}}}{\theta }} \right) ^{{M_2} \times {m_1} - {m_2}}}\end{aligned}$$

(45)

From (42), and $Pr _{{R_2}D}^{out}$ is calculated in (36), we obtain the result of

$\sum \nolimits _{{m_2} = 0}^{{M_2} \times {m_1}} {C_{{M_2} \times {m_1}}^{{m_2}}} \Pr \left[ {{C_2}} \right] \ (\Pr _{{R_2}D}^{out})^{{m_2}}$ as follows:

$$\begin{aligned} \sum \limits _{{m_2} = 0}^{{M_2} \times {m_1}} {C_{{M_2} \times {m_1}}^{{m_2}}} \Pr \left[ {{C_2}} \right] \ (\Pr _{{R_2}D}^{out})^{{m_2}} &= \sum \limits _{{m_2} = 0}^{{M_2} \times {m_1}} {C_{{M_2} \times {m_1}}^{{m_2}}} \\ &\quad \times \, \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{{R_1}{R_2}}}}}{{{P_{th}}}}} \right) ^{{M_2} \times {m_1} - {m_2}}}\left( {1 - \exp \left( { - \frac{{{\lambda _{{R_1}P}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{{R_1}{R_2}}}}}{{{I_P}{\lambda _{{R_1}P}}}}} \right) ^{{M_2} \times {m_1} - {m_2}}}\\ \quad \times \varGamma \left( {{M_2} - {m_2} + 1,{\frac{{{\lambda _{{R_1}P}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {\left( {\frac{{{{\textit{SNR}}_{th}}}}{\theta }} \right) ^{{M_2} \times {m_1} - {m_2}}}\\ &\quad \times \, \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{{R_2}D}}}}{{{P_{th}}}}} \right) ^{{m_2}}}\left( {1 - \exp \left( { - \frac{{{\lambda _{{R_2}P}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{{R_2}D}}}}{{{I_P}{\lambda _{{R_2}P}}}}} \right) ^{{m_2}}}\varGamma \left( {1 + 1,{\frac{{{\lambda _{{R_2}P}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {\left( {\frac{{{{\textit{SNR}}_{th}}}}{\theta }} \right) ^{{m_2}}}\\ &\quad = {X_5} \times {\theta ^{ - {M_2} \times {m_1}}} \end{aligned}$$

(46)

where $X_5$ is denoted as

$$\begin{aligned} {X_5} &= \sum \limits _{{m_2} = 0}^{{M_2} \times {m_1}} {C_{{M_2} \times {m_1}}^{{m_2}}} \times \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{{R_1}{R_2}}}}}{{{P_{th}}}}} \right) ^{{M_2} \times {m_1} - {m_2}}}\left( {1 - \exp \left( { - \frac{{{\lambda _{{R_1}P}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{{R_1}{R_2}}}}}{{{I_P}{\lambda _{{R_1}P}}}}} \right) ^{{M_2} \times {m_1} - {m_2}}}\varGamma \left( {{M_2} - {m_2} + 1,{\frac{{{\lambda _{{R_1}P}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] \\ &\quad \times \, \left[ \begin{array}{ll} {\left( {\frac{{{\lambda _{{R_2}D}}}}{{{P_{th}}}}} \right) ^{{m_2}}}\left( {1 - \exp \left( { - \frac{{{\lambda _{{R_2}P}}{I_P}}}{{{P_{th}}}}} \right) } \right) \\ \quad + {\left( {\frac{{{\lambda _{{R_2}D}}}}{{{I_P}{\lambda _{{R_2}P}}}}} \right) ^{{m_2}}}\varGamma \left( {1 + 1,{\frac{{{\lambda _{{R_2}P}}{I_P}}}{{{P_{th}}}}}} \right) \end{array} \right] {\left( {{{\textit{SNR}}_{th}}} \right) ^{{M_2} \times {m_1}}} \end{aligned}$$

Substituting (41) and (43) into (40), we obtain the asymptotic expressions for $P_2^{out}$ as

$$\begin{aligned} P_2^{out} &= \sum \limits _{{m_1} = 0}^{{M_1}} {C_{{M_1}}^{{m_1}}} {X_4}{\theta ^{ - ({M_1} - {m_1})}}{X_5} \times {\theta ^{ - {M_2} \times {m_1}}}\\ &= \sum \limits _{{m_1} = 0}^{{M_1}} {C_{{M_1}}^{{m_1}}} {X_4}{X_5}{\theta ^{ - ({M_1} - {m_1} + {M_2} \times {m_1})}}\\ &= C_{{M_1}}^0{X_{{4_{{m_1} = 0}}}}{X_{{5_{{m_1} = 0}}}}{\theta ^{ - {M_1}}} + C_{{M_1}}^1{X_{{4_{{m_1} = 1}}}}{X_{{5_{{m_1} = 1}}}}{\theta ^{ - ({M_1} - 1 + {M_2})}}\\ &\quad +\, C_{{M_1}}^2{X_{{4_{{m_1} = 2}}}}{X_{{5_{{m_1} = 2}}}}{\theta ^{ - ({M_1} - 2 + {2M_2})}} \\ &\quad +\, \cdots + C_{{M_1}}^{{M_1}}{X_{{4_{{m_1} = {M_1}}}}}{X_{{5_{{m_1} = {M_1}}}}}{\theta ^{ - ({M_1}{M_2})}} \end{aligned}$$

(47)

Then, the diversity gain for the CT3H protocol can be given as:

$$\begin{aligned} {D_2} &= - \mathop {\lim }\limits _{\theta \rightarrow + \infty } {\frac{{\log \left( {\Pr _2^{out}} \right) }}{{\log \theta }}} \\ &= - \max \left( { - {M_1}, - ({M_1} - 1 + {M_2}), - ({M_1} - 2 + 2{M_2}),\ldots , - ({M_1}{M_2})} \right) \\ &= \min \left( {{M_1},({M_1} - 1 + {M_2}),({M_1} - 2 + 2{M_2}),\ldots ,({M_1}{M_2})} \right)\end{aligned}$$

(48)

Because ${M_1} \le {M_1} - {m_1} + {M_2}{m_1}$; $\forall {m_1} \ge 1,{M_2} \ge 1$

Therefore

$${D_2} = \min \left( {{M_1},({M_1} - 1 + {M_2}),({M_1} - 2 + 2{M_2}),\ldots ,({M_1}{M_2})} \right) = {M_1}$$

(49)