Wireless Information and Power Transfer in Kth Best Relay Selection Systems with Energy Beamforming over Nakagami-m Fading Channels

Abstract

In this paper, we propose and analyze an amplify-and-forward relaying energy harvesting system with Kth best partial relay selection and energy beamforming over Nakagami-m fading channels. The Kth best relay is selected based on criteria such as the Kth best first-hop channel gains (KBFC scheme) and the Kth best second-hop channel gains (KBSC scheme). Both the time-switching relaying and power-splitting relaying protocols are examined. To evaluate the system performance, we derive analytical expressions for the outage probability and throughput in both the delay-limited transmission (DLT) and delay-tolerant transmission (DTT) modes. Then the optimal values of these throughput are determined. The DLT mode is considered in two optimal cases: global-optimal DLT (GODLT), where an optimal pair of the source rate and energy-harvesting ratio is employed, and local-optimal DLT (LODLT), where only optimal energy-harvesting ratio is used. Monte Carlo simulations are presented to corroborate our analysis. The results in terms of throughput show the following. (1) For the LODLT mode, at relatively low SNRs, the KBSC scheme outperforms the KBFC scheme except for $K=1$. (2) For the DTT and GODLT modes, the KBSC scheme is more efficient than the KBFC scheme at high values of K. (3) When the signal quality increases, the throughput for the DTT and GODLT modes is significantly enhanced, whereas that for the LODLT mode reaches an upper limit.

Appendices

Appendix 1: Proof of Proposition 1

The tight lower bound of the outage probability of the proposed system can be calculated as

$$\begin{aligned} {\mathcal {P}}^{\text {low}}(\gamma ) = \Pr \{\gamma _{e2e,[w]}^{\text {up}} < \gamma \}. \end{aligned}$$

(31)

Substituting (14) into (31), we have

$$\begin{aligned} {\mathcal {P}}^{\text {low}}(\gamma ) = 1 - \int \limits _0^{ + \infty } \left( 1 - F_{\left\| \mathbf{h }_2^{( K )} \right\| ^2}\left( \frac{\gamma }{{\mathbb {b}}_{[w]} z} \right) \right) f_{\left\| \mathbf{h }_1^{( K )} \right\| ^2}\left( z + \frac{\gamma }{{\mathbb {a}}_{[w]}} \right) dz, \end{aligned}$$

(32)

where $z = \left( \left\| \mathbf{h }_1^{( K )} \right\| ^2 - \frac{\gamma }{{\mathbb {a}}_{[w]}} \right)$.

Before calculating the outage probability, we need to study the PDF and CDF of the Kth best order of gamma RVs. Consider L gamma RVs, ${\mathcal {X}}_1,\ldots ,{\mathcal {X}}_L$, with parameters $(m,\lambda )$, and the Kth best order RV is determined as ${\mathcal {X}}^{(K)}=Kth \mathop {\arg \max }\limits _{1 \leqslant l \leqslant L} ({\mathcal {X}}_l)$. According to [17], we can respectively obtain the PDF and CDF of ${\mathcal {X}}^{(K)}$ as follows.

$$\begin{aligned}&f_{{\mathcal {X}}^{(K)}}( x;m,\lambda ) = \sum \limits _{n = 0}^{L - K} \left( {\begin{array}{c}L - K\\ n\end{array}}\right) \frac{L!( - 1)^n f(x;m,\lambda )}{(K - 1)!(L - K)!} \left( \frac{\Gamma \left( m,\frac{x}{\lambda } \right) }{\Gamma (m)} \right) ^{K + n - 1}, \end{aligned}$$

(33)

$$\begin{aligned}&F_{{\mathcal {X}}^{(K)}}( x;m,\lambda ) = \sum \limits _{t = 0}^{K - 1} \sum \limits _{n = 0}^{L - t} (- 1)^n \left( {\begin{array}{c}L\\ t\end{array}}\right) \left( {\begin{array}{c}L - t\\ n\end{array}}\right) \left( \frac{\Gamma \left( m,\frac{x}{\lambda } \right) }{\Gamma ( m )} \right) ^{t + n}. \end{aligned}$$

(34)

1.1 Outage Probability for the KBFC Scheme

For the KBFC scheme, the PDF and CDF of $\Vert {\mathbf {h}}_1^{(K)} \Vert ^2$ respectively follow (33) and (34) with parameters $(N_1m_1,\frac{\lambda _1}{m_1})$, the PDF of $\Vert {\mathbf {h}}_2^{(K)} \Vert ^2$ follows (1), and the CDF of $\Vert {\mathbf {h}}_2^{(K)} \Vert ^2$ follows

$$\begin{aligned} F_g ( x;m,\lambda ) = \frac{\gamma \left( m,\frac{x}{\lambda } \right) }{\Gamma ( m )} = 1 - \frac{\Gamma \left( m,\frac{x}{\lambda } \right) }{\Gamma ( m )}, \end{aligned}$$

(35)

with parameters $\left( N_2m_2,\frac{\lambda _2}{m_2} \right)$ [14].

Substituting the PDF of $\Vert {\mathbf {h}}_1^{(K)} \Vert ^2$ and CDF of $\Vert {\mathbf {h}}_2^{(K)} \Vert ^2$ into (32), and using the multinomial theorem given in [18, Eq. (26.4.9)], the outage probability for the KBFC scheme can be calculated as

$$\begin{aligned}&{\mathcal {P}}^{\text {low}}_{{\text {KBSC}},[w]}(\gamma ) = 1 - \sum \limits _{j = 0}^{N_2m_2 - 1} \sum \limits _{n = 0}^{L - K} \sum \limits _{\sum \limits _{i = 0}^{N_1m_1 - 1} p_i^{(1)} = K + n - 1} \sum \limits _{q = 0}^{N_1m_1 + \omega _1 - 1} \frac{L!(- 1)^n}{j! (K - 1)! (L - K)! } \nonumber \\&\quad\times \frac{\left( \frac{m_2\gamma }{{\mathbb {b}}_{[w]}\lambda _2} \right) ^j\left( \frac{m_1}{\lambda _1} \right) ^{N_1m_1 + \omega _1}}{\Gamma ( N_1m_1 )} \left( {\begin{array}{c}L-K\\ n\end{array}}\right) \left( {\begin{array}{c}K+n-1\\ p_0^{(1)}, \ldots ,p_{N_1m_1 - 1}^{(1)}\end{array}}\right) \left( {\begin{array}{c}N_1m_1 + \omega _1 - 1\\ q\end{array}}\right) \nonumber \\&\quad\times \left( \prod \limits _{i = 0}^{N_1m_1 - 1} \left( \frac{1}{i!} \right) ^{p_i^{(1)}} \right) e^{ - \frac{m_1(K + n)\gamma }{{\mathbb {a}}_{[w]} \lambda _1 }} \int \limits _0^{ + \infty } t^{q - j} e^{- \frac{m_2\gamma }{{\mathbb {b}}_{[w]}\lambda _2 t} - \frac{m_1( K + n )t}{\lambda _1}}dt. \end{aligned}$$

(36)

The integration in (36) can be solved using [13, Eq.(3.471.9)]. Then ${\mathcal {P}}^{\text {low}}_{{\text {KBSC}},[w]}(\gamma )$ can be calculated as in (15).

1.2 Outage Probability for the KBSC Scheme

For the KBSC scheme, the PDF and CDF of $\Vert {\mathbf {h}}_1^{(K)} \Vert ^2$ respectively follow (1) and (35) with parameters $(N_1m_1,\frac{\lambda _1}{m_1})$, and the PDF and CDF of $\Vert {\mathbf {h}}_2^{(K)} \Vert ^2$ respectively follow (33) and (34) with parameters $(N_2m_2,\frac{\lambda _2}{m_2})$. Then, the outage probability for the KBSC scheme can be obtained with the same steps as in “Outage Probability for the KBFC Scheme” of Appendix 1.

This ends the proof for Proposition 1.

Appendix 2: Proof of Proposition 2

1.1 Ergodic Capacity for the KBFC Scheme

1.1.1 Calculation for ${\mathcal {C}}_{\gamma _{r,[w]}}$

According to [19] and using the CDF of $\Vert {\mathbf {h}}_1^{(K)} \Vert ^2$ for the KBFC scheme, ${\mathcal {C}}_{\gamma _{r,[w]}}$ can be calculated as

$$\begin{aligned}&{\mathcal {C}}_{\gamma _{r,[w]}} = \frac{1}{2\ln 2}\int \limits _0^{ + \infty } \frac{1 - F_{\left\| \mathbf{h }_1^{(K)} \right\| ^2}( x )}{{\mathbb {a}}_{[w]}^{-1} + x} dx \nonumber \\&= \frac{- 1}{2\ln 2}\sum \limits _{t = 0}^{K - 1} {\mathop{{\mathop{\sum}\limits_{n = 0}}}\limits_{t + n \ne 0}^{L -t}} \sum \limits _{\sum \limits _{i = 0}^{N_1m_1 - 1} p_i^{(1)} = t + n} ( - 1)^n \left( {\begin{array}{c}L\\ t\end{array}}\right) \left( {\begin{array}{c}L - t\\ n\end{array}}\right) \left( {\begin{array}{c}t + n\\ p_0^{(1)}, \ldots ,p_{N_1m_1 - 1}^{(1)}\end{array}}\right) \nonumber \\&\quad \times \left( \prod \limits _{i = 0}^{N_1m_1 - 1} \left( \frac{1}{i!} \right) ^{p_i^{(1)}} \right) \left( \frac{m_1}{\lambda _1} \right) ^{\omega _1} \int \limits _0^{ + \infty } \frac{x^{\omega _1}}{({\mathbb {a}}_{[w]}^{-1} + x )} e^{ - \frac{m_1( t + n )x}{\lambda _1}}dx. \end{aligned}$$

(37)

The integration in (37) can be solved using [13, Eq. (3.383.10)]; hence, ${\mathcal {C}}_{\gamma _{r,[w]}}$ can be expressed as

$$\begin{aligned}{\mathcal {C}}_{\gamma _{r,[w]}}& = \frac{ - 1}{2\ln 2}\sum \limits _{t = 0}^{K - 1} {\mathop{{\mathop{\sum}\limits_{n = 0}}}\limits_{t + n \ne 0}^{L -t}} \sum \limits _{\sum \limits _{i = 0}^{N_1m_1 - 1} {p_i^{(1)}} = t + n} ( - 1 )^n \left( {\begin{array}{c}L\\ t\end{array}}\right) \left( {\begin{array}{c}L - t\\ n\end{array}}\right) \left( {\begin{array}{c}t + n\\ p_0^{(1)}, \ldots ,p_{N_1m_1 - 1}^{(1)}\end{array}}\right) \nonumber \\&\quad \times \left( \prod \limits _{i = 0}^{N_1m_1 - 1} \left( \frac{1}{i!} \right) ^{p_i^{(1)}} \right) \left( \frac{m_1}{{\mathbb {a}}_{[w]}\lambda _1} \right) ^{\omega _1} e^{\frac{m_1( t + n )}{{\mathbb {a}}_{[w]}\lambda _1}}\Gamma ( \omega _1 + 1) \Gamma \left( - \omega _1,\frac{m_1( t + n )}{{\mathbb {a}}_{[w]}\lambda _1} \right) . \end{aligned}$$

(38)

1.1.2 Calculation for ${\mathcal {C}}_{\gamma _{d,[w]}}$

Letting $Y= \Vert \mathbf{h }_1^{(K)} \Vert ^2 \Vert \mathbf{h }_2^{(K)} \Vert ^2$, the PDF of Y for the KBFC scheme can be calculated as

$$\begin{aligned} f_Y( y ) = \int \limits _0^{ + \infty } \frac{1}{x} f_{ \left\| \mathbf{h }_2^{(K)} \right\| ^2} \left( \frac{y}{x} \right) f_{ \left\| \mathbf{h }_1^{(K)} \right\| ^2} ( x ) dx. \end{aligned}$$

(39)

Substituting the PDF of $\Vert \mathbf{h }_1^{(K)} \Vert ^2$ and $\Vert \mathbf{h }_2^{(K)} \Vert ^2$ for the KBFC scheme into (39), and with the help of [13, Eq. (3.471.9)], we have

$$\begin{aligned}&f_Y ( y ) = \sum \limits _{n = 0}^{L - K} \sum \limits _{\sum \limits _{i = 0}^{ N_1 m_1 - 1} p_i^{(1)} = K + n - 1} \frac{ 2L! ( - 1 )^n}{ ( K - 1 )! ( L - K )!\Gamma ( N_1 m_1 ) \Gamma ( N_2 m_2 ) } \left( {\begin{array}{c}L -K\\ n\end{array}}\right) \nonumber \\&\quad\times \left( {\begin{array}{c}K + n - 1\\ p_0^{(1)}, \ldots ,p_{ N_1 m_1 - 1}^{(1)}\end{array}}\right) \left( \prod \limits _{i = 0}^{{N_1}{m_1} - 1} \left( \frac{1}{i!} \right) ^{p_i^{(1)}} \right) \frac{m_1 m_2 }{\lambda _1\lambda _2(K + n )^{N_1 m_1 + \omega _{1}-1}} \nonumber \\&\quad\times z^{\frac{N_1m_1 + \omega _{1} + N_2m_2}{2} - 1} K_{N_1m_1 + \omega _{1} - N_2m_2}\left( 2 \sqrt{z} \right) , \end{aligned}$$

(40)

where $z=\frac{m_2m_1( K + n )y}{\lambda _1\lambda _2}$.

Then, ${\mathcal {C}}_{\gamma _{d,[w]}}$ can be calculated as

$$\begin{aligned} {\mathcal {C}}_{\gamma _{d,[w]}} = \frac{1}{2}\int \limits _0^{ + \infty } \log _2 ( 1 + {\mathbb {b}}_{[w]}y ) f_Y( y ). \end{aligned}$$

(41)

Substituting (40) into (41) and expressing $\ln (1+x)=G_{2,2}^{1,2} \left( {x \big \vert _{1,0}^{1,1}} \right)$ [13, Eq. (07.34.03.0456.01)], we have

$$\begin{aligned}&{\mathcal {C}}_{\gamma _{d,[w]}} = \frac{1}{\ln ( 2)} \sum \limits _{n = 0}^{L - K} \sum \limits _{\sum \limits _{i = 0}^{ N_1 m_1 - 1} p_i^{(1)} = K + n - 1}\frac{L!( - 1 )^n}{ ( L - K)! ( K - 1 )!\Gamma ( N_1m_1 )\Gamma ( N_2m_2)} \nonumber \\&\quad\times \frac{1}{( K + n)^{N_1m_1 + \omega _{1}}} \left( {\begin{array}{c}L - K\\ n\end{array}}\right) \left( {\begin{array}{c}K + n - 1\\ p_0^{(1)}, \ldots ,p_{N_1m_1 - 1}^{(1)}\end{array}}\right) \left( \prod \limits _{i = 0}^{{N_1}{m_1} - 1} \left( \frac{1}{i!} \right) ^{p_i^{(1)}} \right) \nonumber \\&\quad\times \int \limits _0^{ + \infty } z^{\frac{N_1m_1 + \omega _{1} + N_2m_2}{2} - 1} {K_{N_1m_1 + \omega _{1} - N_2m_2}}( 2\sqrt{z} ) G_{2,2}^{1,2} \left( \tfrac{{\mathbb {b}}_{[w]} \lambda _1 \lambda _2}{m_1 m_2 (K+n)} z \bigg \vert {\begin{array}{cc} 1, &{} 1 \\ 1, &{} 0 \end{array}} \right) dz. \end{aligned}$$

(42)

The integration in (42) can be solved using [13, Eq. (7.821.3)]; thus, ${\mathcal {C}}_{\gamma _{d,[w]}}$ can be expressed as

$$\begin{aligned}{\mathcal {C}}_{\gamma _{d,[w]}} &= \frac{1}{2\ln ( 2)} \sum \limits _{n = 0}^{L - K} \sum \limits _{\sum \limits _{i = 0}^{ N_1 m_1 - 1} p_i^{(1)} = K + n - 1}\frac{L!( - 1 )^n}{ ( K - 1 )!( L - K)! \Gamma ( N_1m_1 )\Gamma ( N_2m_2)} \nonumber \\&\quad \times \frac{1}{( K + n)^{N_1m_1 + \omega _{1}}} \left( {\begin{array}{c}L - K\\ n\end{array}}\right) \left( {\begin{array}{c}K + n - 1\\ p_0^{(1)}, \ldots ,p_{N_1m_1 - 1}^{(1)}\end{array}}\right) \left( \prod \limits _{i = 0}^{{N_1}{m_1} - 1} \left( \frac{1}{i!} \right) ^{p_i^{(1)}} \right) \nonumber \\&\quad\times G_{4,2}^{1,4} \left( \tfrac{{\mathbb {b}}_{[w]}\lambda _1\lambda _2}{m_1m_2(K + n)} \bigg \vert {\begin{array}{cccc}{-N_1m_1 - \omega _{1} + 1,}&{}{ - N_2m_2 + 1,}&{}{1,}&{}1 \\ {1,}&{}0&{}{}&{}{} \end{array}} \right) . \end{aligned}$$

(43)

1.1.3 Calculation for ${\mathcal {C}}_{\gamma _{t,[w]}}^{ low }$

For the KBFC scheme, ${\mathcal {C}}_{\gamma _{t,[w]}}^{\text {low}}$ can be determined by evaluating ${\mathbb {E}}\{\ln (\gamma _{r,[w]})\} =\ln ({\mathbb {a}}_{[w]}) +{\mathcal {A}}$ and ${\mathbb {E}}\{\ln (\gamma _{d,[w]})\} =\ln ({\mathbb {b}}_{[w]}) +{\mathcal {A}} +{\mathcal {D}}$, where ${\mathcal {A}}={\mathbb {E}}\{\ln ( \Vert {\mathbf {h}}_1^{(K)} \Vert ^2 )\}$ and ${\mathcal {D}}={\mathbb {E}}\{\ln ( \Vert {\mathbf {h}}_2^{(K)} \Vert ^2 )\}$. Using the PDF of $\Vert {\mathbf {h}}_1^{(K)} \Vert ^2$ and $\Vert {\mathbf {h}}_2^{(K)} \Vert ^2$ given in “Outage Probability for the KBFC Scheme” of Appendix 1 , and with the help of [13, Eq. (4.352.1)], ${\mathcal {A}}$ can be expressed as in (26) and ${\mathcal {D}}$ is given by

$$\begin{aligned} {\mathcal {D}} = \psi \left( N_2m_2 \right) - \ln \left( \frac{m_2}{\lambda _2} \right) . \end{aligned}$$

(44)

From (23), ${\mathcal {C}}_{\gamma _{t,[w]}}^{\text {low}}$ can be calculated as

$$\begin{aligned}&{\mathcal {C}}_{\gamma _{t,[w]}}^{\text {low}} = \frac{1}{2}\log _2 \left( 1 + e ^{( \ln ( {\mathbb {a}}_{[w]} ) + {\mathcal {A}} )} + e^ {\left( \ln ( {\mathbb {b}}_{[w]} ) + {\mathcal {A}} + \psi ( N_2m_2 ) - \ln \left( \frac{m_2}{\lambda _2} \right) \right) } \right) . \end{aligned}$$

(45)

Substituting (38), (43), and (45) into (22), the analytical expression for the ergodic capacity of the KBFC scheme can be obtained.

1.2 Ergodic Capacity for the KBSC Scheme

According to the similar steps as in “Ergodic Capacity for the KBFC Scheme” of Appendix 2, and using the PDF and CDF of $\Vert {\mathbf {h}}_1^{(K)} \Vert ^2$ and $\Vert {\mathbf {h}}_2^{(K)} \Vert ^2$ for the KBSC scheme given in “Outage Probability for the KBSC Scheme” of Appendix 1, the analytical expressions of ${\mathcal {C}}_{\gamma _{r,[w]}}$, ${\mathcal {C}}_{\gamma _{d,[w]}}$, and ${\mathcal {C}}_{\gamma _{t,[w]}}^{\text {low}}$ for the KBSC scheme can be respectively expressed as follows:

$$\begin{aligned}&{\mathcal {C}}_{\gamma _{r,[w]}} = \frac{1}{2\ln 2}\sum \limits _{i = 0}^{N_1m_1 - 1} \frac{1}{i!} {\left( \frac{m_1}{{\mathbb {a}}_{[w]}\lambda _1} \right) ^i} e^{\frac{m_1}{{\mathbb {a}}_{[w]}\lambda _1}} \Gamma (i + 1)\Gamma \left( - i,\frac{m_1}{{\mathbb {a}}_{[w]}\lambda _1} \right) , \end{aligned}$$

(46)

$$\begin{aligned}&{\mathcal {C}}_{\gamma _{d,[w]}} = \frac{1}{2\ln 2} \sum \limits _{n = 0}^{L - K} \sum \limits _{\sum \limits _{j = 0}^{N_2m_2 - 1} p_j^{(2)} = K + n - 1} \frac{(- 1 )^n L!}{ ( L - K )! (K - 1)! \Gamma ( N_1m_1) \Gamma ( N_2m_2 ) } \nonumber \\&\quad\quad \times \frac{1}{( K + n)^{N_2m_2 + \omega _2}}\left( {\begin{array}{c}L - K\\ n\end{array}}\right) \left( {\begin{array}{c}K + n - 1\\ p_0^{(2)}, \ldots ,p_{N_2m_2 - 1}^{(2)}\end{array}}\right) \left( \prod \limits _{j = 0}^{N_2m_2 - 1} \left( \frac{1}{j!} \right) ^{p_j^{(2)}}\right) \nonumber \\&\quad\quad \times G_{4,2}^{1,4} \left( \tfrac{{\mathbb {b}}_{[w]}\lambda _1\lambda _2}{m_1m_2( K + n)} \bigg \vert {\begin{array}{cccc}{-N_1m_1 + 1,}&{}{ - N_2m_2 - \omega _2 + 1,}&{}{1,}&{} 1 \\ {1,}&{} 0&{}{}&{}{} \end{array}}\right) , \end{aligned}$$

(47)

$$\begin{aligned}&{\mathcal {C}}_{\gamma _{t,[w]}}^{\text {low}} = \frac{1}{2} \log _2 \left( 1 + e^{\left( \ln ( {\mathbb {a}}_{[w]}) + \psi ( N_1m_1) - \ln \left( \frac{m_1}{\lambda _1} \right) \right) } \right. \nonumber \\&\quad \left. + e^{\left( \ln ( {\mathbb {b}}_{[w]}) + \psi ( N_1m_1) - \ln \left( \frac{m_1}{\lambda _1} \right) + {\mathcal {B}} \right) } \right) , \end{aligned}$$

(48)

where ${\mathcal {B}}={\mathbb {E}}\{\ln ( \Vert {\mathbf {h}}_1^{(K)} \Vert ^2 )\}$ given by (27).

Substituting (46), (47), and (48) into (22), the analytical expression for the ergodic capacity of the KBSC scheme can be obtained.

This ends the proof for Proposition 2.