The effect of Weibull fading channel on cooperative spectrum sensing network using an improved energy detector

Abstract

This paper analyses the performance of proposed cooperative spectrum sensing (CSS) network in Weibull fading environment. First, we have derived the novel analytic expressions for probabilities of missed detection and false alarm in Weibull fading channel, assuming an improved energy detector (IED), selection combining diversity scheme and multiple antennas at each cognitive radio (CRs). Next, performance is analyzed using complementary receiver operating characteristics curves, total error rate, average channel throughput, and network utility function curves for the proposed CSS network. The optimal performance of CSS network is achieved by optimizing the CSS network parameters. The closed form of expressions for the optimum value of number of CRs, arbitrary power of received signal, and detection threshold at each CR are derived using OR-Rule and AND-Rule at fusion center (FC). The average channel throughput and network utility function analysis are evaluated using $k=1+n$ and $k=N-n$ fusion rules at FC. Finally, the impact of several network parameters such as, multiple antennas at each CR (M), number of CRs (N) in CSS network, Weibull fading parameter (V), arbitrary power of received signal (p), and sensing channel SNR (${\bar{\gamma }})$ on the performance of proposed CSS network are investigated using the simulation results. The performance comparison between conventional energy detector and an IED has been highlighted with the simulations.

Appendix

1.1 Expression for threshold value

The closed form of expression for threshold value using the multiple antennas at each CR can be calculated with the help of Eq. (18). Applying logarithm on both sides to Eq. (18) then it reduces to:

$$\begin{aligned}&\Longrightarrow M \ln \left( {1-\exp \left( {-\left\{ {\frac{\uplambda ^{2\big /P}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) } \right) =\ln \left( {1-P_f } \right) \nonumber \\&\Longrightarrow \exp \left( {{\ln (1-P_f )}\big /M} \right) =1-\exp \left( {-\left\{ {\frac{\lambda ^{2\big /P}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) ,\nonumber \\&\Longrightarrow 1-\exp \left( {{\ln (1-P_f )}\big /M} \right) =\exp \left( {-\left\{ {\frac{\lambda ^{2\big /P}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) ,\nonumber \\&\Longrightarrow \ln \left( {1-\exp \left( {{\ln (1-P_f )}\big /M} \right) } \right) =\left( {-\left\{ {\frac{\lambda ^{2\big /P}\Gamma (P)}{\sigma _n^2 }} \right\} ^{C}} \right) , \nonumber \\\end{aligned}$$

(63)

$$\begin{aligned} \lambda= & {} \left( {\frac{\sigma _n^2 }{\Gamma (\mathrm {P})}\left( {\ln \left( {1\big /{\left( {1-\exp \left( {{\ln (1-P_f )}\big /M} \right) } \right) }} \right) } \right) ^{1\big /C}} \right) ^{p\big /2} \end{aligned}$$

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1.2 Optimization of CSS network parameters

1.2.1 A: Optimization of threshold value using $P_f$ and $P_m$ expressions:

An optimum value of threshold ($\lambda _{opt}$) is required to decide the existance of PU with minimum threshold value. The closed form of expression for optimum value of threshold for single antenna case can be calculated by differentiate Eqs. (9 and 17) w.r.t to $\uplambda$:

$$\begin{aligned}&\frac{\partial P_f }{\partial \lambda }+\frac{\partial P_m }{\partial \lambda }=0 \end{aligned}$$

(65)

$$\begin{aligned}&\Longrightarrow -\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C-1}\nonumber \\&\quad \quad \frac{2\lambda ^{\left( {2\big /p} \right) -1}\Gamma (\mathrm {P})}{p\sigma _n^2 }+\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 \left( {1+\gamma } \right) }} \right\} ^{C}} \right) \nonumber \\&\quad \quad C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\left( {1+\gamma } \right) \sigma _n^2 }} \right\} ^{C-1}\frac{2\lambda ^{\left( {2\big /p} \right) -1}\Gamma (\mathrm {P})}{p\sigma _n^2 \left( {1+\gamma } \right) }=0 \nonumber \\&\Longrightarrow \exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) \nonumber \\&\quad \quad =\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 \left( {1+\gamma } \right) }} \right\} ^{C}} \right) \left\{ {\frac{1}{\left( {1+\gamma } \right) }} \right\} ^{C}, \nonumber \\&\Longrightarrow \left( {\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) =\left( {\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 \left( {1+\gamma } \right) }} \right\} ^{C}} \right) \nonumber \\&\quad \quad +\,C\ln \left( {1+\gamma } \right) , \nonumber \\&\Longrightarrow \left( {\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) \left( {1-\left\{ {\frac{1}{\left( {1+\gamma } \right) }} \right\} ^{C}} \right) \nonumber \\&\quad \quad =C\ln \left( {1+\gamma } \right) , \nonumber \\&{\Longrightarrow }\left( {\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) ={C\ln \left( {1+\gamma } \right) }\Bigg /{\left( {1-\left\{ {\frac{1}{\left( {1+\gamma } \right) }} \right\} ^{C}} \right) },\nonumber \\ \end{aligned}$$

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$$\begin{aligned}&\lambda _{opt} =\left( {\frac{\sigma _n^2 }{\Gamma (\mathrm {P})}\left( {{C\ln \left( {1+\gamma } \right) }\Bigg /{\left( {1-\left\{ {\frac{1}{\left( {1+\gamma } \right) }} \right\} ^{C}} \right) }} \right) ^{1\big /C}} \right) ^{P\big /2}\nonumber \\ \end{aligned}$$

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1.2.2 B: Optimization of arbitrary power of received signal (p) using $P_f$ and $P_m$ expressions:

It is also necessary to optimize the arbitrary power of the received signal. The optimum value of p ($p_{opt}$) can be calculated by differentiating Eqs. (9) and (17) w.r.t p for a single antenna case:

$$\begin{aligned}&\frac{\partial P_f }{\partial p}+\frac{\partial P_m }{\partial p}=0 \end{aligned}$$

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$$\begin{aligned}&\frac{\partial P_m }{\partial p}=-M\left[ {1-\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )}} \right\} ^{C}} \right) } \right] ^{M-1}\nonumber \\&\quad \quad \exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )}} \right\} ^{C}} \right) \nonumber \\&\quad \quad C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (P)}{\sigma _n^2 (1+\gamma )}} \right\} ^{C-1}\frac{2}{p^{2}}\frac{\lambda ^{\left( {2\big /p} \right) }\log \lambda \Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )} \end{aligned}$$

(69)

$$\begin{aligned}&\frac{\partial P_f }{\partial p}=M\left[ {1-\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) } \right] ^{M-1}\nonumber \\&\quad \quad \exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) \nonumber \\&\quad \quad C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C-1}\frac{2}{p^{2}}\frac{\lambda ^{\left( {2\big /p} \right) }\log \lambda \Gamma (\mathrm {P})}{\sigma _n^2 } \end{aligned}$$

(70)

adding Eqs. (69) and (70), then make equal to zero.

$$\begin{aligned}&\Longrightarrow \exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) \nonumber \\&\quad \quad =\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )}} \right\} ^{C}} \right) \left\{ {\frac{1}{(1+\gamma )}} \right\} ^{C}, \nonumber \\&\Longrightarrow \left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}\left( {1-\left( {\frac{1}{(1+\gamma )}} \right) ^{C}} \right) =C\log \left( {1+\gamma } \right) , \nonumber \\&p_{opt} =\frac{2\ln \lambda }{\ln \left( {\left\{ {\frac{C\ln \left( {1+\gamma } \right) }{\left[ {1-\left( {\frac{1}{1+\gamma }} \right) ^{C}} \right] }} \right\} ^{1\big /C}\left( {\frac{\sigma _n^2 }{\Gamma (\mathrm {P})}} \right) } \right) } \end{aligned}$$

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1.3 Optimization of network parameters using OR-Rule at FC

1.3.1 a: Optimization of number of CRs $(N_{opt} ):$

The closed form of expression for $N_{opt}$ value for OR-Rule can be calculated using Eq. (33) as follows:

Total error rate: $\hbox {Z}(\hbox {N})=Q_m +Q_f$

$$\begin{aligned} \Delta \hbox {Z}(\hbox {N})= & {} \hbox {Z}(\hbox {N+1})-\hbox {Z}(\hbox {N})\nonumber \\&{\Longrightarrow }1-[(1-P_f )]^{N+1}-1+[(1-P_f )]^{N}\nonumber \\&+\, [P_m ]^{N+1}-[P_m ]^{N}=0, \nonumber \\&{\Longrightarrow }[P_m ]^{N+1}-[(1-P_f )]^{N+1}-[P_m ]^{N}\nonumber \\&+\,[(1-P_f )]^{N}=0, \nonumber \\&{\Longrightarrow }1-\left( {{(1-P_f )}\big /{P_m }} \right) ^{N+1}-\left( {1\big /{(P_m )}} \right) \nonumber \\&+\,\frac{1}{(P_m )}\left( {{(1-P_f )}\big /{P_m }} \right) ^{N}=0, \nonumber \\&{\Longrightarrow }1-\left( {1\big /{(P_m )}} \right) =\left( {{(1-P_f )}\big /{P_m }} \right) ^{N+1}\nonumber \\&-\,\frac{1}{(P_m )}\left( {{(1-P_f )}\big /{P_m }} \right) ^{N}, \end{aligned}$$

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after simplification, the above expression reduces to

$$\begin{aligned} {\Longrightarrow }\left( {{\left( {1-P_m } \right) }\big /{P_f }} \right) =\left( {{\left( {1-P_f } \right) }\big /{P_m }} \right) ^{N}, \end{aligned}$$

applying logarithm on both sides, finally, the closed form of expression for $N_{opt}$ using OR-Rule at FC is

$$\begin{aligned} N_{opt} \quad = \quad \left\lceil {\frac{\ln \left( {\frac{1-P_m }{P_f }} \right) }{\ln \left( {\frac{1-P_f }{P_m }} \right) }} \right\rceil \end{aligned}$$

(73)

1.3.2 b: Optimization of threshold value $\left( {\lambda _{opt} } \right)$

The closed form of expression for $\lambda _{opt}$ value can be calculated by differentiating Eq. (33) w.r.t to $\uplambda$, and equating to zero.

$$\begin{aligned}&\frac{\partial Q_m }{\partial \lambda }+\frac{\partial Q_f }{\partial \lambda }=0 \end{aligned}$$

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$$\begin{aligned}&{\Longrightarrow }N\left( {1-P_f } \right) ^{N-1}\frac{\partial P_f }{\partial \lambda }+N\left( {P_m } \right) ^{N-1}\frac{\partial P_m }{\partial \lambda }=0 \end{aligned}$$

(75)

For a single antenna case ($M=1$), ${\partial P_f }\big /{\partial \lambda }$ and ${\partial P_m }\big /{\partial \lambda }$ expressions are given in Eq. (24), substituting Eqs. (24) in (75):

$$\begin{aligned} \begin{array}{l} \Longrightarrow \left( {1-P_f } \right) ^{N-1}\left( -\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) \right. \\ \quad \left. C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C-1}\frac{2\lambda ^{\left( {2\big /p} \right) -1}\Gamma (\mathrm {P})}{p\sigma _n^2 } \right) \\ \quad +\,N\left( {P_m } \right) ^{N-1}\left( \exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 \left( {1+\gamma } \right) }} \right\} ^{C}} \right) \right. \\ \quad \left. C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\left( {1+\gamma } \right) \sigma _n^2 }} \right\} ^{C-1}\frac{2\lambda ^{\left( {2\big /p} \right) -1}\Gamma (\mathrm {P})}{p\sigma _n^2 \left( {1+\gamma } \right) } \right) =0, \\ \end{array} \end{aligned}$$

after some algebric simplifications, above equation reduces to

$$\begin{aligned}&\Longrightarrow \left( {1-P_f } \right) ^{N-1}\left( {\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) } \right) \\&\quad \qquad =\left( {P_m } \right) ^{N-1}\left( {\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 \left( {1+\gamma } \right) }} \right\} ^{C}} \right) } \right) \left( {\frac{1}{1+\gamma }} \right) ^{C}, \end{aligned}$$

applying logarithm on both sides and solving algebric expressions, the above expression reduces to

$$\begin{aligned} \lambda _{opt}= & {} \left( \frac{\sigma _n^2 }{\Gamma (\mathrm {P})}\left( \left( {N-1} \right) \left( {\ln \left( {\frac{1-P_f }{P_m }} \right) } \right) \right. \right. \nonumber \\&\left. \left. +\,C\ln \left( {1+\gamma } \right) \Bigg /{\left( {1-\left\{ {\frac{1}{\left( {1+\gamma } \right) }} \right\} ^{C}} \right) } \right) ^{1\big /C} \right) ^{P\big /2} \end{aligned}$$

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1.3.3 c: Optimization of arbitrary power of the received siganl $\left( {p_{opt} } \right)$

The closed form of expression for $p_{opt}$ can be calculated by differentiating Eq. (33) w.r.t to p,

$$\begin{aligned}&\frac{\partial Q_m }{\partial p}+\frac{\partial Q_f }{\partial p}=0 \end{aligned}$$

(77)

$$\begin{aligned}&\Longrightarrow \left( {1-P_f } \right) ^{N-1}\frac{\partial P_f }{\partial p}+\left( {P_m } \right) ^{N-1}\frac{\partial P_m }{\partial p}=0 \end{aligned}$$

(78)

For a single antenna case ($M=1$), ${\partial P_f }\big /{\partial p}$ and ${\partial P_m }\big /{\partial p}$ expressions are given in Eqs. (28) and (29), substituting these equations in Eq. (78):

$$\begin{aligned}&\Longrightarrow {N}\left( {1-P_f } \right) ^{N-1}\left( -M\left[ {1-\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) } \right] ^{M-1}\right. \\&\quad \exp \left. \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C-1}\frac{2}{p^{2}}\frac{\lambda ^{\left( {2\big /p} \right) }\log \lambda \Gamma (\mathrm {P})}{\sigma _n^2 } \right) \\&+\,N\left( {P_m } \right) ^{N-1}\left( -M\left[ {1-\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )}} \right\} ^{C}} \right) } \right] ^{M-1}\right. \\&\quad \left. \exp \left( \! {-\left\{ \! {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )}} \!\right\} ^{C}} \right) C\left\{ \! {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )}} \right\} ^{C-1}\!\!\frac{2}{p^{2}}\frac{\lambda ^{\left( {2\big /p} \!\right) }\log \lambda \Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )} \right) \!=\!0 , \end{aligned}$$

after some algebric simplifications, above equation reduces to

$$\begin{aligned}&\Longrightarrow \left( {1-P_f } \right) ^{N-1}\left( {\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) } \right) \\&\quad =\left( {P_m } \right) ^{N-1}\left( {\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 \left( {1+\gamma } \right) }} \right\} ^{C}} \right) } \right) \left( {\frac{1}{1+\gamma }} \right) ^{C}, \end{aligned}$$

applying logarithm on both sides and solving algebric expressions, the above expression reduces to

$$\begin{aligned} \Longrightarrow \frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }=\frac{\left( {N-1} \right) \left( {\ln \left( {\frac{1-P_f }{P_m }} \right) } \right) \,+\,C\ln \left( {1+\gamma } \right) }{\left( {1-\left( {\frac{1}{1+\gamma }} \right) ^{C}} \right) },\nonumber \\ \end{aligned}$$

apply logarithm on both sides, after simplification, the closed form of expression for $p_{opt}$ is:

$$\begin{aligned} p_{opt} =\frac{2\ln \lambda }{\ln \left( {\left( {\frac{\left( {N-1} \right) \left( {\ln \left( {\frac{1-P_f }{P_m }} \right) } \right) \,+\,C\ln \left( {1+\gamma } \right) }{\left( {1-\left( {\frac{1}{1+\gamma }} \right) ^{C}} \right) }} \right) ^{1\big /C}\frac{\sigma _n^2 }{\Gamma (\mathrm {P})}} \right) } \end{aligned}$$

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1.4 Optimization of network parameters using AND-Rule at FC

1.4.1 a: Optimization of threshold value $\left( {\lambda _{opt} } \right)$

The closed form of expression for $\lambda _{opt}$ value can be calculated using Eqs. (24), (35), (38) and (39):

$$\begin{aligned} \begin{aligned}&\Longrightarrow {N}\left( {P_f } \right) ^{N-1}\left( -\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) \right. \\ {}&\quad \left. C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C-1}\frac{2\lambda ^{\left( {2\big /p} \right) -1}\Gamma (\mathrm {P})}{p\sigma _n^2 } \right) \\&\quad +\,N\left( {1-P_m } \right) ^{N-1}\left( \exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 \left( {1+\gamma } \right) }} \right\} ^{C}} \right) \right. \\ {}&\quad \left. C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\left( {1+\gamma } \right) \sigma _n^2 }} \right\} ^{C-1}\frac{2\lambda ^{\left( {2\big /p} \right) -1}\Gamma (\mathrm {P})}{p\sigma _n^2 \left( {1+\gamma } \right) } \right) =0 \\ \end{aligned} \end{aligned}$$

(80)

after some algebric simplifications, above equation reduces to

$$\begin{aligned}&\Longrightarrow \left( {P_f } \right) ^{N-1}\left( {\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) } \right) \\&\quad =\left( {1-P_m } \right) ^{N-1}\left( {\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 \left( {1+\gamma } \right) }} \right\} ^{C}} \right) } \right) \left( {\frac{1}{1+\gamma }} \right) ^{C}, \end{aligned}$$

applying logarithm on both sides and solving algebric expressions, the above expression reduces to

$$\begin{aligned}&\lambda _{opt} =\left( \frac{\sigma _n^2 }{\Gamma (\mathrm {P})}\left( \left( {N-1} \right) \left( {\ln \left( {\frac{P_f }{1-P_m }} \right) } \right) \right. \right. \nonumber \\&\quad \left. \left. +\,C\ln \left( {1+\gamma } \right) \Bigg /{\left( {1-\left\{ {\frac{1}{\left( {1+\gamma } \right) }} \right\} ^{C}} \right) } \right) ^{1\big /C} \right) ^{P\big /2} \end{aligned}$$

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1.4.2 b: Optimization of arbitrary power of the received siganl $\left( {p_{opt} } \right)$

The closed form of expression for $p_{opt}$ value can be calculated using Eqs. (35), (42), (43), (28) and (29):

$$\begin{aligned}&\Longrightarrow {N}\left( {P_f } \right) ^{N-1}\left( -M\left[ {1-\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) } \right] ^{M-1}\right. \nonumber \\&\quad \left. \exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C-1}\frac{2}{p^{2}}\frac{\lambda ^{\left( {2\big /p} \right) }\log \lambda \Gamma (\mathrm {P})}{\sigma _n^2 } \right) \nonumber \\&+\,N\left( {1-P_m } \right) ^{N-1}\left( -M\left[ {1-\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )}} \right\} ^{C}} \right) } \right] ^{M-1}\right. \nonumber \\&\quad \left. \exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )}} \right\} ^{C}} \right) C\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )}} \right\} ^{C-1}\frac{2}{p^{2}}\frac{\lambda ^{\left( {2\big /p} \right) }\log \lambda \Gamma (\mathrm {P})}{\sigma _n^2 (1+\gamma )} \right) =0\nonumber \\ \end{aligned}$$

(82)

after some algebric simplifications, above equation reduces to

$$\begin{aligned}&\Longrightarrow \left( {P_f } \right) ^{N-1}\left( {\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 }} \right\} ^{C}} \right) } \right) \\&\quad =\left( {1-P_m } \right) ^{N-1}\left( {\exp \left( {-\left\{ {\frac{\lambda ^{2\big /p}\Gamma (\mathrm {P})}{\sigma _n^2 \left( {1+\gamma } \right) }} \right\} ^{C}} \right) } \right) \left( {\frac{1}{1+\gamma }} \right) ^{C}, \end{aligned}$$

applying logarithm on both sides and solving algebric expressions, the above expression reduces to

$$\begin{aligned} p_{opt} =\frac{2\ln \lambda }{\ln \left( {\left( {\frac{\left( {N-1} \right) \left( {\ln \left( {\frac{P_f }{1-P_m }} \right) } \right) \,+\,C\ln \left( {1+\gamma } \right) }{\left( {1-\left( {\frac{1}{1+\gamma }} \right) ^{C}} \right) }} \right) ^{1\big /C}\frac{\sigma _n^2 }{\Gamma (\mathrm {P})}} \right) }\nonumber \\ \end{aligned}$$

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