Cooperative Spectrum Sensing with Distributed/Centralized Relay Selection

Abstract

In this paper, we suggest a new Cooperative Spectrum Sensing (CSS) algorithm using Distributed Relay Selection. Spectrum sensing is performed at Fusion Node (FN) using only the signal from Primary User (PU) when its Signal to Noise Ration (SNR) is larger than threshold $\gamma _{th}$. If SNR of the link between PU and FN is lower than $\gamma _{th}$, a relay is activated using distributed relay selection. All relays compare their SNRs to threshold $\alpha$ and transmit only if the SNR exceeds $\alpha$. If the SNR of all relays are less than $\alpha$, there is an outage event, no relay is selected and only the signal from PU will be used for spectrum sensing. If SNRs of more relays than two relays are larger than $\alpha$, a collision occurs and spectrum sensing uses only the signal from PU. Threshold $\alpha$ is optimized using the Gradient algorithm to yield the largest detection probability. We also compare our results to spectrum sensing with centralized relay selection using opportunistic Amplify and Forward (OAF) or Partial Relay Selection (PRS). In OAF, the end-to-end SNR between PU, relays and FN are sent to the FN to be compared and the relay with largest SNR is activated. In PRS, the SNR at the relays are sent to the FN in order to be compared. Then, the relay with largest SNR of first hop, between PU and relays, is activated. CSS with distributed relay selection is novel and no yet studied.

Appendices

Appendix A

When the SNR at $R_k$ is greater than $\alpha$ and SNRs of all other relay nodes are less than $\alpha$, the SNR at FN is given by [21]

$$\begin{aligned} \Gamma _k=\frac{\Gamma _{max}\Gamma _{R_kFN}}{\Gamma _{max}+\Gamma _{R_kFN}+1} \end{aligned}$$

(37)

where $\Gamma _{max}=max\{\Gamma _{PUR_p}\}$.

We deduce [21]

$$\begin{aligned} \Gamma _k<\Gamma _k^{up}=min(\Gamma _{max},\Gamma _{R_kFN}) \end{aligned}$$

(38)

The CDF of SNR is given by

$$\begin{aligned} P_{\Gamma _k|\Gamma _{PUR_k}>\alpha }(x)>P_{\Gamma _k^{up}|\Gamma _{PUR_k}>\alpha }(x)=1-P(min(\Gamma _{max},\Gamma _{R_kFN})>x|\Gamma _{PUR_k}>\alpha ) \end{aligned}$$

(39)

If $\Gamma _{max}$ and $|\Gamma _{PUR_k}$ are independent we have

$$\begin{aligned} P_{\Gamma _k|\Gamma _{PUR_k}>\alpha }(x)>P_{\Gamma _k^{up}|\Gamma _{PUR_k}>\alpha }(x)=1-P(\Gamma _{max}>x|\Gamma _{PUR_k}>\alpha )P(\Gamma _{R_kFN}>x) \end{aligned}$$

(40)

If $x<\alpha$, $P(\Gamma _{max}>x|\Gamma _{PUR_k}>\alpha )=1$, and we can write

$$\begin{aligned} P_{\Gamma _k|\Gamma _{PUR_k}>\alpha }(x)>P_{\Gamma _k^{up}|\Gamma _{PUR_k}>\alpha }(x)=1-e^{-\frac{x}{\overline{\Gamma }_{R_kFN}}}, x<\alpha \end{aligned}$$

(41)

If $x>\alpha$, we have

$$\begin{aligned}&P_{\Gamma _k|\Gamma _{PUR_k}>\alpha }(x)>P_{\Gamma _k^{up}|\Gamma _{PUR_k}>\alpha }(x)=1-[1-P(\Gamma _{max}<x|\Gamma _{PUR_k}>\alpha )]P(\Gamma _{R_kFN}>x)\nonumber \\&\quad =1-\left[ 1-P(\Gamma _{PUR_k}<x|\Gamma _{PUR_k}>\alpha )\prod _{p=1, p \ne k}^K P(\Gamma _{PUR_p}<x)\right] e^{-\frac{x}{\overline{\Gamma }_{R_kFN}}} \end{aligned}$$

(42)

We have

$$\begin{aligned} P(\Gamma _{PUR_k}<x|\Gamma _{PUR_k}>\alpha )=\int _\alpha ^x \frac{e^{-\frac{(u-\alpha )}{\overline{\Gamma }_{PUR_k}}}}{\overline{\Gamma }_{PUR_k}}du=1-e^{-\frac{(x-\alpha )}{\overline{\Gamma }_{PUR_k}}} \end{aligned}$$

(43)

We denote by $\{i(p)\}_{p=1}^{K-1}=(1,2,\ldots ,k-1,k+1,\ldots ,K)$ to write

$$\begin{aligned} \prod _{p=1, p \ne k}^K P(\Gamma _{PUR_k}<x)=\prod _{p=1}^{K-1} \left( 1-e^{-\frac{x}{\overline{\Gamma }_{PUR_i(p)}}}\right) =\sum _{n=0}^{2^{K-1}} (-1)^{a(n)}e^{-x\sum _{k=1}^{K-1}\frac{b_n(p)}{\overline{\Gamma }_{PUR_i(p)}}} \end{aligned}$$

(44)

where $(b_n(1),b_n(2),\ldots ,b_n(K-1))$ is the binary representation of n and

$$\begin{aligned} a(n)=\sum _{p=1}^{K-1}b_n(p). \end{aligned}$$

(45)

Using (41), (42), (43) and (44), we obtain

$$\begin{aligned} P_{\Gamma _k^{up}|\Gamma _{PUR_k}>\alpha }(x)=\left\{ \begin{array}{l} 1-e^{-\frac{x}{\overline{\Gamma }_{R_kFN}}},\quad x \le \alpha \\ 1-e^{-\frac{x}{\overline{\Gamma }_{R_kFN}}}+\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}e^{-\frac{x}{c_k(n)}}\\ -e^{\frac{\alpha }{\overline{\Gamma }_{PUR_k}}}\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}e^{-\frac{x}{d_k(n)}},\quad x >\alpha \end{array} \right. \end{aligned}$$

(46)

where

$$\begin{aligned} \frac{1}{c_k(n)}= & {} \frac{1}{\overline{\Gamma }_{R_kFN}}+\sum _{p=1}^{K-1}\frac{b_n(p)}{\overline{\Gamma }_{PUR_{i(p)}}} \end{aligned}$$

(47)

$$\begin{aligned} \frac{1}{d_k(n)}= & {} \frac{1}{\overline{\Gamma }_{R_kFN}}+\frac{1}{\overline{\Gamma }_{PUR_k}}+\sum _{p=1}^{K-1}\frac{b_n(p)}{\overline{\Gamma }_{PUR_{i(p)}}} \end{aligned}$$

(48)

By a derivative, we obtain the PDF of SNR

$$\begin{aligned} p_{\Gamma _k^{up}|\Gamma _{PUR_k}>\alpha }(x)=\left\{ \begin{array}{l} \frac{1}{\overline{\Gamma }_{R_kFN}}e^{-\frac{x}{\overline{\Gamma }_{R_kFN}}}=p_1(x),\quad x \le \alpha \\ \frac{1}{\overline{\Gamma }_{R_kFN}}e^{-\frac{x}{\overline{\Gamma }_{R_kFN}}}-\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{1}{c_k(n)}e^{-\frac{x}{c_k(n)}}\\ \quad +\,e^{\frac{\alpha }{\overline{\Gamma }_{PUR_k}}}\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{1}{d_k(n)}e^{-\frac{x}{d_k(n)}}=p_2(x),\quad x >\alpha \end{array} \right. \end{aligned}$$

(49)

Appendix B

Let $Y_k=\Gamma _k+\Gamma _{PUFN}|\Gamma _k>\alpha , \Gamma _{PUFN}<\gamma _{th}$ where $\Gamma _k$ is the SNR of PU-$R_k$-FN link. The PDF of $Y_k$ is written as

$$\begin{aligned}p_{Y_k}(x) &=\int _0^{min(\gamma _{th},x)}p_{\Gamma _{PUFN}|\Gamma _{PUFN}<\gamma _{th}}(u)p_{\Gamma _k|\Gamma _{PUR_k}>\alpha }(x-u)du\nonumber \\&=\int _0^{min(\gamma _{th},x)}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{\Gamma _k|\Gamma _{PUR_k}>\alpha }(x-u)du \end{aligned}$$

(50)

The PDF of an upper bound of $\Gamma _k$, $\Gamma _k^{up}$, is given in “Appendix A”.

From “Appendix A”, we can compute the PDF of $Y_k^{up}=\Gamma _k^{up}+\Gamma _{PUFN}|\Gamma _{PUR_k}>\alpha , \Gamma _{PUFN}<\gamma _{th}$.

Case 1: $\alpha <\gamma _{th}$

• If $0\le x\le \alpha <\gamma _{th}$, from (50), (50) is expressed as

  $$\begin{aligned} p_{Y_k^{up}}(x)=\int _0^{x}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{1}(x-u)du \end{aligned}$$

  (51)

  where $p_1(x)$ is defined in (49).

• If $\alpha \le x\le \gamma _{th}$, we have

  $$\begin{aligned} p_{Y_k^{up}}(x)=\int _0^{x-\alpha }\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{\Gamma _k^{up}}(x-u)du+\int _{x-\alpha }^{x}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{\Gamma _k^{up}}(x-u)du \end{aligned}$$

  (52)

  The first integral is written as

  $$\begin{aligned} \int _0^{x-\alpha }\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{\Gamma _k^{up}}(x-u)du=\int _0^{x-\alpha }\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{T}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_2(x-u)du. \end{aligned}$$

  The second integral is written as

  $$\begin{aligned} \int _{x-\alpha }^x\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{\Gamma _k^{up}}(x-u)du=\int _{x-\alpha }^x\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_1(x-u)du. \end{aligned}$$

  We deduce

  $$\begin{aligned} p_{Y_k^{up}}(x)=\int _0^{x-\alpha }\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_2(x-u)du+\int _{x-\alpha }^x\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_1(x-u)du. \end{aligned}$$

  (53)

• If $\gamma _{th}\le x \le \gamma _{th}+\alpha$, we have

  $$\begin{aligned} p_{Y_k^{up}}(x)=\int _0^{x-\alpha }\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{\Gamma _k^{up}}(x-u)du+\int _{x-\alpha }^{\gamma _{th}}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{\Gamma _k^{up}}(x-u)du \end{aligned}$$

  (54)

  The first integral is equal to

  $$\begin{aligned} \int _0^{x-\alpha }\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{\Gamma _k^{up}}(x-u)du=\int _0^{x-\alpha }\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_2(x-u)du. \end{aligned}$$

  The second integral is equal to

  $$\begin{aligned} \int _{x-\alpha }^{\gamma _{th}}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{\Gamma _k^{up}}(x-u)du=\int _{x-\alpha }^{\gamma _{th}}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_1(x-u)du. \end{aligned}$$

  Therefore, we have

  $$\begin{aligned} p_{Y_k^{up}}(x)=\int _0^{x-\alpha }\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{2}(x-u)du+\int _{x-\alpha }^{\gamma _{th}}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{1}(x-u)du \end{aligned}$$

  (55)

• If $x\ge \alpha +\gamma _{th}$, in (50), we have

$$\begin{aligned} p_{Y_k^{up}}(x)=\int _0^{\gamma _{th}}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{2}(x-u)du \end{aligned}$$

(56)

Using the expressions of $p_1(x)$ and $p_2(x)$ in (49), we obtain

$$\begin{aligned} p_{Y_k^{up}}(x)=\left\{ \begin{array}{l} \frac{e^{-\frac{x}{\overline{\Gamma }_{PUFN}}}-e^{-\frac{x}{\overline{\Gamma }_{R_kFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-\overline{\Gamma }_{R_kFN}\right) },\quad x \le \alpha \\ \frac{\left[ e^{\frac{-x}{\overline{\Gamma }_{PUFN}}}-e^{\frac{-x}{\overline{\Gamma }_{R_kFN}}}\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-\overline{\Gamma }_{R_kFN}\right) }\\ \quad -\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{\left[ e^{\alpha \left( \frac{1}{\overline{\Gamma }_{PUFN}}-\frac{1}{c_k(n)}\right) }e^{\frac{-x}{\overline{\Gamma }_{PUFN}}}-e^{\frac{-x}{c_k(n)}}\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-c_k(n)\right) }\\ \quad +\,e^{\frac{\alpha }{\overline{\Gamma }_{PUR_k}}}\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{\left[ e^{\alpha (\frac{1}{\overline{\Gamma }_{PUFN}}-\frac{1}{d_k(n)})}e^{\frac{-x}{\overline{\Gamma }_{PUFN}}}-e^{\frac{-x}{d_k(n)}}\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-d_k(n)\right) } ,\quad \alpha<x \le \gamma _{th}\\ \frac{e^{\frac{-x}{\overline{\Gamma }_{R_kFN}}}\left( e^{\gamma _{th}\left( \frac{1}{\overline{\Gamma }_{R_kFN}}-\frac{1}{\overline{\Gamma }_{PUFN}}\right) }-1\right) }{(\overline{\Gamma }_{PUFN}-\overline{\Gamma }_{R_kFN})\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) }\\ \quad -\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{\left[ e^{\alpha (\frac{1}{\overline{\Gamma }_{PUFN}}-\frac{1}{c_k(n)})}e^{\frac{-x}{\overline{\Gamma }_{PUFN}}}-e^{\frac{-x}{c_k(n)}}\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-c_k(n)\right) }\\ \quad +\,e^{\frac{\alpha }{\overline{\Gamma }_{PUR_k}}}\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{\left[ e^{\alpha \left( \frac{1}{\overline{\Gamma }_{PUFN}}-\frac{1}{d_k(n)}\right) }e^{\frac{-x}{\overline{\Gamma }_{PUFN}}}-e^{\frac{-x}{d_k(n)}}\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-d_k(n)\right) },\quad \gamma _{th}<x\le \gamma _{th}+\alpha \\ \frac{e^{\frac{-x}{\overline{\Gamma }_{R_kFN}}}\left[ e^{\gamma _{th}(\frac{1}{\overline{\Gamma }_{R_kFN}}-\frac{1}{\overline{\Gamma }_{PUFN}})}-1\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-\overline{\Gamma }_{R_kFN}\right) }\\ \quad -\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{e^{\frac{-x}{c_k(n)}}\left[ e^{\gamma _{th}(\frac{1}{c_k(n)}-\frac{1}{\overline{\Gamma }_{PUFN}})}-1\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-c_k(n)\right) }\\ \quad +\,e^{\frac{\alpha }{\overline{\Gamma }_{PUR_k}}}\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{e^{\frac{-x}{d_k(n)}}\left[ e^{\gamma _{th}(\frac{1}{d_k(n)}-\frac{1}{\overline{\Gamma }_{PUFN}})}-1\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-d_k(n)\right) },\quad x>\alpha +\gamma _{th} \end{array} \right. \end{aligned}$$

(57)

We deduce the DP when $R_k$ is active

$$\begin{aligned}&P_{dk}=\frac{\left[ \overline{\Gamma }_{PUFN}f(2,N,\overline{\Gamma } _{PUFN},\gamma _{th})-\overline{\Gamma }_{R_{k}FN}f(2,N,\overline{\Gamma }_{R_{k}FN},\gamma _{th}) \right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma } _{PUFN}-\overline{\Gamma }_{R_{k}FN}\right) }\nonumber \\&\quad -\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}e^{\alpha (\frac{1}{\overline{\Gamma } _{PUFN}}-\frac{1}{c_{k}(n)})}\overline{\Gamma }_{PUFN}\frac{\left[ g(2,N, \overline{\Gamma }_{PUFN},\alpha )-g(2,N,\overline{\Gamma }_{PUFN},\gamma _{th}+\alpha ) \right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma } _{PUFN}-c_{k}(n)\right) }\nonumber \\&\quad +\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}c_{k}(n)\frac{\left[ g(2,N,c_{k}(n),\alpha )-g(2,N,c_{k}(n),\gamma _{th}+\alpha )\right] }{\left( 1-e^{-\frac{T}{\overline{\Gamma }_{PUFN} }}\right) \left( \overline{\Gamma }_{PUFN}-c_{k}(n)\right) }\nonumber \\&\quad +\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}e^{\alpha (\frac{1}{\overline{\Gamma } _{PUR_{k}}}+\frac{1}{\overline{\Gamma }_{PUFN}}-\frac{1}{d_{k}(n)})} \overline{\Gamma }_{PUFN}\frac{\left[ g(2,N,\overline{\Gamma }_{PUFN},\alpha )-g(2,N,\overline{\Gamma }_{PUFN},\gamma _{th}+\alpha )\right] }{\left( 1-e^{-\frac{\gamma _{th}}{ \overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-d_{k}(n)\right) }\nonumber \\&\quad -\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}e^{\frac{\alpha }{\overline{\Gamma } _{PUR_{k}}}}d_{k}(n)\frac{\left[ g(2,N,d_{k}(n),\alpha )-g(2,N,d_{k}(n),\gamma _{th}+\alpha )\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN} }}\right) \left( \overline{\Gamma }_{PUFN}-d_{k}(n)\right) }\nonumber \\&\quad +\,\frac{(e^{\gamma _{th}(\frac{1}{\overline{\Gamma }_{R_{k}FN}}-\frac{1}{\overline{ \Gamma }_{PUFN}})}-1)\overline{\Gamma }_{R_{k}FN}g(2,N,\overline{\Gamma } _{R_{k}FN},\gamma _{th})}{\left( 1-e^{-\frac{T}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-\overline{\Gamma }_{R_{k}FN}\right) }\nonumber \\&\quad -\,\sum _{n=0}^{2^{K-1}-1}\frac{(-1)^{a(n)}[e^{\gamma _{th}\left( \frac{1}{c_{k}(n)}-\frac{ 1}{\overline{\Gamma }_{PUFN}}\right) }-1]c_{k}(n)g(2,N,c_{k}(n),\gamma _{th}+\alpha )}{ \left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma } _{PUFN}-c_{k}(n)\right) }\nonumber \\&\quad +\,e^{\frac{\alpha }{\overline{\Gamma }_{PUR_{k}}}}\sum _{n=0}^{2^{K-1}-1}\frac{ (-1)^{a(n)}[e^{\gamma _{th}\left( \frac{1}{d_{k}(n)}-\frac{1}{\overline{\Gamma }_{PUFN}} \right) }-1]d_{k}(n)g(2,N,d_{k}(n),\gamma _{th}+\alpha )}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{ \Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-d_{k}(n)\right) } \end{aligned}$$

(58)

Case 2: $\alpha \ge \gamma _{th}$

• If $x\le \gamma _{th} \le \alpha$, (50) becomes

  $$\begin{aligned} p_{Y_k^{up}}(x)=\int _0^{x}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{1}(x-u)du. \end{aligned}$$

  (59)

• If $\gamma _{th}\le x\le \alpha$, we have

  $$\begin{aligned} p_{Y_k^{up}}(x)=\int _0^{T}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{1}(x-u)du. \end{aligned}$$

  (60)

• If $\alpha \le x \le \gamma _{th}+\alpha$, we have

  $$\begin{aligned} p_{Y_k^{up}}(x)=\int _0^{x-\alpha }\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{2}(x-u)du+\int _{x-\alpha }^{\gamma _{th}}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{1}(x-u)du. \end{aligned}$$

  (61)

• If $x\ge \gamma _{th}+\alpha$, (50) becomes

$$\begin{aligned} p_{Y_k^{up}}(x)=\int _0^{\gamma _{th}}\frac{e^{-\frac{u}{\overline{\Gamma }_{PUFN}}}}{\left( 1-e^{-\frac{T}{\overline{\Gamma }_{PUFN}}}\right) \overline{\Gamma }_{PUFN}}p_{2}(x-u)du. \end{aligned}$$

(62)

Using the expressions of $p_1(x)$ and $p_2(x)$ in (49), we obtain

$$\begin{aligned} p_{Y_k^{up}}(x)=\left\{ \begin{array}{l} \frac{e^{-\frac{x}{\overline{\Gamma }_{PUFN}}}-e^{-\frac{x}{\overline{\Gamma }_{R_kFN}}}}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-\overline{\Gamma }_{R_kFN}\right) },\quad x \le \gamma _{th}\\ \frac{e^{-\frac{x}{\overline{\Gamma }_{R_kFN}}}(e^{\gamma _{th}(\frac{1}{\overline{\Gamma }_{R_kFN}}-\frac{1}{\overline{\Gamma }_{PUFN}})}-1)}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-\overline{\Gamma }_{R_kFN}\right) }, \gamma _{th}<x\le \alpha \\ \frac{e^{\frac{-x}{\overline{\Gamma }_{R_kFN}}}\left[ e^{\gamma _{th}(\frac{1}{\overline{\Gamma }_{R_kFN}}-\frac{1}{\overline{\Gamma }_{PUFN}})}-1\right] }{(\overline{\Gamma }_{PUFN}-\overline{\Gamma }_{R_kFN})\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) }\\ \quad -\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{[e^{\alpha \left( \frac{1}{\overline{\Gamma }_{PUFN}}-\frac{1}{c_k(n)}\right) }e^{\frac{-x}{\overline{\Gamma }_{PUFN}}}-e^{\frac{-x}{c_k(n)}}]}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-c_k(n)\right) }\\ \quad +\,e^{\frac{\alpha }{\overline{\Gamma }_{PUR_k}}}\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{\left[ e^{\alpha \left( \frac{1}{\overline{\Gamma }_{PUFN}}-\frac{1}{d_k(n)}\right) }e^{\frac{-x}{\overline{\Gamma }_{PUFN}}}-e^{\frac{-x}{d_k(n)}}\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-d_k(n)\right) } ,\quad \alpha <x\le \gamma _{th}+\alpha \\ \frac{e^{\frac{-x}{\overline{\Gamma }_{R_kFN}}}\left[ e^{\gamma _{th}(\frac{1}{\overline{\Gamma }_{R_kFN}}-\frac{1}{\overline{\Gamma }_{PUFN}})}-1\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-\overline{\Gamma }{R_kFN}\right) }\\ \quad -\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{e^{\frac{-x}{c_k(n)}}\left[ e^{\gamma _{th}(\frac{1}{c_k(n)}-\frac{1}{\overline{\Gamma }_{PUFN}})}-1\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-c_k(n)\right) }\\ \quad +\,e^{\frac{\alpha }{\overline{\Gamma }_{PUR_k}}}\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{e^{\frac{-x}{d_k(n)}}[e^{\gamma _{th}(\frac{1}{d_k(n)}-\frac{1}{\overline{\Gamma }_{PUFN}})}-1]}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-d_k(n)\right) } ,\quad x>\alpha +\gamma _{th} \end{array} \right. \end{aligned}$$

(63)

When $R_k$ amplifies the PU signal to FN, the DP is written as

$$\begin{aligned}&P_{dk}=\frac{\left[ \overline{\Gamma }_{PUFN}f(2,N,\overline{\Gamma } _{PUFN},\gamma _{th})-\overline{\Gamma }_{R_{k}FN}f(2,N,\overline{\Gamma }_{R_{k}FN},\gamma _{th}) \right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma } _{PUFN}-\overline{\Gamma }_{R_{k}FN}\right) }\nonumber \\&\quad +\,\frac{\left( e^{T(\frac{1}{\overline{\Gamma }_{R_{k}FN}}-\frac{1}{\overline{ \Gamma }_{PUFN}})}-1\right) \overline{\Gamma }_{R_{k}FN}g(2,N,\overline{ \Gamma }_{R_{k}FN},\gamma _{th})}{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-\overline{\Gamma }_{R_{k}FN}\right) }\nonumber \\&\quad -\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{e^{\alpha \left( \frac{1}{\overline{ \Gamma }_{PUFN}}-\frac{1}{c_{k}(n)}\right) }\overline{\Gamma }_{PUFN}\left[ g(2,N,\overline{\Gamma }_{PUFN},\alpha )-g(2,N,\overline{\Gamma } _{PUFN},\alpha +\gamma _{th})\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-c_{k}(n)\right) }\nonumber \\&\quad +\,\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{c_{k}(n)\left[ g(2,N,c_{k}(n),\alpha )-g(2,N,c_{k}(n),\alpha +\gamma _{th})\right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN} }}\right) \left( \overline{\Gamma }_{PUFN}-c_{k}(n)\right) }\nonumber \\&\quad +\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{e^{\alpha \left( \frac{1}{\overline{ \Gamma }_{PUR_{k}}}+\frac{1}{\overline{\Gamma }_{PUFN}}-\frac{1}{d_{k}(n)} \right) }\overline{\Gamma }_{PUFN}\left[ g(2,N,\overline{\Gamma } _{PUFN},\alpha )-g(2,N,\overline{\Gamma }_{PUFN},\alpha +\gamma _{th})\right] }{\left( 1-e^{- \frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma }_{PUFN}-d_{k}(n)\right) }\nonumber \\&\quad -\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{e^{\frac{\alpha }{\overline{\Gamma } _{PUR_{k}}}}d_{k}(n)\left[ g(2,N,d_{k}(n),\alpha )-g(2,N,d_{k}(n),\alpha +\gamma _{th}) \right] }{\left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma } _{PUFN}-d_{k}(n)\right) }\nonumber \\&\quad -\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{[e^{T\left( \frac{1}{c_{k}(n)}-\frac{ 1}{\overline{\Gamma }_{PUFN}}\right) }-1]c_{k}(n)g(2,N,c_{k}(n),\alpha +\gamma _{th})}{ \left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma } _{PUFN}-c_{k}(n)\right) }\nonumber \\&\quad +\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{[e^{\gamma _{th}\left( \frac{1}{d_{k}(n)}-\frac{ 1}{\overline{\Gamma }_{PUFN}}\right) }-1]d_{k}(n)g(2,N,d_{k}(n),\alpha +\gamma _{th})}{ \left( 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}}\right) \left( \overline{\Gamma } _{PUFN}-d_{k}(n)\right) } \end{aligned}$$

(64)

The DP in (64) is the same as (58).

Appendix C

PDF of $U^{up}=\Gamma _{PUFN}+\Gamma _{PUR_{sel}FN}^{up}|\Gamma _{PUFN}<\gamma _{th}$.

$$\begin{aligned} p_{U^{up}}(z)=\int _{0}^{min(\gamma _{th},z) }p_{\Gamma _{PUFN}|\Gamma _{PUFN}<\gamma _{th} }(x)p_{\Gamma _{PUR_{sel}FN}^{up}}(z-x)dx \end{aligned}$$

(65)

If $z<\gamma _{th} ,$(19) gives

$$\begin{aligned} p_{U^{up}}(z)= & {} \sum _{1\le k \le K }\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{a _{k,n}}{\overline{\Gamma }^{up}_{PUR_kFN}}\int _{0}^{z}\frac{e^{-\frac{x}{\overline{\Gamma }_{PUFN}}}}{ \overline{\Gamma }_{PUFN}\left[ 1-e^{-\frac{\gamma _{th} }{\overline{\Gamma }_{PUFN}}} \right] }\frac{e^{-\frac{\left( z-x\right) }{a_{k,n}}} }{a_{k,n}}dx \nonumber \\= & {} \sum _{1\le k \le K }\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{a _{k,n}}{\overline{\Gamma }^{up}_{PUR_kFN}}\frac{(e^{-\frac{z}{\overline{\Gamma }_{PUFN}}}-e^{-\frac{z}{a_{k,n}}})}{\left( \overline{\Gamma }_{PUFN}-a_{k,n}\right) \left[ 1-e^{-\frac{\gamma _{th} }{\overline{\Gamma }_{PUFN}}} \right] } \end{aligned}$$

(66)

If $z>\gamma _{th}$, (19) gives

$$\begin{aligned} p_{U^{up}}(z)= & {} \sum _{1\le k \le K }\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{a _{k,n}}{\overline{\Gamma }^{up}_{PUR_kFN}}\int _{0}^{\gamma _{th} }\frac{e^{-\frac{x}{\overline{\Gamma }_{PUFN}}}}{ \overline{\Gamma }_{PUFN}\left[ 1-e^{-\frac{\gamma _{th}}{\overline{\Gamma }_{PUFN}}} \right] }\frac{e^{-\frac{\left( z-x\right) }{a_{k,n}}} }{a_{k,n}}dx \nonumber \\= & {} \sum _{1\le k \le K }\sum _{n=0}^{2^{K-1}-1}(-1)^{a(n)}\frac{a _{k,n}}{\overline{\Gamma }^{up}_{PUR_kFN}}\frac{e^{-\frac{z}{a_{k,n}}}\left[ e^{\gamma _{th} \left( \frac{1}{a_{k,n}}-\frac{1}{\overline{\Gamma }_{PUFN}} \right) }-1\right] }{\left( \overline{\Gamma }_{PUFN}-a _{k,n}\right) \left[ 1-e^{-\frac{\gamma _{th} }{\overline{\Gamma }_{PUFN}}} \right] } \end{aligned}$$

(67)