In this section, we study the performance of primary and secondary networks. For the primary networks, we consider the interference probability. For the secondary networks, we derive the system outage probability and its asymptotic expression at high SNR regime.
Primary networks: interference probability analysis
The interference probability is defined as the probability that the interference from secondary transmitters received at the primary transmitters is greater than the interference threshold, Ip. Considering three communications phases for the secondary networks, we may write the primary interference probability using the theorem of total probability as follows:
$$\begin{array}{*{20}l} {P_{I}} =& \Pr \left({{P_{\mathrm{A}}}{{\left| {{h_{{\text{AP}}}}} \right|}^{2}} > {{{\widetilde I}_{\mathrm{p}}}}} \right) \\ &+ \Pr\left({{P_{\mathrm{A}}}{{\left| {{h_{{\text{AP}}}}} \right|}^{2}} \le {{\widetilde I}_{\mathrm{p}}}} \right) \Pr \left({{P_{\mathrm{B}}}{{\left| {{h_{{\text{BP}}}}} \right|}^{2}} > {{{\widetilde I}_{\mathrm{p}}}}} \right) \\ &+ \Pr \left({{P_{\mathrm{A}}}{{\left| {{h_{{\text{AP}}}}} \right|}^{2}} \le {{\widetilde I}_{\mathrm{p}}}} \right)\Pr \left({{P_{\mathrm{B}}}{{\left| {{h_{{\text{BP}}}}} \right|}^{2}} \le {{\widetilde I}_{\mathrm{p}}}} \right)\\ &\times \Pr \left({{P_{\mathrm{R}}}{{\left| {{h_{{\text{RP}}}}} \right|}^{2}} > {{\widetilde I}_{\mathrm{p}}}} \right). \end{array} $$
(15)
Using the result in [29], we straightforwardly arrive at
$$\begin{array}{*{20}l} {P_{I}} = 0.875, \end{array} $$
(16)
where \(\Pr \left ({\frac {{{{\left | {{h_{{\text {XP}}}}} \right |}^{2}}}}{{{{| {{{\hat h}_{{\text {XP}}}}} |}^{2}}}} > 1} \right) = \frac {1}{2} \) is for Rayleigh fading channels.
To reduce P
I
, we adopt the back-off technique, i.e., reducing the secondary transmit power to obtain acceptable interference. Denoting δX with 0≤δX≤1 as the back-off power control coefficient of node X, the transmit power of node X can be rewritten as
$$\begin{array}{*{20}l} {\bar P_{\mathrm{X}}} = {\delta_{\mathrm{X}}}{P_{\mathrm{X}}} = \frac{{{\delta_{\mathrm{X}}}{{\widetilde I}_{\mathrm{p}}}}}{{{{| {{{\hat h}_{{\text{XP}}}}} |}^{2}}}}. \end{array} $$
(17)
Applying Lemma 1 in [29], after tedious manipulations, we finally obtain the closed-form expression for the interference probability as
$$\begin{array}{*{20}l} {}{P_{I}} &= \frac{{1 + {\varphi_{\mathrm{A}}}}}{2} + \frac{{1 + {\varphi_{\mathrm{B}}}}}{2}\frac{{1 - {\varphi_{\mathrm{A}}}}}{2}\, + \frac{{1 + {\varphi_{\mathrm{R}}}}}{2}\frac{{1 - {\varphi_{\mathrm{A}}}}}{2}\frac{{1 - {\varphi_{\mathrm{B}}}}}{2}\\ &= 1 - \left(\frac{{1 - {\varphi_{\mathrm{A}}}}}{2}\right) \left(\frac{{1 - {\varphi_{\mathrm{B}}}}}{2}\right) \left(\frac{{1 - {\varphi_{\mathrm{R}}}}}{2} \right), \end{array} $$
(18)
where \({\varphi _{\mathrm {X}}} = \frac {{{\delta _{\mathrm {X}}} - 1}}{{\sqrt {{{\left ({{\delta _{\mathrm {X}}} + 1} \right)}^{2}} - 4{\rho ^{2}}{\delta _{\mathrm {X}}}} }}\) with X∈{A, B, R}.
Applying the arithmetic-geometric inequality ([34], Eq. (11.116)) for three positive numbers \(\frac {{1 - {\varphi _{\mathrm {A}}}}}{2}\), \(\frac {{1 - {\varphi _{\mathrm {B}}}}}{2}\), and \(\frac {{1 - {\varphi _{\mathrm {R}}}}}{2}\) in (18), we have
$$\begin{array}{*{20}l} {}\left(\!\frac{{1 - {\varphi_{\mathrm{A}}}}}{2}\!\right)\!\! \left(\!\frac{{1 - {\varphi_{\mathrm{B}}}}}{2}\!\right) \!\!\left(\!\frac{{1 - {\varphi_{\mathrm{R}}}}}{2} \!\right) \!\le \!{\left(\! {\frac{{\frac{{1 - {\varphi_{\mathrm{A}}}}}{2} \,+\, \frac{{1 - {\varphi_{\mathrm{B}}}}}{2} \,+\, \frac{{1 - {\varphi_{\mathrm{R}}}}}{2}}}{3}} \!\right)^{3}}. \end{array} $$
(19)
From (19), it appears that the interference probability can achieve its minimum if and only if δA=δB=δR=δ, where \(\varphi = \frac {{\delta - 1}}{{\sqrt {{{\left ({\delta + 1} \right)}^{2}} - 4{\rho ^{2}}\delta } }}\).
The minimum of P
I
may be given as
$$\begin{array}{*{20}l} {P_{I}} = 1 - {\left({\frac{{1 - \varphi }}{2}} \right)^{3}}, \end{array} $$
(20)
for an equi-correlation coefficient ρ in all interference channels.
Although the back-off power control technique is useful in obtaining satisfactory P
I
, it also degrades the secondary network performance. Thus, it is necessary to calculate the back-off power control peak coefficient for a given P
I
. Using the analysis method as in [29], we can find the back-off power control peak coefficient as follows:
$$\begin{array}{*{20}l} {}{\delta_{\max }} = \frac{{1 + {\kappa^{2}} - 2{\kappa^{2}}{\rho^{2}} - 2\sqrt {{\kappa^{2}} - {\kappa^{2}}{\rho^{2}} - {\kappa^{4}}{\rho^{2}} + {\kappa^{4}}{\rho^{2}}} }}{{1 - {\kappa^{2}}}}, \end{array} $$
(21)
where \(\kappa = 2\sqrt [3]{{1 - {P_{I}}}} - 1\).
Secondary two-way relaying networks
In this section, the performance of cognitive underlay two-way relaying networks over Rayleigh fading channels is studied starting with the instantaneous SNRs at the receiving nodes using the back-off coefficient. In particular, we obtain the SNR at R in the first and second phase, respectively, as follows:
$$\begin{array}{*{20}l} {\gamma^{\prime}_{{\text{AR}}}} = {\delta_{\mathrm{A}}}{\gamma_{{\text{AR}}}}, \end{array} $$
(22)
and
$$\begin{array}{*{20}l} {\gamma^{\prime}_{{\text{BR}}}} = {\delta_{\mathrm{B}}}{\gamma_{{\text{BR}}}}. \end{array} $$
(23)
For the third phase, we have the SNRs at A and B, respectively, as
$$\begin{array}{*{20}l} {\gamma^{\prime}_{{\text{RA}}}} = {\delta_{\mathrm{R}}}{\gamma_{{\text{RA}}}}, \end{array} $$
(24)
and
$$\begin{array}{*{20}l} {\gamma^{\prime}_{{\text{RB}}}} = {\delta_{\mathrm{R}}}{\gamma_{{\text{RB}}}}. \end{array} $$
(25)
For two-way DF relaying, the secondary system outage probability can be written as
$$\begin{array}{*{20}l} \text{OP} = 1 - Pr\left({{{\gamma^{\prime}}_{{\text{AR}}}} > {\gamma_{{\text{th}}}}}, {{{\gamma^{\prime}}_{{\text{BR}}}} > {\gamma_{{\text{th}}}}},{{\gamma_{\mathrm{e}}} > {\gamma_{{\text{th}}}}} \right), \end{array} $$
(26)
where γe is defined as
$$\begin{array}{*{20}l} {\gamma_{\mathrm{e}}} = \min \left\{ {{{\gamma^{\prime}}_{{\text{RA}}}},{{\gamma^{\prime}}_{{\text{RB}}}}} \right\} \end{array} $$
(27)
and γth is the given SNR threshold.
Assuming that γ′AR, γ′BR, and γe are independent of each other, we may write
$$\begin{array}{*{20}l} {}\text{OP} =& 1 - \Pr\left({{{\gamma^{\prime}}_{{\text{AR}}}} > {\gamma_{{\text{th}}}}} \right)\Pr \left({{{\gamma^{\prime}}_{{\text{BR}}}} > {\gamma_{{\text{th}}}}} \right)\Pr \left({{\gamma_{\mathrm{e}}} > {\gamma_{{\text{th}}}}} \right)\\ =& 1 - \left[ {1 - {F_{{{\gamma^{\prime}}_{{\text{AR}}}}}}\left({{\gamma_{{\text{th}}}}} \right)} \right]\left[ {1 - {F_{{{\gamma^{\prime}}_{{\text{BR}}}}}}\left({{\gamma_{{\text{th}}}}} \right)} \right]\\ &\,\,\,\,\,\,\,\,\times\left[ {1 - {F_{{\gamma_{\mathrm{e}}}}}\left({{\gamma_{{\text{th}}}}} \right)} \right], \end{array} $$
(28)
where F
Z
(z) is the CDF of Z.
In order to derive the exact expression of the system outage probability, we consider the following Lemma.
Lemma 1
Let X and Y are exponentially distributed random variables with means λX and λY, respectively, and Z is a random variable defined as \(Z = \frac {X}{{\text {PY} + 1}}\), where \({\mathrm {P}} > 0,\,\,{\mathrm {P}} \in {\mathbb {R}}\). The CDF of Z can be given as
$$\begin{array}{*{20}l} {F_{Z}}\left(z \right) = 1 - \frac{{{\lambda_{X}}}}{{{\mathrm{P}}z{\lambda_{Y}} + {\lambda_{X}}}}{e^{- \frac{z}{{{\lambda_{X}}}}}}. \end{array} $$
(29)
Proof
We have
$$\begin{array}{*{20}l} {F_{Z}}\left(z \right) &= \Pr \left({\frac{X}{{{\text{PY}} + 1}} < z} \right)\\ &= \int\limits_{0}^{\infty} {\Pr \left({X < z + {\mathrm{P}}zy} \right){f_{Y}}\left(y \right)dy} \\ &= 1 - \int\limits_{0}^{\infty} {{e^{- \frac{{z + {\mathrm{P}}zy}}{{{\lambda_{X}}}}}}\frac{1}{{{\lambda_{Y}}}}{e^{- \frac{y}{{{\lambda_{Y}}}}}}dy}\\ &= 1 - \frac{1}{{{\lambda_{Y}}}}{e^{- \frac{z}{{{\lambda_{X}}}}}}\int\limits_{0}^{\infty} {{e^{- \left({\frac{{{\mathrm{P}}z}}{{{\lambda_{X}}}} + \frac{1}{{{\lambda_{Y}}}}} \right)y}}dy}. \end{array} $$
(30)
Applying ([34], eq. (3.31011)), we obtain the CDF of Z as (29). □
Then, the exact closed-form expression for the system OP is stated in Theorem 1.
Theorem 1
For a given ρ, the closed-form expression for the system outage probability over Rayleigh fading channels is
$$\begin{array}{*{20}l} \text{OP} =& 1 - \frac{{{\delta_{\mathrm{A}}}{{I}_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}{{{\gamma_{{\text{th}}}}P{\lambda_{{\text{PR}}}}{\lambda_{{\text{AP}}}}}}{e^{{m_{1}}{k_{1}}}}\text{Ei}\left({ - {m_{1}}{k_{1}}} \right)\\ &\quad\times \frac{{{\delta_{\mathrm{B}}}{I_{\mathrm{p}}}{\lambda_{{\text{BR}}}}}}{{{\gamma_{{\text{th}}}}P{\lambda_{{\text{PR}}}}{\lambda_{{\text{BP}}}}}}{e^{{m_{2}}{k_{2}}}}\text{Ei}\left({ - {m_{2}}{k_{2}}} \right) \frac{\psi }{{{\lambda_{{\text{RP}}}}\left({{x_{2}} - {x_{1}}} \right)}} \\ & \quad\times \left[ - {e^{{x_{1}}\left({\phi + \frac{1}{{{\lambda_{{\text{RP}}}}}}} \right)}}\text{Ei}\left(- {x_{1}}\left({\phi + \frac{1}{{{\lambda_{{\text{RP}}}}}}} \right) \right)\right.\\ &\quad+\left. {e^{{x_{2}}\left({\phi + \frac{1}{{{\lambda_{{\text{RP}}}}}}} \right)}}\text{Ei}\left(- {x_{2}}\left({\phi + \frac{1}{{{\lambda_{{\text{RP}}}}}}} \right) \right) \right], \end{array} $$
(31)
where Ei(·) is the exponential integral [27, Eq. (8.211)], \({m_{1}} = \frac {{{\delta _{\mathrm {A}}}{I_{\mathrm {p}}}{\lambda _{{\text {AR}}}}}}{{{\gamma _{{\text {th}}}}P{\lambda _{{\text {PR}}}}}}\), \({m_{2}} = \frac {{{\delta _{\mathrm {B}}}{I_{\mathrm {p}}}{\lambda _{{\text {BR}}}}}}{{{\gamma _{{\text {th}}}}P{\lambda _{{\text {PR}}}}}}\), \({k_{1}} = \frac {{{\gamma _{{\text {th}}}}}}{{{\delta _{\mathrm {A}}}{I_{\mathrm {p}}}{\lambda _{{\text {AR}}}}}} + \frac {1}{{{\lambda _{{\text {AP}}}}}}\), \({k_{2}} = \frac {{{\gamma _{{\text {th}}}}}}{{{\delta _{\mathrm {B}}}{I_{\mathrm {p}}}{\lambda _{{\text {BR}}}}}} + \frac {1}{{{\lambda _{{\text {BP}}}}}}\), \({x_{1}} = \frac {{{\delta _{\mathrm {R}}}{I_{\mathrm {p}}}{\lambda _{{\text {RA}}}}}}{{{\gamma _{{\text {th}}}}P{\lambda _{{\text {PA}}}}}}\), \({x_{2}} = \frac {{{\delta _{\mathrm {R}}}{I_{\mathrm {p}}}{\lambda _{{\text {RB}}}}}}{{{\gamma _{{\text {th}}}}P{\lambda _{{\text {PB}}}}}}\), \(\phi = \frac {{{\gamma _{{\text {th}}}}}}{{{\delta _{\mathrm {R}}}{I_{\mathrm {p}}}{\lambda _{{\text {RA}}}}}} + \frac {{{\gamma _{{\text {th}}}}}}{{{\delta _{\mathrm {R}}}{I_{\mathrm {p}}}{\lambda _{{\text {RB}}}}}}\), and \(\psi = \frac {{\delta _{\mathrm {R}}^{2}I_{\mathrm {p}}^{2}{\lambda _{{\text {RA}}}}{\lambda _{{\text {RB}}}}}}{{\gamma _{{\text {th}}}^{2}{P^{2}}{\lambda _{{\text {PA}}}}{\lambda _{{\text {PB}}}}}}\).
Proof
From (26), we can see that the explicit expression for the system outage probability in (28) can be defined after determining the CDF of \({\gamma ^{\prime }_{{\text {AR}}}}\), \({\gamma ^{\prime }_{{\text {BR}}}}\), and γe.
We first consider the CDF of \({\gamma ^{\prime }_{{\text {AR}}}}\). Using the conditional probability, we can write the CDF of \({\gamma ^{\prime }_{{\text {AR}}}}\) as
$$\begin{array}{*{20}l} F_{\gamma^{\prime}_{\text{AR}}}(\gamma_{\text{th}}) = \int\limits_{0}^{\infty} F_{{\left. \gamma^{\prime}_{\text{AR}} \right|}{|h_{\text{AP}}|^{2}}}(\gamma_{\text{th}}|x) f_{|h_{\text{AP}}|^{2}}(x)dx \end{array} $$
(32)
where \(F_{{\left. \gamma ^{\prime }_{\text {AR}} \right |} |h_{\text {AP}}|^{2}}(\gamma _{\text {th}}|x)\) is the CDF of \({\gamma ^{\prime }_{{\text {AR}}}}\) conditioned on |hAP|2 and \(f_{|h_{\text {AP}}|^{2}}(x)\) is the PDF of |hAP|2.
Recalling (10) and (22), we can write
$$\begin{array}{*{20}l} F_{{\left. \gamma^{\prime}_{\text{AR}} \right|} |h_{\text{AP}}|^{2}}(\gamma_{\text{th}}|x) = \Pr \left({\frac{{{{\left| {{h_{{\text{AR}}}}} \right|}^{2}}}}{{1 + P{{\left| {{h_{{\text{PR}}}}} \right|}^{2}}}} < \frac{{x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{A}}}{I_{\mathrm{P}}}}}} \right), \end{array} $$
(33)
where \({I_{\mathrm {P}}} = {\tilde I_{\mathrm {p}}}/{N_{0}}\) and P=PPU−Tx/N0.
Applying the result of Lemma 1, where X,Y, and z correspond |hAR|2, |hPR|2, and \( \frac {{x{\gamma _{{\text {th}}}}}}{{{\delta _{\mathrm {A}}}{I_{\mathrm {P}}}}}\), respectively, we have
$$\begin{array}{*{20}l} {F_{{{\gamma^{\prime}}_{{\text{AR}}}}\left| {{{\left| {{{\hat h}_{{\text{AP}}}}} \right|}^{2}}} \right.}}\left({{\gamma_{{\text{th}}}}\left| x \right.} \right) = 1 &- \frac{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}{{{\gamma_{{\text{th}}}}P{\lambda_{{\text{PR}}}}\left({x + \frac{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}{{{\gamma_{{\text{th}}}}P{\lambda_{{\text{PR}}}}}}} \right)}}\\ &\times {e^{- \frac{{x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}}}. \end{array} $$
(34)
Substituting (34) and (2) into (35), we have
$$\begin{array}{*{20}l} {F_{{{\gamma^{\prime}}_{{\text{AR}}}}}}\left({{\gamma_{{\text{th}}}}} \right) =& 1 - \int\limits_{0}^{\infty} \frac{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}{{{\gamma_{{\text{th}}}}P{\lambda_{{\text{PR}}}}{\lambda_{{\text{AP}}}}\left({x + \frac{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}{{{\gamma_{{\text{th}}}}P{\lambda_{{\text{PR}}}}}}} \right)}}\\ &\quad\times {e^{- \left({\frac{{{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}} + \frac{1}{{{\lambda_{{\text{AP}}}}}}} \right)x}}dx. \end{array} $$
(35)
Using ([34], Eq. (3.352.4)), we obtain
$$\begin{array}{*{20}l} {F_{{{\gamma^{\prime}}_{{\text{AR}}}}}}\left({{\gamma_{{\text{th}}}}} \right) = 1 + \frac{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}{{{\gamma_{{\text{th}}}}P{\lambda_{{\text{PR}}}}{\lambda_{{\text{AP}}}}}}{e^{{m_{1}}{k_{1}}}}{\text{Ei}}\left({ - {m_{1}}{k_{1}}} \right), \end{array} $$
(36)
where \({m_{1}} = \frac {{{\delta _{\mathrm {A}}}{I_{\mathrm {p}}}{\lambda _{{\text {AR}}}}}}{{{\gamma _{{\text {th}}}}P{\lambda _{{\text {PR}}}}}}\) and \({k_{1}} = \frac {{{\gamma _{{\text {th}}}}}}{{{\delta _{\mathrm {A}}}{I_{\mathrm {p}}}{\lambda _{{\text {AR}}}}}} + \frac {1}{{{\lambda _{{\text {AP}}}}}}\).
Here, we note that \({\gamma ^{\prime }_{{\text {AR}}}}\) and \({\gamma ^{\prime }_{{\text {BR}}}}\) take the same form. Similarly, the CDF of \({\gamma ^{\prime }_{{\text {BR}}}}\) can be given by
$$\begin{array}{*{20}l} {F_{{{\gamma^{\prime}}_{{\text{BR}}}}}}\left({{\gamma_{{\text{th}}}}} \right) = 1 + \frac{{{\delta_{\mathrm{B}}}{I_{\mathrm{p}}}{\lambda_{{\text{BR}}}}}}{{{\gamma_{{\text{th}}}}P{\lambda_{{\text{PR}}}}{\lambda_{{\text{BP}}}}}}{e^{{m_{2}}{k_{2}}}}{\text{Ei}}\left({ - {m_{2}}{k_{2}}} \right), \end{array} $$
(37)
where \({m_{2}} = \frac {{{\delta _{\mathrm {B}}}{I_{\mathrm {p}}}{\lambda _{{\text {BR}}}}}}{{{\gamma _{{\text {th}}}}P{\lambda _{{\text {PR}}}}}}\) and \({k_{2}} = \frac {{{\gamma _{{\text {th}}}}}}{{{\delta _{\mathrm {B}}}{I_{\mathrm {p}}}{\lambda _{{\text {BR}}}}}} + \frac {1}{{{\lambda _{{\text {BP}}}}}}\).
We are now in a position to derive the CDF of \({F_{{\gamma _{e}}}}\left ({{\gamma _{{\text {th}}}}} \right)\). Making use the fact that γRA and γRB are correlated due to the common random variable \(\left | {{{\hat h}_{{\text {RP}}}}} \right |^{2}\), we can write the CDF of \({F_{{\gamma _{e}}}}\left ({{\gamma _{{\text {th}}}}} \right)\) as
$$\begin{array}{*{20}l} F_{\gamma_{\mathrm{e}}} = \int\limits_{0}^{\infty} F_{\gamma_{\mathrm{e}}||h_{\text{RP}}|^{2}}(\gamma_{\text{th}}|x) f_{|h_{\text{RP}}|^{2}}(x)dx, \end{array} $$
(38)
where \({F_{{\gamma _{\mathrm {e}}}\left | {\left | {{{\hat h}_{{\text {RP}}}}} \right |^{2}} \right.}}\left ({{\gamma _{{\text {th}}}}\left | x \right.} \right)\) denotes the CDF of γe conditioned on \(\left | {{{\hat h}_{{\text {RP}}}}} \right |^{2}\) given by
$$\begin{array}{*{20}l} &{F_{{\gamma_{\mathrm{e}}}\left| {\left| {{{\hat h}_{{\text{RP}}}}} \right|^{2}} \right.}}\left({{\gamma_{{\text{th}}}}\left| x \right.} \right)\\ &\,=\, \Pr \left[ {\min \left({\frac{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{{\left| {{h_{{\text{RA}}}}} \right|}^{2}}}}{{x\left({1 \,+\, P{{\left| {{h_{{\text{PA}}}}} \right|}^{2}}} \right)}},\frac{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{{\left| {{h_{{\text{RB}}}}} \right|}^{2}}}}{{x\left({1 \,+\, P{{\left| {{h_{{\text{PB}}}}} \right|}^{2}}} \right)}}} \right) \!<\! {\gamma_{{\text{th}}}}} \right]\\ &= 1 - \Pr \left({\frac{{{{\left| {{h_{{\text{RA}}}}} \right|}^{2}}}}{{1 + P{{\left| {{h_{{\text{PA}}}}} \right|}^{2}}}} > \frac{{x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}}}} \right)\\ &\times \Pr\left({\frac{{{{\left| {{h_{{\text{RB}}}}} \right|}^{2}}}}{{1 + P{{\left| {{h_{{\text{PB}}}}} \right|}^{2}}}} > \frac{{x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}}}} \right). \end{array} $$
(39)
Applying Lemma 1 for the second term in (39), where X,Y, and z correspond |hRA|2, |hPA|2, and \(\frac {{x{\gamma _{{\text {th}}}}}}{{{\delta _{\mathrm {R}}}{I_{\mathrm {P}}}}}\), respectively, we have
$$\begin{array}{*{20}l} \Pr \left({\frac{{{{\left| {{h_{{\text{RA}}}}} \right|}^{2}}}}{{1 + P{{\left| {{h_{{\text{PA}}}}} \right|}^{2}}}} > \frac{{x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}}}} \right) =& \frac{{{\delta_{\mathrm{R}}}{I_{\mathrm{p}}}{\lambda_{{\text{RA}}}}}}{{{\gamma_{{\text{th}}}}P{\lambda_{{\text{PA}}}}\left({x + {x_{1}}} \right)}}\\ &\times{e^{- \frac{{{\gamma_{{\text{th}}}}x}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{p}}}{\lambda_{{\text{RA}}}}}}}}, \end{array} $$
(40)
where \({x_{1}} = \frac {{{\delta _{\mathrm {R}}}{I_{\mathrm {p}}}{\lambda _{{\text {RA}}}}}}{{{\gamma _{{\text {th}}}}P{\lambda _{{\text {PA}}}}}}\). Similarly, we have
$$\begin{array}{*{20}l} \Pr \left({\frac{{{{\left| {{h_{{\text{RB}}}}} \right|}^{2}}}}{{1 + P{{\left| {{h_{{\text{PB}}}}} \right|}^{2}}}} > \frac{{x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}}}} \right) =& \frac{{{\delta_{\mathrm{R}}}{I_{\mathrm{p}}}{\lambda_{{\text{RB}}}}}}{{{\gamma_{{\text{th}}}}P{\lambda_{{\text{PB}}}}\left({x + {x_{2}}} \right)}}\\ &\times {e^{- \frac{{{\gamma_{{\text{th}}}}x}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{p}}}{\lambda_{{\text{RB}}}}}}}}, \end{array} $$
(41)
where \({x_{2}} = \frac {{{\delta _{\mathrm {R}}}{I_{\mathrm {p}}}{\lambda _{{\text {RB}}}}}}{{{\gamma _{{\text {th}}}}P{\lambda _{{\text {PB}}}}}}\).
Substituting (40), (41), and (39) into (38), we obtain
$$\begin{array}{*{20}l} {F_{{\gamma_{\mathrm{e}}}}}\left({{\gamma_{{\text{th}}}}} \right) =& 1 - \psi \int\limits_{0}^{\infty} {\frac{1}{{\left({x + {x_{1}}} \right)\left({x + {x_{2}}} \right)}}{e^{- \phi x}}} {f_{{\left| {{{\hat h}_{{\text{RP}}}}} \right|^{2}}}}\left(x \right)dx\\ =& 1 \,-\, \frac{\psi }{{{\lambda_{{\text{RP}}}}\left({{x_{2}} - {x_{1}}} \right)}}\int\limits_{0}^{\infty} {\frac{1}{{\left({x + {x_{1}}} \right)}}{e^{- \left({\phi + \frac{1}{{{\lambda_{{\text{RP}}}}}}} \right)x}}} dx\\ &+ \frac{\psi }{{{\lambda_{{\text{RP}}}}\left({{x_{2}} - {x_{1}}} \right)}}\int\limits_{0}^{\infty} {\frac{1}{{\left({x + {x_{2}}} \right)}}{e^{- \left({\phi + \frac{1}{{{\lambda_{{\text{RP}}}}}}} \right)x}}} dx, \end{array} $$
(42)
where \(\phi = \frac {{{\gamma _{{\text {th}}}}}}{{{\delta _{\mathrm {R}}}{I_{\mathrm {p}}}{\lambda _{{\text {RA}}}}}} + \frac {{{\gamma _{{\text {th}}}}}}{{{\delta _{\mathrm {R}}}{I_{\mathrm {p}}}{\lambda _{{\text {RB}}}}}}\) and \(\psi = \frac {{\delta _{\mathrm {R}}^{2}I_{\mathrm {p}}^{2}{\lambda _{{\text {RA}}}}{\lambda _{{\text {RB}}}}}}{{\gamma _{{\text {th}}}^{2}{P^{2}}{\lambda _{{\text {PA}}}}{\lambda _{{\text {PB}}}}}}\). Applying ([34], Eq. (3.352.4)), one has
$$\begin{array}{*{20}l} {F_{{\gamma_{\mathrm{e}}}}}\left({{\gamma_{{\text{th}}}}} \right) =& 1 + \frac{\psi }{{{\lambda_{{\text{RP}}}}\left({{x_{2}} - {x_{1}}} \right)}}{e^{{x_{1}}\left({\phi + \frac{1}{{{\lambda_{{\text{RP}}}}}}} \right)}}\\ &\quad\times {\text{Ei}}\left[ { - {x_{1}}\left({\phi + \frac{1}{{{\lambda_{{\text{RP}}}}}}} \right)} \right]\\ &\quad- \frac{\psi }{{{\lambda_{{\text{RP}}}}\left({{x_{2}} - {x_{1}}} \right)}}{e^{{x_{2}}\left({\phi + \frac{1}{{{\lambda_{{\text{RP}}}}}}} \right)}}\\ &\quad\times{\text{Ei}}\left[ { - {x_{2}}\left({\phi + \frac{1}{{{\lambda_{{\text{RP}}}}}}} \right)} \right]. \end{array} $$
(43)
Substituting (36), (37), and (43) into (28), we obtain an exact expression for the secondary system outage probability, when the interference CSI to the primary receiver is imperfect as in (31). □
Asymptotic system outage probability at high SNR regime
By using \(1 - {e^{- x}}\mathop \approx \limits ^{x \to 0} x\), the approximate CDF expression for \({\gamma ^{\prime }_{{\text {AR}}}}\), \({\gamma ^{\prime }_{{\text {BR}}}}\), and γe over the high SNR region can be calculated, so that the corresponding asymptotic system outage probability can be found.
Theorem 2
Over Rayleigh fading channels, the asymptotic system outage probability at high SNR regime is of the form as (44).
$$ {{\begin{aligned} {}{\text{OP}}\mathop \approx \limits^{{I_{\mathrm{p}}} \to + \infty} & 1 - \left({1 - \frac{{{\gamma_{{\text{th}}}}{\lambda_{{\text{AP}}}}}}{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}\left({1\! + P{\lambda_{{\text{PR}}}}} \right)} \right) \left({1 - \frac{{{\gamma_{{\text{th}}}}{\lambda_{{\text{BP}}}}}}{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{BR}}}}}}\left({1 + P{\lambda_{{\text{PR}}}}} \right)} \right)\\ &\quad\times\left({1 - \frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PA}}}}} \right){\lambda_{{\text{RP}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RA}}}}}} - \frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PB}}}}} \right){\lambda_{{\text{RP}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RB}}}}}}} \right). \end{aligned}}} $$
(44)
Proof
We start the proof from the definition of \(\phantom {\dot {i}\!}{F_{{{\gamma ^{\prime }}_{{\text {AR}}}}}}\left ({{\gamma _{{\text {th}}}}} \right)\) as
$$\begin{array}{*{20}l} {F_{{{\gamma^{\prime}}_{{\text{AR}}}}}}\left({{\gamma_{{\text{th}}}}} \right) =& \Pr \left({\frac{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{{\left| {{h_{{\text{AR}}}}} \right|}^{2}}}}{{{{\left| {{{\hat h}_{{\text{AP}}}}} \right|}^{2}}\left({1 + P{{\left| {{h_{{\text{PR}}}}} \right|}^{2}}} \right)}} < {\gamma_{{\text{th}}}}} \right)\\ =& \int\limits_{0}^{\infty} \int\limits_{0}^{\infty} \Pr \left({{{\left| {{h_{{\text{AR}}}}} \right|}^{2}} < \frac{{x{\gamma_{{\text{th}}}}\left({1 + Py} \right)}}{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}}}} \right)\\ &\times {f_{{{\left| {{{\hat h}_{{\text{AP}}}}} \right|}^{2}}}}\left(x \right){f_{{{\left| {{h_{{\text{PR}}}}} \right|}^{2}}}}\left(y \right)dxdy. \end{array} $$
(45)
Based on the exponential approximate property, we are able to find the approximate CDF for \({\gamma ^{\prime }_{{\text {AR}}}}\) as
$$\begin{array}{*{20}l} {F_{{{\gamma^{\prime}}_{{\text{AR}}}}}}\left({{\gamma_{{\text{th}}}}} \right) \mathop \approx \limits^{{I_{\mathrm{p}}} \to + \infty} &\int\limits_{0}^{\infty} {\int\limits_{0}^{\infty} {\frac{{x{\gamma_{{\text{th}}}}\left({1 + Py} \right)}}{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}}} \\ &\times\frac{1}{{{\lambda_{{\text{AP}}}}}}{e^{- \frac{x}{{{\lambda_{{\text{AP}}}}}}}}\frac{1}{{{\lambda_{{\text{PR}}}}}}{e^{- \frac{y}{{{\lambda_{{\text{PR}}}}}}}}dxdy. \end{array} $$
(46)
Using ([34], Eq. (3.351.3)) for the inner integral of (46), we have
$$\begin{array}{*{20}l} {F_{{{\gamma^{\prime}}_{{\text{AR}}}}}}\left({{\gamma_{{\text{th}}}}} \right)&\mathop \approx \limits^{{I_{\mathrm{p}}} \to + \infty} \frac{{{\gamma_{{\text{th}}}}{\lambda_{{\text{AP}}}}}}{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}\int\limits_{0}^{\infty} {\left({1 + Py} \right)} \frac{1}{{{\lambda_{{\text{PR}}}}}}{e^{- \frac{y}{{{\lambda_{{\text{PR}}}}}}}}dy \\ & = \frac{{{\gamma_{{\text{th}}}}{\lambda_{{\text{AP}}}}}}{{{\delta_{\mathrm{A}}}{I_{\mathrm{p}}}{\lambda_{{\text{AR}}}}}}\left({1 + P{\lambda_{{\text{PR}}}}} \right). \end{array} $$
(47)
Utilizing similar analytical steps, we find the approximate CDF expression for \({\gamma ^{\prime }_{{\text {BR}}}}\) as
$$\begin{array}{*{20}l} {F_{{{\gamma^{\prime}}_{{\text{BR}}}}}}\left({{\gamma_{{\text{th}}}}} \right)\mathop \approx \limits^{{I_{\mathrm{p}}} \to + \infty} \frac{{{\gamma_{{\text{th}}}}{\lambda_{{\text{BP}}}}}}{{{\delta_{\mathrm{B}}}{I_{\mathrm{p}}}{\lambda_{{\text{BR}}}}}}\left({1 + P{\lambda_{{\text{PR}}}}} \right), \end{array} $$
(48)
For the approximate CDF expression for γe, from (39), we have
$$\begin{array}{*{20}l} {F_{\left. {{\gamma_{\mathrm{e}}}} \right|\left| {{{\hat h}_{{\text{AP}}}}} \right|^{2}}}\left({{\gamma_{{\text{th}}}},x} \right) &\,=\, 1 \,-\, \left[ 1 \,-\,\underbrace{\Pr \left({\frac{{{{\left| {{h_{{\text{RA}}}}} \right|}^{2}}}}{{1 \,+\, P{{\left| {{h_{{\text{PA}}}}} \right|}^{2}}}} \!<\! \frac{{x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}}}} \right)}_{I_{1}} \right]\\ &\quad\times\!\! \left[1 \,-\, \underbrace{\Pr \left({\frac{{{{\left| {{h_{{\text{RB}}}}} \right|}^{2}}}}{{1 + P{{\left| {{h_{{\text{PB}}}}} \right|}^{2}}}} \!<\! \frac{{x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}}}} \right)}_{I_{2}} \right]. \end{array} $$
(49)
We first calculate I1 as follows:
$$\begin{array}{*{20}l} {I_{1}} &= \Pr \left({\frac{{{{\left| {{h_{{\text{RA}}}}} \right|}^{2}}}}{{1 + P{{\left| {{h_{{\text{PA}}}}} \right|}^{2}}}} < \frac{{x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}}}} \right)\\ &= \int\limits_{0}^{\infty} {\Pr \left({{{\left| {{h_{{\text{RA}}}}} \right|}^{2}} < \frac{{\left({1 + Py} \right)x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}}}} \right)} {f_{{{\left| {{h_{{\text{PA}}}}} \right|}^{2}}}}\left(y \right)dy\\ &\mathop \approx \limits^{{I_{\mathrm{p}}} \to + \infty} \int\limits_{0}^{\infty} {\frac{{\left({1 + Py} \right)x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RA}}}}}}\frac{1}{{{\lambda_{{\text{PA}}}}}}{e^{- \frac{y}{{{\lambda_{{\text{PA}}}}}}}}dy}\\ &= \frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PA}}}}} \right)}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RA}}}}}}x. \end{array} $$
(50)
Similarly, we have I2 given by
$$\begin{array}{*{20}l} {I_{2}} &= {\Pr \left({\frac{{{{\left| {{h_{{\text{RB}}}}} \right|}^{2}}}}{{1 + P{{\left| {{h_{{\text{PB}}}}} \right|}^{2}}}} < \frac{{x{\gamma_{{\text{th}}}}}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}}}} \right)}\\ &{\approx} \frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PB}}}}} \right)}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RB}}}}}}x. \end{array} $$
(51)
Substituting I1 in (50) and I2 in (51) into (49), we can define the approximate CDF expression for γ
e
as
$$\begin{array}{*{20}l} &{}{F_{{\gamma_{\mathrm{e}}}}}\left({{\gamma_{{\text{th}}}}} \right)\\ &{}= \int\limits_{0}^{\infty} {\left({\frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PA}}}}} \right)}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RA}}}}}}x + \frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PB}}}}} \right)}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RB}}}}}}x} \right)} {f_{{{\left| {{h_{{\text{RP}}}}} \right|}^{2}}}}\left(x \right)dx\\ &{}- \int\limits_{0}^{\infty} {\left({\frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PA}}}}} \right)}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RA}}}}}}\frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PB}}}}} \right)}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RB}}}}}}{x^{2}}} \right)} {f_{{{\left| {{h_{{\text{RP}}}}} \right|}^{2}}}}\left(x \right)dx. \end{array} $$
(52)
With the help of ([34], Eq. (3.351.3)), we derive the closed-form CDF expression for γe in the high SNR region as
$$\begin{array}{*{20}l} {}{F_{{\gamma_{\mathrm{e}}}}}\left({{\gamma_{{\text{th}}}}} \right)\mathop \approx \limits^{{I_{\mathrm{p}}} \to + \infty} \left({\frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PA}}}}} \right)}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RA}}}}}} + \frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PB}}}}} \right)}}{{{\delta_{\mathrm{R}}}{I_{\mathrm{P}}}{\lambda_{{\text{RB}}}}}}} \right){\lambda_{{\text{RP}}}}. \end{array} $$
(53)
Substituting (47), (48), and (53) into (28), we obtain the approximate expression for the system outage probability in the high SNR region as in (44).
From (44), for Ip→+∞, we have
$$\begin{array}{*{20}l} \mathop {{\text{OP}}}\limits^{{I_{\mathrm{p}}} \to + \infty} \sim \frac{1}{{{I_{\mathrm{p}}}}}{\mathrm{M}}, \end{array} $$
(54)
where
$$\begin{array}{*{20}l} {\mathrm{M}} =& \frac{{{\gamma_{{\text{th}}}}{\lambda_{{\text{AP}}}}}}{{{\delta_{\mathrm{A}}}{\lambda_{{\text{AR}}}}}}\left({1 + P{\lambda_{{\text{PR}}}}} \right) + \frac{{{\gamma_{{\text{th}}}}{\lambda_{{\text{BP}}}}}}{{{\delta_{\mathrm{A}}}{\lambda_{{\text{BR}}}}}}\left({1 + P{\lambda_{{\text{PR}}}}} \right)\\ &+ \frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PA}}}}} \right){\lambda_{{\text{RP}}}}}}{{{\delta_{\mathrm{R}}}{\lambda_{{\text{RA}}}}}} + \frac{{{\gamma_{{\text{th}}}}\left({1 + P{\lambda_{{\text{PB}}}}} \right){\lambda_{{\text{RP}}}}}}{{{\delta_{\mathrm{R}}}{\lambda_{{\text{RB}}}}}}. \end{array} $$
(55)
□
Therefore, it can be seen that over the high SNR region, the system outage probability is proportional to \(\frac {1}{{{I_{\mathrm {p}}}}}\). It can be shown that over the high SNR region, the system diversity is one.