On the Outage Performance of Reactive Relay Selection in Cooperative Cognitive Networks Over Nakagami-m Fading Channels

Abstract

This paper evaluates outage performance of reactive relay selection in cognitive radio networks over Nakagami-m fading channels under consideration of imperfect channel information on both interference and transmission channels, peak transmit power constraint, long-term power constraint, interference from primary users (PUs), and direct channel between secondary source and destination. Toward this end, power allocation for secondary transmitters is first proposed to satisfy both power constraints and account for imperfect Nakagami-m channel information and interference from PUs. Then, exact closed-form outage probability representation for secondary destination is suggested to promptly assess system performance and provide useful insights into performance limits. Various results show validity of the proposed expressions, substantial system performance degradation due to imperfect channel information and interference from PUs, error floor in secondary network, performance trade-off between secondary and primary networks, and considerable performance enhancement with respect to the increase in the number of secondary relays and larger fading severity parameter.

Appendices

Appendix 1: Proof of Theorem 1

Both $\eta _{LL1}$ in (9) and $\eta _{LL2}$ in (15) can be uniquely expressed as

$$\begin{aligned} {\eta _{LLp}} = \frac{{{{\left| {{{\hat{h}}_{LLp}}} \right| }^2}{P_L}}}{{{\tau _{LLp}}{P_L} + {{\left| {{h_{iLp}}} \right| }^2}{P_i} + {N_0}}}, \end{aligned}$$

(26)

where $(i,p)=(S,1)$ or $(i,p)=(b,2)$.

The cdf of $\eta _{LLp}$ is represented as

$$\begin{aligned} \begin{aligned} {F_{{\eta _{LLp}}}}\left( x \right)&= \Pr \left\{ {{\eta _{LLp}} \le x} \right\} \\&= \Pr \left\{ {\frac{{{{\left| {{{\hat{h}}_{LLp}}} \right| }^2}{P_L}}}{{{\tau _{LLp}}{P_L} + {{\left| {{h_{iLp}}} \right| }^2}{P_i} + {N_0}}} \le x} \right\} \\&= \Pr \left\{ {{{\left| {{{\hat{h}}_{LLp}}} \right| }^2} \le \frac{x}{{{P_L}}}\left( {{\tau _{LLp}}{P_L} + {{\left| {{h_{iLp}}} \right| }^2}{P_i} + {N_0}} \right) } \right\} \\&= \int \limits _0^\infty {\left[ {\underbrace{\int \limits _0^{\frac{x}{{{P_L}}}\left( {{\tau _{LLp}}{P_L} + {P_i}y + {N_0}} \right) } {{f_{{{\left| {{{\hat{h}}_{LLp}}} \right| }^2}}}\left( w \right) dw} }_{\mathcal {M}}} \right] } {f_{{{\left| {{h_{iLp}}} \right| }^2}}}\left( y \right) dy \end{aligned} \end{aligned}$$

(27)

Substituting (6) into (27), one obtains

$$\begin{aligned} \begin{aligned} {\mathcal {M}}&= \int \limits _0^{\frac{x}{{{P_L}}}\left( {{\tau _{LLp}}{P_L} + {P_i}y + {N_0}} \right) } {\frac{{\beta _{LLp}^{{m_{LLp}}}}}{{\Gamma \left( {{m_{LLp}}} \right) }}{w^{{m_{LLp}} - 1}}{e^{ - {\beta _{LLp}}w}}dw} \\&= \frac{1}{{\Gamma \left( {{m_{LLp}}} \right) }}\gamma \left( {{m_{LLp}};\frac{{{\beta _{LLp}}x}}{{{P_L}}}\left( {{\tau _{LLp}}{P_L} + {P_i}y + {N_0}} \right) } \right) , \end{aligned} \end{aligned}$$

(28)

where $\gamma (a; x)$ is the lower incomplete gamma function [51, Eq. (8.350.1)].

Using the equality

$$\begin{aligned} \gamma \left( {a;x} \right) = \Gamma (a)\left( {1 - {e^{ - x}}\sum \limits _{m = 0}^{a - 1} {\frac{{{x^m}}}{{m!}}} } \right) \end{aligned}$$

(29)

in [52, Eq. (22)] to expand the lower incomplete gamma function in (28) and then substituting the result together with ${f_{{{\left| {{h_{iLp}}} \right| }^2}}}\left( y \right) = {{\Lambda _{iLp}^{{m_{iLp}}}}}{y^{{m_{iLp}} - 1}}{e^{ - {\Lambda _{iLp}}y}}/{{\Gamma \left( {{m_{iLp}}} \right) }}$ in (4) into (27), one obtains

$$\begin{aligned} \begin{aligned} {F_{{\eta _{LLp}}}}\left( x \right)&= \int \limits _0^\infty {\left[ {1 - {e^{ - \frac{{{\beta _{LLp}}\left( {{\tau _{LLp}}{P_L} + {P_i}y + {N_0}} \right) x}}{{{P_L}}}}}\sum \limits _{j = 0}^{{m_{LLp}} - 1} {\frac{1}{{j!}}{{\left( {\frac{{{\beta _{LLp}}\left( {{\tau _{LLp}}{P_L} + {P_i}y + {N_0}} \right) x}}{{{P_L}}}} \right) }^j}} } \right] } \\&\qquad \times \frac{{\Lambda _{iLp}^{{m_{iLp}}}}}{{\Gamma \left( {{m_{iLp}}} \right) }}{y^{{m_{iLp}} - 1}}{e^{ - {\Lambda _{iLp}}y}}dy \\&= \frac{{\Lambda _{iLp}^{{m_{iLp}}}}}{{\Gamma \left( {{m_{iLp}}} \right) }}\int \limits _0^\infty {{y^{{m_{iLp}} - 1}}{e^{ - {\Lambda _{iLp}}y}}} \\&\qquad \times \left[ {1 - {e^{ - \frac{{{\beta _{LLp}}\left( {{\tau _{LLp}}{P_L} + {P_i}y + {N_0}} \right) x}}{{{P_L}}}}}\sum \limits _{j = 0}^{{m_{LLp}} - 1} {\frac{1}{{j!}}{{\left( {\frac{{{\beta _{LLp}}x}}{{{P_L}}}} \right) }^j}\sum \limits _{l = 0}^j {\left( {\begin{array}{l} j \\ l \\ \end{array}} \right) {{\left( {{P_i}y} \right) }^l}{{\left( {{\tau _{LL1}}{P_L} + {N_0}} \right) }^{j - l}}} } } \right] dy, \end{aligned} \end{aligned}$$

(30)

By appropriately arranging terms in (30) and applying the result $\int \nolimits _0^\infty {{x^n}{e^{ - \mu x}}dx} = n!{\mu ^{ - n - 1}}$, the integral in (30) can be solved in closed-form, making (30) same as (18). Therefore, the proof is completed.

Appendix 2: Proof of Theorem 2

Since ${{\eta _{kD2}}}$’s are correlated as mentioned in Sect. 2, ${\mathcal {L}}$ should be evaluated through the conditional probability as

$$\begin{aligned} \begin{aligned} {\mathcal {L}}&= \int \limits _0^\infty {\Pr \left\{ {\left. {\mathop {\max }\limits _{k \in {\mathcal {S}}_i^\ell } {\eta _{kD2}}< {\alpha _s}} \right| Y} \right\} } {f_Y}\left( y \right) dy \\&= \int \limits _0^\infty {\underbrace{\prod \limits _{k \in {\mathcal {S}}_i^\ell } {\underbrace{\Pr \left\{ {\left. {{\eta _{kD2}} < {\alpha _s}} \right| Y} \right\} }_{\mathcal {Y}}} }_{\mathcal {G}}} {f_Y}\left( y \right) dy, \end{aligned} \end{aligned}$$

(31)

where $Y = {\left| {{h_{LD2}}} \right| ^2}$ whose pdf is given in (4) as

$$\begin{aligned} {f_Y}\left( y \right) = \Lambda _{LD2}^{{m_{LD2}}}{y^{{m_{LD2}} - 1}}{e^{ - {\Lambda _{LD2}}y}}/\Gamma \left( {{m_{LD2}}} \right) , \ \ y\ge 0. \end{aligned}$$

(32)

Given ${{\eta _{kD2}}}$ in (13), one immediately obtains

$$\begin{aligned} {\mathcal {Y}} = \Pr \left\{ {{{\left| {{{\hat{h}}_{kD2}}} \right| }^2} \le \frac{{{\alpha _s}\left( {{\tau _{kD2}}{P_k} + {P_L}y + {N_0}} \right) }}{{{P_k}}}} \right\} = \frac{{\gamma \left( {{m_{kD2}};{A_k}y + {B_k}} \right) }}{{\Gamma \left( {{m_{kD2}}} \right) }}, \end{aligned}$$

(33)

where

$$\begin{aligned} \begin{aligned} {A_k}&= {{{\beta _{kD2}}{P_L}{\alpha _s}}}/{{{P_k}}} \\ {B_k}&= {{{\beta _{kD2}}{\alpha _s}\left( {{\tau _{kD2}}{P_k} + {N_0}} \right) }}/{{{P_k}}}. \end{aligned} \end{aligned}$$

(34)

Using (29) to expand the incomplete gamma function in ${\mathcal {Y}}$ and then inserting the result into ${\mathcal {G}}$ in (31), one obtains

$$\begin{aligned} {\mathcal {G}} = \prod \limits _{k \in {\mathcal {S}}_i^\ell } {\left[ {1 - {e^{ - \left( {{A_k}y + {B_k}} \right) }}\sum \limits _{j = 0}^{{m_{kD2}} - 1} {\frac{{{{\left( {{A_k}y + {B_k}} \right) }^j}}}{{j!}}} } \right] }. \end{aligned}$$

(35)

Since relays are closely located as mentioned in Sect. 2, $m_{kD2}=m_{RD2}$, $A_k=A_R$, and $B_k=B_R$, $\forall k \in {\mathcal {K}}$. Therefore, (35) can be simplified as

$$\begin{aligned} \begin{aligned} {\mathcal {G}}&= {\left[ {1 - {e^{ - \left( {{A_R}y + {B_R}} \right) }}\sum \limits _{j = 0}^{{m_{RD2}} - 1} {\frac{{{{\left( {{A_R}y + {B_R}} \right) }^j}}}{{j!}}} } \right] ^{\left| {{\mathcal {S}}_i^\ell } \right| }} \\&= \sum \limits _{a = 0}^{\left| {{\mathcal {S}}_i^\ell } \right| } {\left( {\begin{array}{l} {\left| {{\mathcal {S}}_i^\ell } \right| } \\ a \\ \end{array}} \right) } {\left( { - 1} \right) ^a}{\left( {{e^{ - \left( {{A_R}y + {B_R}} \right) }}\sum \limits _{j = 0}^{{m_{RD2}} - 1} {\frac{{{{\left( {{A_R}y + {B_R}} \right) }^j}}}{{j!}}} } \right) ^a} \end{aligned} \end{aligned}$$

(36)

Applying the multinomial theorem to expand the ${\left( {\sum \nolimits _{j = 0}^{{m_{RD2}} - 1} {\frac{{{{\left( {{A_R}y + {B_R}} \right) }^j}}}{{j!}}} } \right) ^a}$ term in the above, one obtains

$$\begin{aligned} \begin{aligned} {\mathcal {G}}&= \sum \limits _{a = 0}^{\left| {{\mathcal {S}}_i^\ell } \right| } {\left( {\begin{array}{l} {\left| {{\mathcal {S}}_i^\ell } \right| } \\ a \\ \end{array}} \right) a!{{\left( { - 1} \right) }^a}{e^{ - a\left( {{A_R}y + {B_R}} \right) }} } \\&\qquad \times \sum \limits _{{w_0} + {w_1} +\cdots + {w_{{m_{RD2}} - 1}} = a} {\prod \limits _{j = 0}^{{m_{RD2}} - 1} {\left\{ {\frac{1}{{{w_j}!}}{{\left[ {\frac{{{{\left( {{A_R}y + {B_R}} \right) }^j}}}{{j!}}} \right] }^{{w_j}}}} \right\} } } \\&= \sum \limits _{a = 0}^{\left| {{\mathcal {S}}_i^\ell } \right| } {\left( {\begin{array}{l} {\left| {{\mathcal {S}}_i^\ell } \right| } \\ a \\ \end{array}} \right) a!{{\left( { - 1} \right) }^a}{e^{ - a\left( {{A_R}y + {B_R}} \right) }} } \\&\qquad \times \sum \limits _{{w_0} + {w_1} +\cdots + {w_{{m_{RD2}} - 1}} = a} {\left\{ {{{\left( {{A_R}y + {B_R}} \right) }^H}\prod \limits _{j = 0}^{{m_{RD2}} - 1} {\frac{1}{{{w_j}!{{\left( {j!} \right) }^{{w_j}}}}}} } \right\} } \\&= \sum \limits _{a = 0}^{\left| {{\mathcal {S}}_i^\ell } \right| } {\left( {\begin{array}{l} {\left| {{\mathcal {S}}_i^\ell } \right| } \\ a \\ \end{array}} \right) a!{{\left( { - 1} \right) }^a}{e^{ - a\left( {{A_R}y + {B_R}} \right) }} } \\&\qquad \times \sum \limits _{{w_0} + {w_1} +\cdots + {w_{{m_{RD2}} - 1}} = a} {\left\{ {\left[ {\prod \limits _{j = 0}^{{m_{RD2}} - 1} {\frac{1}{{{w_j}!{{\left( {j!} \right) }^{{w_j}}}}}} } \right] \sum \limits _{b = 0}^H {\left( {\begin{array}{l} H \\ b \\ \end{array}} \right) A_R^bB_R^{H - b}{y^b}} } \right\} }, \end{aligned} \end{aligned}$$

(37)

where H is defined in (25).

Inserting (32) and (37) into (31), one obtains the compact representation of ${\mathcal {L}}$ as

$$\begin{aligned} \begin{aligned} {\mathcal {L}}&= \sum \limits _{a = 0}^{\left| {{\mathcal {S}}_i^\ell } \right| } {\left( {\begin{array}{l} {\left| {{\mathcal {S}}_i^\ell } \right| } \\ a \\ \end{array}} \right) \frac{{a!{{\left( { - 1} \right) }^a}}}{{{e^{a{B_R}}}}} } \\&\qquad \times \sum \limits _{{w_0} + {w_1} +\cdots + {w_{{m_{RD2}} - 1}} = a} {\left[ {\prod \limits _{j = 0}^{{m_{RD2}} - 1} {\frac{1}{{{w_j}!{{\left( {j!} \right) }^{{w_j}}}}}} } \right] \sum \limits _{b = 0}^H {\left( {\begin{array}{l} H \\ b \\ \end{array}} \right) A_R^bB_R^{H - b}{\mathcal {J}}} }, \end{aligned} \end{aligned}$$

(38)

where

$$\begin{aligned} \begin{aligned} {\mathcal {J}}&= \int \limits _0^\infty {{e^{ - a{A_R}y}}{y^b}{f_Y}\left( y \right) dy} \\&= \int \limits _0^\infty {{e^{ - a{A_R}y}}{y^b}\frac{{\Lambda _{LD2}^{{m_{LD2}}}}}{{\Gamma \left( {{m_{LD2}}} \right) }}{y^{{m_{LD2}} - 1}}{e^{ - {\Lambda _{LD2}}y}}dy} \\&= \frac{{\Lambda _{LD2}^{{m_{LD2}}}}}{{\Gamma \left( {{m_{LD2}}} \right) }}\int \limits _0^\infty {{y^{{m_{LD2}} + b - 1}}{e^{ - \left( {a{A_R} + {\Lambda _{LD2}}} \right) y}}dy} \\&= \frac{{\Lambda _{LD2}^{{m_{LD2}}}\Gamma \left( {{m_{LD2}} + b} \right) }}{{{{\left( {a{A_R} + {\Lambda _{LD2}}} \right) }^{{m_{LD2}} + b}}\Gamma \left( {{m_{LD2}}} \right) }}. \end{aligned} \end{aligned}$$

(39)

Substituting (39) into (38), ${\mathcal {L}}$ is represented as (24), completing the proof.