Performance Analysis of Energy Detector Over $\alpha -\mu$ Fading Channels with Selection Combining

Abstract

Energy detection is the most widely used technique in cognitive radio networks to enable opportunistic spectrum access. In this paper, the problem of energy detection of an unknown deterministic signal over fading channels is revisited. More particularly, a new closed-form mathematical expression is derived for the average probability of detection of the energy detector (ED) over ${\upalpha }-{\upmu }$ generalized fading channels with selection combining (SC) diversity reception. The derived expression is general and includes as special cases Nakagami-m, Weibull, Gamma, Rayleigh and Exponential distributions. This expression is useful to quantify the performance improvement of the ED with SC diversity reception.

Appendix: Evaluation of \(\bar{P}_{d,\alpha \mu ,sc}\)

The generalized Marcum Q-function $\hbox {Q}_{\mathrm{u}} \left( {\sqrt{2{{\upgamma }}},\sqrt{{{\uplambda }}}} \right)$ in (10) can be rewritten into series representation using [4, eq.(4.74)] as follows:

$$\begin{aligned} Q_u \left( {\sqrt{2\gamma },\sqrt{\lambda }} \right) =\mathop \sum \limits _{n=0}^\infty \frac{{\varGamma } \left( {n+u,\frac{\lambda }{2}} \right) }{n! {\varGamma } (n+u)}\gamma ^{n}\hbox {e}^{-\gamma } \end{aligned}$$

(15)

In addition, the term $\left( {1-\frac{{\varGamma } \left( {\mu ,\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}} \right) }{{\varGamma } \left( \mu \right) }}\right) ^{L-1}$ in (10) can be simplified into series representation by means of binomial and then multinomial expansions, respectively, as follows:

• Binomial expansion

  $$\begin{aligned} \left( {1-\frac{{\varGamma } \left( {\mu ,\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}} \right) }{{\varGamma } \left( \mu \right) }} \right) ^{L-1}=\mathop \sum \limits _{i=0}^{L-1} \left( {-1} \right) ^{i}\left( {{\begin{array}{l} {L-1} \\ i \\ \end{array} }} \right) \left( {\frac{{\varGamma } \left( {\mu ,\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}} \right) }{{\varGamma } \left( \mu \right) }} \right) ^{i}\quad \quad \end{aligned}$$

  (16)

  Using [15, eq. (8.352-4)] with the help of the equality ${\varGamma } \left( \mu \right) =\left( {\mu -1} \right) !$ and taking into consideration integer values only for the $\mu$ parameter, it follows that:

  $$\begin{aligned} \frac{{\varGamma } \left( {\mu ,\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}} \right) }{{\varGamma } \left( \mu \right) }=e^{-\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}}\mathop \sum \limits _{m=0}^{\mu -1} \frac{\left( {\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}} \right) ^{m}}{m!}\quad \quad \end{aligned}$$

  (17)

• Multinomial expansion

  $$\begin{aligned} \left( {e^{-\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}}\mathop \sum \limits _{m=0}^{\mu -1} \frac{\left( {\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}} \right) ^{m}}{m!}} \right) ^{i}=e^{-i\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}}\mathop \sum \limits _{m=0}^{i\left( {\mu -1} \right) } \beta _{mi} \left( \mu \right) \left( {\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}} \right) ^{m}\quad \quad \end{aligned}$$

  (18)

  where $\beta _{mi} \left( \mu \right)$ is the multinomial expansion coefficient which can be computed recursively as illustrated in [4, eq.(9.124)].

Substituting (18) into (16) and then substituting (16) and (15) into (10), the average probability of detection can be expressed as:

$$\begin{aligned} \bar{P}_{d,\alpha \mu ,sc}&= \int \limits _0^\infty \mathop \sum \limits _{n=0}^\infty \frac{{\varGamma } \left( {n+u,\frac{\lambda }{2}} \right) }{n!{\varGamma } (n+u)}\gamma ^{n}e^{-\gamma }\times \frac{L\alpha \mu ^{\mu }}{2{\varGamma } \left( \mu \right) \bar{\gamma }^{\frac{\alpha \mu }{2}}}\mathop \sum \limits _{i=0}^{L-1} \left( {-1} \right) ^{i}\left( {{\begin{array}{l} {L-1} \\ i \\ \end{array} }} \right) e^{-i\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}}\nonumber \\&\mathop \times \sum \limits _{m=0}^{i\left( {\mu -1} \right) } \beta _{mi} \left( \mu \right) \left( {\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}} \right) ^{m}\gamma ^{\frac{\alpha \mu }{2}-1}{e}^{-\mu \left( {\frac{\gamma }{\bar{\gamma } }} \right) ^{\alpha /2}}{d}\gamma \end{aligned}$$

(19)

Following similar manipulation steps to those leading to [9, eq.(7)], the integral in (19) can be put in the form of Laplace transform. Hence, using [12, 2.2.1-22] the average probability of detection in (11) follows.