Another Look at Performance Analysis of Energy Detector with Multichannel Reception in Nakagami-$m$ Fading Channels

Abstract

Energy detection is the most dominant method used for reliable wireless detection owing to its non-coherent structure and low implementation complexity. The main challenge for performance analysis of energy detection comes from the generalized Marcum $Q$-function involved. To overcome this hurdle, this paper revisits the detection problem and introduces a new framework based on the incomplete Toronto function that allows analyzing the performance for non-fading and fading scenarios in a very simple way. Based on newly-derived expressions for the incomplete Toronto function, performance analysis of energy detection is firstly formulated for unfaded Gaussian noise channels. Second, novel closed-from expressions are derived for the average probability of detection over Nakagami-$m$ fading channels. The analyses are based on two approaches: the probability density function-based and the moment generating function-based (MGF). The former uses the canonical series representation of the incomplete Toronto function while the latter employs MGF-derivative method which is based on the $n$th order derivative of the MGF. The analysis is then extended to cases with diversity receptions including maximal ratio combining and switch and stay combining (SSC). For the SSC diversity case, an analytical expression that can be used to determine optimum switching thresholds in a maximum average detection probability is also derived. The analytical results are verified by numerical computations and Monte-Carlo simulations.

Appendices

Appendix 1: Proof of Theorem 1

For a real number $n$, the function ${I_n}\left( x \right)$ can be expressed as [30, eq. (8.445)],

$$\begin{aligned} {I_n}\left( x \right) = \sum \limits _{l = 0}^\infty {\frac{1}{{l!\varGamma \left( {n + l + 1} \right) }}{{\left( {\frac{x}{2}} \right) }^{n + 2l}}}. \end{aligned}$$

(47)

Substituting (47) in (7) and interchanging summation and integration, one obtains,

$$\begin{aligned} {T_B}\left( {m,n,r} \right) = \sum \limits _{l = 0}^\infty {\frac{{2{e^{ - {r^2}}}{r^{2\left( {n + l} \right) - m + 1}}}}{{l!\varGamma \left( {n + l + 1} \right) }}\int \limits _0^B {{t^{2l + m}}{e^{ - {t^2}}}dt} }. \end{aligned}$$

(48)

Using [30, eq. (3.381-8)] and after that [30, eq. (8.354-2)], the ${T_B}\left( {m,n,r} \right)$ can be evaluated as,

$$\begin{aligned} {T_B}\left( {m,n,r} \right) = \sum \limits _{l = 0}^\infty {\sum \limits _{i = 0}^\infty {\frac{{2{{\left( { - 1} \right) }^i}{e^{ - {r^2}}}{r^{2n + 2l - m + 1}}{B^{2l + 2i + m + 1}}}}{{i!l!\left( {2l + 2i + m + 1} \right) \varGamma \left( {n + l + 1} \right) }}} }. \end{aligned}$$

(49)

By recalling that the Pochhammer function is defined as $\varGamma \left( {x + n} \right) = {(x)_n}\varGamma \left( x \right)$, the Gamma function in (49) can be expressed as $\varGamma \left( {n + l + 1} \right) = \varGamma \left( {n + 1} \right) {\left( {n + 1} \right) _l}$. Furthermore, the $\left( {2l + 2i + m + 1} \right)$ term in (49) can be re-written as,

$$\begin{aligned} 2l + 2i + m + 1 = \frac{{2\varGamma \left( {\frac{{m + 3}}{2}} \right) {{\left( {\frac{{m + 3}}{2}} \right) }_{l + i}}}}{{\varGamma \left( {\frac{{m + 1}}{2}} \right) {{\left( {\frac{{m + 1}}{2}} \right) }_{l + i}}}} \end{aligned}$$

(50)

By substituting accordingly in (49) and carrying our basic algebraic manipulations, one obtains

$$\begin{aligned} {T_B}\left( {m,n,r} \right) = \frac{{2{e^{ - {r^2}}}{r^{2n - m + 1}}{B^{m + 1}}}}{{\left( {m + 1} \right) \varGamma \left( {n + 1} \right) }}\sum \limits _{l = 0}^\infty {\sum \limits _{i = 0}^\infty {\frac{{{{\left( {\frac{{m + 1}}{2}} \right) }_{l + i}}}}{{{{\left( {\frac{{m + 3}}{2}} \right) }_{l + i}}{{\left( {n + 1} \right) }_l}}}\frac{{{{\left( {{r^2}{B^2}} \right) }^l}}}{{l!}}\frac{{{{\left( { - {B^2}} \right) }^i}}}{{i!}}} }. \end{aligned}$$

(51)

Notably, the above double series can be expressed in closed-form in terms of the Kampé de Fériet function as in (11), which completes the proof.

Appendix 2: Proof of Theorem 2

By utilizing the tight approximation of the ${I_n}\left( x \right)$ function [33, eq. (19)], namely,

$$\begin{aligned} {I_n}\left( x \right) \simeq \sum \limits _{l = 0}^p {\frac{{\varGamma \left( {p + l} \right) {p^{1 - 2l}}}}{{l!\varGamma \left( {1 + p - l} \right) \varGamma \left( {n + l + 1} \right) }}{{\left( {\frac{x}{2}} \right) }^{n + 2l}}} , \end{aligned}$$

(52)

which is valid for $0 < x < 2p$ ³, and after substituting in (7) one obtains,

$$\begin{aligned} {T_B}\left( {m,n,r} \right) = \sum \limits _{l = 0}^p {\frac{{2{p^{1 - 2l}}{e^{ - {r^2}}}{r^{2\left( {n + l} \right) - m + 1}}\varGamma \left( {l + p} \right) }}{{l!\varGamma \left( {1 + p - l} \right) \varGamma \left( {1 + l + n} \right) }}\int \limits _0^B {{t^{2l + m}}{e^{ - {t^2}}}dt} }. \end{aligned}$$

(53)

The above integral can be expressed in closed-form according to [30, eq. (3.381-8)], namely,

$$\begin{aligned} \int \limits _0^B {{t^{2l + m}}{e^{ - {t^2}}}dt} = \frac{1}{2}\gamma \left( {l + \frac{{m + 1}}{2},{B^2}} \right) . \end{aligned}$$

(54)

Thus, by substituting (54) into (53) and importantly with the aid of the identity $\varGamma \left( {x - n} \right) {\left( {1 - x} \right) _n} = {\left( { - 1} \right) ^n}\varGamma \left( x \right)$, (12) is deduce. This completes the proof.