Performance analysis for visible light communications with Gaussian-plus-Laplacian additive noise

Abstract

In this paper, the performance of an indoor visible light communication (VLC) system in the presence of Gaussian-plus-Laplacian additive noise is investigated. Initially, a realistic VLC system model is established by considering the effects of channel gain and Gaussian-plus-Laplacian additive noise. Theoretical expressions of the average bit error rate (ABER) and average channel capacity are derived, respectively. To show the system performance at a high signal-to-noise ratio, the asymptotic ABER expression is obtained. Numerical results show that all theoretical results agree well with the simulation results. Moreover, the effects of Laplacian noise scale parameter, dimming target, Lambertian order, and the radius of the receiver zone on system performance are also discussed.

Appendices

Appendix A: Proof of Theorem 1

The first part ${{f}_{1}}$ is given by

$$\begin{aligned} {{f}_{1}}=\int _{{{v}_{\min }}}^{{{v}_{\max }}}{{{h}^{-\frac{2}{m+3}-1}}{{e}^{-\frac{\xi PRh}{\lambda }}}}\text {d}h. \end{aligned}$$

(A.1)

Using the upper incomplete Gamma function, eq. (19) holds.

The second part ${{f}_{2}}$ is given by

$$\begin{aligned} {{f}_{2}}=\int _{{{v}_{\min }}}^{{{v}_{\max }}}{{{h}^{-\frac{2}{m+3}-1}}}{{e}^{-\frac{\xi PRh}{\lambda }}}\times \text {erfc}\left( \frac{\xi PRh}{\sqrt{2}\sigma }-\frac{\sigma }{\sqrt{2}\lambda } \right) \text {d}h.\nonumber \\ \end{aligned}$$

(A.2)

Let the integrand function be

$$\begin{aligned} f \!\left( h \right) \!=\!{{h}^{\!-\!\frac{2}{m\!+\!3}\!-\!1}}\!{{e}^{\!-\!\frac{\xi PRh}{\lambda }}}\!\textrm{erfc}\left( \frac{\xi PRh}{\sqrt{2}\sigma }\!-\!\frac{\sigma }{\sqrt{2}\lambda } \right) . \end{aligned}$$

(A.3)

By using the composite Simpson’s rule, ${{f}_{2}}$ is excepressed as

$$\begin{aligned} {{f}_{2}}= & {} \frac{a}{6}\sum \limits _{i=0}^{N-1}\left[ f\left( {{h}_{i}} \right) \!+\!4f\left( {{h}_{i+\frac{1}{2}}} \right) \!+\!f\left( {{h}_{i+1}} \right) \right] \nonumber \\= & {} \frac{a}{6}\left[ f\left( {{v}_{\min }} \right) +4\sum \limits _{i=0}^{N-1}{f\left( {{h}_{i+\frac{1}{2}}} \right) }\right. \nonumber \\{} & {} \left. +2\sum \limits _{i=1}^{N-1}{f\left( {{h}_{i}} \right) }\!+\!f\left( {{v}_{\max }} \right) \right] . \end{aligned}$$

(A.4)

Therefore, eq. (20) holds.

The third part ${{f}_{3}}$ is given by

$$\begin{aligned} {{f}_{3}}=\int _{{{v}_{\min }}}^{{{v}_{\max }}}{{{h}^{-\frac{2}{m+3}-1}}}{{e}^{\frac{\xi PRh}{\lambda }}} \times \text {erfc}\left( \frac{\xi PRh}{\sqrt{2}\sigma }+\frac{\sigma }{\sqrt{2}\lambda } \right) \text {d}h. \nonumber \\ \end{aligned}$$

(A.5)

Because the solution process of ${{f}_{3}}$ is similar to ${{f}_{2}}$, Eq. (21) can be obtained.

The fourth part ${{f}_{4}}$ is given by

$$\begin{aligned} {{f}_{4}}=\int _{{{v}_{\min }}}^{{{v}_{\max }}}{{{h}^{-\frac{2}{m+3}-1}}}\text {erfc}\left( \frac{\xi PRh}{\sqrt{2}\sigma } \right) \text {d}h. \end{aligned}$$

(A.6)

Utilizing [38, Eq. (1.5.1.3)], Eq. (22) holds.

Appendix B: Proof of Lemma 1

When $\lambda \rightarrow 0$, the combined PDF can be expressed as

$$\begin{aligned} \underset{\lambda \rightarrow 0}{\mathop {\lim }}\,{{f}_{n}}\left( n \right)= & {} \frac{1}{2}\!\left[ \underbrace{\underset{\lambda \rightarrow 0}{\mathop {\lim }}\,\frac{1}{\lambda }{{e}^{\frac{{{\sigma }^{2}}}{2{{\lambda }^{2}}}-\frac{n}{\lambda }}}Q\left( \!-\!\frac{n}{\sigma }\!+\!\frac{\sigma }{\lambda } \right) }_{{{L}_{1}}} \right. \nonumber \\{} & {} \left. \!+\underbrace{\underset{\lambda \rightarrow 0}{\mathop {\lim }}\,\frac{1}{\lambda }{{e}^{\frac{{{\sigma }^{2}}}{2{{\lambda }^{2}}}+\frac{n}{\lambda }}}Q\left( \frac{n}{\sigma }\!+\!\frac{\sigma }{\lambda } \right) }_{{{L}_{2}}} \right] . \end{aligned}$$

(B.1)

According to the definition of the Gaussian Q-function, $Q(-\frac{n}{\sigma }+\frac{\sigma }{\lambda })$ is given by

$$\begin{aligned} Q\left( -\frac{n}{\sigma }+\frac{\sigma }{\lambda } \right) =\int _{-\frac{n}{\sigma }+\frac{\sigma }{\lambda }}^{\infty }{\frac{1}{\sqrt{2\pi }}{{e}^{-\frac{{{t}^{2}}}{2}}}}\text {d}t. \end{aligned}$$

(B.2)

By using the L’Hospital rule for ${{L}_{1}}$ in (B.1), we have

$$\begin{aligned} {{L}_{1}}&=\underset{\lambda \rightarrow 0}{\mathop {\lim }}\,\frac{Q\left( -\frac{n}{\sigma }+\frac{\sigma }{\lambda } \right) }{\frac{1}{\frac{1}{\lambda }{{e}^{\frac{{{\sigma }^{2}}}{2{{\lambda }^{2}}}-\frac{n}{\lambda }}}}}=\frac{1}{\sqrt{2\pi }\sigma }{{e}^{-\frac{{{n}^{2}}}{2{{\sigma }^{2}}}}}. \end{aligned}$$

(B.3)

Similarly, ${{L}_{2}}$ is written as

$$\begin{aligned} {{L}_{2}}=\frac{1}{\sqrt{2\pi }\sigma }{{e}^{-\frac{{{n}^{2}}}{2{{\sigma }^{2}}}}}. \end{aligned}$$

(B.4)

Therefore, the combined PDF can be expressed as

$$\begin{aligned} \underset{\lambda \rightarrow 0}{\mathop {\lim }}\,{{f}_{n}}\left( n \right) =\frac{1}{2}\left( {{L}_{1}}+{{L}_{2}} \right) =\frac{1}{\sqrt{2\pi }\sigma }{{e}^{-\frac{{{n}^{2}}}{2{{\sigma }^{2}}}}}. \end{aligned}$$

(B.5)

Appendix C: Proof of Corollary 2

For a M-QAM DCO-OFDM based VLC system, the instantaneous BER is given by [41]

$$\begin{aligned} {{P}_{e}}(\left. e \right| h){} & {} =\frac{2\left( \sqrt{M}-1 \right) }{\sqrt{M}{{\log }_{2}}(\sqrt{M})}Q\left( \sqrt{\frac{3}{M-1}\textrm{SNDR}} \right) \nonumber \\{} & {} +\frac{2\left( \sqrt{M}\!-\!2 \right) }{\sqrt{M}{{\log }_{2}}(\sqrt{M})}Q\!\left( \! 3\sqrt{\frac{3}{M\!-\!1}\textrm{SNDR}} \right) , \end{aligned}$$

(C.1)

where SNDR is the received signal-to-noise-plus-distortion ratio, and it is given by

$$\begin{aligned} \textrm{SNDR}=\frac{P_{{{x}_{\text {c}}}}^{\text {s}}}{{{P}_{{{x}_{\text {s}}}}}-P_{{{x}_{\text {c}}}}^{\text {rd}}-P_{{{x}_{\text {c}}}}^{\text {s}}+\frac{{{\sigma }^{2}}}{{{h}^{2}}}}. \end{aligned}$$

(C.2)

According to (17), the average BER in this scenario is expressed as (C.3), shown at the bottom of this page. By applying the composite Simpson’s rule to ${{f}_{5}}$ and ${{f}_{6}}$, eq. (27) can be obtained.

$$\begin{aligned}{} & {} {{P}_{\text {e}}}\text {(}e)=\frac{2k\left( \sqrt{M}-1 \right) }{\sqrt{M}{{\log }_{2}}(\sqrt{M})}\underbrace{\int _{{{v}_{\min }}}^{{{v}_{\max }}}{{{h}^{-\frac{2}{m+3}-1}}Q\left( \sqrt{\frac{3}{M-1}\frac{P_{{{x}_{\text {c}}}}^{\text {s}}}{{{P}_{{{x}_{\text {s}}}}}-P_{{{x}_{\text {c}}}}^{\text {rd}}-P_{{{x}_{\text {c}}}}^{\text {s}}+\frac{{{\sigma }^{2}}}{{{h}^{2}}}}} \right) }\text {d}h}_{\triangleq {{f}_{5}}} \nonumber \\{} & {} +\frac{2k\left( \sqrt{M}-2 \right) }{\sqrt{M}{{\log }_{2}}(\sqrt{M})}\underbrace{\int _{{{v}_{\min }}}^{{{v}_{\max }}}{{{h}^{-\frac{2}{m+3}-1}}Q\left( 3\sqrt{\frac{3}{M-1}\frac{P_{{{x}_{\text {c}}}}^{\text {s}}}{{{P}_{{{x}_{\text {s}}}}}-P_{{{x}_{\text {c}}}}^{\text {rd}}-P_{{{x}_{\text {c}}}}^{\text {s}}+\frac{{{\sigma }^{2}}}{{{h}^{2}}}}} \right) }\text {d}h}_{\triangleq {{f}_{6}}}. \end{aligned}$$

(C.3)

Appendix D: Proof of Theorem 2

For indoor VLC, the ACC can be expressed as

$$\begin{aligned} \begin{aligned}&{{C}_{\text {ave}}(h)}=\int _{{{v}_{\min }}}^{{{v}_{\max }}}{{{f}_{h}}\left( h \right) C(h)\text {d}h} \\&\!=\underbrace{-kH\left( n \right) \int _{{{v}_{\min }}}^{{{v}_{\max }}}{{{h}^{\!-\!\frac{2}{m \!+ \!3}\!-1}}}\text {d}h}_{\triangleq {{y}_{1}}} \\&\!+\!\frac{k}{2}\!\underbrace{\int _{{{v}\!_{\min }}}^{{{v}\!_{\max }}}\!{{{h}^{\!-\!\frac{2}{m \!+\!3}\!-\!1}}} \!\ln \! \left( \! {{e}^{2H\left( \! n \! \right) }}\!+\!{{\left( hR\xi Pe \right) }\!^{2}}\! \right) \!\text {d} h}_{\triangleq {{y}_{2}}}. \end{aligned}\nonumber \\ \end{aligned}$$

(D.1)

For the first term ${{y}_{1}}$, we have

$$\begin{aligned} {{y}_{1}}=-H(n). \end{aligned}$$

(D.2)

Let ${{h}^{2}}=t$, ${{y}_{2}}$ in (D.1) can be simplified as

$$\begin{aligned} {{y}_{2}}{} & {} \!=\!\left( m+3 \right) H\left( n \right) \left[ v_{\min }^{-\frac{2}{m+3}}-v_{_{\text {max}}}^{-\frac{2}{m+3}} \right] \nonumber \\{} & {} \!+\!\frac{1}{2}\!\underbrace{\int _{v \!_{\min }^{2}}^{v \!_{\max }^{2}}\!{{{t}^{\!-\!\frac{1}{m \!+\!3}\!-\!1}}}\!\ln \! \left( \! 1\!+\!\frac{{{R}^{2}}{{\xi }^{2}}{{P}^{2}}{{e}^{2}}t}{{{e}^{2H\left( n \right) }}}\! \right) \text {d}t}_{\triangleq {{I}_{1}}}. \end{aligned}$$

(D.3)

By using (11) and (26) in [42], the following equality holds

$$\begin{aligned}{} & {} \int _{a}^{b}{\ln (1+cy){{y}^{d-1}}\text {d}y} \nonumber \\{} & {} =\!{{b}\!^{d}}\!G_{3,}^{1,}{}_{3}^{3}\!\left[ b\!c\left| \begin{matrix} 1\!,\!1\!,\!1\!-\!d \\ 1\!,\!-d\!,\!0 \\ \end{matrix} \right. \right] \! -\!{{a}\!^{d}}\!G_{3,}^{1,}{}_{3}^{3}\!\left[ ac\!\left| \! \begin{matrix} 1\!,\!1\!,\!1\!-\!d \\ 1\!,\!-d\!,0 \\ \end{matrix} \right. \! \right] \!. \end{aligned}$$

(D.4)

By using (D.4), ${{I}_{1}}$ can be written as (D.5), shown at the bottom of this page.

$$\begin{aligned} \begin{aligned} {{I}_{1}}&=v_{\max }^{2\left( -\frac{1}{m+3} \right) }G_{3,}^{1,}{}_{3}^{3}\left[ v_{\max }^{2}\frac{{{R}^{2}}{{\xi }^{2}}{{P}^{2}}{{e}^{2}}}{{{e}^{2H\left( n \right) }}}\left| \begin{matrix} 1,1,1-\left( -\frac{1}{m+3} \right) \\ 1,-\left( -\frac{1}{m+3} \right) ,0 \\ \end{matrix} \right. \right] \\&-v_{\min }^{2\left( -\frac{1}{m+3} \right) }G_{3,}^{1,}{}_{3}^{3}\left[ v_{\min }^{2}\frac{{{R}^{2}}{{\xi }^{2}}{{P}^{2}}{{e}^{2}}}{{{e}^{2H\left( n \right) }}}\left| \begin{matrix} 1,1,1-\left( -\frac{1}{m+3} \right) \\ 1,-\left( -\frac{1}{m+3} \right) ,0 \\ \end{matrix} \right. \right] . \end{aligned}\nonumber \\ \end{aligned}$$

(D.5)

Therefore, ${{y}_{2}}$ can be finally given by

$$\begin{aligned} {{y}_{2}}=\mu \left( m+3,{{v}_{\min }},{{v}_{\max }},{{e}^{2-2H\left( n \right) }} \right) , \end{aligned}$$

(D.6)

where $\mu (a,b,c,d)$ is defined by (36).