Fundamental Analysis for Visible Light Communication with Input‐Dependent Noise

Recently, visible light communication (VLC) has drawn much attention. In literature, the noise in VLC is often assumed to be independent of the input signal. This assumption neglects a fundamental issue of VLC: due to the random nature of photon emission in the lighting source, the strength of the noise depends on the signal itself. Therefore, the input‐dependent noise in VLC should be considered. Given this, the fundamental analysis for the VLC with input‐dependent noise is presented in this chapter. Based on the information theory, the theoretical expression of the mutual information is derived. However, the expression of the mutual information is not in a closed form. Furthermore, the lower bound of the mutual information is derived in a closed form. Moreover, the theoretical expression of the bit error rate is also derived. Numerical results verify the accuracy of the derived theoretical expressions in this chapter

Comparison principle for stochastic heat equations driven by α\alpha-stable white noises

For a class of non-linear stochastic heat equations driven by $\alpha$-stable white noises for $\alpha\in(1,2)$ with Lipschitz coefficients, we first show the existence and pathwise uniqueness of $L^p$-valued c\`{a}dl\`{a}g solutions to such a equation for $p\in(\alpha,2]$ by considering a sequence of approximating stochastic heat equations driven by truncated $\alpha$-stable white noises obtained by removing the big jumps from the original $\alpha$-stable white noises. If the $\alpha$-stable white noise is spectrally one-sided, under additional monotonicity assumption on noise coefficients, we prove a comparison theorem on the $L^2$-valued c\`{a}dl\`{a}g solutions of such a equation. As a consequence, the non-negativity of the $L^2$-valued c\`{a}dl\`{a}g solution is established for the above stochastic heat equation with non-negative initial function

Existence of weak solutions to stochastic heat equations driven by truncated α\alpha-stable white noises with non-Lipschitz coefficients

We consider a class of stochastic heat equations driven by truncated $\alpha$-stable white noises for $1<\alpha<2$ with noise coefficients that are continuous but not necessarily Lipschitz and satisfy globally linear growth conditions. We prove the existence of weak solution, taking values in two different spaces, to such an equation using a weak convergence argument on solutions to the approximating stochastic heat equations. For $1<\alpha<2$ the weak solution is a measure-valued c\`{a}dl\`{a}g process. However, for $1<\alpha<5/3$ the weak solution is a c\`{a}dl\`{a}g process taking function values, and in this case we further show that for $0<p<5/3$ the uniform $p$-th moment for $L^p$-norm of the weak solution is finite, and that the weak solution is uniformly stochastic continuous in $L^p$ sense and satisfies a flow property