This paper addresses the nonlinear memory effects in the response of typical illumination light emitting diodes (LEDs), in order to enhance the performance of visible light communication (VLC) systems. These LEDs have a limited bandwidth of only several MHz. To reflect the physical mechanisms in the quantum well, we describe the LED transient response by a nonlinear dynamic differential equation. Three different mechanisms of the nonlinearity are relevant in the double hetero-structure LEDs, which result in dynamic nonlinearities, that is, a mixture of nonlinearities and memory effects. Hitherto, generic pre-distorter and non-linear equalizers have been studied for the LEDs. Yet this paper shows that recombination rates of photon generation can be translated into an equivalent discrete-time circuit that can be inverted. This allows us to develop a new pre-distorter with a simpler and more efficient structure than previously studied and overly generic approaches. The novel pre-distorter along with a parameter estimation can effectively overcome LED nonlinearity for high-speed VLC with amplitude-based single carrier modulations, including ON-OFF keying and pulse amplitude modulation-4 systems, and with the multi-carrier orthogonal frequency-division multiplexing. We report experimentally obtained eye-diagrams, first to justify our choice for the LED model on which our nonlinear pre-distorter have been based, and second to verify the effectiveness in enhancing the VLC link performance to the extent predicted by our model.