This paper presents an analysis of the relationship between noise and bandwidth in visible light communication (VLC) systems. In the past few years, pre-emphasis and post-equalization techniques were proposed to extend the bandwidth of VLC systems. However, these bandwidth extension techniques also influence noise and sensitivity of the VLC systems. In this paper, first, we build a system model of VLC transceivers and circuit models of pre-emphasis and post-equalization. Next, we theoretically compare the bandwidth and noise of three different transceiver structures comprising a single pre-emphasis circuit, a single post-equalization circuit and a combination of pre-emphasis and post-equalization circuits. Finally, we validate the presented theoretical analysis using experimental results. The result shows that for the same resonant frequency, and for high signal-to-noise ratio (S/N), VLC systems employing post-equalization or pre-emphasis have the same bandwidth extension ability. Therefore, a transceiver employing both the pre-emphasis and post-equalization techniques has a bandwidth √2 times the bandwidth of the systems employing only the pre-emphasis or post-equalization. Based on the theoretical analysis of noise, the VLC system with only active pre-emphasis shows the lowest noise, which is a good choice for low-noise systems. The result of this paper may provide a new perspective of noise and sensitivity of the bandwidth extension techniques in VLC systems.