A detailed analysis of the noise spectrum of a semiconductor optical amplifier excited by an amplitude-modulated input signal is presented. This extends the well-established theory for the case of continuous-wave input signals and is relevant for various applications within optical signal processing. One important example is the analysis of noise in microwave photonic elements based on slow-light propagation in semiconductor optical amplifiers. Expressions for the noise spectra are derived and the dependence on important operation parameters, such as input power, modulation depth, and modulation frequency, is investigated and explained. We find several interesting modifications to the spectra compared with the continuous-wave case. In particular, the side-bands present in the input-signal lead via four-wave mixing effects to additional structure in the spectra as well as additional noise components.