We present a study of the generation and detection of terahertz acoustic phonons in GaAs/AlAs superlattices through femtosecond pump and probe optical pulses, where the spectral contributions of the generation and detection processes are decoupled and identified. The processes are spatially separated in different superlattices grown with an intermediate 1-µm-thick GaAs layer. Pumping and probing occur on opposite superlattices. A thickness gradient present on one of the superlattices allows us to identify the features in the spectra resulting either from the generation or from the detection. A Raman scattering characterization of the sample is also presented.