A multiple quantum-well semiconductor saturable absorber mirror (MQW-SESAM) structure has been investigated by femtosecond pump-probe laser spectroscopy at a central wavelength of around 1050 nm. Coherent acoustic phonons are generated and detected over a wide frequency range from ~15 GHz to ~800 GHz. In the optical absorption region, i.e., in the multiple quantum wells (In0.27Ga0.73As), acoustic frequency combs centered at ~365 GHz, with a comb spacing of ~33 GHz, are generated. Most importantly, in the transparent region, i.e., in the distributed Bragg reflector, which is formed by a non-doped long-period semiconductor GaAs/Al0.95Ga0.05As superlattice, the mini-Brillouin-zone center, as well as zone-edge acoustic modes, are observed. The mini-zone-center modes with a fundamental frequency of 32 GHz can be attributed to the spatial modulation of the pump optical interference field with a period very close to that of the distributed Bragg reflector, in combination with the periodic spatial modulation of the electrostriction coefficient in the distributed Bragg reflector. The excitation of mini-zone-edge modes is attributed to the stimulated subharmonic decay of the fundamental center modes. Their subsequent back-folding to the mini-Brillouin-zone center makes them Raman active for the probe light.