Any popular experiment in chemistry that digital signal processing played a crucial role in?
Many spectroscopic experiments record analog data, which is subsequently Fourier transformed to give a frequency-domain spectrum. For example, in a nuclear magnetic resonance experiment, the signal emitted by an excited nucleus is picked up as a voltage in an RF coil for 1-2 sec. This time-domain signal is processed then converted to the frequency domain by FT. 

As for 'popular', Richard Ernst won the 1991 Nobel in Chemistry for his contributions to this field of research.