The challenge of Multiuser Detection (MUD) is that of demodulating mutually interfering signals given that at any time instant the number of active users is typically small. The promise of compressed sensing is the demodulation of sparse superpositions of signature waveforms from very few measurements. This paper considers signature waveforms that are are drawn from a Gabor frame. It describes a MUD architecture that uses subsampling to convert analog input to a digital signal, and then uses iterative matching pursuit to recover the active users. Compressive demodulation requires K logN samples to recover K active users whereas standard MUD requires N samples. The paper provides theoretical performance guarantees and consistent numerical simulations. © 2012 IEEE.