In this paper we consider the problem of sampling far below the Nyquist rate signals that are sparse linear superpositions of shifts of a known, potentially wide-band, pulse. This signal model is key for applications such as Ultra Wide Band (UWB) communications or neural signal processing. Following the recently proposed Compressed Sensing methodology, we study several acquisition strategies and show that the approximations recovered via $\ell_1$ minimization are greatly enhanced if one uses Spread Spectrum modulation prior to applying random Fourier measurements. We complement our experiments with a discussion of possible hardware implementation of our technique.