We consider the problem of designing low-complexity digital receivers for CDMA systems operating over channels with single or multiple propagation paths. We specifically exploit a finite rate of innovation property of CDMA signals and develop a method that leads to an efficient solution of the combined problem of multipath channel estimation and signal detection from a low-dimensional subspace of a received signal. Unlike existing schemes that typically resort to chip rate sampling and manipulate with large correlation matrices, our framework allows for digital CDMA receivers to operate at a significantly lower sampling rate, chosen to be close to a rate of innovation of a received signal. Our approach, therefore, can considerably reduce computational requirements compared to existing solutions while providing similar performances, and allows to use slower A/D converters, thus reducing power consumption.