The conventional cyclic-prefix code division multiple access (CP-CDMA) system usually requires a chip-rate frequency-domain equaliser (FDE) at the cost of a tremendous discrete Fourier transform (DFT) size. To reduce the required DFT size in the CP-CDMA uplink receiver, this work studies a new symbol-rate equaliser moving the code despreader ahead of the FDE. The authors formulate the composite effect of the chip-rate channel impulse response (CIR) and code despreading as an equivalent symbol-rate channel model together with a small additive noise. Then, an iterative two-stage multiuser interference (MUI) cancellation algorithm can be employed with the decision feedback equalisation method. In addition, a novel compressive sensing (CS)-based interpolation method is proposed for compressible CIR estimation to be used in the MUI cancellation algorithm. With the CS method, the DFT size is dramatically reduced for channel estimation and no high-resolution interpolation kernels are required. The numerical results show that the new receiver is effective on MUI cancellation and only a couple of iterations are required to achieve a performance similar to the chip-rate equaliser.