We investigate a compressive sensing system in which the sensors introduce a distortion to the measurements in the form of unknown gains. We focus on blind calibration, using measures performed on a few unknown (but sparse) signals. We extend our earlier study on real positive gains to two generalized cases (signed real-valued gains; complex-valued gains), and show that the recovery of unknown gains together with the sparse signals is possible in a wide variety of scenarios. The simultaneous recovery of the gains and the sparse signals is formulated as a convex optimization problem which can be solved easily using off-the-shelf algorithms. Numerical simulations demonstrate that the proposed approach is effective provided that sufficiently many (unknown, but sparse) calibrating signals are provided, especially when the sign or phase of the unknown gains are not completely random.