Compressed sensing theory shows that sparse signals can be sampled at a much lower rate than required by the Nyquist-Shannon theorem. Unfortunately, in some practical situations, it is sometimes not possible to perfectly know the exact characteristics of the sampling sensor. In many applications dealing with distributed sensors or radars, the location or intrinsic parameters of the sensors are not exactly known, which in turn results in unknown phase shifts and/or gains at each sensor. Similarly, applications with microphone arrays are shown to require calibration of each microphone to account for the unknown gain and phase shifts introduced. Unlike additive perturbations in the measurement matrix, this multiplicative perturbation may introduce significant distortion if ignored during signal reconstruction. Joint sensor calibration and signal recovery enables many application fields to benefit from compressive sensing and sparsity while simultaneously estimating the sensor parameters.
For further details, please refer to the following publications: Blind Sensor Calibration in Sparse Recovery Using Convex Optimization and Blind Phase Calibration in Sparse Recovery.