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. A related problem is that of cable permutation arising in manually built arrays with many sensors.
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:
Convex Optimization Approaches for Blind Sensor Calibration using Sparsity (IEEE Transactions on Signal Processing 2014)
see also the associated technical report Balancing Sparsity and Rank Constraints in Quadratic Basis Pursuit and related workshop/conference presentations on our Project Publications page.
Compressed sensing with unknown sensor permutation (ICASSP 2014) 
