  The polarization analysis of the light is typically carried out using
modulation schemes. The light of unknown polarization state is passed through a
set of known modulation optics and a detector is used to measure the total
intensity passing the system. The modulation optics is modified several times
and, with the aid of such several measurements, the unknown polarization state
of the light can be inferred. How to find the optimal demodulation process has
been investigated in the past. However, since the modulation matrix has to be
measured for a given instrument and the optical elements can present problems
of repeatability, some uncertainty is present in the elements of the modulation
matrix and/or covariances between these elements. We analyze in detail this
issue, presenting analytical formulae for calculating the covariance matrix
produced by the propagation of such uncertainties on the demodulation matrix,
on the inferred Stokes parameters and on the efficiency of the modulation
process. We demonstrate that, even if the covariance matrix of the modulation
matrix is diagonal, the covariance matrix of the demodulation matrix is, in
general, non-diagonal because matrix inversion is a nonlinear operation. This
propagates through the demodulation process and induces correlations on the
inferred Stokes parameters.
