  We develop the pseudo-Cl method for reconstructing the Cosmic Microwave
Background (CMB) temperature and polarization auto- and cross-power spectra,
and estimate the pseudo-Cl covariance matrix for a realistic experiment on the
cut sky. We calculate the full coupling equations for all six possible CMB
power spectra, relating the observed pseudo-Cl's to the underlying all-sky
Cl's, and test the reconstruction on both full-sky and cut-sky simulated CMB
data sets. In particular we consider the reconstruction of the Cl from upcoming
ground-based polarization experiments covering areas of a few hundred sq.
degrees and find that the method is fast, unbiased and performs well over a
wide range of multipoles from l=2 to l=2500. We then calculate the full
covariance matrix between the modes of the pseudo-temperature and polarization
power spectra, assuming that the underlying CMB fields are Gaussian randomly
distributed. The complexity of the covariance matrix prohibits its rapid
calculation, required for parameter estimation. Hence we present an
approximation for the covariance matrix in terms of convolutions of the
underlying power spectra. The coupling matrices in these expressions can be
estimated by fitting to numerical simulations, circumventing direct and slow
calculation, and further, inaccurate analytic approximations. We show that
these coupling matrices are mostly independent of cosmology, and that the full
covariance matrix for all six pseudo-Cl power spectra can be quickly and
accurately calculated for any given cosmological model using this method. We
compare these semi-analytic covariance matrices against simulations and find
good agreement, the accuracy of which depends mainly on survey area and the
range of cosmological parameters considered.
