  We apply the iterative Expectation-Maximization algorithm (EM) to estimate
the power spectrum of the CMB from multifrequency microwave maps. In addition,
we are also able to provide a reconstruction of the CMB map. By assuming that
the combined emission of the foregrounds plus the instrumental noise is
Gaussian distributed in Fourier space, we have simplified the EM procedure
finding an analytical expression for the maximization step. By using the
simplified expression the CPU time can be greatly reduced. We test the
stability of our power spectrum estimator with realistic simulations of Planck
data, including point sources and allowing for spatial variation of the
frequency dependence of the Galactic emissions. Without prior information about
any of the components, our new estimator can recover the CMB power spectrum up
to scales l~1500 with less than 10% error. This result is significantly
improved if the brightest point sources are removed before applying our
estimator. In this way, the CMB power spectrum can be recovered up to l~1700
with 10% error and up to l~2100 with 50% error. This result is very close to
the one that would be obtained in the ideal case of only CMB plus white noise,
for which all our assumptions are satisfied. Moreover, the EM algorithm also
provides an straightforward mechanism to reconstruct the CMB map. The recovered
cosmological signal shows a high degree of correlation (r = 0.98) with the
input map and low residuals.
