  Over the last decade, measurements of the CMB anisotropy has spearheaded the
remarkable transition of cosmology into a precision science. However,
addressing the systematic effects in the increasingly sensitive, high
resolution, `full' sky measurements from different CMB experiments pose a stiff
challenge. The analysis techniques must not only be computationally fast to
contend with the huge size of the data, but, the higher sensitivity also limits
the simplifying assumptions which can then be invoked to achieve the desired
speed without compromising the final precision goals. While maximum likelihood
is desirable, the enormous computational cost makes the suboptimal method of
power spectrum estimation using Pseudo-C_l unavoidable for high resolution
data. We provide a (semi)analytic framework to estimate bias in the power
spectrum due to the effect of beam non-circularity and non-uniform sky coverage
including incomplete/masked sky maps and scan strategy. The approach is
perturbative in the distortion of the beam from non-circularity, allowing for
rapid computations when the beam is mildly non-circular. We advocate that it is
computationally advantageous to employ `soft' azimuthally apodized masks whose
spherical harmonic transform die down fast with m. We numerically implement our
method for non-rotating beams. We present preliminary estimates of the
computational cost to evaluate the bias for the upcoming CMB anisotropy probes
l_max~3000, with angular resolution comparable to the Planck surveyor mission.
We further show that this implementation and estimate is applicable for
rotating beams on equal declination scans and possibly can be extended to
simple approximations to other scan strategies.
