  The measurements of the angular power spectrum of the Cosmic Microwave
Background (CMB) anisotropy has proved crucial to the emergence of cosmology as
a precision science in recent years. In this remarkable data rich period, the
limitations to precision now arise from the the inability to account for finer
systematic effects in data analysis. The non-circularity of the experimental
beam has become progressively important as CMB experiments strive to attain
higher angular resolution and sensitivity. We present an analytic framework for
studying the leading order effects of a non-circular beam on the CMB power
spectrum estimation. We consider a non-circular beam of fixed shape but
variable orientation. We compute the bias in the pseudo-$C_l$ power spectrum
estimator and then construct an unbiased estimator using the bias matrix. The
covariance matrix of the unbiased estimator is computed for smooth,
non-circular beams. Quantitative results are shown for CMB maps made by a
\emph{hypothetical} experiment with a non-circular beam comparable to our fits
to the WMAP beam maps described in the appendix and uses a \emph{toy} scan
strategy. We find that significant effects on CMB power spectrum can arise due
to non-circular beam on multipoles comparable to, and beyond, the inverse
average beam-width where the pseudo-$C_l$ approach may be the method of choice
due to computational limitations of analyzing the large datasets from current
and near future CMB experiments.
