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Precise measurements of the Cosmic Microwave Background (CMB) anisotropy havebeen one of the foremost concerns in modern cosmology as it provides valuableinformation on the cosmology of the universe. However, an accurate estimationof the CMB power spectrum faces many challenges as the CMB experimentssensitivity increases. Furthermore, for polarization experiments the precisionof measurement is complicated by the fact that the polarisation signal is veryfaint compared to the measured total intensity, and could be impossible todetect in the presence of high level of systematics. One of the most importantsource of errors in CMB polarization experiment is the beam asymmetry. Forlarge data set the estimation of the CMB polarization power spectrum withstandard optimal Maximum Likelihood (ML) is prohibitive for high resolution CMBexperiments due to the enormous required computation time. In this paper, wepresent a semi-analytical framework using the pseudo-$C_{l}$ estimator tocompute the power spectrum TE of the temperature anisotropy and the E-componentof the polarization radiation field using non-circular beams. We adopt a modelof beams obtained from a perturbative expansion of the beam around a circular(axisymmetric) one in harmonic space, and compute the resulting bias matrixwhich relates the true power spectrum with the observed one using an efficientalgorithm for rapid computation. We show that for a multipole up to$l_{max}=500$, the bias matrix can be computed in less than one second with asingle CPU processor at 2.53 GHz. We find that the uncertainties induced by thebeam asymmetry in the polarization power spectrum at the peak of the biasmatrix for WMAP and Planck experiments can be as large as a few 10 to 20%(upper limit for Planck LFI 30 GHz)

CMB polarization TE power spectrum estimation with non-circular beam