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A new rigorous formulation to evaluate the performance of optical receivers with arbitrary optical filtering and partially polarised noise is proposed. The formulation is based on the calculation of the moment generating function (MGF) of the current at the receiver output and is validated by comparison of the probability density function (PDF) of the current obtained from the MGF with the PDF obtained by Monte Carlo simulation. It is shown that there are always two orthogonal directions over which the noise-polarised component can be projected so that the noise projections are statistically uncorrelated. As a consequence, it is concluded that the MGF of the current depends only on the noise degree of polarisation and on the angle between signal and noise polarisations. An explicit dependence of the mean and variance of the signal-amplified spontaneous emission (ASE) and ASE-ASE beat noises on those two factors is presented. It is shown that, when the noise polarisation is parallel to the signal polarisation, the receiver sensitivity is maintained (in comparison with the unpolarised ASE noise case), even in case of highly polarised noise. If the noise polarisation is orthogonal to the signal polarisation, the sensitivity degrades with the degree of polarisation increase, mainly due to signal power reduction.