We review the time domain, frequency domain, and Fourier series Karhunen–Loéve series expansion (KLSE) methods for exact BER evaluation in intensity- and phase-modulated direct-detection optically amplified systems. We compare their complexity and computational efficiency, and discuss the most relevant implementation issues. We show that the method based on a Fourier series expansion has the simplest implementation and requires the minimum number of eigenvalues to converge to the exact BER value for various kind of optical filters. For this method, we also introduce an equivalent form of the moment generating function, that avoids the singularity for eigenvalues equal to zero, and derive an alternative expansion where signal and noise are expanded on the same orthonormal basis.
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