Derivation of Analytical Expressions for the Bit-Error Probability in Lightwave Systems with Optical Amplifiers

We describe a relatively simple derivation of the bit-error probability for a lightwave communications system using an amplitude-shift-keying (ASK) pulse modulation format and employing optical amplifiers such that amplified spontaneous emission noise dominates all other noise sources. This noise may be either polarized in the same direction as the signal or it may be unpolarized. Mathematically, it is represented as a Fourier series expansion with Fourier coefficients that a re assumed to be independent Gaussian random variables. Prior to square-law detection, signal and noise passed through an optical bandpass filter. The detected current is finally filtered by temporal integration over the time slot occupied by one bit (integrate-and-dump receiver). The bit-error probability is given in a closed analytical form that is derived by the approximate evaluation of several integrals appearing in the analysis. Finally, we use our theory to derive the wellknown Gaussian approximation and find that it overestimates the biterror rate by one to two orders of magnitude. Derivations of the biterror probability of binary ASK signals a re not new, the contribution of this paper consists in its simplified approach (Fourier series expansion of the noise) and in the closed analytical form in which the final result is presented in terms of elementary functions.