 ratio between the power of the receiver noise and the power of the optical noise The BER, as in the unrepeated case, is expressed as (6.61) where b is the estimate of the received bit Npat is the number of patterns In order to estimate with a suitable approximation the BER, it seems that the specific form of the decision variable can be exploited to reduce the problem to a Gaussian probability. As a matter of fact, it seems that, after the execution of the square of the field quad- ratures, the term with the ASE noise squared can be neglected with respect to the beat between the ASE noise and the signal. With this approximation the problem is effectively reduced to a Gaussian problem, but the solution is affected by an error on the order of magnitude of 3 dB: that is, in order to reach a low BER (e.g., 10-12), the needed power is estimated to be half of the correct value. This is due to the fact that the beat term is much greater than the square noise term in average, but when an error occurs it means that the instantaneous noise power is on the order of magnitude of the signal power and in this condition it is not true that square noise is negligible. Neglecting it means more or less neglecting half the noise, when a wrong bit estimate occurs from which the magnitude of the error in the Gaussian approximation. The terms of the sum (6.61) can be estimated correctly using the so-called saddle point approximation, obtaining a closed form expression for the error probability. It is not within the scope of this section to demonstrate this approximation, the inter- ested reader is encouraged to start from the bibliography regarding this point [27,28]. From an operative point of view, the saddle point approximation requires to find a real solution xo (if it exists) of the equation (6.62) where Cth is the threshold value. If this solution is found, defining the function O(x)=In[F(x)]+Cthx-In(x) = (6.63) Transmission Systems Architectures and Performances the error probability is approximated as (6.64) and with a simila