assume that all of the noise processes that affect the 1 and 0 bits are Gaussian, as illustrated schematically in Figure 6.14. In that case, the probability density functions (Kartalopoulos, 2004) of the pho- tocurrents are given by (6.13) In equation 6.13, I1,0 are the mean photocurrents of the 1s or Os, according to the sub- script, and are the corresponding variances (i.e., the squares of the standard deviations) due to all of the noise sources that act on them. It is important to stress that the equation does not apply in all cases. For example, shot noise is a Poissonian process that can only be approximated by Gaussian statistics (Kartalopoulos, 2004) and the inaccuracy can be sig- nificant in APD-based receivers (Kobayashi & Mikawa, 2002). Furthermore, nonlinear trans- mission effects can impose bit distortions that are not appropriately modeled by Gaussian probability distribution functions. In some amplified systems it is preferable to replace equa- tion 6.13 with a x2 function (Chan & Conradi, 1997) and in others accurate results may require numerical models together with more demanding computation techniques. BER is calculated using the following equation: Bit Error Rate Logic 0 Logic 1 Logic 0 Decision Level Bit Period (tp) Probability Density Fig. 6.14 Schematic of the probability density functions for the detected photocurrents of 1 and 0 bits. It is assumed that Gaussian functions apply. The arrow indicates the overlap region corresponding to bit errors. (6.14) in which P(0|1) is the probability of deciding 0 when the bit is a 1 and P(1|0) is the opposite. The two factors of 1/2 result from the fact that in most transmission schemes there are equal numbers of 1s and 0s in the datastream and therefore they are equally probable. (Refer to the description of bit scrambling in the section Synchronous Digital Hierarchy.) The functions P(0|1) and P(1|0) are shown in Figure 6.14 as the small shaded area (indicated by the arrow) where the probability density functions cross to the "wrong" side of the decis