ver a time long with respect to the symbol time: the result is an average field whose polar- ization fluctuations are only due to the fiber propagation. In this case, simply observing the variations of such an average vector, the fiber-induced polarization rotations can be compensated [35]. The last class of algorithms has the very important advantage not to require electron- ics speed comparable to the symbol rate, but it is not applicable to all the constellations, requiring an average Stokes vector different from zero. 9.3.4 Ideal Performances of Multilevel Systems In this paragraph, we will compare different multilevel modulation formats considering only the limitation deriving from ASE. This approach is justified by the fact that in optically amplified systems, generally an optical preamplifier is placed in front of the receiver, SO that the received power is sufficiently high to neglect both receiver thermal noise and shot noise. Naturally, in real systems, several other elements thwart the system performances; how- ever, a first approach based only on noise consideration is quite useful to have a first com- parison among different systems. As in the case of binary systems, also the performances of multilevel systems are evalu- ated considering the bit error probability. As a matter of fact, in real applications, the original message is almost always binary and multilevel coding is used only to ease the transmission. Let us assume, as usual, that errors are independent from the transmitted symbol and from the pattern transmitted in the previous symbol intervals. The bit error probability Pe and the symbol error probability Ps are related by the equation (9.47) The average number of wrong bits <e> consequent to a symbol error depends on the cod- ing, that is, on the association between bit patterns and points in the multilevel constella- tion. This association is generally called tagging of the constellation. In order to have the minimum error probability, it is useful to tag a constellation in such a way th