ed from coding is composed of statistically independent symbols. In general, the multilevel coding is performed by dividing the incoming bit stream in words of n bits and coding each word with a symbol from an alphabet of M = 2n symbols. Even if this is a particular class of multilevel modulations, it is still sufficiently rich to contain remarkably different modulation formats. 9.3.1 Optical Instantaneous Multilevel Modulation In Chapter 6, we have introduced the so-called quadrature representation of the single mode electromagnetic field that propagates in an optical fiber. Let us call p+ and p- two orthogonal unitary polarization vectors (e.g., the output polar- ization principal states or two orthogonal linear polarizations). In general, once the slowly varying component of the optical field is neglected, the transversal shape that is constant during propagation (see Chapter 4), can be written as E(t)=[e1(t)+ie2(t)]p++[e3(t)+ie4(t)]p (9.8) The four real functions e;(t) (j = 1, 2, 3, 4) are the so-called four quadratures of the opti- cal field in the base (p+p) and they are the independent degrees of freedom that can be exploited to modulate the field. The optical field power P(t) can be expressed as the sum of the square of the quadratures, that is, =e}(t)+e2(t)+e3(t)+e4(t) (9.9) Exploiting (9.9), it is possible to introduce the so-called quadrature space, that is, the space having the quadratures as coordinates. This is a Euclidean four dimensional space, where the instantaneous value of the field is represented by a four-vector whose norm is the field power. In this space, the set of field states having the same power are thus spherical sur- faces whose radius is the square root of the power itself. A general instantaneous multilevel modulation of the field can be represented as following (9.10) where g;(t) is the pulse used to modulate the jth quadrature Sj,k is one of the set of amplitudes (S1, SNj) adopted for the modulation of e;(t), where we can assume that Nj = 2nj, SO that M = NN2N3N4 = 2n with n=n1+