ure space that associates to each symbol repre- sented by the point Pk a rotation matrix Yk that transforms the reference point (1, 0, 0, 0) in Pk. The matrixes Yk depends only on the used constellations and can be stored in a look-up table. In the presence of a small frequency difference between the signal and the LO, the sig- nals after JMI can be written as C1(tj) = (9.40) where ti indicates the discrete time after sampling in the electronics front-end. The four dimensional vector whose coordinates are Ck(t;) is multiplied in each time inter- val by the Yk thus obtaining only noise on the third and fourth components, while the first and second components contains the frequency deviation. Adding c(tj) and c2(tj), (9.41) is obtained, where all the noise terms are collected into ND(t) that results, with a very good approximation, a white Gaussian noise. This function is now independent from the transmitted signal and it can be used to drive the frequency tracking circuit. 9.3.3.2 M-DPSK Optimum Receiver Although the quadrature receiver can work with M-DPSK signals once a suitable decision rule is implemented, it is not the optimum receiver, which is the receiver assuring the minimum error probability in the presence of optical noise. Moreover, the quadrature receiver is very complicated in architecture and requires high speed electronics, with all the consequences in terms of costs and power consumption. Thus, it is also practically very important to design a simpler receiver for M-DPSK signals. The receiver scheme is shown in Figure 9.32 and is the analogous of the binary DPSK receiver shown in Chapter 6, but for the fact that to better use the received power a bal- anced configuration has been adopted. The idea is to use the signal as a sort of LO by delaying it of a symbol interval and lever- aging on the differential encoding at the transmitter. If the transmitted signal has the form (9.16), at the receiver the signal writes as E(t,k) = (9.42) where k indicates the considered symbol interval 0(k) depends on t