therefore,
is always less than unity so that 
converges for any x value. For example, 
(100%modulation) and 
gives 
=2.0, 
(10
modulation) and 
gives 
. The Mach-Zehnder intensity modulator has a sinusoidal transmission as a function of the applied voltage, and the driving voltage is sinusoidal. Therefore, the transmission function has a steeper edge than sinusoidal modulation which is simply assumed here. In an actual situation, this asymmetric modulation function can remove the noise more efficiently.
Fig.7
Fig. 7 shows the noise reduction effect through synchronous modulation. In Fig. 7(a), the solid and dashed lines are analytic values, the circles and squares represent numerical simulations where a is infinite and 
= 0.01(20 dB modulation). When the synchronous modulation is not applied, the noise accumulates linearly as shown by the dashed line and squares. Synchronous modulation greatly reduces noise accumulation as shown by the solid line and circles. The noise level saturates at twice the input value after only a few shapings. Fig. 7(b) shows the same calculations for various modulation depths. The converged value calculated with (9) is also indicated by the squares in the figure. Even when an intensity modulator with a small extinction ratio was used, it still converges the noise to a very low level.
When coherent ASE noise is added to a soliton pulse, the soliton waveform and spectrum are modified, and this eventually prevents a stable soliton pulse from being transmitted over long distances. If the noise is not too large, the perturbation still yields a stable soliton and a dispersive wave with a small amplitude. Therefore, it is important to install a modulator before the coherent noise builds up and interacts strongly with the soliton.
The principle of the noise reduction is based on the fact that a nondispersive pulse (soliton) and dispersive noise (linear) behave differently. Therefore, this scheme does not work for linear transmission systems operating at a zero-GVD wavelength, in which the noise and the signal pulse behave in the same way.
