# A Pulse Shaping Based Optical Transmission System of 128 QAM for DWDM with N \(\times\) 904 Gbps
## 4 Simulations and Performance Analysis
From Figure 8, it can be obtained that the EVM output from NTS-128 QAM is seriously influenced by the bandwidth of the GOF. The EVM of the transmission channel will be worse when the bandwidth of the GOF that is on a wavelength-link diverges from 70 GHz. The EVM of transmission channel will be deteriorated seriously when the bandwidth of the GOF that is on a wavelength-link is less than 67 GHz. The constellations output from UDSP will be arranged orderly and clearly when the bandwidth of the GOF is configured as 70 GHz. However, the constellations output from the UDSP will become blurred when the bandwidth of the GOF diverges from 70 GHz. According to Figures 7 and 8, we can learn that BER and EVM are similarly influenced by the bandwidth of the GOF; the degree of influence shows a trend of near synchronization with the change of the bandwidth of the GOF. This also further verifies that there is a certain mathematical function relationship between EVM and BER. Consequently, the bandwidth of the GOF plays a key role in the overall performances of the transmission system, so it is very important to configure the bandwidth of the GOF properly. When the transmitted digital symbols of the coherent photonic transmitter are far greater than the number of the constellations (that is, the transmitted digital symbols are far greater than 128), the signal-noise-ratio (SNR) of the transmission system can be expressed as [26]:  
\[EVM\approx\left[\frac{1}{SNR}\right]^{\frac{1}{2}}=\left[\frac{N_{0}}{E_{s}} \right]^{\frac{1}{2}} \tag{15}\]  
where \(N_{0}\) is the energy-spectrum of the noise signal and \(E_{s}\) is the energy per symbol of the useful signal. Then, the relationship between EVM and SNR can be further expressed as:  
\[SNR\approx\frac{1}{EVM^{2}} \tag{16}\]  
Figure 8: Constellation-diagrams and diagram of curve between the bandwidth of GOF and error vector magnitude (EVM). According to the relationship of the mathematical function between SNR and BER, the following mathematical model can be obtained from [26]:  
Figure 6:0  
where \(E_{b}\) is the energy per bit of the useful signal and \(M\) is the number of binary digits. For example, the sequence-number of 128 QAM is 128 (that is, \(M=128\)), and \(erfc\) is the error function. According to \(\frac{E_{s}}{N_{0}}=\frac{E_{b}}{N_{0}}\cdot\log_{2}M\) and Equations (16) and (17) can be further described by:  
Figure 6:1  
Substituting \(M=128\), and consequently, Equation (18) can be further expressed as:  
Figure 6:2  
Equation (19) can be described by:  
Figure 6:3  
where \(k=2\Big{(}1-\frac{1}{\sqrt{128}}\Big{)}\); \(f(EVM)=erfc\Big{(}\sqrt{\frac{3}{254}\cdot\frac{1}{EVM^{2}}}\Big{)}\), and it is a mathematical function with respect to EVM, where EVM is a variable. From Equation (20), it can be obtained that there is a mathematical function relationship between BER and EVM in the system, which further demonstrates that the simulation results are reliable from the trend of curve changes in Figures 7 and 8. The transmission speed of the communication system is configured as 900 Gbps and, the bandwidth of the GOF is set as 70 GHz. We can then obtain the relationship diagrams between BER and OSNR, and EVM and OSNR, which are shown in Figures 9 and 10, respectively. From Figure 9, it can be obtained that OSNR has a large impact on BER, and BER decreases rapidly with the increase of OSNR, where BER will reach the optimum value when the value of OSNR is greater than 26 dB. From Figure 10, it can be obtained that the value of EVM shows a large downward trend with the increase of OSNR, where EVM will reach the optimum value when the value of OSNR is greater than 24 dB. According to Figures 9 and 10, it can be seen that BER and EVM are similarly influenced by OSNR; the degree of influence shows a trend of near synchronization with the change of OSNR. This also further verifies that there is a certain mathematical function relationship between EVM and BER, and also further explains the correctness of Equation (20).