# A Pulse Shaping Based Optical Transmission System of 128 QAM for DWDM with N \(\times\) 904 Gbps
## 3 System Architecture and Transceiver Units
After the stage of adaptive equalizer, the signal stream is re-sampled further to \(1\times sample/symbol\). The \((N/2+1)\) sampled signal is used. The imbalances of amplitude and phase within the quadrature (Q) signals and in-phase (I) signals are alleviated by using the compensation of QI. The Gram-Schmidt orthogonalization procedure (GSOP) [18] is used for correcting non-orthogonalization. Two non-orthogonal components from the  
Figure 4: Setup for photonic heterodyne coherent receiver (PIN-PD: positive-intrinsic-negative photodiode; ES: Electrical Subtractor; PS: Polarization Splitter; EA: Electrical Amplifier). Figure 5: Algorithm design of the ultra-fast digital signal processors (UDSP). received signal by \(r_{I}(t)\) and \(r_{Q}(t)\) can be defined, the results of the GSOP for a new orthonormal signals can be described by \(I^{o}(t)\) and \(Q^{o}(t)\), as follows:  
\[I^{o}(t)=\frac{r_{I}(t)}{\sqrt{P_{I}}},\ Q^{\prime}(t)=r_{Q}(t)-\rho\cdot I^{o}( t),\ Q^{o}(t)=\frac{Q^{\prime}(t)}{\sqrt{P_{Q}}} \tag{6}\]  
where \(\rho=E\big{\{}r_{I}(t),r_{Q}(t)\big{\}}\) is the correlation coefficient; \(P_{I}=E\big{\{}r_{I}^{2}(t)\big{\}}\); \(P_{Q}=E\big{\{}Q^{\prime 2}(t)\big{\}}\) and \(E\{.\}\) is the ensemble average operator. Digital filtering can be used for compensating the chromatic dispersion (CD) that is resulting from propagation over fiber. The filter of dispersion compensating can be used in either frequency domain or time domain [19]. The mathematical model of the transfer function for dispersion in the frequency domain can be characterized as follows:  
\[G(z,w)=\exp\biggl{(}-j\frac{D\cdot\lambda^{2}\cdot z}{4\cdot\pi\cdot\mathrm{c }}\cdot w^{2}\biggr{)} \tag{7}\]  
where \(z\) is the propagation distance, \(\lambda\) is the wavelength, \(w\) is the angular-frequency, \(c\) is the speed of light, \(j\) is the imaginary-unit, and \(\mathrm{D}=D_{0}+S\cdot(\lambda-\lambda_{0})\) is the dispersion-coefficient of the optical fiber for the wavelength of \(\lambda\), \(S\) is the dispersion-slope, and \(\lambda_{0}\) is the reference wavelength. A finite impulse response filter (FIR) of the time domain with \(N\) taps is utilized. The tap weights are described by:  
\[a_{k}=\sqrt{\frac{j\cdot c\cdot T^{2}}{D\cdot\lambda^{2}\cdot z}}\cdot\exp \biggl{(}-j\cdot\frac{\pi\cdot c\cdot T^{2}}{D\cdot\lambda^{2}\cdot z}\cdot k ^{2}\biggr{)}\ \ (-\biggl{|}\frac{N}{2}\biggr{|}\leq k\leq\biggl{|}\frac{N}{2}\biggr{|} \biggr{)} \tag{8}\]  
where \(\mathrm{T}=\pi/w_{n}\), \(w_{n}\) is the Nyquist-frequency. Nonlinear compensation with a digital back propagation (BP) method [20] is performed. Back propagation needs the inverse nonlinear Schrodinger equation (NLSE) [20] to be solved for the optical fiber link. For a single polarization and without spatial domain, the NLSE can be described by:  
\[\frac{\partial E}{\partial(-z)}=(D+N)E \tag{9}\]  
where \(N\) is the nonlinear-operator, \(D\) is the differential operator accounting, and \(E\) is the complex field, which are given by:  
\[\left\{ \begin{array}{l}D=\frac{j}{2}\cdot\beta_{2}\cdot\frac{\partial^{2}}{ \partial t^{2}}-\frac{\alpha}{2}\\ N=j\gamma\left|E\right|^{2}\end{array} \right. \tag{10}\]  
where \(\alpha\) is the attenuation coefficient, \(\beta_{2}\) is the group velocity dispersion factor, and \(\gamma\) is the nonlinearity factor. Timing recovery is used for the synchronization of the symbols. The sampling phase and sampling frequency must be determined. The digital square and filter algorithm is utilized [21]. The adaptive equalizer (AE) is used for the compensation of polarization mode dispersion (PMD), the compensation of residual chromatic dispersion and reducing inter-symbol interference. Two-stage Constant modulus algorithm-radius directed (CMA-RD) algorithm is used [22; 23].