# A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
## 5 Applications of NFT and INFT on an Optical Communication System  
In this section, we will apply our proposed implementation of the NFT/INFT algorithms on a modulated signal used in optical communications. The information is encoded in continuous scattering data at the transmitter side and is decoded on the receiver side using both NFT and INFT numerical methods (as explained in the previous sections). We call this a naive implementation as it is in principle simple, but it has some disadvantages that we are going to note below. The second modulation that we will investigate is similar to the previous one, with the difference of better controlling the pulses in the time and frequency domains. Lastly, we will apply our NFT/INFT algorithm to the recently proposed method called b-modulation, where only \(b(\lambda)\) will be used to encode the information at the transmitter side. The nonlinear Fourier transform transmission system will exploit its orthogonality property to allow different users to transmit at the same time in optical fiber medium without any interference with each other. This is similar to the traditional OFDM that is used in almost all new wireless and wired technologies. The scenario that we are going to investigate will involve only the continuous part of the nonlinear spectrum, as the discrete part will increase the complexity at the receiver side. The use of a root searching algorithm is needed, and it is difficult to control their number. In a numerical simulation, we will make sure that there is no discrete part of the nonlinear spectrum. The first modulation where we are going to test our implementation of NFT/INFT algorithm is originally proposed in [16]. Suppose we have the following quadrature phase shift keying (QPSK) modulated signal expressed in the time domain as:  
\[s(t,0)=\sum_{k}s_{k}p(t-kT) \tag{60}\]  
where \(s_{k}\in\{1\pm i,-1\pm i\}\) complex symbols, \(T=0.02[ns]\) is the pulse duration, and \(p(t)\) is the root raised cosine function with unit energy and roll off factor \(\rho=0.2\). Let us also define: \(S(f)=\mathcal{F}\{s(t)\}\), the linear Fourier transform of our modulated signal. In the first modulation, the linear spectrum \(S(f)\) is mapped directly to the continuous nonlinear spectrum defined as the ratio between scattering data, called the reflection coefficient: \(\eta_{c}(\lambda,0)=\frac{b(\lambda,0)}{a(\lambda,0)}=-\frac{1}{2\pi}S\Big{(} -\frac{\lambda}{\pi}\Big{)}\). In order to compute the numerical sample in the time  
Figure 5: The magnitude of \(|u(t,0)|\) in blue and \(2|F(2t,0)|\) in red. domain, we use the INFT algorithm as explained in the previous sections. The transmitted signal in the optical fiber will be:  
\[\eta_{c}(t,0)=INFT(\eta_{c}(\lambda,0)) \tag{61}\]  
At the receiver side, the NFT algorithm is used to obtain:  
\[\eta_{c}(\lambda,z)=NFT(\eta_{c}(t,z)) \tag{62}\]  
The channel equalization is also applied to obtain the transmitted non linear spectrum as:  
\[\eta_{c}(\lambda,0)=\eta_{c}(\lambda,z)e^{-4j\lambda^{2}z} \tag{63}\]  
The equalization is performed by inverting the effect of the channel, which is simply a phase shift in the nonlinear Fourier transform. An inverse Fourier transform followed by a match filter are used to recover the received symbols:  
\[\hat{s_{k}}=\int_{-\infty}^{\infty}s(t,z)p^{*}(t-kT)dt \tag{64}\]  
where \(\hat{s_{k}}\) is an estimation of \(s_{k}\) in case the channel is affected by white Gaussian noise. In the second modulation that we are going to investigate, the reflection coefficient \(\eta_{c}(\lambda,0)\) is not obtained directly from mapping the linear Fourier transform, but from the following transformation to ensure that the signals have the same energy in linear and nonlinear space:  
\[\eta_{c,n}(\lambda,0)=\left(e^{|S(\lambda,0)|^{2}}-1\right)^{1/2}e^{jarg(S( \lambda,0))} \tag{65}\]  
The third modulation to test our algorithm is called b-modulation.