# Overcoming Challenges in Large-Core SI-POF-Based System-Level Modeling and Simulation
## 4 System Level Modeling: Two-Step and One-Step Approaches
Thus, for a given source with a particular spatial distribution, the frequency response of the POF layout can be derived as [29]:  
\[H(\omega)=2\pi\int\limits_{\theta=0}^{\theta_{max}}p_{out}(\theta,\omega)\sin( \theta)d\theta \tag{28}\]  
where \(p_{out}(\theta\),\(\omega)\) is the output intensity vector in the frequency domain, whose discretized version is \(\mathbf{P_{out}(\omega)}\) from Equation (27), that is shown here as a function of the propagation angle, as well as of the frequency. Therefore, the spatial aspects of the system can be modeled as an equivalent linear system whose frequency response is \(H(\omega)\). Once the equivalent frequency response of the POF system has been obtained, the temporal aspects of the system design are introduced during the second step, allowing for the calculation of bit error rate (BER) and eye diagrams using different modulation formats. We implemented the first step using MATLABTM  
outputs of all the components constituting the POF system. Figure 7 shows schematically the general idea of the one-step approach. The one-step example is shown in the frequency domain, although it will be equivalent to work in the temporal domain. Optical intensity vectors \(\mathbf{P}(\omega)\) at the output of the system components are represented in 2-D according to their dimensions: frequency \(\omega\) and propagation angle \(\theta\). As can be observed from Figure 7, in the one-step simulation approach, the optical intensity vector at the output of the transmitter, which is propagated throughout the system components, contains not only the spatial but also the frequency properties of the transmitted signal. Therefore, time-domain signals can be easily extracted at any point of the link from the corresponding intensity vectors in order to obtain the desired system-level performance measurements such as BER or eye diagram [61; 62]. The derivation from the two-step to the one-step methodology can be done very easily by adding the signal frequency properties as explained in the following. Considering that a data signal \(s(t)\) is to be transmitted through a POF link, the output data signal \(r(t)\) can be obtained using the inverse Fourier transform and according to the two-step approach as:  
\[r(t)=\mathrm{FT}^{-1}(S(\omega)\cdot H(\omega))=\mathrm{FT}^{-1}\!\left(S( \omega)\cdot 2\pi\int\limits_{\theta=0}^{\theta_{\mathrm{max}}}p_{out}( \theta,\omega)\sin(\theta)d\theta\right) \tag{29}\]  
where \(S(\omega)\) is the Fourier transform of the input data signal \(s(t)\) and \(H(\omega)\) is the equivalent frequency response of the POF system, that is obtained from Equation (28). Since in Equation (29), the integral is computed in the spatial domain, it can be rewritten as  
\[r(t)=\mathrm{FT}^{-1}\!\left(2\pi\int\limits_{\theta=0}^{\theta_{\mathrm{max}} }p_{out,sig}(\theta,\omega)\sin(\theta)d\theta\right) \tag{30}\]  
where the new variable \(p_{out,sig}(\theta,\omega)\), that has been introduced, accounts for the output intensity containing both the spatial characteristics of the output signal and its frequency characteristics. This intensity can be calculated taking into account Equations (27) and (29):  
\[\mathbf{P_{out,sig}}(\omega)=S(\omega)\cdot\mathbf{P_{out}}(\omega)=\mathbf{M_ {det}}\cdot\mathbf{F}(\omega)\cdot S(\omega)\cdot\mathbf{P_{s}}(\omega)= \mathbf{M_{det}}\cdot\mathbf{F}(\omega)\cdot\mathbf{P_{s,sig}}(\omega) \tag{31}\]  
Figure 7: System-level simulation of a POF link according to the one-step methodology. so that in the intensity vector \(\mathbf{P_{s,sig}}(\omega)\) the frequency information of the transmitted data signal has been merged with the spatial information of the optical source and can be propagated in a block-by-block basis to perform system-level simulation of the POP link.