# Electrical Equalization Analysis of PAM-4 Transmission in Short-Reach Optical Systems
## 2 General Theory  
When using a direct modulation, the information is encoded on semiconductor lasers by current modulation. This action, in ideal terms, can be modeled as a square action derived from the square ratio between the electric current and the electric field. According to Figure 1, we can express the signal along the communication system using Equations (1)-(6) discussed in the following:  
\[s(t)=f(\sum_{n=0}^{\infty}I_{n}h_{TX}(t-nT)), \tag{1}\]  
where \(I_{n}\) is a single symbol transmitted at time \(n\); \(T\) is the symbol period; \(h_{TX}\) represents the impulse response of the transmitter, which includes both the modulator driver and the modulator impulse responses; and \(f(\cdot)\) represents the modulator electrical current input to optical field output transfer function. The modulator output is followed by a single mode fiber (SMF) with impulse response \(c(t)\); thus, the receiver optical input field is presented by Equation (2), where * denotes convolution. \[r(t)=s(t)*c(t) \tag{2}\]  
In the direct detection case, the photodiode square-law operation is followed by a low pass filter (LPF), and the received electrical signal can be expressed as [6]:  
\[y(t)=\Big{(}|r(t)|^{2}+n(t)\Big{)}*h_{RX}(t), \tag{3}\]  
where \(n(t)\) is an additive white Gaussian noise (AWGN) and \(h_{RX}(t)\) represents the impulse response of the receiver's LPF. Placing Equations (1) and (2) at Equation (3) yields Equation (4):  
\[y(t)=\Big{|}f(\sum_{n=0}^{\infty}I_{n}h_{TX}(t-nT))*c(t)\Big{|}^{2}*h_{RX}(t)+ z(t), \tag{4}\]  
where \(z(t)=n(t)*h_{RX}(t)\). The sampled signal \(y(kT)\) (\(k=0\), \(1\ldots\)) is fed into the equalizer to mitigate ISI introduced by the dispersion of the SMF, and the transmitter and receiver components' imperfection response. Assuming zero distance fiber, \(c(t)\ =1\), and \(y(kT)\) is [13]:  
\[y(kT)=\Big{|}f(\sum_{n=0}^{\infty}I_{n}h_{TX}(kT-nT))\Big{|}^{2}*h_{RX}(kT)+z( kT), \tag{5}\]  
In this case, we obtain a linear system. Assuming that the modulator modulates at its linear regime, the optical field output is simply the square-root of the electrical modulator  
Figure 1: Block diagram of an optical communication system. input current; therefore, \(f=\sqrt{(\cdot)}\)[14] and the discrete time equivalent model can be expressed as:  
\[y_{k}=h_{TX,0}\hat{I}_{k}h_{RX,k}+\sum_{ \begin{array}{c}n=0\\ n\
eq k\end{array} }^{\infty}I_{n}h_{TX,k-n}\;*h_{RX,k}+z_{k}, \tag{6}\]  
The second expression in Equation (6) represents the ISI among a number of adjacent symbols [13], the first expression represents the desirable symbol, compatible to sample \(k\), and \(z_{k}\) is the \(k\)th sample of \(z(t)\). The equalizer's purpose is to compensate for the ISI by finding the reverse channel response while considering the noise power. The ISI is in the model in Equation (6), while \(c(t)=1\) is linear. However, in practice, we do have a fiber, so, definitively, \(c(t)\
eq 1\), and the ISI, as the system itself, is \"non-linear\". In a case of non-linear ISI, the use of the FFE or DFE is suboptimal compared to the maximum-likelihood sequence-estimation (MLSE) equalizer. In this research, we used the mean square error (MSE) criterion through the least mean square (LMS) algorithm for the coefficients' adaptation and optimization. As shown in [13], the filtered gradient LMS algorithm is given by Equation (7).