## 4 Equalization Technologies
### Ffe/Dfe  
#### 4.1.1 Conventional FFE/DFE  
The feed-forward equalizer is an effective method for linear impairments compensation and widely used nowadays. The most basic components of FFE is the finite impulse response (FIR) filter, whose structure is shown in Figure 7a. The output of FIR is expressed as [67]  
\[y(k)=\sum_{l=0}^{n-1}h_{l}x(k-l) \tag{2}\]  
where \(x(k)\) and \(y(k)\) are the input and output signal of FIR at the sampling instant \(k\), respectively. \(\mathrm{h}=[h_{0}h_{1}h_{2}\dots h_{n-1}]\) is the array of tap weights, while \(n\) is the number of taps.  
Figure 7b depicts the structure of decision-directed FFE, where the FIR filter is noted as \(H(z)\). The tap weights can be updated by the zero-forcing (ZF) algorithm, least mean squares (LMS) algorithm, recursive least squares (RLS) algorithm and so on [67; 68; 69]. Different convergence algorithms just affect the speed of obtaining optimal tap weights and are not the focus of this review. Here, we only illustrate one of the most common techniques called direct decision LMS (DD-LMS) algorithm. The \((k+1)\)-th update of the filter tap weights is given by [68]  
\[h(k+1)=h(k)+2\mu e(k)R(k) \tag{3}\]  
where \(\mu\) is the step size and the error \(e(k)=\delta(k)-z(k)\) is between the desired signal \(\delta(k)\) and the output signal \(z(k)\). The \(R(k)=[r(k),r(k-1),r(k-2),\dots,r(k-n+1)]\) is the vector of the input signal. The FFE can boost the power of high-frequency components that undergo large losses due to the system bandwidth limitations. It should be noted that FFE can operate at symbol rate sampling or higher. Compared with symbol-spaced FFE, the fractionally-spaced FFE which is sampled at several times the symbol rate allows the matched filter to be realized digitally and can take care of phase recovery. In addition, it has a lower residual error at the cost of computational complexity. The destructive effect of the frequency notches is unable to be compensated for easily with an FFE but could be efficiently mitigated using a DFE. Unlike FFE, the input of DFE is the signal after decision, as shown in Figure 8a.  
Figure 7: The structure of **(a)** FIR filter: H(z); **(b)** FFE.  
In the case of DFE, the signal before decision is calculated by [69]  
\[z(k)=r(k)-\sum_{l=0}^{n-1}h_{l}\delta(k-l) \tag{4}\]  
The \((k+1)\)th update of the filer tap weights is given by [69]  
\[h(k+1)=h(k)+2\mu e(k)Z(k) \tag{5}\]  
where \(Z(k)=[z(k),z(k-1),z(k-2),\dots,z(k-n+1)]\). DFE is usually operated at one sample per symbol. It should be noted that DFE can successfully equalize frequency notches by pole insertion, but it may suffer from error propagation and is unstable due to the decision feedback scheme. Moreover, only post-cursor ISI can be deal with DFE, while the CD-induced channel impulse response contains both pre-cursor and post-cursor. Therefore, the best choice for practical implementation is a combination of an FFE and a DFE, which can be seen from Figure 8b. The FFE and DFE can be placed in the transmitter for pre-compensation or in the receiver for post-compensation [60]. To solve the problem of error propagation, a transmitter-side DFE called Tomlinson-Harashima pre-coding has recently been proposed and investigated in various PAM4 short-reach optical links [70; 71; 72; 73]. In addition, many novel equalizers based on conventional FFE/DFE are proposed and investigated to deal with the problems in practical implementation and improve the transmission performance. Next, three recently proposed equalizers including DD-FTN, ID-FFE/ID-DFE, and CR-FFE will be illustrated as examples.