# A Low BER DB-PAM4 Adaptive Equalizer for Large Channel Loss in Wireline Receivers
## 2 The Principle and Realization of the Adaptive Equalizer
### Theoretical Analysis and Implementation of Adaptive Equilibrium
#### 2.2.1 Pre-Equalization Process  
Duobinary uses known intersymbol interference to reduce the required channel bandwidth [5], which makes it a specific form of the partial response signal. Specifically, the current symbol is added to the previous symbol through a delay filter: \(H(z)=1+z^{-1}\). Figure 2a shows the power spectral density of the PAM4 signal at the transmitter, The DB-PAM4 signal is generated at the receiver by shaping the total channel response into the form \(1+z^{-1}\). Figure 2b shows the single impulse response through the DB channel. Figure 2c shows the power spectrum of the signal at the receiving end, the bandwidth of which is reduced to half of the original signal [6]. In Figure 3, the red curve shows the power spectrum of the ideal DB channel, but it is obviously unrealistic to shape the full frequency band for a low-pass actual channel, so we use a low-pass filter to fit the DB channel. As shown by the blue line in Figure 3, with the combination of channel and CTLE is this type of Bessel low-pass filter [7], the processing result of the DB channel can also be satisfied. Although the high-frequency signal is lost, it is more conducive to implementation.  
Figure 1: Adaptive equalizer overall architecture.  
Since the 112 Gb/s PAM4 signal has a similar power spectrum to the 56 Gb/s NRZ signal, and the latter is more conducive to judging whether the DB channel is generated at the receiving end, we use the NRZ signal for processing in the pre-equalization stage. When the transmitter sends an NRZ signal, the receiver will generate three levels through the DB channel, and these level transitions do not include the highest-to-lowest direct transition. Using this feature, the unknown channel can be converted into a DB path, and the pre-equalization process is shown in Figure 4. The transmitter sends a 56 Gb/s NRZ signal (equivalent to only sending the highest and lowest levels of pre-PAM4), and when it reaches the receiver through the channel, the \"2\" \"1\" and \"0\" signals are judged by the decider and the error signal is calculated.  
In order for the algorithm to actually be implemented, certain information must be constrained. Since the ideal levels of \"2\" and \"0\" are uncertain until CTLE stabilizes, and the DB signal is symmetrical about the middle level, the error when dropping the \"2\" and  
Figure 4: Ideal \(1+z^{-1}\) channel and actual fitting method.  
Figure 3: Ideal \(1+z^{-1}\) channel and actual fitting method.  
Figure 2: (a) 112 Gb/s PAM4 power spectral density (PSD) at TX. (b) Shape total channel response to \(1+z^{-1}\). (c) 112 Gb/s DB-PAM4 Power spectral density (PSD) at RX.  
\"0\" signals will not affect the convergence accuracy of the algorithm, and the symbol value is used instead of the ideal level of the output. The error signal _e(n)_ is:  
\[\left\{\begin{array}{ll}e(n)=0-x(n)&x(n)\in\{1\}\\ e(n)=0&x(n)\in\{2,0\}\end{array}\right. \tag{1}\]  
The updated formula for the bias voltage dlev is expressed as [8]:  
\[\text{dlev}_{n+1}=\text{dlev}_{n}+\mu e(n)\cdot d(n) \tag{2}\]  
Since the eye diagram is completely closed at the beginning of the algorithm, and the error is large, in order to avoid CTLE over-equalization and ensure the reliability of adjustment, DFE is added to the algorithm. The DFE of 1 tap mainly provides the determined symbol value and error for the algorithm. In addition, the initial value of the DFE tap starts from 0. The ideal resistance value of CTLE to adapt to the DB channel and the initial tap coefficient of DFE can be obtained through adjustment. After stabilization, the output result of the DB channel is shown in the lower right corner of Figure 4. The eye diagram of the receiver can clearly see the DouBinary eye diagram.