# A 60 GS/s Two-Stage Sampler with a Linearity Calibration Loop for PAM-8 Receivers
## 3 Linearity Calibration Feedback Loop
By removing the zero mean value terms, which are filtered, we can express in first approximation the voltage \(v_{corr}(k)\) as:  
\[v_{corr}(k)\approx G\dfrac{[a_{3}C_{3}G_{ch}^{2}+a_{5}C_{5}G_{ch}^{4}+a_{7}C_{7 }C_{ch}^{6}]}{2}\approx G\dfrac{[a_{3}C_{3}G_{ch}^{2}]}{2} \tag{5}\]  
where \(C_{3},\ldots C_{7}\) are coefficients dependent on the impulse response of the channel and filter and are equal to:  
\[0.5G_{ch}^{i}C_{i}\approx E\{[x(k)^{i}\otimes h_{FFE}(k)]u(k)\}, \tag{6}\]  
By solving this equation, we can find terms with an even exponent that have a mean value different than zero, meaning there are nonfiltered positive terms that multiply the distortion coefficient. Moreover, the terms of the fifth and seventh order can be in first approximation neglected because they are multiplied for a higher order gain, which is smaller than one \(G_{ch}\approx 0.3\). For example, in the particular case of an ideal channel and thus no FFE, there is a third-order coefficient equal to:  
\[E\{[x(k)^{3}\otimes h_{FFE}(k)]u(k)\}= E\{[(G_{ch}u(k)\otimes h_{ch}(k))^{3}\otimes h_{FFE}(k)]u(k)\}= \tag{7}\] \[E\{[(G_{ch}u(k)h_{ch}(0))^{3}\otimes h_{FFE}(k)]u(k)\}=\] Figure 4:0  
After sensing the value of \(a_{3}\), the loop minimizes it, using \(v_{corr}\), thus reducing the third-order harmonic \(H3\):  
Figure 4:1To obtain the value of the third harmonic, the Fourier transform of a sinusoid \(G_{ch}sin(2\pi f_{in}t)\) distorted by the TH circuit can be calculated:  
\[ \begin{split} S(f)=&-\frac{1}{128}((64G_{ch}+48a_{3} G_{ch}^{3}+40a_{5}G_{ch}^{5}+35a_{7}G_{ch}^{7})\delta(f-f_{in})\\ &+(16a_{3}G_{ch}^{3}+20a_{5}G_{ch}^{5}+21a_{7}G_{ch}^{7})\delta(f- 3f_{in})+...).\end{split}  \tag{9}\]  
Then, the third harmonic component \((16a_{3}G_{ch}^{3}+20a_{5}G_{ch}^{5}+21a_{7}G_{ch}^{7})\delta(f-3f_{in})\) can be normalized resulting in (8). Normally, the FFE is not capable of eliminating all the ISI major components, meaning the convolution between the channel and the FFE is not equal to a Kronecker delta (\(h_{e}=h_{ch}\otimes h_{FFE}\
eq\delta\)). Knowing that \(\sum_{i}h_{e}(i)=1\), this results in \(h_{e}(1)<1\), where \(h_{e}(1)\) is the component that multiplies the symbol \(u(k)\). In this case, if for simplicity we neglect \(a_{5}\) and \(a_{7}\), we obtain after the filtering:  
\[v_{corr}(k)\approx\frac{G[h_{e}(1)-1]}{2}+\frac{GG_{ch}^{2}a_{3}C_{3}}{2}. \tag{10}\]  
There is a term that is not canceled by the subtraction of \(u(k)\) that makes the system set to a wrong value of \(v_{corr}\). The same problem could arise if the gain control is not accurate enough, leading once again to a residual component proportional to \(u(k)\), which is not filtered, and which does not let the system settle to the desired value. To overcome these problems, we introduce the error elaboration block shown in Figure 6, different to the one in [8] allows us to remove not only the residual ISI component but also the gain error. For the simplicity of calculation, we assume \(a_{3}\) is the only distortion component different from zero, \(d(k)\) is then:  
\[d(k)=G_{r}h_{e}(1)u(k)+G_{nor}[a_{3}x^{3}(k)+n(k)]\otimes h_{FFE}(k), \tag{11}\]  
where \(G_{r}\) is the residual gain of the system, which is different from 1, in the case when the system can not exactly match the gain of the channel (or the gain variations of the analog  
Figure 6: The block that processes the difference between the input and the output of the slicer allowing cancellation of the residual ISI and gain error in blue (in red the non-robust to ISI and gain elaboration block). circuit components) \(G_{nor}\
eq 1/G_{ch}\), and the first order terms given by the residual ISI are neglected because they will be filtered after being multiplied or divided by \(u(k)\).