## 2 Materials and Methods
### Passive Equalization  
The mathematical expression relates the transfer function of the EQ as the inverse of the line's transfer function as in the relation [2]:  
\[\mathrm{H_{EQ}(f)}=\mathrm{H_{line}^{-1}(f)}, \tag{1}\]  
or, if the transfer function of the line is known actually as the IL of the line or the \(\mathsf{S_{21}}\) scattering parameter, the EQ transfer function is defined by the relation:  
\[\mathrm{H_{EQ}(f)}=\mathrm{H_{comp}(f)}/\mathrm{H_{line}(f)}, \tag{2}\]  
The meaning of \(\mathrm{H_{comp}(f)}\) is the overall transfer function of the system after EQ is part of transmitter, channel, or receiver system and is represented at theoretical level in Figure 8c. In the frequency domain, \(\mathrm{H_{comp}(f)}\) is obtained by multiplication of \(\mathrm{H_{EQ}(f)}\) and \(\mathrm{H_{line}(f)}\) and in time domain, as convolution of associated time domain impulse functions for line and EQ. Amplitude and phase distortion in the new circuit with the EQ added needs to reverse the effects of the interconnect to a perfectly original signal form. This is the ideal case, but in practice, there is no perfect EQ and throughout of the following work, we will demonstrate that there are various implementations, discrete and distributed, that compensate the line's imperfections for multi-gigabits per second signaling systems. Points of insertion of the EQ are either at the receiver end, post-emphasis, or to the transmitter, pre-emphasis. In the first case, the driver's output current is increased over the nominal value when the bit is transitioning [2]. The analysis done in the pre-emphasis block is based on single bit or multiple bits from the data stream. Firstly, the current is reduced by a predetermined amount only when at transitions 0 to 1 or 1 to 0. For bits with same polarity, the nominal current value is lower. The high-frequency transitions are, in conclusion, pre-emphasis relative to the back-to-back same polarity bits (series of 1 s or 0 s) thus high frequency-like domain is transmitted with more energy than low frequency domain [2]. Figure 8: Transfer functions: (**a**) lossy line (**b**) EQ and (**c**) compensated line (dashed trace). Based on the location of the emphasis system in the chain we shall have as depicted in the below Figure 9:  
* Pre-emphasis
* Pre- and post-emphasis  
Often, high-speed designs use the eye diagram to evaluate system performance. The construction of the eye diagram assumes slices into sections of the time domain signal waveform followed by overlaying process [5; 10]. These sections are considered a small number of symbols in length. Folding the input signal on a time segment of length equal to the symbol period allows voltage amplitude representation on the vertical axis, while on the horizontal is the time variation that includes typically one or two symbols. Signal distortions are easy to be demonstrated on the eye diagrams [10]. The closer the eye, the more distortion we have on the signal and larger eye opening implies more margin to the voltage and timing requirements. There are several metrics that can be achieved by representation of the eye: a mask can evidentiate unwanted crossings, corners, transitions, and also gives a feedback of the jitter measurements. The height of the eye should be large enough to ensure the voltage thresholds for high and low levels assuming additive noise additionally degrades the system performance. Furthermore, the eye should be wide enough to provide adequate timing margins for the receiver. The variations in voltage and timing are noise and jitter and using these metrics one can calculate the system's budget. The mask mentioned earlier in this section represents a forbidden region that should not be crossed over imposed in the eye diagram representation (in black represented in Figure 10a). It is a method to find the minimum voltage height and time width at the receiver end in a signaling systems affecting the amplitude and phase due to loss and mismatch. It is based on peak distortion analysis method.