Welcome to Tim’s Blackboard! This is the place to find discussions on interesting topics related to signal integrity and power integrity.

This week on Tim’s Blackboard is “Eye-opening Experience with CTLE,” where we study one of the equalization techniques. This post has an associated ADS workspace. Download it now!

# CTLE Opens Closed Eyes

In the previous post, we discussed how **frequency-dependent** loss of a channel causes the eye to close and concluded with the use of equalization to open the eye.

*Fig. 1: A statistical channel simulation in Keysight ADS to demonstrate how CTLE of different DC attenuation opens closed PAM4 eyes.*

Today, we will take a close look at Continuous-Time Linear Equalization (CTLE) and how it opens closed eyes for us, see Fig. 1.

# Concept of Equalization

As I am writing this section, I ask myself,

“What does equalization imply in a non-technical context?”

And I am pleasantly surprised by Merriam-Webster Dictionary.

Recall that a lossy channel distorts the spectrum of the original single pulse **unevenly**. Seen in the time domain, the sharp transitions of the pulse spread out at the beginning and the end, as demonstrated in Fig. 2.

Merriam-Webster is right! To **equalize** is to make the frequency-dependent loss** evenly distributed** throughout a wide range of frequencies.** **

*Fig. 2: Because the lossy channel attenuates higher frequency components more than lower frequency ones, the sharp transitions at the beginning and the end of the single pulse spread out.*

# Continuous-Time Linear Equalization Technique

Fig. 3 shows a collection of Continuous-Time Linear Equalization (CTLE) responses for a reference receiver according to IEEE 802.3bs Draft Standard for Ethernet (October 10^{th} 2017).

*Fig. 3: A collection of CTLE responses for a reference receiver according to IEEE 802.3bs standard for Ethernet. To illustrate the behavior of the CTLE response, the x-axis of the graph is normalized to the Nyquist frequency.*

Plotted against the Nyquist frequency, the curves of CTLE response give us insights on how CTLE evenly distributes the loss. While the CTLE response peaks at frequency close to the Nyquist to preserve content at higher frequencies, there is loss to attenuate spectral content at lower frequencies.

The construction of the CTLE response is that of a peaking filter with three poles and two zeros, defined by

where is the CTLE gain, are the CTLE poles, is the CTLE zero and are the CTLE low frequency pole and zero. An excel spreadsheet of the reference CTLE coefficients can be downloaded here. The coefficients are taken from Table 120E-2 of the October draft standard.

On a system level, we are adding an equalizer block after the channel. Applying the multiplication property, in the frequency domain, we can view the channel and equalizer block together as the response of an equalized channel, as demonstrated in Fig. 4.

*Fig. 4: Illustration of the combined equalized channel response consisting of channel and equalizer.*

# Application of CTLE

Fig. 5 shows the insertion loss of a 10-inch stripline channel from Wild River Technology’s ISI-32 platform. We can see the level of insertion loss increases with frequency. In other words, channel loss is **unevenly** distributed throughout frequencies.

*Fig. 5: Left: Wild River Technology's loss characterization ISI-32 platform. Right: the insertion loss of a 10-inch stripline channel from the platform.*

Because the goal of equalization is to provide a more evenly distributed loss through a wider bandwidth than the original channel, we would expect the equalizer to improve the unevenness of the original channel.

Shown on the left of Fig. 6 is a comparison between the 10-inch stripline channel and the CTLE response. As channel loss drops with frequency, CTLE provides a peak to counteract the effect.

*Fig. 6: Left: channel response and CTLE response comparison. As the S _{21} of the channel drops, CTLE picks up to even out the increasing loss. Right: comparison between the original channel and equalized channel. The equalized channel has a more even frequency response throughout the frequencies below Nyquist.*

Shown on the right of Fig. 6 is the equalized channel. The CTLE has successfully created more even loss curve than the original.

But how do I know for sure the loss of the equalized channel is really more even for wider bandwidth than the original?

To compare the responses of the channel before and after equalization on an equal footing, we normalize the equalized channel response to have 0 dB of loss at low frequency. Fig. 7 is the result of the two curves. Allowing the two responses to have identical loss at low frequency, we observe that, indeed, the equalized channel provides a more even frequency response for a wider frequency range.

*Fig. 7: Comparison between original channel and the equalized channel response (normalized). The comparison demonstrates that the equalized channel provides a more even frequency response up to close to Nyquist. *

# Single Pulse Response with CTLE

Since CTLE improves the evenness of the frequency response, we should consequently expect the single pulse response in time domain to improve as well. In particular, we expect a restoration of the transitioning edges which was distorted by the original high frequency loss. After equalization, the spread of the single pulse should be reduced.

*Fig. 8: After equalization, the single pulse spectrum is restored and results in the reduction of spread of the single pulse in the time domain.*

Fig. 8 shows simulation results consistent with our expectation. As we apply more and more DC attenuation to restore the spectrum, the spread of the single pulse keeps decreasing.

However, to my surprise, the maximum eye opening **does not** happen at maximum DC attenuation at 9 dB.

From the animation above, one observes both the reduction of the spread and reduction of amplitude. Until 6.5 dB of CTLE DC attenuation, the spread of the single pulse is positive and reaches almost zero at 6.5 dB. As the DC attenuation increases to more than 6.5 dB, the single pulse spectrum is restored too much, resulting in a negative dip at the end of the pulse.

# Achieve Maximum Eye-opening

Because the single pulse response is a special case of an eye diagram, we would also expect the eye to exhibit the same behavior. The eye opening should reach a maximum at around 6.5 dB of DC attenuation.

*Fig. 9: ADS statistical channel simulation of an eye to show the eye opening with different CTLE configurations.*

In Fig. 9, one can somewhat make out the increase of eye opening as the eye amplitude decreases. To identify the precise eye width and height, we plot the width and height measurements against the CTLE configurations, see Fig. 10.

*Fig. 10: Eye width and eye height for different CTLE configuration. As expected, the maximum of eye width and eye height occurs at 6.5 dB of DC attenuation.*

As expected, at 6.5 dB of CTLE DC attenuation, both eye height and eye width are at the maximum. However, this might not be always the case. Every channel is a little different, and every eye is a little different. Therefore, it is important and necessary to perform analyses in both frequency and time domain, view the single pulse response and review eye diagrams.

# More Equalization Techniques

Although the IEEE reference CTLE curves are all passive and have maximum 0 dB gain, depending on the need, CTLE implementations can also be active and have positive gain. As the name “continuous-time” suggests, one implements CTLE with analog components. Nonetheless, equalization can also be done in discrete-time with digital signal processing.

In the next two posts, we will discuss equalization in the discrete-time such as Feed Forward Equalization and Decision Feedback Equalization. Make sure you apply for a free trial and download the attached workspace to apply IEEE reference CTLE's to your own channel!

That's this week's Tim's Blackboard. See you next time!