# Tim’s Blackboard #9: Eye-opening Experience with FFE

Blog Post created by Tim Wang Lee on Jan 12, 2018

Welcome to Tim’s Blackboard! This is the place to find discussion on topics related to signal integrity and power integrity. Find other cool posts here!

# Introduction

Last time on Tim’s Blackboard, I talked about Continuous-Time Linear Equalization (CTLE). This week, I will take things to discrete-time and discuss Feed-Forward Equalization (FFE).

All the ADS content shown is in the attached workspace.

# FFE Opens Closed Eyes

Illustrated in Fig. 1, Feed-Forward Equalization (FFE) can open a closed eye. However, unlike Continuous-Time Linear Equalization (CTLE), where the equalization is done with analog components in continuous-time, FFE happens digitally in discrete-time.

Fig. 1: A statistical channel simulation in Keysight ADS to demonstrate Feed-Forward Equalization (FFE) opening a closed eye.

Today, we will take a closer look at Feed-Forward Equalization and how it opens closed eyes for us.

# Concept of Feed-Forward Equalization Technique

In the time domain, FFE creates a pre-distorted pulse at the transmitter by combining delayed pulses multiplied by different weights. By choosing the correct weights to multiply each delayed pulse, one reduces the overall Inter-Symbol Interference (ISI) and opens the eye.

Knowing how the frequency-dependent loss of a channel spreads out the single pulse response, FFE generates a channel-specific pre-distorted pulse at the transmitter to compensate for the spread. An example FFE pulse is shown in Fig. 2.

Fig. 2: The overlay of FFE pulse to be transmitted and channel single pulse response shows FFE inserts pulses of negative amplitudes around the main pulse to cancel out positive amplitude ISI.

Transmitted, the FFE pulse travels down the channel. As the positive main pulse spreads, the negative amplitudes surrounding the main pulse “cancel” out the spreading.

Fig. 3: The equalized single pulse is more contained because FFE reduces the ISI spreading.

Shown in Fig. 3 is the single pulse response of Wild River Technology’s 10-inch stripline channel before and after FFE. Compared to the original channel single pulse response, the equalized single pulse response shows reduced ISI.

How does one generate the pre-distorted pulses?

# Realization of Feed-Forward Equalization

Demonstrated in Fig. 4 is the transmitter architecture of FFE. Mathematically, the structure of FFE is identical to finite impulse response (FIR) filter, where a signal goes through delay elements, and each delayed signal is multiplied by a coefficient of different weight. The important difference is the coefficients.

Fig. 4: An animation of FFE generating a pre-distorted pulse at the transmitter.

We refer to the delay element as tap spacing, and one tap spacing is usually one Unit-Interval. The weighting coefficients are known as tap weights, or just taps.

The “cursor tap”, C0, has the largest magnitude of all the taps and is the main contributor of the entire FFE pulse. The subscript of each tap, then, indicates the location of the taps relative to the cursor tap. For example, the tap, C-1, is one tap spacing before the cursor tap. We further use the term pre-cursor taps to group the taps before the cursor tap, and post-cursor taps for the ones after.

One can also apply the cursor categorization to a channel single pulse response. Fig. 5 is an example of a channel single pulse response with corresponding pre- and post- cursors.

Fig. 5: Application of cursor categorization to channel single pulse response. The cursor is the point where maximum amplitude occurs. Before the cursor are the pre-cursors, and after, post-cursors.

Marking the channel single pulse response sheds different light on our understanding of the channel. Shown in Fig. 6 are single pulse responses of a simulated lossless channel and a lossy one.

Fig. 6: Comparison between a simulated lossless channel and simulated lossy channel reveals more post-cursor ISI than pre-cursor.

One observes frequency-dependent loss of the channel causes ISI in the pre-cursor and post-cursor. Moreover, for the simulated channel in Fig. 6, there exists more post-cursor ISI than pre-cursor. Consequently, in this case, our selection of the tap values would focus on correcting post-cursor ISI.

In practice, user informs FFE algorithm how many pre-cursor and post-cursor taps to use. FFE algorithm then calculates proper values for pre-cursor and post-cursor taps to eliminate ISI.

How do we compute the tap values!?

# Algorithms to Identify Tap Values

No hard work required! Keysight ADS has FFE algorithms built-in to compute taps that optimize the eye opening. Nonetheless, "Advanced Signal Integrity For High-Speed Digital Designs” provides a good pencil-and-paper example of FFE Zero-Forcing solution [1].

There are also adaptive equalization techniques that compare the desired equalizer output and the actual equalizer output. Adaptive FFE techniques such as Least-Mean-Square (LMS) and Recursive-Least-Squares (RLS) are available in Keysight ADS.

Shown in Fig. 7 is the adaptive algorithm window where user specifies different parameters for different adaptive algorithms.

# Comparison Between CTLE and FFE

In the time domain, we should expect both techniques to correct for pre-cursor ISI and post-cursor ISI. However, because of the continuous-time, analog nature of CTLE, we expect CTLE to provide only limited improvement in pre-cursor ISI. On the other hand, operating digitally in discrete-time, we expect FFE to reduce ISI in both pre-cursor and post-cursor.

Fig. 8: Equalized single pulse response comparison between CTLE and FFE. Because of its digital nature, FFE corrects ISI beyond the first pre-cursor.

Shown in Fig. 8 is the single pulse response after CTLE and after FFE. As expected, CTLE barely provides ISI reduction beyond the first pre-cursor. In contrast, FFE can correct more pre-cursor ISI based on the number of the taps specified by the user.

Seen the frequency domain, we expect CTLE to have a high-pass filter characteristic as specified by the zeroes and poles of the filter. In the case of FFE, because of the nature of finite impulse response filter, we expect amplification and attenuation of different harmonics of Nyquist Frequency.

Fig. 9: Comparison between CTLE and FFE spectrum. As CTLE uses the high-pass filter response to counter the channel low-pass response, FFE amplifies the odd harmonics of the Nyquist frequency to equalize the channel.

Shown in Fig. 9 is the frequency response of CTLE and FFE. From the comparison, we see as CTLE focuses on boosting frequency content at the Nyquist frequency, FFE algorithm is selecting taps that effectively amplify the odd harmonics to achieve desired equalization result.

# Feed-Forward Equalization Summary

By selecting proper taps, FFE uses delayed pulses to cancel out ISI in the time domain. Viewed in the frequency domain, FFE effectively amplifies the odd harmonics of the Nyquist frequency and reduces ISI.

So far, both CTLE and FFE are linear equalizers. In the next post, we will cover a non-linear equalization technique: Decision Feedback Equalization (DFE).

Make sure you apply for a free trial and download the attached workspace to apply FFE to your own channel!

That's this week's Tim's Blackboard. Find other cool posts here!

See you next time!

# References

[1]       S. H. Hall, Advanced signal integrity for high-speed digital designs. 2009.