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 “Root Cause of Eye Closure,” where we study one of the root causes of eye closure.

After reading the post, download the attached ADS workspace to experiment with Fourier Transform and channel simulator!

# Introduction

Even when you do everything right, i.e., using controlled impedance lines and termination strategy, loss remains a problem when traces are long and when transmitting in the Gigabit regime.

Specifically, it is the **frequency-dependent** **loss** that significantly degrades the signal quality at the receiver. Fig. 1 demonstrates the resulting eye diagrams of frequency-dependent loss and constant loss with the same loss at Nyquist frequency.

*Fig. 1: ADS simulation of two different channels with the same loss at Nyquist frequency. The eye closure of the channel with frequency-dependent loss is more prominent than the channel with constant loss (Eye Diagrams are offset to illustrate the eye closure).*

Given the same transmitter, receiver and same loss at the Nyquist frequency, the channel with frequency-dependent loss introduces more Inter-Symbol Interference (ISI) and degrades the eye horizontally more than the channel with constant loss.

But how?

In this post, we will transmit a single pulse and use our knowledge of time domain and frequency domain transformation to learn the impact of frequency-dependent loss.

# Single Pulse in Frequency Domain

To view the pulse in frequency domain, we perform Fourier Transform to decompose the input signal, channel, and output signal into their corresponding frequency spectrum. Fig. 2 shows the mathematical relationship of the input, the channel, and the output.

*Fig. 2: Illustration of performing Fourier Transform on the time domain input-channel-output relationship. *

After the transformation, the time domain convolution corresponds to multiplication in the frequency domain. The output spectrum is the product of the input spectrum and the channel frequency response after multiplication.

Fig. 3 demonstrates a single pulse going through a channel in both time and frequency domain. Since the frequency domain and time domain are two sides of the same coin, if we have the frequency spectrum of a time domain signal, we could apply Inverse Fourier Transform on the spectrum to retrieve the time domain representation.

*Fig. 3: Sending a single pulse through a channel.*

We see difference in the shapes of the input and output single pulse spectrum in Fig. 3, and we know the channel ought to change the input spectrum, but

How does the channel frequency response cause the spread of the output time domain waveform?

To answer the question, let’s take a closer look at the time and frequency domain relationship.

# Reconstruction of Waveform from Spectrum

Frequency domain representation shows how different frequency components interact with each other to create the time domain waveform. The constructive and destructive interference of sine waves of different frequencies works together to form the time domain waveform.

Thus, the shape of the frequency spectrum is important when one wants to reconstruct and maintain the shape of the original time domain waveform. For example, if we were to divide the the amplitude of the entire spectrum by two, we should expect the resulting time domain waveform to still be a single pulse, but with half of the original amplitude.

*Fig. 4: Time domain and frequency domain representation of the original and modified waveforms in ADS. Because different frequency components work together to produce the shape of the original pulse, if the relative strengths of all components are identical, the shape of the time domain waveform is the same*.

Fig. 4 shows a modified spectrum of the same shape and the result of Inverse Fourier Transform. As we expect, because the same modification, dividing by two, is done to the entire spectrum, the relationship between different frequencies is the same. Consequently, the shape of the single pulse waveform is maintained in the time domain, and the peak amplitude is indeed half of the original.

However, if we don’t treat the spectrum as a whole, and we alter only a small part of the spectrum, we expect to see a small change in the spectrum to produce a dramatic change in the shape of single pulse waveform, as shown in Fig. 5.

*Fig. 5: Although the spectrum is modified a little bit, the relative strengths of different components are different. The new shape of the spectrum no longer corresponds to the original single pulse.*

Although Fig. 5 is an extreme case where a small part of spectrum is removed, it underscores the importance of treating a given spectrum in its entirety to maintain the corresponding time domain waveform.

To see how the single pulse would look after going through the channel, let’s take a look at how the channel treats different frequency components.

# Channel Frequency Response Changes Spectrum

We can see from Fig. 6 the frequency response of the channel modifies the spectrum differently at different frequencies. Therefore, we expect the shape of the reconstructed pulse to be different from the original.

*Fig. 6: Channel frequency response shows more attenuation at higher frequencies than lower frequencies.*

Specifically, because the channel attenuates higher frequency components that make up the sharp transition more than the lower frequency ones, at the output of the channel, the rising and falling transitions of the pulse will spread.

A comparison of lossy channel and lossless channel in Fig. 7 shows consistent result with our expectation. The lossy channel distorts the spectrum of the original input pulse unevenly. Seen in the time domain, the sharp transitions of the original pulse spread out at the beginning and the end.

*Fig. 7: 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.*

The spreading of the single pulse is known as the inter-symbol interference (ISI) because the current pulse interferes with the one pulse before and the one after. To reduce ISI is to reduce eye closure.

# How to Avoid Eye Closure

Because of frequency-dependent loss closes the eye, to open the eye, we do the following:

- Reduce the amount of loss,
- Remove the frequency dependence of the loss.

Given a fixed data rate, to reduce the amount of loss, we can:

- Keep trace as short as possible,
- Use substrate with lower Dk and Df,
- Use smoother conductor and as low resistance as budget allows.

To remove frequency dependence of the loss, we can **equalize** the spectrum with different equalization techniques:

- CTLE: Continuous Time Linear Equalizer,
- FFE: Feed-Forward Equalizer,
- DFE: Decision Feedback Equalizer.

Fig. 8 shows an example of applying equalization to open an eye.

*Fig. 8: Equalization result of ADS channel simulation (Eye diagrams are offset to illustrate the opening).*

The next blog talks more about the different equalization techniques and how to perform them in ADS.

Make sure you apply for a free trial and download the attached workspace to experiment with Fourier Transform and channel simulator!

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