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Any signal-integrity engineer will tell you that analyzing closed eye diagrams has never been easy. A transmission channel’s frequency response (or lack of bandwidth) causes inter-symbol interference (ISI), the primary eye-closing culprit. While I’ve written about closed eye analysis techniques before, this time we’ll measure the ISI left over after equalization, the so-called residual ISI. In the process, we’ll see the simple guts of decision feedback equalization (DFE).

The pulse response is gaining popularity in technology standards that define high-speed serial buses. You can produce a pulse with a pattern generator by transmitting a long string of zeros, a one, and then another long string of zeros, that is, a pulse is a non-return-to zero (NRZ) bit and the pulse response is the same as the SBR (single bit response).

Like the impulse response, the pulse response includes everything there is to know about the circuit, the impedances of every trace, connector, cable, pin, ball, and so on. It’s all built in, both magnitude and phase. You can even produce crosstalk pulse responses by transmitting a pulse on an aggressor and measuring *SBRx(t)* on the victim.

**Figure 1** shows how the impulse (a) and the pulse response (b) are equivalent.

The pulse response, *SBR(t)*, is related to the impulse response, *h(t)* by

where

The pulse response, *SBR(t)*, can be measured in the frequency domain on a vector network analyzer (VNA), with time domain reflectometry/time domain transmissivity (TDR/TDT), or with an oscilloscope. It’s also easy to extract from simulations.

Because pulse-response measurements provide everything there is to know about the channel—everything about the channel that is linear and time invariant, which ought to be everything we need to worry about—the standards use it in measurements and calculations of specified performance variables like channel-operating margin (COM) and signal-to-noise distortion ratio (SNDR).

In real systems, the receiver samples the waveform discretely once per symbol at the baud rate (i.e., the bit rate for NRZ and half the bit rate for PAM4).

where the sum is carried out across the duration of the pulse response. The granularity of

The sampling point, *t _{sp}*, is one UI following the initial rise of

The residual ISI, let’s call it *ResISI(n)*, is the ISI that remains at each UI after equalization. To calculate *ResISI(n)*, we need to include transmitter equalization—de-emphasis or transmitter feed-forward equalization (FFE) in the transmitted pulse. We also need to include the effects of the receiver continuous time linear equalization (CTLE), which is easy to do in an IBIS simulator like ADS (Keysight’s Advanced Design System). The DFE can be put in by hand:

*ResISI(n)* is the difference between the pre- and post-equalized pulse response; perfect equalization would mean *ResISI(n)*=0 for all *n*. The cool part (I think it’s cool) is how DFE is included explicitly through its taps, *b(n)*; simple as can be in that middle term, totally obvious, right? But still magic.

To get a single parameter measure of the residual ISI, add its components like you would the sides of a right triangle, the root-sum-of-squares.

You can see in **Figure 3** how the three equalizers affect the pulse.

You can play the same game with *SBRx(t)* to calculate how your equalization scheme affects crosstalk. With *SBR* and *SBRx* from all of the aggressors, you can calculate the post-equalization shape of any waveform by including the DFE explicitly the way we did for *ResISI*. That is, you can see what the waveform looks at the slicer, deep inside the receiver.

*—Ransom Stephens is a technologist, science writer, novelist, and Raiders fan, which explains why he hasn’t fired up a cigar for quite some time.*

**Related articles:**

- Analyzing closed eyes: CRJ and CDJ
- Why FEC plays nice with DFE
- Optimize equalization for FFE, CTLE, DFE, and crosstalk
- The tradeoff between crosstalk and loss
- See the crosstalk in 100GbE
- Boost resolution of crosstalk measurements