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.

impulse and bit width
Figure 1 The limited bandwidth of a transmission channel elongates both (a) impulses, h(t), and (b) single bits, SBR(t). Graphic courtesy of Anritsu

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

single bit response
where pulse(t) is the long string of zeros, a 1 (or a 3 for PAM4) and another long string of zeros.

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 SBR(t) and h(t) is typically M=32 samples per unit interval (UI), as shown in Figure 2
Pulse response M32
Figure 2 SBR(t) sampled M times per UI produces a waveform with more detail than sampling once per UI. Graphic courtesy of Anritsu

The sampling point, tsp, is one UI following the initial rise of SBR(t).

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.

equalizer effect of pulse
Figure 3 ADS IBIS simulation of a pulse response with and without equalization (Graphic Courtesy of Wild River Technology and Keysight Technologies).

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.

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