Analyzing closed eyes: CRJ and CDJ

-January 12, 2016

ISI (inter-symbol interference) closes eye diagrams at high data rates. The skin effect and dispersion—the frequency dependence of the dielectric "constant"—combine to cause the messy low-pass nature of channels, or equivalently, in the time domain, to smear the pulse response across many bits. As data rates increase and higher frequency components are introduced, ISI gets worse.

Closed Eye (photo by Ransom)

The combination of one, two, or three types of equalization open the eyes enough for the decision circuit to identify symbols—FFE (feed forward equalization) at the transmitter, CTLE (continuous time equalization) at the receive, and/or DFE (decision feedback equalization) also at the receiver. For NRZ (non-return to zero) PAM2 (2-level pulse amplitude modulation) signals, there’s just the one eye with the baseband symbol for 1s high and 0s low. For PAM4 (4-level pulse amplitude modulation) there are three eyes, one separating each of the four symbol levels, since PAM4 encodes two bits in each symbol. In both cases, at high rates, the signal that enters the receiver has closed eyes.

Because ISI closes the eyes, the trick to analyzing them is a combination of "embedding" the receiver equalization scheme(s) and using carefully chosen test patterns. The test patterns are chosen to control ISI. "Embedding" receiver equalization amounts to emulating it within your test equipment or applying the equalization to a captured waveform offline. With the eye opened, you can see how much ISI is left uncorrected and the impact of other signal impairments—just like you do for open eyes.

Emerging 56 Gbit/s specifications offer new approaches to gauging signal impairments independent of ISI.

Start with a square wave clock-like signal, 1010… for NRZ-PAM2 or (11)(00)(11)(00) for PAM4, and measure CRJrms (rms clock random jitter), and CDJpp (peak-to-peak clock deterministic jitter). Since there’s no signal on the data, no symbols to interfere, CRJrms and CDJpp measure RJ (random jitter) and DJ (deterministic jitter) independent of ISI, but retain the random noise and other uncorrelated impairments like crosstalk, PJ (periodic jitter), and any other EMI (electromagnetic interference). Apply the transmitter equalization scheme, if there is one, and embed the receiver CTLE and/or DFE as appropriate so that the square waveform also includes the equalized version of all the signal impairments that are left over.

Because the jitter distribution—think of a histogram of the time-delay of an eye diagram crossing point—includes both RJ and DJ, they must be separated. One separation technique gaining favor from the high rate standards uses the dual-Dirac model to extract CRJrms and CDJpp in a way that’s a bit upside down.

Example of a jitter distribution, this one has mostly DJ in the form of sinusoidal jitter
(photo by Ransom).

The dual-Dirac model is usually used to estimate TJ(BER), the total jitter defined at a BER (bit-error ratio) or, equivalently, the eye width at a BER, EW(BER)—TJ is the eye closure and EW is the eye opening—from rms RJ and peak-to-peak DJ that has been extracted from the jitter distribution. Each test and measurement company uses its own technique for extracting RJ and DJ from oscilloscope or BERT data. With RJ and DJ in hand, the dual Dirac model is used to extrapolate down to low BERs, usually BER ≤1E-12, and get TJ(BER) or EW(BER) from:

TJ(BER) = 2Q(BER) x RJrms + DJpp

Where Q(BER) is a parameter that comes from the Gaussian nature of RJ that you can calculate from the inverse error function, Q(BER) = ¯2 × inverse-erf(1-BER) or look up in a table.

But to get CRJrms and CDJpp, we turn that approach upside down. Instead of using RJ and DJ to estimate TJ, we measure TJ at two approachable BERs, 1E-5 and 1E-6, and then solve the resulting dual Dirac expressions:

TJ(1E-5) = 8.83 x CRJrms + CDJpp
TJ(1E-6) = 9.78 x CRJrms + CDJpp


CRJrms = 1.05 x [TJ(1E-6) – TJ(1E-5)]
CDJpp = -9.3 x TJ(1E-6) + 10.3 x TJ(1E-5)

The resulting CRJrms and CDJpp give the underlying RJ and DJ, independent of the ISI that closes the eye. The dual-Dirac model they're extracted from assumes that RJ follows a Gaussian distribution and that the non-ISI peak-to-peak DJ is bounded. For the model to work, the DJ must be bounded at BERs considerably larger than 1E-5. Since ISI has been removed, and since it is what causes peak-to-peak DJ to have long, low BER tails, assuming that DJ is bounded at high BER is perfectly reasonable.

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