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To perform an eye mask test, engineers used to draw a mask on the display of an analog oscilloscope with a grease pencil. Next, they had to trigger the oscilloscope with clock signal, turn up the persistence, and if the interior of the sketch remained dark, the signal passed. From there, we went to well-defined masks on both real-time and equivalent-time digital oscilloscopes; pixels that lit up within the mask were bad news. Modern mask tests specify a maximum allowed number of these “violations” over a large number of waveforms.

While eye-mask measurements give you a quick way to gauge if a transmitter is working, they don’t tell you much about the only thing that matters: the bit-error ratio (BER, the number of bit errors to the total number of transmitted bits). **Figure 1** shows a mask in an eye diagram where no waveforms enter the mask.

In his presentation at DesignCon 2018, Teledyne LeCroy’s Marty Miller made a subtle point that relates eye mask measurements to BER-contour measurements.

**BER contours**

BER contours are like topographical maps, which show contours of constant elevation. BER contour measurements show an eye diagram with contours of equal BER.

Another way to think of them is as a three-dimensional bathtub plot. A bathtub plot (**Figure 3**) measures BER as a function of the time-delay position of the sampling point, BER(*t*). Because a BER contour includes the BER variation for different positions of the sampling point in both time-delay and voltage (or power). BER contours are the solutions to BER(*t*, *V*) = constant.

Dr. Miller pointed out that the BER of a real sampler is not the BER at any *point* (*t*, *V*). It’s not the BER at a point along or even a point within a BER contour. The relevant BER is the total BER encompassed by the sampler’s timing resolution and voltage sensitivity.

The receiver’s bandwidth sets the timing resolution—the length of time it takes to sample a bit—what we think of as setup and hold. The voltage sensitivity is the smallest peak-to-peak voltage swing between logic 1s and 0s that a sampler can reliably distinguish. Picture a little ellipse with semi-major axis given by the sampler’s timing resolution and semi-minor axis given by its voltage sensitivity—the actual geometry of the sampler resolution/sensitivity varies by part, but an ellipse is an obvious guess. Typical voltage slicers in high speed serial applications have timing resolutions of a few picoseconds and voltage sensitivities around 30 mV (**Figure 4**).

**Mask-error ratio**

Dr. Miller defines the mask error ratio (MER) as the ratio of the number of times that single-bit waveforms, each one unit-interval wide, enter the mask to the total number of bits transmitted (**Figure 5**). Because the waveform of a single bit can cause many mask violations but only one mask error, MER is not the same as the number of mask violations.

We can think of the MER as the BER that would be experienced by a real logic decoder whose timing resolution and voltage sensitivity are defined by the area of the mask.

To relate the BER-contour to the MER, Dr. Miller studied four cases. First, a mask thin in the time coordinate that extends from the center of the eye vertically up to the 1E-4 BER contour. This mask corresponds to a sampler with zero (i.e., perfect) timing resolution and voltage sensitivity given by the voltage swing from the center of the eye to the 1E-4 contour. In this special case, BER is about the same as MER.

Second, extend that same mask downward so that it’s essentially a thin vertical rectangle from the bottom to the top of the BER=1E-4 contour. For this mask, MER=2E-4, twice the value of the BER contour. At the top of the mask 1s are mistaken for 0s and at the bottom of the mask 0s are mistaken for 1s; over a large number of samples we get twice as many errors as we would for a perfect sampling point set at either the top or bottom of the 1E-4 contour.

In the third experiment, rotate that mask 90 degrees. Essentially a sampler with perfect voltage sensitivity but timing resolution that extends across the BER=1E-4 contour horizontally. For similar reasons, this experiment also gives MER=2E-4.

So far, all we’ve done is measure a few simple cases that show how nonzero timing resolution and voltage sensitivity cause real samplers to have BERs greater than or equal to the BER-contours associated with them.

In the fourth test Dr. Miller defined a mask that coincided with the BER=1E-4 contour; the mask is defined by the closed curve that the BER=1E-4 contour circumscribes. He measured MER=6.27E-4. It’s unlikely that the factor 6.27 expresses anything other than the degree of independence of separate bit errors.

Let’s reconsider the question of how we can relate the good old mask test to the BER performance of a system. Suppose we have a set of receivers that pass the same mask test. Dr. Miller concluded that there is no obvious way to use a mask test to predict maximum BER performance, essentially validating the shift of recently released standards from mask tests to eye height (EH) and eye width (EW) measurements defined with respect to a BER.

“Mask testing,” said Miller, “is more complex and perhaps in its common form is not really an objective metric for specifying channel performance (e.g. effective BER). If anything, the experiments regarding thin vertical and horizontal masks indicate the notions of eye height and eye width at BER are perhaps a better metric.”

—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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