Getting error-free data across a high-speed serial interface can be a challenge, whether that interface is a Gigabit Ethernet physical layer connecting a client to a router, or a low-voltage differential signaling port sending high-definition video content to a monitor. However, by determining the quality of the serial channel through bit-error-rate (BER) testing to establish the error rate, and using eye patterns to provide a visual representation of the stability and margins of the physical channel, you can maximize the data-transfer speed while minimizing the number of errors.

From a user’s point of view, the fundamental performance metric for a digital communication system is the bit-error rate (BER). Generally, users are more interested in the digital information than in the causes of information loss. (Information loss and the improvement of methods for data transfer are left for engineers to figure out.) For users, the simplest and most accurate test for overall performance of a communication system remains the BER test, which provides in a single statistic a useful gauge of fidelity for the entire system.

Unfortunately, BER testing has some drawbacks: the equipment needed for measurements tends to be expensive, the effect of process and temperature on signal timing can give false readings, the test time is directly proportional to test quality, and low BER readings provide no indication of what caused the problem. Again, users focus on how well the system works, but improving the system is a job for engineers.

Many designs (non-wireless) target a BER of one per trillion (BER = 10-12) \[1\]. With all of the trillions of bits moving around the Internet, it would be beneficial to further improve on this number. Yet despite the complexity and the time and cost expended to perform a valid BER test, such tests provide no clue as to the causes of information loss. BER tests are great for users, but engineers intent on understanding the cause of a bit-error problem frequently use another tool that adds an analog supplement—eye diagrams—to the digital BER tests. However, the analog domain doesn’t have the luxury of error checking and error correction that’s possible in the digital domain.

Eye diagrams have become ubiquitous among digital communication/network engineers, especially since the advent of digital oscilloscopes. After looking at several eye diagrams, a trained communications engineer can often make an accurate guess as to the source of the problem. Eye diagrams have been used to gauge the performance of digital transmission systems since the days of RS-232 communications, and they continue to provide guidance and suggest routes to improvement. Forty years later, digital communication protocols still use eye diagrams to judge signal integrity.

USB specifications require an eye diagram to meet certain rise and fall times, as well as timing-jitter tolerance \[2\]. Digital broadcast television uses 8-vestigial sideband modulation (8VSB) to produce an eight-level eye pattern for estimating performance. Similarly, eye patterns on high-speed Ethernet and Sonet networks can provide visual feedback to digital diagnostic tools that can help pinpoint potential performance problems. Now, many BER-test manufacturers are adding eye-diagram capability to their machines \[1\].

Eye Diagram Measurements

The distinct advantage of eye diagrams over the BER test is the indication of the source of problems and paths for improvement. Eye diagrams are produced by sampling random data streams and then superimposing the samples in every trigger interval. Because each sample is random, the superposition produces all sorts of patterns and their permutations on top of each other, producing what faintly resembles an eye (Fig. 1).

In the early days of analog oscilloscopes, engineers sketched jitter variations using various input signals \[3\]. Today’s digital scopes perform this task while providing additional capabilities. The CSA8000 from Tektronix, for instance, lets you set the length of time for sampling (persistence), take a histogram for timing jitter and amplitude variation, and list statistical data for each parameter, such as the mean, median, and standard deviation. In short, it gives you the precise quantitative data necessary for statistical purposes in estimating the BER. The CSA8000 normalizes statistical data as Gaussian variables.

In an ideal channel with no timing jitter, the transition point occurs at the same instant in every time interval. Due to jitter, however, there’s some variation in the transition point (Fig. 2). Jitter consists of random jitter (RJ) and deterministic jitter (DJ). Random jitter is unbounded, and can be described by a Gaussian random variable. Deterministic jitter has many sources, and is bounded. The histogram in Figure 2 measures total jitter (TJ), which is the sum of random jitter and deterministic jitter (TJ = RJ + DJ).

Various techniques are available for isolating the random component of jitter. Both random jitter and deterministic jitter should be taken into account when estimating the BER. However, n othing beats the accuracy of a full-blown BER test. Therefore, eye-diagram estimates should not be used as substitutes for a BER test.

BER Estimation Using Eye Diagrams

Lab experience tells us that an open eye indicates low data loss and smooth operation. Thus, an ideal eye diagram is one in which every transition point occurs at the same instant with respect to every trigger interval. Functionally, we can represent that requirement by an ideal impulse function (Fig. 3)*.* The actual random jitter, which causes the transition points to vary in time, can sometimes be described by a random variable. The most common modeling of random jitter is Gaussian. We model the variation as Gaussian random variables because real systems model well with Gaussian distributions \[1\]; the mathematics for Gaussian random variables are easily understood; and many digital sampling scopes (such as the CSA8000) normalize the data into Gaussian statistics.

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Due to jitter, the transition point can be represented as a probability and represented by the Gaussian probability density function (Fig. 3, again). In another approach, you can model the sampling points as Gaussian random variables and find the conditional probability of error. Both methods give the same answer. The probability density function (PDF) for a2 in Figure 3 is:

, (1)

where a2 is the mean transition point, z is the random variable, and s is the standard deviation or RMS value. To find the probability that our random variable has no errors, integrate Equation 1 over the limits shown. The probability of error then becomes the area under the curve (Fig. 4). That area represents transitions by a2 that are sampled as a1 or a3, or transitions of a1 and a3 that are sampled as a2.

The area under the curve for random variable a2 is:

(2)

and

(3)

The total probability of error is the sum of the two equations multiplied by 2. A factor of 2 shows up because of the conditional probability associated with a1 and a3, which we assume is symmetrical with respect to the conditional probability of a2.

(4)

To solve for the probability of error for a2, the integral in Equation 4 has limits from sampling point a1 to infinity, and from sampling point a0 to negative infinity. Due to symmetry, this equation simplifies to Equation 5. Graphically, it represents the area shaded under the curve in Figure 4:

(5)

You may have forgotten how to solve Equation 5, but (thankfully) you don’t have to. The CSA8000 histogram gives you statistical data normalized as Gaussian random variables. Gaussian statistics are easy to use because you need only two parameters: the mean and the standard deviation. Usually, you can reduce this to a single parameter by normalizing the mean to zero (Fig. 5)*.*

The standard deviation represents random jitter, and you want (ideally) to separate random jitter from the deterministic jitter. To do that, you must feed a known pattern into the system and then cancel out the random jitter by averaging the known samples. We assume the noise and random jitter behave as Gaussian random variables with zero mean, and therefore an average of the samples cancels out the random jitter, leaving you with only the deterministic jitter. You can then modify the standard deviation to include the deterministic jitter, and proceed with an estimate of the BER using the new standard deviation. For a more detailed analysis of the computation of random and deterministic jitter, see reference \[4\].

Once the standard deviation is found, you calculate the number of standard deviations from the mean to the next sampling interval as the z number. Then, statistics will give you the probability as a function of distance from the mean (Fig. 6). Due to exponential drop-off, 6-sigma (6s) gives a probability of error close to one in a billion, while 7-sigma comes close to one in a trillion.

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If a sigma table isn’t available, you can solve Equation 5 within the proper limits by an approximation. You can normalize Equation 5 to zero-mean Gaussian by making a substitution:

Let .

Then dz = sdu, and Equation 5 simplifies to Equation 6:

(6)

For random variables beyond 3-sigma, the equation (fortunately) can be approximated:

(7)

Equation 7 can then be used to estimate the bit-error rate given a single variable (x). The value of x is the mean distance from the transition point to center, divided by the standard deviation (Fig. 7).

Example

Figure 8 shows an eye diagram based on the CSA8000 oscilloscope and a histogram taken at a transition point. The histogram gives statistical parameters such as the mean and standard deviation, shown on the right. The mean is normalized to zero, and the distance from the mean to the next sampling point, obtained through the use of cursors, is measured to be 710 ps. The standard deviation shown is 69.83 ps. The value of x is 10.2, and if you plug that value into Equation 7, it yields an estimate of the bit-error rate.

If you solve for the BER in Figure 8, you get an error probability that’s infinitesimally small. We have to remember that an open eye like that of Figure 8 indicates good signal quality across the channel. Different data rates can have the same BER, if the limitation is in the clock-data recovery (CDR) circuitry of the receiver. (The above analysis doesn’t take into account the jitter tolerance of the CDR circuitry.) Otherwise, bit errors are also caused by factors that include amplitude noise, bandwidth limitations, and signal distortions like overshoot and undershoot. As an engineer, you must understand the limitations of an estimate and know how to interpret it.

Zeeshawn Shameem, customer applications engineer, received a BSEE from the University of California at Los Angeles. He can be contacted at [email protected]

References:

1. Strassberg, Dan, “Eyeing Jitter,” *EDN*, pp. 42-52, May 2003.

2. “Universal Serial Bus Specification Revision 2.0,” pp. 131-165, April 2000.

3. Lauterbach, Michael, “Getting More out of Eye Diagrams,” *IEEE Spectrum*, pp. 61-63, March 1997.

4. Application Note 1181, “Measuring Random Jitter on a Digital Sampling Oscilloscope,”http://www.maxim-ic.com/appnotes.cfm/appnote_number/1181, September 2002.

5. Sklar, Bernard, “Digital Communications Fundamentals and Applications 2nd Edition,” pp. 105-136, January 2001.