Let’s say you need to measure a repetitive signal with respect to time. This is usually a straightforward task, simply set the oscilloscope trigger in the right place and stop the sample where you want it. Much to the annoyance of engineers making any measurements, this gets quite difficult and expensive as the speed goes up. A 2GHz DSO is a pretty normal desktop scope for an electrical engineer, but 2GHz ≲ 50GHz.

This is not to say that you cannot buy a 50GHz DSO, it’s just that you may need to sell your house, with a normal MSRP of around $600,000 — excluding the rest of the measurement setup, which you need an RF engineer to design. There are quite literally no probes that go this fast, so you’re gonna need to figure that one out on your own, also.

Understandably this makes some companies uncomfortable, especially if they only need to measure digital interfaces — and moreover do not need real-time analysis. It is one thing to make a live measurement on an InfiniBand link — for whatever freak needs to do that — but it is a slightly easier one to make a longer measurement, disregarding the individual bits, and instead focusing on the rise and fall of the signals.

Eye Eye, Captain!

I always feel like a real grownup when I get the eye diagram going on my DSO. For those not familiar with this glorious form of data representation, it is a very efficient format for expressing the behavior of a digital line.

A real-time measurement of an HDMI link made on an obscenely expensive oscilloscope. Credit: Diodes Incorporated

If you wanted to garner the line data from just this representation, you are more or less out of luck. The eye diagram is the accumulation of a large number of samples of a high-frequency digital signal (1.2M samples per “unit interval” in this case, where a UI is roughly just the width of time from crossing to crossing).

In interfaces such as HDMI, there is a positive and negative state, each correlating to a 1 or a 0. It follows that if you repetitively sample this signal in time with the data, regardless of the value of the data, you will end up with this eye shape. Usually, you will either have a clock signal or be able to recover the clock from the data lines using a technique insightfully called clock recovery.

So let’s say I want to make an eye diagram just like this one, but I rather not cough up $500K for it. We can abstract this to a data line, and a clock line that is in time with the data. Normally, with a real-time oscilloscope, we would need to measure the voltage values seen at the data line and maybe the clock as well, usually using a wickedly fast ADC. That’s the expensive part, so let’s change the architecture.

Instead of measuring constantly, let’s only start measurement at one point in time, like when our clock rises past 50%. When we measure the data line at that instant, we will get some voltage value. We can mark that value down, it’s okay to take as long as we need to do this, as the data line will continue to be in phase with the clock for as long as we need it to. Here’s the magic. To make the next measurement, instead of measuring at the same point again, let’s delay the measurement from this trigger. Just a tiny, itty bitty amount. Let’s do this by a predictable interval every single time until we have a pretty good picture of the signal.

This is pretty good! Decreasing the delay time, then, will allow us to measure with higher effective bandwidth without needing to resort to exponentially more expensive ADCs. In fact, our time domain resolution is mostly limited by how finely I can delay the signal, and the certainty of the “sampling window” of the sampling system, often called a sample and hold. This sampling window is the window of time during which the sample is taken. There will be some error in the amplitude measurement by virtue of this window, as we are uncertain where within this window the reported amplitude is — only that it is contained within this window.

Just one more thing — since the data is unpredictable, we will also get falling edges and no edges at all with pretty decent frequency. That will simply become another part of our eye diagram. Finally, we have some form of simplistic eye diagram.

As a bonus exercise, let us calculate the bandwidth of this system. Let us consider that each point is spread by 500ps, meaning that each time we incremented by a repeatable 500ps — disregarding sampling aperture jitter. Thus, since our signal is going from -1 to 1 unit, our 10% point is 0.2 units, and our 90% is 1.8 units. That would give us a 10-90 rise time of approximately 564ps. The bandwidth of the system from this information can be approximated by the equation:

This gives a bandwidth of 620MHz. The wave under measurement has an edge bandwidth of 1.1GHz by the same approximation, in comparison. This relationship would, of course, scale to the unit interval chosen — as well as by a smaller step size. Modern systems can reach into the 100GHz range. Now that’s fast.

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Approximating Via Characteristics