The Fifth Harmonic: Tradeoffs Between Sampling and Real-Time Oscilloscopes

Choosing the right oscilloscope for your application is a complicated process, especially in today’s economy when budgets are tight and purchases must be exact. There are multiple types of oscilloscopes, including real-time (RT) and equivalent-time (ET) oscilloscopes. Within each type of scope, many different specifications must be considered. One of the most important specifications is bandwidth—if there’s not enough bandwidth to at least satisfy Nyquist, the signal will experience significant aliasing.

When choosing which bandwidth to purchase, there are no simple answers. Vendors will argue a “fifth harmonic” rule of thumb. The vendor will say that the fifth harmonic is all you need to determine how much bandwidth you need. However, it’s important to understand under typical conditions at the receiver that even if the oscilloscope has enough bandwidth to theoretically capture the fifth harmonic, it’s possible the oscilloscope won’t capture any fifth-harmonic content at all.

Given the economy of today, avoid purchasing an oscilloscope to capture the fifth harmonic based on a rule of thumb, considering that with its extra price it can only capture to the third harmonic. You could purchase a lower bandwidth oscilloscope that captures the same harmonic content for less money. That’s why it is necessary to know exactly how much bandwidth is truly needed and what features are required by a designer.

Finally, it’s important to understand the benefits and disadvantages of different oscilloscope types, including RT and ET oscilloscopes. Once you understand all of these specifications (including noise floor, dynamic range, and bandwidth) and how they affect measurements, you can then be sure you’re purchasing the correct oscilloscope for your application and needs.

THE IMPORTANCE OF NOISE FLOOR Data rates continue to climb—what was once state of the art a few years ago is now considered old technology. Data rates reaching 5 Gbits/s are becoming common. As a result, there’s no longer time to transition a bit from 0 to 1 with a 2-V signal. A number of serial technologies have peak-to-peak voltages of 800 mV or less, meaning their eyes are shrinking rapidly.

Every oscilloscope has an intrinsic noise floor. The higher the volt per division setting, the higher the oscilloscope noise floor. An 800-mV peak-to-peak signal will require a voltage per division setting of at least 100 mV/div to see the necessary details of the signal (Fig. 1).

If the intrinsic oscilloscope noise at 100 mV/div is 50 mV, then 8% of the signal is now oscilloscope noise. Even worse, if the noise floor is 100 mV, 13% of the signal is oscilloscope noise. Assuming that a signal’s eye is shrinking, 13% extra noise can cause an open eye to appear closed. Thus, designers may be tempted to overdesign. Typically, ET oscilloscopes have a lower noise floor than RT oscilloscopes. However, with proper research, you can find RT oscilloscopes with noise floors that are less than 5% of the signal discussed above.

DYNAMIC RANGE AND SIGNAL-TO-NOISE RATIO An important impact of a high noise floor is that it will affect an oscilloscope’s dynamic range and the signal-to-noise ratio (SNR). An oscilloscope’s SNR can be defined as the ratio of the largest signal level that an oscilloscope can handle to the smallest signal it can still distinguish between its noise floor, since a digital signal’s dynamic range and SNR can be used virtually interchangeably.

Dynamic range is important to understand, because it directly affects the amount of frequency content an oscilloscope can capture (this will be discussed in detail in the next section). Typically, an oscilloscope’s dynamic range, which is the difference between the largest spur and the scope’s noise floor, is defined in dB.

An oscilloscope’s dynamic range can also be determined by looking at an FFT of a sine-wave input into a scope. To make this measurement, a memory depth of less than 1000 points must be applied to the FFT of the oscilloscope. At memory depths greater than 1000 points, the scope will show higher dynamic range, since it’s now mimicking averaging mode or high-resolution mode (which are ways to increase the dynamic range).

One rule of thumb for calculating an oscilloscope’s dynamic range is its effective bits. The effective bits are directly tied to the bits with the oscilloscope’s ADC. These vary by the bandwidth of the oscilloscope—the higher the bandwidth of the scope, the lower the effective bits and the dynamic range. Ideally, RT oscilloscopes would have a dynamic range of around 50 dB (assuming 8 bits). This is calculated by finding the SNR of the oscilloscope.

SNR = 6.02N + 1.76 (where N is the effective bits)

ET oscilloscopes (typically 12 to 14 bits) could have a dynamic range as large as 85 dB. Typically, though, RT scopes will feature a dynamic range between 35 and 45 dB, while ET scope dynamic range falls between 50 and 70 dB.

A number of other factors can affect the dynamic range of an oscilloscope. Scope companies will boost their bandwidth with DSP to achieve higher bandwidths. If an oscilloscope is boosting the top bandwidth by greater than 20% from their highest analog bandwidth, it will significantly impact dynamic range at the greater bandwidth. While the increase will provide the oscilloscope more bandwidth, the dynamic range could shrink by as much as 10 dB. This can be seen with the oscilloscope’s FFT. An oscilloscope’s dynamic range can also be affected by interleaving an ADC. If the vendor has a number of interleaving errors in its ADC, it will affect the dynamic range.

FREQUENCY CONTENT Fourier synthesis states that all complex signals in their simplest forms can be constructed by adding sine waves of different frequencies and phases together. These frequencies can be seen in the frequency domain by plotting frequency against amplitude. The amount of sine waves that’s captured is known as the frequency content of a signal.

When looking at the frequency content of a 100-MHz square wave, there are multiple spurs known as harmonics and each is well defined. Also, each harmonic has a different amplitude (the first being the largest and gradually getting smaller). When these sine waves are added back together, we see the square wave. This means that the more harmonics (harmonic content) an oscilloscope is able to capture, the more accurately a signal can be displayed when reconstructed in the time domain.

Harmonics occur at every 0.5 times the bit rate. For example, a 4-Gbit/s signal would have significant harmonics at 2, 4, 6, 8, 10 GHz, going to infinity. In a perfect square wave with an instrument that has infinite dynamic range, getting the fifth harmonic requires an oscilloscope with at least 10 GHz of bandwidth. However, perfect square waves don’t exist in real life, especially in today’s high-speed digital signals. A number of factors erode the strength of the harmonics and can spread out the harmonic content. The most important factor is the signal’s rise time.

An ideal square wave has a rise time of near 0 ps. In real-world applications under the best designs, however, rise times at the receiver very rarely exceed 30 ps. Rise times are slowed by PCB material, connectors, traces, and distance. FR-4 material, popular due to its low cost, is one of the most common materials used in PCB material. FR-4 has an absolute maximum rise time of 45 ps; however, it will typically see maximum rise times of only 50 to 60 ps (80/20).

Consider the SATA 6G HFTP signal spectrum analysis with a rise time of 45 ps (Fig. 2). It has a bit rate of 6 Gbits/s (fifth harmonic at 15 GHz). However, the fifth harmonic has an amplitude that’s 45 dB less than the first harmonic. For an oscilloscope to display any of its content in the time domain, the oscilloscope must have a dynamic range of at least 45 dB. RT oscilloscopes have real-world dynamic ranges of 44 dB or less, meaning even RT scopes with 15 GHz or more of bandwidth will miss the fifth harmonic.

For the fifth harmonic to significantly influence the measurement of an RT scope (about 35 dB below the carrier frequency), the rise time must be faster than 30 ps. Such fast rise times don’t occur frequently in real-life signals. On the other hand, an ET oscilloscope has the dynamic range to capture the entire fifth harmonic of a SATA 6G signal. At this point, a designer must decide between using an RT oscilloscope that can capture through the third harmonic or an ET oscilloscope that can capture harmonic content equivalent to its bandwidth.

Another example to consider is PCI Express Gen 2. The Gen 2 signal has a data rate of 5 Gbits/s. This data rate would require an oscilloscope of at least 12.5 GHz to capture the fifth harmonic. However, the signal will typically have rise times from between 50 and 70 ps at the receiver. The result is that the fifth harmonic is 48 dB down from the zero crossing, meaning that a 12.5-GHz RT scope (dynamic range of 35 to 45 dB) won’t capture any of the fifth-harmonic content.

Incredibly, 8- and 12.5-GHz RT oscilloscopes will capture the same harmonic content of the PCI Express Gen II signal. For this signal to capture the fifth-harmonic content, it’s necessary to use an ET oscilloscope with its higher dynamic range. It should be noted that the PCI Express Sig currently requires at least 12.5 GHz for PCI Express Gen 2 testing. This is evident by the two real-time eyes shown in (Figs. 3a and 3b). Note that despite the extra bandwidth, the real-time eyes look identical at 60-ps rise times.

TRADEOFFS If the third harmonic is enough harmonic content, RT oscilloscopes offer a number of features not available with ET oscilloscopes. The most significant of these is that an RT scope will offer a sampling rate and memory depth significantly higher than an ET scope. An RT scope also doesn’t require an external trigger, because it will trigger itself. An ET scope, on the other hand, requires an outside trigger and takes multiple single-point acquisitions to display the data.

RT oscilloscopes tend to have more automated compliance software applications, adding convenience and speed to measurements. For PCI Express Gen 2, Agilent, LeCroy, and Tektronix RT scopes have automated software for measuring a number of key specifications. In contrast, ET scopes have more bandwidth and less noise. In addition to bandwidth, ET scopes provide a module that will allow for TDR measurements.

CONCLUSION Choosing the right oscilloscope for your application is, of course, very important. A general rule of thumb has been to use the fifth harmonic as a measure for how much bandwidth to purchase. However, for rise times slower than 30 ps, RT oscilloscopes won’t capture a significant amount of harmonic content from the fifth harmonic.

In fact, at high data rates (6 Gbits/s and faster) at the receiver in most cases, RT scopes won’t capture the fifth harmonic at all regardless of bandwidth. This is because the fifth harmonic’s content is below the dynamic range of an RT scope. To capture the fifth harmonic, it’s necessary to use an ET oscilloscope. RT scopes offer features such as deeper memory and sampling rate.

This all means that an oscilloscope purchaser must decide between capturing up to the third harmonic and getting the features of an RT scope or capturing the fifth harmonic with an ET scope. It should be noted that as rise times decrease below 30 ps, the fifth-harmonic content will become much more prevalent. As a result, in the future, RT scopes will also be able to capture the fifth harmonic.