Oscilloscope Bandwidth
All oscilloscopes exhibit a low-pass frequency response that rolls-off at higher frequencies, as shown in Figure 1. Most scopes with bandwidth specifications of 1GHz and below typically have what is called a Gaussian response, which exhibits a slow roll-off characteristic beginning at approximately one-third the -3dB frequency. Oscilloscopes with bandwidth specifications greater than 1GHz typically have a maximally-flat frequency response, as shown in Figure 2. This type of response usually exhibits a flatter in-band response with a sharper roll-off characteristic near the -3dB frequency.

There are advantages and disadvantages to each of these types of oscilloscope frequency responses. For instance, oscilloscopes with a maximally-flat response attenuate in-band signals less than scopes with a Gaussian response, meaning that scopes with maximally-flat responses are able to make more accurate measurements on in-band signals. However, a scope with a Gaussian response attenuates out-of-band signals less than a scope with maximally-flat response, meaning that scopes with Gaussian responses typically have a faster rise time than scopes with maximally-flat response, given the same bandwidth specification. But, sometimes it is advantageous to attenuate out-of-band signals to a higher degree in order to help eliminate higher-frequency components that can contribute to aliasing in order to satisfy Nyquist criteria (f_MAX< f_S).¹

Whether your scope has a Gaussian response, maximally-flat response, or somewhere in between, the lowest frequency at which the input signal is attenuated by 3dB is considered the scope's bandwidth. Oscilloscope bandwidth and frequency response can be tested with a swept frequency using a sine wave signal generator. Signal attenuation at the -3dB frequency translates into approximately -30% amplitude error. So you cannot expect to make accurate measurements on signals that have significant frequencies near your scope's bandwidth.

Closely related to an oscilloscope's bandwidth specification is its rise time specification. Scopes with a Gaussian-type response will have an approximate rise time of 0.35/f_BW based on a 10- to 90-percent criterion. Scopes with a maximally-flat response typically have rise time specifications in the range of 0.4/f_BW, depending on the sharpness of the frequency roll-off characteristic. But it is important to remember that a scope's rise time is not the fastest edge speed that the oscilloscope can accurately measure. It is the fastest edge speed the scope can possibly produce if the input signal has a theoretical infinitely fast rise time (0 ps). Although this theoretical specification is impossible to test (since pulse generators don't have infinitely fast edges) from a practical perspective, you can test your oscilloscope's rise time by inputting a pulse that has edge speeds that are 3 to 5 times faster than the scope's rise time specification.

Digital Applications
As a rule of thumb, your scope's bandwidth should be at least five times higher than the fastest digital clock rate in your system under test. If your scope meets this criterion, it will capture up to the fifth harmonic with minimum signal attenuation. This component of the signal is very important in determining the overall shape of your digital signals. But if you need to make accurate measurements on high-speed edges, this simple formula does not take into account the actual highest-frequency components embedded in fast rising and falling edges.

Rule of thumb: f_BW≤5 x f_clk

A more accurate method to determine required oscilloscope bandwidth is to ascertain the maximum frequency present in your digital signals, which is not the maximum clock rate. The maximum frequency will be based on the fastest edge speeds in your designs. So, the first step is to determine the rise and fall times of your fastest signals. You can usually obtain this information from published specifications for devices used in your designs.

You can then use a simple formula to compute the maximum "practical" frequency component. Dr. Howard W. Johnson refers to this frequency component as the "knee" frequency (f_knee).² All fast edges have an infinite spectrum of frequency components. However, there is an inflection (or "knee") in the frequency spectrum of fast edges where frequency components higher than f_knee are insignificant in determining the shape of the signal. To calculate f_knee:
f_knee=0.5/RT (10-90 percent)
f_knee=0.4/RT (20-80 percent)
For signals with rise time characteristics based on 10- to 90-percent thresholds, f_knee is equal to 0.5 divided by the rise time of the signal. For signals with rise time characteristics based on 20- to 80-percent thresholds, which is very common in many of today's device specifications, f_knee is equal to 0.4 divided by the rise time of the signal. (Don't confuse these rise times with a scope's specified rise time; we are talking about actual signal edge speeds.)

The third step is to determine the oscilloscope bandwidth required to measure this signal, based on your desired degree of accuracy when measuring rise times and fall times. Table 1 shows multiplying factors for various degrees of accuracy for scopes with a Gaussian or a maximally-flat frequency response. (Remember, most scopes with bandwidth specifications of 1GHz and below typically have a Gaussian-type response, and most scopes with bandwidths greater than 1GHz typically have a maximally-flat type response.)

Here is an example: Determine the minimum required bandwidth of an oscilloscope with an approximate Gaussian frequency response to measure a 500-ps rise time (10 to 90%)

If the signal has an approximate rise/fall time of 500 ps (based on a 10- to 90-percent criteria), then the maximum practical frequency component (f_knee) in the signal would be approximately 1GHz:

f_knee=(0.5/500ps) = 1 GHz

If you are able tolerate up to 20 percent timing errors when making parametric rise time and fall time measurements on your signals, then you could use a 1-GHz bandwidth oscilloscope for your digital measurement applications. But if you need timing accuracy in the range of 3 percent, then a scope with 2-GHz bandwidth would be the better choice. Let's now make some measurements on a digital clock signal with characteristics similar to this example, using scopes with various bandwidths.

Digital Clock Measurement Comparisons
Figure 3 shows the waveform results when measuring a 100MHz digital clock signal with 500-ps edge speeds (10 to 90 percent) using an Agilent MSO6014A 100-MHz bandwidth oscilloscope. As you can see, this scope primarily just passes through the 100-MHz fundamental of this clock signal, thus representing our clock signal as an approximate sine wave. A 100-MHz scope may be a good solution for many 8-bit, MCU-based designs with clock rates in the 10- to 20-MHz range, but 100-MHz bandwidth is clearly insufficient for this 100-MHz clock signal.

A 500-MHz bandwidth oscilloscope is able to capture up to the fifth harmonic (Figure 4), which was our first rule-of-thumb recommendation. But when we measure the rise time, we see that the scope measures approximately 750 ps. In this case, the scope is not making a very accurate measurement on the rise time of this signal. It is actually measuring something closer to its own rise time (700 ps), not the input signal's rise time, which is closer to 500 ps. We need a higher-bandwidth scope for this digital measurement application if timing measurements are important.

With a 1GHz bandwidth scope, we have a much more accurate picture of this signal, as shown in Figure 5. When we select a rise-time measurement on this scope, we measure approximately 550 ps. This measurement is providing us with approximately 10 percent measurement accuracy and may be a very acceptable measurement solution, especially if capital funding is an issue. However, even this measurement using a 1-GHz bandwidth scope might be considered borderline. If we want to make edge-speed measurements with greater than 3-percent accuracy on this signal with 500-ps edge speeds, we really need to use a scope with 2-GHz bandwidth or higher, as we determined in the walk-through example earlier.

With a 2-GHz bandwidth scope, now we are seeing an accurate representation of this clock signal along with a very accurate rise time measurement of approximately 495 ps, as shown in Figure 6.

Some high-bandwidth oscilloscopes have upgradeable bandwidth, so, if 2-GHz bandwidth is sufficient for today, you can initially use an entry level 2-GHz scope and then upgrade all the way up to 13-GHz if you need additional bandwidth.

Analog Applications
Years ago, most oscilloscope vendors recommended that your scope's bandwidth should be at least three times higher than the maximum signal frequency. Although this 3X multiplying factor would not apply to digital applications based on clock rates, it still applies to analog applications, such as modulated RF. To understand where this 3-to-1 multiplying factor comes from, let's look at an actual frequency response of a 1GHz bandwidth scope.

Figure 7 shows a swept response test (20 MHz to 2 GHz) on a 1-GHz bandwidth oscilloscope. As you can see, at exactly 1 GHz, the input is attenuated by about 1.7 dB, which is well within the -3-dB limitation that defines a scope's bandwidth. However, to make accurate measurements on analog signals, you need to use the scope in the portion of the frequency band where it is still relatively flat with minimal attenuation. At approximately one-third the scope's 1-GHz bandwidth, this particular scope exhibits virtually no attenuation (0 dB). Recognize, however, that not all scopes exhibit this type of response.

The swept frequency response test shown in Figure 8 was performed on a 1.5-GHz bandwidth scope from another scope vendor. This is an example of a very non-flat frequency response. The characteristics of this response are neither Gaussian nor maximally-flat. It appears to be very peaked, which can result in severe waveform distortion for both analog and digital signals.

Unfortunately, a scope's bandwidth specification, which is the 3-dB attenuation frequency, says nothing about the attenuation or amplification at other frequencies. Even at one-fifth this scope's bandwidth, signals are attenuated by approximately 1 dB (10 percent). So, in this case, following the 3X rule of thumb would not be wise. When you are selecting a scope, it is a good idea to choose a reputable scope vendor and pay close attention to the relative flatness of the scope's frequency response.

For digital applications, consider selecting a scope with a bandwidth that is at least five times higher than the fastest clock rate in your design. However, if you need to make accurate edge-speed measurements on your signals, you will need to determine the maximum practical frequency present in your signal.

For analog applications, select a scope that has a bandwidth that is at least three times higher than the highest analog signal frequency in your designs. But this rule-of-thumb recommendation only applies to scopes that have a relatively flat response in their lower frequency band.

And when you are considering a scope for today's applications, don't forget about tomorrow. If your budget is flexible, buying a little extra margin today may save you money in the future.

References:

1. For a deeper understanding of Nyquist's sampling theory, refer to Agilent Application Note 1587, "Evaluating Oscilloscope Sample Rates vs. Sampling Fidelity."
2. High-Speed Digital Design, A Handbook of Black Magic, Howard Johnson, Martin Graham, 1993, Prentice Hall PTD, Prentice-Hall, Inc, Upper Saddle River, New Jersey 07458
3. For more information about selecting the right oscilloscope bandwidth, view a video demo.

About the author
Johnnie Hancock is a Signal Integrity Applications Engineer within Agilent Technologies Electronic Products Group. He began his career with Hewlett-Packard in 1979 as an embedded hardware designer, and holds a patent for digital oscilloscope amplifier calibration. Johnnie is currently responsible for worldwide application support activities that promote Agilent's digitizing oscilloscopes and he regularly speaks at technical conferences worldwide. Johnnie graduated from the University of South Florida with a degree in electrical engineering. In his spare time, he enjoys restoring his 112-year-old Victorian home located in Colorado Springs. Johnnie can be reached at johnnie_hancock@agilent.com.