Oscilloscope Basics 2: Sampling Rate

1. Nyquist Sampling Theorem

What sampling rate is required for digital measurement applications? Some engineers believe in Nyquist's theorem and believe that a sampling rate of twice the bandwidth of the oscilloscope is sufficient. Other engineers do not believe in digital filtering techniques based on the Nyquist criterion and prefer to use an oscilloscope with a sampling rate of 10 to 20 times the bandwidth specification. The actual situation is somewhere in between. To understand why, you must understand Nyquist's theorem and its relationship to the frequency response of an oscilloscope. Dr. Harry Nyquist (Figure 1) posited that:

Nyquist sampling theorem For a finite bandwidth signal with a maximum frequency fMAX, the equally spaced sampling frequency fS must be greater than twice the maximum frequency fMAX in order to uniquely reconstruct the signal without aliasing.

The Nyquist sampling theorem can be summarized into two simple rules, however, it is not that simple for DSO technology.

1. The highest frequency component collected must be less than half the sampling rate.

2. The second rule is that samples must be taken at equal intervals, which is often forgotten.

What Nyquist calls fMAX is what we usually refer to as the Nyquist frequency (fN), which is different from the oscilloscope bandwidth (fBW). If the oscilloscope bandwidth is specified exactly at Nyquist (fN), it means that the oscilloscope has an ideal brickwall response that is completely attenuated at this same frequency (as shown in Figure 2). Frequency components below the Nyquist frequency are completely passed (gain = 1), and frequency components above the Nyquist frequency are completely rejected. However, this frequency response filter cannot be implemented in hardware.

Figure 2 - Ideal brick-wall frequency response

Most oscilloscopes with bandwidth specifications of 1 GHz and below have a response type called a Gaussian frequency response. As the signal input frequency approaches the specified bandwidth of the oscilloscope, the measured amplitude will slowly decrease. The signal will be attenuated by 3 dB (~30%) at the bandwidth frequency. If the bandwidth of the oscilloscope is specified exactly as Nyquist (fN) (as shown in Figure 3), the components of the input signal above this frequency will be sampled (red shaded area) even though they are attenuated by more than 3 dB, especially when the input signal contains fast edges (when measuring digital signals). This phenomenon violates the first rule of the Nyquist sampling theorem.

Figure 3 - Typical oscilloscope Gaussian frequency response with bandwidth (fBW) specified as the Nyquist frequency (fN)

Most oscilloscope vendors do not specify the bandwidth of their oscilloscope at the Nyquist frequency (fN), although some do. However, waveform recorder/digitizer vendors often specify the bandwidth of their instruments at the Nyquist frequency. Now let's look at what happens if the bandwidth of an oscilloscope is the same as the Nyquist frequency (fN).

Figure 4 shows a 500-MHz bandwidth oscilloscope sampling at exactly 1 GSa/s when operating in three- or four-channel mode. Although the fundamental frequency (clock frequency) of the input signal is within the Nyquist range, the signal edges contain significant frequency components that fall well outside the Nyquist frequency (fN). A closer look reveals that the signal edges have varying degrees of preshoot, overshoot, and a variety of edge speeds, which appear to be “erratic.” This is a sign of aliasing, and it clearly shows that simply using an oscilloscope with a bandwidth twice the sampling rate is not enough to make reliable digital signal measurements.

Figure 4 - Aliased edges resulting from sampling with a 1 GSa/s sampling rate and 500-MHz bandwidth oscilloscope

So how does the definition of an oscilloscope's bandwidth (fBW) relate to the oscilloscope's sampling rate (fS) and Nyquist frequency (fN)? To minimize the acquisition of frequency components beyond the Nyquist frequency (fN), most oscilloscope vendors specify the bandwidth of their oscilloscopes with a typical Gaussian frequency response as 1/4 to 1/5 or less of the real-time sampling rate (as shown in Figure 5). Although sampling at a rate many times greater than the oscilloscope's bandwidth can further reduce the likelihood of acquiring frequency components outside the Nyquist frequency (fN), a 4:1 sampling rate to bandwidth ratio is sufficient to obtain reliable digital measurements.

Figure 5 - Limiting the oscilloscope bandwidth (fBW) to 1/4 the sampling rate (fS/4) reduces frequency components above the Nyquist frequency (fN).

Oscilloscopes with bandwidth specifications in the 2-GHz and higher range typically have a steeper frequency rolloff response/characteristic. We call this type of frequency response a "maximally flat" response. Since an oscilloscope with a maximally flat response approaches the ideal characteristics of a brick-wall filter, where frequency components beyond Nyquist are more attenuated, it can provide a good display of digitally filtered input signals without taking multiple samples. In theory, a vendor could specify the bandwidth of an oscilloscope with this type of response (assuming the front-end analog hardware is capable of that) as fS/2.5.

Figure 6 shows a 500-MHz bandwidth oscilloscope capturing a 100-MHz clock signal with edge speeds in the 1 ns (10% - 90%) range. The 500 MHz bandwidth specification is the minimum recommended bandwidth to accurately capture this digital signal. This particular oscilloscope is capable of sampling at 4 GSa/s in dual-channel mode, or 2 GSa/s in triple or quad-channel mode. Figure 6 shows an oscilloscope sampling at 2 GSa/s, which has a sampling frequency that is twice the Nyquist frequency (fN) and four times the bandwidth frequency (fBW). This graph shows that an oscilloscope with a 4:1 sampling rate to bandwidth ratio can provide a very stable and accurate representation of the input signal. And, with the help of Sin(x)/x waveform reconstruction/interpolation digital filtering technology, this oscilloscope can achieve waveform and measurement resolutions on the order of tens of picoseconds. Compared to our previous example shown in Figure 4 (using an oscilloscope of the same bandwidth, but sampling at only twice the bandwidth (fN), the difference in waveform stability and accuracy is immediately apparent.

Figure 6 - Using a Keysight 500-MHz bandwidth oscilloscope sampling at 2 GSa/s, this 100-MHz clock signal with an edge speed of 1 ns can be accurately measured.

So what happens if we double the sample rate to 4 GSa/s and sample with the same 500-MHz bandwidth oscilloscope (fBW x 8)? You might intuitively think that this oscilloscope would produce better waveforms and measurements. But as Figure 7 shows, you can only achieve a small improvement. If you look closely at the two waveform plots (Figures 6 and 7), you will see that there is only a slight preshoot and overshoot in the displayed waveform when sampling at 4 GSa/s (fBW x 8). However, the rise time measurement shows the same result (1.02 ns). The key to the slight improvement in waveform fidelity is that no additional error sources were introduced when the oscilloscope's sample rate to bandwidth ratio was increased from 4:1 (2 GSa/s) to 8:1 (4 GSa/s). This brings us to the topic of our discussion: What happens if Nyquist's second rule is violated? Nyquist emphasized that samples must be taken at equal intervals. Users often overlook this important rule when evaluating digital storage oscilloscopes.

Figure 7 - Sampling at 4 GSa/s with a Keysight 500-MHz bandwidth oscilloscope provides minimal improvement in measurement performance compared to sampling at 2 GSa/s

2. Cross-cutting real-time sampling

How can oscilloscope vendors create oscilloscopes with higher sampling rates when ADC technology has reached its limits in terms of maximum sampling rate? The pursuit of higher sampling rates may simply be to satisfy the oscilloscope user's perception that "higher is better", or the user believes that a higher sampling rate may actually be required to obtain higher bandwidth real-time oscilloscope measurements. However, making an oscilloscope with a higher sampling rate is not as simple as selecting an off-the-shelf analog-to-digital converter with a higher sampling rate.

A common technique used by all major oscilloscope vendors is to interleave multiple real-time ADCs. However, this interleaved sampling technique should not be confused with the repetitive acquisition technique, which we call “equivalent-time” sampling.

Figure 8 shows the block diagram of a real-time interleaved ADC system consisting of two ADCs using phase-delayed sampling techniques. In this example, ADC 2 always samples ½ clock cycle after ADC 1 samples. After each real-time acquisition cycle is completed, the oscilloscope’s CPU or waveform processing ASIC retrieves the data stored in each ADC’s ​​acquisition memory and then interleaves the samples to obtain a real-time digitized waveform with double the sample density (twice the sampling rate).

Oscilloscopes with real-time interleaved sampling features must comply with two requirements. First, to achieve accurate interleaving without distortion, the vertical gain, offset, and frequency response of each ADC must be strictly matched. Second, the phase-delayed clock must be calibrated with high precision to meet the requirements of Nyquist Rule II, which is equally spaced sampling. In other words, the sampling clock of ADC 2 must be delayed exactly 180 degrees after the sample ADC 1. Both conditions are very important for accurate interleaving. However, in order to have a more intuitive understanding of the errors caused by poor interleaving, the following article will focus on the errors caused by poor phase-delay timing alone.

Figure 8 - Real-time sampling system consisting of two interleaved ADCs

The timing diagram shown in Figure 9 illustrates that interleaved sampling can have timing errors if the two interleaved ADC phase-delayed clock systems are not exactly ½ sample period delayed from each other. The diagram shows the points digitized in real time (red dots) relative to where the input signal actually converted. However, due to the imperfect phase-delay timing alignment (purple waveform), these digitized points are not sampled equally spaced, thus violating Nyquist's second rule.

Figure 9 - Timing diagram for non-equally spaced sampling

When the oscilloscope’s waveform processing engine retrieves the data stored in each ADC acquisition memory, it first assumes that the samples in each memory device are equally spaced. When you try to reconstruct the shape of the original input signal, the signal represented by the oscilloscope’s Sin(x)/x reconstruction filter will be severely distorted (as shown in Figure 10).

Real-time acquisition distortion (sometimes called “sampling noise”) can be misinterpreted as random noise when you view repetitive acquisitions because the phase relationship between the input signal and the oscilloscope’s sample clock is random. However, this phase relationship is not completely random, but rather has some determinism and is directly related to the oscilloscope’s sample clock.

Figure 10 - This timing diagram shows a waveform distorted by poor phase delay timing, reconstructed using a Sin(x)/x filter.

3. Crossover distortion test

Oscilloscope vendors do not provide customers with specifications in their DSO data sheets that can directly quantify the digital processing of the oscilloscope. However, there are still a variety of tests that can be easily performed to not only measure the effects of sampling distortion, but also to determine and quantify sampling distortion. Below is a list of tests that can be performed on oscilloscopes to detect and compare crossover distortion:

Crossover distortion test

1. Effective number of bits analysis using sine wave

2. Sine wave comparison test

3. Spectrum Analysis

4. Measurement stability

Effective digit analysis

Some oscilloscope vendors offer the most stringent specification for quantifying sampling fidelity, the effective number of bits (ENOB). However, ENOB is a composite specification made up of several error components, including input amplifier harmonic distortion and random noise. While the ENOB test can provide a good baseline comparison of the overall accuracy between different oscilloscopes, the ENOB concept is not an easy concept to understand and requires a lot of complex calculations to be performed on the digitized data before it is imported into a PC. Basically, the ENOB test extracts a theoretical best fit sinusoid from the digitized sinusoid. This sinusoid curve fitting algorithm will remove the errors introduced by the oscilloscope amplifier gain and offset errors. The test then calculates the RMS error of the digitized sinusoid relative to the ideal/extracted sinusoid over a period of time. This RMS error is then compared to the theoretical RMS error produced by an "N" bit ideal ADC. For example, if the oscilloscope acquisition system has an accuracy of 5.3 effective bits, then an ideal 5.3 bit ADC system should produce the same amount of RMS error.

You can also perform a more intuitive and simple test: simply input a sine wave generated by a high-quality signal generator (with a frequency close to the bandwidth of the oscilloscope being tested) to see if the oscilloscope will produce ADC interleaving distortion. Then make a judgment on the filtered digitized waveform.

Additionally, you can measure the ADC distortion due to uncalibrated ADCs in the frequency domain using the scope's Fast Fourier Transformation capability. With a pure sine wave input, the ideal/undistorted spectrum should consist of a single frequency component at the input frequency. Anything else in frequency is a distortion component. You can also use this technique for digital clock signals, but the spectrum becomes more complicated, so you need to know where to start.

Another simple test that can be performed is to compare the stability of parametric measurements such as rise time, fall time, or standard deviation of Vp-p between oscilloscopes with similar bandwidths. If crossover distortion is present, it will produce unstable measurements just like random noise.

Sine wave comparison test

Figure 11 shows the simplest and most intuitive comparison test—the sine wave test. The waveform shown in Figure 11a is a single-shot capture of a 200-MHz sine wave using a Keysight InfiniiVision 1-GHz bandwidth oscilloscope with a sample rate of 4 GSa/s. This oscilloscope uses non-interleaved ADC technology, which gives a sample rate to bandwidth ratio of 4:1. The waveform shown in Figure 11b is a single-shot capture of the same 200-MHz sine wave using a LeCroy's 1-GHz bandwidth oscilloscope with a sample rate of 10 GSa/s. This oscilloscope uses interleaved ADC technology, which gives a maximum sample rate to bandwidth ratio of 10:1.

We would intuitively think that for oscilloscopes of the same bandwidth, the one with the higher sampling rate should produce more accurate measurements, but from this comparison we can see that the oscilloscope with the lower sampling rate actually represents the 200 MHz input sine wave more accurately. This is not because lower sampling rates are better, but because a poorly calibrated interleaved real-time ADC will negate the advantage of the higher sampling rate.

Accurately calibrated interleaved ADC techniques become more important for oscilloscopes with higher bandwidths and higher sample rates. While a fixed amount of phase-delayed clock error may not be important at lower sample rates, at higher sample rates (lower sample periods), the same amount of phase-delayed clock error becomes very important. Now we will compare a higher bandwidth oscilloscope that uses real-time interleaving techniques with a higher bandwidth oscilloscope that does not use this technique.

Figure 11a - 200-MHz sine wave captured using a Keysight 1-GHz bandwidth oscilloscope at 4 GSa/s sampling rate

Figure 11b - 200-MHz sine wave captured using a LeCroy 1-GHz bandwidth oscilloscope sampling at 10 GSa/s

Figure 12 shows a screenshot of two sine wave tests comparing a 2.5 GHz sine wave captured by a Keysight 3-GHz bandwidth oscilloscope at 20 GSa/s sampling rate (non-interleaved) and 40 GSa/s sampling rate (interleaved). This particular DSO uses a single-chip 20 GSa/s ADC behind each of the four channels. However, if only two channels of the oscilloscope are used, the instrument automatically interleaves the ADC pairs to provide a real-time sampling rate of no less than 40 GSa/s.

On the surface, we cannot observe much difference between the quality of these two waveforms. Both waveforms appear to be relatively pure sine waves with only minimal distortion. However, when we perform Vp-p statistical measurements, we find that the higher sample rate measurement results in slightly more stable measurements, which is consistent with what we would expect.

Figure 12a - 2.5-GHz sine wave captured using a Keysight Infiniium oscilloscope at 20 GSa/s (non-interleaved) sampling rate

Figure 12b - 2.5-GHz sine wave captured using a Keysight Infiniium oscilloscope at 40 GSa/s (interleaved) sampling rate

Figure 13 shows a set of sine wave tests comparing a Tektronix 2.5-GHz bandwidth oscilloscope capturing a 2.5 GHz sine wave at a 10 GSa/s sample rate (non-interleaved) versus capturing the same 2.5 GHz sine wave at a 40 GSa/s sample rate (interleaved). This particular DSO uses a single-chip 10 GSa/s ADC behind each of the four channels. However, if only one channel of the oscilloscope is used, the instrument automatically interleaves the four ADCs to provide a real-time sample rate of no less than 40 GSa/s in one channel.

In this sine wave test, we can see a significant difference in waveform fidelity between each sample rate setting. When the scope is sampling at 10 GSa/s (Figure 13a) without an interleaved ADC, its display of the input sine wave is quite good—although the Vp-p measurement is only one-quarter as stable as that of a similar bandwidth Keysight oscilloscope. When sampling at 40 GSa/s (Figure 13b) with the interleaved ADC technique, we can clearly see the waveform distortion produced by the Tek DSO, and the stability of the Vp-p measurement is similarly poor. This is counter-intuitive: most engineers would expect to get more accurate and stable measurements when sampling at a higher sample rate with the same oscilloscope. The main reason this measurement is inconsistent with intuitive expectations is poor vertical and/or timing alignment of the real-time interleaved ADC system.

Figure 13a - 2.5-GHz sine wave captured using a Tektronix 2.5-GHz sine wave sampled at 10 GSa/s (non-interleaved)

Figure 13b - 2.5-GHz sine wave captured using a Tektronix 2.5-GHz sine wave sampled at 40 GSa/s (interleaved)

Spectrum analysis comparison test

Sine wave testing does not truly identify the source of distortion, but only shows the effects of the various errors/components of distortion. However, spectrum/FFT analysis can correctly determine the components of distortion, including harmonic distortion, random noise, and cross-sampling distortion. When using a sine wave generated by a high-quality signal generator, there should be only one frequency component in the input signal. Any frequency components other than the fundamental frequency detected by performing an FFT analysis on the digitized waveform are distortion components introduced by the oscilloscope.

Figure 14a shows the results of an FFT analysis of a 2.5 GHz sine wave captured at 40 GSa/s using a single shot capture on a Keysight Infiniium oscilloscope. The worst distortion spur is measured at approximately 90 dB below the fundamental frequency. This distortion component is actually second harmonic distortion, most likely generated by the signal generator. Its magnitude is extremely negligible, even below the in-band noise floor of the oscilloscope.

Figure 14b shows the results of an FFT analysis of the same 2.5-GHz sine wave captured in a single shot using a Tektronix oscilloscope, also sampling at 40 GSa/s. The worst distortion spur in this FFT analysis is measured approximately 32 dB below the fundamental frequency. This high level of distortion explains why the sine wave test (Figure 13b) produced a distorted waveform. This distortion frequency occurs at 7.5 GHz, which is exactly 10 GHz below the input signal frequency (2.5 GHz), but folded back into the positive domain. The next highest distortion component occurs at 12.5 GHz. This is exactly 10 GHz above the input signal frequency (2.5 GHz). Both of these distortion components are directly related to the 40-GSa/s sampling clock and its crossover clock frequency (10 GHz). These distortion components are not caused by random or harmonic distortion, but rather by real-time crossover ADC distortion.

Figure 14a - FFT analysis of a 2.5-GHz sine wave captured with a Keysight Infiniium oscilloscope at 40 GSa/s.

Figure 14b - FFT analysis of a 2.5-GHz sine wave captured with a Tektronix oscilloscope at 40 GSa/s.

Digital clock measurement stability comparison test

As a digital designer, you might say that you don't really care about distortion of analog signals such as sine waves. However, it is important to remember that all digital signals can be decomposed into an infinite number of sine waves. If the fifth harmonic of the digital clock is distorted, then the resulting digital waveform will also be distorted.

Although it is difficult to test the sampling distortion of digital clock signals, it can still be done. However, we do not recommend visual distortion testing of digital signals. This is because there is no absolutely "pure" digital clock generator. Even a digital signal generated by a high-performance pulse generator will have varying degrees of overshoot or perturbation and will have different edge speeds. In addition, due to the pulse response characteristics of the oscilloscope and the frequency response that may not be flat, the front-end hardware of the oscilloscope may cause distortion of the pulse waveform of the digitized signal.

However, some tests can be performed using a high-speed clock signal to compare the measurement quality of the oscilloscope ADC system. One such test can compare the stability of parameter measurements, such as the standard deviation of rise and fall times. Cross-sampling distortion will cause unstable edge measurements and add deterministic jitter components to the high-speed edges of digital signals.

Figure 15 shows two oscilloscopes with similar bandwidths capturing and measuring the rise time of a 400 MHz clock signal with edge speeds in the 250 ps range. Figure 15a shows a Keysight 3 GHz bandwidth oscilloscope interleaved with a pair of 20-GSa/s ADCs sampling the signal at 40 GSa/s, resulting in a repetitive rise time measurement with a standard deviation of 3.3 ps. Figure 15b shows an image of a Tektronix 2.5 GHz bandwidth oscilloscope interleaved with four 10-GSa/s ADCs also sampling the signal at 40 GSa/s. In addition to showing more instability, the rise time of this digital signal has a standard deviation of 9.3 ps. The more accurate ADC calibration in the Keysight oscilloscope, combined with the lower noise floor, allows the Keysight oscilloscope to more accurately capture the higher frequency harmonics in this clock signal, providing a more stable measurement.

Figure 15a - 400-MHz clock signal captured using a Keysight Infiniium 3-GHz oscilloscope at 40 GSa/s sampling rate

Figure 15b - 400-MHz clock signal captured using a Tektronix 2.5-GHz oscilloscope at 40 GSa/s sampling rate

When using FFT to analyze the frequency components of a digital clock signal, its spectrum is much more complex than testing the spectrum of a simple sine wave. A pure digital clock pulse generated by a high-quality pulse generator should consist of a fundamental frequency component and its odd harmonics. If the duty cycle of the clock pulse is not exactly 50%, the spectrum will also contain low-amplitude even harmonics. However, if you know what to measure and what to ignore, you can use the oscilloscope's FFT math function to measure the cross-sampling distortion of digital signals in the frequency domain.

Figure 16a shows the spectrum of a 400-MHz clock signal captured with a Keysight 3-GHz bandwidth oscilloscope at 40 GSa/s sampling rate. The only frequency spurs that can be observed are the fundamental frequency component, the third, fifth, and seventh harmonics, and some even harmonics. All other spurious signals in the spectrum are well below the in-band noise floor of the oscilloscope.

Figure 16b shows the spectrum of a 400-MHz clock signal captured using a Tektronix 2.5-GHz bandwidth oscilloscope, also at 40 GSa/s sampling rate. In this FFT analysis, we can see not only the fundamental frequency component and its associated harmonics, but also several spurious signals at higher frequencies (around 10 GHz to 40 GHz). These spurious signal images are directly related to a poorly calibrated interleaved ADC system.

Figure 16a - FFT analysis of a 400-MHz clock signal using a Keysight Infiniium 3-GHz bandwidth oscilloscope

Figure 16b - FFT analysis of a 400-MHz clock signal using a Tektronix 2.5-GHz bandwidth oscilloscope

4. Conclusion

There are many factors other than sampling rate that affect the fidelity of an oscilloscope's signal. In some cases, an oscilloscope with a lower sampling rate can produce more accurate measurements.

Meeting the Nyquist condition requires the oscilloscope to sample at a rate three to five times higher than the oscilloscope's bandwidth specification, depending on the oscilloscope's frequency attenuation characteristics. To achieve higher sampling rates, oscilloscope vendors often require interleaving of multiple real-time ADCs. However, if real-time interleaving is used, it is critical that the interleaved ADCs are vertically matched and that the timing of the phase-delayed clock pulses is precise. It is important to note that the issue is not the number of interleaved ADCs, but the accuracy of the interleaving. Otherwise, the second rule of Nyquist (equally spaced sampling) is violated, which will produce distortion and negate the expected benefit of a higher sampling rate oscilloscope.

If you compare the waveform fidelity of oscilloscopes of similar bandwidth, you will find that Keysight real-time oscilloscopes, with their ultra-high precision ADC technology, can provide a super-fidelity display of the input signal.