evaluating oscilloscope sample rates 5989-5732en

8/11/2019 Evaluating Oscilloscope Sample Rates 5989-5732EN

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How much sample rate do you need foryour digital measurement applications?Some engineers have total trustin Nyquist and claim that just 2X

sampling over the scopes bandwidthis sufficient. Other engineers donttrust digital filtering techniques basedon Nyquist criteria and prefer that theirscopes sample at rates that are 10Xto 20X over the scopes bandwidthspecification. The truth actually liessomewhere in between. To understandwhy, you must have an understandingof the Nyquist theorem and howit relates to a scopes frequencyresponse. Dr. Harry Nyquist (Figure 1)postulated:

Nyquists Sampling Theorem

Nyquist Sampling TheoremFor a limited bandwidth signal witha maximum frequency f MAX , theequally-spaced sampling frequencyf S must be greater than twice themaximum frequency f MAX , in orderto have the signal be uniquelyreconstructed without aliasing.

Figure 1: Dr. Harry Nyquist, 1889-1976,articulated his sampling theorem in 1928

Nyquists sampling theorem can besummarized into two simplerulesbut perhaps not-so-simple forDSO technology.

1. The highest frequency componentsampled must be less than half thesampling frequency.

2. The second rule, which is oftenforgotten, is that samples mustbe equally spaced.

What Nyquist calls fMAX is what weusually refer to as the Nyquist fre-quency (fN), which is not the same asoscilloscope bandwidth (f BW). If anoscilloscopes bandwidth is specifiedexactly at the Nyquist frequency (f N),this implies that the oscilloscope hasan ideal brick-wall response that fallsoff exactly at this same frequency, asshown in Figure 2. Frequency compo-nents below the Nyquist frequencyare perfectly passed (gain =1), and fre-quency components above the Nyquistfrequency are perfectly eliminated.Unfortunately, this type of frequencyresponse filter is impossible to imple-ment in hardware.

Figure 2: Theoretical brick-wall frequency response


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Most oscilloscopes with bandwidthspecifications of 1 GHz and belowhave what is known as a Gaussianfrequency response. As signal input

frequencies approach the scopesspecified bandwidth, measuredamplitudes slowly decrease. Signalscan be attenuated by as much as 3 dB(~30%) at the bandwidth frequency.If a scopes bandwidth is specifiedexactly at the Nyquist frequency (f N),as shown in Figure 3, input signalfrequency components above thisfrequencyalthough attenuated bymore than 3 dBcan be sampled(red hashed area)especially whenthe input signal contains fast edges,as is often the case when you aremeasuring digital signals. This is aviolation of Nyquists first rule.

Most scope vendors dont specifytheir scopes bandwidth at the Nyquistfrequency (f N)but some do. However,it is very common for vendors ofwaveform recorders/digitizersto specify the bandwidth of theirinstruments at the Nyquist frequency.Lets now see what can happen whena scopes bandwidth is the same asthe Nyquist frequency (f N).

Figure 4 shows an example of a500-MHz bandwidth scope samplingat just 1 GSa/s while operatingin a three- or four-channel mode.Although the fundamental frequency(clock rate) of the input signal is wellwithin Nyquists criteria, the signalsedges contain significant frequencycomponents well beyond the Nyquistfrequency (f N). When you view them

repetitively, the edges of this signalappear to wobble with varyingdegrees of pre-shoot, over-shoot, andvarious edge speeds. This is evidenceof aliasing, and it clearly demonstratesthat a sample rate-to-bandwidth ratioof just 2:1 is insufficient for reliabledigital signal measurements.

Nyquists Sampling Theorem (continued)

Figure 3: Typical oscilloscope Gaussian frequency response with bandwidth (f BW)specified at the Nyquist frequency (f N)

Figure 4: 500-MHz bandwidth scope sampling at 1 GSa/s producesaliased edges

Aliasing

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So, where should a scopes bandwidth(fBW) be specified relative to thescopes sample rate (f S) and theNyquist frequency (f N)? To minimize

sampling significant frequencycomponents above the Nyquistfrequency (f N), most scope vendorsspecify the bandwidth of their scopesthat have a typical Gaussian frequencyresponse at 1/4th to 1/5th, or lower,than the scopes real-time samplerate, as shown is Figure 5. Althoughsampling at even higher rates relativeto the scopes bandwidth would furtherminimize the possibility of samplingfrequency components beyond theNyquist frequency (f N), a sample rate-to-bandwidth ratio of 4:1 is sufficient toproduce reliable digital measurements.

Figure 5: Limiting oscilloscope bandwidth (f BW) to 1/4 the sample rate (f S/4)

reduces frequency components above the Nyquist frequency (f N)


Oscilloscopes with bandwidthspecifications in the 2-GHz andhigher range typically have asharper frequency roll-off response/characteristic. We call this type offrequency response a maximally-flat response. Since a scope with amaximally-flat response approachesthe ideal characteristics of a brick-wall

filter, where frequency components

beyond the Nyquist frequency areattenuated to a higher degree, notas many samples are required toproduce a good representation of theinput signal using digital filtering.Vendors can theoretically specify thebandwidth of scopes with this typeof response (assuming the front-endanalog hardware is capable) at f S/2.5.

However, most scope vendors have notpushed this specification beyond f S/3.

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Figure 6: Agilent 500-MHz bandwidth scope sampling at 2 GSa/sshows an accurate measurement of this 100-MHz clock with a 1-nsedge speed


Figure 6 shows a 500-MHz bandwidthscope capturing a 100-MHz clocksignal with edge speeds in the rangeof 1 ns (10 to 90%). A bandwidth

specification of 500 MHz would be theminimum recommended bandwidth toaccurately capture this digital signal.This particular scope is able to sampleat 4 GSa/s in a 2-channel mode ofoperation, or 2 GSa/s in a three- orfour-channel mode of operation.Figure 6 shows the scope sampling at2 GSa/s, which is twice the Nyquistfrequency (f N) and four times thebandwidth frequency (f BW). This showsthat a scope with a samplerate-to-bandwidth ratio of 4:1produces a very stable and accuraterepresentation of the input signal.And with Sin(x)/x waveformreconstruction/interpolation digitalfiltering, the scope provides waveformand measurement resolution in the 10sof picoseconds range. The differencein waveform stability and accuracy issignificant compared to the examplewe showed earlier (Figure 4) with ascope of the same bandwidth samplingat just twice the bandwidth (f N).

So what happens if we double thesample rate to 4 GSa/s in this same500-MHz bandwidth scope (f BW x 8)?You might intuitively believe that thescope would produce significantlybetter waveform and measurementresults. But as you can see in Figure7, there is some improvement, but itis minimal. If you look closely at thesetwo waveform images (Figure 6 andFigure 7), you can see that when yousample at 4 GSa/s (f BW x 8), there is

slightly less pre-shoot and over-shootin the displayed waveform. But therise time measurement shows thesame results (1.02 ns). The key tothis slight improvement in waveformfidelity is that additional error sourceswere not introduced when the sample-

rate-to-bandwidth ratio of this scopeincreased from 4:1 (2 GSa/s) to 8:1(4 GSa/s). And this leads us into ournext topic: What happens if Nyquistsrule 2 is violated? Nyquist says thatsamples must be evenly spaced. Usersoften overlook this important rulewhen they evaluate digital storageoscilloscopes.

Figure 7: Agilent 500-MHz bandwidth scope sampling at 4 GSa/s pro-duces minimal measurement improvement over sampling at 2 GSa/s

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Interleaved Real-Time Sampling

When ADC technology has beenstretched to its limit in terms of maxi-mum sample rate, how do oscilloscopevendors create scopes with even higher

sample rates? The drive for highersample rates may be simply to satisfyscope users' perception that more isbetteror higher sample rates mayactually be required to produce higher-bandwidth real-time oscilloscopemeasurements. But producing highersample rates in oscilloscopes is not aseasy as simply selecting a higher sam-ple rate off-the-shelf analog-to-digitalconverter.

A common technique adopted by allmajor scope vendors is to interleavemultiple real-time ADCs. But dontconfuse this sampling technique withinterleaving samples from repetitiveacquisitions, which we call "equivalent-time" sampling.

Figure 8 shows a block diagram ofa real-time interleaved ADC systemconsisting of two ADCs with phase-delayed sampling. In this example,ADC 2 always samples clock periodafter ADC 1 samples. After each real-time acquisition cycle is complete, thescopes CPU or waveform processingASIC retrieves the data stored in eachADC acquisition memory and theninterleaves the samples to produce thereal-time digitized waveform with twicethe sample density (2X sample rate).

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

Scopes with real-time interleavedsampling must adhere to two require-ments. For accurate distortion-freeinterleaving, each ADCs vertical gain,offset and frequency response mustbe closely matched. Secondly, thephase-delayed clocks must be alignedwith high precision in order to satisfyNyquists rule 2 that dictates equally-

spaced samples. In other words,the sample clock for ADC 2 must bedelayed precisely 180 degrees afterthe clock that samples ADC 1. Bothof these criteria are important foraccurate interleaving. However, for amore intuitive understanding of thepossible errors that can occur due topoor interleaving, the rest of this paperwill focus on errors due to poor phase-delayed clocking.

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Testing for Interleave Distortion (continued)

Visual sine wave comparison tests

Figure 11 shows the simplest andmost intuitive comparative testthe

visual sine wave test. The waveformshown in Figure 11a is a single-shotcapture of a 200 MHz sine wave usingan Agilent InfiniiVision 1-GHz band-width scope sampling at 4 GSa/s. Thisscope has a sample-rate-to-bandwidthratio of 4:1 using non-interleaved ADCtechnology. The waveform shown inFigure 11b is a single-shot capture ofthe same 200 MHz sine wave usingLeCroys 1-GHz bandwidth scopesampling at 10 GSa/s. This scope hasa maximum sample-rate-to-bandwidthratio of 10:1 using interleaved technol-ogy.

Although we would intuitively believethat a higher-sample-rate scope of thesame bandwidth should produce moreaccurate measurement results, we cansee in this measurement comparisonthat the lower sample rate scope actu-ally produces a much more accuraterepresentation of the 200 MHz inputsine wave. This is not because lowersample rates are better, but becausepoorly aligned interleaved real-timeADCs negate the benefit of highersample rates.

Precision alignment of interleaved ADCtechnology becomes even morecritical in higher bandwidth and highersample rate scopes. Although a fixedamount of phase-delayed clock errormay be insignificant at lower samplerates, this same fixed amount of tim-ing error becomes significant at higher

sample rates (lower sample periods).Lets now compare two higher-band-width oscilloscopes with and withoutreal-time interleaved technology.

Figure 11a: 200-MHz sine wave captured on an Agilent 1-GHz bandwidthoscilloscope sampling at 4 GSa/s

Figure 11b: 200-MHz sine wave captured on a LeCroy 1-GHz bandwidthoscilloscope sampling at 10 GSa/s

Interleave Distortion

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Figure 12 shows two screen-shots ofa visual sine wave test comparing anAgilent 3-GHz bandwidth scopesampling at 20 GSa/s (non-interleaved)

and 40 GSa/s (interleaved) capturinga 2.5 GHz sine wave. This particularDSO uses single-chip 20 GSa/s ADCsbehind each of four channels. Butwhen using just two channels of thescope, the instrument automaticallyinterleaves pairs of ADCs to provide upto 40 GSa/s real-time sampling.

Visually, we cant detect much differ-ence between the qualities of thesetwo waveforms. Both waveformsappear to be relatively pure sinewaves with minimal distortion. Butwhen we perform a statistical Vp-pmeasurement, we can see that thehigher sample rate measurementproduces slightly more stablemeasurementas we would expect.

Figure 12a: 2.5-GHz sine wave captured on the Agilent Infiniium oscilloscopesampling at 20 GSa/s (non-interleaved)

Figure 12b: 2.5-GHz sine wave captured on the Agilent Infiniium oscilloscopesampling at 40 GSa/s (interleaved)

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Figure 13 shows a visual sine wavetest comparing the Tektronix 2.5-GHzbandwidth scope sampling at10 GSa/s (non-interleaved) and

40 GSa/s (interleaved) capturing thesame 2.5 GHz sine wave. This particu-lar DSO uses single-chip 10 GSa/sADCs behind each of four-channels.But when you use just one channel ofthe scope, the instrument automatical-ly interleaves its four ADCs to provideup to 40 GSa/s real-time sampling ona single channel.

In this visual sine wave test we cansee a big difference in waveform fidel-ity between each of these sample ratesettings. When sampling at 10 GSa/s(Figure 13a) without interleaved ADCs,the scope produces a fairly goodrepresentation of the input sine wave,although the Vp-p measurement isapproximately four times less stablethan the measurement performed onthe Agilent scope of similar bandwidth.When sampling at 40 GSa/s (Figure13b) with interleaved ADC technology,we can clearly see waveform distortionproduced by the Tek DSO, as well asa less stable Vp-p measurement. Thisis counter-intuitive. Most engineerswould expect more accurate and stablemeasurement results when samplingat a higher rate using the same scope.The degradation in measurementresults is primarily due to poor verticaland/or timing alignment of the real-time interleaved ADC system.

Figure 13a: 2.5-GHz sine wave captured on a Tektronix 2.5-GHz bandwidthoscilloscope sampling at 10 GSa/s (non-interleaved)

Figure 13b: 2.5-GHz sine wave captured on a Tektronix 2.5-GHz bandwidthoscilloscope sampling at 40 GSa/s (interleaved)

Interleave Distortion

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Spectrum analysis comparisontests

The visual sine wave test doesnt reallyprove where the distortion is comingfrom. It merely shows the effect of vari-ous error/components of distortion.However, a spectrum/FFT analysis willpositively identify components ofdistortion including harmonic distor-tion, random noise, and interleavedsampling distortion. Using a sine wavegenerated from a high-quality signalgenerator, there should be only one fre-quency component in the input signal.Any frequency components other thanthe fundamental frequency detected in

an FFT analysis on thedigitized waveform are oscilloscope-induced distortion components.

Figure 14a shows an FFT analysis of asingle-shot capture of a 2.5 GHz sinewave using Agilents Infiniium oscil-loscope sampling at 40 GSa/s. Theworst-case distortion spur measuresapproximately 90 dB below the funda-mental. This component of distortionis actually second harmonic distortion,most likely produced by the signal

generator. And its level is extremelyinsignificant and is even lower than thescopes in-band noise floor.

Figure 14b shows an FFT analysis of asingle-shot capture of the same 2.5-GHzsine wave using a Tektronix oscillo-scopealso sampling at 40 GSa/s. Theworst-case distortion spur in this FFTanalysis measures approximately 32 dBbelow the fundamental. This is a sig-nificant level of distortion and explainswhy the sine wave test (Figure 13b)produced a distorted waveform. Thefrequency of this distortion occurs at7.5 GHz. This is exactly 10 GHz belowthe input signal frequency (2.5 GHz),but folded back into the positivedomain. The next highest componentof distortion occurs at 12.5 GHz. This isexactly 10 GHz above the input signalfrequency (2.5 GHz). Both of these

Figure 14a: FFT analysis of 2.5-GHz sine wave captured on anAgilent Infiniium oscilloscope sampling at 40 GSa/s

Figure 14b: FFT analysis of 2.5-GHz sine wave captured on a Tektronixoscilloscope sampling at 40 GSa/s

components of distortion are directlyrelated to the 40-GSa/s samplingclock and its interleaved clock rates(10 GHz). These components ofdistortion are not caused by random orharmonic distortion. They are causedby real-time interleaved ADC distortion.

10 GSa/s Distortion(-32 dB) {

{

40 GSa/s Distortion

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Digital clock measurementstability comparison tests

As a digital designer, you may say that

you really dont care about distortionon analog signals, such as on sinewaves. But you must remember thatall digital signals can be decomposedinto an infinite number of sine waves.If the fifth harmonic of a digital clockis distorted, then the composite digitalwaveform will also be distorted.

Although it is more difficult to performsampling distortion testing on digitalclock signals, it can be done. But mak-ing a visual distortion test on digitalsignals is not recommended. There isno such thing as a pure digital clockgenerator. Digital signals, even thosegenerated by the highest-performancepulse generators, can have varyingdegrees of overshoot and perturba-tions, and can have various edgespeeds. In addition, pulse shapes ofdigitized signals can be distorted bythe scopes front-end hardware due tothe scopes pulse response character-istics and possibly a non-flatfrequency response.

But there are a few tests you can per-form using high-speed clock signals tocompare the quality of a scopes ADCsystem. One test is to compare para-metric measurement stability, such asthe standard deviation of rise timesand fall times. Interleave sampling dis-tortion will contribute to unstable edgemeasurements and inject a determin-istic component of jitter into the high-speed edges of digital signals.

Figure 15 shows two scopes with simi-lar bandwidth capturing and measuringthe rise time of a 400 MHz digital clocksignal with edge speeds in the rangeof 250 ps. Figure 15a shows an Agilent3 GHz bandwidth scope interleavingtwo 20-GSa/s ADC in order to samplethis signal at 40 GSa/s. The resultantrepetitive rise time measurement has astandard deviation of 3.3 ps. Figure 15b

Figure 15a: 400-MHz clock captured on an Agilent Infiniium 3-GHzoscilloscope sampling at 40 GSa/s

Figure 15b: 400-MHz clock captured on a Tektronix 2.5-GHz oscilloscopesampling at 40 GSa/s

shows a Tektronix 2.5 GHz bandwidthscope interleaving four 10 GSa/sADCs in order to also sample at40 GSa/s. In addition to a more unsta-ble display, the rise time measurementon this digital clock has a standarddeviation of 9.3 ps. The more tightly

aligned ADC interleaving in the Agilentscope, along with a lower noise floor,makes it possible for the Agilent scopeto more accurately capture the higher-frequency harmonics of this clocksignal, thereby providing more stablemeasurements.

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When you view the frequency compo-nents of a digital clock signal using FFTanalysis, the spectrum is much morecomplex than when you test a simple

sine wave. A pure digital clock gener-ated from a high-quality pulse genera-tor should consist of the fundamentalfrequency component and its odd har-monics. If the duty cycle of the clock isnot exactly 50%, then the spectrum willalso contain lower-amplitude even har-monics. But if you know what to lookfor and what to ignore, you can mea-sure interleave sampling distortion ondigital signals in the frequency domainusing the scopes FFT math function.

Figure 16a shows the spectrum of a400-MHz clock captured on an Agilent3-GHz bandwidth scope sampling at40 GSa/s. The only observable frequen-cy spurs are the fundamental, thirdharmonic, fifth harmonic, and seventhharmonicalong with some minoreven harmonics. All other spurs in thespectrum are well below the scopesin-band noise floor.

Figure 16b shows the spectrum of a400 MHz clock captured on a Tektronix2.5 GHz bandwidth scopealsosampling at 40 GSa/s. In this FFTanalysis, we not only see thefundamental frequency component andits associated harmonics, but we alsosee several spurs at higher frequenciesclustered around 10 GHz and40 GHz. These imaging spurs aredirectly related to this scopes poorlyaligned interleaved ADC system.

Figure 16a: FFT analysis on a 400-MHz clock using an Agilent Infiniium3-GHz bandwidth oscilloscope

Figure 16b: FFT analysis on a 400-MHz clock using a Tektronix 2.5-GHz

bandwidth oscilloscope

10 GSa/s Distortion(27 dB below5th harmonic)

{

{

40 GSa/s Distortion

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Glossary

ADC Analog-to-digital converter

Aliasing Waveform errors produced by a digital filter when reconstructing a sampled signal thatcontains frequency components above the Nyquist frequency (f S)

Brick-wall frequency response A theoretical hardware or software filter that perfectly passes all frequency componentsbelow a specific frequency and perfectly eliminates all frequency components above thesame frequency point

DSO Digital storage oscilloscope

Equivalent-time sampling A sampling technique that interleaves samples taken from repetitive acquisitions

FFT Fast Fourier transform

Gaussian frequency response A low-pass frequency response that has a slow roll-off characteristic that begins at approxi-mately 1/3 the 3 dB frequency (bandwidth). Oscilloscopes with bandwidth specifications of1 GHz and below typically exhibit an approximate Gaussian response.

In-band Frequency components below the 3 dB (bandwidth) frequency

Interleaved real-time sampling A sampling technique that interleaves samples from multiple real-time ADCs using phase-delayed clocking

Maximally-flat response A low-pass frequency response that is relatively flat below the -3 dB frequency and thenrolls-off sharply near the 3 dB frequency (bandwidth). Oscilloscopes with bandwidthspecifications greater than 1 GHz typically exhibit a maximally-flat response

Nyquist sampling theorem States that for a limited bandwidth (band-limited) signal with maximum frequency (f max), theequally-spaced sampling frequency f S must be greater than twice the maximum frequencyfmax, in order to have the signal be uniquely reconstructed without aliasing

Oscilloscope bandwidth The lowest frequency at which input signal sine waves are attenuated by 3 dB(-30% amplitude error)

Out-of-band Frequency components above the 3 dB (bandwidth) frequency

Real-time sampling A sampling technique that acquires samples in a single-shot acquisition at a high rate .

Sampling noise A deterministic component of distortion related to a scopes sample clock


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