August 2007 59U.S. Government work not protected by U.S. Copyright.

Many modern high-speed oscillo-scopes are well suited for precisemeasurements of microwave wave-forms, modulated signals, and non-linear phenomena. These oscillo-

scopes have bandwidths up to 100 GHz and are avail-able with nominal 50- input impedances. Like theirlow-frequency counterparts, these oscilloscopes areflexible, easy to use, and relatively inexpensive; thismakes them ideal for testing a wide variety ofmicrowave sources and other microwave components.

As with all microwave instruments, characterizingand correcting for the oscilloscope’s imperfections are keyto making accurate measurements. Important imperfec-tions in high-speed sampling oscilloscopes include jitter,drift, and distortion in the oscilloscope’s timebase, as well

as imperfections in the impulse response of the oscillo-scope’s sampling circuitry and the oscilloscope’s imped-ance match. With suitable corrections, however, thesehigh-speed sampling oscilloscopes can be used not onlyto debug circuits but also to perform metrology-grademeasurements at the highest microwave frequencies.Most of the equipment required is readily available inmost microwave labs: a vector network analyzer, amicrowave signal generator, and, of course, a samplingoscilloscope! Here, we summarize many of the correc-tions discussed in [1] and [2] that are necessary formetrology-grade measurements and illustrate the appli-cation of these oscilloscopes to the characterization ofmicrowave signals (see “Comparison of MicrowaveSignal Measurement Methods” and “Measurement of RFPower and Modulation Envelope”).

© DIGITALSTOCK

Dylan Williams (dylan@boulder.nist.gov), Paul Hale, and Kate A. Remley are with National Institute of Standardsand Technology (NIST), Boulder, Colorado, USA.

Dylan Williams, Paul Hale,and Kate A. Remley

Digital Object Identifier 10.1109/MMM.2007.900110

60 August 2007

Types of Digital OscilloscopesThe first oscilloscope that you used may have beenan analog instrument. However, modern oscillo-scopes generally process, store, and display informa-tion digitally [3]. Modern digital oscilloscopes aresold with widely varying features, but themicrowave engineer can roughly group these instru-ments into two categories based on their internaloperation: real-time oscilloscopes and sampling (orequivalent-time) oscilloscopes.

Real-time oscilloscopes repetitively sample a wave-form at a high rate and store the measurements in acircular memory buffer. By setting trigger events, theuser can determine which part of the waveform toview. Trigger events might be limited to simple edgetriggers in lower-end models, but higher-end modelscan use sophisticated digital processing to trigger onanomalous events in complicated digital ormicrowave signals.

General-purpose real-time oscilloscopes typicallyhave a high input impedance and are designed to non-invasively measure voltages inside operating electricalcircuits in real time. Due to parasitics in the input cir-cuitry, the bandwidth of these oscilloscopes is limitedto about 500 MHz. High-end real-time oscilloscopescircumvent this bandwidth limitation by embeddingthe sampling circuitry in a 50- transmission line ter-minated in a 50- load. These high-end real-time oscil-loscopes achieve bandwidths comparable to those oflow-end sampling oscilloscopes; they can be connecteddirectly to the output port of a microwave circuit and

measure the voltage that the circuit generates acrossthe oscilloscope’s nominal 50- input impedance. Thisapproximates the matched environment in which mostmicrowave components and circuits are designed tooperate.

Real-time oscilloscopes use an interleaved analog-to-digital converter technology to achieve sampling rates ashigh as 50 Gsamples/s and bandwidths as high as 20GHz. Real-time oscilloscopes are often capable of acquir-ing tens or hundreds of millions of samples and canprocess these waveforms to obtain information on vari-ous properties of the signal, including jitter ormicrowave modulation.

Although real-time oscilloscopes are extremely ver-satile, they also have some properties that can limittheir utility for microwave applications. For example,real-time oscilloscopes use high-speed analog-to-digitalconverters and must move huge amounts of data intoand out of memory, so their resolution is typically lim-ited to 8 b. Also, it is difficult to perfectly match thegain, response, and delay of the different interleavedanalog-to-digital converters, which reduces fidelity andlimits bandwidth.

High-Speed Sampling OscilloscopesHigh-speed sampling oscilloscopes, which will be thefocus of the remainder of this article, use an equiva-lent-time sampling strategy to achieve useful band-widths as high as 100 GHz. Most of these have a nom-inal 50- input impedance and are designed to mea-sure repetitive input signals. The use of equivalent-time sampling allows for greater fidelity than is possi-ble in real-time oscilloscopes.

Figure 1 shows a schematic of a high-speed sam-pling oscilloscope. After being triggered, the oscillo-scope uses a programmable delay generator to momen-tarily close the switch. This samples the voltage at theoscilloscope’s input. The net charge that moves throughthe closed switch to the hold capacitor is proportionalto the voltage at the oscilloscope’s input when the

switch closes. A sensitive ampli-fier and analog-to-digital con-verter are then used to measurethis charge and thus the voltagethat was present at the oscillo-scope’s input when the switchwas closed.

The switches in a samplingoscilloscope are typically con-structed with fast sampling diodesand are “opened” and “closed” byfast electrical strobe pulses. Thesestrobe pulses momentarily placethe normally reverse-biased (off)sampling diodes into a conductiveforward-biased (on) state. Mostmodern sampling oscilloscopesFigure 1. The operation of a sampling oscilloscope.

Sampling Oscilloscope

50 ΩOutputInput

++++

Many modern high-speedoscilloscopes are well suited forprecise measurements of microwavewaveforms, modulated signals, andnonlinear phenomena.

August 2007 61

use nonlinear transmission lines to sharpen these strobepulses and can “open” and “close” the diode switches inthe sampling gates in roughly 2–20 ps. This, combinedwith an accurate timebase, allows the sampling oscillo-scope to measure the voltage at the input of the oscillo-scope very precisely in time.

Oscilloscope TimebaseIt usually takes a few microseconds or more for a sam-pling oscilloscope to acquire a voltage sample; thisrestricts sampling oscilloscopes to the measurement ofrepetitive signals. The typical measurement strategyused to measure a waveform is called “equivalent timesampling.” In this measurement approach, the signal is

repeated over and over, and the oscilloscope’s timebaseis configured to close the switch just a little bit later ineach cycle of the signal. Each repetition of the signalallows a new sample to be taken, adding an additionalvoltage sample to the measured waveform.

Types of Oscilloscope TimebasesThere are three basic classes of oscilloscope timebases,with a number of variations: conventional triggered time-bases, synchronized timebases, and a hybrid timebase [4]that marries some of the best attributes of the first two.

Conventional oscilloscope timebases use trigger cir-cuits and a programmable delay generator to time theacquisition of voltage samples [5], [6]. These timebases

Accurately characterizing high-speed electrical wave-forms from fundamental principles is a major challenge.NIST has developed a sophisticated electro-optic sampling system to perform these fundamental electri-cal measurements (see “NIST Electro-Optic SamplingSystem”). This electro-optic sampling systems is, inessence, a very fast sampling oscilloscope based onelectro-optic interactions. The electro-optic samplingsystems at NIST, NPL, and PTB have measurementbandwidths of many hundreds of gigahertz. (At lowerfrequencies, the “nose-to-nose” calibration is some-times used to calibrate sampling oscilloscopes [16],[23]. However, the assumptions concerning the equiva-lence of a sampler’s kickout pulses and impulseresponse break down at higher frequencies [24].)

Figure 2 illustrates the traceability chain currentlyused at NIST. Photodiodes calibrated on NIST’s electro-

optic sampling system are used to calibrate the magni-tude and phase response of the oscilloscopes [14]. Inprinciple, calibrated power meters could be used toimprove the characterization of the magnitude responseof the oscilloscope, but NIST uses them only to set theoverall magnitude scaling of the oscilloscope calibration.The calibrated oscilloscopes are used to characterizepulse sources, step sources, comb generators [18], [25],microwave mixers [26], and modulated microwavesources [2], [20], [27]. These can, in turn, be used tocalibrate a variety of instruments, including other oscillo-scopes, vector signal analyzers, microwave receivers,LSNAs, and peak power meters [27].

NIST’s waveform calibration services are ideally suit-ed for microwave applications.NIST provides calibratedphotodiodes and will calibrate oscilloscope plug-ins andpulse sources for a fee. NIST performs mismatch correc-

tions on all measurementsand provides reflection coeffi-cients of oscilloscopes andsources. This makes it possi-ble to fully mismatch correctmicrowave measurementsperformed with oscilloscopescalibrated at NIST and allowsusers of these services todevelop complete Théveninequivalent circuits for sourcescalibrated at NIST. NIST alsoprovides covariance matricesto describe the uncertaintiesin most measurements (see“Representing Uncertaintieswith Covariance Matrices”).This allows uncertainties tobe expressed in either thetemporal or frequencydomains.Figure 2. The NIST waveform traceability chain.

Photodiode

Oscilloscope

Power Sensor

Vector NetworkAnalyzer

Adapter

Sources

Fundamental Calibrations Derived Calibrations

Waveform Calibration Services at NIST

62 August 2007

are extremely flexible and easy to use, though they areprone to jitter and timebase distortion.

Synchronized timebases use an oscillator lockedto, and slightly offset from, a reference signal fromthe source. Many microwave engineers may be famil-iar with the implementation of this timebase inmicrowave transition analyzers and sampling down-converters. These timebases are more costly and lessflexible because they can only lock to and trigger onperiodic signals. On the other hand, they are also lesssusceptible to jitter and drift, virtually eliminate time-base distortion, and sample more quickly than con-ventional timebases.

The hybrid approach [4], [7] uses a conventionaltrigger with programmable delay generator to takesamples, but simultaneously measures a set of refer-ence sinusoids that are synchronized with the signalbeing acquired to correct the oscilloscope’s timebase.

This allows simultaneous correction for jitter, drift,and timebase distortion in a number of less-expensiveconventionally triggered oscilloscopes (see “NISTTimebase Correction Software”).

Jitter, Drift, and Timebase DistortionImperfections in the oscilloscope’s timebase include jit-ter (the random component of the error in the time atwhich the oscilloscope samples voltages), drift (a slowdrift in the timebase between successive measurementsweeps), and timebase distortion (a systematic andrepeatable distortion in the oscilloscope’s timebase).The amount of jitter and drift depend greatly on thetriggering scheme employed and usually are verytightly linked to the hardware.

Timebase distortion typically depends on theoscilloscope’s internal clocks and does not dependon the triggering scheme employed. Vandersteen et

Figure 3 sketches NIST’s electro-optic sampling system[28]–[30]. The mode-locked fiber laser emits a series ofshort optical pulses approximately 100 fs in duration thatare split by the beam splitter into an optical “excitationbeam” and an optical “sampling beam.” The optical excita-tion beam excites the photodiode, which generates a fastelectrical pulse measured by the system. This electricalpulse is coupled by the wafer probe onto a coplanarwaveguide (CPW) fabricated on an electro-optic substrate.

The optical sampling beam is used to reconstruct therepetitive electrical waveform generated by the photodi-ode at the on-wafer reference plane in the CPW. This is

done by passing the sampling beam through a variableoptical delay, polarizing it, and then passing it throughone of the gaps of the CPW. Since the substrate is elec-tro-optic, the electric field between the CPW conductorschanges the polarization of the optical sampling beampassing through it. The polarization analyzer detects thischange, which is proportional to the voltage in the CPWat the instant at which the optical pulse arrived there.This process does not perturb the electrical signal onthe CPW. Changing the delay of the sampling beamallows us to map out the voltage at the reference planein the CPW as a function of time.

The final step in the cali-bration is to use vector net-work analysis to character-ize the photodiode andresistor reflection coeffi-cients, as well as the scat-tering parameters of theprobe head. These reflec-tion coefficients and scat-tering parameters are usedto calculate the electricalwaveform at the photodi-ode’s coaxial connector.Despite the high accuracyof the network analyzer,these reflection coefficientsand scattering parametersare some of the largestsources of uncertainty wehave been able to identifyin these measurements.Figure 3. Sketch of NIST’s electro-optic sampling system.

PulsedLaser

Photodiode

Photodiode

ExcitationBeam

MeasurementReference

Plane

CPW

Resistor

Sampling-Beam Laser

Spot

OpticalPolarization-

State Analyzer

LiTaO3 Wafer

Sampling Beam Variable Optical Delay

ProbeHead

ProbeHead

NIST Electro-Optic Sampling System

August 2007 63

al. [8], Rolain et al. [9], Stennbakken et al. [10], andWang et al. [11] pioneered modern methods for mea-suring and correcting for timebase distortion. Theseare based on measuring sinusoids with the oscillo-scope to characterize distortion in the oscilloscope’stimebase and using deviations of the measured sig-nals from sinusoids to infer distortion in the oscillo-scope’s timebase.

The hybrid method of [4] is an outgrowth of thesetimebase-distortion characterization methods. It mea-sures the sinusoids in real time to simultaneously cor-rect for jitter, drift, and timebase distortion in conven-tionally triggered oscilloscopes.

Mismatch CorrectionAs microwave engineers know all too well, imped-ances are difficult to control at microwave frequenciesand multiple reflections between sources and loadsmust be accounted for with microwave mismatch cor-rections if good accuracy is to be achieved. We havefound vector network analyzers to be extremely usefultools for mismatch correcting oscilloscope measure-ments. We use network analyzers to characterize ourmicrowave sources, adapters, and oscilloscopes togreat advantage and apply mismatch correctionswhenever possible. These mismatch corrections aredescribed in greater detail in [1].

Mismatch generally results in reflections at specificpoints in time. Applying mismatch corrections effec-tively requires great accuracy in the oscilloscope’s time-base to accurately measure these temporal locations.Thus, errors in the oscilloscope’s timebase usually needto be corrected before mismatch corrections can beproperly applied.

Mismatch correcting oscilloscope measurementswith vector network analyzer measurements alsorequires linear time-invariant behavior. The princi-pal issue is that the vector network analyzer mea-sures the impedance of the oscilloscope when itssampling gate is open. The oscilloscope, therefore,must be configured to close and then open its sam-pling gate quickly enough that reflections off theclosed gate are not re-reflected off of elements exter-nal to the oscilloscope and measured by the sam-pling gate before it opens again. This issue is usually

Figure 4 plots the temporal impulse response of aphotodiode calculated from the Fourier transform of itscomplex mismatch-corrected spectrum as measuredon NIST’s electro-optic sampling system. These datawere first presented in [22]. The principal reflectionfrom the CPW load in the electro-opticsampling system is quite large andoccurs at about 400 ps, but it is notseen in Figure 4 because it has beencorrected out of the measurement.

The figure also plots the uncertainty inthis impulse response calculated from acovariance matrix [22]. This formulationaccounts for the correlations in the fre-quency-domain data when it maps thefrequency-domain data into the timedomain. This uncertainty peaks near themaximum of the photodiode’s impulseresponse, as expected.

Less obvious is the smaller peak inuncertainty at 400 ps. The raw measure-ment of the photodiode’s impulseresponse has a large reflection at thatpoint, which is removed almost com-

pletely by the frequency-domain mismatch correctionwe employ. While the mismatch correction is very effec-tive at eliminating this artifact of the measurement sys-tem near 400 ps, imperfections in the mismatch correc-tion nevertheless visibly raise the uncertainty there.

Figure 4. Temporal impulse response of a photodiode measured inNIST’s electro-optic sampling system.

4 0.10

0.08

0.06

0.04

0.02

0

3

2

Impu

lse

Res

pons

e (V

/pC

)

Impulse ResponseStandard Uncertainty

Sta

ndar

d U

ncer

tain

ty (

V/p

C)

1

0

0 200 400 600Time (ps)

800 1,000−1

Representing Uncertainties with Covariance Matrices

Although real-time oscilloscopes areextremely versatile, they also havesome properties that can limit theirutility for microwave applications.

64 August 2007

Imperfections in the timebase, suchas jitter and timebase distortion, cancause significant measurement errorsat microwave frequencies. NIST’sTimebase Correction Software [31]corrects for both random and system-atic timebase errors with measure-ments of two quadrature sinusoidsmade simultaneously and synchro-nized with the waveform being char-acterized. The method is described indetail in [4], and a similar implemen-tation is presented in [7].

A typical measurement configura-tion for characterizing a modulatedmicrowave signal with a sampling oscilloscope is shown inFigure 5. In the standard configuration, the oscilloscopewould be triggered by the 10-MHz reference from thesource and the modulated signal measured on channel 3,as shown in Figure 5. To improve the timebase, we alsomeasure two reference sinusoids on channels 1 and 2with the oscilloscope. Though there may be a great deal ofdistortion and jitter in the oscilloscope timebase, these twoquadrature reference sinusoids allow the actual times atwhich the samples were taken to be reconstructed withgreat accuracy. This is because they are made simultane-ously with, and synchronized to, the modulated signalmeasured on channel 3 of the oscilloscope.

A simple illustration of the timebase correction concept isshown in Figure 6; this plots uncorrected measurements (cir-cles) of a reference sinusoid with an estimate of the distort-ed sinusoid (solid curve). The estimated sinusoid is foundby minimizing the average “distance” between the samplesand the sinusoid. If we assume, for illustrative purposes, that

there is no additive noise, we can estimate the total errordue to timebase distortion and jitter by drawing a horizontalline between each measurement (circles) and the distortedsinusoid. The length of each line represents the differencebetween the nominal (oscilloscope) time at which themeasurement was taken and the time as determined bythe distorted sinusoidal fit. The time at which each lineintersects the distorted sinusoid is the corrected time foreach sample.

Figure 7 shows an actual measurement of a sinusoidmeasured simultaneously with two reference sinusoids(not shown) that were used to calculate the timebaseerror. The estimated jitter before the correction was about3.3 ps, and the effects of timebase distortion are clearlyvisible at 4 ns. After correction for timebase error, theresidual error for this example is only about 0.2 ps.

Figure 5. A typical measurement configuration used for characterizing amodulated signal with the NIST’s Timebase Correction Software [31].

Modulated Signal Source Oscilloscope

Reference

Modulator

10 MHz ReferenceTrigger

Channel 1

Channel 2

Channel 3Modulated Signal

90°Hybrid 90°

0°

Figure 6. The circles show the sampled signal and the solidcurve shows the estimated signal. The horizontal bars showthe differences between the times estimated from the curveand the nominal oscilloscope timebase. (From [4].)

Figure 7. Portion of five sinusoids measured on chan-nel 3 before (bottom) and after (top) correction fortimebase errors. The offset to the corrected curve hasbeen added for clarity. (From [4].)

0.4

0.3

0.2

0.1

Vol

tage

(V

)

0

3.9 3.95 4

Time (ns)

4.05 4.1

−0.1

−0.2

NIST Timebase Correction Software

August 2007 65

Sampling oscilloscopes have measurementbandwidths up to 100 GHz; they can beused to characterize a variety of waveformsat microwave frequencies, including pulses,modulated signals [7], [20], and signalsrich in harmonics. Microwave engineersmay be more familiar with instrumentssuch as the microwave transition analyzer(MTA) and the LSNA [12], [13], which weredeveloped specifically to acquire waveformsrich in harmonics while preserving the mag-nitude and phase relationships betweenharmonic-frequency components [2]. Infact, the phase references for an LSNA areusually characterized by calibrated samplingoscilloscopes (see “Waveform CalibrationServices at NIST”).

While the LSNA and MTA are oftenused to characterize harmonic distortioncreated by nonlinear circuits such aspower amplifiers, they are also capable ofcharacterizing waveforms such as the 1-GHz square-wave pulse train shown inFigure 8. Because these instrumentsemploy temporal measurements, they areable to measure both the magnitude andphase of modulated microwave signals.Figures 9 and 10 show the frequency-domain magnitude and phase measuredby the three instruments.

The measurements of the waveform’sharmonics made by the LSNA, MTA, andoscilloscope are in good agreement. Theagreement between the oscilloscope and theLSNA measurements is better than theagreement with the MTA at the higher har-monics. One of the advantages of using anLSNA is that it automatically mismatch cor-rects its measurements. For the scope mea-surements used in this comparison, we car-ried out timebase distortion and impulseresponse correction of the scope, but did notcarry out mismatch correction.

Figure 9. Comparison of magnitude measurements.

Figure 8. Comparison of the temporal waveform measurements.

0.2

0.1

0

0 0.5 1Time (ns)

ScopeLSNAMTA

1.5 2

−0.1

Am

plitu

de (

V)

−0.2

−0.3

−0.4

Comparison of Microwave Signal Measurement Methods

Figure 10. Comparison of phase measurements.

200

150

100

50

0

−50

−100

−150

−2000 5 10 15 20 25

Frequency (GHz)

ScopeLSNAMTA

Pha

se (

°)

20151050

10

0

−10

−20

−30

−40

Mag

nitu

de (

dBm

)

−50

−60

−70

−80

−90

Frequency (GHz)

ScopeLSNAMTA

66 August 2007

resolved by designing in short lengths of transmis-sion line in the oscilloscope’s front end. This con-straint is discussed in greater detail in [1].

The large-signal network analyzers (LSNA) [12],[13] can be looked at as a sort of hybrid oscilloscope

and network analyzer that combines the functionalityof the two. Like a network analyzer, an LSNA usescouplers and multiple sampling circuits to allow forsimultaneous measurement of the forward and back-ward waves at each port and to perform mismatchcorrections there. The samplers themselves are config-ured to perform temporal measurements of the large-

signal waveforms, which may be distorted, at theseports. In fact, very early versions of the LSNA wereconstructed with microwave couplers and samplingoscilloscopes.

Impulse-Response CharacterizationEven the fastest high-speed sampling oscilloscopeshave an impulse response of finite duration. The oscil-loscope measures a convolution of the input signaland this impulse response. The oscilloscope’s impulseresponse must be characterized and deconvolved toperform the most accurate measurements [1], [14].This problem presents the most fundamental andchallenging aspect of the calibration of high-speedoscilloscopes.

The National Physical Laboratory (NPL) in theUnited Kingdom, the Physikalisch-Technische Bun-desanstalt (PTB) in Germany, and the NationalInstitute of Standards and Technology (NIST) in theUnited States maintain complex measurement systemsbased on electro-optic sampling to characterize fastelectrical pulse sources (see “NIST Electro-OpticSampling System”). While these systems differ in thedetails of their construction and application, the keyidea is to exploit the extremely high speed of certain

by David Humphreys

Sampling oscilloscopes can measure the modulationenvelope of RF signals; these signals can be used to calibrate wideband RF power meters. Modulated sig-nals can be measured directly (see “Comparison ofMicrowave Signal Measurement Methods” and [7]and [20]), or the oscilloscope can be triggered onthe modulation envelope so that the underlying RFsignal appears as random samples.

Triggering on the modula-tion envelope simplifies themeasurement setup. In thisapproach, the variance of themeasured samples is propor-tional to the RF power in theenvelope [27], and the modu-lation envelope and powercan be determined from thestatistics of the samples. Thisapproach is closely related tothat used in swept-sine oscil-loscope calibrations [15]–[19].

Figure 11 illustrates theprocedure. The signal fromthe source is modulated witha 9-µs-long rectangular pulse.While the exact amplitude

and phase cannot be determined from the statistics ofthe signal, the modulation envelope and power can bedetermined from the variance of the measured samples.This has been applied to the calibration of microwavepeak power meters at NPL in the United Kingdom [27].

David Humphreys is with the National PhysicalLaboratory, Teddington, U.K.

Figure 11. Measuring the RF power envelope through use of a sampling oscilloscope.

3

2

1

0

−1

−2

−3

Sig

nal (

V)

Time (µs)

10 dBm, 1024 Points

Trace 1

Trace 2

Trace 3

0 10 20 30 40

Measurement of RF Power and Modulation Envelope

One of the best reasons to use ahigh-speed sampling oscilloscope inyour work is its ability to quickly,accurately, and inexpensively acquireand display detailed temporalwaveforms at microwavefrequencies.

August 2007 67

electro-optic interactions to characterize fast electricalreference pulses. These reference pulses can then beused to characterize the impulse response of even thefastest oscilloscopes (see “Waveform CalibrationServices at NIST”).

Sometimes these electro-optic sampling systems areaugmented by “swept-sine” amplitude calibrationsbased on measurements of sinusoids with knownamplitudes. These amplitude-only calibrations can bemade traceable to extremely accurate calorimetricpower measurements [15]–[19].

Other Oscilloscope ImperfectionsIn our laboratory, we usually correct for imperfectionsin the oscilloscope’s timebase, impulse response, andinput impedance. These are not the only imperfectionsto watch out for.

Nonlinear ResponseNonlinearity in the oscilloscope’s sampling circuitry isdifficult to characterize and correct for. The moststraightforward approach to dealing with this problemis to limit the amplitude of the input signals to the oscil-loscope so that they do not exceed about 150 mV and touse averaging to increase the dynamic range. The abil-ity to average the measurements can be greatlyimproved by correcting for jitter, drift, and timebasedistortion before averaging, though this usuallyrequires external processing of the data. We are able toachieve a dynamic range of about 60 dB in our labora-tory when we use timebase correction software.

Strobe LeakageA portion of the strobe pulse used to switch the sam-pling diodes into forward conduction and close thesampling gate is coupled out of the front end of theoscilloscope. Fortunately, the strobe leakage occurs atthe same time at which the sample is taken. This doesnot allow the leakage signal enough time to affect thecircuit being tested before the sample is taken. Strobeleakage is nevertheless undesirable and is usuallyminimized by the use of two or more samplingdiodes in a balanced configuration. This allows forfirst-order cancellation of the strobe leakage from thesampling circuit.

Low-Frequency Leakage into the Sampling CircuitryIt is difficult to completely prevent low-frequencyleakage through the sampling diodes in high-speedoscilloscopes, even when they are in their reverse-biased (off) state. This leads to a phenomenon that isoften referred to as “blow-by” in the industry andcauses low-frequency leakage through the samplingcircuit to the hold capacitor. Blow-by manifests itselfas slow settling or oscillations in the oscilloscope mea-surements on microsecond time scales. Blow-by is

sometimes corrected for with compensation circuits inthe sampling circuits themselves. Blow-by can also becorrected for by first performing a measurement withthe strobe off to characterize the blow-by and thensubtracting this from later measurements with thestrobe activated.

Transformations Between Time and FrequencyWe also often need to relate the temporal waveformsdirectly acquired by the oscilloscope to typicalmicrowave frequency-domain quantities, such as theamplitude and phases of the components of a modulat-ed microwave signal [20]. This requirement arises bothwhen performing mismatch corrections and whenevaluating other aspects of signal quality, such as har-monic distortion.

The transformations used depend on the signaltype. The Fourier transform of a repetitive signal withfinite power can be directly derived from a discretenumerical Fourier transform. Approximations to thecontinuous Fourier transform of single pulses withfinite energy can be constructed from discrete numeri-cal Fourier transforms in a similar manner. These trans-formations are discussed in greater detail in [1]. Wehave also developed a software package that simplifiesthese calculations [21].

Transforming measurement uncertainties betweentemporal and frequency-domain representations is lessobvious. The transformations of uncertainties betweenthe two domains depend greatly on how the measure-ments from different times or frequencies are correlat-ed. For example, we know that white noise in onedomain transforms into white noise in the otherdomain. The energy in an error signal that takes theform of a ripple in the frequency domain, however,tends to bunch up at a single point in time. This leadsto large errors at specific temporal locations. The cor-relations in the uncertainties must be captured to pre-dict how uncertainties in one domain will transform tothe other domain.

To address this problem, we developed a rigorousapproach that is uniquely suited to measurements ofinterest to the microwave community. The approach isbased on using covariance matrices to represent ouruncertainties and capture correlations between them[1], [22] (see [1] and “Representing Uncertainties withCovariance Matrices”).

High-speed sampling oscilloscopesuse an equivalent-time samplingstrategy to achieve useful bandwidthsas high as 100 GHz.

68 August 2007

Start Using One Today!One of the best reasons to use a high-speed samplingoscilloscope in your work is its ability to quickly, accu-rately, and inexpensively acquire and display detailedtemporal waveforms at microwave frequencies. Thisoften greatly speeds troubleshooting and lends itself tothe development of an intuitive feel for what is goingon in a circuit.

With the addition of timebase, impulse response,and mismatch corrections, you can turn your high-speed sampling oscilloscope into a precision microwaveinstrument. For much the same reasons that we find alow-frequency oscilloscope on almost every electricalengineer’s bench, perhaps the time has come to startputting high-speed sampling oscilloscopes on ourmicrowave measurement benches!

References[1] D.F. Williams, T.S. Clement, P.D. Hale, and A. Dienstfrey,

“Terminology for high-speed sampling-oscilloscope calibration,”in Automatic RF Techniques Group Conf. Dig., vol. 68, Dec. 2006, pp.9–14.

[2] K.A. Remley and P.D. Hale, “Magnitude and phase calibrationsfor RF, microwave, and high-speed digital signal measurements,”in RF and Microwave Handbook, 3rd ed. Piscataway, NJ: IEEE Press,2007.

[3] M. Kahrs, “50 Years of RF and microwave sampling,” IEEE Trans.Microwave Theory Tech., vol. 51, no. 6, pp. 1787–1805, June 2003.

[4] P.D. Hale, C.M. Wang, D.F. Williams, K.A. Remley, and J.Wepman, “Compensation of random and systematic timing errorsin sampling oscilloscopes,” IEEE Trans. Instrum. Meas., vol. 55, no.6, pp. 2146–2154, Dec. 2006.

[5] J. Verspecht, “Calibration of a measurement system for high fre-quency nonlinear devices,” Ph.D. thesis, Free University ofBrussels, 1995.

[6] J.B. Retting and L. Dobos, “Picosecond time interval measure-ments,” IEEE Trans. Instum. Meas., vol. 44, pp. 284–287, Mar. 1995.

[7] D.A. Humphreys, “Vector measurement of modulated RF signalsby an in-phase and quadrature referencing technique,” IEE Proc.Sci. Meas. Tech., vol. 153, no. 6, pp. 210–216, Nov. 2006.

[8] G. Vandersteen, Y. Rolain, and J. Schoukens, “System identifica-tion technique for data acquisition characterization,” IEEE Trans.Instrum. Meas. Tech. Conf., vol. 2, pp. 1211–1216, May 1998.

[9] Y. Rolain, J. Schoukens, and G. Vandersteen, “Signal reconstrcu-tion for non-equidistant finite length sample sets: A ‘KIS’approach,” IEEE Trans. Instrum. Meas., vol. 47, no. 5, pp.1046–1052, Oct. 1998.

[10] G.N. Stenbakken and J.P. Deyst, “Timebase nonlinearity deter-mination using iterated sine-fit analysis,” IEEE Trans. Instum.Meas., vol. 47, no. 5, pp. 1056–1061, Oct. 1998.

[11] C.M. Wang, P.D. Hale, and K.J. Coakley, “Least-squares estima-tion of time-base distortion of sampling oscilloscopes,” IEEE

Trans. Instrum. Meas., vol. 48, no. 6, pp. 1324–1332, Dec. 1999.[12] J. Verspecht, “Large-signal network analysis,” IEEE Microwave

Mag., vol. 6, no. 4, pp. 82–92, Dec. 2005.[13] W. Van Moer and Y. Rolain, “A large-signal network analyzer:

Why is it needed?,” IEEE Microwave Mag., vol. 7, no. 6, pp. 46–62,Dec. 2006.

[14] T.S. Clement, P.D. Hale, D.F. Williams, C.M. Wang, A. Dienstfrey,and D.A. Keenan, “Calibration of sampling oscilloscopes withhigh-speed photodiodes,” IEEE Trans. Microwave Theory Tech., vol.54, no. 8, pp. 3173–3181, Aug. 2006.

[15] D. Henderson, A.G. Roddie, and A.J.A. Smith, “Recent develop-ments in the calibration of fast sampling oscilloscopes,” IEE Proc.-A, vol. 139, no. 5, pp. 254–260, Sept. 1992.

[16] J. Verspecht and K. Rush, “Individual characterization of broad-band sampling oscilloscopes with a nose-to-nose calibration pro-cedure,” IEEE Trans. Instum. Meas., vol. 43, no. 2, pp. 347–354, Apr.1994.

[17] P.D. Hale, T.S. Clement, K.J. Coakley, C.M. Wang, D.C. DeGroot,and A.P. Verdoni, “Estimating the magnitude and phase responseof a 50 GHz sampling oscilloscope using the ‘nose-to-nose’method,” in Automatic RF Techniques Group Conf. Dig., vol. 55, pp.35–42, June 2000.

[18] F. Verbeyst, “Contributions to large-signal network analysis,”Ph.D. thesis, Vrije Universiteit Brussel, 2006.

[19] J.B. Scott, “Rapid millimetre-wave response characterization to wellbeyond 120 GHz using an improved nose-to-nose method,” in IEEEMTT-S Int. Microwave Symp. Dig., vol. 3, pp. 1511–1514, June 2003.

[20] K.A. Remley, P.D. Hale, D.I. Bergman, and D.A. Keenan,“Comparison of multisine measurements from instrumentationcapable of nonlinear system characterization,” in ARFTGConference Dig., vol. 66, pp. 34–43, Dec. 2005.

[21] D.F. Williams and A. Dienstfrey, “Fourier-transform helper,”2006 [Online]. Available: http://www.boulder.nist.gov/dylan

[22] D.F. Williams, A. Lewandowski, T.S. Clement, C.M. Wang, P.D.Hale, J.M. Morgan, D.A. Keenan, and A. Dienstfrey, “Covariance-based uncertainty analysis of the NIST Electro-optic SamplingSystem,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 1, pp.481–491, Jan. 2006.

[23] K. Rush, S. Draving, and J. Kerley, “Characterizing high-speedoscilloscopes,” IEEE Spectr., pp. 38–39, Jan. 1990.

[24] D.F. Williams, T.S. Clement, K.A. Remley, and P.D. Hale,“Systematic error of the nose-to-nose sampling oscilloscope cali-bration,” Submitted to IEEE Trans. Microwave Theory Tech., to bepublished.

[25] F. Verbeyst, “Commercial solutions for advanced componentcharacterization, possibly in non-50 Ohm environments,” in Proc.2006 Int. Microwave Symp., San Francisco, CA, 2006.

[26] D.F. Williams, H. Khenissi, F. Ndagijimana, K.A. Remley, J.P.Dunsmore, P.D. Hale, C.M. Wang, and T.S. Clement, “Sampling-oscilloscope measurement of a microwave mixer with single-digitphase accuracy,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 3,pp. 1210–1217, Mar. 2006.

[27] D.A. Humphreys and J. Miall, “Traceable RF peak power mea-surements for mobile communications,” IEEE Trans. Instrum.Meas., vol. 54, no. 2, pp. 680–683, Apr. 2005.

[28] T.S. Clement, P.D. Hale, D.F. Williams, and J.M. Morgan,“Calibrating photoreceiver response to 110 GHz,” in 15th AnnualMeeting of the IEEE Lasers and Electro-Optics Society Conf. Dig., pp.877–878, Nov. 2002.

[29] D.F. Williams, P.D. Hale, T.S. Clement, and J.M. Morgan,“Mismatch corrections for electro-optic sampling systems,” inAutomatic RF Techniques Group Conf. Dig., vol. 56, pp. 141–145,Nov. 2000.

[30] D.F. Williams, P.D. Hale, T.S. Clement, and J.M. Morgan,“Calibrating electro-optic sampling systems,” in IEEE MTT-S Int.Microwave Symp. Dig., vol. 3, pp. 1527–1530, May 2001.

[31] P.D. Hale, C.M. Wang, D.F. Williams, K.A. Remley, and J.Wepman, “Time Base Correction (TBC) software package,” 2005[Online]. Available: http://www.boulder.nist.gov/div815/HSM_Project/Software.htm

Conventional oscilloscope timebases use trigger circuits

and a programmable delay generator to time the acquisition

of voltage samples.