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NIST Technical Note 1470

NIST Measurement Services: Fast

Repetitive Pulse Transition

Parameters, 50 fl

ICO

9,00 5

C2-

Donald R. Larson

Nicholas G. Paulter, Jr.

National Institute of Standards and Technology • Technology Administration • U.S. Department of Commerce

NIST Technical Note 1470

NIST Measurement Services: Fast

Repetitive Pulse Transition

Parameters, 50 H

Donald R. Larson

Nicholas G. Paulter, Jr. Electronics and Electrical Engineering laboratory

Quantum Electrical Metrology Division

May 2005

/ W \

T3F

U.S. Department of Commerce

Carlos M. Gutierrez, Secretary'

Technology Administration

Phillip J. Bond, Under Secretaryfor Technology

National Institute of Standards and Technology

Hratch G. Semerjian, Acting Director

Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept adequately. Such

identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the

entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Technical Note 1470

Natl. Inst. Stand. Technol. Tech.. Note 1470, 56 pages (May 2005)

CODEN: NTNOEF

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WASHINGTON: 2005

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Mail: Stop SSOP, Washington. DC 20402-0001
Table of Contents

List of Tables 4

List of Figures 4

1

.

Description of Measurement Service 5

1 .

1

Quantities Measured, Ranges, and Expanded Uncertainties 6

1 .2 Types of Artifact or Transfer Standards 8

1 .3 Shipping and Handling Instructions 9

2. Design Philosophy 10

2.1 Physical Principles of Sampler Structures 10

2.2 History 11 2.3 Impulse Response using the Nose-to-Nose Technique 14

2.4 Timebase Characterization 15

2.5 Vertical Axis Calibration 16

2.6 Physical Principles of Pulse Generators 16

3. Description of System 17

3.1 Trigger Generator 17

3.2 Splitter/Delay Line 18

3.3 Microwave Synthesizer 18

3.4 Sampler 18

3.5 Pulse Generator 19

3.6 Computer 19

3.7 Software 19

4. Operating Procedure 24

4.

1

Preparation 24

4.2 Data Acquisition 25

4.3 Data Analysis 26

4.4 Preparation of the Uncertainty Analysis 26

4.5 Preparation of the Report Of Special Test 27

4.6 Miscellaneous Tasks 27

5. Uncertainty Analysis 29

5 .

1

Pulse Amplitude Uncertainty 29

5.2 Transition Duration Uncertainty 31

5.3 Overshoot Uncertainty 34

5.4 Undershoot (preshoot) Uncertainty 35

6. Procedures for Quality Control 36
7. References 37

Appendices 39

Al. Sample Report of Special Test 39

A2. Pulse Parameter Uncertainty Analysis, Metrologia, Vol. 39, pp. 143-155, 2002. . . 39
List of Tables

Table 1-1. Uncertainty for Calibration of Fast Repetitive Pulse Transition Parameters 8

Table 2-1. Personnel and Organizational History 12

List of Figures

Figure 1-1 . Typical waveform measured under 65200S 5

Figure 1-2. Definition of pulse amplitude and transition duration 6

Figure 1-3. Definition of overshoot and undershoot 7

Figure 3-1. Waveform Parameter Analysis System 17

Figure 3-2. FPOCS Main Menu 19

Figure 3-3. FPOCS Menu Selections 20

Figure 3-4. FPOCS Data Acquisition and Temperature Logging Menu 20

Figure 3-5. FPOCS Waveform Plot and Step Parameters Menu 21

Figure 3-6. FPOCS Timebase Characterization Menu 21

Figure 3-7. FPOCS Timebase Correction Menu 22

Figure 3-8. FPOCS Step-Step Deconvolution Menu 22

Figure 3-9. Typical FPOCS Instructions/Help Menu 23

Figure 5-1. The percent change in pulse amplitude with temperature relative to 15 °C 29

Figure 5-2. Timebase errors 31
NIST Measurement Services: Fast Repetitive Pulse Transition Parameters, 50 D.

Donald R. Larson and Nicholas G. Paulter, Jr.

ABSTRACT This document describes the Special Test 65200S, "Fast Repetitive Pulse

Transition Parameters, 50 Q." This special test service is used to acquire the parameters:' pulse

transition and amplitude of fast repetitive pulses.

1. Description of Measurement Service

The 65200S is optimized for measuring certain parameters of waveforms corresponding to very

fast transition pulses, where the transition durations of these pulses are less than 350 ps (or

equivalently, the 3 dB attenuation bandwidth of the pulse is greater than 1 GHz). These pulses can also represent the step response of high-speed samplers. An example of a measured

waveform is shown in Figure 1-1.

0.05

-0.25 =

0.0 4.0x10 8.0x10'

lu 1.2x1(

Time (s)

1.6x10 2.0x10

Figure 1-1. Typical waveform measured under 65200S.

5
Figure 1-2. Definition of pulse amplitude and transition duration.

A brief description of this measurement service is provided in [1]. The service is currently

performed using the Waveform Parameter Analysis System (WPAS).

1.1 Quantities Measured, Ranges, and Expanded Uncertainties

The parameters provided by this service are: waveform or transition amplitude, transition

duration, overshoot and undershoot. The amplitude is typically reported in volts, the transition

duration (both 10 % to 90 % and 20 % to 80 %) is reported in seconds, and the undershoot and

overshoot are in percent of waveform amplitude. The parameters are best described by Figures

1-2 and 1-3. These parameters conform to the definitions and descriptions contained in EEEE

Standard 181-2003 [2].

The definition of amplitude as given in the IEEE Standard 181-2003 is "the level of the state succeeding a transition minus the level of the state preceding the same transition." Transition

duration is defined as the difference between two instants on the waveform, these instants

occurring when the waveform crosses certain user-defined reference levels, such as the 1 % and

90 % reference levels.
Post-Transition Overshoot

Post-Transition Undershoot

Figure 1-3. Definition of overshoot and undershoot.

The parameters of undershoot and overshoot have historically been subject to interpretation and,

without further clarification, are considered deprecated terms in Reference 2. The most common

usage of undershoot and overshoot refer to what is now defined as the pre-transition undershoot

and post-transition overshoot [2]. There is also pre-transition overshoot and post-transition

undershoot; this gives a total of four "shoof'-related terms. In Figure 1-3, the Post-Transition

Aberration Region is defined in [2] to be that period between the 50 % reference level instant

(t50%) and t 50o/o + 3td where td is the transition duration. The Pre-Transition Aberration Region is similarly defined (t50% - 3^. Overshoot and undershoot (the deprecated terms) were once

included in this service but were dropped for lack of a suitable uncertainty analysis. The four

"shoof'-related terms were recently introduced into the 65200S after completion of a new

uncertainty analysis, automation of the uncertainty analysis calculation, and analysis of a

sufficient number of 65200S tests. The algorithms used to obtain these parameters also conform

to the IEEE standard. The parameters, ranges, and expanded uncertainties for the 65200S are

listed in Table 1-1 . Notes are shown after the table.
Table 1-1. Uncertainty for Calibration of Fast Repetitive Pulse Transition Parameters

Parameter Parameter Range Typical Expanded Uncertainty

Pulse Amplitude (A) -400 mV < A < 400 mV 1.0mV+1.4AA'

Transition Duration (tj) 5 ps ^ td < 100 ns 1.25 ps + 0.1 At 2

Pulse Duration (tP) (between

50 % reference level instants)

10 ps < tP < 100 ns 1.8ps + 0.14At

Pulse Overshoot < 50 % 2% 3

Pulse Undershoot < 50 % 2% 3

Notes:

'AA is the amplitude discretization interval and is calculated using the full-scale amplitude range

set on the sampler (for example, the full scale amplitude range is 100 mV for an amplitude

sensitivity setting of 10 mV/div and a full scale display of 10 vertical divisions) and effective

number of bits of the analog-to-digital converter at the input of the sampler. The effective

number of bits is based on the actual number of bits of the converter and signal averaging where

the noise level exceeds the range of the least significant bit of the converter.

2 At is the sampling interval, that is, the interval between sampling instances, used during

acquisition of the DUT waveform. For example, a waveform epoch of 1 ns where the waveform

contains 1000 elements gives a sampling interval of 1 ps.

3 By definition, undershoot and overshoot are positive values, therefore, the lower uncertainty

bound is limited to a value such that undershoot and overshoot are greater than %.

1.2 Types of Artifact or Transfer Standards

To be acceptable for calibration, the customer's pulse generator must meet the criteria outlined in

the following list.

1

.

The device must provide a trigger output with a repetition rate between 10 Hz and 2 GHz.

Alternatively, NIST can provide a range of trigger signals, however, in this case, the

customer's pulse generator must be able to be driven at a rate between 10 Hz and 2 GHz.

2. The customer's device must have a nominal output impedance of 50 Q.

3

.

The customer's device must have a precision coaxial output connector, such as, an SMA,

GPC-7, Type N, GPC-3.5, etc. Note, the NIST WPAS input uses a 2.4 mm (female)

coaxial connector. Consequently, the measured response of the submitted device using

any other coaxial connector will contain the effects of any necessary adapters.
4. The maximum input signal amplitude (including overshoot and undershoot) that is measurable without attenuators is 800 mV. For pulses with greater pulse amplitudes, the

customer shall supply an attenuator to decrease the pulse amplitude to 800 mV or less. The permissible dc offset (without attenuators) is ± 500 mV.

5. The minimum pulse transition duration must be greater than or equal to 5 ps.

6. The pulse duration is only measured for rectangular pulses or impulse-like pulses.

If the customer's device is a sampling head, it must either be compatible with existing NISTowned oscilloscope mainframes (Tektronix 1 1801, TDS 8000, HP 54120, HP 54750, and Agilent

86 100A)

1 or the customer must provide a compatible mainframe. If the customer provides the

mainframe, the customer must make provision for transfer of the data acquired by the

oscilloscope to the PC compatible computer of the WPAS for analysis.

1.3 Shipping and Handling Instructions

All aspects of shipping (both receipt from and returning to the customer) are arranged through

the responsible NIST personnel.

Pulse generators and samplers are static sensitive and should only be handled on electro-static

discharge (ESD) protective surfaces and while wearing a wrist strap or other means of controlling

static discharge. The WPAS sits on an ESD work surface and ESD wrist straps are located at multiple points around and on this work surface.

'Certain commercial equipment, instruments, or materials are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the

National Institute of Standards and Technology, nor does it imply that the materials or equipment

identified are necessarily the best available for the purpose.
2. Design Philosophy

The purpose of the 65200S is to provide pulse parameters (pulse amplitude, transition duration,

overshoot, and undershoot), as defined by national and international standards [2,3], of

waveforms containing one or two fast transitions and two states (such as the high and low states).

The methods used in the 65200S for computing the values of these parameters are also defined

by national and international standards [2,3]. The waveforms obtained with the 65200S can be

either those taken from the output of a customer's pulse generator and measured with a WPAS

sampler or those taken from the output of a WPAS pulse generator and measured with a customer's sampler. To measure the step response of high-speed samplers, the 65200S uses pulse generators with known outputs. To measure the output of high-speed pulse generators, the

65200S uses samplers with known impulse responses.

Instruments used to measure dc and steady-state values, such as voltmeters, signal analyzers, and

spectrum analyzers, cannot be used in the 65200S because the 65200S is used to measure

transition-related parameters of pulse waveforms. Instruments for measuring high speed

transitions must be used to accomplish the requirements for the 65200S, and the only instruments

that satisfy these requirements are high-speed waveform recorders (real time and sampling

oscilloscopes). In particular, the 65200S uses high-speed equivalent-time samplers, with a step

response transition duration < 5 ps (or, equivalently, a 3 dB attenuation bandwidth > 70 GHz).

Current state-of-the-art real time oscilloscopes do not have the bandwidth (impulse response)

necessary to measure high-speed pulse parameters.

2.1 Physical Principles of Sampler Structures

The voltage pulse that is input to the WPAS sampler is sampled as it propagates down a transmission line. The sampling elements are in contact with the transmission line but have a much higher electrical impedance (> 1 kD.) than that of the transmission line (nominally 50 Q).

Consequently, the effect of the sampling element is negligibly small (100 times less than the

published uncertainty values) and, if desired, can be computed based on a simple voltage divider

network. The sampler and the transmission line are incorporated into one unit and are not

typically separable. The voltage or current of the pulse is sampled as it propagates down the

transmission line, and this sampled voltage or current is used to construct the pulse's waveform.

The sampled waveform of the pulse is presently obtained using an equivalent-time sampling

process. In this process, the sampling element is activated by a trigger pulse at specific instants

relative to the pulse being measured. Typically only one instant on the waveform is obtained for

each trigger pulse, and the waveform datum corresponding to this instant may be the result of

more than one sampled event. The waveform is constructed by concatenating the waveform data

from successive sampling instants. This sampling process requires a repetitive stationary pulse

signal. Deviation of the pulse from stationarity results in an increase in trigger jitter and

amplitude noise. The effect ofjitter is compensated and the effect of both the jitter and noise are considered components in the uncertainty analysis (see Section 5).

Sampling structures include, but are not limited to, semiconductor diodes, superconducting

10
Josephson junction devices, electrooptic and photoconductive devices. Semiconductor diodes

are the mainstay of sampling instruments because they can be implemented using all solid state technology, whereas electrooptic and photoconductive methods require, in addition to solid state technology, ultrashort pulsed laser technology. Semiconductor diodes provide a sampled voltage

waveform of the pulse. This voltage waveform is obtained by strobing the diode with an

electrical pulse. The instant the strobe is being applied to the diode, the forward resistance of the

diode decreases and a replica of the pulse at that instant passes through the diode to a hold

circuit. The magnitude and duration of the replica is dependent on the magnitude and duration of

the strobe pulse, the switching behavior of the diode, and the material properties of the diode (see

[4,5] for example, for information on diode operation).

Josephson junctions are metal-insulator-metal devices that are cooled until the metal becomes

superconducting. If the insulator, usually an oxide, is sufficiently thin, current may tunnel

through the insulating barrier. This device may switch between zero and finite voltage states in a few picoseconds [6]. Hypres, Inc. manufactured and marketed a digital sampling oscilloscope

using a Josephson junction based sampler from 1987 until 1990. The impulse response duration

was approximately 5 ps, the 3dB attenuation bandwidth was 70 GHz and the sampler was cooled

using liquid helium.

Electro-optic devices provide a sampled voltage waveform of the pulse. This voltage waveform

is obtained through the interaction of the electric field of the propagating pulse with laser light in an electrooptic material. The applied electric field modulates the refractive index of the

electrooptic material usually through the Pockels Effect. This index modulation can be measured

with an appropriate optical system by observing a change in optical intensity [7]. This process

has resulted in several different sampling instruments[8-10], one of which is used routinely for

high-speed pulse measurements [11].

Photoconductive devices provide a sampled current waveform of the pulse. The current

waveform is obtained by extracting charge from the propagating pulse. The charge is extracted

when the sampling gate is excited (turned on) (for examples, see [12-15]). The amount of charge

extracted and the effect of the sampling process on the propagating pulse are dependent on the

on-state resistance of the photoconductor [16]. Several measurement systems have been

developed using this technology [12,17,18].

2.2 History

NIST has offered the transition duration special test service since before July 24, 1970 when the

first special test was performed. Dr. Norris S. Nahman, Section Chief of the Pulse and Time

Domain Section in the Electromagnetics Division (272.20), oversaw the introduction of the pulse

parameter measurements at NIST (then NBS). R. M. Jickling performed this first test on a Tektronix model 067-0513-00 tunnel diode pulse generator. Rockwell was the customer and

they were charged $130 for the service. The reported transition duration of that first test was

21 ps ±3 ps. Today, the fee for this service, provided "at cost," is typically $2400. For the

majority of the tests performed, the transition durations are approximately 15 ps ± 1.25 ps. Table

11
2-1 lists the personnel and the managing NBS/NIST divisions.

Table 2-1. Personnel and Organizational History.

Calibration Leader (CL) or Authorized User (AU) Date NBS/NIST Division

R.M. Jickling (CL) 1970- 1972 272

W. L. Gans (CL) 1973 - 1995 272,276,724,723,813,

811

J. R. Andrews (AU) 1976-1980 276, 724

L. A. Terrell (AU) 1988 723

S. M. Chesnut (AU) 1989 723

C. A. Jones (AU) 1990 723

N. G. Paulter (AU) 1990 -present 723,813,811,817

J. P. Deyst (CL) 1995 - 1998 811

D. R. Larson (CL) 1998 -present 811,817

At the inception of the 65200S service in 1970, the decision was made to use commercially

available oscilloscopes to perform the measurements. Although commercially available

oscilloscopes are still used in the performance of the 65200S, because they are easy to operate

and the measured waveforms may be easily transferred to computers for data analysis, this is not

a requirement for the 65200S. If used, the commercial oscilloscope must be fully characterized

to determine the uncertainty in the measurement results. The most important characteristic of the

sampling head is its impulse response. Almost all commercial sampling oscilloscopes have used

semiconductor diodes in their samplers. The measured waveform is the convolution of the input

signal and the impulse response of the sampling head. Since every sampling head has a finite

duration sampling aperture and thus, a finite bandwidth, the measured transition duration will be

slower than the transition duration of the input signal. If the impulse response of the sampling

head can be determined, it can be deconvolved from the measured signal to yield a more accurate

estimate of the input signal. Over the years, the 65200S has used either the modeling or

characterization of the sampler to determine its impulse response. In addition to the impulse

response, the vertical gain, the horizontal axis or timebase error, and the temperature dependence

of the impulse response and vertical gain need to be characterized. The pulse generator, which

provides a known output pulse, requires calibration of its output pulse amplitude, transition

duration, and aberrations.

In 1970, a Hewlett-Packard (HP, now Agilent) 181 A oscilloscope with an HP 1815 sampling

12
head and later an HP 1817 sampling head was used.

In 1972, NBS was using an HP 140A oscilloscope with an 1430B sampling head and the system

was referred to as the APMS (Automatic Pulse Measurement System). The 3 dB attenuation

bandwidth of the sampler was estimated to be 18 GHz. It was specially modified by NBS to allow waveform data transfer to a mini-computer.

From the first test in 1970 until the inauguration of the AWAMS in 1989, the impulse response

of the sampling heads used in this service were obtained by modeling the sampling head

circuitry. This work was done by a guest researcher at NBS, Dr. Sedki Riad [19]. The circuit

model was verified by comparing predicted measurement results with actual for a variety of input

signals.

In 1989, the Automatic Waveform Acquisition and Measurement System or AWAMS,

incorporating an HP 54120 oscilloscope with an HP 54121 A sampling head was characterized

and used for performing calibrations.

The impulse response of the sampling head used in the AWAMS was determined using a

different technique than the modeling techniques previously used. In the AWAMS sampler

impulse estimation process, waveforms from three step generators were acquired using a commercial, superconducting sampling oscilloscope (Hypres, Inc. Picosecond Signal Processor,

model PSP-1000). These Hypres-acquired waveforms were presumed to be perfect estimates of

the signals produced by the step generators because the manufacturer claimed that the bandwidth

of this sampler was about 70 GHz which was three and half times the bandwidth of the step

generators and of the AWAMS sampling head (20 GHz). The impulse response of the AWAMS

sampling head was determined by acquiring waveforms from the three step generators using the

AWAMS and then deconvolving the assumed perfect step pulse estimates obtained from the

superconducting sampling oscilloscope. This introduced a bias in the result and the reported

uncertainty was non-symmetrical, reflecting this bias.

In 2001, to meet the above sampler requirements, NIST began using an HP 54750A oscilloscope

with HP 54752A sampling heads (50 GHz bandwidth). Use of this equipment is not mandatory

and any instruments with equivalent specifications can also be used. The present measurement

system used in the 65200S is referred to as the Waveform Parameter Analysis System (WPAS).

The impulse response estimate of the sampling heads used in the WPAS has been obtained using

the "Nose-to-Nose" technique (see Section 2.4). This measurement technique is performed

periodically to check the impulse response estimate of the samplers.

In 2004, NIST replaced the HP 54750A/54752A with the Agilent 86100A oscilloscope and

Agilent 861 1 8A sampling head which has a bandwidth greater than 70 GHz. This instrument has

also been characterized using the "Nose-to-Nose" technique.

2.3 Impulse Response using the Nose-to-Nose Technique

13
This technique is only applicable when using particular diode-based sampling circuits provided

by certain manufacturers. In these particular sampling circuits, if the offset level of the sampler

is a value other than V, the sampling head generates or "kicks out" a pulse each time it samples a signal [20,21]. It has been assumed that the pulse so generated is equivalent to the

impulse response of the sampling head. We have checked that assumption and found that it introduces an uncertainty of only about 0.3 ps in the impulse response estimate of the sampling

head. The Nose-to-Nose technique uses three sampling heads and two oscilloscope mainframes.

The inputs of two sampling heads, each in a separate mainframe, are connected together in such a way as to make them appear "nose-to-nose." One sampling head is made to kick-out a pulse and

the other sampling head is used to acquire the kick-out pulse. However, the strobe pulse, which

is generated by the trigger and initiates the sampling event, feeds through the sampling diode and

appears as part of the measured kick-out waveform. This can be removed since the polarity of

the strobe pulse contribution does not change with the offset voltage polarity whereas the kickout pulse contribution does. Two kick-out waveforms are acquired, one with an offset voltage

polarity opposite from the other, and a difference waveform is calculated by dividing the

difference of the two waveforms by two. The resulting difference waveform is an estimate of the

desired kick-out pulse. Each of the three sampling heads, indicated below by the letters a, b and

c, are used sequentially as a sampler and as a pulse generator. The following equations represent

how the impulse response estimate is obtained from these measurements. The measurements

yield three difference waveforms, ntnab, where sampler a acts as the pulse generator and sampler

b as the receiver; ntnbc, where sampler b acts as the pulse generator and sampler c as the receiver;

and ntnca , where sampler c acts as the pulse generator and sampler a as the receiver. The "*"

indicates the convolution operator:

ntnab = a

g *br , ntnbc = b

g *cr

, (1)

ntnca = Cz* ar>

where the "g" and "r" subscripts denote generator and receiver. The sampler impulse response is computed via a deconvolution process which is usually performed in the frequency domain. The

spectra of the difference waveforms is given by:

^=4v4

v>f{v4 (2)

C„A. = F{cn *ar .,

where F indicates a Fourier transform. The deconvolution process involves multiplication of the

spectra of two difference waveforms and division by a third difference waveform spectrum:

14
A

l Cg Ar^r

C

g

Br A BBC

C

g > g i Bg Ar

The estimates of the impulse response of the three samplers are then found by performing the

inverse Fourier transform on the A, B, and C thus found.

2.4 Timebase Characterization

In the AWAMS, which preceded the WPAS, the timebase was calibrated by applying a sinewave

of known frequency to the input of the oscilloscope, measuring the time interval between zero

crossings, and then comparing the length of these intervals to the expected value calculated from

the known frequency. This gave the cumulative timebase error between zero crossings. The

waveform was then corrected using linear interpolation. However, this process assumes the

timebase errors are uniform between zero crossings.

The WPAS timebase uses a more accurate characterization method than was used for the

AWAMS. The WPAS timebase is calibrated using nominally single- frequency sinusoidal

signals provided by a microwave synthesized frequency source. Waveforms of these signals are acquired and errors in the timebase are derived from sinewave curve fitting techniques [22]. The

synthesizer is the artifact standard and the manufacturer-claimed uncertainty in frequency

uncertainty (lOuHz/Hz) is used. These errors may be mapped to the time-domain via the

following analysis.

T=NAt = X~, (4)

Js

where T is the duration of the waveform epoch, N is the number of samples (data elements) in the

waveform, At is the interval between sample instants,^ is the input frequency provided by the source, and X is the number of cycles oifs observed in the waveform epoch. To determine the

uncertainty in T caused by uncertainty, ur infs

, we take the appropriate partial derivative and

analyze the result. The effect of uj on the measurement is given by:

uAD-^-^-rf. (S)

The uncertainty in^ relative tofs is about lOuHz/Hz. Therefore, the uncertainty in the duration of

15
the waveform epoch due to uncertainties in the reference frequency is about lOuHz/Hz. Since

the waveform epoch contains between about 1 000 and 4000 elements, this epoch error corresponds to about 0.04 of a sample interval. This is less than the limit imposed by amplitude

noise.

2.5 Vertical Axis Calibration

The amplitude gain of the WPAS is based on comparing WPAS measurements made on a given

pulse source with those made using a high-accuracy voltmeter, the NIST Sampling Waveform

Analyzer (SWA [23]), on the same pulse source. The voltage accuracy of the SWA is approximately 100 times greater than that of the WPAS. Furthermore, the amplitude accuracy of

the voltmeter has been compared to that of thermal voltage converters [24] on which many

national metrology laboratories base ac voltage measurements. The pulse measured by the SWA

is the transfer standard upon which calibration of the WPAS sampler amplitude gain is based.

The step response of the WPAS sampler is determined by measuring a reference fast-transition

electrical pulse and deconvolving this known pulse from its measurement [25,26,27]. This

process is also known as a waveform reconstruction process, and more will be said about the

effect of this process on measurement uncertainty in Section 5. There are several ways to obtain

the known electrical pulse [1 1,20,21]. This known electrical pulse is the artifact standard on

which the step response of the WPAS sampler is calibrated. The pulse generator of the WPAS is calibrated using a sampler with a known step or impulse response. The process used in this

calibration is similar to that for the sampler step response, however, in this case the sampler is the known quantity.

2.6 Physical Principles of Pulse Generators

Pulse generators can also be achieved with different technologies and methods, but the most

common methods use semiconductor diodes [5]. A less commonly used method, but one that

provides pulses of very short duration and with very high bandwidth is the photoconductive

method [12]. Both of these methods operate similarly to those described or referenced in Section

2.1.

16
3. Description of System

The WPAS is depicted diagrammatically in figure 3-1. Described in this section are the

performance requirements of each of the components given in Figure 3-1 and the manufacturer

name and model that we are presently using to satisfy the stated requirements.

Computer / IEEE-488

crr:~. Microwave Synthesizer

888888 I

• « •

Sampling Oscilloscope

Trigger Generator

Splitter/Delay Line Pulse Generator

r Delayed Trigger

Pulse Signal

Trigger Pulse

Figure 3-1. Waveform Parameter Analysis System.

This service provides characterizations of both pulse generators and sampling heads from several

manufacturers. The equipment usage is slightly different depending on which is being

characterized.

3.1 Trigger Generator

The trigger generator must exhibit a fast transition to minimize the introduction ofjitter caused

by noise on the trigger signal. It also must have sufficient amplitude to drive both the

oscilloscope trigger and the pulse generator trigger after passing through the delay line/splitter.

Present performance requirements:

output pulse transition duration: < 150 ps

output pulse amplitude: ^ 2 V

output impedance: 50 Q. ± 1 Q

The trigger generator presently used to meet these requirements is a Berkeley Nucleonics

17
Corporation model 6040 mainframe with a model 201 E pulse generator plug-in.

3.2 Splitter/Delay Line

Present performance requirements:

delay: 22 ns to 24 ns output impedance: 50 Cl ± 1 Q.

The presently-used splitter/delay line is a packaged collection of passive coaxial devices that split

the trigger pulse into two signals and delays one of the resultant trigger signals by a fixed

amount. The undelayed trigger pulse is used to trigger the oscilloscope. The delayed trigger

pulse is used to trigger the Pulse Generator. The delay is necessary since the oscilloscope waits a

certain interval after receiving a trigger pulse before sampling. The presently-used oscilloscope

requires a 22 ns trigger delay. By using this splitter/delay line instrument, the sampling head and

pulse generator can be made to operate from a common trigger pulse, thereby reducing system

trigger jitter.

3.3 Microwave Synthesizer

Section 2.5 outlined the application of the microwave synthesizer to the timebase

characterization. Continuous wave operation is required from the microwave synthesizer.

Present performance requirements:

Frequency Range: 0.01 GHz to 26.5 GHz

Spectral Purity (Harmonics): < -55 dBc, 0.05 to 40 GHz

(Subharmonics): < -55 dBc

Leveled Output: > +1 dBm

The current microwave synthesizer is a Gigatronics model 7000 that has been specially modified

by the manufacturer to include a 100 MHz reference output. The microwave synthesizer is used

to measure the error in the timebase of the oscilloscope. The 100 MHz reference output is used

to trigger the oscilloscope and provides a lower jitter trigger signal than a 10 MHz reference

output typically used on microwave synthesizers. However, the frequencies used for the

timebase characterization are low enough that a 10 MHz reference would be sufficient. Other

microwave synthesizers have been used and yield suitable results.

3.4 Sampler

The performance requirements for the sampler and pulse generator are near the "state-of-the-art"

in commercial sampler and pulse generator technology.

Present performance requirements for the 65200S sampler are: transition duration ( 1 % to 90 %)

< 5 ps

amplitude range: ^ -400 mV to ^ 400 mV

trigger jitter: <, 1 ps

18
The sampler presently used in the 65200S consists of a sampling head and oscilloscope

mainframe which houses the sampling head and provides the timebase, the two most critical

elements in the measurement process. The WPAS currently uses an Agilent 86100A

oscilloscope mainframe and an Agilent 861 18A dual channel sampling head plug-in module.

The nominal 3 dB attenuation bandwidth of the sampling head is 80 GHz. The sampler selection

is subject to change as the "state-of-the-art" advances. It is evident from the history provided in section 2.2 that system components have been upgraded in the past and it is anticipated that this

process of upgrading will continue.

3.5 Pulse Generator

Present performance requirements for the 65200S pulse generator are: transition duration (10 % to 90 %): < 5 ps

amplitude: ^ 0.2 V

trigger jitter: < 1 ps

Typically, the pulse generator used is a Tektronix 067-1338-00 Calibration Reference Step

Generator or the kick-out pulse from an Agilent 861 18A sampler.

3.6 Computer

The computer controller includes an IEEE 488 interface card that enable communications with

all the equipment used in the WPAS. It is used to set up the oscilloscope and microwave

synthesizer, to retrieve the waveforms acquired by the oscilloscope, and store them. It is also

used to run the LabVIEW Vis (programs) that are used to calculate the deconvolution, determine

the waveform parameters, and estimate the uncertainties.

3.7 Software

LabVIEW Vis have been written for many aspects of the service. Many of these have been

gathered into a collection of libraries referred to as the NIST Fast Pulse Oscilloscope Calibration

System (FPOCS). The FPOCS is currently used at NIST and at the U.S. Air Force Primary

Standards Laboratory. The user interface for the main menu of the FPOCS software is depicted

in Figure 3-2. From the main menu, the operator selects a sub-menu (see item 3 on the main

menu, Figure 3-3) to perform the available operations. The menus for performing the data

acquisition and temperature logging, pulse parameter calculation, timebase characterization and

correction, and deconvolution are included (Figures 3-4 through 3-8). Each menu also includes an "Instructions and Help" menu containing instructions for operating this equipment and

running the software (Figure 3-9).

19
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UNCERTAINTY ANALYSIS

Figure 3-3. FPOCS Menu Selections.

20
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21
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Figure 3-7. FPOCS Timebase Correction Menu.

22
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Figure 3-9. Typical FPOCS

Instructions/Help Menu.

23
4. Operating Procedure

The following operating procedures must be strictly followed to obtain consistent, repeatable

measurement results.

4.1 Preparation

Prior to beginning the measurements, several records need to be started or updated and the

equipment to be used must be prepared for the tests.

4.1.1 Documentation

Once the device to be calibrated has been delivered to the laboratory by the Division Office staff, a manila folder is created as a repository of the test results and all notes on the device under test and test conditions and results, shipping information, etc. This folder is maintained by the group

performing the 65200S test and is presently located in a filing cabinet located outside of the

shielded room in which the tests are performed. A new test sheet is also prepared and inserted in

the three ring binder located near the laboratory entrance, also outside the shielded room. This

test sheet is used to maintain a record of the important dates associated with each test which may

not be recorded elsewhere, such as the date the device was delivered for testing and the date the

test report was finished. It is also used to record the fee charged for the test. The Calibration

Services office staff will prepare a blue calibration folder and the Division Office staff will create an entry in the Information System to Support Calibrations (ISSC), a web-based calibration

tracking system. Once entered in the ISSC, an email will be sent that contains the due date, the

customer purchase order number and the calibration folder number. The ISSC system will also

automatically send an email reminder to the individual performing the test when the test due date

is one week away. At the same time, the ISSC system will also send an email to the Group

Leader of the individual performing the test.

4.1.2 Preparation of Measurement System Components

All connectors are inspected and cleaned prior to connecting the equipment. The connectors are inspected for bent or mis-shapen center or outer conductors. The cleaning procedure is as follows. Blow out the connector with clean, dry compressed gas. Then using a cellularurethane-foam tipped cleaning swab dipped in an alcohol solution, wipe out the mating surfaces

of the connector. Blow out again with clean compressed gas to remove loose particles and

excess alcohol. A new swab is used for each connector. The head of the swab should be small

enough to fit between the center conductor and the interior of the threaded nut of the connector

without applying force to the center conductor. The foam swabs that we are presently using are Chemtronics "Flextips Mini" foam swabs, part number CXM50. It has a 4 mm (3/16 inch) wide,

open celled urethane foam head. The alcohol solution that we are using is Radio Shack

"Professional Tape Head Cleaner Fluid," catalog number 44-1 113. It is composed of

Isopropanol (49 %), Ethanol (48 %), and Methanol (3 %). The clean, dry compressed gas

presently used is Chemtronics "Ultra Jet" part number ESI 020. The following steps must be

performed.

Inspect and clean connectors on DUT.

24
Inspect and clean connectors on sampler.

Inspect and clean connectors on microwave synthesizer.

Inspect and clean adapter (if any) used between DUT and Sampler.

Inspect and clean connectors on BNC trigger source. Inspect and clean connectors on the trigger delay line. Inspect and clean connectors on all cables to be used.

Connect all equipment and cables, using a torque wrench to apply 0.90 Nm to each connection.

Power-on all equipment and let warm-up for at least 2 h before continuing with the test. The two

hour time period is not arbitrary, it is a minimum established by monitoring the temperature of

the oscilloscope mainframe.

4.2 Data Acquisition

1

.

Record the temperature and humidity at the beginning of the measurements.

2. Perform Internal Calibration of Sampler Plug-In, found in the Utility menu of the

oscilloscope. This sets the vertical gain and offset values.

3. Acquire sinewave data for timebase calibration (use the Gigatronics synthesizer and 100

MHz ref output as the trigger signal to the oscilloscope for best signal to noise). 4.0 GHz

and 3.8 GHz are the frequencies that are used most often. At least two frequencies must

be used and they must not be harmonically related. At each frequency, at least two phases

must be used since the low slope segments of each sinewave are omitted and complete

epoch coverage is required. The phase may be adjusting by changing the trigger level a few millivolts.

4. Calculate the timebase error is calculated by running the timebase characterization

operation (see Figure 3-6) with the sinewaves just acquired as inputs.

5. Change the trigger source from the Gigatronics synthesizer "100 MHz ref output to the

output of the BNC 6040/20 IE pulse generator.

6. Examine the timebase error and identify a region of low or zero slope. When measuring

the DUT waveforms or when sampling the reference input (kick-out pulse), the transition

should be on a low slope region of the timebase error. The transition instant can be

positioned by adding a cable or an adapter to the trigger signal path.

7. Acquire DUT waveforms using the FPOCS data acquisition menu. Typical settings are: 200 ps/div

transition at 1.5 division (300 ps) from start. 40 mV/div, -125 mV offset (or as appropriate for the DUT)

trigger + slope, 2.5 GHz trigger bandwidth, trigger level 280 mV, best throughput setting

512 averages, 1024 points (4096 points for sampler calibrations)

25
8. Acquire 3 waveforms, disconnect, re-connect and repeat until at least 12 waveforms

acquired.

9. The temperature before and after each waveform is acquired will be automatically logged

by the FPOCS Data Acquisition Menu.

1 0. Measure the jitter by increasing the time resolution to 10 ps/div and using the built-in

histogram capabilities of the oscilloscope. A 1.0 mV window should be used for

consistent results. Average at least three jitter measurement results.

1 1

.

Record the temperature and humidity at the end of the test.

12. Make a record of the test conditions including humidity, temperature, start time and end

time, oscilloscope settings, filenames of acquired waveforms, microwave synthesizer

settings and trigger generator settings. Also record the use of any adapters and whether or not the customer included a power supply, cable or other items. Note any abnormal or unusual occurrences such as broken parts or unexpected device properties.

4.3 Data Analysis

The pulse parameters are found and tabulated for each acquired waveform using the Waveform

Plot and Step Parameters routine in the FPOCS (fig. 3-5) and using the software program

"Pulsepar.VI." Each waveform is processed independently of all the others.

The amplitude is found by constructing a histogram of the vertical data. This histogram is then

divided into two sub-histograms, an upper and a lower histogram, the division taking place at the

mean value of the maximum and minimum waveform values. The modes are found for the upper

and lower histograms. The two resulting mode values are taken as the low state (Sj) and high

state (S2 ), the amplitude being the difference between these two states.

The 10 %, 20 %, 50 %, 80 %, and 90 % reference levels are then calculated using the waveform

amplitude just found. The reference level instants (instant that the waveform intersects the

reference level) are then found for each reference level. Since the discrete waveform values

often do not occur at the sought reference level, linear interpolation of the data is used. The 10 %

to 90 % transition duration is the difference between the 10 % and 90 % reference level instants.

The 20 % to 80 % transition duration is defined similarly.

The acquired waveforms are processed to remove the effects of the finite bandwidth of the

sampler. The deconvolution of sampler impulse response is done using the routine in the FPOCS

and latest sampler impulse response estimate. The deconvolution algorithm is described in

reference [25] and was developed by Tamas Daboczi while a guest researcher at NIST. The

deconvolved waveforms are also stored on the computer.

The pulse parameters for these deconvolved waveforms are found as previously described.

26
4.4 Preparation of the Uncertainty Analysis

The uncertainty analysis has been automated using LabVIEW Vis. Update the Jitter Control

Chart, the Gain Control Chart (if applicable), and the Temperature data in the Uncertainty Vis.

A listing file for the acquired data and another for the deconvolved data is also created.

Calculate the uncertainty using the LabVIEW Uncertainty Vis for Amplitude and Transition

Duration. Print the front panel for each parameter's uncertainty.

4.5 Preparation of the Report Of Special Test

The values reported are: amplitude, transition duration (both 10 % to 90 % and 20 % to 80 %

transition durations), undershoot and overshoot. The reported amplitude value is the average of

all acquired waveforms amplitudes. The transition duration, undershoot and overshoot parameter

values are the average of the respective parameters of all the deconvolved waveforms.

The Report of Special Test is generated by retrieving the last test report and changing the entries as needed. The assigned folder number, customer name, device serial number, and list of test conditions and notes need particular attention in the updating process.

Example waveforms are also included in the report. Select an acquired waveform with a

transition duration closest to the reported, average transition duration for inclusion in the Report

of Special Test. Also select a deconvolved waveform using the same criteria for inclusion.

Presently, the AXUM graphing software package is used for plotting the waveform data and

importing into the Report. An example of the Report of Special Test is found in Appendix Al

Clean the connectors of the DUT and repack in shipping container provided by the customer.

In addition to the Report of Special Test, an attachment outlining the uncertainty analysis and

measurement system has been prepared and is provided to the customer.

4.6 Miscellaneous Tasks

After the report is finished and signed, it must be reviewed by the Project Leader, Group Leader

and Division Chief. The Project Leader must initial the "Check Copy" of the Report which will

be archived by the Division Office staff. Two copies (the Check Copy and Customers Copy)

should be placed in the blue calibration folder (prepared and delivered by Division Office staff)

along with the attachment. A third copy is filed with the test notes in the manila folder prepared

at the beginning of the test.

The blue folder also contains the return shipping documents and information. These should be

placed with the device, and left on the work surface near the door to the hallway for pickup by

the Division Office personnel. They will arrange return of the device to the customer.

The ISSC must be notified that the calibration is complete or it will automatically send email

messages to the operator and his supervisors when the calibration estimated completion date

nears. The operator of this service must visit the ISSC web site at: http://www-i.nist.gov/cal27
bin/calibration.cgi, enter the "Technical Staff Functions" (username and password required) and

"Close Test and Enter Calibration Data." The only function currently being used by this service

is the calibration date; no reports are generated or fees calculated.

28
5. Uncertainty Analysis

The uncertainty analysis follows NIST recommended practice as outlined in [28]. Originally,

both Type A and B uncertainties were identified. After extensive measurement system

characterization, most Type B uncertainties have been replaced by Type A. In many cases, this

resulted in a reduction in reported uncertainty. The following section outlines the uncertainty

analysis associated with each reported parameter; for more detail, see Appendix A2.

5.1 Pulse Amplitude Uncertainty

Calculating the uncertainty in the pulse amplitude requires having an equation that describes the

reported pulse amplitude, VA. For our system:

VA = ^4^L =(r Af *V.T )/g, (6)

where the horizontal bars represent the arithmetic mean, V^c is the average of the set of M, pulse

amplitudes corrected for sampler offset errors, VAT is the average of the amplitude corrections

required for a change in measurement temperature, and g is the transient gain correction of the

sampler. Ideally, g = 1 if the sampler exhibits no pulse gain or attenuation and the sampler has

settled within the waveform epoch. The V^c is given by:

Va,c = __./__ Vs2,c - Vsi.c

v (7) = Vs2,m - Vs2,off,m ~l VSl.m ~ Vsi.off.m I,

where Vofr is the voltage offset. The "c" and "m" in the subscripts refer to corrected and

measured voltage values. We have observed that for the presently-available, high-bandwidth

samplers, the voltage offset error is the same for both the top. line (S2) and bottom line (SI

)

voltage levels, therefore, the offset voltage contribution can be ignored since it is zero. The temperature correction term is obtained by measuring the change in the observed pulse

amplitude with temperature (see Fig. 5-1) [29] and is given by:

Vat = S A y/AjlTmeas — Tref I, (8)

where Tmeas is the average of MT temperature values taken during the pulse measurement process,

S is the slope of a straight line fit through a set of previously-acquired amplitude versus temperature data, and Tref is the average temperature of the sampling head taken when the

sampler impulse response was determined.

The transient gain term, g , is obtained by taking the ratio of the amplitude of the reference pulse

as measured using the sampler and the amplitude of the reference pulse as measured using the

SWA reference instrument. As mentioned earlier, the gain term is obtained from a control chart

and is given by:

29
M

M M — —

1 _ v* 1 ^

Vs2,m,r,i ~ Vsi.m.r.i

g i=l Mg tl VS 2,rjr,i-Vsi,r,r,i

(9)

where the "r,r" in the subscript denotes the reference measurement instrument with reference

pulse generator, "m,r" denotes test instrument and reference pulse generator, and there are Mg independent gain measurements.

2

-0

-2

-4

-6

-8

-10

-12

1

1

i

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1 ^1

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1

1

1

i

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00

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o —+ 5 •••••• J .-.*-- J ------- 5 —¥— 5

v. ^ .. ...'._ "^

i 1 1 14 16 18 20 22 24 26 28 30

temperature (°C)

32 34 36

Figure 5-1. The percent change in pulse amplitude with temperature relative to 15 °C.

In addition to measurement-related uncertainties, the reported amplitude values are also subject

to uncertainties from the method used to calculate these values, which in this case, is a histogram

method. The histogram-derived amplitude values, for example for VS2 , are given by:

V -V • V -N _max__Vmm_ v S2 S2 N bins (10)

30
where Nbjns is the number of histogram bins and NS2 is the bin number of the topline mode bin in

the histogram.

Each of these variables contribute to the pulse amplitude. To determine the uncertainty in the

estimate of the pulse amplitude, the partial derivatives of equations 6-10 are taken with respect to the appropriate variables. These partial derivatives are listed in Tables la-lc in Appendix A2,

"Pulse Parameter Uncertainty Analysis," Metrologia, Vol. 39, ppl43-155, 2002. As previously

stated, the calculation of the uncertainty has been automated using a LabVIEW VI. The

Uncertainty, Amplitude VI determines the value of each of these partial derivatives and combines

them in quadrature [28]. The coverage factors are calculated and the expanded uncertainty with

95 % confidence bounds is calculated and reported with the estimate of the pulse amplitude.

5.2 Transition Duration Uncertainty

The reported (and therefore, reconstructed) waveform transition duration, td

, is the average

transition duration extracted from M, reconstructed pulse waveforms. The td is related to the

transition duration of the measured waveform, tim, and the transition duration of the sampler step

response, tdr :

M,

*d =

2-1* d,R,i W (11)

~ fdecV^m'td.r J + At d)AT ,

where T is temperature and tiR , tdm, and tir are the transition durations of the reconstructed,

measured, and sampler step response waveforms. The specific deconvolution functional

relationship, fdec , between td, t4m, and tdr is dependent on the type of waveforms used. For

example, for a Gaussian waveform, td is equal to the square root of the difference of the squares

of transition durations of the measured and step response waveforms. The AtdAT is the

temperature-induced incremental change in transition duration [29].

Since we do not know a priori the functional relationship between td

, tim, and tdr, we obtain an

empirical relationship for the three parameters. We obtain this relationship by fitting a curve

(such as a polynomial) to td R versus t4m data and separately to tiR versus t4r data where both td>m and tdr are varied within expected values and the td is obtained from the reconstructed

waveforms. The tim and tir can be put in terms of the sampling intervals:

t -YSt (12)

where

^

and XL are the real-valued (non-integer) number of sampling intervals describing the

transition duration for the measured sampler step response waveforms (which includes jitter) and

5t is the duration of the equi-spaced sampling interval. The 5t is the average duration of the

31
sampling interval over the waveform epoch [30] and is measured using sine-fit techniques [22]

during the timebase error measurement process. Figure 5-2 shows the timebase errors taken from

two different sampling oscilloscopes. X^ is the result of the convolution of the measurement

jitter and the sampler step response:

Xq - yXr +Xj

(13)

where Xr and X, are the number of sampling intervals in the sampler step response and equivalent

1/3 Q.

10

8

6

4 h

2

-0

-2

-4

-6

-8

-10

-12

4 5 6

time (ns)

10

Figure 5-2. Timebase errors.

jitter step response transition durations. Although X, may not accurately be described by a Gaussian waveform, XT and X= are added in quadrature (Gaussian approximation) to get X^.

Errors are associated with this approximation [31,32] and the uncertainty bounds are adjusted

accordingly. The Xj includes drift of the sampling aperture with respect to its trigger.

The temperature dependent change in transition duration can be expanded:

Atd,AT ~ ^At/AT^meas ~Tre f

J- (14)

32
The curve, SAt/AT , is a straight line fit to the Ms measured transition duration versus temperature

data pairs that are measured independently of the M, acquired waveforms. Using (12) and (14)

in (11) gives:

*d = ^decC^m'^rj^ + ^At/AT^me33 _ Tre f

J. (15)

For samplers and pulse generators presently in use at NIST, the value of SAt/AT is approximately

zero or is much less than the reported uncertainties, however, it will be maintained here for

completeness.

The values of Xm, Xr

, and Xj are determined by linear interpolation to obtain the instant in time

(the reference level instant) corresponding to the given reference level. The value of Xm

(and

analogously for Xr and Xj), is: Xm

= ^K (16)

where t L1 and t u are the time instances corresponding to the first (LI) and second (L2) reference

levels of the transition duration. For example, in the 10 % to 90 % transition duration, LI is the

10 % reference level instant and L2 is the 90 % reference level instant. The t

L1 is given by:

t L , = t L1_ + i^fi^(Ll-Ll_), (17)

where the "+" and "-" subscripts denote the actual data values found immediately above and

below the reference level and, for the time variables, the subscripts correspond to sampling

instances of the those data. The tu can be expanded similarly to that done for t^. The values of

LI and L2 can be expanded:

Ll = Vsl>m +Pl(VS2>m -Vsl)m ), L2 = Vslim + P2(Vs2 , m -Vsl , m ),

where PI and P2 are the percent reference values, such as 10 % and 90 % or 20 % and 80 %.

Each of these variables contribute to the transition duration. To determine the uncertainty in the

estimate of the transition duration, the partial derivatives of Equations 1 1-18 are taken with

respect to the appropriate variables. These partial derivatives are listed in Tables 2a-2d in

Appendix A2. The calculation of the uncertainty has been automated using a LabVIEW VI. The

Uncertainty, Transition Duration VI determines the value of each of these partial derivatives and

combines them in quadrature [28]. The coverage factors are calculated and the expanded

uncertainty with 95 % confidence bounds is calculated and reported with the estimate of the

33
transition duration.

5.3 Overshoot Uncertainty

Voltage offset errors will not be considered here because they will cancel as they did for the

uncertainty calculation of VAc . The equation describing the calculation for the overshoot is:

OS

^max,R ~~ Vs2,R

VA.R

(19)

where VmaxR is the maximum voltage in the appropriate aberration region of the reconstructed

waveform, VS2R is the voltage corresponding to State 2 of the reconstructed waveform and VAR

the amplitude of the reconstructed waveform. Overshoot is usually presented as a percentage.

is Vmax R can be written as: Vv max,R = VV S2,R +T Vv OS,R' (20)

where

VOS.R

bu V t OS v OS,m L d,m

"d,R

(21)

and pos is a correction factor that is determined experimentally and V0Sm is the overshoot of the

acquired waveform. Equation (21) describes an empirical relationship between the overshoot

and transition duration of the reconstructed (reported) waveform and that of the measured

waveform. In (21), we assume that the product of the overshoot voltage and transition duration

is not affected by an all-pass filter, which is how the sampler impulse response is expected to behave for an input signal that has a bandwidth lower than that of the sampler. Vos m can be

expanded:

V~r. Y = V - Vp, OS,m T max.m T S2,m ' (22)

The tiR can be expanded similar to that of t4m, and using this expansion and (12), (20), (21), and

(22) in (19) yields for OS:

OS =

1

V ^S2,R - VS1R j V VS2,m + Pos[ymax,m S2,m

X,

X — VS2,R (23)

where XR

is the real-valued number of sampling intervals describing the transition duration of the

reconstructed waveform. The correction factor Pos is determined by fitting a curve to a set of M9 os kmVos.m versus t are obtained from the acquired waveforms and 4RV0SiR data. The t4m and V

34
the td R and Vos R parameters are obtained from the reconstructed waveforms. The uncertainty in Pos is the standard deviation in the fitted curve relative to the set of corresponding tdmV0Sm

versus td RV0S R data, and the coverage factor is determined by the number of Poss.

Equation 23 identifies the variables that contribute to the overshoot. To determine the

uncertainty in the estimate of the overshoot, the partial derivative of equation 23 is taken with

respect to each of these variables. These partial derivatives are listed in Table 3 that appears in Appendix A2. The calculation of the uncertainty has been automated using a LabVIEW VI. The

Uncertainty, Overshoot VI determines the value of each of these partial derivatives and combines

them in quadrature [28]. The coverage factors are calculated and the expanded uncertainty with

95 % confidence bounds is calculated and reported with the estimate of the overshoot.

5.4 Undershoot (preshoot) Uncertainty

The undershoot uncertainty calculation is performed similarly to the overshoot uncertainty

calculation (see Sec. 5.3) with the appropriate change in variables. This uncertainty estimate

yields:

US =

( 1 V x ^

V Vs2,R ~ VS1R j

^Sl,m + P US [^min,m ~ ^Sl,m J ~ VS1,R (24)

-R J

where |3 US is a correction factor that is determined experimentally as is done for (3 0S . Equation

(20) provides an empirical relationship between the undershoot and transition duration of the

reconstructed (reported) waveform and that of the measured waveform. The correction factor (3 L

, S is determined by fitting a curve to a set of M9 t4RVUSR versus tim VUSm data. The uncertainty in

pus is the standard deviation in the fitted curve relative to the set of corresponding timVus m

versus t4RVUSiR data, and the coverage factor is determined by the number of (3 uss.

Equation 24 identifies the variables that contribute to the undershoot. To determine the

uncertainty in the estimate of the undershoot, the partial derivative of Equation 23 is taken with

respect to each of these variables. These partial derivatives are listed in Table 4 that appears in

Appendix A2. The calculation of the uncertainty has been automated using a LabVIEW VI. The

Uncertainty, Undershoot VI determines the value of each of these partial derivatives and

combines them in quadrature [28]. The coverage factors are calculated and the expanded

uncertainty with 95 % confidence bounds is calculated and reported with the estimate of the

undershoot.

35
6. Procedures for Quality Control

The quality of the measurement service has benefitted from having just one person responsible

for the measurements for many years. The previous operators of the service have been

conscientious and skilled. Other quality control measures are incorporated in the service.

6.1 Control Charts

Several control charts are maintained as part of the service. These are used in the uncertainty

analysis and any significant change will negatively impact (increase) the uncertainty of the

measurement results. The impulse response estimate is obtained using the Nose-to-Nose

technique and is done periodically. This estimate is added to a Impulse Response Control Chart

and compared to previous estimates. The single most important piece of equipment is the

sampler and its condition is indicated by the impulse response estimate. A change in the gain of

the sampler would also indicate a degradation of the sampler and the Gain Control Chart is monitored for changes.

6.2 Check Standard

No check standards are maintained by this service, however, as part of each calibration, the

transition duration and amplitude estimate is compared with prior estimates obtained for that

particular device. The DUT then become a "check standard." If the new estimate is outside the

uncertainty bounds of the prior estimate, then an attempt is made to identify the problem. If the

conclusion is that the DUT has changed, the customer is notified.

6.3 Procedure to Amend a Calibration

If an error is found in a calibration report that has already been delivered to the customer, the

Project Leader is notified. After discussing the error and possible routes to correct the error, a proposed route is brought to the Group Leader. The Group Leader approves the correction.

36
7. References

1N.G. Paulter and D.R. Larson, "NIST's High-Speed Pulse Parameter Measurement

Service," IEEE Instrumentation and Measurement Technology Conference, Anchorage,

AK, USA 21-23, May 2002.

2. IEEE Standard 181-2003.

3. Pulse Techniques and Apparatus, Part 1 : Pulse Terms and Definitions; Part 2: Pulse

Measurements and Analysis, General Considerations, EEC Publications 469-1 and 469-2,

Intl. Electrotech. Com. (IEC), Switzerland (1974).

4. S.M. Sze, Physics of Semiconductor Devices, Second Edition, John Wiley & Sons, New

York, 1981.

5. B.G. Streetman, Solid State Electronic Devices, Second Edition, Prentice Hall, Inc., Englewood Cliffs, NJ, 1980.

6. C. Hamilton, D. McDonald, J. Sauvageau, and S. Whiteley, "Standards and High-Speed

Instrumentation," Proc. IEEE, Vol. 77, pp. 1224-1232, 1989.

7. A. Yariv and P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser

Radiation, John Wiley & Sons, New York, 1984.

8. D.H. Auston and M.C. Nuss, "Electrooptic generation and detection of femtosecond

electrical transients," IEEE J. Quantum Electronics, Vol. 24, pp. 184-197, 1988.

9. J. A. Valdmanis and G. Mourou, "Subpicosecond electrooptic sampling: Principles and

applications," IEEE J. Quantum Electronics, Vol QE-22, pp. 69-78, 1986.

10. B.H. Kolner and D.M. Bloom, "Subpicosecond electroopic sampling in GaAs integrated

circuits," IEEE J. Quantum Electronics, Vol. QE-22, pp. 79-93, 1986.

11. D. Henderson, A.G. Roddie, and AJ.A. Smith, "Recent developments in the calibration

of fast sampling oscilloscopes," IEE Proceedings-A, Vol. 139, pp. 254-260, September

1992.

12. D.H. Auston, "Picosecond optoelectronic switching and gating in silicon," Appl. Phys.

Lett., vol. 26, pp. 101-103, 1975.

13. F.J. Leonberger and P.F. Moulton, "High-speed InP optoelectronic switch," Appl. Phys.

Lett., vol. 35, pp. 712-714, 1979.

14. G. Mourou and W. Knox, "High-power switching with picosecond precision," Appl.

Phys. Lett., vol. 35, pp. 492-495, 1979.

1 5

R.B. Hammond, N.G. Paulter, A.E. Iverson, and R.C. Smith, "Sub- 1 00 ps bulkrecombination-limited InP:Fe photoconductive detector," pp. 157-160, IEEE Int. Electron

Devices Meet. Tech. Dig., 1981.

16. N.G. Paulter, "Selecting a short-pulse laser system for photoconductive generation of

high-speed electrical pulses," Optical Engineering, Vol. 35, pp. 3296-3300, November

1996.

17. D.E. Cooper and S.C. Moss, "Picosecond optoelectronic measurement of the highfrequency scattering parameters of a GaAs FET," IEEE J. Quantum Electronics, Vol. QE22, pp. 94-100, 1986.

18. N.G. Paulter, High-speed optoelectronic pulse generation and sampling system," IEEE

Trans. Instrum. Meas., Vol. 37, pp. 449-453, September 1988.

37
19. S.M. Riad, "Modeling of the HP-1430A feedthrough wideband (28 ps) sampling head,"

IEEE Trans. Instrum. Meas., Vol. 31, pp. 110-115, June 1982.

20. K. Rush, S. Draving, and J. Kerley, "Characterizing high-speed oscilloscopes," IEEE

Spectrum, pp. 38-39, September 1990.

21

.

J. Verspecht, "Broadband sampling oscilloscope characterization with the 'nose-to-nose'

calibration procedure: A theoretical and practical analysis," IEEE Trans. Instrum. Meas.,

Vol. 44, pp. 991-997, December 1995.

22. G.N. Stenbakken and J.P. Deyst, "Time-base nonlinearity determination using iterated

sine-fit analysis," IEEE Trans. Instrum. Meas., Vol. 47, pp. 1056-1061, October 1998.

23. T.M. Souders, B.C. Waltrip, O.B. Laug, and J.P. Deyst, "A wideband sampling

voltmeter," IEEE Trans. Instrum. Meas., Vol. 46, pp. 947-953, August 1997.

24. P.S. Filipski, C.J. van Mullen, D. Janik, M. Klonz, J.R. Kinard, T.E. Lipe, and B.C.

Waltrip, "Comparison of high-frequency ac-dc voltage transfer standards at NRC, VSL,

PTB, and NIST," IEEE Trans. Instrum. Meas., Vol. 50, pp. 349-352, April 2001

.

25. T. Dabozi and I. Kollar, "Multiparameter optimization of inverse filtering algorithms,"

IEEE Trans. Instrum. Meas., Vol. 45, pp. 417 to 421, April 1996.

26. N.S. Nahman and M.E. Guillaume, "Deconvolution of time domain waveforms in the

presence of noise," National Bureau of Standards Technical Note 1047, U.S. Department

of Commerce, Washington, DC, 1981.

27. N.G. Paulter, "A causal regularizing deconvolution filter for optimal waveform

reconstruction," IEEE Trans. Instrum. Meas., Vol. 43, pp. 740-747, October 1994.

28. B.N. Taylor and C.E. Kuyatt, NIST Technical Note 1297, "Guidelines for evaluating and

expressing the uncertainty of NIST measurement results," U.S. Dept. Commerce, 1994.

29. D.R. Larson and N.G. Paulter, "Temperature effects on the high-speed response of

digitizing sampling oscilloscopes," NCSL International, 2000 Workshop and Symposium,

Toronto, Canada, 16-20 July 2000.

30. N.G. Paulter and D.R. Larson, "Time-base setting dependence of pulse parameters using

high-bandwidth digital sampling oscilloscope," NCSL International, 2000 Workshop and

Symposium, Toronto, Canada, 16-20 July 2000.

31. C. Mittermayer and A. Steininger, "On the determination of dynamic errors for rise time

measurement with an oscilloscope," IEEE Trans. Instrum. Meas., Vol. 48, December

1999, pp. 1103-1107.

32. W.L. Gans, "The measurement and deconvolution of time jitter in equivalent-time

waveform samplers," IEEE Trans. Instrum. Meas., Vol. IM-32, pp. 126-133, March 1983.

Other Applicable Documents

33. Dynamic Calibration of Waveform Recorders and Oscilloscopes Using Pulse Standards,

W. L. Gans, IEEE Trans. Instrum. Meas, Vol. 39 (6), pp. 952- 957 (Dec. 1990).

34. D.R. Larson and N.G. Paulter, "The effect of offset voltage on the kick-out pulses used in

the nose-to-nose sampler impulse response characterization method," Instrumentation and

Measurement Technology Conference, MTC 2000, Baltimore, USA, pp. 1425-1428, 1-4

May 2000.

38
Appendices

Al. Sample Report of Special Test.

A2. N.G. Paulter and D.R. Larson, "Pulse Parameter Uncertainty Analysis," Metrologia, Vol. 39,

pp. 143-155,2002.

39
REPORT OF SPECIAL TEST

Special Test No. 65200S: Fast Repetitive Pulse Parameters

PULSE GENERATOR HEAD

Manufacturer

Model number, Serial number

Submitted by:

Customer

Somewhere, Worldwide

The pulse amplitude and the pulse first transition durations (both the 20 % to 80 % and

10 % to 90 % definitions) were measured using the NIST Waveform Parameter Analysis

System (WPAS). The results are presented in Table 1. The expanded uncertainty

bounds shown correspond to a 95 % level of confidence. (See Attachment for further

details.)

TABLE 1

Pulse Amplitude 245.3 mV± 1.5 mV

Pulse First Transition Duration (20 % to 80 %) 10.5 ps±1.25 ps

Pulse First Transition Duration (10 % to 90 %) 15.6ps±1.25ps

Page 1 of 4

Test Report No. 817/123456-xy

Date of Test: month, day, year
Pulse Generator Head

manufacturer.

Model number, serial number

Test Conditions and Notes for this Special Test

1The pulse generator head was received on January 27, 2004 in good condition.

No power supply was shipped with the subject pulse generator, NIST provided

the necessary power supply.

2. Laboratory ambient temperature = 22.6 °C ± 1 .0 °C.

3. Laboratory ambient relative humidity = 19 % ± 10 %.

4. The APC-3.5 male output connector of the subject pulse generator head was

connected to a precision APC-3.5 to APC-2.4 connector adapter. This was

connected to the APC-2.4 connector of the sampler. The necessary trigger

pulse for the subject pulse generator head was furnished by a NIST-supplied

rectangular pulse generator. The trigger pulse amplitude was 1 .0 V (0.0 V low

state, 1 .0 V high state) and the repetition rate was 100.0 kHz. The equipment

was allowed to operate for at least two hours prior to measurement.

5. The oscilloscope trigger delay was adjusted to place the 50 % pulse amplitude

reference level instant of the first transition approximately 300 ps from the start of

the 2.0 ns pulse waveform epoch.

6. Figure 1 is a plot of one of the subject pulse generator waveforms as acquired

with the WPAS. This plot shows the pulse waveform before any corrections for

voltage-axis and time-axis gain and linearity, jitter deconvolution when

necessary, and deconvolution of the sampler impulse response. Figure 2 is a

plot of the waveform after the necessary corrections have been made.

For the Director, Special Test performed by:

National Institute of Standards

and Technology

Donald R. Larson

(301)975-2437

Gerald FitzPatrick, Group Leader

Quantum Electrical Metrology Division

Page 2 of 4

Test Report No. 817/123456-xy

Date of Test: month, day, year

Reference No. 12345
Pulse Generator Head

Manufacturer

Model number, serial number

50

-50

>

E

V -100

T3

Q.

E

CO -150

-200

-250

filename

; mj\^—-i-

.

\

; _J

i i i i i

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

time (ns)

Figure 1. Measured waveform, Customer, Manufacturer, model number, serial number.

Page 3 of 4

Test Report No. 817/123456-xy

Date of Test: month, day, year
Pulse Generator Head

Manufacturer

Model number, serial number

50

-50

>

E

^ -100

Q.

E

TO •150

-200

-250

; \

filename

N^^-•****>-*•-.

:

;

;

J

i.i. i i ; 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

time (ns)

Figure 2. Deconvolved waveform, Customer, Manufacturer, model number, serial

number.

Page 4 of 4

Test Report No. 817/123456-xy

Date of Test: month, day, year
metrologia

Pulse parameter uncertainty analysis

N. G. Paulter and D. R. Larson

Abstract. A detailed uncertainty analysis is presented for the pulse parameter measurement service of the National

Institute of Standards and Technology (N1ST, USA). It relates to the new pulse parameter measurement and

extraction processes. Uncertainties for pulse amplitude, transition duration, overshoot and undershoot (preshoot)

are given. Effects of temperature variation, impulse response estimate, pulse parameter extraction algorithms,

time-base distortion, calibration procedures and the waveform reconstruction process are included.

1. Introduction

The NIST supports a measurement service for highspeed (transition durations <20 ps) pulse generators

that provides an estimate of the pulse parameters

of amplitude and transition duration [1], Overshoot

and undershoot (preshoot) parameters were previously

provided as well. However, support for these parameters was discontinued because of the lack of a viable uncertainty analysis [2], which is addressed by this paper.The NIST is one of two national laboratories that provide a pulse parameter measurement service; the other being the National Physical Laboratory (NPL)

in the United Kingdom. The NIST and NPL are performing a comparison of pulse parameter results, which includes measured data, corrected data (if applicable) and reconstructed data. The results to date

indicate that both national laboratories are in close agreement.

The NIST measurement service currently uses commercial, high-bandwidth sampling oscilloscopes (3 dB

attenuation, bandwidths of approximately 50 GHz)

and pulse generators (3 dB attenuation, bandwidths

of approximately 20 GHz) to measure the pulse

parameters of short-transition-duration (high-speed)

pulse generators and the step response of high-speed

samplers. The purpose of this paper is to present our new uncertainty analysis for these parameters, which

are pulse amplitude, transition duration, overshoot and

undershoot. For brevity, not all variables are described at the point of first use, but a list of variables and their description is provided in the glossary. Also, it should

be pointed out that the step response of a device is equal

to the convolution integral of the impulse response of

that device with an ideal step. Both terms are used in

this paper because transition durations and bandwidths

of electronic systems are typically calculated from the

step responses and, in waveform reconstruction, impulse

responses are typically deconvolved.

2. Background

The pulse parameter measurement process that is used

to acquire waveforms is briefly described in this section. The measurement process consists of a set of measurements of the customer's pulse generator or sampler (the device under test, or DUT) and a set of instrument calibration measurements. Some of the instrument calibration measurements are made during

the DUT measurement sequence and some are not. The calibration measurements include time-base errors, sampler gain, jitter and sampler step response. Figure 1

is a diagram of the NIST pulse parameter measurement

system.

PC INSTRUMENT CONTROL / DATA ACQUISITION / ANALYSIS

""~g r-4_J l

]]

1 Printer

MICROWAVE

SYNTHESIZER \1 n

TRIGGER

PULSE GENERATOR

DELAY LINE

(56 re) PULSE GENERATOR

SAMPLER

Trigger input

Signal Input

12

OSCILLOSCOPE MAINFRAME

DUT1 Of DUT2

Calibrate DUT 1 using sampler of DUT 2 using pulse generator

N. G. Paulter and D. R. Larson: Electricity Division, National

Institute of Standards and Technology (NIST), 1 00 Bureau

Drive, Gaithersburg, MD 20899, USA.

Figure 1. Diagram of NIST pulse measurement system.

The dotted lines indicate insertion of instruments used in time-base calibration.

Melroloffia, 2002, 39, 143-155 143
N. G. Paulfer unci D. R. Larson

An estimate of the step response of the NIST

50 GHz sampler used to measure the DUT is obtained

using the "nose-to-nose" method [3, 4], the results of

which have been compared with results using swept

frequency and optoelectronic methods [5]. We are currently examining the "nose-to-nose" method and

its limitation in sampler calibration |6]. Measurements

that are used to estimate the sampler step response, system jitter and dynamic gain of the sampler are taken routinely, but not necessarily as part of the DUT measurement sequence, and a control chart is maintained for the mean value and standard deviation

of the transition durations of the sampler step response and the equivalent jitter step response and the sampler

dynamic gain. The reason that the measurements of

these parameters are not part of the DUT measurement

sequence is that the sampler step response, the system

jitter and the sampler dynamic gain are stable (small

observable variation) and the variations in these parameters are represented in their associated control chart data. The DUT jitter is intentionally excluded

from the system jitter because DUT jitter affects the DUT parameters of transition duration, overshoot

and undershoot and will be exhibited in customer

measurements.

Several sets of data are acquired for the customer's DUT. For brevity, the discussion that follows assumes

that the DUT is a pulse generator. A set of data

consists of Mi sampler-acquired DUT waveforms and one measurement of the time-base errors. Measurements

of the time-base errors [7-10] are a routine part of the DUT measurement procedure. The DUT measurement

sequence is as follows:

1. measure time-base error: one independent measurement;

2. acquire waveforms: M\ independent measurements

of DUT output.

The DUT waveforms are subsequently corrected for gain and time-base errors only if these errors are large

relative to the reported uncertainties. The corrected or uncorrected waveforms are then used in a reconstruction process to obtain a waveform that is an accurate estimate of the pulse measured by the sampler. The

accuracy of this estimate (the reconstructed waveform)

is dependent on the reconstruction process and the accuracy of the estimate of the sampler impulse

response. The waveform reconstruction process uses an

iterative deconvolution of the sampler impulse response from the measured data. From each reconstructed waveform, pulse parameter values are computed. The

set of pulse parameter values thus computed is used to determine the mean value and standard deviation for the given parameter.

The pulse parameter computations are based on histogram methods [11]. The first step in the calculations is to compute the histogram of the waveform. Next the top-line, VS2, and bottom-line, Vsi,

values are obtained from the histogram. Then, using

Vs2 and Vgi, the pulse parameters are obtained for the waveform.

This uncertainty analysis, because it is applied

to acquired waveforms, is applicable to both the measurement of the output of pulse generators and

the step response of samplers with the appropriate

change in reference measurements and waveforms.

When measuring the output of a pulse generator it is assumed that the sampler step response is the reference,

and when measuring the sampler step response it is assumed that the output of the pulse generator is the reference.

3. Uncertainty analysis

The reported pulse parameters are an average of the

particular pulse parameters obtained from a set of

Mi pulse waveforms measured using the NIST pulse

measurement systems. The average of a parameter.

W for example, is given by

^ 1 M,

w = wXMi

w^< (1) i=\

where Mi is the number of values for the parameter

W, one value for each waveform, and W is dependent

on a number of variables, ayThe uncertainty for this average, W for example, is given by

UW — ^eff

= k (2a)

— &eff (2b)

where it is assumed in (2a) that the a} values are uncorrelated, which is the reason why there are no

cross terms in the partial derivatives with respect to the aj . In (2b) it is further assumed that the Uj values are

the same for every W{\ that is, the uncertainties in the variables for a given parameter are the same for every waveform. The term ken is the statistical weight [12]

applied to the uncertainties of variables obtained from

a limited number of trials. For a number of variables with different degrees of freedom, /ceff is found by first calculating the effective degrees of freedom using [12]

"off = M

i=\

(3)

144 Meiruhfitcj, 2002. 39. 143-155
Pulse parameter uncertainty analysis

Table la. Variables affecting pulse amplitude uncertainty.

Variable

ft/ Uncenaint\

'

Partial derivative dW(ct\ .a-2 ftp On,

Type Degrees of freedom

I/. 'SI .in

T)

*=>->...,

0\'s>. m OH nil

Vsi.-

cn-'s,,,, OH ui i

'»-. +

v yfjy Mr " T ""/'iiium Tr ,. f a '

r .. r

Sav/at Sl.JT

1 1 ^A77 77

1 1

5\V'/AV

5A i' /I'''

r,,,^ - rr„f

A/, - I

A/, - 1

Win - 1

.V, - i

Mr, - 1 M(

; - 2

1. The letter V is used to represent amplitude-related uncertainty contributions to </, see (8) and Table Ic. The letter H is used to represent histosram-related uncertainty contributions, see (11) and Table lb.

Table lb. Variables affecting the uncertainty in Vso obtained using a histosram method.

Variable a,

Uncertainty

Ui

ffV"mM

(=0)

0Vmm

(=0)

»N«

(-0)

"Nbta. (< ±l/10JVbi „

Partial derivative 9V^ 2 (ai .«2 ftp)Type

<9a,

Degrees of freedom

» max

Mil in

^52

Wbins

7Vs-2 Wbins

7Vs2 Whins V - V • ' max * mm

M.ins V•max - VMinn wbins

Table lc. Variables affecting g.

Variable

fti Uncertainty 1 Partial derivative 0Wf Qi ,a-2 ftp

Oct,

Type Degrees of freedom

i',

Vs2 .n,,

Vs,,,i,

Vs2.r

Vsi,

aVs2.m .r. OH

-U H

1

^S1 .n,, + OH

-U H

f M\o

1

1

fMio

V'sa.r - Vs.,

1 Vs2, -Vsi.r

9Vs2. r a// -UH

, ,' dVsl , T

1 |Vs2.m.r ~ Vsi.m.r|

rM^ (Vs-i, -Vsi,)

2 J |Vs2..n.r ~ Vsi.m.rl rW^ (Vs2,-Vsi , r )

2 OH

1. The letter // is used to represent histogram-related uncertainty contributions see (11) and Table lb.

M2 - 1

Mi - 1

M-, - 1

M: , - 1

Metrologia, 2002. 39. 143-155 145
N. G. Pauller and D. R. Larson

Table 2a. Variables affecting pulse transition duration uncertainty.

Variable Uncertainty

a.

3artial derivative d\V(n\ ,a> rtp) On,

Type Degrees of freedom

u

A',,,

A",

c)A"„,

I -JJf"i !

(see Tables 2b and 2d)

)XT

1 dli "

(see Table 2b)

0X„, IrF'"'

'a

hi

I a77 E

1 |y.l.H.

on^'.i.ni.

^V,?, (o7

cttr a/, l.r.i

" { im A, n " M, V,=i Ufl Otjj

(sec Table 2b)

A'/,:, - 1

A/, - 1

A/t - 1

A/,2 - 1

Table 2b. Variables affecting X value uncertainty.

Variable Uncertainty a, it,

mLl ^L,

(=0)

m,. 1 + '"'l, + (=0)

mL .2 _ a«u..2 _ (=0)

m.L 2 +

ffmla+ (-0)

Li p» + (Ms)

(see Table 2c) U_ *f„_

(=0)

*i +

"•> + (=«)

L2 V<>

+ (^7"-)

(see Table 2) L2 _

<r/. 2 _ (=0)

L2 + »la+ (=0)

Partial derivative #ir(ai.a2 , ftp)

ctai

Type Degrees of freedom

£.+ -Li

£l+ -ii_

Li - ti_

L, + -Li_

L,

+ -L2 L2+ -L2 _ L2 -•L2_

L2+ - L2 _

ii + -it

(m/ , -mi. ) =

L2+ - L2 (m/. - mL2 _ ) — -J

(L,2 + - L2 _ )

(m/J2+ - rn L2 _ )

Is} — L2 _ (L2+ — L2 _

where i/, is the number of degrees of freedom for the parameters (shown in Tables 1 to 4) and c, are the partial differential equations (shown in Tables 1 to 4). Weight

fceff is then found from ucS using the £ -distribution [12]. The uncertainties in the variables, U{ (where the subscript refers to parameter i), are obtained from

independent measurements that provide values for these

particular variables. The tables list the source of Uj for

the appropriate variables. To calculate the uncertainty of

the different parameters, the partial derivatives of these parameters with respect to the independent variables must be calculated. These partial derivatives are also

146 Metmlogia, 2002. 39. 143-155
Pulse parameter uncertainty analysis

Table 2c. Variables affecting uncertainty of L\ and L_.

Variable

n,

Uncertainty Partial derivative

c)\V((\] .n-2 op)

On,

Type Degrees of freedom

Vsi..„ see Table la

see Table la P%

PV,

Table 2d. Variables affecting uncertainty of A',

Variable Uncertainty Partial derivative cMl (ft'j . rt-2 ftp ) On,

Type Degrees of freedom

T (7 '/"„„. n» A M8 -2

A M4 - 1 A Ms - 1

Table 3. Variables affecting uncertainty in pulse overshoot.

Variable Uncertainty 1 a, u,

Partial derivative dW{n\ . Q2, . . . . ftp 9a,Type Degrees of freedom

(=0)

Vs-i,,,

Ksi

xd

do

r Vsa.„.

KS2R ^S2.R

ff vsl . R

(see Tables 2b and 2c)

a'^ +

{-0H U "

(see Tables 2b and 2c) "in

1 MiA',,,, 1

i \

/r; V Ar.j v A . n

1 v,„, 1

TT A

1 mT

i n£ W'A.R.- /io[W,„,-vS2 IK"- '-*» '^*y

s

M,\J^J] (/3o[K,a x.m.i-^2.m.i

Am.i 1

l R.!

V A.K., ^ E (V—» > - V«.„. jl^T^j)

Mn -2

1. The prime notation indicates that the uncertainty for these parameters must include histogram-dependent uncertainties, as for the example

Vs2 in Section 3.1. That is, the prime notation indicates

OV

(33)

Metrologia, 2002, 39, 143-155 147
N. G. Paulter and D. R. Larson

Table 4. Variables affecting uncertainty in pulse undershoot.

Variable Uncertainty 1 'artial derivative

U\\ (a\ . (X) a|> ) On,

Type Degrees of freedom

xA

ftv

a \,(=0)

^

(see Tables 2b and 2c)

f lwH//

(see Tables 2b and 2c)

1

\;£(,"' 'k~v^-.

\.=

*M;£

i Mi

A/i

\;£

V

i m7

i a77

\f.

^K^Tidir

8« -[v,, -r.,„„] A',,,. AV

I i ^('[Imiii.ni.i _ ^'Sl.ni.iJ-Tr-

\\ H , / V AH.i

£(/*< ['mi,,,„.i-Vs,,,,i] T

1 1

A'h.j K\.B.i

X, I M I 1 iiiii.Mi.i - ^Sl.m.i

Km.I 1 ,ElH -.^. iip'v 1

M, , - 2

I. The prime notation indicates that the uncertainty for these parameters must include histogram-dependem uncertainties, as for the example

Ys-2 in Section 3.1 (see also (33)).

shown in the tables. The tables for each variable include

its type of uncertainty [12] and degrees of freedom, v. For measured data, the number of degrees of freedom

is given by u = Mk — 1, where M* is the number of

data elements used to compute the value of the k-th

variable. For fits to data, v is given by v — Mk - m*,

where rrik is the number of coefficients used to fit the

data. The number of degrees of freedom for certain variables is equal to infinity {u — oo) because the calculation of the value of these variables is based on a

specific fixed waveform. Accordingly, every (an infinite set) computation of the value of that variable for that waveform yields the same result. The variation in measurements, represented by the symbol a in the tables and text, is unless otherwise

indicated the standard deviation of a set of measurement

values of a given parameter or the standard deviation of

the residuals of a curve fitted to the data. For example,

of the first case, ayS2 m

in Table la is the standard

deviation of M\ values, one Vs2,m value taken from

each of the M\ measured (acquired) waveforms. For example, of the second case, the ctSav//a7. in Table la

is the standard deviation of the residuals to a fit to the amplitude versus temperature data.

3.1 Pulse amplitude

The pulse amplitude is obtained using a histogrambased algorithm (see Section 2). The pulse amplitude

is the difference between Vs2 and Vsi. Calculating the uncertainty in the pulse amplitude requires an equation

that describes the reported pulse amplitude, VA :

VA =

V,A,c/r (4)

where the horizontal bars indicate the arithmetic mean,

^A.c/r is tne mean of the set of M\ pulse amplitudes

corrected for sampler offset errors and waveform

reconstruction errors, and g is the mean of the transient amplitude gain correction of the sampler. A common

practice in oscilloscope calibrations is to use or include

a static level gain-correction term in (4). However,

since the signals being measured are transients (steps or

impulses), as opposed to static levels, a transient gain

term, the g in (4), should be used. The transient gain

is affected by the impulse response of the sampler and

the waveform epoch because of the settling response

of the sampler. For example, if the sampler response

148 Metmlogia, 2002, 39, 143-155
Pulse parameter uncertainty analysis

has not settled by the end of the epoch, then g will be less than one for that epoch. Ideally, Jj = 1 if the sampler exhibits no pulse gain or attenuation and the sampler has settled within the waveform epoch. The

VAtC/t is given by

V^., r =TA .X = {VS2x -FS l.c)6r = n S'2.111 _ ^S2,off.m ~~ SI. in _ I Sl.off.ni) +

VAT]br , (5)

where VQn is the voltage offset, which is a bias in the observed voltage, V' A t is the mean of the set of M\

pulse amplitudes corrected for sampler offset errors, V±t * s tne mean of the amplitude corrections required

for a change in measurement temperature, and X reflects the error in the amplitude of the reconstructed waveform caused by the reconstruction process. Ideally,

r should be 1 because the sampler impulse response

estimates integrate to 1. However, the reconstruction process introduces an error in the amplitude. This

scaling error is exactly corrected by rescaling the jmlse

amplitude of the reconstructed waveform to equal V' A c

. The subscripts c and m refer to corrected and measured

voltage values. We have observed that for the currently

available high-bandwidth samplers, the voltage offset error is the same for both the top-line (S2) and bottomline (SI) voltage levels; therefore, the offset voltage

contribution can be ignored.

The temperature-correction term is obtained by

measuring the change in the__observed pulse amplitude

with temperature [13]. The VAT term is therefore

VAT = SA\'/A7V(Tmeas r Tref), (6)

where Tmeas is the average of M4 sampler temperature

values taken during the pulse measurement process, S is the slope of a straight-line fit through a set of

previously acquired amplitude versus temperature data,

and Tre f is the average reference sampler temperature

that is taken to be the mean of M5 temperature values

of the sampling head measured when the sampler

impulse response was determined. The amplitude versus temperature data consist of a set of MG data pairs and

are recorded over a temperature range between T2 and

2\; the difference between these two temperatures is AT. Every sampler may exhibit a unique temperaturedependent response. Amplitude VA can now be rewritten using (5) and (6) in (4):

VA = 1 . U)

The transient gain term, g, is obtained by taking

the ratio of the amplitude of the reference pulse as measured using the sampler and the amplitude of

Mctroloxia, 2002, 39, 143-155

the reference pulse as measured using a reference instrument [14], which provides more accurate pulse

amplitude measurements than high-speed samplers. As

mentioned above, the gain term is obtained from a control chart and is given by

1

A/„i

A/io _ 1 V^ ^/

S2,m.r.i ~ "si.in,

" Ml° h FS2.r.i - FS i, r ,i (8)

where the subscript r denotes the reference pulse

measurement and there are M10 independent gain

terms. Using the ratio of average amplitudes in (8) is numerically more stable than an average of

the amplitude ratios. Furthermore, the ratio of the average amplitudes relaxes the requirement that a reference measurement be made for each sampleracquired measurement. A set of M2 measurements of

the reference pulse are used to obtain the reference- instrument-measured parameters and a set of M3

measurements of the reference pulse are used to obtain the sampler-measured parameters. Temperaturedependent gain effects are included in Vat. which

describes the change in pulse amplitude with change

in temperature relative to Tre f. The gi were computed

from data taken at Trof. The variables shown in (8) contribute to the uncertainty in VA . The uncertainty in VA , uyA

, is dependent on the uncertainties of all the variables

from which VA is dependent, and these variables are

listed in Tables la and lb. The uncertainties, u,, in the

variables are obtained from independent measurements

that provide values for these particular variables. Table la lists the source of ut for the appropriate

variables. The partial derivatives of VA with respect to

the independent variables are also shown in Table la, as are its associated degrees of freedom and uncertainty

type [11]. rmeas, although obtained from Mi averages

of the average of M4 temperature measurements taken

during each waveform acquisition, has v = Mi — 1. The

uncertainties for Sav/at require an empirical formula

relating the pulse amplitude of the acquired waveform

to temperature (see Figure 2) which is obtained by

fitting a curve, typically a line, to the data.

In addition to measurement-related uncertainties,

the reported amplitude values are also subject to uncertainties from the method used to calculate these

values, in this case a histogram. The histogram-derived

amplitude values, for example for Vs2, are given by

VS2 = Ns2&V + Vmin ,

where

AV

/max - Vr mm

AT,bins

(9)

(10)

149
N. G. Paulrcr and D. R. Larson

2

" -

"I

n | 1 ==*

U

-2

-4

-6

"" J

fv

v. -8

10

• SG2

•-•«- - SHI

I -----• SH2 — *-— SH3 r r

T\ V 12

i

i

14 16 18 20 22 24 26 28 30 32 34 36 Temperature / °C

Figure 2. The percentage change in pulse amplitude with temperature relative to 15 °C. The designation SG refers

to step generator and SH to sampling head. The number

following the designation refers to models of devices made

by different manufacturers.

and AI' is the histogram bin size and A/bins is the number of histogram bins. The uncertainty contribution

associated with Vs2, for example, is then

dVcS2 1 dli

UH

Nuns

\

(V^-V^yu^+N^x

ffV.„.. + aVm,

i »

f N

22 v «»ax ^niinj WA'| )1115 (ID

The term H is used as a place holder to represent all the histogram-based dependences of a variable, such as Vs2

in the example of (11). Table lb, used as an example,

lists variables affecting the Vs2 amplitude variables. All amplitude values listed in the first column of Table la have an analogous list of variables. The degrees of

freedom are infinite for yVS2 and 7Vt,i„s because they are extracted using a given pulse parameter algorithm and

the output of this algorithm will not vary for a given

waveform. The degrees of freedom are infinite for Vmax and V,ni„ because these values, for a given waveform,

are fixed once a waveform has been acquired.

3.2 Transition duration

The transition duration is the difference between the occurrences of user-defined amplitude reference levels, for example, the 10 % and 90 % amplitude reference

levels. The times at which the waveform crosses

these reference levels are called reference-level instants. Accordingly, the 10 % to 90 % transition duration is the difference between the 90 % and 10 % reference

level instants. The reported waveform transition duration, t d , is the average transition duration extracted from M\

reconstructed pulse waveforms, f ci.R- Duration t,\ is related to the transition duration of the acquired

waveform, t d „,, and the transition duration of the estimated sampler step response, t d T

:

U

a/, Mi77 X>»-

;=1

/deckel, in + ^'d.AT'il.r) + &t<\,TQC- (12)

where A^.at is the temperature-induced incremental

change in transition duration [13] (described below).

A/<i.rec is the bias in the transition duration caused by

the reconstruction process, and /( | PC is used to indicate

the deconvolution functional relationship between t^.Kfcd.m and ^d, r

- Duration t^ r is found from a waveform

that is obtained by deconvolving a waveform with

transition duration / c |. r from a waveform with transition duration tj ni . The specific functional relationship, /dec , between t^, td, ni , and £d.r is dependent on the type of

waveform used and can only be derived for certain ideal waveforms. For example, for Gaussian waveforms, t d

is equal to the square root of the difference of the squares of transition durations of the measured and

step response waveforms. For general waveforms, we

can obtain an empirical relationship relating the three parameters. The first step in obtaining this empirical

relationship is to vary either t dm or ta.r i n the waveform

reconstruction process, keeping the other constant, and

noting the variation in i dR. This provides two sets of data, one relating t<j,R to t d<m for a fixed td, T and

another relating t d ,R to t dT for a fixed t dm . We obtain our empirical relationship by fitting a curve (such as a polynomial) to these sets of data, ia,m versus td< R

and ^d,r versus ^.r- The reconstruction process we

now use is described in [15]. However, we are also

investigating various filtering methods, one of which

is discussed in [16]. Using our present reconstruction

algorithms on known waveforms, we have observed a

bias, A^drec, and a noise-dependent variation in t d<R . The noise-dependent variation is also contained in the empirical relationship, but the bias is not. Therefore, the uncertainty for the reconstruction process is computed

by adding the absolute value of the bias to the computed

uncertainty.

The parameters t d ^n and id,r can be expressed in terms of the sampling intervals:

£d,m — ^m&£,

td,r - XTjM,

(13)

where Xm and Xti are the real-valued (non-integer)

number of sampling intervals describing the transition duration for the measured sampler step response waveforms (including jitter) and 8/ is the duration of

the equispaced sampling interval. XTi is computed from

waveforms that are the result of the convolution of the

150 Mctrvlogta, 2002. 39. 143-155
Pulse parameter uncertainty analysis

f: 7^ '.:": 4 \r j ,v Jp v_ A'

;

-

e

-' \ i/'jP

V ;'h > "' '"-

: !V J/"

. ..

--• V

SGI 1 SH3J

1 1 ! .

-2-4

-6

-8 -10

-12 0123456789 10 Time / ns

Figure 3. Time-base errors. The designation SG refers to step generator and SH to sampling head. The number

following the designation refers to different models of devices made by different manufacturers.

system jitter, represented by a Gaussian waveform, and

the sampler step response, obtained from the sampler

calibration method. The result of this convolution,

estimated using the Gaussian approximation, is XT i

Xr2 + X2 (14)

where Xr and Xj are the number of sampling intervals

in the sampler step response and equivalent jitter step response transition durations. However, there are errors associated with this approximation [17]. For this uncertainty analysis the convolutions of the two waveforms are performed numerically and the

resultant waveform used in subsequent processing. The

measurement jitter typically has a normal distribution. We approximate Xj using a direct measurement taken

by the sampler. We have observed that estimates of

the jitter obtained employing a geometric method [18] are very sensitive to the transition region from which

the estimate is made. Term Xj includes drift of the sampling aperture with respect to its trigger. Term ht is the average duration of the sampling intervals that span

either the transition region of the waveform or the entire waveform [19] and is measured using sine-fit techniques

[9] during the time-base calibration process. Figure 3 shows an example of time-base errors (vertical axis) versus measurement time. One time-base calibration is performed for each waveform; therefore the variation

in 8i is the variation of bt among the Mj acquired

waveforms. Similarly, the variation in Xm is dependent on the set of Mj acquired waveforms. The variation in XT is dependent on the set of M7 acquired reference waveforms. The variation in Xj is dependent on the set of M\2 jitter measurements. The temperature-dependent

change in transition duration can be expanded:

A*d,AT = 5At/ATK(T1 rmcas . refj (15)

--""

„.

,.~H'Jl

--•"- -**- ?

;•*"'

?:**'''' •-'''

1 ' -•

|

> SG> •• SG2 —•— SHI

--••-•• SH2 —•*•— SH3 -*-- 3Hd

i l 22 24 26 28 Temperature /°C

30 32

Figure 4. The change in transition duration, Ar, .9o, with

temperature relative to 15 °C. The designation SG refers

to step generator and SH to sampling head. The number

following the designation refers to models of devices made

by different manufacturers.

where SAr/ ^T is the slope of a straight-line fit to Mg transition-duration-temperature data pairs (see

Figure 4), measured independently of the Mi acquired

waveforms. Using (13) and (15) in (12) gives

^d — fdec[Xm bt +

SA*/Ar(7\noas ~ Trcf)Xrfit] + A^j.rcc- (16)

For the samplers and pulse generators currently in use at the NIST, the value of Aid >AT is approximated as zero because it is much less than the reported uncertainties; however, it is retained here for completeness. As we did

in the pulse amplitude uncertainty analysis, we generate

a table for the variables contributing to the transition duration value (see Table 2a). The values of A

rm , XT and A^j are determined

by linearly interpolating to obtain the instant in time

(the reference-level instant) corresponding to the given

reference level. The value of Xm (and analogously

for XT and Xj), is X

t L2 - t Li (17)

where t Ll and t Li are the time instances corresponding

to the first (Li) and second (L2) reference levels of

the transition duration. For example, in the 10 % to 90 % transition duration, L x is the 10 % referencelevel instant and L2 is the 90 % reference-level instant. Instant ^is given by

tr., = t (18)

where the subscripts + and - denote the actual sampling instances found on either side of the referencelevel instant (either t L or t^) and the data values

Melrologia, 2002. 39, 143-155 151
N. G. Puulter and D. R. Larson

corresponding to the reference-level instants. Similarly,

t Ll is given by

^Li. — tL-2 j tL,=t L , +—

±

_^(L2 -L2 _).

Using (18) and (19) in (17) yields

t-L,, -tL,

(19)

Xm t L; +

L2+ - L2 _ (L2 -L2 _)-

'- M-Ly L1+ -L, = mL2 _ - mLl _ + (m. L _1+ - mL.,_ ) x

/U

L>2 — L>2_

L2+ — L2 _ (mi rn Li .(20)

where the rn L terms are the time indices corresponding

to the actual data found immediately above and below

the reference levels. Using (20), we can obtain the uncertainty expansion for Xm (and, similarly, for Xr and Xj), which is shown in Table 2b. All the variables listed in Table 2b are extracted using our pulse

parameter algorithm and, consequently, the degrees of

freedom are infinite. The values of L\ and L2 can be

expanded:

Li = VS l, m + Pl(Vs2,m-VSl , m), L2 = Vsi,m + P2(Vs2,m -VS l, m), (21)

where Px and P2 are the percentage reference values,

such as 10 % and 90 % or 20 % and 80 %. Table 2c

provides an uncertainty assessment for L\ and L2 .

3.3 Overshoot

Overshoot is, for waveforms with positive-going

transitions, the maximum positive amplitude excursion

relative to VS2 that the waveform makes near the

transition region. On the other hand, for waveforms with

negative-going transitions, overshoot is the maximum

negative amplitude excursion relative to V$2 that the waveform makes near the transition region. For

overshoot, we currently define "near the transition region" to be that period between the 50 % referencelevel instant (t5Q%) and t 50% + 3td (IEEE TC-10

Subcommittee on Pulse Techniques). Voltage offset errors are not considered here because they cancel, as they did for the uncertainty calculation of Va,c

- The

equation describing the calculation for the overshoot is

=

V,max.R -vfS2,R

VjA,R

+ AOr (22)

where AOrcc is the reconstruction-induced bias in the overshoot of the reconstructed waveform. Vmax r may

be written

v' max R — ^S2,R + lo.R, (23)

where

l

7o,n =

floVc),mt-d.m

^rl.R (24)

and fio ' s a correction factor that is needed because of

the reconstruction process and is found by fitting a curve

to an Mcj-element set of ^d.mVo.m versus id.R^'o.R

data. The t&, m and \'

C) m parameters are obtained

from the acquired waveforms and ^d,R and Vq ,r from

the reconstructed waveforms. Equation (24) describes an empirical relationship between the overshoot and

transition duration of the reconstructed (reported)

waveform and those of the acquired waveform. As with

transition duration (see Section 3.2), we have observed

a bias and a noise-dependent variation in Vq,r caused

by the waveform reconstruction: the noise-dependent

variation is contained in the empirical relationship but

the bias is not. Therefore, the uncertainty in overshoot

is computed by adding the computed uncertainty to the absolute value of the bias. In (24), we assume that the product of the overshoot voltage and transition duration

is not affected by an all-pass filter, which is how the sampler impulse response is expected to behave for an input signal that has a 3 dB attenuation bandwidth

lower than that of the sampler. The uncertainties in

td, r are included in the uncertainty estimate of VQ .r by propagation of uncertainties through ia,R- Vo.m can

be expanded:

Vo,m = Vn v<S2,m- (25)

Parameter £<j,r can be expanded in a similar way to *d.m ( 13 )

:

^d,R — Xn&t, (26)

where X^ is the non-integer number of sampling

intervals describing the transition duration of the reconstructed waveform. Using (13), (23), (24), (25)

and (26) in (22) yields for O:

Vs2,R — Vsi,R

\(Po[Vmax, m -VS2, m]^]

AOrec . (27)

The uncertainty in fio is the standard deviation in the fitted curve relative to the set of corresponding

id.m^b.m versus td.RVb.R data and the coverage

factor is determined by the number of 0o values. Table 3 shows the uncertainty-related parameters for

the variables affecting O.

152 Meirologia, 2002. 39. 143-155
Pulse parameter uncertainty analysis

3.4 Undershoot (preshoot)

Undershoot is, for waveforms with positive-going

transitions, the maximum negative amplitude excursion

relative to Vs\ that the waveform makes near the

transition region. On the other hand, for waveforms with

negative-going transitions, undershoot is the maximum

positive amplitude excursion relative to Y'si that the waveform makes near the transition region. For

undershoot, we currently define "near the transition region" to be that period between the t^% — 3t (

\ and

t biyy, (IEEE TC-10 Subcommittee on Pulse Techniques).

Voltage offset errors are not considered here because

they cancel, as they did for the uncertainty calculation

of \~a.c- The equation describing the calculation for the undershoot is

U =

r„ m.R ^Sl.R VA u + AUT, (28)

where A£/rcc is the reconstruction-induced bias in the undershoot of the reconstructed waveform. Vm; n .R may

be written

^min.R = Vsi,r + Vj/.R,

where

VUtR =

^d,R

(29)

(30)

and j3u is a correction factor that is determined

experimentally as for /fo- Equation (30) provides an

empirical relationship between the undershoot and

transition duration of the reconstructed (reported)

waveform and that of the acquired waveform. As with

overshoot (see Section 3.3), we have observed a bias and a noise-dependent variation in Vjy.R caused by the waveform reconstruction. The uncertainty in undershoot

is computed by adding the computed uncertainty to the absolute value of the bias. We assume that the product

of the undershoot voltage and transition duration is not

affected by an all-pass filter, which is how the sampler

impulse response is expected to behave for an input

signal that has a 3 dB attenuation bandwidth lower

than that of the sampler. Vy, m can be expanded:

Vn m = V„ VS l,r (31)

Using (13), (26), (29), (30), and (31) in (28) yields

for U: ^r^V^[Knin , m -^1)in ]|^ | i Vs2,R

ALT (32)

The correction factor 0u is determined by fitting a curve to an Mg-element set of ta,mVu,m versus ^.rV^r

data. The uncertainty in (3u is the standard deviation in the fitted curve relative to the set of corresponding

td,mVu,m versus fd.R^t/.R data and the coverage

factor is determined by the number of f3v values. Table 4 shows the uncertainty-related parameters for

the variables affecting U.

4. Summary

A detailed uncertainty analysis of the parameters

transition duration, overshoot, undershoot and amplitude of step-like waveforms was performed. This

analysis included a consideration of effects that can

affect the value of the reported parameters, such

as temperature, computation algorithms, history of

instrument performance, equipment limitations and

estimates of the response characteristics of the instrument. Our present published uncertainties, which

are the result of the new measurement process and associated uncertainty analysis, for high-speed

(7 ps < 1^ < 350 ps) pulse generators and samplers are ±2 mV for pulse amplitude and ±1.5 ps for transition duration. Our measured uncertainties, however, are

smaller.

Acknowledgements. We would like to thank W. F. Guthrie and G. N. Stenbakken of the NIST, Gaithersburg, Md., and D. C. DeGroot of the NIST, Boulder,

Colo., for technical comments and B. A. Bell of the NIST, Gaithersburg, for administrative support.

Note: Electricity Division, Electronics and Electrical Engineering Laboratory, Technology Administration,

Department of Commerce. Official contribution of the National Institute of Standards and Technology, not

subject to copyright in the USA.

References

1. NIST Calibration Services Users Guide, NIST Spec. Publ. SP250, Washington, D.C., U.S. Dept. of Commerce, 1998,

189-193.

2. Paulter N. G., Larson D. R., Improving the uncertainty

analysis of NIST's pulse parameter measurement service, Automatic Radio Frequency Techniques Group, 56th Conference, Boulder, Colo., 30 November to 1 December

2000.

3. Rush K., Draving S., Kerley J., IEEE Spectrum, 1990,

38-39.

4. Verspecht J., IEEE Trans. Instrum. Meas., 1995, 44, 991-997.

5. Henderson D., Roddie A. G., Smith A. J. A., IEE

Proceedings-A, 1992, 139, 254-260.

6. Larson D. R., Paulter N. G., Proc. 17th IEEE Instrumentation and Measurement Technology Conference, IMTC

2000, Baltimore, USA, 1-4 May 2000, 1425-1428.

7. Verspecht J., IEEE Trans. Instrum. Meas., 1994, 43, 210-215.

8. Pintelon R., Schoukens J., IEEE Trans. Instrum. Meas.,

1996, 45, 588-593.

Metrologia, 2002. 39, 143-155 153
N. G. Paulter and D. R. Larson

9. Stenbakken G. N., Deyst J. P., IEEE Trans, lustrum. Meas., 1998, 47, 1056-1061.

10. Wang C. M., Hale P. D., Coakley K. J., IEEE Trans, lustrum. Meas., 1999, 48, 1324-1332.

1 1IEC 469-2: International Electrotechnical Commission.

Pulse Techniques and Apparatus, Part 2: Pulse Measurement and Analysis, General Considerations, 1987.

12. Taylor B, N., Kuyatt C. E., Guidelines for evaluating and

expressing the uncertainty of NIST measurement results, N1ST Technical Note 1297, Washington, D.C.. U.S. Dept. of Commerce, 1994.

13. Larson D. R.. Paulter N. G., Temperature effects on the high-speed response of digitizing sampling oscilloscopes,

NCSL International. 2000 Workshop and Symposium,

Toronto, Canada, 16-20 July 2000.

14. Souders T. M., Waltrip B. C, Laug 0. B„ Deyst J. P., IEEE Trans. Instrum. Meas., 1997, 46, 947-953.

15. Dabozi T., KollaY I., IEEE Trans. Instrum. Meas.. 1996,

45, 417-421.

16. Roy S., Souders T. M., Proc. 17th IEEE Instrumentation and Measurement Technology Conference. IMTC 2000,

Baltimore, USA, 1-4 May 2000, 1429-1434.

17. Gans W. L„ IEEE Trans. Instrum. Meas., 1983. IM-32.

126-133.

18. Mittermayer C, Steininger A.. IEEE Trans. Instrum. Meas., 1999. 48, 1103-1107.

19. Paulter N. G.. Larson D. R., Time-base setting dependence

ofpulse parameters using high-bandwidth digital sampling

oscilloscopes, NCSL International, 2000 Workshop and

Symposium, Toronto, Canada, 16-20 July 2000.

Received on 27 September 2001 and in revised form

on 3 January 2002.

Glossary of terms

Synonymous meanings: measured, acquired, or sampled waveform

Not-reported variables

c, partial differential equation

ij transient amplitude gain-correction term

L\ low reference level (for example, the 10 % reference level) L\_ data value found at r/.

j L\ data value found at ti , L? high reference level (for example, the 90 % reference level) L>_ data value found at t ;, ., L>^ data value found at /./ „ M\ number of waveforms in DUT measurement set M> number of reference pulse waveforms measured with sampler and used in sampler gain calibration M\ number of reference pulse waveforms measured with reference instrument and used in sampler gain calibration M-i number of temperature measurements taken during DUT measurement process Mr, number of temperature measurements performed during sampler impulse response characterization Ma number of amplitude-temperature data pairs used to determine temperature effects on sampler's amplitude response

or pulse generator's output amplitude

Mj number of independent waveforms used to estimate transition duration of sampler step response, from control chart Ms number of transition duration-temperature data pairs used to determine temperature effects on sampler's transition duration response or pulse generator's output transition duration A/« number of td.m^b.m versus /h.r^o.r data pairs used to calculate (io curve Mm number of gain terms, from control chart Mn number of r,i.,„V(:, m versus /h.rVcr data pairs used to calculate (ii curve M\ > number of jitter measurements, from control chart M\:\ number of amplitude versus time data pairs used to calculate duration of sampling interval mi, , time index for data having value closest to but < Li

7/iy.

j time index for data having value closest to but > L\

mi, 2 time index for data having value closest to but < Lo

TtiL-2 dme index for data having value closest to but > La

/Vbj ns number of bins in histogram

N<,\ bin number in histogram corresponding to lower state level mode bin Ns-2 bin number in histogram corresponding to upper state level mode bin Pi percentage of pulse amplitude for reference level L\ P> percentage of pulse amplitude for reference level La Sht/AT temperature-dependent change in pulse transition duration ^_\\ /a/ temperature-dependent change in pulse amplitude

T temperature

T,,i,.;is temperature at which a particular waveform was recorded

T,,.f average temperature over which a set of waveforms was recorded

fd.in transition duration, acquired waveform

t ( \ , transition duration, sampler step response estimate convolved with jitter tj.H transition duration, reconstructed waveform

//.

, Li reference-level instant

t/,, data instant preceding £/,,

f./., data instant following II^

ti.., La reference-level instant

154 Metmlogia, 2002. 39. 143-155
Pulse parameter uncertainty analysis

1

1

.., data instant preceding f./ ,.,

/./ ., data instant following ti .., V.\ , pulse amplitude, corrected waveform V\ ,,, pulse amplitude, acquired waveform \\ .,„ , pulse amplitude, acquired waveform, reference pulse

\'\ n pulse amplitude, reconstructed waveform

\'\ , pulse amplitude, reference-instrument acquired waveform, reference pulse

\ s i ., lower state level, corrected waveform

\'si.„, lower state level, acquired waveform

I'st.m.i lower Mate level, acquired waveform, using reference pulse and sampler

I'si.h lower state level, reconstructed waveform

I si.,-., lower state level, acquired waveform, using reference pulse and reference measurement instrument

V'niax.iii maximum value, acquired waveform

1 i,,i,x i< maximum value, reconstructed waveform

I'iniii.n, minimum value, acquired waveform

I'min.H minimum value, reconstructed' waveform

ls-j.1- upper state level, corrected waveform

l'S2.m upper state level, acquired waveform

Js-j.m.i- upper state level, acquired waveform, using reference pulse and sampler Vsa.H upper state level, reconstructed waveform

l'S2.r.r upper state level, acquired waveform, using reference pulse and reference measurement instrument

\'o „, overshoot value, acquired waveform

l-'O.H overshoot value, reconstructed waveform

Ij-.m undershoot value, acquired waveform

I

i n undershoot value, reconstructed waveform

\ "^

/ temperature-induced incremental change in pulse amplitude

A',i non-integer number of sampling intervals in transition duration of reconstructed waveform

A'j non-integer number of sampling intervals in transition duration of effective jitter step response

A',,, non-integer number of sampling intervals in transition duration of acquired waveform

A'r non-integer number of sampling intervals in transition duration of sampler step response waveform

fto overshoot correction factor relating transition duration of measured and reconstructed waveforms #C undershoot correction factor relating transition duration of measured and reconstructed waveforms

Atd._v/- temperature-induced incremental change in transition duration AV histogram bin size 8/ sampling interval

;/,.rr effective degrees of freedom

;/, degrees of freedom

Reported variables O overshoot value U undershoot (preshoot) value

(d transition duration

V,\ pulse amplitude, reported

u, uncertainty of the ?-th variable

Meimlogia, 2002. 39. 143-155 155
Institute ok Physics Publishing MhTROLOCLA

Metrologia 43 (2006) 121-128 doi: 1 0. 1 088/0026- 1 394/4 3/1/017 Some effects of temperature variation on

sampling oscilloscopes and pulse

generators

Donald R Larson 1 and Nicholas G Paulter Jr 1

Quantum Electrical Metrology Division, NIST, Gaithersburg, MD 20899-8172, USA

Received 15 September 2005

Published 22 December 2005

Online at stacks.iop.org/Met/43/ 1 2

Abstract

The effects of temperature variation on the timebase errors and impulse

responses of two 50 GHz bandwidth sampling oscilloscopes and on the pulse parameters of two pulse generators commonly used for oscilloscope

calibrations are reported. The observed variations are significant for high accuracy measurements and contribute to the uncertainty of any measurements performed.

1. Introduction

Equivalent time, sampling oscilloscopes are now commercially

available with 3 dB attenuation bandwidths exceeding 50 GHz.

Pulse generators with transition durations of less than 16 ps are also commercially available. These high

speed sampling oscilloscopes and pulse generators are used to make measurements needed to characterize high

speed communications networks and components. These measurements are often made in locations where the temperature may change by several degrees Celsius between

measurements. We have observed that the sampling heads and pulse generators may show differences due to changes in the ambient temperature. Furthermore, the uncertainties in the measurements may mean the difference between meeting

specifications and failing an expensive network component.

2. Measurement set-up

We tested two different manufacturers' oscilloscope mainframes together with four different sampling heads and two

different step generators. One sampling oscilloscope main- frame is identified as SMI and the two sampling heads used

with it are identified as SHI and SH2. SHI is a 50 GHz

(3dB attenuation bandwidth) sampling head and SH2 is a

1 Quantum Electrical Metrology Division, Electronics and Electrical Engineering Laboratory, National Institute of Standards and Technology, Technology Administration, Department of Commerce. Official contribution of the National Institute of Standards and Technology, not subject to copyright

in the USA.

20 GHz bandwidth sampling head. Similarly, the other sampling oscilloscope mainframe is identified as SM2 and the sampling heads used with it are identified as SH3 and SH4. SH3 and SH4 are, respectively, a 50 GHz (3 dB attenuation bandwidth) sampling head and a 20 GHz bandwidth sampling

head. Each of these oscilloscopes was tested with two different manufacturers' step generators having a nominal bandwidth of 20 GHz. The step generators are identified as SGI and SG2.

Before making any measurements, all instruments were

allowed to warm up for at least two hours after applying power. This warm-up period was determined by attaching

a temperature sensor to the cases of several instruments and

observing the temperature of the instruments as a function of time. The measurements were performed by placing the instrument under test (oscilloscope mainframe, step generator

or sampling head) inside an environmental chamber. This

environmental chamber is located inside a shielded room

where the temperature is controlled to within 1 °C over the measurement period. Only the temperature of the instrument under test was intentionally varied. The other components were kept at room temperature, 23.0 °C ± 1 °C. The sampling

heads were connected to the oscilloscope mainframes using cabled extender modules purchased from the manufacturers. The step generator was connected to the sampling head

using a high bandwidth (approximately 26 GHz) coaxial cable approximately 0.5 m long. When the step generators were

being evaluated, a 50 GHz bandwidth sampling head was

used. The trigger signal source was also kept at room

temperature. The trigger signal input was located in the mainframe of one oscilloscope model and in the sampling

heads of the other model. The temperature sensor was

0026- 1394/06/0101 2 1+O8S30.O0 © 2006 BIPM and IOP Publishing Ltd Printed in the UK
D R Larson and N G Paulter Jr

80

60

40

> 20 E—

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-40

-60

-80

-\ /~\

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10 20 30 40 50 60

time/ps

70 80 90 100

Figure 1. Waveform distortion caused by timebase error, 15 GHz sine wave.

a J type thermocouple and attached to the case of the instrument under test. The temperatures used in this work

represent the manufacturers' narrowest specified operating temperature range (15 °C to 35 °C). The temperature was

incremented in 5 °C steps. The instrument under test was

kept at the target temperature for at least 30min before measurements were made. In all cases, the temperature

of the instrument varied by less than 0.2 °C during the measurements. At each set temperature, multiple waveforms

were acquired, the parameters of interest determined and

the mean and standard deviations of the parameters were

calculated.

3. Sampling oscilloscope timebase errors

The timebase of a sampling oscilloscope generates an

impressive range of epochs that can be varied from picoseconds

to seconds, nine to eleven orders of magnitude. For the oscilloscope mainframes examined, the timebase consists of

a startable oscillator and a time interpolator vernier (fine delay

ramp) that has a delay range equal to one period of the

startable oscillator. The timebase can be viewed as a repeated concatenation of the time interpolator vernier at every cycle of

the startable oscillator until the desired epoch is achieved [1]. Unless the range of the time interpolator vernier is exactly one period of the startable oscillator, the sampling instant immediately after the concatenation occurs may differ from

the intended sampling instant by several picoseconds. This

timing error may produce a visible error in the waveform

(figure 1 ). The time interpolator vernier, which has a range of several nanoseconds, is not perfectly linear and timing

errors are also seen in that range. The non-linearity of

the time interpolator vernier will also distort the waveform

but this distortion is usually not obvious when observing an

acquired waveform. The timing errors due to the non-linearity of the time interpolator vernier can be separated into two

components, a fixed error in the reported sampling interval and a variable timing error that varies throughout the range

of the time interpolator vernier. This first error component

leads to the overall slope seen in the timebase error (figure 2). Although the second error component varies over the range

of the time interpolator vernier, it varies the same way in each concatenation and can be seen as the repeating pattern

in figure 2. NIST has developed a method to characterize these timing

errors [2] which has been used to obtain the results presented

here. To summarize this technique, the single-frequency output from a synthesized sine-wave source is connected to the sampling oscilloscope and two or more unique waveforms

are acquired. Each acquired waveform has a different phase

relative to the trigger. The acquired waveforms are then compared to a theoretical sine wave and the residuals, divided by the derivative of the theoretical sine wave, yield the timebase

error. The timebase error is the deviation of the actual time from the sampler reported time (intended time) as a function

of the sampler reported time. It is evident from figures I and 2 that these timebase errors may impact measured pulse parameters [3]. To characterize the temperature dependence of the timebase error, an oscilloscope mainframe was placed in the environmental chamber while the 50 GHz bandwidth

sampling head remained outside the environmental chamber at room temperature. The oscilloscope mainframe was allowed

to come to thermal equilibrium by waiting 45 min before

acquiring timebase data. A set of sine waves were acquired

(two frequencies and multiple phases) and then the temperature was set to the next target temperature. This procedure was

followed for both the mainframes tested. A subset of the results obtained is depicted in figures 3 and 4. The measurement

122 Metmlogia, 43 (2006) 121-128
Temperature variation on sampling oscilloscopes and pulse generators

Q -0

time/ns

Figure 2. Timebase errors for SMI and SM2 sampling oscilloscopes.

Figure 3. Timebase error for SMI oscilloscope at three temperatures.

uncertainty corresponding to a 95% confidence interval is ±0.08 ps. The concatenation error for SMI (see figure 3) is seen to go through a minimum and change sign near 30 °C. Over the range of temperatures used, the timebase error for SM2 (see figure 4) did not go through a similar minimum but decreased monotonically. The error in the reported sampling

interval changes for both mainframes as indicated by the tilt of the graph of the error estimate. The variation of the nonlinearity appears to be nearly constant with temperature for both mainframes.

4. Pulse parameters

The pulse amplitude, high state, low state and transition durations (10% to 90% and 20% to 80%) were determined

according to the procedures outlined in IEEE Standard 181-2003 [4]. A histogram of the data is first created and

the two maxima of the resulting bimodal distribution define

the high state and the low state. The number of histogram bins used for the data presented here was 4096. The amplitude

is the difference between the high state and the low state.

Meiwlogia, 43 (2006) 121-128 123
D R Larson and N G Paulter Jr

10

JjWjjfU*j3i! 35 °C (vertically offset by +6 ps) y"

Figure 4. Timebase error for SM2 oscilloscope at three temperatures.

22 24 26 28

temperature /°C

Figure 5. Change in transition duration (10% to 90%) as a function of temperature.

36

The 10%, 20%, 80% and 90% reference levels are calculated and their occurrence instants found by linear interpolation. The

transition duration is the difference between the occurrence

instants of the appropriate percentage reference levels. The change in transition duration (10% to 90%) of all eight devices as a function of temperature is depicted in figure 5. Sampling heads SHI, SH2 and SH4 exhibited a small increase in transition duration (decrease in bandwidth)

with increasing temperature. The transition duration of

the waveform from step generator SGI decreased significantly

with temperature. Step generator SG2 exhibited only a small

increase in transition duration with increasing temperatures. As previously mentioned, a set of data was obtained at each

temperature; the maximum standard deviation observed in these sets of data was 0.206 ps. Although the 10% to 90% transition duration is the most

quoted pulse parameter, the 20% to 80% transition duration

is included here and in the calibrations we perform. The

20% to 80% transition duration is affected less by aberrations than the 10% to 90% transition duration and, consequently.

124 Metrologia, 43 (2006) 121-128
Temperature variation on sampling oscilloscopes and pulse generators

at

20 22 24 26 28

temperature /°C

36

Figure 6. Change in transition duration (20% to 80%) as a function of temperature.

22 24 26 28

temperature/°C

Figure 7. Change in amplitude (percentage) as a function of temperature.

often exhibits a smaller standard deviation. Figure 6 depicts

the temperature dependence of the transition duration (20%

to 80%) for all devices tested. These results were similar

to the results for the 10% to 90% transition duration. The

maximum standard deviation observed in any of the (20% to 80%) transition duration data sets is 0. 130ps. Another parameter used to describe a pulse is the pulse

amplitude. When the temperature is varied, both sampler gain and offset can vary. Figure 7 displays the amplitude changes we measured. SH3 displayed an unusually large decrease

in amplitude with increasing temperature. To confirm this

behaviour, a second sampling head of the same make and model was also tested with similar results. The other sampling

heads and step generators exhibited increasing step amplitudes

with increasing temperature. SHI and SH2 (from the same

manufacturer) were almost temperature invariant. For all the amplitude data reported here, the maximum standard deviation of a data set was 0.455 mV for a nominal pulse amplitude

of245mV.

The changes in the voltage level associated with the high

state and the voltage level associated with the low state were

also examined and are depicted in figures 8 and 9. For sampling

Meimlogia, 43 (2006) 121-128 125
D R Larson and N G Paulter Jr

25

15

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16 20 22 24 26 28

temperature/°C

30 32 34 36

Figure 8. Changes in the high state, S2, as a function of temperature.

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i i 14 16 20 22 24 26 28 30 32 34

temperature/°C

36

Figure 9. Changes in the low state, SI , as a function of temperature.

heads SHI, SH2 and SH4, the low states shifted to lower

values as the temperature increased. However, because the high states exhibited a shift nearly equal to the low state shift and in the same direction, the change in amplitude (figure 7)

for waveforms measured with these sampling heads is small. SH3, on the other hand, exhibited both a relatively large change

in amplitude and offset. A waveform measured with this sampling head exhibited a decrease in amplitude and offset magnitude as the temperature increased. The change in pulse parameters as a function of

temperature for each device tested is summarized in table I

Each entry is the slope of a straight line fit to the data for

that particular step generator, sampling head or oscilloscope mainframe.

5. Transition occurrence instant

The position of the pulse in the epoch was also observed

to change with temperature when the sampling oscilloscope

mainframe, sampling head or pulse generator temperature

varied. Figure 10 depicts the change in the transition occurrence instant (50% reference level instant) as a function

126 Metwlogia, 43 (2006) 121-128
Temperature variation on sampling oscilloscopes and pulse generators

Table 1. Change in pulse parameters as a function of temperature, linear fit to data.

High state slope/ (mV/°C) Low state slope/ (mV/°C)

Amplitude slope/ (mV/°C)

10% to 90%

transition duration

slope/(ps/°C)

20% to 80%

transition duration

slope/(ps/°C) SGI 0.033 -0.026 0.059 -0.137 -0.071 SG2 -0.006 -0.043 0.037 0.020 0.019 SHI -0.976 -0.983 0.007 0.076 0.080 SH2 -0.206 -0.214 0.008 0.032 0.016 SH3 -0.435 1.043 -1.478 -0.003 -0.003 SH4 -0.385 -0.477 0.093 0.068 0.041 SMI -0.037 0.024 -0.061 -0.017 -0.007 - SM2 -0.040 -0.027 -0.014 0.003 0.003

20

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14 16 18 20 22 24 26 28

temperature/°C

30 32 34 36

Figure 10. Change in transition occurrence instant (50% reference level instant) as a function of temperature.

of temperature. The maximum observed standard deviation

for all instruments was 0.44 ps. For the sampling heads and pulse generators, this is independent of the temperature dependence of the timebase error or trigger signal since

the oscilloscope mainframe temperature was held constant

for those measurements. However, the large change in transition occurrence instant observed for SH4 is probably due to the trigger signal being routed through the sampling

head. The changes observed for SMI and SM2 are much

too large to be the result of the change in the timebase

error. Under certain conditions, the observed change in transition occurrence instant will have a significant effect on the transition duration. For example, if the temperature changes while a pulse waveform is being acquired and further,

if the embedded waveform averaging routine is being used, the

transition duration will increase as a result of this temperature- induced change in transition occurrence instant. It should be

noted that the transition duration results presented previously were obtained at set temperatures and the small standard

deviation of the measurements indicates that the change in transition occurrence instant was not a significant factor. This

is indicative of a well-controlled environment, a necessity for a

metrology facility involved in calibrating pulse generators and

samplers.

6. Jitter

When making pulse waveform measurements, the acquired waveforms are impacted by the presence of trigger or system jitter. As with drift of the transition occurrence

instant, averaging waveforms acquired in the presence of

jitter will increase the transition duration. The effect on

transition duration may be removed by measuring the jitter, modelling it with a Gaussian distribution and deconvolving

the Gaussian distribution model from the acquired waveforms

before determining the pulse parameters [5], Because the measurements reported here are comparative, we did not remove jitter from any of the waveforms used. Consequently,

the transition duration values will exhibit effects of the temperature dependence of jitter. However, the observed changes in jitter were much less than the uncertainty in our

transition duration values. For SMI, the average jitter was

less than 1 .5 ps with a maximum standard deviation of 0.06 ps and varied less than 0.2 ps over the range of temperatures

Metrologia, 43 (2006) 121-128 127
D R Larson and N G Paulter Jr

examined. For SM2, the average jitter was less than l.Ops

with a maximum standard deviation of 0.02 ps and varied less than 0.03 ps over the temperature range used.

7. Summary

Changes in the error of the oscilloscope timebase have been

characterized for two manufacturers' sampling oscilloscopes, both commonly used to characterize high speed digital communications networks and components. The changes in pulse parameters with temperature have been determined for two step generators, four sampling heads and two sampling

oscilloscope mainframes. The transition duration measured

by each sampling head increased (a bandwidth decrease) as the temperature increased except SH3 which displayed almost no change. Step generator SGI produced a pulse with a

faster transition duration (bandwidth increase) with increasing temperature. The step amplitudes for SGI, SG2, SHI, SH2,

SH4, SMI and SM2 were nearly insensitive to temperature

changes, although level shifts were observed. This indicates a change in offset with temperature. No statistically significant changes in trigger or system jitter were observed. Changes

in the pulse aberrations of overshoot and undershoot were

also observed but are not reported here. Temperature-induced

changes in pulse parameters and transition occurrence instant can contribute significantly to the uncertainty estimate [6,7]. The results indicate a need for a well-controlled environment,

typical of good metrology facilities, for pulse parameter measurements with low uncertainties. Although the results from the two sampling heads of the same model were

similar, they were sufficiently different that each instrument had to be individually characterized. We note that when

comparing measurement results from different laboratories using identical equipment or from the same laboratory but

at different temperatures, the relative temperature differences must be known in order to explain the differences in results. Both sampling oscilloscopes examined here incorporated an embedded sampling head calibration routine. These routines were executed only once, before any measurements were made.

Acknowledgments

We wish to thank T M Souders for his technical comments.

This work was partially funded by the LIS Air Force Calibration Coordination Group.

References

[1] Dobos L 1994 Advancements in picosecond resolution time

interval measurement techniques Conf. on Precision Electromagnetic Measurements (Boulder, CO,

27 June-1 July 1994) pp 13-14

[2] Stenbakken G N and Deyst J P 1998 Time-base nonlinearity determination using iterated sine-fit analysis IEEE Trans. Instrum. Meas. 47 1056-61

[3] Paulter N G Jr and Larson D R 2000 Time-base setting dependence of pulse parameters using high-bandwidth digital sampling oscilloscopes National Conf. ofStandards

Laboratories International (Toronto, ON, Canada, 16-20 July)

[4] IEEE Std 181-2003 IEEE Standard on Transitions, Pulses and

Related Waveforms (New York: IEEE) pp 19, 24

[5] Gans W L 1983 The measurement and deconvolution of time

jitter in equivalent-time waveform samplers IEEE Trans. Instrum. Meas. EvI-32 126-33

[6] Larson D R and Paulter N G Jr 2000 Temperature effects on the high speed response of digitizing sampling oscilloscopes National Conf. ofStandards Laboratories International

(Toronto, ON, Canada, 16-20 July)

[7] Paulter N G and Larson D R 2002 Pulse parameter uncertainty

analysis Metrologia 39 143-55

128 Melwlogia, 43 (2006) 1 2 1 - 1 28