Electronics Guide

Type A and Type B Evaluation

Type A uncertainty evaluation uses statistical analysis of repeated measurements. Taking n readings under the same conditions gives their experimental standard deviation s, a direct estimate of the random scatter. When the mean of independent readings is reported, its standard uncertainty is the standard deviation of the mean:

u = s / √n

Because this falls only as 1/√n, halving the uncertainty takes four times as many readings. The approach observes variability directly, but a standard deviation computed from only a few readings is itself uncertain, and the GUM then calls for a larger coverage factor.

Type B evaluation relies on other sources of information, including calibration certificates, manufacturer specifications, published data, and engineering judgment. When statistical sampling is impractical or impossible, these sources provide the basis for the estimate. Type B contributions often dominate in digital measurements, where systematic effects such as timebase error, gain and offset error, and the quantization of a steady, noise-free signal cannot be reduced by repeating the measurement.

The combined standard uncertainty merges Type A and Type B contributions by root-sum-square combination when the contributions are independent. Correlated components require an additional term that accounts for their interdependence, described under Measurement Correlation below. The expanded uncertainty U multiplies the combined standard uncertainty by a coverage factor, most often k = 2, which gives an interval with a coverage probability of approximately 95 percent when the combined distribution is close to normal and the effective number of degrees of freedom is large.

Uncertainty Budgets

An uncertainty budget systematically documents all identified uncertainty sources, their evaluated magnitudes, and how they combine to produce the total measurement uncertainty. This structured approach ensures that no significant contributors are overlooked and provides traceability for the uncertainty claim. For digital measurements, the budget typically includes contributions from the analog front end, the conversion process, timing systems, and data processing.

Sensitivity coefficients describe how uncertainty in each input quantity propagates to the measurement result. When the measurand y is a function f of input quantities x_i, the sensitivity coefficient of each input is the partial derivative c_i = ∂f/∂x_i, and for uncorrelated inputs the GUM's law of propagation of uncertainty gives the combined standard uncertainty u_c(y):

u_c²(y) = ∑ c_i²u²(x_i)

The law is a first-order approximation, valid when the measurement function is close to linear over a few standard uncertainties around each input. For strongly nonlinear functions, the GUM adds the most important higher-order terms, and Monte Carlo simulation avoids the approximation altogether. The general treatment, including correlated inputs and Monte Carlo methods, is set out in Measurement Uncertainty.

Nyquist Considerations

The Nyquist-Shannon sampling theorem establishes that a bandlimited signal can be perfectly reconstructed from samples taken at greater than twice the signal's highest frequency component. However, practical measurements rarely achieve this theoretical ideal. Signals with frequency content above the Nyquist limit alias into the measurement band, appearing as spurious lower-frequency components that corrupt the measurement. Anti-aliasing filters reduce but cannot eliminate this contamination, and their imperfect response contributes additional uncertainty.

For signals with energy extending above the Nyquist frequency, aliasing uncertainty depends on the spectral content of the signal and the anti-aliasing filter characteristics. Signals with significant high-frequency components require either higher sampling rates or steeper filter rolloff to maintain acceptable aliasing uncertainty. The uncertainty contribution can be estimated from the filter rejection at frequencies that would alias into the measurement band.

Spectral Leakage

Finite-duration sampling windows cause spectral leakage, spreading energy from each frequency component across a range of frequencies in the computed spectrum. This effect introduces uncertainty in both frequency and amplitude measurements. The leakage pattern depends on the window function applied to the sampled data, with different windows trading off main lobe width against side lobe suppression.

Rectangular windows provide the narrowest main lobe but have high side lobes that can mask small signals near larger ones. Windowing functions such as Hann (often called Hanning), Hamming, and Blackman reduce side lobes at the cost of broader main lobes. The choice of window affects frequency resolution and amplitude accuracy, with the uncertainty contribution depending on the signal characteristics and measurement requirements.

The window also sets the amplitude error for a tone that falls between two frequency bins, known as scalloping loss. A tone midway between bins reads 3.92 dB low with a rectangular window and 1.42 dB low with a Hann window. Flat-top windows hold this error to a small fraction of a decibel at the cost of a much wider main lobe, which is why spectrum analyzers and FFT-based instruments offer them for amplitude measurements, while narrower windows serve when resolving closely spaced tones matters more.

Coherent sampling, where the sampling window contains an integer number of signal cycles, eliminates leakage for periodic signals. However, achieving exact coherence requires either precise control of the signal frequency or adaptive adjustment of the sampling parameters. When coherent sampling is impractical, windowing provides the best mitigation of leakage effects.

Sample Rate Uncertainty

Uncertainty in the actual sampling rate directly affects time and frequency measurements. The sample rate is typically derived from a crystal oscillator, whose frequency uncertainty depends on initial tolerance, temperature stability, and aging, while its phase noise adds jitter from one sample to the next. High-precision measurements may require external frequency references with calibration traceability to national standards.

For time-interval measurements, sample rate uncertainty contributes directly to the timing uncertainty. Frequency measurements reflect it equally, because the frequency scale derives from the assumed sample rate: a sample clock running 100 ppm fast makes measured time intervals read 100 ppm long and measured frequencies 100 ppm low. The sample rate uncertainty typically appears as a systematic contribution in the uncertainty budget.

Quantization Noise

For signals that span many quantization levels, quantization error approximates a uniform random distribution with peak-to-peak amplitude equal to one least significant bit (LSB). The standard deviation of this uniform distribution, the RMS quantization noise, establishes the fundamental quantization noise floor:

σ_q = q / √12 ≈ 0.29q

where q is the size of one LSB, the full-scale range divided by 2^N for an N-bit converter. The model assumes that the error is uncorrelated with the signal, a condition best satisfied by signals that span many codes and bear no simple relation to the sampling clock.

The signal-to-quantization-noise ratio (SQNR) for an ideal converter driven by a full-scale sine wave follows the relationship

SQNR = 6.02N + 1.76 dB

where N is the number of bits. Each additional bit therefore contributes about 6 dB. A 12-bit converter theoretically achieves about 74 dB SQNR, while a 16-bit converter reaches approximately 98 dB. Actual converters fall short of these ideals due to differential nonlinearity, integral nonlinearity, and noise contributions beyond quantization.

For DC or slowly varying signals that remain within one or a few quantization levels, the simple uniform distribution model breaks down. The quantization error becomes systematic rather than random, and averaging repeated measurements does not reduce the uncertainty below the quantization limit. Dithering techniques add small amounts of noise to decorrelate the quantization error, restoring the statistical properties that enable averaging to improve resolution.

Effective Number of Bits

Effective number of bits (ENOB) provides a practical measure of converter performance that accounts for all noise and distortion sources. ENOB is calculated from the measured signal-to-noise-and-distortion ratio (SINAD) in decibels by inverting the ideal SQNR relation, and represents the number of bits an ideal converter would need to achieve the same SINAD:

ENOB = (SINAD − 1.76) / 6.02

A 16-bit converter measuring a SINAD of 86 dB therefore has an ENOB of 14.0 bits. The difference between nominal resolution and ENOB indicates how much converter imperfections degrade performance below the theoretical limit.

ENOB varies with input frequency, typically degrading at higher frequencies where converter bandwidth limitations and aperture uncertainty become significant. Specifying uncertainty based on ENOB rather than nominal resolution provides a more realistic assessment of actual measurement capability. For noise and distortion together, the 16-bit converter above behaves like an ideal 14-bit converter, whose quantization noise is four times that of an ideal 16-bit one, so a budget built on the nominal bit depth would understate this contribution fourfold.

Differential and Integral Nonlinearity

Differential nonlinearity (DNL) describes variations in quantization step size from the ideal value of one LSB. DNL causes some codes to represent wider or narrower voltage ranges than others, distorting the transfer function, and the quantization interval of the code actually read is its own width, one LSB plus that code's DNL, rather than the nominal one. A code whose DNL reaches −1 LSB has zero width and never appears whatever the input voltage, a missing code. A converter whose DNL stays greater than −1 LSB at every code is therefore guaranteed to have no missing codes.

Integral nonlinearity (INL) accumulates DNL errors across the converter range, representing the deviation of the actual transfer function from the ideal straight line. INL causes systematic measurement errors that vary with signal amplitude. For measurements spanning the full converter range, INL contributes directly to uncertainty. Converters with specified INL allow this contribution to be included in uncertainty budgets.

Both DNL and INL can be reduced through calibration. Measuring the actual code transitions and compensating for their deviations from ideal improves effective linearity. Digital correction applies the inverse of the measured nonlinearity to converter output, approaching ideal performance at the cost of calibration complexity and potential sensitivity to drift.

Aperture Jitter Sources

Aperture jitter arises from multiple sources within the acquisition system. Clock jitter in the sampling clock source contributes directly, with phase noise in the clock oscillator translating to timing uncertainty. The sample-and-hold circuit adds aperture uncertainty from variations in the opening time of the sampling switch. Noise in the comparator or track-and-hold amplifier causes additional variation in the effective sampling instant.

A converter's own aperture jitter is typically specified as an RMS value. High-performance converters achieve aperture jitter below 100 femtoseconds, while general-purpose devices may have several picoseconds of jitter. The external sampling clock contributes its own jitter on top of the converter's internal aperture uncertainty, and because the two are independent they add in root-sum-square fashion to give the total jitter t_j:

t_j = √(t_clk² + t_ap²)

where t_clk is the RMS jitter of the clock at the converter input and t_ap is the converter's aperture jitter. A clock with 100 fs of RMS jitter driving a converter with 60 fs of aperture jitter gives a total of about 117 fs, so in high-speed systems the clock, not the converter, often sets the limit.

Voltage Uncertainty from Aperture Jitter

The voltage uncertainty caused by aperture jitter depends on the signal's rate of change at the sampling instant. For a sinusoid of frequency f and peak amplitude A, v(t) = A sin(2πft), the rate of change is greatest at the zero crossings, where the slew rate is 2πfA. Multiplying this slew rate by the RMS jitter t_j gives the RMS voltage error of samples taken at a zero crossing, 2πfA t_j. Averaged over a whole cycle the slew rate is lower, and the RMS error is smaller by a factor of √2.

This relationship establishes a fundamental limit on achievable signal-to-noise ratio for sampled signals. Dividing the signal's RMS value, A/√2, by the cycle-averaged RMS error cancels the amplitude, so even with infinite converter resolution, jitter alone limits the SNR, in decibels, to

SNR = −20 log₁₀(2πf t_j)

where f is the signal frequency and t_j is the total RMS jitter. A system with 1 picosecond of RMS jitter measuring a 100 MHz signal achieves at best 64 dB SNR from this effect alone, the ideal figure for a converter of about 10.3 bits. Each tenfold increase in signal frequency costs 20 dB, so the same jitter allows 84 dB at 10 MHz.

For non-sinusoidal signals, the appropriate slew rate depends on the specific waveform. Step transitions and pulse edges produce the highest slew rates and thus suffer the most from aperture uncertainty. The uncertainty contribution should be evaluated using the actual signal characteristics rather than assuming sinusoidal behavior.

Minimizing Aperture Effects

Reducing aperture uncertainty requires attention to the entire signal path from clock source through the sampling circuit. Low-jitter clock oscillators with good phase noise performance establish the timing reference. Clean power supplies and proper grounding prevent noise injection into sensitive timing circuits. Layout techniques minimize coupling between noisy digital signals and the sampling clock.

When measuring signals with very high slew rates, track-and-hold amplifiers placed before the converter can reduce effective aperture uncertainty. These circuits capture the signal at a well-defined instant and hold it stable during the conversion process. The track-and-hold's aperture jitter then determines timing uncertainty rather than the converter's internal sample-and-hold.

Trigger Jitter

Trigger jitter describes variation in the actual trigger instant relative to the intended trigger condition. Noise on the input signal causes the trigger threshold crossing to vary in time, with the jitter magnitude inversely proportional to the signal slew rate at the crossing:

σ_t ≈ σ_v / (dv/dt)

where σ_v is the RMS noise voltage and dv/dt is the slew rate at the threshold. An edge slewing at 1 V/ns with 2 mV of RMS noise triggers with about 2 ps of RMS jitter; slow the edge to 0.1 V/ns and the jitter rises to about 20 ps. Slow edges or noisy signals therefore produce more trigger jitter than fast, clean edges.

The trigger circuit itself contributes jitter from comparator noise and hysteresis uncertainty. Even with an ideal input signal, these internal sources cause some variation in trigger timing. High-performance instruments specify trigger jitter under defined conditions, allowing this contribution to be included in uncertainty analysis.

For repetitive measurements using averaging or equivalent-time sampling, trigger jitter causes different portions of successive acquisitions to be combined, smearing waveform features and reducing effective bandwidth. When the trigger instants scatter with a Gaussian distribution of standard deviation σ, averaging convolves the waveform with that distribution, a low-pass filter whose −3 dB frequency is about 0.13/σ. With 10 ps of RMS trigger jitter the averaged waveform is limited to roughly 13 GHz, and faster edges are reported slower than they are, however many acquisitions are averaged.

Trigger Level Uncertainty

Uncertainty in the actual trigger threshold voltage affects both timing and amplitude measurements. If the trigger level differs from its nominal setting, measurements referenced to the trigger point reflect this error. Calibration of trigger level accuracy can characterize this contribution, which typically appears as a systematic uncertainty.

Temperature dependence of the trigger comparator causes the effective trigger level to drift over operating conditions. Instruments with internal calibration can track and compensate for this drift, while those without require the user to account for potential trigger level variation in the uncertainty budget.

Trigger Holdoff and Delay

Trigger holdoff and delay settings introduce additional timing uncertainties. Holdoff prevents triggering for a specified interval after each trigger, with uncertainty in this interval affecting measurements that depend on trigger timing. Trigger delay shifts the acquisition window relative to the trigger point, with delay accuracy contributing to timing uncertainty.

When measurements require precise knowledge of the time between trigger and acquired data, the trigger delay accuracy specification determines the uncertainty contribution. High-resolution timing measurements may require calibration of the actual trigger delay rather than relying on nominal settings.

Crystal Oscillator Characteristics

Most digital instruments derive their timebase from crystal oscillators, whose frequency accuracy depends on several factors. Initial frequency tolerance, typically specified in parts per million (ppm), describes how closely the oscillator frequency matches the nominal value when manufactured. Temperature coefficient describes frequency variation with ambient temperature, while aging rate characterizes long-term frequency drift.

An instrument's crystal timebase might specify 25 ppm initial tolerance, 10 ppm of variation over the operating temperature range, and 5 ppm per year of aging. Added as worst-case limits, these reach 40 ppm after one year of aging and grow by a further 5 ppm for each year without adjustment. Combined by root-sum-square as independent effects, the one-year figure is about 27 ppm. In a GUM budget each limit is usually taken as the half-width of a rectangular distribution and divided by √3, giving standard uncertainties of 14.4, 5.8, and 2.9 ppm and a combined standard uncertainty of about 15.8 ppm. Whichever treatment is used, the result applies to all time measurements as a scaling factor: 40 ppm of a 1 ms interval is 40 ns.

Temperature-compensated crystal oscillators (TCXOs) narrow the temperature term with circuitry that corrects for the crystal's temperature coefficient, reaching about 0.5 to 5 ppm. Oven-controlled crystal oscillators (OCXOs) provide significantly better stability by maintaining the crystal at a constant elevated temperature. OCXO specifications of 0.01 to 0.1 ppm are common, reducing timebase uncertainty by orders of magnitude. The improvement comes at the cost of warm-up time, power consumption, and instrument cost.

External Reference Synchronization

High-precision measurements may require synchronization to external frequency references with calibration traceability. Averaged over a day, a GPS-disciplined oscillator agrees with national standards to about one part in 10¹² or better, effectively eliminating timebase uncertainty as a significant contributor; over shorter averaging times, the stability of its local oscillator sets the figure. A cesium-beam standard offers similar or better accuracy without an external signal, while a free-running rubidium standard, though far more stable than quartz, drifts slowly as it ages and is typically accurate to parts in 10¹⁰ to 10¹¹.

When using external references, the uncertainty of the reference source replaces the internal oscillator uncertainty in the budget. The instrument locks to the reference through a phase-locked loop, which adds a little phase noise but no frequency error once locked, and a constant cable delay does not affect frequency or time-interval results at all. Cable and distribution delays matter only when absolute time is transferred, as with a one-pulse-per-second signal, and must then be measured or corrected.

Time Interval Accuracy

Time interval measurements combine timebase accuracy with quantization and interpolation uncertainties. The basic time resolution equals the sample period, but interpolation techniques can estimate event timing between samples. Interpolation accuracy depends on the algorithm used and the signal characteristics, adding uncertainty beyond the fundamental quantization.

The timebase term grows with the interval, while the others do not. An interval T measured against a timebase with fractional error ε is in error by εT, so a 25 ppm timebase adds 2.5 ps to a 100 ns interval but 25 µs to a 1 s interval. Short intervals are therefore dominated by edge timing jitter and interpolation, long ones by the timebase.

For high-resolution time measurements, dedicated time-to-digital converters (TDCs) provide picosecond-level resolution through techniques like vernier interpolation or delay-line measurement. TDC uncertainty specifications include both resolution and linearity contributions, which must be combined with any timebase uncertainty for the overall time measurement uncertainty.

Common Mode Uncertainty

Measurements made with the same instrument at similar times often share systematic uncertainties. Timebase, gain, and offset errors push every such reading in the same direction: an offset by the same amount, a gain or timebase error by the same proportion. When computing differences between nearly equal readings, these common-mode uncertainties largely cancel, potentially yielding much lower uncertainty than the individual measurement uncertainties would suggest.

Summing correlated measurements works the other way: common-mode effects add linearly rather than combining as root-sum-square. Two time measurements of equal magnitude, each carrying a 1% uncertainty from a shared timebase, produce a ratio with essentially zero uncertainty from that source. Their sum, however, still carries the full 1% common-mode uncertainty, worse than the roughly 0.7% that would result if the two contributions were independent. Measurements made on different instruments share no such common error, so comparing them brings no cancellation, and their uncertainties combine as independent contributions.

Correlation Coefficients

The correlation coefficient quantifies the degree of statistical dependence between uncertainty contributions, ranging from −1 for perfect negative correlation through 0 for independence to +1 for perfect positive correlation. For a measurand that depends on two inputs, the law of propagation of uncertainty gains a cross term:

u_c²(y) = c₁²u²(x₁) + c₂²u²(x₂) + 2c₁c₂u(x₁)u(x₂)r(x₁, x₂)

where r(x₁, x₂) is the correlation coefficient. In a sum, the sensitivity coefficients share a sign and positive correlation increases the combined uncertainty. In a difference or a ratio they have opposite signs, and positive correlation reduces it, to zero when r = +1 and the two contributions are equal in size.

Determining correlation coefficients requires understanding the physical origins of uncertainty contributions. Sources derived from the same physical quantity are typically highly correlated. Independent noise sources are uncorrelated. Partial correlation occurs when sources share some common influence but also have independent components.

Differential Measurements

Differential measurement techniques exploit correlation to achieve lower uncertainty than absolute measurements. By measuring the difference between two signals using the same instrument, common-mode errors cancel. This approach is particularly effective for comparing similar signals, where the difference may be orders of magnitude smaller than the absolute values.

Time interval measurements benefit from differential techniques when both events are measured against the same timebase. When both edges fall within one oscilloscope acquisition, the trigger instant and the fixed delays of the input path shift them together, so trigger jitter and trigger delay cancel in the interval. What remains is the uncorrelated timing noise on each edge, set by the signal noise and slew rate at each crossing, together with interpolation error and any skew between channels when the edges arrive on different inputs. The timebase error does not cancel, because it scales the interval, but for short intervals it is tiny. A counter with separate start and stop channels behaves similarly, except that its two trigger comparators contribute independent jitter. Either way, the interval uncertainty can be much lower than the absolute timing uncertainty of either event.

Identifying Uncertainty Sources

A systematic approach to identifying uncertainty sources considers the complete measurement chain from input signal through displayed result. The input signal itself may have uncertainty in its true value. Input conditioning including attenuation, impedance matching, and filtering introduces additional uncertainty. The analog-to-digital conversion adds quantization, nonlinearity, and aperture effects. Timing systems contribute timebase and trigger uncertainties. Signal processing and computation may introduce numerical errors or algorithmic assumptions.

Cause-and-effect diagrams help organize the identification process, tracing all influences on the measurement result. Literature review and comparison with similar measurement procedures can reveal uncertainty sources that might otherwise be overlooked. When in doubt about whether a source is significant, including it with a conservative estimate is preferable to ignoring it.

Estimating Magnitudes

Manufacturer specifications provide the primary source for estimating many uncertainty contributions. Specifications should be interpreted carefully, understanding whether they represent maximum limits, typical values, or statistical distributions. Maximum specifications are often treated as the half-width a of a rectangular distribution, whose standard uncertainty is a/√3, while typical values may suggest normal distributions. A calibration certificate's expanded uncertainty is converted by dividing by its stated coverage factor, u = U/k. Treated this way, the ±0.5 LSB limit of an ideal quantizer gives 0.5/√3 ≈ 0.29 LSB, the same result as the q/√12 of the quantization noise model.

When specifications are unavailable or insufficient, engineering judgment based on similar equipment, physical principles, or conservative assumptions provides estimates. The basis for each estimate should be documented in the uncertainty budget to support later review and refinement.

Calibration data can reduce uncertainty by providing measured values rather than relying on specifications. A calibrated offset correction has uncertainty determined by the calibration process rather than the much larger manufacturer's offset specification. The calibration uncertainty and any drift since calibration then become the relevant contributions.

Uncertainty Reduction Strategies

Understanding uncertainty sources suggests strategies for improving measurement quality. Increasing the number of averaged measurements reduces the random contributions, which are usually evaluated by Type A methods, but does not affect systematic contributions; for quantization it helps only when noise or dither makes the error random. Calibration can reduce systematic uncertainties by replacing specifications with measured values. Environmental control reduces temperature-dependent contributions.

Selecting appropriate instruments for the measurement task ensures that instrument uncertainties do not dominate unnecessarily. Using a 16-bit converter for measurements requiring only 8-bit accuracy adds cost without benefit, while an 8-bit converter cannot deliver 12-bit accuracy: averaging and dither can refine its resolution, but not correct its nonlinearity. Matching instrument capability to measurement requirements optimizes both performance and cost.

Differential and ratiometric techniques exploit correlation to achieve lower uncertainty than absolute measurements. When comparing two signals or measuring a ratio, common-mode errors cancel, potentially achieving much better uncertainty than the individual measurement specifications would suggest.

Guide to the Expression of Uncertainty in Measurement

The Guide to the Expression of Uncertainty in Measurement (GUM), first published in 1993 and now maintained by the Joint Committee for Guides in Metrology (JCGM) as JCGM 100:2008, is the internationally accepted reference for uncertainty evaluation. The GUM defines terminology, describes evaluation methods for Type A and Type B contributions, explains uncertainty propagation, and specifies how to report results. Following GUM methodology ensures that uncertainty claims from different sources can be meaningfully compared.

The GUM supplements and amendments address specific applications and advanced topics. GUM Supplement 1 (JCGM 101:2008) describes Monte Carlo methods for uncertainty propagation when analytical methods are inadequate. GUM Supplement 2 (JCGM 102:2011) extends the framework to any number of output quantities, which may be correlated. Amendment 1 to JCGM 100:2008, issued in 2026, addresses nonlinearity in measurement models. These documents provide tools for handling complex measurement situations beyond the basic GUM framework.

Industry-Specific Guidelines

Various industries have developed specific guidelines for uncertainty evaluation in their measurement domains. For digital measurement, IEEE Std 1241, for analog-to-digital converters, and IEEE Std 1057, for digitizing waveform recorders and oscilloscopes, define the test methods behind figures such as SINAD, ENOB, and aperture uncertainty, so that a specification can be traced to a stated procedure. Calibration laboratories accredited to ISO/IEC 17025 must evaluate the measurement uncertainty of their results, and accreditation bodies publish guidance on doing so, such as EA-4/02 from the European co-operation for Accreditation.

These guidelines often provide worked examples and typical uncertainty contributions for common measurements. While not replacing careful analysis of specific measurement situations, they offer valuable starting points and benchmarks for comparison.

Documentation and Reporting

Complete uncertainty documentation includes identification of all significant sources, the evaluation method and data for each, sensitivity coefficients, correlation treatments, and the combination method used. This documentation supports review, enables refinement as additional information becomes available, and provides traceability for the uncertainty claim.

A reported result should state the measured value and either its combined standard uncertainty or its expanded uncertainty, and an expanded uncertainty needs its coverage factor and approximate coverage probability beside it. The form "x = (1.234 ± 0.005) V, where the expanded uncertainty U = 0.005 V is based on a coverage factor k = 2, giving a coverage probability of approximately 95 percent" clearly communicates both the result and its quality. The uncertainty is normally quoted to no more than two significant figures, and the result is rounded to the same decimal place.