Traceable frequency measurements with counters

Received signals from Global Navigation Satellite Systems (GNSS) are nowadays widely used by industry laboratories for ensuring metrological traceability for their respective range of calibration services in the field of time and frequency. Usually, a local frequency standard is steered by continuous GNSS signal reception providing at its output stable and accurate reference signals for the laboratory measurement equipment, in general for synthesizers and counters. Reception of GNSS signals is surely an adequate and practical tool for the purpose, however further steps are needed to establish traceability in a strict metrological sense. Based on already available guidelines and publications, this paper is a contribution to the discussion how metrological traceability to internationally accepted standards can be established in a calibration laboratory. We restrict the discussion to equipment in common use which may not necessarily be of the highest sophistication. In this spirit, we develop a detailed scheme for an uncertainty budget comprising all links of the traceability chain from the device under test to the SI second, the scale-unit of Coordinated Universal Time. Then we go through and apply this scheme step by step to a demonstration setup for frequency measurements with a counter with varying operational parameters. In this framework, a novel approach to distinguish between components of statistical measurement uncertainty is introduced. Furthermore, the limiting uncertainty contributions are discussed and based on a suitable set of parameters an expression for the best measurement capability is given. With this scheme at hand a user may develop an uncertainty budget adapted to his own setup, especially if acceptance from a national accreditation body is sought.

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Introduction
The reception of signals from Global Navigation satellite Systems (GNSS) offers the possibility to control oscillators almost wherever users need a stable and accurate frequency and time reference for their purpose.One example is the use of GNSS DOs as frequency references which enables calibration services in the field of time and frequency to be offered by calibration laboratories.Here the reception of GNSS signals facilitates a continuous link between an operator's own reference and an external standard, thereby ensuring continuous monitoring and control of the laboratory standard.
There is a whole bunch of products available that are technically adapted to this specific application.Reception of GNSS signals is surely an adequate and practical tool for the purpose, further steps are however needed to establish traceability in a strict metrological sense.According to section 2.41 of the Vocabulaire international de métrologie (VIM) [1] metrological traceability means a 'property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty'.An operator thus needs to work out and write down a budget regarding the uncertainty contributions from every single link of the traceability chain from his own setup to the primary frequency and time reference if traceability is required e.g. by a national accreditation body or by other institutions.Examples are industry calibration laboratories deserving an accreditation according to the standard DIN EN ISO/IEC 17025 [2] (for a commented issue in German language see [3]).
Frequency measurement with counters is a universal tool for a wide range of applications.Accordingly, there are many studies on the properties of counters as well as the corresponding averaging methods and their effect on the results.Basic features of the construction of and measurements with frequency counters can be found in the relevant literature, which includes textbooks [4,5], selected manufacturers' manuals which clearly collect the basic properties [6][7][8][9][10][11], and recent technical publications (see e.g.[12] or references given in the next lines).During the last decades, in various papers statistical properties of frequency counters were analyzed and methods of evaluation developed.For example, Rubiola [13] discusses the operation of a frequency counter, highlighting the use of interpolation and averaging for achieving high resolution in measuring input frequency over time.Dawkins et al [14] investigated the discrepancy in the measurement of the Allan variance, a widely used time-domain measure of frequency stability, when employing modern high-resolution frequency counters.They analyze the operation of these counters in the Fourier domain, revealing significant distortions in reported results and suggesting strategies to either avoid or correct for these inaccuracies.Modern processes use the time-to-digital concept as described in [15] and references therein.It introduces a frequency counter, i.e. a frequency-todigital converter-based on the linear regression algorithm on time stamps.Other processes using the time-to-digital concept are e.g.[16].The achievable frequency range has also been drastically extended, from the THz region [17] to the optical domain [18].Reference frequencies (typically 5 or 10 MHz) are often traced via GNSS-DOs to international agreed standards.A typical configuration used is the common view architecture, in which traceability can be ensured either by the user or by the national metrology institute.See e.g.discussions and descriptions in [19,20].
This work is a contribution to the discussion how metrological traceability of frequency measurements to internationally accepted standards can be established.We understand this work as a concretization and completion of already published contributions of both other and our own work.Significant documentation on traceability to US standards was published earlier [21,22].Those form together with previous papers of this work's authors [23][24][25] the foundation of this report.Based on already available guidelines [26,27] we illustrate in the next section a full scheme for traceable frequency measurements with counters in a calibration laboratory.We choose the GNSS-DO setup, which is often used in industry laboratories, because of the above-mentioned advantage.Other configurations are possible, e.g. a scheduled periodic calibration of the local standard, but are not discussed in this work.Then we develop and introduce a detailed scheme for an uncertainty budget comprising all links of the traceability chain from the DUT to the SI second, available (with very low uncertainty) as the scale unit of Coordinated Universal Time (UTC) [28].Aspects regarding the measurement equipment (particularly counter and synthesizer) are discussed in section 3. We go through these and apply this scheme to a demonstration setup for frequency measurements with a counter with varying operational parameters.In this framework, a novel approach to distinguish between different components of statistical measurement uncertainty is introduced.Aspects regarding the operation of the local frequency reference including the chain links for monitoring the relation to UTC are summarized in section 4. In section 5 the limiting uncertainty contributions are summarized and based on a suitable set of parameters an expression for the best measurement capability is given.Finally, in section 6 a summary and outlook are presented.With this scheme at hand a user may develop an uncertainty budget adapted to his own setup.

Architecture for traceable frequency measurements
The traceability from frequency measurement devices to units prescribed by national laws and ultimately to UTC comprises (hardware) components as illustrated in figure 1.In this scheme the measurement devices (frequency counter and synthesizer) are fed with suitable distribution amplifiers by an external reference frequency provided by a so-called DO which usually comprises a temperature stabilized crystal oscillator or a Rb atomic frequency standard.This local reference is controlled by the reception of signals broadcast by remote systems delivering accurate frequency and time services.Nowadays the most used means is the GPS.Alternatives are other GNSS, e.g. the European Galileo Satellite System [29] or the traditional (and regional) low frequency transmitters, e.g. the German DCF77 [30].Users of such signals can rely on the fact that these time signals are monitored continuously by national authorities, which are National Metrology Institutes, like the PTB in Germany, or so-called Designated Institutes in some countries.At PTB the timeof-arrival of the received timing signals from GPS, Galileo and DCF77 are measured with respect to the local representation of UTC, which is called UTC(PTB).The results are published and thus available to users in the weekly issued PTB TSB.Access can be provided via an SFTP server inquiring user ID and password from PTB by sending an email to time@ptb.de.PTB itself ensures the traceability of its reference time scale UTC(PTB) by participation in the CIPM MRA Key Comparison CCTF-K001.UTC.In the monthly publication Circular T [31] of the BIPM, among other information, computed time deviations between the timescales of contributing institutes, such as UTC(PTB), and UTC are provided.Relevant data published in the Circular T are included in PTB's TSB afterwards.
On the operator side, primarily inevitable devices are a frequency counter and a synthesizer.They facilitate both measuring frequency signals from and provision of frequency signals to devices under test (DUT), respectively.Both are usually connected to a single external frequency reference.Such configuration allows a full characterization and testing of the equipment in use (see field (a) in figure 1).And exactly these two steps are mandatory pieces for establishing and maintaining traceability.With both devices the operator develops the uncertainty budget for practical and applicable 'standard' parameter configurations, which are in particular frequency signal form (we only deal with sine-wave signals throughout this work), signal amplitude, and frequency range.The latter is defined by the overlapping frequency ranges of counter and synthesizer.It is clear that the operator cannot make any commitment about calibration capabilities in frequency ranges which are not accessible during the uncertainty evaluation and the obligatory intermediate inspections.
The second part is the operation of the local frequency reference (see field (b) in figure 1), which is a Rb atomic clock in the example presented in this paper.In general, it is controlled or disciplined by external means as described above.The operator needs to take measures to achieve appropriate estimations about the uncertainty contributions.
The third constituent is in external responsibility, the monitoring of the external frequency and time reference with respect to international accepted standards (see field (c) in figure 1).In this work, PTB's reference time scale UTC(PTB) plays this part.As already described in previous publications [23,25], monitoring results are published in the PTB TSB and the operator should collect them for his own analysis of the contributing uncertainties.All three parts together form the basis for the uncertainty budget, which is developed step by step in the subsequent sections.
As briefly mentioned before, the architecture presented here is not the only solution for establishing traceability.Alternatives are for example to relinquish the use of a DO and to operate a free running frequency reference which is calibrated periodically by an accredited laboratory offering such service.The simplest way along this line would be to work with a fully calibrated counter using its internal oscillator as reference.However, this is only an alternative if the requirements for the best measurement capabilities are not very stringent as the user would have to rely on the manufacturer's specification of oscillator aging and drift in time.Another disadvantage is that continuous monitoring is being given up, and a potential malfunction may be overseen and could be detected only during the next scheduled recalibration of the device.

Uncertainty contributions from the counter/synthesizer part
The characteristics of the chosen frequency counter in interaction with the properties of the input frequency signal are usually decisive for the smallest achievable uncertainty, provided the frequency standard used as a reference is sufficiently good.In the following sections, the important influencing factors and their impact on measurement uncertainties are developed and experimentally determined using a counter and a synthesizer widely used in metrological applications.At the end of this chapter there will be an exemplary estimation of the smallest uncertainty in a common frequency range for a customary setup.This estimation can then be used as a starting point for extensions of the range or for alternative measuring devices.

Statistics u A
The statistical uncertainty contributions classified as u A depend mainly but not only on the characteristics of the frequency counter used.They are also determined to some extent by the quality of the input signal.Figure 2 depicts the main components of a frequency counter and at the bottom the relevant signal and noise characteristics are indicated.In legacy application notes [6] the functioning is explained as follows: 'The input signal is initially conditioned to a form that is compatible with the internal circuitry of the counter.The conditioned signal appearing at the door of the main gate is a pulse train where each pulse corresponds to one cycle or event of The introduction of reciprocal counters transferred the simple counting of events into a time interval measurement.At a given gate time (GT), the reciprocal frequency counter determines the corresponding time interval from a number of zero crossings (events) of the frequency to be measured.The reciprocal value is the measured frequency.In this paper, we consider only the reciprocal frequency counters, which are mainly used today.
3.1.1.Proposition.In this section, the three relevant contributions to the statistical uncertainty are derived.They are the threshold timing jitter T th , the single shot timing jitter T ss and the least significant digit of the frequency measurement result.The commonly most significant contribution to statistical uncertainty is the threshold timing jitter.A real signal, whose frequency should be determined, is always affected by noise.When applying a simple vector model, such a signal can be separated into a pure signal portion s and a pure noise portion n (cf figure 3).The noise part causes a random variation (jitter) of the phase ϕ (e.g. the positive zero crossing of a sinusoidal amplitude trace).In the presence of noise its exact position is thus indeterminable.
Between the maximum phase angle ∆φ of the noisy signal and its signal to noise ratio the following relation can be written: The ∆ accents the fact that ∆φ is a phase angle variation in relation to an ideal noise free signal.n and s are noise and signal amplitudes, respectively.For s/n > 6 dB corresponding n/s < 0.5 one can write approximately The phase angle variation is equal to a variation in time of the zero crossing with respect to the period of the frequency signal.The following is valid: where ∆t is the phase time variation and T is the period of a signal.With the frequency f = 1/T this results in: A phase time variation ∆t in a measurement interval GT (gate time) is equal to a relative frequency deviation, and it follows that The next step is to move from a snapshot to an integrated estimate of the uncertainty.Usually, frequency counters refer to zero crossings of periodic signals with symmetrical positive and negative amplitudes.For sinusoidal signals, which is the only type of signals we discuss in this paper, the slew rate of their amplitude becomes maximum at the zero crossing.That is, at this point, a pure signal with a peak amplitude s, which is superimposed on average by the rms (root mean square) amplitude of the noise signal n/ √ 2. It means, that s and n are still considered as peak amplitudes for the moment.An additional factor √ 2 results from the trigger or threshold uncertainty at the beginning and end of the measurement and cancels the √ 2 just introduced.This results in a threshold uncertainty due to the noise at the trigger point: The quantities s and n are now optionally amplitudes or could be considered alternatively and typically both together as rms amplitudes and we replace s by the input signal rms-Voltage V rms and n by the quadratic sum of relevant noise voltages V sn is the rms noise voltage of the input signal and V in is the noise of the input amplifier.We get: The slew rate at the positive zero crossing, which is usually the threshold or trigger point, of a sinusoidal signal is: We thus obtain: and the threshold timing jitter contained in it is: It describes the input signal dependent random trigger variation due to existent noise sources and/or interference.

Single shot timing jitter.
The so called 'single shot timing jitter' T ss [9] is the ultimate lower limit of measurement noise and independent from input signal characteristics.The 'timing resolution between a single pair of start/stop edges' virtually characterizes the same attribute if multiplied by √ 2. It is not always clearly explained which property is meant in a specific case, and one should be careful to interpret manufactures specification.The related uncertainty contribution is inversely proportional to the gate time: The least significant digit defines the resolution of the frequency counting measurement result as displayed on the front panel of the device or recorded in data files.Its uncertainty contribution is given by: N is the effective number of digits.Equation ( 12) consists of the smallest increment divided by 2 and a factor of 1/√3, which is based on the assumption that the true values which are rounded to the least significant digit are equally distributed.
In fact, the underlying time interval measurement of the frequency counter has its own limitation by the quantization of the 'internal' least significant digit.Its uncertainty is not readily available but can be estimated from the corresponding specification, which is 4 ps for time interval measurements in one example [8].It is usually assumed being small enough to be neglected but is mentioned here for completeness.
3.1.1.4.u A summary.All three contributions to the statistical uncertainty can be quadratically summarized: We substitute the three components by equations ( 9), ( 11) and (12), respectively.This results in the following: In the next section we describe how the individual quantities in ( 14) can be determined making tailored measurements.

Demonstration.
3.1.2.1.Determination of V in and Tss.We determine V in and T ss of a widely used counter model SR620 from Stanford Research Systems [8] with a simple setup, which is depicted in figure 4 below.A 10 MHz signal originating from UTC(PTB) is provided with a dedicated FDA and feeds the counter as an external time base.A 50 Ω input impedance was selected and the trigger level (threshold) was adjusted to 0 V for counting at the highest slew rate.The signal amplitude to be measured is reduced after an initial measurement by a series of passive attenuation devices (3 dB, 6 dB, … 21 dB) to avoid any noise contribution by further active components.
What can we measure with this setup?We start with equation ( 14) and apply two simplifications: First, we measure a sinusoidal signal-as used throughout this work-which is directly derived from UTC(PTB).This is a time scale based on a state-of-the-art active hydrogen maser, exhibiting very low phase noise.Compared with the internal noise voltage the signal noise can be neglected.Also, the uncertainty attached to  the least significant digit, which is in this case 2.9 × 10 −12 is about a factor of 2-3 smaller than the observed lowest standard deviation at the highest signal level.Equation ( 14) becomes: We vary the slew rate at threshold SR th by inserting the series of attenuators and record 10 000 samples of frequency values.Mean and standard deviations (SD) relative to the nominal frequency f 0 = 10 MHz are computed and plotted versus the respective slew rate of the input signal as squares in figure 5.The standard deviations represent u A as given in (15).For easier interpretation of the plot a few data points are labeled with the corresponding rms signal amplitudes.
From a fit through the data (red line) we get T ss = 6.2 ± 0.5 ps and V in = 233 ± 6 µV.Compared with the specifications (T ss = 25 ps/ √ 2 ≈ 18 ps and V in = 350 µV [8]) it can be stated that for the device in hands the effective properties are significantly better than specified.Equation (15) with specified parameter values results in an upper limit (dash dotted line) with some margin.

Synthesizer's contribution.
When seeking after the best measurement capabilities of a frequency counter setup the part of the signal source cannot be neglected.The impact of an available and representative synthesizer must be included in the uncertainty evaluation, if the measurement scope comprises a full and continuous range of frequencies, in contrast to selected standard frequencies (e.g. 5 MHz, 10 MHz), when no synthesizer is required for the characterization.The synthesizer is a representative for a so called 'device under test' (DUT) and should be selected in a way which causes minimal deterioration of the counter noise characteristics.In particular, the averaged noise voltage V sn in the respective bandwidth of the counter input should be as small as possible.We use a Marconi signal generator model 2024 in the setup as depicted in figure 6.The external time base of the synthesizer is connected to the FDA in parallel to the counter.So, both are fed with the same high quality reference signal.Then the output of the synthesizer is used directly as the input signal for the counter.
To get a clear picture of the frequency measurement noise we vary the GT for two setups: first by using the direct FDA output (figure 4) and second with the synthesizer included (figure 6).The sinusoidal signals all have a frequency of 10 MHz and a rms amplitude of 1 V in 50 Ω input impedance.The number of recorded samples decreases depending on the GT from 10 000 (GT = 1 ms) to 100 (GT = 500 s).From these samples mean value and standard deviation of the single measurements are computed.The latter is a measure of the statistical uncertainty and depicted in figure 7. The 'direct' measurements (squares) are underlaid with a plot of equation ( 14) using the parameters (T ss = 6.2 ps, V in = 233 µV, N = 14, and V sn = 0 V).For the direct measurement the signal voltage noise is neglected.The impact of the u lsd for the largest number of displayed digits (N = 14, dashed red line) has a clear impact on the longest gate time (500 s).Note, that for shorter gate times the used counter subsequently reduces the number of displayed digits every decade, with the effect that there is a saw-tooth like pattern appearing just perceptible in the trace of the standard deviation results.
We keep the above listed parameters T ss , V in , and N fixed and use equation ( 14) (with the term representing u lsd ) to fit the 'synthesizer' measurements with V sn as the only free parameter.The result is a rms signal noise voltage of 1.6 mV and for almost the complete range of gate times the standard deviation follows 2.6 × 10 −11 s/GT.Note that now the impact of the least significant digit is covered by the noise introduced by the synthesizer.

Systematics u B
There is not much to be discussed about the systematic uncertainty contributions.It can be considered as to be based on a fundamental internal delay asymmetry T da inside the electronic circuit opening and closing the gate: The counter manufacturer's specifications for T da are 100 ps as typical value and 350 ps as the maximum [8].In figure 8 moduli (absolute values) of averaged relative frequencies are shown for both cases: The results with the 'direct' signals (black open circles) and the 'synthesizer' signals (black crosses) are similar, only deviating from a 1/GT line at very short gate times ⩽ 1 ms.Compared with the specifications it is obvious that the selected unit surpasses the typical uncertainty (blue dotted line) but complies with the maximum (blue straight line).A fit through the direct measurements, which seems to be a bit higher than the synthesizer measurements give T da = 227 ps.We analyzed two other counter units of the same model and found that the specified typical value is hardly or not met, but the maximum value is met for both units.Our conclusion is that the typical value is too optimistic, but the maximum value could be used as a good measure for the systematic uncertainty.The discussion in the next section supports this decision.
As an addition, the statistical uncertainties as depicted in figure 7 and discussed above are also shown here (gray symbols and lines).It is obvious, that in this setting (f 0 = 10 MHz) the systematic uncertainty is dominant, and the statistical part can be practically neglected.We will analyze in the next section, how this relation changes when the frequency to be measured is changed over a specific range.

How do u A and u B vary in dependence of signal frequency?
So far, we only used a nominal frequency f 0 = 10 MHz to analyze the various uncertainty components affecting a frequency measurement with a dedicated counter.If we consider relative frequencies, the systematic uncertainty is believed to be independent of the measured frequency and the statistical uncertainty follows a 1/f 0 line and should show clearly such a profile.We use the setup of figure 6 with the standard signal (V rms = 1 V, 50 Ω input impedance) and vary the frequency over the available overlapping frequency range of both the synthesizer and the counter (in its standard measurement mode) from 9 kHz to 300 MHz.This will be exactly the frequency range for which the best measurement capability could be reported by a calibration laboratory using a similar equipment set-up.Whereas the counter is capable to measure lower frequencies, the smallest output frequency of the synthesizer defines the lower limit and the highest frequency of the counter in its standard mode limits the upper frequency in this examination.
In figure 9 the uncertainty contributions of the statistical and systematic uncertainty contributions in dependence of the nominal frequency to be measured are depicted.Four representative gate times were chosen, and results are displayed in four separate plots to allow a distinct judgment of the contributions and to characterize the specific limitations at different nominal frequencies and gate times.The four plots illustrate the limits at the shortest gate time providing meaningful results for our purpose (GT = 1 ms), performance at practical gate times for many routine tasks (1 s and 100 s), and specific results at the longest practically applicable gate time for the determination of the best measurement capability (500 s).The following discussion concerns all four graphs if not otherwise specified.
The relative standard deviations of the measured frequency values are depicted as red squares.They follow and agree with the expected lines using respective parameters as derived in section 3.1 over a wide frequency range (dotted black line).Only at the highest frequencies (f 0 ⩾ 100 MHz) there is a clear deviation from the expectation visible (detail 'a').However, the corresponding systematic uncertainty (blue dots) is significant higher in this regime and the excessive standard deviation (deviation from the expected trace) can be neglected when calculating the overall combined uncertainty.As discussed above, the so-called 'typical value' for the systematic uncertainty may be too optimistic for a realistic estimation of the uncertainty.In fact, taking the maximum value (u B = 350 ps/GT) we get reasonable agreement with the measured values.They are usually below that limit but touch the maximum value for the highest frequencies 100 MHz and 300 MHz.The significant deviation of the measurement results at low frequencies (detail 'b') are fortunately completely covered by the increase of u A in direction to the lower frequencies when calculating the combined uncertainty u.
Statistical and systematic uncertainty are usually combined as follows: shown as black solid line.It is clearly visible that u A dominates measurement results at the low frequency range up to about 1 MHz and u B dominates the uncertainty at higher frequencies.This is the same for all gate times.It is also obvious that longer gate times generally lead to lower uncertainty for both statistical and systematic parts.In fact, both are proportional to 1/GT.However, standard work practice collecting data will also limit the gate time.We consider GT = 500 s as the maximum gate time, because in a measurement task, one would need a sufficient number of measurements for a meaningful statistical analysis.A minimum number of 100 samples would already lead to an overall measurement time of more than half a day for a single result.In this case the uncertainty reads This expression is taken as the contribution of the frequency counter/synthesizer setup to the best measurement capability.Uncertainty of frequency measurements dependent on the nominal frequency for selected gate times from 1 ms to 500 s.The number of single measurements for each data point ranges from 10 000 down to 100, respectively.More details are provided in the text.
The contributions of the frequency reference are discussed in the next chapter.

Uncertainty contributions from DO operation and common view
In this section the uncertainty contributions relevant for the provision of the reference frequency are summarized and discussed for the case of PTB as a representation of a metrology institute operating a physical realization of UTC and for a setup which can be found in many commercial laboratories offering calibration services to industry customers.As sketched in the introduction and section 2, in addition to the measurement setup the traceability chain is formed with two more links.It is first the operation of a local frequency reference, which is in case of a calibration laboratory often a GNSS DO.Operation means that the necessary operational parameters are monitored and recorded including the steering status and parameters of the disciplinary function (often a phase lock loop).Besides the case discussed here, it is also possible to operate a standalone oscillator as a frequency reference which is calibrated with a suitable defined interval (e.g.once per year).The second link is established by the continuous monitoring and recording of GNSS signals done at metrology institutes.They can thus provide computed time scale differences, e.g.UTC(PTB)-GPS time and the link between UTC(PTB) and UTC in our case.These links are discussed in the subsequent chapters.

Operation of a DO
In our set-up we use a Rubidium atomic clock combined with and steered by a single frequency, multi-channel GPS receiver connected to an active antenna on the roof.We analyze the frequency instability in dependence of the averaging time by computing the overlapping Allan standard deviation of secondly measured phases between 10-MHz signals of UTC(PTB) and the DO. 10 d of data were used.In figure 10 the frequency instability in dependence of the averaging time is depicted (data were previously shown in [23,32]).Due to the steering process a local maximum at τ ≈ 1000 s exists.This has consequences for the applicable statistical uncertainty.
In dependence of the gate time chosen for the frequency counter measurement one needs to take into account the respective frequency instability at the same averaging time.In figure 10 three exemplary gate/averaging times are selected.At τ = 1 s the respective frequency instability of 7.4 × 10 −12 is directly the statistical uncertainty.In the 'region' of the valley between τ ≈ 10 s and τ ≈ 1000 s the Allan deviation does not correctly describe the statistical uncertainty because the low instability is inaccessible due to the frequency variations induced by the steering process.The minimum uncertainty in this region is at the level of the maximum which is u A = 2.1 × 10 −12 .The absolute mean frequency offset of the 10 d of data was found as 3.5 × 10 −14 [32] and has no significant impact on the DO's combined uncertainty which is for a gate time GT = 500 s practically equal to the u DO = 2.1 × 10 −12 .The operator needs to take measures to achieve appropriate estimations about the uncertainty contributions.It means that sometimes the operator has no other choice and needs to rely on manufacturers specifications.Nevertheless, he shall always strive for an accurate characterization of the DO in use.Certainly, the correct functioning of the steering mechanism shall be monitored.Thus, only devices providing access to respective operational parameters (e.g.oscillator control voltage) are suitable for the purpose.

Monitoring UTC(PTB)-GPS time
PTB monitors GPS time with suitable GNSS receivers with respect to its own time scale UTC(PTB).From the observations, daily averages of UTC(PTB)-GPS time are computed and reported in the PTB TSB.Its current format is described in [25] (for availability information see section 2).How the data are prepared for the TSB is described in [24] and here summarized as follows.The used GNSS receiver provides CGGTTS formatted data which are averaged after a suitable MAD filter removed outliers.The results for the year 2020 are depicted in the left graph of figure 11.The uncertainty in the recorded time offset is dictated by the delay determination for GPS signals of the respective receiver.Values were provided by BIPM [33] and the uncertainty amounts to 1.5 ns.Daily frequency values can simply calculated from time differences of consecutive days as shown in the right graph of figure 11.The time uncertainty is irrelevant in this case.
We consider the rms value as the standard uncertainty u CV for a 'perfect syntonization' between UTC(PTB) and GPS time.For calendar year 2020, frequency mean and standard deviation (SD) are −0.24× 10 −15 and −7.273 × 10 −15 , respectively.The rms value of all data for 2020 is u CV = −7.277× 10 −15 .The uncertainties (statistical and systematic) which are stated in the TSB do not apply here because the statistical part directly translates into the distribution of the reported values and is thus already a part of the rms distribution.The systematic is considered as varying only very slightly over long periods of time and is thus considered to be negligible for the calculation of the daily frequency values.
In table 1 the standard uncertainty u CV of monitoring GPS time with respect to UTC(PTB) for the last seven years is summarized.There has been a remarkable improvement of the uncertainty after 2016.This is mostly due to an apparent noise reduction of the GNSS monitoring data at PTB.

UTC(PTB) and UTC
The differences between UTC and its physical realizations UTC(k) of the contributing laboratories are reported for every fifth day in the BIPM Circular T [31].Hence, values UTC − UTC(PTB) are provided and available for each MJD ending with a last digit '4' or '9'.The values are accompanied by uncertainty estimations for u A and u B for a whole computation month.In our uncertainty estimation, we do not consider these values as discussed above.
The published data of calendar year 2020 are depicted in the left graph of figure 12. UTC(PTB) 'meanders' very smoothly around UTC, except a small step by the end of the year.This was due to a rather 'offside' but valid value used for steering the frequency of an active hydrogen maser as physical source of UTC(PTB) signals [34].It happened that the value was based on the ensemble of PTB's thermal-beam cesium clocks when no fountain data were available on one day of the year.Nevertheless, the deviation of UTC(PTB) never exceeded 4 ns during the whole year.In the right graph of figure 12 the corresponding frequency values are depicted.The step mentioned above translates in a single frequency value outlying but valid either.In 2020, the mean frequency UTC(PTB) deviation from UTC was −1.56 × 10 −16 with a SD = 6.13 × 10 −16 , which gives a rms u k = 6.33 × 10 −16 .
The results of the last seven years are given in table 2. The combined uncertainty u k was always below 9 × 10 −16 and the steadiness of this value represents the firm quality of UTC(PTB).
UTC itself is a compromise between an ultra-stable smooth time scale and the best available representation of the SI second.In its realization such compromise manifested itself in a significant deviation (called 'd') of the frequency difference between the ensemble of primary clocks and secondary representations of the second and UTC which prevailed for several months (see [31]).The smoothness requirement does not allow a rapid change of the UTC frequency.Nevertheless, in the long term the frequency of UTC is expected to wind closely around an imaginary ideal timescale with the exact unit '1 s' as its scale interval.In every Circular T the difference d is reported for the respective month.The quadratic sum of the mean and SD of all 2020 values of d is 4.8 × 10 −16 .To each published value of d an uncertainty is stated.The uncertainty varies only slightly over a calendar year and its mean is taken as the uncertainty for one year.The yearly averages of d values of the last five years together with the mean of their uncertainty are shown in figure 13 as black squares with error bars, respectively.The long term 'winding' of d around zero is apparent reflecting the very smooth steering of UTC.The standard deviation of the monthly data around the yearly mean (red error bars) does not blur this shape much.
The combined uncertainty of d for 2020 is u CT = 5.1 × 10 −16 Note that the uncertainty of d of u A and u B usual, but (as discussed above) only the systematic part would be needed to be considered.Note, the uncertainty of d is dependent on various factors (e.g. the number and quality of the contributing primary and secondary frequency standards) but the overall level is so low that a more detailed analysis is not relevant in the context of this work.In table 3 the results of the last seven years are summarized.It can be stated that the uncertainty of d (u d ) is the smallest contribution to the combined uncertainty.Long term steering of UTC and variations of d within each calendar year seem to be more pronounced.However, all values are in the range of 10 −16 and thus practically negligible for our purpose.

Best measurement capability
In this chapter the achieved results are summarized and merged together into a single expression stating the practically lowest achievable uncertainty with the specific setup used in this work.This is commonly called the best measurement capability.In our setup the requirements to achieve this uncertainty are the provision of a sinusoidal signals in the frequency range f 0 = 9 kHz-300 MHz with a rms amplitude V rms = 1 V (corresponding to a peak-to-peak amplitude of about V pp ≈ 3 V) at an input impedance of 50 Ω.The chosen gate is GT = 500 s considered as the longest practically applicable gate time.In table 4 the uncertainty contributions are summarized.Two cases are distinguished.First the case of a national metrology institute as PTB is.In this case the operation of a DO as a reference is not necessary and consequently related uncertainties do not apply.The second case is a calibration laboratory operating a DO.Note, that the uncertainty of the DO operation is not negligible and must be taken into account.In the notation of relative frequencies most of the contributions are independent of the frequency and only one is frequency dependent, which is the type A uncertainty of the frequency counter.
The single uncertainty contributions are illustrated in figure 14 for both cases mentioned above.In the left graph the case of PTB is shown and it is obvious that the uncertainty is solely limited by the frequency counter and synthesizer chosen.In the case of a calibration lab the uncertainty of the DO in use becomes dominant at higher frequencies.In consequence a careful characterization of the DO is mandatory.The black line is the quadratic sum of all components reflecting the overall uncertainty and best measurement capability.
Derived from the values in table 4 the combined standard uncertainty in the case of a calibration laboratory is: The equivalent expression in absolute frequency values is: This equation has the typical form which occurs in spread sheet fields to report the best measurement capabilities of calibration labs.Especially in dedicated annexes of accreditation certifications issued by e.g. the German accreditation body DAkkS such expressions are used.It is somewhat unwieldy, and an easier-to-read version could be the arithmetically equivalent expression as the magnitude of a row vector: We neglected aspects, which are part of the Guide to the Uncertainty of Measurement GUM [35], especially the distribution function of the data and the degrees of freedom.All frequency measurements used in this work are assumed to be dominated by white frequency noise at the relevant averaging times.This is not completely true, as the bump in the frequency instability plot of the DO (cf figure 10) suggests, but the actual distribution of measurement data is not easy to determine.In consequence we do not multiply the standard Table 3. Summary of the frequency difference between the ensemble of primary and secondary representations of the second and UTC.Yearly mean values, standard deviations of the monthly values around the mean, yearly averages of the uncertainty as stated in the Circular T, and the quadratic sum of all are stated for the past seven years.

Year
Mean/10    3 a Not applicable in an institute maintaining a physical realization of UTC as PTB.uncertainty results by a factor 1/√N but keep the corresponding weighting factor 1. In addition, we commonly used at least 100 samples, much more in many cases, and the total degree of freedom is surely >50.This justifies the application of a coverage factor k = 2.However, no result presented in this paper has been multiplied by k.

Summary/outlook
In this paper we derived a complete uncertainty budget for a traceable frequency measurement with a measurement set-up that is often employed in industrial calibration laboratories.This uncertainty evaluation could be used as a guide to inspect the laboratories' own setup and procedures to provide traceable calibrations to their customers.One focus was on the synthesizer/counter part, dominating the uncertainty especially at lower frequencies.Special care must be taken to choose the suitable frequency standard for the purpose, because its instability may limit the uncertainty especially at higher frequencies.The aim of this paper was to establish a coherent and consistently structured uncertainty budget for a standard procedure.We have not considered so called enhanced resolution counters, which may improve the frequency estimation due to more sophisticated statistical processes [13,14], and leave it for future considerations.Further work could focus on extending the frequency range in the scope of calibration, or on the discussion of procedures to measure specific reference frequencies (e.g. 5 MHz, 10 MHz or 100 MHz).Also, traceable time interval measurements would be of some interest, as such measurements gain importance in highly accurate velocity and acceleration measurements.

Figure 1 .
Figure 1.Scheme for traceable frequency measurements or signal generation with instruments connected to an external frequency reference of a disciplined oscillator (DO) which receives broadcast time and frequency signals.In this example the traceability is established to UTC via PTB.

Figure 2 .
Figure 2. Basic block diagram of a frequency counter with the relevant signal and noise parameters.These are the slew rate of the signal at the threshold voltage SR th , the noise voltage on the signal to be measured Vsn, the noise voltage of the counter input V in and the single shot timing jitter Tss.

Figure 3 .
Figure 3. Simple vector scheme to separate pure signal s and pure noise n portion of a noisy signal.∆φ is the phase angle variation between actual and pure signals.

Figure 4 .
Figure 4. Setup to measure the effective internal noise voltage V in and single shot timing jitter Tss of the frequency counter.

Figure 5 .
Figure 5.Standard deviation of relative frequency values (f 0 = 10 MHz) recorded in dependence of the slew rate at threshold SR th of the input signal.

Figure 6 .
Figure 6.Setup of synthesizer/counter combination to evaluate the best measurement capability.

Figure 7 .
Figure 7. Standard deviation of the relative frequency versus gate time.

Figure 8 .
Figure 8. Moduli (absolute values) of measured relative frequency values (10 MHz) compared with systematic uncertainty specifications and statistical uncertainties.

Figure 9 .
Figure 9.Uncertainty of frequency measurements dependent on the nominal frequency for selected gate times from 1 ms to 500 s.The number of single measurements for each data point ranges from 10 000 down to 100, respectively.More details are provided in the text.

Figure 10 .
Figure 10.Frequency instability (overlapping Allan standard deviation) of the 10 MHz output of a GPS disciplined Rb oscillator.For exemplary averaging times corresponding to respective gate times of the same magnitude the effective statistical uncertainties for frequency measurements are given.

Figure 11 .
Figure 11.Left: daily values of UTC(PTB)-GPS time for the year 2020 as published in PTB Time Service Bulletin.Right: daily frequency values derived from the data shown in the left graph.The modified Julian date (MJD) is based on a continuous count of days, in which MJD 58850 corresponds to 2 January 2020.

Figure 12 .
Figure 12.Timescale differences UTC − UTC(PTB) taken from Circular T of year 2020 (left) and corresponding relative frequency values for five days intervals (right).MJD 58850 corresponds to 2 January 2020.

Figure 13 .
Figure 13.Yearly averages of the frequency difference between the ensemble of primary and secondary representations of the second and UTC.

Figure 14 .
Figure 14.Illustration of the uncertainty contributions over the chosen setup's frequency range (f 0 = 9 kHz-300 MHz).Left: case of a metrology institute like PTB, right: case of a calibration lab equipped with a GPS disciplined DO.

Table 1 .
Seven years summary of the standard uncertainty u CV for frequency measurements from monitoring GPS time with respect to UTC(PTB).

Table 2 .
Seven years summary of uncertainty contributions (1 σ) for the difference in rate of UTC − UTC(PTB).Yearly means, standard deviation (SD), and combined (rms) are given.N is the number of data points per year as published in Circular T.

Table 4 .
−16SD/10−16u d /10 −16 combined u CT /10 −16 Summary of uncertainty contributions for two cases.First, the case of a national metrology institute as PTB and second the case of a calibration lab.The displayed acronyms are introduced in the respective section.