Primary calibration of mechanical sensors with digital output for dynamic applications


ACTA IMEKO 
ISSN: 2221-870X 
September 2021, Volume 10, Number 3, 177 - 184 

 

ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 177 

Primary calibration of mechanical sensors with digital output 
for dynamic applications 

Benedikt Seeger1, Thomas Bruns1 

1 Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany 

 

 

Section: RESEARCH PAPER  

Keywords: Primary calibration; MEMS; complex frequency response; digital interface; dynamic measurement  

Citation: Benedikt Seeger, Thomas Bruns, Primary calibration of mechanical sensors with digital output for dynamic applications, Acta IMEKO, vol. 10, no. 3, 
article 24, September 2021, identifier: IMEKO-ACTA-10 (2021)-03-24 

Section Editor: Francesco Lamonaca, University of Calabria, Italy 

Received February 26, 2021; In final form July 9, 2021; Published September 2021 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Funding: This project 17IND12 Met4FoF has received funding from the EMPIR programme co-financed by the Participating States and from the European 
Union’s Horizon 2020 research and innovation programme. 

Corresponding author: Thomas Bruns, e-mail: thomas.bruns@ptb.de  

 

1. INTRODUCTION 

More and more applications and appliances in both the 
industrial and the consumer fields are integrating micro-
mechanical sensors with a direct digital output. That means the 
measured value of a quantity like acceleration, air pressure or 
force is accessible via a digital interface like SPI, I2C or CAN in 
terms of a binary number. Those values can be directly fed into 
control units or digital displays for further processing without 
any need for traditional signal conditioning by amplifiers or 
sampling by discrete AD converters.  

This is very beneficial for the integration of measurement 
capabilities in all kinds of devices and systems, from simple 
telephones to UAVs1 and beyond. Accordingly, the market for 
MEMS sensors has been booming for several years. 

 In condition monitoring applications, e.g., on production 
machines, these sensors can be used to measure vibrations, also 
using information from the frequency domain [1], which makes 

 
1 Unmanned aerial vehicles also known as drones. 

dynamic calibration and absolute timestamping advantageous [2], 
[3].  

From the metrology perspective, such sensors are rather 
inconspicuous as long as the focus is on a static measurement. 
As soon as the focus shifts onto dynamic signals, however, the 
timing of signal acquisition is of high importance, and the 
integrated sampling and signal conditioning of such sensors 
creates new challenges for the calibration lab.  

This starts with the fact that existing documentary standards 
and guidelines for dynamic calibration, like the current 
ISO 16063 series or [4] rely on the synchronised timing of the 
(analogue input) data acquisition channels, which is under the 
control of the calibration system. Recent work on the calibration 
of sensors for other mechanical quantities like force, torque or 
pressure relies on the same principles. 

While the calibration of the magnitude of a sensor's complex 
transfer function only needs the technical interfaces and a 
stationary excitation signal [5], the phase or group delay 
calibration is strongly dependent on either the synchronicity of 
the channels or other precise knowledge of the sample timing.  

ABSTRACT 
This article tackles the challenge of the dynamic calibration of modern sensors with integrated data sampling and purely digital output 
for the measurement of mechanical quantities like acceleration, angular velocity, force, pressure, or torque. Based on the established 
calibration methods using sine excitation, it describes an extension of the established methods and devices that yields primary 
calibration results for the magnitude and phase of the complex transfer function. The system is demonstrated with a focus on primary 
accelerometer calibrations but can easily be transferred to the other mechanical quantities. Furthermore, it is shown that the method 
can be used to investigate the quality and characteristics of the timing for the internal sampling of such digital output sensors. Thus, it 
is able to gain crucial information for any subsequent phase-related measurements with such sensors. 

mailto:thomas.bruns@ptb.de


 

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2. EXTENSION TO CONVENTIONAL PRIMARY CALIBRATION 

A conventional analogue calibration system (ACS) in 
compliance with ISO 16063-11 or ISO 16063-21 is based on two 
data acquisition channels with common timing usually provided 
by a central clock signal in a single data acquisition system. The 

two channels record the reference quantity (𝑦ref ), like 
acceleration, velocity, or displacement) and the calibration 
quantity (voltage, charge) as the output of the device under test 
(DUT).  

The clock is usually quite accurate and the sampling of the 
time series from the acquisition can be considered equidistant. 

Using a mono-frequent sinusoidal drive signal with angular 

frequency 𝜔 for the mechanical excitation (e.g., via a shaker), a 
linear sine approximation is used to quantify the reference input 

(𝑦ref) and the response (𝑦DUT) of the DUT. Via linear least 
squares fitting, the parameters in the following equations are 
determined: 

𝑦ref(𝑡𝑖 ) = 𝐴ref sin(𝜔𝑡𝑖) + 𝐵ref cos(𝜔𝑡𝑖) + 𝐶ref 

𝑦DUT (𝑡𝑖 ) = 𝐴DUT sin(𝜔𝑡𝑖 ) + 𝐵DUT cos(𝜔𝑡𝑖 ) + 𝐶DUT , 
(1) 

where 𝐴 and 𝐵 can be considered as quadrature components of 
each signal, while 𝐶 is introduced to cover any potential bias 
introduced into the measurement. 

The complex transfer function S of the DUT is then given in 

terms of magnitude �̂� and phase Δ𝜑 by 

𝑆(𝜔) = �̂�(𝜔) ⋅ e−𝑖 Δ𝜑(𝜔)    with  

�̂�(𝜔) =  √
𝐴DUT

2 + 𝐵DUT
2

𝐴ref
2 + 𝐵ref

2
   and (2) 

𝜑(𝜔) = arctan (
𝐵DUT
𝐴DUT

) − arctan (
𝐵ref
𝐴ref

) .  

Although not self-evident from the equations, it is crucial for 
the correct calibration of the phase delay that the time base for 

the data acquisition (represented by 𝑡𝑖  in equation (1) is the same 
for both channels. 

In Figure 1, such an ACS is represented on the top left. The 
channel for the DUT called "sync" in this schematic for reasons 
that will become clear, soon. 

2.1. Sample timestamping 

For the case of a digital output sensor (DOS), the prerequisite 
last mentioned in the previous paragraph is no longer fulfilled. 
Neither is the sample clock frequency (or the actual instance of 
sampling) under the control of the laboratory's ACS nor is the 
exact instance of sampling even known to the measurement 
system. 

A reliable phase calibration under such circumstances requires 
an extension of the ACS in order to mitigate the lack of 
knowledge and control.  

For the rest of this section it is presumed that the DOS 
provides a "data ready signal", that is, it provides a means to 
signal to any connected recording system that a new sample has 
been acquired and is ready for delivery. This signal is active 
whenever a new sample was acquired by the sensor. 

The term "sample" in relation to the DOS relates to one set 
of values of the measurands of the DOS. These values are 
nominally associated with the same single point in time, e.g., for 
a three-component accelerometer, a sample could be a tuple 

(𝑎𝑥 , 𝑎𝑦 , 𝑎𝑧 ). 
The complications compared to a conventional analogue 

sensor are threefold: 
1. The data interface is digital and cannot be read out by the 

classic ACS. 
2. The precise sample clock is unknown and asynchronous 

in relation to the ACS's internal clock. 
3. The sampling instances of the DOS are not necessarily 

equidistant. 
To overcome these complications and accomplish a phase 

calibration of a DOS with data ready output, the ACS is extended 
with a separate digital acquisition unit (DAU) as shown in the 
top right part of Figure 1. 

 

Figure 1. The conventional analogue calibration system (ACS) on the top left extended by the digital acquisition unit (DAU) on the top right. At the bottom the 
respective signal traces over time as acquired by the two systems.  

 
 
 

   

   

 
 
 

                             

   
    

   
    

         

          
       

    

   
   

                        

   
    

       
     

      
    
     

   
    



 

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The core of this DAU is the sampling and timestamping of 
the data from the digital interface of the DOS related to a 
common time base, which it also uses for the sampling of an 
additional analogue channel called "sync" (orange arrows) in 
Figure 1.  

In the implementation used for the investigations in this 
publication, the sampling of DOS data and sync were performed 
synchronously. This, however, is not a strict requirement for the 
procedure. 

Whenever the DOS signals the availability of a new sample 
via the data ready output (upper green arrow in Figure 1, the 

DAU immediately samples the sync signal, (𝑦sync) (right orange 

input to DAU in Figure 1), reads the digital data sample, (𝑦DOS) 
from the DOS interface (lower green arrow in Figure 1) and 
marks this data tuple with a precise timestamp based on a GPS-
aligned clock of the microprocessor used.  

Repeating this procedure, the DAU acquires a time series of 
digital and analogue data with a sample clock as close to the DOS 
internal sample clock as possible but with timing information as 
accurate as the external clock provided to the DAU. 

The relation to the reference signal for the calibration, which 
is still acquired with the ACS (blue arrow in Figure 1), is provided 
by a sync signal (orange arrow) and ACS input in Figure 1, hence 
the naming of this channel.  

The sync signal is a mono-frequent sinusoidal voltage with the 
identical frequency to the excitation signal. It is fed into the DUT 
channel of the ACS and to the sync input of the DAU. 

Such a signal can be easily derived directly from the excitation 
signal or taken from the reference channel of the ACS. 

As a consequence of this setup and procedure, the originally 
independent systems of the ACS and the DAU can now be linked 
in time via the common sync signal which is sampled by both 
systems based on their respective time bases. This is depicted in 
the time traces in the lower part of Figure 1, where the same sync 
signal (in orange) is given twice: once as acquired by the ACS and 
secondly as acquired by the DAU. 

2.2. Implementation details of the DAU 

The implementation of the DAU used in this study was based 
on an STM32F767® microcontroller and a related development 
board (NUCLEO-F767ZI®). 

The acquisition of DOS output and the ADC was handled in 
a single interrupt service routine, which was triggered by the data 
ready pin of the DOS. The timing information was generated by 
an internal 108 MHz counter, which, in turn, was synchronised 
with an externally provided pulse-per-second (PPS) clock from a 
GPS receiver. The details of the implementation are available 
under open access conditions [11], [12] 

The uncertainty of the absolute timestamp was evaluated by 
measuring a reference PPS signal stabilised to PTB's atomic clock 
with the DAU. Figure 2 shows the deviation of the timestamps 
from the expected value as determined by the DAU. That 
deviation is mainly caused by the short-term drift of the 
microcontroller's phase locked loop and the intrinsic jitter of the 
GPS's PPS signal. Overall, an accuracy better than 150 ns was 
achieved. 

2.3. Data analysis for sample time stamping 

Based on the described signal generation and data acquisition 
scheme, the system generates four times series of timestamped 
samples related to three different signal sources. The amplitudes 
and initial phases of these can be determined by appropriate sine 

approximation methods (described below). Those time-series 
are:  

i. The reference signal acquired by the ACS: 

𝑦𝑖,ref(𝑡𝑖 ) = �̂�ref ⋅ sin(𝜔 𝑡𝑖,ACS − 𝜑ref). 

ii. The sync signal acquired by the ACS: 

𝑦𝑖,sync(𝑡𝑖,ACS) = �̂�sync,ACS ⋅ sin(𝜔 𝑡𝑖,ACS − 𝜑sync,ACS). 

iii. The sync signal acquired by the DAU: 

𝑦𝑘,sync(𝑡𝑘,DAU) = �̂�sync,DAU ⋅ sin(𝜔 𝑡𝑘,DAU − 𝜑sync,DAU). 

iv. The DOS signal acquired by the DAU: 

𝑦𝑙,DOS(𝑡𝑙,DOS) = �̂�DOS ⋅ sin(𝜔 𝑡𝑙,DOS − 𝜑DOS) . 

Here, the index i indicates that (i) and (ii) have the same time 
base and are synchronously sampled (by the ACS). In contrast 
(iii) and (iv) do need the same time base (from the DAU) but not 
necessarily synchronous sampling, hence the indices are k and l. 

The magnitude of the transfer function of the DOS is 
independent of the actual timing situation and can be directly 
calculated from (i) and (iv) as 

�̂�(𝜔) =  
�̂�DOS
�̂�ref

 . (3) 

Since (ii) and (iii) are records of the same source with only 

apparent but no real existing delay between 𝜑sync,ACS and 

𝜑sync,DAU, the time bases of ACS and DAU and thus the initial 
phases of the reference and the DOS can be linked. 

The phase difference between 𝑦ref  and 𝑦DOS is accordingly 
given by 

Δ𝜑 =  𝜑DOS − 𝜑sync,Dau + 𝜑sync,ACS − 𝜑ref . (4) 

It should be noted that the accuracy of that link is dependent 
on the timestamping accuracy of the DAU and not on the 
precision of the DOS's sample clock. As well as the uncertainty 
of the sync signal (iii) sampled by the DAU. The transfer function 
of the DAU ADC must be calibrated and taken into account [13]. 
On the contrary, the measurement scheme described above 
provides additional information on the precision of the DOS's 
sample clock, as described in Section 3. 

2.4. Series timestamping 

Some digital sensors/measuring systems do not provide a 
data ready signal for each sample but instead record a full time 
series of data after a given trigger signal. To calibrate such a kind 
of sensor (we will call it "time series sensor”, TSS), precise 
knowledge of the relative time between the trigger and the 
excitation signal is needed. If the ACS does not provide such a 

 

Figure 2. Distribution of the timestamp deviations determined using the DAU 
with reference to an atomic clock stabilised PPS signal over a monitoring 
period of 45 hours.  

  

   

          
         



 

ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 180 

synchronised trigger signal, separate analogue trigger logic, e.g. 
from a digital storage oscilloscope (DSO), can be used to derive 
and generate the trigger signal from an analogue sync signal like 
that mentioned in Section 2.1.  

Figure 3 shows a schematic diagram of a setup to calibrate 
such a TSS. The ACS again acquires the generator signal 

𝑦sync (𝑡𝑖 ), see Section 2.1, as for an analogue DUT (orange arrow 
in Figure 3 into the ACS). The DSO (in our case a Rigol 
MSO5072) triggers, upon zero crossing of the same analogue 
sync signal (orange arrow in Figure 3 into the DSO) and 
generates a trigger signal which is connected to the TSS's trigger 
input (brown arrow in Figure 3). This starts the recording of the 

time series 𝑦TSS(𝑡TSS). 
Note that due to noise, the DSO does not trigger perfectly at 

the zero crossing of the sync signal, and there is some additional 
processing delay within the DSO. However, the resulting actual 
trigger delay can be determined and corrected for, e.g. by sine 
approximation of the sync signal acquired simultaneously by the 
DSO and determining the actual trigger phase 

Δ𝜑DSO,TriggerNoise. 
To this end, in a dedicated setup, the DSO is used to sample 

its own trigger output simultaneously to the sine wave of the sync 
signal, which causes that trigger output. Based on the two 
simultaneously acquired traces, the delay between the zero 
crossing of the sync signal (e.g. determined by sine 
approximation) and the rising edge of the trigger can be 
determined. Due to inevitable noise, there will be a certain 
variation in the determined trigger delay. 

In the actual setup, the trigger signal is emitted with a mean 

delay of 𝜏DSO = 236(1) ns. This was previously calibrated with a 
rectangle generator and the feedback of the trigger output to a 
second input channel of the DSO.  

The use of a rectangle generator, providing a steep rising edge, 
was considered more precise than the sine approximation 
approach applied to the sync signal. However, it requires 
additional equipment. 

2.5. Data analysis for series timestamping 

Subsequent to the data acquisition described, a four-
parameter sine approximation (FPSA) according to [6] is used to 

determine the angular frequency 𝜔, amplitude �̂�𝑇𝑆𝑆, initial phase 
𝜑𝑇𝑆𝑆  and offset 𝐶𝑇𝑆𝑆 of the TSS's output 𝑦TSS,𝜔 (𝑡TSS) (Note 
that due to the operation principle of a TSS, the time instances 
are unknown.) With these parameters, the magnitude response 

�̂�(𝜔) of its complex transfer function at the excitation frequency 
𝜔 can be obtained directly from the amplitude of the reference 

signal �̂�ref(𝜔) and the amplitude of the time-series signal 

�̂�TSS(𝜔) 

�̂�(𝜔) =
�̂�TSS(𝜔)

�̂�ref(𝜔)
 . (5) 

Note that the internal sampling frequency of the TSS, 𝑓s,TSS 
may differ from its specified nominal sampling frequency 

𝑓s,TSS,nominal. Such differences lead to a deviation of the apparent 
excitation frequency determined by the FPSA as compared to the 

applied excitation frequency 𝜔.  
A reduced internal sample rate 𝑓s,TSS leads to an apparent 

increase of 𝜔TSS, and accordingly an increased 𝑓s,TSS leads to an 
apparent reduction of this frequency. The actual mean internal 

sample rate 𝑓s,TSS can then be determined as follows: 

𝑓s,TSS =  𝑓s,TSS,nominal ⋅
𝜔

𝜔TSS
 , (6) 

provided that the excitation frequency from the ACS, 𝜔, is well 
known. 

The initial phase of the TSS signal 𝜑TSS was determined using 
the four-parameter approximation and is thus linked to the actual 
period of the excitation signal. Therefore, no correction related 
to sample rate deviations of the measured initial phase is 
necessary. 

 

Figure 3. The ACS with a digital storage oscilloscope (DSO) as a trigger generator for the TSS. The DSO triggers on the zero crossing of the sync signal. The 
trigger output of the DSO, in turn, triggers the data acquisition of the TSS. The traces are the reference signal (blue), the analogue sinusoidal sync signal 
(orange), the trigger (brown) and the sampled time series of the TSS (green).  



 

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Consequently, the phase response Δ𝜑 of the complex transfer 
function of a TSS can be calculated as follows, 

Δ𝜑 =  𝜑TSS − Δ𝜑DSO − Δ𝜑ACS , (7) 

using the phase difference Δ𝜑ACS between the reference and 
sync signal obtained by the analogue calibration system2. 

Consequently, the complex transfer function �̂�(𝜔) is given by: 

𝑆(𝜔) = �̂�(𝜔) ⋅ e−𝑖 Δ𝜑 =
�̂�TSS (𝜔)

�̂�ref(𝜔)
⋅ 𝑒 −𝑖(𝜑TSS−Δ𝜑DSO−Δ𝜑ACS). (8) 

3. SAMPLE RATE PROBLEMS AND MITIGATION STRATEGIES 

Typical DOSs generate their internal sample rate with an 
internal clock unit based e.g. on an RC oscillator. Such simple 
clock generators are subject to drift depending on the operational 
temperature or supply voltage etc. Accordingly, the stability and 
precision of such clocks should be checked.  

In [7] the effect of a biased sample frequency was described, 
and a rather elaborate correction method was provided. The 
calibration procedures presented in this article are independent 
of the internal sample clock by design, because they make use of 
external GPS-aligned timestamping. Even more, these 
procedures provide the direct means for a quantitative 
determination of sample clock accuracy and stability. This can be 
achieved at least in two ways: 

• Direct sample interval evaluation  

Based on the timestamps 𝑡𝑙,DOS, the sample interval between 

each DOS sample pair can be calculated as 𝑡𝑙+1,DOS − 𝑡𝑙,DOS 
and compared to the nominal sample interval. This 
procedure and the resulting insights, of course, depend on 
the availability of independent precise and stable timestamps 
for each sample, which makes it unfeasible for the series 
timestamping. 

• Nominal frequency evaluation  
To get a more general estimate of the DOS sample rate 
precision and stability, it is sufficient to have a known, stable 
and precise single frequency excitation for the DOS. 
Accordingly, this method can be applied even without the 
complete calibration set-up. If the internal clock of the DOS 

deviates by a factor 𝜆 from the laboratory clock, the DOS 
senses an external excitation with frequency 𝑓vib with an 
apparent frequency 𝑓vib,DOS. The relation between the two is 

𝑓vib,DOS =
𝑓vib

𝜆
. (9) 

This is the equivalent situation to that described by 
equation (6) for series timestamping. 
The apparent frequency sensed by the DOS can easily be 
determined by applying an FPSA [4] to the time series 
measured by the DOS. This algorithm does not only fit the 
amplitude and phase of a sinusoidal signal but iteratively 
reduces apparent frequency deviations starting with an initial 
frequency guess. 
Note that in the case of sample timestamping the timestamps 

𝑡𝑙  used for that fit have to be the nominal (deviating) 

 
2 It should be noted that any additional phase delays within 

the ACS known from conventional calibration have to be 
considered in the same fashion for the digitally enhanced system. 

timestamps calculated from the nominal sample rate of the 
DOS. Then, equation (9) yields the following for the real 

mean sample rate 𝑓DOS,real of the DOS  

𝑓DOS,real =
𝑓DOS,nom 

𝜆
 . (10) 

4. FIRST PRIMARY CALIBRATION RESULTS 

4.1. The transfer function 

In order to test the above-mentioned concept, the DAU was 
used to perform a primary calibration of the z-axis of a 6-degree-
of-freedom MEMS sensor of the type MPU-9250 following [6]. 
Figure 4 shows the situation in the ACS, i.e. the DOS mounted 
on the shaker's armature and the connected DAU. For the 
determined complex transfer function, the reference acceleration 
was measured by laser interferometry on top of the MEMS's 
housing. A second laser was targeted at the surface of the carrier 
breakout board right next to the DOS in order to investigate any 
relative motion due to the solder joints. This, however, is of no 
concern in this manuscript. 

The generator signal driving the power amplifier for the 
shaker was used as the sync signal to link the ACS and the DAU. 
The calibration was performed in the frequency range from 
10 Hz to 200 Hz with single frequency sine excitation and 
excitation levels between 2.5 m/s² and 100 m/s². 

The MPU-9250 sensor is one of those that provide a "data 
ready" signal whenever the internal sampling process is finished. 
According to the previous description, the DAU acquired a 
sample of three acceleration values, three angular rate values and 
one temperature value for each trigger from the data ready signal 
and timestamped this set of values with an absolute time value. 
The sensor was running at a nominal sampling rate of 1000 S/s. 

 

Figure 4. Primary calibration setup for z-axis of an MPU-9250 sensor. The 
bottom half shows the DAU with the STM32F7 development board in 
piggyback. In the top half, the MPU-9250-breakout board mounted on a steel 
adapter and screwed to the armature of an SE-09 shaker is shown.  

   

      

   

   

   

    



 

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The analysis focused on the single (nominal) axis of excitation, 
which was the z-axis. 

By using the sync signal as described in Section 2.3 and sine 
fitting of the z-axis values related to the absolute timestamps, it 
was possible to attain a traceable primary calibration of the 
magnitude and phase of the respective complex transfer 
function. The results are depicted in Figure 5 for three repeated 
measurements of the same unit. 

A linear regression to the phase results for the lower 
frequencies suggests a group delay of the sensor of 8,1 ms. 

4.2. The internal sampling 

In addition to this classic evaluation, the timestamping 
enables the evaluation of the stability of the sensor's internal 
sample clock and possible influencing quantities. In direct 
comparison to a 10 MHz reference frequency distributed within 
PTB and traceable to the atomic fountain clocks of PTB, the 
utilised GPS-based timestamping revealed a mean sample 
interval of 996,96 µs for a measurement time of approx. 8000 s. 
Over this time the sample interval length yielded a standard 
deviation of 21 ns and a spread of 335 ns, where admittedly the 
quantization of the individual timestamps is 10 ns due to the 
counting frequency of 108 MHz of the DAU. The distribution 
of the sample intervals is shown in the histogram in Figure 6. 

From the visual appearance, it is evident that the distribution 
is not Gaussian and therefore not the usually supposed normal 
distribution. Rather than that, certain values of the sample 
interval are apparently strongly favoured over others. One cause 
for such an effect may be found in sensor-internal digital 
compensation processes in relation to temperature. As the MPU-
9250 also features an internal temperature probe, the 

investigation of the correlation of the sample interval versus the 
temperature is an obvious next step. 

Figure 7 depicts the relation of the sample interval over the 
apparent temperature as measured internally. It is evident that 
three to four of the preferred sample durations can be associated 
with a certain temperature and that, in general, the average 
sampling frequency falls (slightly) with rising temperature. For an 
in-depth understanding of this peculiar relation, a detailed 
knowledge of the internal processes of the sensor is probably 
necessary. 

Nevertheless, it remains to be noted that the application of 
sample timestamping with the newly presented DAU could 
demonstrate that the statistical properties of the sampling unit of 
this sensor are far from the typically expected uniform or normal 
distribution. 

In order to compare this finding with some other DOS, a 
Bosch BMA280 was calibrated in an identical setting, and the 
measured sample intervals were subject to the same processing. 
Figure 8 depicts the results. 

In this case the distribution is split into three parts with a 
prioritised centre part which has three orders of magnitude more 
counts than the side lobes. Such a shape makes the usual 
characterisation with a standard deviation almost meaningless. 
Why such a peculiar distribution is generated becomes clearer on 
closer inspection of the time series of sample intervals given in 
Figure 9. 

 

Figure 5. Magnitude (top) and phase (bottom) of the complex transfer 
function of an MPU-9250 sensor measured in 3 repetitions (#1, #2, #3). The 
insert show details at low frequencies.  

 

Figure 6. Distribution of sample intervals of an MPU-9250 DOS determined 
by sample timestamping. Counts are in logarithmic scale.  

 

Figure 7. Relation of sample interval and sensed temperature of the 
MPU9250 sensor.  

 

   

   

   

   

 

     

 
 
 
 
  
 
  
  
 
  
   
   
  
 
   
 
 
 
  
 
 

               

  

  

  
    

    

 

    

    

        

    

    

    

    

    

    

   

 

     

 
 
 
  
  
  
  
 
  
   
   
   
  
 
 
  
 
 

               

  

  

  

   

   

   

   

   

   

   

              



 

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As seen in the upper chart, the vast majorities of the sample 
intervals are close to the mean value of 485 µs. However, 
periodically, i.e. approximately every 1.1 s or 2200 samples, there 
is a double spike. The close-up of one of those features in the 
lower chart of the figure discloses a combination of a delayed 
followed by a premature sample. Presumably, some internal 
process of the digital part of the sensor, like an adaptive digital 
filter or an adjustable compensation, takes some excessive 
computation time, slowing down either the sampling process or 
the delivery (data ready) of the analogue quantity. In order to 
keep the mean sample rate constant, the delay is followed by an 
abridged sample interval compensating the delay. 

5. MEASUREMENT UNCERTAINTY 

This article is not dedicated to an in-depth discussion and 
evaluation of the measurement uncertainty budget associated 
with the new hardware and methodology. This will need more 
research in the various components involved and probably is 
worth an article on its own. Nevertheless, we want to briefly 
address a few issues that were found during the research work so 
far. 

5.1. Magnitude of the transfer function 

The magnitude calibration is a rather straight forward process 
for the DOS and in terms of measurement uncertainty very 
similar to conventional work with AD-converted output 
voltages. A significant difference is, however, that opposed to 
conventional set-ups, no gain adjustment by using external 

amplifiers is possible. This results in a fixed signal resolution that 
is linked to the embedded ADC and has to be considered in the 
measurement uncertainty of the DOS especially for excitations 
in the lower amplitude range. 

5.2. Phase of the transfer function 

The phase of the transfer function is the actual challenge in 
calibration and therefore the focus of this article. Internally the 
measurement uncertainty of the phase is relying on the quality of 
the embedded oscillator and some firmware aspects of the DOS. 
This was demonstrated by the measurement-analysis of the 
previous section 4. While the influence of a low-quality oscillator 
may be mitigated by precise timestamping during calibration, 
such an approach is usually not feasible for the typical application 
of DOS. Here, the nominal sampling rate is assumed.  

In addition to these components of measurement uncertainty 
of phase intrinsic to the DOS, the hardware of the DAU and the 
methodology add their own components. 

Of course, the phase or group delay associated with the 
sampling of the sync-signal by the DAU has to be calibrated, as 
this is no longer controlled by the ACS. The best estimate of this 
group delay can be corrected, but an uncertainty remains from 
this calibration. This was briefly addressed in sections 2.2 and 2.4 
for the work presented here. 

Additional phase measurement uncertainty is inflicted by the 
noise on the sync signal which links the time-bases of ACS and 
DAU. This is strongly dependent on the source available as sync 
signal and in particular its signal-to-noise ratio. This noise 
degrades the phase accuracy on both sides, the ACS and the 
DAU. However, in the described set-up it can hardly be avoided. 

6. CONCLUSION AND OUTLOOK 

Conventional analogue accelerometer calibration systems are 
not capable of calibrating modern digital output sensors. 
However, an extension providing the necessary digital interface 
and analogue input for a suitable reference signal can extend the 
capabilities to the magnitude and phase calibration of the 
complex frequency response of such sensors. The typically 
independent internal sampling of the DOS poses a special 
challenge in the calibration as well as a new source of 
measurement uncertainty and a new specification that has to 
become part of the calibration routine and the calibration result. 

With two different approaches, it is possible to calibrate 
devices that deliver separate samples as well as devices that buffer 
a series of samples after a single trigger and deliver the whole 
series at once. Provided that the calibration laboratory is 
equipped with a precise and stable timestamp source, like a GPS-
aligned clock, simple procedures lead to detailed insights 
concerning the internal timing of the DOS and their 
dependencies on environmental conditions. 

It turns out that simple classical assumptions about 
characteristics like the accuracy and jitter of the internal sampling 
units can be misleading. However, the application of external 
timestamping, as demonstrated, gives a clear and traceable 
quantification of these properties.  

Within the European project titled Met4FoF, a cross 
validation in terms of an interlaboratory comparison between 
different participants is planned. In order to address the question 
of consistency of the results, a further evaluation of additional 
measurement uncertainty components involved with the use of 
the DOS will be included. An existing calibration system will be 
equipped with a GPS based absolute time base so that the phase 
of the transfer function can be determined without a reference 

 

Figure 8. Distribution of sample interval lengths of a BMA280 DOS 
determined by sample timestamping. Counts are in logarithmic scale.  

 

Figure 9. Sample interval length over time plot for the BMA280 sensor. The 
Lower plot is a zoom-in on the upper overview.  



 

ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 184 

signal, but by means of the absolute time. By comparison with 
this system, the solution presented here will be evaluated with 
respect to uncertainty. 

The intrinsic timing properties, in particular, add a new 
dimension to measurement uncertainty estimation. In addition, 
most MEMS accelerometers on the market are multi-component 
devices. The question of dynamic crosstalk may therefore add 
another complication, where the static (DC) crosstalk has already 
been studied in other publications such as [7], [8]. 

ACKNOWLEDGEMENT 

This project 17IND12 Met4FoF has received funding from 
the EMPIR programme co-financed by the Participating States 
and from the European Union’s Horizon 2020 research and 
innovation programme 

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https://ieeexplore.ieee.org/document/7151267
https://cfmetrologie.edpsciences.org/articles/metrology/pdf/2019/01/metrology_cim2019_22003.pdf
https://cfmetrologie.edpsciences.org/articles/metrology/pdf/2019/01/metrology_cim2019_22003.pdf
https://doi.org/10.3390/s21062019
https://doi.org/10.7795/550.20190425EN
https://doi.org/10.5194/jsss-7-245-2018
https://standards.ieee.org/standard/1057-2017.html
https://doi.org/10.1088/1681-7575/ab5505
https://doi.org/10.1088/1681-7575/ab79be
https://www.imeko.org/publications/tc22-2017/IMEKO-TC22-2017-015.pdf
https://www.imeko.org/publications/tc22-2017/IMEKO-TC22-2017-015.pdf
https://github.com/Met4FoF/Met4FoF-SmartUpUnit
https://circuitmaker.com/Projects/Details/Benedikt-Seeger-2/Met4FoF-Interface-Board
https://circuitmaker.com/Projects/Details/Benedikt-Seeger-2/Met4FoF-Interface-Board
http://dx.doi.org/10.21014/acta_imeko.v9i5.1008