Instrumentation & Measurement Magazine 25-3 - 8

Statistical Analysis for the Concluded
Results
From (2), (3), (12) and (13), the effective parameters that may affect
the ADC instrument's uncertainty budget are the closed
interval [a-b], N-bit wordlengths where N is a positive integer,
and the digital display resolution r. It's worth mentioning that Xj
does not appear in the above-mentioned equations. So, the statistical
investigation was conducted to evaluate the effects of each
of those parameters on the uncertainty budget of the resolution.
The value of r (the digital display resolution) has a significant effect
on both rj
u and ur. However, r does not affect the difference
between the uncertainty calculated by the proposed approach
and the conventional approach as per (2) and (13) (Table 1).
From Table 1 and Table 2, it is observed that neither Xj
nor r
affects the difference of uncertainty between the conventional
uncertainties and the uncertainty estimated by the proposed
approach, where a is defined as the minimum value that can be
identified, usually equal to 0. As for parameters b and N, they
need to be further investigated as discussed below.
Table 3 is an implementation of the proposed algorithm based
on the previously mentioned equations. This algorithm estimates
the evaluated perfect precise resolution standard uncertainty rj
u
associated with a user-input indication Xj that has an index j as
well as the difference between the modeled and the conventional
uncertainties. The estimation takes into consideration the wordlength
N of the ADC of the digital instrument and the lower and
Table 1 - Values and the differences between the uncertainty calculated by the proposed approach
and the conventional uncertainty at a chosen specified value of N=32 and b =4
r
0.000001
0.00001
0.0001
0.001
u
rj 
2.88943984E-07
2.887020E-06
2.88677823E-05
2.886754034E-04
ur
2.88675135E-07
2.886751E-06
2.88675135E-05
2.886751346E-04
E(s)
2.688496619E-10
2.688496404E-10
2.688496664E-10
2.688496858E-10
Table 2 - Values and differences between the uncertainty calculated by the proposed
approach and the conventional uncertainty at specified value N=16 and b =3
r
0.000001
0.00001
0.0001
u
rj 
1.3503376E-05
1.6101452E-05
4.2082214E-05
ur
2.8867513E-07
2.8867513E-06
2.8867513E-05
E(s)
1.3214701E-05
1.3214701E-05
1.3214701E-05
Table 3 - Proposed algorithm for determining the perfect precise expected resolution
uncertainty and the conventional uncertainty value ur
the difference
b
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
8
N
32
32
32
24
24
24
16
16
16
8
8
8
4
4
4
different values of b and N and at constant value of r (where r = 1E − 06)
rj 
δ
4.656613E-10
6.984919E-10
9.313226E-10
1.192093E-07
1.788139E-07
2.384186E-07
3.051804E-05
4.577707E-05
6.103609E-05
7.843137E-03
1.176471E-02
1.568627E-02
1.333333E-01
2.000000E-01
2.666667E-01
u
2.888096E-07
2.888768E-07
2.889440E-07
3.230879E-07
3.402943E-07
3.575007E-07
9.098476E-06
1.350338E-05
1.790828E-05
2.264407E-03
3.396467E-03
4.528526E-03
3.849031E-02
5.773532E-02
7.698032E-02
ur
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
2.88675135E-07
IEEE Instrumentation & Measurement Magazine
at
uur
rj  
1.3442488E-10
2.0163729E-10
2.6884958E-10
3.4412760E-08
5.1619140E-08
6.8825520E-08
8.8098004E-06
1.3214701E-05
1.7619601E-05
2.2641187E-03
3.3961781E-03
4.5282374E-03
3.8490018E-02
5.7735027E-02
7.6980036E-02
May 2022
Instrumentation & Measurement Magazine 25-3

Table of Contents for the Digital Edition of Instrumentation & Measurement Magazine 25-3
