Instrumentation & Measurement Magazine 25-3 - 4

Fig. 2. The rectangular probability distribution function (PDF).
Fig. 1. The differences among the original analog signals, the quantized
signals and, the signal reconstructed from the quantized signals, for a 4-bit
analog-to-digital converter (ADC) digitizing a sine wave, from [10], used with
permission.
the results of the digital amplitude close to the original analog
amplitude. Fig. 1 shows the divergence between the original
analog signal, the reconstructed signals from the signals being
quantized, and the quantized signals [10].
Different approaches were developed to identify the
quantization effects on the resolution uncertainty. These
approaches were addressed in many studies [10], [11], especially
for high-precision measuring instruments. All of these
studies deal with the resolution effects on the determination
of the measurand value and its estimated uncertainty.
Most of these approaches describe this effect using complex
models and neglect the accurate determination of the actual
quantized signal based on approximation techniques. The
approximation techniques' (rounding) is not quite accurate,
especially for high-precision measurements such as international
comparisons.
The current study recommends the revision of related resolution
uncertainty and proposes a simple model to quantify
the quantization error effects and hence improve the indicators'
measurement reliability.
GUM Traditional Mechanism for
Estimating the Uncertainty of Digital
Indicator Resolution
Based on GUM, the digital indicator with a display device that
has a resolution r and indicates a value of X, the stimulus value
that corresponds to the indication is modeled as a continuous
random variable S and is supposed to be rectangular distributed
in the range
Xr s X r /2 (Fig. 2) [12] where X, r,
  
/ 2
and s are in the same unit of the display device.
The conventional estimation of the resolution uncertainty
value ur
can be calculated from the following equation [13]:
  
r
urs
12
The application of the conventional GUM rule in evaluating
the resolution uncertainty of digital indicators includes
4
Fig. 3. The perfect ADC versus ideal ADC (from [16], used with permission).
IEEE Instrumentation & Measurement Magazine
May 2022
(1)
some advantages and disadvantages that can be easily observed
or deduced from its statistical model analysis. The
advantage of the GUM mechanism is simplicity and no need
for complex mathematics. The predictable value of the stimulus
is equal to the indicated value [14]. The disadvantages of
GUM are the lack of error identification, where the only error
taken into account is the rounding-for-display error despite
many other errors arising from other sources such as filtering
due to quantization (quantization error) and the impact of
noise in the stimulus signal on the resolution uncertainty [15].
Quantization Errors
Perfect ADC: Fig. 3 illustrates a quantization pattern detected
with full transfer functions compared with an ideal transfer
function. The perfect ADC is amounting to the ideal ADC on
the precise mid-step of every step. This means that the perfect
ADC ideally approximates input values to the closest output
point values and the quantization of the inputs is a uniform
step [16].
Offset Errors: Offset errors of the ADC are defined as the deviation
of the actual ADC's transfer function from the perfect
ADC's transfer function at the point of zero to the transition
measured in the Least Significant Bit (LSB). Offset error can be
positive or negative. It is identified as a fixed distinction, over
the full scale of the ADC, between the true output values and
the idealistic output values [17] (Fig. 4).

Instrumentation & Measurement Magazine 25-3

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