Random noise test of analog-to-digital converters


ACTA IMEKO 
ISSN: 2221-870X 
June 2023, Volume 12, Number 2, 1 - 8 

 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 1 

Random noise test of analog-to-digital converters 

Francisco Alegria1 

1 Instituto de Telecomunicações and Instituto Superior Técnico/Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisbon,Portugal  

 

 

Section: RESEARCH PAPER  

Keywords: analog to digital converter; testing; noise; estimation; error analysis; random; applied mathematics 

Citation: Francisco Alegria, Random noise test of analog-to-digital converters, Acta IMEKO, vol. 12, no. 2, article 27, June 2023, identifier: IMEKO-ACTA-
12 (2023)-02-27 

Section Editor: Laura Fabbiano, Politecnico di Bari, Italy 

Received March 11, 2023; In final form May 7, 2023; Published June 2023 

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. 

Corresponding author: Francisco Alegria, e-mail: francisco.alegria@tecnico.ulisboa.pt  

 

1. INTRODUCTION 

The standard deviation of the random noise present in an 
analog to digital converter (ADC) is an important parameter used 
to describe the performance of ADCs and to choose the proper 
ADC to use in a given application. The knowledge of the noise 
standard deviation in a test setup is also needed when performing 
other ADC tests, namely the Standard Static Test [1], [2] and the 
Standard Histogram Test [3]-[11]; the Small Waves Histogram 
Test [12]-[16]; the Sinefitting Test [17]-[23] and the Jitter Test; 
[24]-[26] for the determination of the error; and precision of the 
ADC parameters estimated with them. In some cases, random 
noise is used, with great advantage, as a stimulus signal [27]. 

The issue of quantifying the uncertainty of the estimates of 
ADC characteristics does not concern only the tests 
recommended in the IEEE 1057 standard but are common to all 
ADC tests, like the ones in the IEC 62008 standard [28]-[29], and 
the IEEE 1241_2000 standard [30], for example. 

In [31] an analysis off the precision of the estimates obtained 
with this test has been carried out. There, an expression for the 
minimum number of samples required to guarantee a certain 
bound on the uncertainty of the results, was presented. This is 
important in order to minimize the duration of the test because 
the exact number of samples required can be calculated. 

The presence of phase noise in the stimulus signal generator 
as well as jitter in the sampling instants also affect electronic 
systems in general and are important to quantify [1], [31] [34].  

The test, as described in [1], section 8.6.2, consists in 
synchronously acquiring two sets of a certain number of samples 

(𝑀). The noise standard deviation (𝜎) is then estimated from the 
root mean square of the difference between the output codes of 
those two sets (mse – mean square error). If the noise standard 
deviation is high enough, a null input voltage is sufficient to 
perform the test, if not, a triangular stimulus signal should be 
used. 

In section 3, the error in the estimation of the random noise 
standard deviation due to the use of the heuristically derived 
estimator recommended in the IEEE standard is analyzed. In 
section 4 we show how to use a sinusoidal stimulus signal instead 
of a triangular shaped one. A study about the influence of the 
stimulus signal amplitude and offset on the estimation of the 
random noise standard deviation is then presented. This will 
allow the derivation of an expression for the minimum value of 
stimulus signal amplitude that guarantees an upper bound on the 
random noise estimator bias. An expression that allows the exact 
computation of the random noise standard deviation when 
necessary is going to be presented. This approach is 
computationally intensive and not appropriate for hand 
computation but can be easily implemented in a computer if 
necessary. 

2. DC STIMULUS SIGNAL 

Random additive noise in ADCs, as described in [1], is a non-
deterministic fluctuation of the ADC output and is described by 

ABSTRACT 
The random noise test of analog to digital converters recommended by the IEEE 1057 Standard for digitizing waveform recorders is 
studied. The heuristically derived expression presented for the estimation of the random noise standard deviation is experimentally 
validated. The standard suggests a triangular stimulus signal. Here we will show how to use a sinusoidal stimulus signal to carry out the 
test. The influence of stimulus signal offset and amplitude on the estimation error is also analyzed and an expression is presented to 
compute the minimum amplitude value that guarantees an upper bound on the estimator bias. 

mailto:francisco.alegria@tecnico.ulisboa.pt


 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 2 

its frequency spectrum and statistical properties. It is usually 
considered that the noise present is white (flat frequency 
spectrum), with a stationary probability density function. Also, it 
considered that the noise is additive and independent of the 
stimulus signal. 

Due to the presence of random noise at the ADC input, the 

output code (𝑘) can be considered a discrete random variable 
which can assume any value between 0 and 2𝑛𝑏 − 1 for a 𝑛𝑏-bit 
ADC. 

When the additive noise standard deviation is higher than the 

ADC ideal code bin width (𝑄) the suggested method in [1] is to 
short circuit the ADC input and acquire two sets of samples (𝑘𝑎𝑗 

and 𝑘𝑏𝑗) and subtract the codes obtained. This eliminates fixed 
errors of the ADC but preserves the random nature of the output 
codes. 

Considering that the additive noise present has a null mean (a 
non-null value would be accounted as an offset error), the 
average of the output codes will also be null. Therefore, the mean 

square error (𝑚𝑠𝑒) is just: 

𝑚𝑠𝑒 =
1

𝑀
∑(𝑘𝑎𝑗 − 𝑘𝑏𝑗)

2
𝑀−1

𝑗=0

 . (1) 

The mean square error obtained is twice the variance of each 
set since they are independent of each other. According to [20] 
the expected value of the mean square error determined by (1) is 
twice the variance of the output codes: 

𝐸{𝑚𝑠𝑒} = 𝜎𝑘𝑎
2 + 𝜎𝑘𝑏

2 = 2𝜎𝑘
2 . (2) 

Taken this into account, and considering that the variance of 
the output codes is equal to that of the additive noise, the 
estimated variance of the latter is just 

𝜎𝑟
2̂ =

𝑚𝑠𝑒

2
 , (3) 

where 𝜎𝑟 is the normalized noise standard deviation 𝜎𝑟 = 𝜎/𝑄. 
However, this only works when the additive noise standard 
deviation is higher than the ADC ideal code bin width as we will 
show next. 

It is chosen here to use normalized voltages, expressed in LSB 
(least significant bits) units, by dividing the voltages with the ideal 

ADC code bin width 𝑄. The normalized stimulus signal voltage 
is represented by 𝑦 and the normalized random noise voltage is 
represented by 𝑟. The normalized sampled voltage at instant 𝑡𝑗 is 
thus given by 

𝑢(𝑡𝑗) = 𝑦(𝑡𝑗) + 𝑟(𝑡𝑗) . (4) 

Considering that the normalized additive noise has a null 

mean and a standard deviation represented by 𝜎𝑟, we have for 
the sampled voltage, which is also a random variable: 

𝜇𝑢 = 𝑦,      𝜎𝑢 = 𝜎𝑟 . (5) 

Being the additive noise normally distributed, the sampled 
voltage probability density function is 

𝑓𝑢(𝑢|𝑦) =
1

√2 π 𝜎𝑟
⋅ 𝑒

−
(𝑢−𝑦)2

2 𝜎𝑟
2

 (6) 

and its distribution function is 

𝐹𝑢(𝑈|𝑦) = ∫ 𝑓𝑢(𝑢) ⋅ d𝑢
𝑈

−∞

=
1

√2 π 𝜎𝑟
⋅ ∫ 𝑒

−
(𝑢−𝑦)2

2⋅𝜎𝑟
2
⋅ d𝑢

𝑈

−∞

                    

=
1

2
+
1

2
⋅ erf(

𝑈 − 𝑦

√2 ⋅ 𝜎𝑟
) . 

(7) 

Due to the subtraction of the codes obtained with the two 
sample sets in (1), we can consider the ADC as having an ideal 
behavior since any fixed errors were eliminated by the 
subtraction and random errors can be considered as being 
present in the stimulus signal input. 

The probability 𝑝𝑘 of a sample having output code 𝑘 is equal 
to the probability of the sampled voltage being equal to or lower 

than the transition voltage 𝑇[𝑘 + 1] and equal to or greater than 
transition voltage 𝑇[𝑘] (for the middle codes): 

𝑝𝑘 = 𝑃{𝑈[𝑘] ≤ 𝑢 ≤ 𝑈[𝑘 + 1]}   ,   𝑘 = 1, . . . ,2
𝑛𝑏 − 2 (8) 

where we used the normalized transition voltage 𝑝0 = 𝑈[𝑘] =
𝑇[𝑘]/𝑄. Additionally, for the first code we have 

𝑝0 = 𝑃{𝑢 ≤ 𝑈[1]} (9) 

and for the last code we have 

𝑝2𝑛𝑏−1 = 𝑃{𝑈[2
𝑛𝑏 − 1] ≤ 𝑢} . (10) 

The probability 𝑝𝑘 can thus be expressed with the help of the 
sampled voltage distribution function: 

𝑝𝑘(𝑦)

= {

𝐹𝑢(𝑈[1]|𝑦)                                   , 𝑘 = 0

𝐹𝑢(𝑈[𝑘 + 1]|𝑦) − 𝐹𝑢(𝑈[𝑘]|𝑦), 𝑘 = 1,2, . . . ,2
𝑛𝑏 − 2

1 − 𝐹𝑢(𝑈[2
𝑛𝑏 − 1]|𝑦)               , 𝑘 = 2𝑛𝑏 − 1 .

 
(11) 

In Figure 1, the noise probability function and the probability 

of occurring code 𝑘 (shaded area) is represented. 
The mean, second moment and variance of the output codes 

are, by definition [20]: 

𝜇𝑘|𝑦 = ∑ 𝑘 ⋅ 𝑝𝑘(𝑦)

2𝑛𝑏−1

𝑘=0

,   𝑚2𝑘|𝑦 = ∑ 𝑘
2 ⋅ 𝑝𝑘(𝑦)

2𝑛𝑏−1

𝑘=0

 

𝜎𝑘|𝑦
2 = 𝑚2𝑘|𝑦 − 𝜇𝑘|𝑦

2  . 

(12) 

In Figure 2 the standard deviation of 𝑘 as a function of the 
noise standard deviation for two different values of stimulus 
signal is represented. It was calculated using (7), (11) and (12). 

For values of noise standard deviation much greater than the 

ideal code bin width (𝜎𝑟 ≫ 1), the standard deviation of the 

U[k+1]U[k]
u

f
u

0
 

Figure 1. Representation of the noise probability function and the probability 
of occurring code k (shaded area).  



 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 3 

output codes is the same as the standard deviation of noise (not 
clearly seen in Figure 2 because the horizontal axis only goes to 

1 LSB). This corresponds to the case 𝑄 → 0. In these 
circumstances the discrete variable 𝑘 becomes a continuous 
variable whose probability density function tends to the 
probability density function of noise. 

In this case the test can be used to estimate the noise standard 
deviation: 

𝜎𝑟 → ∞ ⇒ 𝜎𝑘|𝑦 → 𝜎𝑟 . (13) 

In the opposite situation, when the noise standard deviation 

is lower than 1 LSB (𝜎𝑟 < 1), it can be observed in Figure 2 that 
𝜎𝑘 is not equal to 𝜎𝑟 anymore, becoming even independent of 
𝜎𝑟 when it goes to 0 (the curve becomes horizontal). This makes 
it impossible to estimate the noise standard deviation from the 
output code’s standard deviation. 

With the goal of justifying the analytical expression presented 

in [1] for the relation between 𝜎𝑟 and 𝜎𝑘 we are going to analyze 
the case 𝜎𝑟 ≪ 1. In Figure 3 it can be seen the case where only 
two output codes, 𝑘 − 1 and 𝑘, have a significant probability of 
occurring for any value 𝑦 of stimulus signal.  

The index of the transition voltage closest to 𝑦 determines the 
value of 𝑘 and consequently which two codes occur for that 
particular value of 𝑦. Since the normalized transition voltages are 
integer values (considering an ADC transfer function with no 

true zero), the difference 𝑈[𝑘] − 𝑦 will always be between -0.5 
and 0.5: 

𝑈[𝑘] − 𝑦 = 𝑦 − ⌊𝑦 + 0.5⌋ , (14) 

where ⌊𝑥⌋ represents the highest integer lower than 𝑥. 
The probability that the output code of a sample is 𝑘 − 1 is 

𝑝𝑘−1(𝑦) = ∫ 𝑓𝑢(𝑢|𝑦) ⋅ d𝑢
𝑈[𝑘]

𝑈[𝑘−1]

 . (15) 

Because we are considering the case where the probability 
density function of the sampled voltage is negligible for voltages 

lower than 𝑈[𝑘 − 1], we can write 

𝑝𝑘−1(𝑦) = ∫ 𝑓𝑢(𝑢|𝑦) ⋅ 𝑑𝑢
𝑈[𝑘]

−∞

= 𝐹𝑢(𝑈[𝑘]|𝑦) . (16) 

The probability of occurring code 𝑘 is obviously 

𝑝𝑘(𝑦) = 1 − 𝑝𝑘−1(𝑦) = 1 − 𝐹𝑢(𝑈[𝑘]|𝑦) (17) 

since we considered that only codes 𝑘 and 𝑘 − 1 could occur. 
The mean of the output codes is thus 

𝜇𝑘|𝑦 = (𝑘 − 1) ⋅ 𝑝𝑘−1(𝑦) + 𝑘 ⋅ 𝑝𝑘(𝑦) (18) 

and the variance 

𝜎𝑘|𝑦
2 = (𝑘 − 1)2 ⋅ 𝑝𝑘−1(𝑦) + 𝑘

2 ⋅ 𝑝𝑘(𝑦) − 𝜇𝑘|𝑦
2              

= 𝑝𝑘−1(𝑦) − 𝑝𝑘−1
2 (𝑦) . 

(19) 

From (7) and (16) this variance can be written as 

𝜎𝑘|𝑦
2 = [

1

2
+
1

2
⋅ erf(

𝑈[𝑘] − 𝑦

√2 ⋅ 𝜎𝑟
)]

⋅ [
1

2
−
1

2
⋅ erf(

𝑈[𝑘] − 𝑦

√2 ⋅ 𝜎𝑟
)], 𝜎𝑟 ≪ 1 , 

(20) 

which can be rewritten as 

𝜎𝑘|𝑦
2 =

1

4
−
1

4
⋅ erf2 (

𝑈[𝑘] − 𝑦

√2 ⋅ 𝜎𝑟
)   ,   𝜎𝑟 ≪ 1 . (21) 

This conditional variance is represented in Figure 4 as a 
function of the DC input voltage. Note that this function is 

periodic because the transition voltage 𝑈[𝑘] in [21] is the closest 
to 𝑦, and has a period of 1 because the voltage 𝑦 is normalized 
to the ideal code bin width (𝑄).  

To justify the values of code standard deviation in the absence 

of noise (𝜎𝑟 → 0) consider the following two situations. First 
the DC input voltage is equal to one of the transition voltages 

(𝑈[𝑘]), as represented in Figure 5a.  
Half of the represented curve is to the left of the transition 

voltage and the other to the right. The probability of having code 

𝑘 − 1 is equal to the probability of having code 𝑘 (50%) so the 
variance is 1/4 and the standard deviation is 1/2 as seen by the 

dashed line in Figure 2 for 𝜎𝑟 ≪ 1. In Figure 5b, the case where 

 

Figure 2. Representation of the code variance as a function of the additive 
noise standard deviation. The dashed line represents the situation where the 
input DC voltage is exactly equal to one of the ADC transition voltages and 
the solid line represents the case where the DC input voltage is exactly 
between two consecutive transition voltages.  

U[k+1]U[k]
u

f
u

U[k−1]

p
k-1

y

p
k

 

Figure 3. Representation of the probability density function of noise.  

 

Figure 4. Representation of the conditional variance of the output codes as 
a function of the DC input voltage.  



 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 4 

the DC input voltage is exactly between two consecutive 
transition voltages is represented. When the noise standard 

deviation goes to 0, the probability of having code 𝑘 is 1, so the 
standard deviation is 0 (solid line in Figure 2). For intermediate 
cases the curve is between the two lines in Figure 2. 

In conclusion, the use of a DC input signal is not appropriate 
when estimating small values of additive noise because the 
output codes of the acquired samples do not vary enough to 
make their standard deviation directly related to the additive 
noise standard deviation, as stated in [1]. 

3. TRIANGULAR STIMULUS SIGNAL 

To overcome the problem referred in the previous section, 
the IEEE standard [1] suggests the use of a triangular stimulus 
signal that spans several ADC codes (about 10). Due to the 
subtraction of the codes obtained from the two sets of samples, 
the mean square error is still twice that of the additive noise if 
the sets are acquired synchronously in order for the samples of 
both sets to have the same phase. 

The variance of the output codes can be calculated from the 

amplitude distribution, 𝑓𝑦, of the stimulus signal [20]. 

𝜎𝑘
2 = ∫ 𝜎𝑘|𝑦

2 (𝑦) ⋅ 𝑓𝑦(𝑦) d𝑦
∞

−∞

 . (22) 

For a triangular stimulus signal, with an amplitude 𝐴 and an 
offset 𝐶, normalized by the ideal code bin width (𝐴𝑄 = 𝐴/𝑄 and 

𝐶𝑄 = 𝐶/𝑄) the amplitude distribution is 

𝑓𝑦(𝑦) = {

1

2 𝐴𝑄
   ,   |𝑦 − 𝐶𝑄| < 𝐴𝑄

0         ,    otherwise .

 (23) 

The output codes variance is thus 

𝜎𝑘
2 =

1

2 𝐴𝑄
∫ 𝜎𝑘|𝑦

2 (𝑦) d𝑦

𝐶𝑄+𝐴𝑄

𝐶𝑄−𝐴𝑄

 . (24) 

Inserting (21) leads to 

𝜎𝑘
2 =

1

2 𝐴𝑄
∫ [

1

4
−
1

4
⋅ erf2 (

𝑈[𝑘] − 𝑦

√2 ⋅ 𝜎𝑟
)]d𝑦

𝐶𝑄+𝐴𝑄

𝐶𝑄−𝐴𝑄

,  

𝜎𝑟 ≪ 1 . 

(25) 

The corresponding standard deviation is represented in 
Figure 6 by a solid line. 

Equation (24) states that the variance 𝜎𝑘
2 is the average of the 

curve in Figure 4, which is the representation of (21), in an 

interval with length 2𝐴𝑄 centered at𝐶𝑄.  

3.1. Low Noise Approximation 

The integral in (24) does not have a closed form. We can, 
however, obtain an approximated expression. To achieve that we 
look into the case where we have a small value of noise standard 
deviation. In this situation the individual bell-shaped curves in 

Figure 4 do not overlap each other. Furthermore, when 𝐴𝑄 goes 

to  the result of (24) is the same as integrating the curve in 

Figure 4 from −1 to 1. In this case the result in the value of (24) 

is independent on the triangular wave offset 𝐶. Mathematically 
this statement can be expressed as 

𝜎𝑘
2 =

1

2𝐴𝑄
∫ 𝜎𝑘|𝑦

2 (𝑦) d𝑦

𝐶𝑄+𝐴𝑄

𝐶𝑄−𝐴𝑄

|

𝐴𝑄→∞

=
1

2
∫𝜎𝑘|𝑦

2 (𝑦) d𝑦

1

−1

 . 

(26) 

Using (21) we can write 

lim
𝐴𝑄→∞

𝜎𝑘
2 =

√2 ⋅ 𝜎𝑟
4

∫[1 − erf2(𝑥)]𝑑𝑥

∞

−∞

   ,   𝜎𝑟 ≪ 1 . (27) 

This integral can be numerically calculated leading to a value 

of √8 π⁄ . The variance is thus 

lim
𝜎𝑟→0
𝐴𝑄→∞

𝜎𝑘
2 =

1

√π
𝜎𝑟 . (28) 

This approximation is represented in Figure 6 by the dashed-
dot line. It can be seen that, in fact, it approximates the solid thin 

line for small values of 𝜎𝑟. 

3.2. High Noise Approximation  

We will look now into the case of a noise standard deviation 

much higher than the ideal code bin width(𝜎𝑟 → ∞). When 
carrying out the test with a DC stimulus signal with normalized 

voltage 𝑦, the variance of 𝑘 conditional to 𝑦 does not depend on 
the value of 𝑦 and is equal to the random noise variance as stated 
in (13). In the case of a triangular stimulus signal this leads, from 
(24), to 

lim
𝜎𝑟→∞

𝜎𝑘
2 = 𝜎𝑟

2 . (29) 

U[k+1]U[k]
u

U[k−1]

f
u

p
k-1

=0.5 p
k
=0.5

      U[k+1]U[k]
u

f
u

p
k
=1

 
 a) b) 

Figure 5. Representation of the probability density function of noise and the 

probability of occurring code k − 1 and code k.  

 

Figure 6. Representation of the code standard deviation as a function of the 
additive noise standard deviation (solid line) for a triangular stimulus signal 
with amplitude Q/2. The dashed and the dot-dashed lines represent the 
approximation for small and large additive noise standard deviation given by 
(28) and (29) respectively. The circles represent the uncertainty of the codes 
standard deviation obtained experimentally.  



 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 5 

This situation is represented by the dotted line in Figure 6. 
The goal of the Random Noise Test is to be able to estimate 

the random noise standard deviation, 𝜎�̂� from the measured 
mean square error (𝑚𝑠𝑒). 

3.3. IEEE 1057 Noise Estimator 

As we have demonstrated in the previous sections, the 
expected value of the mean square error is a function of the 
random noise standard deviation. Combining (28) and (29) with 
(2), leads to 

lim
𝜎𝑟→0
𝐴𝑄→∞

𝜇mse =
2

√𝜋
𝜎𝑟 lim

𝜎𝑟→∞
𝜇mse = 𝜎𝑟

2 . 
(30) 

From these approximations the IEEE standard [1] presents a 
heuristically derived expression for the random noise standard 
deviation estimation:  

𝜎�̂� = [(√
𝑚𝑠𝑒

2
)

−𝑅

+ (√π
𝑚𝑠𝑒

2
)
−𝑅

]

−
1
𝑅

,    𝑅 = 4. (31) 

Note that the variables 𝜎 and 𝑚𝑠𝑒 used in [1] are expressed 
in volt and squared volt respectively while the variables 𝜎𝑟 and 
𝑚𝑠𝑒 used here are dimensionless. 

The error of the additive noise standard deviation estimation, 
represented in Figure 7, is given by 

𝑒𝜎�̂� = 𝜎�̂� − 𝜎𝑟 . (32) 

It can be seen that the error is always smaller, in absolute 

value, than 0.022 LSB. When 𝜎𝑟 → ∞ the error goes to 0. The 
factor 4 used in (31) is very close to the value of 3.8203 which 
leads to the optimum approximation for an expression of the 
kind of (31) and corresponds to the position of the minimum in 
Figure 8. 

We have, so far, showed how the expression for the 
estimation of the random noise standard deviation from the 
measured mean square error recommended in the IEEE 1057-
94 standard was derived. the remaining of this paper we will go 
a step further by showing how this test can be carried out with a 
sinusoidal stimulus signal as well. We will also focus on the 
influence that the stimulus signal amplitude and offset have on 
the test results. As we saw in this section, the expression for the 
random noise estimator was heuristically derived from 
approximations for which an infinite stimulus signal amplitude is 
assumed. Furthermore, in the IEEE standard a value of 5 LSB 
for the stimulus signal amplitude is recommended without any 

justification. In test conditions where the stimulus signal 
generator has low accuracy on the wave amplitude and has offset 
error or the ADC under test has high resolution, it is difficult to 
have a stimulus signal amplitude exactly equal to 5 LSB. It is thus 
important to know how different values of stimulus signal 
amplitude and offset affect the bias of the estimated random 
noise standard deviation. This will be carried out here for both 
types of stimulus signal shapes – triangular and sinusoidal. As it 
will be shown, the higher the amplitude the lower the bias. The 
goal will be to have an expression that will tell us what should be 
the minimum value to use in order to guarantee an upper bound 
on the estimator bias.   

4. SINUSOIDAL STIMULUS SIGNAL 

Although the IEEE 1057 standard recommends the use of a 
triangular wave signal for the cases where the random noise to 
be measured is small compared to the quantization step, this test 
can also be carried out with a sine wave as we will show here. 
Since sine wave generators are more common than triangular 
wave ones, this will benefit those wanting to measure small 
values of random noise but without having to look for a 
triangular wave generator. 

The variance of the output codes can be calculated the same 
way as in the case of triangular wave stimulus, using (22), but 

with a different amplitude distribution, 𝑓𝑦. For a sine wave, one 
has [8]: 

𝑓𝑦(𝑦) =

{
 

 
1

π√𝐴𝑄
2 − (𝑦 − 𝐶𝑄)

2
   ,   |𝑦 − 𝐶𝑄| < 𝐴𝑄

0         ,    otherwise .

 (33) 

Inputting this into (22) leads to 

𝜎𝑘
2 = ∫ 𝜎𝑘|𝑦

2 (𝑦)
1

𝜋√𝐴𝑄
2 − (𝑦 − 𝐶𝑄)

2
d𝑦

𝐶𝑄+𝐴𝑄

𝐶𝑄−𝐴𝑄

 . 
(34) 

The strategy used in the case of the triangular stimulus signal 
was to derive two expressions for the asymptotic cases of high 
and low random noise standard deviation and then heuristically 
construct an expression for the estimator. As it turns out, those 
two asymptotic expressions are the same regardless of the 
stimulus signal shape.  

For large values of random noise we concluded earlier that 
the variance of the output codes is independent of the DC 
stimulus signal value. Using eq. (13) in (34) leads to  

Figure 7. Representation of the estimation error of the additive noise 
standard deviation, obtained with (31), as a function of the actual standard 
deviation.  

 

Figure 8. Representation of the estimation error of the additive noise 
standard deviation, obtained with (31), as a function of the parameter R.  



 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 6 

𝜎𝑘
2|𝜎𝑟→∞ = 𝜎𝑟

2 ⋅ ∫
1

π√𝐴𝑄
2 − (𝑦 − 𝐶𝑄)

2
d𝑦

𝐶𝑄+𝐴𝑄

𝐶𝑄−𝐴𝑄

= 𝜎𝑟
2 

(35) 

which is the same as the one obtained for the triangular shaped 
stimulus signal (eq. (29)). 

For small values of random noise standard deviation, we 
made an extra approximation that was to consider an infinite 
stimulus signal amplitude. The ADC input range will thus only 
cover the middle of the stimulus signal range. There the 
sinusoidal shape can be approximated by a straight line which 
makes it similar to a triangular wave and thus will lead to the same 
approximation for the output codes variance:  

𝜎𝑘
2|𝜎𝑟→0
𝐴𝑄→∞

=
1

√π
𝜎𝑟 . (36) 

It is thus shown that the same heuristically derived expression 
for the triangular stimulus case, eq. (31), can be employed when 
using a sinusoidal stimulus signal. 

5. EXPERIMENTAL VALIDATION 

A triangular stimulus signal, with amplitude 39.216 mV and 
frequency 10 Hz, was applied to a 12-bit data acquisition board 

(Keithley DAS 1601) in the 10 V range and two sets of 10000 
samples were acquired at a sampling frequency of 100 kHz. Only 
the 8 most significant bits were used so that the ADC would 
present an almost ideal behavior. This was validated by testing 
the data acquisition board (using all 12 bits) with the Standard 
Histogram Test and obtaining an integral non-linearity (INL) and 
differential non-linearity (DNL) lower than 0.5 LSB. 

A Stanford DS360 function generator was used to generate 
the additive noise and a Wavetek 9100 calibrator was used to 
generate the triangular wave. The data acquisition board used 
possesses 16 single-ended or 8 differential inputs. It was one of 
those differential inputs that was used (CH0). The Stanford 
DS360 function generator was connected between pins 37 
(“CH0 HI IN”) and 19 (“LL GND”) and the Wavetek 9100 
calibrator was connected between pins 18 (“CH0 LOW IN”) and 
19 (“LL GND”). This way the gaussian noise and the triangular 
wave were added (actually, subtracted) internally. 

The results are represented in Figure 6 were a good agreement 
with the theoretical curve, represented by the thin solid line is 
observed. 

6. STIMULUS SIGNAL AMPLITUDE 

In the previous sections we addressed the problem of 
determining an estimator for the random noise standard 
deviation when using a triangular and a sinusoidal stimulus signal.  

As observed in equation (24), the code standard deviation 
depends on the stimulus signal amplitude. This can be verified in 

Figure 9 where two cases are plotted, one for 𝐴𝑄 = 0.5 (dotted 

line) and another for 𝐴𝑄 = 0.75 (solid line). The difference is 
highest for values of noise standard deviation smaller than 
0.5 LSB. 

That dependence can be also observed in Figure 10 for two 
different values of triangular wave offset. The case of null 

amplitude (𝐴𝑄 = 0) corresponds to the DC input case 
represented in Figure 2. The higher the triangular wave 
amplitude the less influence it has on the output code standard 

deviation. Whenever the amplitude 𝐴 is a multiple of 𝑄/2, that 
is, 2𝐴𝑄 is integer, the code standard deviation has the same value. 

When performing the test to estimate the random noise 
standard deviation, a value of triangular wave amplitude has to 
be chosen. This choice will influence the estimated value, 
however, observing Figure 10, there are some values that seem 
more advantageous than others, namely the odd multiples of 

𝑄/2 (𝐴𝑄 = 0.5,1.5,2.5,…) because the slope of the curve in 
those points is smaller making the estimate less sensible to errors 
in the triangular wave amplitude that in practical conditions are 
sure to exist. Also, in that case, the output code variance becomes 
practically independent of the stimulus signal offset. The higher 
the triangular wave amplitude, the higher will be the additive 
noise introduced by the function generator (because of its 
different ranges with higher amplification) making very high 
values of amplitude not advisable. 

The IEEE standard [1] suggests the use of 𝐴𝑄 = 5 without 
explaining the reason. We hope the work presented here sheds 
some light on the subject. As seen here, a marginally better choice 

would be, for instance, 𝐴𝑄 = 5.5 since the slope of the curve in 

that point is smaller than at 𝐴𝑄 = 5 and consequently the test 
would be less affected by eventual inaccuracies in the signal 
amplitude. 

 

Figure 9. Representation of the code standard deviation as a function of the 
additive noise standard deviation for a triangular stimulus signal with 
amplitude Q/2 (dotted line) and an amplitude 0.75Q (solid line).  

 

Figure 10. Representation of the code standard deviation as a function of the 
triangular wave amplitude divided by the ideal code bin width. The solid line 
represents the situation of an offset equal to an ADC transition voltage and 
the dotted line a situation where the offset value is exactly between two 
consecutive transition voltages. The dotted line represents the value taken 
by the code standard deviation when the triangular amplitude is Q/2. A 
normalized value of noise standard deviation of 0.1 (one tenth of the ideal 
ADC code bin width) was used.  

 

0

0,1

0,2

0,3

0,4

0,5

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

A Q  (LSB)

sk  (LSB)



 

ACTA IMEKO | www.imeko.org June 2023 | Volume 12 | Number 2 | 7 

7. CONCLUSIONS 

In this paper, the test method proposed by the IEEE 1057 
standard [1] to estimate the random noise standard deviation was 
analyzed in detail. The expression presented there to calculate the 
additive noise standard deviation from the ADC output codes 
mean square error is explained here from a different perspective. 
The influence of both triangular wave amplitude and offset on 
the estimation error was studied. In Figure 11, the maximum 
estimation error for different values of triangular stimulus signal 
amplitude and offset is represented. It can be seen that for small 
values of amplitude the maximum error depends strongly on the 
triangular wave offset. The actual error on estimating the noise 
standard deviation when using the expression presented in [1] is 
calculated and it was concluded that it was smaller than 2.2% of 
the ideal ADC code bin width (0.022 LSB). The choice of 
triangular stimulus amplitude is studied and a justification for the 
value proposed in [1] is given.  

We suggest here that in the case were a triangular or sinusoidal 
wave generator is not available the test be carried out with a DC 
stimulus signal, with any value inside the ADC range, and that 
the noise standard deviation be estimated as being equal to the 
code root mean square error, divided by 2. If the value obtained 
is smaller than Q/2 the value used as the noise standard deviation 
estimative should be Q/2. Of course, the standard deviation 
could be smaller, but, as shown here, it could not be higher. In 
summary, we propose, just for the case where a triangular or sine 
wave generator cannot be used, the following expression for the 
estimation: 

𝜎�̂� = min(
1

2
,√
𝑚𝑠𝑒

2
) . (37) 

Experimental results were presented that validate the 
approach taken. 

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