Capability of detection

Background noise exists ubiquitously in analytical instruments, whether or not a sample is applied to the instrument. This document is concerned with mathematical methodologies for estimating the minimum detectable value in case that the most predominant source of measurement uncertainty is background noise. The minimum detectable value can directly and mathematically be derived from the stochastic characteristics of the background noise. This document specifies basic methods to — extract the stochastic properties of the background noise, — use the stochastic properties to estimate the standard deviation (SD) or coefficient of variation (CV) of the response variable, and — calculate the minimum detectable value based on the SD or CV obtained above. The methods described in this document are useful for checking the detection of a certain substance by various types of measurement equipment in which the background noise of the instrumental output predominates over the other sources of measurement uncertainty. Feasible choices are visible and ultraviolet absorption spectrometry, atomic absorption spectrometry, atomic fluorescence spectrometry, luminescence spectrometry, liquid chromatography and gas chromatography.

Capacité de détection

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Publication Date
04-Sep-2018
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6060 - International Standard published
Start Date
17-Jul-2018
Completion Date
05-Sep-2018
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INTERNATIONAL ISO
STANDARD 11843-7
Second edition
2018-09
Capability of detection —
Part 7:
Methodology based on stochastic
properties of instrumental noise
Capacité de détection —
Partie 7: Méthodologie basée sur les propriétés stochastiques du bruit
instrumental
Reference number
ISO 11843-7:2018(E)
ISO 2018
---------------------- Page: 1 ----------------------
ISO 11843-7:2018(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2018

All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may

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on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address

below or ISO’s member body in the country of the requester.
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Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2018 – All rights reserved
---------------------- Page: 2 ----------------------
ISO 11843-7:2018(E)
Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions ..................................................................................................................................................................................... 2

4 Quantitative analysis and background noise .......................................................................................................................... 3

4.1 Error sources of analysis ................................................................................................................................................................ 3

4.2 Random processes in background ......................................................................................................................................... 3

5 Theories for precision .................................................................................................................................................................................... 4

5.1 Theory based on auto-covariance function .................................................................................................................... 4

5.2 Theory based on power spectrum ......................................................................................................................................... 6

6 Practical use of FUMI theory .................................................................................................................................................................... 9

6.1 Estimation of noise parameters by Fourier transform ......................................................................................... 9

6.2 Estimation of noise parameters by autocovariance function ......................................................................11

6.3 Procedures for estimation of SD ...........................................................................................................................................11

Annex A (informative) Symbols and abbreviated terms used in this document..................................................14

Annex B (informative) Derivation of Formula (7) ................................................................................................................................15

Annex C (informative) Derivation of Formulae (14) to (16) .....................................................................................................16

Bibliography .............................................................................................................................................................................................................................18

© ISO 2018 – All rights reserved iii
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ISO 11843-7:2018(E)
Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards

bodies (ISO member bodies). The work of preparing International Standards is normally carried out

through ISO technical committees. Each member body interested in a subject for which a technical

committee has been established has the right to be represented on that committee. International

organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.

ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of

electrotechnical standardization.

The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/patents).

Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.

For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO's adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see www .iso

.org/iso/foreword .html.

This document was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,

Subcommittee SC 6, Measurement methods and results.

This second edition cancels and replaces the first edition (ISO 11843-7:2012), which has been technically

revised.
The main changes compared to the previous edition are as follows:
— created a new 6.2;
— 6.2 of the first edition is renumbered 6.3.
A list of all parts in the ISO 11843 series can be found on the ISO website.

Any feedback or questions on this document should be directed to the user’s national standards body. A

complete listing of these bodies can be found at www .iso .org/members .html.
iv © ISO 2018 – All rights reserved
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ISO 11843-7:2018(E)
Introduction

The series of ISO 11843 is based on the probability distributions of the net state variable (measurand)

for both the linear and nonlinear calibration situations. The focus is implicitly, though sometimes

explicitly, on the uncertainty associated with an estimate of the measured response predominantly

coming from the baseline noise in instrumental analysis. In many, if not most, analytical instruments,

the baseline noise is considered the prime cause of uncertainty when the sample amount is as low as the

minimum detectable value. Within its domain of applicability, the method given in this document can

dispense with the repetition of real samples, thus helping to improve global environments by saving

time and energy that would be required by repetition.

The basic concept of ISO 11843-7 is the mathematical description of the probability distribution of

the response variable in terms of mathematically well-defined random processes. This description

straightforwardly leads to the minimum detectable value. As for the relation of the response and

measurand, linear and nonlinear calibration functions can be applied. In this manner, compatibility

with ISO 11843-2 and ISO 11843-5 is ensured.

The definition and applicability of the minimum detectable value are described in ISO 11843-1 and

ISO 11843-2; the definition and applicability of the precision profile are described in ISO 11843-5. The

precision profile expresses how the precision changes depending on the net state variable. ISO 11843-7

specifies the practical use of the fundamental concepts in ISO 11843 in case of the background noise

predominance in instrumental analysis.

The minimum detectable value, x , is generally expressed in the unit of the net state variable. If the

calibration function is linear, the SD or CV of the response variable estimated in this document can

linearly be transformed to the SD or CV of the net state variable, which in turn can be used for the

estimation of the minimum detectable value, x .

If the calibration function is nonlinear, the precision profile of the response variable in this document

needs to be transformed to the precision profile of the net state variable as shown in ISO 11843-5. In

this situation, the contents of ISO 11843-5 can be used for this purpose without modification.

© ISO 2018 – All rights reserved v
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INTERNATIONAL STANDARD ISO 11843-7:2018(E)
Capability of detection —
Part 7:
Methodology based on stochastic properties of
instrumental noise
1 Scope

Background noise exists ubiquitously in analytical instruments, whether or not a sample is applied

to the instrument. This document is concerned with mathematical methodologies for estimating the

minimum detectable value in case that the most predominant source of measurement uncertainty is

background noise. The minimum detectable value can directly and mathematically be derived from the

stochastic characteristics of the background noise.
This document specifies basic methods to
— extract the stochastic properties of the background noise,

— use the stochastic properties to estimate the standard deviation (SD) or coefficient of variation (CV)

of the response variable, and
— calculate the minimum detectable value based on the SD or CV obtained above.

The methods described in this document are useful for checking the detection of a certain substance

by various types of measurement equipment in which the background noise of the instrumental

output predominates over the other sources of measurement uncertainty. Feasible choices are visible

and ultraviolet absorption spectrometry, atomic absorption spectrometry, atomic fluorescence

spectrometry, luminescence spectrometry, liquid chromatography and gas chromatography.

2 Normative references

The following documents are referred to in the text in such a way that some or all of their content

constitutes requirements of this document. For dated references, only the edition cited applies. For

undated references, the latest edition of the referenced document (including any amendments) applies.

ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in

probability
ISO 3534-2, Statistics — Vocabulary and symbols — Part 2: Applied statistics
ISO 3534-3, Statistics — Vocabulary and symbols — Part 3: Design of experiments

ISO 5725-1, Accuracy (trueness and precision) of measurement methods and results — Part 1: General

principles and definitions
ISO 11843-1, Capability of detection — Part 1: Terms and definitions

ISO 11843-2, Capability of detection — Part 2: Methodology in the linear calibration case

ISO 11843-5, Capability of detection — Part 5: Methodology in the linear and non-linear calibration cases

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ISO 11843-7:2018(E)
3 Terms and definitions

For the purposes of this document, the terms and definitions given in ISO 3534-1, ISO 3534-2,

ISO 3534-3, ISO 5725-1, ISO 11843-1, ISO 11843-2, ISO 11843-5 and the following apply. A list of symbols

and abbreviated terms used in this document is provided in Annex A.

ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https: //www .iso .org/obp
— IEC Electropedia: available at http: //www .electropedia .org/
3.1
precision profile

mathematical description of the standard deviation (SD) of the response variable

[σ (X)] or net state variable [σ (X)] as a function of the net state variable
Y X

Note 1 to entry: The coefficient of variation (CV) of the response variable or net state variable as a function of the

net state variable is also referred to as a precision profile.

Note 2 to entry: Precision means the SD or CV of the observed response variable or SD or CV of the net state

variable when estimated by the calibration function (see ISO 11843-5).

[SOURCE: ISO 11843-5:2008, 3.4, modified — “coefficient of variation” has been removed and Note 1 to

entry has been added instead. Note 2 to entry has also been added.]
3.2
minimum detectable value of the net state variable

value of the net state variable in the actual state that will lead, with probability 1 – β, to the conclusion

that the system is not in the basic state

Note 1 to entry: Under the assumption that the SD, σ (X), of the net state variable is constant [(σ (X) = σ ], the

X X X
minimum detectable value, x , is defined as
xk=+ k σ (1)
dc d X
where

k denotes a coefficient to specify the probability of an error of the first kind;

k is a coefficient to specify the probability of an error of the second kind.

If the SD, σ , of the response variable is assumed to be constant [σ (X) = σ ], then the minimum detectable value

Y Y Y
can be calculated by the following Formula (2):
xk=+ k σ //ddYX (2)
()()
dc d Y

where |dY/dX| denotes the absolute value of the slope of the linear calibration function and is constant.

Note 2 to entry: If the net state variable is normally distributed, the coefficients k = k = 1,65 specify the

c d

probabilities of an error of the first and second kinds (= 5 %) and Formula (1) can simply be written as x = 3,30σ .

d X

Note 3 to entry: If k = k = 1,65, Formula (1) takes the form that σ / x = 1/3,30 = 30 %. Therefore, x can be

c d X d d

found in the precision profile (3.1). x is located at X, the CV of which is 30 %.

Note 4 to entry: Different types of precision profiles (3.1) are defined, but they can be transformed to each other.

For example, the SD, σ (X), of the response variable can be transformed to the SD, σ (X), of the net state variable

Y X

by means of the absolute value of the derivative, |dY/dX|, of the calibration function [Y = f(X)]: σ (X) = ⌠ (X)/|dY/

X Y

dX| (see ISO 11843-5). This treatment is an approximation, the extent of which depends on local curvature,

2 2
involving d Y/dX .
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ISO 11843-7:2018(E)

[SOURCE: ISO 11843-5:2008, 3.2, modified — Notes to entry 1 to 4 have been added.]

4 Quantitative analysis and background noise
4.1 Error sources of analysis

The quantitative analysis to obtain a measurand from a sample is generally considered to consist

of preparation, instrumental analysis, data handling and calibration. These steps of analysis are

mechanically independent of each other and so are probabilistically independent as well.

This document applies only to instrumental analysis. However, the errors from the other steps affect

the error of the final value of the measurand, as well. That is, the combined uncertainty associated with

an estimate of the measurand depends on the propagation of all uncertainties relating to the relevant

steps. The following conditions are necessary for the use of ISO 11843-7.

At concentrations near the minimum detectable value in chromatography, the error from the sample

injection into a chromatograph is even less important (e.g. CV = 0,3 % in a recent apparatus) than the

background noise (CV = 30 % by definition). If the importance of a factor other than noise is comparable

to that of the noise, the methodologies of this document are not applicable.

Data handling is usually a process to extract a signal component from noisy instrumental output such

as peak height or area, which is a relative height of a summit of a peak-shaped signal or integration of

intensities over a signal region, respectively. The statistical influences of this process are the major

concern of this document. The use of a digital or analogue filter can also be taken into account, if the

noise after the filtration is analysed for this purpose.
4.2 Random processes in background

Typical examples of the response variable are area and height measured in chromatography. In this

document, intensity difference [Formula (6)] and area [Formulae (10) and (11)] are taken as the

difference and summation of intensities Y of instrumental output. The response variables are usually

independent of each other even if they are obtained from consecutive measurement by the same

instrument. On the other hand, the consecutive intensities Y are formulated as a time-dependent

[1]
random process, and in many cases, can be considered 1/f noise .

The power spectrum, P( f ), of 1/f noise has a slope inversely proportional to frequency, f:

Pf ∝ (3)
when f is near zero.

The simplest model of random processes is the white noise. Let w denote the random variable of the

white noise at point i. By definition, the mean of the white noise is zero and the SD, w , of the white noise

is constant at every point i. A prominent feature of the white noise is that the noise intensities, w and

w , are independent of each other, if i ≠ j.

The autoregressive process [AR(1)-process] of first order is a mathematical model in which the

intensities, M and M , are not independent of each other (i ≠ j). The AR(1)-process is treated as a major

i j

component of time-dependent changes of instrumental output [see Formula (9)]. The AR(1)-process at

point i is defined to take the form
MM=+ρ m (4)
ii−1 i
where
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ISO 11843-7:2018(E)
m denotes the random variable of the white noise at point i;
ρ is a constant parameter (−1 < ρ < 1).
5 Theories for precision
5.1 Theory based on auto-covariance function
[2][3][4]
A theory has been proposed based on an auto-covariance function:
 
ψ τ ≡ EY Y (5)
st +τ t
 
 00s 

where E[·] denotes the mean of a random variable inside the square brackets over t .

Figure 1 — Signal (upper line) and noise (lower line) with intensity difference

The upper part of Figure 1 depicts the signal as (an approximation to) a rectangular pulse. Noise

(constituting background) on the signal is depicted as the oscillatory curve in the lower part of the

figure. t denotes a time value on the background portion of the signal and t + τ denotes a time value

0 0 s

on the signal itself. The measurement (signal reading) is the difference in intensities at times t and

t + τ . The value of the signal would be zero at t in the absence of background noise. The signal has a

0 s 0

finite value at t + τ when a sample is measured. In the ISO 11843-7 measurement model, the signal and

0 s

noise are superimposed, and this mixed random process takes the value Y at time t . The intensities at

i i

times t and t + τ are described as Y and Y , respectively, and the intensity difference is given

0 0 s
t t +τ
0 0 s
by Formula (6).
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ISO 11843-7:2018(E)
Figure 2 — Auto-covariance function of noise

The difference between the values of the auto-covariance function, ψ(τ), at 0 and τ gives the right side

of Formula (7).

Near the minimum detectable value, which is dictated by the background fluctuation, intensity

difference often applies in analytical optical spectrometry. The definition of intensity difference, e.g.

[2][3][4]
signal reading corrected for background, is :
ΔYY=−Y (6)
tt+τ
00s

Here, ΔY corresponds to the response variable Y. The variance of the intensity difference is written as

[2][3][4]
shown in Formula (7) . [For the derivation of Formula (7), see Annex B.]
 
σψ= 20 −ψτ (7)
() ()
ΔY s
 

Formula (7) is of practical use when the actual auto-covariance functions, ψ(0) and ψ(τ ), are known

from the observation of background noise as shown in Figure 2. The substitution of Formula (7) for

Formula (2) (σ = σ ) leads to the minimum detectable value.
Y ΔY
[5]

Use can be made of the Wiener-Khintchine theorem , which relates the auto-covariance function to

the power spectral density through the Fourier transform:
ψ ττ= Sf Gf cos 2πffd (8)
() () () ()
sb s
where
S ( f ) denotes the power spectrum of the observed background noise;
G( f ) is the frequency response of the (linear) read-out system.

Formula (8) indicates the estimation of the measurement SD, Formula (7), through the noise power

spectrum.
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ISO 11843-7:2018(E)
5.2 Theory based on power spectrum

A theory based on the power spectrum of the baseline, which is called FUMI (function of mutual

[6][7][8]

information) , provides the SD values of the measured area and height in instrumental analysis.

These measured values are the integration of the instrumental output over the integration region, as

illustrated in Figure 3. If the signal (shape and size) is invariant, the error of measured area or height,

as long as it comes from the noise alone, is equal to the area created by the noise over the integration

region. That is, the measurement error is the same as the noise-created area. The SD of the noise-

created areas coincides with the SD of measured heights or areas.
Key
Z zero window
S signal region
I integration region
NOTE Additional symbols are explained in Annex A.
Figure 3 — Signal and noise with zero window and integration area
The number of data points over the integration region is k − k .
f c

In the FUMI theory, the noise intensity, Y , at point i is described as the mixed random processes of the

white noise and AR(1)-process:
Yw=+M (9)
ii i

The purpose of the FUMI theory is to estimate the SD of the noise-created areas, A , over the integration

region (see Figure 3).

In practice, especially chromatography, different modes of integration are adopted as illustrated in

Figure 4. The measurement is of the integrated intensities above the baseline, which is horizontal or

oblique within the domain [k + 1, k ]. The horizontal baseline is horizontally drawn from the intensity

c f

(corrected) at the zero point, and the oblique line is drawn between the intensities at the edges of

6 © ISO 2018 – All rights reserved
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ISO 11843-7:2018(E)

the signal region [0, k ]. The latter is often useful for a slowly changing background, called “drift” in

chromatography.

Figure 4 — Integration modes over horizontal line (dashed line) and oblique line (solid line)

between t and t
k +1 k
c f

The noise-created area, which is the area between the random path and horizontal line over the

integration region without signal, can be written as
AY= Δt (10)
ik=+1
where
Y the noise intensity is described by Formula (9);
Δt is the time interval between consecutive data.
Here, A means the response variable Y.

If the oblique line is used as the baseline, the noise-created area takes the form:

 
 
AY= ΔtA− (11)
Fi T
 
 
ik=+1
 

where A denotes the area of the trapezoid created by the oblique line, horizontal line and vertical

lines at the edges of the integration region from k + 1 to k (see Figure 4). The area of the trapezoid in

c f

Figure 4 is taken with a negative sign. The area is taken with a positive sign if the oblique line lies above

the horizontal line in the integration region and with a negative sign otherwise.

The general expression of the SD, σ , of noise-created areas is:
12/
 2 
 
σ = EA (12)
 
 
 
 

where E[·] denotes the ensemble mean of a random variable inside the square brackets. It should

be noted that E[A ] = 0, since, by definition, the ensemble mean of the noise-created area over the

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ISO 11843-7:2018(E)

horizontal line [the first term in the right side of Formula (11)] is zero and the ensemble mean of the

trapezoid, A , is zero (also see Annex C).

The above derivation is based on the assumption that there exists no uncertainty of the zero level,

i.e. Y = 0. In practice, however, this type of uncertainty exists and should be taken into account. The

measurement in chromatography is usually performed relative to the zero level, which is the average of

background intensities over a region referred to here as a zero window (see Figure 3).

The squared SD, σ , of measured areas (noise-created areas) within the zero window takes the

[6][7][8]
form :
22 2
σσ=+σ (13)
YZ F
2 2

where σ denotes the variance originating from the zero window and σ is the variance from the

Z F

measured area [σ given by Formula (12)]. The summary of the derivation of Formulae (14) and (15) is

given in Annex C. The minimum detectable value can be obtained by the substitution of Formula (13)

for Formula (2).
2 [6][7][8]
The variance, σ , can be described as
2 2
bb2
 
kk− kk−
() ()
1 − ρ 1 − ρ
fc fc
2 2 2 2
 
 
σ = w + b −2ρ + ρ m (14)
2 2
 
b 1 − ρ
1 − ρ
b 1 − ρ  
[6][7][8]

and the influence of the signal integration over the signal region takes the form :

σ =−kk w (first term, 15)
F fc
 2 kk− 
kk− ()
1 1 − ρ 1 − ρ
2 2
 
+ kk−−2ρ + ρ m (second term, 15)
2  2 
1 − ρ
()1 − ρ 1 − ρ
 
 
2kk −k
 
cf c
1 − ρ 1 − ρ
2 2
  
+ρ m (third term, 15)
 
1 − ρ
1 − ρ
 
+aw (fourth term, 15)
 kk−+1 −i 
 
kk−  
2kk −k −2k
   
kk−−i
ef c c
 
1− ρ  1− ρ kk+−1 1− ρ 1− ρ ec 
 
2 2
  ec   
+ α −2α ρ + ρ m
 
  
2 −2
   
1− ρ 1− ρ
 1− ρ 1− ρ   
 i=1 
   
 
 
 
(fifth term, 15)
where
kk− kk++1
()()
fc fc
α = (16)
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ISO 11843-7:2018(E)
denotes the SD of the white noise;

is the SD of the white noise included in the autoregressive process of first order;

is the constant parameter of the autoregressive process of first order;
k , k and k are defined in Figures 3 and 4;
c f e
b denotes the number of consecutive data points in the zero window [−b + 1, 0].

The five terms in Formula (15) denote the following stochastic contributions to the measurement

uncertainty:

— first term: the error from the white noise in the integration domain (k − k data points);

f c

— second term: the error from the AR(1)-process in the integration domain (k − k data points);

f c

— third term: the influence of k data points between the zero point and the starting point of integration;

— fourth term: the effect of the white noise in the oblique baseline;
— fifth term: the effect of the AR(1)-process in the oblique baseline.

NOTE The applicability of the FUMI theory is rather wide, but there are two typical situations where it is not.

One is that the predominant error source is not background noise. In mass spectroscopy, if the ionization process

produces much more error than the noise, the FUMI theory underestimates the SD of measured areas.

The other situation is where actual instrumental noise includes the noise that cannot successfully be

approximated by the mixed processes of the white noise and AR(1)-process. An example is spike noise of high

intensity.

Baseline shifts of fixed patterns which are beyond the stationary assumption are often observable

in gradient chromatography, but the integration mode of the oblique baseline can assure, though

restricted, the robustness of the approach.
6 Practical use of FUMI theory
6.1 Estimation of noise parameters by Fourier transform

All the parameters necessary for applying the FUMI theory, i.e. Formulae (13) to (16), can be uniquely

determined from the experimental data. The signal parameters (b, k , k , k ) can be set according to the

c f e
 

shape of a target peak, as shown in Figure 3. On the other hand, the noise parameters ()wm,,ρ are

automatically determined from the power spectral density of the noise, as described below.

The noise power spectral density results from the Fourier transform of noise data, Y . The Fourier and

inverse Fourier transforms are:
N−1
YY= W (17)
i=0
N−1
−ki
Y = YW (18)
k=0
where
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ISO 11843-7:2018(E)

N denotes the number of data points involved in the region of the Fourier transform;

W exp[−j(2π/N)];
j is the imaginary unit.
The power spectral density, P(k), of the random process, Y , is defined as
Pk = (19)
ˆ ˆ
where Y is the conjugate number of Y .
...

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