Ci a ion: Duc, M.L.; Bilik, P.;
Ma inek, R. Ha monics Signal
Fea u e Ex ac ion Techniques: A
Re iew. Ma hema ics 2023,11, 1877.
h ps://doi.o g/10.3390/
ma h11081877
Academic Edi o : Song ing Luo
Recei ed: 16 Ma ch 2023
Re ised: 11 Ap il 2023
Accep ed: 14 Ap il 2023
Published: 15 Ap il 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
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A ibu ion (CC BY) license (h ps://
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ma hema ics
Re iew
Ha monics Signal Fea u e Ex ac ion Techniques: A Re iew
Minh Ly Duc 1,2,* , Pe Bilik 2and Radek Ma inek 2
1Facul y o Comme ce, Van Lang Uni e si y, 69/68 Dang Thuy T am, Wa d 13, BinhThanh Dis ic ,
Ho Chi Minh Ci y 70000, Vie nam
2
Depa men o Cybe ne ics and Biomedical Enginee ing, VSB–Technical Uni e si y o Os a a, 17. Lis opadu
15, 708 33 Os a a, Czech Republic; pe [email p o ec ed] (P.B.); [email p o ec ed] (R.M.)
*Co espondence: [email p o ec ed]
Abs ac :
Ha monic es ima ion is essen ial o mi iga ing o supp essing ha monic dis o ions in
powe sys ems. The mos impo an idea is ha spec um analysis, wa e o m es ima ion, ha monic
sou ce classi ica ion, sou ce loca ion, he de e mina ion o ha monic sou ce con ibu ions, da a
clus e ing, and il e -based ha monic elimina ion capaci y a e also conside ed. The ea u e ex ac ion
me hod is a undamen al componen o he op imiza ion ha imp o es he e ec i eness o he
Ha monic Mi iga ion me hod. In his s udy, echniques o ex ac undamen al equencies and
ha monics in he equency domain, he ime domain, and he spa ial domain include 67 li e a u e
e iews and an o e all assessmen . The combina ions o signal p ocessing wi h a i icial in elligence
(AI) echniques a e also e iewed and e alua ed in his s udy. The bene i o he ea u e ex ac ion
me hods is ha he analysis ex ac s he powe ul basic in o ma ion o he eedback signals om he
senso s wi h he mos edundancy, ensu ing he highes e iciency o he nex sampling p ocess o
algo i hms. This s udy p o ides an o e iew o he undamen al equency and ha monic ex ac ion
me hods o ecen yea s, an analysis, and a p esen a ion o hei ad an ages and limi a ions.
Keywo ds: ha monic; equency domain; ime domain; undamen al equency
MSC: 49M25
1. In oduc ion
Dis ibu ed ene gy esou ces ha e inc eased he pene a ion a e o enewable ene gy
sou ces bu ha e also led o in e mi ency and poo powe quali y [
1
]. To add ess his,
a mic og id combining pa ially dis ibu ed ene gy esou ces wi h a u ili y g id [
2
] has
been p oposed. Ha monic dis o ion has been p oposed o inc ease he addi ional losses o
elec ical equipmen , o e hea ing i and educing equipmen e iciency and u iliza ion. The
ha monic p oblem o he mic og id has become a majo issue wi h wo main sou ces: elec-
onic powe de ices and nonlinea loads [
3
]. Elec onic powe de ices, such as in e e s,
ec i ie s, and s a ic compensa o s, which gene a e high- equency ha monics ha can be
supp essed by LC o LCL il e s [
4
], a e widely used. Nonlinea loads a e he main eason
o gene a ing ou pu ol age d op, which leads o he dis o ion o he in e e ou pu
ol age wa e o m. To educe ha monics and imp o e sys em e iciency, an h opological
compensa ion s a egies ha e been s udied [3].
Ha monics a ec powe quali y and inc ease sys em losses by up o 27%. Powe
quali y issues a e mani es ed in ol age, cu en , o equency de ia ions, esul ing in
he ailu e o mal unc ion o equipmen [
5
]. Common powe issues a e empo a y o
s eady-s a e ol age o equency a ia ions such as impulsi e o oscilla o y ansien s and
ol age sags. Vol age sags and dips a e caused by sho ci cui aul s and mo o s a ing [
6
].
Ha monics de a e ans o me s and a ec high- equency con olle s, while ansien s and
ol age sag in luence p o ec ion and con ol equipmen . Al e na ing cu en d i es ide
h ough in e up ions, bu induc ion mo o s a e s and DC d i e con ac o s equi e backup
RC ci cui s [7].
Ma hema ics 2023,11, 1877. h ps://doi.o g/10.3390/ma h11081877 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2023,11, 1877 2 o 36
Ex ac ing he undamen al componen o ha monics using adi ional and mode n
echniques is a esea ch end. I de e mines he exac ha monic ype and is an inpu o he
con ol algo i hms o selec he app op ia e compensa ing cu en o he los cu en in
he sou ce [
3
]. The shun adap i e powe il e (SAPF) is a sui able choice o he end o
using mode n op imiza ion echniques in he selec ion o compensa ing cu en s, p o iding
high e iciency o compensa ing he cu en loss caused by ha monics [
8
]. Reac i e powe
compensa ion is he adminis a ion o eac i e ene gy o imp o e he pe o mance o
he AC sys em. I is seen in wo ways: load and ol age suppo . The aim is o achie e
an imp o ed powe ac o and eal powe balance, while ol age suppo is necessa y
o educe ol age luc ua ions a a gi en e minal [
9
]. In bo h cases, he eac i e powe
ha lows h ough he mic og id mus be e ec i ely con olled and compensa ed. Ac i e
ha monic il e s wo k on he p inciple o measu ing he magni ude and equency o he
cu en s ( om 1s o de o 50 h o de ) o he load [
3
,
8
]. The p ocesso will analyze he da a
and send a signal o con ol IGBT opening and closing and o b ing ha monic cu en s om
he 2nd o de o 50 h o de wi h he same magni ude and opposi e di ec ion as he sys em
ha monic cu en o elimina e all ha monic cu en s a e he posi ion o connec ion o he
elec ical sys em o he ac i e ha monic il e (Figu e 1). The p ocesso pe o ms analysis
and ex ac ion algo i hms o ha monics mo e accu a ely and as e , and he ha monic
emo al e iciency inc eases acco dingly.
Ma hema ics 2023, 11, 1877 2 o 39
ide h ough in e up ions, bu induc ion mo o s a e s and DC d i e con ac o s equi e
backup RC ci cui s [7].
Ex ac ing he undamen al componen o ha monics using adi ional and mode n
echniques is a esea ch end. I de e mines he exac ha monic ype and is an inpu o
he con ol algo i hms o selec he app op ia e compensa ing cu en o he los cu en
in he sou ce [3]. The shun adap i e powe il e (SAPF) is a sui able choice o he end
o using mode n op imiza ion echniques in he selec ion o compensa ing cu en s,
p o iding high e iciency o compensa ing he cu en loss caused by ha monics [8]. Re-
ac i e powe compensa ion is he adminis a ion o eac i e ene gy o imp o e he pe -
o mance o he AC sys em. I is seen in wo ways: load and ol age suppo . The aim is
o achie e an imp o ed powe ac o and eal powe balance, while ol age suppo is
necessa y o educe ol age luc ua ions a a gi en e minal [9]. In bo h cases, he eac i e
powe ha lows h ough he mic og id mus be e ec i ely con olled and compensa ed.
Ac i e ha monic il e s wo k on he p inciple o measu ing he magni ude and equency
o he cu en s ( om 1s o de o 50 h o de ) o he load [3,8]. The p ocesso will analyze
he da a and send a signal o con ol IGBT opening and closing and o b ing ha monic
cu en s om he 2nd o de o 50 h o de wi h he same magni ude and opposi e di ec ion
as he sys em ha monic cu en o elimina e all ha monic cu en s a e he posi ion o
connec ion o he elec ical sys em o he ac i e ha monic il e (Figu e 1). The p ocesso
pe o ms analysis and ex ac ion algo i hms o ha monics mo e accu a ely and as e ,
and he ha monic emo al e iciency inc eases acco dingly.
Figu e 1. Shun adap i e powe il e (SAPF) in h ee-phase powe supply.
Syn he ic e alua ion s udies on he equency domain and ime domain ha monic
componen ex ac ion me hods ha e been ca ied ou (Table 1). Howe e , he e a e s ill
many me hods o ex ac ing ha monic componen s in he signal ha ha e no been e al-
ua ed by esea che s and g adua e s uden s. This s udy conduc s a li e a u e e iew o
s udies on he equency domain ( he Adap i e Ha monic Wa ele T ans o m (AHWT)
me hod and Sliding Disc e e Wa ele T ans o m (SDWT) me hod), ime domain ( he Em-
pi ical Mode Decomposi ion (EMD) me hod, Sliding Window EMD (LWEMD) me hod,
Adap i e Ha monic Decomposi ion (AHD) me hod, and Adap i e Model-Based Scheme
Wi h Sho Sliding Analysis Window (AMS) me hod), and space domain (Head-Rela ed
Figu e 1. Shun adap i e powe il e (SAPF) in h ee-phase powe supply.
Syn he ic e alua ion s udies on he equency domain and ime domain ha monic com-
ponen ex ac ion me hods ha e been ca ied ou (Table 1). Howe e , he e a e s ill many
me hods o ex ac ing ha monic componen s in he signal ha ha e no been e alua ed
by esea che s and g adua e s uden s. This s udy conduc s a li e a u e e iew o s udies
on he equency domain ( he Adap i e Ha monic Wa ele T ans o m (AHWT) me hod
and Sliding Disc e e Wa ele T ans o m (SDWT) me hod), ime domain ( he Empi ical
Mode Decomposi ion (EMD) me hod, Sliding Window EMD (LWEMD) me hod, Adap i e
Ha monic Decomposi ion (AHD) me hod, and Adap i e Model-Based Scheme Wi h Sho
Sliding Analysis Window (AMS) me hod), and space domain (Head-Rela ed T ans e Func-
ions (HRTF) me hod) ha monic ex ac ion o p o ide he mos comp ehensi e o e iew
and a documen o u u e esea che s.
Ma hema ics 2023,11, 1877 3 o 36
Table 1.
A b ie o e iew o p e ious li e a u e e iew documen s on me hods o ex ac ing ha mon-
ics in signals.
Domain Me hods o Ex ac ing Fea u e Re Yea s
Time
domain
Selec i e Ha monic Elimina ion Pulse-Wid h Modula ion
(SHEPWM) Powe signal [10] 2017
S a is ical Time-Domain Fea u es me hod, Uppe and Lowe
Bound o His og am me hod, Au o eg essi e (AR) Coe icien s
me hod, Hjo s’ Pa ame e s me hod, Singula Value
Decomposi ion (SVD) me hod, Piecewise Agg ega e
App oxima ion (PAA), and Adap i e Piecewise Cons an
App oxima ion (APCA) me hod
Vib a ion signal [11] 2017
Ma hema ical Mo phology (MM) Ope a o s me hod Elec oencephalog am
(EEG) signal [11] 2017
SRF algo i hm, pq Theo y algo i hm Powe signal [12] 2017
Fi ing a Sum o Exponen ials me hod, Fi ing a S aigh Line o
he La e S age me hod
Pulsed Eddy Cu en
(PEC) signal [13] 2019
ADALINE Technique,
Sel -Tuning Fil e (STF) me hod Powe signal [14] 2019
Con olu ional Neu al Ne wo k (CNN) Raw signal [15] 2020
Pa icle Swa m Op imiza ion (PSO), Powe signal [16] 2020
Ze o-C ossing Ra e (ZCR) me hod, Sho Time Ene gy (STE)
me hod, Au o-Co ela ion-Based Fea u es me hod,
Rhy hm-Based me hod
Audio signal [17] 2020
F equency Domains
Fas Fou ie T ans o m (FFT) Me hod, Eigen ec o me hods
(EM), Wa ele T ans o m (WT), and Au o-Reg essi e
me hod (ARM)
Elec oencephalog am
(EEG) signal [18] 2014
Wa ele T ans o ms (WT) me hod, Independen Componen
Analysis (ICA) me hod, P incipal Componen Analysis
(PCA) me hod
Elec oencephalog am
(EEG) signal [19] 2015
S a is ical F equency-Domain Fea u es, Spec al Skewness,
Spec al Ku osis, Spec al En opy and Shannon En opy
Fea u e me hods, Sho -Time Fou ie T ans o m (STFT) me hod,
Wa ele T ans o m and Wa ele Decomposi ion me hods,
Wigne –Ville Dis ibu ion (WVD) me hod
Vib a ion signal [11] 2017
Lea ning Techniques (Gene ic Algo i hm (GA)) and A i icial
Neu al Ne wo k (ANN), Sliding Window Fou ie
Analysis (SWFA)
Powe signal [12] 2017
Sho -Time Fou ie T ans o m (STFT) me hod, Con inuous
Wa ele T ans o m (CWT) me hod, Hilbe –Huang T ans o m
(HHT) me hod
Elec oencephalog am
(EEG) signal [20] 2017
Linea P edic i e Coding (LPC) Coe icien s me hod, Code
Exci ed Linea P edic ion (CELP) me hod, Linea Spec al
F equency me hod
Audio signal [17] 2020
Pe mu a ion En opy (PE) me hod, Dispe sion En opy (DE)
me hod, Empi ical Wa ele T ans o m (EWT) me hod, Re e se
Dispe sion En opy (RDE) me hod
Raw signal [21] 2020
Fou ie T ans o m (FT) me hod, Fas Fou ie T ans o m (FFT)
me hod, S-T ans o m me hod, Wa ele -T ans o m (WT) me hod
Powe signal [22] 2021
Space domain
F ac al Dimension me hod, Co ela ion Dimension me hod,
App oxima e En opy me hod, La ges Lyapuno Exponen
me hod, Kolmogo o –Smi no Tes me hod
Vib a ion signal [11] 2017
Ma hema ics 2023,11, 1877 4 o 36
The main cha ac e is ics o he powe signal include he phase angle
(θ)
and magni-
ude ampli ude. The phase o igin is calcula ed om he undamen al componen alues
o he ol age signal (
Vabc
o h ee-phase sou ces). In p ac ice, he powe supply sup-
plies many loads, including linea and non-linea loads. Non-linea loads include powe
equency con e e s, powe supply swi ches, and LED ligh ing sys ems [
22
]. They a e
he main sou ce o ha monics in he powe supply. Ha monics gene a ed in he powe
supply cause phenomena such as ans o me explosion, hea ing on he su ace o elec ical
equipmen , and he educed ope a ing e iciency o elec ical equipmen using powe due
o sou ce quali y supplied wi h poo quali y powe [
23
]. The e o e, de e mining he los
cu en co ec ly and accu a ely and selec ing he co ec , su icien amoun o cu en o
compensa e o he numbe o cu en losses in he powe supply gene a ed by he ha mon-
ics is a p omising u u e s udy a ea o esea che s. Analyzing, ex ac ing, and de ec ing
ha monic cha ac e is ics play an impo an ole in ha monic mi iga ion [
24
]. The e has
been much esea ch on ha monic mi iga ion me hods in he las ew decades. Howe e , he
e ec i eness o hese s udies is s ill limi ed. Finding a me hod o implemen he ha monic
mi iga ion ha b ings he mos op imal e ec is s ill an open issue o u u e esea che s.
In an applica ion using a shun adap i e powe il e (SAPF) in a h ee-phase powe
supply (Figu e 1), he los sou ce cu en
(iL)
is compensa ed by he cu en ex ac ed
om he SAPF
(iF)
; he supply cu en
(iS)
is a ec ed by he ha monics a ising om
he non-linea load. The ha monic p ocessing block p o ides an algo i hm o handling
load cu en
(iL)
and ex ac ing undamen al equency and ha monic cu en
(iL,ha m)
.
The ol age/cu en con olle hen gene a es a PWM pulse om he e e ence ol age
signal
V e 
ed o he SAPF il e . In some cases, ha monic ol age needs o be de ec ed
by me hods such as a se ies adap i e powe il e o hyb id adap i e powe il e and
dis ibu ed gene a ion o implemen powe quali y imp o emen [25,26].
Two me hods o de ec ing ha monics in powe supply by ex ac ing ha monics a e
conside ed:
•
De ec ing he o e all ha monic, i.e., pe o ming he emo al o he undamen al
equency componen o he load cu en (IL), which ex ac s only he ha monics in
he o m o a signal [27].
•
Selec i e ha monic de ec ion is he p ac ice o isola ing ha monics in o a se o signals
and ex ac ing hem a he ou pu [28,29].
The e alua ion o he o e all ha monic de ec ion me hod, he selec i e ha monic
de ec ion me hod, has many ad an ages:
•
Con olling in es men cos s o ha monic compensa ion and imp o ing powe quali y
in a easonable way [5,30].
•
The compensa ion sys em has delay ime, igge ime, and delay ime co esponding
o di e en delay angles o di e en ha monic ypes. Selec i e ha monic compensa ion
allows indi idual signals o be co ec ed since each ha monic pa ame e is adjus ed
om a single o se angle ela i e o he hys e esis angle [28].
•
The shun adap i e powe il e can be ins alled in combina ion wi h a passi e il e
ha pe o ms he sys em’s hyb id compensa ion unc ion. The SAPF ensu es a low-
o de ha monic unc ion and he low-pass il e (LPF) pe o ms a high-o de ha monic
compensa ion unc ion [
31
]. Howe e , he LPF is la ge and does no espond o
low-o de ha monics. The SAPF has he limi a ion o no esponding o high-o de
ha monics (high equency) and he SAPF has a high swi ching equency [
32
]. This
inc eases he elec omagne ic in e e ence and insula ion s ess. In he case o using
a hyb id compensa ion me hod, i is necessa y o selec he co esponding e e ence
signal o he SAPF [33].
The as , accu a e ha monic ex ac ion me hod helps he il e o iden i y and p o ide
sui able compensa ion o he los cu en in he powe supply quickly because ha monics
change equen ly in he powe supply [
34
]. F om di e en de ices, he issues o di e en
g id phases and, co espondingly, di e en ha monic de ec ion a ise bu hey a e bo h
Ma hema ics 2023,11, 1877 5 o 36
closely ela ed o eal- ime ha monic ex ac ion and de ec ion in he powe supply. The
mesh phase de ec ion me hod pe o ms he sepa a ion o he posi i e sequence in o he
basic signal componen om he noisy signal [
35
]. The selec i e ha monic de ec ion me hod
pe o ms he unc ion o ex ac ing indi idual ha monics om he cu en o ol age signal
o he dis u bed signal [36]. The main objec i es o his s udy include:
•A e iew o s udies ela ed o ha monic ea u e ex ac ion in he las ew decades.
•
The equency and ime domain ha monic ea u e ex ac ion me hods and hyb id
me hods in ha monic ea u e ex ac ion ope a ion a e sys ema ically e iewed.
•
The o mulas and ma hema ical models used o he ex ac ion o ha monics in he
equency and ime domains a e s udied and e alua ed in de ail in his s udy.
•
The o e all e alua ion and compa ison o he p ocessing ime e iciency o each
ha monic ex ac ion me hod in he equency domain and he ime domain.
•
The iden i ica ion o he limi a ions o he ha monic ex ac ion me hods in he e-
quency domain and in he ime domain. A he same ime, i aises open issues o
u u e esea ch.
This esea ch pape is s uc u ed as ollows: Sec ion 2p esen s he e ec s o ha monics.
Sec ion 3shows he con en o ha monic signal analysis. Sec ion 4de ails he ha monic
ea u e ex ac ion echnique. Sec ion 5demons a es he compa ison high ligh on he
ha monic ea u e ex ac ion echnique and he u u e esea ch opic and Sec ion 6desc ibes
he con en o he conclusion.
2. E ec s o Ha monics
Ha monics a e a o m o in e e ence ha di ec ly a ec powe quali y and ha e a
e y bad e ec on he equipmen and machine y used in a ac o y. Ha monics cause
cables o o e hea , damaging insula ion [
37
]. Ha monics educe mo o li e, cause mo o
o e hea ing, and induce a loud ope a ing noise [
38
]. Ha monics gi e ise o CB o e load,
o e hea ing, and ans o me explosion (while he amoun o elec ici y used is s ill less
han a ed). Ha monics cause ci cui b eake s, ap oma s, and uses o be a ec ed o
unknown easons [
39
]. Ha monics cause se ious ha m o he capaci o by damaging he
dielec ic, bulging he capaci o , educing he li e o he capaci o , and e en causing an
abno mal capaci o explosion [
40
]. Ha monic in e e ence a ec s elecommunica ions
equipmen and au oma ion sys ems. Ha monics cause measu ing equipmen o ope a e
inco ec ly, and hey cause ene gy was e oo (Figu e 2). Ha monics occu when he diode
ec i ie has no passing cu en and he cu en goes di ec ly o he in e e s while he AC
ol age a he inpu is less han he DC ol age a he capaci o . The sinusoidal shape o
he sou ce cu en is comple ely dis o ed when a case o ha monics a ises o only a ew
seconds. The ha monics gene a ed in he powe supply cause hea ing o he conduc o s, he
insula ion is b oken, he pe o mance o he elec ical equipmen is educed, and he li e o
he elec ical equipmen is educed o e ime. The mo o o elec ical equipmen ope a es
wi h noise and i is easy o gene a e hea ha monics ha a e no well con olled and can
damage he dielec ics in he capaci o s, sho en he li e o he capaci o s, and po en ially
blow up he capaci o s. Ha monic cu en s in o a ing machines cause hea ing e ec s such
as eddy cu en losses p opo ional o he squa e o he equency [
3
,
8
]. Ha monic cycles
can cause addi ional losses by inducing highe equency cu en s and nega i e o ques in
machine o o s. Ha monic cu en s can lead o he o e loading o powe ac o co ec ion
capaci o s and he de a ing o cables. Ha monic componen s add phan om powe o he
o al powe consump ion o a ans o me , causing i o o e load, hea up, and bu n. They
hea up and bu n conduc o s, causing se ious losses in he elec ical sys em. In a h ee-
phase sys em, he neu al conduc o is hea ed o bu ned o c ea e a s able sys em. The N-G
(neu al-ea h) ol age is oo la ge. The b eake jumps o unknown easons. This causes
he ailu e o he PF eac i e powe compensa ion capaci o . Noise in communica ion
sys ems can lead o he o e load o capaci o s and ans o me s due o he weakening
o ha monic cu en s, esul ing in he o ma ion o an LC ci cui . As o sys ems using
backup gene a o s o used on ships and d illing igs, when unning, due o he gene a o ’s

Ma hema ics 2023,11, 1877 6 o 36
induc ance cha ac e is ics ha a e highe han con en ional ans o me s, ha monics will be
ampli ied mo e se iously, om 3 o 4 imes, and he se iousness o he sys em equipmen is
g ea e and can e en cause a gene a o i e, which is e y dange ous and cos ly. Ha monics
also cause losses on he coil and s eel co e o he mo o o inc ease, dis o he o que
o m, educe machine e iciency, and cause noise o a ec he e o o measu ing de ices,
leading o e oneous measu emen esul s. Mo e dange ously, he highe -o de ha monic
wa es can also gene a e mo o sha o que o cause mechanical esonance oscilla ions
ha damage mechanical componen s in he engine, causing he licke ing o elec ical
equipmen and ligh ing, a ec ing people, and causing elec omagne ic wa es o p opaga e
in space, a ec ing anscei e s.
Ma hema ics 2023, 11, 1877 7 o 39
Figu e 2. E ec s o ha monics.
The load in he dis ibu ion powe supply gene a es many ypes o ha monics, a -
ec ing he quali y o use and he pe o mance o powe -using equipmen , educing i s
e iciency. Much elec ical equipmen damage, such as i e o explosion, is also caused by
ha monic sou ces [37]. The highe equency ha monic cu en causes elec ons o low o
he ou side o he conduc o , which educes he cu en -ca ying capaci y, esul ing in a
dec ease in powe a ing causing hea gain and damage o he insula ion. Ha monic dis-
o ion has a di ec e ec on he powe ac o [38,39]. Many ha monics ha e a low powe
ac o alue. The hea losses gene a ed by he ha monics shi ing o use and pay o he
eac i e powe and ha monic cu en s can cause he capaci o o ail [33,39].
T ans o me aging o hea ing on he su ace o he ans o me body is mainly caused
by ha monics in he powe supply [38]. The ans o me s uc u e is o med by winding
se e al coils placed close o each o he and sepa a ed by insula ion; when he powe lows
h ough he windings wi h ha monics gene a ed in hem, o e ol age esul s [34]. Load
occu s in he ans o me , gene a ing hea in he ans o me body, educing he ope a ing
e iciency o he ans o me , and educing he insula ion s eng h o he windings in he
ans o me . Eddy cu en s due o s ay lux losses cause o e hea ing. A empe a u e in-
c ease o 7–10 deg ees can educe he li e o an insula ing ma e ial by hal .
The p o ec ion o elec ical equipmen is p o ided in he elec oly ic powe supply
de ices ha pe o m o e load p o ec ion, sho ci cui p o ec ion, o p o ec ion om o e -
hea ing gene a ed in elec ical ci cui s, elimina ing all he e ec s po en ially a ec ing he
pe o mance o elec ical equipmen . Today, indus ial plan s use a lo o swi ching de-
ices such as in e e s and swi ches o de ices ha con ol dynamic mechanisms in in-
dus ial machines [32,35]. The ac o y loo uses a lo o high-in ensi y discha ge (HID)
bulbs o ligh up he ac o y. The powe sou ce gene a es ha monics om he abo e-men-
ioned de ices and he ha monics hemsel es educe he pe o mance o hose de ices.
Elimina ing ha monics, o minimizing ha monics gene a ed in he powe supply, equi es
new esea ch o imp o e elec ical equipmen he esponse le el o which does no c ea e
ha monics in he powe supply; his is a di icul equi emen o esea che s. C ea ing
me hods o elimina e ha monics in powe supplies by compu e p og ams combined wi h
high- ech equipmen is also a p omising esea ch di ec ion.
In he e a o he 4.0 indus ial e olu ion, many nonlinea loads a e p oduced and
ope a ed in he dis ibu ion powe sys em. Nonlinea loads such as LEDs, compu e mon-
i o s, powe supply swi ches, and ans o me s pe o m he communica ion be ween he
Figu e 2. E ec s o ha monics.
The load in he dis ibu ion powe supply gene a es many ypes o ha monics, a -
ec ing he quali y o use and he pe o mance o powe -using equipmen , educing i s
e iciency. Much elec ical equipmen damage, such as i e o explosion, is also caused by
ha monic sou ces [
37
]. The highe equency ha monic cu en causes elec ons o low
o he ou side o he conduc o , which educes he cu en -ca ying capaci y, esul ing in
a dec ease in powe a ing causing hea gain and damage o he insula ion. Ha monic
dis o ion has a di ec e ec on he powe ac o [
38
,
39
]. Many ha monics ha e a low powe
ac o alue. The hea losses gene a ed by he ha monics shi ing o use and pay o he
eac i e powe and ha monic cu en s can cause he capaci o o ail [33,39].
T ans o me aging o hea ing on he su ace o he ans o me body is mainly caused
by ha monics in he powe supply [
38
]. The ans o me s uc u e is o med by winding
se e al coils placed close o each o he and sepa a ed by insula ion; when he powe lows
h ough he windings wi h ha monics gene a ed in hem, o e ol age esul s [
34
]. Load
occu s in he ans o me , gene a ing hea in he ans o me body, educing he ope a ing
e iciency o he ans o me , and educing he insula ion s eng h o he windings in he
ans o me . Eddy cu en s due o s ay lux losses cause o e hea ing. A empe a u e
inc ease o 7–10 deg ees can educe he li e o an insula ing ma e ial by hal .
The p o ec ion o elec ical equipmen is p o ided in he elec oly ic powe supply
de ices ha pe o m o e load p o ec ion, sho ci cui p o ec ion, o p o ec ion om
o e hea ing gene a ed in elec ical ci cui s, elimina ing all he e ec s po en ially a ec ing
he pe o mance o elec ical equipmen . Today, indus ial plan s use a lo o swi ching
de ices such as in e e s and swi ches o de ices ha con ol dynamic mechanisms in
Ma hema ics 2023,11, 1877 7 o 36
indus ial machines [
32
,
35
]. The ac o y loo uses a lo o high-in ensi y discha ge (HID)
bulbs o ligh up he ac o y. The powe sou ce gene a es ha monics om he abo e-
men ioned de ices and he ha monics hemsel es educe he pe o mance o hose de ices.
Elimina ing ha monics, o minimizing ha monics gene a ed in he powe supply, equi es
new esea ch o imp o e elec ical equipmen he esponse le el o which does no c ea e
ha monics in he powe supply; his is a di icul equi emen o esea che s. C ea ing
me hods o elimina e ha monics in powe supplies by compu e p og ams combined wi h
high- ech equipmen is also a p omising esea ch di ec ion.
In he e a o he 4.0 indus ial e olu ion, many nonlinea loads a e p oduced and
ope a ed in he dis ibu ion powe sys em. Nonlinea loads such as LEDs, compu e moni-
o s, powe supply swi ches, and ans o me s pe o m he communica ion be ween he
powe sou ce and he loads. These de ices gene a e ha monics in he powe supply, ha -
monics causing signi ican damage o he pe o mance and ope abili y o he loads [
37
,
38
].
Ha monic cu en s a ise in he powe supply and se iously a ec he communica ion
sys em [
41
]. A magne ic couplings in elephones o in o ma ion ansmission sou ces,
ha monics will cause in e e ence and he in o ma ion ansmi ed will no mee he equi e-
men s o he ansmission speed will be delayed [
39
]. The me hod ha communica ion
equipmen supplie s use o minimize ha monics a ec ing communica ion lines consis s in
using equipmen o shield he amoun o induc ance in pa allel conduc o s and building a
de ice o measu e and con i m he in o ma ion in e e ence sys em. The maximum alue
o he ha monic cu en can be much highe han he sine wa e shape a he undamen al
equency, causing alse ipping [40].
Au oma ion de ices use a lo o mo o s, and he pe o mance o he mo o s is se e ely
a ec ed by he ha monics gene a ed by he cu en . Many ypes o mo o s ope a e ac-
co ding o he mechanism o using he PWM me hod o adjus he ope a ing mechanism;
ha monics cause he mechanism o ope a e no as desi ed, e.g., o que ipples c ea ed by
wa e in e ac ion ha monics cause his mechanical oscilla ion [
35
]. The ha monics gene -
a ed by he PWM in e e s a ec he e iciency o he elec ic mo o s much mo e han he
powe supply [
34
]. Nonlinea loads in he dis ibu ion powe supply c ea e le els ha
nega i ely a ec he pe o mance o ans o me s. The ans o me eeds he ec i ie six
pulses wi h a DC load and powe dissipa ion ac o s such as o al ha monic dis o ion
(THD) comp omise e iciency in he ans o me . Squi el-bed synch onous mo o s ope a e
on he lux densi y a he clea ances o inc ease he o que p ope ies o he mo o [
35
,
39
].
Howe e , he ha monics gene a ed a hose gaps a ec he magne ic ield o he s a o
and he o o nega i ely, hus impac ing he o que o he mo o . Resea che s calcula e
he lux densi y a he gaps using he Fini e Elemen Analysis (FEA) o mula. Usually,
pa allel capaci o s a e used o pe o m he unc ion o il e ing high-o de ha monics o
a single- uned ha monic il e . High- equency ol age componen s cause eddy cu en
losses in he co e o he AC mo o . These losses inc ease he ope a ing empe a u e o he
aul as well as he coil a ound he co e and can cause undesi ed o que spikes. Excessi e
ha monic dis o ion will cause a lo o ze o in e e ence o he cu en wa e o m, a ec ing
he iming o he ol age egula o . This may cause he gene a o o s op wo king.
3. Ha monic Signal Analysis
Ha monic componen ex ac ion analysis o he signal is pe o med in ou s ages
(
Figu e 3
). S age 1 pe o ms no maliza ion o he signals in he equency domain o in
he ime domain. Signal no maliza ion includes many di e en unc ions, depending on
he ype o senso , so he e is no single de ice ha can p o ide comple e no maliza ion
o all senso s. Time- equency ep esen a ion (TFR) desc ibes pa ame e s pe o med
o e ime including he ins an aneous RMS cu en pa ame e , ins an aneous undamen al
RMS cu en pa ame e , o al ha monic dis o ion (THD) pa ame e , and pa ame e o
ins an aneous TnHD. The cha ac e is ics o TFR a e empo ally in o ma i e and spec ally
in o ma i e (Equa ions (2)–(6)). The signal is analyzed acco ding o he equency shown
h ough he spec al shapes; he ime- a ying equency is shown speci ically acco ding o
Ma hema ics 2023,11, 1877 8 o 36
he ime- a ying spec al in o ma ion shape. Time- equency ep esen a ion is conside ed
a use ul ool o moni o ing he signal being analyzed by equency. S age 2 pe o ms he
es ima ion o basic signal componen s and pa ame e s. Ins an aneous oo -mean-squa e
(RMS) ol ages and Ins an aneous oo -mean-squa e (RMS) undamen al ol ages a e
he squa e oo s o he mean o e one cycle o he squa e o he ins an aneous ol age
(
Equa ions (7) and (8)
). A quan i a i e uni is used o measu e ha monic dis o ion in a
signal sou ce. Ha monic dis o ion o o al ha monic dis o ion (THD) is measu ed as he
a io alue o he o al powe o all ha monic componen s o he powe o he undamen al
equency. The lowe he THD alue, he mo e comple e he sys em’s ou pu signal wa e-
o m is in he sine wa e shape and he less noise o dis o ion he e is. The THD alue
index is used as an indica o o powe quali y assessmen acco ding o he IEEE 519:2014
s anda d. The smalle he THD alue, he less hea gene a ion he powe sys em has, and
he lowe he he mal powe emissions in he ield. This p o es ha he powe sou ce is o
good quali y and imp o es he pe o mance o elec ical equipmen . The moni o ing and
e alua ion o powe quali y can be ca ied ou acco ding o he IEEE 519:2014 s anda d.
S age 3 pe o ms he classi ica ion o signal cha ac e is ics. The ins an aneous o al ha -
monic dis o ion (THD) pa ame e o he ha monics is calcula ed acco ding o he measu e
o he ha monic con en in a wa e o m and exp ess alue acco ding o Fo mula (9) and
he ins an aneous o al non-ha monic dis o ion TnHD( ) pa ame e o he ha monics is
calcula ed acco ding o Fo mula (10). In addi ion, S age 4 ha monizes he classi ica ion
o signal ypes. A de e minis ic classi ica ion me hod used in p ac ical applica ions is a
ule-based classi ie ha is easy o implemen and elies on h eshold se ings and expe
ules. The classi ica ion o ha monic signals is based on pa ame e s o e icien inpu
h eshold se ings and expe ules ha mee IEEE 519:2014 (Figu e 4). The ha monic signal
in he powe supply is no malized o an in ensi y signal ha is analyzed in he equency
and ime domains [
42
,
43
]. The basic pa ame e s in he ha monic ex ac ion analysis sys em
include RMS undamen al ol age, o al wa e o m dis o ion, ins an aneous RMS ol age,
o al non-ha monic dis o ion, and calcula ed o al ha monic dis o ion [
38
,
39
]. The abo e-
s a ed pa ame e s a e used as inpu pa ame e s o he ha monic componen classi ica ion
and analysis sys em.
The ha monic signal model is analyzed o ex ac he signal o he undamen al
componen s o he ha monics acco ding o he IEEE 519:2014 s anda d [
36
–
39
], which is
buil acco ding o Fo mula (1) acco ding o he exponen ial signal complex shape.
xwd( )=ej2π 0 +A·ej2π 1 (1)
whe e
0
is he undamen al signal equency,
1
is he ha monic o in e ha monic equency
and is he ime. As o ha monics,
A=
0.25
and 1=
250
Hz
. As o in e ha monics,
A=0.25 and 1=275 Hz.
Signal ime- equency dis ibu ion is a me hod o ep esen ing a signal in e ms o
ime equencies ha include componen s such as Spec um, Gabo T ans o m, and S-
T ans o m [42,43].
The spec um cha implemen s he dis ibu ion o he undamen al componen s o he
signal in e ms o equency and ime. The Hanning window pe o ms a na ow analysis
o signal componen s by he equency wi h a window leng h o 512, and he equency
and ime esolu ion o he signal a e pe o med acco ding o Fo mula (2).
Px( , )=Z∞
−∞x( )w(τ− )·e−j2π d 
2
(2)
whe e x( ): signal, w(n): he p esence o whi e noise, : a unc ion o he equency.
The Gabo T ans o m me hod pe o ms he analysis o he local p ope ies o a se o
signals wi h equency and ime domain cha ac e is ics [
43
]. The esolu ion o he signals
Ma hema ics 2023,11, 1877 9 o 36
in he equency domain and in he ime domain, always ha ing he same Gabo T ans o m
alue o all equencies, is shown using Fo mula (3):
C(n,k)=Z∞
−∞x(τ)h∗(n,k)dτ(3)
whe e x( ): signal, h∗(n,k): a dual basis o bio hogonal basis.
Ma hema ics 2023, 11, 1877 9 o 39
s anda d. The smalle he THD alue, he less hea gene a ion he powe sys em has, and
he lowe he he mal powe emissions in he ield. This p o es ha he powe sou ce is
o good quali y and imp o es he pe o mance o elec ical equipmen . The moni o ing
and e alua ion o powe quali y can be ca ied ou acco ding o he IEEE 519:2014 s and-
a d. S age 3 pe o ms he classi ica ion o signal cha ac e is ics. The ins an aneous o al
ha monic dis o ion (THD) pa ame e o he ha monics is calcula ed acco ding o he
measu e o he ha monic con en in a wa e o m and exp ess alue acco ding o o mula
(9) and he ins an aneous o al non-ha monic dis o ion TnHD( ) pa ame e o he ha -
monics is calcula ed acco ding o o mula (10). In addi ion, S age 4 ha monizes he classi-
ica ion o signal ypes. A de e minis ic classi ica ion me hod used in p ac ical applica-
ions is a ule-based classi ie ha is easy o implemen and elies on h eshold se ings
and expe ules. The classi ica ion o ha monic signals is based on pa ame e s o e icien
inpu h eshold se ings and expe ules ha mee IEEE 519:2014 (Figu e 4). The ha monic
signal in he powe supply is no malized o an in ensi y signal ha is analyzed in he
equency and ime domains [42,43]. The basic pa ame e s in he ha monic ex ac ion
analysis sys em include RMS undamen al ol age, o al wa e o m dis o ion, ins an ane-
ous RMS ol age, o al non-ha monic dis o ion, and calcula ed o al ha monic dis o ion
[38,39]. The abo e-s a ed pa ame e s a e used as inpu pa ame e s o he ha monic com-
ponen classi ica ion and analysis sys em.
Figu e 3. Flow cha o ha monic signal de ec ion and classi ica ion.
The S-T ans o m (ST) me hod is conside ed as a ime- equency spec al localiza ion
me hod ha is made by combining wo me hods, namely Sho -Time Fou ie T ans o m
(STFT) and Wa ele T ans o m. The ST me hod also uses he window model, bu ST
implemen s he me hod o expanding he windows in Gaussian o m and pe ec ing he
signal esolu ion in he equency domain ep esen ed by he eal dis ibu ion spec a and
i ual shows de ailed acco ding o Fo mulas (4)–(6).
ST(τ, )=Z∞
−∞h( )| |
√2πe−(τ− )2 2
2·e−j2π d (4)
Ma hema ics 2023,11, 1877 16 o 36
•
A e each di e ence i e a ion, he IMF p o ides di e en alues a he da a blocks. I
is necessa y o ensu e ha he IMF alue is con inuously connec ed o he da a blocks
by selec ing he numbe o i e a ions oge he .
•
The bulk da a pe o med by he sc eening p ocess make i di icul o eal- ime
analysis o he inal signals. The e o e, disca ding he inal signal is necessa y.
•
The e is a selec i ely ixed numbe o i e a ions o signal il e ing. Howe e , low-
equency signals a e s ill p esen inside he da a blocks. The e o e, his low- equency
signal ejec ion solu ion should be s udied and implemen ed when emo ing he
ha monic signal om he ca ie in he X( ) signal.
The LWEMD me hod is implemen ed in o
X( )
signal analysis o sepa a e ha monics
om he ca ie . The LWEMD me hod imp o es he il e ing p ocess om he EMD me hod
by sho ening he numbe o signal i e a ions by applying he He mi e in e pola ion o
gene a e an in lec ion poin signal a ze o in e sec ions. F om he e, he en elopes a e
calcula ed and all low- equency signals a e cu o o he du a ion o he algo i hm. Some
ad an ages disco e ed when implemen ing he LWEMD algo i hm in ha monic ex ac ion
a e as ollows:
•The il e ing p ocess is s eamlined and educed when implemen ing he algo i hm.
•The de ec ion o low- equency ha monics in da a blocks is gua an eed.
•
The execu ion ime o ex ac ha monics om he signal is less han he adi ional
EMD me hod.
The LWEMD algo i hm is de ailed s ep-by-s ep (Table 6) and he low cha applying
LWEMD o ha monic ex ac ion is shown in Figu e 7.
Table 6. The LWEMD algo i hm.
S ep No. Explained in De ail S ep-by-S ep
S ep 1: C ea ing da a block om bu e 1 da ase .
S ep 2: Implemen ing he He mi e spline in e pola ion me hod o he bu e egion o he
signal X( ).
S ep 3: Calcula ing he a e age alue acco ding o Fo mula (20).
m1( )=Xmax ( )−Xmin ( )
2
(20)
S ep 4: Finding he IMF alue a he i s da a block o he signal acco ding o Fo mula (21).
im i( )=X( )−m1( )
(21)
S ep 5:
Calcula ing he esidual alue o he signal in he i s signal da a block acco ding o Fo mula
(22).
1( )=X( )−im 1( )
(22)
S ep 6: Ob aining he alue o 1( )
S ep 7:
Applying in e pola ion He mi e spline o de ec ex emes and calcula ing con ou s acco ding
o Fo mula (23).
d 1( )=d 1( )
d
(23)
S ep 8: Ex ac ing he alue a ime i whe e no d 1( ) alue exis s.
S ep 9:
Calcula ing he mean alue o signal m2( )acco ding o he esidual alue o signal 1( )
acco ding o Fo mula (24).
m2( )= 1max ( )− 1min ( )
2
(24)
S ep 10: Finding he IMF alue a signal alue m2( )acco ding o Fo mula (25).
im 2( )= 1( )−m2( )
(25)
S ep 11: Calcula ing he esidual alue a signal alue m2( )acco ding o Fo mula (26).
2( )= 1( )−im 2( )
(26)
S ep 12: Cu ing o he block signal a ime i.
S ep 13:
The da a a e sa ed in he las block and a leas se en ex eme poin s a e sa ed o duplica e
da a blocks.

Ma hema ics 2023,11, 1877 17 o 36
Ma hema ics 2023, 11, 1877 18 o 39
Figu e 7. Flow cha o LWEMD o ha monic ex ac ion.
Table 6. The LWEMD algo i hm.
S ep No. Explained in De ail S ep-by-S ep
S ep 1: C ea ing da a block om bu e 1 da ase .
S ep 2: Implemen ing he He mi e spline in e pola ion me hod o he bu e egion o
he signal 𝑋(𝑡).
S ep 3:
Calcula ing he a e age alue acco ding o Fo mula (20).
𝑚(𝑡)=𝑋(𝑡)−𝑋(𝑡)
2
(20)
Figu e 7. Flow cha o LWEMD o ha monic ex ac ion.
4.1.3. Adap i e Ha monic Decomposi ion (AHD) Me hod
Vib a ion signals om o a ing mo o s con ain e o pulses ha gi e ise o equency
modula ion e ec s. In he equency domain a a uni o m in e al, he e is a connec ion
be ween wo pulses including he aul -causing pulse and he pulse o a ha monic clus e .
In a de ini e esonance sequence, his symphony helps de e mine he placemen o samples
and spec um dis ibu ion o he signal and de ec s pe iodic pulses ha cause e o s. The
adap i e ha monic decomposi ion me hod de ec s e o pulses wi h a high noise a io in
Ma hema ics 2023,11, 1877 18 o 36
he ime domain and disc e e e o spec a e en in he case o low signal- o-noise (SNR).
The e o pulse de ec ion algo i hm is de ailed in he ollowing s eps (Table 7).
Table 7. The e o pulse de ec ion algo i hm.
S ep No. Explained in De ail in he Following S eps
S ep 1:
The camco de gene a es ib a ion pulses ha o m e o pulses and is ma hema ically modeled o he e o
pulse acco ding o Fo mula (27).
g( )=s( )+n( )=A( ).cosh2πR
0 (τ)dτ+ϕ0i+n( )=
=
p
∑
p=1
Sp( )+n( )=
p
∑
p=1
ap.cos2π p( )+θp+n( )(27)
Whe e s( )is an impulse signal consis ing o ha monics
.
P has a equency in he esonan se ies g( )wi h IA and IF
modula ion le els. n( ) ep esen s addi ional noise signals and noise componen s a ising om he sou ce o he
came a ib a ions.
S ep 2:
An e o pulse de ec ion model is designed by sepa a ing he Sp( )ha monics in he esonan ange o he g( )
angula signal and pu ing hem back oge he . This p oposed ha monic ex ac ion me hod can ex ac ha monics
wi h an SNR cap ha esponds o componen decomposi ion and bandwid h sh inkage.
S ep 3:
The adap i e ha monic decomposi ion me hod pe o ms equency shi ing o he ha monic componen s. Gi en
an ini ial equency po he se ies esonan o he P h ha monic pa
,
he Sp( )
ha monic o he momen wi h he
P h ha monic componen is ans o med acco ding o Fo mula (28).
Sp( )=Up( ).cos2π p( )+Vp( ).sin2π p( )(28)
The wo ha monic displacemen componen s a e desc ibed using Fo mula (29)



Up( )=ap.cosh2π p−e
p( )+θpi
Vp( )=−ap.sinh2π p−e
p( )+θpi(29)
S ep 4:
whe e he es ima ed equency e
pis close o he o iginal equency p alue. The wo shi ing ha monics Up( ),
Vp( ) o m good a ia ion pa e ns in he ime domain and noise o ze o- equency ends in he equency
domain. The Sp( )ha monic, a he ime o ha ing he P h ha monic componen , is econs uc ed o he o iginal
ampli ude and phase acco ding o Fo mulas (30) and (31).
a(p|e
p= p)=qU2
p( )−V2
p( )(30)
θ(p|e
p= p)= an−1h−Vp( )
Up( )i(31)
S ep 5:
Based on he a o emen ioned equency change ope a ion, op imiza ion is pe o med ollowed by disc e iza ion
acco ding o he es ima ed equency alue and he Sp( )
ha monic componen is ep oduced in he P h ha monic
acco ding o Fo mula (32):
min
Up,Vp,ne
ponτϑUp,Vp,e
po=
min
Up,Vp,ne
pon∅ω2
p2+∅V2
P2+ϑg−CpUp+Sp.Vp2
2o
(32)
whe e ∅second o de de ia ion ope a o is used o calcula e a quan i a i e alue o he smoo hness o ha monic
displacemen componen s Up( ),Vp( ).ϑPenal y coe icien and disc e e a iables g sampled o e ime
{ 0, 1, . . . , i, . . . , l−1}a e calcula ed acco ding o Fo mulas (33)–(36).
g=[g( 0),g( 1), . . . , g( l−1)]T(33)
UP=Up( 0),Up( 1), . . . , Up( l−1)T(34)
VP=Vp( 0),Vp( 1), . . . , Vp( l−1)T(35)
Cp=diaghcos2πe
p( 0),cos2πe
p( 1), . . . , cos2πe
p( l−1)i (36)
Sp=diaghsin2πe
p( 0),sin2πe
p( 1), . . . , sin2πe
p( l−1)i (37)
Analyzing he Sp( )ha monic componen a he ime o he P h ha monic by upda ing, and upda ing he de ails
o he op imal equa ion acco ding o Fo mula (37).
4.1.4. Adap i e Model-Based Scheme wi h Sho Sliding Analysis Window (AMS)
This imp o es powe quali y by p o iding he co ec and su icien amoun o cu en
loss compensa ion in he powe supply. I ully and in de ail de e mines he undamen al
and ha monic componen s o a powe supply ha condi ion he e iciency o he powe
Ma hema ics 2023,11, 1877 19 o 36
supply’s lossy cu en compensa ion ope a ion. The weakness o he equency domain
me hod o ha monic ex ac ion is ha i gene a es a sampling delay o a leas one cycle and
depends on he equency esolu ion. The weakness o he me hod o ex ac ing ha monic
componen s in he powe sou ce by he ime domain me hod is ha i does no gua an ee
he s abili y and de ia ion o he in e e ence.
The p oposed adap i e model-based scheme wi h a sho sliding analysis window
me hod ul ills he ea u es o online sampling and di ec ly co ec s analysis models in he
da a o ex ac he ha monic componen in eal ime and moni o he equency o se a
each cycle o he sample da a. I ex ac s he ha monic componen accu a ely, p o iding
imely compensa ion o he loss o cu en , imp o ing he powe quali y, and imp o ing
he wo king e iciency o he new me hod. I ex ac s he undamen al and ha monic
componen s o sample da a online. I se s he poin o he ha monic signal o ollow he
sine wa e shape.
The powe signal (S) in disc e e ime (
Sn
) o m o he amoun o sample collec ed (N)
du ing he (∆ )pe iod is p esen ed as a sine componen (H) acco ding o Fo mula (38):
Sn=
H
∑
h=1
ah.cos(nhw1∆ +θh),n=0, 1, . . . , N−1 (38)
whe e ah: ampli ude, θh: ini ial phase angle, w1=2π 1: undamen al angula equency.
To simpli y he calcula ion, Fo mula (1) is analyzed acco ding o Fo mula (39):
Sn=
H
∑
h=1Ah.ejnhw1∆ +A∗
h.e−jnhw1∆ =
H
∑
h=1Ah.xn
h+A∗
h.(xn
h)∗(39)
whe e
Ah=ahejnhw1∆
2
: complex ampli ude,
xh=ejnhw1∆
, and
(∗)
: complex conjuga e
calcula ion.
The ampli ude alue (A) calcula ed by minimizing he e o be ween he ac ual numbe
o samples, sn, and i s es ima e is p esen ed using Fo mula (40):
A=a gmin N−1
∑
n=0Sn−ˆ
Sn2!(40)
Complex ampli ude es ima ion is acco ding o Fo mula (41):
ˆ
A=XTX−1.XT.S(41)
whe e T: T anspose o a ma ix.
The ampli ude and phase angle o he h- h ha monic a e a gumen s o he complex
ampli ude and a e exp essed by Equa ions (42) and (43):
ah=2|Ah|(42)
θh=a g{Ah}(43)
Ha monic componen ex ac ion gene a es a minimal e o because he undamen al
equencies o he signals in he powe supply a e ime-biased and de ia ed om hei
nominal alues due o he powe imbalance be ween he powe supply and he loads on
demand. The e a e many me hods used o modi y he X ma ix when equency bias occu s.
The F equency Domain In e pola ion (FDI) me hod analyzes and de ec s he undamen al
equency in a be e manne . Howe e , equency esolu ion and delay o a leas one cycle
a e incu ed depending on he ini e amoun o he analysis window in FDI. The Kalman
il e ing me hod and he PLL-based echnique pe o m es ima ion e o acking and ime-
domain pa ame e uning o he sys em o pe o m he synch oniza ion o he esul s om
Ma hema ics 2023,11, 1877 20 o 36
he measu emen . Howe e , he weakness o his me hod is ha he me hod o de e mining
he pa ame e s is no sui able o main aining he s abili y o he signal quan i y and
imp o ing he accu acy and con e gence speed. I p e en s long du a ions a ising in
equency domain me hods and c ea es nume ical ins abili y in ime domain me hods. The
sho sliding window echnique pe o ms signal analysis (
sn
) om Equa ion (2) o o m a
low-pass il e acco ding o Equa ion (44), desc ibed as ollows:
s1−n=A1xn
1+A∗
1.(xn
1)∗(44)
obse ing h ee consecu i e da a samples and desc ibing hem in de ail acco ding o
Equa ion (45):
s1−n−2=A1.xn−2
1+A∗
1.xn−2
1∗
s1−n−1=A1.xn−1
1+A∗
1.xn−1
1∗
s1−n=A1xn
1+A∗
1.xn
1∗
(45)
The assumed linea ela ionship o h ee consecu i e samples is shown in
Equa ion (46)
.
The i ing pa ame e (
ε
) is conside ed he e o es ima ion pa ame e desc ibed by
Equa ion (47), wi h he es ima ed sample being ˆ
s1−n.
ˆ
s1−n=s1−n−2+ε.s1−n−1(46)
ε=a gmin(E)=a gminN
∑
n=3|s1−n−ˆ
s1−n|2=
=a gminN
∑
n=3|s1−n−s1−n−2−ε.s1−n−1|2(47)
The minimum e o es ima e (E) is p esen ed by Equa ion (48) and hen he linea
es ima ion pa ame e is econs uc ed acco ding o Equa ion (49):
dE
dε=2
N
∑
n=3
(s1−n−s1−n−2−ε.s1−n−1).(−s1−n−1)=0 (48)
ε=∑N
n=3(s1−n−1).(s1−n−s1−n−2)
∑N
n=3s2
1−n−1(49)
Rep esen a i e equa ions o h ee samples a e econs uc ed acco ding o Equa ion (50):
A1.xn−2
1+A∗
1.xn−2
1∗+ε.A1.xn−1
1+ε.A∗
1.xn−1
1∗=
=A1.xn
1+A∗
1.xn
1∗=>x2
1−ε.x1−1=0(50)
The undamen al equency in o ma ion o he h ee sample signals is shown a
x1
acco ding o Equa ion (51):
x1=ε±j√ε2+4
2=ejw1∆ =cos(w1∆ )+jsin(w1∆ )(51)
The undamen al equency componen s o he h ee sample signals con aining he
ma ching pa ame e (e) a e shown by Equa ion (52):
1=
cos−1∑N
n=3(s1−n−1)(s1−n−s1−n−2)
2∑N
n=3s2
1−n−1
2π∆ (52)
Ma hema ics 2023,11, 1877 21 o 36
Fundamen al equency alue (
1
) pe o ms an analysis model uning ope a ion con-
duc ing he X-ma ix modi ica ion unc ion, which imp o es he accu acy o he undamen-
al equency and ha monic componen ex ac ion in he signal.
The bene i s o he AMS solu ion a e ha i moni o s equencies by sliding window
unc ion and pe o ms analysis model modi ica ion o imp o e he accu acy o eal- ime
a ying undamen al equency and ha monic ex ac ion. The sliding window in N on-
line acquisi ion samples helps in no mal equency domain signal analysis and equency
moni o ing. The ha monic ex ac ion is pe o med quickly, ega dless o equency eso-
lu ion. The undamen al equency componen (
1
) p e en s he nume ical imbalance o
con en ional ime domain echniques and helps o de e mine he app op ia e pa ame e s
(Figu e 8).
Ma hema ics 2023, 11, 1877 23 o 39
con en ional ime domain echniques and helps o de e mine he app op ia e pa ame e s
(Figu e 8).
Figu e 8. Solu ion p ocedu e o he p oposed AMS.
The limi a ion o he AMS me hod is ha i uses a low-pass il e ha de ines he
pa ame e s o ma ch he equency esponse o he undamen al signal due o he a enu-
a ion o he signal magni ude. The s udy o modi ying equency de ec ion by a new
me hod ha is be e han he low-pass il e me hod is p omising.
4.2. Ha monic Fea u e Ex ac ion Technique in he F equency Domain
4.2.1. Adap i e Ha monic Wa ele T ans o m (AHWT)
The componen s o a wa e signal include equency con en o ime equency.
Adap i e ha monic wa ele ans o m uses a ime- equency sepa a ion echnique o ex-
ploi highly e icien esponse ea u es and ou pe o ms empi ical mode decomposi ion
(EMD) me hods. The AHWT me hod uses a de e minis ic basis o ex ac he ea u es o
he signal in he ime- equency domain. The c oss-compa ed wa ele and AHWT cha -
ac e is ics con i m impo an ea u es in wa e signals o de e mine he e iciency and
damage o he wa e o m s uc u e. The AHWT me hod implemen s one il e bank; a
each il e , a speci ic equency ange (𝑚2𝜋,𝑛2𝜋),0≤𝑚≤𝑛 is designed, which is called
he pa ame e le el. The size o each il e is designed o be small and comple e in he
equency domain, also known as he ideal sequence pass il e . A comple e il e o ms
an o hogonal wa ele and is de ailed by Equa ion (53):
𝑤(𝑡)=𝑤𝑡− 𝑘
𝑛−𝑚=𝑒𝑥𝑝󰇣𝑖𝑛2𝜋󰇡𝑡−𝑘𝑛−𝑚
󰇢󰇤−𝑒𝑥𝑝󰇣𝑖𝑚2𝜋󰇡𝑡−𝑘𝑛−𝑚
󰇢󰇤
(𝑛−𝑚)𝑖2𝜋(𝑡) (53)
The alue o he gene alized ha monic wa ele is ob ained by he in e se Fou ie
ans o m acco ding o Fo mula (54):
Figu e 8. Solu ion p ocedu e o he p oposed AMS.
The limi a ion o he AMS me hod is ha i uses a low-pass il e ha de ines he pa-
ame e s o ma ch he equency esponse o he undamen al signal due o he a enua ion
o he signal magni ude. The s udy o modi ying equency de ec ion by a new me hod
ha is be e han he low-pass il e me hod is p omising.
4.2. Ha monic Fea u e Ex ac ion Technique in he F equency Domain
4.2.1. Adap i e Ha monic Wa ele T ans o m (AHWT)
The componen s o a wa e signal include equency con en o ime equency. Adap-
i e ha monic wa ele ans o m uses a ime- equency sepa a ion echnique o exploi
highly e icien esponse ea u es and ou pe o ms empi ical mode decomposi ion (EMD)
me hods. The AHWT me hod uses a de e minis ic basis o ex ac he ea u es o he signal
in he ime- equency domain. The c oss-compa ed wa ele and AHWT cha ac e is ics
con i m impo an ea u es in wa e signals o de e mine he e iciency and damage o he
wa e o m s uc u e. The AHWT me hod implemen s one il e bank; a each il e , a speci ic
equency ange
(m2π,n2π)
, 0
≤m≤n
is designed, which is called he pa ame e le el.
The size o each il e is designed o be small and comple e in he equency domain, also

Ma hema ics 2023,11, 1877 22 o 36
known as he ideal sequence pass il e . A comple e il e o ms an o hogonal wa ele and
is de ailed by Equa ion (53):
wmnk( )=wmn −k
n−m=exphin2π −k
n−mi−exphim2π −k
n−mi
(n−m)i2π( )(53)
The alue o he gene alized ha monic wa ele is ob ained by he in e se Fou ie
ans o m acco ding o Fo mula (54):
wmnk(w)=(1
(n−m)2πe−1wk
n−m,m2π≤w≤n2π
0 , o he wise (54)
whe e he in ege Kis he displacemen pa ame e in he egion (m,n) and each le el o
he K alue ep esen s a equency ange in he equency domain. The ad an age o he
ha monic wa ele me hod is ha he signal is analyzed wi hin a limi ed ange o speci ic
equency anges.
The disc e e ha monic wa ele ans o m is based on he Fas Fou ie T ans o m
(FFT) me hod, which esponds well o he signals o senso s ope a ing in eal-wo ld
en i onmen s ha collec ime se ies signal da a
{x( ), =0, 1, 2, . . . , N−1}
; Fou ie
coe icien s
{F(q),q=0, 1, . . . , N−1}
and
F(q)
a e calcula ed using he Fas Fou ie
T ans o m (FFT) Fo mula (55).
F(q)=1
N
N−1
∑
=0
x( ).exp−i2π q
N(55)
The ha monic wa ele coe icien {amnk}is calcula ed using Fo mula (56).
amnk =
n−m−1
∑
l=0
x( ).exp−i2πkl
n−m,k=0, 1, . . . , n−m−1 (56)
This s udy econs uc s he o iginal ime se ies om he pa ame e s o he ha monic
wa ele s unc ion. Howe e , in disc e e ans o m, con inuous wa ele unc ions a e
eplaced by co esponding ci cula con inuous unc ions acco ding o Fo mula (57):
WC
mnk( )=1
(n−m)
n−1
∑
l=m
expi2πl
n−k
n−m (57)
The signal S( )is de e mined in he ime uni in e al acco ding o Fo mula (58):
S( )=
n−1
∑
k=mnamnk.WC
mnk( )+amnk.WC
mnk( )o(58)
The selec ion o
{(m0,n0),(m1,n1), . . . , (ml−1,nl−1)}
pa ame e pai s mus begin wi h
he
m0=
0 alue and con inue wi h each pai ha ouches each o he un il
nl−1=N
.
N
is he Nyquis equency and lis he o al numbe o le els.
The s eng h o he ha monic wa ele lies in he lexible selec ion o pa ame e pai s
(m, n) as he basis o he possible subha monics. In a case whe e a wa ele le el (m,
n) is de e mined in a equency band, ha signal is sepa a ed by he Wa ele T ans o m
me hod. This demons a es ha he Ha monic Wa ele T ans o m me hod has he po en ial
o pe o m he same de ec ion. The poin o his issue is wha me hod o use o choose he
pa ame e pai (m, n) acco dingly. Acco ding o he signal p ocessing heo y, a signal whose
signal ene gy is spa sely concen a ed in a ew basic unc ions is conside ed a good signal.
The me hod o Shannon en opy acco ding o Fo mula (59) is implemen ed by he o iginal
Ha monic Wa ele T ans o m Hyb id Imp o emen me hod o selec a sui able pai o (m,
Ma hema ics 2023,11, 1877 23 o 36
n) pa ame e s. Each pai o
{(m0,n0),(m1,n1), . . . , (ml−1,nl−1)}
pa ame e s selec ed o
p ocessing in he algo i hm is conside ed an
∅={0, 1, . . . , N−1}
elemen and sea ches
o he bes egion ha mee s he wa ele coe icien wi h he minimum en opy alue:
H(Z)=−∑
j
Pj.logPj(59)
whe e Pj=|Zj|2
kZk2and Pj.logPj=0when Pj=0.
The Shannon en opy alue is a measu e o spa si y alue and he smalle he Shannon
en opy alue, he be e he sea ch a ea. The sea ch loop occu s ypically wo o h ee
imes. Eigh Fou ie coe icien s a e used in he loop, and he numbe o i e a ions o 16
elemen s is used o he algo i hm o ma hema ical equa ions a le el 2 (Figu e 9).
Ma hema ics 2023, 11, 1877 25 o 39
The Shannon en opy alue is a measu e o spa si y alue and he smalle he Shan-
non en opy alue, he be e he sea ch a ea. The sea ch loop occu s ypically wo o h ee
imes. Eigh Fou ie coe icien s a e used in he loop, and he numbe o i e a ions o 16
elemen s is used o he algo i hm o ma hema ical equa ions a le el 2 (Figu e 9)
Figu e 9. A sample bina y sea ch ee o pa i ion selec ion used o AHWT (N = 16).
In he algo i hm lowcha (Figu e 10), each Fou ie coe icien is se o he i s
g oup, whe e in he Shannon en opy alue is se as he i s en opy alue. Each pa am-
e e pai (m, n) ep esen s a subg oup. In he second i e a ion, he FFT is pe o med on
each g oup o wo adjacen Fou ie coe icien s, and he alues o he en opy a e calcu-
la ed and compa ed wi h he sum o he co esponding ini ial en opy. Th ough he whole
p ocess o sea ching by mul iple loops, he pai ing p ocess p ice is he bes . The AHWT
coe icien is upda ed. A he same ime, he wa ele ’s basic unc ion eimplemen s he
signal econs uc ion.
Figu e 9. A sample bina y sea ch ee o pa i ion selec ion used o AHWT (N = 16).
In he algo i hm lowcha (Figu e 10), each Fou ie coe icien is se o he i s g oup,
whe e in he Shannon en opy alue is se as he i s en opy alue. Each pa ame e pai
(m,n) ep esen s a subg oup. In he second i e a ion, he FFT is pe o med on each g oup o
wo adjacen Fou ie coe icien s, and he alues o he en opy a e calcula ed and compa ed
wi h he sum o he co esponding ini ial en opy. Th ough he whole p ocess o sea ching
by mul iple loops, he pai ing p ocess p ice is he bes . The AHWT coe icien is upda ed.
A he same ime, he wa ele ’s basic unc ion eimplemen s he signal econs uc ion.
4.2.2. Sliding Disc e e Wa ele T ans o m (SDWT)
The ac i e powe il e (APF) has a con e sion equency om 10 kHz o 20 kHz. A
he ou pu o he APF, an LCL il e is used o a good esponse o he g oup o con e ed
ha monics. Howe e , he LCL il e has complex design pa ame e s, o en gene a ing
esonance poin s, and he complex ci cui design and con ol algo i hm o he APF be-
comes di icul . Designing SiC-MOSFET in o he sou ce de ice o he APF o inc ease he
swi ching equency o 50 kHz and using an L il e ins ead o he LCL il e helps supp ess
subha monics a he swi ching swi ch o a minimum, making he ci cui design simple and
he algo i hm in he APF easy and simple. A he same ime, he sampling equency and
swi ching equency a e as e and inc ease exponen ially. As a esul , ha monic de ec ion
achie es highe accu acy and be e ou pu cu en con ol. The ha monic ex ac ion algo-
i hm applied o he APF is ep esen ed by a sliding window disc e e Fou ie ans o m
(SDFT) and equency domain analyzed low con ol algo i hm. The modi ica ion o he
Ma hema ics 2023,11, 1877 24 o 36
SDWT algo i hm hal cycle and RC hal cycle educes algo i hm delay by hal and APF
dynamic esponse ime om wo imes o hal o he powe equency cycle.
Ma hema ics 2023, 11, 1877 26 o 39
Figu e 10. Flow cha o he i e a i e sea ching algo i hm.
4.2.2. Sliding Disc e e Wa ele T ans o m (SDWT)
The ac i e powe il e (APF) has a con e sion equency om 10 kHz o 20 kHz. A
he ou pu o he APF, an LCL il e is used o a good esponse o he g oup o con e ed
ha monics. Howe e , he LCL il e has complex design pa ame e s, o en gene a ing es-
onance poin s, and he complex ci cui design and con ol algo i hm o he APF becomes
di icul . Designing SiC-MOSFET in o he sou ce de ice o he APF o inc ease he swi ch-
ing equency o 50 kHz and using an L il e ins ead o he LCL il e helps supp ess
subha monics a he swi ching swi ch o a minimum, making he ci cui design simple
and he algo i hm in he APF easy and simple. A he same ime, he sampling equency
and swi ching equency a e as e and inc ease exponen ially. As a esul , ha monic de-
ec ion achie es highe accu acy and be e ou pu cu en con ol. The ha monic ex ac-
ion algo i hm applied o he APF is ep esen ed by a sliding window disc e e Fou ie
ans o m (SDFT) and equency domain analyzed low con ol algo i hm. The
Figu e 10. Flow cha o he i e a i e sea ching algo i hm.
The ha monic ex ac ion algo i hm plays a decisi e ole in he ha monic mi iga ion
model. P ecise ha monic ex ac ion helps he APF o p o ide accu a e and as compensa -
ing cu en s. The SDWT me hod is a commonly used ha monic ex ac ion and is de eloped
om he adi ional DWT me hod. The o mulas used o he ha monic ex ac ion p ocess
a e shown below.
1. The desc ip ion o he n h ha monic componen is made acco ding o Fo mula (60):
in(k)=An(k).cos2πnk
N+Bn(k).sin2πk
N(60)
Ma hema ics 2023,11, 1877 25 o 36
2. The AA and BB coe icien s a e calcula ed acco ding o Fo mula (61):



An(k)=An(k−1)+2
N.cos2πnk
N[i(k)−i(k−N]
Bn(k)=Bn(k−1)+2
Nsin2πnk
N[i(k)−i(k−N](61)
whe e Nis he numbe o sampling poin s in one cycle and Kis he la es cu en sam-
pling poin .
The main di e ence be ween he SDWT me hod and he DWT me hod is in he
di e en upda e me hod o he coe icien s
An(k)
,
Bn(k)
. The DWT me hod (Figu e 11)
equi es a ull da a sampling cycle o calcula e and upda e he coe icien s. The SDWT
me hod (Figu e 12) akes a new sample each ime and he co esponding sample alue o
one cycle be o e i is disca ded and eplaced wi h he newly acqui ed sample alue and
upda ed coe icien s, which inc eases he ime e iciency o he sys em.
Ma hema ics 2023, 11, 1877 27 o 39
modi ica ion o he SDWT algo i hm hal cycle and RC hal cycle educes algo i hm delay
by hal and APF dynamic esponse ime om wo imes o hal o he powe equency
cycle.
The ha monic ex ac ion algo i hm plays a decisi e ole in he ha monic mi iga ion
model. P ecise ha monic ex ac ion helps he APF o p o ide accu a e and as compen-
sa ing cu en s. The SDWT me hod is a commonly used ha monic ex ac ion and is de el-
oped om he adi ional DWT me hod. The o mulas used o he ha monic ex ac ion
p ocess a e shown below.
1. The desc ip ion o he n h ha monic componen is made acco ding o Fo mula (60):
𝑖(𝑘)=
𝐴
(𝑘).𝑐𝑜𝑠2𝜋𝑛𝑘
𝑁+𝐵(𝑘).𝑠𝑖𝑛2𝜋𝑘
𝑁 (60)
2. The AA and BB coe icien s a e calcula ed acco ding o Fo mula (61):

𝐴
(𝑘)=
𝐴
(𝑘−1)+2
𝑁.𝑐𝑜𝑠2𝜋𝑛𝑘
𝑁󰇟𝑖(𝑘)−𝑖(𝑘−𝑁󰇠
𝐵(𝑘)=𝐵(𝑘−1)+2
𝑁𝑠𝑖𝑛2𝜋𝑛𝑘
𝑁󰇟𝑖(𝑘)−𝑖(𝑘−𝑁󰇠 (61)
whe e N is he numbe o sampling poin s in one cycle and K is he la es cu en sampling
poin .
The main di e ence be ween he SDWT me hod and he DWT me hod is in he di -
e en upda e me hod o he coe icien s 𝐴(𝑘), 𝐵(𝑘). The DWT me hod (Figu e 11) e-
qui es a ull da a sampling cycle o calcula e and upda e he coe icien s. The SDWT
me hod (Figu e 12) akes a new sample each ime and he co esponding sample alue o
one cycle be o e i is disca ded and eplaced wi h he newly acqui ed sample alue and
upda ed coe icien s, which inc eases he ime e iciency o he sys em.
Figu e 11. Da a sampling cycle o he DWT me hod.
Figu e 12. Da a sampling cycle o he SDWT me hod.
Howe e , SDWT equi es addi ional memo y space o s o e sample alues o one
cycle, and his in oduces i s inhe en delay o e one cycle.
The ex ac ion o hal -cycle ha monics by he SDWT me hod is desc ibed by an ex-
ponen ial unc ion and n h ha monic exp ession acco ding o Fo mula (62):
𝑖(𝑘)=𝐼(𝑘)𝑒
 (62)
The 𝐼(𝑘) coe icien is calcula ed acco ding o Fo mula (63):
𝐼(𝑘)=1
𝑁 𝑖(𝑙)𝑒


 =
=𝐼(𝑘−1)+1
𝑁𝑖(𝑘)𝑒
−1
𝑁𝑖(𝑘−𝑁)𝑒
.𝑒
(63)
Figu e 11. Da a sampling cycle o he DWT me hod.
Ma hema ics 2023, 11, 1877 27 o 39
modi ica ion o he SDWT algo i hm hal cycle and RC hal cycle educes algo i hm delay
by hal and APF dynamic esponse ime om wo imes o hal o he powe equency
cycle.
The ha monic ex ac ion algo i hm plays a decisi e ole in he ha monic mi iga ion
model. P ecise ha monic ex ac ion helps he APF o p o ide accu a e and as compen-
sa ing cu en s. The SDWT me hod is a commonly used ha monic ex ac ion and is de el-
oped om he adi ional DWT me hod. The o mulas used o he ha monic ex ac ion
p ocess a e shown below.
1. The desc ip ion o he n h ha monic componen is made acco ding o Fo mula (60):
𝑖(𝑘)=
𝐴
(𝑘).𝑐𝑜𝑠2𝜋𝑛𝑘
𝑁+𝐵(𝑘).𝑠𝑖𝑛2𝜋𝑘
𝑁 (60)
2. The AA and BB coe icien s a e calcula ed acco ding o Fo mula (61):

𝐴
(𝑘)=
𝐴
(𝑘−1)+2
𝑁.𝑐𝑜𝑠2𝜋𝑛𝑘
𝑁󰇟𝑖(𝑘)−𝑖(𝑘−𝑁󰇠
𝐵(𝑘)=𝐵(𝑘−1)+2
𝑁𝑠𝑖𝑛2𝜋𝑛𝑘
𝑁󰇟𝑖(𝑘)−𝑖(𝑘−𝑁󰇠 (61)
whe e N is he numbe o sampling poin s in one cycle and K is he la es cu en sampling
poin .
The main di e ence be ween he SDWT me hod and he DWT me hod is in he di -
e en upda e me hod o he coe icien s 𝐴(𝑘), 𝐵(𝑘). The DWT me hod (Figu e 11) e-
qui es a ull da a sampling cycle o calcula e and upda e he coe icien s. The SDWT
me hod (Figu e 12) akes a new sample each ime and he co esponding sample alue o
one cycle be o e i is disca ded and eplaced wi h he newly acqui ed sample alue and
upda ed coe icien s, which inc eases he ime e iciency o he sys em.
Figu e 11. Da a sampling cycle o he DWT me hod.
Figu e 12. Da a sampling cycle o he SDWT me hod.
Howe e , SDWT equi es addi ional memo y space o s o e sample alues o one
cycle, and his in oduces i s inhe en delay o e one cycle.
The ex ac ion o hal -cycle ha monics by he SDWT me hod is desc ibed by an ex-
ponen ial unc ion and n h ha monic exp ession acco ding o Fo mula (62):
𝑖(𝑘)=𝐼(𝑘)𝑒
 (62)
The 𝐼(𝑘) coe icien is calcula ed acco ding o Fo mula (63):
𝐼(𝑘)=1
𝑁 𝑖(𝑙)𝑒


 =
=𝐼(𝑘−1)+1
𝑁𝑖(𝑘)𝑒
−1
𝑁𝑖(𝑘−𝑁)𝑒
.𝑒
(63)
Figu e 12. Da a sampling cycle o he SDWT me hod.
Howe e , SDWT equi es addi ional memo y space o s o e sample alues o one
cycle, and his in oduces i s inhe en delay o e one cycle.
The ex ac ion o hal -cycle ha monics by he SDWT me hod is desc ibed by an
exponen ial unc ion and n h ha monic exp ession acco ding o Fo mula (62):
in(k)=In(k)ej2πk
N(62)
The In(k)coe icien is calcula ed acco ding o Fo mula (63):
In(k)=1
N
k
∑
l=k−N+1
i(l)e−j2πnk
N=
=In(k−1)+1
Ni(k)e−j2πnk
N−1
Ni(k−N)e−j2πnk
N.ej2nπ
(63)
I you mul iply bo h sides o Equa ion (5) by he exp ession
ej2πk
N
, you ob ain he esul
o Equa ion (64):
in(k)=in(k−1)ej2πk
N+1
Ni(k)−1
Ni(k−N)ej2nπ(64)
I you con e Fo mula (6) o he Z domain, you ob ain he SDFT ans e unc ion
acco ding o Fo mula (65):
Hs(Z)=Zin(k)
i(k)=1
N.1−Z−N
1−ej2nπ
N.Z−1(65)
Ma hema ics 2023,11, 1877 32 o 36
•
The ex ac ion ime o he ha monic componen in he signal con ibu es o imp o ing
he e iciency o ha monic il e ing de ices. Cu en me hods do no mee he ha monic
il e ing a e equi ed by compu e p ocessing because he e is no pe ec me hod o
pe o m he ex ac ion o all ha monic componen s in he signal.
•
The e iciency o ex ac ing ha monic componen s in he signal is he deciding ac o
in he success o ailu e o he ha monic il e de ice. Accu a e, comple e, and imely
ha monic componen ex ac ion is a sui able inpu o classi ie s and signal selec o s
ha compensa e o ha monic losses quickly. A new me hod o applying a i icial
in elligence algo i hms is conside ed pe ec o u u e esea che s.
6. Conclusions
Signals a e ecognized and classi ied based on he ea u e composi ion o each co -
esponding signal ype. The accu acy o signal ea u e ex ac ion me hods con ibu es o
imp o ed signal-p ocessing pe o mance. The nonlinea signals a e ea u e-ex ac ed based
on he nonlinea dynamic analysis me hod and his nonlinea dynamic analysis me hod is
widely used in signal p ocessing. The selec ion o signal-p ocessing algo i hms and ea u es
de e mines he pe o mance o he signal-p ocessing sys em. The main con ibu ions o
his s udy a e as ollows:
•
This pape p esen s ou me hods (EMD, LWEMD, AHD, and AMS) o ex ac ha -
monic componen s based on he ime domain. The AMD me hod pe o ms ull
ha monic ex ac ion by binning, bu i is ime-consuming in e ms o signal p ocessing
and many low- equency signals exis . The LWEMD me hod pe o ms a small amoun
o signal ex ac ion du ing il e ing, which imp o es he e iciency o signal ex ac ion
and sho ens he signal p ocessing ime, e ec i ely esponding o low- equency signal
moni o ing. The AHD me hod de ec s pulses wi h a high noise a e. The AMS me hod
implemen s he analysis model modi ica ion in he sliding window o imp o e he
ha monic ex ac ion accu acy.
•The s udy also p esen s, in de ail, wo ex ac ion me hods (AHWT, SDWT) o ou pu
ha monics in he equency domain. The AHWT me hod uses a c oss-compa ison
o wa ele ea u es o moni o he e iciency and damage o he signal wa e o m
s uc u e. The SDWT me hod pe o ms hal -cycle co ec ion, which educes he signal
p ocessing ime bu gene a es high-gain second ha monics.
•
This pape p esen s a me hod (HRTF) o spa ial domain ha monic componen ex ac-
ion. The HRTF s a es ha sphe ical wa ele s a e local unc ions and ha e di icul y
expanding signals in sphe es, while sphe ical ha monics ep esen coa se s uc u es o
low esolu ion. Bo h a e used o model spa ial de ails.
This pape makes an o e iew and p esen s in de ail some me hods used o ex ac
he ha monic and undamen al wa e componen s in signals in he ime domain, equency
domain, and space domain. In p inciple, all h ee applica ions o ha monic and unda-
men al equency ex ac ion a e di e en . In some cases o di e en signals, ex ac ing
he ha monic componen gi es be e esul s. Howe e , he equency domain ex ac-
ion me hod equi es a mo e complex me hod and uses a lo o complex ma hema ical
equa ions, which leads o highe compu a ional cos s and an inc eased sampling delay
o a minimum o one sampling cycle and depends on he esolu ion o he equency. A
me hod is used o ex ac he ha monic componen in he equency domain and hal e he
sampling pe iod o sho en he ha monic ex ac ion ime. Howe e , i gene a es second
ha monics. Along wi h he de elopmen o swi ching de ices as well as equency mod-
ula ion de ices, his leads o an inc easing amoun o digi al signal p ocessing o e ime
and he de elopmen o new echnologies and p ocessing me hods. Signal managemen is
inc easingly demanding in e ms o he cha ac e is ics o ha monic ex ac ion e iciency;
he signal ex ac ion p ocessing ime mus be as . A me hod o signal ex ac ion in he
ime domain is used by many de elope s o ha monic ex ac ion applica ions because hey
help in he eal- ime moni o ing o he signal. In he ime domain, he signal ex ac ion
me hod is used in combina ion wi h me hods such as eedback con ol echniques o

Ma hema ics 2023,11, 1877 33 o 36
signals, il e ing, and canceling echniques acco ding o he use ’s equi emen s. Howe e ,
he eedback signal uning echnique is a di icul echnique ha equi es high accu acy o
he uning pa ame e s; he e o e, he signal-p ocessing sys em needs o pe o m he co ec
pa ame e co ec ion o achie e high signal p ocessing e iciency ha p o ides op imal
signal-p ocessing pe o mance when he e a e changes in signal p ocessing. Ex ac ing he
ha monic componen signal in he ime domain has one weakness: i does no gua an ee
he s abili y and du abili y o he gene a ed noise signals.
Fil e s and ha monic componen signal ex ac ion always come wi h many challenges
ha signal ex ac ion o p ocessing echniques ace, such as he delay o leng hening o
he eedback signal when p ocessing signal il e ing; o pe o m il e cancella ion, he
delay signal mus be wi hin a ce ain ame o e e ence. Se e al me hods ha e been
de eloped o elimina e he a o emen ioned p oblems, and hose me hods a e implemen ed
en i ely in a s a iona y ame o e e ence ha ensu es he comple e dynamics o hei
ope a ion. Ma hema ical equa ions a e also conside ed o use o simula e il e ing o
canceling p ocesses o de e ed signals because he ope a o s o he di e ence a e close
o he ma hema ical model. The ope a o models used o cancel he e lec ed signals in
his pe iod a e implemen ed a disc e e impulse esponse il e s. Techniques ha e been
de eloped o cancel he signal delay using i s -o de il e s o ou h-o de il e s and hey
a e widely used in digi al signal p ocessing and deli e solid pe o mance. Mo eo e , e o
s a es a ise when signal p ocessing is lowe o ze o; he esponse a e has he sho es ime
o mee he goal o ex ac ing he mos accu a e ha monic componen o il e sys em
selec ion o compensa e o he bes ha monic losses and imp o e he pe o mance o he
signal sys em.
Ex ac ing ha monics om he signal by a i icial in elligence echniques is inc easingly
being conside ed. Using me a-heu is ic op imiza ion echniques in signal p ocessing wi h
suppo om compu e science is always a p omising esea ch di ec ion.
Au ho Con ibu ions:
Concep ualiza ion, M.L.D. and P.B.; me hodology, M.L.D.; so wa e, M.L.D.;
alida ion, M.L.D. and P.B.; o mal analysis, M.L.D.; in es iga ion, M.L.D.; esou ces, M.L.D.; da a
cu a ion, M.L.D.; w i ing—o iginal d a p epa a ion, M.L.D.; w i ing— e iew and edi ing, M.L.D.;
isualiza ion, M.L.D.; supe ision, P.B. and R.M.; p ojec adminis a ion, P.B. and R.M.; unding
acquisi ion, P.B. and R.M. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
This wo k was suppo ed in pa by he Minis y o Educa ion o he Czech Republic
(P ojec No. SP2023/090).
Da a A ailabili y S a emen : No applicable.
Acknowledgmen s:
The au ho s a e ex emely g a e ul o VSB–Technical Uni e si y o Os a a,
Czechia o inancial suppo . They would also like o exp ess hei g a i ude o Van Lang Uni e si y,
Vie nam, o suppo ing his esea ch.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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