applied
sciences
A icle
Modi ied Fil e ed-X Hie a chical LMS Algo i hm wi h
Sequen ial Pa ial Upda es o Ac i e Noise Con ol
Ped o Ramos Lo en e 1,* , Raúl Ma ín Fe e 2, Fe nando A anz Ma ínez 1and Guille mo Palacios-Na a o 1


Ci a ion: Ramos Lo en e, P.; Ma ín
Fe e , R.; A anz Ma ínez, F.;
Palacios-Na a o, G. Modi ied
Fil e ed-X Hie a chical LMS
Algo i hm wi h Sequen ial Pa ial
Upda es o Ac i e Noise Con ol.
Appl. Sci. 2021,11, 344. h ps://
doi.o g/10.3390/app11010344
Recei ed: 26 No embe 2020
Accep ed: 28 Decembe 2020
Published: 31 Decembe 2020
Publishe ’s No e: MDPI s ays neu-
al wi h ega d o ju isdic ional clai-
ms in published maps and ins i u io-
nal a ilia ions.
Copy igh : © 2020 by he au ho s. Li-
censee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and con-
di ions o he C ea i e Commons A -
ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
1Depa men o Elec onic Enginee ing and Communica ions, Uni e si y o Za agoza, 44003 Te uel, Spain;
a anz@uniza .es (F.A.M.); guille mo.palacios@uniza .es (G.P.-N.)
2Depa men o Compu e Science and Sys ems Enginee ing, Uni e si y o Za agoza, 44003 Te uel, Spain;
ama @uniza .es
*Co espondence: p amos@uniza .es
Abs ac :
In he ield o ac i e noise con ol (ANC), a popula me hod is he modi ied il e ed-x LMS
algo i hm. Howe e , i has wo d awbacks: i s compu a ional complexi y highe han ha o he
con en ional FxLMS, and i s con e gence a e ha could s ill be imp o ed. The e o e, we p opose an
adap i e s a egy which aims a speeding up he con e gence a e o an ANC sys em dealing wi h
pe iodic dis u bances. This algo i hm consis s in combining he o ganiza ion o he il e weigh s in a
hie a chy o sub il e s o sho e leng h and hei sequen ial pa ial upda es (PU). Ou con ibu ion is
h ee old: (1) we p o ide he heo e ical basis o he exis ence o a equency-dependen pa ame e ,
called gain in s ep-size. (2) The heo e ical uppe bound o he s ep-size is compa ed wi h he limi
ob ained om simula ions. (3) Addi ional expe imen s show ha his s a egy esul s in a as
algo i hm wi h a compu a ional complexi y close o ha o he con en ional FxLMS.
Keywo ds:
adap i e signal p ocessing; ac i e con ol o pe iodic noise; modi ied il e ed-x LMS;
hie a chical il e ; sequen ial pa ial upda es
1. In oduc ion
A enua ion o acous ic dis u bances has ecei ed widesp ead a en ion in ecen
decades since noise se iously a ec s human heal h [
1
–
3
]. Thus, noise con ol s a egies
ha e been applied in di e en scena ios, such as ai c a , oad ehicles, o he p oximi y o
ai condi ioning duc s, whe e he noise le el has o be educed o imp o e in elligibili y.
Apa om passi e echniques based on he abso p ion and e lec ion p ope ies o
ma e ials [
4
,
5
], acous ic noise educ ion can be done by using ac i e noise con ol (ANC)
echniques based on he p inciple o des uc i e wa e in e e ence. Thus, o cancel he
annoying noise a a gi en loca ion, an an i-noise is gene a ed wi h he same ampli ude
as he undesi ed dis u bance, bu wi h an app op ia e phase shi . This is ca ied ou
by means o seconda y sou ces, gene a ing a zone o silence a ound an acous ical senso .
As he p ope ies—powe , equency, e c.—o he undesi ed acous ic dis u bance may be
ime- a ian , adap i e con ol sys ems ha e o be implemen ed o a enua e he noise [6].
One may ind in [
7
] a e iew o ANC echniques o noise cancella ion inside
au omobiles— ha is ou ield o in e es —du ing he pas 15 yea s, including comme cial
de elopmen s a ailable in mass p oduc ion ehicles.
The mos popula adap i e algo i hm used in DSP-based implemen a ions o ANC
sys ems is he il e ed-x LMS (FxLMS) algo i hm, o iginally p oposed by Mo gan [
8
].
Figu e 1shows he way he elec o-acous ic elemen s a e a anged and he block diag am
o his solu ion.
Appl. Sci. 2021,11, 344. h ps://doi.o g/10.3390/app11010344 h ps://www.mdpi.com/jou nal/applsci
Appl. Sci. 2021,11, 344 2 o 24
Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 o 25
(a)
(b)
Figu e 1. Single-channel ac i e noise con ol sys em using he FxLMS algo i hm. (a) Physical a angemen o he elec o-
acous ic elemen s. (b) Equi alen block diag am.
The p ima y pa h 𝑃(𝑧) comp ises he elemen s om he e e ence mic ophone o he
e o mic ophone, whe eas he seconda y pa h 𝑆(𝑧) includes he elemen s om he sec-
onda y sou ce o he e o mic ophone, namely he D/A con e e , he powe ampli ie ,
he loudspeake , he acous ic pa h, he e o mic ophone, and he A/D con e e . The
adap i e con ol il e is deno ed as 𝑊(𝑧). Due o he p esence o a seconda y pa h ha is
ed by he ou pu o he ANC sys em, de i ing he LMS-based ANC solu ion leads o a
speci ic ecu si e equa ion. Indeed, he adap i e il e aps a e upda ed by adding a
weigh ed e m de ined as he p oduc o he e e ence signal il e ed by he seconda y
pa h and he so-called e o (The e o e(n) is de ined as he di e ence be ween he an i-
noise and he undesi ed dis u bance). The e o e, he FxLMS-based solu ion equi es an
accu a e es ima e 𝑆
󰆻(𝑧) o he seconda y pa h [8]. Mo eo e , he con e gence o he adap-
i e il e depends on he s ep size.
In [9] (The e sion o he FxLMS wi h leakage add essed in [9] is o en used in p ac-
ical implemen a ions o cons ain he powe o he ou pu y(n) o he cancelle . Then, he
leaky FxLMS algo i hm educes undesi able e ec s due o nume ical e o s in ini e-p e-
cision machines, o e load o he seconda y sou ce, e c.), a s ochas ic analysis o he FxLMS
based on he i s and second o de momen s o he weigh -e o ec o makes i possible
o de i e he uppe s ep-size bound, whe eas a con e gence condi ion o he FxLMS wi h
de e minis ic e e ence can be ound in [10]. A comple e s a is ical con e gence analysis
o he FxLMS algo i hm wi hou assuming a speci ic model o he e e ence signal can be
ound in [11].
E en i he compu a ional complexi y o he FxLMS is qui e low, i needs o be e-
duced as much as possible o be implemen ed in DSP-based eal ime applica ions. In ad-
di ion, he FxLMS algo i hm su e s om slow con e gence mainly due o he ou pu de-
lay caused by 𝑆(𝑧). Mo eo e , e o s in he es ima e o he seconda y pa h esul in ins a-
bili y o he FxLMS algo i hm [12–14]. The e o e, a ious me hods ha e been p oposed o
a oid he abo e d awbacks.
Thus, o educe he compu a ional complexi y o he con ol algo i hm, he delayed-x
LMS [15,16] can be conside ed. This con ol s a egy is based on he hypo hesis ha he sec-
onda y pa h model o he FxLMS me hod does no ha e o be accu a e and can be ep e-
sen ed by a delay. To e ec i ely emo e he delay o he seconda y pa h wi hin he coe i-
cien upda es, he modi ied FxLMS (Mod FxLMS) algo i hm [17,18] has been p oposed. I is
based on he es ima ion o he undesi ed noise by il e ing he ou pu o he ANC by he
es ima e o he seconda y pa h 𝑆
󰆻(𝑧) and by adding he esul ing ou pu 𝑦(𝑛) o he e o
measu ed by he e o mic ophone. Ha ing es ima ed he undesi ed noise 𝑑
󰆻(𝑛), he sec-
onda y pa h and he adap i e il e a e swapped in he upda es pa h. Then, he e o signal
o he adap i e algo i hm is calcula ed as he di e ence be ween he es ima ed noise and
he ou pu o he adap i e il e . Hence, he beha io o he sys em is simila o ha o he
Figu e 1.
Single-channel ac i e noise con ol sys em using he FxLMS algo i hm. (
a
) Physical a angemen o he elec o-
acous ic elemen s. (b) Equi alen block diag am.
The p ima y pa h
P(z)
comp ises he elemen s om he e e ence mic ophone o
he e o mic ophone, whe eas he seconda y pa h
S(z)
includes he elemen s om he
seconda y sou ce o he e o mic ophone, namely he D/A con e e , he powe ampli ie ,
he loudspeake , he acous ic pa h, he e o mic ophone, and he A/D con e e . The
adap i e con ol il e is deno ed as
W(z)
. Due o he p esence o a seconda y pa h ha
is ed by he ou pu o he ANC sys em, de i ing he LMS-based ANC solu ion leads o
a speci ic ecu si e equa ion. Indeed, he adap i e il e aps a e upda ed by adding a
weigh ed e m de ined as he p oduc o he e e ence signal il e ed by he seconda y pa h
and he so-called e o (The e o e(n) is de ined as he di e ence be ween he an inoise
and he undesi ed dis u bance). The e o e, he FxLMS-based solu ion equi es an accu a e
es ima e
e
S(z)
o he seconda y pa h [
8
]. Mo eo e , he con e gence o he adap i e il e
depends on he s ep size.
In [
9
] (The e sion o he FxLMS wi h leakage add essed in [
9
] is o en used in p ac ical
implemen a ions o cons ain he powe o he ou pu y(n) o he cancelle . Then, he leaky
FxLMS algo i hm educes undesi able e ec s due o nume ical e o s in ini e-p ecision
machines, o e load o he seconda y sou ce, e c.), a s ochas ic analysis o he FxLMS based
on he i s and second o de momen s o he weigh -e o ec o makes i possible o
de i e he uppe s ep-size bound, whe eas a con e gence condi ion o he FxLMS wi h
de e minis ic e e ence can be ound in [
10
]. A comple e s a is ical con e gence analysis o
he FxLMS algo i hm wi hou assuming a speci ic model o he e e ence signal can be
ound in [11].
E en i he compu a ional complexi y o he FxLMS is qui e low, i needs o be educed
as much as possible o be implemen ed in DSP-based eal ime applica ions. In addi ion,
he FxLMS algo i hm su e s om slow con e gence mainly due o he ou pu delay caused
by
S(z)
. Mo eo e , e o s in he es ima e o he seconda y pa h esul in ins abili y o he
FxLMS algo i hm [
12
–
14
]. The e o e, a ious me hods ha e been p oposed o a oid he
abo e d awbacks.
Thus, o educe he compu a ional complexi y o he con ol algo i hm, he delayed-x
LMS [
15
,
16
] can be conside ed. This con ol s a egy is based on he hypo hesis ha he
seconda y pa h model o he FxLMS me hod does no ha e o be accu a e and can be
ep esen ed by a delay. To e ec i ely emo e he delay o he seconda y pa h wi hin
he coe icien upda es, he modi ied FxLMS (Mod FxLMS) algo i hm [
17
,
18
] has been
p oposed. I is based on he es ima ion o he undesi ed noise by il e ing he ou pu o he
ANC by he es ima e o he seconda y pa h
e
S(z)
and by adding he esul ing ou pu
e
y(n)
o
he e o measu ed by he e o mic ophone. Ha ing es ima ed he undesi ed noise
e
d(n)
,
he seconda y pa h and he adap i e il e a e swapped in he upda es pa h. Then, he
e o signal o he adap i e algo i hm is calcula ed as he di e ence be ween he es ima ed
noise and he ou pu o he adap i e il e . Hence, he beha io o he sys em is simila
Appl. Sci. 2021,11, 344 3 o 24
o ha o he con en ional LMS algo i hm. Ne e heless, he seconda y pa h impulse
esponse is assumed o be accu a ely es ima ed (A equency domain analysis abou he
beha iou o he Mod FxLMS algo i hm in he p esence o seconda y pa h modelling
e o s is p oposed in [
19
]). The con e gence a e o he Mod FxLMS algo i hm is inc eased
a he cos o an addi ional compu a ional complexi y, which u ns ou o be he main
d awback o he app oach. A ade-o has hence o be ound be ween con e gence a e
and compu a ional complexi y [
20
]. Reduced-complexi y implemen a ions o he Mod
FxLMS ha e been p oposed in [
21
,
22
], bu he con e gence speed can s ill be imp o ed.
In [
23
] a new delay-less equency-domain ANC algo i hm is p oposed. The p oposal
no only emo es he delay in he weigh adap a ion (as he modi ied il e -x scheme
implies) bu also emo es he delay in he signal pa h. The p oposed s a egy exhibi s
lowe compu a ional complexi y han o he s a e-o - he-a equency-domain FxLMS
algo i hms [24].
In his pape , we p opose o combine he Mod FxLMS s uc u e and he hie a chical
LMS (HLMS) algo i hm, ini ially de eloped in he ield o channel equaliza ion by Woo [
25
].
One may ind a pe o mance analysis o he HLMS algo i hm in [
26
]. In [
27
], he mean-
squa ed e o in a wo-le el HLMS algo i hm is analyzed; in his example, he HLMS is
used as a p edic i e s a egy ha can signi ican ly speed up he con e gence a e du ing
he ini ial s age o he algo i hm.
In he HLMS adap i e algo i hm, he il e coe icien s a e o ganized in o a hie a chy
o sub il e s o sho e leng h dis ibu ed in
α
le els (de ails on he hie a chical a angemen
o sub il e s a e p o ided in Sec ion 2). The ou pu signals o he sub il e s a le el
(l−1)
a e he inpu signals o he sub il e s placed a he nex le el l, wi h l a ying om 1 o
α
. Then, he numbe o sub il e s pe le el is di ided om le el (l
−
1) o le el lby a
ac o gi en by he leng h o he sub il e s a le el (l
−
1). A he las le el o he hie a chy,
namely le el
α
, he e is only one sub il e . Since he sub il e s ha e sho e leng h han
a con en ional FIR il e , hey can con e ge as e , as ecalled in he Appendix A.3 o
he Appendix A. Howe e , he compu a ional complexi y associa ed o his mul i-le el
s uc u e is highe han ha o he con en ional LMS algo i hm.
To add ess he abo e p oblem, we sugges using pa ial upda es (PU) o he adap i e
il e coe icien s. A widely used PU algo i hm is he sequen ial PU LMS algo i hm wi h
decima ion ac o N[
28
]. This algo i hm upda es a subse o size L/N, ou o Lcoe icien s—
wj(n), 1 ≤j≤L—pe i e a ion o a con en ional L-leng h FIR il e acco ding o
wj(n+1) = wj(n) + µx(n−j+1)e(n)i (n−j+1)mod N=0
wj(n)o he wise (1)
whe e
µ
is he s ep-size o he algo i hm, x(n) he inpu signal, and e(n) he e o . Ne e he-
less, he highe he decima ion ac o Nis, he lowe he con e gence a e will be. In [
29
],
we ha e shown ha , in he con ex o a con en ional adap i e FIR il e , his lowe con-
e gence a e can be compensa ed, unde he assump ion o a pe iodic inpu signal, by an
a o dable inc ease in he s ep-size
µ
. As he maximum s ep-size ha ensu es con e gence
wi h a sequen ial PU algo i hm is N imes la ge han he maximum s ep-size o a ull
upda es adap i e algo i hm, one can in oduce a pa ame e called gain in s ep-size, ha
de e mines he ac o by which he s ep-size
µ
can be mul iplied o imp o e he con e -
gence a e o he sequen ial PU adap i e algo i hm. No e ha he heo e ical analysis o he
s a egy de eloped in [
29
] excludes he use o ce ain equencies co esponding o no ches
appea ing in he gain in s ep-size whose wid h and exac loca ion depend on he sys em
pa ame e s, namely he decima ion ac o , he sampling equency and he leng h o he
adap i e il e .
In his pape , ou pu pose is hence o s udy he ele ance o he combina ion o he
Mod FxLMS, he HLMS and he sequen ial PU LMS wi h gain in s ep size. The esul ing
ANC app oach is called he modi ied il e ed-x hie a chical sequen ial PU LMS algo i hm
wi h gain in s ep-size (G
µ
—Mod Fx H Seq LMS). The o he con ibu ions o his pape
consis in:
Appl. Sci. 2021,11, 344 4 o 24
(1) de i ing he heo e ical gain in s ep-size o he con ol s a egy. I is de ined as he
a io be ween he uppe bounds on he s ep-sizes e alua ed in he wo ollowing cases:
when only a subse o he weigh s o he hie a chical il e p oposed by Woo [
25
] is upda ed
a each i e a ion and when e e y ap— ega dless he posi ion o he weigh in he hie a chy
o sub il e s—is upda ed a e e y cycle. We will see ha he equency esponse o his
gain in he s ep-size exhibi s no ches. Thei wid h and loca ion depend on he leng h o
he slowes sub il e o he hie a chy, he decima ion ac o , and he sampling equency.
The e o e, his phenomenon has o be aken in o accoun when he inpu signal con ains
ha monics a equencies co esponding o he loca ion o he no ches;
(2) ca ying ou compu e -based expe imen s o con i m ha he p edic ed heo e ical
gain in s ep o he G
µ
—Mod Fx H Seq LMS algo i hm is well sui ed o he maximum
a o dable inc ease in s ep-size ob ained by simula ions;
(3) comple ing addi ional compu e -based simula ions o es he pe o mance o he
Gµ—Mod Fx H Seq LMS algo i hm o ac i e a enua ion o pe iodic dis u bances.
The pape is o ganized as ollows. In Sec ion 2, we p opose he modi ied il e ed-x
hie a chical sequen ial PU LMS algo i hm wi h gain in s ep-size. Sec ion 3deals wi h he
con e gence analysis o he p oposed algo i hm. The app oach consis s in applying o
he hie a chical il e used in ou p oposal he esul s p o ided in he Appendix A o a
con en ional adap i e FIR il e . Resul s o compu e -based simula ions a e p o ided in
Sec ion 4. We ca y ou a compa ison be ween he heo e ical p edic ion and he expe imen-
al beha io o he p oposed algo i hm. The expe imen s also include a compa a i e s udy
o a ious ANC s a egies in e ms o con e gence a e and compu a ional complexi y.
Sec ion 5is de o ed o discussion.
2. Modi ied FX Hie a chical Sequen ial PU LMS Algo i hm wi h Gain in S ep-Size
In his sec ion, we p opose he G
µ
-Mod Fx H Seq LMS algo i hm by combining he
Mod FxLMS, he HLMS, and he sequen ial PU LMS wi h gain in s ep-size. Ou goal is o
de i e an ANC adap i e algo i hm wi h a as e con e gence a e han he con en ional
FxLMS wi h a simila compu a ional complexi y. Figu e 2shows he block diag am o he
p oposed algo i hm.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 o 25
(1) de i ing he heo e ical gain in s ep-size o he con ol s a egy. I is de ined as he
a io be ween he uppe bounds on he s ep-sizes e alua ed in he wo ollowing cases:
when only a subse o he weigh s o he hie a chical il e p oposed by Woo [25] is up-
da ed a each i e a ion and when e e y ap— ega dless he posi ion o he weigh in he
hie a chy o sub il e s—is upda ed a e e y cycle. We will see ha he equency esponse
o his gain in he s ep-size exhibi s no ches. Thei wid h and loca ion depend on he
leng h o he slowes sub il e o he hie a chy, he decima ion ac o , and he sampling
equency. The e o e, his phenomenon has o be aken in o accoun when he inpu signal
con ains ha monics a equencies co esponding o he loca ion o he no ches;
(2) ca ying ou compu e -based expe imen s o con i m ha he p edic ed heo e i-
cal gain in s ep o he Gμ—Mod Fx H Seq LMS algo i hm is well sui ed o he maximum
a o dable inc ease in s ep-size ob ained by simula ions;
(3) comple ing addi ional compu e -based simula ions o es he pe o mance o he
Gμ—Mod Fx H Seq LMS algo i hm o ac i e a enua ion o pe iodic dis u bances.
The pape is o ganized as ollows. In Sec ion 2, we p opose he modi ied il e ed-x
hie a chical sequen ial PU LMS algo i hm wi h gain in s ep-size. Sec ion 3 deals wi h he
con e gence analysis o he p oposed algo i hm. The app oach consis s in applying o he
hie a chical il e used in ou p oposal he esul s p o ided in he Appendix o a con en-
ional adap i e FIR il e . Resul s o compu e -based simula ions a e p o ided in Sec ion
4. We ca y ou a compa ison be ween he heo e ical p edic ion and he expe imen al
beha io o he p oposed algo i hm. The expe imen s also include a compa a i e s udy o
a ious ANC s a egies in e ms o con e gence a e and compu a ional complexi y. Sec-
ion 5 is de o ed o discussion.
2. Modi ied FX Hie a chical Sequen ial PU LMS Algo i hm wi h Gain in S ep-Size
In his sec ion, we p opose he Gμ-Mod Fx H Seq LMS algo i hm by combining he
Mod FxLMS, he HLMS, and he sequen ial PU LMS wi h gain in s ep-size. Ou goal is o
de i e an ANC adap i e algo i hm wi h a as e con e gence a e han he con en ional
FxLMS wi h a simila compu a ional complexi y. Figu e 2 shows he block diag am o he
p oposed algo i hm.
Figu e 2. Block diag am o he modi ied il e ed-x hie a chical LMS algo i hm wi h sequen ial
pa ial upda es.
Acco ding o Figu e 2, he il e ed e e ence 𝑥′(𝑛) is he inpu o he con ol adap i e
il e whe eas he e e ence signal 𝑥(𝑛) is il e ed by a sla e il e , which is a copy o he
Figu e 2.
Block diag am o he modi ied il e ed-x hie a chical LMS algo i hm wi h sequen ial pa ial
upda es.
Acco ding o Figu e 2, he il e ed e e ence
x0(n)
is he inpu o he con ol adap i e
il e whe eas he e e ence signal
x(n)
is il e ed by a sla e il e , which is a copy o he
con ol adap i e il e . By cascading he sla e il e and he es ima e o he seconda y pa h
Appl. Sci. 2021,11, 344 5 o 24
e
S(z)
, and hen by passing he e e ence signal
x(n)
h ough he esul ing il e , one can
ob ain an es ima ion
e
y(n)
o he an inoise and hence an es ima ion
e
d(n)
o he undesi ed
noise. Inaccu acy o he seconda y pa h es ima e and i s e ec s a e discussed in [
12
–
14
]
in he con ex o il e ed x LMS algo i hms. A ha s age, he ou pu
z(n)
o he adap i e
con ol il e is di ec ly sub ac ed om he es ima ed noise
e
d(n)
o p o ide he nume ical
e o en(n).
One may ind in [
30
] a comple e e iew o echniques o es ima ion o he e o signal
using signal p ocessing algo i hms. As his e o is used o upda e he adap i e con ol
il e , he limi a ions imposed on he s ep-size
µ
o he s anda d e sion o he FxLMS
algo i hm a e now o e come. Le us now ocus ou a en ion on he hie a chical il e .
Gi en he numbe Lo aps a he i s le el o he hie a chy, he numbe o sub il e s a he
l h le el is gi en by
Nl=L
l
∏
=1
β
=
α
∏
=1
β
l
∏
=1
β
=
α
∏
=1+1
β (2)
whe e
βl
deno es (wi h his no a ion, we implici ly assume ha he sub il e s a he same
le el ha e he same numbe o aps and his numbe may a y om one le el o ano he )
he numbe o weigh s o a sub il e a le el l, a ying om 1 o
α
. As he sub il e leng h
may a y om one le el o ano he , he s ep-size bound o e e y sub il e can be di e en .
In he sequel, he coe icien s o he i h hie a chically a anged sub il e impulse esponse
a he l h le el a e deno ed as
wl
i(n) = hwl
i1(n)wl
i2(n)··· wl
iβl(n)i, 1 ≤l≤α, 1 ≤i≤Nl(3)
whe e
wl
ij
deno es he weigh o he j h ap o he i h sub il e a he l h le el. In addi ion,
zl
ij
and
yl
ij
, espec i ely, deno e he inpu signals o he he j h ap o he i h sub il e a
he l h le el o he adap i e and he sla e hie a chical il e s. The ou pu s o he adap i e
and he sla e hie a chical il e s, espec i ely deno ed as
z(n)
and
y(n)
, a e gi en by he
las loop o he mul ile el il e ing, ha is,
z(n)=zα+1
11 (n)
and
y(n)=yα+1
11 (n)
. The e o
signal o he i h sub il e a he l h le el is deno ed as
el
i
. These e o signals a e ob ained by
sub ac ing he ou pu o e e y sub il e om he es ima ed noise
e
d(n)
. I should be no ed
ha he necessi y o using he es ima ed noise
e
d(n)
o upda e he sub il e s placed a he
in e media e le els o he hie a chy is al eady sol ed as we use he Mod FxLMS e sion o
he ANC algo i hm. Figu e 3shows he a chi ec u e o a 2-le el hie a chical il e . In his
example, he numbe o sub il e s a le els 1 and 2 a e
N1=L/β
and
N2=
1, espec i ely.
The numbe o coe icien s o e e y sub il e a le els 1 and 2 a e βand L/β, espec i ely.
The main d awback o he HLMS is he high compu a ional complexi y inhe en ly
associa ed o i s mul i-le el s uc u e.
Sequen ial PU o he coe icien s o he hie a chical il e a e used o educe he com-
pu a ional complexi y. PU a e applied o e e y coe icien a e e y le el, o he hie a chical
o ganiza ion o aps, om he i s ap o he i s sub il e o he las ap o he las sub-
il e . Fo ins ance, in Figu e 3, he shadowed coe icien s o he hie a chical il e a e he
N-equally-spaced aps ha ha e o be upda ed a a gi en ime n. A he ollowing i e a ions
o he upda ing p ocess, namely n+ 1, n+ 2,
. . .
,n+N
−
1, he nex subse s o equally
spaced coe icien s o he hie a chical il e a e upda ed.

Appl. Sci. 2021,11, 344 6 o 24
Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 o 25
Figu e 3. Two-le el hie a chical il e .
The main d awback o he HLMS is he high compu a ional complexi y inhe en ly
associa ed o i s mul i-le el s uc u e.
Sequen ial PU o he coe icien s o he hie a chical il e a e used o educe he com-
pu a ional complexi y. PU a e applied o e e y coe icien a e e y le el, o he hie a -
chical o ganiza ion o aps, om he i s ap o he i s sub il e o he las ap o he las
sub il e . Fo ins ance, in Figu e 3, he shadowed coe icien s o he hie a chical il e a e
he N-equally-spaced aps ha ha e o be upda ed a a gi en ime n. A he ollowing i e -
a ions o he upda ing p ocess, namely n + 1, n + 2, …, n + N − 1, he nex subse s o equally
spaced coe icien s o he hie a chical il e a e upda ed.
Due o PU, he algo i hm su e s om a educ ion in con e gence a e as N inc eases.
Then, by using he gain in s ep-size, he slowe con e gence a e o he sequen ial PU
adap i e algo i hm can be compensa ed. The s a egy hence gi es he same pe o mance
as ha o he ull upda es algo i hm in e ms o con e gence a e, bu wi h lowe compu-
a ional complexi y. In p e ious wo ks [29], his s a egy is analyzed in he con ex o a
con en ional adap i e FIR il e .
The Gμ—Mod Fx H Seq LMS deals wi h pe iodic dis u bances. These pe iodic noises,
such as engine noise, a e e y o en he subjec o cancella ion in ANC applica ions. This
is due o wo easons. Fi s , hese dis u bances a e he mos annoying and, second, i is
usually easy o ind a good e e ence signal o cancel hem. In he Algo i hm 1, he Gμ—
Mod Fx H Seq LMS algo i hm is gi en:
Figu e 3. Two-le el hie a chical il e .
Due o PU, he algo i hm su e s om a educ ion in con e gence a e as Ninc eases.
Then, by using he gain in s ep-size, he slowe con e gence a e o he sequen ial PU
adap i e algo i hm can be compensa ed. The s a egy hence gi es he same pe o mance
as ha o he ull upda es algo i hm in e ms o con e gence a e, bu wi h lowe compu-
a ional complexi y. In p e ious wo ks [
29
], his s a egy is analyzed in he con ex o a
con en ional adap i e FIR il e .
The G
µ
—Mod Fx H Seq LMS deals wi h pe iodic dis u bances. These pe iodic noises,
such as engine noise, a e e y o en he subjec o cancella ion in ANC applica ions. This
is due o wo easons. Fi s , hese dis u bances a e he mos annoying and, second, i
is usually easy o ind a good e e ence signal o cancel hem. In he Algo i hm 1, he
Gµ—Mod Fx H Seq LMS algo i hm is gi en:
Appl. Sci. 2021,11, 344 7 o 24
Algo i hm 1 Gµ—Mod Fx H Seq LMS algo i hm
o i = 1 o # i e a ions /* MAIN LOOP*/
y1
∀i∀j(n) = x(n)/*Fi s le el o sla e hie a chical il e is illed wi h
x(n)*/
/* SLAVE HIERARCHICAL FILTER */
o l = 1 o α/* F om i s o op (α)le el o he hie a chy o*/
o i = 1 o α
∏
=l+1
β /* F om i s o las sub il e a each le el */
/* Compu ing he ou pu o e e y
sub il e */
yl+1
p q (n) = wl
iT(n)yl
ip=di
βle,q=i−bi−1
βlcβl
end o o (i)
end o o (l)→y(n) = yα+1
1 1 (n)/* END OF SLAVE HIERARCHICAL FILTER */
e
y(n) = ~
sT(n)yα+1
11 (n)
/* Compu ing he an inoise signal whe e */
/* es(n) = [es1es2. . . esL]Tand */
/* yα+1
1 1 (n) =
hyα+1
11 (n). . . yα+1
1 1 (n−Ls+1))iT*/
/* Measu ed e o em(n) = d(n)−y0(n)*/
e
d(n) = e
y(n) + em(n)/* Compu ing he es ima ed noise */
x0(n) = ~
sT(n)x(n)/* Fil e ing he e e encex(n) =
[x(n). . . x(n−Ls+1))]T*/
z1
∀i∀j(n) = x0(n)/* Fi s le el o adap i e hie a chical il e is illed
wi h x0(n)
*//* ADAPTIVE HIERARCHICAL FILTER */
o l = 1 o α/* F om i s o op le el o he hie a chy */
o i = 1 o α
∏
=l+1
β /* F om i s o las sub il e a each le el */
/* Compu ing he ou pu o e e y
sub il e */
zl+1
p q (n) = wl
iT(n)zl
ip=di
βle,q=i−bi−1
βlcβl
el
i(n) = e
d(n)−wl
iT(n)zl
i(n)/* Compu ing he e o o e e y sub il e */
o j = 1 o βl/* Fo e e y ap, Sequen ial pa ial upda es */
i
(k−((i−1)βl+j) + 1)mod N== 0
wl
i j(k+1) = wl
i j(k) + Gµµlel
i(n)zl
i j(n)
else
wl
i j(k+1) = wl
i j(k)
end o i
end o o (j)
end o o (i)
end o o (l)→z(n) = zα+1
1 1 (n)/* END OF ADAPTIVE HIERARCHICAL
FILTER */
end o o (n)/* END OF MAIN LOOP */
3. Con e gence Analysis
In he i s pa o his sec ion, we es ablish he assump ions aken in o accoun in
he con e gence analysis. In he second sub-sec ion, we de i e he gain in s ep-size o he
Gµ—Mod Fx H Seq LMS algo i hm.
3.1. Assump ions in he Con e gence Analysis
In [
29
], we ha e de i ed an uppe bound on he s ep-size o he Fx sequen ial PU LMS
algo i hm upda ing a con en ional FIR il e . This analysis is based on wo assump ions,
Appl. Sci. 2021,11, 344 8 o 24
namely he independence heo y be ween he e e ence signal and he il e weigh s, and
he slow con e gence condi ion (The eade is e e ed o [
29
] o mo e in o ma ion on he
assump ion o independence heo y and he slow con e gence condi ion. Despi e he ac
ha such assump ions migh be ini ially ques ionable when dealing wi h pe iodic inpu s,
we con i m in [
29
] he easibili y o assuming bo h condi ions in he analysis o a FIR-based
Fx sequen ial PU LMS s a egy o a enua e pe iodic dis u bances).
When using a mo e complex il e ing s uc u e [
25
] based on hie a chically a anged
sub il e s
wl
i(n)
, 1
≤l≤α
, 1
≤i≤N
, he o e all con e gence o he hie a chical s uc u e
is assumed o be cons ained by he hie a chical a anged sub il e
wslow(n)
ha con e ges
wi h he slowes con e gence a e. By applying o
wslow(n)
he con e gence analysis o a
con en ional FIR il e ecalled in he Appendix A, one can ob ain he analy ical exp ession
o he gain in s ep size o he Gµ—Mod Fx H Seq LMS algo i hm.
Le us now gi e he c i e ia o ecognize he slowes sub il e
wslow(n)
o he a ange-
men . The con e gence condi ions o
wl
i(n)
, 1
≤l≤α
, 1
≤i≤N
, depends on i s leng h
and on i s inpu signal. The la ge he sub il e is, he smalle he maximum s ep-size is.
The e o e, he la ge he sub il e is, he slowe he con e gence will be (See Appendix A.3
o he Appendix A, whe e he dependence o he s ep-size bound on he leng h o a il e
is de i ed). As a as he inpu signal is conce ned, Woo [
25
] s a es ha he in luence o
he inpu on he con e gence is ela ed o he le el o he sub il e in he hie a chy. The
eigen alue sp ead o he inpu -signal au oco ela ion ma ix becomes smalle om le el
l o le el l+ 1 because he hie a chical s uc u e ends o a e age he eigen alues o he
ela ed inpu -signal au oco ela ion ma ix. Then, assuming ha he numbe o aps
β
o
e e y sub il e is he same, ega dless he posi ion in he hie a chy, he bo le-neck in he
con e gence p ocess is loca ed a he sub il e s o he i s le el.
Ne e heless, in ou app oach, e e y hie a chically a anged sub il e , and mo e
pa icula ly
wslow(n)
, is upda ed by he sequen ial PU algo i hm. The e o e, we ha e o
conside ha he logical sub il e (we conside a logical sub il e as he se o N-equally-
spaced aps o a il e upda ed a e e y i e a ion o he upda ing p ocess acco ding o
a sequen ial LMS algo i hm wi h decima ion ac o N(see Appendix A)) is o med by
he subse o
β
/N coe icien s o he
β
-leng h sub il e
wslow(n)
. These
β
/N coe icien s
a e upda ed in one i e a ion o he sequen ial LMS algo i hm wi h decima ion ac o N.
The e o e, he con e gence condi ion o he hie a chical a angemen is es ablished on he
basis o he join con e gence o he Nlogical sub il e s in o which he slowes sub il e
wslow(n)
is decomposed. Ha ing de e mined he elemen ha limi s he con e gence a e,
we de i e he gain in s ep-size o he hie a chical s uc u e in he nex sec ion.
3.2. Gain in S ep-Size o he Gµ—Mod Fx H Seq LMS Algo i hm
Resul s de i ed in he Appendix A o a con en ional FIR il e ( heo e ical de i a ion
o he gain in s ep-size o a con en ional adap i e FIR il e whose coe icien s a e pa ially
upda ed acco ding o he sequen ial LMS algo i hm can be ound in he Appendix A.4 o
he Appendix), a e ex ended o he
β
-leng h slowes sub il e o he hie a chy
wslow(n)
,
when sequen ial PU wi h decima ion ac o Na e applied o he hie a chical il e . The ole
o his slowes sub il e can be played by any o he sub il e s loca ed a he i s le el o he
hie a chy.
To ob ain mo e easily he ac o by which he s ep-size pa ame e
µ
o he p oposed
algo i hm can be inc eased wi h ega d o he s ep-size o he ull upda es app oach, we
impose he use o he same alue o
µ
o e e y sub il e o he hie a chy. Then, we
conside he bound on he s ep-size o he hie a chical il e as he maximum alue o
µ
ha ensu es con e gence in all he sub il e s.
Since
β
is sho e han he o al numbe Lo aps a he i s le el o he hie a chy
(
β=α
√L
ypically), and, p o ided ha he decima ion ac o N> 1, he numbe
β
/N
o sub il e coe icien s ha a e e ec i ely upda ed pe i e a ion is small. When
β
/N is
no in ege , his a io mus be ounded o he nea es in ege ei he owa ds ze o,
β/N
,
o owa ds in ini y,
β/N
. In ha case, he
β
-leng h hie a chical sub il e is decomposed
Appl. Sci. 2021,11, 344 9 o 24
in o
β−Nβ/N
logical sub il e s o leng h
β/N
, and
N−β+Nβ/N
logical sub il e s o
leng h
β/N
. The
β/N
-leng h logical sub il e s ha e a slowe con e gence a e han ha o
he
β/N
-leng h logical sub il e s (in Appendix A.3 o he Appendix A, i is p o ed ha
he la ge an adap i e il e , he smalle he bound on i s s ep-size). As he e is a leas
one
β/N
-leng h logical sub il e pe i e a ion a he i s le el, he con e gence a e o a
β/N
-leng h logical sub il e de e mines he con e gence o he hie a chy o he sub il e s.
Fo he sake o simplici y o ob ain he dependence o he gain in s ep-size on he
leng h
β
and on he decima ion ac o N, le a single one o no malized equency
0
be he
inpu signal o he hie a chical s uc u e. Thus, he gain in s ep-size is gi en by
Gµ(1, 0,β,N)=maxn1
4hβ±sin(β2π 0)
sin(2π 0)io
max(1
4"lβ
Nm±sinlβ
Nm2πN 0
sin(2πN 0)#). (4)
We ha e ca ied ou se e al compu e -based simula ions o compa e he heo e ical
p edic ion gi en by Equa ion (4), wi h he expe imen al esul s. This s udy con i ms ha
his wo s -case hypo hesis makes i possible o accu a ely p edic he beha io o he
expe imen al con e gence p ocess. In Sec ion 4.1, we p o ide a compa ison be ween he
heo e ical gain in s ep-size, gi en by Equa ion (4), wi h he a o dable inc ease in s ep-size
ob ained by MATLAB simula ion.
4. Simula ion Resul s
The pu pose o his sec ion is wo old. Fi s , we compa e he heo e ical p edic ion o
he gain in s ep-size—Equa ion (4)—wi h compu e -based esul s. Then, we analyze he
ele ance o he G
µ
—Mod Fx H Seq LMS algo i hm in an ANC sys em when dealing wi h
ha monic dis u bances.
4.1. Gain in S ep-Size: Simula ion s. Theo y
Le us conside he ollowing simula ion p o ocol: i co esponds o he 1
×
1
×
1
a angemen , ha is, 1 e e ence senso , 1 e o mic ophone and 1 seconda y sou ce (see
Figu e 1). The i s le el o he hie a chical il e consis s o 384 coe icien s o ganized in
16 sub il e s o 24 aps. The e o e, in he second le el o he hie a chy, one has a 16-leng h
sub il e . The e e ence is a single sinusoidal signal whose equency a ies in 41.6 Hz s eps
om 41.6 o 4000 Hz. The sampling equency is 8000 samples/s. P ima y and seconda y
pa hs a e se o il e s modelling eal-wo ld ac i e noise con ol sys ems. We use il e s
25 h o de IIR il e s p o ided by Kuo and Mo gan in [
6
] (in he book-a ached loppy
disc ea u ing C and assembly p og ams o implemen ing ANC sys ems). Plan models
om his well-known e e ence a e conside ed among esea che s in he opic as a alid
benchma k. Figu es 4and 5show he magni ude and phase o he p ima y and seconda y
pa hs, espec i ely.
Appl. Sci. 2021,11, 344 16 o 24
when he sequen ial PU LMS algo i hm is used wi h decima ion ac o (N<L; in addi ion,
and o he sake o simplici y, we assume h oughou he pape ha he a io L/Nis in ege )
N, he weigh s o he L-leng h il e a e upda ed by means o he ollowing ecu sion
w(n+1) = w(n) + µe(n)I(N)
1+nmodNx(n)(A3)
whe e he L×Lma ix I(N)
p, wi h p=1+n mod N ≤Nis de ined by
I(N)
p=diag 0··· 0
|{z }
p−1
10··· 0
| {z }
N−1
10··· 0
| {z }
N−1
1 0 ···0 1 0··· 0
| {z }
N−p!(A4)
As one has
I(N)
px(n)
=
0··· 0
| {z }
p−1
x(n−p+1)0··· 0
| {z }
N−1
x(n−p−N+1)0··· 0x(n−L+N−p+1)0··· 0
| {z }
N−p

T(A5)
Equa ion (A3) leads o
wp+αN(n+1) = wp+αN(n) + µe(n)x(n−p−αN+1) o α=0, . . . , L
N−1
wp+αN(n)o he wise.(A6)
A ha s age, le us in oduce he p h L/N-leng h “logical sub il e ”, wi h pcyclically
a ying om 1 o N. I is de ined by he se o L/N equally-spaced aps
wpwp+Nwp+2N. . . wp+L−N
o he L-leng h il e esponse ec o
w(n)
. A ime
n+p−1
, whe e
p=
1
+n mod N
, he p h logical sub il e is upda ed by means o
Equa ion (A6). The e o e, he Nsub il e s equi e he same signal inpu samples o be
upda ed. They a e s o ed in he ec o
x(N)(n)
ha co esponds o he N-decima ed e sion
o x(n).
x(N)(n) = [x(n)x(n−N)x(n−2N)··· x(n−L+N)]T. (A7)
A ime
n+N
,
x(N)(n)
is shi ed, inse ing a new sample a he posi ion o me ly
gi en by he index n, while he olde da a is los . See Table A1, which shows he subse s o
coe icien s o he il e esponse ec o
w(n)=[w1w2. . . wL]T
o be upda ed acco ding
o Equa ion (A3) du ing
N+
1 consecu i e i e a ions and hei co esponding samples o
he inpu ec o .
Table A1.
Coe icien s o be upda ed—de ining a logical sub il e —du ing N + 1 consecu i e i e a ions and hei co esponding
samples o he inpu ec o .
# I e a ion Coe icien s Tha Fo m he Logical Sub il e
Upda ed a he Cu en I e a ion
Samples o he Inpu Vec o Used o Upda e he
Logical Sub il e
1[w1w1+Nw1+2N. . . wL−N+1]T[x(n)x(n−N)x(n−2N). . . x(n−L+1))]T
.
.
..
.
..
.
.
N[wNw2Nw3N. . . wL]T[x(n)x(n−N)x(n−2N). . . x(n−L+1))]T
N+ 1 [w1w1+Nw1+2N. . . wL−N+1]T[x(n+N)x(n)x(n−N). . . x(n−L+N+1))]T
In ligh o Equa ion (A7), he (L/N)
×
(L/N) Toepli z au oco ela ion ma ix
R(N)
o
he decima ed inpu ec o x(N)(n)is gi en by
R(N)=Ehx(N)(n)x(N)T(n)i. (A8)
In he sequel, he con e gence p ope ies o he (L/N)-leng h logical sub il e s is
analyzed based on he eigen alues o he au oco ela ion ma ix R(N).

Appl. Sci. 2021,11, 344 17 o 24
Appendix A.2. Eigen alues o he Au oco ela ion Ma ix o a Pe iodic Signal Consis ing o K
Ha monics
Le us assume ha he inpu signal x(n)o an adap i e il e is de ined as ollows
x(n) =
K
∑
k=1
Ckcos(2πk 0n+ϕk)(A9)
whe e
0
is he no malized undamen al equency,
{φk}k=1,...,K
he ini ial andom phases
mu ually independen and uni o mly dis ibu ed om 0 o 2
π
and
{Ck}k=1,...,K
he ampli-
udes o he ha monics. Gi en Equa ion (A9), he au oco ela ion unc ion o inpu signal
x(n)can be exp essed as
xx(τ) =
K
∑
k=1
C2
k
2cos(2πk 0τ). (A10)
The e o e, he au oco ela ion ma ix o he inpu ec o
x(n)
can be exp essed as he
sum o Kma ices Rko size L×Las ollows
R=
K
∑
k=1
C2
kRk(A11)
whe e
Rk=1
2

















1 cos(2πk 0)··· ··· cos[2πk(L−1) ]
cos(2πk 0)1....
.
.
.
.
...........
.
.
.
.
.......cos(2πk 0)
cos[2πk(L−1) ]··· ··· cos(2πk 0)1

















. (A12)
The la ges eigen alue λk,max(k 0)o each ma ix Rkis gi en by [31]
λk,max(k 0) = max1
4L±sin(L2πk 0)
sin(2πk 0) (A13)
whe e he subsc ip k e e s o he index o he subma ix Rk.
Acco ding o he iangle inequali y [
32
], appendix E, he la ges eigen alue o a sum
o ma ices is bounded by he sum o he la ges eigen alues o each o i s componen s.
The e o e, he la ges eigen alue λ o ,max o Ris bounded by
λ o ,max ≤
K
∑
k=1
C2
kλk,max(k 0) =
K
∑
k=1
C2
kmax1
4L±sin(L2πk 0)
sin(2πk 0) (A14)
whe e he subsc ip o e e s o he whole au oco ela ion ma ix R.
As a as he sequen ial PU LMS algo i hm wi h decima ion ac o Nis conce ned, he
con e gence condi ion o he whole il e migh be ansla ed o he pa allel con e gence
o Nlogical sub il e s o leng h L/N upda ed by a N-decima ed inpu signal
x(N)(n)
[
29
].
Adjus ing he abo e app oach o he case o sequen ial PU LMS, whe e he size o he
au oco ela ion ma ix is L/N and he sampling equency is di ided by N, we deal wi h K
ma ices R(N)
ko size (L/N)×(L/N) de ined by
Appl. Sci. 2021,11, 344 18 o 24
R(N)
k=1
2





















1 cos(2πNk 0)··· ··· cosh2πNkL
N−1 i
cos(2πNk 0)1....
.
.
.
.
...........
.
.
.
.
.......cos(2πNk 0)
cosh2πNkL
N−1 i··· ··· cos(2πNk 0)1





















. (A15)
The e o e, he la ges eigen alue
λ(N)
k,max (k 0)
o each ma ix
R(N)
k
can be exp essed as
ollows
λ(N)
k,max(k 0) = max


1
4
L
N±sinL
N2πkN 0
sin(2πkN 0)


. (A16)
Conside ing he iangle inequali y, he la ges eigen alue
λ(N)
o ,max
o he (L/N)
×
(L/N)
ma ix R(N)=K
∑
k=1
C2
kR(N)
kis bounded by
λ(N)
o ,max ≤
K
∑
k=1
C2
kλ(N)
k,max(k 0) =
K
∑
k=1
C2
kmax


1
4
L
N±sinL
N2πkN 0
sin(2πkN 0)


. (A17)
I should be no iced ha o N= 1 he sequen ial PU LMS algo i hm educes o
he con en ional ull upda es LMS algo i hm and Equa ions (A16) and (A17) educe o
Equa ions (A13) and (A14), espec i ely.
Appendix A.3. E ec o he Leng h o he Fil e on he S ep-Size Bound
In his sec ion, we analyze he e ec o he leng h o an adap i e il e on he maximum
alue o he s ep-size ha ensu es con e gence o he adap i e algo i hm. The analysis
deals wi h he case o he e e ence signal de ined in Equa ion (A.9). The dependence o
he s ep-size bound on he numbe o coe icien s o he il e is s udied no only o he ull
upda es LMS algo i hm, bu also o he sequen ial PU LMS algo i hm wi h decima ion
ac o N.
(a)
Full upda es LMS algo i hm
Le he inpu signal o he LMS algo i hm be he pe iodic signal gi en by
Equa ion (A9)
.
The con e gence in mean o he weigh s o he il e is gua an eed i he s ep-size sa is ies
he inequali y [33]
0<µLMS <2
λ o ,max
. (A18)
Thus, combining Equa ions (A14) and (A18), we ob ain a mo e es ic i e bound on
he s ep-size ha ensu es con e gence in mean.
0<µLMS <2
K
∑
k=1
C2
kmaxn1
4hL±sin(L2πk 0)
sin(2πk 0)io. (A19)
Hence, he bound on
µLMS
depends on he equency
0
, he leng h o he il e L, and
he weigh s
Ck
o he inpu signal. To simpli y he g aphical ep esen a ion o he bound,
he inpu signal o a L-leng h adap i e il e upda ed by he con en ional LMS algo i hm is
de ined by he ollowing single one o no malized equency 0.
x(n) = cos(2π 0n+ϕ). (A20)
Appl. Sci. 2021,11, 344 19 o 24
Due o Equa ions (A19) and (A20), he bound on he s ep-size is hen gi en by
0<µLMS <2
maxn1
4hL±sin(L2π 0)
sin(2π 0)io=2
λ o ,max
. (A21)
Figu e A1 shows he size bound o he con en ional LMS algo i hm o L= 16, 32, 64,
and 128. The no malized equency o he inpu ec o a ies om 0 o 0.5.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 20 o 25
Hence, he bound on 𝜇 depends on he equency 𝑓, he leng h o he il e L,
and he weigh s 𝐶 o he inpu signal. To simpli y he g aphical ep esen a ion o he
bound, he inpu signal o a L-leng h adap i e il e upda ed by he con en ional LMS
algo i hm is de ined by he ollowing single one o no malized equency 𝑓.
. )2cos()( 0
φ
π
+= n nx (A20)
Due o Equa ions (A19) and (A20), he bound on he s ep-size is hen gi en by
()
()
.
2
2sin
2sin
4
1
max
2
0
max, o
0
0
λ
π
π
μ
=











±
<<
L
L
LMS
(A21)
Figu e A1 shows he size bound o he con en ional LMS algo i hm o L = 16, 32, 64,
and 128. The no malized equency o he inpu ec o a ies om 0 o 0.5.
.
Figu e A1. S ep-size bound o he con en ional ull upda es LMS algo i hm o di e en il e
leng hs, L = 16, 32, 64, and 128. The inpu ec o is a single one whose no malized equency a -
ies om 0 o 0.5.
Acco ding o Figu e A1, and conside ing ha a la ge s ep-size gua an ees as con-
e gence a e, we conclude ha a sho e il e can con e ge as e han a la ge one.
(b) Sequen ial PU LMS algo i hm.
Le he L × 1 inpu ec o o he sequen ial PU LMS algo i hm be gi en by he N-deci-
ma ed e sion 𝐱()(𝑛) o he pe iodic signal 𝐱(𝑛) exp essed by Equa ion (A9). A simila
analysis as he one ca ied ou in he p e ious sec ion o he con en ional LMS algo i hm
yields a mo e es ic i e bound on he s ep-size ha ensu es con e gence in mean o he
case o he sequen ial PU LMS algo i hm.
Figu e A1.
S ep-size bound o he con en ional ull upda es LMS algo i hm o di e en il e leng hs,
L= 16, 32, 64, and 128. The inpu ec o is a single one whose no malized equency a ies om 0
o 0.5.
Acco ding o Figu e A1, and conside ing ha a la ge s ep-size gua an ees as con e -
gence a e, we conclude ha a sho e il e can con e ge as e han a la ge one.
(b)
Sequen ial PU LMS algo i hm.
Le he L
×
1 inpu ec o o he sequen ial PU LMS algo i hm be gi en by he N-
decima ed e sion
x(N)(n)
o he pe iodic signal
x(n)
exp essed by Equa ion (A9). A
simila analysis as he one ca ied ou in he p e ious sec ion o he con en ional LMS
algo i hm yields a mo e es ic i e bound on he s ep-size ha ensu es con e gence in
mean o he case o he sequen ial PU LMS algo i hm.
0<µSeqLMS <2
K
∑
k=1
C2
kmax1
4L
N±sin(L
N2πkN 0)
sin(2πkN 0)<2
λ(N)
o ,max
. (A22)
This bound on
µSeqLMS
depends on he equency
0
, he leng h o he il e L, he
weigh s o he inpu signal
Ck
, and he decima ion ac o N. As we did in he p e ious
sec ion o simpli y he g aphical ep esen a ion o he bound, we educe he numbe o
ha monics o he inpu signal o K= 1. In so doing, he L
×
1 inpu ec o o he sequen ial
Appl. Sci. 2021,11, 344 20 o 24
PU LMS algo i hm is gi en by he N-decima ed e sion
x(N)(n)
o a sinusoidal signal
de ined in Equa ion (A20). The bound on he s ep-size is hen educed o
0<µSeqLMS <2
max1
4L
N±sin(L
N2πN 0)
sin(2πN 0)=2
λ(N)
o ,max
. (A23)
Figu es A2 and A3 show he size bound o he sequen ial PU LMS algo i hm gi en by
Equa ion (A23) o decima ion ac o s N= 2 and N= 4, espec i ely. Resul s a e gi en o
L= 16, 32, 64, and 128. The no malized equency o he inpu ec o a ies om 0 o 0.5.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 21 o 25
()
.
2
2sin
2sin
4
1
max
2
0)(
max, o
10
0
2
N
K
kk
SeqLMS
kN
kN
N
L
N
L
C
λ
π
π
μ
<































±
<<

=
(A22)
This bound on 𝝁𝑺𝒆𝒒𝑳𝑴𝑺 depends on he equency 𝒇𝟎, he leng h o he il e L, he
weigh s o he inpu signal 𝑪𝒌, and he decima ion ac o N. As we did in he p e ious
sec ion o simpli y he g aphical ep esen a ion o he bound, we educe he numbe o
ha monics o he inpu signal o K = 1. In so doing, he L × 1 inpu ec o o he sequen ial
PU LMS algo i hm is gi en by he N-decima ed e sion 𝐱(𝑵)(𝒏) o a sinusoidal signal
de ined in Equa ion (A20). The bound on he s ep-size is hen educed o
()
.
2
2sin
2sin
4
1
max
2
0)(
max, o
0
0
N
SeqLMS
N
N
N
L
N
L
λ
π
π
μ
=































±
<<
(A23)
Figu es A2 and A3 show he size bound o he sequen ial PU LMS algo i hm gi en by
Equa ion (A23) o decima ion ac o s N = 2 and N = 4, espec i ely. Resul s a e gi en o L
= 16, 32, 64, and 128. The no malized equency o he inpu ec o a ies om 0 o 0.5.
Figu e A2. S ep-size bound o he sequen ial PU LMS algo i hm wi h decima ion ac o N = 2 o
di e en il e leng hs, L = 16, 32, 64, and 128. The inpu ec o is an N-decima ed single one
whose no malized equency a ies om 0 o 0.5.
Figu e A2.
S ep-size bound o he sequen ial PU LMS algo i hm wi h decima ion ac o N= 2 o
di e en il e leng hs, L= 16, 32, 64, and 128. The inpu ec o is an N-decima ed single one whose
no malized equency a ies om 0 o 0.5.
Acco ding o Figu es A2 and A3, and conside ing ha a la ge s ep-size gua an ees
as con e gence a e, we conclude ha a sho e il e can con e ge as e han a la ge one
when he sequen ial PU LMS is used.
Appl. Sci. 2021,11, 344 21 o 24
Appl. Sci. 2021, 11, x FOR PEER REVIEW 22 o 25
Figu e A3. S ep-size bound o he sequen ial PU LMS algo i hm wi h decima ion ac o N = 4 o
di e en il e leng hs, L = 16, 32, 64, and 128. The inpu ec o is an N-decima ed single one
whose no malized equency a ies om 0 o 0.5.
Acco ding o Figu es A2 and A3, and conside ing ha a la ge s ep-size gua an ees
as con e gence a e, we conclude ha a sho e il e can con e ge as e han a la ge
one when he sequen ial PU LMS is used
Appendix A.4. The Gain in S ep-Size
By de ining he gain in s ep-size Gμ as he a io be ween he bounds on he s ep-sizes
in wo di e en cases—N > 1 (sequen ial PU LMS) and N = 1 (con en ional LMS)—we
ob ain he ac o by which he s ep-size pa ame e can be mul iplied when he adap i e
algo i hm uses sequen ial PU
()
{}
{}
{}
{}
()
()
()
.
2sin
2sin
4
1
max
2sin
2sin
4
1
max
)(
)(
max
2
max
2
,,,
10
0
2
10
0
2
1
0
)(
max,
2
1
0max,
2
max, o
)(
max, o
0




=
=
=
=































±











±
=
===
K
kk
K
kk
K
k
N
kk
K
kkk
N
LMS
SeqLMS
kN
kN
N
L
N
L
C
k
k L
LC
k C
k C
bound
bound
NL KG
π
π
π
π
λ
λ
λ
λ
μ
μ
μ
(A24)
To mo e easily isualize he dependence o he gain in s ep-size on he leng h o he
il e L and on he decima ion ac o N, he numbe o ha monics o he inpu signal is se
Figu e A3.
S ep-size bound o he sequen ial PU LMS algo i hm wi h decima ion ac o N= 4 o
di e en il e leng hs, L= 16, 32, 64, and 128. The inpu ec o is an N-decima ed single one whose
no malized equency a ies om 0 o 0.5.
Appendix A.4. The Gain in S ep-Size
By de ining he gain in s ep-size G
µ
as he a io be ween he bounds on he s ep-sizes
in wo di e en cases—N> 1 (sequen ial PU LMS) and N= 1 (con en ional LMS)—we
ob ain he ac o by which he s ep-size pa ame e can be mul iplied when he adap i e
algo i hm uses sequen ial PU
Gµ(K, 0,L,N)=bound{µSeqLMS}
bound{µLMS}=
2
maxλ(N)
o ,max
2
max{λ o ,max}
=
K
∑
k=1
C2
kλk,max(k 0)
K
∑
k=1
C2
kλ(N)
k,max(k 0)
=
K
∑
k=1
C2
kmax1
4L±sin(L2πk 0)
sin(2πk 0)
K
∑
k=1
C2
kmax(1
4"L
N±sin(L
N2πkN 0)
sin(2πkN 0)#).
(A24)
To mo e easily isualize he dependence o he gain in s ep-size on he leng h o he
il e Land on he decima ion ac o N, he numbe o ha monics o he inpu signal is se
o K= 1. Now, he gain in s ep-size, ha is, he a io be ween he bounds on he s ep-size
when N> 1 and N= 1, is gi en by
Gµ(1, 0,L,N)=boundµSeqLMS
bound{µLMS}=maxn1
4hL±sin(L2π 0)
sin(2π 0)io
max1
4L
N±sin(L
N2πN 0)
sin(2πN 0). (A25)
Figu es A4 and A5 show, espec i ely, he gain in s ep-size o a single one when
di e en decima ion ac o s and di e en il e leng hs a e conside ed. Acco ding o
Figu es A4 and A5 show ha he s ep-size can be mul iplied by Nas long as ce ain
equencies, a which a no ch in he gain in s ep-size appea s, a e a oided. The loca ion

Appl. Sci. 2021,11, 344 22 o 24
o hese c i ical equencies, as well as he numbe and wid h o he no ches, will be
analyzed as a unc ion o he sampling equency
Fs
, he leng h o he adap i e il e L,
and he decima ion ac o N. Acco ding o Equa ions (A24) and (A25), wi h inc easing
decima ion ac o N, he s ep-size can be mul iplied by Nand, as a esul o ha a o dable
compensa ion, he sequen ial PU LMS algo i hm con e gence is as as as he ull upda es
LMS algo i hm as long as he undesi ed dis u bance is ee o componen s loca ed a he
no ches o he gain in s ep-size.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 23 o 25
o K = 1. Now, he gain in s ep-size, ha is, he a io be ween he bounds on he s ep-size
when N > 1 and N = 1, is gi en by
()
{}
{}
()
()
()
.
2sin
2sin
4
1
max
2sin
2sin
4
1
max
,,,1
0
0
0
0
0































±











±
==
N
N
N
L
N
L
L
L
bound
bound
NL G
LMS
SeqLMS
π
π
π
π
μ
μ
μ
(A25)
Figu es A4 and A5 show, espec i ely, he gain in s ep-size o a single one when
di e en decima ion ac o s and di e en il e leng hs a e conside ed. Acco ding o Fig-
u es A4 and A5 show ha he s ep-size can be mul iplied by N as long as ce ain equen-
cies, a which a no ch in he gain in s ep-size appea s, a e a oided. The loca ion o hese
c i ical equencies, as well as he numbe and wid h o he no ches, will be analyzed as a
unc ion o he sampling equency s
F, he leng h o he adap i e il e L, and he deci-
ma ion ac o N. Acco ding o Equa ions (A24) and (A25), wi h inc easing decima ion ac-
o N, he s ep-size can be mul iplied by N and, as a esul o ha a o dable compensa ion,
he sequen ial PU LMS algo i hm con e gence is as as as he ull upda es LMS algo i hm
as long as he undesi ed dis u bance is ee o componen s loca ed a he no ches o he
gain in s ep-size.
Figu e A4. Gain in s ep-size o a single one and di e en decima ion ac o s, N = 1, 2, 4, and 8.
The leng h o he il e is se o L = 256 aps.
Figu e A4.
Gain in s ep-size o a single one and di e en decima ion ac o s, N= 1, 2, 4, and 8. The
leng h o he il e is se o L= 256 aps.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 24 o 25
Figu e A5. Gain in s ep-size o a single one and di e en il e leng hs, L = 8, 32, and 128 wi h
decima ion ac o N = 2.
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Heal h 2018, 15, 208, doi:10.3390/ije ph15020208.
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Figu e A5.
Gain in s ep-size o a single one and di e en il e leng hs, L= 8, 32, and 128 wi h
decima ion ac o N= 2.
Appl. Sci. 2021,11, 344 23 o 24
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