91
1 INTRODUCTION
Theobjectiveofthispaperistoshowuniversalityof
the Volterra and Wiener series in description of
nonlinear systems and phenomena, and in solving
numerous nonlinear problems occurring in diverse
engineeringdisciplines,rangingfromelectronicsand
telecommunications to such ones as navigation and
transportation. This is possible beca
use the Volterra
series is a natural extension of the convolution
integraldescriptionforlinearsystemstothenonlinear
case, but the Wiener series exploits the powerful
orthogonalityprincipleappliedtotheVolterraseries
todescribenonlinearsystems with stochastic inputs.
It follows from the material presented in this paper
how powerful are these two m
athematical tools in
considerationofnonlinearproblemsofengineering.
2 NONLINEARSYSTEMSANDPHENOMENA
Whatarethenonlinearsystemsandphenomena?The
simplest answer to this question is the following:
thesearetheonesthatarenotlinear.Inotherwords,
their description (model) cannot be formulat
ed with
theuseofoneorasetoflinearalgebraicequations,or
linear operators, or ordinary or partial differential
equations,orcombinationsofthem.Oneveryuseful
and,ontheotherhand, alsofundamentalcriterionfor
recognition whether a given system or phenomenon
behaveslinearlyisinvestigationofit
sresponsetoan
amplified or attenuated sum of two external signals
(excitations)appliedatitsinput.Ifthisresponseisa
caseof applying them separately to the system, and
amplified or attenuated exactly in the same way as
weretheinputsignals.Mat
hematically,usingsystem
On Modelling of Nonlinear Systems and Phenomena
with the Use of Volterra and Wiener Series
A.Borys
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT: This is a short tutorial on Volterra and Wiener series applications to modelling of nonlinear
systemsandphenomena,andalsoasurveyoftherecentachievementsinthisarea.Inparticular,weshowhere
howthephilosophiesstandingbehindeachoftheabovetheoriesdifferfromeachother.Ontheotherhand,we
discussalsom
athematicalrelationshipsbetweenVolterraandWienerkernelsandoperators.Also,theproblem
ofabestapproximationoflargescalenonlinearsystemsusingVolterraoperatorsinweightedFockspacesis
described.Examplesofapplicationsconsideredarethefollowing:Volterraseriesuseindescriptionofnonlinear
distortionsinsat
ellitesystemsandtheirequalizationorcompensation,exploitingWienerkernelstomodelling
of biological systems, the use of both Volterra and Wiener theories in description of ocean waves and in
magneticresonancespectroscopy.Moreover,connectionsbetweenVolterraseriesandneuralnetworkmodels,
andalsoinputoutputdescriptionsofquant
umsystemsbyVolterraseriesarediscussed.Finally,weconsider
application of Volterra series to solving some nonlinear problems occurring in hydrology, navigation, and
transportation.
http://www.transnav.eu
the International Journal
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Volume 9
Number 1
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DOI:10.12716/1001.09.01.11
92
descriptionbyoperators,wecanexpresstheaboveas
follows
12 1 2
H
x x Hx Hx


(1)
where H denotes an operator describing the system.
This operator works on a set of admissible input
signals,producingresponsesatthesystemoutput.In
(1),
1
x
and
2
x
meansomeinputsignals,members
of the above set. Usually in applications, they are
functions of time or position, or both of them.
Moreover,α and β are real numbers expressing
amplificationorattenuationfactorsmentionedabove.
Notefurtherthatthecondition(1)assumesthesame
formwhenH,
1
x
,and
2
x
areassumedtobevectors.
Then,αandβremainscalars.
In(1),weassumedtheusage of ordinaryalgebra
with the common understanding of addition
operation “+” and multiplication operation
”.
However, in this context, note there are some other
algebras in which the condition (1), with another
understanding of the aforementioned algebraic
operations, is fulfilled. Examples of such systems of
interest in the areas of signal processing and
networking are considered in (Oppenheim, A. V.
1965) and (Boudec, J.Y.
& Thiran P. 2004),
respectively.Obviously,then,thesesystemslinearin
newalgebrasbehavenonlinearlyinordinaryone.
In this paper, we do not study dynamics of
nonlinearsystemsorphenomena,which,bytheway,
areveryinterestingbecausegettingricherthanthose
oflinearones.Here,rather,we
focusonsearchingfor
inputoutputrelations.Forthispurpose,theVolterra
series (Volterra V. 1959), named so in honor of its
founder an Italian mathematician Vito Volterra,
turned out to be very useful in solving many
nonlinear engineering problems. However, among
following: convergence problems occurring for
signalsofhigheramplitudes(similarlyasinaTaylor
series)andproblemswithmeasuringitskernels.For
circumventingthis,NorbertWienerdevisedarelated
mathematical tool by orthogonalization of
components of the Volterra series leading to
an
expansion named after him a Wiener expansion
(WienerN.1942,WienerN.1958).
This paper is organized as follows. In sections 2
and 3, respectively, the Volterra series and Wiener
series are presented. The next section describes
shortlytheproblemofabestapproximationoflarge
scale nonlinear systems using
Volterra operators in
weighted Fock spaces. Finally, the last section 5
presents a list of interesting applications of the
VolterraandWienertheoriesindifferentengineering
disciplines.
3 VOLTERRASERIES
3.1 BasicsofVolterraseriesfortimeinvariantsystems
withmemory
LetusbeginwithconsiderationofaVolterraseries
of
continuous time for description of nonlinear time‐
invariant (stationary) systems with memory. To this
end, assume that an inputoutput behavior a
nonlinear system considered can be described by a
nonlinearoperator;thatisbysuchanoperatorHthat
does not obey (1). Volterra shown that under some
conditions
thisoperatorcanbeexpandedinaseriesof
thesocalledVolterraoperatorsas
 







00
nn
nn
yt H xt H xt y xt




, (2)
where
x
t
and
y
t
are the input and output
signal, respectively. Moreover, by
nn
yxt Hxt
, we define the partial nth
order system’s response, where

n
Hxt
means
thenthorderVolterraoperator.Further,notethatfor
a fixed value of time t this operator is simply a
functional,calledrespectivelythenthorderVolterra
functional.
ThesuccessiveVolterraoperatorsaregivenbythe
followingiteratedintegrals
(0) (0)
()
y
th
, (3a)
(1) (1)
() ( ) ( )yt h xt d



, (3b)
(2) (2)
12 1 2 1 2
() ( , ) ( ) ( )yt h xt xt dd


 


,(3c)
......... .,
() ()
123
123 123
( ) ... ( , , ,..., )
( ) ( ) ( )... ( ) ...
nn
n
nn
yt h
x
txtxt xtdddd



   



, (3d)
......... .,
where
(0)
h
is the system impulse response of the
zerothorder(intermsofcurrentsorvoltages,itisthe
dccomponentintheexpansion).Further, thefunction
()
123
( , , ,..., ), 1, 2,3,...,
n
n
hn

means the nth
order nonlinear impulse response of a nonlinear
systemconsidered.Notethatforn=1thisisastandard
linearimpulseresponse.
Looking at (3b), and then at (2) with the next
components in this expansion given by (3c), and
generallyby(3d),weseethattheVolterra
series(2)is
anextension of the wellknown convolution integral
forlineartimeinvariant(LTI)systems.
Obviously, for description of nonlinear systems
aTaylorseries.
Furthermore,it can be shown (Schetzen M. 1980)
that for the stability reasons of the
Volterra series
descriptionasufficientconditionisthefollowing:
()
123 1 2 3
... ( , , ,..., ) ...
n
nn
hdddd


   

(4)
93
forn=1,2,3,....Itisnotanecessaryonefor
2n
.
In his papers (Sandberg I. W. 1985, Sandberg I. W.
1990), Sandberg showed that in the above case for
nonlinear impulse responses that are physically
realizable,ithastheform
1
()
11
sup .. ( ,.., ) ..
n
n
nn
JJ
hdd




J
, (5)
whereΨmeans a set of all general nvectors
1
[ . . ]
n
J
J having elements being finite sums of
boundedsubintervalsoftheset
0, ) .
Moreover, for causal nonlinear systems, we have
(SchetzenM.1981)
()
12
( , ,.., ) 0 for any , 1,2,..., ,
and 1, 2,.... .
n
ni
hin
n


(6)
Finally, it can be shown (Borys A. 2007) that the
Volterraseriesconvergesifthefollowing:
()
12 1 2
1
lim .. ( , ,.., ) ..
n
n
nn
n
x
hddd



  

(7)
holds,where
x
meansthenormofaninputsignal.
Inderivationof(7)in(BorysA.2007),itwasassumed

sup
df
t
x
xt

.
3.2 Volterraseriesfortimevaryingsystemswithmemory
Inpractice,thereoccuralsosituationswherewehave
to with nonlinear physical systems of which
parameterschangewithtime.Obviously,theycannot
be treated as stationary in this case. Then, when
describingthembyaVolterraseries,wemustassume
thattheir nonlinear impulse responses depend upon
time.Andthisisacorrectapproach.
Concluding, we can say that the structure of
equations(2)and(3)remainsunchangedinthiscase,
but we shall have
,'Hxt t
,
,'
n
Hxtt
, and
()
123
( , , ,..., , ' ),
n
n
ht

0,1, 2,3,...,n
dependent