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
sumoftwooutputsignals(responses)receivedinthe
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
ofabestapproximationoflarge‐scalenonlinearsystemsusingVolterraoperatorsinweightedFockspacesis
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,
andalsoinput‐outputdescriptionsofquant
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
on Marine Navigation
and Safety of Sea Transportation
Volume 9
Number 1
March 2015
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
newalgebrasbehavenon‐linearlyinordinaryone.
In this paper, we do not study dynamics of
nonlinearsystemsorphenomena,which,bytheway,
areveryinterestingbecausegettingricherthanthose
oflinearones.Here,rather,we
focusonsearchingfor
descriptionsoftheirsteadystates,havinginmindthe
input‐outputrelations.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
advantages,ithasalsosomedrawbacks.Thesearethe
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 BasicsofVolterraseriesfortime‐invariantsystems
withmemory
LetusbeginwithconsiderationofaVolterraseries
of
continuous time for description of nonlinear time‐
invariant (stationary) systems with memory. To this
end, assume that an input‐output behavior a
nonlinear system considered can be described by a
nonlinearoperator;thatisbysuchanoperatorHthat
does not obey (1). Volterra shown that under some
conditions
thisoperatorcanbeexpandedinaseriesof
theso‐calledVolterraoperatorsas
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 n‐th
order system’s response, where
n
Hxt
means
then‐thorderVolterraoperator.Further,notethatfor
a fixed value of time t this operator is simply a
functional,calledrespectivelythen‐thorderVolterra
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
zero‐thorder(intermsofcurrentsorvoltages,itisthe
dccomponentintheexpansion).Further, thefunction
()
123
( , , ,..., ), 1, 2,3,...,
n
n
hn
means the n‐th
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 well‐known convolution integral
forlineartime‐invariant(LTI)systems.
Obviously, for description of nonlinear systems
withoutmemory,insteadofaVolterraseries,weuse
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)
whereΨmeans a set of all general n‐vectors
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 Volterraseriesfortime‐varyingsystemswithmemory
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
uponanadditionaltimevariable
't
.
Oneveryprominentexampleofsuchthesystems
as sketched above are wireless communication
channels, whose characteristics vary with time and
position, and which are additionally, in most cases,
nonlinearones,asforexamplesatellitechannels.
Formoredetailsregardingmodellingofnonlinear
time‐varyingsystemsbyVolterraorrelated
series,see
papersofSandberg(SandbergI.W.1982,SandbergI.
W.1983)andcitedtherein.
3.3 Volterraseriesfordiscrete‐timenonlinearsystems
withmemory
A variant of the Volterra series for discrete‐time
(digital) nonlinear systems is named the discrete
Volterra series (Borys A. 2000). For nonlinear time‐
invariantsystems,ithasthefollowingform:
00
nn
nn
yk Hxk H xk y xk
(8)
with
(0) (0)
() = yk h
, (9a)
(1) (1)
() = () ( )
i
yk hixki
, (9b)
12
(2) (2)
12 1 2
() (, )( )( )
ii
y
khiixkixki
, (9c)
123
(3) (3)
123 1 2 3
() (, , )( )( )( )
ii i
y
khiiixkixkixki
,(9d)
......... .,
1
() ()
11
( ) ... ( ,.., ) ( )... ( )
n
nn
nn
ii
yk hiixki xki
, (9e)
......... ,
wherekmeansadiscretetimeand
0
h
isthezero‐th
order impulse response (constant component).
Moreover,
ih
1
is the i‐th sample of the system
firstorderimpulseresponse(linearone).Andfurther,
n
n
iiih ,...,,
21
,
,...2
n
, mean the corresponding
samples of the multidimensional impulse responses
of orders greater than 1, related with the Volterra
operatorsofhigherorderterms
2n
inequation
(8).
ConditionsforstabilityoftheVolterraoperatorsin
the discrete Volterra series given by (8), for their
causality,andfinallyfortheconvergenceofthewhole
series(8)areanalogoustothosegivenrespectivelyby
(4) or (5), by (6), and by (7), for the case of
a
continuoustime.Theyanddetailsoftheirderivation
can be found, for example, in (Borys A. 2000) and
referencecitedtherein.
Similarly,extensionofthediscreteVolterraseries
(8)forstationarysystemsofthediscretetimetothat
fornon‐stationaryonescanbeeasilydoneinasimilar
way
as shortly described in subsection 2.2 for the
continuoustimecase.
4 WIENERSERIES
4.1 Reasonsforsearchingforanorthogonalseries
We can view the Volterra series as a mathematical
toolofgeneraltypeforapproximationofbehaviorof
nonlinearsystemsinsteadystate.Thatmeansthatin
94
thiscase,wedonotadjusttheabovedescriptiontoa
certain type (class) of input signals from a set of
admissible ones. The only limitation here is the
amplitude of these signals of which increase causes
convergence problems. Moreover, in the case of
descriptionofanonlinearsystemwith
memorybya
Volterraseries,in almostallcases,thestructureand
elements of this system are known. From this, it
possible to deduce the form of functions describing
system’s nonlinear impulses, or equivalently in the
multidimensionalfrequencydomain,ofitsnonlinear
transfer functions of the corresponding orders
(Bussgang J.
J. & Ehrman L. & Graham J. W. 1974,
BedrosianE.&RiceS.O.1971).
Anotherapproximationphilosophystandsbehind
anexpansionwecallheretheWienerseries(Schetzen
M. 1980). In opposite to the previous approach,
sketchedinsection2,weadjustinthiscasetheform
of the
series components to a specific class of input
signalsusedinagivenapplication‐toachievebetter
convergence properties and adjustment to measured
data. In other words, having records of data
measuredatinputandoutputofagivensystemand
knowing nothing about its internal structure, we
approximatebehavior
ofthissystemsogoodasonly
itispossibleforaclassofinputsignalschosen.
Basic ideas of the above two schemes of
approximation can be illustrated by comparison of
approximationofagivenfunctionof,sayonevaria ble
t, on an interval
12
ttt (obviously having no
memory)bypolynomials.Wehavetwochoices:1.we
can expand this function in a Taylor series and
truncateitatthen‐thcomponent(ndependingupona
required accuracy) or 2. we can expand the
considered function in a series of the first m
orthogonal
polynomials, as for example, Legendre,
HermiteorChebyshevpolynomials(mdepending,as
before,uponarequiredaccuracy).Andnownotethat
the first approach (1.) corresponds with the
approximationof a nonlinear operators (systems)
withmemorybyaVolterraseries,butthesecond(2.)
with a Wiener series. As we
shall see in the next
subsection, the Wiener series uses Hermite
polynomials for orthogonalization of Volterra series
components.
4.2 NotionofWienerG‐functionals
To define the so‐called Wiener G‐functionals (G‐
operators), we need first to explain the notion of
nonhomogeneousVolterraoperators.Andtothisend,
note
that a Volterra operator of the n‐th order is
homogeneous if the following:
n
Hcxt
n
n
cH xt
holdswithcmeaningaconstant.If
this does not hold, a given Volterra operator is a
nonhomogeneousone.
Using similar nomenclature as in (Schetzen M.
1981), we define a nonhomogeneous Volterra
operatorofthefirstdegree(order),
1
g
,as
1101 1 01
01
(1)
,;
() ( )
ghhxt Hxt h
hxtdh
, (10)
wherethedoublesuperscript(0)1at
01
h
meansthat
thezero‐thorder(thewords“degree”and“order”are
used interchangeably in this paper) homogeneous
Volterra operator is a component of the first order
nonhomogeneous Volterra operator
1
g
. We see
that the operator
1
g is a sum of two
components, of a homogeneous Volterra operator of
thefirstorderandof
01
h
(aconstantcomponent).
Similarly,the nonhomogeneous Volterra operator
ofthesecondorderwillhavetheform
221202 2
12 02
(2)
12 1
,,;
(, ) ( )
ghhhxt Hxt
Hxth h xt
(11)
02
(1)2
212
() () ()
x
tdd h xtdh
,
where now
2
g
is a sum of three homogeneous
Volterra operators:
2
Hxt
and
12
Hxt
being, respectively, second and first order
convolutions, and
02 02
H
xt h
being a
constant.
So,ingenerally,wecanwrite
10
1
0
1
, ,.., ;
nnnn n n
in n
in
ghh hxt Hxt
Hxth
. (12)
For orthogonalization of the Volterra series,
Wiener chose (Schetzen M. 1981) the Hermite
polynomials; they have, after normalization, the
followingform:
2
01 2
2
232
3
3
11
1, ,
2
1
, 3 , ..... ,
6
H
xHxxHx x
Hx x x
(13)
where a subscript by H denotes degree (order) of a
given polynomial. A recursion formula describing
thesepolynomialsisgivenby
2
1
1
1
nnn
d
Hx xHx Hx
dx
n
. (14)
As the orthonormal polynomials, they satisfy the
followingequality:
95
2
1 for
0 for
mn
mn
HxHxwxdx
mn
(15)
with
wx
meaning a weighting function. In the
case of (15), that is of the Hermite polynomia ls, the
weightingfunction
wx
issuchthat
2
2
2
1
exp
2
2
x
wx
, (16)
where
is a constant. In the means of
approximation by the Wiener G‐functionals, x in
equations(13– 16)standsforawhiteGaussiantime
function applied as the input signal at the system’s
input.Theparameter
in(16)playsaroleofatime
varianceoftheinputsignal,thatis
22
2
1
lim
2
T
T
T
Av xt xt dt
T
, (17a)
wheretheoperationofcalculationofthetimeaverage
is denoted by the symbol Av. Moreover, it was
assumedin(17a)thatthesignalaverage,
Av x t ,
isequaltozero.Thatis
1
lim 0
2
T
T
T
A
vxt xtdt
T
. (17b)
The property of Gaussianity of the system input
signalmeansthatitsamplitudedistributionintimeis
describedby the bell‐shapedGaussian function (16).
More, the property of being “white” means that its
autocorrelation function, let denote it
xx
R
, is
equal to the Dirac impulse
multiplied by a
constant,say
0
N ,thatis
0
1
lim
2
T
xx
T
T
RxtxtdtN
T
. (18a)
Then, the Fourier transform of
xx
R
meaning
the so‐called power density spectrum, let denote it
xx
Gj
,isconstant.Thatis
0xx
Gj N
, (18b)
wherevariable
meanstheangularfrequency.
For this class of input signals, being white and
Gaussian,Wiener coined his G‐functionals.They are
defined as a set of nonhomogeneous Volterra
functionals
10
,,..,;
nnnn n
g
kk k xt
for which
thefollowingorthogonalityprinciple
0
,.., ; 0 for
mnnn
A
vH xt g k k xt m n
(19)
holds(SchetzenM.1981,RughW.J.1981),where,as
mentioned before,
x
t
is assumed to be Gaussian
and having the autocorrelation function given by
(18a). Moreover,
m
Hxt in (19) means any m‐
th order homogeneous Volterra operator. In what
follows,wewilldenotethenonhomogeneousVolterra
functionals
n
g
satisfying (19) by a capital letter
G. For this reason, they will be called Wiener G‐
functionals(foragivent)orWienerG‐operators(for
allvaluesoft,thatisconsideredasafunctionoft).
Assuming that the first Wiener G‐operator,
0
G
,
equals a constant and applying condition (19) for
successive
1,2,3,....n
,wegeta setofG‐operators.
Theprocedureisdescribedinmoredetailin(Schetzen
M. 1980, Rugh W. J. 1981). Here, we present for
illustration the first four Wiener G‐operators. They
havethefollowingform:
00 0
;Gkxt k
, (20a)
11
(1)
;() ()Gkxt k xt d
, (20b)
22
(2)
12 1
(2)
212 0
;(,) ()
() (,)
Gkxt k xt
xt d d N k d
, (20c)
33 3 13
(3)
123 1 2
(1)3
3123 1 11
;
(, , ) ( )( )
() ()()
Gkxt Kxt K xt
kxtxt
xt d d d k xt d
, (20d)
where
(1)3
1
()k
in(20d)isgivenby
13
(3)
101222
3(,,)kNkd
. (20e)
Note that in equations (20) in
;
nn
Gkxt
,
0,1,2, 3n
, only the leading component
n
k
is
shown in the square brackets, for shortening the
notation. The functions
n
k
are called the Wiener
kernels(SchetzenM.1981).Furthermore,observethat
the numerical coefficients accompanying the
componentsontherighthandsidesofequations(20)
are the same as those by the consecutive terms of
Hermite polynomials (13). This is not fortuitous; for
moredetailssee,forexample,(SchetzenM.
1980).
96
Using relation (19), it can be shown that the
following orthogonality relationship between the
WienerG‐operators
;;0
mm nn
Av G k x t G k x t
(21)
holdsforall
mn .
4.3 Wienerdescriptionofanonlinearsystem
Using the properties of his G‐operators, which were
presentedintheprevioussubsection,Wiener showed
that the response
yt
of a nonlinear system to a
white Gaussian signal
x
t
can be described by an
orthogonalseriesoftheform
0
;
nn
n
y
tGkxt
. (22)
Theexpansiongivenby(22)wasnamed,afterhis
founder,theWienerseries.
Means of modelling of nonlinear systems driven
byinputsignalbeingrealizationsofstochasticwhite
GaussianprocessesisillustratedinFig.1.
Figure1. Nonlinear system modelling with the use of the
Wieneroperatorsandinputsignalsbeingrealizationsofthe
whiteGaussianprocesses.
Let a true output signal at the nonlinear system
outputbez(t), and itsapproximate by the truncated
Wiener series (given by (22)) with the first p
components (including
0
G
)
p
yt. Then, the
mean‐square value of the error
p
et between the
system’s output signal z(t) and
p
yt
can be
expressedinthefollowingway(SchetzenM.1981):
22
2
2
2
011
1
00
! ... ,...,
pp
p
n
n
z
nn
n
Av e t Av z t Av y t
nN k d d
. (23)
In (23),
2
2
2
z
Av z t Av z t
is the
varianceofthetruesystem’sresponse.
Obviously, in accordance with the rules of
orthogonal approximation, the approximation error
2
p
Av e t
decreases with the increase of the
numberofelementsused,thatiswiththeincreaseof
the upper index p in the sum symbol in (23). Its
smallestvalueisgivenby
2
2
lim
p
p
Av e t Av e t
. (24)
4.4 OrthogonalexpansionoftheWienerkernelsinthe
Wienerseries
Observefrom(23)thattheWienerkernelssatisfythe
followingequality
2
11
00
.. ,.., , 1, 2, 3,..
n
nn
kddn
. (25)
The condition given by (25) is sufficient for
expandingtheWienerkernels in a setof orthogonal
functions. However, the orthonormal Laguerre
functionsareusuallychosenintheliteraturebecause
they can be easily physically realized, as all‐pass
filters.Forexample,see(SchetzenM.1980).
Theexpansion
oftheWienerkernel
n
k
withthe
use of Laguerre functions
( ), 0,1,2,...,
m
lt m
has
thefollowingform(SchetzenM.1981):
11
12
1..1
00
,.., ..
nn
n
nmmmmn
mm
kall
, (26)
wherethecoefficients
1
..
n
mm
a aregivenby
11
.. 1 1 1
00
.. ,..,
nn
n
mm n m m n n
aklldd
. (27)
In particular, the Laguerre expansion of the first
orderWienerkernel
1
k
,restrictedtothefirst
1p
components,assumestheform
1
0
p
mm
m
kal
(28a)
with
1
0
mm
ak ld
. (28b)
4.5 TheWienermodel
Using the results of derivations presented in the
previoussubsections 3.1–3.4,itcanbefurthershown
thataverygeneralmodelfordescriptionofnonlinear
00
;Gkxt
11
;Gkxt
22
;Gkxt
outputsignal
y(t)
inputsignal
x(t)
+
97
systemsfollowsfromtheseoutcomes.Thismodel or
its variants were used in a vast number of research
papersdealingwiththenonlinearsystems.Itiscalled
theWienermodel(SchetzenM.1981)anditsstructure
ispresentedinFig.2.
Figure2.TheWienermodelofanonlinearsystem.
InFig.2,thefirstpartofthemodelconsistingofN
blocks of linear subsystems having (linear) impulse
responses denoted
1
,...,
N
ht h t is a single‐input
multi‐output system. Elements of the above set of
impulseresponsesareorthonormal.Thenextpartof
themodelisamulti‐input(Ninputs)multi‐output(M
outputs) nonlinear system without memory, using
multidimensionalHermitefunctions.Andfinally,the
lastpartofthemodel
inFig.2,consistsofasetofM
multipliers,
1
,...,
M
, and a summing unit.
Furthermore, note that all the memory of a given
nonlinear system that is modelled according to the
structure of Fig. 2 is concentrated solely in its first
(linear)part.
4.6 RemarksonstochasticfunctionalFourierseries,
Cameron‐Martintypeexpansionandsomeother
relatedones
Obviously,
the scheme of modelling of nonlinear
systemsexcitedbysignalsbeingrealizationsofwhite
Gaussian stochastic processes can be extended for
other ones, for example Poisson processes
(MarmarelisV.Z.&BergerT.W.2005).
It is interesting that formulation of a stochastic
version of the Fourier series is possible on
the basis
orthogonalfunctionalsinarandomenvironment(for
randomprocesses).ThiswasdonebyYasuiin(Yasui
S. 1979). In this paper, the relationships existing
between the Wieneer kernels, Volterra kernels
(nonlinearimpulseresponses),andcoefficientsinthe
so‐called Cameron‐Martin functional expansion
(CameronR.H.&MartinW.
T.1947)arefoundand
discussedverythoroughly.
Itisalsoworthnotinghereanalgebraicapproach
to nonlinear functional expansions (Fliess M. &
Lamnabhi M. & Lamnabhi‐Lagarrigue F. 1983)
leadingtotheexpansionsoftheVolterraseriestype.
This method relies upon the use of a formal power
series in several non‐commutative variables and of
iterated integrals. For more details, see (Fliess M. &
LamnabhiM.&Lamnabhi‐LagarrigueF.1983).
5 BESTAPPROXIMATIONOFLARGE‐SCALE
NONLINEARSYSTEMSUSINGVOLTERRA
OPERATORSINWEIGHTEDFOCKSPACES
One of the important problems with the Volterra
series(aswellas
withtheWienerseries)applications
is that the number of calculations to be performed
grows exponentially with the order (degree) of
system’s nonlinearities, which have to be taken into
account to achieve good enough accuracy of the
approximation. The number of the needed
calculations grows also in a similar way
with the
increase of system input and/or output
dimensionalities. The above facts cause that the
Volterraseriesapplicationsarelimitedtoratherlow‐
dimensional systems and/or such ones with mild
nonlinearities.
Asshownin(DeFigueiredoR.J.P.&DwyerIIIT.
A. W. 1980), the above problem can be
largely
circumvented by reformulating the Volterra series
with the use of a special mathematical tool called a
reproducing kernel Hilbert space (RKHS). In the
above paper, this tool was used in an appropriately
chosenweightedFockspace.Formoredetails,see(De
Figueiredo R. J. P. & Dwyer III T.
A. W. 1980) and
referencescitedtherein.
6 SOMEINTERESTINGAPPLICATIONSOFTHE
VOLTERRAANDWIENERTHEORIES
In this final section, because of lack of space, we
present only examples of some interesting
applications of the Volterra and Wiener theories in
telecommunications, biological sciences, oceanology,
andphysics.
Telecommunications.Volterra series
and an
orthogonalseriesderivedfromithavebeenusedfor
description of nonlinear distortions occurring in
satellite communication channels. On this basis, the
corresponding schemes for equalization of these
nonlinear channels and compensation of distortions
havebeenworkedoutin(BenedettoS.&BiglieriE.&
Daffara R. 1979, Gutierrez
A. & Ryan W. 2000).
Another examples of applications for solving
nonlinear problems in radio communication are
presented in (BedrosianE. & RiceS. O. 1971,
BussgangJ.J.&EhrmanL.&GrahamJ.W.1974).
Biological sciences.Examples of applications ofthe
Wiener theory in this area can
be found in articles
(Dijk P. & Wit H. P. & Segenhout J. M. 1994,
MarmarelisP.Z.&NakaK.‐I.1972, Marmarelis V.
Z.&Zhao,SclabassiR.J.&RischH.A.&HinmanC.
L.&KroinJ.S.&EnnsN.F.&NamerowN.
S.1977).
Oceanology.Nonlinearoceanwavemodellingwith
theuseoftheVolterraandothermathematicaltoolsis
describedin(MaltzF.2009).
Physics.TheuseoftheVolterraandWienerseries
in magnetic resonance spectroscopy has been
exploited in (Blümich B 1985). Very interesting and
promisingistheapplication
oftheVolterraseriesto
description of objects and phenomena in quantum
1
ht
2
ht
N
ht
y
(
t
)
x
(
t
)
+
α1
α2
αM
nonlinear
system
without
memory
98
physics(ZhangJ.&LiuY.&,WuR.‐B.&JacobsK.&
OzdemirS.K.&LanY.&TarnT.‐J.&NoriF.2014).
Hydrology.InterestingapplicationsoftheVolterra
seriesarepresented,forexample,in(NapiórkowskiJ.
J.&StrupczewskiW.G.)andpaperscited
therein.
Navigation. Application of the Volterra filters in
solving nonlinear problems of navigation can be
found,forexample,in(ParkS.H.2007).
Transportation. Nonlinear problems of tra nsport‐
tationaretackledwiththeuseofWienermeasurein
(FeyelD.&A.S.Üstünel2004).
7 CONCLUSIONS
First, a concise
introduction to the Volterra and
Wienerserieshasbeenmadeinthispaper.Second,a
generalmodelofnonlinearsystems,calledtheWiener
modelafterhisfounder,hasbeenpresented.Also,a
modelfordescriptionofverylargenonlinearsystems,
based on the Volterra series and the so‐called
reproducing
kernelHilbertspace,hasbeendescribed.
Finally, numerous applications of the above
mathematical tools in such areas as tele‐
communications, biological sciences, oceanology,
physics, hydrology, navigation, and transportation
have been enumerated. However, because a lack of
space,theyarenotpresentedhereinmoredetail,with
some needed illustrations. This will
be done during
an oral presentation at the conference. Nevertheless,
we hope, all the examples given witness strongly
greatusefulnessof the VolterraandWiener theories
inengineering.
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