387
1 INTRODUCTION
In research papers and textbooks on systems theory
andsignalprocessing,incaseofusingamathematical
concept: the Dirac distribution (called also a Dirac
delta or a Dirac impulse) in them – in different
contexts–it isassumedthat thisdistribution can be
usedboth
asanoperatorandasasignal(PrandoniP.,
VetterliM.2008),(VetterliM.,KovacevicJ.,GoyalV.
K.2014),(IngleK.,ProakisJ.G.2012),(OppenheimA.
V., Schafer R. W., Buck J. R. 1998), (Dąbrowski A.
2008), (HowellK.B.2001), (Gasquet C., Witomski P.
1998),(OsgoodB.
2014).Thatisitcanplayaroleofan
operator,butalsoa roleofasignal.Anditseemsthat
in engineering sciences, particularly in systems
theory, this way of thinking has its roots in the fact
that any linear (non‐pathological) system can be
describedbythe
followingconvolutionintegral:
yt h xt d
, (1)
where y(t) and x(t) mean an output and an input
signal of a system, respectively. This system is
assumedtobe characterizedbyitssystem’sfunction
(calledalsoitsimpulseresponse)h(t).Variabletin(1)
standsforacontinuoustime.
Mathematically, (1) can beviewed asan
operator
thatmapsinputsignalsx(t)’sintooutputsignalsy(t)’s
ofagivensystem.Moreover,itiswellknownthat(1)
iswelldeterminedforalltheimpulseresponsesh(t)’s
aswellassignalsx(t)’swhichoccurinengineering.
Furthermore,thefunctionsdenotedash(t)andx(t)
in(1),
andwhichoccurinengineering,canactasboth:
system’sfunctionsaswellassysteminputsignals.To
see this, let us introduce an auxiliary variable
ʹtt
in(1).Thisleadsto
''' ' ''
yt httxtdt xthttdt
. (2)
Andfinally,namingtheauxiliaryvariablet’by
,
weobtainfrom(2)thefollowing:
yt x ht d
. (3)
Notethatnowin(3)x(t)playsaroleofasystem’s
function, while h(t) a role of an input signal of this
Dirac Delta as a Useful Technical Tool in Modelling
Signals but Hard to Think About It as a Physical Signal
Itself
A.Borys
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Inthispaper,weshowthattheDiracdeltaisausefultechnicaltoolinmodellingsignalsbuthard
tothinkaboutitasaphysicalsignalitself.Thisthesisissupportedherebyanexamplecomingfromthefieldof
measuringphysicalquantitiesand
measurementtheory.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 18
Number 2
June 2024
DOI:10.12716/1001.18.02.16
388
system.Thatistheoppositeofwhatwasbefore,in(1).
Or,inotherwords,thisshowswearenotabletosay
whatisasystem’sfunctionandwhatitsinputsignal
ina pair: h(t),x(t)– knowingonly theoutputsignal
y(t).Thatish(t)and
x(t)commutewitheachother(in
theirroles)intheconvolutionintegraloperatorgiven
by(1).
The so‐called Dirac delta is an object that is also
used in the systems theory and signal processing
(more generally, in engineering). And, in the
engineering literature, it is most often denoted by
a
symbol
(t),whatsuggeststhatitcouldbetreatedasa
function(althoughweknowverywellthatitisnot).
However,duetothisbelief(thatitcanbehandledasa
function),itisusedbyengineersinbothrolesin(1).
That is as a system’s function as
well as an input
signal– in the same way as h(t)and x(t) considered
above.
Let us take now a closer look at this issue. And,
considerfirstthecasewhenh(t)in(1)isassumedtobe
aDiracimpulse.Sothisallowsustorewrite(1)inthe
followingform:
yt xt d xt
. (4)
Further, note that the outcome on the right‐hand
sideofequality(4)resultsfromapplyingtheso‐called
sifting property of the Dirac delta therein. And this
allowsus to conclude that the Dirac delta makesan
identityoperatorfromaconvolutionone.
Thenotationoftheconvolution
integralcontaining
aDiracdelta,asin(4),requiresoneexplanationmore
becausesuchanintegraldoesnotinfactexist,neither
in the Riemannʹs sense nor in the Lebesgueʹs sense.
Nevertheless, because of the convenience and habit,
this notation is used by engineers for denoting
somethingwhat
must bemathematicallyunderstood
asadistribution.Andthisterminologicalconvention
willbeusedinwhatfollows.
Letuscheckwhether
(t)andx(t)in(4)commute
with each other. And, to this end, assume that the
operation given by (4) also possesses the properties
whichwereexploitedintransforming(1)to(3)–with
the intermediate step shown in (2). So performing
nowthesamemanipulationsasthoseindicatedfrom
(1)
to(3)above,thatis
''' ' '',
yt ttxtdt xt ttdt
(5)
weget
yt x t d xt
. (6)
Comparison of (4) with (6) allows us to say that
really
(t)andx(t)commutewitheachother(intheir
roles of a system’s function and a system’s input
signal).Atleastmathematically,thisseemstobefully
trueandcorrect.
Inthenext section, we examinewhetherphysical
systemsexhibitthisproperty.
2 DIRACDELTAANDINPUTSIGNALDONOT
ALWAYS
COMMUTEINDESCRIPTIONSOF
LINEARPHYSICALSYSTEMS
Theflagshipexamplegiveninengineeringtextbooks
tojustifythevalidity,oreventhenecessity,ofusing
theDiracdeltaconceptindescribingvariousphysical
phenomenaistheprocessofmeasuringtemperature,
voltage,current,forexample.Theauthorofthispaper
analyzed critically
the process of measuring any
physical quantity in general as well as measuring
temperature in particular – in the following papers:
(BorysA.2020a),(BorysA.2020b),(BorysA.2020c)–
intermsofthevalidityofusingDiracdeltasintheir
descriptions.Theresultsachievedtherewillserveasa
startingpointfortheconsiderationsofthissection.
It has been shown in (Borys A. 2020c) that the
processofmeasuringofaphysicalquantity(suchas,
forexample,temperature)intimecanbedescribedin
a similar way as one describes the sampling of a
signal of a continuous
time. This is shown
schematicallyinFig.1.
Figure1. Illustrationtomodellingameasuringprocess via
thedescriptionofsamplingsignalsofacontinuoustimeas
discussedin(BorysA.2020c).Inthisapproach,weassume
thatthemeasuringdeviceʺdeliversʺvaluesofthemeasured
physicalquantityinastepwisemanner.The lengthofeach
stepisassumed
tobeequaltoTsseconds.Further,during
eachofthesesteps,itisassumedthatthemeasuringdevice
averagesthemeasured quantityatimeequaltoTa<Ts.The
averagedvaluesareassignedsuccessivelytotimeintervals
starting from the beginning of a given step (lasting T
s
seconds)toitsend(thesamevalueofthemeasuredphysical
quantity applies to all time instants belonging to a given
interval).(Thisfigureisbasedonaone,whichwasusedin
discussionspresentedin(BorysA.2020c)).
Denote now by x(t) a waveform according to
which a physical quantity changes in time (for
example, the temperature mentioned above).
Obviously, because of the reasons mentioned above,
themeasuringdeviceisunabletoprovideuswiththis
waveforminanundistortedmanner.Here,wemodel
itsbehaviorasillustratedin
Fig.1andasdescribedin
the caption to this figure. And we look for an
analyticalexpressiondescribingasignalregisteredby
ourmeasuringdevice;wedenoteitbyy(t).
Let us start with calculation of the value of the
signaly(t)thatisapplicableintheinterval
0
s
tT
.
Denote it by
0
asaaa
yTTyT
it will be given
by
0
a
T
aa
yT xt tdt
, (7)
where
(t)meansanaveragingfunction.
In the next step, to illustrate the averaging
operation in time given by (7), let us choose the
simplest possible form of
(t) therein that fulfilsthe
389
conditionsforsuchfunctionsformulatedin(Strichartz
R.1994).Thatfunctionhasthefollowingform:
1 for 0
0 elsewhere .
aa
TtT
t
(8)
Substituting(8)into(7)gives
0
1
a
T
aa
a
yT xtdt
T
. (9)
Nowwewillshowthatasthefollowingproperty:
a
ttT
holds in the case of the function
(8),(7)canbeexpressedequivalentlyasaconvolution
integral.Tothisend, werewrite(7) inthefollowing
way:
0
0
0
.
a
a
a
T
aa a a
T
T
aa
yT xt tT dt xT d
xT d xT d
(10)
In derivation of the final result in (10), we have
additionally used an auxiliary variable
=‐(t‐Ta) and
thefactthatthefunction
(t)givenby(8)isidentically
equaltozerooutsidetheinterval
(0,
a
T
.
Inthenextstep,notethata similarrelationas(9)
for y
a(Ta) can be written for every ya(kTs+Ta), where
k=…,‐1,0,1,….Thatisthefollowingone:
sa
s
kT T
asa s
kT
ykT T xt tkTdt
. (11)
Further,togetasimilarexpressionas(10),weuse
thefactthat
ssa
tkT tkT T
holdsforthe
function
(t)givenby(8).So(11)canbere‐writtenas
0
0
=
.
sa
s
a
a
kT T
asa sa
kT
T
sa sa
T
sa
ykT T xt tkT T dt
xkT T d xkT T d
xkT T d
(12)
Note that in derivation of thefinal result in (12),
we have used an auxiliary variable
s
a
tkTT
and,similarlyasbefore,thefactthatthefunction
(t)
given by (8) is identically equal to zero outside the
interval
(0,
a
T
.
Havingderivedtheresults(12)and(10)(wherethe
latterisaspecialcaseof(12)fork=0),weareablenow
toexpressthesignaly(t)foralltimes.Itwillbegiven
by
asa sa
yt ykTT xkTT d
(13)
for t belonging to the successive time intervals
1
s
s
kT t k T
when k assumes successively the
valuesk=…,‐1,0,1,….
Further,observethatthefunctiongivenby(13)isa
stepfunctionwiththevaluesofitsstepsequaltothe
corresponding
's,
asa
ykT T
k=…,‐1,0,1,….
occurring in the successive time segments
1
s
s
kT t k T
,k=…,‐1,0,1,….
Itisinterestingtonotethatthefunctiongivenby
(13)canbeexpressedalsoinanotherway,asasumof
some functions. And, to see this, let us start with
defining first these functions; we define them in the
followingway:
for 1
and
0 outside the above range of 's
s
asa sa
kT
ss
ykT T xkT T d
yt
kT t k T
t
(14)
with k’s in (14) that may take the following values:
…,‐1,0,1,….So,withthehelpofthefunctionsgivenby
(14),wecanexpressy(t)from(13)inacompactwayas
follows:
s
kT
k
yt y t
. (15)
As already said in Introduction, in various
technicaldisciplineswhichusedescriptionsinformof
convolutionintegrals,itisassumedthatwhatstands
ontheleft‐hand side underaconvolution integralis
relatedwithsomeoperator(operation)performedon
asignal(physicalquantity)varyingintime–the
latter
standing on the right‐hand side under the above
integral.Obviously,thismatterofoccupiedpositionis
a matter of convention, but it has its justification in
what the convolution integral is used for in
engineeringsciences.Figurativelyspeaking,wecould
express this in the following way: a convolution
integralweavestogethertworoles:ofatransforming
operation(performedbyasystemconsidered)andof
beingasignal(physicalquantity),whichissubjected
totheactionof theformer.Andasalreadysaid, the
firstroleiscustomarilyassignedtotheleft‐handside
under the integral, and the second
to its right‐hand
side.
So, now with regard to the convolution integrals
occurringintheexpressions(13)and(14),thefunction
(t) is playing therein a role of a transforming
operation, but the function x(t) a role of a physical
quantity(forexample,ofatemperaturevaryingwith
time).And,asalreadyknownfromtheconsiderations
presentedinIntroduction, as longas these functions
remainʺdecentʺ (what we mean under this term
is
explainedbelow),theycanperformbothroles.Thatis
they can stand in a convolution integral on both
positions:beingtheleft‐handsideaswellastheright‐
hand side of the expression under the integral.
Unfortunately, when considering concrete physical
systemsthisisnot alwaysthe case.
Inwhatfollows,
weexplainthispointonan example ofmeasuringa
390
time‐varyingtemperature;thisexampleisconsidered
throughoutthepaper.
Asweknow,temperatureasaphysicalquantityis
bounded. For example, let us consider the
temperature on Earth. We can say that this
temperature does not exceed the lower limit of‐100
degrees Celsius and the upper limit of
+100 degrees
Celsius. Considerit as changing with the passageof
time: opassing hours, days, years. So it will be
represented by a function of time. Further, let us
identify it with the function x(t) introduced
previously.Soitwillbeaboundedfunctionforwhich
wecanwrite
for every
x
tM t
, (16)
whereMdenotestheboundingconstraintimposedon
thefunctionx(t).
Letusnowtakesucha
(t)functionoccurringin
(13) and (14) which does not exhibit the constraint
givenin(16).Thatistherearepossibleabsolutevalues
of
(t) which exceed the value of M. In this case,
obviously, the functions x(t) and
(t) cannot change
their roles in (13) and (14) because
(t) so chosen is
notaphysicallyreasonablefunctionthatdescribesthe
temperature changes on Earth. In other words, the
above functions x(t) and
(t) do not commute (their
rolesdonotcommute)intheintegralsin(13)and(14)
becauseofphysicalreasons.
Of course, bydropping the condition(16) forthe
functionx(t),weʺrestoreʺthecommutativityproperty
of the functions x(t) and
(t) in the integrals in (13)
and(14),butatthecostthatthefunctionx(t)willnot
beabletobeinterpretedasafunctionthatdetermines
temperaturechangesonEarth.
As we will see further on, the lack of
commutativitypropertyofcertainfunctions
(t)with
the functiondescribing temperature changes on
Earth will manifest itself in full as we move in the
formulas (13) and (14) from a finite‐time averaging
operation(i.e.withafinite)toʺidealʺaveragingin
time,i.e.withthevalueoftheparameterT
agoingto
zero.
The result presented in this section, which
indicates possibility of the lack of commutativity
property between an input signal at the input of a
linearsystemanditsso‐calledsystem’sfunction–ina
description of that system, may seem a little bit
strange. We are
accustomed to the fact that the
aforementionedproperty takes place. However, note
that the fact that this is not always the case has
already been pointed out by others, for example by
IrwinSandberginthefollowingpapers:(SandbergI.
2008) and (Sandberg I. 2000). So, really, the
commutativity property is
not obligatory in linear
systems.
3 IDEALAVERAGING
Let us now consider the case of a temperature
measurement, as in the example of the previous
section, where the averagingoperation is performed
at ever shorter time intervals. Note that such a
scenarioisreferredtointheliterature,forexample
in
(StrichartzR.1994)tojustifytheneedfortheuseofa
Dirac delta. So, now, we will assume that in our
averagingfunction
(t),givenby(8),theparameterTa
goes to zero. Thus, this function will approach the
Diracʹs delta – in the sense of the series‐based
distribution theory (see, for example, (Hoskins R. F.
2010), (Strichartz R. 1994) – in the integrals in the
expressions (13) and (14). And these formulas will
takethenthefollowing
forms:
iais s s
y t y kT x kT d x kT
(17)
for t belonging to the successive time intervals
1
s
s
kT t k T
when k assumes successively the
valuesk=…,‐1,0,1,…,and
for 1
and
0 outside the above range of 's .
s
ai s s s
kT i
ss
y kT x kT d x kT
yt
kT t k T
t
(18)
In(17)and(18),thevaluesof
'
ai s
ykTs
standfor
thecorresponding
'
as
ykT s
calculatedinthecaseof
consideringanidealaveraging;thatiswiththeonein
which the parameter
0
a
T
. Obviously, the latter
means that the system’s function in this case
tt
(in the sense explained, for example, in
(HoskinsR.F.2010)and(StrichartzR.1994)).Andjust
becauseofthisreason,wespeakhereaboutanideal
averaging(extendingthesubscriptaat
'
as
ykT s
to
ai). Moreover, for the same reasons, the letterʺiʺ is
also added to as an subscript at y(t) in (17) and for
extending subscripts at
's
s
kT
yt in (18), i.e. to
visualize
i
yt y t
andk=…,‐1,0,1,….
Takingintoaccount the above changesin indices
requires(15)toberewritten,too;namelyas
s
ikTi
k
yt y t
. (19)
Furthermore,notethat thefunction y
i(t) givenby
(17)or(19)remainsastepfunction(asits“non‐ideal”
version given by (13) or (15)). Its steps in the
successive time intervals:
1,
s
s
kT t k T
,
k=…,‐1,0,1,…, will be equal to the values of the
function x(t) at the successive time instants kT
s,
k=…,‐1,0,1,….
Wedrawalsothereaderʹsattentiontothefactthat
thefunctiony
i(t)duetoitsshapeasastepfunctionis
notidenticalwiththefunctionx(t).Inotherwords,the
following:
i
yt xt
(20)
holds.
Finally in this section, note that the function x(t)
cannot replace in any way that action of the Dirac
delta(i.e.theactionofperforminganidealaveraging),
which we see in (17) or (18). Simply because of the
391
constraint(16)imposedonthisfunction,whichmakes
itimpossibletoassumethatitcangrowtoinfinityfor
some times – as it was possible with the function
tt
in (13) and (14) (in the sense of the
series‐based distribution theory (see, for example,
(Hoskins R. F. 2010), (Strichartz R. 1994))). And see
thatthisfurtherreinforceswhatwediscoveredinthe
previous section. Namely that the roles of the
functions
(t) and x(t) in description of the
temperaturemeasurementwiththeuseoftherelation
(13) or (15) do not commutate with each other. In
general,thefunctionx(t)shouldnotbeinterpretedin
thiscaseasasystem’sfunctionand
(t)asasignalat
theinputofthemeasurementdevice(system).Always
the opposite should occur. That is the function x(t)
should be identified with the signal applied to a
system’sinput,but
(t)shouldbeidentifiedwiththe
system’sfunctionofthissystem.
4 DIRACDELTAASATECHNICALMEANSTO
DETERMINETHEIMPULSERESPONSEOFA
LINEARSYSTEM
Acommonlyusedmethodintheliterature(VlachJ.,
Singhal K. 1983), (Sandberg I. 2003) for determining
the system’s function (called also the impulse
responseofasystem)ofsystemshavingdescriptions
informoftheconvolutionintegralistoapplya Dirac
delta that is assumed then to be an input signal.
However,aswellknown,suchsignalsarenot really
encountered in engineering. So the Dirac impulse
shouldbethentreatedmore
asatechnicalmeansfor
calculating a system’s function rather than a real
signal.Formally,seethatapplying
x
tt
in(1),
weget
yt h t d ht
. (21)
Thatisthentheoutputsignalofasystemisequal
toitssystem’sfunctionh(t),which,justbecauseofthe
application of the Dirac impulse at the input of a
system,iscalleditsimpulseresponse.
In the context of the above, note that to get the
resultgivenby(21),weassumedinfact,tacitly,thatin
the natural description of a linear system by a
convolutionintegraltheunboundedinputsignalsare
admissibletherein.Andjustthisassumptionallowed
ustousex(t)=
(t)in(1)toget(21).
But what to do when the input signals in the
convolutionintegraldescription(1)arenotallowable
to exceed some values? As, for instance, in the
example analyzed in the previous section (see the
constraint (16)). Is it possible to determine the
system’s function from
(1) despite the above
restrictionorrathernot?
Wecanreasoninthiscaseasfollows.Letusinsert
into(1)successivefunctionscomingfromasequence
approximatingthe Dirac impulse in the sense ofthe
series‐based distribution theory (see, for example,
(Hoskins R. F. 2010), (Strichartz R. 1994))
and check
eachtimewhetherthecalculatedconvolutionintegral
exists. And finally, check whether this infinite
sequenceofintegralspossessesitslimitforeverytime
instant(i.e.alimitfunction).Ifyes,onemustconclude
thatthisproceduremakessense.And,wegetauseful
result that provides us with the
system’s function,
accordingto(21).
Note,however,thattheproceduredescribedabove
is only partially applicable in (13) and (14): only at
those places, where the calculation of the values of
s
a
kT T
isperformed.Inmoredetail,wegetthen
from(13)and(14)
sa sa
yt kT T kT T d
(22)
for t belonging to the successive time intervals
1
s
s
kT t k T
when k assumes successively the
valuesk=…,‐1,0,1,…,and
for 1
and
0 outside the above range of 's ,
s
sa sa
kT
ss
kT T kT T d
yt
kT t k T
t
(23)
respectively. The indices k’s in (23) may take the
followingvalues:…,‐1,0,1,….
Further,from(15),weobtain
s
kT
k
yt y t
(24)
in the case considered. Moreover, note that the
functiony
(t)in(22)orin(24)meansy(t)in(13)orin
(15) for a particular x(t)=
(t). Similarly,
's
s
kT
yt in
(23) stand for
's
s
kT
yt in (14) for a particular
x(t)=
(t).
In the next step, see that the function y
(t)
calculatedin(22)orin(24),whenthefunction
(t)is
givenby(8),assumesthefollowing form:
1
for 0
and
0 outside the above range of 's .
a
a
s
T
T
tT
yt
t
(25)
Comparisonofthefunctiony
(t)givenby(25)with
the function
(t) given by (8) shows that they differ
fromeachother.
One may ask why this happens. The answer is
rather obvious. A mapping of the signal x(t) to the
measured one, y(t), performed by a measuring
equipment consists not only of a locally performed
convolutionoperations(convolutionintegrals).It
also
includes a momentary (delayed) holding of the
ʺworkedoutʺaveragevalueinthemeasuringdevice.
392
5 CONCLUSIONS
Under assumption of an ideal operation of some
systems, the so‐called Dirac deltas (Dirac impulses)
appear in their descriptions.Unfortunately, in many
textbooks and papers, they become a source of
misinterpretationsanderrors.Oneofsuchbasicerrors
lies in the fact that the Dirac impulse is
uncritically
assumedtobea one ofthe possiblesignals thatcan
appearintheidealdescriptionofagivensystem.That
isitcanbetreatedinterchangeablywithitsso‐called
impulse response. But it cannot, and this is pointed
out in this paper. An example of a device,
which
measurestemperature,wasusedheretoillustratethe
analysis, derivations and discussion presented.
Anotherexampleofthistype,comingfromthetheory
of sampling ideally analog signals, is discussed in
another work (Borys A. 2023) of the author of this
paper.
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