393
1 INTRODUCTION
Usingaschemesimilartothatonewhichisshownin
Fig.1,PrandoniandVetterliintheirbook(Prandoni
P. & Vetterli M. 2008; on page 284) explained the
behaviorandoperationofanA/Dconverter.
Figure1.AbasicideaofanA/Dconverterillustratedwith
theuseofaFETtransistorswitchconnectedtoacapacitor
holdingthevalueofananaloginputsignaltakenatthetime
ofswitching.
AsimilarschemeasthatshowninFig.1(inablock
form)canbealsofoundinawell‐knownmonograph
ofvandePlassche(vandePlasscheR.1994;onpage
4).(Thatisinabookspeciallydedicatedtothetheory
ofA/DandD/Aconversionsandto
thediscussionof
integratedcircuitsthatperformtheseoperations.)
TheblocklabeledS/HinFig.2implementstheso‐
called operation of taking a sample value at a time
instantkT,k=…,‐2,‐1,0,1,2,…andmaintainingitfora
timeperiodT.Suchanoperationisperformedby
that
electroniccircuit,whichisshowninFig.1,andwhich
consistsofaFETtransistorandacapacitorconnected
toits“source”terminal.TheFETtransistor,controlled
by pulses applied to its “gate” terminal, opens
regularlyeveryTsecondsandchargesthecapacitorto
acurrentvalueofthe
voltageatitsʺsourceʺterminal.
The task of capacitor is to sustain this voltage value
for a time equal at least to T. The voltage values,
A New Result on Description of the Spectrum of a
Sampled Signal
A.Borys
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Itisexplained,inintroductionofthispaper,whythedescriptionoftheoutputsignalatanA/D
converterintheformthatispresentedinsuchrespectedtextbooksas:aonewrittenbyPrandoniandVetterli,
andanotheronebyvandePlasscheis
appropriateandcorrect.Unlikeallothers,especiallythoseusinginitthe
so‐calledcombofDiracdeltas.Thelatteronesdonotleadtogettingacorrectformulaforthespectrumofthe
outputwaveformofanA/Dconverter,ortheyyieldnoformulaatall.Usingthedescription
oftheA/Doutput
signalinformofastepfunction(asinthetextbooksmentionedabove),anew,correctformulaforcalculating
thespectrumofthesampledsignalisderivedinthispaper.Itisarevisedversionoftheformulacurrentlyused
intheliterature, thatisof the
so‐calledDiscrete‐TimeFourierTransform (DTFT),and itisa product of this
DTFTandacertain correctionfactor.Finally,someliteratureitemsare referredtoinwhichthedesignersof
integrated circuits (containing A/D converters) point out discrepancies that arise in designs when the
multiplyingfactormentionedabove
isnottakenintoaccount.
http://www.transnav.eu
the International Journal
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Volume 18
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DOI:10.12716/1001.18.02.1
7
394
whichchangeinstepsonthecapacitor,aresubjected
toaquantizationprocess,andfurtherthesequantized
valuesaremappedintonumbers.
Figure2.AblockdiagramofanA/Dconverterconsistingof
three blocks: a sample and hold (S/H) unit, an amplitude
quantization unit (Q), and a mapper (coder) of quantized
valuestonumbers.
It followsfrom Figs. 1 and 2, and descriptions of
theequivalentcircuitsofanA/Dconvertershownin
these figures that its output waveforms (understood
asfunctionsofacontinuoustimet)havetheformofa
slightly disturbed step function presented in Fig. 3.
Theshapeofactual
waveformsattheoutputsofA/D
convertersismorerichthanthestepfunctionshown
in Fig. 3 and depends upon the architecture and
technology in which a given converter is
implemented. This shape is characterized by such
parameters as: settling time, acquisition time,
aperture,aperturejitter,holdmodesettlingtime,
hold
mode feedthrough, droop; see, for example, (van de
PlasscheR.1994;page74).However,fromthepointof
viewofadesignerofsignalprocessingsystems,most
of these parameters are of secondary importance
(whichdoesnotmeanatallthattheyarenotrelevant
to designers of their
integratedstructures in specific
semiconductor technologies). In the description
visualized in Fig. 3, we restricted ourselves to
pointingoutthatineachtimeinterval<kT,(k+1)T>,k=
…,‐2,‐1,0,1,2,… we have its initial segment (the so‐
calledtrackpartandbeginningoftheholdpart)rich
in changes
and the second one (covering the almost
entire hold part) already stabilized on the hold
voltage level of a given time interval. The former
segment is denoted on the waveform of Fig. 3 by a
diagonal dash, while the latter is marked with a
longerdash,paralleltothetimeaxis
–ateachofthe
aforementionedtimeintervals.
Figure3.Sketchillustratingtheformofthewaveformofan
examplesampledsignal,denotedbyx
s(t)and related with
an un‐sampled one x(t) (not shown here). (This figure is
basedonaone,whichwasusedindiscussionspresentedin
(BorysA.2022)).
In Fig. 3,
,
3
sH
Q
xT
and
,
sH
Q
xT
, where the
lower index Q means the operation of amplitude
quantization, stand for illustration of the quantized
values of the sampled signal x
s(t). These values are
assigned to the following instants: ‐3T and T,
respectively,andworkedoutintheholdpartsofthe
corresponding time intervals (mentioned above). In
Fig. 3, it is assumed that the track part (including
beginningoftheholdpart,too)lasts
seconds,and
thetrackandholdpartstogetherlastTseconds.
As we know from the literature, the idealized
version of the signal sampling process neglects the
switchingtimeinthisprocess,assumingthatthetime
ismuchsmallerthanT.Inotherwords,inanideal
case,
=0isassumed.Thenthewaveformshownin
Fig.3takestheformwhichisvisualizedinFig.4.
Figure4. Sketch illustrating an idealized version of the
waveformshowninFig.3;itisdenotedherebyx
st(t).(This
figure is based on a one, which was used in discussions
presentedin(BorysA.2022)).
Furthermore, note that the coder of quantized
values shown in Fig. 2 plays, in addition to
performing the conversion of these values into
numbers,aroleofanelementthatholdsanumberit
generatedatagiveninstant,asanencoder,forexactly
T seconds – before feeding it further
into a signal
processororasignalprocessorbuffer.Therefore,the
waveformattheoutputofthedecoderisexactlythe
same as the one shown in Fig. 4 (in this idealized
version), except that the quantized values are now
ʺscaledʺtonumbers. Forcompletenessofthepicture
ofwhat
appearsasthefinalresultattheoutputofan
A/Dconverter,thewaveforminFig.4 isredrawntoa
“scaled”oneshowninFig.5.
Figure5. Sketch illustrating an idealized version of the
waveform shown in Fig. 3 after performing amplitude
quantizationandcodingintonumbers;itisdenotedhereby
x
sic(t). (This figure is based on a one, which was used in
discussionspresentedin(BorysA.2022)).
InFig.5,
,
3
sH
QC
xT
and
,
sH
QC
xT
,wherethe
lower index QC means performing both the
operations: amplitude quantization and coding (one
aftertheother),areexamplevaluesof thequantized
andcodedsignal
sic
x
t
.
Inordertomakefurthercomparisonsoftheideal
descriptions of the sampled signal presented above,
letusalsoaddtothemtheonedescribingthesampled
signalimmediately before performing the amplitude
quantization on it. But we give up here a graphical
illustrationofitsinceawaveformin
thefigurewould
havethesameformasthatoneshowninFig.4,with
the only difference in that the values of theʺstair
stepsʺ on it would differ slightly from the
corresponding ones in Fig. 4. Furthermore, these
395
values would belong to the set of real numbers.
Finally,letuscallthiswaveformas
siq
x
t
.
Now,usingtheabovesignalandnotation,wesee
thatthesignaldeterminingthequantizationerrore
q(t)
canbeexpressedaccordingtothefollowingequation
(OppenheimA.V.,SchaferR.W.,BuckJ.R.1998):
si siq q
x
txtet
. (1)
Thatiswegetfrom(1a)
qsisiq
et xt x t
. (2)
Notethatinourconsiderationspresentedherethe
signal
siq
x
t
couldbealsointerpretedotherwise.For
example,asaonewiththevaluesaveragedineachof
the time intervals <kT, (k+1)T>, k = …,‐2,‐1,0,1,2,… –
overapartorinthewholetimeinterval(ofthelength
T).Suchaproposalwasmade,forinstance,byVetterli
M., Kovacevic J., and Goyal V. K. in their book
(VetterliM., KovacevicJ.,GoyalV.K.2014; onpage
45). Further, note that there are also other possible
interpretationsofthesignaldenotedhereas
siq
x
t
.
But we see here clearly that all of them differ or
would differ, more or less, in values of the voltage
levels on the correspondingʺstair stepsʺ of a step
functionthatdescribestheirshape.Thatislikesucha
function as the one visualized in Fig. 4 but before
quantizationoperation.Sowe can characterizeallof
themthroughthefollowingequation:
ssiqsiq
et x t x tTT
, (3)
where e
s(t) stands for an error related with the
samplingprocess(asithasbeendepictedhere).And
meansthefloorfunctionin(3).
Eq.(3)canberewrittenintheform
siq siq s
x
tx tTTet
, (4)
whichallowsustosaythatallthedescriptionsofthe
sampled operation immediately before performing
the amplitude quantization, mentioned just above,
differfromeachotheronlyintheerrorfunctione
s(t).
(Becauseall the values
siq
x
tT T
for these
functionsarethesame.)
Introducing(4)into(1)gives
si siq s q
x
tx tTTetet
. (5)
Andthisresultshowsusthatthesampledsignal,
beforeperformingitscoding,issubjecttotwoerrors:
e
s(t) and eq(t) both related with the uncertainty in
amplitude.
Obviously,intheprocessofsamplingasignal,we
have also to do with uncertainty in determining the
sampling time instants. This, however, has no
significantimpactontheshapeofthex
s(t)waveforms.
At most, they are shifted on the time axis by a
constant value, and with a small time jitter the
changesof widths of “stair steps” of a step function
(as,forexample,oftheoneinFig.4)canbeneglected.
2 CALCULATIONOFTHESAMPLEDSIGNAL
SPECTRUM
Discussion of the cases presented in the previous
sectionshowedthatthemostappropriatedescription
ofthewaveformattheoutputofanA/Dconverterisa
step function – independently whether we
understand by it the signal immediately before
performingthequantizationoperationorthealready
quantizedone,or
thecodedlatterone(locally,ateach
of theʺstair stepsʺ of this function). And, as already
recognized, the cases mentioned above differ from
each other only by small deviations in the values of
thevoltagelevelsofthecorrespondingʺstairstepsʺof
theirstepfunctions.However,thisisirrelevantto
the
problemconsideredinthispaper.Whatisimportant
here is the staircasecharacter of thefunction,which
describes what happens at the output of an A/D
converter.Andthisallowsustodescribeallofthese
casesanalyticallythrougha single(generic)formula,
asfollows:
for 1
and with
gg
xt xkT kTt k T
ktT
, (6)
wherex
g(t)standsforthevalueofthek‐th“stairstep”
of the step function
siq
x
t
or
si
x
t
, or also
sic
x
t
.
Moreover,forthesakeofclarity,itisworthrecalling
atthispointthattheerrorfunctionse
s(t)andeq(t)are,
obviously,alsostepfunctions.
Letus nowcalculatetheFouriertransformofthe
waveform given by (6). We will do this in detail
startingwith
0
0
2
exp 2
... exp 2 0
exp 2
exp 2 ...
ggg
T
gg
T
T
g
T
Xf xt xt jftdt
xT jftdtx
jftdt
xT j ftdt
, (7)
where
gg
Xt xt
means the Fourier transform
ofthesignalx
g(t),fisthefrequency,and
1
j
.
Inthenextstep,calculatingintegralsin(7),weget
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0
0
2
... exp 2
2
0
exp 2
22
exp 2 ...=
= ... exp 2 +
2
+exp20 0
exp 2 0 + 0 exp 2
exp 2
exp 2 2 ... = ... 2
2
g
g
T
T
gg
T
T
g
gg
g
gg
g
xT
Xf jft
jf
xT xT
jft
jf jf
jft
j
xT jfT
f
xT jfT xT
jfT xT jfT
xT j fT xT
j
j
fT x T
f
exp 2 + 0
exp 2 0 0
exp 2 2
exp 2 2 ... .
ggg
gg
gg
xT jfT xTxT
jfT xTxT
jfT xTxT
jfT
(8)
Inacompactform,(8)canbeexpressedas
1
=1
2
1
exp 2 DTFT
2
1 ,
ggg
k
df
g
g
Xf xkTxk T
jf
jfkT xkT
jf
xk T
(9)
where
DTFT
means the so‐called Discrete Time
FourierTransform(OppenheimA.V.,SchaferR.W.,
BuckJ.R.1998),(VetterliM.,KovacevicJ.,GoyalV.K.
2014). (Moreover, note that(9) includes, at the same
time,alsothedefinitionofthistransform.)
Further,observethat(9)canrewrittenas
1
=DTFT
2
DTFT 1 .
gg
g
Xf xkT
jf
xk T
(10)
Moreover, we can transform
DTFT 1
g
x
kT
occurringin(10)tothefollowingform:
DTFT 1 = 1
exp 2
exp 2 1 exp 2
exp 2
DTFT exp 2 .
gg
k
g
k
g
k
g
x
kT xkT
jfkT xkT
jfk T jfT
xkT j fkT
xkT j fT
(11)
Notethattogetafinalresultin(11)weintroduced
first an auxiliary variable
1
kk
there, then
performedsomemanipulations, and finallydropped
prime symbol at
k
. Substituting next this result to
(10),weget
DTFT
1exp 2
2
DTFT
exp exp
2exp
sin
DTFT exp
DTFT sinc exp .
g
g
g
g
g
xkT
Xf jfT
jf
TxkT
jfT jfT
jfT jfT
fT
TxkTjfT
fT
TxkT fTjfT
(12)
In this way, we arrived at the end of our
calculations of the spectrum of a sampled signal.
Hence,ourfinalresultthatfollowsfrom(14)hasthe
followingform:
DTFT sinc exp
g
g
Xf
TxkT fTjfT
. (13)
3 DISCUSSIONOFTHERESULTACHIEVED
Evidently,theresultgivenby(13)differsfromtheone
tellingusthattheDTFTofasequenceofanalogsignal
samples is a Fourier transform of this sequence. In
otherwords, the DTFT is identifiedwith (or plays a
roleof)a
Fouriertransform(spectrum)ofthesampled
versionofananalogsignal,anditturnsoutherethat
thisisnotaquitecorrectidentification.Furthermore,
this identification is done without any justification,
simplyassumingaprioriitasadefinition.
Moreover, as shown in Introduction, the above
approachisdifficult
tojustifywhenconfronted with
signalsthatappearattheoutputofanA/Dconverter.
Undoubtedly, these are continuous‐time signals so
calculationoftheirFouriertransforms(spectra)canbe
done within the framework of a classical Fourier
theory–withoutanyproblems.Itisnotnecessaryto
reachforsome
newtools,suchas,forexample,DTFT.
Theresultobtainedinthispapershowsthattheabove
ispossible.Besides,ourtacklingwiththeproblemis
firmlyʺanchoredʺinrealitiesofphysicaloperationsof
sampling analog signals. Additionally, it also turns
outthattheoutcomescanbeexpressedinterms
ofthe
DTFTandacertainmultiplyingfunction,see(13).
Note, however, that for very small values of the
frequency f, compared to the value of the sampling
frequency f
s=1/T, i.e. for
1
s
s
f
fff
, it can be
assumedthatthevaluesofthemultiplyingfunctionin
(13)areapproximatelyequalto1.Andthisallowsus,
for these frequency ranges, to assume that the
followingformula:
DTFT
gg
X
fT xkT (14)
isvalidapproximately.
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Bytheway,notetheoccurrenceofascalingfactor
Tinequations(12),(13),and(14)above.Itsexistence
is due to the fact that the DTFT is only a weighted
sumofsignalsamples,whileaʺrealʺsignalspectrum
is related to integration over time, orʺsummation
timestime.ʺ
Obviously, the expression given by (13) for
calculationofthespectrumofasampledsignaldiffers
alsofromthefollowinghighlycelebratedformula:
1
s
ds
k
X
fXfkf
T
, (15)
where X(f) means the Fourier transform of an un‐
sampled signal x(t), but X
sd(f) is identified with the
spectrum of its sampled version x
sd(f) (described in
another way as x
g(t) here). To see this, let us use in
(15)therelationship(OppenheimA.V.,SchaferR.W.,
BuckJ.R.1998),(VetterliM.,KovacevicJ.,GoyalV.K.
2014) according to which the DTFT equals the
expression occurring on the right‐hand side of (15).
Evidently, we see then that
gsd
X
fTXf
holds.
Moreover, we get then the equivalent of (13) in the
followingform:
sinc exp
.
g
s
k
X
ffTjfT
Xf kf
(16)
So we can say that the formulas (13) and (16)
postulatetheneedtointroduceacorrectingcoefficient
(correcting function):
sinc exp
f
TjfT
into the
expression on the right‐hand side of (15), which is
currentlyinforceintheliterature.
4 CONCLUSIONS
The need to develop a new, better description of
output waveforms of A/D converters have been
indirectlyconfirmedinmanyplacesintheliterature.
For example, see (de la Rosa
J., Perez‐Verdu B.,
Medeiro F., del Rio R., Rodriguez‐Vazquez A. 2001)
and(delaRosaJ.,Perez‐VerduB.,MedeiroF.,delRio
R.,Rodriguez‐VazquezA.2004).Amongothers,these
authorsconfirmthattheeffectwhichisdiscussedin
thispaperismostannoyedatthelargeratios
ofsignal
frequencies to sampling frequencies (comparable to
eachother).Andthiseffectactuallydisappearsatvery
high sampling frequencies (that is, when the above
frequencyratioisrelativelysmall;see(14)).
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Borys A. 2022. Sampled signal description that is used in
calculation of spectrum of this signal needs revision,
IEEE Signal Processing Letters,
submitted for
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