Tài liệu 01 Fourier Series, Fourier Transforms, and the DFT doc

Jenkins, W.K. “Fourier Series, Fourier Transforms, and the DFT”
Digital Signal Processing Handbook
Ed. Vijay K. Madisetti and Douglas B. Williams
Boca Raton: CRC Press LLC, 1999
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1999byCRCPressLLC
1
Fourier Series, Fourier Transforms,
and the DFT
W. Kenneth Jenkins
University of Illinois,
Urbana-Champaign
1.1Introduction
1.2FourierSeriesRepresentationofContinuousTime
PeriodicSignals
ExponentialFourierSeries
•
TheTrigonometricFourierSeries
•
Convergence of the Fourier Series
1.3TheClassicalFourierTransformforContinuousTime
Signals
PropertiesoftheContinuousTimeFourierTransform
•
Fourier Spectrum of the Continuous Time Sampling Model
•
FourierTransform ofPeriodicContinuousTimeSignals
•
The
Generalized Complex Fourier Transform

1.4TheDiscreteTimeFourierTransform
Properties ofthe Discrete Time FourierTransform
•
Relation-
ship between the Continuous and Discrete Time Spectra
1.5TheDiscreteFourierTransform
Properties of the Discrete Fourier Series
•
Fourier Block Pro-
cessing in Real-Time Filtering Applications
•
Fast Fourier
Transform Algorithms
1.6FamilyTreeofFourierTransforms
1.7SelectedApplicationsofFourierMethods
Fast Fourier Transform in Spectral Analysis
•
Finite Impulse
Response Digital Filter Design
•
Fourier Analysis of Ideal and
Practical Digital-to-Analog Conversion
1.8Summary
References
1.1 Introduction
Fourier methods are commonly used for signal analysis and system design in modern telecommu-
nications, radar, and image processing systems. ClassicalFourier methods such as the Fourier series
andtheFourier integral areusedforcontinuoustime(CT)signalsandsystems,i.e.,systemsinwhich
a characteristic signal, s(t),isdefinedatallvaluesoft on the continuum −∞ <t<∞ .Amore
recentlydevelopedsetofFouriermethods,includingthediscretetimeFouriertransform(DTFT)and

the discrete Fourier transform (DFT),are extensions of basic Fourier concepts that apply to discrete
time (DT) signals. A characteristic DT signal, s[n], is defined only for values of n where n is an
integer in the range −∞ <n<∞. The following discussion presents basic concepts and outlines
important properties forboth the CT and DT classes ofFouriermethods, with aparticular emphasis
ontherelationshipsbetweenthese twoclasses. The classofDT Fouriermethodsisparticularly useful
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asabasisfordigitalsignal processing(DSP)becauseit extendsthetheory of classical Fourieranalysis
to DT signals and leads to many effective algorithms that can be directly implemented on general
computers or special pur pose DSP devices.
TherelationshipbetweentheCTandtheDTdomainsischaracterizedbytheoperationsofsampling
and reconstruction. If s
a
(t) denotes a signal s(t) that has been uniformly sampled every T seconds,
then the mathematical representation of s
a
(t) is given by
s
a
(t) =
∞

n=−∞
s(t)δ(t − nT ) (1.1)
where δ(t) is a CT impulse function defined to be zero for all t = 0, undefined at t = 0, and has
unit area when integ rated from t =−∞to t =+∞. Because the only places at which the product
s(t)δ(t−nT ) isnotidenticallyequaltozeroareatthesamplinginstances,s(t)in(1.1)canbereplaced
with s(nT ) without changing the overall meaning of the expression. Hence, an alternate expression
for s

a
(t) that is often useful in Fourier analysis is given by
s
a
(t) =
∞

n=−∞
s(nT )δ(t − nT ) (1.2)
The CT sampling model s
a
(t) consists of a sequence of CT impulse functions uniformly spaced at
intervalsofT secondsandweightedbythevaluesofthesignals(t)atthesamplinginstants,asdepicted
in Fig. 1.1. Note that s
a
(t) is not defined at the sampling instants because the CT impulse function
itself is not defined at t = 0. However, the values of s(t) at the sampling instants are imbedded as
“area under the curve” of s
a
(t), and as such represent a useful mathematical model of the sampling
process. In the DT domain the sampling model is simply the sequence defined by taking the values
of s(t) at the sampling instants, i.e.,
s[n]=s(t)|
t=nT
(1.3)
Incontrast to s
a
(t), which is notdefinedatthesamplinginstants,s[n] iswelldefinedatthesampling
instants, as illustratedin Fig. 1.2. Thus, it is nowclearthats
a

(t) ands[n] aredifferent but equivalent
models of the sampling process in the CT and DT domains, respectively. They are both useful for
signal analysis in their corresponding domains. Their equivalenceis established by the fact that they
have equal spectra in the Fourier domain, and that the underlying CT signal from which s
a
(t) and
s[n] are derived can be recovered from either sampling representation, provided a sufficiently large
sampling rate is used in the sampling operation (see below).
1.2 Fourier Series Representation of Continuous Time Periodic
Signals
It is convenient to begin this discussion with the classical Fourier series representation of a p eriodic
timedomainsignal,andthenderive the Fourierintegralfromthisrepresentationbyfinding the limit
of the Fourier coefficient representationas the period goes toinfinity. The conditionsunderwhicha
periodic signal s(t) can be expanded in a Fourier series are known as the Dirichet conditions. They
require that in each period s(t) has a finite number of discontinuities, a finite number of maxima
and minima, and that s(t) satisfies the following absolute convergence criterion [1]:

T/2
−T/2
|s(t)| dt < ∞ (1.4)
Itis assumed in the following discussion that these basic conditions are satisfied byall functions that
will be represented by a Fourier ser ies.
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FIGURE1.1:CTmodelofasampledCTsignal.
FIGURE1.2:DTmodelofasampledCTsignal.
1.2.1 ExponentialFourierSeries
IfaCTsignals(t)isperiodicwithaperiodT,thentheclassicalcomplexFourierseriesrepresentation
ofs(t)isgivenby

s(t)=
∞

n=−∞
a
n
e
jnω
0
t
(1.5a)
whereω
0
=2π/T,andwherethea
n
arethecomplexFouriercoefficientsgivenby
a
n
=(1/T)

T/2
−T/2
s(t)e
−jnω
0
t
dt (1.5b)
Itiswellknownthatforeveryvalueoftwheres(t)iscontinuous,theright-handsideof(1.5a)
convergestos(t).Atvaluesoftwheres(t)hasafinitejumpdiscontinuity,theright-handside
of(1.5a)convergestotheaverageofs(t

−
)ands(t
+
),wheres(t
−
)≡lim
→0
s(t−)ands(t
+
)≡
lim
→0
s(t+).
Forexample,theFourierseriesexpansionofthesawtoothwaveformillustratedinFig.1.3ischar-
acterizedbyT=2π,ω
0
=1,a
0
=0,anda
n
=a
−n
=Acos(nπ)/(jnπ)forn=1,2, ,.The
coefficientsoftheexponentialFourierseriesrepresentedby(1.5b)canbeinterpretedasthespec-
tralrepresentationofs(t),becausethea
n
-thcoefficientrepresentsthecontributionofthe(nω
0
)-th
frequencytothetotalsignals(t).Becausethea

n
arecomplexvalued,theFourierdomainrepresen-
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tationhasbothamagnitudeandaphasespectrum.Forexample,themagnitudeofthea
n
isplotted
inFig.1.4forthesawtoothwaveformofFig.1.3.Thefactthatthea
n
constituteadiscretesetis
consistentwiththefactthataperiodicsignalhasa“linespectrum,”i.e.,thespectrumcontainsonly
integermultiplesofthefundamentalfrequencyω
0
.Therefore,theequationpairgivenby(1.5a)
and(1.5b)canbeinterpretedasatransformpairthatissimilartotheCTFouriertransformfor
periodicsignals.ThisleadstotheobservationthattheclassicalFourierseriescanbeinterpreted
asaspecialtransformthatprovidesaone-to-oneinvertiblemappingbetweenthediscrete-spectral
domainandtheCTdomain.Thenextsectionshowshowtheperiodicityconstraintcanberemoved
toproducethemoregeneralclassicalCTFouriertransform,whichappliesequallywelltoperiodic
andaperiodictimedomainwaveforms.
FIGURE1.3:PeriodicCTsignalusedinFourierseriesexample.
FIGURE1.4:MagnitudeoftheFouriercoefficientsforexampleofFigure1.3.
1.2.2 TheTrigonometricFourierSeries
AlthoughFourierseriesexpansionsexistforcomplexperiodicsignals,andFouriertheorycanbe
generalizedtothecaseofcomplexsignals,thetheoryandresultsaremoreeasilyexpressedforreal-
valuedsignals.Thefollowingdiscussionassumesthatthesignals(t)isreal-valuedforthesakeof
simplifyingthediscussion.However,allresultsarevalidforcomplexsignals,althoughthedetailsof
thetheorywillbecomesomewhatmorecomplicated.
Forreal-valuedsignalss(t),itispossibletomanipulatethecomplexexponentialformoftheFourier

seriesintoatrigonometricformthatcontainssin(ω
0
t)andcos(ω
0
t)termswithcorrespondingreal-
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valued coefficients [1]. The trigonometric form of the Fourier series for a real-valued signal s(t) is
given by
s(t) =
∞

n=0
b
n
cos(nω
0
t) +
∞

n=1
c
n
sin(nω
0
t) (1.6a)
where ω
0
= 2π/T .Theb

n
and c
n
are real-valued Fourier coefficients determined by
FIGURE 1.5: Periodic CT signal used in Fourier series example 2.
FIGURE 1.6: Fourier coefficients for example of Figure 1.5.
b
0
= (1/T )

T/2
−T/2
s(t)dt
b
n
= (2/T )

T/2
−T/2
s(t)cos(nω
0
t)dt, n = 1, 2, , (1.6b)
c
n
= (2/T )

T/2
−T/2
s(t)sin(nω
0

t)dt, n = 1, 2, ,
An arbitrary real-valued signal s(t) can be expressed as a sum of even and odd components, s(t) =
s
even
(t) + s
odd
(t),wheres
even
(t) = s
even
(−t) and s
odd
(t) =−s
odd
(−t), and where s
even
(t) =
[s(t) + s(−t)]/2 and s
odd
(t) =[s(t) − s(−t)]/2. For the trigonometric Fourier series, it can be
shownthats
even
(t)isrepresentedbythe(e ven)cosinetermsintheinfiniteseries,s
odd
(t)isrepresented
by the (odd) sine terms, and b
0
is the DC level of the signal. Therefore, if it can be determined by
inspectionthata signalhasDClevel,orifit isevenorodd,thenthecorrectformof thetrigonometric
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seriescanbechosentosimplifytheanalysis.Forexample,itiseasilyseenthatthesignalshownin
Fig.1.5isanevensignalwithazeroDClevel.Thereforeitcanbeaccuratelyrepresentedbythecosine
serieswithb
n
=2Asin(πn/2)/(πn/2),n=1,2, ,asillustratedinFig.1.6.Incontrast,notethat
thesawtoothwaveformusedinthepreviousexampleisanoddsignalwithzeroDClevel;thus,itcan
becompletelyspecifiedbythesinetermsofthetrigonometricseries.Thisresultcanbedemonstrated
bypairingeachpositivefrequencycomponentfromtheexponentialserieswithitsconjugatepartner,
i.e.,c
n
=sin(nω
0
t)=a
n
e
jnω
0
t
+a
−n
e
−jnω
0
t
,wherebyitisfoundthatc
n
=2Acos(nπ)/(nπ)for
thisexample.Ingeneralitisfoundthata

n
=(b
n
−jc
n
)/2forn=1,2, ,a
0
=b
0
,anda
−n
=a
∗
n
.
ThetrigonometricFourierseriesiscommoninthesignalprocessingliteraturebecauseitreplaces
complexcoefficientswithrealonesandoftenresultsinasimplerandmoreintuitiveinterpretation
oftheresults.
1.2.3 ConvergenceoftheFourierSeries
TheFourierseriesrepresentationofaperiodicsignalisanapproximationthatexhibitsmeansquared
convergencetothetruesignal.Ifs(t)isaperiodicsignalofperiodT,ands

(t)denotestheFourier
seriesapproximationofs(t),thens(t)ands

(t)areequalinthemeansquaresenseif
MSE=

T/2
−T/2

|s(t)−s(t)

|
2
dt=0 (1.7)
Evenwith(1.7)satisfied,meansquareerror(MSE)convergencedoesnotmeanthats(t)=s

(t)
ateveryvalueoft.Inparticular,itisknownthatatvaluesoft,wheres(t)isdiscontinuous,the
Fourierseriesconvergestotheaverageofthelimitingvaluestotheleftandrightofthediscontinuity.
Forexample,ift
0
isapointofdiscontinuity,thens

(t
0
)=[s(t
−
0
)+s(t
+
0
)]/2,wheres(t
−
0
)and
s(t
+
0
)weredefinedpreviously.(Notethatatpointsofcontinuity,thisconditionisalsosatisfiedby

thedefinitionofcontinuity.)BecausetheDirichetconditionsrequirethats(t)haveatmostafinite
numberofpointsofdiscontinuityinoneperiod,thesetS
t
,definedasallvaluesoftwithinone
periodwheres(t)=s

(t),containsafinitenumberofpoints,andS
t
isasetofmeasurezerointhe
formalmathematicalsense.Therefore,s(t)anditsFourierseriesexpansions

(t)areequalalmost
everywhere,ands(t)canbeconsideredidenticaltos

(t)fortheanalysisofmostpracticalengineering
problems.
Convergencealmosteverywhereissatisfiedonlyinthelimitasaninfinitenumberoftermsare
includedintheFourierseriesexpansion.IftheinfiniteseriesexpansionoftheFourierseriesis
truncatedtoafinitenumberofterms,asitmustbeinpracticalapplications,thentheapproximation
willexhibitanoscillatorybehavioraroundthediscontinuity,knownastheGibbsphenomenon[1].
Lets

N
(t)denoteatruncatedFourierseriesapproximationofs(t),whereonlythetermsin(1.5a)
fromn=−Nton=NareincludedifthecomplexFourierseriesrepresentationisused,orwhere
onlythetermsin(1.6a)fromn=0ton=NareincludedifthetrigonometricformoftheFourier
seriesisused.Itiswellknownthatinthevicinityofadiscontinuityatt
0
theGibbsphenomenon
causess


N
(t)tobeapoorapproximationtos(t).ThepeakmagnitudeoftheGibbsoscillationis13%
ofthesizeofthejumpdiscontinuitys(t
−
0
)−s(t
+
0
)regardlessofthenumberoftermsusedinthe
approximation.AsNincreases,theregionthatcontainstheoscillationbecomesmoreconcentrated
intheneighborhoodofthediscontinuity,until,inthelimitasNapproachesinfinity,theGibbs
oscillationissqueezedintoasinglepointofmismatchatt
0
.
Ifs

(t)isreplacedbys

N
(t)in(1.7),itisimportanttounderstandthebehavioroftheerrorMSE
N
asafunctionofN,where
MSE
N
=

T/2
−T/2
|s(t)−s


N
(t)|
2
dt (1.8)
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AnimportantpropertyoftheFourierseriesisthattheexponentialbasisfunctionse
jnω
0
t
(orsin(nω
0
t)
andcos(nω
0
t)forthetrigonometricform )forn=0,±1,±2, (orn=0,1,2, forthe
trigonometricform)constituteanorthonormalset,i.e.,t
nk
=1forn=k,andt
nk
=0forn=k,
where
t
nk
=(1/T)

T/2
−T/2

(e
−jnω
0
t
)(e
jkω
0
t
)dt (1.9)
AstermsareaddedtotheFourierseriesexpansion,theorthogonalityofthebasisfunctionsguarantees
thattheerrordecreasesinthemeansquaresense,i.e.,thatMSE
N
monotonicallydecreasesasNis
increased.Therefore,apractitionercanproceedwiththeconfidencethatwhenapplyingFourierseries
analysismoretermsarealwaysbetterthanfewerintermsoftheaccuracyofthesignalrepresentations.
1.3 TheClassicalFourierTransformforContinuous
TimeSignals
TheperiodicityconstraintimposedontheFourierseriesrepresentationcanberemovedbytakingthe
limitsof(1.5a)and(1.5b)astheperiodTisincreasedtoinfinity.Somemathematicalpreliminaries
arerequiredsothattheresultswillbewelldefinedafterthelimitistaken.Itisconvenienttoremove
the(1/T)factorinfrontoftheintegralbymultiplying(1.5b)throughbyT,andthenreplacing
Ta
n
bya

n
inboth(1.5a)and(1.5b).Becauseω
0
=2π/T,asTincreasestoinfinity,ω
0

becomes
infinitesimallysmall,aconditionthatisdenotedbyreplacingω
0
withω.Thefactor(1/T)in(1.5a)
becomes(ω/2π).Withthesealgebraicmanipulationsandchangesinnotation(1.5a)and(1.5b)
takeonthefollowingformpriortotakingthelimit:
s(t)= (1/2π)
∞

n=−∞
a

n
e
jnωt
ω (1.10a)
a

n
=

T/2
−T/2
s(t)e
−jnωt
dt (1.10b)
ThefinalstepinobtainingtheCTFouriertransformistotakethelimitofboth(1.10a)and(1.10b)
asT→∞.Inthelimittheinfinitesummationin(1.10a)becomesanintegral,ωbecomesdω,
nωbecomesω,anda


n
becomestheCTFouriertransformofs(t),denotedbyS(jω).Theresult
issummarizedbythefollowingtransformpair,whichisknownthroughoutmostoftheengineering
literatureastheclassicalCTFouriertransform(CTFT):
s(t)= (1/2π)

∞
−∞
S(jω)e
jωt
dω (1.11a)
S(jω)=

∞
−∞
s(t)e
−jωt
dt (1.11b)
Often(1.11a\)iscalledtheFourierintegraland(1.11b)issimplycalledtheFouriertransform.The
relationshipS(jω)=F{s(t)}denotestheFouriertransformationofs(t),whereF{·}isasymbolic
notationfortheFouriertransformoperator,andwhereωbecomesthecontinuousfrequencyvariable
aftertheperiodicityconstraintisremoved.Atransformpairs(t)↔S(jω)representsaone-to-
oneinvertiblemappingaslongass(t)satisfiesconditionswhichguaranteethattheFourierintegral
converges.
From(1.11a)itiseasilyseenthatF{δ(t−t
0
)}=e
−jωt
0
,andfrom(1.11b)thatF

−1
{2πδ(ω−
ω
0
)}=e
jω
0
t
,sothatδ(t−t
0
)↔e
−jωt
0
ande
jω
0
t
↔2πδ(ω−ω
0
)arevalidFouriertransform
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pairs. UsingtheserelationshipsitiseasytoestablishtheFouriertransformsofcos(ω
0
t)andsin(ω
0
t),
as well as many other useful waveforms that are encountered in common signal analysis problems.
A number of such transforms are shown in Table 1.1.

The CTFT is useful in the analysis and design of CT systems, i.e., systems that process CT signals.
FourieranalysisisparticularlyapplicabletothedesignofCTfilterswhicharecharacterizedbyFourier
magnitude and phase spectra, i.e., by |H(jω)| and arg H(jω),whereH(jω) is commonly called
the frequency response of the filter. For example, an idealtransmissionchannel isone which passes
a signal without distorting it. The signal may be scaled by a real constant A and delayed by a fixed
time increment t
0
, implying that the impulse response of an ideal channel is Aδ(t − t
0
), and its
corresponding frequency response is Ae
−jωt
0
. Hence, the frequency response of an ideal channel is
specifiedbyconstantamplitudeforallfrequencies,anda phasecharacteristicwhichislinearfunction
given by ωt
0
.
1.3.1 Properties of the Continuous Time Fourier Transform
The CTFT has many properties that make it useful for the analysis and design of linear CT systems.
Some of the more useful properties are stated below. Amore complete list of the CTFT properties is
giveninTable1.2. Proofs of these properties can be found in [2] and [3]. In the following discus-
sion F {·} denotes the Fourier transform operation, F
−1
{·} denotes the inverse Fourier transform
operation, and ∗ denotes the convolution operation defined as
f
1
(t) ∗ f
2

(t) =

∞
−∞
f
1
(t − τ)f
2
(τ ) dτ
1. Linearity (superposition): F {af
1
(t) + bf
2
(t)}=aF {f
1
(t)}+bF{f
2
(t)}
(a and b, complex constants)
2. Time shifting: F{f(t − t
0
)}=e
−jωt
0
F {f(t)}
3. Frequency shifting: e
jω
0
t
f(t)= F

−1
{F(j(ω− ω
0
))}
4. Time domain convolution: F {f
1
(t) ∗ f
2
(t)}=F {f
1
(t)}F {f
2
(t)}
5. Frequency domain convolution: F{f
1
(t)f
2
(t)}=(1/2π)F {f
1
(t)}∗F {f
2
(t)}
6. Time differentiation: −jωF(jω) = F{d(f (t))/dt}
7. Time integration: F{

t
−∞
f(τ)dτ}=(1/j ω)F (j ω ) + πF(0)δ(ω)
The above properties are particularly useful in CT system analysis and design, especially when the
system characteristics are easily specified in the frequency domain, as in linear filtering. Note that

properties 1, 6, and7 areuseful for solving differential or integralequations. Property 4 provides the
basis for many signal processing algorithms because many systems can be specified directly by their
impulseorfrequencyresponse. Property3isparticularlyusefulinanalyzingcommunicationsystems
inwhichdifferentmodulationfor matsarecommonlyusedtoshiftspectralenergytofrequencybands
that are appropriate for the application.
1.3.2 Fourier Spectrum of the Continuous Time Sampling Model
Because the CT sampling model s
a
(t), given in (1.1), is in its ownright a CT signal, it is appropriate
to apply the CTFT to obtain an expression for the spectrum of the sampled signal:
F {s
a
(t)}=F

∞

n=−∞
s(t)δ(t − nT )

=
∞

n=−∞
s(nT )e
−jωTn
(1.12)
Becausetheexpressionontheright-hand sideof(1.12)isafunctionofe
jωT
itiscustomarytodenote
the transform as F(e

jωT
) = F {s
a
(t)}. Later in the chapter this result is compared to the result of
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TABLE1.1 SomeBasicCTFTPairs
FourierSeriesCoefficients
Signal FourierTransform (ifperiodic)
+∞

k=−∞
a
k
e
jkω
0
t
2π
+∞

k=−∞
a
k
δ(ω
k
ω
0
)a

k
e
jω
0
t
2πδ(ω+ω
0
)
a
1
=1
a
k
=0, otherwise
cosω
0
tπ[δ(ω−ω
0
)+δ(ω+ω
0
)]
a
1
=a
−1
=
1
2
a
k

=0, otherwise
sinω
0
t
π
j
[δ(ω−ω
0
)−δ(ω+ω
0
)]
a
1
=−a
−1
=
1
2j
a
k
=0, otherwise
x(t)=12πδ(ω)
a
0
=1,a
k
=0,k=0

hasthisFourierseriesrepresentationforany
choiceof

T
0
>0

Periodicsquarewave
x(t)=



1, |t|<T
1
0,T
1
<|t|≤
T
0
2
+∞

k=−∞
2sinkω
0
T
1
k
δ(ω
k
ω
0
)

ω
0
T
1
π
sinc

kω
0
T
1
π

=
sinkω
0
T
1
kπ
and
x(t+T
0
)=x(t)
+∞

n=−∞
δ(t−nT)
2π
T
+∞


k=−∞δ

ω−
2πk
T

a
k
=
1
T
forallk
x(t)=

1, |t|<T
1
0, |t|>T
1
2T
1
sinc

ωT
1
π

=
2sinωT
1

ω
—
W
π
sinc

Wt
π

=
sinWt
πt
X(ω)=

1, |ω|<W
0, |ω|>W
—
δ(t) 1—
u(t)
1
jω
+πδ(ω)
—
δ(t−t
0
)e
jωt
0
—
e

−at
u(t),Re{a}>0
1
a+jω
—
te
−at
u(t),Re{a}>0
1
(a+jω)
2
—
t
n−1
(n−1)!
e
−at
u(t),
Re{a}>0
1
(a+jω)
n
—
c
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TABLE1.2 PropertiesoftheCTFT
Name IfFf(t)=F(jω),then
Definition f(jω)=


∞
−∞
f(t)e
jωt
dt
f(t)=
1
2π

∞
−∞
F(jω)e
jωt
dω
Superposition F[af
1
(t)+bf
2
(t)]=aF
1
(jω)+bF
2
(jω)
Simplificationif:
(a)
f(t)iseven F(jω)=2

∞
0
f(t)cosωtdt

(b)f(t)isodd F(jω)=2j

∞
0
f(t)sinωtdt
Negativet Ff(−t)=F
∗
(jω)
Scaling:
(a)Time F
f(at)=
1
|a|
F

jω
a

(b)Magnitude Faf(t)=aF(jω)
Differentiation F

d
n
dt
n
f(t)

=(jω)
n
F(jω)

Integration F


t
−∞
f(x)dx

=
1
jω
F(jω)+πF(0)δ(ω)
Timeshifting Ff(t−a)=F(jω)e
jωa
Modulation Ff(t)e
jω
0
t
=F[j(ω−ω
0
)]
{
Ff(t)cosω
0
t=
1
2
F[j(ω−ω
0
)]+F[j(ω+ω
0

)]}
{
Ff(t)sinω
0
t=
1
2
j[F[j(ω−ω
0
)]−F[j(ω+ω
0
)]}
Timeconvolution F
−1
[F
1
(jω)F
2
(jω)]=

∞
−∞
f
1
(τ)f
2
(τ)f
2
(t
τ

)dτ
Frequencyconvolution F[f
1
(t)f
2
(t)]=
1
2π

∞
−∞
F
1
(jλ)F
2
[j(ω
λ
)]dλ
operatingontheDTsamplingmodel,namelys[n],withtheDTFouriertransformtoillustratethat
thetwosamplingmodelshavethesamespectrum.
1.3.3 FourierTransformofPeriodicContinuousTimeSignals
WesawearlierthataperiodicCTsignalcanbeexpressedintermsofitsFourierseries.TheCTFT
canthenbeappliedtotheFourierseriesrepresentationofs(t)toproduceamathematicalexpression
forthe“linespectrum”characteristicofperiodicsignals.
F{s(t)}=F

∞

n=−∞
a

n
e
jnω
0
t

=2π
∞

n=−∞
a
n
δ(ω−nω
0
) (1.13)
ThespectrumisshownpictoriallyinFig.1.7.Notethesimilaritybetweenthespectralrepresentation
ofFig.1.7andtheplotoftheFouriercoefficientsinFig.1.4,whichwasheuristicallyinterpreted
asa“linespectrum”.Figures1.4and1.7aredifferentbutequivalentrepresentationsoftheFourier
c
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spectrum.NotethatFig.1.4isaDTrepresentationofthespectrum,whileFig.1.7isaCTmodelof
thesamespectrum.
FIGURE1.7:SpectrumoftheFourierseriesrepresentationofs(t).
1.3.4 TheGeneralizedComplexFourierTransform
TheCTFTcharacterizedby(1.11a)and(1.11b)canbegeneralizedbyconsideringthevariablejω
tobethespecialcaseofu=σ+jωwithσ=0,writing(1.11a)intermsofu,andinterpretingu
asacomplexfrequencyvariable.TheresultingcomplexFouriertransformpairisgivenby(1.14a)
and(1.14b)
s(t)= (1/2πj)


σ+j∞
σ−j∞
S(u)e
jut
du (1.14a)
S(u) =

∞
−∞
s(t)e
−jut
dt (1.14b)
Thesetofallvaluesofuforwhichtheintegralof(1.14b)convergesiscalledtheregionofconvergence
(ROC).BecausethetransformS(u)isdefinedonlyforvaluesofuwithintheROC,thepathof
integrationin(1.14a)mustbedefinedbyσsothattheentirepathlieswithintheROC.Insome
literaturethistransformpairiscalledthebilateralLaplacetransformbecauseitisthesameresult
obtainedbyincludingboththenegativeandpositiveportionsofthetimeaxisintheclassicalLaplace
transformintegral.[Notethatin(1.14a)thecomplexfrequencyvariablewasdenotedbyurather
thanbythemorecommons,inordertoavoidconfusionwithearlierusesofs(·)assignalnotation.]
ThecomplexFouriertransform(bilateralLaplacetransform)isnotoftenusedinsolvingpractical
problems,butitssignificanceliesinthefactthatitisthemostgeneralformthatrepresentsthepoint
atwhichFourierandLaplacetransformconceptsbecomethesame.Identifyingthisconnection
reinforcesthenotionthatFourierandLaplacetransformconceptsaresimilarbecausetheyarederived
byplacingdifferentconstraintsonthesamegeneralform.
1.4 TheDiscreteTimeFourierTransform
ThediscretetimeFouriertransform(DTFT)canbeobtainedbyusingtheDTsamplingmodeland
consideringtherelationshipobtainedin(1.12)tobethedefinitionoftheDTFT.LettingT=1so
thatthesamplingperiodisremovedfromtheequationsandthefrequencyvariableisreplacedwith
c

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1999byCRCPressLLC
anormalizedfrequencyω

=ωT,theDTFTpairisdefinedin(1.15a).Notethatinordertosimplify
notationitisnotcustomarytodistinguishbetweenωandω

,butrathertorelyonthecontextofthe
discussiontodeterminewhetherωreferstothenormalized(T=1)ortheunnormalized(T=1)
frequencyvariable.
S(e
jω

) =
∞

n=−∞
s[n]e
−jω

n
(1.15a)
s[n]=(1/2π)

π
−π
S(e
jω

)e

jnω

dω

(1.15b)
ThespectrumS(e
jω

)isperiodicinω

withperiod2π.Thefundamentalperiodintherange
−π<ω

≤π,sometimesreferredtoasthebaseband,istheusefulfrequencyrangeoftheDTsystem
becausefrequencycomponentsinthisrangecanberepresentedunambiguouslyinsampledform
(withoutaliasingerror).Inmuchofthesignalprocessingliteraturetheexplicitprimednotationis
omittedfromthefrequencyvariable.However,theexplicitprimednotationwillbeusedthroughout
thissectionbecausethepotentialexistsforconfusionwhensomanyrelatedFourierconceptsare
discussedwithinthesameframework.
Bycomparing(1.12)and(1.15a),andnotingthatω

=ωT,itisestablishedthat
F{s
a
(t)}=DTFT{s[n]} (1.16)
wheres[n]=s(t)
t=nT
.Thisdemonstratesthatthespectrumofs
a
(t),ascalculatedbytheCTFourier

transformisidenticaltothespectrumofs[n]ascalculatedbytheDTFT.Therefore,althoughs
a
(t)
ands[n]arequitedifferentsamplingmodels,theyareequivalentinthesensethattheyhavethesame
Fourierdomainrepresentation.
AlistofcommonDTFTpairsispresentedinTable1.3.JustastheCTFouriertransformisuseful
inCTsignalsystemanalysisanddesign,theDTFTisequallyusefulinthesamecapacityforDT
systems.ItisindeedfortuitousthatFouriertransformtheorycanbeextendedinthiswaytoapply
toDTsystems.
InthesamewaythattheCTFouriertransformwasfoundtobeaspecialcaseofthecomplexFourier
transform(orbilateralLaplacetransform),theDTFTisaspecialcaseofthebilateralz-transform
withz=e
jω

t
.Themoregeneralbilateralz-transformisgivenb y
S(z)=
∞

n=−∞
s[n]z
−n
(1.17a)
s[n]=(1/2πj)

C
S(z)z
n−1
dz (1.17b)
whereCisacounterclockwisecontourofintegrationwhichisaclosedpathcompletelycontained

withintheregionofconvergenceofS(z).RecallthattheDTFTwasobtainedbytakingtheCTFourier
transformoftheCTsamplingmodelrepresentedbys
a
(t).Similarly,thebilateralz-transformresults
bytakingthebilateralLaplacetransformofs
a
(t).Ifthelowerlimitonthesummationof(1.17a)is
takentoben=0,then(1.17a)and(1.17b)becometheone-sidedz-transform,whichistheDT
equivalentoftheone-sidedLTforCTsignals.Thehierarchicalrelationshipamongthesevarious
conceptsforDTsystemsisdiscussedlaterinthischapter,whereitwillbeshownthatthefamily
structureoftheDTfamilytreeisidenticaltothatoftheCTfamily.ForeveryCTtransforminthe
CTworldthereisananalogousDTtransformintheDTworld,andviceversa.
c
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TABLE1.3 SomeBasicDTFTPairs
Sequence FourierTransform
1.δ[n] 1
2.
δ[n−n
0
] e
−jωn
0
3.1 (−∞<n<∞)
∞

k=−∞
2πδ(ω+2πk)
4.a

n
u[n] (|a|<1)
1
1−ae
−jω
5.u[n]
1
1−e
−jω
+
∞

k=−∞
πδ(ω+2πk)
6.(n+1)a
n
u[n] (|a|<1)
1
(1−ae
−jω
)
2
7.
r
2
sinω
p
(n+1)
sinω
p

u[n] (|r|<1)
1
1−2rcosω
p
e
−jω
+r
2
e
j2ω
8.
sinω
c
n
πn
Xe
jω
=

1, |ω|<ω
c
0,ω
c
<|ω|≤π
9.x[n]−

1, 0≤n≤M
0,
otherwise
sin[ω(M+1)/2]

sin(ω/2)
e
−jωM/2
10.e
jω
0
n
∞

k=−∞
2πδ(ω−ω
0
+2πk)
11.cos(ω
0
n+φ) π
∞

k=−∞
[e
jφ
δ(ω−ω
0
+2πk)+e
−jφ
δ(ω+ω
0
+2πk)]
1.4.1 PropertiesoftheDiscreteTimeFourierTransform
BecausetheDTFTisacloserelativeoftheclassicalCTFouriertransformitshouldcomeasnosurprise

thatmanypropertiesoftheDTFTaresimilartothosepresentedfortheCTFouriertransforminthe
previoussection.Infact,formanyofthepropertiespresentedearlierananalogouspropertyexists
fortheDTFT.ThefollowinglistparallelsthelistthatwaspresentedintheprevioussectionfortheCT
Fouriertransform,totheextentthatthesamepropertyexists.AmorecompletelistofDTFTpairsis
giveninTable1.4.(Notethattheprimednotationonω

isdroppedinthefollowingtosimplifythe
notation,andtobeconsistentwithstandardusage.)
1.Linearity(superposition):DTFT{af
1
[n]+bf
2
[n]}=aDTFT{f
1
[n]}+bDTFT{f
2
[n]}
(aandb,complexconstants)
2.Indexshifting:DTFT{f[n−n
0
]}=e
−jωn
0
DTFT{f[n]}
3.Frequencyshifting:e
jω
0
n
f[n]=DTFT
−1

{F(e
j(ω−ω
0
)
)}
4.Timedomainconvolution:DTFT{f
1
[n]∗f
2
[n]}=DTFT{f
1
[n]}DTFT{f
2
[n]}
5.Frequencydomainconvolution:DTFT{f
1
[n]f
2
[n]}=(1/2π)DTFT{f
1
[n]}∗DTFT{f
2
[n]}
6.Frequencydifferentiation:nf[n]=DTFT
−1
{dF(e
jω
)/dω}
Notethatthetime-differentiationandtime-integrationpropertiesoftheCTFTdonothaveanalogous
counterpartsintheDTFTbecausetimedomaindifferentiationandintegrationarenotdefinedforDT

c
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TABLE1.4 PropertiesoftheDTFT
Sequence FourierTransform
x[n] X(e
jω
)
y[n] Y(e
jω
)
1.ax[n]+by[n] aX(e
jω
)+bY(e
jω
)
2.x[n−n
d
] (n
d
aninteger) e
−jωn
d
X(e
jω
)
3.e
jω
0
n

x[n] X(e
j(ω−ω
0
)
)
4.x[−n] X(e
−jω
) ifx[n]isreal
X
∗
(e
jω
)
5.nx[n] j
dX(e
jω
)
dω
6.x[n]∗y[n] X(e
jω
)Y(e
jω
)
7.x[n]y[n]
1
2π

x
−x
X(e

jθ
)Y(e
j(ω−θ)
)dθ
Parseval’sTheorem
8.
∞

n=−∞
|x[n]|
2
=
1
2π

π
−π
|X(e
jω
)|
2
dω
9.
∞

n=−∞
x[n]y
∗
[n]=
1

2π
π
inf
−π
X(e
jω
)Y
∗
(e
jω
)dω
signals.WhenworkingwithDTsystemspractitionersmustoftenmanipulatedifferenceequations
inthefrequencydomain.Forthispurposeproperty1andproperty2areveryimportant.Aswith
theCTFT,property4isveryimportantforDTsystemsbecauseitallowsengineerstoworkwith
thefrequencyresponseofthesystem,inordertoachievepropershapingoftheinputspectrumor
toachievefrequencyselectivefilteringfornoisereductionorsignaldetection.Also,property3is
usefulfortheanalysisofmodulationandfilteringoperationscommoninbothanaloganddigital
communicationsystems.
TheDTFTisdefinedsothatthetimedomainisdiscreteandthefrequencydomainiscontinuous.
ThisisincontrasttotheCTFTthatisdefinedtohavecontinuoustimeandcontinuousfrequency
domains.ThemathematicaldualoftheDTFTalsoexists,whichisatransformpairthathasa
continuoustimedomainandadiscretefrequencydomain.Infact,thedualconceptisreallythe
sameastheFourierseriesforperiodicCTsignalspresentedearlierinthechapter,asrepresented
by(1.5a)and(1.5b).However,theclassicalFourierseriesarisesfromtheassumptionthattheCT
signalisinherentlyperiodic,asopposedtothetimedomainbecomingperiodicbyvirtueofsampling
thespectrumofacontinuousfrequency(aperiodictime)function[8].ThedualoftheDTFT,the
discretefrequencyFouriertransform(DFFT),hasbeenformulatedanditspropertiestabulatedas
aninterestingandusefultransforminitsownright[5].AlthoughtheDFFTissimilarinconcept
totheclassicalCTFourierseries,theformalpropertiesoftheDFFT[5]servetoclarifytheeffects
offrequencydomainsamplingandtimedomainaliasing.Theseeffectsareobscuredintheclassical

treatmentoftheCTFourierseriesbecausetheemphasisisontheinherent“linespectrum”that
resultsfromtimedomainperiodicity.TheDFFTisusefulfortheanalysisanddesignofdigitalfilters
thatareproducedbyfrequencysamplingtechniques.
1.4.2 RelationshipbetweentheContinuousandDiscreteTimeSpectra
BecauseDTsignalsoftenoriginatebysamplingCTsignals,itisimportanttodeveloptherelationship
betweentheoriginalspectrumoftheCTsignalandthespectrumoftheDTsignalthatresults.First,
c
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1999byCRCPressLLC
theCTFTisappliedtotheCTsamplingmodel,andthepropertieslistedaboveareusedtoproduce
thefollowingresult:
F{s
a
(t)}=F

s(t)
∞

n=−∞
δ(t−nT)

= (1/2π)S(jω)∗F

∞

n=−∞
δ(t−nT)

(1.18)
Inthissectionitisimportanttodistinguishbetweenωandω


,sotheexplicitprimednotation
isusedinthefollowingdiscussionwhereneededforclarification.Becausethesamplingfunction
(summationofshiftedimpulses)ontheright-handsideoftheaboveequationisperiodicwithperiod
TitcanbereplacedwithaCTFourierseriesexpansionasfollows:
S(e
jωT
)=F{s
a
(t)}=(1/2π)S(jω)∗F

∞

n=−∞
(1/T)e
j(2π/T)nt

ApplyingthefrequencydomainconvolutionpropertyoftheCTFTyields
S(e
jωT
)=(1/2π)
∞

n=−∞
S(jω)∗(2π/T)δ(ω−(2π/T)n)
Theresultis
S(e
jωT
)=(1/T)
∞


n=−∞
S(j[ω−(2π/T)n])=(1/T)
∞

n=−∞
S(j[ω−nω
s
]) (1.19a)
whereω
s
=(2π/T)isthesamplingfrequencyexpressedinradianspersecond.Analternateform
fortheexpressionof(1.19a)is
S(e
jω

)=(1/T)
∞

n=−∞
S(j[(ω

−n2π)/T]) (1.19b)
whereω

=ωTisthenormalizedDTfrequencyaxisexpressedinradians.NotethatS(e
jωT
)=
S(e
jω


)consistsofaninfinitenumberofreplicasoftheCTspectrumS(jω),positionedatintervals
of(2π/T)ontheωaxis(oratintervalsof2πontheω

axis),asillustratedinFig.1.8.Notethatif
S(jω)isbandlimitedwithabandwidthω
c
,andifTischosensufficientlysmallsothatω
s
>2ω
c
,
thentheDTspectrumisacopyofS(jω)(scaledby1/T)inthebaseband.Thelimitingcaseof
ω
s
=2ω
c
iscalledtheNyquistsamplingfrequency.WheneveraCTsignalissampledatorabove
theNyquistrate,noaliasingdistortionoccurs(i.e.,thebasebandspectrumdoesnotoverlapwiththe
higher-orderreplicas)andtheCTsignalcanbeexactlyrecoveredfromitssamplesbyextractingthe
basebandspectrumofS(e
jω

)withanideallow-passfilterthatrecoverstheoriginalCTspectrumby
removingallspectralreplicasoutsidethebasebandandscalingthebasebandbyafactorofT.
1.5 TheDiscreteFourierTransform
ToobtainthediscreteFouriertransform(DFT)thecontinuousfrequencydomainoftheDTFT
issampledatNpointsuniformlyspacedaroundtheunitcircleinthez-plane,i.e.,atthepoints
c
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1999byCRCPressLLC
FIGURE1.8:IllustrationoftherelationshipbetweentheCTandDTspectra.
ω
k
=(2πk/N),k=0,1, ,N−1.TheresultistheDFTpairdefinedby(1.20a)and(1.20b).
Thesignals[n]iseitherafinitelengthsequenceoflengthN,oritisaperiodicsequencewithperiod
N.
S[k]=
N−1

n=0
s[n]e
−j2πkn/N
k=0,1, ,N−1 (1.20a)
s[n]=(1/N)
N−1

k=0
S[k]e
j2πkn/N
n=0,1, ,N−1 (1.20b)
Regardlessofwhethers[n]isafinitelengthorperiodicsequence,theDFTtreatstheNsamplesof
s[n]asthoughtheyareoneperiodofaperiodicsequence.ThisisanimportantfeatureoftheDFT,
andonethatmustbehandledproperlyinsignalprocessingtopreventtheintroductionofartifacts.
ImportantpropertiesoftheDFTaresummarizedinTable1.5.Thenotation((k))
N
denoteskmodulo
N,andR
N
[n]isarectangularwindowsuchthatR

N
[n]=1forn=0, ,N−1,andR
N
[n]=0
forn<0andn≥N.Thetransformrelationshipgivenby(1.20a)and(1.20b)isalsovalidwhen
s[n]andS[k]areperiodicsequences,eachofperiodN.Inthiscasenandkarepermittedtorange
overthecompletesetofrealintegers,andS[k]isreferredtoasthediscreteFourierseries(DFS).The
DFSisdevelopedbysomeauthorsasadistincttransformpairinitsownright[6].Whetherthe
DFTandtheDFSareconsideredidenticalordistinctisnotveryimportantinthisdiscussion.The
importantpointtobeemphasizedhereisthattheDFTtreatss[n]asthoughitwereasingleperiod
ofaperiodicsequence,andallsignalprocessingdonewiththeDFTwillinherittheconsequencesof
thisassumedperiodicity.
1.5.1 PropertiesoftheDiscreteFourierSeries
MostofthepropertieslistedinTable1.5fortheDFTaresimilartothoseofthez-transformandthe
DTFT,althoughsomeimportantdifferencesexist.Forexample,property5(time-shiftingproperty),
holdsforcircularshiftsofthefinitelengthsequences[n],whichisconsistentwiththenotionthatthe
DFTtreatss[n]asoneperiodofaperiodicsequence.Also,themultiplicationoftwoDFTsresultsin
thecircularconvolutionofthecorrespondingDTsequences,asspecifiedbyproperty7.Thislatter
propertyisquitedifferentfromthelinearconvolutionpropertyoftheDTFT.Circularconvolution
istheresultoftheassumedperiodicitydiscussedinthepreviousparagraph.Circularconvolution
issimplyalinearconvolutionoftheperiodicextensionsofthefinitesequencesbeingconvolved,in
whicheachofthefinitesequencesoflengthNdefinesthestructureofoneperiodoftheperiodic
extensions.
Forexample,supposeonewishestoimplementadigitalfilterwithfiniteimpulseresponse(FIR)
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TABLE1.5 PropertiesoftheDFT
Finite-LengthSequence(LengthN) N-PointDFT(LengthN)
1.x[n] X[k]

2.x
1
[n],x
2
[n] X
1
[k],X
2
[k]
3.ax
1
[n]+bx
2
[n] aX
1
[k]+bX
2
[k]
4.X[n] Nx[((−k))
N
]
5.x[((n
m
))
N
] W
km
N
X[k]
6.W

−ln
N
x[n] X[((k−l))
N
]
7.
N−1

m=0
x
1
(m)x
2
[((n
m
))
N
] X
1
[k]X
2
[k]
8.x
1
[n]x
2
[n]
1
N
N−1


l=0
X
1
(l)X
2
[((k−l)
N
]
9.x
∗
[n] X
∗
[((−k))
N
]
10.x
∗
[((−n))
N
] X
∗
[k]
11.Re{x[n]} X
ep
[k]=
1
2
{X[((k))
N

]+K
∗
[((−k))
N
]}
12.jIm{x[n]} X
op
[k]=
1
2
{X[((k))
N
]−X
∗
[((−k))
N
]}
13.x
ep
[n]=
1
2
{x[n]+x
∗
[((−n))
N
]} Re{X[k]}
14.x
op
[n]=

1
2
{x[n]−x
∗
[((−n))
N
]} jIm{X[k]}
Properties15–17applyonlywhenx[n]isreal
15.Symmetryproperties









X[k]=X
∗
[((−k))
N
]
Re{X[k]} = Re{X[((−k))
N
]}
Im{X[k]} = −Im{X[((−k))
N
]}
|X[k]| = |X[((−k))

N
]|
<){X[k]} = −<){X[((−k))
N
]}
16.x
ep
[n]=
1
2
{x[n]+x[((−n))
N
]} Re{X[k]}
17.x
op
[n]=
1
2
{x[n]−x[((−n))
N
]} jIm{X[k]}
h[n].Theoutputy(n)inresponsetoinputs[n]isgivenby
y[n]=
N−1

k=0
h[k]s[n−k] (1.21)
wherey(n)isobtainedbytransformingh[n]ands[n]intoH[k]andS[k]usingtheDFT,multiplying
thetransformspoint-wisetoobtainY[k]=H[k]S[k],andthenusingtheinverseDFTtoobtain
y[n]=DFT

−1
{Y[k]}.Ifs[n]isafinitesequenceoflengthM,thentheresultsofthecircular
convolutionimplementedbytheDFTwillcorrespondtothedesiredlinearconvolutioniftheblock
lengthoftheDFT,N
DFT
,ischosensufficientlylargesothatN
DFT
≥N+M−1andbothh[n]
ands[n]arepaddedwithzeroestoformblocksoflengthN
DFT
.
1.5.2 FourierBlockProcessinginReal-TimeFilteringApplications
InsomepracticalapplicationseitherthevalueofMistoolargeforthememoryavailable,ors[n]may
notactuallybefiniteinlength,butratheracontinualstreamofdatasamplesthatmustbeprocessed
byafilteratreal-timerates.Twowell-knownalgorithmsareavailablethatpartitions[n]intosmaller
blocksandprocesstheindividualblockswithasmaller-lengthDFT:(1)overlap-savepartitioning
and(2)overlap-addpartitioning.Eachofthesealgorithmsissummarizedbelow.
Overlap-SaveProcessing
InthisalgorithmN
DFT
ischosentobesomeconvenientvaluewithN
DFT
>N.Thesignal
s[n]ispartitionedintoblockswhichareoflengthN
DFT
andwhichoverlapbyN−1datapoints.
Hence,thekthblockiss
k
[n]=s[n+k(N
DFT

−N+1)],n=0, ,N
DFT
−1.Thefilterimpulse
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response is augmented with N
DFT
− N zeroes to produce
h
pad
[n]=

h[n],n= 0, ,N − 1
0,n= N, ,N
DFT
− 1

(1.22)
TheDFTisthenusedtoobtainY
pad
[n]=DFT{h
pad
[n]}·DFT{s
k
[n]},andy
pad
[n]=IDFT{Y
pad
[n]}.

Fromthe y
pad
[n] array the values that correctly correspond to the linear convolution are saved; val-
ues that are erroneous due to wrap-around error caused by the circular convolution of the DFT are
discarded. The kth block of the filtered output is obtained by
y
k
[n]=

y
pad
[n],n= N − 1, ,N
DFT
− 1
0,n= 0, ,N − 2

(1.23)
Fortheoverlap-save algorithm,eachtimea block is processed there are N
DFT
− N + 1 points saved
and N − 1 points discarded. Each block moves forward by N
DFT
− N + 1 data points and overlaps
the previous block by N − 1 points.
Overlap-Add Processing
This algorithm is similar to the previous one except that the kth input block is defined as
s
k
[n]=


s[n + kL],n= 0, ,L− 1
0,n= L, ,N
DFT
− 1

(1.24)
where L = N
DFT
− N + 1. The filter function h
pad
[n] is augmented with zeroes, as before, to
createh
pad
[n], and the DFTprocessingisexecuted as before. In each block y
pad
[n] that is obtained
attheoutputthefirstN − 1 pointsareerroneous,thelastN − 1 pointsareerroneous,andthemiddle
N
DFT
− 2(N − 1) points correctly correspond to the linear convolution. However, if the last N − 1
points from block k are overlappedwith the first N − 1 points from block k + 1 and added pairwise,
correct results corresponding to linear convolution are also obtained from these positions. Hence,
after this addition the number of correct points produced per block is N
DFT
− N + 1, which is the
same as that for the overlap-save algorithm. The overlap-add algorithm requires approximately the
sameamountofcomputationastheoverlap-savealgorithm,althoughtheadditionoftheoverlapping
portionsofblocksisextra. Thisfeature,togetherwiththeaddeddelayofwaitingforthe nextblockto
be finished before the previous one is complete, has resulted in more popularity for the overlap-save
algorithm in practical applications.

Block filtering algorithms make it possible to efficiently filter continual data streams in real time
because the fast Fourier transform (FFT) algorithm can be used to implement the DFT, thereby
minimizing the total computation time and permitting reasonably high overall data rates. However,
block filtering generates data in bursts, i.e., a delay occurs during which no filtered data appear, and
then an entire block is suddenly generated. In real-time systems buffering must be used. The block
algorithms are particularly effective for filtering very long sequences of data that are prerecorded on
magnetic tape or disk.
1.5.3 Fast Fourier Transform Algorithms
The DFT is typically implemented in practice with one of the common forms of the FFT algorithm.
The FFT is not a Fourier transform in its own right, but simply a computationally efficient algo-
rithm that reduces the complexity of computing the DFT from order {N
2
} to order {N log
2
N}.
When N is large, the computational savings provided by the FFT algorithm is so great that the FFT
makesreal-timeDFTanalysispracticalin manysituationsthat wouldbeentirelyimpracticalwithout
c
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1999 by CRC Press LLC
it.FastFouriertransformalgorithmsabound,includingdecimation-in-time(D-I-T)algorithms,
decimation-in-frequency(D-I-F)algorithms,bit-reversedalgorithms,normallyorderedalgorithms,
mixed-radixalgorithms(forblocklengthsthatarenotpowersof2),primefactoralgorithms,and
Winogradalgorithms[7].TheD-I-TandtheD-I-Fradix-2FFTalgorithmsarethemostwidelyused
inpractice.DetaileddiscussionsofvariousFFTalgorithmscanbefoundin[3,6,7],and[10].
TheFFTiseasilyunderstoodbyexaminingthesimpleexampleofN=8.TheFFTalgorithmcan
bedevelopedinnumerousways,allofwhichdealwithanesteddecompositionofthesummation
operatorof(1.20a).Thedevelopmentpresentedhereiscalledanalgebraicdevelopmentofthe
FFTbecauseitfollowsstraightforwardalgebraicmanipulation.First,thesummationindices(k,n)
in(1.20a)areexpressedasexplicitbinaryintegers,k=k

2
4+k
1
2+k
0
andn=n
2
4+n
1
2+n
0
,
wherek
i
andn
i
arebitsthattakeonthevaluesofeither0or1.Iftheseexpressionsaresubstituted
into(1.20a),alltermsintheexponentthatcontainthefactorN=8canbedeletedbecause
e
−j2πl
=1foranyintegerl.Upondeletingsuchtermsandregroupingtheremainingterms,the
productnkcanbeexpressedineitheroftwoways:
nk = (4k
0
)n
2
+(4k
1
+2k
0

)n
1
+(4k
2
+2k
1
+k
0
)n
0
(1.25a)
nk = (4n
0
)k
2
+(4n
1
+2n
0
)k
1
+(4n
2
+2n
1
+n
0
)k
0
(1.25b)

Substituting(1.25a)into(1.20a)leadstotheD-I-TFFT,whereassubstituting(1.25b)leadstothe
D-I-FFFT.OnlytheD-I-TFFTisdiscussedfurtherhere.TheD-I-Fandvariousrelatedformsare
treatedindetailin[6].
TheD-I-TFFTdecomposesintolog
2
Nstagesofcomputation,plusastageofbitreversal,
x
1
[k
0
,n
1
,n
0
]=
1

n
2
=0
s[n
2
,n
1
,n
0
]W
4k
0
n

2
8
(stage1) (1.26a)
x
2
[k
0
,k
1
,n
0
]=
1

n
1
=0
x[k
0
,n
1
,n
0
]W
(4k
1
+2k
0
)n
2

8
(stage2) (1.26b)
x
3
[k
0
,k
1
,k
2
]=
1

n
0
=0
x[k
0
,k
1
,n
0
]W
(4k
2
+2k
1
+k
0
)n

0
8
(stage3) (1.26c)
S[k
2
,k
1
,k
0
]=x
3
[k
0
,k
1
,k
2
] (bitreversal) (1.26d)
Ineachsummationaboveoneofthen
i
issummedoutoftheexpression,whileatthesametimeanew
k
i
isintroduced.Thenotationischosentoreflectthis.Forexample,instage3,n
0
issummedout,
k
2
isintroducedasanewvariable,andn
0

isreplacedbyk
2
intheresult.Thelastoperation,called
bitreversal,isnecessarytocorrectlylocatethefrequencysamplesX[k]inthememory.Itiseasyto
showthatifthesamplesarepairedcorrectly,anin-placecomputationcanbedonebyasequenceof
butterflyoperations.Theterm“in-place”meansthateachtimeabutterflyistobecomputed,apair
ofdatasamplesisreadfrommemory,andthenewdatapairproducedbythebutterflycalculation
iswrittenbackintothememorylocationswheretheoriginalpairwasstored,therebyoverwriting
theoriginaldata.Anin-placealgorithmisdesignedsothateachdatapairisneededforonlyone
butterfly,andthusthenewresultscanbeimmediatelystoredontopoftheoldinordertominimize
memoryrequirements.
Forexample,instage3thek=6andk=7samplesshouldbepaired,yieldinga“butterfly”
computationthatrequiresonecomplexmultiply,onecomplexadd,andonesubtract:
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x
3
(1, 1, 0) = x
2
(1, 1, 0) + W
3
8
x
2
(1, 1, 1) (1.27a)
x
3
(1, 1, 1) = x
2

(1, 1, 0) − W
3
8
x
2
(1, 1, 1) (1.27b)
Samplesx
2
(6) andx
2
(7) arereadfromthememory, thebutterflyisexecutedonthepair, andx
3
(6)
and x
3
(7) are written back to the memory, overwriting the original values of x
2
(6) and x
2
(7).In
general, N/2 butterflies are found in each stage and there are log
2
N stages, so the total number of
butterflies is (N/2) log
2
N. Because one complex multiplication per butterfly is the maximum, the
total number of multiplications is bounded by (N/2) log
2
N (some of the multiplies involve factors
of unity and should not be counted).

Figure 1.9 shows the signal flow graph of the D-I-T FFT for N = 8. This algorithm is referred to
FIGURE 1.9: D-I-T FFT algorithm with normally ordered inputs and bit-reversed outputs.
as an in-place FFT with normally ordered input samples and bit-reversed outputs. Minor variations
that include both bit-reversed inputs and normally ordered outputs and non-in-place algorithms
with normally ordered inputs and outputs are possible. Also, when N is not a power of 2, a mixed-
radix algorithm can be used to reduce computation. The mixed-radix FFT is most efficient when
N is highly composite, i.e., N = p
r
1
1
p
r
2
2
···p
r
L
L
, where the p
i
are small prime numbers and the r
i
are positive integers. It can be shown that the order of complexity of the mixed radix FFT is order
{N(r
1
(p
1
−1)+r
2
(p

2
−1)+···+r
L
(p
L
−1)}. Becauseofthelackofuniformityofstructureamong
stages, this algorithm has not received much attention for hardware implementation. However, the
mixed-radix FFT is often used in software applications, especially for processing data recorded in
laboratory experiments in which it is not convenient to restrict the block lengths to be powers of 2.
Many advancedFFTalgorithms, such as higher-radixforms, the mixed-radixform, theprime-factor
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1999 by CRC Press LLC
algorithm, and the Winograd algorithm are described in [9]. Algorithms specialized for real-valued
data reduce the computational cost by a factor of two. A radix-2 D-I-T FFT program, written in C
language, is listed in Table 1.6.
1.6 Family Tree of Fourier Transforms
It is now possible to illustrate the functional relationships among the various forms of Fourier
transformsthathavebeendiscussedintheprevioussections. ThefamilytreeofCTFouriertransform
isshowninFig.1.10,wherethemostgeneral,andconsequentlythemostpowerful,Fouriertransform
is the classical complex Fourier transform (or equivalently, the bilateral Laplace transform). Note
that the complex Fourier transform is identical to the bilateral Laplace transform, and it is at this
level that the classical Laplace transform and Fourier transform techniques become identical. Each
specialmemberoftheCTFourier family isobtainedbyimpressingcertainconstraints on the general
form, thereby producing special transforms that are simpler and more useful in practical problems
where the constraints are met.
The analogous family of DT Fourier techniques is presented in Fig. 1.11, in which the bilateral
z-transform is analogous to the complex Fourier transform, the unilateral z-transform is analogous
to the classical (one-sided) Laplace transform, the DTFT is analogous to the classical Fourier (CT)
transform, and the DFT is analogous to the classical (CT) Fourier series.

1.7 Selected Applications of Fourier Methods
1.7.1 Fast Fourier Transform in Spectral Analysis
An FFT program is often used to perform spectral analysis on signals that are sampled and recorded
aspart of laboratoryexperiments,or in certain typesofdataacquisitionsystems. Several issues must
be addressed when spectral analysis is performed on (sampled) analog waveforms that are observed
over a finite interval of time.
Windowing
The FFT treats the block of data as though it were one period of a periodic sequence. If the
underlying waveform is not per iodic, then harmonic distortion may occur because the periodic
waveform created by the FFT may have sharp discontinuities at the boundaries of the blocks. This
effectisminimizedbyremovingthemeanofthedata (it can alwaysbereinserted)andbywindowing
the data so the ends of the block are smoothly tapered to zero. A good rule of thumb is to taper
10% of the data on each end of the block using either a cosine taper or one of the other common
windows shown in Table1.7. An alternateinterpretationofthisphenomenonisthatthefinitelength
observation has already windowed the true waveform with a rectangular window that has large
spectral sidelobes (see Table 1.7). Hence, applying an additional window results in a more desirable
window that minimizes frequency domain distortion.
Zero Padding
An improve d spectral analysis is achieved if the block length of the FFT is increased. This
can be done by taking more samples within the observation interval, increasing the length of the
observation inter val, or augmenting the original data set with zeroes. First, it must be understood
that the finite observation interval results in a fundamental limit on the spectral resolution, even
before the signals are sampled. The CT rectangular window has a (sin x)/x spectrum, which is
convolvedwiththetrue spectrumof the analog signal. Therefore,thefrequencyresolutionislimited
bythewidth of the mainlobe in the (sin x)/x spectrum, which is inversely proportional to the length
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TABLE 1.6 An In-Place D-I-T FFT Program in C Language
/*****************************************************

* fft: in-place radix-2 DFT of a complex input
*
* input:
* n: length of FFT: must be a power of two
* m: n = 2**m
* input/output:
* x: float array of length n with real part of data
* y: float array of length n with image part of data
******************************************************/
fft(n,m,x,y)
tnt n,m;
float x[ ], y[ ]:
{
int i,j,k,nl,n2:
float c,s,e,a,t1,t2;
j = 0; /*BIT-REVERSE */
n2 = n/2;
for (i=1; 1 < n-1; i++) /*bit-reverse counter */
{
nl = n1/2;
while(j>=n1)
{
j=j-n1;
nl = n1/2;
}
j=j+nl;
if (i < j) /*swap data */
{
t1 = x[i]; x[i] = x[j]; x[j] = t1;
t1 = y[i]; y[i] = y[j]; y[j] = t1;

}
}
n1=0;n2=1; /*FFT*/
for(i=0;i<m;i++) /*state loop */
{
n1 = n2; n2 = n2 + n2;
e = -6.283185307179586/n2;
a = 0.0;
for (j=0; j < n1; j++) /*flight loop */
{
c = cos(a); s=sin (a);
a=a+e;
for (k=j;k<n;k=k+n2) /*butterfly loop */
{
t1 = c*x[k+n1] - s*y[k+n1];
t2 = s*x[k+n1] + c*y[k+n1];
x[k+n1] = x[k] - t1;
y[k+n1] = y[k] - t2;
x[k] = x[k] + t1;
y[k] = y[k] + t2;
}
}
}
return;
}
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1999 by CRC Press LLC
FIGURE 1.10: Relationships among CT Fourier concepts.
of the observation interval. Sampling causes a certain degree of aliasing, although this effect can

be minimized by sampling at a high enough rate. Therefore, lengthening the obser v ation interval
increasesthefundamentalresolutionlimit,whiletakingmoresampleswithintheobservationinterval
minimizesaliasingdistortionandprovidesabetterdefinition(moresamplepoints)ontheunderlying
spectrum.
PaddingthedatawithzeroesandcomputingalongerFFTdoesgivemorefrequencydomainpoints
(improved spectral resolution), but it does not improve the fundamental limit, nor does it alter the
effects of aliasing error. The resolution limits are established by the observation interval and the
sampling rate. No amount of zero padding can improve these basic limits. However, zero padding
is a useful tool for providing more spectral definition, i.e., it allows a better view of the (distorted)
spectrum that results once the observation and sampling effects have occurred.
Leakage and the Picket Fence Effect
An FFT with block length N can accurately resolve only frequencies ω
k
= (2π/N)k, k =
0, ,N− 1 that are integer multiples of the fundamental ω
1
= (2π/N). An analog waveformthat
issampledandsubjectedtospectr alanalysismayhavefrequencycomponentsbetweentheharmonics.
Forexample,acomponentatfrequencyω
k+1/2
= (2π/N)(k+1/2) willappearscatteredthroughout
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1999 by CRC Press LLC
TABLE 1.7 Common Window Functions
Peak Minimum
Side-Lobe Stopband
Amplitude Mainlobe Attenuation
Name Function (dB) Width (dB)
Rectangular ω(n) = 1. 0 ≤ n ≤ N − 1 −13 4π/N −21

Bartlett ω(n) =

2/N, 0 ≤ n ≤ (N − 1)/2
22n/N, (N − 1)/2 ≤ n ≤ N − 1
−25 8π/N −25
Hanning ω(n) = (1/2)[1 − cos(2πn/N)]−31 8π/N −44
0 ≤ n ≤ N − 1 −43 8π/N −53
Hamming ω(n) = 0.54 − 0.46cos(2πn/N), −43 8π/N −53
0 ≤ n ≤ N − 1
Backman ω(n) = 0.42 − 0.5cos(2πn/N) −57 12π/N −74
+ 0.08 cos(4π n/N )
,
0 ≤ n ≤ N − 1
thespect rum. TheeffectisillustratedinFig.1.12forasinusoidthatisobservedthrougharectangular
window and then sampledatN points. The picketfenceeffectmeans that not all frequenciescan be
seen bythe FFT. Harmonic components areseen accurately, but other components “slipthrough the
picketfence” while their energyis“leaked” into the harmonics. Theseeffectsproduceartifacts in the
spectral domain that must be carefully monitored to assure that an accurate spectrum is obtained
from FFT processing.
1.7.2 Finite Impulse Response Digital Filter Design
AcommonmethodfordesigningFIRdigitalfiltersisbyuseofwindowingandFFTanalysis. Ingeneral,
windowdesignscanbecarriedoutwiththeaidofahandcalculatorandatableofwell-knownwindow
functions. Let h[n] be the impulse response that corresponds to some desired frequency response,
H(e
jω
).IfH(e
jω
) has sharp discontinuities, such as the low-pass example shown in Fig. 1.13, then
h[n] willrepresent an infinite impulse response (IIR) function. Theobjective is to time limit h[n] in
such a way as to not distort H(e

jω
) any more than necessary. If h[n] is simply truncated, a ripple
(Gibbs phenomenon) occurs around the discontinuities in the spectrum, resulting in a distorted
filter (Fig. 1.13).
Supposethatw[n] isawindowfunctionthattimelimitsh[n] tocreateanFIRapproximation,h

[n];
i.e., h

[n]=w[n]h[n]. Then if W(e
jω
) is the DTFT of w[n], h

[n] will have a Fourier transform
given by H

(e
jω
) = W(e
jω
) ∗ H(e
jω
),where∗ denotes convolution. Thus, the ripples in H

(e
jω
)
resultfromthesidelobesof W(e
jω
). Ideally, W(e

jω
) shouldbesimilartoanimpulse sothatH

(e
jω
)
is approximately equal to H(e
jω
).
Special Case. Let h[n]=cos nω
0
, for all n. Then h[n]=w[n] cosnω
0
, and
H

(e
jω
) = (1/2)W (e
j(ω+ω
0
)
) + (1/2)W(e
j(ω−ω
0
)
) (1.28)
as illustrated in Fig. 1.14. For this simple class, the center frequency of the bandpass is controlled
by ω
0

, and both the shape of the bandpass and the sidelobe structure are strictly determined by the
choice of the window. While this simple class of FIRs does not allow for very flexible designs, it is a
simple technique for determining quite useful low-pass, bandpass, and high-pass FIRs.
GeneralCase. Specifyanidealfrequencyresponse,H(e
jω
),andchoosesamplesatselected
values of ω. Use a long inverse FFT of length N

to find h

[n], an approximation to h[n], where if
N is the desired length of the final filter, then N

 N . Then use a carefully selected window to
truncate h

[n] to obtain h[n] by letting h[n]=ω[n]h

[n]. Finally, use an FFT of length N

to find
H

(e
jω
).IfH

(e
jω
) is a satisfactory approximation to H(e

jω
), the design is finished. If not, choose
anewH(e
jω
) oraneww[n] and repeat. Throughout the designprocedure it is important to choose
N

= kN, withk aninteger that is typicallyintherange of 4 to 10. Because this designtechnique is a
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1999 by CRC Press LLC