Chapter5.FourierAnalysisforDiscrete-Time

SignalsandSystems

ChapterObjec@ves

1.  Learntechniquesforrepresen3ngdiscrete-)meperiodic

signalsusingorthogonalsetsofperiodicbasisfunc3ons.

2.  Studyproper3esofexponen)al,trigonometricandcompact

Fourierseries,andcondi3onsfortheirexistence.

3.  LearntheFouriertransformfornon-periodicsignalasan

extensionofFourierseriesforperiodicsignals

4.Studytheproper)esoftheFouriertransform.Understandthe

conceptsofenergyandpowerspectraldensity.

5.2Exponen@alFourierSeries(EFS)

!x(t) = ckejkω0t

k=−∞

∞

∑

Synthesisequa@on:

Analysisequa@on:

ck =1T0

!x(t)e− jkω0t dtt0

t0+T0∫

Con@nue-TimeFourierSeries

!x[n]= ckej(2π /N )kn

k=0

N−1

∑

Synthesisequa@on:

Analysisequa@on:

ck =1N

x[n]e− j(2π /N )kn

n=0

N−1

∑

Discrete-TimeFourierSeries

Linearity

5.2.7Proper@esofFourierSeries

x(t) ℑ← →⎯ ck

Wherea1anda2areanytwoconstants

Con@nue-TimeFourierSeries Discrete-TimeFourierSeries

y(t) ℑ← →⎯ dk

a1x(t)+ a2y(t)ℑ← →⎯ a1ck + a2dk

x[n] ℑ← →⎯ ck

y[n] ℑ← →⎯ dk

a1x[n]+ a2y[n]ℑ← →⎯ a1ck + a2dk

TimeshiL

5.2.7Proper@esofFourierSeries

Con@nue-TimeFourierSeries Discrete-TimeFourierSeries

!x(t) = ckejkw0t

k=−∞

∞

∑

!x(t −τ ) = [cke− jkw0τ ]e jkw0t

k=−∞

∞

∑

!x[n]= ckej(2π /N )kn

k=0

N−1

∑

!x[n−m]= ckej(2π /N )kn

k=0

N−1

∑ e− j(2π /N )km

5.3AnalysisofNon-periodicCon@nuous-TimeSignals

Discrete-TimeFourierTransform

X(Ω) = x[n]e− jΩn

n=−∞

∞

∑x[n]= 12π

X(Ω)e jΩn dΩ−π

π∫

Synthesisequa@on(inverse): Analysisequa@on(forward):

2πk2M +1

− >Ω

5.3AnalysisofNon-periodicCon@nuous-TimeSignals

X(ω) = x(t)e− jwt dt−∞

∞∫

x(t) = 12π

X(ω)e jwt dw−∞

∞∫

Synthesisequa@on(inverse):

Analysisequa@on(forward):

Con@nue-TimeFourierTransform Discrete-TimeFourierTransform

X(Ω) = x[n]e− jΩn

n=−∞

∞

∑

x[n]= 12π

X(Ω)e jΩn dΩ−π

π∫

Synthesisequa@on(inverse):

Analysisequa@on(forward):

IsitalwayspossibletodeterminetheFourierseriescoefficients?

5.3.2ExistenceofFourierTransform

²  Absolutesummable:

x[n] <∞n=−∞

∞

∑

²  Square-summable:

x[n] 2 <∞n=−∞

∞

∑

Linearity:

5.3.5Proper@esofFourierTransform

x1[n]ℑ← →⎯ X1(Ω) and

Wherea1anda2areanytwoconstants

Periodicity:

x2[n]ℑ← →⎯ X2(Ω)

α1x1[n]+α2x2[n]ℑ← →⎯ α1X1(Ω)+α2X2(Ω)

X(Ω+ 2πr) = X(Ω)

forallintegersr

5.3.5Proper@esofFourierTransform

TimeShiLing:

x[n] ℑ← →⎯ X(Ω) x[n−m] ℑ← →⎯ X(Ω)e− jΩm

FrequencyShiLing:

x[n]e− jΩ0n ℑ← →⎯ X(Ω−Ω0 )x[n] ℑ← →⎯ X(Ω)

Convolu@onProperty:

5.3.5Proper@esofFourierTransform

x1[n]ℑ← →⎯ X1(Ω)

x1[n]* x2[n]ℑ← →⎯ X1(Ω)X2(Ω) X1(Ω)*X2(Ω)

ℑ← →⎯ x1[n]x2[n]

x2[n]ℑ← →⎯ X2(Ω)

Parseval’sTheorem:

5.4EnergyandPowerinFrequencyDomain

Foraperiodicpowersignalx(t)

1T0

x(t) 2 dt = ck2

k=−∞

∞

∑t0

t0+T0∫

Foranon-periodicpowersignal

x(t) 2 dt =−∞

∞∫ X( f ) 2 df

−∞

∞∫

Con@nue-Time

1N

x[n] 2

k=0

N−1

∑ = ck2

k=0

N−1

∑

Discrete-Time

Con@nue-Time Discrete-Time

x[n] 2

k=0

N−1

∑ =12π

X(Ω) 2 dΩ−π

π∫

PowerSpectralDensity:

5.4EnergyandPowerinFrequencyDomain

Sx (Ω) = 2π ck2δ(Ω− kΩ0 )

k=−∞

∞

∑

Autocorrela@onFunc@on:

5.4EnergyandPowerinFrequencyDomain

Foraenergysignalx(t)theautocorrela@onfunc@onis

rxx[m]= x[n]x[n+m]n=−∞

∞

∑

Systemfunc@on(frequencyresponse)

5.5SystemFunc@onConcept

Impulseresponse(h[n]) Systemfunc3on(H(Ω))FourierTransform

H (Ω) =ℑ h[n]{ }= h[n]e− jΩn

n=−∞

∞

∑

Ingeneral,H(w)isacomplexfunc3onofw,andcanbewriJeninpolarformas:

H (Ω) = H (Ω) e jΘ(Ω)

5.8DiscreteFourierTransform

x[n]= ckej(2π /N )kn

k=0

N−1

∑

Synthesisequa@on(inverse):

ck =1N

x[n]e− j(2π /N )kn

n=0

N−1

∑

DTFS DTFT

X(Ω) = x[n]e− jΩn

n=−∞

∞

∑

x[n]= 12π

X(Ω)e jΩn dΩ−π

π∫

Analysisequa@on(forward):

DFT

x[n]= 1N

X[k]e j(2π /N )kn

k=0

N−1

∑

X[k]= x[n]e− j(2π /N )kn

n=0

N−1

∑

k=0,1,….,N-1

n=0,1,….,N-1

k=0,1,….,N-1

n=0,1,….,N-1

5.8DiscreteFourierTransform

DTFT

X(Ω) = x[n]e− jΩn

n=−∞

∞

∑

DFT

X[k]= x[n]e− j(2π /N )kn

n=0

N−1

∑

Rela@onshipoftheDFTtotheDTFT

TheDFTofalength-NsignalisequaltoitsDTFTevaluatedatasetofNangularfrequenciesequallyspacedintheinterval[0,2π).Letanindexedsetofangularfrequenciesbedefinedas

Ωk =2πkN, k = 0,1,.....,N −1

X[k]= X(Ω) = x[n]e− j(2π /N )kn

n=0

N−1

∑

5.8DiscreteFourierTransform

WhydoweneedDFT?

²  Thesignalx[n]anditsDFTX[k]eachhaveNsamples,makingthediscreteFouriertransformprac3calforcomputerimplementa3on.

²  Fastandefficientalgorithm,knowasfastFouriertransforms(FFTs),

areavailableforthecomputa3onoftheDFT.

²  DFTcanbeusedforapproxima3ngotherformsofFourierseriesandtransformsforbothcon3nuous-3meanddiscrete-3mesystem.

²  Dedicatedprocessorsareavailableforfastandefficient.

computa3onoftheDFTwithminimalornoprogrammingneeded.