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.