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FourierSeriesandTransforms
9/2/08 Comp665–RealandSpecialSignals 1
Website is now online at: http://www.unc.edu/courses/2008fall/comp/665/001/
“Discrete”Exponen8alFunc8on
• DiscreteConvolu?on:
• Convolu?onwithanexponen?alsignal,:
• Ifwedefine:then:
9/2/08 Comp665–FourierSeriesandTransforms 2
€
y(n) = h(k)x(n − k)k =−∞
∞
∑
€
y(n) = h(k)eω(n−k )
k =−∞
∞
∑
= h(k)e−ω k
k =−∞
∞
∑
e
ωn
€
H (eω ) = h(k)eω k
k =−∞
∞
∑
y(n) = H (eω )eω n
€
x(n) = eω n
Eigenvector Eigenvalue
That’s not exactly the same definition of
convolution that he
used before…
but by commutivity
it is identical
SinusoidsasExponen8als
• Euler’sRela?on:• Subs?tu?ng:• Interpreta?on:
“TheresponseofaLSIsystemtoasinusoidinputwithfrequencyω,isascaledsinusoidofthesamefrequency”
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€
eiωt = cos(ωt) + isin(ωt)
€
y(n) = H (eiω )eiω n
h(n)
What’s the value of H(πn/12)?
SolvingforH(ejω)
• Recallfromlast?methattherealandimaginarycomponentsofacomplexexponen?alcanbeequivalentlyinterpretedasthemagnitudeandphase‐shiWofsinusoid
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€
H = Re(He jω )2 + Im(He jω )2
tan(ϕ ) =Im(He jω )Re(He jω )
WhatweKnow
• Theinputandtheoutput
• Thus
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€
x[n] = sin( πn12
) y[n] = 0.3sin( πn12
+ π3)
€
H = 0.3ϕ = π
3
Re(H ) = H cos(ϕ ) = 0.3(12)
Im(H ) = H sin(ϕ ) = 0.3( 32
)
H πn12
= 0.3(12) + i0.3( 3
2)
Mul8pleSinusoids
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h(n)
Fourier’sConjecture
• JosephFourier,an18thcenturyFrenchmathema?cianandphysicist,claimedthatanyfunc?onofavariable,whethercon?nuousordiscon?nuous,couldbeexpandedintoaseriesofsinusoidswithperiodsthataremul?plesofthevariable
• Thoughnotstrictlycorrect,heiscreditedwithinven?ngadecomposi?onofsignalsintoseriesofsinusoidscalledtheir“FourierSeries”
9/2/08 Comp665–FourierSeriesandTransforms 7
SignalsasSumsofSinusoids
• Howdowe“transform”anarbitrarysignaltoasumofsinusoids?
• Eachtermisjusta“dotproduct”ofaserieswithacomplexsinusoid
9/2/08 Comp665–FourierSeriesandTransforms 8
€
X [k ] = x(n)e−i 2πkN
n
n=0
N −1
∑ k = 0,…,N −1
€
X [k ] = x(n) cos(2πkN
n) − isin(2πkN
n)[ ]n=0
N −1
∑ k = 0,…,N −1
DotProductsasProjec8ons
• Theelement‐wisesum‐of‐productsofserieselementsorvectorcomponentsisoWencalledthe“inner”or“dot”product.
• Adotproductcanbeinterpretedasthelengthofonevectorprojectedontotheother
9/2/08 Comp665–FourierSeriesandTransforms 9
a b
€
a ⋅ b
Coordinates
• Coordinatesaremerelyaseriesofprojec?onsontoaspecificsetofvectors,eachcalleda“basis”vector
• ThesamethingisgoingonwhenweFouriertransformasignal,weprojecttheoriginalsignal(apoint)ontoan“N‐dimensional”basis
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x2
x1
b
a
FourierBasisFunc8ons
9/2/08 Comp665–FourierSeriesandTransforms 11
InverseMapping
• Onceasignalismappedfromaseriestoaweightedsumofcomplexsinusoids,itcanbemappedbacktoaseriesasfollows:
• ComplexnumbersX[k]representtheamplitudeandphaseofthesinusoidalcomponentsoftheinput"signal"x[n].
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€
x[n] =1N
X [k ]ei 2πnN
k
k =0
N −1
∑ n = 0,…,N −1
MakingThingsConcrete
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0 N-1
x[ ]
Spatial Domain Frequency Domain
N uniformly spaced samples
0 N/2
Re(X[ ])
N/2 + 1 coefficients (cosine amplitudes)
0 N/2
Im(X[ ])
N/2 + 1 coefficients (sine amplitudes)
ForwardDFT
InverseDFT
SomeContext
• Whydowecare?– Mappingsignalsbackandforthbetweenspa?alandfrequency
domainssimplifiesanalysis(convolu?oninpar?cular)
– Wehaveintui?onforperiodicfunc?ons– Providesano?onof“scale”forcharacterizingsignals
• Largescale=lowfrequency• Smallscale=highfrequency
• Assumesthatsignalsareperiodic– Perhapstheyreallyare– wecan“pretend”theyareoutsideofourdomainofinterest
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FourierDomainProper8es
• Linearity
• ShiWing
• Symmetry
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€
a x[n] + b y[n]↔ a X [k ] + bY [k ]
€
x[n + n0]↔ e−i 2πkN
n0 X [k ]
€
x[n],real↔ Re(X [k ]) = Re(X [N − k ]), k > 0Im(X [k ]) = −Im(X [N − k ]), k > 0
Graphically
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Each successive basis function represents a higher frequency sinusoid, that is related to the original signal’s sampling rate by:
Once k>N/2 the number of samples per period are less than 1, and the kth basis “aliases” as one with lower frequency
Observation:
€
period(X [k ]) =2πkN
MoreProper8es
• Convolu?on
• Modula?on
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€
x[n]∗h[n]↔ X [k ]H [k ]
€
x[n] h[n]↔ X [k ]∗ H [k ]