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”

9/2/08 Comp665–FourierSeriesandTransforms 3

€

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

9/2/08 Comp665–FourierSeriesandTransforms 4

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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

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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].

9/2/08 Comp665–FourierSeriesandTransforms 12

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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

9/2/08 Comp665–FourierSeriesandTransforms 15

€

a x[n] + b y[n]↔ a X [k ] + bY [k ]

€

x[n + n0]↔ e−i 2πkN

n0 X [k ]

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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 ]