Magnitude and Phase The Fourier Transform: Examples ...

1 The Fourier transform : Examples , Properties, Common PairsThe Fourier transform : Examples , Properties, Common PairsCS 450: Introduction to Digital Signal and Image ProcessingBryan MorseBYU Computer ScienceThe Fourier transform : Examples , Properties, Common PairsMagnitude and PhaseRemember: complex numbers can be thought of as (real,imaginary)or ( Magnitude , Phase ). Magnitude :|F|=[<(F)2+= (F)2]1/2 Phase : (F)= tan 1=(F)<(F)Real partHow much of a cosine of that frequency you needImaginary part How much of a sine of that frequency you needMagnitudeAmplitude of combined cosine and sinePhaseRelative proportions of sine and cosineThe Fourier transform : Examples , Properties, Common PairsExample: Fourier transform of a Cosinef(t) =cos(2 st)F(u) =Z f(t)e i2 utdt=Z cos(2 st)e i2 utdt=Z cos(2 st) [cos( 2 ut) +isin( 2 ut)]dt=Z cos(2 st)cos( 2 ut)dt+iZ cos(2 st)sin( 2 ut)dt=Z cos(2 st)cos(2 ut)dt iZ cos(2 st)sin(2 ut)dt0 except whenu= s0 for allu=12 (u s) +12 (u+s)The Fourier transform .

2 Examples , Properties, Common PairsExample: Fourier transform of a CosineSpatial DomainFrequency Domaincos(2 st)12 (u s) +12 (u+s) Fourier transform : Examples , Properties, Common PairsOdd and Even FunctionsEvenOddf( t) =f(t)f( t) = f(t)SymmetricAnti-symmetricCosinesSinesT ransform is real transform is imaginary for real-valued signalsThe Fourier transform : Examples , Properties, Common PairsSinusoidsSpatial Domain Frequency Domainf(t)F(u)cos(2 st)12[ (u+s) + (u s)]sin(2 st)12i[ (u+s) (u s)]The Fourier transform : Examples , Properties, Common PairsConstant FunctionsSpatial Domain Frequency Domainf(t)F(u)1 (u)aa (u)The Fourier transform : Examples , Properties, Common PairsDelta FunctionsSpatial Domain Frequency Domainf(t)F(u) (t)1 The Fourier transform : Examples , Properties, Common PairsSquare PulseSpatial DomainFrequency Domainf(t)F(u){1 if a/2 t a/20 otherwisesinc(a u) =sin(a u)a uThe Fourier transform : Examples , Properties, Common PairsSquare PulseThe Fourier transform : Examples , Properties, Common PairsTriangleSpatial DomainFrequency Domainf(t)F(u){1 |t|if a t a0 otherwisesinc2(a u)The Fourier transform .}}

3 Examples , Properties, Common PairsCombSpatial Domain Frequency Domainf(t)F(u) (tmodk) (umod 1/k)The Fourier transform : Examples , Properties, Common PairsGaussianSpatial Domain Frequency Domainf(t)F(u)e t2e u2 The Fourier transform : Examples , Properties, Common PairsDifferentiationSpatial Domain Frequency Domainf(t)F(u)ddt2 iuThe Fourier transform : Examples , Properties, Common PairsSome Common Fourier transform PairsSpatial DomainFrequency Domainf(t)F(u)Cosinecos(2 st)Deltas12[ (u+s) + (u s)]Sinesin(2 st)Deltas12i[ (u+s) (u s)]Unit1 Delta (u)ConstantaDeltaa (u)Delta (t)Unit1 Comb (tmodk)Comb (umod 1/k)The Fourier transform : Examples , Properties, Common PairsMore Common Fourier transform PairsSpatial DomainFrequency Domainf(t)F(u)Square1 if a/2 t a/20 otherwiseSincsinc(a u)Triangle1 |t|if a t a0otherwiseSinc2sinc2(a u)Gaussiane t2 Gaussiane u2 DifferentiationddtRamp2 iuThe Fourier transform : Examples , Properties, Common PairsProperties.

4 NotationLetFdenote the Fourier Transform: F=F(f)LetF 1denote the Inverse Fourier Transform: f=F 1(F)The Fourier transform : Examples , Properties, Common PairsProperties: LinearityAdding two functions together adds their Fourier Transforms together:F(f+g) =F(f) +F(g)Multiplying a function by a scalar constant multiplies its FourierTransform by the same constant:F(af) =aF(f)The Fourier transform : Examples , Properties, Common PairsProperties: TranslationTranslating a function leaves the Magnitude unchanged and adds aconstant to the (t a)F1=F(f1)F2=F(f2)then|F2|=|F1| (F2) = (F1) 2 uaIntuition: Magnitude tells you how much , Phase tells you where.

5 The Fourier transform : Examples , Properties, Common PairsChange of Scale: Square Pulse RevisitedThe Fourier transform : Examples , Properties, Common PairsRayleigh s TheoremTotal energy (sum of squares) is the same in either domain: |f(t)|2dt= |F(u)|2du