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Fourier TransformΒΆ

Complex form or exponential form of Fourier Integral

𝔉[f(t)]=βˆ«βˆ’βˆžβˆžeβˆ’iΟ‰t f(t) dt

assume that

βˆ«βˆ’βˆžβˆž|f(x)|dx

converges

  • Comparison

    1. compare to Laplace transform

      sβ†’iΟ‰βˆ«0βˆžβ†’βˆ«βˆ’βˆžβˆž
    2. compare to Fourier Series Fourier Transform can be regarded as Fourier series whose pβ†’βˆž.


Amplitude SpectrumΒΆ

The amplitude spectrum of f(t) is a graph of |𝔉[f(Ο‰)]|


Fourier & Inverse Fourier TransformΒΆ

  • Fourier transform

    𝔉[f(x)]=βˆ«βˆ’βˆžβˆžeβˆ’iΟ‰x f(x) dx
  • inverse Fourier transform

    π”‰βˆ’1[F(Ο‰)]=12Ο€βˆ«βˆ’βˆžβˆžF(Ο‰) eiΟ‰x dΟ‰=f(x)

Fourier Sine TransformΒΆ

  • Fourier sine transform

    𝔉s[f(x)]=∫0∞f(x) sin⁑(Ο‰x) dx=F(Ο‰)
  • inverse Fourier sine transform

    𝔉sβˆ’1[F(Ο‰)]=2Ο€βˆ«0∞F(Ο‰)sin⁑(Ο‰x) dΟ‰=f(x)

Fourier Cosine TransformΒΆ

  • Fourier cosine transform
𝔉c[f(x)]=∫0∞f(x)cos⁑(Ο‰x) dx=F(Ο‰)
  • inverse Fourier cosine transform
𝔉cβˆ’1[F(Ο‰)]=2Ο€βˆ«0∞F(Ο‰) cos⁑(Ο‰x) dΟ‰=f(x)

Fourier Transform propertiesΒΆ

LinearityΒΆ

Ξ±,Ξ²βˆˆβ„π”‰[Ξ±f(t)+Ξ²g(t)]=α𝔉[f(t)]+𝔉[Ξ²g(t)]

Time ShiftingΒΆ

𝔉[f(tβˆ’t0)]=eβˆ’iΟ‰t0 π”‰[f(t)]
  • Inverse Time Shifting

    π”‰βˆ’1[eβˆ’iΟ‰t0F(Ο‰)](t)=f(tβˆ’t0)

Frequency ShiftingΒΆ

𝔉[eiΟ‰0t f(t)]=F(Ο‰βˆ’Ο‰0)
  • Inverse Frequency Shifting

    π”‰βˆ’1[F(Ο‰βˆ’Ο‰0)]=eiΟ‰0t f(t)

ScalingΒΆ

𝔉[f(ct)]=1|c|F(Ο‰c)
  • Inverse Scaling
π”‰βˆ’1[F(Ο‰c)]=|c| f(ct)

Time ReversalΒΆ

𝔉[f(βˆ’t)]=F(βˆ’Ο‰)π”‰βˆ’1[F(βˆ’Ο‰)]=f(βˆ’t)

SymmetryΒΆ

𝔉[F(t)]=2Ο€f(βˆ’Ο‰)

modulationΒΆ

𝔉[f(t)cos⁑(Ο‰0t)]=12[F(Ο‰+Ο‰0)+F(Ο‰βˆ’Ο‰0)]𝔉[f(t)sin⁑(Ο‰0t)]=i2[F(Ο‰+Ο‰0)βˆ’F(Ο‰βˆ’Ο‰0)]

particular exampleΒΆ

prof. Laplace Transform

𝔉[H(t)eβˆ’at]=1a+iΟ‰

Frequency DifferentiationΒΆ

𝔉[tnf(t)]=(i)nF(n)

Transforms of DerivativesΒΆ

  1. for Fourier transform

    𝔉[fβ€²(x)]=iΟ‰F(Ο‰)𝔉[f(n)]=(iΟ‰)n F(Ο‰)
  2. for Fourier sine transform

    𝔉s[fβ€²(x)]=βˆ’Ο‰ π”‰c[f(x)]𝔉s[fβ€³(x)]=βˆ’Ο‰2 π”‰s[f(x)]+Ο‰f(0)
  3. for Fourier cosine transform

    𝔉c[fβ€²(x)]=ω𝔉s[f(x)]βˆ’f(0)𝔉c[fβ€³(x)]=βˆ’Ο‰2𝔉c[f(x)]βˆ’fβ€²(0)

Transform of IntegralΒΆ

𝔉[βˆ«βˆ’βˆžtf(Ο„)dΟ„]=1iΟ‰F(Ο‰)

ConvolutionΒΆ

  • Definition
f(t)βˆ—g(t)=βˆ«βˆ’βˆžβˆžf(Ο„) g(tβˆ’Ο„) dΟ„
  • Property
fβˆ—g=gβˆ—f(Ξ±f+Ξ²g)βˆ—h=Ξ±(fβˆ—h)+Ξ²(gβˆ—h)
  • Theorem
βˆ«βˆ’βˆžβˆž[f(t)βˆ—g(t)] dt=βˆ«βˆ’βˆžβˆžf(t)dtβ‹…βˆ«βˆ’βˆžβˆžg(t)dt𝔉[f(t)βˆ—g(t)]=F(Ο‰) G(Ο‰)𝔉[f(t) g(t)]=12Ο€F(Ο‰)βˆ—G(Ο‰)

Dirac Delta FunctionΒΆ

  • Definition

    Ξ΄(t)=limaβ†’0[12aH(t+a)βˆ’H(tβˆ’a)]

Fourier TransformΒΆ

𝔉[Ξ΄(t)]=𝔉[limaβ†’0[12aH(t+a)βˆ’H(tβˆ’a)]]=limaβ†’0sin⁑(aΟ‰)aΟ‰=1

FilteringΒΆ

βˆ«βˆ’βˆžβˆžf(t)Ξ΄(tβˆ’t0) dt=f(t0)

Discrete Fourier TransformΒΆ

Let u={uj}j=0Nβˆ’1

D[u]=Uk=βˆ‘j=0Nβˆ’1ujexp⁑(βˆ’i Ο‰0j TN)=βˆ‘j=0Nβˆ’1ujexp⁑(βˆ’i 2Ο€ j 1N)

LinearityΒΆ

D[au+bv]=aUk+bVk

PeriodicityΒΆ

Uk+N=Uk

Inverse N-point DFTΒΆ

uj=1Nβˆ‘j=0Nβˆ’1Ukexp⁑(i Ο‰0j TN)=1Nβˆ‘j=0Nβˆ’1Ukexp⁑(i 2Ο€ j 1N)

Complex Fourier CoefficientsΒΆ

dkβ‰ˆ1Nβ‹…Uk

note that the coefficient of complex Fourier Cω

CΟ‰=12π⋅𝔉[f(t)]

Sampled Fourier SeriesΒΆ

SM(jTn)β‰ˆ1Nβˆ‘k=0Nβˆ’1Vk eiΟ‰0jkT/N

in which

Vk={Ukfor k=0,1,…,M0for k=M+1,…,Nβˆ’M+1Ukfor k=Nβˆ’M,…,Nβˆ’1

Solving the BVPΒΆ

  • BVP (Boundary Value Problem)

There are three possible condition below

  1. βˆ’βˆž<x<∞

    β‡’ Fourier transform

  2. 0<x<∞ and u(x,y)|x=0=0

    β‡’ Fourier sine transform

  3. 0<x<∞ and βˆ‚βˆ‚xu(x,y)|x=0=0

    β‡’ Fourier cosine transform


Method of Separation of VariableΒΆ

kernel : assume that 1.

u(x,y)=X(x) Y(y)

2.

"f(x)" = "f(y)" = βˆ’Ξ»