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Lecture 3. Fourier Transform. f(t) – Function F( w ) – Fourier harmonic Time / Frequency exp(i k r) Co-ordinate / Wave-number Temporal (Spatial) spectrum: Spectral Analysis F = F( w ) : Spectral Distribution, Spectrum. Sunspot Data. Time Series Spectrum. FT: Properties. - PowerPoint PPT Presentation
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Fourier Transformf(t) – Function
F() – Fourier harmonic
Time / Frequency
exp(i k r)
Co-ordinate / Wave-number
Temporal (Spatial) spectrum: Spectral Analysis
F = F() : Spectral Distribution, Spectrum
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Discrete Fourier Transform
Continuous: (- … )
Discrete: ( h1 … hN )
k = 0 … (N-1)
Discrete Spectrum
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DFT constraints
Length-Scales
Domain Size: L = X1-XN
Step Size: X = Xk – Xk-1
Wave-Numbers:Maximal wave-number: K = 2/ XMinimal wave-number: K = 2/ L
Number of Fourier Harmonics: N(sampling rate, resolution)
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Fast Fourier Transform
DFT:
O(N2) calculation process
FFT Algorithm: N = 2m
O(N log2N) calculation process
(Cooley, Turkey 1960, … Gauss 1805)
N = 106 FFT : 30sec
FT : 2 week
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PDE: Spectral MethodPDE:
Fourier decomposition in SPACE
ODE solvera = a(k,t)
Inverse Fourier transformA = A(x,t)
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Spectral Simulations
Discrteize PDE: Sampling
DFT: mostly FFT
1. Transform PDE to spectral ODE
2. Solve ODE (e.g., R-K)
3. Inverse transform to reconstruct solutions
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Spectral Method: Features
Initial Value Problem- Calculate initial values in k-space
Boundary Value Problem- Integrate boundaries into k-space
Spatial Inhomogeneities- Introduce numerical variables to homogenize- Integrate during reconstruction
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Spectral Method: Problems
1. ShocksDiscontinuity: 0
Kcr = 1/ 2 Kmax < Kcr
2. Complex BoundariesIll-known numerical instabilities;
3. Nonlinearities
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Comparison
Spectral Method:
Linear combination of continuous functions;
Global approach;
Finite Difference, Flux conservation:
Array of piecewise functions;
Local approach;
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+ / -
+ (very) fast for smooth solutions
+ Exponential convergence
+ Best for turbulent spectrum
- Shocks
- Inhomogeneities
- Complex Boundaries
- Need for serial reconstruction (integration)