Fast Fourier Transforms lecture notes
Lecture 11 Fast Fourier Transform (FFT)
Weinan E1,2 and Tiejun Li2
1
Department of Mathematics,
Princeton University, weinan@princeton.edu 2
School of Mathematical Sciences,
Peking University, tieli@pku.edu.cn No.1 Science Building, 1575
Examples
Fast Fourier Transform
Outline
Examples
Fast Fourier Transform
Applications
Applications
Examples
Fast Fourier Transform
Applications
Signal processing
Filtering: a polluted signal
1.5
1
0.5
0
−0.5
−1
−1.5
0
200
400
600
800
1000
1200
High pass and low pass filter (signal and noise)
1.5
1
0.5
0
−0.5
−1
−1.5
0
200
400
600
800
1000
1200
How to obtain the high frequency and low frequency quickly?
Examples
Fast Fourier Transform
Solving PDEs on rectangular mesh
Solving the Poisson equations
−∆u = f in Ω u = 0 on ∂Ω in the rectangular domain
After discretization we will obtain the linear system with about N 2 unknowns −
ui+1,j + ui−1,j + ui,j+1 + ui,j−1 − 4ui,j
= fij
4h2
The FFT would give a fast algorithm to solve the system above with computational efforts O(N 2 log2 N ).
Applications
Examples
Fast Fourier Transform
Applications
Computing convolution (
)
Suppose
2π
f (x − y)g(y)dy
h(x) =
0
is the convolution of f and g, where f (x), g(x) ∈ C2π are period 2π functions. Take xj = jδ, j = 0, 1, . . . , N − 1, δ =
2π
N
and apply simple rectangular
discretization
N −1
f (xi − xj )g(xj ) · δ
h(xi ) ≈
i = 0, 1, . . . , N − 1
j=0
Define fi = f (xi ), gi = g(xi ), and let fi is period N respect to the subscript i, define
N −1
fi−j gj · δ
hi = j=0 The direct computation is O(N 2 ).
i = 0, 1, . . . , N − 1
Examples
Fast Fourier Transform
Fast Fourier Transform
Fast Fourier Transform is one of the top 10 algorithms in 20th century.
But its idea is quite simple, even for a high school student!
Applications
Examples
Fast Fourier Transform
Outline
Examples
Fast Fourier Transform
Applications
Applications
Examples
Fast Fourier Transform
Weinan E1,2 and Tiejun Li2
1
Department of Mathematics,
Princeton University, weinan@princeton.edu 2
School of Mathematical Sciences,
Peking University, tieli@pku.edu.cn No.1 Science Building, 1575
Examples
Fast Fourier Transform
Outline
Examples
Fast Fourier Transform
Applications
Applications
Examples
Fast Fourier Transform
Applications
Signal processing
Filtering: a polluted signal
1.5
1
0.5
0
−0.5
−1
−1.5
0
200
400
600
800
1000
1200
High pass and low pass filter (signal and noise)
1.5
1
0.5
0
−0.5
−1
−1.5
0
200
400
600
800
1000
1200
How to obtain the high frequency and low frequency quickly?
Examples
Fast Fourier Transform
Solving PDEs on rectangular mesh
Solving the Poisson equations
−∆u = f in Ω u = 0 on ∂Ω in the rectangular domain
After discretization we will obtain the linear system with about N 2 unknowns −
ui+1,j + ui−1,j + ui,j+1 + ui,j−1 − 4ui,j
= fij
4h2
The FFT would give a fast algorithm to solve the system above with computational efforts O(N 2 log2 N ).
Applications
Examples
Fast Fourier Transform
Applications
Computing convolution (
)
Suppose
2π
f (x − y)g(y)dy
h(x) =
0
is the convolution of f and g, where f (x), g(x) ∈ C2π are period 2π functions. Take xj = jδ, j = 0, 1, . . . , N − 1, δ =
2π
N
and apply simple rectangular
discretization
N −1
f (xi − xj )g(xj ) · δ
h(xi ) ≈
i = 0, 1, . . . , N − 1
j=0
Define fi = f (xi ), gi = g(xi ), and let fi is period N respect to the subscript i, define
N −1
fi−j gj · δ
hi = j=0 The direct computation is O(N 2 ).
i = 0, 1, . . . , N − 1
Examples
Fast Fourier Transform
Fast Fourier Transform
Fast Fourier Transform is one of the top 10 algorithms in 20th century.
But its idea is quite simple, even for a high school student!
Applications
Examples
Fast Fourier Transform
Outline
Examples
Fast Fourier Transform
Applications
Applications
Examples
Fast Fourier Transform