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Signal Processing Course: Fourier Processing

Gabriel Peyré
January 01, 2012

Signal Processing Course: Fourier Processing

Gabriel Peyré

January 01, 2012
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  1. Fourier
    Processing
    Gabriel Peyré
    http://www.ceremade.dauphine.fr/~peyre/numerical-tour/

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  2. Overview
    •Continuous Fourier Basis
    •Discrete Fourier Basis
    •Sampling
    •2D Fourier Basis
    •Fourier Approximation

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  3. m
    (x) = em
    (x) = e2i mx
    Continuous Fourier Bases
    Continuous Fourier basis:

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  4. m
    (x) = em
    (x) = e2i mx
    Continuous Fourier Bases
    Continuous Fourier basis:

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  5. Fourier and Convolution

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  6. Fourier and Convolution
    x− x+
    x 1
    2
    1
    2
    f
    f
    ∗1[− 1
    2
    , 1
    2
    ]

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  7. Fourier and Convolution
    x− x+
    x 1
    2
    1
    2
    f
    f
    ∗1[− 1
    2
    , 1
    2
    ]
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1

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  8. Fourier and Convolution
    x− x+
    x 1
    2
    1
    2
    f
    f
    ∗1[− 1
    2
    , 1
    2
    ]
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1

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  9. Overview
    •Continuous Fourier Basis
    •Discrete Fourier Basis
    •Sampling
    •2D Fourier Basis
    •Fourier Approximation

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  10. Discrete Fourier Transform

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  11. Discrete Fourier Transform

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  12. Discrete Fourier Transform
    ˆ
    g[m] = ˆ
    f[m] · ˆ
    h[m]

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  13. Discrete Fourier Transform
    ˆ
    g[m] = ˆ
    f[m] · ˆ
    h[m]

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  14. Overview
    •Continuous Fourier Basis
    •Discrete Fourier Basis
    •Sampling
    •2D Fourier Basis
    •Fourier Approximation

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  15. Infinite continuous domains:
    Periodic continuous domains:
    Infinite discrete domains:
    Periodic discrete domains:
    f0
    (t), t R
    f0
    (t), t ⇥ [0, 1] R/Z
    The Four Settings
    ˆ
    f[m] =
    N 1
    n=0
    f[n]e 2i
    N
    mn
    ˆ
    f0
    ( ) =
    +⇥

    f0
    (t)e i tdt
    ˆ
    f0
    [m] =
    1
    0
    f0
    (t)e 2i mtdt
    ˆ
    f( ) =
    n Z
    f[n]ei n
    Note: for Fourier, bounded periodic.
    .. . .. .
    .. .
    .. .
    f[n], n Z
    f[n], n ⇤ {0, . . . , N 1} ⇥ Z/NZ

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  16. ˆ
    f[m] =
    N 1
    n=0
    f[n]e 2i
    N
    mn
    Fourier Transforms
    Discrete
    Infinite Periodic
    f[n], n Z f[n], 0 n < N
    Periodization
    Continuous
    f0
    (t), t R f0
    (t), t [0, 1]
    f0
    (t) ⇥
    n
    f0
    (t + n)
    Sampling
    ˆ
    f0
    ( ) ⇥ { ˆ
    f0
    (k)}k
    Discrete
    Infinite
    Periodic
    Continuous
    Sampling
    ˆ
    f[k], 0 k < N
    ˆ
    f0
    ( ), R ˆ
    f0
    [k], k Z
    Fourier transform
    Isometry f ⇥ ˆ
    f
    ˆ
    f0
    ( ) =
    +⇥

    f0
    (t)e i tdt
    ˆ
    f0
    [m] =
    1
    0
    f0
    (t)e 2i mtdt
    ˆ
    f( ) =
    n Z
    f[n]ei n
    ˆ
    f(⇥), ⇥ [0, 2 ]
    Periodization
    ˆ
    f(⇥) =
    k
    ˆ
    f0
    (N(⇥ + 2k )) f[n] = f0
    (n/N)

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  17. Sampling and Periodization
    (a)
    (c)
    (d)
    (b)
    1
    0

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  18. Sampling and Periodization: Aliasing
    (b)
    (c)
    (d)
    (a)
    0
    1

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  19. Uniform Sampling and Smoothness

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  20. Uniform Sampling and Smoothness

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  21. Uniform Sampling and Smoothness

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  22. Uniform Sampling and Smoothness

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  23. Overview
    •Continuous Fourier Basis
    •Discrete Fourier Basis
    •Sampling
    •2D Fourier Basis
    •Fourier Approximation

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  24. 2D Fourier Basis
    em
    [n] =
    1
    N
    e
    2i
    N0
    m1n1+ 2i
    N0
    m2n2 = em1
    [n1
    ]em2
    [n2
    ]

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  25. 2D Fourier Basis
    em
    [n] =
    1
    N
    e
    2i
    N0
    m1n1+ 2i
    N0
    m2n2 = em1
    [n1
    ]em2
    [n2
    ]

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  26. Overview
    •Continuous Fourier Basis
    •Discrete Fourier Basis
    •Sampling
    •2D Fourier Basis
    •Fourier Approximation

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  27. 1D Fourier Approximation

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  28. 1D Fourier Approximation
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1

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  29. 2D Fourier Approximation

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