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「フーリエ級数」から「高速フーリエ変換」まで全部やります!【2019.07.20更新】

kenyu
July 09, 2019
81k

 「フーリエ級数」から「高速フーリエ変換」まで全部やります!【2019.07.20更新】

このスライドでは,
・フーリエ級数
・複素フーリエ級数
・フーリエ変換(連続)
・離散フーリエ変換(DFT)
・高速フーリエ変換(FFT)
を解説しています.

ブログはこちら
【フーリエ解析05】高速フーリエ変換(FFT)とは?内側のアルゴリズムを解説!【解説動画付き】
https://kenyu-life.com/2019/07/08/what_is_fft/

Twitter → https://twitter.com/kenyu0501_?lang=ja

Youtube → https://youtu.be/zWkQX58nXiw

kenyu

July 09, 2019
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Transcript

  1. ͚ΜΏʔ!LFOZV@ 5XJUUFS

     山口大学大学院 博士課程/ 学術研究員/
    ৴߸ॲཧΛֶ΅͏ʂ
    dϑʔϦΤղੳฤʂʂd ྑ͚Ε͹ϑΥϩʔ
    ͯ͠Լ͍͞
    ໨࣍
    ೾ܗʹ͍ͭͯ෮श
    ϑʔϦΤڃ਺
    ෳૉϑʔϦΤڃ਺
    ࿈ଓͳϑʔϦΤม׵
    ཭ࢄϑʔϦΤม׵ %'5

    ߴ଎ϑʔϦΤม׵ ''5

    ͜ΕΒʹ͍ͭͯ΍͍͖ͬͯ·͢ʂ
    جૅ͔Βͱ͜ͱΜղઆ͍͖ͯ͠·
    ͢ɽ

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  2. ͪ͜ΒͷຊΛࢀߟʹͯ͠·͢
    ࢀߟࢿྉ
    ಓ۩ͱͯ͠ͷϑʔϦΤղੳ
    ༚Ҫྑ޾༚Ҫఃඒ
    Ϟʔυղੳೖ໳
    ௕দতஉ
    Ͳͬͪ΋ྑॻͰ͢

    View full-size slide

  3. ৴߸ʹ͍ͭͯ ϑʔϦΤڃ਺΁ͷ༠͍

    x
    y
    f(x, y)
    ۭؒతͳؔ਺ɿը૾ͱ͔
    t
    ۭ࣌ؒతͳؔ਺ɿө૾
    f(x, y, t)
    f(t)
    ࣌ؒతͳؔ਺ɿԹ౓ͱ͔
    f(t)
    ࣍ݩͷ৔߹Λߟ͑ΔͱେมͳͷͰɼ࣍ݩͰߟ͑Δ͜ͱʹ͢Δ
    ৭ʑͳ৴߸ʹ͍ͭͯ
    ࣍ݩͷؔ਺Λʮແݶݸͷࡾ֯ؔ਺ͷ࿨Ͱදݱ͢Δʯ
    ෳࡶ ͔΋͠Εͳ͍Α͏
    ͳؔ਺ˠ୯७ͳؔ਺Λ଍ͨ͠΋ͷ
    ෳࡶͳ৴߸Λཧղ͢ΔͨΊͷπʔϧʂͲͷΑ͏ͳಛੑ͕͋Δͷ͔͕Θ͔Δ

    View full-size slide

  4. पظT

    f =
    1
    T
    पظؔ਺
    t + T
    t
    ࣍ݩͷ࣌ؒؔ਺ ೾ܗ
    ʹؔ͢Δجຊ༻ޠͷ෮श
    प೾਺ ৼಈ਺
    f
    ɹ୯Ґ࣌ؒʹؚ·ΕΔपظT ͷݸ਺
    f(t + T) = f(t)
    ϑʔϦΤม׵ޙͷۭؒͰ͋Δप೾਺ྖҬ͕
    ৴߸ॲཧΛߦ͏্Ͱͱͯ΋େࣄͳ΋ͷͰ͋Δ
    ৴߸ʹ͍ͭͯ ϑʔϦΤڃ਺΁ͷ༠͍

    ͜ͷࢿྉͰ͸ɼ࣌ؒతͳؔ਺ ೾
    Λѻ͍·͢ɽ
    ͦͷͨΊɼجຊతͳ༻ޠ΍ه߸ʹ͍ͭͯ෮श͓͖ͯ͠·͠ΐ͏ʂ
    ೾ܗͷॲཧΛ͢Δ্Ͱ͔ͳΓେࣄͳ֓೦Λͪΐͬͱ͚ͩઌʹऔΓ্͛·͢ɽ

    View full-size slide

  5. θ
    cos θ
    sin θ
    P(x, y)
    x
    y ϑʔϦΤม׵͸༩͑Βͨؔ਺Λࡾ֯ؔ਺ ਖ਼ݭ೾
    Ͱද͢
    ਖ਼ݭ೾͸ԁͱਂؔ͘ΘΔ
    ВΛಈతʹଊ͑ɼಈܘOP͕୯Ґ࣌ؒ͋ͨΓʹਐΉ֯Λ֯प೾਺ω ͱ͢Δͱɼ
    ҎԼͷؔ܎͕੒Γཱͭ
    O
    y = sin θ
    x = cos θ
    θ = ωt
    ֯प೾਺ω Ͱಈܘ͕ճస͢Δͱ͖ɼ
    ͦͷಈܘͷ୯Ґ࣌ؒ͋ͨΓͷճస਺͸ৼಈ਺ प೾਺
    fͳͷͰ
    f =
    ω

    ֯प೾਺ω ͸पظT ͱͷؔ܎ͩͱɼҎԼʹͳΔ
    f =
    1
    T
    ω =

    T
    ୯Ґԁ
    ֯प೾਺
    ৴߸ʹ͍ͭͯ ϑʔϦΤڃ਺΁ͷ༠͍

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  6. ϑʔϦΤղੳɹͦͷ
    dϑʔϦΤڃ਺ɾ௚ަجఈd
    f(t) = a0
    + a1
    cos ω0
    t + b1
    sin ω0
    t + a2
    cos 2ω0
    t + b2
    sin 2ω0
    t + ⋯
    ◼ϑʔϦΤڃ਺ͬͯԿʁʁʁͬͯਓ

    2
    T
    − 2
    T
    1 ⋅ sin nω0
    tdt =

    2
    T
    − 2
    T
    1 ⋅ cos nω0
    tdt = 0

    2
    T
    − 2
    T
    sin nω0
    t cos nω0
    tdt = 0
    ◼௚ަجఈͬͯԿʁʁʁͬͯਓ
    f(t) = a0
    +


    n=1
    {an
    cos
    2πn
    T0
    t + bn
    sin
    2πn
    T0
    t}
    ͚ΜΏʔ!LFOZV@ 5XJUUFS

     山口大学大学院 博士課程/ 学術研究員/

    View full-size slide

  7. ϑʔϦΤڃ਺ͱ͸
    ਺΍ؔ਺ͷྻΛແݶʹՃ͑߹Θͤͨ΋ͷ
    f(t)
    a0
    a1
    cos ω0
    t
    a2
    cos 2ω0
    t
    b2
    sin 2ω0
    t
    b1
    sin ω0
    t
    ɾ
    ɾ
    ɾ
    ɾ
    ɾ
    ɾ
    ؔ਺f(t) ͕͋Δ஋ͷഒ਺ͷ֯प೾਺ω0
    Λ΋ͭਖ਼ݭ೾ʹ෼ղ͞ΕΔ
    n = 0
    n = 1
    n = 2
    f(t) = a0
    + a1
    cos ω0
    t + b1
    sin ω0
    t + a2
    cos 2ω0
    t + b2
    sin 2ω0
    t + ⋯
    ϑʔϦΤڃ਺Ͱද͍ͨؔ͠਺
    ͦͷؔ਺͔ΒܾΊΒΕΔఆ਺
    ֯प೾਺͸ఆ਺ഒʹͳΔˠ ഒͱ͔ɼഒͱ͔ɼ൒୺ͳ਺ࣈʹͳΒͳ͍

    ϑʔϦΤڃ਺
    ɹɹɹɾɾɾʮؔ਺͕ແݶݸͷࡾ֯ؔ਺ͷ࿨ͰදΘͤΔʯ
    ɹɹɹɾɾɾʮάϥϑ͕ਖ਼ݭ೾ͷॏͶ߹ΘͤͰදΘͤΔʯ

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  8. ϑʔϦΤڃ਺ͷجຊपظͱجຊप೾਺ʹ͍ͭͯ
    ɹɹɹɾɾɾʮجຊप೾਺ͷ੔਺ഒͷਖ਼ݭ೾͔͠ग़ͯ͜ͳ͍͜ͱʹͳΔʯ
    ϑʔϦΤڃ਺ͷجຊप೾਺ͷܾ·Γ
    a0
    a1
    cos ω0
    t
    a2
    cos 2ω0
    t
    b2
    sin 2ω0
    t
    b1
    sin ω0
    t
    ɾ
    ɾ
    ɾ
    ɾ
    ɾ
    ɾ
    n = 0
    n = 1
    n = 2
    f(t)
    جຊपظ͕TͳΒɼجຊप೾਺͸)[ʹͳΔɽ

    ͦͷ੔਺ഒ͔͠ݱΕͳ͍ͷͰɼ)[ )[ )[ͷ੒෼ɼɼɼʹͳΔɽ
    ͳͥͳΒɼجຊपظͷதͰɼ੔਺ݸͷ೾͕ऩ·Βͳ͍ͱ͍͚ͳ͍ͨΊʂ
    ଍͠߹Θͤͯपظతͳؔ਺ʹ͢ΔͨΊʂ
    جຊपظT0
    ω0
    =

    T0
    f0
    =
    1
    T0
    جຊपظT0
    ʹΑͬͯɼ
    جຊप೾਺f0
    ͕ܾ·Δ
    جຊप೾਺f0
    ʹΑͬͯ
    جຊ֯प೾਺ω0
    ͕ܾ·Δ

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  9. ϑʔϦΤڃ਺ల։ͷΠϝʔδਤ
    f(t)
    t
    a0
    a1
    a2
    b2
    b1
    a3
    b3
    ɾ

    ɾ

    ɾ
    v =
    n
    T
    ωn
    =
    2nπ
    T
    ֯प೾਺
    प೾਺
    1, cos
    2nπt
    T
    , sin
    2nπt
    T
    1, cos ωn
    t, sin ωn
    t
    ࣌ؒྖҬ
    प೾਺ྖҬ
    ϑʔϦΤڃ਺ͷ࿨Λߏ੒͢Δجຊ೾


    ͱ

    ֯
    ؔ

    ͷ

    ܎

    Λ
    ܾ
    Ί
    Δ
    cos ω0
    t
    cos 2ω0
    t
    sin 2ω0
    t
    sin ω0
    t
    cos 3ω0
    t
    sin 3ω0
    t
    ఆ਺
    ͷࡾ֯ؔ਺
    ω0
    = 0
    ͷࡾ֯ؔ਺

    View full-size slide

  10. ϑʔϦΤڃ਺ͷཧ࿦ͱ܎਺ʹ͍ͭͯ
    f(t)
    T
    2

    T
    2
    ͷؒͷؔ਺Λߟ͑Δ

    T
    2
    ≤ t ≤
    T
    2
    ϑʔϦΤڃ਺͸༗ݶ۠ؒͰߟ͑Δɽର৅͸पظؔ਺Ͱ͋Δɽ
    f(t) = a0
    + (a1
    cos
    2πt
    T
    + b1
    sin
    2πt
    T
    ) + (a2
    cos
    4πt
    T
    + b2
    sin
    4πt
    T
    ) + ⋯ + (an
    cos
    2nπt
    T
    + bn
    sin
    2nπt
    T
    ) + ⋯
    ෯͕T Ͱ͋Ε͹ԿͰ΋ྑ͍

    ɾ
    ɾ
    ɾ
    a0
    =
    1
    T ∫
    2
    T
    − 2
    T
    f(t)dt an
    =
    2
    T ∫
    2
    T
    − 2
    T
    f(t)cos
    2nπt
    T
    dt
    bn
    =
    2
    T ∫
    2
    T
    − 2
    T
    f(t)sin
    2nπt
    T
    dt
    ϑʔϦΤڃ਺ͷཧ࿦
    ϑʔϦΤڃ਺ల։ɾɾɾؔ਺ΛϑʔϦΤڃ਺Ͱද͢͜ͱ
    Ͱఆٛ͞ΕͨϑʔϦΤ܎਺ a0, a1, a2, ɾɾɾ, b1, b2, ɾɾɾ
    ΛٻΊΔ

    T
    2
    ≤ t ≤
    T
    2
    ˠޙ΄Ͳৄ͘͠ಋग़͢Δ
    f(t) = a0
    +


    n=1
    {an
    cos
    2πn
    T0
    t + bn
    sin
    2πn
    T0
    t}

    View full-size slide

  11. ϑʔϦΤڃ਺ల։͞Εͨؔ਺͸पظT Ͱ܁Γฦ͢
    f(t) = a0
    + (a1
    cos
    2πt
    T
    + b1
    sin
    2πt
    T
    ) + (a2
    cos
    4πt
    T
    + b2
    sin
    4πt
    T
    ) + ⋯ + (an
    cos
    2nπt
    T
    + bn
    sin
    2nπt
    T
    ) + ⋯
    a0
    a1
    a2
    b2
    b1
    a3
    b3
    ɾ

    ɾ

    ɾ
    पظT ͷपظؔ਺ ܁Γฦ͢

    f(t)
    T
    2

    T
    2
    a0
    a1
    a2
    b2
    b1
    a3
    b3
    ɾ

    ɾ

    ɾ
    a0
    a1
    a2
    b2
    b1
    a3
    b3
    ɾ

    ɾ

    ɾ
    ϑʔϦΤڃ਺͸ɼपظT ͷؔ਺ʹ͢Δ͜ͱ
    f(t)
    ϑʔϦΤڃ਺͸पظT ͷपظؔ਺Ͱ͋Δ

    View full-size slide

  12. ؔ਺f(t) Λ௚ަ͢ΔجఈͱͳΔ
    ؔ਺
    Ͱදͨ͠ͷ͕ϑʔϦΤڃ਺
    ϑʔϦΤڃ਺ͷ௚ަجఈͷΠϝʔδਤ
    ɾؔ਺Λۭؒͷ఺ͱ͢Δ
    ɾ఺Λࢦ͢ϕΫτϧ͕ʮجఈʯͱݺ͹ΕΔ૊ͷϕΫτϧͷҰ࣍݁߹ʹͳΔ
    ฏ໘ϕΫτϧ͸ɼ௚ަ͢
    ΔʮجఈϕΫτϧʯͷҰ
    ࣍݁߹Ͱද͞ΕΔ
    P
    e1
    e2
    O
    OP = a1
    e1
    + a2
    e2
    f(t) = a0
    + ⋯ + an
    cos nω0
    t + bn
    sin nω0
    t + ⋯
    f(t)
    cos ω0
    t
    sin ω0
    t
    1, cos nω0
    t, sin nω0
    t
    ϑʔϦΤڃ਺ͷཧ࿦͸ɼؔ਺ۭؒͰΠϝʔδ͢Δͱ෼͔Γ΍͍͢

    View full-size slide

  13. ௚ަੑͷؔ܎ɿؔ਺ͷੵͷੵ෼=಺ੵ
    e1
    e2
    e1
    ⋅ e2
    = 0

    a
    b
    f(t)g * (t)dt
    ؔ਺ͷ಺ੵ (b ≤ t ≤ a)
    ؔ਺ͷ಺ੵͬͯ
    ฏ໘ϕΫτϧͱ਺ֶతʹҰॹ
    g * (t) = g(t)
    ͕࣮਺ͷ৔߹
    g(t)
    ௚ަͱ͸ʂʁ
    ʮجఈʯͱͳΔؔ਺ηοτ͸௚ަੑΛ࣋ͨͳ͚Ε͹͍͚ͳ͍ɽ
    ೋͭͷϕΫτϧ͕௚ަͰ͋Δͱ͖ɼ಺ੵ͸
    ؔ਺ͷ಺ੵͬͯͲ͏ॻ͘ͷ͔ʂʁ
    Ҏ্ͷؔ਺ͷੵ෼͕ͷͱ͖ɼ֤ؔ਺͸௚ަ͍ͯ͠Δ
    ΞελϦεΫɹ͸ɼෳૉڞ໾ɽ
    ෳૉڞ໾ΛऔΔͱ͍͏͜ͱ͸ɼෳ
    ૉฏ໘্Ͱɼ࣮࣠ʹରͯ͠ର৅ͳ
    Ґஔʹಈ͔͢ɽ
    *
    ϑʔϦΤڃ਺Ͱ͸ؔ܎ͳ͍ɽ
    ෳૉϑʔϦΤڃ਺Ͱେࣄɽ

    View full-size slide


  14. a
    b
    f(t)g(t)dt = 0
    ௚ަੑ (b ≤ t ≤ a)
    ؔ਺ͷ಺ੵ͸ͳͥੵ෼Λ͢Δͷ͔ʂʁ
    ʮ֤ؔ਺ͷֻ͚ࢉͷੵ෼஋= 0Ͱ௚ަ͍ͯ͠ΔʯΛղऍ͢Δ
    ֤ؔ਺Λʮແݶ࣍ݩϕΫτϧʯͱݟͳ͢
    f(t) = ( f1
    , f2
    , f3
    , f4
    , f5
    , f6
    , f7
    )
    g(t) = (g1
    , g2
    , g3
    , g4
    , g5
    , g6
    , g7
    )
    f(t)
    a b
    f1
    f2
    f3
    f4
    f5
    f6
    f7
    g(t)
    a b
    g1
    g2
    g3
    g4
    g5
    g6
    g7
    ྫʼ࣍ݩͰݟͳ͢ͱʜ ؔ਺ΛҎԼͷΑ͏ʹϕΫτϧͰݟͳ͢
    ಺ੵͬͯɼ֤ཁૉ͝ͱʹֻ͚ͯ
    ૯࿨ΛͱΔΑͶʁ
    7

    i=1
    fi
    gi
    ͜Μͳײ͡
    ࠓ͸࣍ݩϕΫτϧ͚ͩͲɼ͜ΕΛແݶ࣍ݩϕΫτϧ
    ͱͨ͠Βɼ૯࿨͕ʮੵ෼ʯʹͳΔɽ
    ͭ·Γɼੵ෼ͩͱɼϕΫτϧͷ಺ੵͱΈͳͤΔ
    ௚ަੑͷؔ܎ɿؔ਺ͷੵͷੵ෼಺ੵ

    View full-size slide

  15. f(t)
    a b
    f1
    ֤ؔ਺Λʮແݶ࣍ݩϕΫτϧʯͱݟͳ͢
    ྫʼ࣍ݩ ྫʼ࣍ݩ ྫʼແݶ࣍ݩ
    f(t)
    a b
    f1
    f2
    f3
    f4
    f(t)
    a b
    f1
    f2
    f3
    f4
    f5
    f6
    f7
    f(t) = ( f1
    , f2
    , f3
    , f4
    , f5
    , f6
    , f7
    )
    f(t) = ( f1
    , f2
    , f3
    , f4
    ) f(t) = ( f1
    , ⋯, f∞
    )
    f∞
    f(t) = ( f1
    , ⋯, f∞
    )
    g(t) = (g1
    , ⋯, g∞
    )
    f(t) = ( f1
    , f2
    , f3
    , f4
    , f5
    , f6
    , f7
    )
    g(t) = (g1
    , g2
    , g3
    , g4
    , g5
    , g6
    , g7
    )
    ಺ੵͬͯɼ֤ཁૉ͝ͱʹֻ͚ͯ૯࿨ΛͱΔ
    7

    i−1
    fi
    gi
    ͜Μͳײ͡
    ྫʼ࣍ݩ


    i=1
    fi
    gi ∫

    i=1
    fi
    gi
    dt
    ྫʼແݶ࣍ݩ
    ແݶ࣍ݩΛߟ͑Δͱ
    ݁ہੵ෼ʹͳΔʂ
    ϑʔϦΤڃ਺ɿؔ਺Λແݶ࣍ݩϕΫτϧͱͯ͠ද͢Πϝʔδਤ

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  16. ؔ਺f(t) Λ௚ަ͢ΔجఈͱͳΔ
    ؔ਺
    Ͱදͨ͠ͷ͕ϑʔϦΤڃ਺
    ϑʔϦΤڃ਺ͷେࣄͳࣜ
    f(t) = a0
    + ⋯ + an
    cos nω0
    t + bn
    sin nω0
    t + ⋯
    f(t)
    cos ω0
    t
    sin ω0
    t
    1, cos nω0
    t, sin nω0
    t

    2
    T
    − 2
    T
    1 ⋅ sin nω0
    tdt =

    2
    T
    − 2
    T
    1 ⋅ cos nω0
    tdt = 0

    2
    T
    − 2
    T
    sin nω0
    t cos nω0
    tdt = 0

    2
    T
    − 2
    T
    sin nω0
    t sin mω0
    tdt = 0

    2
    T
    − 2
    T
    cos nω0
    t cos mω0
    tdt = 0

    2
    T
    − 2
    T
    12dt = T

    2
    T
    − 2
    T
    sin n2ω0
    tdt =

    2
    T
    − 2
    T
    cos n2ω0
    tdt =
    T
    2
    ʮϑʔϦΤ܎਺ͷಋग़ʹඞཁͳੑ࣭ʯ
    ʮ௚ަجఈʯ

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  17. ϑʔϦΤ܎਺ͷಋग़a0
    ʮϑʔϦΤ܎਺ a0, an, bn

    ͷಋग़ʯ
    cos nω0
    t, sin nω0
    t
    a0
    ʮશͯͷ߲ʹΛֻ͚ͯੵ෼ʯˠ௚ަجఈͷੑ࣭ ੵ෼஋
    ͔Βಋग़Ͱ͖Δ

    2
    T
    − 2
    T
    f(t)dt =

    2
    T
    − 2
    T
    a0
    dt +

    2
    T
    − 2
    T
    a1
    cos ω0
    tdt +

    2
    T
    − 2
    T
    b1
    sin ω0
    tdt + ⋯ +

    2
    T
    − 2
    T
    an
    cos nω0
    tdt +

    2
    T
    − 2
    T
    bn
    sin nω0
    tdt + ⋯

    2
    T
    − 2
    T
    1 ⋅ sin nω0
    tdt =

    2
    T
    − 2
    T
    1 ⋅ cos nω0
    tdt = 0
    ʮ௚ަجఈʯͷੑ࣭
    ͚ͩ͜͜࢒Δ

    2
    T
    − 2
    T
    f(t)dt = a0[t]
    2
    T
    − 2
    T
    a0
    =
    1
    T ∫
    2
    T
    − 2
    T
    f(t)dt

    ಉ༷ʹan, bn
    ΋ɼͦΕͧΕಋग़Ͱ͖Δ
    f(t) = a0
    + a1
    cos ω0
    t + b1
    sin ω0
    t + ⋯ + a2
    cos nω0
    t + b2
    sin nω0
    t + ⋯
    Λશମʹֻ͚ͯੵ෼
    ٻΊ͍ͨϑʔϦΤ܎਺Ҏ֎͕ফ͑ΔΑ͏ʹ
    ޻෉ͯࣜ͠มܗΛߦ͏͚ͩʂʂ
    पظͷதʹͪΐ͏Ͳ੔਺
    ݸͷ೾͕ऩ·ΔͷͰੵ෼
    ͢ΔͱzzͱΠϝʔδ͢Δ
    ͱ෼͔Γ΍͍͢
    ݩͷ৴߸f(t) ͷ࣌ؒతͳฏۉ஋ˠ௚ྲྀ੒෼ͱͳΔ
    a0
    :

    View full-size slide

  18. ʮϑʔϦΤ܎਺ a0, an, bn

    ͷಋग़ʯ
    +⋯ +

    2
    T
    − 2
    T
    an
    cos nω0
    t cos nω0
    tdt +

    2
    T
    − 2
    T
    bn
    sin nω0
    t cos nω0
    tdt + ⋯
    an
    ʮશͯͷ߲ʹΛֻ͚ͯੵ෼

    2
    T
    − 2
    T
    f(t)cos nω0
    tdt =

    2
    T
    − 2
    T
    a0
    cos nω0
    tdt +

    2
    T
    − 2
    T
    a1
    cos ω0
    t cos nω0
    tdt +

    2
    T
    − 2
    T
    b1
    sin ω0
    t cos nω0
    tdt
    ʮ௚ަجఈʯͷੑ࣭
    ͚ͩ͜͜࢒Δ
    an
    =
    2
    T ∫
    2
    T
    − 2
    T
    f(t)cos nω0
    tdt


    f(t) = a0
    + a1
    cos ω0
    t + b1
    sin ω0
    t + ⋯ + a2
    cos nω0
    t + b2
    sin nω0
    t + ⋯
    cos nω0
    t

    2
    T
    − 2
    T
    f(t)cos nω0
    tdt =

    2
    T
    − 2
    T
    an
    cos nω0
    t cos nω0
    tdt

    2
    T
    − 2
    T
    sin n2ω0
    tdt =

    2
    T
    − 2
    T
    cos n2ω0
    tdt =
    T
    2

    2
    T
    − 2
    T
    sin nω0
    t cos nω0
    tdt = 0

    2
    T
    − 2
    T
    cos nω0
    t cos mω0
    tdt = 0
    ϑʔϦΤ܎਺ͷಋग़an

    View full-size slide

  19. ʮϑʔϦΤ܎਺ a0, an, bn

    ͷಋग़ʯ
    +⋯ +

    2
    T
    − 2
    T
    an
    cos nω0
    t sin nω0
    tdt +

    2
    T
    − 2
    T
    bn
    sin nω0
    t sin nω0
    tdt + ⋯
    bn
    ʮશͯͷ߲ʹΛֻ͚ͯੵ෼

    2
    T
    − 2
    T
    f(t)sin nω0
    tdt =

    2
    T
    − 2
    T
    a0
    sin nω0
    tdt +

    2
    T
    − 2
    T
    a1
    cos ω0
    t sin nω0
    tdt +

    2
    T
    − 2
    T
    b1
    sin ω0
    t sin nω0
    tdt
    ʮ௚ަجఈʯͷੑ࣭
    ͚ͩ͜͜࢒Δ
    bn
    =
    2
    T ∫
    2
    T
    − 2
    T
    f(t)sin nω0
    tdt


    f(t) = a0
    + a1
    cos ω0
    t + b1
    sin ω0
    t + ⋯ + a2
    cos nω0
    t + b2
    sin nω0
    t + ⋯
    sin nω0
    t

    2
    T
    − 2
    T
    f(t)sin nω0
    tdt =

    2
    T
    − 2
    T
    bn
    sin nω0
    t sin nω0
    tdt

    2
    T
    − 2
    T
    sin n2ω0
    tdt =

    2
    T
    − 2
    T
    cos n2ω0
    tdt =
    T
    2

    2
    T
    − 2
    T
    sin nω0
    t cos nω0
    tdt = 0

    2
    T
    − 2
    T
    sin nω0
    t sin mω0
    tdt = 0
    ϑʔϦΤ܎਺ͷಋग़bn

    View full-size slide

  20. ϑʔϦΤڃ਺ల։ͷผͷදهํ๏
    cosͱsin ͸Ґ૬͕1/4 ͣΕͯΔ͚ͩͳͷͰɼ·ͱΊΔ͜ͱ͕Ͱ͖Δ
    n ഒͰ͔͠ݱΕͳ͍
    f(t) = a0
    + a1
    cos ω0
    t + b1
    sin ω0
    t + ⋯ + an
    cos nω0
    t + bn
    sin nω0
    t + ⋯
    an
    cos nω0
    t + bn
    sin nω0
    t
    a2
    n
    + b2
    n (
    an
    a2
    n
    + b2
    n
    cos nω0
    t +
    bn
    a2
    n
    + b2
    n
    sin nω0
    t
    )
    θn
    (
    an
    a2
    n
    + b2
    n
    ,
    bn
    a2
    n
    + b2
    n
    )
    tan θn
    =
    an
    bn
    θn
    = tan−1
    an
    bn
    a2
    n
    + b2
    n (
    sin θn
    cos nω0
    t + cos θn
    sin nω0
    t
    )
    An
    sin
    (
    nω0
    t + θn)
    ಉ͡प೾਺ͷcos ͱsin ͕ͭͷsin Ͱهड़͢Δ͜ͱ͕Ͱ͖Δ
    ϑʔϦΤڃ਺ల։͸ɼల։͞Ε֤ͨsin ೾ͦΕͧΕͷৼ෯ͱҐ૬
    ΛٻΊΔ͜ͱͰ΋දݱ͢Δ͜ͱ͕Ͱ͖Δɽ
    An
    = a2
    n
    + b2
    n
    y
    x

    View full-size slide

  21. ϑʔϦΤڃ਺͓͞Β͍
    f(t) = a0
    + ⋯ + an
    cos nω0
    t + bn
    sin nω0
    t + ⋯
    f(t)
    cos ω0
    t
    sin ω0
    t
    ؔ਺f(t) Λ௚ަ͢ΔجఈͱͳΔ
    ؔ਺
    Ͱදͨ͠ͷ͕ϑʔϦΤڃ਺
    1, cos nω0
    t, sin nω0
    t

    2
    T
    − 2
    T
    1 ⋅ sin nω0
    tdt =

    2
    T
    − 2
    T
    1 ⋅ cos nω0
    tdt = 0

    2
    T
    − 2
    T
    sin nω0
    t cos nω0
    tdt = 0

    2
    T
    − 2
    T
    sin nω0
    t sin mω0
    tdt = 0

    2
    T
    − 2
    T
    cos nω0
    t cos mω0
    tdt = 0
    ʮ௚ަجఈʯ

    2
    T
    − 2
    T
    12dt = T
    ʮϑʔϦΤ܎਺ͷಋग़ʹඞཁͳੑ࣭ʯ

    2
    T
    − 2
    T
    sin n2ω0
    tdt =

    2
    T
    − 2
    T
    cos n2ω0
    tdt =
    T
    2
    f(t) = a0
    + (a1
    cos
    2πt
    T
    + b1
    sin
    2πt
    T
    ) + (a2
    cos
    4πt
    T
    + b2
    sin
    4πt
    T
    ) + ⋯ + (an
    cos
    2nπt
    T
    + bn
    sin
    2nπt
    T
    ) + ⋯
    a0
    =
    1
    T ∫
    2
    T
    − 2
    T
    f(t)dt
    an
    =
    2
    T ∫
    2
    T
    − 2
    T
    f(t)cos
    2nπt
    T
    dt
    bn
    =
    2
    T ∫
    2
    T
    − 2
    T
    f(t)sin
    2nπt
    T
    dt

    View full-size slide

  22. ϑʔϦΤղੳɹͦͷ
    dϑʔϦΤڃ਺ɾ௚ަجఈ ࿅श໰୊ฤ
    d
    f(t) = a0
    + a1
    cos ω0
    t + b1
    sin ω0
    t + a2
    cos 2ω0
    t + b2
    sin 2ω0
    t + ⋯
    ϑʔϦΤڃ਺ͬͯԿʁʁʁͬͯਓ

    2
    T
    − 2
    T
    1 ⋅ sin nω0
    tdt =

    2
    T
    − 2
    T
    1 ⋅ cos nω0
    tdt = 0

    2
    T
    − 2
    T
    sin nω0
    t cos nω0
    tdt = 0
    ௚ަجఈͬͯԿʁʁʁͬͯਓ
    ͚ΜΏʔ!LFOZV@
     山口大学大学院 博士課程/ 学術研究員/
    ࿅श໰୊ฤ

    ղઆ͸:PVUVCFʹʂ

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  23. ϑʔϦΤղੳɹͦͷ
    dෳૉϑʔϦΤڃ਺ͱ͸d
    f(t) = ⋯ + c−n
    e−i 2πnt
    T + ⋯ + c−1
    e−i 2πt
    T + c0
    + c1
    ei 2πt
    T + ⋯ + c+n
    ei 2πnt
    T + ⋯
    ◼ෳૉϑʔϦΤڃ਺ͬͯԿʁʁʁ

    2
    T
    − 2
    T
    ei 2πm
    T
    te−i 2πn
    T
    tdt =
    ◼ෳૉϑʔϦΤڃ਺ͷ௚ަجఈͬͯԿʁʁʁ
    {
    0 (m ≠ n)
    T (m = n) eiθ = cos θ + i sin θ
    ΦΠϥʔͷެࣜ΋࢖͏Αʂ
    ͚ΜΏʔ!LFOZV@ 5XJUUFS

     山口大学大学院 博士課程/ 学術研究員/

    View full-size slide

  24. ʮϑʔϦΤڃ਺ʯͱʮෳૉϑʔϦΤڃ਺ʯͷҧ͍
    f(t) = a0
    + a1
    cos ω0
    t + b1
    sin ω0
    t + ⋯ + an
    cos nω0
    t + bn
    sin nω0
    t + ⋯
    f(t) = ⋯ + c−n
    e−iω0
    nt + ⋯ + c−1
    e−iω0
    t + c0
    + c1
    eiω0
    t + ⋯ + cn
    eiω0
    nt + ⋯
    ω0 2ω0
    nω0
    0


    a0
    a1
    a2
    an
    b1
    b2
    bn
    ω
    cosͱsin Λ༻͍ͯɼω0
    ͷ੔਺ഒͷप೾਺Λ࣋ͭ੒෼ΛݟΔ

    ⋯ ω0 2ω0
    0
    −ω0
    −2ω0
    nω0
    −nω0
    |c0
    |
    |c1
    |
    |c2
    |
    |cn
    |
    |c−1
    |
    |c−2
    |
    |c−n
    |


    ω0 2ω0
    0
    −ω0
    −2ω0
    nω0
    −nω0 ∠c0
    ∠c1
    ∠c2 ∠cn
    ∠c−1
    ∠c−2
    ∠c−n
    ω
    ω
    f(t)
    t
    ʮϑʔϦΤڃ਺ʯͱʮෳૉϑʔϦΤڃ਺ʯͷΠϝʔδ
    ෳૉࢦ਺ؔ਺Λ࢖ͬͯɼ֯प೾਺੒෼ͷৼ෯ͱҐ૬Λ໌ࣔతʹࣔ͢
    e−iω0
    t
    ਺ֶతʹ
    औΓѻ͍ʹ͍͘
    ਺ֶతʹ
    औΓѻ͍қ͍
    ෳૉ਺ͷੈքͷల։͢Δ
    ɹɾల։͕ࣜ៉ྷʹͳΔ
    ɹɾϑʔϦΤม׵ʹͭͳ͕Δ

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  25. ෳૉϑʔϦΤڃ਺ͷزԿֶతཧղͷͨΊʹ
    ࡾ֯ؔ਺ͷΠϝʔδ
    sin ω0
    t
    Im
    Re
    t
    Im
    Re
    t
    ෳૉࢦ਺ؔ਺ͷΠϝʔδ
    t
    eiω0
    t
    e−iω0
    t
    ϑʔϦΤڃ਺΋ෳૉϑʔϦΤڃ਺΋ɼ୯ʹ೾ͷ଍͠߹Θͤ
    ࣮࣠ͱڏ࣠Λճస͠ͳ͕
    Βɼ࣌ؒ࣠ํ޲ʹਐΜͰ
    ͍͘ͷͰཐટʹͳΔ
    ্Լʹৼಈ͠ɼ
    ࣌ؒ࣠ํ޲ʹਐΜͰ͍͘

    View full-size slide

  26. ϑʔϦΤڃ਺ˠෳૉϑʔϦΤڃ਺΁ͷ༠͍
    ͜Ε·ͰͷϑʔϦΤڃ਺ͷཧ࿦὎࣮਺ͷੈքͷల։
    ࣮͸͜ͷ··ͷల։ͩͱɼ਺ֶతʹѻ͍ʹ͍͘ɽ
    f(t) = a0
    +


    n=1
    (an
    cos nω0
    t + bn
    sin nω0
    t)
    f(t) =


    n=0
    An
    sin(nω0
    t + θn)
    eiθ = cos θ + i sin θ
    θn
    (
    an
    a2
    n
    + b2
    n
    ,
    bn
    a2
    n
    + b2
    n
    )
    y
    x
    θn
    cos θn
    + i sin θn
    Im
    Re
    i
    1
    ͜Ε·Ͱ࣮਺্Ͱ͔͠ఆٛ͞Ε͍ͯͳ͔ͬͨ
    ؔ਺Λෳૉ਺ʹ޿͛ΔΑ͏ʹఆٛ͠௚͢ɽ
    ΦΠϥʔͷެࣜ
    eiθ ͸ɼෳૉฏ໘ͷ୯Ґԁ্ͷภ֯ВͷҐஔ

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  27. ৄࡉͳࣜมܗ
    ΦΠϥʔͷެࣜΛ࢖༻ͯ͠ɼϑʔϦΤڃ਺ΛෳૉϑʔϦΤڃ਺΁ͱ͢Δ
    eiθ = cos θ + i sin θ
    e−iθ = cos θ − i sin θ
    cos θ =
    eiθ + e−iθ
    2
    sin θ =
    eiθ − e−iθ
    2i
    f(t) = a0
    +


    n=1
    (an
    cos nω0
    t + bn
    sin nω0
    t)
    f(t) = a0
    +


    n=1
    (an
    einω0
    t + e−inω0
    t
    2
    + bn
    einω0
    t − e−inω0
    t
    2i )
    f(t) = a0
    +


    n=1
    (
    an
    − ibn
    2
    einω0
    t +
    an
    + ibn
    2
    e−inω0
    t
    )
    f(t) =


    n=−∞
    cn
    einω0
    t
    −1 → − ∞ +1 → + ∞
    −∞ → + ∞
    Ұͭʹ·ͱΊΔ
    cn ɿϑʔϦΤ܎਺
    f(t) = ⋯ + c−n
    e−i 2πnt
    T + ⋯ + c−1
    e−i 2πt
    T + c0
    + c1
    ei 2πt
    T + ⋯ + cn
    ei 2πnt
    T + ⋯
    ෳૉϑʔϦΤڃ਺͸ʮෳૉࢦ਺ؔ਺ʯͰల։͢Δ΋ͷ͚ͩͷ΋ͷ

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  28. ෳૉϑʔϦΤڃ਺ͷ௚ަੑ͸ʂʁ
    ෳૉࢦ਺ؔ਺΋௚ަੑΛ࣋ͭ

    ɹෳૉࢦ਺ؔ਺͸ࡾ֯ؔ਺͔Βग़དྷ͍ͯΔͷͰɼࡾ֯ؔ਺ͷ௚ަੑΛҾ͖ܧ͍Ͱ͍Δ
    f(t) = ⋯ + c−n
    e−i 2πnt
    T + ⋯ + c−1
    e−i 2πt
    T + c0
    + c1
    ei 2πt
    T + ⋯ + cn
    ei 2πnt
    T + ⋯
    ʮ௚ަجఈʯͰ͋Δඞཁ͕͋Δˠ͓ޓ͍ͷؔ਺ͷੵ෼஋͕

    ฏ໘ϕΫτϧͰ͍͏಺ੵ

    P
    e1
    e2
    O
    OP = a1
    e1
    + a2
    e2
    f(t)
    cn
    ei 2πn
    T
    t
    c−n
    e−i 2πn
    T
    t
    f(t) = ⋯ + c−n
    e−i 2πnt
    T + ⋯ + cn
    ei 2πnt
    T + ⋯
    ؔ਺ۭؒͷ఺f(t) ͸

    ௚ަجఈͱͳΔؔ਺
    ηοτͰల։͞ΕΔ
    ฏ໘ϕΫτϧ͸ɼ
    ௚ަ͢Δʮجఈϕ
    ΫτϧʯͷҰ࣍݁
    ߹Ͱද͞ΕΔ

    View full-size slide

  29. ؔ਺ͷ಺ੵͷܭࢉʹ͍ͭͯ
    ෳૉࢦ਺ؔ਺΋௚ަੑΛ࣋ͭ

    ɹෳૉࢦ਺ؔ਺͸ࡾ֯ؔ਺͔Βग़དྷ͍ͯΔͷͰɼࡾ֯ؔ਺ͷ௚ަੑΛҾ͖ܧ͍Ͱ͍Δ

    a
    b
    f(t)g * (t)dt
    ؔ਺ͷ಺ੵ (b ≤ t ≤ a)
    g * (t) = g(t)
    ͕࣮਺ͷ৔߹
    g(t)
    ؔ਺ͷ಺ੵͬͯͲ͏ॻ͘ͷ͔ʂʁ
    ΞελϦεΫɹ͸ɼෳૉڞ໾ɽ
    *
    ෳૉϑʔϦΤڃ਺Ͱ͸ɼڏ਺ه߸
    iͷූ߸͕ೖΕସΘΔΑʂ

    ࣮࣠ʹରͯ͠ର৅͔ͩΒͶʂ

    2
    T
    − 2
    T
    ei 2πm
    T
    te−i 2πn
    T
    tdt

    2
    T
    − 2
    T
    ei 2πm
    T
    t{ei 2πn
    T
    t} * dt
    θn
    eiθ = cos θn
    + i sin θn
    Im
    Re
    i
    1
    e−iθ = cos θn
    − i sin θn
    {eiθ} *
    ෳૉ਺ͷϕΫτϧͰɼ಺ੵΛܭࢉΛ͢Δͱ͖ʹ͸ɼยํ͸ෳૉڞ໾ʹ͢Δͷ͕ϧʔϧ
    ʮෳૉڞ໾Λֻ͚ͯੵ෼ʂʯ

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  30. 2
    T
    − 2
    T
    ei 2πm
    T
    te−i 2πn
    T
    tdt = {
    0 (m ≠ n)
    T (m = n)
    f(t) = a0
    + ⋯ + an
    cos nω0
    t + bn
    sin nω0
    t + ⋯
    f(t)
    cos ω0
    t
    sin ω0
    t

    2
    T
    − 2
    T
    1 ⋅ sin nω0
    tdt =

    2
    T
    − 2
    T
    1 ⋅ cos nω0
    tdt = 0

    2
    T
    − 2
    T
    sin nω0
    t cos nω0
    tdt = 0

    2
    T
    − 2
    T
    sin nω0
    t sin mω0
    tdt = 0

    2
    T
    − 2
    T
    cos nω0
    t cos mω0
    tdt = 0
    ʮ௚ަجఈʯ
    f(t) = a0
    + (a1
    cos
    2πt
    T
    + b1
    sin
    2πt
    T
    ) + (a2
    cos
    4πt
    T
    + b2
    sin
    4πt
    T
    ) + ⋯ + (an
    cos
    2nπt
    T
    + bn
    sin
    2nπt
    T
    ) + ⋯
    ෳૉࢦ਺ؔ਺ͷʮ௚ަجఈʯ
    ϑʔϦΤڃ਺ͷ෮शɾɾɾͪͳΈʹϑʔϦΤڃ਺ͷ௚ަجఈ͸ͪ͜Β
    ఆٛҬͷ֓೦͸͍࣋ͬͯΔ
    पظT

    (n = 0, ± 1, ± 2, ⋯)
    ௚ަجఈͷੑ࣭

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  31. ෳૉϑʔϦΤڃ਺ͷ܎਺Λಋग़͢Δ
    ʮల։ࣜͷ྆ลʹɼӈ͔ΒɹɹɹɹΛ͔͚Δʯˠʮͦͷޙ۠ؒT Ͱੵ෼ʯ
    f(t) = ⋯ + c−n
    e−i 2πnt
    T + ⋯ + c−1
    e−i 2πt
    T + c0
    + c1
    ei 2πt
    T + ⋯ + cn
    ei 2πnt
    T + ⋯
    cn

    2
    T
    − 2
    T
    f(t)ei−2πnt
    T dt = ⋯ + cn−1 ∫
    2
    T
    − 2
    T
    ei 2π(n − 1)t
    T e−i 2πnt
    T dt + cn ∫
    2
    T
    − 2
    T
    ei 2πnt
    T e−i 2πnt
    T dt + cn+1 ∫
    2
    T
    − 2
    T
    ei 2π(n + 1)t
    T e−i 2πnt
    T dt + ⋯
    e−i 2πnt
    T


    ͚ͩ͜͜࢒Δ

    2
    T
    − 2
    T
    f(t)e−i 2πnt
    T dt = cn ∫
    2
    T
    − 2
    T
    ei 2πnt
    T e−i 2πnt
    T dt
    T ʹͳΔ
    cn
    =
    1
    T ∫
    2
    T
    − 2
    T
    f(t)e−i 2πnt
    T dt

    2
    T
    − 2
    T
    ei 2πm
    T
    te−i 2πn
    T
    tdt = {
    0 (m ≠ n)
    T (m = n)
    ʮ௚ަجఈʯͷੑ࣭
    ෳૉϑʔϦΤ܎਺ͷಋग़cn

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  32. ෳૉϑʔϦΤڃ਺ల։ͷϚΠφεଆΛߟ͑Δ
    ΦΠϥʔͷެ͔ࣜΒղऍ͢Δ
    cos θ =
    eiθ + e−iθ
    2
    ϓϥεଆͷθ ϚΠφεଆͷθ
    ଍ͯ͠2 ͰׂΔͱcos θ ͕ͰΔ
    θ
    eiθ
    Im
    Re
    i
    1
    e−iθ
    ڏ෦ଆ͕ͪΐ͏Ͳ
    ফ͑ͯɼ࣮෦͕࢒Δ
    f(t) = ⋯ + c−n
    e−i 2πnt
    T + ⋯ + c−2
    e−i 4πt
    T + c−1
    e−i 2πt
    T + c0
    + c1
    ei 2πt
    T + c2
    ei 4πt
    T + ⋯ + cn
    ei 2πnt
    T + ⋯
    ࣮਺f(t)ͷ৔߹΋ಉ༷ʹߟ͑Δ
    ଍͠߹Θ͞Εͯɼͪΐ͏Ͳڏ਺෦͕ফ͑Δ
    |cn
    | = |c−n
    |
    t = 0 ͷ࣌ ॳظҐஔ
    ΋ߟ͑ͯɼภ֯Λಋग़͢Δ
    ڞ໾ͳؔ܎Ͱ͓ޓ͍͕ৗʹଧͪফ͠߹͏ͷͰɼ
    Im
    Re
    t
    Im
    Re
    t
    ∠cn
    = − ∠c−n
    c+1
    eiω0
    t
    c−1
    e−iω0
    t
    େ͖͕͞ಉ͡Ͱූ߸͕ٯ

    View full-size slide

  33. ɹɹͷΈͰৼ෯ͱ
    Ґ૬Λද͢͜ͱ͕
    Ͱ͖ͯخ͍͠ʂ
    ࣮਺஋f(t)ʹର͢ΔෳૉϑʔϦΤڃ਺ల։͸ɼɹɹɹ͕େࣄͳ৘ใͰ͋Δ
    ෳૉϑʔϦΤ܎਺ͷҙຯ߹͍

    ⋯ ω0 2ω0
    0
    −ω0
    −2ω0
    nω0
    −nω0
    |c0
    |
    |c1
    |
    |c2
    |
    |cn
    |
    |c−1
    |
    |c−2
    |
    |c−n
    |


    ω0 2ω0
    0
    −ω0
    −2ω0
    nω0
    −nω0 ∠c0
    ∠c1
    ∠c2 ∠cn
    ∠c−1
    ∠c−2
    ∠c−n
    ω
    ω
    f(t)
    t
    ʮϑʔϦΤڃ਺ʯͱʮෳૉϑʔϦΤڃ਺ʯͷΠϝʔδ
    ෳૉࢦ਺ؔ਺Λ࢖ͬͯɼ֯प೾਺੒෼ͷৼ෯ͱҐ૬Λ໌ࣔతʹࣔ͢
    e−iω0
    t
    0 ≤ n
    0 ≤ n
    ˞ෳૉ਺஋f(t)Ͱ͋Ε͹஫ҙʂ
    ৼ෯εϖΫτϧ
    Ґ૬εϖΫτϧ
    f(t) = ⋯ + c−n
    e−i 2πnt
    T + ⋯ + c−2
    e−i 4πt
    T + c−1
    e−i 2πt
    T + c0
    + c1
    ei 2πt
    T + c2
    ei 4πt
    T + ⋯ + cn
    ei 2πnt
    T + ⋯
    cn
    |cn
    |2
    ύϫʔεϖΫτϧ

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  34. ϑʔϦΤղੳɹͦͷ
    dෳૉϑʔϦΤڃ਺ ࿅श໰୊
    ͱ͸d
    f(t) = ⋯ + c−n
    e−i 2πnt
    T + ⋯ + c−1
    e−i 2πt
    T + c0
    + c1
    ei 2πt
    T + ⋯ + c−n
    ei 2πnt
    T + ⋯
    ◼ෳૉϑʔϦΤڃ਺ͬͯԿʁʁʁ

    2
    T
    − 2
    T
    ei 2πm
    T
    te−i 2πn
    T
    tdt =
    ◼ෳૉϑʔϦΤͷ௚ަجఈͬͯԿʁʁʁ
    ͚ΜΏʔ!LFOZV@
     山口大学大学院 博士課程/ 学術研究員/
    {
    0 (m ≠ n)
    T (m = n) ࿅श໰୊ฤ

    f(t) = {
    0 (−π ≤ t < 0)
    1 (0 ≤ t ≤ π)
    ◼࣍ͷؔ਺ΛෳૉϑʔϦΤڃ਺Ͱදͤɽ
    −π π t
    f(t)
    1
    ղઆ͸:PVUVCFʹʂ

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  35. ϑʔϦΤղੳɹͦͷ
    dϑʔϦΤม׵ͱ͸d
    ◼ϑʔϦΤม׵ͬͯԿʁʁʁ
    ࣌ؒྖҬ प೾਺ྖҬ
    प೾਺ྖҬΛ֬ೝ
    Ͱ͖ͨΒԿ͕خ͍͠ͷʁ
    F(ω) =


    −∞
    f(t)e−iωtdt
    ϑʔϦΤม׵ͬͯͳʹʁʁʁ
    ͚ΜΏʔ!LFOZV@ 5XJUUFS

     山口大学大学院 博士課程/ 学術研究員/
    ཭ࢄϑʔϦΤม׵Ͱ͸ͳ͍Αʂ

    ͜ͷϑʔϦΤม׵͸࿈ଓͷؔ਺ʹؔ͢ΔϑʔϦΤม׵

    View full-size slide

  36. ෳૉϑʔϦΤڃ਺ ਺ֶͷ͓࿩

    ෳૉϑʔϦΤڃ਺ͷ܎਺Λಋग़͢Δ
    ʮల։ࣜͷ྆ลʹɼӈ͔ΒɹɹɹɹΛ͔͚Δʯˠʮͦͷޙ۠ؒT Ͱੵ෼ʯ
    f(t) = ⋯ + c−n
    e−i 2πnt
    T + ⋯ + c−1
    e−i 2πt
    T + c0
    + c1
    ei 2πt
    T + ⋯ + cn
    ei 2πnt
    T + ⋯
    cn

    2
    T
    − 2
    T
    f(t)ei−2πnt
    T dt = ⋯ + cn−1 ∫
    2
    T
    − 2
    T
    ei 2π(n − 1)t
    T e−i 2πnt
    T dt + cn ∫
    2
    T
    − 2
    T
    ei 2πnt
    T e−i 2πnt
    T dt + cn+1 ∫
    2
    T
    − 2
    T
    ei 2π(n + 1)t
    T e−i 2πnt
    T dt + ⋯
    e−i 2πnt
    T


    ͚ͩ͜͜࢒Δ

    2
    T
    − 2
    T
    f(t)e−i 2πnt
    T dt = cn ∫
    2
    T
    − 2
    T
    ei 2πnt
    T e−i 2πnt
    T dt
    T ʹͳΔ
    cn
    =
    1
    T ∫
    2
    T
    − 2
    T
    f(t)e−i 2πnt
    T dt

    2
    T
    − 2
    T
    ei 2πm
    T
    te−i 2πn
    T
    tdt = {
    0 (m ≠ n)
    T (m = n)
    ʮ௚ަجఈʯͷੑ࣭

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  37. ϑʔϦΤม׵ ਺ֶͷ͓࿩

    f(t) = ⋯ + c−n
    e−inω0
    t + ⋯ + c−1
    e−iω0
    t + c0
    + c1
    eiω0
    t + ⋯ + cn
    einω0
    t + ⋯
    cn
    =
    1
    T ∫
    2
    T
    − 2
    T
    f(t)e−inω0
    tdt
    ෳૉϑʔϦΤڃ਺ͷ৔߹
    f(t)
    T
    ω0
    =

    T
    ֯प೾਺

    ࣌ؒྖҬ
    प೾਺ྖҬ
    ֯प೾਺ͷഒ਺ͷ

    ϑʔϦΤ܎਺Λಋग़͢Δ
    ؔ਺Λෳૉࢦ਺ؔ਺Ͱల։͢Δ
    einω0
    t
    ɾఆ͕ٛ۠ؒݶఆ
    पظؔ਺

    F(ω) =


    −∞
    f(t)e−iωtdt
    ΋ͱ΋ͱܾΊ͍ͯͨ֯प೾਺Ͱ͸ͳ͘ɼ
    ࣌ؒ೾ܗͷؔ਺͔Β௚઀ɼप೾਺৘ใͷؔ਺Λ
    औΓग़͢ɽ
    ϑʔϦΤม׵ͷ৔߹
    n ൪໨ͷറΓ

    εϖΫτϧ͸ͱͼͱͼ

    ɾఆ͕ٛ۠ؒແݶ
    ඇपظؔ਺

    c0
    c−1
    c−2
    c2
    c1
    c−3
    c3
    ɾ

    ɾ

    ɾ

    View full-size slide

  38. ω0
    =

    T
    ֯प೾਺


    ؒ

    Ҭ




    Ҭ
    n ൪໨ͷറΓ

    εϖΫτϧ͸ͱͼͱͼ

    f(t)
    t
    f(t)
    T
    einω0
    t
    0
    ෳૉϑʔϦΤڃ਺
    ɾ

    ɾ

    ɾ
    F(ω)
    −∞
    +∞
    ֯प೾਺͸࿈ଓతͳ஋ɽ
    ͦͯ͠࿈ଓؔ਺ʹͳΔɽ
    ϑʔϦΤม׵
    eiωt
    ෳૉϑʔϦΤڃ਺ͱϑʔϦΤม׵ͷҧ͍
    c0
    c−1
    c−2
    c2
    c1
    c−3
    c3
    ɾ

    ɾ

    ɾ

    View full-size slide

  39. ෳૉϑʔϦΤڃ਺ల։͔ΒɼϑʔϦΤม׵΁
    ෳૉϑʔϦΤڃ਺ల։

    ⋯ ω0 2ω0
    0
    −ω0
    −2ω0
    nω0
    −nω0
    |c0
    |
    |c1
    |
    |c2
    |
    |cn
    |
    |c−1
    |
    |c−2
    |
    |c−n
    |
    ω
    f(t) =


    n=−∞
    cn
    eiω0
    nt
    ω0
    =

    T
    पظؔ਺ͳͷͰɼपظT ͕େࣄͳཁૉͩͬͨ
    ͦΕʹΑͬͯɼجຊ֯प೾਺ω0
    ͕ग़ͯ͘Δ
    ෳૉϑʔϦΤ܎਺cnʹɼෳૉࢦ਺ؔ਺
    Λֻ͚ͨ΋ͷΛ଍͠߹Θͤɼ૯࿨ΛͱΔɽ
    eiω0
    nt
    cn
    eiω0
    nt

    ⋯ ω0 2ω0
    0
    −ω0
    −2ω0
    nω0
    −nω0
    |c0
    |
    |c1
    |
    |c2
    |
    |cn
    |
    |c−1
    |
    |c−2
    |
    |c−n
    |
    ω
    cn
    eiω0
    nt
    ω0
    ω0
    cn
    eiω0
    nt
    ໘ੵ
    ω0
    =

    T ω0
    पظT Λແݶେʿʹʂ ͕ඇৗʹখ͘͞ͳΔ
    प೾਺εϖΫτϧ͕
    ͱͼͱͼͰ͸ͳ͘ɼ࿈ଓʹͳΔ
    ઢ प೾਺εϖΫτϧ
    ͷִؒ
    Λڀۃʹڱ͘͠ɼ໘ੵͱͯ͠
    औΓѻ͏Α͏ʹߟ͑Δ
    ૯࿨Λܭࢉ͢Δ໰୊ˠ໘ੵΛܭࢉ͢Δ໰୊΁

    View full-size slide

  40. ϑʔϦΤม׵ͷ਺ࣜΛಋग़͢ΔɿٯϑʔϦΤม׵
    f(t) =


    n=−∞
    cn
    eiω0
    nt
    ෳૉϑʔϦΤڃ਺ల։͔ΒɼٯϑʔϦΤม׵
    =


    n=−∞
    ω0
    cn
    eiω0
    nt
    ω0
    =
    1



    n=−∞
    ω0
    2πcn
    eiω0
    nt
    ω0
    =
    1



    n=−∞
    ω0
    2πcn
    eiω(n)t
    ω0
    =
    1



    n=−∞
    ω0
    F(ω(n))eiω(n)t
    F(ω(n)) =
    2πcn
    ω0
    ω(n) = ω0
    n
    f(t) =
    1



    n=−∞
    F(ω(n))eiω(n)tΔω
    T0
    → ∞ ͷۃݶΛߟ͑Δ
    f(t) =
    1
    2π ∫

    −∞
    F(ω)eiωtdω
    cn
    eiω0
    nt
    ω0
    ω0
    cn
    eiω0
    nt
    ໘ੵ
    Λ͘͘Γग़͢
    1/2π
    جຊ֯प೾਺ͷ
    n ഒͷ஋Λ࣋ͭ
    ֯प೾਺
    प೾਺εϖΫτ
    ϧΛද͢৽ͨͳ
    ม਺Λఆٛ
    Δω = ω0 ͱஔ͖׵͑Δ
    ω(n) → ω
    nഒͷ஋Λ࣋ͭ֯प೾਺͕
    ࣮਺ω ͷؔ਺ʹͳΔ
    Δω → dω
    ແݶখʹͳΔ
    ٯϑʔϦΤม׵
    ޙʑͷϑʔϦΤม׵ͷࣜΛ
    γϯϓ
    ϧʹهड़͢ΔͨΊ

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  41. ϑʔϦΤม׵ͷ਺ࣜΛಋग़͢ΔɿϑʔϦΤม׵
    cn
    =
    1
    T ∫
    2
    T
    − 2
    T
    f(t)e−iω(n)tdt
    =

    ω0
    1
    T ∫
    2
    T
    − 2
    T
    f(t)e−iω(n)tdt
    F(ω(n)) =
    2πcn
    ω0
    ω0
    =

    T
    =

    2
    T
    − 2
    T
    f(t)e−iω(n)tdt
    ϑʔϦΤม׵΋ಉ༷ʹಋग़
    F(ω) =


    −∞
    f(t)e−iωtdt
    ৽ͨʹఆٛͨ͠प೾਺εϖΫτϧ
    ෳૉϑʔϦΤ܎਺Λ୅ೖ͢Δ
    ੵ෼ͷલʹ͋Δ
    ܎਺Λফڈ
    ω(n) → ω
    nഒͷ஋Λ࣋ͭ֯प೾਺͕
    ࣮਺ω ͷؔ਺ʹͳΔ
    T0
    → ∞ ͷۃݶΛߟ͑Δ
    ϑʔϦΤม׵

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  42. ࣌ؒྖҬ
    प೾਺ྖҬ
    F(ω) =


    −∞
    f(t)e−iωtdt
    f(t) =
    1
    2π ∫

    −∞
    F(ω)eiωtdω
    ϑʔϦΤม׵
    ٯϑʔϦΤม׵
    P
    e1
    e2
    O
    OP = a1
    e1
    + a2
    e2
    f(t)
    eiω1
    t
    eiω2
    t
    ؔ਺ۭؒͷ఺f(t) ͸

    ௚ަجఈͱͳΔؔ਺
    ηοτͰల։͞ΕΔ
    ฏ໘ϕΫτϧ͸ɼ
    ௚ަ͢Δʮجఈϕ
    ΫτϧʯͷҰ࣍݁
    ߹Ͱද͞ΕΔ
    F(ω) =


    −∞
    f(t)e−iωtdt
    ࿈ଓ͍ͯ͠Δ
    ͍͔ͳΔ֯प೾਺΋PL
    ϑʔϦΤม׵ ਺ֶͷ͓࿩

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  43. ࣌ؒྖҬ
    प೾਺ྖҬ
    F(ω) =


    −∞
    f(t)e−iωtdt
    ϑʔϦΤม׵
    ٯϑʔϦΤม׵
    ϑʔϦΤม׵ͷجఈ eiωt
    F(ω) =


    −∞
    f(t)e−iωtdt ෳૉ਺ͷ಺ੵ͸ɼ
    ҰํΛڞ໾ͳෳૉ਺ʹֻ͚ͯ͠߹ΘͤΔ
    ෳૉਖ਼ݭ೾
    ϑʔϦΤม׵ ਺ֶͷ͓࿩

    f(t) =
    1
    2π ∫

    −∞
    F(ω)eiωtdω

    View full-size slide

  44. ·ͱΊɿϑʔϦΤڃ਺ల։ͱٯϑʔϦΤม׵ͷಛ௃
    f(t) =


    n=−∞
    cn
    eiω0
    nt
    ෳૉϑʔϦΤڃ਺
    c0
    c−1
    c−2
    c2
    c1
    c−3
    c3
    einω0
    t
    ɾपظT ͷ࣌ؒ৴߸Λѻ͏
    ɾෳૉࢦ਺ؔ਺ͷʮ૯࿨ʯͰදݱ
    ɾप೾਺੒෼͸ͱͼͱͼ ແݶݸɼ੔਺ഒ

    ٯϑʔϦΤม׵
    f(t) =
    1
    2π ∫

    −∞
    F(ω)eiωtdω
    F(ω)
    eiωt
    ɾपظతͱ͸ݶΒͳ͍࣌ؒ৴߸Λѻ͏
    ɾෳૉࢦ਺ؔ਺ͷʮੵ෼ʯͰදݱ
    ɾप೾਺͸࿈ଓؔ਺ͱͳΔ ࣮਺

    ϑʔϦΤڃ਺ల։ʹରԠ͢Δͷ͸ɼٯϑʔϦΤม׵ʂʂ
    ʮϑʔϦΤม׵ʯ͸ɼʮϑʔϦΤڃ਺ͷ܎਺ʯΛಋग़͢ΔํʹରԠʂʂ
    F(ω) =


    −∞
    f(t)e−iωtdt cn
    =
    1
    T ∫
    2
    T
    − 2
    T
    f(t)e−iω0
    ntdt

    View full-size slide

  45. ࿅श໰୊ฤ

    f(t) = {
    0 (t < 0)
    e−t (0 ≤ t)
    [
    ◼࣍ͷؔ਺ΛϑʔϦΤม׵ͯ͠ಘΒΕΔؔ਺͸ʁ
    t
    f(t)
    1
    0
    ϑʔϦΤղੳɹͦͷ
    dϑʔϦΤม׵ ࿅श໰୊
    ͱ͸d
    ͚ΜΏʔ!LFOZV@
     山口大学大学院 博士課程/ 学術研究員/
    f(t) =
    {
    1 (− ϵ
    2
    ≤ t ≤ ϵ
    2
    )
    0 (otherwise)
    f(t)
    1
    0

    ϵ
    2
    ϵ
    2
    ◼࣍ͷؔ਺ΛϑʔϦΤม׵ͯ͠ಘΒΕΔؔ਺͸ʁ
    ղઆ͸:PVUVCFʹʂ

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  46. ϑʔϦΤղੳɹͦͷ
    d཭ࢄϑʔϦΤม׵ %'5
    ͱ͸d
    ◼σ
    Οδλϧ৴߸ ཭ࢄ৴߸
    ͷϑʔϦΤม׵
    x(t) x(t)
    0 D 2D 3D 4D
    t t
    ཭ࢄ৴߸
    D αϯϓϦϯάपظ
    T = ND
    ͚ΜΏʔ!LFOZV@ 5XJUUFS

     山口大学大学院 博士課程/ 学術研究員/

    View full-size slide

  47. ཭ࢄϑʔϦΤม׵%'5 ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩

    ࣮ݧͳͲͰηϯγϯάͨ͠஋ʹ͸ɼ཭ࢄϑʔϦΤม׵Λ࢖͏ʂ
    w w w w w w w w
    ϑʔϦΤม׵͸ؔ਺ʹରͯ͠࢖͏΋ͷ ਺ֶ

    w w w w w w
    x(t) x(t)
    0 D 2D 3D 4D
    ϑʔϦΤม׵Ͱٞ࿦͍ͯͨ͠

    ؔ਺͸࿈ଓత
    ࣮ݧσʔλ͸ҰఆִؒͰ
    ਺஋Խ͞ΕͨσʔλʹͳΔ
    t t
    ཭ࢄ৴߸
    D αϯϓϦϯάपظ
    T = ND
    ཭ࢄϑʔϦΤม׵͸ɼ༗ݶͷ࣌ؒT Λѻ͏ͷͰɼ

    Ͳ͔ͬͪͱ͍͏ͱෳૉϑʔϦΤڃ਺ʹ͍ۙͧʂ
    ཭ࢄ৴߸͸

    ੵ෼Ͱ͖Δͷʂʁ

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  48. ৴߸ॲཧ
    "%ม׵ "OBMPHUPEJHJUBMDPOWFSTJPO

    ࢦઌ຺೾ηϯαʔ
    t t
    τ t t
    ඪຊԽ αϯϓϦϯά
    ྔࢠԽ
    αϯϓϦϯάִؒ
    0000
    0001
    0010
    0011
    0100
    0101
    0111
    0110
    1000
    Ξφϩά৴߸ σ
    Οδλϧ৴߸
    (0011,0100,0110,1000,0011,0011,0011,0101,0111,0100,0011,⋯)
    Ξφϩά৴߸
    ɹ࣌ؒతʹ࿈ଓͰมԽ͢Δ৴߸
    σΟδλϧ৴߸ ෆ࿈ଓͳσʔλͷू߹

    ɹҰఆִ࣌ؒؒͷͱͼͱͼͷ஋Ͱදݱ
    ͨ͠৴߸
    ܭଌػثͷεϖοΫ
    ͬͯେࣄͳͷͰ͢
    Ϗοτɿਐ਺ͷܻͷ୯Ґ

    View full-size slide

  49. ߴ଎ϑʔϦΤม׵''5
    φΠΩετͷඪຊԽఆཧ
    ͋Δਖ਼ݭ೾ΛඪຊԽ͢Δࡍʹ͸ɼͦͷपظͷ൒෼ΑΓ΋୹͍ඪຊԽִؒΛ༻͍ͳ͚Ε͹ͳΒͳ͍
    पظT
    पظͷ
    ൒෼T/2
    ඪຊԽִؒ

    αϯϓϦϯάִؒ

    τ
    T
    τ
    T
    τ
    T
    τ
    [
    τ <
    T
    2
    τ = T
    T
    2
    < τ < T
    τ =
    T
    2
    ௚ઢʹͳΔ
    पظ͕େ͖͍೾͕
    ؒҧͬͯग़ͯ͘Δ
    ྵ఺͹͔Γͩͱ
    ௚ઢ͕ग़ͯ͘Δ
    ඪຊԽ఺਺Λ௨ա͢Δ೾ͷ͏ͪͰ
    ࠷΋Ώͬ͘Γͱͨ͠೾ͱͯ͠ɼݸ
    ͚ͩਖ਼͘͠࠶ݱͰ͖Δ

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  50. ཭ࢄϑʔϦΤม׵%'5 ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩

    x(t)
    t
    ࣌ؒྖҬ
    प೾਺ྖҬ
    F(ω) =


    −∞
    x(t)e−iωtdt
    x(t)
    0 D 2D 3D 4D
    t
    ཭ࢄ৴߸
    D αϯϓϦϯάपظ
    T = ND
    ω
    F(ω)
    ω
    F(ω) ৗʹ
    ͱͼͱͼͷ৴߸ͳͷͰ

    ੵ෼
    ੵ෼
    ཭ࢄ৴߸ʹؔͯ͠͸޻෉͕ඞཁ
    ཭ࢄϑʔϦΤม׵͸ੵ෼Ͱ͖ͳ͍ʂʁ
    w w w w w w w w

    View full-size slide

  51. x(t)
    0 D 2D 3D 4D
    t
    T = 4D
    ཭ࢄϑʔϦΤม׵%'5 ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩

    0 ≤ t < T
    ཭ࢄϑʔϦΤม׵Ͱप೾਺ྖҬʹ΋͍ͬͯ͘޻෉
    ༗ݶ۠ؒͰղੳ T͸ؚ·ͳ͍

    x(t) = ⋯ + c−n
    ei 2π
    T
    (−nt) + ⋯ + c−1
    ei 2π
    T
    (−t) + c0
    + c1
    ei 2π
    T
    (nt) + ⋯ + cn
    ei 2π
    T
    (+nt) + ⋯
    x0
    x1
    x2
    x3
    x(t) = c0
    1 + c1
    ei 2π
    4D
    (1t) + c2
    ei 2π
    4D
    (2t) + c3
    ei 2π
    4D
    (3t)
    ͭͷෳૉਖ਼ݭ೾͚ͩ࢖༻Ͱ͖Ε͹ྑ͍
    ͜Ε͕ͭͷ཭ࢄ৴߸ͷ཭ࢄϑʔϦΤม׵ͷجຊͷࣜʹͳΓ·͢
    ཭ࢄ৴߸
    c(t) =
    N

    n=0
    x(t)ei 2π
    T
    (nt)

    View full-size slide

  52. x(t)
    1
    ei 2π
    4D
    (nt)
    ؔ਺ۭؒͷ఺x(t) ͸ɼ௚ަ
    ͢Δͭͷෳૉਖ਼ݭ೾ͷҰ࣍
    ݁߹Ͱද͞ΕΔ͜ͱʹͳΔ
    ཭ࢄϑʔϦΤม׵%'5 ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩

    x(t)
    0 D 2D 3D 4D
    t
    T = 4D
    x0
    x1
    x2
    x3
    x(t) = c0
    1 + c1
    ei 2π
    4D
    (1t) + c2
    ei 2π
    4D
    (2t) + c3
    ei 2π
    4D
    (3t)
    ෳૉϑʔϦΤڃ਺ͱಉͩ͡Ͷʂ
    ཭ࢄ৴߸
    ཭ࢄ৴߸x0 ~ x3
    ͔Βɼप೾਺৘ใc0 ~ c3
    ͕ಘΒΕΔ
    ల։͞Ε֤߲ͨ ෳૉਖ਼ݭ೾
    ͸΋ͪΖΜ௚ަ͍ͯ͠Δ

    View full-size slide

  53. ཭ࢄϑʔϦΤม׵%'5 ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩

    ཭ࢄ৴߸x0 ~ x3
    ͔Βɼप೾਺৘ใc0 ~ c3
    ͕ಘΒΕΔ
    x(D) = c0
    + c1
    ei 2π
    4 + c2
    ei 4π
    4 + c3
    ei 6π
    4
    t ΁0, D, 2D, 3D Λ୅ೖͯ͠ɼ࿈ཱํఔࣜΛཱͯΔ
    x(0) = c0
    + c1
    + c2
    + c3
    x(t) = c0
    1 + c1
    ei 2π
    4D
    (1t) + c2
    ei 2π
    4D
    (2t) + c3
    ei 2π
    4D
    (3t)
    x(2D) = c0
    + c1
    ei 4π
    4 + c2
    ei 8π
    4 + c3
    ei 12π
    4
    x(3D) = c0
    + c1
    ei 6π
    4 + c2
    ei 12π
    4 + c3
    ei 18π
    4
    Ҏ্ͷ࿈ཱํఔࣜΛղ͍ͯɼ཭ࢄ৴߸x0 ~ x3
    ͔Βɼ
    प೾਺৘ใc0 ~ c3
    ͷؔ܎Λಋग़͢Δ

    View full-size slide

  54. ཭ࢄϑʔϦΤม׵%'5 ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩

    x0
    x1
    x2
    x3
    =
    1 1 1 1
    1 ei 2π
    4 ei 4π
    4 ei 6π
    4
    1 ei 4π
    4 ei π
    4 ei 12π
    4
    1 ei 6π
    4 ei 12π
    4 ei 18π
    4
    c0
    c1
    c2
    c3
    x(D) = c0
    + c1
    ei 2π
    4 + c2
    ei 4π
    4 + c3
    ei 6π
    4
    x(0) = c0
    + c1
    + c2
    + c3
    x(2D) = c0
    + c1
    ei 4π
    4 + c2
    ei 8π
    4 + c3
    ei 12π
    4
    x(3D) = c0
    + c1
    ei 6π
    4 + c2
    ei 12π
    4 + c3
    ei 18π
    4
    Ҏ্ͷ࿈ཱํఔࣜΛղ͍ͯɼ཭ࢄ৴߸x0 ~ x3
    ͔Βɼ
    प೾਺৘ใc0 ~ c3
    ͷؔ܎Λಋग़͢Δ
    ߦྻܗࣜʹมߋ
    ֤ߦϕΫτϧ͸΋ͪΖΜ௚ަ͍ͯ͠Δ

    View full-size slide

  55. ཭ࢄϑʔϦΤม׵%'5 ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩

    x0
    x1
    x2
    x3
    =
    1 1 1 1
    1 ei 2π
    4 ei 4π
    4 ei 6π
    4
    1 ei 4π
    4 ei π
    4 ei 12π
    4
    1 ei 6π
    4 ei 12π
    4 ei 18π
    4
    c0
    c1
    c2
    c3
    1 1 1 1
    1 e−i 2π
    4 e−i 4π
    4 e−i 6π
    4
    1 e−i 4π
    4 e−i π
    4 e−i 12π
    4
    1 e−i 6π
    4 e−i 12π
    4 e−i 18π
    4
    x0
    x1
    x2
    x3
    =
    1 1 1 1
    1 e−i 2π
    4 e−i 4π
    4 e−i 6π
    4
    1 e−i 4π
    4 e−i π
    4 e−i 12π
    4
    1 e−i 6π
    4 e−i 12π
    4 e−i 18π
    4
    1 1 1 1
    1 ei 2π
    4 ei 4π
    4 ei 6π
    4
    1 ei 4π
    4 ei π
    4 ei 12π
    4
    1 ei 6π
    4 ei 12π
    4 ei 18π
    4
    c0
    c1
    c2
    c3
    ӈลͷਖ਼ํߦྻͷ֤੒෼Λڞ໾ͳෳૉ਺ʹͯ͠సஔͨ͠ߦྻΛ྆ลʹ͔͚Δ
    4
    =

    View full-size slide

  56. ཭ࢄϑʔϦΤม׵%'5 ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩

    ཭ࢄ৴߸͕ͭͷ৔߹ͷ཭ࢄϑʔϦΤม׵ͷެࣜ
    4
    c0
    c1
    c2
    c3
    =
    1 1 1 1
    1 e−i 2π
    4 e−i 4π
    4 e−i 6π
    4
    1 e−i 4π
    4 e−i π
    4 e−i 12π
    4
    1 e−i 6π
    4 e−i 12π
    4 e−i 18π
    4
    x0
    x1
    x2
    x3
    4
    c0
    c1
    c2
    c3
    =
    X0
    X1
    X2
    X3
    X0
    X1
    X2
    X3
    =
    1 1 1 1
    1 e−i 2π
    4 e−i 4π
    4 e−i 6π
    4
    1 e−i 4π
    4 e−i π
    4 e−i 12π
    4
    1 e−i 6π
    4 e−i 12π
    4 e−i 18π
    4
    x0
    x1
    x2
    x3
    ཧ࿦ల։͸ϑʔϦΤڃ਺ͱಉ͡Ͱ͋Δ

    View full-size slide

  57. ཭ࢄϑʔϦΤม׵%'5 ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩

    1 1 1 1
    1 e−i 2π
    4 e−i 4π
    4 e−i 6π
    4
    1 e−i 4π
    4 e−i π
    4 e−i 12π
    4
    1 e−i 6π
    4 e−i 12π
    4 e−i 18π
    4
    x0
    x1
    x2
    x3
    = 4
    c0
    c1
    c2
    c3
    4
    c0
    c1
    c2
    c3
    =
    X0
    X1
    X2
    X3
    x0
    x1
    x2
    x3
    =
    1 1 1 1
    1 ei 2π
    4 ei 4π
    4 ei 6π
    4
    1 ei 4π
    4 ei π
    4 ei 12π
    4
    1 ei 6π
    4 ei 12π
    4 ei 18π
    4
    c0
    c1
    c2
    c3
    c0
    c1
    c2
    c3
    =
    1
    4
    X0
    X1
    X2
    X3
    ݩʑͷࣜʹ୅ೖ͢Δ
    ஔ͖׵͑
    ٯ཭ࢄϑʔϦΤม׵ͷಋग़
    x0
    x1
    x2
    x3
    =
    1
    4
    1 1 1 1
    1 ei 2π
    4 ei 4π
    4 ei 6π
    4
    1 ei 4π
    4 ei π
    4 ei 12π
    4
    1 ei 6π
    4 ei 12π
    4 ei 18π
    4
    X0
    X1
    X2
    X3
    ٯ཭ࢄϑʔϦΤม׵΋݁ہ͸ɼ
    ཭ࢄϑʔϦΤม׵ͱಉ͡ʂ

    View full-size slide

  58. ཭ࢄϑʔϦΤม׵%'5 ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩

    X0
    X1
    X2
    X3
    =
    1 1 1 1
    1 e−i 2π
    4 e−i 4π
    4 e−i 6π
    4
    1 e−i 4π
    4 e−i π
    4 e−i 12π
    4
    1 e−i 6π
    4 e−i 12π
    4 e−i 18π
    4
    x0
    x1
    x2
    x3
    X0
    X1
    X2
    X3
    =
    1 1 1 1
    1 −i −1 i
    1 −1 1 −1
    1 i −1 −i
    x0
    x1
    x2
    x3
    ΦΠϥʔͷެࣜ
    eiθ = cos θ + i sin θ
    e−iθ = cos θ − i sin θ
    e−i π
    2 = cos
    π
    2
    − i sin
    π
    2
    e−i π
    2 = − i


    e−i π
    2 = − i
    ei π
    2 = i
    eiπ = 1 e−iπ = − 1
    ͜ͷਖ਼ํߦྻʹ͸͋Δنଇੑ͕͋Δ
    ͜ͷنଇੑΛ্ख͘ར༻ͨ͠ͷ͕ʮߴ଎ϑʔϦΤม׵ FFT
    ʯ

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  59. ߴ଎ϑʔϦΤม׵''5΁ͷ༠͍ɽɽɽ
    X0
    X1
    X2
    X3
    =
    1 1 1 1
    1 −i −1 i
    1 −1 1 −1
    1 i −1 −i
    x0
    x1
    x2
    x3
    ◼ΦΠϥʔͷެࣜʹΑΓॻ͖௚͢ ΦΠϥʔͷެࣜ
    eiθ = cos θ + i sin θ
    e−iθ = cos θ − i sin θ
    e−i π
    2 = − i
    ei π
    2 = i
    eiπ = 1 e−iπ = − 1
    X0
    X2
    X1
    X3
    =
    1 1 1 1
    1 −1 1 −1
    1 −i −1 i
    1 i −1 −i
    x0
    x1
    x2
    x3
    ߦ໨ͱߦ໨ΛೖΕସ͑Δ
    E1
    = [
    1 1
    1 −i]
    E2
    = [
    1 −i
    1 i ]
    −E2
    [
    E1
    E1
    E2
    −E2
    ]
    %'5ͷม׵ߦྻ͕
    ؆୯ʹද͞ΕΔ
    ͜ͷΑ͏ͳײ͡Ͱɽɽɽ

    View full-size slide

  60. ϑʔϦΤղੳɹͦͷ
    d཭ࢄϑʔϦΤม׵ ࿅श໰୊
    ͱ͸d
    ࿅श໰୊ฤ

    ◼཭ࢄ৴߸\x0, x1, x2, x3
    ^͕\1, 1, 0, 0^ͷͱ͖ɼ཭ࢄϑʔϦΤม׵ %'5
    ͷ
    ͷ܎਺X0 ~ X3
    ΛٻΊΑɽ
    ͚ΜΏʔ!LFOZV@ 5XJUUFS

     山口大学大学院 博士課程/ 学術研究員/

    View full-size slide

  61. ϑʔϦΤղੳɹͦͷ
    dߴ଎ϑʔϦΤม׵ ''5
    ͱ͸d
    ◼ߴ଎ϑʔϦΤม׵''5ͱ͸ɼ཭ࢄϑʔϦΤม׵%'5Λߴ଎ʹ͢Δ΋ͷʂ
    X0
    X1
    X2
    X3
    =
    1 1 1 1
    1 e−i 2π
    4 e−i 4π
    4 e−i 6π
    4
    1 e−i 4π
    4 e−i π
    4 e−i 12π
    4
    1 e−i 6π
    4 e−i 12π
    4 e−i 18π
    4
    x0
    x1
    x2
    x3
    %'5ͷม׵ߦྻ
    प೾਺৘ใΛಘΔ

    ཭ࢄ৴߸Λೖྗ
    ͜ͷਖ਼ํߦྻʹ͸͋Δنଇੑ͕͋Δ
    ͜ͷنଇੑΛ্ख͘ར༻ͨ͠ͷ͕ʮߴ଎ϑʔϦΤม׵ FFT
    ʯ
    ͚ΜΏʔ!LFOZV@ 5XJUUFS

     山口大学大学院 博士課程/ 学術研究員/

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  62. ߴ଎ϑʔϦΤม׵''5
    ''5Λཧղ͍ͯͧ͘͠ʂ
    ɾσʔλ਺͕ͷ΂͖৐ʹैΘͳ͚Ε͹͍͚ͳ͍
    ɾσʔλ਺͕ ͷͱ͖Ͱߟ͑ͯɼҰൠԽ·Ͱ
    ɾόλϑϥΠԋࢉʹΑΔܭࢉͷ޻෉ɽ
    ɾϏοτϦόʔεʹΑΓܭࢉͷฒͼସ͑ɽ
    ۩ମతʹ਺஋ΛऔΓ্
    ͛ͯߟ͍͖͑ͯ·͢
    όλϑϥΠԋࢉͱϏοτ
    Ϧόʔε͕෼͔Ε͹0,ʂ
    ͨͩ୯ʹ࢖͍ͬͯΔ͚ͩͰ͸ͳͯ͘ɼ
    ಺ଆͷΞϧΰϦζϜΛཧղ͠·͠ΐʔʂ

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  63. ཭ࢄϑʔϦΤม׵%'5ͷ͓͞Β͍
    X0
    X1
    X2

    XN−1
    =
    1 1 1 ⋯ 1
    1 e−i 2π
    N e−i 4π
    N ⋯ e−i 2π(N − 1)
    N
    1 e−i 4π
    N e−i 8π
    N ⋯ e−i 4π(N − 1)
    N
    ⋮ ⋮ ⋮ ⋱ ⋮
    1 e−i 2π(N − 1)
    N e−i 4π(N − 1)
    N ⋯ e−i 2π(N − 1)(N − 1)
    N
    x0
    x1
    x2

    xN−1
    X0
    X1
    X2
    X3
    =
    1 1 1 1
    1 e−i 2π
    4 e−i 4π
    4 e−i 6π
    4
    1 e−i 4π
    4 e−i π
    4 e−i 12π
    4
    1 e−i 6π
    4 e−i 12π
    4 e−i 18π
    4
    x0
    x1
    x2
    x3
    %'5ͱ͸Nݸͷσʔλ\x0, x1, x2,ɾɾɾ, xN-1
    ^ͱɼͦͷσʔλ௕ʹରԠ͢Δ
    ෳૉਖ਼ݭ೾ͷ܎਺\X0, X1, X2,ɾɾɾ, XN-1
    ^Λ݁ͼ͚ͭΔม׵
    Nʹ4ͷͱ͖
    ҰൠԽ͢Δ ม׵ߦྻ
    x0
    x1
    x2
    ⋯ xn−1
    t


    X0
    X1
    X2
    ⋯ Xn−1
    ω

    ม׵ߦྻ
    Nݸͷ཭ࢄ৴߸ͷσʔλΛ
    ॲཧ͢Δʹ͸ɼN2 ճͷෳૉ
    ਺ͷੵͷܭࢉ͕ඞཁʹͳΔ
    N2ճͷܭࢉճ਺Λେ෯ʹݮগͤ͞Δ޻෉͕͋Δˠ''5

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  64. ߴ଎ϑʔϦΤม׵''5
    X0
    X1
    X2

    XN−1
    =
    1 1 1 ⋯ 1
    1 e−i 2π
    N e−i 4π
    N ⋯ e−i 2π(N − 1)
    N
    1 e−i 4π
    N e−i 8π
    N ⋯ e−i 4π(N − 1)
    N
    ⋮ ⋮ ⋮ ⋱ ⋮
    1 e−i 2π(N − 1)
    N e−i 4π(N − 1)
    N ⋯ e−i 2π(N − 1)(N − 1)
    N
    x0
    x1
    x2

    xN−1
    ߴ଎ϑʔϦΤม׵''5ͷ֓ཁ
    ม׵ߦྻ
    N2ճͷܭࢉճ਺Λେ෯ʹݮগͤ͞Δ޻෉͕͋Δˠ''5
    ܭࢉͷॱংΛม͑ͯɼಉྨͷখ͞ͳܭࢉʹখ෼͚͢Δ
    N2 →
    N
    2
    (log2
    N − 1)
    ۩ମతͳ৐ࢉ ֻ͚ࢉ
    ճ਺ͷ஋
    ʻྫ͑͹ʼ N = 210 = 1024
    N2 = 1024 × 1024 ≒ 1000000
    N
    2
    (log2
    N − 1) = 512 log2
    210 − 1 ≒ 4600
    %'5

    ''5

    ɾ೥ʹ$PPMFZͱ5VLFZ͕޿Ίͨɽ
    ɾʮप೾਺ؒҾ͖ܕ''5ʯͱʮ࣌ؒؒҾ͖ܕ''5ʯͷλΠϓ͕͋Δɽ
    ɾʮप೾਺ؒҾ͖ܕ''5ʯ͸ϑʔϦΤม׵ޙͷ஋Λฒ΂ସ͑Δɽ
    ɾʮ࣌ؒؒҾ͖ܕ''5ʯ͸ϑʔϦΤม׵લͷ࣌ؒσʔλΛฒͼସ͑Δ
    ''5ͷσʔλ਺͸ɼͷ΂͖৐ʹͳΔ

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  65. ߴ଎ϑʔϦΤม׵''5
    X0
    X1
    X2

    XN−1
    =
    1 1 1 ⋯ 1
    1 e−i 2π
    N e−i 4π
    N ⋯ e−i 2π(N − 1)
    N
    1 e−i 4π
    N e−i 8π
    N ⋯ e−i 4π(N − 1)
    N
    ⋮ ⋮ ⋮ ⋱ ⋮
    1 e−i 2π(N − 1)
    N e−i 4π(N − 1)
    N ⋯ e−i 2π(N − 1)(N − 1)
    N
    x0
    x1
    x2

    xN−1
    ճసҼࢠΛಋೖͯ͠ɼࣜΛݟ΍͘͢͢Δ
    WN
    = e−i 2π
    N
    WN
    X0
    X1
    X2

    XN−1
    =
    1 1 1 ⋯ 1
    1 WN
    W2
    N
    ⋯ WN−1
    N
    1 W2
    N
    W4
    N
    ⋯ W2(N−1)
    N
    ⋮ ⋮ ⋮ ⋱ ⋮
    1 W(N−1)
    N
    W2(N−1)
    N
    ⋯ W(N−1)(N−1)
    N
    x0
    x1
    x2

    xN−1
    ෳૉࢦ਺ؔ਺ͷ΂͖৐͕ෳૉฏ໘৐Ͱ͸ճసΛ͢Δ
    WN
    = e−i 2π
    N
    W8
    = e−i 2π
    8
    W0
    8
    = (e−i 2π
    8 )0 = 1
    W1
    8
    = e−i 2π
    8 =
    1
    2
    − i
    1
    2
    e−iθ = cos θ − i sin θ
    W2
    8
    W3
    8
    W4
    8
    W5
    8
    W6
    8
    W7
    8
    Re
    Im
    W0
    8
    = W8
    8
    = W16
    8
    = ⋯
    W1
    8
    = W9
    8
    = W17
    8
    = ⋯
    Nʹ8ͷͱ͖
    ճసҼࢠͰهड़͢Δ͜ͱʹΑͬͯɼ
    ܭࢉΛ࡟ݮ͢ΔͨΊͷ޻෉͕Ͱ͖Δɽ

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  66. ߴ଎ϑʔϦΤม׵''5
    N = 2
    ͷႈʹै͏σʔλ਺ͱγάφϧϑϩʔਤͷؔ܎
    N = 4
    (
    X0
    X1
    ) = (
    1 1
    1 W2
    ) (
    x0
    x1
    )
    W2
    = e−i 2π
    2 = e−iπ = − 1
    W0
    2
    = W2
    2
    = W4
    2
    = 0
    W1
    2
    = W3
    2
    = W5
    2
    = − 1 Re
    Im
    X0
    X1
    X2
    X3
    =
    1 1 1 1
    1 W1
    4
    W2
    4
    W3
    4
    1 W2
    4
    W4
    4
    W6
    4
    1 W3
    4
    W6
    4
    W9
    4
    x0
    x1
    x2
    x3
    W4
    = e−i 2π
    4 = e−i π
    2 = − i
    Re
    Im
    W0
    4
    = W4
    4
    = W8
    4
    = 1
    W1
    4
    = W5
    4
    = W9
    4
    = − i
    W2
    4
    = W6
    4
    = − 1
    W3
    4
    = i
    N = 8 W8
    = e−i 2π
    8 = e−i π
    4 = +
    1
    2

    1
    2
    i
    X0
    X1
    X2

    X7
    =
    1 1 1 ⋯ 1
    1 W8
    W2
    8
    ⋯ W7
    8
    1 W2
    8
    W4
    8
    ⋯ W14
    8
    ⋮ ⋮ ⋮ ⋱ ⋮
    1 W7
    8
    W14
    8
    ⋯ W49
    8
    x0
    x1
    x2

    x7
    W0
    8
    = (e−i 2π
    8 )0 = 1
    W1
    8
    = e−i 2π
    8 =
    1
    2
    − i
    1
    2
    W2
    8
    W3
    8
    W4
    8
    W5
    8
    W6
    8
    W7
    8
    Re
    Im

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  67. ߴ଎ϑʔϦΤม׵''5
    όλϑϥΠԋࢉͱγάφϧϑϩʔਤ
    a
    b
    a + b
    a − b

    ''5ͷܭࢉ͸ɼͭͷσʔλΛՃݮ͠ɼ͞ΒʹճసҼࢠͷk ৐Λ͔͚Δͱ͍͏جຊԋࢉ
    WN
    = e−i 2π
    N
    a
    b
    a + b
    Wk
    N
    (a − b)
    − Wk
    N
    ௏ͷӋͷܗʹࣅ͍ͯΔͷͰɼ
    όλϑϥΠԋࢉ
    ͜ͷԋࢉͷਤ͸
    γάφϧϑϩʔਤ
    ਤ" ਤ#
    ͭͷσʔλa bΛೖྗͨ͠ͱ͖ɼ
    ͭͷ࿨ΛҰํʹɼͭͷࠩΛଞํʹग़ྗ
    ͭͷσʔλa bΛೖྗͨ͠ͱ͖ɼ
    ͭͷ࿨ΛҰํʹɼͭͷࠩʹճసҼࢠΛ
    ͔͚ͨ஋Λଞํʹग़ྗ
    ͞Βʹɼ''5ͷܭࢉ͸ʮϏοτϦόʔεʯͱ͍͏ɼॱংʹσʔλΛฒ΂ସ͑Δૢ࡞Λ͢Δ

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  68. ߴ଎ϑʔϦΤม׵''5
    όλϑϥΠԋࢉͷྫ
    N = 2
    (
    X0
    X1
    ) = (
    1 1
    1 W1
    2
    ) (
    x0
    x1
    )
    σʔλ਺N = 21
    ճసҼࢠ͸ɼN = 1ͳͷͰ͜͏ͳΓ·͢
    WN
    = e−i 2π
    N
    W2
    = e−i 2π
    2
    W0
    2
    = W2
    2
    = 1
    e−iθ = cos θ − i sin θ
    W1
    2
    = W3
    2
    = − 1 Re
    Im
    Nʹ2ͷͱ͖
    ( = e−iπ = − 1)
    X0
    = (x0
    + x1
    )
    X1
    = (x0
    − x1
    )
    (
    X0
    X1
    ) = (
    1 1
    1 −1) (
    x0
    x1
    )
    x0
    x1
    x0
    + x1
    = X0
    x0
    − x1
    = X1

    x0
    x1
    X0
    X1
    W2
    γάφϧϑϩʔਤ
    ্ͷγάφϧϑϩʔਤͷॲཧ
    Λ͜ͷΑ͏ʹॻ͍͓ͯ͘

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  69. ߴ଎ϑʔϦΤม׵''5
    N = 4
    X0
    X1
    X2
    X3
    =
    1 1 1 1
    1 W1
    N
    W2
    N
    W3
    N
    1 W2
    N
    W4
    N
    W6
    N
    1 W3
    N
    W6
    N
    W9
    N
    x0
    x1
    x2
    x3
    σʔλ਺N = 22
    ճసҼࢠ͸ɼN = 4ͳͷͰ͜͏ͳΓ·͢
    WN
    = e−i 2π
    N
    W4
    = e−i 2π
    4
    W0
    4
    = W4
    4
    = W8
    4
    = 1
    e−iθ = cos θ − i sin θ
    W1
    4
    = W5
    4
    = W9
    4
    = − i
    W2
    4
    = W6
    4
    = − 1
    W3
    4
    = i
    Re
    Im
    Nʹ4ͷͱ͖
    ( = e−i π
    2 = − i)
    X0
    X1
    X2
    X3
    =
    1 1 1 1
    1 W1
    N
    −1 W3
    N
    1 −1 1 −1
    1 W3
    N
    −1 W1
    N
    x0
    x1
    x2
    x3
    ͱΓ͋͑ͣɼ࣮෦Λ୅ೖ
    όλϑϥΠԋࢉͷྫ

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  70. ߴ଎ϑʔϦΤม׵''5
    ஋ ਐ਺ ٯస ϏοτϦόʔε




    X0
    X2
    X1
    X3
    =
    1 1 1 1
    1 −1 1 −1
    1 W1
    N
    −1 W3
    N
    1 W3
    N
    −1 W1
    N
    x0
    x1
    x2
    x3
    ஋ͷ਺Λਐ਺ʹͯ͠ɼ
    ͦͷͱΛશͯ൓స͞
    ͤͨ΋ͷ

    प೾਺৘ใΛ΋ͱʹϏοτϦόʔεΛߦͳͬͯɼࣜมܗΛߦ͏
    㲔 㲔

    X0
    X1
    X2
    X3
    =
    1 1 1 1
    1 W1
    N
    −1 W3
    N
    1 −1 1 −1
    1 W3
    N
    −1 W1
    N
    x0
    x1
    x2
    x3
    ϑʔϦΤม׵લͷૢ࡞
    N = 4

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  71. ߴ଎ϑʔϦΤม׵''5
    X0
    X2
    X1
    X3
    =
    1 1 1 1
    1 −1 1 −1
    1 W1
    N
    −1 W3
    N
    1 W3
    N
    −1 W1
    N
    x0
    x1
    x2
    x3

    X0
    = x0
    + x1
    + x2
    + x3
    X2
    = x0
    − x1
    + x2
    − x3
    X1
    = x0
    + x1
    W1
    4
    − x2
    + x3
    W3
    4
    X3
    = x0
    + x1
    W3
    4
    − x2
    + x3
    W1
    4
    όλϑϥΠԋࢉͱࣅͯͳ͍ʁ
    X0
    = (x0
    + x2
    ) + (x1
    + x3
    )
    X2
    = (x0
    + x2
    ) − (x1
    + x3
    )
    X1
    = (x0
    − x2
    ) + W1
    4
    (x1
    + x3
    W2
    4
    ) = W0
    4
    (x0
    − x2
    ) + W1
    4
    (x1
    − x3
    )
    X3
    = (x0
    − x2
    ) + W1
    4
    (x1
    W2
    4
    + x3
    ) = W0
    4
    (x0
    − x2
    ) − W1
    4
    (x1
    − x3
    )
    N = 4
    N = 2 ͷ࣌ͷ

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  72. ߴ଎ϑʔϦΤม׵''5
    X0
    = (x0
    + x2
    ) + (x1
    + x3
    )
    X2
    = (x0
    + x2
    ) − (x1
    + x3
    )
    X1
    = (x0
    − x2
    ) + W1
    4
    (x1
    + x3
    W2
    4
    ) = W0
    4
    (x0
    − x2
    ) + W1
    4
    (x1
    − x3
    )
    X3
    = (x0
    − x2
    ) + W1
    4
    (x1
    W2
    4
    + x3
    ) = W0
    4
    (x0
    − x2
    ) − W1
    4
    (x1
    − x3
    )
    (x0
    + x2
    )

    (x1
    + x3
    )
    (x0
    + x2
    ) + (x1
    + x3
    ) = X0
    (x0
    + x2
    ) − (x1
    + x3
    ) = X2
    W0
    4
    (x0
    − x2
    )

    W1
    4
    (x1
    − x3
    )
    W0
    4
    (x0
    + x2
    ) + W1
    4
    (x1
    − x3
    ) = X1
    W0
    4
    (x0
    + x2
    ) − W1
    4
    (x1
    − x3
    ) = X3
    W2
    W2
    N = 4

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  73. ߴ଎ϑʔϦΤม׵''5
    (x0
    + x2
    )

    (x1
    + x3
    )
    (x0
    + x2
    ) + (x1
    + x3
    ) = X0
    (x0
    + x2
    ) − (x1
    + x3
    ) = X2
    W0
    4
    (x0
    − x2
    )

    W1
    4
    (x1
    − x3
    )
    W0
    4
    (x0
    + x2
    ) + W1
    4
    (x1
    − x3
    ) = X1
    W0
    4
    (x0
    + x2
    ) − W1
    4
    (x1
    − x3
    ) = X3
    W2
    W2
    N = 4
    x0
    x1
    X0
    X1
    W4
    x2
    x3
    X2
    X3
    x0

    (x0
    + x2
    )
    (x1
    + x3
    )

    W0
    4
    (x0
    − x2
    )
    W1
    4
    (x1
    − x3
    )
    x1
    x2
    x3
    W0
    4
    W1
    4

    X0
    X2
    W2

    X1
    X3
    W2
    N = 2ͷγάφϧϑϩʔਤ
    ͕಺ଆʹ͋Δ

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  74. ߴ଎ϑʔϦΤม׵''5
    ϏοτϦόʔε
    N = 8 σʔλ਺N = 23
    ઌʹγάφϧϑϩʔਤ͔Β



    X1
    X5

    X3
    X7


    W0
    4
    W1
    4

    X0
    X4

    X2
    X6
    W2
    x4
    x5
    x6
    x7
    x0
    x1
    x2
    x3
    W4
    W4
    W2
    W8
    W0
    4
    W1
    4
    W2
    W2
    ϏοτϦόʔε
    σʔλ਺͕

    N = 22 ͷ࣌ͷॲཧ
    σʔλ਺͕

    N = 2ͷ࣌ͷॲཧ




    W0
    8
    W1
    8
    W2
    8
    W3
    8

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  75. ߴ଎ϑʔϦΤม׵''5
    N = 8
    X0
    X1
    X2

    X7
    =
    1 1 1 ⋯ 1
    1 W8
    W2
    8
    ⋯ W7
    8
    1 W2
    8
    W4
    8
    ⋯ W14
    8
    ⋮ ⋮ ⋮ ⋱ ⋮
    1 W7
    8
    W14
    8
    ⋯ W49
    8
    x0
    x1
    x2

    x7
    %'5ͷม׵ࣜ
    W0
    8
    = (e−i 2π
    8 )0 = 1
    W1
    8
    = e−i 2π
    8 =
    1
    2
    − i
    1
    2
    W2
    8
    W3
    8
    W4
    8
    W5
    8
    W6
    8
    W7
    8
    Re
    Im
    W8
    = e−i 2π
    8 =
    1
    2
    − i
    1
    2
    ճసҼࢠ
    ஋ ਐ਺ ٯస ϏοτϦόʔε








    㲔 㲔
    Xn
    ɹ͸Ϗοτ
    Ϧόʔεॱ
    X0
    = x0
    + x1
    + x2
    + x3
    + x4
    + x5
    + x6
    + x7
    X4
    = x0
    − x1
    + x2
    − x3
    + x4
    − x5
    + x6
    − x7
    X2
    = x0
    + x1
    W2
    8
    − x2
    − x3
    W2
    8
    + x4
    + x5
    W2
    8
    − x6
    − x7
    W2
    8
    X6
    = x0
    − x1
    W2
    8
    − x2
    + x3
    W2
    8
    + x4
    − x5
    W2
    8
    − x6
    + x7
    W2
    8
    X1
    = x0
    + x1
    W8
    + x2
    W2
    8
    + x3
    W3
    8
    − x4
    − x5
    W8
    − x6
    W2
    8
    − x7
    W3
    8
    X5
    = x0
    − x1
    W8
    + x2
    W2
    8
    − x3
    W3
    8
    − x4
    + x5
    W8
    − x6
    W2
    8
    + x7
    W3
    8
    X3
    = x0
    + x1
    W3
    8
    − x2
    W2
    8
    − x3
    W5
    8
    − x4
    − x5
    W3
    8
    + x6
    W2
    8
    + x7
    W5
    8
    X7
    = x0
    − x1
    W3
    8
    − x2
    W2
    8
    + x3
    W5
    8
    − x4
    + x5
    W3
    8
    + x6
    W2
    8
    − x7
    W5
    8
    Re
    Im
    W0
    4
    = 1
    W1
    4
    = − i
    W2
    4
    = − 1
    W3
    4
    = i
    W4
    = e−i 2π
    4 = − i
    W0
    4
    = W0
    8
    W1
    4
    = W2
    8
    Λ࢖ͬͯࣜมܗ

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  76. ߴ଎ϑʔϦΤม׵''5
    X0
    = x0
    + x1
    + x2
    + x3
    + x4
    + x5
    + x6
    + x7
    X4
    = x0
    − x1
    + x2
    − x3
    + x4
    − x5
    + x6
    − x7
    X2
    = x0
    + x1
    W2
    8
    − x2
    − x3
    W2
    8
    + x4
    + x5
    W2
    8
    − x6
    − x7
    W2
    8
    X6
    = x0
    − x1
    W2
    8
    − x2
    + x3
    W2
    8
    + x4
    − x5
    W2
    8
    − x6
    + x7
    W2
    8
    X1
    = x0
    + x1
    W8
    + x2
    W2
    8
    + x3
    W3
    8
    − x4
    − x5
    W8
    − x6
    W2
    8
    − x7
    W3
    8
    X5
    = x0
    − x1
    W8
    + x2
    W2
    8
    − x3
    W3
    8
    − x4
    + x5
    W8
    − x6
    W2
    8
    + x7
    W3
    8
    X3
    = x0
    + x1
    W3
    8
    − x2
    W2
    8
    − x3
    W5
    8
    − x4
    − x5
    W3
    8
    + x6
    W2
    8
    + x7
    W5
    8
    X7
    = x0
    − x1
    W3
    8
    − x2
    W2
    8
    + x3
    W5
    8
    − x4
    + x5
    W3
    8
    + x6
    W2
    8
    − x7
    W5
    8
    N = 8 N = 4 ͕ग़ͯ͘ΔΑ͏ʹมߋ͢Δ
    X0
    = {(x0
    + x4
    ) + (x2
    + x6
    )} + {(x1
    + x5
    ) + (x3
    + x7
    )}
    X4
    = {(x0
    + x4
    ) + (x2
    + x6
    )} − {(x1
    + x5
    ) + (x3
    + x7
    )}
    X2
    = W0
    4
    {(x0
    + x4
    ) − (x2
    + x6
    )} + W1
    4
    {(x1
    + x5
    ) − (x3
    + x7
    )}
    X6
    = W0
    4
    {(x0
    + x4
    ) − (x2
    + x6
    )} − W1
    4
    {(x1
    + x5
    ) − (x3
    + x7
    )}
    X1
    = {W0
    8
    (x0
    − x4
    ) + W2
    8
    (x2
    − x6
    )} + {W1
    8
    (x1
    − x5
    ) + W3
    8
    (x3
    − x7
    )}
    X5
    = {W0
    8
    (x0
    − x4
    ) + W2
    8
    (x2
    − x6
    )} − {W1
    8
    (x1
    − x5
    ) + W3
    8
    (x3
    − x7
    )}
    X3
    = W0
    4
    {(W0
    8
    x0
    − x4
    ) − W2
    8
    (x2
    − x6
    )} + W1
    4
    {W1
    8
    (x1
    − x5
    ) − W3
    8
    (x3
    − x7
    )}
    X7
    = W0
    4
    {(W0
    8
    x0
    − x4
    ) − W2
    8
    (x2
    − x6
    )} − W1
    4
    {W1
    8
    (x1
    − x5
    ) − W3
    8
    (x3
    − x7
    )}



    X1
    X5

    X3
    X7


    W0
    4
    W1
    4

    X0
    X4

    X2
    X6
    W2
    x4
    x5
    x6
    x7
    x0
    x1
    x2
    x3
    W4
    W4
    W2
    W8
    W0
    4
    W1
    4
    W2
    W2
    (x0
    + x4
    )
    (x1
    + x5
    )
    (x2
    + x6
    )
    (x3
    + x7
    )
    W0
    8
    (x0
    − x4
    )
    W1
    8
    (x1
    − x5
    )
    W2
    8
    (x2
    − x6
    )
    W3
    8
    (x3
    − x7
    )
    W0
    8
    W1
    8
    W2
    8
    W3
    8




    N/2ͷॲஔʹʂ

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  77. ߴ଎ϑʔϦΤม׵''5
    ''5ͷҰൠԽΛ͢Δ
    ̍ɽ্൒෼͔ΒɼԼ൒෼ʹࣼΊઢΛ
    ̎ɽԼ൒෼͔Βɼ্൒෼΋ಉ༷ʹ
    ̏ɽ্൒෼͸࿨ԋࢉ
    ̐ɽԼ൒෼͸ࠩԋࢉ͠ɼճసࢠΛఴ෇
    ̑ɽγάφϧϑϩʔਤʹैͬͯܭࢉ
    xN
    2
    xN
    2
    +1
    x0
    x1
    x2
    x3
    xN
    2
    +2
    xN
    2
    +3
    N
    2
    N
    2
    ⋯ ⋯
    ⋯ ⋯




    W0
    N
    W1
    N
    W2
    N
    W3
    N
    ্൒෼ͱԼ൒෼ͷܭࢉ݁ՌΛͦΕͧΕ
    /ݸͷ཭ࢄ৴߸ͷͨΊͷ''5ॲཧʹ
    Ҿ͖౉͢
    ݩʑͷσʔλ਺͸ͷ΂͖৐ͳͷͰɼ
    Ҏ্Λ܁Γฦ͢͜ͱʹΑͬͯɼ࠷ऴత
    ʹݸͷσʔλʹͳΔɽ

    ͜Ε͕''5ͷܭࢉͰ͋Δɽ
    W
    N
    2
    ͷγάφϧ
    ϑϩʔਤ
    W
    N
    2
    ͷγάφϧ
    ϑϩʔਤ


    Xn
    ɹɹ͸Ϗοτ
    Ϧόʔεॱ

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  78. ߴ଎ϑʔϦΤม׵''5



    X1
    X5

    X3
    X7


    W0
    4
    W1
    4

    X0
    X4

    X2
    X6
    x4
    x5
    x6
    x7
    x0
    x1
    x2
    x3
    W0
    4
    W1
    4
    ৐ࢉճ਺
    ৐ࢉճ਺͸Ͳͷ͘Β͍ʁ
    ෳૉ਺ͷܭࢉ͸େมʁ
    W1
    8

    W2
    8

    W3
    8
    W4
    8


    ճͷόλϑϥΠԋࢉ
    ճͷόλϑϥΠԋࢉ

    ճͷෳૉ਺ͷ৐ࢉ
    ճͷόλϑϥΠԋࢉ

    ճͷෳૉ਺ͷ৐ࢉ
    log2
    N = log2
    8 = 3 ஈ
    όλϑϥΠͷஈ਺
    N
    2
    ճ
    N
    2
    ճ
    N
    2
    ճ
    ʻྫ͑͹ʼ N = 210 = 1024
    N2 = 1024 × 1024 ≒ 1000000
    N
    2
    (log2
    N − 1) = 512 log2
    210 − 1 ≒ 4600
    %'5

    ''5

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  79. ৴߸ॲཧ
    ࢀߟࢿྉ
    ಓ۩ͱͯ͠ͷϑʔϦΤղੳ
    ༚Ҫྑ޾༚Ҫఃඒ
    Ϟʔυղੳೖ໳
    ௕দতஉ
    Ͳͬͪ΋ྑॻͰ͢

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  80. ৴߸ॲཧ
    ࢀߟ8FCαΠτ
    ɾ΍Δ෉ͰֶͿσ
    Οδλϧ৴߸ॲཧ
    IUUQXXXJDJTUPIPLVBDKQdTXLMFDUVSFZBSVPETQUPDIUNM

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