「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

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
2. ͪ͜ΒͷຊΛࢀߟʹͯ͠·͢ ࢀߟࢿྉ ಓ۩ͱͯ͠ͷϑʔϦΤղੳ ༚Ҫྑ޾༚Ҫఃඒ Ϟʔυղੳೖ໳ ௕দতஉ Ͳͬͪ΋ྑॻͰ͢
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਺
   ˠ
   ୯७ͳؔ਺Λ଍ͨ͠΋ͷ ෳࡶͳ৴߸Λཧղ͢ΔͨΊͷπʔϧʂͲͷΑ͏ͳಛੑ͕͋Δͷ͔͕Θ͔Δ
4. पظ
   T ඵ f = 1 T पظؔ਺ t + T t ࣍ݩͷ࣌ؒؔ਺ ೾ܗ ʹؔ͢Δجຊ༻ޠͷ෮श प೾਺ ৼಈ਺
   f
   ɹ୯Ґ࣌ؒʹؚ·ΕΔपظ
   T 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
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 2π ֯प೾਺
   ω ͸पظ
   T ͱͷؔ܎ͩͱɼҎԼʹͳΔ f = 1 T ω = 2π T ୯Ґԁ ֯प೾਺ ৴߸ʹ͍ͭͯ
   ϑʔϦΤڃ਺΁ͷ༠͍
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/
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਺ഒʹͳΔ
   ˠ
   ഒͱ͔ɼഒͱ͔ɼ൒୺ͳ਺ࣈʹͳΒͳ͍ ϑʔϦΤڃ਺
   ɹɹɹɾɾɾʮؔ਺͕ແݶݸͷࡾ֯ؔ਺ͷ࿨ͰදΘͤΔʯ
   ɹɹɹɾɾɾʮάϥϑ͕ਖ਼ݭ೾ͷॏͶ߹ΘͤͰදΘͤΔʯ
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) جຊपظ͕
   T
   ͳΒɼجຊप೾਺͸
   )[
   ʹͳΔɽ
 ͦͷ੔਺ഒ͔͠ݱΕͳ͍ͷͰɼ)[
   )[
   )[ͷ੒෼ɼɼɼʹͳΔɽ
   ͳͥͳΒɼجຊपظͷதͰɼ੔਺ݸͷ೾͕ऩ·Βͳ͍ͱ͍͚ͳ͍ͨΊʂ
   ଍͠߹Θͤͯपظతͳؔ਺ʹ͢ΔͨΊʂ جຊपظ
   T0 ω0 = 2π T0 f0 = 1 T0 جຊपظ
   T0 ʹΑͬͯɼ
   جຊप೾਺
   f0
   ͕ܾ·Δ
   جຊप೾਺
   f0
   ʹΑͬͯ
   جຊ֯प೾਺
   ω0
   ͕ܾ·Δ
   
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೾ ఆ ਺ ͱ ࡾ ֯ ؔ ਺ 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਺
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 ) + ⋯ ෯͕
    T 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 ˠ
    ޙ΄Ͳৄ͘͠ಋग़͢Δ f(t) = a0 + ∞ ∑ n=1 {an cos 2πn T0 t + bn sin 2πn T0 t}
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 पظ
    T 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਺͸ɼपظ
    T ͷؔ਺
    
    
    
    
    
    
    
    ʹ͢Δ͜ͱ f(t) ϑʔϦΤڃ਺͸पظ
    T ͷपظؔ਺Ͱ͋Δ
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΍͍͢
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
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෼
    
    ಺ੵ
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
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
17. ϑʔϦΤ܎਺ͷಋग़
    a0 ʮϑʔϦΤ܎਺ a0, an, bn ͷಋग़ʯ cos nω0 t, sin nω0 t a0 ʮશͯͷ߲ʹΛֻ͚ͯੵ෼ʯ
    ˠ
    ௚ަ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     ಉ༷ʹ
    an, 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 :
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 ʮશͯͷ߲ʹ
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    Λֻ͚ͯੵ෼
    ∫ 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܎਺ͷಋग़
    an
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 ʮશͯͷ߲ʹ
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    Λֻ͚ͯੵ෼
    ∫ 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܎਺ͷಋग़
    bn
20. ϑʔϦΤڃ਺ల։ͷผͷදهํ๏ cos
    ͱ
    sin ͸Ґ૬͕
    1/4 ͣ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೾਺ͷ
    cos ͱ
    sin ͕ͭͷ
    sin Ͱهड़͢Δ͜ͱ͕Ͱ͖Δ ϑʔϦΤڃ਺ల։͸ɼల։͞Ε֤ͨ
    sin ೾ͦΕͧΕͷৼ෯ͱҐ૬ ΛٻΊΔ͜ͱͰ΋දݱ͢Δ͜ͱ͕Ͱ͖Δɽ An = a2 n + b2 n y x
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
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
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/
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 cos
    ͱ
    sin 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
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
26. ϑʔϦΤڃ਺
    ˠ
    ෳૉϑʔϦΤڃ਺΁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
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
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௚ަجఈʯͰ͋Δඞཁ͕͋Δ
    
    ˠ
    
    
    ͓ޓ͍ͷؔ਺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
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
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਺ͷ෮श
    ɾɾɾͪͳΈʹϑʔϦΤڃ਺ͷ௚ަجఈ͸ͪ͜Β ఆٛҬͷ֓೦͸͍࣋ͬͯΔ
    
    पظ
    T (n = 0, ± 1, ± 2, ⋯) ௚ަجఈͷੑ࣭
31. ෳૉϑʔϦΤڃ਺ͷ܎਺Λಋग़͢Δ ʮల։ࣜͷ྆ลʹɼӈ͔ΒɹɹɹɹΛ͔͚Δʯ
    ˠ
    ʮͦͷޙ۠ؒ
    T 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܎਺ͷಋग़
    cn
32. ෳૉϑʔϦΤڃ਺ల։ͷϚΠφεଆΛߟ͑Δ ΦΠϥʔͷެ͔ࣜΒղऍ͢Δ cos θ = eiθ + e−iθ 2 ϓϥεଆͷθ ϚΠφεଆͷθ ଍ͯ͠
    2 ͰׂΔͱ
    cos 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
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
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
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׵
36. ෳૉϑʔϦΤڃ਺
    ਺ֶͷ͓࿩ ෳૉϑʔϦΤڃ਺ͷ܎਺Λಋग़͢Δ ʮల։ࣜͷ྆ลʹɼӈ͔ΒɹɹɹɹΛ͔͚Δʯ
    ˠ
    ʮͦͷޙ۠ؒ
    T 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
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 = 2π 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
38. ω0 = 2π T ֯प೾਺
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    ࣌ ؒ ྖ Ҭ प ೾ ਺ ྖ 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
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 = 2π T पظؔ਺ͳͷͰɼपظ
    T ͕େࣄͳཁૉͩͬͨ
    ͦΕʹΑͬͯɼجຊ֯प೾਺
    ω0 ͕ग़ͯ͘Δ ෳૉϑʔϦΤ܎਺
    cn
    ʹɼෳૉࢦ਺ؔ਺
    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 = 2π T ω0 पظ
    T Λແݶେʿʹʂ ͕ඇৗʹখ͘͞ͳΔ प೾਺εϖΫτϧ͕
    ͱͼͱͼͰ͸ͳ͘ɼ࿈ଓʹͳΔ ઢ प೾਺εϖΫτϧ ͷִؒ Λڀۃʹڱ͘͠ɼ໘ੵͱͯ͠ औΓѻ͏Α͏ʹߟ͑Δ ૯࿨Λܭࢉ͢Δ໰୊
    ˠ
    ໘ੵΛܭࢉ͢Δ໰୊΁
40. ϑʔϦΤม׵ͷ਺ࣜΛಋग़͢ΔɿٯϑʔϦΤม׵ f(t) = ∞ ∑ n=−∞ cn eiω0 nt ෳૉϑʔϦΤڃ਺ల։͔ΒɼٯϑʔϦΤม׵ = ∞ ∑ n=−∞ ω0 cn eiω0 nt ω0 = 1 2π ∞ ∑ n=−∞ ω0 2πcn eiω0 nt ω0 = 1 2π ∞ ∑ n=−∞ ω0 2πcn eiω(n)t ω0 = 1 2π ∞ ∑ n=−∞ ω0 F(ω(n))eiω(n)t F(ω(n)) = 2πcn ω0 ω(n) = ω0 n f(t) = 1 2π ∞ ∑ 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 n
    ഒͷ஋Λ࣋ͭ֯प೾਺͕
    ࣮਺
    ω ͷؔ਺ʹͳΔ Δω → dω ແݶখʹͳΔ ٯϑʔϦΤม׵ ޙʑͷϑʔϦΤม׵ͷࣜΛ γϯϓ ϧʹهड़͢ΔͨΊ
41. ϑʔϦΤม׵ͷ਺ࣜΛಋग़͢ΔɿϑʔϦΤม׵ cn = 1 T ∫ 2 T − 2 T f(t)e−iω(n)tdt = 2π ω0 1 T ∫ 2 T − 2 T f(t)e−iω(n)tdt F(ω(n)) = 2πcn ω0 ω0 = 2π 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 n
    ഒͷ஋Λ࣋ͭ֯प೾਺͕
    ࣮਺
    ω ͷؔ਺ʹͳΔ T0 → ∞ ͷۃݶΛߟ͑Δ ϑʔϦΤม׵
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࿩
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
44. ·ͱΊɿϑʔϦΤڃ਺ల։ͱٯϑʔϦΤม׵ͷಛ௃ f(t) = ∞ ∑ n=−∞ cn eiω0 nt ෳૉϑʔϦΤڃ਺ c0 c−1 c−2 c2 c1 c−3 c3 einω0 t ɾपظ
    T 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
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
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/
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༗ݶͷ࣌ؒ
    T Λѻ͏ͷͰɼ
 Ͳ͔ͬͪͱ͍͏ͱෳૉϑʔϦΤڃ਺ʹ͍ۙͧʂ ཭ࢄ৴߸͸
 ੵ෼Ͱ͖Δͷʂʁ
48. ৴߸ॲཧ "%ม׵
    "OBMPHUPEJHJUBM
    DPOWFSTJPO 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
49. ߴ଎ϑʔϦΤม׵
    ''5 φΠΩετͷඪຊԽఆཧ ͋Δਖ਼ݭ೾ΛඪຊԽ͢Δࡍʹ͸ɼͦͷपظͷ൒෼ΑΓ΋୹͍ඪຊԽִؒΛ༻͍ͳ͚Ε͹ͳΒͳ͍ पظ
    T 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
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
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)
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೾਺৘ใ
    c0 ~ c3 ͕ಘΒΕΔ ల։͞Ε֤߲ͨ ෳૉਖ਼ݭ೾ ͸΋ͪΖΜ௚ަ͍ͯ͠Δ
53. ཭ࢄϑʔϦΤม׵
    %'5
    ࣮ࡍʹݱ৔Ͱ࢖͏͓࿩ ཭ࢄ৴߸
    x0 ~ x3 ͔Βɼप೾਺৘ใ
    c0 ~ 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೾਺৘ใ
    c0 ~ c3 ͷؔ܎Λಋग़͢Δ
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೾਺৘ใ
    c0 ~ c3 ͷؔ܎Λಋग़͢Δ ߦྻܗࣜʹมߋ ֤ߦϕΫτϧ͸΋ͪΖΜ௚ަ͍ͯ͠Δ
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 =
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
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
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
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
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/
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 㲔 ཭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/
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
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 ⋯ Nݸ X0 X1 X2 ⋯ Xn−1 ω ⋯ ม׵ߦྻ Nݸͷ཭ࢄ৴߸ͷσʔλΛ ॲཧ͢Δʹ͸ɼN2 ճͷෳૉ ਺ͷੵͷܭࢉ͕ඞཁʹͳΔ N2ճͷܭࢉճ਺Λେ෯ʹݮগͤ͞Δ޻෉͕͋Δ
    ˠ
    ''5
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׵
    ''5
    ͷ֓ཁ ม׵ߦྻ N2ճͷܭࢉճ਺Λେ෯ʹݮগͤ͞Δ޻෉͕͋Δ
    ˠ
    ''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
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 ճసҼࢠ
    
    
    
    
    
    Λಋೖͯ͠ɼࣜΛݟ΍͘͢͢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
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
67. ߴ଎ϑʔϦΤม׵
    ''5 όλϑϥΠԋࢉͱγάφϧϑϩʔਤ a b a + b a − b − ''5ͷܭࢉ͸ɼͭͷσʔλΛՃݮ͠ɼ͞ΒʹճసҼࢠͷ
    k ৐Λ͔͚Δͱ͍͏جຊԋࢉ WN = e−i 2π N a b a + b Wk N (a − b) − Wk N ௏ͷӋͷܗʹࣅ͍ͯΔͷͰɼ
    όλϑϥΠԋࢉ ͜ͷԋࢉͷਤ͸
    γάφϧϑϩʔਤ ਤ" ਤ# ͭͷσʔλ
    a
    b
    Λೖྗͨ͠ͱ͖ɼ
    ͭͷ࿨ΛҰํʹɼͭͷࠩΛଞํʹग़ྗ ͭͷσʔλ
    a
    b
    Λೖྗͨ͠ͱ͖ɼ
    ͭͷ࿨ΛҰํʹɼͭͷࠩʹճసҼࢠΛ
    ͔͚ͨ஋Λଞํʹग़ྗ ͞Βʹɼ''5ͷܭࢉ͸ʮϏοτϦόʔεʯͱ͍͏ɼॱংʹσʔλΛฒ΂ସ͑Δૢ࡞Λ͢Δ
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
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
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 㲔 प೾਺৘ใΛ΋ͱʹϏοτϦόʔεΛߦͳͬͯɼࣜมܗΛߦ͏ 㲔 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
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 㲔 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
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
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
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
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
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
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
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
79. ৴߸ॲཧ ࢀߟࢿྉ ಓ۩ͱͯ͠ͷϑʔϦΤղੳ ༚Ҫྑ޾༚Ҫఃඒ Ϟʔυղੳೖ໳ ௕দতஉ Ͳͬͪ΋ྑॻͰ͢
80. ৴߸ॲཧ ࢀߟ8FCαΠτ ɾ΍Δ෉ͰֶͿσ Οδλϧ৴߸ॲཧ IUUQXXXJDJTUPIPLVBDKQdTXLMFDUVSFZBSVPETQUPDIUNM