逆FM音源

与えられた楽器の音色に近いFM音源パラメータの探索を勾配法で解く方法を解説します
これは2020年2月8日に行われた カーネル/VM探検隊@関西 10回目 での発表資料です
サンプルコード: https://github.com/Fadis/ifm

Transcript

1. ٯFMԻݯ NAOMASA MATSUBAYASHI INVERTED FREQUENCY MODULATION SYNTHESIZER https://github.com/Fadis/ifm αϯϓϧίʔυ
2. Իָͷԋ૗͸೾ͷੜ੒ εϐʔΧʔͷ ΞΠίϯ ిѹͷ೾ ۭؾͷ೾ Ի֊ Իྔ υ ίϯϐϡʔλͰԻָΛ૗ͰΔʹ͸ Ի֊ʹରԠ͢Δप೾਺ͷ೾Λੜ੒͠ͳ͚Ε͹ͳΒͳ͍ ?
3. Իݯνοϓͷొ৔ εϐʔΧʔͷ ΞΠίϯ ిѹͷ೾ ۭؾͷ೾ Ի֊ Իྔ υ ͜ͷੜ੒Λߦ͏ϋʔυ΢ΣΞ͕ొ৔ͨ͠
4. υ Ϩ ϛ ϑΝ ι ԣ࣠: प೾਺ ॎ࣠: ࣌ؒ(ͦΕͧΕͷԻ֊ʹ͍ͭͯ໐Γ࢝Ί͔Β2ඵ෼)
5. 65.406Hz 130.81Hz 196.22Hz 261.63Hz 327.03Hz 392.44Hz 457.84Hz 523.25Hz 588.66Hz υ Ϩ ϛ ϑΝ ι
6. 65.406Hz 130.81Hz 196.22Hz 261.63Hz 327.03Hz 392.44Hz 457.84Hz 523.25Hz 588.66Hz υ Ϩ ϛ ϑΝ ι Ի֊Λָ࣋ͭثͷԻ͸ ͋Δप೾਺ͷ೾ͱ ͦͷ੔਺ഒͷप೾਺ͷ೾ͷॏͶ߹Θͤ ͜ͷ͏ͪ࠷΋௿͍प೾਺ͷ೾Λ جԻͱݺͿ Ի֊্͕͕Δͱ جԻͷप೾਺্͕͕͍ͬͯ͘
7. 65.406Hz 130.81Hz 196.22Hz 261.63Hz 327.03Hz 392.44Hz 457.84Hz 52 υ Ϩ ϛ ϑΝ ι جԻͷ੔਺ഒͷ प೾਺Ͱग़Δ೾Λ ഒԻͱݺͿ جԻͷप೾਺্͕͕Δͱ ഒԻͷप೾਺΋্͕Δ
8. άϥϯυϐΞϊ ΞϧταοΫε ϏϒϥϑΥϯ όΠΦϦϯ ϚϦϯό
9. 261.63Hz 523.25Hz 784.88Hz 1046.5Hz 1308.1Hz 1569.8Hz 1831.4Hz 2093.0Hz 2354.6Hz άϥϯυϐΞϊ ΞϧταοΫε ϏϒϥϑΥϯ όΠΦϦϯ ϚϦϯό
10. 261.63Hz 523.25Hz 784.88Hz 1046.5Hz 1308.1Hz 1569.8Hz 1831.4Hz 2093.0Hz 2354.6Hz άϥϯυϐΞϊ ΞϧταοΫε ϏϒϥϑΥϯ όΠΦϦϯ ϚϦϯό Ի͕4ΦΫλʔϒ໨ͷυʹͳΔ৚͕݅ 261.63HzͷجԻΛ࣋ͭ͜ͱͳͷͰ Ͳͷָث΋جԻʹ͸ େ͖ͳҧ͍͸ݟΒΕͳ͍
11. 1046.5Hz 1308.1Hz 1569.8Hz 1831.4Hz 2093.0Hz άϥϯυϐΞϊ ΞϧταοΫε ϏϒϥϑΥϯ όΠΦϦϯ ϚϦϯό 2616.3Hz 2877.8Hz 3139.5Hz 3401.1Hz ഒԻͷҧ͍͕ ָثͷԻ৭ͷҧ͍Λ ੜΈग़͢
12. άϥϯυϐΞϊ ΞϧταοΫε ϏϒϥϑΥϯ όΠΦϦϯ ϚϦϯό
13. άϥϯυϐΞϊ ԣ࣠: प೾਺ ॎ࣠: ࣌ؒ t ͋Δ࣌ࠁʹ͓͚Δશͯͷप೾਺ͷ੒෼ͷ૯࿨ΛऔΔͱ ͓͓Αͦͦͷ࣌ࠁͷԻͷৼ෯( Իྔ)͕ಘΒΕΔ ≃
14. Τϯϕϩʔϓ ָثͷԻͷৼ෯͸࣌ؒมԽ͢Δ ಛʹϐΞϊ΍ϚϦϯόͷΑ͏ͳ ࠷ॳʹՃ͑ͨྗ͚ͩͰ࠷ޙ·Ͱ໐Δָث͸ ៉ྷͳݮਰৼಈͷۂઢΛඳ͘
15. άϥϯυϐΞϊ ࠷ॳʹՃ͑ͨྗ͚ͩͰ࠷ޙ·Ͱ໐ΔָثͰ͸ प೾਺ͷߴ͍ഒԻ΄Ͳૣ͘ফ͑Δ܏޲ʹ͋Δ όΠΦϦϯ ܧଓతʹྗ͕Ճ͑ΒΕΔָثͰ͸ ໐Β͢ͷΛ΍ΊΔ·Ͱಉׂ͡߹ͷഒԻ͕ग़Δ܏޲ʹ͋Δ
16. ཁٻ͞ΕͨԻ֊ʹରԠ͢ΔجԻͷαΠϯ೾ͱ ָثʹԠͨ͡ഒԻͷαΠϯ೾Λੜ੒͢Δ ͦΕΒͷৼ෯͕࣮ࡍͷָثͱಉ͡Α͏ʹ࣌ؒมԽ͍ͯ͠Ε͹ ͦͷָثͰཁٻ͞ΕͨԻ֊ͷԻΛ໐Βͨ͠Α͏ʹฉ͑͜Δ ϐΞϊͷԻͰ υ ഒ཰ ৼ෯                     ∑
17. ϐΞϊͷԻͰ υ ໰୊఺ αΠϯ೾ͷܭࢉ͸҆͘ͳ͍ ഒ཰ ৼ෯                     ∑
18. प೾਺มௐ f (t) = sin (2πωc t + B sin (2πωm t))
19. ωc = ωm 2ωc = ωm 3ωc = ωm 4ωc = ωm f (t) = sin (2πωc t + B sin (2πωm t))
20. ഒԻ੒෼ͷݮਰ f (t) = Ec (t) sin (2πωc t + Em (t) B sin (2πωm t)) ৼ෯ͷ࣌ؒมԽΛ ίϯτϩʔϧ͢Δ ഒԻ੒෼ͷྔͷ࣌ؒมԽΛ ίϯτϩʔϧ͢Δ άϥϯυϐΞϊ
21. '.Իݯ͸
    ΞίʔεςΟοΫͳָثͷ
    ಛ௃Λগͳ͍ύϥϝʔλͰදݱग़དྷΔ
    ѱ͘ͳ͍ϞσϧͰ͋Δ
22. OΦϖϨʔλͷ'.Իݯͷܭࢉࣜ O
    ΦϖϨʔλ਺
    &J
    Τϯϕϩʔϓ
    TJ
    جԻͷप೾਺ʹର͢Δഒ཰
    ЋJK
    ΦϖϨʔλK͕ΦϖϨʔλJΛมௐ͢Δڧ͞
    ЌJ
    ग़ྗʹؚ·ΕΔΦϖϨʔλJͷ৴߸ͷׂ߹ ୈ12ճΧʔωϧ/VM୳ݕୂ Ҩ఻తFMԻݯ(2015೥)ΑΓ '.Իݯͷ໰୊఺
23. Ͳ͜Λ͍ͬͨ͡Β
    ཉָ͍͠ثͷԻ৭ʹۙ෇͘ͷ͔
    ͬ͞ͺΓΘ͔ΒΜ OΦϖϨʔλͷ'.Իݯͷܭࢉࣜ ୈ12ճΧʔωϧ/VM୳ݕୂ Ҩ఻తFMԻݯ(2015೥)ΑΓ '.Իݯͷ໰୊఺ ૂͬͨԻ৭Λ
    ࡞Δͷ͕
    ͱͯ΋೉͍͠
24. ୈੈ୅ ୈੈ୅ ୈੈ୅ Ҩ఻తΞϧΰϦζϜ ͜ͷૢ࡞Λ܁Γฦ͢͜ͱʹΑͬͯ
    ༩͑ΒΕͨ৚݅ʹΑ͘ద߹͢ΔݸମΛ
    ݟ͚ͭग़͢͜ͱ͕Ͱ͖Δ ୈ12ճΧʔωϧ/VM୳ݕୂ Ҩ఻తFMԻݯ(2015೥)ΑΓ ͦ͜Ͱ
    Ҩ఻తΞϧΰϦζϜͰ
    '.Իݯύϥϝʔλͷ
    ୳ࡧΛࢼΈͨ
25. Ҩ఻తFMԻݯ ར఺
    &OE
    UP
    &OEͰԻ͔Β'.Իݯύϥϝʔλ͕ಘΒΕΔ
    ࣮༻ʹ଱͑Δ඼࣭ͷग़ྗ͕ಘΒΕΔ
    ܽ఺
    ηοτͷύϥϝʔλΛݟ͚ͭΔͷʹ(16Ͱ͔͔࣌ؒۙ͘Δ
    ෳ਺ͷ୳ࡧ݁Ռ͕ઢܗิؒͰ͖Δอূ͕ͳ͍
26. ୈ8ճΧʔωϧ/VM୳ݕୂ@ؔ੢ χϡʔϥϧFMԻݯ(2017೥)ΑΓ χϡʔϥϧωοτϫʔΫͰ
    '.Իݯύϥϝʔλͷ
    ୳ࡧΛࢼΈͨ Caffe Φʔϓϯιʔεͳ σΟʔϓϥʔχϯάϑϨʔϜϫʔΫͷ1ͭ http://caffe.berkeleyvision.org/
27. χϡʔϥϧFMԻݯ ར఺
    &OE
    UP
    &OEͰԻ͔Β'.Իݯύϥϝʔλ͕ಘΒΕΔ
    ֶशࡁΈͷωοτϫʔΫ͕͋Ε͹਺ඵͰύϥϝʔλΛಘΒΕΔ
    ܽ఺
    ग़ྗͷ඼࣭͕࣮༻ʹ଱͑ͳ͍
    ෳ਺ͷ୳ࡧ݁Ռ͕ઢܗิؒͰ͖Δอূ͕ͳ͍
28. ωc = ωm 2ωc = ωm 3ωc = ωm 4ωc = ωm 5ωc = ωm f (t) = A cos (2πωc t + B sin (2πωm t))
29. ഒԻ੒෼ͷग़ํʹ͸نଇੑ͕͋ΔΑ͏ʹݟ͑Δ ωc = ωm 2ωc = ωm 3ωc = ωm 4ωc = ωm 5ωc = ωm f (t) = A cos (2πωc t + B sin (2πωm t))
30. Ͳ͜Λ͍ͬͨ͡Β
    ཉָ͍͠ثͷԻ৭ʹۙ෇͘ͷ͔
    ͬ͞ͺΓΘ͔ΒΜ OΦϖϨʔλͷ'.Իݯͷܭࢉࣜ ຊ౰ʹ? ୈ12ճΧʔωϧ/VM୳ݕୂ Ҩ఻తFMԻݯ(2015೥)ΑΓ
31. sin (x) 2πωm t + sin (x) × B × A ग़ྗ 2πωc × × 2ΦϖϨʔλ௚ྻͷ৔߹ Τϯϕϩʔϓ͸ ඍখͳ࣌ؒͷൣғͰ͸ఆ਺ͱݟ၏ͤΔͨΊ ͱ ʹؚ·ΕΔ A B ͋Δ࣌ࠁ෇ۙͰͷFMԻݯͷग़ྗΛߟ͑Δ
32. ΩϟϦΞΛ ʹஔ͖׵͑Δ cos cos (x) 2πωm t + sin (x) × B × A ग़ྗ 2πωc × × f (t) = A cos (2πωc t + B sin (2πωm t)) Ґ૬͕ ͣΕΔ͕ৼ෯ʹ͸Өڹ͕ͳ͍ π 2
33. f (t) = A cos (2πωc t + B sin (2πωm t)) ࡾ֯ؔ਺ͷՃ๏ఆཧ f (t) = A cos (2πωc t) cos (B sin (2πωm t)) −A sin (2πωc t) sin (B sin (2πωm t)) cos (a + b) = cos a cos b − sin a sin b
34. f (t) = A cos (2πωc t) cos (B sin (2πωm t)) −A sin (2πωc t) sin (B sin (2πωm t)) cos (z sin θ) = J0 (z) + 2 ∞ ∑ k=1 J2k (z) cos (2kθ) sin (z sin θ) = 2 ∞ ∑ k=0 J2k+1 (z) sin ((2k + 1) θ) Olver, Frank W. NIST handbook of mathematical functions. Cambridge New York: Cambridge University Press NIST, 2010. p.226
35. cos (z sin θ) = J0 (z) + 2 ∞ ∑ k=1 J2k (z) cos (2kθ) sin (z sin θ) = 2 ∞ ∑ k=0 J2k+1 (z) sin ((2k + 1) θ) Olver, Frank W. NIST handbook of mathematical functions. Cambridge New York: Cambridge University Press NIST, 2010. p.226 f (t) = A cos (2πωc t) ( J0 (B) + 2 ∞ ∑ k=1 J2k (B) cos (2k (2πωm t))) −A sin (2πωc t) ( 2 ∞ ∑ k=0 J2k+1 (B) sin ((2k + 1) (2πωm t)))
36. f (t) = AJ0 (B) cos (2πωc t) +2A ∞ ∑ k=1 J2k (B) cos (2k (2πωm t)) cos (2πωc t) −2A ∞ ∑ J2k+1 (B) sin ((2k + 1) (2πωm t)) sin (2πωc t) ࣜมܗͯ͠ f (t) = A cos (2πωc t) ( J0 (B) + 2 ∞ ∑ k=1 J2k (B) cos (2k (2πωm t))) −A sin (2πωc t) ( 2 ∞ ∑ k=0 J2k+1 (B) sin ((2k + 1) (2πωm t)))
37. cos a cos b = 1 2 (cos (a − b) + cos (a + b)) sin a sin b = 1 2 (cos (a + b) − cos (a − b)) ࡾ֯ؔ਺ͷੵ࿨ެࣜ f (t) = AJ0 (B) cos (2πωc t) +2A ∞ ∑ k=1 J2k (B) cos (2k (2πωm t)) cos (2πωc t) −2A ∞ ∑ k=0 J2k+1 (B) sin ((2k + 1) (2πωm t)) sin (2πωc t)
38. ࡾ֯ؔ਺ͷੵ࿨ެࣜ f (t) = AJ0 (B) cos (2πωc t) +A ∞ ∑ k=1 J2k (B) cos ((2k (2πωm) − 2πωc) t) +A ∞ ∑ k=1 J2k (B) cos ((2k (2πωm) + 2πωc) t) −A ∞ ∑ k=0 J2k+1 (B) cos (((2k + 1) (2πωm) + 2πωc) t) +A ∞ ∑ k=0 J2k+1 (B) cos (((2k + 1) (2πωm) − 2πωc) t) 2 ( ( ) ( ))
39. +A∑ k=1 J2k (B) cos ((2k (2πωm) − 2πωc) t) +A ∞ ∑ k=1 J2k (B) cos ((2k (2πωm) + 2πωc) t) −A ∞ ∑ k=0 J2k+1 (B) cos (((2k + 1) (2πωm) + 2πωc) t) +A ∞ ∑ k=0 J2k+1 (B) cos (((2k + 1) (2πωm) − 2πωc) t) ࣜΛ੔ཧ͢Δͱ f (t) = A ∞ ∑ n=−∞ Jn (B) cos (2πt (nωm + ωc))
40. f (t) = A ∞ ∑ n=−∞ Jn (B) cos (2πt (nωm + ωc)) ͜ͷৼ෯Ͱ ͜ͷप೾਺ͷ೾Λ શͯ଍͠߹ΘͤΔͱ FMԻݯ͕ग़ྗ͢Δ೾ͱ౳Ձ प೾਺มௐͷల։
41. ͱ͜ΖͰ
42. f (t) = A ∞ ∑ n=−∞ Jn (B) cos (2πt (nωm + ωc)) ͬͯԿ? J
43. ୈҰछϕοηϧؔ਺ Jn (x) = 2 π ∫ π 2 0 sin (x sin ω) sin nωdω (n = 2k + 1|k ∈ ℕ) Jn (x) = 2 π ∫ π 2 0 cos (x sin ω) cos nωdω (n = 2k|k ∈ ℕ) ͕ح਺ͷ࣌ n ͕ۮ਺ͷ࣌ n ஫ҙ: ͜͜Ͱࣗવ਺ ͸ ΛؚΉ ℕ 0
44. ϙΠϯτ ࣍਺͕ߴ͍΄Ͳ ্ཱ͕ͪΓ͕؇΍͔ʹͳΔ ϙΠϯτ ͕େ͖͘ͳΔͱ पظؔ਺ͱݟ၏ͤΔ x
45. ͕ෛͷ৔߹ n
46. f (t) = A ∞ ∑ n=−∞ Jn (B) cos (2πt (nωm + ωc)) ͕͜͜ෛͷ࣌ प೾਺ ͷ೾͸Ґ૬͕ ͣΕͯग़ͯ͘Δ 2πt (nωm + ωc) π ͔͠͠ਓؒͷௌ֮͸Իͷप೾਺ຖͷৼ෯͚ͩΛௌ͍͍ͯͯ Ґ૬ͷҧ͍Λ஌֮͢Δ͜ͱ͸Ͱ͖ͳ͍
47. ࣮ࡍʹ஌֮͞ΕΔԻͷৼ෯͸ ϕοηϧؔ਺ͷઈର஋ʹͳΔ ∴
48. f (t) = A ∞ ∑ n=−∞ Jn (B) cos (2πt (nωm + ωc)) ͷ৔߹ ෛͷप೾਺ͷ೾Λ࣋ͭ͜ͱʹͳΔ nωm + ωc < 0 ͳͷͰ ͋Δෛͷप೾਺ ͷ೾ؚ͕·Ε͍ͯΔ࣌ ͦΕ͸प೾਺ ͷ೾ͱͯ͠؍ଌ͞ΕΔ cos (x) = cos (−x) ω |ω| ෛͷप೾਺੒෼
49. 2ωc = ωm 3ωc = ωm 0 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 0 1 2 3 4 5 6 −1 −2 −3 −4 −5 −6 ͭ·ΓFMԻݯͷεϖΫτϧͷҙຯ͸͜͏ 2ωc 2ωc 3ωc 3ωc
50. ྔࢠԽͯ͠ ઢܗิؒͯ͠ ૭ؔ਺Λ͔͚ͯ FFTׂ͖ͯͨ͠ʹ͸ Α͘Ұக͍ͯ͠Δ ͷ৔߹ͷ ల։݁ՌͱFFTͷൺֱ 3ωc = ωm
51. ༩͑ΒΕͨεϖΫτϧʹ͍ۙ݁Ռ͕ಘΒΕΔ
    ͱ ͱ ͱ ΛٻΊ͍ͨ A B ωc ωm f (t) = A ∞ ∑ n=−∞ Jn (B) cos (2πt (nωm + ωc))
52. ͸ ͷ஋͔͠ฦ͞ͳ͍ҝ
    ͢΂ͯͷ ͷ૯࿨Ͱׂ͓͚ͬͯ͹ ͸ܭଌͨ͠ৼ෯ͦͷ΋ͷʹͳΔ cos −1 ≤ x ≤ 1 J A f (t) = A 1 ∑∞ n=−∞ |Jn (B)| ∞ ∑ n=−∞ Jn (B) cos (2πt (nωm + ωc)) A (0.5) = 0.1
53. ָثͷԻͷৼ෯͸جԻ෇ۙͰ࠷΋େ͖͘ ԕ͍ഒԻ΄Ͳখ͘͞ͳΔ܏޲͕͋Δ جԻͷৼ෯Λ ʹͯ͠ ԕ͍ഒԻ΄Ͳ࣍਺ͷߴ͍ Λ࢖͏Α͏ʹ͢Δͷ͕ಘࡦ J0 J
54. ָثͷԻͷৼ෯͸جԻ෇ۙͰ࠷΋େ͖͘ ԕ͍ഒԻ΄Ͳখ͘͞ͳΔ܏޲͕͋Δ جԻͷৼ෯Λ ʹͯ͠ ԕ͍ഒԻ΄Ͳ࣍਺ͷߴ͍ Λ࢖͏Α͏ʹ͢Δͷ͕ಘࡦ J0 J ͭ·Γ ͸جԻͷप೾਺ ͸ ͷ੔਺ഒͷ஋ʹͳΔ ωc ωm ωc
55. Λେ͖͗͘͢͠Δͱप೾਺ͷߴ͍ഒԻ͕ՄௌҬΛ௒͑Δ αϯϓϦϯάप೾਺ʹରͯ͠ߴ͗͢ΔഒԻ ͸جԻͷ੔਺ഒͰͳ͍ͱ͜ΖʹϐʔΫΛ࡞Δ ωm 261.63Hz 20kHz 20ωc = ωm 10ωc = ωm 30ωc = ωm ਓؒͷௌ֮Ͱ
    ฉ͖औΕΔݶք ͜͏ͳΔͱԻ֊Λָ࣋ͭثͱͯ͠੒ཱ͠ͳ͍ҝ ͱ ͷൺ͸ҎԼͰे෼ ωm ωc
56. ͱ ͸ఆ·ͬͨ
    ͸ߴʑ௨ΓͳͷͰ
    ֤ Ͱ༩͑ΒΕͨεϖΫτϧʹ࠷΋ۙ͘ͳΔ ΛٻΊΕ͹ྑ͍ A ωc ωm ωm B χϡʔϥϧωοτϫʔΫͱಉ͡ඇઢܗ࠷దԽ໰୊ͳͷͰ
    ޯ഑๏Ͱ߈ΊΔ
57. جԻͷ੔਺ഒͷ೾ͷৼ෯ͷΈʹண໨͢Δ 3ωc = ωm άϥϯυϐΞϊ J0 (B) J1 (B) J1 (B) J2 (B) J2 (B) J3 (B) J3 (B) J4 (B) J4 (B) J5 (B) J5 (B) J6 (B) D10 = J3 (B)2 − E2 10 L10 = D10 D3 = 0 L3 = E3 ੜ੒෺ ظ଴͢Δग़ྗ Λޡࠩٯ఻೻͢Δ͜ͱͰ ͷमਖ਼ͷྔΛܾఆ͢Δ Dn B
58. جԻͷ੔਺ഒͷ೾ͷৼ෯ͷΈʹண໨͢Δ 3ωc = ωm άϥϯυϐΞϊ J0 (B) J1 (B) J1 (B) J2 (B) J2 (B) J3 (B) J3 (B) J4 (B) J4 (B) J5 (B) J5 (B) J6 (B) D10 = J3 (B)2 − E2 10 L10 = D10 D3 = 0 L3 = E3 ΋ޙͰ࢖͏ͷͰٻΊ͓ͯ͘ 50 ∑ n=1 Ln ʹΑΒͣੜ͡Δޡࠩ B
59. Adam α = 0.001 β1 = 0.9 β2 = 0.999 ϵ = 10−8 mt = β1 mt−1 + (1 − β1) gt vt = β2 vt−1 + (1 − β2) g2 t ̂ mt = mt 1 − βt 1 ̂ vt = vt 1 − βt 2 θt = θt−1 − α ̂ mt ̂ vt + ϵ χϡʔϥϧωοτϫʔΫͷֶशͰ
    Α͘༻͍ΒΕΔ࠷దԽΞϧΰϦζϜ ࣮૷͕؆୯ͳׂʹͦͦ͜͜ੑೳ͕ྑ͍ ʹ͍ͭͯޡࠩͷޯ഑ ʹରͯ͠
    Λ ͚ͩมԽͤ͞Δ θ g θ −α ̂ mt ̂ vt + ϵ
60. Jn (x) = 2 π ∫ π 2 0 sin (x sin ω) sin nωdω (n = 2k + 1|k ∈ ℕ) Jn (x) = 2 π ∫ π 2 0 cos (x sin ω) cos nωdω (n = 2k|k ∈ ℕ) ޡࠩٯ఻೻Λޮ཰Α͘ߦ͏ʹ͸ର৅ͷؔ਺Λඍ෼͢Δඞཁ͕͋Δ ͍ͭ͜͸ඍ෼Ͱ͖Δͷ͔
61. dJn (x) dx = 1 2 (Jn−1 (x) − Jn+1 (x)) Good News! ୈछϕοηϧؔ਺ʹ͸ղੳతʹඍ෼Ͱ͖Δࣄ͕஌ΒΕ͍ͯΔ
62. dJn (x) dx = 1 2 (Jn−1 (x) − Jn+1 (x)) Bad News! ୈछϕοηϧؔ਺ͷඍ෼ʹ͸ୈछϕοηϧؔ਺ؚ͕·Ε͍ͯΔ
63. Jn (x) = 2 π ∫ π 2 0 sin (x sin ω) sin nωdω (n = 2k + 1|k ∈ ℕ) Jn (x) = 2 π ∫ π 2 0 cos (x sin ω) cos nωdω (n = 2k|k ∈ ℕ) ͜ͷ෦෼ʹ໘౗ͳੵ෼ؚ͕·Ε͍ͯΔ ୆ܗެࣜΛ࢖ͬͯ਺஋తʹղ͘͜ͱ͸Ͱ͖Δ͕
    ΛճٻΊΔͷʹେྔͷܭࢉ͕ඞཁʹͳΔҝ
    ద੾ͳ Λ୳͢൓෮ܭࢉͷதͰ΍Γͨ͘ͳ͍ Jn (x) B ͳͥ໰୊͔
64. ϕοηϧؔ਺ͷۙࣅ Tumakov, D.N. Lobachevskii J Math (2019) 40: 1725. https://doi.org/10.1134/S1995080219100287 The Faster Methods for Computing Bessel Functions of the First Kind of an Integer Order with Application to Graphic Processors p0 (x) = 8 x q0 (x) = p (x)2 t0 (x) = x − 0.78539816 J0 (x) = 0.079577472p0 (x) ((0.99999692 − 0.0010731477q0 (x)) cos (t0 (x)) + (0.015624982 + (−0.00014270786 + 0.0000059374342q0) q0 (x)) p0 (x) sin (t0 (x))) p1 (x) = 0.63661977 x q1 (x) = p (x)2 J1 (x) = p (1 − (0.46263771 − 1.1771851q1 (x)) q1 (x)) cos (x − 2.3561945 (0.46263771 − 1.1771851q1 (x)) p1 (x)) Jn (x) = 2 (n − 1) x Jn−1 (x) − Jn−2 (x)
65. ϕοηϧؔ਺ͷۙࣅ Tumakov, D.N. Lobachevskii J Math (2019) 40: 1725. https://doi.org/10.1134/S1995080219100287 The Faster Methods for Computing Bessel Functions of the First Kind of an Integer Order with Application to Graphic Processors Jn (x) = 2 (n − 1) x Jn−1 (x) − Jn−2 (x) ͜ͷۙࣅͷDPPMͳͱ͜Ζ
    ͭԼͷ࣍਺ ͱ ͭԼͷ࣍਺ ͕ط஌ͷ৔߹
    ΋ͷ͘͢͝؆୯ͳܭࢉͰ ͕ٻ·Δ Jn−1 (x) Jn−2 (x) Jn (x) Լͷ࣍਺͔Βܭࢉ͢Δͱര଎Ͱϕοηϧؔ਺ͷ஋͕ٻ·Δ ∴
66. J0 (x)
67. J1 (x)
68. J2 (x)
69. J6 (x)
70. J10 (x) ۙ๣ͰਧͬඈͿ x = 0
71. p0 (x) = 8 x q0 (x) = p (x)2 t0 (x) = x − 0.78539816 J0 (x) = 0.079577472p0 (x) ((0.99999692 − 0.0010731477q0 (x)) cos (t0 (x)) + (0.015624982 + (−0.00014270786 + 0.0000059374342q0) q0 (x)) p0 (x) sin (t0 (x))) p1 (x) = 0.63661977 x q1 (x) = p (x)2 J1 (x) = p (1 − (0.46263771 − 1.1771851q1 (x)) q1 (x)) cos (x − 2.3561945 (0.46263771 − 1.1771851q1 (x)) p1 (x)) Jn (x) = 2 (n − 1) x Jn−1 (x) − Jn−2 (x) ͜͜ ͜͜ ͜͜ ֻ͕͔͍ͬͯΔҝ ۙ๣ͰਧͬඈͿఆΊʹ͋Δ 1 x x = 0
72. Jn (x) = 2(n − 1) x Jn−1 (x) − Jn−2 (x) (x ≥ 2 (n − 1)) Jn (x) = Jn−1 (x) (x < 2 (n − 1)) ݩͷ࿦จͰఏҊ͞Ε͍ͯΔճආํ๏ ͩͬͨΒ1ͭԼͷ࣍਺ͷϕοηϧؔ਺Λͦͷ··࢖͏ x < 2 (n − 1)
73. J0 (x)
74. J1 (x)
75. J2 (x)
76. J6 (x)
77. J10 (x) ਧͬඈͼʹ͘͘͸ͳΔ
    Ͱ΋ਧͬඈͿ
    Ͱͷ஋͕߹Θͳ͍ x < 2 (n − 1)
78. ద੾ͳ ͷ୳ࡧ͸
    ͜ͷลΓ͔ΒղΛ୳͢ҝ
    ͜ͷ··Ͱ͸͜ͷۙࣅ͸࢖͑ͳ͍ B
79. ϕοηϧؔ਺ͷۙࣅ ͔Β ·Ͱͷ஋Λଟ߲ࣜͰิؒ͢Δ x = 0 x = n Jn (x) = 2(n − 1) x Jn−1 (x) − Jn−2 (x) (x ≥ n) Jn (x) = an x4 + bn x3 (x < n) an = − Jn (n) n5 + 1 n4 dJn dx (n) bn = Jn (n) 2n3 − 1 n2 dJn dx (n) ͨͩ͠
80. ଟ߲ࣜͷ܎਺Λܾఆ͢ΔͨΊʹ
    ͱ ͕ཁΔ Jn (n) dJn dx (n) ͜ͷͭͷ஋Λ ͋ͨΓ·Ͱ
    ࣄલʹٻΊ͓ͯ͘ n = 60 an = − Jn (n) n5 + 1 n4 dJn dx (n) bn = Jn (n) 2n3 − 1 n2 dJn dx (n) n 0 1.000000 0.000000 1 0.440051 0.383960 2 0.351942 0.175971 3 0.308550 0.073205 4 0.279146 0.032248 5 0.255199 0.011946 6 0.245656 0.002787 7 0.233618 -0.004337 8 0.221478 -0.008807 9 0.214564 -0.010998 10 0.207680 -0.012935 Jn (n) dJn dx (n)
81. J0 (x) = 1 − x2 4 + x4 64 − x6 2304 (x < 1) J1 (x) = x 2 − x3 16 + x5 384 − x7 18432 + x9 1474560 (x < 4) ͱ Ͱ ͕খ͍͞৔߹ ୯७ͳςʔϥʔల։ʹ੾Γସ͑Δ J0 J1 x ͜ΕΒΛ࢖ͬͯϕοηϧؔ਺ͷۙࣅ͕ ਧͬඈͿ෦෼ʹ֖Λ͢Δ
82. J0 (x)
83. J1 (x)
84. J2 (x)
85. J6 (x)
86. ࣮༻ʹ଱͑Δۙࣅ J10 (x)
87. const auto t0 = std::chrono::steady_clock::now(); for( unsigned int b_ = 0.f; b_ != 400; ++b_ ) { float b = b * 0.1f; for( unsigned int n = 0; n != 60; ++n ) { ifm::bessel_kind1( b, n ); } } const auto t1 = std::chrono::steady_clock::now(); for( unsigned int b_ = 0.f; b_ != 400; ++b_ ) { float b = b * 0.1f; float l1 = 0; float l2 = 0; for( unsigned int n = 0; n != 60; ++n ) { float y = ifm::bessel_kind1_approx_2019( b, n, l1, l2, pre ); l2 = l1; l1 = y; } } const auto t2 = std::chrono::steady_clock::now(); float e0 = float( std::chrono::duration_cast< std::chrono::microseconds >( t1 - t0 ).count() ); float e1 = float( std::chrono::duration_cast< std::chrono::microseconds >( t2 - t1 ).count() ); $ ./bessel_benchmark -b bessel2.mp ୆ܗެࣜ: 0.0484659Mbps(24000 samples in 495194microseconds) ۙࣅࣜ: 269.663Mbps(24000 samples in 89microseconds) 1570෼ׂͷ ୆ܗެࣜʹΑΔܭࢉͱൺֱͯ͠ 5500ഒߴ଎ Mega Bessel Per Second Intel(R) Core(TM) i5-6600 3.3GHz Linux 5.3.7 + gcc 8.3.0 -O2 -march=nativeͰ1ίΞͷΈ࢖༻
88. ϐΞϊͷ৔߹
89. ϐΞϊͷ৔߹ น
90. ϚϦϯόͷ৔߹
91. ϚϦϯόͷ৔߹ น
92. ϚϦϯόͷ৔߹
93. ࣌ؒมԽ Τϯϕϩʔϓ͸؇΍͔ʹมԽ͢Δҝ ࣌ࠁ ʹ͍ͭͯٻΊͨ ͸ ࣌ࠁ ͷ ΛٻΊΔࡍͷॳظ஋ͱͯ͠࢖͑Δ t B t + 1 B t t − 1 t − 2 t − 3 t + 1 t + 2 t + 3 Bt Bt Bt+1 Bt+2 Bt−1 Bt−2
94. ࣌ؒมԽ Τϯϕϩʔϓ͸؇΍͔ʹมԽ͢Δҝ ࣌ࠁ ʹ͍ͭͯٻΊͨ ͸ ࣌ࠁ ͷ ΛٻΊΔࡍͷॳظ஋ͱͯ͠࢖͑Δ t B t + 1 B t t − 1 t − 2 t − 3 t + 1 t + 2 t + 3 Bt Bt Bt+1 Bt+2 Bt−1 Bt−2 ҰൠతͳFMԻݯ͸ Λ࣌ؒͰมԽͤ͞ΒΕͳ͍ͷͰ ࣌ࠁ Ͱ࠷΋௿͍ Λग़ͨ͠ Λͦͷ··Ҿ͖ܧ͙ ωm t L ωm ωm ωm ωm ωm ωm ωm
96. ΞίʔεςΟοΫͳָثͷৼ෯ͷมԽ͸
    छྨʹ෼ྨͰ͖Δ ܧଓతʹྗ͕Ճ͑ΒΕΔ
    ඇݮਰৼಈ ࠷ॳ͚ͩྗ͕Ճ͑ΒΕΔ
    ݮਰৼಈ
97. Τϯϕϩʔϓਪఆ f (t) = (1 − β) e−αt + β ݮਰৼಈ߲ ඇݮਰৼಈ߲ ೖྗ೾ܗ͔ΒٻΊͨΤϯϕϩʔϓ ͱ ͱ
    ͜ͷؔ਺ͷग़ྗͷ͕ࠩ࠷খʹͳΔΑ͏ʹ
    ͱ ΛٻΊΔ A (t) B (t) α β
98. f (t) = (1 − β) e−αt + β ferror (t) = (fexpected (t) − f (t)) 2 dferror (t) dα = 2(1 − β)te−αt ((1 − β)(−e−αt) − β + fexpected (t)) dferror (t) dβ = 2 (e−αt − 1) ((1 − β)(−e−αt) − β + fexpected (t)) ͜Ε΋൓෮ͯ͠ ͱ Λमਖ਼͍͚ͯ͠͹ղ͚Δ α β
99. ղ͚ͨ FFT͔ΒಘͨΤϯϕϩʔϓ͸ ࣌ʑ૯࿨͕1Λ௒͑ΔͷͰ࠷େ஋͕1ʹͳΔΑ͏ʹ εέʔϧ͓ͯ͘͠
100. Τϯϕϩʔϓਪఆ σΟέΠதؒϨϕϧ αεςΠϯϨϕϧ ΞλοΫதؒϨϕϧ ΞλοΫ1 ΞλοΫ2 σΟέΠ1 σΟέΠ2 αεςΠϯ ϦϦʔε Α͋͘Δ'.ԻݯͷΤϯϕϩʔϓ͸
     Λஈ֊ͷઢܗͳؔ਺Ͱۙࣅ͢Δ e−αt
101. FYQͱEFDBZʹғ·Εͨ໘ੵͱ
     FYQͱEFBZʹғ·Εͨ໘ੵͱ
     FYQͱTVTUBJOʹғ·Εͨ໘ੵͷ
     ࿨͕࠷খͱͳΔ఺ ͱ ΛٻΊΔ n m n m
102. Τϯϕϩʔϓਪఆ flinear (t, n, m) = (f (m) − f (n)) m − n t + nf (m) − mf (n) n − m fconst (t, n, m) = nf (m) − mf (n) n − m ferror (n, m, l) = ∫ n 0 flinear (t,0,n) − f (t) dt + ∫ m n flinear (t, n, m) − f (t) dt + ∫ l m fconst (t, m, l) − f (t) dt ͱ ʹ͍ͭͯͷޯ഑ΛٻΊ
     ஋Λमਖ਼͍͚ͯ͠͹ྑ͍ n m ݮਰৼಈ͢Δָثͷ৔߹
     Λ ʹ Λαϯϓϧ௕ݻఆ͢Δ fconst 0 l
103. ୈ12ճΧʔωϧ/VM୳ݕୂ Ҩ఻తFMԻݯ(2015೥)ΑΓ ຊ෺ͷϐΞϊͷԻ '.ԻݯͷϐΞϊͷԻ ܭࢉ࣌ؒ 2࣌ؒ43෼42ඵ/keyframe ܭࢉ؀ڥ
 Intel Core i5-6600(Skylake) 3.3GHz 4ίΞ nvidia GeForceGTX1070 1.68GHz 1920CUDAίΞ FMԻݯͷ࢓༷ 4ΦϖϨʔλϑΟʔυόοΫ෇͖ ιϑτ΢ΣΞFMԻݯ
104. ࣮ࡍͷϐΞϊͷԻ χϡʔϥϧFMԻݯ Ҩ఻తFMԻݯ ϐΞϊͬΆ͘ฉ͑͜Δ෦෼΋͋Δ͕ Ҩ఻తFMԻݯͱൺ΂Δͱ͠ΐͬͺ͍ ϐΞϊ ݁Ռ ୈ8ճΧʔωϧ/VM୳ݕୂ@ؔ੢ χϡʔϥϧFMԻݯ(2017೥)ΑΓ ܭࢉ࣌ؒ 3ඵ/keyframe ܭࢉ؀ڥ
 Intel Core i5-6600(Skylake) 3.3GHz 4ίΞ nvidia GeForceGTX1070 1.68GHz 1920CUDAίΞ FMԻݯͷ࢓༷ 8ΦϖϨʔλϑΟʔυόοΫ෇͖ ιϑτ΢ΣΞFMԻݯ
105. ຊ෺ͷϐΞϊ 2ΦϖϨʔλFMԻݯϐΞϊ ܭࢉ࣌ؒ 4෼1ඵ/keyframe ܭࢉ؀ڥ
 Intel Core i5-6600(Skylake) 3.3GHz 4ίΞ CPUͷΈ࢖༻ FMԻݯͷ࢓༷ 2ΦϖϨʔλϑΟʔυόοΫͳ͠ ιϑτ΢ΣΞFMԻݯ
106. ຊ෺ͷϚϦϯό 2ΦϖϨʔλFMԻݯϚϦϯό
107. ຊ෺ͷόΠΦϦϯ 2ΦϖϨʔλFMԻݯόΠΦϦϯ
108. 3ΦϖϨʔλͷ৔߹ sin (x) cos (x) 2πω1 t + × E1 × E0 ग़ྗ 2πω0 × × 2πω2 E2 × sin (x) × + f (t) = E0 cos (2πω0 t + E1 sin (2πω1 t) + E2 sin (2πω2 t)) f (t) = E0 ∞ ∑ n1 =−∞ ∞ ∑ n2 =−∞ Jn1 (E1) Jn2 (E2) cos (2πt (n1 ω1 + n2 ω2 + ω0)) ҰԠղ͚Δ ΋ͬͱෳࡶͳFMԻݯʹ͍ͭͯղ͚ͳ͍ͷ?
109. f (t) = E0 ∞ ∑ n1 =−∞ ∞ ∑ n2 =−∞ Jn1 (E1) Jn2 (E2) cos (2πt (n1 ω1 + n2 ω2 + ω0)) ͜Εͱ ͜Εͱ ͜Εͱ ͜ΕΛ ݟ͚ͭΔඞཁ͕͋Δ
110. ͜ͷํ๏ͩͱඞཁͳࢼߦճ਺͕(op਺-1)৐Ͱ૿͑Δ
111. ͜ͷ͋ͨΓ͕ͬͦ͝Γফ͍͑ͯΔͷ͸ԿͰ
112. ഒԻ͕ظ଴͢Δप೾਺͔Β͔ᷮʹͣΕͯग़͍ͯΔ ޡࠩ͸ظ଴͢Δप೾਺ͷৼ෯͚ͩΛ࢖ͬͯٻΊ͍ͯΔͨΊ ͣΕͯग़͍ͯΔഒԻ͸ग़͍ͯͳ͍ͷͱಉ͡ѻ͍ʹͳΔ ΞίʔεςΟοΫͳָثͳΒ ͲΜͳʹ͖ͪΜͱௐ཯ͯ͋ͬͯ͠΋ ߴप೾ͷഒԻͰ͸͍͘Β͔ͣΕΔ
113. ·ͱΊ FMԻݯ͸ղ͚Δ ඇઢܗ࠷దԽ໰୊ͳͷͰ χϡʔϥϧωοτϫʔΫͷֶशͱಉ͡ख๏͕ޮ͘ ͔͠͠3ΦϖϨʔλҎ্ʹରԠ͢Δʹ͸ ΋͏গ͠޻෉͕ඞཁ