逆FM音源 - Speaker Deck

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ٯFMԻݯ NAOMASA MATSUBAYASHI INVERTED FREQUENCY MODULATION SYNTHESIZER https://github.com/Fadis/ifm αϯϓϧίʔυ

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Իָͷԋ૗͸೾ͷੜ੒ εϐʔΧʔͷ ΞΠίϯ ిѹͷ೾ ۭؾͷ೾ Ի֊ Իྔ υ ίϯϐϡʔλͰԻָΛ૗ͰΔʹ͸ Ի֊ʹରԠ͢Δप೾਺ͷ೾Λੜ੒͠ͳ͚Ε͹ͳΒͳ͍ ?

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Իݯνοϓͷొ৔ εϐʔΧʔͷ ΞΠίϯ ిѹͷ೾ ۭؾͷ೾ Ի֊ Իྔ υ ͜ͷੜ੒Λߦ͏ϋʔυ΢ΣΞ͕ొ৔ͨ͠

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υ Ϩ ϛ ϑΝ ι ԣ࣠: प೾਺ ॎ࣠: ࣌ؒ(ͦΕͧΕͷԻ֊ʹ͍ͭͯ໐Γ࢝Ί͔Β2ඵ෼)

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65.406Hz 130.81Hz 196.22Hz 261.63Hz 327.03Hz 392.44Hz 457.84Hz 523.25Hz 588.66Hz υ Ϩ ϛ ϑΝ ι

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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೾਺্͕͕͍ͬͯ͘

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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

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άϥϯυϐΞϊ ΞϧταοΫε ϏϒϥϑΥϯ όΠΦϦϯ ϚϦϯό

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261.63Hz 523.25Hz 784.88Hz 1046.5Hz 1308.1Hz 1569.8Hz 1831.4Hz 2093.0Hz 2354.6Hz άϥϯυϐΞϊ ΞϧταοΫε ϏϒϥϑΥϯ όΠΦϦϯ ϚϦϯό

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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

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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

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άϥϯυϐΞϊ ΞϧταοΫε ϏϒϥϑΥϯ όΠΦϦϯ ϚϦϯό

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άϥϯυϐΞϊ ԣ࣠: प೾਺ ॎ࣠: ࣌ؒ t ͋Δ࣌ࠁʹ͓͚Δશͯͷप೾਺ͷ੒෼ͷ૯࿨ΛऔΔͱ ͓͓Αͦͦͷ࣌ࠁͷԻͷৼ෯( Իྔ)͕ಘΒΕΔ ≃

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Τϯϕϩʔϓ ָثͷԻͷৼ෯͸࣌ؒมԽ͢Δ ಛʹϐΞϊ΍ϚϦϯόͷΑ͏ͳ ࠷ॳʹՃ͑ͨྗ͚ͩͰ࠷ޙ·Ͱ໐Δָث͸ ៉ྷͳݮਰৼಈͷۂઢΛඳ͘

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άϥϯυϐΞϊ ࠷ॳʹՃ͑ͨྗ͚ͩͰ࠷ޙ·Ͱ໐ΔָثͰ͸ प೾਺ͷߴ͍ഒԻ΄Ͳૣ͘ফ͑Δ܏޲ʹ͋Δ όΠΦϦϯ ܧଓతʹྗ͕Ճ͑ΒΕΔָثͰ͸ ໐Β͢ͷΛ΍ΊΔ·Ͱಉׂ͡߹ͷഒԻ͕ग़Δ܏޲ʹ͋Δ

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ཁٻ͞ΕͨԻ֊ʹରԠ͢ΔجԻͷαΠϯ೾ͱ ָثʹԠͨ͡ഒԻͷαΠϯ೾Λੜ੒͢Δ ͦΕΒͷৼ෯͕࣮ࡍͷָثͱಉ͡Α͏ʹ࣌ؒมԽ͍ͯ͠Ε͹ ͦͷָثͰཁٻ͞ΕͨԻ֊ͷԻΛ໐Βͨ͠Α͏ʹฉ͑͜Δ ϐΞϊͷԻͰ υ ഒ཰ ৼ෯ ∑

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ϐΞϊͷԻͰ υ ໰୊఺ αΠϯ೾ͷܭࢉ͸҆͘ͳ͍ ഒ཰ ৼ෯ ∑

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प೾਺มௐ f (t) = sin (2πωc t + B sin (2πωm t))

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ωc = ωm 2ωc = ωm 3ωc = ωm 4ωc = ωm f (t) = sin (2πωc t + B sin (2πωm t))

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ഒԻ੒෼ͷݮਰ f (t) = Ec (t) sin (2πωc t + Em (t) B sin (2πωm t)) ৼ෯ͷ࣌ؒมԽΛ ίϯτϩʔϧ͢Δ ഒԻ੒෼ͷྔͷ࣌ؒมԽΛ ίϯτϩʔϧ͢Δ άϥϯυϐΞϊ

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'.Իݯ͸ ΞίʔεςΟοΫͳָثͷ ಛ௃Λগͳ͍ύϥϝʔλͰදݱग़དྷΔ ѱ͘ͳ͍ϞσϧͰ͋Δ

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OΦϖϨʔλͷ'.Իݯͷܭࢉࣜ OΦϖϨʔλ਺ &JΤϯϕϩʔϓ TJجԻͷप೾਺ʹର͢Δഒ཰ ЋJKΦϖϨʔλK͕ΦϖϨʔλJΛมௐ͢Δڧ͞ ЌJग़ྗʹؚ·ΕΔΦϖϨʔλJͷ৴߸ͷׂ߹ ୈ12ճΧʔωϧ/VM୳ݕୂ Ҩ఻తFMԻݯ(2015೥)ΑΓ '.Իݯͷ໰୊఺

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Ͳ͜Λ͍ͬͨ͡Β ཉָ͍͠ثͷԻ৭ʹۙ෇͘ͷ͔ ͬ͞ͺΓΘ͔ΒΜ OΦϖϨʔλͷ'.Իݯͷܭࢉࣜ ୈ12ճΧʔωϧ/VM୳ݕୂ Ҩ఻తFMԻݯ(2015೥)ΑΓ '.Իݯͷ໰୊఺ ૂͬͨԻ৭Λ ࡞Δͷ͕ ͱͯ΋೉͍͠

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ୈੈ୅ ୈੈ୅ ୈੈ୅ Ҩ఻తΞϧΰϦζϜ ͜ͷૢ࡞Λ܁Γฦ͢͜ͱʹΑͬͯ ༩͑ΒΕͨ৚݅ʹΑ͘ద߹͢ΔݸମΛ ݟ͚ͭग़͢͜ͱ͕Ͱ͖Δ ୈ12ճΧʔωϧ/VM୳ݕୂ Ҩ఻తFMԻݯ(2015೥)ΑΓ ͦ͜Ͱ Ҩ఻తΞϧΰϦζϜͰ '.Իݯύϥϝʔλͷ ୳ࡧΛࢼΈͨ

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Ҩ఻తFMԻݯ ར఺ &OEUP&OEͰԻ͔Β'.Իݯύϥϝʔλ͕ಘΒΕΔ ࣮༻ʹ଱͑Δ඼࣭ͷग़ྗ͕ಘΒΕΔ ܽ఺ ηοτͷύϥϝʔλΛݟ͚ͭΔͷʹ(16Ͱ͔͔࣌ؒۙ͘Δ ෳ਺ͷ୳ࡧ݁Ռ͕ઢܗิؒͰ͖Δอূ͕ͳ͍

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ୈ8ճΧʔωϧ/VM୳ݕୂ@ؔ੢ χϡʔϥϧFMԻݯ(2017೥)ΑΓ χϡʔϥϧωοτϫʔΫͰ '.Իݯύϥϝʔλͷ ୳ࡧΛࢼΈͨ Caffe Φʔϓϯιʔεͳ σΟʔϓϥʔχϯάϑϨʔϜϫʔΫͷ1ͭ http://caffe.berkeleyvision.org/

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χϡʔϥϧFMԻݯ ར఺ &OEUP&OEͰԻ͔Β'.Իݯύϥϝʔλ͕ಘΒΕΔ ֶशࡁΈͷωοτϫʔΫ͕͋Ε͹਺ඵͰύϥϝʔλΛಘΒΕΔ ܽ఺ ग़ྗͷ඼࣭͕࣮༻ʹ଱͑ͳ͍ ෳ਺ͷ୳ࡧ݁Ռ͕ઢܗิؒͰ͖Δอূ͕ͳ͍

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ωc = ωm 2ωc = ωm 3ωc = ωm 4ωc = ωm 5ωc = ωm f (t) = A cos (2πωc t + B sin (2πωm t))

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ഒԻ੒෼ͷग़ํʹ͸نଇੑ͕͋ΔΑ͏ʹݟ͑Δ ωc = ωm 2ωc = ωm 3ωc = ωm 4ωc = ωm 5ωc = ωm f (t) = A cos (2πωc t + B sin (2πωm t))

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Ͳ͜Λ͍ͬͨ͡Β ཉָ͍͠ثͷԻ৭ʹۙ෇͘ͷ͔ ͬ͞ͺΓΘ͔ΒΜ OΦϖϨʔλͷ'.Իݯͷܭࢉࣜ ຊ౰ʹ? ୈ12ճΧʔωϧ/VM୳ݕୂ Ҩ఻తFMԻݯ(2015೥)ΑΓ

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sin (x) 2πωm t + sin (x) × B × A ग़ྗ 2πωc × × 2ΦϖϨʔλ௚ྻͷ৔߹ Τϯϕϩʔϓ͸ ඍখͳ࣌ؒͷൣғͰ͸ఆ਺ͱݟ၏ͤΔͨΊ ͱ ʹؚ·ΕΔ A B ͋Δ࣌ࠁ෇ۙͰͷFMԻݯͷग़ྗΛߟ͑Δ

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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

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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

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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

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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)))

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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)))

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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)

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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 ( ( ) ( ))

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+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))

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f (t) = A ∞ ∑ n=−∞ Jn (B) cos (2πt (nωm + ωc)) ͜ͷৼ෯Ͱ ͜ͷप೾਺ͷ೾Λ શͯ଍͠߹ΘͤΔͱ FMԻݯ͕ग़ྗ͢Δ೾ͱ౳Ձ प೾਺มௐͷల։

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ͱ͜ΖͰ

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f (t) = A ∞ ∑ n=−∞ Jn (B) cos (2πt (nωm + ωc)) ͬͯԿ? J

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ୈ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

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ϙΠϯτ ࣍਺͕ߴ͍΄Ͳ ্ཱ͕ͪΓ͕؇΍͔ʹͳΔ ϙΠϯτ ͕େ͖͘ͳΔͱ पظؔ਺ͱݟ၏ͤΔ x

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͕ෛͷ৔߹ n

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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

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࣮ࡍʹ஌֮͞ΕΔԻͷৼ෯͸ ϕοηϧؔ਺ͷઈର஋ʹͳΔ ∴

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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೾਺੒෼

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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

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ྔࢠԽͯ͠ ઢܗิؒͯ͠ ૭ؔ਺Λ͔͚ͯ FFTׂ͖ͯͨ͠ʹ͸ Α͘Ұக͍ͯ͠Δ ͷ৔߹ͷ ల։݁ՌͱFFTͷൺֱ 3ωc = ωm

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༩͑ΒΕͨεϖΫτϧʹ͍ۙ݁Ռ͕ಘΒΕΔ ͱ ͱ ͱ ΛٻΊ͍ͨ A B ωc ωm f (t) = A ∞ ∑ n=−∞ Jn (B) cos (2πt (nωm + ωc))

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͸ 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

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ָثͷԻͷৼ෯͸جԻ෇ۙͰ࠷΋େ͖͘ ԕ͍ഒԻ΄Ͳখ͘͞ͳΔ܏޲͕͋Δ جԻͷৼ෯Λ ʹͯ͠ ԕ͍ഒԻ΄Ͳ࣍਺ͷߴ͍ Λ࢖͏Α͏ʹ͢Δͷ͕ಘࡦ J0 J

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ָثͷԻͷৼ෯͸جԻ෇ۙͰ࠷΋େ͖͘ ԕ͍ഒԻ΄Ͳখ͘͞ͳΔ܏޲͕͋Δ جԻͷৼ෯Λ ʹͯ͠ ԕ͍ഒԻ΄Ͳ࣍਺ͷߴ͍ Λ࢖͏Α͏ʹ͢Δͷ͕ಘࡦ J0 J ͭ·Γ ͸جԻͷप೾਺ ͸ ͷ੔਺ഒͷ஋ʹͳΔ ωc ωm ωc

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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

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ͱ ͸ఆ·ͬͨ ͸ߴʑ௨ΓͳͷͰ ֤ Ͱ༩͑ΒΕͨεϖΫτϧʹ࠷΋ۙ͘ͳΔ ΛٻΊΕ͹ྑ͍ A ωc ωm ωm B χϡʔϥϧωοτϫʔΫͱಉ͡ඇઢܗ࠷దԽ໰୊ͳͷͰ ޯ഑๏Ͱ߈ΊΔ

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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

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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

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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

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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

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dJn (x) dx = 1 2 (Jn−1 (x) − Jn+1 (x)) Good News! ୈछϕοηϧؔ਺ʹ͸ղੳతʹඍ෼Ͱ͖Δࣄ͕஌ΒΕ͍ͯΔ

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dJn (x) dx = 1 2 (Jn−1 (x) − Jn+1 (x)) Bad News! ୈछϕοηϧؔ਺ͷඍ෼ʹ͸ୈछϕοηϧؔ਺ؚ͕·Ε͍ͯΔ

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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໰୊͔

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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)

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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 ∴

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J0 (x)

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J1 (x)

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J2 (x)

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J6 (x)

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J10 (x) ۙ๣ͰਧͬඈͿ x = 0

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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

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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)

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J0 (x)

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J1 (x)

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J2 (x)

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J6 (x)

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J10 (x) ਧͬඈͼʹ͘͘͸ͳΔ Ͱ΋ਧͬඈͿ Ͱͷ஋͕߹Θͳ͍ x < 2 (n − 1)

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ద੾ͳ ͷ୳ࡧ͸ ͜ͷลΓ͔ΒղΛ୳͢ҝ ͜ͷ··Ͱ͸͜ͷۙࣅ͸࢖͑ͳ͍ B

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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) ͨͩ͠

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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)

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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

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J0 (x)

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J1 (x)

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J2 (x)

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J6 (x)

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࣮༻ʹ଱͑Δۙࣅ J10 (x)

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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࢖༻

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ϐΞϊͷ৔߹

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ϐΞϊͷ৔߹ น

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ϚϦϯόͷ৔߹

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ϚϦϯόͷ৔߹ น

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ϚϦϯόͷ৔߹

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࣌ؒ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

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࣌ؒ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

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ΞίʔεςΟοΫͳָثͷৼ෯ͷมԽ͸ छྨʹ෼ྨͰ͖Δ ܧଓతʹྗ͕Ճ͑ΒΕΔ ඇݮਰৼಈ ࠷ॳ͚ͩྗ͕Ճ͑ΒΕΔ ݮਰৼಈ

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Τϯϕϩʔϓਪఆ f (t) = (1 − β) e−αt + β ݮਰৼಈ߲ ඇݮਰৼಈ߲ ೖྗ೾ܗ͔ΒٻΊͨΤϯϕϩʔϓ ͱ ͱ ͜ͷؔ਺ͷग़ྗͷ͕ࠩ࠷খʹͳΔΑ͏ʹ ͱ ΛٻΊΔ A (t) B (t) α β

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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

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ղ͚ͨ FFT͔ΒಘͨΤϯϕϩʔϓ͸ ࣌ʑ૯࿨͕1Λ௒͑ΔͷͰ࠷େ஋͕1ʹͳΔΑ͏ʹ εέʔϧ͓ͯ͘͠

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Τϯϕϩʔϓਪఆ σΟέΠதؒϨϕϧ αεςΠϯϨϕϧ ΞλοΫதؒϨϕϧ ΞλοΫ1 ΞλοΫ2 σΟέΠ1 σΟέΠ2 αεςΠϯ ϦϦʔε Α͋͘Δ'.ԻݯͷΤϯϕϩʔϓ͸ Λஈ֊ͷઢܗͳؔ਺Ͱۙࣅ͢Δ e−αt

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FYQͱEFDBZʹғ·Εͨ໘ੵͱ FYQͱEFBZʹғ·Εͨ໘ੵͱ FYQͱTVTUBJOʹғ·Εͨ໘ੵͷ ࿨͕࠷খͱͳΔ఺ ͱ ΛٻΊΔ n m n m

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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

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ୈ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

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࣮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

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ຊ෺ͷϐΞϊ 2ΦϖϨʔλFMԻݯϐΞϊ ܭࢉ࣌ؒ 4෼1ඵ/keyframe ܭࢉ؀ڥ  Intel Core i5-6600(Skylake) 3.3GHz 4ίΞ CPUͷΈ࢖༻ FMԻݯͷ࢓༷ 2ΦϖϨʔλϑΟʔυόοΫͳ͠ ιϑτ΢ΣΞFMԻݯ

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ຊ෺ͷϚϦϯό 2ΦϖϨʔλFMԻݯϚϦϯό

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ຊ෺ͷόΠΦϦϯ 2ΦϖϨʔλFMԻݯόΠΦϦϯ

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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?

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f (t) = E0 ∞ ∑ n1 =−∞ ∞ ∑ n2 =−∞ Jn1 (E1) Jn2 (E2) cos (2πt (n1 ω1 + n2 ω2 + ω0)) ͜Εͱ ͜Εͱ ͜Εͱ ͜ΕΛ ݟ͚ͭΔඞཁ͕͋Δ

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͜ͷํ๏ͩͱඞཁͳࢼߦճ਺͕(op਺-1)৐Ͱ૿͑Δ

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͜ͷ͋ͨΓ͕ͬͦ͝Γফ͍͑ͯΔͷ͸ԿͰ

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ഒԻ͕ظ଴͢Δप೾਺͔Β͔ᷮʹͣΕͯग़͍ͯΔ ޡࠩ͸ظ଴͢Δप೾਺ͷৼ෯͚ͩΛ࢖ͬͯٻΊ͍ͯΔͨΊ ͣΕͯग़͍ͯΔഒԻ͸ग़͍ͯͳ͍ͷͱಉ͡ѻ͍ʹͳΔ ΞίʔεςΟοΫͳָثͳΒ ͲΜͳʹ͖ͪΜͱௐ཯ͯ͋ͬͯ͠΋ ߴप೾ͷഒԻͰ͸͍͘Β͔ͣΕΔ

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·ͱΊ FMԻݯ͸ղ͚Δ ඇઢܗ࠷దԽ໰୊ͳͷͰ χϡʔϥϧωοτϫʔΫͷֶशͱಉ͡ख๏͕ޮ͘ ͔͠͠3ΦϖϨʔλҎ্ʹରԠ͢Δʹ͸ ΋͏গ͠޻෉͕ඞཁ