学生実験におけるフーリエ解析

ֶੜ࣮ݧʹ͓͚ΔϑʔϦΤղੳ

େ࡚ਖ਼༤

ۄ઒େֶ޻ֶ෦ιϑτ΢ΣΞαΠΤϯεֶՊ, 194–8610 ౦ژ౎ொాࢢۄ઒ֶԂ 6–1–1 College of Engineering, Tamagawa University,

6–1–1 Tamagawa-gakuen, Machida-shi, Tokyo 194–8610

Abstract

Here we give some troubles in teaching and their solutions occured during the Software Science Experiment course, which is opened for the 4th semester in the Department of Software Science. One of the subjects of this experiment course is Fourier analysis using MyPC. Some students are not familiar with calculating the integration of sinusoidal function, and also some need support for drawing graphs with MS Excel. Typical mistakes and their settlements are given.

Keywords: Students expreriment, Fourier analysis, troubles and their solutions.

1 ͸͡Ίʹ ຊֶ޻ֶ෦ιϑτ΢ΣΞαΠΤϯεֶՊͰ͸ ʮιϑτ΢ΣΞαΠΤϯε࣮ݧIʢҎԼɼ࣮ݧIʣʯ ͱʮιϑτ΢ΣΞαΠΤϯε࣮ݧIIʢҎԼɼ࣮ݧ IIʣʯΛඞमՊ໨ͱ࣮ͯ͠ࢪ͖ͯͨ͠ɽͦΕͧΕ̐ ηϝελʔɼ̑ηϝελʔͰ։ߨ͞Εɼ࣮ݧʹΑ Δݱ৅೺ѲͱɼͦΕΛϨϙʔτʹදٕ͢ज़ͷमಘ Λ໨ࢦ͍ͯͨ͠ɽ֤࣮ݧ͸̏ςʔϚͰߏ੒͞Εɼ ࣮ݧ̞Ͱ͸ʮύιίϯʹΑΔ৴߸ղੳʯͱ୊ͯ͠ ϑʔϦΤղੳͷॳาΛमಘ͢Δ΋ͷΛஶऀ͸୲౰ ͨ͠ɽ ຊߘͰ͸ɼͦͷ࣮ݧʹ͓͍ͯमಘΛ໨ࢦ߲ͨ͠ ໨ɼͦͷख๏ɼͦ͜Ͱੜͨ͡໰୊ͱͦͷղܾํ๏ ʹ͍ͭͯड़΂Δɽͦ͜ʹݟΒΕΔز͔ͭͷ޻෉͕ ࣮ݧʹݶΒͣɼϑʔϦΤղੳʹؔ࿈͢Δߨٛʹ͓ ͍ͯ΋໾ཱͯ͹޾͍Ͱ͋Δɽ 2 ਺ֶతجૅ ϑʔϦΤղੳͷୈҰา͸पظ৴߸ͷϑʔϦΤڃ ਺ల։Ͱ͋ΔɽͦͷͨΊʹ͸ʮपظ৴߸ͷఆٛʯ ͱͦͷ୅දྫͱͯ͠ͷʮਖ਼ݭ೾৴߸ͷੑ࣭ʯ͕ඞ ཁෆՄܽͰ͋ΔɽͦͷͨΊɼҎԼͷΑ͏ʹهͨ͠ɽ पظ৴߸ ৴߸f(t)͕पظT [sec.]Λ࣋ͭ৔߹ɼ೚ҙͷ࣌ ࠁtʹରͯ͠ҎԼͷؔ܎͕੒Γཱͭɽ f(t) = f(t + T ). (1) ౰વɼmΛ੔਺ͱͯ࣍͠ͷؔ܎΋੒Γཱͭɽ f(t) = f(t + mT ). (2) ͜͜Ͱ࠷΋୹͍पظT ΛجຊपظͱݺͿɽ ਖ਼ݭ೾৴߸ Ұൠʹਖ਼ݭ೾৴߸͸ৼ෯Aɼप೾਺fɼॳظҐ ૬θΛ༻͍ͯ࣍ࣜͰද͞ΕΔɽ f(t) = A sin[2πft + θ]. (3) ͜͜Ͱ໌Β͔ʹपظT = 1/f͕੒Γཱͭɽ ͞ΒʹϑʔϦΤڃ਺ల։͸ʮࡾ֯ؔ਺ͷ௚ަੑʯ Λࠜڌͱ͠ɼपظ৴߸ͷ࣌ؒ೾ܗΛجຊप೾਺ͱ ͦͷߴௐ೾੒෼͔ΒͳΔप೾਺εϖΫτϧʹ෼ղ ͢ΔɽͦͷͨΊʮϕΫτϧͷ಺ੵ͕θϩʹͳΔͱ

௚ަ͢Δʯͱ͍͏֓೦΍ɼ۩ମతͳʮࡾ֯ؔ਺ͷ ੵ෼ʯͷٕज़͕ඞཁʹͳΔɽ͔͠͠୅਺ֶ͸ʮͦ ͏ݴ͏΋ͷͩʯͱ͍֮͑ͯͯ΋ɼղੳֶʢಛʹࡾ ֯ؔ਺ͷੵ෼ʣ͸͍֮͑ͯͳ͍͔ɼͦ΋ͦ΋शͬ ͨ͜ͱ͕ແֶ͍ੜ΋࢒೦ͳ͕ΒຊֶՊʹ͸ଘࡏ͢ ΔɽͦͷͨΊ࣮ݧIͰ͸ෆఆੵ෼ͷެ͕ࣜར༻Ͱ ͖ΔΑ͏ʹͳΔ͜ͱΛ໨తͱͨ͠ɽ ۩ମతͳ಺༰ͱͯ͠͸sin 2πf tͱsin 2π(2f )t ͷҰपظʹΘͨΔੵ෼Λݟͤɼͦͷաఔͷཧղ Λଅ͢͜ͱͰ௚ަੑͷྫͱͨ͠ɽͦΕʹଓ͍ͯ ֶੜࣗ਎͕cos 2πf tͱcos 2π(2f )tɼsin 2πf tͱ

cos 2πf tͷ௚ަੑΛ֬ೝ͢Δ͜ͱΛԋशIͱͯ͠ ՝ͨ͠ɽ ௚ަੑ ਖ਼ݭ೾৴߸͸ҟͳΔप೾਺੒෼Λؚ·ͳ͍ɽಉ͡ पظT [sec.]ͷਖ਼ݭ೾৴߸ͱͯ͠sin[2πf t]ʢf = 1/Tʣͱsin[2π(2f )t]ʢجຊपظ͸T/2ʣΛߟ͑ Δɽ͜ͷͱ͖྆ऀͷੵΛҰपظʹΘͨΓੵ෼͢ Δͱ࣍ࣜΛಘΔɽ  _T/2 −T/2sin[2πf t]× sin[2π(2f)t]dt =  _T/2 −T/2 1 2(cos[(2πf t)− {2π(2f)t}] − cos[(2πft) + {2π(2f)t}]) dt = 1 2  _T/2 −T/2(cos[−2πft] − cos[3 · 2πft]) dt = 1 2  −1 2πf sin[−2πft] − 1 6πf sin[6πf t] T 2 −T 2 = −1 4πf  sin  −2πfT 2  − sin  −2πf−T 2  − 1 12πf  sin  6πfT 2  − sin  6πf−T 2  ɹ = 0. (4) ͨͩ͠ _{f ·} T₂ = _T1 · T₂ = 1₂, sin[±π] = sin[±3π] = 0Λ༻͍ͨɽࣜ(4)ΑΓɼsin[2πf t] ͱsin[2π(2f )t]͸૬खͷ੒෼Λؚ·ͳ͍͜ͱ͕ ൑Δɽ͜ͷੑ࣭Λʮޓ͍ʹ௚ަ͢ΔʯͱݺͿɽ ԋशI

֤ࣗͰcos[2πf t]ͱcos[2π(2f )t]ɼcos[2πf t]ͱ

sin[2πf t]ͷ৔߹ʹ͍ͭͯޓ͍ʹ௚ަ͢Δ͜ͱΛ ͔֬ΊΑʢඞཁͰ͋Ε͹෇࿥Λࢀরʣɽ ͜͜Ͱ໰୊ͱͳͬͨͷ͸ʮࡾ֯ؔ਺ͷੵɾ࿨ʯ ͷެ͕ࣜୈҰาͰ͋Γɼ͍֮͑ͯͳͯ͘΋Ճ๏ఆ ཧ͔Βಋग़Ͱ͖Δ͜ͱΛ఻͕͑ͨεϚϗͰެࣜΛ ௐ΂Δֶੜ͕̍ʗ̏ఔ౓ډͨɽߋʹsin x, cos xͷ حؔ਺ɼۮؔ਺ʹ͍ͭͯͷ஌ࣝ΋ࣄલʹ஌͍ͬͯ Δֶੜ͸গ਺Ͱ͋ΓɼͦΕΛ༻͍ͨ؆ུԽ͸ল͍ ͨɽͦͯ͠sin[±nπ] = 0 (n͸੔਺)Λ͍֮͑ͯ Δֶੜ΋গͳ͔ͬͨɽΑͬͯ͜ΕΒͷ಺༰Λ෇࿥ ʹࡌͤΔ͜ͱͱͨ͠ɽ ෇࿥ʢҰ෦ʣ ࡾ֯ؔ਺ͷجຊެࣜ sin[−θ] = − sin θ, cos[−θ] = cos θ. ੵɹˠɹ࿨ɾࠩ sin α cos β = 1 2{sin[α + β] + sin[α − β]} , cos α sin β = 1 2{sin[α + β] − sin[α − β]} , sin α sin β =−1 2{cos[α + β] − cos[α − β]} , cos α cos β = 1 2{cos[α + β] + cos[α − β]} . ಛผͳҰൠ֯ͷ஋ʢn, ͸੔਺ʣ sin[nπ] = 0, cos[nπ] = (−1)n. ෆఆੵ෼ެࣜ  xa_{dx =} 1 a + 1xa+1, (a= −1),  adx = ax,  eaxdx = 1 aeax,  sin[ax]dx =−1 acos[ax],  cos[ax]dx = 1 asin[ax],

ఆੵ෼  _b a f(x)dx = [F (x)]b_a= F (b)− F (a),  _b a f(x)dx = −  _a b f(x)dx,  _c a f(x)dx =  _b a f(x)dx +  _c b f(x)dx,  _b a f(x) · g _{(x)dx = [f (x)· g(x)]}b a −  _b a f(x)· g(x)dx. ΋͏Ұ఺ɼԋशIͷղ౴͕ʮੵ෼݁Ռ͕θϩʹ ͳͬͨʯͱ͜ΖͰࢭΊͯ͠·͏ֶੜ͕൒਺͍ۙ͘ ͨɽ͜ͷ՝୊ͷ໨త͸௚ަੑΛࣔ͢͜ͱͰ͋Γɼ ࠷ޙʹʮΑͬͯ***ͱ***͸௚ަ͢ΔɽʯͱͷҰจ Λॻ͘Α͏ʹࢦಋΛߦͬͨɽ 3 ϑʔϦΤڃ਺ల։ ௚ަੑͷ֬ೝʹΑͬͯϑʔϦΤ܎਺ɼϑʔϦΤ ڃ਺ల։ͷઆ໌͕Մೳͱͳͬͨɽ࣮ݧॻʹ͸ҎԼ ͷ಺༰Λهͨ͠ɽ ϑʔϦΤ܎਺ पظT ͷपظ৴߸f(t)͸༷ʑͳप೾਺੒෼Λ ͍࣋ͬͯΔɽ͔͠͠प೾਺f(= 1/T )[Hz]ͷ੒ ෼͸cos[2πf t]Λf(t)ʹֻ͚ͯੵ෼ͯ͠ಘΒΕ ͨྔa₁ͱɼsin[2πf t]Λf(t)ʹֻ͚ͯੵ෼ͯ͠ ಘΒΕͨྔb₁ͷΈͰ͋Δɽಉ༷ʹप೾਺mf[Hz] ʢm͸੔਺ʣͷ੒෼ʹ͍ͭͯ΋a_m, b_mͱͯ͠ಘ ΒΕΔɽ a_m = 2 T  _T/2 −T/2f(t) · cos[2π(mf)t]dt, (5) b_m = 2 T  _T/2 −T/2f(t) · sin[2π(mf)t]dt. (6) ͨͩ͠ɼੵ෼ͷલͷ܎਺2/T ͸ਖ਼نԽͷͨΊɽ ͜ͷa_m, b_mΛʮϑʔϦΤ܎਺ʯͱݺͼɼपظ৴ ߸f(t)ʹؚ·ΕΔcos[2πmf t]ͱsin[2πmf t]ͷ ৼ෯Λද͍ͯ͠Δɽ ϑʔϦΤڃ਺ల։ ϑʔϦΤ܎਺͕ٻΊΒΕΕ͹पظ৴߸f(t)͸Ҏ ԼͷΑ͏ʹදݱͰ͖Δɽ f(t) = a0 2 + ∞  m=1 (a_mcos[2π(mf )t] +b_msin[2π(mf )t]) . (7) ্ࣜʹ͓͍ͯm͸ແݶେ·Ͱ଍͠߹ΘͤΔඞཁ ͕͋Δ͕ɼ༗ݶͷmͰଧͪ੾ͬͯ΋f(t)ʹۙ ͍೾ܗ͕ಘΒΕΔɽ ࣮ͦͯ͠ࡍʹϑʔϦΤ܎਺ΛٻΊΔԋशͱͯ͠ ҎԼͷԋशIIΛֶੜʹ՝ͨ͠ɽ ԋशII पظ T (ඞཁͰ͋Ε͹T = 10−3_{[sec.]) ͷۣܗ}͘ ͚ ͍ ೾ɼࡾ֯೾ͷ࣌ؒ೾ܗ͸ͦΕͧΕࣜ(8), (9)Ͱ ද͞ΕΔɽ·֤ͣ࣌ؒ೾ܗΛ࣮ݧϊʔτʹඳ͖ɼ ͦΕʹଓ͍ͯͦΕͧΕͷϑʔϦΤ܎਺a_m, b_mΛ m = 10·Ͱܭࢉ͢ΔʢඞཁͰ͋Ε͹෇࿥Λࢀ রʣɽ • ۣܗ೾ f(t) =  1 |t| ≤ T/4, −1 T/2 > |t| > T/4. (8) • ࡾ֯೾ f(t) =  4 Tt + 1 −T/2 ≤ t ≤ 0, −_T4t + 1 0 < t < T/2. (9) ԋशIIͷղ๏ʹ͓͍ͯɼੵ෼۠ؒͷ෼ׂΛΠ ϝʔδ͠қ͘͢ΔͨΊʹ࣌ؒ೾ܗΛϊʔτʹॻ͔ ͤΔ͜ͱʹͨ͠ɽ͔͠͠ɼ͜͜Ͱ໰୊ʹͳͬͨͷ ͸ઈର஋ͷ֎͠ํΛ๨Ε͍ͯΔֶੜׂ͕̔΄Ͳډ ͨ͜ͱͰ͋Δɽ͢ͳΘͪɼʮઈର஋ͷத਎͕ਖ਼Ͱ ͋Ε͹ͦͷ··֎͠ɼෛͰ͋Ε͹ϚΠφεΛ෇͚ ͯ֎͢ʯͱ͍͏͜ͱ͕࣮ߦͰ͖ͳֶ͍ੜ͕ଟ͔ͬ ͨɽ·ͨʮۣܗ೾ʯ͕ಡΊͳֶ͍ੜ΋ଟ਺͍ͨͨ ΊϧϏΛଧͭ͜ͱͱͨ͠ɽ ۣܗ೾ͷϑʔϦΤ܎਺ಋग़͸ެࣜΛݟͳ͕Β΍ Ε͹େ෦෼ͷֶੜ͕࣮ߦͰ͖ͨɽ͔͠͠ࡾ֯೾ͷ ৔߹͸෦෼ੵ෼ͷٕज़ʢʁʣ͕ඞཁͰ͋ΓɼҰൠ ࿦ͱͯ͠ͷެࣜ͸෇࿥ʹࡌͤͯ͋Δɽ

෇࿥ʢҰ෦ʣ ෦෼ੵ෼  _b a f(x) · g _{(x)dx (10)} = [f (x)· g(x)]b_a−  _b a f(x)· g(x)dx. ͔͠͠ɼͦΕΛ༻͍ͯ۩ମతʹੵ෼Λ࣮ࢪͰ͖Δ ֶੜ͸ׂ̎ఔ౓Ͱ͋ͬͨɽ࣮ࡍͷܭࢉ͸ɼࣜ(9) ΑΓҎԼͷखॱͰ͋Δɽ a_m = 2 T  ₀ −T/2  4 Tt + 1  cos[2π(mf )t]dx (11) +2 T  _T/2 0  −_T4t + 1  cos[2π(mf )t]dx ໰ ୊ ͱ ͳ Δ ͷ ͸ t cos[2π(mf)t]dx Ͱ ͋ Γɼ f(t) = t, g_{(t) = cos[2π(mf )t]ͱஔ͍ͯܭࢉΛਐ} ΊΔඞཁ͕͋Δɽ ͦͯ͠࠷ऴతʹϑʔϦΤ܎਺ΛҎԼͷදͷܗࣜ ʹ·ͱΊΔΑ͏ɼࢦࣔΛग़͕ͨ͠MS WordΛ༻ ͍ͨ࡞දʹ΋໰୊͕ଟ਺ੜͨ͡ɽ ද1: ϑʔϦΤ܎਺ͷ·ͱΊํ m 1 2 3 · · · 10 a_{m · · · ·} b_{m · · · ·} MS Wordʹ͓͍ͯදΛ࡞੒͢Δʹ͸Wordࣗ ਎ͷػೳΛ࢖͏৔߹ͱExcelͰ࡞දͨ͠ϞϊΛ WordʹషΓ෇͚Δ৔߹ͷ̎छྨͷํ๏͕͋Δɽ WordͰ࡞දͨ͠৔߹ʹ͸ʮ਺ࣜʯͳͲΛ༻͍ͯ ਺ࣜ΍ه߸ΛೖྗͰ͖Δ͕Excel͔ΒషΓ෇͚Δ ৔߹ʹ͸ηϧʹʮ਺ࣜʯ͕ೖྗ͞ΕͣɼςΩετ ϘοΫεͱͯ͠ೝࣝ͞Εͯ͠·͏ɽΑͬͯWord ʹΑΔ࡞දΛࢦࣔͨ͠ɽ ·ͨOffice 2007Ҏ߱ͷWordʹ෇ଐͷʮ਺ࣜʯ Ͱ͸࿦ཧతʹਖ਼͍͕ࣜ͠ॻ͖ͮΒ͘ɼϑΥϯτ͕ ࣼΊʢΠλϦοΫʣʹͳΔ৔߹ͱཱͭʢϩʔϚϯʣ ৔߹ͷ৚͕݅ෆ໌֬Ͱ͋ΔɽΑͬͯʮૠೖʯλϒ͔ ΒʮΦϒδΣΫτʯΛબͼɼʮMicrosoft ਺ࣜ3.0ʯ ͷར༻ΛקΊͨɽ ֶੜͷதʹ͸਺ࣜπʔϧΛ࢖ΘͣʹϑΥϯτα Πζͷมߋ͚ͩͰ্෇͖ɼԼ෇͖Λදͦ͏ͱͨ͠ ऀ΋ډͨɽ͔͠͠ಠཱࣜͱͷෆ౷ҰΛࢦఠ͠ɼ͢ ΂ͯͷ৔߹ʹಉ͡਺ࣜπʔϧΛ࢖͏͜ͱΛࢦࣔ ͨ͠ɽ ͳ͓ɼΠλϦοΫͱϩʔϚϯͷॻ͖෼͚͸ʮม ਺͸ΠλϦοΫʯɼʮఆ਺͸ϩʔϚϯʯͱͯ͠ࢦࣔ ͸ग़͕ͨ͠ɼ࣮ݧॻʹ͸໌ه͠ͳ͔ͬͨɽ͜ͷล Γ͸ֶձ΍ݸਓͷߟ͑ํͰํ਑͕ҟͳΔ৔߹͕͋ ΔͨΊ໌จԽ͠ͳ͔ͬͨɽ 4 ExcelʹΑΔ೾ܗ߹੒ ͔͜͜Β͸2೔໨ͷ࣮ݧʹͳΔɽઌʹٻΊͨਖ਼ ݭ೾ͱࡾ֯೾ͷϑʔϦΤ܎਺ʹج͖ͮɼExcelͰ ඳ͍ͨߴप೾੒෼Λ଍͠߹ΘͤΔ͜ͱͰਖ਼ݭ೾΍ ࡾ֯೾ʹ͍ۙ೾ܗ͕߹੒Ͱ͖Δ͜ͱΛ֬ೝ͢Δɽ ۩ମతʹ͸ҎԼͷखॱΛ࣮ݧॻʹهड़ͨ͠ɽ Excelͷجૅ஌ࣝI ࿈ଓσʔλͷ࡞੒ʢAྻʹ1͔Β1000·Ͱͷ ਺஋ΛೖΕΔʣ (1) A1ͷηϧΛબ୒͠ɼ“1”Λೖྗ ˎ ࣮ࡍͷೖྗ࣌ʹ_“ɹ_”͸ෆཁɽ (2) ӈͷεΫϩʔϧόʔͷ্෦ʹ͋Δʮ෼ׂϘο ΫεʯΛϓϧμ΢ϯ͠ɼϫʔΫγʔτΛ্ Լʹ̎෼ׂ͢Δɽ (3) ԼଆͷϫʔΫγʔτΛεΫϩʔϧ͠ɼ1000 ߦ໨෇ۙΛදࣔͤ͞Δɽ (4) A1ͷηϧΛΫϦοΫ͠ɼʮSHIFTʯΩʔΛ ԡ͠ͳ͕ΒA1000ͷηϧΛΫϦοΫɽ ˎ ͜ΕͰA1͔ΒA1000·Ͱͷηϧ͕બ୒͞ ΕΔɽ (5) ʮϗʔϜʯλϒͷʮฤूʯˠʮϑΟϧʯˠ ʮ࿈ଓσʔλͷ࡞੒ʯΛબ୒ɽ (6) ʮ૿෼஋ʯ͕ʮ̍ʯͰ͋Δ͜ͱΛ֬ೝͯ͠ ʮ̤̠ʯɽ ͨͩ͠ϫʔΫγʔτΛ̎෼ׂ͢Δʮ෼ׂϘοΫ εʯ͸Office2013Ҏ߱͸ແ͘ͳ͓ͬͯΓɼͦͷ୅ ΘΓʹʮදࣔʯλϒ͔Βʮ΢Οϯυ΢ʯ࿮ͷʮ෼

ׂʯΛબ୒͢Δඞཁ͕͋Δɽ ଓ͍ͯσʔλͷೖྗΛҎԼͷࢦࣔʹΑΓߦ͏ɽ Excelͷجૅ஌ࣝII ࿈ଓσʔλͷ࡞੒ʢBྻͷ1͔Β1000·Ͱʹ −π͔Βπ·Ͱͷ਺஋Λۉ౳ʹೖΕΔʣ (1) B1ͷηϧΛબ୒͠ɼ “=A1*2*PI()/1000-PI()”ͱೖྗɽ ˎ ͜ͷ͕ࣜԿΛҙຯ͍ͯ͠Δ͔ߟ͑Δ͜ͱɽ (2) B1͔ΒB1000·ͰͷηϧΛબ୒ɽ (3) ʮϗʔϜʯλϒͷʮฤूʯˠʮϑΟϧʯˠ ʮԼํ޲΁ίϐʔʯΛબ୒ɽ (4) B1 ʹ-3.13531ɼͦ ͜ ͔ Β ૿ ͑ ͯ ͍ͬͯ B1000ʹ3.141593͕ೖ͍ͬͯΔ͜ͱΛ֬ ೝ͢Δɽ Excelͷجૅ஌ࣝIII άϥϑͷ࡞੒ʢIʣ (1) B1ͷηϧΛΫϦοΫͯ͠ɼʮૠೖʯλϒ͔ Βʮάϥϑʯˠʮࢄ෍ਤʯˠʮࢄ෍ਤʢ௚ ઢʣʯΛબ୒ɽ (2) ඞཁͰ͋Ε͹άϥϑΛΫϦοΫ͠ɼʮάϥϑ πʔϧʯͷʮσβΠϯʯλϒ͔Βʮσʔλʯ ˠʮσʔλͷબ୒ʯͰඞཁͳσʔλΛ௥Ճɼ ࡟আͰ͖Δɽ ˎ ʮσʔλൣғʯʹ “= Sheet1! $B$1 : $B$1000 ”ͱೖྗ͞Ε͍ͯΔɽಉ࣌ʹຌྫ ߲໨ͷʮฤूʯΛΫϦοΫ͢Ε͹ʮܥྻ̍ʯ ͳͲͱ੒͍ͬͯΔຌྫͷ಺༰΋มߋͰ͖Δɽ (3) ಉ༷ʹʮϨΠΞ΢τʯλϒ͔Βॎ࣠ɼԣ࣠ ͷॻࣜɼϥϕϧͳͲ͕มߋͰ͖Δɽ Ҏ্ͷ݁Ռɼਤ1ΛಘΔɽ ͜͜Ͱ͸ࢄ෍ਤΛ༻͍͕ͨɼֶੜ͸ંΕઢάϥ ϑΛબͼ͕ͨΔɽࢄ෍ਤͱંΕઢάϥϑͷҧ͍͸ ԣ࣠ͷબఆ͕Ͱ͖Δ͔Ͳ͏͔Ͱ͋Δɽ۩ମతʹ͸ ද2ͷσʔλΛߟ͑ɼંΕઢάϥϑͱࢄ෍ਤʹ͠ ͯΈΔͱਤ2ͱਤ3ΛಘΔɽ͢ͳΘͪંΕઢάϥ ϑͷ৔߹͸ԣ࣠͸ৗʹ1, 2, 3, Ͱ͋Γɼͦͷதԝ ਤ1: Excelͷجૅ஌ࣝI,II,III ʹϚʔΧʔ͕ݱΕΔɽͦͯ͠AྻͱBྻ͸ͦΕ ͧΕҟͳΔ̎ຊͷάϥϑʹͳΔɽҰํͰࢄ෍ਤʹ ͓͍ͯ͸AྻΛԣ࣠ɼBྻΛॎ࣠ͱͯ͠ϚʔΧʔ ͕ଧͨΕΔɽ͢ͳΘͪy = f(x)ͷ༷ʹɼ༩͑Β Εͨxͷ஋ʹରͯ͠yͷ஋͕ఆ·Δ৔߹ʹ͸ࢄ෍ ਤΛ࢖Θͳ͍ͱਖ਼͘͠ඳըͰ͖ͳ͍ɽΑ্ͬͯʹ ࣔͨ͠Α͏ʹૢ࡞खॱΛॻ͖ද͢͜ͱͰඞͣʮࢄ ෍ਤʯΛબͿΑ͏ʹ࢓޲͚ͨɽֶ͔͠͠ੜ͸ʮં Εઢʯʹऒ͔ΕΔΑ͏Ͱ͋Δɽ ද2: ExcelͰάϥϑΛඳͨ͘Ίͷσʔλྫ A B 1 0.1 1 2 0.2 2 3 0.3 3 4 0.4 4 5 0.5 5 6 0.6 6 7 0.7 7 8 0.8 8 9 0.9 9 10 1.0 10 ͦΕҎ֎ʹ΋Excelͷάϥϑʹ͸͍͔ͭ͘໰୊ ͕͋Δɽ·ͣ͸উखʹάϥϑͷλΠτϧΛਤͷத ʹॻ͘͜ͱͰ͋Δɽιϑτ΢ΣΞαΠΤϯεֶՊ

0 2 4 6 8 10 12 1 2 3 4 5 6 7 8 9 10 ⣔ิ1 ⣔ิ2 ਤ2: ંΕઢͷ৔߹ 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 ⣔ิ1 ਤ3: ࢄ෍ਤͷ৔߹ Ͱ͸ʮਤ൪߸ɼλΠτϧ͸ਤͷԼʯͱࢦࣔΛͯ͠ ͍Δɽ͜Ε͸ࢲୡͷؔ܎͢ΔֶձͷதͰ͸IEEE ͕ࢦ਑ͱ͓ͯ͠Γɼ೔ຊͷిࢠ৘ใ௨৴ֶձ΋ͦ Εʹ४͍ͯ͡Δ͜ͱΛड͚ͯͷࢦಋͰ͋Δɽͪͳ Έʹʮද൪߸ɼλΠτϧ͸දͷ্ʯ͕΍͸Γجຊ Ͱ͋Δɽ͍ͣΕʹͯ͠΋ਤ൪߸ɼλΠτϧΛਤͷ தʹؚΊΔͷͰ͸ͳ͘ɼਤͱ͸ಠཱͯ͠Word಺ Ͱૠೖ͢΂͖Ͱ͋Δɽ͜Ε͸ฤूաఔʹ͓͍ͯਤ ൪߸ɼλΠτϧͷมߋ͕ඞཁʹͳͬͨ৔߹ʹExcel ͳͲͷਤ·Ͱ໭ͬͯมߋ͢ΔͷͰ͸ͳ͘ɼWord ಺ͷॲཧͰࡁ·ͤΔ͜ͱΛ໨తͱ͍ͯ͠Δɽ·ͨɼ ਤʹλΠτϧ͕෇͘ͷͰҰຊ͔͠ઢ͕ແ͍৔߹͸ ຌྫʢʮܥྻ̍ʯͳͲʣ͸ඞཁͳ͍ɽߋʹExcel͸ ԣ࣠ӈଆʹ༨നΛ෇͚ͯ͘ΕΔɽਤ3ͷ৔߹͸ࠨ ʹ0͔Β0.1·Ͱͷ༨ന͕͋Δ͔Β໨ཱͨͳ͍͕ɼ ࠨ୺͔Βઢ͕Ҿ͔Ε͍ͯͯ΋ӈ୺ʹۭനʢਤ3Ͱ ͸1͔Β1.2ʣ͕ଘࡏ͠ɼόϥϯε͕ѱ͍ɽͰ΋ ֶੜ͸ʮExcelͷग़ྗ͕͜͏ͳ͔ͬͨΒʯͱݴͬ ͯͦͷ··షͬͯ͘Δྫ͕ଟ͍ɽղܾͷͨΊʹ͸ ԣ࣠ͷ਺஋ΛΫϦοΫͨ͠ޙɼϦϘϯͷʮάϥϑ πʔϧʯͷʮϨΠΞ΢τʯλϒΛબ୒͠ɼࠨ୺ʹ͋ Δʮબ୒ର৅ͷॻࣜઃఆʯΛΫϦοΫ͢Ε͹ʮ࣠ ͷॻࣜઃఆʯ૭্ཱ͕͕ͪΓɼ࣠ͷΦϓγϣϯͰ ʮ࠷େ஋ʯΛ1.0ʹ߹ΘͤΕ͹ྑ͍ɽ ଓ͍ͯҎԼͷԋशΛߦ͍ɼߴप೾੒෼ͷඳըͷ ४උΛߦ͏ɽ ԋशI CྻɼDྻͷͦΕͧΕͷ1ߦ໨͔Β1000ߦ໨ ʹ_−2π͔Β2πɼ_−3π ͔Β3π·Ͱͷ਺஋Λۉ ౳ʹೖΕΑɽ ͜ͷԋशʹ͓͍ͯ͸1࣍ؔ਺y = ax + bͷy ੾ยbͱ܏͖aΛߟ͑ͯؔ਺Λೖྗͯ͠ཉ͔ͬ͠ ͕ͨɼؾֶ͍ͮͨੜ͸C1ͷηϧʹ“=2*B1”ͱ ೖྗ͠ɼD1ͷηϧʹ“=3*B1”ͱೖྗ͍ͯͨ͠ɽ ࣍ʹɼಘΒΕͨྻͷ஋Λ༻͍࣮ͯࡍʹ༨ݭ೾ ʢجຊ೾ʣΛඳ͘ɽ Excelͷجૅ஌ࣝIV άϥϑͷ࡞੒ʢIIʣ (1) E1ͷηϧΛબ୒͠ɼʮ਺ࣜʯλϒ͔Βʮؔ ਺ϥΠϒϥϦʯͷʮ਺ֶʗࡾ֯ʯΛϓϧμ ΢ϯ͠ɼʮCOSʯʢ৔߹ʹΑͬͯ͸ʮSINʯʣ Λબ୒͢Δɽ (2) ʮؔ਺ͷҾ਺ʯ૭͕։͍ͨΒʮB1ʯͱೖྗ ͠ʮOKʯɽ ˎ ݁Ռతʹ“ʹCOS(B1)”ͱೖྗ͞ΕΔɽ (3) E1͔ΒE1000·ͰͷηϧΛબ୒ɽ (4) ʮϗʔϜʯλϒͷʮฤूʯˠʮϑΟϧʯˠ ʮԼํ޲΁ίϐʔʯΛબ୒ɽ (5) E1ͷηϧΛΫϦοΫͯ͠ɼʮૠೖʯλϒ͔ Βʮάϥϑʯˠʮࢄ෍ਤʯˠʮࢄ෍ਤʢ௚ ઢʣʯΛબ୒ɽ (6) ඞཁʹԠͯ͡ຌྫ΍ॎ࣠ɼԣ࣠Λݟ΍͘͢ มߋ͢Δɽ ࠷ޙͷ߲໨ʹ͓͚Δຌྫɼॎ࣠ɼԣ࣠ͷมߋ΋ ر๬ͷ߲໨ΛΫϦοΫ͠ɼϦϘϯͷʮάϥϑπʔ ϧʯͷʮϨΠΞ΢τʯλϒΛબ୒͠ɼࠨ୺ʹ͋Δ ʮબ୒ର৅ͷॻࣜઃఆʯ͔ΒՄೳͱͳΔɽ ্هͷʮάϥϑͷ࡞੒ʢIIʣʯʹΑͬͯͰجຊ पظͷ༨ݭ೾Λඳ͍ͨɽଓ͍ͯԋशIIͰৼ෯Λ มߋͨ͠ΓɼԋशIIIͰߴௐ೾Λඳ͍ͨΓ͢Δɽ

ԋशII ্هͷखॱͰಘΒΕͨ༨ݭ(ίαΠϯ)೾ͷάϥ ϑͷৼ෯Λ̎ഒʹ͢Δํ๏Λߟ࣮͑ͯߦ͠ɼά ϥϑΛ࡞੒ͤΑɽ ͜Ε͸ࣜ(7)ʹ͓͚Δcos 2πmf tͷৼ෯Λa_m ͱ͢Δ४උͰ͋Δɽ ԋशIII ʮExcelͷجૅ஌ࣝIVʯͰಘΒΕͨ༨ݭ೾ͷά ϥϑ͸1͔Β1000·ͰΛҰपظͱ͢Δ༨ݭ೾ cos[2πf t]Ͱ͋Δɽ͢ͳΘͪԣ࣠Λ࣌ؒͱ͠ɼͦ ͷ஋ʮ1ʯ͔Βʮ1000ʯ·ͰΛ10−3[sec.]ͱݟͳ ͢ͱप೾਺1kHzͷ༨ݭ೾͕ඳ͚ͨ͜ͱʹͳΔɽ (1) ͜ͷάϥϑͷԣ࣠Ͱt = 0ʹରԠ͢ΔҐஔ ͸Ͳ͔͜ɼߟ͑Αɽ (2) ԋशIͷ݁ՌΛར༻ͯ͠प೾਺͕m(= 2, 3, · · · , 10)ഒͷ༨ݭ೾cos[2π(mf )t]ͷάϥϑ ΋࡞੒ͤΑɽ ͜͜Ͱߴௐ೾੒෼ͷάϥϑΛ࡞੒͢Δ͕ɼઌͷ ંΕઢάϥϑͱࢄ෍ਤͰઆ໌ͨ͠໰୊͕ಉ༷ʹى ͜Δɽ͢ͳΘͪɼଟ͘ͷֶੜ͸_{m = 1, 2, · · · , 10} ͷσʔλΛExcelͷྻʹฒ΂ͯάϥϑͷ४උΛ͢ Δɽ͔͠͠ࠨ୺ͷྻʹ͸m = 1ͷσʔλ͕ೖͬͯ ͓ΓɼͦΕ͕Excelʹ͸ʮx࣠ͷ஋ʯͰ͋Δͱ൑அ ͞Εͯ͠·͏ɽͦͷ݁ՌɼಘΒΕΔάϥϑ͕ਤ4 ͷΑ͏ʹͳͬͯ͠·͏͜ͱ͕ଟ͍ɽͪͳΈʹਖ਼ղ ͸ਤ5ͷΑ͏ʹ੒Δ΂͖Ͱ͋Δɽ͢ͳΘͪɼબ୒ ͢Δσʔλͷࠨ୺ʹx࣠ʹରԠ͢Δ஋ΛೖΕ͓ͯ ͘ඞཁ͕͋Δɽͳ͓͜ͷྫͰ͸x࣠Λ1͔Β1000 ·Ͱͱ͕ͨ͠ɼ࠲ඪม׵Λͯ͠_−0.5͔Β0.5ʹ ͠ɼ୯ҐΛ[ms]ͱ͢Ε͹1kHzͷपظ৴߸Λѻͬ ͍ͯΔ͜ͱͱͳΔɽ͜ͷ࠲ඪม׵͸࣮ݧ։࢝౰ॳɼ ֶੜʹٻΊ͕ͨҙຯΛཧղ͢Δֶੜ͕গͳ͔ͬͨ ͨΊɼ̏೥໨͔Β͸ֶੜʹ՝͞ͳ͘ͳͬͨɽ ·ͨɼਤ4ɼ5ʹ͓͍ͯຌྫ͕ʮܥྻ1ɼܥྻ 2ɼɾɾɾʯͱͳ͍ͬͯΔ͕౰વɼมߋ͢΂͖Ͱ͋Δɽ ͦΕ͸มߋ͍ͨ͠άϥϑΛબ୒ͨ͠ޙɼʮάϥϑ πʔϧʯͷʮσβΠϯʯλϒ͔Βʮσʔλͷબ୒ʯ ΛΫϦοΫ͠ɼࠨʹ͋Δʮຌྫ߲໨ʯͷத͔Βر ๬ͷσʔλΛબͼʮฤूʯϘλϯΛԡ͢͜ͱͰر ๬ͷจݴʹมߋ͠ɼ̤̠Λԡͤ͹࣮ݱͰ͖Δɽ ͦͯ͠߹੒೾ܗͱൺֱ͢ΔͨΊʹݩͷ࣌ؒ೾ܗ -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 ⣔ิ1 ⣔ิ2 ⣔ิ3 ⣔ิ4 ⣔ิ5 ⣔ิ6 ⣔ิ7 ⣔ิ8 ⣔ิ9 ਤ 4: ߴௐ೾੒෼ͷඳըʢࣦഊྫʣ -1.5 -1 -0.5 0 0.5 1 1.5 0 200 400 600 800 1000 ⣔ิ1 ⣔ิ2 ⣔ิ3 ⣔ิ4 ⣔ิ5 ⣔ิ6 ⣔ิ7 ⣔ิ8 ⣔ิ9 ⣔ิ10 ਤ 5: ߴௐ೾੒෼ͷඳըʢ੒ޭྫʣ ΋ԋशIVͱͯ͠ඳ͔ͤͨɽ ԋशIV ࣜ(8), (9)Ͱࣔ͞Εۣͨܗ೾ɼࡾ֯೾ͷ࣌ؒ೾ ܗ΋1͔Β1000ΛҰपظͱͯ͠ExcelͰ࡞੒ ͤΑɽ ͜ͷ৔߹΋ۣܗ೾͸༰қʹඳ͚Δ͕ࡾ֯೾͸ख ͣ͜Δֶੜ͕ଟ͔ͬͨɽ͢ͳΘۣͪܗ೾ͷ৔߹͸ ʮ1͔Β250·Ͱ₋₁ɼ251͔Β750·Ͱ1ɼͦ͠ ͯ751͔Β1000·Ͱʹ−1ʯΛೖྗ͢Ε͹ඳ͚ Δɽ͔͠͠ࡾ֯೾ͷ৔߹͸ʮ1ͷͱ͖ʹ_−1(+α)ɼ ͔ͦ͜Β1࣍ؔ਺ͱͯ͠૿Ճ͠ɼ500ͷͱ͖ʹ1ɽ ͦͯ͠Ұ࣍ؔ਺ͱͯ͠ݮগ͠1000ͷͱ͖ʹ₋₁ʯ ͱͳΔඞཁ͕͋ΔɽͦͷͨΊ1͔Β500·Ͱʹ͸ “=2*A1/500-1”ͳͲͱೖྗ͠ɼ501͔Β1000· Ͱʹ͸“=(-2)*A501/500+3”ͳͲͱೖྗ͢Δඞ ཁ͕͋Δɽࣗ෼Ͱ͜ΕΛߟ͑ͯೖྗͰֶ͖ͨੜ͸ ൒਺ະຬͰ͋ͬͨɽ ͜͜·Ͱͷ४උ͕ऴΘͬͯɼ͍Α͍Α೾ܗ߹੒ ͷ࣮ݧͰ͋Δɽ ࣮ݧ಺༰ 1) पظT ͷۣܗ೾ɼࡾ֯೾ͷϑʔϦΤ܎਺͘ ͚ ͍ ʢa_m, b_{m, m = 0, 1, 2, · · · , 10}ʣ͸طʹٻΊ

ͯ͋Δɽ 2) ԋश III Ͱ࡞੒ͨ͠༨ݭ೾ cos[2π(mf )t] ʢm = 0, 1, 2,· · · , 10ʣͷ ৼ ෯ Λ ϑ ʔ Ϧ Τ ܎ ਺ a_m ʹ ม ߋ ͠ ɼExcel ্ ʹ a_mcos[2π(mf )t]ͷσʔλΛ४උ͢Δɽ ˌ ඞཁͰ͋Ε͹b_msin[2π(mf )t]ͷσʔλ΋ ४උ͢Δɽ 3) ࣜ(7)ͷϑʔϦΤڃ਺ల։Λ༗ݶͷप೾਺ ੒෼ʢM = 5, 10ʣ·Ͱ଍͠߹Θͤͨ৔߹ ʹಘΒΕΔ೾ܗf_{(t)͕ݩͷ࣌ؒ೾ܗf(t)} ʹ͍͔ۙͲ͏͔ΛExcelͰάϥϑʹͯ͠؍ ࡯͢Δɽ f(t) = a0 2 + M  m=1 (a_mcos[2π(mf )t] +b_msin[2π(mf )t]) . (12) ˌ M ͷҧ͍͕߹੒͞Εͨ೾ܗf_(t)ʹͲ͏Ө ڹ͢Δ͔ཧղ͢Δɽ ͜ͷ࣮ݧ಺༰Λॲཧ͢Δखॿ͚ͱͯ࣍͠ͷجૅ ஌ࣝ΋ܝࡌͨ͠ɽ Excelͷجૅ஌ࣝV _M m=1xm(= x1+ x2+· · · + xM)ͷܭࢉ (1) M_m=1x_mͷ݁ՌΛೖྗ͍ͨ͠ηϧΛબ୒ ͢Δɽ (2) ʮ਺ࣜʯλϒ͔Βʮؔ਺ϥΠϒϥϦʯͷʮ਺ ֶʗࡾ֯ʯΛϓϧμ΢ϯ͠ɼʮSUMʯΛબ୒ ͢Δɽ (3) ʮؔ਺ͷҾ਺ʯ૭͕։͍ͨΒɼʮ਺஋̍ʯʹ x₁ͷ஋͕ೖ͍ͬͯΔηϧɼʮ਺஋̎ʯʹx₂ ͷ஋͕ೖ͍ͬͯΔηϧɼͱͯ͠x_M ͷ஋͕ ೖ͍ͬͯΔηϧ·ͰΛॱ࣍બ୒͢Δɽ (4) ʮOKʯΛԡ͢͜ͱʹΑͬͯܭࢉ݁Ռ͕࠷ ॳͷηϧʹදࣔ͞ΕΔɽ ˎ ඞཁͰ͋Ε͹ʮԼํ޲ʹίϐʔʯ্ͯ͠ه ͷܭࢉΛԼͷηϧʹ΋൓өͤ͞Δɽ ͜ΕʹΑͬͯݩͷ೾ܗɼ߹੒೾ʢM = 5, 10ʣ ͷάϥϑ͕ग़དྷ্͕Δɽ·ͣ͸ظ଴͞ΕΔ݁ՌΛ ਤ6ͱਤ7ʹࣔ͢ɽ -1.5 -1 -0.5 0 0.5 1 1.5 0 200 400 600 800 1000 M=5 M=10 ▴ᙧἼ ਤ6: ۣܗ೾ͷ߹੒೾ܗͱݩ೾ܗ -1.5 -1 -0.5 0 0.5 1 1.5 0 200 400 600 800 1000 M=5 M=10 ୕ゅἼ ਤ7: ࡾ֯೾ͷ߹੒೾ܗͱݩ೾ܗ ͜͜·Ͱͷάϥϑ࡞੒Ͱྑ͘ݟΒΕͨΤϥʔ ͸ɼۣܗ೾ͷϑʔϦΤ܎਺͕શͯਖ਼ͷ஋ʹ੒ͬͯ ͓Γ߹੒೾͕ෆࢥٞͳܗʹ੒͍ͬͯΔ৔߹͕͋Δɽ Τϥʔάϥϑʹ׳Εͯ͘Δͱʮ͜ͷਤܗ͸͜ͷΤ ϥʔʯͱ൑அͰ͖ΔΑ͏ʹͳΔɽ·ͨࡾ֯೾ͷ৔ ߹ʹ͸ΤϥʔͰ͸ͳ͍͕̏ຊͷઢ͕΄΅ॏͳΔͨ Ίɼάϥϑͷଠ͞΍લޙͷॱ൪Λ޻෉͢Δ͜ͱͰ ݟ΍͍͢άϥϑ͕࡞੒Ͱ͖Δɽ ·ͨCݴޠͷϓϩάϥϜʹΑͬͯϑʔϦΤ܎ ਺a_mͱcos[2π(mf )t]ͷ஋Λܭࢉ͠ɼࣜ(12)ʹ ͓͍ͯM = 20΍30ͷ৔߹ͷσʔλΛٻΊΔ͜ ͱ΋ՄೳͰ͋Δɽͦͯ͠csvܗࣜͰϑΝΠϧग़ྗ ͠ɼExcelͰಡΈࠐΉ͜ͱʹΑͬͯਤ6ɼ7ͱॏͶ ͯඳ͘͜ͱ΋ՄೳͰ͋ΔɽϓϩάϥϜྫΛҎԼʹ ࣔ͢ɽ #include <stdio.h> #include <math.h> int main(void) {

double sum,am; double pi = 3.14159265; int i,m; FILE *fp; fp=fopen("data1.csv","w"); for(i=1;i<=1000;i++){ sum=0; for(m=1;m<=30;m++){ am=4.0/(pi*m)*sin(pi*m/2.0); printf("%d,%f\n",m,am); sum=sum +am*cos(m*(2.0*pi *i/1000.0-pi)); } fprintf(fp,"%d,%f\n",i,sum); } fclose(fp); return 0; } ࣮ࡍʹϓϩάϥϜͰσʔλΛ࡞ͬͯάϥϑʹࡌͤ ֶͨੜ͕աڈʹ਺໊ډΔɽ ͜ΕΒͷ಺༰͕֬ೝͰ্͖ͨͰൃలԋशͱͯ͠ ҎԼͷ಺༰Λهͨ͠ɽ ൃలԋशI ҎԼʹࣔ͢೾ܗʹ͍ͭͯ΋ϑʔϦΤ܎਺Λಋग़ ͠ɼͦͷৼ෯ͷਖ਼ݭ೾ʢ΋͘͠͸༨ݭ೾ʣΛ଍ ্͛͠Δ͜ͱʹΑͬͯݩͷ೾ܗ͕࠶ݱ͞ΕΔ͜ ͱΛ֬ೝͤΑɽ • Ұपظ͕࣍ࣜͰఆٛ͞ΕΔۣܗ೾ f(t) =  −1 −T/2 ≤ t < 0, 1 0≤ t < T/2. (13) • Ұपظ͕࣍ࣜͰఆٛ͞ΕΔ೾ܗʢf = 1/Tʣ f(t) = cos2πf t + π 4  , −T/2 < t < T/2. (14) ൃలԋशII ݩͷ࣌ؒ೾ܗf(t)ͱࣜʢ12ʣʹج͍ͮͯ߹੒ ͞Εͨ೾ܗf_{(t)ͱͷࠩҟ͕Ͳͷఔ౓͋Δͷ͔} ΛͦΕͧΕͷ೾ܗʹ͍ͭͯ਺஋Ͱࣔͤɽ • ࠩҟΛٻΊΔҰྫͱͯ͠_{|f(t) − f}(t)|ͷੵ ෼ɼ͢ͳΘͪ _−T/2T/2 _{|f(t) − f}(t)|dtͰධՁ Ͱ͖Δɽ • ࣮ࡍʹ͸Excel্Ͱ֤࣌ࠁt_iʹରԠ͢Δ཭ ࢄతσʔλΛͦΕͧΕf(t_i)ͱf_(t i)ͱ͠ ͯಘ͍ͯΔͷͰ1000_{i=1 |f(ti})− f(t_i)|Λܭ ࢉ͢Ε͹Α͍ɽ • ্هͱ͸ҟͳΔධՁํ๏ʹ͍ͭͯ΋ߟ͑ͯ ΈΑɽ ࢒೦ͳ͕ΒൃలԋशIΛղֶ͘ੜ͸1೥ʹҰਓ ͔ೋਓɼൃలԋशIIʹࢸͬͯ͸աڈ10೥΄Ͳͷ ؒʹยखͰ଍ΓΔ਺͔͠ଘࡏ͠ͳ͔ͬͨɽಛʹ͜ ͷ5೥΄Ͳ͸օແͰ͋Δɽ 5 ·ͱΊ ιϑτ΢ΣΞαΠΤϯε࣮ݧͰߦ͖ͬͯͨϑʔ ϦΤղੳͷमಘ໨ඪͱͦͷ࣮ݧ಺༰Λࣔͨ͠ɽ֤ ߲໨Ͱੜ͡Δ໰୊఺ͱͦΕʹର͢Δղܾࡦʹ͍ͭ ͯ۩ମతʹࣔͨ͠ɽ͜ΕΒͷ಺༰͕কདྷͷֶੜ࣮ ݧɼ΋͘͠͸ϑʔϦΤղੳͷߨٛʹ໾ཱͭͱ޾͍ Ͱ͋Δɽ ࢀߟจݙ [1] ౔ࢁ຀෉ɼफ૾ษɼࢁ࡚ߒҰɼখ઒ߊɼେ࡚ ਖ਼༤ɼιϑτ΢ΣΞαΠΤϯε࣮ݧ̞ࢦಋॻɼ ୈ̍൛ɼ2009೥9݄11೔ [2] ౔ࢁ຀෉ɼफ૾ษɼࢁ࡚ߒҰɼখ઒ߊɼେ࡚ ਖ਼༤ɼιϑτ΢ΣΞαΠΤϯε࣮ݧ̞ࢦಋॻɼ 2013೥൛ɼ2013೥9݄18೔ ̎̌̍̒೥݄̎̎̕೔ݪߘड෇ Received, February 29, 2016 ̎̌̍̒೥݄̐̍೔ݪߘमਖ਼ Revised, April 1st, 2016 2016 年２月 29 日原稿受付，2016 年３月 14 日採録決定 Received, February 29, 2016; accepted, March 14, 2016