学生実験におけるフーリエ解析II

(1)

ֶੜ࣮ݧʹ͓͚ΔϑʔϦΤղੳ

II

Some studies on Fourier analysis in students experiment II

େ࡚ਖ਼༤

Masao Osaki

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

In succession of the paper I wrote last year, here we give some troubles in teaching and their solutions occured during the Software Science Experiment II course, which is opened for the 5th semester in the Department of Software Science. One of the subjects of the experiment course is understanding the fundamentals of PCM (Pulse Code Modulation). They deal with the sampling theorem and bandwidth of communications. Some students are still not familiar with Fourier analysis while the explanation of the sampling theorem is done at the frequency region. Typical mistakes and their settlements are given.

Keywords: Students expreriment, Fourier analysis, Pulse Code Modulation, Sampling theorem.

1 ͸͡Ίʹ ຊֶ޻ֶ෦ιϑτ΢ΣΞαΠΤϯεֶՊͰ͸ ʮιϑτ΢ΣΞαΠΤϯε࣮ݧIʢҎԼɼ࣮ݧIʣʯ ͱʮιϑτ΢ΣΞαΠΤϯε࣮ݧIIʢҎԼɼ࣮ݧ IIʣʯΛඞमՊ໨ͱ࣮ͯ͠ࢪ͖ͯͨ͠ɽͦΕͧΕ̐ ηϝελʔɼ̑ηϝελʔͰ։ߨ͞Εɼ࣮ݧʹΑ Δݱ৅೺ѲͱɼͦΕΛϨϙʔτʹදٕ͢ज़ͷमಘ Λ໨ࢦ͍ͯͨ͠ɽ֤࣮ݧ͸̏ςʔϚͰߏ੒͞Εɼ ࣮ݧIIͰ͸ʮPCMʯͱ୊ͯ͠ύϧεූ߸มௐͷ جૅ஌ࣝΛमಘ͢ΔςʔϚΛஶऀ͸୲౰ͨ͠ɽ ຊߘͰ͸ɼͦͷ࣮ݧʹ͓͍ͯमಘΛ໨ࢦ߲ͨ͠ ໨ɼͦͷख๏ɼͦ͜Ͱੜͨ͡໰୊ͱͦͷղܾํ๏ ʹ͍ͭͯड़΂Δɽͦ͜ʹݟΒΕΔز͔ͭͷ޻෉͕ ࣮ݧʹݶΒͣɼεϖΫτϧղੳʹؔ࿈͢Δߨٛʹ ͓͍ͯ΋໾ཱͯ͹޾͍Ͱ͋Δɽ 2 ਺ֶతجૅ طʹ࣮ݧIΛཤम͓ͯ͠Γʢ୯Ґͷऔಘ͸อূ ͷݶΓͰ͸ͳ͍ʣɼϑʔϦΤղੳʹ͍ͭͯ࿩Λฉ ͍ͨ͜ͱ͸͋Δֶੜ͕ର৅Ͱ͋ΔɽͦͷͨΊɼج ૅ͔Βͷઆ໌͸লུ͠ɼ࣮ݧ಺Ͱ༻͍ΔͰ͋Ζ͏ ҎԼͷ߲໨ʹ͍ͭͯԋश໰୊ͱͯ͠՝ͨ͠ɽ 1) प೾਺_f₁ ͷਖ਼ݭ೾_{f(t) = A cos 2πf}₁_tΛෳ ૉϑʔϦΤڃ਺ల։͠ɼಘΒΕͨεϖΫτϧ ΛਤࣔͤΑɽ 2) पظ _T ͷۣܗ೾͸ࣜ͘ ͚ ͍ (1)Ͱද͞ΕΔɽ·ͣ ࣜ(1)ͷ࣌ؒ೾ܗΛ࣮ݧϊʔτʹඳ͖ɼଓ͍ ۣͯܗ೾ͷϑʔϦΤ܎਺_a_m_{, b}_mΛಋग़ͤΑɽ f(t) = 1, |t| ≤ τ/2, 0, T/2 > |t| > τ/2. (1) 1)͸ඪຊԽͷఆཧʹ͓͚Δ৘ใ৴߸_s(t)Λ૝ ఆ͍ͯ͠Δɽ͜ͷෳૉϑʔϦΤ܎਺ಋग़ʹ͓͍ͯɼ ΦΠϥʔͷؔ܎ࣜͱࡾ֯ؔ਺ͷ௚ަੑΛ༻͍Δɽ ͢ͳΘͪҎԼͷల։Ͱ͋Δɽ cm = _T1 _T/2 −T/2f(t) · exp [−2πimf1t] dt

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= 1 T

_T/2

−T/2A cos 2πf1t · exp [−2πimf1t] dt

= A T _T/2 −T/2 cos 2πf₁t {cos[2πmf₁t] −i sin[2πmf1t]} dt = A T _T/2 −T/2cos[2πf1t] cos[2πmf1t]dt −iA_T _T/2 −T/2 cos[2πf₁t] sin[2πmf₁t]dt. ͦͯ͠ڏ਺෦෼ʢୈೋ߲ʣΛߟ͑Δͱࡾ֯ؔ਺ͷ ௚ަੑ͔Βશͯͷ_mʹ͍ͭͯθϩͰ͋Δɽ࣍ʹ ࣮਺෦෼ʢୈҰ߲ʣΛߟ͑Δͱ_{m = ±1}Ҏ֎͸ࡾ ֯ؔ਺ͷ௚ަੑ͔ΒθϩʹͳΔɽ c±1 = A_T _T/2 −T/2cos[2πf1t] cos[2πf1t]dt = A T _T/2 −T/2 cos[2· 2πf₁t] + cos[0] 2 dt = A 2T _T/2 −T/2cos[4πf1t]dt + A 2T _T/2 −T/21· dt = A 2T 1 4πf₁ sin[4πf1t] _T/2 −T/2 + A 2T [t] T/2 −T/2 = A 2T · 1 4πf₁ sin 4πf₁T 2 − sin 4πf₁−T 2 + A 2T T 2 − −T 2 = A 8πT f₁ {sin[2πT f1] + sin[2πT f1]} + A 2T · T. f1 = 1/T Ͱ͋Δ͜ͱɼsin 2π = 0Ͱ͋Δ͜ͱΛ ༻͍ΔͱୈҰ߲͸θϩͱͳΓɼ࠷ऴతʹҎԼͷ݁ ՌΛಘΔɽ c±1= A₂. (2) ΑͬͯɼෳૉϑʔϦΤڃ਺ల։͸࣍ࣜͱͳΔɽ f(t) = A 2 exp[2πi(−f1)t] + A 2 exp[2πif1t]. (3) ͜ΕΛਤࣔ͢Δͱਤ1ʹࣔ͢εϖΫτϧΛಘΔɽ ͔͠͠ɼΦΠϥʔͷؔ܎ࣜΛ͍֮͑ͯΔֶੜ͸ 3ׂ΄Ͳɼ௚ަੑ͸஌͍ͬͯͯ΋࢖͍͜ͳͤΔֶ ੜ͸2ׂ΄Ͳʹݮগ͢Δɽͦͷ݁Ռɼ΄ͱΜͲશ ͯͷաఔΛهड़ͯ͠ݟͤΔඞཁ͕༗ͬͨɽ஌ࣝͷ F(f) f -f ₁ f 1 0 A/2 ਤ1: A cos 2πf₁tͷप೾਺εϖΫτϧ ఆணΛਤΔʹ͸܁Γฦ͕͠ඞཁͰ͋ΔͱݴΘΕΔ ͕ɼݱঢ়ΑΓ΋ߋʹසൟʹࡾ֯ؔ਺ͳͲͷܭࢉΛ ܁Γฦ͢ඞཁ͕͋Δͱߟ͑Δɽ ࣍ʹ2)͸PCM৴߸͕઎༗͢ΔଳҬ෯ʹ͍ͭ ͯཧղ͢ΔͨΊʹඞཁͳ஌ࣝͰ͋Δɽ·ͣ͸࣮ݧ Iͱಉ༷ʹࣜ(1)ͷઈର஋Λ֎͢͜ͱ͕ࠔ೉ͳֶ ੜ͕4ׂ΄Ͳډͨɽͦͷઆ໌ʹج͖ͮ࡞ਤ͠ɼਤ 2ΛಘΔɽͦͯ͜͠ͷਤ2Λݩʹੵ෼۠ؒΛ෼ׂ ͠ɼϑʔϦΤ܎਺Λಋग़͢Δɽ f(t) t T/2 -T/2 -τ/2 τ/2 0 1 ਤ2: ۣܗ೾ͷ࣌ؒ೾ܗ am = _T2 _T/2 −T/2f(t) · cos[2π(mf1)t]dt = 2 T _−τ/2 −T/2 0· cos[2π(mf1)t]dt +2 T _τ/2 −τ/21· cos[2π(mf1)t]dt +2 T _T/2 τ/2 0· cos[2π(mf1)t]dt = 2 T _τ/2 −τ/2 cos[2π(mf₁)t]dt = 2 T 1 2π(mf₁)sin[2π(mf1)t] _τ/2 −τ/2

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= 2 T · 2π(mf1) sin 2π(mf₁)τ 2 − sin 2π(mf₁)· −τ 2 = 1 πmT f1 {sin [πmf1τ] + sin [πmf1τ]} = 2 sin[πmf1τ] πmT f1 . f1= 1/Tͷؔ܎Λ༻͍Δͱ࠷ऴతʹ࣍ࣜΛಘΔɽ am= 2 sin[πm(τ/T )]_πm . (4) ·ͨɼ_a₀= 2τ/T΋ఆٛʹ͕ͨͬͯ͠ܭࢉ͢Ε͹ ಘΒΕΔɽ ࣍ʹ_b_mΛಋग़͢Δɽ bm = _T2 _T/2 −T/2f(t) · sin[2π(mf1 )t]dt = 2 T _−τ/2 −T/2 0· sin[2π(mf1)t]dt +2 T _τ/2 −τ/21· sin[2π(mf1)t]dt +2 T _T/2 τ/2 0· sin[2π(mf₁)t]dt = 2 T _τ/2 −τ/2sin[2π(mf1)t]dt = 0. (5) ࠷ޙ͸sinx͕حؔ਺Ͱ͋Δੑ࣭Λ࢖ͬͨɽҎ্ Λ·ͱΊΔͱ࣍ͷ݁ՌΛಘΔɽ am= 2 sin[πm(τ/T )]_πm , bm= 0. (6) ࣮ࡍͷܭࢉʹ͓͍ͯ_b_m͕θϩʹ੒Βͳֶ͍ੜ ͕3ׂ΄ͲݟΒΕͨɽ΍͸Γsinxͱcosxͷੵ෼ ͷؔ܎ʹೃછΊ͍ͯͳֶ͍ੜ͕গͳ͔ΒͣډΔɽ 3 ඪຊԽͷఆཧ ඪຊԽͷఆཧΛจষͰॻ͚͹ʰप೾਺_f₁[Hz]Ҏ ্ͷप೾਺੒෼Λ࣋ͨͳ͍Α͏ʹଳҬ੍ݶ͞Εͨ ৴߸͸ɼ 1 2f1[s]ΑΓ΋খִ͍ؒ͞ͷ౳ִؒඪຊ஋ ʹΑͬͯҰҙతʹܾఆͰ͖ΔʱͰ͋Δɽ͢ͳΘͪ ૹΓ͍ͨ৘ใ৴߸_s(t)ͷ࠷ߴप೾਺͕_f₁[Hz]ͷ ৔߹ɼͦͷ2ഒҎ্ͷඪຊԽप೾਺_f_s[Hz]Ͱඪຊ Խ͢Ε͹৘ใ৴߸_s(t)͸׬શʹ࠶ݱͰ͖Δ͜ͱʹ ͳΔɽ ҎԼͰ͸Ξφϩά৴߸Λσδλϧ৴߸ʹม׵ ͢Δաఔʹ͓͍ͯଘࡏ͢ΔʮඪຊԽग़ྗʯͱݺ͹ ΕΔ৴߸ʢਤ3ʣͷ࣌ؒ೾ܗͱप೾਺εϖΫτϧ Λղੳ͠ɼඪຊԽͷఆཧ͕ҙຯ͢Δͱ͜ΖΛཧղ ͢Δɽ f sam (t) t ਤ3: ඪຊԽग़ྗ_f_sam(t)ͷྫ 3.1 ඪຊԽग़ྗͷ࣌ؒ೾ܗ ४උͱͯ͠Πϯύϧε৴߸_δ(t)Λཧղ͓ͯ͠ ͘ɽ͜Ε͸_{t = 0}Ͱߴ͕͞ແݶେɼ෯͕θϩɼ໘ ੵ͕1ͷؔ਺Ͱ͋Δɽ͢ͳΘͪ_{t = 0}Ҏ֎Ͱؔ਺ ͷ஋͸0ͱͳΔɽ͜ͷΠϯύϧε৴߸͕पظ_T_sͰ ܁Γฦ͢৴߸͕Πϯύϧεྻ৴߸_δ_T_s(t)Ͱ͋Δɽ δTs(t) = ∞ n=−∞ δ(t − nTs). (7) ԋशI ͜ͷΠϯύϧεྻ৴߸_δ_T_s(t)Λ্࣌ؒ࣠Ͱਤࣔ ͤΑɽ ͜ͷԋशʹؔͯ͠͸Ҿֻ͔ͬΔֶੜ͸গͳ͔ͬ ͨɽ࣍ͷਤ4ΛಘΔɽ δ (t) t T -2T -T 0 2T Ts s s s s ਤ4: Πϯύϧεྻ৴߸_δ_T_s(t) ࣍ʹ৘ใ৴߸_s(t)ʹΠϯύϧεྻ৴߸_δ_T_s(t)Λ ֻ͚߹ΘͤͯඪຊԽग़ྗ_f_sam(t)ΛಘΔɽ fsam(t) = s(t) · δTs(t). (8)

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͢ͳΘͪਤ3͕ಘΒΕΔ͜ͱʹͳΔɽ 3.2 ඪຊԽग़ྗͷप೾਺εϖΫτϧ ଓ͍ͯ͜ͷ৴߸_f_sam(t)ͷϑʔϦΤม׵Λߦ͍ɼ प೾਺εϖΫτϧ_F_sam(f)Λಋग़͢Δɽ৘ใ৴߸ s(t)ͷϑʔϦΤม׵Λ_S(f)Ͱද͠ɼΠϯύϧε ྻ৴߸_δ_T_s(t)ͷϑʔϦΤม׵͕2π_T s · δfs(f)Ͱ͋Δ ͜ͱΛ༻͍Δͱ࣍ࣜΛಘΔɽ fsam(t) ⇒ F_sam(f) = 1 2π S(f) ∗ 2_Tπ sδfs(f) = 1 Ts[S(f) ∗ δfs(f)] . (9) ͜͜Ͱه߸ʮ_∗ʯ͸৞ΈࠐΈੵ෼Λද͓ͯ͠ΓɼҰ ൠʹؔ਺_F₁(f)ͱ_F₂(f)ͷ৞ΈࠐΈੵ෼͸࣍ࣜͰ ఆٛ͞ΕΔɽ F1(f) ∗ F2(f) = _∞ −∞F1(f − ξ) · F2(ξ)dξ. (10) ·ͨɼؔ਺_δ_f_s(f)͸࣍ࣜͰද͞ΕΔप೾਺্࣠ͷ Πϯύϧεྻؔ਺Λҙຯ͢Δɽ δfs(f) = ∞ n=−∞ δ(f − nfs). (11) ͜͜Ͱ_f_s= _T1 s ͕੒Γཱͭɽ ԋशII ͜ͷΠϯύϧεྻؔ਺_δ_f_s(f)Λप೾਺্࣠Ͱਤ ࣔͤΑɽ ͜ͷԋशʹؔͯ͠͸ԋशIͱಉ౳ͩͬͨͨΊ΄ ͱΜͲͷֶੜ͕ਖ਼ղʢਤ5ʣΛಘΒΕͨɽͨͩɼԣ ͕࣠_tͷֶੜ΋1ׂఔ౓ډͨɽ δ (f) f f −2fs −fs 0 s 2fs f_s ਤ 5: Πϯύϧεྻؔ਺_δ_f_s(f) ԋशIII ৘ใ৴߸͕_{s(t) = A cos 2πf}₁_tͰ͋ͬͨ৔߹ɼ fs > 2f1ΛԾఆͯࣜ͠(9)ʹࣔ͢प೾਺εϖΫ τϧ_F_sam(f)Λप೾਺্࣠ͰਤࣔͤΑɽͨͩ͠ ʰ৴߸͕ଘࡏ͢Δप೾਺ʱͷΈʹ஫໨͠ɼॎ࣠ͷ ஋·Ͱ͸ཁٻ͠ͳ͍ɽ ৘ใ৴߸_s(t)ͷεϖΫτϧ_S(f)͸طʹ2ষʹ ͓͍ͯਤ1ͱͯ͠ಘ͍ͯΔɽͦͯ͠Πϯύϧεྻ ؔ਺_δ_f_s(f)΋ਤ5ͱͯ͠ಘΒΕ͍ͯΔɽΑͬͯ͜ ͷԋशͰ͸྆ऀͷ৞ΈࠐΈੵ෼ͷ࣮ߦ͢Δ͚ͩͰ ͋Δɽͨͩ͠_S(f)Λ_F₁(f − ξ)ͱͯ͠දͨ͠৔ ߹ʹม਺_f͔Βม਺_ξʹมΘͬͯɼ͔ͭϚΠφε ͷූ߸͕෇͍͍ͯΔ͜ͱɼͦͯ͠֎෦ύϥϝʔλ ͱͳͬͨ_f͕มԽ͢Δ͜ͱʹΑͬͯԿ͕ى͖͍ͯ Δ͔Λཧղͤ͞Δඞཁ͕͋ΔɽΑͬͯҎԼͷΑ͏ ͳղઆΛࢼΈͨɽ 1) ؔ਺_{f(x) = ax + b}Λద౰ʹඳ͘ʢ_{a, b > 0} ΛԾఆɼਤ6ʣɽ f (x) x 0 b 1 a ਤ6: f(x) = ax + bͷάϥϑʢ_{a, b > 0}ΛԾఆʣ 2) ؔ਺_f(−x)Λඳ͘ɽ_{a > 0}ΛԾఆ͍ͯ͠Δͷ Ͱ_xΛ૿΍͢ͱ_f(−x)͸ݮগ͍ͯ͘͠ɽ͢ ͳΘͪɼ_y࣠ରশʹࠨӈΛೖΕସ͑ͨάϥϑ ͱͳΔʢਤ7ʣɽ 3) ͦͯؔ͠਺_{f(c − x) = f{−(x − c)}}Λඳ͘ ʢ_{c > 0}ΛԾఆʣɽྫ͑͹_{f(c − x) = b}ͱͳΔ ৚݅͸_{c − x = 0}ɼ͢ͳΘͪ_{x = c}ͷͱ͖Ͱ ͋Δɽ͢ͳΘͪάϥϑ͸_x࣠ɼਖ਼ʢӈʣͷํ ޲ʹ_c͚ͩҠಈ͢Δʢਤ8ʣɽ 4) ͜ΕʹΑΓؔ਺f(x)Λf(c − x)ͱͨ͠৔߹ ʹɼಘΒΕΔάϥϑ͕ʮ_y࣠ରশʹࠨӈΛೖ Εସ͑ɼ_cͷ෼͚ͩӈʹҠಈ͢Δʯ͜ͱΛཧ ղ͢Δɽ

(5)

f (−x) x 0 b 1 a ਤ7: f(−x)ͷάϥϑʢ_{a, b > 0}ΛԾఆʣ f (c-x) x 0 b c ਤ 8: f(c − x)ͷάϥϑʢ_{c > 0}ΛԾఆʣ Ҏ্ͷղઆΛͨ͠ޙɼ৘ใ৴߸ͷεϖΫτϧ_S(f) Λ_{S(f − ξ)}ͱͯ͠_ξͷؔ਺ͱͯ͠ද͢ʢਤ9ʣɽ 0 S(f - ξ) ξ -f₁ A/2Ts f₁ S(-ξ) f ਤ9: S(f − ξ)ͷάϥϑ ͦͯ͠Πϯύϧεྻؔ਺_δ_f_s(ξ)ͱ্Լʹฒ΂ͯ ඳ͖ɼ྆ऀͷΠϯύϧε͕ॏͳΔ_fͷ৔߹ʹͷΈ Fsam(f)͕஋Λ࣋ͭ͜ͱΛࣔ͢ʢਤ10ʣɽ ԋशIV ্هͷԋशIIIͰਤࣔͨ͠_F_sam(f)ʹ͓͍ͯɼ प೾਺_f₁Λ૿Ճ͍ͯ͘͠ͱ֤प೾਺εϖΫτϧ ͸ࠨӈͷͲͪΒʹಈ͔͘ߟ͑Αɽ_f₁ͷ૿Ճʹ൐ ͍_f_s = 2f₁Λܦͯ_f_s _{< 2f}₁ ͱͳΔͱप೾਺ε 0 F (f) f -f₁ f₁ A/2Ts sam f_s -f_s -2f_s 2f_s 3f_s f_s -f₁ f_s +f₁ 0 δ (ξ) ξ f fs -fs -2fs 2fs 3fs 0 S(f - ξ) ξ -f₁ A/2Ts f₁ S(-ξ) f s ਤ10: S(f − ξ), δ_f_s(ξ),ͦͯ͠_F_sam(f)ͷάϥϑ ϖΫτϧ_F_sam(f)ʹ͸ͲͷΑ͏ͳݱ৅͕ݟΒΕ Δ͔આ໌ͤΑɽ ݁Ռ͸ͦΕͧΕਤ11, 12, 13ͱͳΔɽ͢ͳΘ ͪਤ11ʹ͓͍ͯ͸_f₁ͱ_f_s_{− f}₁͕_f_s_/2ʹ޲͔ͬ ͖ͯۙͮɼਤ12Ͱ͸_f_s_/2 ʹ͓͍ͯ྆ऀ͕ॏͳ Δɽͦͯ͠ਤ13ɼ͢ͳΘͪ_f_s _{< 2f}₁ͷ৔߹ʹ͸ f1ͱfs− f1ͷҐஔ͕ೖΕସΘΓɼf1ͷप೾਺Λ ඪຊԽͯ͠΋_f_s_{− f}₁ͷप೾਺͕ಘΒΕΔͱ༧૝ ͞ΕΔɽ࣮ݧͰ͸ΦγϩείʔϓͷFFTػೳΛ ༻͍ͯεϖΫτϧͷೖΕସΘΓΛ֬ೝͨ͠ɽߋʹ εϐʔΧʔ͔Βฉ͑͜ΔԻ΋ඪຊԽप೾਺ͷ൒෼ Ͱ͋Δ4kHz·Ͱ্͕ͬͯߦͬͨޙɼ_f₁ͷ্ঢʹ ΋͔͔ΘΒͣԼ͕͍ͬͯ͘͜ͱ͕֬ೝͰ͖ͨɽఏ ग़͞Ε࣮ͨݧϨϙʔτ͔Βֶੜͷ಺༰ཧղΛਪ࡯ ͢ΔͱɼඪຊԽͷఆཧͷඞཁੑͱɼͦΕ͕ຬ଍͞ Εͳ͍৔߹ʹى͖Δݱ৅ͷཧղ͸ग़དྷ͍ͯͨͱݴ ͑Δɽ 4 PCM৴߸ͷ఻ૹଳҬ ΞφϩάมௐํࣜͰ͋ΔAM΍FMʹ͓͍ͯɼ ͦͷ఻ૹଳҬ͸৘ใ৴߸ͷ࠷ߴप೾਺Λ_f₁Ͱද ͨ͠৔߹ʹͦΕͧΕ_BW_AM = 2f₁, BW_FM =

(6)

0 F (f) f -f₁ f₁ sam f_s f_s -f₁ f_s +f₁ ਤ11: f₁Λ૿Ճͤͨ͞৔߹ͷ_F_sam(f)ɽ 0 F (f) f -f₁ f₁ sam f_s f_s -f₁ fs +f1 ਤ12: f_s= 2f₁৔߹ͷ_F_sam(f)ɽ 0 F (f) f f₁ sam f_s f_s -f₁ 2f_s -f₁ -f_s +f₁ ਤ13: f_s< 2f₁৔߹ͷ_F_sam(f)ɽ 2(m_f+ 1)f₁ͱͳΔ͜ͱ͕஌ΒΕ͍ͯΔɽͨͩ͠ mf ͸FMʹ͓͚Δมௐࢦ਺Λද͢ɽ͢ͳΘͪɼ ఻ૹଳҬ͕৘ใ৴߸ͷ࠷େप೾਺Ͱܾఆ͞ΕΔɽ ҰํͰप೾਺ར༻ޮ཰ͷ఺͔Βݴ͑͹ɼ఻ૹଳҬ ͸ڱ͚Ε͹ڱ͍΄Ͳޮ཰͕ྑ͘ͳΔɽͦ͜Ͱσδ λϧ௨৴ͷ୅දͰ͋ΔPCMʹ͓͍ͯඞཁͱ͞Ε Δ఻ૹଳҬΛಋग़͠ɼΞφϩά௨৴ͷͦΕͱൺֱ ͢Δ͜ͱʹΑͬͯप೾਺ར༻ޮ཰Λٞ࿦͢Δɽ ҎԼͰ͸ɼ·ͣPCMͷ఻ૹଳҬΛಋग़͠ɼͦ Εʹଓ͍֤ͯมௐํࣜؒͷൺֱΛߦ͏ɽ 4.1 PCM఻ૹଳҬͷಋग़ σδλϧ৴߸ͷූ߸पظͱͯ͠_TΛԾఆ͢Δͱɼ 8bitූ߸Խͷ৔߹ʹ୯Ұύϧεͷ෯_τ͸_{τ = T/8} Ͱද͞ΕΔɽ͢ͳΘͪූ߸ʮ00000001ʯͷ৔߹ͷ εϖΫτϧ͸2ষͷࣜ(6)ʹ͓͍ͯ_{τ = T/8}Λ ୅ೖ͢Ε͹ಘΒΕΔɽ·ͨɼ͜ͷͱ͖ͷجຊप೾ ਺Λ_f₀ = 1/T Ͱද͢ɽಛʹPCMʹ͓͍ͯ͸ූ ߸पظ͸ඪຊԽपظʹ౳͘͠ɼ݁Ռͱͯ͠ඪຊԽ प೾਺ͱجຊप೾਺͸Ұக͢Δʢ_f₀ = f_sʣɽ΋ ͪΖΜɼҎԼʹࣔ͢Α͏ʹූ߸ࣗମʹ܁Γฦ͕͠ ༗Δ৔߹ʹ͸ූ߸पظ͸ҰఆͰ͋ͬͯ΋جຊप೾ ਺͸มԽ͠ɼඪຊԽप೾਺ͱͷҰக͸ࣦΘΕΔɽ ͢ͳΘͪɼʮ00010001ʯͷ৔߹͸_T=T/2ͱͳͬ ͯجຊप೾਺͸_f₀ = 1/T = 2f₀ͱͳΔɽͦͯ͠ τ/T _{= 1}_/4_{ΛಘΔɽҰํͰʮ}_00000011_ʯ_ʢ_4bit_ͷ ʮ0001ʯʹ૬౰ʣͷ৔߹ʹ΋_τ =T/4Ͱ͋Δ͕ج ຊप೾਺͸_f₀Ͱ͋Δɽ ԋशVI 8bitූ߸ʹ͓͍ͯɼ্هͷූ߸ΛؚΊɼͦΕҎ ֎ʹʮ00001111ʯͱʮ01010101ʯͷύϫʔεϖ ΫτϧΛཧ࿦͔ΒٻΊɼͦΕͧΕΛਤࣔͤΑɽ ͜ͷ໰୊ʹؔͯ͠͸_τ ͱ_T ͷ૊߹ͤͰϑʔϦ Τ܎਺͕มԽ͠ɼ͔ͭූ߸ͷपظੑʹΑͬͯجຊ प೾਺΋มԽ͢ΔͨΊɼຊ౰ʹཧղ͍ͯ͠Δֶੜ ͸1ׂఔ౓Ͱ͋ͬͨɽ ͜ͷ࣮ݧͷઆ໌ͱͯ͠ɼ·ͣ͸ϑʔϦΤ܎਺ͷ ෼ྨΛߦ͏ɽ্هͷঢ়گ͔ΒϑʔϦΤ܎਺_a_mΛ ҎԼͷΑ͏ʹఆٛ͢Δɽ • 00000001ɿ_{τ/T = 1/8}ͷ৔߹ am= 2 sin(_πmπm/8) (12) • 00000011ɿ_τ = 2τ ͢ͳΘͪ_τ_{/T = 1/4}ͷ ৔߹ a m= 2 sin(_πmπm/4) (13) ͜Ε͸_T = T/2͢ͳΘͪ_τ/T = 1/4ͷ৔ ߹(00010001)΋ಉ༷Ͱ͋Δɽ • 00001111ɿ_τ = 4τ͢ͳΘͪ_τ_{/T = 1/2}ͷ ৔߹ a m= 2 sin(πm/2) πm (14) ͜Ε͸_T=T/4͢ͳΘͪ_τ/T= 1/2ͷ৔ ߹(01010101)ɼ͞Βʹτ = 2τ, T=T/2ͷ ৔߹(00110011)΋ಉ༷Ͱ͋Δɽ

(7)

·ͨ4Ϗοτ఻ૹͷ৔߹ɼ8Ϗοτ఻ૹͷූ߸ͱ ͸ҎԼͷؔ܎͕੒Γཱͭɽ (0001)₄ ↔ (00000011)₈ (15) (0011)₄ ↔ (00001111)₈ (16) (0101)₄ ↔ (00110011)₈ (17) Ҏ্ͷٞ࿦ʹج͖ͮ۩ମతʹϑʔϦΤ܎਺ΛٻΊ Δͱද1ΛಘΔɽ ද1: ֤৚݅ԼͰͷϑʔϦΤ܎਺ m 0 1 2 3

am 1₄ 2 sin[π/8]_π 2 sin[π/4]_2π 2 sin[3π/8]_3π a

m 12 2 sin[π/4]π 2 sin[π/2]2π 2 sin[3π/4]3π a

m 1 2 sin[π/2]π 0 2 sin[3π/2]3π

4 5 6 7

2 sin[π/2]

4π 2 sin[5π/8]5π 2 sin[3π/4]6π 2 sin[7π/8]7π 0 2 sin[5π/4]_5π 2 sin[3π/2]_6π 2 sin[7π/4]_7π

0 2 sin[5π/2]_5π 0 2 sin[7π/2]_7π 8 9 · · · 0 2 sin[9π/8]_9π · · · 0 2 sin[9π/4]_9π · · · 0 2 sin[9π/2]_9π · · · ͨͩ͠ɼݫີʹ͸ࣜ(6)ͰಘΒΕͨϑʔϦΤ܎ ਺͸࣌ؒ೾ܗΛۮؔ਺ͱͯ͠ѻ͓ͬͯΓɼ্هͷ ූ߸ྻ͕ۮؔ਺Ͱ͋Δอূ͸ͳ͍ɽ͔͠͠ύϫʔ εϖΫτϧͷΈʹ஫໨͢Ε͹ɼ݁Ռ͸Ұக͢Δͨ Ίৄࡉͷઆ໌Λলུ͠ɼΦγϩείʔϓͷFFT ػೳͰ؍ଌ͞ΕΔ݁ՌͱͷҰகͷΈʹ஫໨࣮ͯ͠ ݧΛਐΊͨɽ ࣍ʹجຊप೾਺_f₀Λߟ͑ΔɽPCM࣮ݧ૷ஔʹ ͓͍ͯඪຊԽप೾਺͸_f_s=8[kHz]ͰݻఆͰ͋Δɽ Αͬͯූ߸ʹ܁Γฦ͕͠ແ͍৔߹͸جຊप೾਺ f0 = 8[kHz]Ͱ͋Δɽද1ͷϑʔϦΤ܎਺Λ૊Έ ߹ΘͤΔͱ(000000001), (00000011), (00001111) ͷप೾਺εϖΫτϧΛਤ14͔Βਤ16 ʹࣔ͢ɽ ࣍ʹ2ճͷ܁Γฦ͕͠༗Δ৔߹͸_f₀ = 16[kHz] ͱͳΓɼද1ͱ߹Θͤͯ(00010001), (00110011) ͷप೾਺εϖΫτϧͱͯ͠ਤ17 ͱਤ18ΛಘΔɽ 0 8 16 24 32 40 48 56 64 72 0.2 0. 0.2 0.4 0.6 [kHz]

a

m ਤ14: (00000001)ͷप೾਺εϖΫτϧ 0 8 16 24 32 40 48 56 64 72 0.2 0. 0.2 0.4 0.6 [kHz]

a

m ਤ15: (00000011)ͷप೾਺εϖΫτϧ 0 8 16 24 32 40 48 56 64 72 0.2 0. 0.2 0.4 0.6 [kHz]

a

m ਤ16: (00001111)ͷप೾਺εϖΫτϧ ͦͯ͠ 4 ճͷ܁Γฦ͕͠༗Δ৔߹͸ _f₀ = 32[kHz]ͱͳΓɼ(01010101)ͷप೾਺εϖΫτϧ ͸ਤ19ͱͯ͠ಘΒΕΔɽ ͜ΕΑΓʮ00000001ʯͷ৔߹ɼ_{m = 8}ͰॳΊ ͯ_a_m = 0ͱͳΔɽ͢ͳΘͪ఻ૹଳҬ͸64kHz Ͱ͋Δɽ࣍ʹʮ00000011ʯͷ৔߹͸m = 4Ͱॳ Ίͯ_a_m = 0ͱͳΓɼ32kHzͷଳҬ͕ඞཁͱͳ Δɽ͞Βʹʮ00001111ʯͷ৔߹͸_{m = 2}ͰॳΊ ͯ_a_m = 0ͱͳΓɼ16kHzͷଳҬ͕ඞཁͱͳΔɽ ٯʹ8Ϗοτதʹ2ճ܁Γฦ͢ූ߸ͷʹ͓͍ͯ͸ɼ ʮ00010001ʯͷ৔߹ʹ64kHzɼʮ00110011ʯͷ৔߹ ʹ͸32kHzͷଳҬ͕ඞཁͰ͋Δɽͦͯ͠8Ϗοτ தʹ4ճ܁Γฦ͢ූ߸ɼ͢ͳΘͪʮ01010101ʯͷ ৔߹͸΍͸Γ64kHzͷ఻ૹଳҬ͕ඞཁͱݴ͑Δɽ

(8)

0 8 16 24 32 40 48 56 64 72 0.2 0. 0.2 0.4 0.6 [kHz]

a

m ਤ17: (00010001)ͷप೾਺εϖΫτϧ 0 8 16 24 32 40 48 56 64 72 0.2 0. 0.2 0.4 0.6 [kHz]

a

m ਤ18: (001110011)ͷप೾਺εϖΫτϧ ্ͨͩ͠هͷଳҬ͸ยଆଳҬ͚ͩΛߟ͓͑ͯΓɼ ͜ͷσδλϧ৴߸Λ༻͍ͯৼ෯ภҠมௐʢASKʣ Λߦͨ৔߹ɼͦͷଳҬ෯͸2ഒʹͳΔɽ ࣮ࡍͷ࣮ݧʹ͓͍ͯΦγϩείʔϓͷFFTػ ೳʹΑΓύϫʔεϖΫτϧΛଌఆͨ͠ɽ४උͨ͠ ූ߸ʹԠͯ͡ཧ࿦Ͳ͓Γͷप೾਺ʹεϖΫτϧ͕ ཱͭ͜ͱΛ֬ೝ͕ͨ͠ɼFFTͷεέʔϧͷௐ੔ ʹखؒऔΔֶੜ͕3ׂఔ౓ډͨɽͦͯ͠ཧ࿦ͱ࣮ ݧͷ݁Ռ͔Βɼಉ͡ඪຊԽप೾਺Ͱ͋ͬͯ΋ྔࢠ ԽϏοτ਺͕গͳ͍ͱ఻ૹଳҬ͕ڱ͘ͳΓɼٯʹ ࠷খύϧε෯͕ಉ͡Ͱ͋Ε͹ූ߸ͷ܁Γฦ͠ճ਺ ͕มԽͯ͠΋ඞཁͳ఻ૹଳҬ͸มԽ͠ͳ͍ͱ͍͏ ࣄ͕1ׂఔ౓ͷֶੜʹཧղ͞Εͨͱࢥ͏ɽ 0 8 16 24 32 40 48 56 64 72 0.2 0. 0.2 0.4 0.6 [kHz]

a

m ਤ19: (01010101)ͷप೾਺εϖΫτϧ 4.2 Ξφϩάมௐͱͷ఻ૹଳҬͷൺֱ ઌʹࣔͨ͠Α͏ʹɼ৘ใ৴߸ͷ࠷ߴप೾਺ Λ_f₁ Ͱදͨ͠৔߹ʹৼ෯มௐʢAMʣͱप೾਺ มௐʢFMʣͦΕͧΕͷ఻ૹଳҬ͸ _BW_AM = 2f₁, BW_FM = 2(m_f + 1)f₁ ͱͳΔɽҰํͰύϧ εූ߸มௐʢPCMʣʹ͓͍ͯ͸ɼඪຊԽͷఆཧ ΑΓඪຊԽप೾਺_f_s= 2f₁Λཁ੥͞Εɼ_nϏοτ ྔࢠԽΛߦͬͨ৔߹ʹ͸_BW_PCM = 2· 2f₁· nͱ ͳΔɽి࿩ϨϕϧͷԻ࣭Ͱ͋Δ_{n = 8}ͷ৔߹ʹ͓ ͍ͯ΋AMͷ16ഒɼFMͷ4ഒʢ_m_f = 4ͱ͠ ͨʣͷ఻ૹଳҬ͕ඞཁͱͳΔɽ͢ͳΘͪɼσδλ ϧมௐͰ͋ΔPCMΛ༻͍Δํ͕఻ૹଳҬ͸޿͘ ͳΔ͜ͱ͕൑Δɽ ͔͠͠ͳ͕Βσδλϧσʔλ͸ࡶԻʹڧ͘ɼߋ ʹѹॖٕज़΍ɼଟ஋มௐͷར༻ʹΑΓ఻ૹଳҬΛ ڱ͘͢Δ͜ͱ͕Ͱ͖Δɽͦͷ݁Ռɼ࣮༻Խ͞Εͯ ͍Δσδλϧ௨৴͸Ξφϩά௨৴ΑΓ΋प೾਺ར ༻ޮ཰ͷߴ͍γεςϜͱ੒͍ͬͯΔɽ 5 ·ͱΊ ιϑτ΢ΣΞαΠΤϯε࣮ݧIIͰߦ͖ͬͯͨ PCM࣮ݧʹ͓͚Δमಘ໨ඪͱͯ͠ʮඪຊԽͷఆ ཧʯͱʮ఻ૹଳҬʯΛऔΓ্͛ͨɽͦ͜Ͱ͸ϑʔ ϦΤղੳΛ༻͍ͯݱ৅Λཧ࿦తʹ༧ଌ͠ɼ࣮ࡍʹ ଌఆɼ؍࡯͢Δ͜ͱͰཧ࿦ͷཪ෇͚ΛऔΓݱ৅ཧ ղΛਐΊͨɽ֤߲ͦͯ͠໨Ͱੜ͡Δ໰୊఺ͱͦΕ ʹର͢Δղܾࡦʹ͍ͭͯ۩ମతʹࣔͨ͠ɽ͜ΕΒ ͷ಺༰͕কདྷͷֶੜ࣮ݧɼ΋͘͠͸ϑʔϦΤղੳ ͷߨٛʹ໾ཱͭͱ޾͍Ͱ͋Δɽ ࢀߟจݙ [1] ౔ࢁ຀෉ɼफ૾ษɼࢁ࡚ߒҰɼখ઒ߊɼେ࡚ ਖ਼༤ɼιϑτ΢ΣΞαΠΤϯε࣮ݧ̞Iࢦಋ ॻɼୈ̍൛ɼ2009೥9݄11೔ [2] ౔ࢁ຀෉ɼफ૾ษɼࢁ࡚ߒҰɼখ઒ߊɼେ࡚ ਖ਼༤ɼιϑτ΢ΣΞαΠΤϯε࣮ݧ̞Iࢦಋ ॻɼ₂₀₁₅೥൛ɼ2015೥4݄3೔ ̎̌̍̓೥݄̏̒೔ݪߘड෇ Received, March 1, 2017 2017年3月6日原稿受付， 2017年4月13日採録決定 Received, March 6, 2017, accepted, April 13, 2017