ᐔᚑ 14 ᐕᐲቇ჻ቇ૏⺰ᢥ

㜞ᄙ㊀ൻㅢାߩߚ߼ߩ
㓞ធ๟ᵄᢙ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼ
A Self-Interference Canceller
for a High Density Frequency Division Multiplexing

1030278

㜞ᯅ ᘕม

ᜰዉᢎຬ

ỳ᧛ ᣽ೣ

2003 ᐕ 2 ᦬ 24 ᣣ㜞⍮Ꮏ⑼ᄢቇ ᖱႎࠪࠬ࠹ࡓᎿቇ⑼ⷐᣦ㜞ᄙ㊀ൻㅢାߩߚ߼ߩ
㓞ធ๟ᵄᢙ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼ㜞ᯅ ᘕม៤Ꮺ㔚⹤߿ή✢ LAN ߥߤߩ᥉෸ߦࠃࠅ㧘ࠗࡦ࠲࡯ࡀ࠶࠻ធ⛯⠪߿㜞ㅦࡑ࡞࠴ࡔ࠺ࠖࠕή
✢ㅢାߩ㔛ⷐߪჇടߒ⛯ߌߡ޿ࠆ㧚৻ᣇߢ㧘৻⥸ή✢ㅢାߢ೑↪ߢ߈ࠆ๟ᵄᢙ⾗Ḯߦߪ㒢ࠅ߇޽ࠆߚ߼㧘๟ᵄᢙ೑↪ല₸ߩ㜞޿ㅢାᣇᑼ߇᳞߼ࠄࠇࠆ㧚⃻࿷㧘๟ᵄᢙ೑↪ല₸ߩ㜞޿ㅢାᣇᑼߣߒߡ㧘⋥੤๟ᵄᢙಽഀᄙ㊀㧔OFDM㧕߇ᵈ⋡ࠍ㓸߼ߡ޿ࠆ㧚OFDM ߪ㧘㓞ធߔࠆ๟ᵄᢙߩାภ㧔એਅ㧘㓞ធାภ㧕ߦ⋥੤ᕈࠍᜬߚߖࠆߎ
ߣߦࠃࠅ㧘ାภࠬࡍࠢ࠻࡞ߩඨಽࠍ㊀ߨߡ㈩⟎ߔࠆߎߣࠍน⢻ߦߒߡ޿ࠆ㧚OFDM ࠃࠅ߽ାภࠬࡍࠢ࠻࡞ࠍ㊀ߨߚ႐ว㧘⋥੤ᕈ߇ᄬࠊࠇࠆߚ߼㧘㓞ធାภ߇ᐓᷤାภߣߥࠅㅢାߩᕈ
⢻߇ഠൻߔࠆ㧚ᧄ⺰ᢥߢߪ㧘ߎߩ㓞ធାภߩᓇ㗀ࠍシᷫߢ߈ࠆ㧘㓞ធ๟ᵄᢙ࠺࡯࠲ࠠࡖࡦ࠮
࡞ᣇᑼࠍឭ᩺ߒ㧘ࠪࡒࡘ࡟࡯࡚ࠪࡦߦࠃࠅߘߩᕈ⢻ࠍ⹏ଔߒߡ޿ࠆ㧚ឭ᩺ᣇᑼࠍណ↪ߔࠆߎߣߢ㧘․ߦ ∆f T㧔๟ᵄᢙ㈩⟎㑆㓒 × ࠪࡦࡏ࡞㐳㧕= 0.6 ߩߣ߈ߦߪ BER = 10−3 ߦ߅ߌࠆEb /N0 ߢ⚂ 5 dB㧘∆f T = 0.5 ߩߣ߈ߦߪ⚂ 3 dB ߩᡷༀ߇ᓧࠄࠇ㧘ឭ᩺ᣇᑼ߇๟ᵄᢙ೑↪
ല₸ߩะ਄ߦኻߒߡ᦭ലߢ޽ࠆߎߣ߇᣿ࠄ߆ߣߥߞߚ㧚ࠠ࡯ࡢ࡯࠼

OFDM㧘ᄙ㊀ൻ㧘๟ᵄᢙ೑↪ല₸㧘࠺࡯࠲ࠠࡖࡦ࠮࡞㧘エ್ቯ—i—Abstract
A Self-Interference Canceller
for a High Density Frequency Division Multiplexing
Shinji TAKAHASHI

Orthogonal frequency division multiplexing has been attracted recently because of
its high spectral eﬃciency and its easy conﬁguration of DFT (discrete Fourier transform)
based transceiver. OFDM makes it possible to overlap the half of signal spectra using
a certain distinct frequency for subcarriers. When we choose a narrower frequency
diﬀerence between two adjacent subcarriers than the OFDM, a large self-interference
appears. Then, communication quality will deteriorate.
This paper proposes a self-interference cancellation method for a higher spectral
density frequency division multiplexing. The proposed method ﬁrst estimates the amplitude and phase angle of adjacent frequency signal to the signal of interest, and thendemodulates the signal of interest after subtracting the replica of adjacent signal fromthe received signal.
As a result, it is shown that a high density FDM can be realized by using the
proposed canceller with an approximately twice spectral eﬃciency, with a slight extra
power.
key words

OFDM, Multiplexing, Spectral Eﬃciency, Interference Canceller, Soft

Decision— ii —

⋡ᰴ
╙1┨

߹߃߇߈

1

╙2┨

PSK

3

2.1

ᄌ⺞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.2

BPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.3

QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

OFDM

7

3.1

OFDM ߣߪ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.2

OFDM ߩේℂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.3

DFT ࠍ↪޿ߚ OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

╙3┨

╙4┨

㜞ᄙ㊀ൻㅢା

11

4.1

ⶄ⚛ࡌ࡯ࠬࡃࡦ࠼ାภ⴫⃻ߩዉ಴ . . . . . . . . . . . . . . . . . . . . . .

11

4.2

ㅍାᯏࡕ࠺࡞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

4.3

ฃାᯏࡕ࠺࡞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

4.4

㜞ᄙ㊀ൻㅢାߩᕈ⢻ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

4.4.1

BPSK ᣇᑼߦ߅ߌࠆᕈ⢻ . . . . . . . . . . . . . . . . . . . . . . .

15

4.4.2

QPSK ᣇᑼߦ߅ߌࠆᕈ⢻ . . . . . . . . . . . . . . . . . . . . . . .

17

╙5┨

㓞ធ๟ᵄᢙ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼ

20

5.1

ࠕ࡞ࠧ࡝࠭ࡓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

5.2

エ್ቯ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

5.3

ᕈ⢻⹏ଔ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

ࡆ࠶࠻⺋ࠅ₸․ᕈ . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

5.3.1

— iii —

⋡ᰴ

5.3.2
╙6┨

૶↪Ꮺၞ᏷ߦኻߔࠆ๟ᵄᢙ೑↪ല₸. . . . . . . . . . . . . . . . .

26
28

߹ߣ߼

⻢ㄉ

29

ෳ⠨ᢥ₂

30

ઃ㍳ A

ࡑ࠶࠴࠻ࡈࠖ࡞࠲ [4]

31

A.1

ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߣߪ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

A.2

ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߩේℂ . . . . . . . . . . . . . . . . . . . . . . . . . . .

32— iv —

࿑⋡ᰴ
2.1

BPSK ାภߩ⊒↢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

BPSK ᵄߩ଀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.3

QPSK ାภߩ⊒↢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.4

QPSK ାภߩ૏⋧⁁ᘒ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.5

QPSK ᵄߩ଀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.1

OFDM ߩ๟ᵄᢙ․ᕈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.2

QPSK ࠍ↪޿ߚ OFDM ㅍାᯏ . . . . . . . . . . . . . . . . . . . . . . . .

8

3.3

QPSK ࠍ↪޿ߚ OFDM ฃାᯏ . . . . . . . . . . . . . . . . . . . . . . . .

8

3.4

IDFT ࠍ↪޿ߚ OFDM ㅍାᯏ . . . . . . . . . . . . . . . . . . . . . . . .

9

3.5

DFT ࠍ↪޿ߚ OFDM ฃାᯏ . . . . . . . . . . . . . . . . . . . . . . . . .

10

4.1

࠳࠙ࡦࠦࡦࡃ࡯࠲ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

4.2

ㅍାᯏ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

4.3

ฃାᯏ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

4.4

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = 0㧕 . . . . . . . . . . . . . . . . . . . .

16

4.5

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = π/4㧕 . . . . . . . . . . . . . . . . . . .

16

4.6

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = π/2㧕 . . . . . . . . . . . . . . . . . . .

17

4.7

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = 0㧕 . . . . . . . . . . . . . . . . . . . .

18

4.8

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = π/4㧕 . . . . . . . . . . . . . . . . . . .

18

4.9

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = π/2㧕 . . . . . . . . . . . . . . . . . . .

19

5.1

࠺࡯࠲ࠠࡖࡦ࠮࡞ฃାᯏ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

5.2

エ್ቯᦛ✢ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

5.3

エ್ቯ࠺࡯࠲ࠠࡖࡦ࠮࡞ฃାᯏ . . . . . . . . . . . . . . . . . . . . . . . .

22

—v—

࿑⋡ᰴ

5.4

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔OFDM ߣ ∆f T = 0.8 ߩᲧセ㧕 . . . . . . . . . . . . .

24

5.5

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔OFDM ߣ ∆f T = 0.7 ߩᲧセ㧕 . . . . . . . . . . . . .

24

5.6

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔OFDM ߣ ∆f T = 0.6 ߩᲧセ㧕 . . . . . . . . . . . . .

25

5.7

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔OFDM ߣ ∆f T = 0.5 ߩᲧセ㧕 . . . . . . . . . . . . .

25

5.8

૶↪Ꮺၞ᏷ߦኻߔࠆ๟ᵄᢙ೑↪ല₸ . . . . . . . . . . . . . . . . . . . . . .

26

A.1 ࡄ࡞ࠬߩ଀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

A.2 ࡈࠖ࡞࠲ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

A.3 ࡈࠖ࡞࠲ߩᔕ╵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35— vi —

╙1┨߹߃߇߈៤Ꮺ㔚⹤߿ή✢ LAN㧔Local Area Network㧕ߥߤߩ᥉෸ߦࠃࠅ㧘ࠗࡦ࠲࡯ࡀ࠶࠻ធ⛯⠪߿㜞ㅦࡑ࡞࠴ࡔ࠺ࠖࠕή✢ㅢାߩ㔛ⷐߪჇടߒ⛯ߌߡ޿ࠆ㧚৻ᣇߢ㧘๟ᵄᢙ⾗Ḯߪ৻⷗ή㒢ߩࠃ߁ߦᕁ߃ࠆ߇㧘ታ㓙ߦߪ৻⥸ή✢ㅢାߢ೑↪ߢ߈ࠆ๟ᵄᢙ⾗Ḯߦߪ㒢ࠅ߇޽ࠆߚ߼㧘๟ᵄᢙ೑↪ല₸ߩ㜞޿ㅢାᣇᑼ߇᳞߼ࠄࠇࠆ㧚⃻࿷㧘ⶄᢙߩାภࠍ๟ᵄᢙゲ਄ߦ㈩⟎ߔࠆᣇᴺߣߒߡ๟ᵄᢙಽഀᄙ㊀㧔Frequency Division

Multiplexing: FDM㧕߇⍮ࠄࠇߡ޿ࠆ㧚߹ߚ㧘ᦨㄭߦߥߞߡ㧘๟ᵄᢙ೑↪ല₸ࠍ㜞߼ࠆߚ߼㧘⋥੤ᕈࠍ೑↪ߒߡ㓞ធߔࠆାภࠬࡍࠢ࠻࡞ߩඨಽࠍ㊀ߨߡ㈩⟎ߔࠆߎߣࠍน⢻ߦߔࠆ㧘⋥੤๟ᵄᢙಽഀᄙ㊀㧔Orthogonal FDM: OFDM㧕߇ᵈ⋡ࠍ㓸߼ߡ޿ࠆ [1]㧘[2]㧚OFDM ࠃࠅ߽ାภࠬࡍࠢ࠻࡞ࠍ㊀ߨߚ႐ว㧘⋥੤ᕈ߇ᄬࠊࠇࠆߚ߼㧘㓞ធߔࠆ๟ᵄᢙߩାภ㧔એਅ㧘㓞ធାภ㧕߇ᐓᷤାภߣߥࠅㅢାߩᕈ⢻߇ഠൻߔࠆ㧚ߘߎߢ㧘ᧄ⺰ᢥߢߪ㧘㓞ធାภߩᓇ㗀ࠍシᷫߢ߈ࠆ㓞ធ๟ᵄᢙ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼࠍឭ᩺ߒ㧘ࡆ࠶࠻⺋ࠅ₸ߩᡷༀߣ๟ᵄᢙ೑↪
ല₸ߩะ਄ߩⷰὐ߆ࠄ㧘ߘߩ᦭ലᕈࠍ᣿ࠄ߆ߦߔࠆ㧚એਅ㧘ᧄ⺰ᢥߩ᭴ᚑࠍㅀߴࠆ㧚߹ߕ㧘╙ 2 ┨㧘╙ 3 ┨ߢߪ㧘ឭ᩺ᣇᑼߦኻߔࠆ೨⍮⼂ߣߒߡ㧘૏⋧ࠪࡈ࠻ࠠ࡯ࠗࡦࠣ㧔Phase Shift Keying: PSK㧕ߣ OFDM ߦߟ޿ߡㅀߴࠆ㧚PSK
ߪ OFDM ߦ߽૶ࠊࠇࠆ৻⥸⊛ߥᄌ⺞ᣇᑼߢ㧘ឭ᩺ᣇᑼߦ߽↪޿ߡ޿ࠆ㧚߹ߚ㧘OFDM ߪឭ᩺ᣇᑼߣߩᕈ⢻Ყセߦ↪޿ࠆ㧚ᰴߦ㧘╙ 4 ┨ߢߪ㧘㜞ᄙ㊀ൻㅢାߩㅍฃାᯏࡕ࠺࡞ࠍ␜ߔ
ߣ౒ߦ㧘OFDM ࠃࠅ߽ାภࠬࡍࠢ࠻࡞ࠍ㊀ߨࠆߎߣߦࠃߞߡߤߩ⒟ᐲㅢାᕈ⢻߇ഠൻߔࠆ
ߩ߆ࠍ㧘ࡆ࠶࠻⺋ࠅ₸㧔Bit Error Rate: BER㧕․ᕈߢ⴫ߔ㧚ߘߒߡ㧘╙ 5 ┨ߢߪ㧘㓞ធ๟ᵄᢙ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼߩឭ᩺ࠍⴕ޿㧘╙ 4 ┨ߢ᳞߼ߚࡆ࠶࠻⺋ࠅ₸․ᕈߣᲧセߒߡㅢାᕈ⢻ߩഠൻࠍᛥ೙ߔࠆߎߣ߇ߢ߈ࠆߎߣࠍ␜ߔ㧚ᦨᓟߦ㧘╙ 6 ┨ߢߪ㧘ᧄ⺰ᢥߩ߹ߣ߼ߣߒ

—1—ߡ⚿⺰ࠍㅀߴࠆ㧚

—2—

╙2┨

PSK
2.1

ᄌ⺞

ᄌ⺞ߣߪ㧘ࡌ࡯ࠬࡃࡦ࠼ାภߩᄌൻߦวࠊߖߡ៝ㅍᵄࠍᄌൻߐߖ㧘ࡌ࡯ࠬࡃࡦ࠼ାภߩᖱ
ႎࠍ៝ㅍᵄߦਸ਼ߖࠆߎߣߢ޽ࠆ㧚ᄌ⺞ߦߪ៝ㅍᵄᄌ⺞ߩ߶߆ߦࡄ࡞ࠬᄌ⺞߽޽ࠆ߇㧘৻⥸ߦߚߛ‫ޟ‬ᄌ⺞‫߁޿ߣޠ‬႐วߪ៝ㅍᵄᄌ⺞ߩߎߣࠍᜰߔ㧚៝ㅍᵄᄌ⺞ߢߪ㧘៝ㅍᵄ๟ᵄᢙ fc ߩ៝ㅍᵄ cos(2πfc t) ߩࡄ࡜ࡔ࡯࠲ߩ৻ㇱࠍࡌ࡯ࠬࡃࡦ
࠼ାภߦኻᔕߐߖࠆ㧚ߎߩߣ߈ᄌ⺞ᵄߪᰴᑼߢ⴫ߐࠇࠆ㧚s(t) = A(t) cos(2πfc t + φ(t))(2.1)ߎߎߢ㧘ߤߩࡄ࡜ࡔ࡯࠲ࠍࡌ࡯ࠬࡃࡦ࠼ାภߦኻᔕߐߖࠆ߆ߦࠃߞߡ㧘ᝄ᏷ᄌ⺞㧘૏⋧ᄌ⺞㧘๟ᵄᢙᄌ⺞ߩ 3 ߟߦᄌ⺞ᣇᑼࠍಽ㘃ߔࠆߎߣ߇ߢ߈ࠆ [1]㧚ᧄ⺰ᢥߢߪ㧘៝ㅍᵄߩ૏⋧ φ(t) ࠍࡌ࡯ࠬࡃࡦ࠼ାภߦኻᔕߐߖࠆ૏⋧ᄌ⺞ᣇᑼߩ߁ߜ㧘࠺ࠖࠫ࠲࡞ᄌ⺞ᣇᑼߢ޽ࠆ૏⋧ࠪࡈ࠻ࠠ࡯ࠗࡦࠣ㧔Phase Shift Keying: PSK㧕ࠍ↪޿ࠆ㧚

PSK ߪᝄ᏷ A(t) ࠍ৻ቯ୯ A ߣߒߡ૏⋧ φ(t) ࠍࡌ࡯ࠬࡃࡦ࠼ାภߦኻᔕߐߖࠆ߽ߩߢ㧘ߘߩᄌ⺞ᵄߪᰴߩࠃ߁ߦ⴫ߐࠇࠆ㧚s(t) = A cos(2πfc t + φn (t))(2.2)ߚߛߒ㧘φn (t) ߪ n ⇟⋡ߩࠪࡦࡏ࡞ߦኻᔕߒߚ૏⋧ߢ㧘1 ࠪࡦࡏ࡞㐳ࠍ T ߣߔࠆߣ㧘t = nTߏߣߦ৻ቯߩ୯ࠍߣࠆ߽ߩߣߔࠆ㧚߹ߚ㧘ᑼ (2.2) ߪᰴߩࠃ߁ߦᄌᒻߔࠆߎߣ߇ߢ߈ࠆ㧚AA
s(t) = √ cos(φn (t)) cos(2πfc t) − √ sin(φn (t)) sin(2πfc t)
2
2

—3—

(2.3)

2.2 BPSKߎࠇߪ㧘੕޿ߦ૏⋧߇ π/2[rad] ߛߌ⇣ߥࠆ៝ㅍᵄࠍ sin ෸߮ cos ߢᄌ⺞ߒ㧘ߘࠇࠄࠍวᚑߔ
ࠆߎߣߢᄌ⺞ᵄࠍᓧࠆߎߣ߇ߢ߈ࠆߎߣࠍ⴫ߒߡ޿ࠆ㧚ߜߥߺߦ㧘PSK ߢߪ౉ജ࠺࡯࠲ log2 (M) ࡆ࠶࠻ࠍ 1 ߟߩ૏⋧ߦኻᔕߐߖࠆ㧚ߎߩ߁ߜM = 2 ߩ 2 ⋧ PSK㧔Binary PSK: BPSK㧕ߣ M = 4 ߩ 4 ⋧ PSK㧔Quadri PSK: QPSK㧕ߪᐢߊ৻⥸ߦ↪޿ࠄࠇߡ߅ࠅ㧘ᧄ⺰ᢥਛߢ߽૶↪ߒߡ޿ࠆߚ߼㧘એਅߦߘߩ․ᓽࠍㅀߴࠆ㧚

2.2

BPSK

BPSK ାภߪ૏⋧ φn (t) ߇ 0 ߹ߚߪ π ߣߥࠆ㧚ߎࠇࠍᑼ (2.3) ߦઍ౉ߔࠆߣ
s(t) = Ad(t) cos(2πfc t),(d(t) = ±1)

(2.4)

ߣߥࠆ㧚
ᑼ (2.4) ߆ࠄಽ߆ࠆࠃ߁ߦ㧘BPSK ାภߪ㧘៝ㅍᵄߣ౉ജ࠺࡯࠲ d(t) ߣߩਸ਼▚ߦࠃࠅ⊒↢ߐߖࠆߎߣ߇ߢ߈ࠆ㧚ߎߩ᭽ሶࠍ࿑ 2.1 ߦ␜ߔ㧚࿑ 2.1 BPSK ାภߩ⊒↢BPSK ᵄߩ଀ࠍ࿑ 2.2 ߦ␜ߔ㧚—4—

2.3 QPSK࿑ 2.2 BPSK ᵄߩ଀

2.3

QPSK

QPSK ାภߪ㧘±π/4㧘±3π/4 ߩ 4 ߟߩ૏⋧⁁ᘒࠍߣࠆ㧚ߎࠇࠍᑼ (2.3) ߦઍ౉ߔࠆߣ
AA
s(t) = √ d0 (t) cos(2πfc t) − √ d1 (t) sin(2πfc t),22(d0 (t), d1 (t) = ±1)

(2.5)

ߣߥࠆ㧚

QPSK ାภߩ⊒↢ᣇᴺࠍ࿑ 2.3 ߦ␜ߔ㧚QPSK ାภࠍ⊒↢ߐߖࠆߦߪ㧘߹ߕ౉ജ࠺࡯࠲
ࠍ 2 ࡆ࠶࠻߹ߣ߼ߡ⋥ਗ೉ᄌ឵ߔࠆ㧚ߘߩᓟ㧘ߘࠇߙࠇࠍ੕޿ߦ૏⋧߇ π/2[rad] ߛߌ⇣ߥࠆ៝ㅍᵄߢਸ਼▚ߒߡ㧘⿷ߒ޽ࠊߖࠆ㧚࿑ 2.3 QPSK ାภߩ⊒↢QPSK ାภߩ૏⋧ߣ౉ജ࠺࡯࠲ d0 㧘d1 ߩ⚵วߖߩኻᔕࠍ࿑ 2.4㧘QPSK ᵄߩ଀ࠍ࿑ 2.5
ߦ␜ߔ㧚

—5—

2.3 QPSK࿑ 2.4 QPSK ାภߩ૏⋧⁁ᘒ

࿑ 2.5 QPSK ᵄߩ଀

—6—

╙3┨

OFDM
3.1

OFDM ߣߪ

⋥੤๟ᵄᢙಽഀᄙ㊀㧔Orthogonal Frequency Division Multiplexing: OFDM㧕ߪ㧘ㅍା࠺࡯࠲ࠍⶄᢙߩ⋥੤ߒߚᦨዊ๟ᵄᢙ㈩⟎㑆㓒ߩ೽៝ㅍᵄߦࠃࠅᄙ㊀ൻߔࠆᄙ㊀ൻᣇᑼߢ޽ࠆ㧚೽៝ㅍᵄߩᄌ⺞ߦߪ㧘PSK ߿⋥੤ᝄ᏷ᄌ⺞㧔Quadrate Amplitude Modulation:

QAM㧕߇↪޿ࠄࠇࠆ㧚ߘߩ๟ᵄᢙ․ᕈߪ࿑ 3.1 ߩࠃ߁ߦߥࠆ㧚࿑ 3.1 OFDM ߩ๟ᵄᢙ․ᕈ

3.2

OFDM ߩේℂ

࿑ 3.2 ߪᄌ⺞ߦ QPSK ࠍ↪޿ߚ OFDM ㅍାᯏߢ޽ࠆ㧚౉ജ࠺࡯࠲ߪ⋥ਗ೉ᄌ឵ߐࠇ㧘2ࡆ࠶࠻߹ߣ߼ߡ 1 ࠪࡦࡏ࡞ߣߒߡ೽៝ㅍᵄߦഀࠅᒰߡࠄࠇࠆ㧚ߎߩࠪࡦࡏ࡞ߦࠃࠅ೽៝ㅍᵄ
ߪᄌ⺞ߐࠇࠆ㧚߹ߚ㧘೽៝ㅍᵄߪ㧘ߘࠇߙࠇࠪࡦࡏ࡞㐳 T ߩㅒᢙߕߟ๟ᵄᢙ߇ߕࠄߐࠇߡ
޿ࠆ㧚߹ߚ㧘࿑ 3.3 ߪ QPSK ࠍ↪޿ߚ OFDM ฃାᯏߢ޽ࠆ㧚ฃାାภߪฦ೽៝ㅍᵄ๟ᵄᢙߢQPSK ᄌ⺞ߐࠇ㧘ⶄ⚛ࡌ࡯ࠬࡃࡦ࠼ାภߣߥࠆ㧚—7—

3.2 OFDM ߩේℂ࿑ 3.2 QPSK ࠍ↪޿ߚ OFDM ㅍାᯏ࿑ 3.3 QPSK ࠍ↪޿ߚ OFDM ฃାᯏ—8—

3.3 DFT ࠍ↪޿ߚ OFDM

3.3

DFT ࠍ↪޿ߚ OFDM

࿑ 3.2㧘3.3 ߩࠃ߁ߥㅍฃାᯏߪ㧘೽៝ㅍᵄߩᢙ߇ᄙ޿႐วߦߪታ⃻߇㔍ߒ޿߇㧘㔌ᢔࡈ࡯࡝ࠛᄌ឵㧔Discrete Fourier Transform: DFT㧕ࠍ↪޿ࠆߎߣߢߎࠇࠍ⸃᳿ߢ߈ࠆ㧚n ⇟⋡ߩ೽៝ㅍᵄߩ๟ᵄᢙࠍ fn = (nf0 )㧘ࠪࡦࡏ࡞㐳ࠍ T = 1/f0 㧘೽៝ㅍᵄᢙࠍ N 㧘ࠨࡦࡊ࡝ࡦࠣ㑆㓒ࠍ T /N ߣߔࠆߣ㧘㔌ᢔㅒࡈ࡯࡝ࠛᄌ឵㧔Inverse DFT: IDFT㧕ߦࠃߞߡ↢ᚑߐࠇࠆାภߪᰴᑼߢ⴫ߐࠇࠆ㧚


s

p
N f0


=



N
−1


dn

n=0

p
N f0





2πpn
exp j
N


,

(p = 0, 1, . . . , N − 1)

(3.1)

ߎߎߢ㧘p ߪାภߩࠨࡦࡊ࡝ࡦࠣ⇟ภߢ޽ࠆ㧚ฃାᯏߢߪ DFT ࠍ↪޿ߡᓳ⺞ߔࠆ㧚ᑼߢ⴫ߔߣ


dˆ

p
N f0


=

N
−1 N
−1


i=0 n=0


d

p
N f0





2π(fn − fi )p
exp j
N


(3.2)

ߣߥࠆ㧚
࿑ 3.4 ߦ IDFT ࠍ↪޿ߚ OFDM ㅍାᯏ㧘࿑ 3.5 ߦ DFT ࠍ↪޿ߚ OFDM ฃାᯏࠍ␜ߔ㧚ߎࠇࠄߪ࿑ 3.2㧘3.3 ߩㅍฃାᯏߣ╬ଔߥ᭴ᚑߢ޽ࠆ㧚࿑ 3.4 IDFT ࠍ↪޿ߚ OFDM ㅍାᯏ—9—

3.3 DFT ࠍ↪޿ߚ OFDM

࿑ 3.5 DFT ࠍ↪޿ߚ OFDM ฃାᯏ— 10 —

╙4┨㜞ᄙ㊀ൻㅢା
4.1

ⶄ⚛ࡌ࡯ࠬࡃࡦ࠼ାภ⴫⃻ߩዉ಴߹ߕ㧘๟ᵄᢙߕࠇ ∆f ߩ޽ࠆ RF ାภߩⶄ⚛ࡌ࡯ࠬࡃࡦ࠼ାภ⴫⃻ߦߟ޿ߡ߹ߣ߼ࠆ㧚ฃାᯏߩၮḰ๟ᵄᢙ߆ࠄ ∆f ߛߌ๟ᵄᢙߩߕࠇߚฃାାภ g(t) ࠍg(t) ==√

2P b(t) cos(2π(fc + ∆f )t + α)√

2P b(t) cos(2πfc t + 2π∆f t + α)(4.1)

ߣߔࠆ㧔P ߪฃା㔚ജ [W]㧘α ߪೋᦼ૏⋧㧕㧚ߎߎߢ㧘b(t) ߪb(t) =∞
d(n)a(t − nT )

(4.2)

n=−∞



d(n) =
a(t) =±1(BPSK)±1 ± j(QPSK)

1,

0<t<T

0,

otherwise

(4.3)

(4.4)

ߢ޽ࠆ㧔T ߪࠪࡦࡏ࡞㐳㧕㧚ฃା஥ߢߪߎࠇࠍ㧘࿑ 4.1 ߩࠃ߁ߦห⋧ᚑಽߣ⋥੤ᚑಽߦಽߌ
ߡ⁛┙ߦᬌ಴ߒߚᓟวᚑߔࠆ㧚ߎߎߢߪ㧘ห⋧ᚑಽࠍⶄ⚛ᢙߩታㇱ㧘⋥੤ᚑಽࠍⶄ⚛ᢙߩ⯯ㇱߣߒߡ⴫ߒߡ޿ࠆ㧚࿑ 4.1 ߩ x(n) ߪ√
1 (n+1)Tx(n) =
g(t) 2 cos(2πfc t)dt
T nT
√
1√
=
2P b(t) cos(2πfc t + 2π∆f t + α) 2 cos(2πfc t)dt
 T


T 

1
ߣߔࠆߣ
fc

— 11 —

4.1 ⶄ⚛ࡌ࡯ࠬࡃࡦ࠼ାภ⴫⃻ߩዉ಴

࿑ 4.1

࠳࠙ࡦࠦࡦࡃ࡯࠲

 (n+1)T1
2√{cos(2π∆f t + α)}dt=
P d(n)T2nt
 (n+1)T1√=
P d(n)
cos(2π∆f t + α)dt
T
nt
1√
1
(n+1)T=[sin(2π∆f t + α)]nTP d(n)
T
2π∆f t√sin(π∆f T )= P d(n) cos(2π∆f T n + π∆f T + α)π∆f T√
= P d(n)Sa(π∆f T ) cos(2π∆f T n + π∆f T + α)(4.5)ߣ⴫ߐࠇࠆ㧚ߎߎߢ㧘Sa(x) = sin(x)/x ߢ޽ࠆ㧚ห᭽ߦ㧘y(n) ߪ1y(n) =T (n+1)TnT√−g(t) 2 sin(2πfc t)dt

 (n+1)T1√=
P d(n)sin(2π∆f t + α)dt
T
nt
1√
1
(n+1)T[− cos(2π∆f t + α)]nT=P d(n)
T
2π∆f t√= P d(n)Sa(π∆f T ) sin(2π∆f T n + π∆f T + α)(4.6)ߣ⴫ߐࠇࠆ㧚ࠃߞߡx(n) + jy(n) =√P d(n)Sa(π∆f T ) exp(j(2π∆f T n + π∆f T + α))(4.7)ߣߥࠅ㧘ߎࠇ߇๟ᵄᢙߕࠇߩ޽ࠆ RF ାภߩⶄ⚛ࡌ࡯ࠬࡃࡦ࠼⴫⃻ߢ޽ࠆ㧚ߜߥߺߦ㧘OFDM ߢߪ๟ᵄᢙࠬࡍࠢ࠻࡞ߩඨಽࠍ㊀ߨߡ㈩⟎ߔࠆ㧚ߎࠇߪ㧘๟ᵄᢙߕࠇ

∆f ߣାภࠬࡍࠢ࠻࡞ߩඨಽߩ᏷ 1/T ߇╬ߒ޿㧘ߔߥࠊߜ ∆f T = 1 ߣ޿߁ߎߣߢ޽ࠆ㧚ߎ— 12 —

4.2 ㅍାᯏࡕ࠺࡞

ࠇࠍᑼ (4.7) ߦઍ౉ߔࠆߣ㧘Sa(π) = 0 ࠃࠅ㧘x(n) + jy(n) = 0 ߣߥࠆ㧚ߎࠇߪ㓞ធߔࠆ๟ᵄᢙߩାภߩᓇ㗀ࠍోߊฃߌߥ޿ߣ޿߁ߎߣࠍ⴫ߒߡ޿ࠆ㧚OFDM ߪߎߩᕈ⾰ࠍ೑↪ߔࠆߎߣߢ๟ᵄᢙ೑↪ല₸ࠍ㜞߼ߡ޿ࠆ㧚߹ߚ㧘ߎߎߢߪ㔀㖸ߩࡕ࠺࡞ൻߦߟ޿ߡߪ⋭⇛ߔࠆ߇㧘࠳࠙ࡦࠦࡦࡃ࡯࠲ߩห⋧ᚑಽ߅ࠃ߮⋥੤ᚑಽߘࠇߙࠇߦኻߒߡ㧘ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߩࡕ࠺࡞㧔ઃ㍳ A㧕߇ㆡ↪ߢ߈ࠆ㧚

4.2ㅍାᯏࡕ࠺࡞

࿑ 4.2 ߪ㧘๟ᵄᢙᏅࠍ೑↪ߒߚᄙ㊀ൻᣇᑼߦ߅ߌࠆㅍାᯏࡕ࠺࡞ߢ޽ࠆ㧚౉ജ࠺࡯࠲ߪ߹ߕ⋥ਗ೉ᄌ឵ߐࠇ㧘ߘࠇߙࠇ⇣ߥߞߚ៝ㅍᵄߦࠃࠅᄌ⺞ߐࠇࠆ㧚ߘߩᓟ⿷ߒวࠊߐࠇ㧘ㅍାߐࠇࠆ㧚

࿑ 4.2

ㅍାᯏ

— 13 —

4.3 ฃାᯏࡕ࠺࡞

4.3

ฃାᯏࡕ࠺࡞

࿑ 4.3 ߪ㧘๟ᵄᢙᏅࠍ೑↪ߒߚᄙ㊀ൻᣇᑼߦ߅ߌࠆฃାᯏࡕ࠺࡞ߢ޽ࠆ㧚

࿑ 4.3

ฃାᯏ

߹ߕ㧘ฃାߐࠇߚାภߪ㧘K ୘ߩାภฃାㇱߦㅍࠄࠇ㧘࠳࠙ࡦࠦࡦࡃ࡯࠲ߦࠃߞߡⶄ
⚛ࡌ࡯ࠬࡃࡦ࠼ାภߣߥࠆ㧚ߎߩⶄ⚛ࡌ࡯ࠬࡃࡦ࠼ାภࠍ A-D ᄌ឵ߒߚ߽ߩࠍ౉ജାภ

zk (n) ߣߔࠆ㧚ᰴߦ౉ജାภߣ૏⋧ផቯ୯ wk∗ (n) ߩⓍ ck (n) ࠍ᳞߼㧘࠺࡯࠲ᬌ಴ࠍⴕ߁㧚ᦨᓟߦ㧘ߔߴߡߩାภฃାㇱ߆ࠄ಴ജߐࠇߚ࠺࡯࠲ࠍਗ⋥೉ᄌ឵ߒߡ಴ജߔࠆ㧚߹ߚ㧘૏⋧ផቯ୯ߪ㧘ෳᾖାภ dˆk ߣ㧘౉ജାภߣ૏⋧ផቯ୯ߩౝⓍ ck (n) ߣߩᏅ߇ᦨዊߣߥࠆࠃ߁ㆡᔕ⊛ߦ೙ᓮߐࠇࠆ㧚ᧄ⺰ᢥߢߪ㧘ㆡᔕࠕ࡞ࠧ࡝࠭ࡓߣߒߡᱜⷙൻ LMS
㧔Normalized Least Mean Squares: N-LMS㧕ࠕ࡞ࠧ࡝࠭ࡓ [3] ࠍ↪޿ߡ޿ࠆ㧚— 14 —

4.4 㜞ᄙ㊀ൻㅢାߩᕈ⢻4.4

㜞ᄙ㊀ൻㅢାߩᕈ⢻ାภࠬࡍࠢ࠻࡞ࠍ㊀ߨߡ㈩⟎ߒߚ႐ว㧘ฃାᤨߦ㓞ធߔࠆ๟ᵄᢙߩାภ㧔એਅ㧘㓞ធାภ㧕߇ᐓᷤାภߣߥࠅᕈ⢻߇ഠൻߔࠆ㧚ߘߎߢ㧘ାภࠬࡍࠢ࠻࡞ࠍ OFDM ࠃࠅ߽㊀ߨࠆߎߣߦ
ࠃࠅ㧘ࡆ࠶࠻⺋ࠅ₸㧔Bit Error Rate: BER㧕߇ߤߩࠃ߁ߦᄌൻߔࠆߩ߆ࠍ␜ߔ㧚◲නߩߚ߼㧘2 ᄙ㊀ߢᬌ⸛ࠍⴕ޿㧘ㅢା〝ߪ⊕⦡ࠟ࠙ࠬ㔀㖸ㅢା〝 [4] ߣߒߚ㧚

4.4.1

BPSK ᣇᑼߦ߅ߌࠆᕈ⢻࿑ 4.4㧘4.5㧘4.6 ߦᄌ⺞ᣇᑼߣߒߡ BPSK ࠍ↪޿ߚ႐วߦ߅ߌࠆࡆ࠶࠻⺋ࠅ₸․ᕈࠍ␜ߔ㧚࿑ 4.4㧘4.5㧘4.6 ߢߪ㧘ᚲᦸାภߩ૏⋧ α0 ࠍ 0[rad] ߢ࿕ቯߒ㧘㓞ធାภߩ૏⋧ α1 ࠍ 0[rad]㧘

π/4[rad]㧘π/2[rad] ߣᄌൻߐߖߚ㧚ߎࠇࠄߩ࿑ࠍᲧߴࠆߣ㧘∆f T = 0.5 ߩ႐วߩࡆ࠶࠻⺋ࠅ₸․ᕈߛߌ߇㓞ធାภߩ૏⋧ߦࠃߞߡᄢ߈ߊ⇣ߥࠆߎߣ߇ಽ߆ࠆ㧚․ߦ㧘α1 = 0[rad] ߩ႐ว㧘㓞ធାภߩᓇ㗀ࠍฃߌߡ޿ߥ޿㧚ߎࠇߪ㧘ାภߩᄌ⺞ᣇᑼߣߒߡ BPSK ࠍ↪޿ߡ޿ࠆ੐ߦ⿠࿃ߔࠆ㧚ᑼ (4.7) ߦ P = 1㧘∆f T = 0.5㧘α = 0 ࠍઍ౉ߔࠆߣ㧘x(n) + jy(n) = d(n)Sa(π/2) exp(j(2πn + π/2))(4.8)ߣߥࠆ㧚ߎߩᑼ߆ࠄ㧘㓞ធାภߩห⋧ᚑಽ߇ᚲᦸାภߩ⋥੤ᚑಽ㧘㓞ធାภߩ⋥੤ᚑಽ߇ᚲ
ᦸାภߩห⋧ᚑಽߦᐓᷤߣߒߡ⃻ࠇࠆ੐߇ࠊ߆ࠆ㧚BPSK ߢߪାภߩห⋧ᚑಽߦߩߺ࠺࡯
࠲߇ሽ࿷ߔࠆߚ߼㧘ᚲᦸାภߩ⋥੤ᚑಽߦ⃻ࠇߚ㓞ធାภߩᓇ㗀ࠍήⷞߔࠆߎߣ߇ߢ߈㧘߹ߚ㧘ᚲᦸାภߩห⋧ᚑಽߦߪోߊᓇ㗀߇⃻ࠇߥ޿ߣ޿߁ߎߣߦߥࠆ㧚ߎࠇࠍ೑↪ߔࠆߎߣߢ㧘BPSK ߦ߅޿ߡߪ ∆f T = 0.5 ߢ QPSK ࠍ↪޿ߚ OFDM ߣห╬ߩ๟ᵄᢙ೑↪ല₸ࠍታ⃻ߢ߈ࠆ㧚

∆f T = 0.5 એᄖߢߪ㧘๟ᵄᢙ㈩⟎㑆㓒߇⁜ߊߥࠇ߫ߥࠆ߶ߤㅢାߩᕈ⢻ߪഠൻߒߡ޿ࠆ㧚— 15 —

4.4 㜞ᄙ㊀ൻㅢାߩᕈ⢻࿑ 4.4ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = 0㧕

࿑ 4.5

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = π/4㧕

— 16 —

4.4 㜞ᄙ㊀ൻㅢାߩᕈ⢻

࿑ 4.6

4.4.2

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = π/2㧕

QPSK ᣇᑼߦ߅ߌࠆᕈ⢻࿑ 4.7㧘4.8㧘4.9 ߦᄌ⺞ᣇᑼߣߒߡ QPSK ࠍ↪޿ߚ႐วߩࡆ࠶࠻⺋ࠅ₸․ᕈࠍ␜ߔ㧚BPSK ߩࠃ߁ߦ㓞ធାภߩᓇ㗀ࠍฃߌߥ޿ߣ޿߁ߎߣߪߥ޿߇㧘QPSK ߦ߅޿ߡ߽
∆f T = 0.5 ߩ႐วߩࡆ࠶࠻⺋ࠅ₸․ᕈߛߌ߇㓞ធାภߩ૏⋧ߦࠃߞߡ⇣ߥߞߡ޿ࠆ㧚․ߦ㧘
α1 = π/4[rad] ߩߣ߈ߦࡆ࠶࠻⺋ࠅ₸․ᕈ߇⪺ߒߊഠൻߒߡ޿ࠆ㧚ߎࠇߪ㧘㓞ធାภߩห⋧ᚑಽߣ⋥੤ᚑಽߩࡌࠢ࠻࡞๺߇㧘ᚲᦸାภߩห⋧ᚑಽߣ⋥੤ᚑಽߩߤߜࠄ߆ߦߩߺᏱߦᐓᷤ
ߣߒߡ⃻ࠇࠆ߆ࠄߢ޽ࠆ㧚— 17 —

4.4 㜞ᄙ㊀ൻㅢାߩᕈ⢻࿑ 4.7

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = 0㧕

࿑ 4.8

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = π/4㧕

— 18 —

4.4 㜞ᄙ㊀ൻㅢାߩᕈ⢻࿑ 4.9

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔α0 = 0, α1 = π/2㧕

— 19 —

╙5┨㓞ធ๟ᵄᢙ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼ
5.1

ࠕ࡞ࠧ࡝࠭ࡓ

ߎࠇ߹ߢߦㅀߴߚࠃ߁ߦ㧘㜞ᄙ㊀ൻㅢାߦ߅޿ߡߪ㓞ធାภߩᓇ㗀ߢㅢାߩᕈ⢻߇ഠൻߔࠆ㧚ߎࠇࠍᡷༀߔࠆߚ߼ߦߪ㧘ฃା஥ߢ㓞ធାภߩᓇ㗀ࠍシᷫߔࠆᔅⷐ߇޽ࠆ㧚ߘߎߢ㧘࿑

5.1 ߩࠃ߁ߥฃାᯏࠍ↪޿ࠆ㓞ធ๟ᵄᢙ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼࠍ⠨߃ߚ㧚ߥ߅㧘ߎߎߢߪ◲නߩߚ߼ାภฃାㇱࠍ 2 ߟߣߒߡ޿ࠆ㧚࿑ 5.1

࠺࡯࠲ࠠࡖࡦ࠮࡞ฃାᯏ

ߎߩᣇᑼߩ․ᓽߪ㧘ฦାภࠬࡍࠢ࠻࡞ߦኻᔕߔࠆฃାㇱ߇࠺࡯࠲ᬌ಴ࠍⴕ߁㓙ߦ㧘㓞ធାภߦኻᔕߔࠆฃାㇱ߆ࠄㅍࠄࠇߚ࠺࡯࠲ߩផቯ୯ dˆk (n) ߣ૏⋧ߩផቯ୯ wk (= exp(jαk )) ࠍ

— 20 —

5.2 エ್ቯ↪޿ߡ㓞ធାภߩ࡟ࡊ࡝ࠞࠍ૞ࠅ㧘ᷫ▚ߔࠆߎߣߢᓇ㗀ࠍシᷫߔࠆߣ޿߁ߎߣߢ޽ࠆ㧚ౕ૕⊛ߥ଀ࠍ␜ߔ㧚2 ᄙ㊀㧘ฃା㔚ജࠍ 1W ߣߔࠆߣ㧘0 ⇟ฃାㇱߩ౉ജାภߪ

z0 (n) = d0 (n) exp(jα0 ) + d1 (n)Sa(π∆f T ) exp(j(2π∆f T n + π∆f T + α1 )) (5.1)
1 ⇟ฃାㇱߩ౉ജାภߪ
z1 (n) = d1 (n) exp(jα1 ) + d0 (n)Sa(−π∆f T ) exp(j(−2π∆f T n − π∆f T + α0 )) (5.2)ߢ⴫ߐࠇࠆ㧚
߹ߕ㧘1 ⇟ฃାㇱߢ઒ฃାࠍⴕ޿㧘࠺࡯࠲ d1 (n) ߩផቯ୯ dˆ1 (n) ࠍᓧࠆߣ౒ߦ㧘ㆡᔕࠕ࡞ࠧ࡝࠭ࡓߦࠃࠅᣂߒ޿૏⋧ߩផቯ୯ w1 ࠍᓧࠆ㧚ߎߎߢߪᣂߒ޿૏⋧ߩផቯ୯ߪ 0 ⇟ฃାㇱߩᐓᷤផቯㇱ߳ㅍࠆߛߌߢ㧘ᦝᣂߪⴕࠊߥ޿㧚ᰴߦ㧘0 ⇟ฃାㇱߢฃାࠍⴕ߁߇㧘ߎߩߣ߈ߦవ߶ߤᓧߚ 2 ߟߩ୯ dˆ1 (n)㧘w1 ࠍ↪޿ߡ㓞ធାภߩ࡟ࡊ࡝ࠞࠍ૞ࠅ㧘౉ജାภ߆ࠄᷫ▚ߔࠆ㧚ߎࠇࠍᑼߢ⴫ߔߣ

ẑ0 (n) = z0 (n) − dˆ1 (n)Sa(π∆f T ) exp(j(2π∆f T n + π∆f T ))w1

(5.3)

ߣߥࠆ㧚ߎߩ ẑ0 (n) ࠍ 0 ⇟ฃାㇱߩ౉ജାภߣߒߡ࠺࡯࠲ dˆ0 (n) ࠍᬌ಴ߒฃାߔࠆߣ౒ߦ㧘૏⋧ߩផቯ୯ w0 ࠍ᳞߼㧘ᦝᣂߔࠆ㧚ᦨᓟߦ㧘1 ⇟ฃାㇱߢ dˆ0 (n)㧘w0 ࠍ↪޿ߡ 0 ⇟ฃାㇱߣห᭽ߦฃାࠍⴕ޿㧘૏⋧ߩផቯ୯ࠍᦝᣂߔࠆ㧚

5.2

エ್ቯ࠺࡯࠲ࠠࡖࡦ࠮࡞ࠍⴕ߁㓙㧘࠺࡯࠲ߩផቯ୯ dˆk (n) ߇⺋ߞߡ޿ࠆߣ㧘ᐓᷤࠍჇ᏷ߒߡߒ߹
޿㧘ㅒലᨐߣߥࠆ㧚࠺࡯࠲ᬌ಴ߦ↪޿ࠆ⎬್ቯߢߪ㧘౉ജ߇ᱜ߆⽶߆ߛߌࠍ⷗ߡ಴ജࠍ᳿߼
ࠆߚ߼㧘౉ജ߇ 0 ઃㄭߩߣ߈ߩା㗬ᕈ߇ૐߊߥࠆ㧚ߘߎߢ㧘ࠃࠅ㜞޿♖ᐲߩ࠺࡯࠲ࠠࡖࡦ࠮
࡞ࠍታ⃻ߔࠆߚ߼㧘࠺࡯࠲ߩផቯ୯ࠍ᳞߼ࠆ㓙ߦエ್ቯࠍⴕ߁ᣇᑼߦߟ޿ߡᬌ⸛ߒߚ㧚エ್ቯߦߪࠪࠣࡕࠗ࠼㑐ᢙࠍᔕ↪ߒߚ㧚

f (x) = 2

1
1 + exp(−ax)— 21 —

− 1,(a > 0)(5.4)

5.2 エ್ቯߎߎߢ a ߪエ್ቯᦛ✢ߩ௑߈ࠍ᳿߼ࠆࡄ࡜ࡔ࡯࠲ߢ޽ࠆ㧚エ್ቯᦛ✢ࠍ࿑ 5.2 ߦ␜ߔ㧚

࿑ 5.2

エ್ቯᦛ✢

エ್ቯࠍ↪޿ࠆ႐วߩฃାᯏࡕ࠺࡞ࠍ࿑ 5.3 ߦ␜ߔ㧚࿑ 5.3エ್ቯ࠺࡯࠲ࠠࡖࡦ࠮࡞ฃାᯏ

— 22 —

5.3 ᕈ⢻⹏ଔ

5.3

ᕈ⢻⹏ଔ

ឭ᩺ᣇᑼߩᕈ⢻ࠍ⹏ଔߔࠆߚ߼㧘ࡆ࠶࠻⺋ࠅ₸․ᕈߣ૶↪Ꮺၞ᏷ߦኻߔࠆ๟ᵄᢙ೑↪ല₸
ࠍ␜ߔ㧚

5.3.1

ࡆ࠶࠻⺋ࠅ₸․ᕈឭ᩺ᣇᑼߦ߅ߌࠆࡆ࠶࠻⺋ࠅ₸․ᕈࠍ࿑ 5.4㧘5.5㧘5.6㧘5.7 ߦ␜ߔ㧚ᄙ㊀ൻᢙ 2㧘ฃା㔚ജ 1W㧘ห૏⋧㧘ㅢା〝ߪ⊕⦡ࠟ࠙ࠬ㔀㖸ㅢା〝ߣߒߚ㧚߹ߚ㧘エ್ቯᦛ✢ߩ௑߈ࠍ᳿߼ࠆ
ࡄ࡜ࡔ࡯࠲ a ߦߟ޿ߡߪ㧘ฦ ∆f T ߦኻߒߡ 1 ߆ࠄ 6 ߹ߢߩᢛᢙߩ޿ߕࠇ߆ߣߒߡࠪࡒࡘ
࡟࡯࡚ࠪࡦࠍⴕ޿㧘ߘߩ⚿ᨐ㧘ᦨ߽ലᨐ߇⹺߼ࠄࠇߚ߽ߩߦߟ޿ߡ࿑␜ߒߡ޿ࠆ㧚࿑ 5.4 ߪ㧘OFDM㧔∆f T = 1㧕ߣ ∆f T = 0.8 ߩ႐วࠍᲧセߒߚ߽ߩߢ޽ࠆ㧚OFDM ߢߪ㧘㓞ធାภ߇⋥੤ߒߡ޿ࠆߚ߼⋧੕ᐓᷤߪߥߊ㧘Eb /N0 = 6.8 dB ߢࡆ࠶࠻⺋ࠅ₸߇⚂10−3 ߣߥࠆ㧚৻ᣇ㧘∆f T = 0.8 ߩ႐วߪ㧘㓞ធାภߩᓇ㗀ߦࠃࠅ㧘Eb /N0 = 9 dB ߢࡆ࠶࠻⺋ࠅ₸߇⚂ 10−3 ߣߥࠆ㧚ߒ߆ߒ㧘࠺࡯࠲ࠠࡖࡦ࠮࡞ࠍⴕ߁ߎߣߦࠃࠅߘߩᓇ㗀ࠍシᷫߢ߈ࠆ㧚࿑ߦ␜ߐࠇࠆࠃ߁ߦ㧘エ್ቯࠍ↪޿ߚ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼߦࠃߞߡ㧘∆f T = 0.8 ߩࡆ࠶࠻⺋ࠅ₸․ᕈߪ OFDM ߣ߶߷ห⒟ᐲߦ߹ߢᡷༀߐࠇࠆ㧚࿑ 5.5 ߪ㧘OFDM ߣ ∆f T = 0.7 ߩ႐วࠍᲧセߒߚ߽ߩߢ޽ࠆ㧚೨ㅀߒߚ ∆f T = 0.8 ߣᲧߴߡ㧘๟ᵄᢙ㈩⟎㑆㓒߇⁜޿ಽߛߌ㓞ធାภߩᓇ㗀ࠍᒝߊฃߌ㧘ࡆ࠶࠻⺋ࠅ₸․ᕈ߇ࠃࠅ
ഠൻߔࠆߚ߼㧘࠺࡯࠲ࠠࡖࡦ࠮࡞ߥߒߢߪ Eb /N0 = 11.2 dB ߢࡆ࠶࠻⺋ࠅ₸߇⚂ 10−3 ߣߥࠆ㧚ߒ߆ߒ㧘࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼߢߪ Eb /N0 = 8.5 dB㧘ߐࠄߦエ್ቯࠍ↪޿ߚ႐วߪ

Eb /N0 = 7.8 dB ߢࡆ࠶࠻⺋ࠅ₸߇⚂ 10−3 ߣߥࠆ㧚࿑ 5.6 ߪ㧘OFDM ߣ ∆f T = 0.6 ߩ႐วࠍᲧセߒߚ߽ߩߢ޽ࠆ㧚࠺࡯࠲ࠠࡖࡦ࠮࡞ߥߒߢ
ߪ 15.2 dB㧘࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼߢߪ 12.2 dB㧘ߐࠄߦエ್ቯࠍ↪޿ߚ႐วߪ 10 dB ߢࡆ࠶࠻⺋ࠅ₸߇⚂ 10−3 ߣߥࠆ㧚⚂ 5 dB ߣᄢ߈ߊᡷༀߐࠇߡ߅ࠅ㧘ߎߎߢ␜ߒߚ 4 ߟߩ଀ߩ߁ߜߢߪᦨ߽ Eb /N0 ߩᡷༀലᨐ߇㗼⪺ߦ⴫ࠇߡ޿ࠆ㧚࿑ 5.7 ߪ㧘OFDM ߣ ∆f T = 0.5 ߩ႐วࠍᲧセߒߚ߽ߩߢ޽ࠆ㧚∆f T = 0.5 ߩ႐ว㧘ࡆ࠶— 23 —

5.3 ᕈ⢻⹏ଔ

࿑ 5.4

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔OFDM ߣ ∆f T = 0.8 ߩᲧセ㧕

࿑ 5.5ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔OFDM ߣ ∆f T = 0.7 ߩᲧセ㧕

— 24 —

5.3 ᕈ⢻⹏ଔ

࿑ 5.6

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔OFDM ߣ ∆f T = 0.6 ߩᲧセ㧕

࿑ 5.7

ࡆ࠶࠻⺋ࠅ₸․ᕈ㧔OFDM ߣ ∆f T = 0.5 ߩᲧセ㧕

— 25 —

5.3 ᕈ⢻⹏ଔ࠻⺋ࠅ₸߇⚂ 10−3 ߣߥࠆ Eb /N0 ߇㧘࠺࡯࠲ࠠࡖࡦ࠮࡞ߥߒߢߪ 15.3 dB㧘エ್ቯࠍ↪޿ߥ޿࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼߢߪ 15 dB ߣ㧘߶ߣࠎߤᡷༀ߇⷗ࠄࠇߥ޿㧚ߎࠇߪ㧘๟ᵄᢙ㈩⟎㑆㓒߇⁜ߊߥࠅ㧘㓞ធାภߩᓇ㗀߇ᄢ߈ߊߥࠅߔ߉ߚߚ߼ߦ㧘࠺࡯࠲ផቯߦ⺋ࠅ߇㗫಴ߔࠆߎߣ߇ේ࿃ߢ޽ࠆߣ⠨߃ࠄࠇࠆ㧚৻ᣇ㧘エ್ቯࠍ↪޿ߚ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼߢߪ㧘12 dB ߢࡆ࠶࠻⺋ࠅ₸߇⚂ 10−3 ߣߥࠅ㧘3 dB ߶ߤᡷༀߐࠇߡ޿ࠆ㧚ߎߩߎߣ߆ࠄ㧘エ್ቯࠍ↪޿ࠆߎߣߢ࠺࡯࠲ផቯߩ⺋ࠅߦࠃࠆᓇ㗀ࠍᛥ߃㧘ࠃࠅ♖ᐲߩ㜞޿࠺࡯࠲ࠠࡖࡦ࠮࡞ࠍታ⃻ߔࠆߎߣ߇ߢ߈ࠆߣ⸒߃ࠆ㧚

5.3.2

૶↪Ꮺၞ᏷ߦኻߔࠆ๟ᵄᢙ೑↪ല₸૶↪ߢ߈ࠆᏪၞ᏷ߦኻߔࠆ๟ᵄᢙ೑↪ല₸ߦߟ޿ߡ㧘OFDM ߣ ∆f T = 0.5 ߣࠍᲧセߒߚ߽ߩࠍ࿑ 5.8 ߦ␜ߔ㧚๟ᵄᢙ೑↪ല₸ߪ㧘KR/L㧔೽៝ㅍᵄᢙ × ࠺࡯࠲ㅍାㅦᐲ/૶↪Ꮺၞ᏷㧕[bits] ߢ᳞߼ࠄࠇࠆ㧚࿑ 5.8

૶↪Ꮺၞ᏷ߦኻߔࠆ๟ᵄᢙ೑↪ല₸࿑ߩࠃ߁ߦ㧘OFDM ߩඨಽߩ๟ᵄᢙ㈩⟎㑆㓒ߢାภࠬࡍࠢ࠻࡞ࠍ㈩⟎ߒߚ႐ว㧘OFDM— 26 —

5.3 ᕈ⢻⹏ଔߩ⚂ 2 ୚ߩ๟ᵄᢙ೑↪ല₸߇ᓧࠄࠇࠆ㧚— 27 —

╙6┨߹ߣ߼ᧄ⺰ᢥߢߪ㜞ᄙ㊀ൻㅢାߩߚ߼ߩ㓞ធ๟ᵄᢙ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼࠍឭ᩺ߒ㧘ߘߩᕈ⢻ߩ⹏ଔࠍⴕߞߚ㧚ߘߩ⚿ᨐ㧘ᄌ⺞ᣇᑼ QPSK㧘ᄙ㊀ൻᢙ 2㧘ห૏⋧㧘∆f T = 0.6 ߩ႐ว㧘

BER = 10−3 ߦ߅ߌࠆ Eb /N0 ߢ⚂ 5 dB ߩᡷༀ߇ᓧࠄࠇߚ㧚߹ߚ㧘OFDM ߩ⚂ 2 ୚ߩ๟ᵄᢙ೑↪ല₸߇ᓧࠄࠇࠆ ∆f T = 0.5 ߩ႐วߦ߅޿ߡ߽㧘⚂ 3 dB ߩᡷༀࠍᓧࠆߎߣ߇ߢ߈
ߚ㧚ߎࠇߪ㧘エ್ቯࠍ↪޿ߚ࠺࡯࠲ࠠࡖࡦ࠮࡞ᣇᑼߦࠃࠅ㧘㓞ធାภߩᓇ㗀ࠍല₸ࠃߊシᷫߔࠆߎߣ߇ߢ߈ࠆߎߣࠍ␜ߒߡ޿ࠆ㧚ߒߚ߇ߞߡ㧘ឭ᩺ᣇᑼߪ㧘㓞ធାภߩ๟ᵄᢙ㈩⟎㑆㓒ࠍ OFDM ࠃࠅ⁜ߊߒߚᤨߩࡆ࠶࠻⺋ࠅ₸․ᕈߩഠൻࠍᄢ᏷ߦᛥ߃ࠆߎߣ߇น⢻ߢ޽ࠅ㧘๟ᵄᢙ೑↪ല₸ߩะ਄ߦኻߒߡ᦭ലߢ޽ࠆߣ⸒߃ࠆ㧚੹ᓟߩ⺖㗴ߣߒߡߪ㧘ᄙ㊀ൻᢙࠍᄢ߈ߊ
ߒߚ႐วߩᬌ⸛㧘ᄌ⺞ᣇᑼߩᄙ୯ᄙ⋧ൻ㧘๟ᵄᢙ㈩⟎㑆㓒ߏߣߩエ್ቯᦛ✢ߩ௑߈ߩᦨㆡൻ
ߥߤ߇᜼ߍࠄࠇࠆ㧚— 28 —

⻢ㄉ㜞⍮Ꮏ⑼ᄢቇᖱႎࠪࠬ࠹ࡓᎿቇ⑼㧘ỳ᧛᣽ೣ⻠Ꮷߦߪ㧘ᧄ⎇ⓥߦ߅޿ߡᄙߊߩᓮᜰዉ㧘ᓮ㖊ᠤࠍ⾦ߞߚ㧚ߎߎߦᢘᗧࠍ⴫ߒ㧘ෘߊᓮ␞↳ߒ਄ߍࠆ㧚߹ߚ㧘ጤ↰⺈ᢎ᝼㧘⩵ᳰ⼾ഥᢎ᝼ߦߪ೽ᩏࠍ߅ᒁ߈ฃߌ޿ߚߛ߈㧘ᧄ⎇ⓥߦߟ޿ߡߩ᦭⋉ߥᓮഥ⸒ࠍ޿ߚߛ޿ߚ㧚ߎߎߦ⸥ߒߡ⻢ᗧࠍ⴫ߔࠆ㧚ᦨᓟߦ㧘౒ߦ⎇ⓥࠍⴕ޿㧘⼏⺰ࠍ੤ߒ㧘ബ߹ߒวߞߡ߈ߚ⎇ⓥቶߩખ㑆ߦᗵ⻢ߔࠆ㧚— 29 —ෳ⠨ᢥ₂[1] ᢧ⮮ ᵗ৻㧘“࠺ࠖࠫ࠲࡞ή✢ㅢାߩᄌᓳ⺞”㧘㔚ሶᖱႎㅢାቇળ㧘1996㧚
[2] ૒⮮ ᜏᦶ㧘“OFDM ᛛⴚߩၮ␆߆ࠄᔕ↪߹ߢ”㧘࡝ࠕ࡜ࠗ࠭␠㧘1999㧚
[3] S. ࡋࠗࠠࡦ㧔ᱞㇱ ᐙ ⸶㧕㧘“ㆡᔕࡈࠖ࡞࠲౉㐷”㧘⃻ઍᎿቇ␠㧘1987㧚
[4] B. P. ࡜ࠪࠖ㧔ጊਛ ᗉਯഥ㧘ቝ૒⟤ ⥝৻ ౒⸶㧕㧘“ㅢାᣇᑼ”㧘ࡑࠣࡠ࠙ࡅ࡞㧘1977㧚— 30 —ઃ㍳ Aࡑ࠶࠴࠻ࡈࠖ࡞࠲ [4]A.1ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߣߪ

ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߣߪ㧘ߘߩࡈࠖ࡞࠲಴ജߢߩାภኻ㔀㖸Ყࠍᦨᄢߦߔࠆߎߣࠍ⋡⊛ߣߔࠆࡈࠖ࡞࠲ߩߎߣߢ޽ࠆ㧚ᧄ⺰ᢥߢᛒ߁ࠃ߁ߥ࠺ࠖࠫ࠲࡞࠺࡯࠲ㅢାߢߪ㧘᦭㒢୘ߩᣢ⍮ߩᵄᒻࠍㅍ಴ߒ㧘ߘࠇࠍฃାᯏ஥ߢᬌ಴ߔࠆ㧚ㅢାߩㅜਛߢ㧘ାภߦߪㅢା〝ߩ㔀㖸߇ടࠊࠅ㧘ߎߩ㔀㖸ߩᓇ㗀ߦࠃߞߡฃାᤨߩ್ቯߦ⺋ࠅ߇↢ߓࠆ㧚߹ߚ㧘⺋ࠅ߇↢ߓࠆഀว㧔⺋ࠅ₸㧕ߪ㔀㖸߇ᄢ߈ߊߥࠆߦ
ߒߚ߇ߞߡჇടߔࠆߣ⠨߃ࠄࠇࠆ㧚⺋ࠅ₸ߪ࠺ࠖࠫ࠲࡞ㅢାߩାภᬌ಴ߢߪ㊀ⷐߥၮḰߢ޽ࠆ㧚⺋ࠅ₸߇ૐߌࠇ߫ૐ޿߶ߤ㧘ࠪࠬ࠹ࡓߩା㗬ᕈߪ㜞޿ߣ⸒߃ࠆ߇㧘ࠪࠬ࠹ࡓߦࠃߞߡߪ
ᔅߕߒ߽ାภኻ㔀㖸Ყ㧔Signal to Noise Ratio: SN Ყ㧕ࠍᄢ߈ߊߔࠆߎߣ߇⺋ࠅ₸ࠍዊߐߊߔࠆߣߪ㒢ࠄߥ޿㧚ߒ߆ߒ㧘⊕⦡㔀㖸ࠍ઻߁߶ߣࠎߤߩㅢାࠪࠬ࠹ࡓߦ߅޿ߡ㧘SN Ყߪ㊀ⷐߥⷐ⚛ߢ޽ࠅ㧘ߎࠇࠍᦨᄢߣߔࠆࡈࠖ࡞࠲ߪᦨ߽ၮᧄ⊛ߥ߽ߩߢ޽ࠆ㧚ߘߎߢ㧘એਅߢߪࡑ࠶࠴࠻ࡈࠖ࡞࠲ߩේℂߦߟ޿ߡㅀߴࠆ㧚ߎߎߢߪ৻଀ߣߒߡ㧘2 ㅴ╓ภㅢା㧘ߔߥࠊߜ

2 ߟߩ╓ภߒ߆↪޿ߥ޿ㅢାࠍขࠅᛒ߁㧚2 ㅴ╓ภㅢାߢߪ㧘ㅢାାภߪ 2 ߟߩ╓ภߛߌߢવㅍߐࠇࠆ㧚╓ภߩ৻ᣇࠍ࿑ A.1 ߩࠃ߁ߥࡄ࡞ࠬߢ⴫ߒ㧘߽߁৻ᣇࠍࡄ࡞ࠬ߇ሽ࿷ߒߥ޿⁁ᘒߢ⴫ߔ㧚ࡄ࡞ࠬ s(t) ߇ሽ࿷ߔࠆᤨ㑆ࠍ τ ⑽ߣߒ㧘ࡄ࡞ࠬ߇ሽ࿷ߒߥ޿ᤨ㑆߽หߓߊ τ ⑽ߣߔࠆ㧚ߎߎߢߪ㧘ࡄ࡞ࠬߩᵄᒻߩᒻ⁁ߩᬌ಴ߪ㊀ⷐߢߥ޿㧚࠺ࠖࠫ࠲࡞ㅢାᣇᑼߢߪ㧘ࡄ࡞ࠬߩᵄᒻߪᣢ⍮ߢ޽ࠆߚ߼㧘ߘߩ᦭ήߛߌࠍ್ቯߔࠇ߫ࠃ޿㧚— 31 —

A.2 ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߩේℂ࿑ A.1 ࡄ࡞ࠬߩ଀

A.2

ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߩේℂ

వߦㅀߴߚࠃ߁ߦ㧘ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߩേ૞ߪ್ቯߩ⍍㑆ߦାภᚑಽࠍჇᒝߒ㧘㔀㖸ᚑಽࠍᷫ⴮ߐߖࠆߎߣߢ޽ࠆ㧚ߎࠇߪ޽ࠆ⍍㑆ߩ಴ജߦ߅ߌࠆାภᝄ᏷ߣ㔀㖸ᝄ᏷ߣߩᲧࠍᦨᄢ
ߦߔࠆߎߣߦ╬ߒ޿㧚ᝄ᏷ߦߪᱜ⽶߇޽ࠅ㧘2 ਸ਼ࠍ↪޿ࠆ߶߁߇ଢ೑ߢ޽ࠆߩߢ㧘ାภᝄ᏷߅ࠃ߮㔀㖸ᝄ᏷ߩ 2 ਸ਼ߩᲧ߇ᦨᄢߦߥࠆࠃ߁ߥࡈࠖ࡞࠲ࠍ⸳⸘ߔࠆ㧚࿑ A.2 ߦ␜ߔࠃ߁ߦ㧘౉ജାภࠍ s(t) + n(t) ߣߔࠆ㧚ߎߎߢ s(t) ߪାภࡄ࡞ࠬ㧘n(t) ߪㅢା〝ߩ㔀㖸ߢ޽ࠆ㧚߹ߚ㧘ࡈࠖ࡞࠲಴ജࠍ so (t) + no (t) ߣߔࠆ㧚so (t) ߪ಴ജାภᚑಽ㧘no (t) ߪ಴ജ㔀㖸ᚑಽߢ޽ࠆ㧚ߎߎߢ㧘޽ࠆ⍍㑆 t = tm ߦ߅ߌࠆ s2o (t)/n2o (t) ߩ୯ࠍᦨᄢߦ
ߒߚ޿㧚ߒ߆ߒ㧘no (t) ߪ࡜ࡦ࠳ࡓାภߢ޽ࠅᱜ⏕ߦߪ᳿ቯߢ߈ߥ޿ߚ߼㧘ߘߩ 2 ਸ਼ᐔဋ୯n2o (t) ࠍ↪޿ߡρ=s2o (t)(A.1)

n2o (t)ߢਈ߃ࠄࠇࠆ ρ ࠍᦨᄢߦߔࠆ㧚࿑ A.2 ࡈࠖ࡞࠲s(t) ߩࡈ࡯࡝ࠛᄌ឵ࠍ S(ω)㧘ᦸ߹ߒ޿ᦨㆡࡈࠖ࡞࠲ߩવ㆐㑐ᢙࠍ H(ω) ߣߔࠆߣ㧘಴ജାภᚑಽߪso (t) = F−1

1[S(ω)H(ω)] =2π ∞H(ω)S(ω) exp(jωt)dω−∞— 32 —(A.2)

A.2 ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߩේℂ

߹ߚ

 ∞

1
so (tm ) =2π−∞

H(ω)S(ω) exp(jωtm )dω

(A.3)

ߣߥࠆ㧚
㔀㖸ାภߩ 2 ਸ਼ᐔဋ୯ߪ㧘಴ജߩ㔀㖸㔚ജኒᐲࠬࡍࠢ࠻࡞ߢ⴫ߔߎߣ߇ߢ߈ࠆ㧚Sn (ω) ࠍ౉ജ㔀㖸ାภ n(t) ߩ㔚ജኒᐲࠬࡍࠢ࠻࡞ߣߔࠆߣ㧘|H(ω)|2 Sn (ω) ߪ no (t) ߩ㔚ജኒᐲࠬࡍ
ࠢ࠻࡞ߦߥࠆ㧚ߒߚ߇ߞߡ㧘಴ജ㔀㖸ᚑಽߩ 2 ਸ਼ᐔဋߪn2o (t) =

1
2π

 ∞

−∞

|H(ω)|2 Sn (ω)dω(A.4)

no (t) ߩ 2 ਸ਼ᐔဋ୯ߪ t ߩ㑐ᢙߢߥ޿߆ࠄ ∞12
no (tm ) =|H(ω)|2 Sn (ω)dω
2π −∞

(A.5)

ߣߥࠆ㧚ㅢା〝ߩ㔀㖸 n(t) ࠍ㔚ജኒᐲ N /2 ࠍ߽ߟ⊕⦡㔀㖸ߣߔࠆߣSn (ω) =߆ࠄ

n2o (tm ) =

N
4π

N
2

(A.6)

|H(ω)|2 dω

(A.7)

 ∞
−∞

ߣߥࠆ㧚
ᑼ (A.1) ߦᑼ (A.3)㧘(A.5) ࠍઍ౉ߔࠆߣ


2
 ∞

 −∞ H(ω)S(ω) exp(jωtm )dω 
s2o (tm )∞=ρ=
πN −∞ |H(ω)|2 dω
n2o (tm )(A.8)ߣߥࠆ㧚so (t) ߪታᢙߢ޽ࠆ߆ࠄ㧘s2o (t) = |so (t)|2 ߣ⴫ߔ㧚ߎߎߢ Schwarz ߩਇ╬ᑼࠍ↪޿ࠆߣ ∞2  ∞ ∞2
H(ω)S(ω) exp(jωtm )dω  ≤|H(ω)| dω|S(ω)|2 dω

−∞

−∞

(A.9)

−∞

ߣߥࠆ㧚ᑼ (A.8) ߦᑼ (A.9) ߩਇ╬ᑼࠍઍ౉ߔࠆߣ

s2o (tm )1≤πNn2o (tm ) ∞−∞— 33 —|S(ω)|2 dω

(A.10)

A.2 ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߩේℂ

ߒߚ߇ߞߡ

1
ρmax =πN ∞−∞|S(ω)|2 dω

(A.11)

ߣߥࠆ㧚ߎߩᦨᄢ୯ߪᑼ (A.9) ߩ╬ภ߇ᚑࠅ┙ߟߣ߈ߦᓧࠄࠇ㧘H(ω) = kS ∗ (ω) exp(−jωtm ) = kS(−ω) exp(−jωtm )(A.12)

ߩߣ߈ߛߌᚑ┙ߔࠆ㧚ߎߎߢ k ߪછᗧߩቯᢙߢ޽ࠆ㧚ᦨㆡࡈࠖ࡞࠲ߩࠗࡦࡄ࡞ࠬᔕ╵ h(t) ߪh(t) = F −1 [H(ω)] = F −1 [kS(−ω) exp(−jωtm )](A.13)

ߢ⴫ߐࠇࠆ㧚ߎߎߢ㧘S(−ω) ߩㅒࡈ࡯࡝ࠛᄌ឵ߪ s(−t) ߢ޽ࠅ㧘exp(−jωtm ) ߪ tm ⑽ߩᤨ
㑆ゲߩ⒖േࠍ⴫ߔߎߣࠃࠅ㧘h(t) = ks(tm − t)

(A.14)

ߣߥࠆ㧚ߎߎߢߪ㧘ଢቱ਄ k = 1 ߣߔࠆ㧚ାภ s(tm − t) ߪ s(t) ࠍု⋥ゲߢ᛬ࠅ㄰ߒ㧘tm ⑽ߛߌฝߦ⒖േߔࠆߎߣߢᓧࠄࠇࠆ㧚ߎߎߢ㧘ࠗࡦࡄ࡞ࠬᔕ╵ h(t) ߪ tm < τ ߢߪ࿃ᨐᓞࠍḩ⿷ߒߥ޿ߚ߼㧘ታ⃻ਇน⢻ߢ޽ࠆ㧚߹ߚ㧘tm ߪ್ቯࠍߢ߈ࠆߛߌᣧߊߔࠆߚ߼ߦዊߐ޿ߎߣ߇ᦸ߹ࠇࠆߚ߼㧘tm > τ ࠃࠅ tm = τߩ߶߁߇ࠃ޿㧚ߘߩߚ߼㧘ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߩࠗࡦࡄ࡞ࠬᔕ╵ߪ㧘ାภ s(t) ࠍု⋥ゲߢ᛬ࠅ㄰ߒ㧘ߘࠇࠍ τ ⑽ߛߌฝߦ⒖േߐߖߚ߽ߩߢ޽ࠆ㧚ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߢᓧࠄࠇࠆ SN Ყߪ㧘ାภ s(t) ߇ฃାᯏߦ౉ࠅ⚳ࠊࠆᤨ㑆 tm ߩ⍍㑆ߦᦨᄢߦߥࠆ㧚ߘߩᦨᄢ୯ߪᑼ (A.11) ߢਈ߃ࠄࠇࠆ㧚ାภ s(t) ߩࠛࡀ࡞ࠡ࡯ Es ߪ

 ∞

1
Es =
s (t)dt =
2π
−∞
2

 ∞
−∞

|S(ω)|2 dω

(A.15)

ߢ⴫ߐࠇࠆ㧚ߒߚ߇ߞߡ

ρ=s2o (tm )n2o (tm )

=

Es
2Es
=
N /2
N

ߣߥࠆ㧚

— 34 —(A.16)

A.2 ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߩේℂ

಴ജାภᚑಽߩᝄ᏷ so (tm ) ߪ㧘ᑼ (A.3) ߦᑼ (A.12) ࠍઍ౉ߔࠆߣᓧࠄࠇ

1so (tm ) =
2π

 ∞
−∞

|S(ω)|2 dω = Es

(A.17)

ߣߥࠆ㧚ߎࠇࠃࠅ಴ജାภᚑಽߩᝄ᏷ߪ t = tm ߢᦨᄢߦߥࠅ㧘ߘߩᄢ߈ߐߪାภ s(t) ߩࠛࡀ࡞ࠡ࡯ Es ߦߥࠆ㧚ߎߩ᭽ሶࠍ࿑ A.3 ߦ␜ߔ㧚ߎࠇߪ㧘ᦨᄢᝄ᏷ߪ S(t) ߩᵄᒻߣߪή㑐ଥߢ㧘ߘߩࠛࡀ࡞ࠡ࡯ߩᄢ߈ߐߛߌߦଐሽߔࠆߎߣࠍ⴫ߒߡ޿ࠆ㧚࿑ A.3 ࡈࠖ࡞࠲ߩᔕ╵಴ജ㔀㖸ᚑಽߩ 2 ਸ਼ᐔဋ୯ߪ㧘ᑼ (A.16) ߦᑼ (A.17) ࠍઍ౉ߔࠆߣᓧࠄࠇ

n2o (tm ) =

N Es
2

ߣߥࠆ㧚
߹ߚ㧘ࡑ࠶࠴࠻ࡈࠖ࡞࠲ߦࠃࠆᬌ಴ߪၮᧄ⊛ߦߪหᦼᬌᵄߢ޽ࠆ㧚— 35 —(A.18)