PgTR iP h s j

國 立 中 央 大 學

數 學 研 究 所

碩 士 論 文

雙正交凌波函數於血壓與 交感神經活性訊號分析之應用

研 究 生：羅 文 仁

指導教授：單 維 彰 博士

中 華 民 國 九 十 二 年 六 月 十 九 日

國立中央大學圖書館 碩博士論文授權書

(91 年 5 月最新修正版)

本授權書所授權之論文全文與電子檔，為本人於國立中央大學，撰寫 之碩/博士學位論文。(以下請擇一勾選)

( v )同意 (立即開放)

( )同意 (一年後開放)，原因是：

( )同意 (二年後開放)，原因是：

( )不同意，原因是：

以非專屬、無償授權國立中央大學圖書館與國家圖書館，基於推動讀 者間「資源共享、互惠合作」之理念，於回饋社會與學術研究之目的，

得不限地域、時間與次數，以紙本、光碟、網路或其它各種方法收錄、

重製、與發行，或再授權他人以各種方法重製與利用。以提供讀者基 於個人非營利性質之線上檢索、閱覽、下載或列印。

研究生簽名: 羅 文 仁

論文名稱: 雙正交凌波函數於血壓與交感神經活性訊號分析之應用 指導教授姓名： 單 維 彰

系所 ： 數 學 所 o博士 R碩士班 學號： 90221006

日期：民國 92 年 6 月 19 日

備註：

1. 本授權書請填寫並親筆簽名後，裝訂於各紙本論文封面後之次頁（全文電 子檔內之授權書簽名，可用電腦打字代替）。

2. 請加印一份單張之授權書，填寫並親筆簽名後，於辦理離校時交圖書館（以 統一代轉寄給國家圖書館）。

3. 讀者基於個人非營利性質之線上檢索、閱覽、下載或列印上列論文，應依 著作權法相關規定辦理。

¿b

¦9Š0ä$Ê 0.02 ƒ 1.7 Hz íä0¸ˇq, >>ÿ%x|wŠ0×üí?‰
ú

>>ÿ%º4dÔìä0‰, @v}¨A°ä0í¦9‰
¥ÿukªWû˝í{æ

Í7mUTÜí{æ³Þ, ¦šìÜDZ sž²u.e x, °vZ sž²6u׶

}mUTÜj¶íŽ-
ĤÊøı³, ÅH7¦šìÜ£w½b41‚àbM }j¶V Rû×àZ sž²ít
Ê˙íªŸjÞ, 6úàSUà MATLAB VªW0§Z sž²õTj¶#7ypüí·H

jZ sž²ÊmU}&,@à˜, O~šƒbÄxe_@4í ”¢” DœQíl

µÆ, ZAÑmU}&íÇø_²Ï
ÊVÖ~šƒb2, £>~šƒbQ¡ú˚íÔ 4, ªUmU}&!‹.ßÞR, Ĥ²Ï£>~šƒbVªW¦9D>>ÿ%º4m U}&
FJÊùı³, z£>~šƒbóÉø…dø<cÜ
ßZ, 6½hcÜ Battle- Lemar´e ¸ Meyer ~šƒb
7Ñ7\Mdíêc4, Êúı³6>H7ÞÓõðDm U¦)j¶
ƒ7ûı, zø<@àÊä$}&DvÈåóÉ$l¾, $ød7j„, c ÜD}&
|(, Bb×ÛbW}&j¶D!‹n

ñ“

øı ¦šìÜDZ sž² 1

1.1 Nyquist ¦šìÜ . . . 1

1.2 Z sž² . . . 2

1.3 MATLAB, FFT( ), IFFT( ) . . . 5

ùı £>~šƒb 6 2.1 Battle-Lemari´e ~šƒb . . . 6

2.2 Meyer ~šƒb . . . 11

2.3 MRA (Multiresolution Analysis) . . . 13

2.4 £>~šƒb . . . 17

2.5 ê1½í‘K . . . 18

2.6 ˙š[bt . . . 22

úı ÞÓõðDmU¦) 28 3.1 AÑŒé¶} . . . 28

3.2 ÌAÑŒé¶} . . . 29

ûı ä$}&DvÈå 31 4.1 Cross-Correlation . . . 31

4.2 Cross-Covariance . . . 31

4.3 CSD D PSD . . . 32

4.4 Phase ƒbDž²ƒb (Transfer Function) . . . 33

4.5 Coherence . . . 35

4.6 Bartlett D Welch j¶ . . . 36

üı bW}&!‹Dn 39

5.1 }&j¶ . . . 39

5.2 ˙š[b¦) . . . 40

5.3 ~š}jƒ . . . 40

5.4 !‹Dn . . . 41

¡5d. 48

øı ¦šìÜDZ sž²

mUTÜí{æ³, ¦šìÜDZ sž²u.e!… x, 6umUTÜíŽ-
FJÊ¥

øı³, Bbܤsá3æJ£àSUà MATLAB ªWõT

1.1 Nyquist ¦šìÜ

Ñ)ƒ.ÜöíbPmU, úéªmU¦šÿéí½b
¦šä0 (sampling frequency) ¬ QímU, ³Ÿ¶ømU½V; ¦šM/vÈ (duration) ¬s, ̶×)œQíä$

¦šìÜÉ[ƒsK9, øu¦šä0íòQ, ùu¦šM/vÈíÅs

cìbPmU|ת¿ä0 (ä ) Ñ f_max, |üª¿ä0Ñ fmin, ¦šìÜÿuJ-s_

‘K:

1. ¦šä0 ≥ 2f_max

2. ¦šM/vÈ ≥ 1/f_min

2f_max ¢˚ Nyquist ä0
ÉbÅ—,Hs_‘K, éªmUÿ?J¦š(íbPmU½

õT,, b¨|Dä$ú@íä0dW, ÿÛbnj¦šìÜ
7%(ÊRû×àZ sž² tv, 6TXÇøimV ú@ä0Wj¶

1.2 Z sž²

#ìƒb f (x) ∈ L¹(R)T L²(R) †wZ sž²ÿu f (ω) =ˆ

Z ∞

−∞

f (x)e^−iωxdx

7

f (x) = 1 2π

Z ∞

−∞

f (ω)eˆ ^iωxdω ÿu ˆf (ω) íZ sLž²

%â Euler t

e^±iωx= cos(ωx) ± i sin(ωx)

)ø, cos ÊÀPvÈ ([0, 1]) qíËÓä0u _2π^w , ¥šíä0M}¨A˙ªŸví˚×

I ω 7→ 2πω ªW‰b‰², ठω ÿ}uö£íËÓä0
½ŸZ sž²t

f (ω) =ˆ Z ∞

−∞

f (x)e^−i2πωxdx

Lž²t

f (x) = Z ∞

−∞

f (ω)eˆ ^i2πωxdω

Z sž²DLž²ít(, Zªªø¥Rû×àZ sž² (DFT) ít

cqç |x| > ^A₂ v f (x) = 0 , † f (ω) =ˆ

Z ∞

−∞

f (x)e^−i2πωxdx = Z ^A₂

−^A

2

f (x)e^−i2πωxdx

z [−^A₂,^A₂] } N ¨, I N ÑXb, U) ∆x = _N^A , I

N N

°vcq˛ø f (x_n) , Ñ7¯U,íjZ, I

g(x) = f (x)e^−2iπωx

Í(@àG$¶dbM }, ZªJû| }Mí¡N,lu Z ^A₂

−^A

2

g(x)dx ≈ ∆x

2 {g(−A 2) + 2

N 2−1

X

n=−^N₂+1

g(x_n) + g(A 2)}

Ü,ø_õMíZ‰, . à }M, ]ªJÑ7líjZ, 7I g(−^A₂) = g(^A₂), FJ

f (ω) =ˆ Z ^A₂

−^A

2

g(x)dx

≈ ∆x

N 2

X

n=−^N₂+1

g(x_n)

= A N

N 2

X

n=−^N₂+1

f (xn)e^−2iπωxn

ƒ¤ ˆf (ω) ZªJúLSí ω V°M

Í(.â²ìÖý¸¨<í ω b\Uà
âk¦šä0u ^N_A, / ^N_A ≥ 2f_max
I Ω =

N

A, † f_max = ^Ω₂ FJ ω í¸ˇu [−^Ω₂,^Ω₂]
¢ÄѦšM/vÈÑ A, FJ 1/f_max = A, ]|üª¿ä0Ñ ¹

A
Uà|üª¿ä0çTä0–Èí~’È

∆ω = 1 A

† ˆf (ω) É?Ê

ω_k= k∆ω, k = −N 2 . . .N

2

,¦M
ÄÑÉ N _õíM\àÊ },l,, ¥#7ø_ßÜâé ω ¦ N _Mÿß

FJ¦

k = −N

2 + 1 . . .N 2

/

Ω = N ∆ω = N

A, ∆x∆ω = 1 N

I

f_n = f (x_n), F_k =

N 2

X

n=−^N₂+1

f_ne^−i2πnkN

†

f (ωˆ _k) ≈ A N

N 2

X

n=−^N₂+1

f (x_n)e^−i2πωkxn

= A N

N 2

X

n=−^N₂+1

fne^−i2πnkN

= A NF_k

FJ, #ì N _õM f_n, DFT ÿuø f_nž²A F_k í(4ø¦
7¥< F_k (ÿªR

 ˆf (ω_k) Ï.Öÿu _N^AF_k, ƒ¤ZêA×àZ sž²tíRû

cqÊ [^−Ω₂ ,^Ω₂] q,lZ sLž²í }, ×àZ sLž²t (IDFT) 6ªJ%âó

°íRû¬˙1)ƒéN×àZ sž²tí!‹:

f (x_n) ≈ Ω N

N 2

X

k=−^N₂+1

f (ωˆ _k)e^i2πxnωk

= Ω N

N 2

X

k=−^N₂+1

f (ωˆ k)e^i2πnkN

ʤZ.y;H

1.3 MATLAB, FFT( ), IFFT( )

BbªJòQà,øRû|í DFT t, ÊÚ72Ÿ|ž²DLž²˙
Jø&Ñ N í²¾ f bªW DFT, UàRûtFŸ|í˙ø O(N²) íl¾
lFÛí vÈ}óçªh, 7F‚0§Z sž² (FFT) ÿu DFT 0§Æ¶

MATLAB ³s_ƒ, }u fft( ) D ifft( ), ªàVúbWªW0§Z sž²DLž

²
ʤ, BbHUà¥s_ƒV6ŒªŸ˙
ç˙QYƒ²¾ f (, }lŒ&

N u´ÑXb, à‹.Í, †f f 1¢l, yVZãÊ fft( ) ƒªW«
Ä fft( ) ƒ…™íŸ¶, FŠíä0A}·}D£íä0A}>²P0æ[
FJʪW fft( ) ( ÿ.âzŠíä0A}½h”V, Í(y ,ø_b _N^A
¥šÿ}D,øFRûí!‹

ó°

f=fft(f);

f=[f(N/2+2:N) f(1:N/2+1)];

f=A*f/N;

DFT ˙Ò¨

J,Uà MATLAB FŸ|í˙Ò¨
M)·<íu, ªWLž²v, JbLž²í²¾u â fft( ) F`¨íu, †.âyz£Šä0ú|(yªWLž²
´†øû_.£üí!‹

7É B (?¹,øí Ω) .ÎJ N íŸÄÊk ifft( ) ˛z _N¹ ¨ÖÊq7

f=[f(N/2:N) f(1:N/2-1)];

f=ifft(f);

f=B*f;

IDFT ˙Ò¨

ùı £>~šƒb

ÖÍ£>í~šƒbnuBbí3i, OuBb6Y“ Battle-Lemari´e ¸ Meyer ~šƒ b, 1½hcÜøZ

2.1 Battle-Lemari´e ~šƒb

Battle-Lemari´e wavelets ÿu‚à£>“íŸÜ, ø!…š‘ƒb (B-spline) ¨A wavelets

JI φ(x) u Battle-Lemari´e wavelets íAƒb (scaling function)
I φ_s(x) Ñ p ¼

!…š‘ƒb, /I Suppφ_s(x) ú˚k x = 0 (J p uXb), Cú˚k x = ¹₂ (J p ÑJ b)
Ĥ {φ_s(x − k)}_k∈Z u(4Ö
ÑjZ–c, ¥³í)U p D [1] øš; ç p = 1 v, ÿuøOíɼš‘ƒb, 6ÿuj]ƒb (box function)

!…š‘ƒb˛uAƒb, ;W [1, 4” 3.4], Jbé {φ_s(x − k)} $Aø £†!, . âÅ—

∞

X

k=−∞

| ˆφs(ω + 2kπ)|² = 1 (2.1.1)

˛ø

φˆ_s(ω) = e^−iω2σsin(^ω₂)

ω 2

p

w2

σ =







1, p uJb 0, p uXb

ç p ≥ 2 ív`, (2.1.1) u.A í, FJI H(ω) =

∞

X | ˆφ_s(ω + 2kπ)|²

† H(ω) }u 2π /0£íU‚ƒb, ĤI ˆφ D ˆφ_s ÈíÉ[Ñ

φ(ω) =ˆ 1

pH(ω)φˆ_s(ω)

* [5, p88] )ø

H(2ω) = −sin^2p(ω) (2p − 1)!

 d^2p−1

dω^2p−1cot(ω)

Wà

p = 2, H(ω) = ¹₃ +²₃cos²(^ω₂)

p = 3, H(ω) = ₁₅² + ¹¹₁₅cos²(^ω₂) + ₁₅² cos⁴(^ω₂)

I m₀(ω) u φ(x) íÔƒb, †

m₀(ω) =

φ(2ω)ˆ φ(ω)ˆ

= pH(ω) ˆφ_s(2ω) pH(2ω) ˆφs(ω)

= 1 2

∞

X

k=−∞

c_ke^−ikω

¥šÿªJvƒ ck 7
c_k ̤Ö_ÝÉ[b, O˘kÖá¾à, c_k ú˚k x = 0 C x = ¹₂
J-Ü, Ë,7 p = 2, 3, 4, 5 í c_k, w¦Ÿíj¶Ñ

|

n

X

k=−n

c_k− 2| ≤ 5 × 10⁻⁶

k c_k k c_k 1 1.15632663044579 10 0.00072356251301 2 0.56186292858765 11 -0.00031720285555 3 -0.09772354847998 12 -0.00017350463597 4 -0.07346181335547 13 0.00007828566487 5 0.02400068439163 14 0.00004244222575 6 0.01412883469138 15 -0.00001954273439 7 -0.00549176158313 16 -0.00001052790655 8 -0.00311402901546 17 0.00000492117905 9 0.00130584362611

p=2

k c_k k c_k

1 0.96218850337647 14 0.00005149189024 2 0.19510923625013 15 -0.00049372120202 3 -0.17654342735281 16 -0.00001800199152 4 -0.02934097454448 17 0.00019929573525 5 0.05937163254037 18 0.00000645605761 6 0.00599366802040 19 -0.00008104108370 7 -0.02137287194074 20 -0.00000236237046 8 -0.00158880221461 21 0.00003314796576 9 0.00806884459996 22 0.00000087836323 10 0.00047350887755 23 -0.00001362322068 11 -0.00312788862797 24 -0.00000033091386 12 -0.00015251802585 25 0.00000562112308 13 0.00123483234718

p=3

k c_k k c_k 1 1.08347151256866 17 0.00079186999511 2 0.61365927344264 18 0.00065352962214 3 -0.07099595988486 19 -0.00040359352543 4 -0.15561584376755 20 -0.00032858869439 5 0.04536924029542 21 0.00020653439292 6 0.05949363315412 22 0.00016635055029 7 -0.02429097832036 23 -0.00010606378924 8 -0.02543084221422 24 -0.00008468217554 9 0.01228286171785 25 0.00005463412644 10 0.01159864029621 26 0.00004330399578 11 -0.00615725880956 27 -0.00002821716465 12 -0.00549057846550 28 -0.00002222839431 13 0.00309247829086 29 0.00001460738679 14 0.00266173875568 30 0.00001144675909 15 -0.00156092382332 31 -0.00000757744078 16 -0.00131125702104

p=4

k c_k k c_k 1 0.94230165749406 22 0.00004787742388 2 0.23935992558848 23 -0.00049967097120 3 -0.19570263414422 24 -0.00002541971835 4 -0.06300189335224 25 0.00029121541201 5 0.09290180706792 26 0.00001365185942 6 0.02154391441627 27 -0.00017028132917 7 -0.04843594992729 28 -0.00000740280973 8 -0.00821303447623 29 0.00009984794930 9 0.02609527570506 30 0.00000404735857 10 0.00342535357307 31 -0.00005869053564 11 -0.01436521645402 32 -0.00000222859178 12 -0.00153122737290 33 0.00003457175320 13 0.00803102176857 34 0.00000123475901 14 0.00072110789898 35 -0.00002040281402 15 -0.00454212409729 36 -0.00000068786968 16 -0.00035284122449 37 0.00001206099474 17 0.00259182385834 38 0.00000038506747 18 0.00017766463859 39 -0.00000714042396 19 -0.00148934274931 40 -0.00000021649660 20 -0.00009143746086 41 0.00000423299342 21 0.00086066342350

p=5

2.2 Meyer ~šƒb

I m₀(ω) uAƒb φ(x) íÔƒb, † m₀(ω) u 2π U‚ƒb, ú Meyer ~šƒb7 k, wÔƒbì2Ñ

m0(ω) =















1, |ω| ≤ ^π₃ β(ω), ^π₃ ≤ |ω| ≤ ^2π₃

0, ^2π₃ ≤ |ω| ≤ π

;W [1, 4” 3.5], Û²Ï β(ω) U) m₀(ω) Å—

|m₀(ω)|²+ |m₀(ω + π)|² = 1, ∀ω ∈ R

J φ(x) Å—

φ(ω) = mˆ ₀(ω 2) ˆφ(ω

2)

† {φ(x − k)} u£†!
FJø,Þí./RÆ, ªJ)ƒ

φ(ω) =ˆ

∞

Y

p=1

m₀(ω 2^p)

âk

m₀(^ω₂) = 0, |ω| ≥ ^4π₃

m₀(₂^ω2) = m₀(₂^ω3) = m₀(₂^ω4) = . . . = 1, |ω| ≤ ^4π₃

FJ ˆφ(ω) ÿªJ\“A

φ(ω) =ˆ







m₀(^ω₂), |ω| ≤ ^4π₃ 0, |ω| > ^4π₃

I φ(x) u ˆφ(ω) íZ sLž², † {φ(x − k)} ÿ}u£†!

°ší;¶, I m₁(ω) u ψ(x) íÔƒb, U) ˆψ(ω) Å—

ψ(ω) = mˆ ₁(ω 2) ˆφ(ω

2)

;W [1, (3.77)] )ø

m₁(ω) = −e^−iωm₀(ω + π)

†

ψ(ω) =ˆ







m₁(^ω₂)m₀(^ω₄), |ω| ≤ ^8π₃ 0, |ω| > ^8π₃

]

ψ(ω) =ˆ























0, |ω| ≤ ^2π₃ m₁(^ω₂), ^2π₃ ≤ |ω| ≤ ^4π₃ e^−iω2m₀(^ω₄), ^4π₃ ≤ |ω| ≤ ^8π₃

0, |ω| > ^8π₃

I ψ(x) Ñ ˆψ(ω) íZ sLž², † {ψ(x − k)} 6uø £†!

%âJ,íRû, ªc Meyer ~ší£†!u.ñøí, ²Ï_çí β(ω) ¯¯ m0(ω) í ì2, ÿªJû|ø £†!
J-uø_™ÄíWä

I

β(ω) = cos[^π₂v(^3|ω|_π − 1)], ^π₃ ≤ |ω| ≤ ^2π₃

w2 v(x) uËí¼Gƒb, Å—J-s_‘K:

v(x) =







0, x ≤ 0 1, x ≥ 1

¸

v(x) + v(1 − x) = 1

²Ï

v(x) = x⁴(35 − 84x + 70x²− 20x³)

)ƒ

φ(ω) =ˆ















1, |ω| ≤ ^2π₃ cos[^π₂v(_2π³ |ω| − 1)], ^2π₃ ≤ |ω| ≤ ^4π₃

0, |ω| ≥ ^4π₃

ψ(ω) =ˆ























0, |ω| ≤ ^2π₃ e^iω2 sin[^π₂v(_2π³ |ω| − 1)], ^2π₃ ≤ |ω| ≤ ^4π₃ e^iω2 cos[^π₂v(_4π³ |ω| − 1)], ^4π₃ ≤ |ω| ≤ ^8π₃

0, otherwise

ø ˆφ(ω) £ ˆψ(ω) ªWZ sLž², ÿªJ)ƒ φ(x) ¸ ψ(x)
¥s_ƒbuÌÌ°Qí, φ(x) ú˚k x = 0 , ψ(x) ú˚k x = ¹₂, .¬ÓO |x| íÚÓ, …bíMÿ}Q¡ 0

Ê‡Þ Battle-Lemari´e Aƒbnƒ p M , Ou Meyer ~š³F‚í p M, ÄÑ ψˆ^(k)(0) = 0, FJ R x^kψ(x)dx = 0, ∀k = 0, 1, 2, . . ., 6ÿu Meyer ~š£>kFÖá

, ĤÌân p M

2.3 MRA (Multiresolution Analysis)

MRA, C˚ÑÖ½j&˛È}&, Níuø£ä˛Èí {Vj}_j∈Z . . . V−2 ⊂ V−1 ⊂ V₀ ⊂ V₁ ⊂ V₂. . .

/

∀f ∈ V_j ⇔ f (x 2^j) ∈ V₀

∪_j∈ZV_j = L²(R), ∩j∈ZV_j = {0}

µó V_j 2ø} Riesz !
ʤ, BbÉn£†!
JI φ ∈ V₀, † {φ(x − n)} } Ê V₀ 2$Aø £†!

ì2

φjk =

√

2^jφ(2^jx − k)

†ú©_ j ∈ Z, {φjk}_k∈Z } A V_j 2íø £†!

z V_j ;dA L²(R) í.° íV¡˛È; †úL<#ìí f ∈ L²(R) I Pj(f ) ∈ V_j u f Ê V_j 2íI , †

P_j(f ) =X

k∈Z

hf, φ_jkiφ_jk

¢ÄÑ φ ∈ V₀ ⊂ V₁, FJ φ .âÅ—J-$í

φ(x) =X

n∈Z

c_nφ(2x − n) (2.3.1)

,H˚ÑA (scaling equation), 7Å—Aíƒb φ ˚ÑAƒb (scaling function), c_n ÑA[b (scaling coefficients)

%â {φ(x − k)} í£>4”ZªR) c_n .âÅ—J-

X

n

c_nc_n+2k = 2δ_k0

Í(D¥_ MRA óÉ:í~šƒbÿªJì2A ψ(x) =X

dnφ(2x − n), dn= (−1)ⁿc−n+1 (2.3.2)

¥³í c_n ªJuõbCuµb, Ñ7jZ, IFí c_n ·uõb

°šË, Jì2

ψ_jk(x) =√

2^jψ(2^jx − k), j, k ∈ Z

† {ψ_jk} }u L²(R) 2íø £†!

ú©ø_ ìí j, {ψ_jk} A W_j ˛Èíø £†!, 7~š[b hf, ψ_jki H[Oƒb f Ê V_j D V_j+1 2I íÏæ, 6ÿu

V_j+1 = V_j⊕ W_j P_j+1(f ) = P_j(f ) +X

k∈Z

hf, ψ_jkiψ_jk

Ê V_j ˛È2, ªJl f íI [bD~š[b hf, ψ_jki = ^√1

2

X

n

d_2k+nhf, φ_j+1,ni hf, φ_jki = ^√1

2

X

n

c_2k+nhf, φ_j+1,ni

I h_l = ^√1

2c_l, g_l = ^{(−1)√ l}

2 c_−l+1, FJ

hf, ψjki =X

n

g2k+nhf, φj+1,ni

hf, φ_jki =X

n

h_2k+nhf, φ_j+1,ni

(2.3.3)

,Ht¨ÖL (convolution) í-Z
TXàS*œü íI [bVlœ×  I [bD~š[bj¶, u˘k}jít
7*œ× I [b Aœü I [b í¯Atÿu

hf, φ_j+1,mi =X

k

hf, φ_jkihφ_jk, φ_j+1,mi + hf, ψ_jkihψ_jk, φ_j+1,mi

=X

k

h_m+2khf, φ_jki + g_m+2khf, ψ_jki

(2.3.4)

Í(ø (2.3.3) Hp (2.3.4), cÜ)ƒ

hf, φ_j+1,mi =X

n

n X

k

h

h_m+2kh_n+2k+ g_m+2kg_n+2kio

hf, φ_j+1,ni

J;bê1í½ÿ.â m = n /

X

k

h

h_m+2kh_n+2k+ g_m+2kg_n+2ki

= δ_mn

y‹, g_n= (−1)ⁿh−n+1 , FJ,Hª“A

X

l

h_lh_l+2m = 1 2

X

l

c_lc_l+2m = δ_m0

Ê [1, (3.40)] )ø, ƒb ψ bAÑø_”~š‚ƒb”.âbÅ—

Z | ˆψ(ω)|²

|ω| dω < ∞

w2 ˆψ u ψ íZ sž²

Hp ω = 0, ‹, R φ(x)dx = 1 ¥_‘K, ÿ)ƒ

ψ(0) =ˆ Z

ψ(x)dx = 0

ªc ψ }uø_òä˙šƒb
/

X

n

(−1)ⁿc_n= 0

Ĥ

X

n

g_n = 0

ÄÑ (2.3.1), °vúsiªWZ sž², êÛ φ(ω) = mˆ ₀(ω

2) ˆφ(ω 2) =

∞

Y

j=1

m₀(ω 2^j)

m0(ω) u 2π U‚ƒb, Ñ φ(x) íÔƒb m₀(ω) = 1 2

X

n

c_ne^−inω

FJê1½í‘KÿªŸA

|m₀(ω)|²+ |m₀(ω + π)|² = 1

Hp ω = 0 ÿ})ƒ

m₀(π) = 0, |m₀(0)|² = 1

/

φ(0) =ˆ Z

φ(x)dx = 1

ªc φ(x) }uø_Qä˙šƒb

;W‡Þ (2.3.2), y‹, kψk = 1 ¥_‘K, † {ψ_jk} }uø Ê L²(R) í£†!
1 /ªJ/qí„p

f (x) =X

j,k

hf, ψ_jkiψ_jk

2.4 £>~šƒb

ƒñ‡Ñ¢, Fní£>~šƒb·uUà°ø QäDòä˙š[b h_n, g_n VªW}j

¸½
*£>~šƒbªJRƒÂ£>~šƒb
6ÿu, Uàs úX! (dual ba- sis) ψ_jk ¸ ψe_jk ©ø ·uâÀø_ƒb ψ ¸ eψ í3òF¨A

£>~š!}ª£>~š!V)µÆ, ÄÑ}s Öµj&˛È

. . . ⊂ V₋₂ ⊂ V₋₁ ⊂ V₀ ⊂ V₁ ⊂ V₂ ⊂ . . .

. . . ⊂ eV₋₂ ⊂ eV₋₁ ⊂ eV₀ ⊂ eV₁ ⊂ eV₂ ⊂ . . .

I W_j ˛Èu V_j D V_j+1 ˛ÈíÏæ, Éu W_j 6⊥ V_j
Ê£>!í8”-, }

X

k

|hf, φ_j+1,ki|² =X

k

|hf, φ_jki|² + |hf, ψ_jki|²

Ê£>í8$-, ÄÑ W_j 6⊥ V_j , FJÉ}æÊs_õb A, B Å— 0 < A ≤ 1 ≤ B U )

AX

k

|hf, φ_jki|²+ |hf, ψ_jki|² ≤ X

k

|hf, φ_j+1,ki|²

≤ BX

k

|hf, φ_jki|²+ |hf, ψ_jki|²

B¤ªø, c,-äÉ?\„ {ψ_jk} }uø L²(R) 2í Riesz !, º.?\„Uà¥

!ªJê1í½
!k¥_Üâ, kuùªúX!í–1
óúk W_j , °ší6}

 fW_j Ñ Ve_j D Ve_j+1 ÈíÏæ, 7/J-p£>y í4”: fW_j ⊥ V_j, W_j ⊥ eV_j
%(B by}õƒ

f =X

j,k

hf, eψ_jkiψ_jk =X

j,k

hf, ψ_jki ˜ψ_jk

2.5 ê1½í‘K

ʤBb;b¨| 4 ˙š[b, }u

h = (hn)_n∈Z, g = (gn)_n∈Z, ˜h = (˜hn)_n∈Z, ˜g = (˜gn)_n∈Z

w2ís {h, g} àV}j, ÇÕs {˜h, ˜g} àV¯A
l*ø mU c⁰ = (c⁰_n)_n∈Z Çá, z h, g “Vú c⁰ªWL «, )ƒ}jt

c¹_n=X

k

h_2n−kc⁰_k

d¹_n=X

k

g_2n−kc⁰_k

(2.5.1)

yV}z ˜h, ˜g “Vú c¹_n, d¹_n ªWL «Í(ó‹, )ƒ¯At

˜

c⁰_l =X

n

h˜h_2n−lc¹_n+ ˜g_2n−ld¹_ni

(2.5.2)

z (2.5.1) Hp (2.5.2)

˜

c⁰_l =X

k

h X

n

h˜_2n−lh_2n−k+ ˜g_2n−lg_2n−ki c⁰_k

7ê1í½ÿu ˜c⁰_l = c⁰_l, ]b°

X

n

h˜h_2n−lh2n−k+ ˜g2n−lg2n−k

i

= δlk

I z = e^−iω, ªJzmUD˙š[b[Aø_ z ƒb (z-function)

h(z) = X

n

hnzⁿ, c⁰(z) =X

n

c⁰_nzⁿ, etc, . . .

I

¯

a(z) =X

n

a−nzⁿ=X

n

a_nz⁻ⁿ

†ú |z| = 1 ¸ a_n∈ R } a(z) = ¯a(z)

7J,’m, ZªZŸ}jtÑ

c¹(z²) = 1 2 h

h(z)c⁰(z) + h(−z)c⁰(−z)i

zd¹(z²) = 1 2 h

g(z)c⁰(z) − g(−z)c⁰(−z) i

7¯AtÑ

˜

c⁰(z) = 1 2

h¯˜h(z)h(z) + ¯˜g(z)g(z) i

c⁰(z) + 1 2

h¯˜h(z)h(−z) − ¯g(z)g(−z)˜ i

c⁰(−z)

*,Võê1½ÿu

1 2 h

h(z)¯˜h(z) + g(z)¯˜g(z)i

= 1 (2.5.3a)

1 2 h

h(−z)¯˜h(z) − g(−z)¯˜g(z) i

= 0 (2.5.3b)

ÄÑ (2.5.3a) FJBbø− h(−z) ¸ g(−z) 1.}°vÑ 0, 6ÿu…b.}u°É;, C˚Ñ”
* (2.5.3b) êÛ, J g(−z) = 0, † h(−z) = 0 C ¯˜h(z) = 0, ‹, g(−z) D h(−z) ”, FJ h(z) = 0
°Ü, ç h(−z) = 0 v, 6})ƒ ¯¯˜ g(z) = 0 6ÿu˜







g(−z) D ¯˜h(z) u°íÉ;

h(−z) D ¯g(z) u°íÉ;˜

FJ

¯˜

h(z) = g(−z)p(z)

¯˜

g(z) = h(−z)q(z)

(2.5.4)

¥³í p, q ·u z ícbŸjÖá

$,, Ê (2.5.4) í ¯g(z) ¥_si° g(−z), )ƒ˜

¯˜

g(z) · g(−z) = h(−z)q(z) · g(−z) ·p(z)

p(z) = q(z)

p(z) ·¯˜h(z)h(−z)

ÄÑ (2.5.3b) FJ q(z) = p(z) 6ÿu

¯˜

h(z) = g(−z)p(z)

¯

(2.5.5)

yzR)í ¯˜h, ¯g H (2.5.3a), )ƒ˜

p(z) h

h(z)g(−z) + h(−z)g(z) i

= 2

7,í p(z) ñøª?íj¹Ñ

p(z) = αz^k FJ

h(z)g(−z) + h(−z)g(z) = 2α⁻¹z^−k

α uµb, k ucb
yz p(z) Hp (2.5.5) )ƒ

¯˜

h = αz^kg(−z), ¯g = αz˜ ^kh(−z)

ԁVz, I k = 0, α = −1 )ƒ

¯˜

h = (−1)g(−z), ¯g = (−1)h(−z)˜

C6

g_n= (−1)ⁿ⁺¹˜h−n, ˜g_n = (−1)ⁿ⁺¹h−n (2.5.6)

Í(Hp (2.5.3a), )ƒ

h(z)h(z) + h(−z)¯˜ h(−z) = 2¯˜

6ÿu

X

n

hn˜hn+2k = δk0

2.6 ˙š[bt

*_"£>~š!víAƒb φ ÇáVì2 eφ
•à‡Þíì2j

φ(x) =X

n

c_nφ(2x − n) =√ 2X

n

h_nφ(2x − n)

φ(x) =e X

n

˜

c_nφ(2x − n) =e √ 2X

n

˜h_lφ(2x − n)e

I m₀ ¸

me₀ }Ñ φ ¸ eφ íÔƒb m0(ω) = 1

2 X

n

cne^−inω = 1

√2 X

n

hne^−inω

me₀(ω) = 1 2

X

n

˜

c_ne^−inω = 1

√2 X

n

˜h_ne^−inω

ú φ ¸ eφ dZ sž²

φ(ω) = mb ₀(ω 2) bφ(ω

2) =

∞

Y

j=1

m₀(ω 2^j)

be

φ(ω) =me₀(ω 2)bφ(e ω

2) =

∞

Y

j=1

me₀(ω 2^j)

Éb m0(0) = 1 = me0(0) , 6ÿu bφ(0) = 1 = bφ(0) v, †,Hís_ÌÌ ÿ}Y¹
e

¥H[ φ ¸ φ øšuQä˙šƒb
e

°ší, ªJਣ>~šƒb ψ íj¶V¨|£>í ψ ¸ eψ

ψ(x) =X

n

dnφ(2x − n)

ψ(x) =e X

n

d˜nφ(2x − n)e w2 d_n = (−1)^lc˜−l+1, ˜d_n = (−1)^lc−l+1 /I g_n= ^√1

2d_n, ˜g_n= ^√1

2

d˜_n y‹, (2.5.6) ¥ _‘K

ψ(x) = √ 2X

g_nφ(2x − n) =√ 2X

(−1)ⁿ⁺¹h˜−nφ(2x − n)

ψ(x) =e √ 2X

n

˜

g_nφ(2x − n) =e √ 2X

n

(−1)ⁿ⁺¹h_−nφ(2x − n)e

øšú ψ ¸ eψ UsidZ sž²

ψ(ω) = eb ^iω2 me0(ω

2 + π) bφ(ω 2) be

ψ(ω) = e^iω2 m₀(ω

2 + π)bφ(e ω 2)

Hp ω = 0, )ƒ me₀(π) = 0 = m₀(π), 6ÿu bψ(0) = 0 = bψ(0), ¥H[ ee ψ ¸ ψ øšu òä˙šƒb

à°‡ÞFì2í ψ_jk øš, ªJì2

ψ_jk =

√

2^jψ(2^jx − k)

ψe_jk =

√

2^jψ(2e ^jx − k)

;W [4, ìÜ 3.2], ÉbæÊ/_b C ¸  U) bφ ¸ bφ Å—e

|bφ(ω)| ≤ C(1 + ||)^−12−

|bφ(ω)| ≤ C(1 + ||)e ^−12−

6ÿu φ, bb φ ∈ Le ²(R), ÿ} ψ, eψ ∈ L²(R), Í(úL< f ∈ L²(R)

f = X

j,k∈Z

hf, eψ_jkiψ_jk = X

j,k∈Z

hf, ψ_jki eψ_jk

{ψ_jk}, { eψ_jk} Ì} Aø L²(R) 2í Riesz !
7 { eψ_jk} u {ψ_jk} íúX!, , HíµsBb, .BbuUàµø !Vú f d}j, ·ªJyâÇø úX!

ø f ½V

Q-VbÜyÖÉÔƒb m₀ í4”
JÔƒb m₀ Å—

m₀(ω) = e^−iλω|m₀(ω)|, ∀λ ∈ R

ÿ˚ m₀ x(4óPÏ (linear phase)
âk m₀ u 2π U‚ƒb, FJ}FU λ ∈ Z, Ô

Vz, ªJI λ = 0
‹,Éb c_n uõb, ÿ}U),ŸA

m₀(−ω) = m₀(ω)

¥šíú˚G, ú˚2-Ñ 0 (ú˚k c₀) FJ¢˚ÑXú˚

,Þú m₀ ín˛%§Î7 Haar ƒbí8”, ¥uÄÑ Haar íAƒb φ uú˚k ¹₂
7.u 0 6ÿu

φ_Haar(1 − x) = φ_Haar(x)

ú˚k ¹

2 íAƒb φ Fú@íÔƒb m₀ .Å—,HXú˚í, 7Å—

e^−iω2 m₀(−ω

2 ) = e^iω2 m₀(ω 2)

e^iωm₀(ω) u 2π U‚ƒb, /ú˚k c₁, C˚ÑJú˚

ʤcqF\níAƒb φ, eφ wÔƒb m₀, me₀ .uXú˚ÿuJú˚; àå

íimVõ, 6ÿuåú˚k c₀ .Íÿuú˚k c₁

²ìAƒb φ,  m₀ (, Bb´Ûb²ìø_me₀ VU)-HA

m₀(ω)me₀(ω) + m₀(ω + π)me₀(ω + π) = 1 (2.6.1)

me₀ ¸ m₀ øšxó°íXú˚CJú˚í4”

ÇøjÞ, m₀, me₀ @b?\ (1 + e^−iω)^L(1 + e^−iω)^L˜ FcÎ, L, ˜L ≥ 1
Jb°L, ˜L ?

;W [4, 4” 6.2] )ø, J m₀,me₀ uXú˚, † m₀,me₀ ÿª[A

m₀(ω) = cosω

2

2l

p₀(cos ω)

me₀(ω) = cosω

2

2˜l

ep₀(cos ω)

JÑJú˚, †ªJŸA

m₀(ω) = e^−iω2  cosω

2

2l+1

p₀(cos ω)

me₀(ω) = e^−iω2  cosω

2

2˜l+1

pe₀(cos ω) w2 p₀ uøÖá, p₀(−1) 6= 0, l, ˜l ∈ N

.uXú˚CuJú˚, Hp (2.6.1) ·})ƒ-

 cosω

2

2k

p₀(cos ω)ep₀(cos ω) + sinω

2

2k

p₀(− cos ω)ep₀(− cos ω) = 1

ÊXú˚v k = l + ˜l; ÊJú˚v k = l + ˜l + 1
Jà ^{1−cos ω}₂ = sin^{2 ω}₂ Hp,, )ƒ

 cosω

2

2k

P (sin² ω 2) +

 sinω

2

2k

P (cos² ω 2) = 1 C6u

(1 − x)^kP (x) + x^kP (1 − x) = 1, x = sin²ω

2 (2.6.2)

;W [4, ìÜ 6.3] J p₁, p₂ ÑŸb}u n₁, n₂ íÖá, / p₁, p₂ ³u°íÉ; (

”); †}æÊñøí, |òŸb}Ñ n₂− 1, n₁− 1 íÖá q₁, q₂ U)

p₁(x)q₁(x) + p₂(x)q₂(x) = 1 (2.6.3)

A
@à¥_ìÜ, I p₁(u) = (1 − u)^k, p₂(u) = u^k, ÄÑ p₁(1 − u) = p₂(u), à 1 − u VH u Hp (2.6.3) )ƒ

p₂(u)q₁(1 − u) + p₁(u)q₂(1 − u) = 1

I qe₁(u) = q₂(1 − u), qe₂(u) = q₁(1 − u) ;W q₁, q₂ íñø4, )ƒ q₂(u) = q₁(1 − u) U) P (u) ≡ q₁(u) íü}u (2.6.3) íø_j
ʤ8”-, ½Ÿ (2.6.2)

P (u) = (1 − u)^−k − u^k(1 − u)^−kP (1 − u)

ÊU¬Giú P (u) dœ Ç, ˛ø P íŸbu k − 1, FJÉbLJÞí k áÿß, )ƒ

P (u) =

k−1

X

n=0

 k + n − 1 n

 uⁿ

°v, P (u) 6uŸb|üíj

FJ

P (sin² ω

2) = p₀(cos ω)pe₀(cos ω) =

k−1

X

n=0

 k − 1 + n n



(sin² ω 2)ⁿ

P (cos² ω

2) = p0(− cos ω)pe0(− cos ω) =

k−1

X

n=0

 k − 1 + n n



(cos² ω 2)ⁿ

J-BbÜ CDF [4]5Ôy£>~šƒbí¨¶
I φ_N Ѽb N íš‘ƒb (c 2.1

), * [4, p540] )øú@íÔƒb m^N₀ 6ªJŸA m^N₀ (ω) = (^1+e₂^−iω)N e^iωbN2c

= e^−iσω2(cos^ω₂)^N

=

N −b^N₂c

X

n=−b^N₂c

2^−N

 N

n + b^N₂c

 e^−inω

= 1 2

X

n

c_ne^−inω

/

m^2L₀ (−ω) = m^2L₀ (ω), m^2L+1₀ (−ω

2) = e^iωm^2L+1₀ (ω 2)

I l = L, p₀ ≡ 1 †Å— (2.6.1) í me₀ ÿ}u

me^{N, ˜}₀ ^N = e^−iσω2(cos ω

2)^N˜hX^k−1

n=0

 k − 1 + n n

 (sinω

2)²ⁿi

= 1 2

X

n

˜ cne^−inω

N ≥ 1, N + ee N = 2k ÑXb, 7

σ =







0, if eN ÑXb 1, if eN ÑJb

Êüıøà N = 6, eN = 8 í˙š[b, dbW}&

úı ÞÓõðDmU¦)

rÖ-¦Sè¸A3ÿ%Í$º477óÉ, JbòQW¿qDA3ÿ%º4, .â%¬a ï4œ×íGX, úA7k, ÿÜ »W…<
FJú@æ»ç7k, ê|aï4ü/?

õ¥@A3ÿ%º4íj¶ÿé)½b
Ñ7\Mdıíêc4, ÔËø [2] ³FTƒ5Ó õðj¶Y“ʤ, JX¡5

3.1 AÑŒé¶}

ýÁ Wistar 43 (½¾ 380 B 450 s) â:5·¦ pentobarbital (50 mg/kg), ,}(, d

$ Ó0žJZ¿¾09¸“ÓíÓ0·¦
d−ž1Q,A ãÜÂ, øã−í ù“ïÖ¾−„Ê 3.5% B 4.0% 5È
%âìÅÚ&MÓíÅk 38 ± 1^◦C

QO, ~Ç43:5, A˝‡,jí!ì -2}×|‡>>ÿ%, J¼äø‡>>ÿ%5±

-«H4ú;, z¡-«0k.é•Â”Ú”,, kò¦>kÚ”¸ÿ%¶ˇàJ ìÿ%¸

Ú”1Ž¤"í£Ê¢ÿ%,
ÿ%íÚP‰“mU%¬øä Ñ 30 ∼ 3K Hz 5¦˙

šÂªW˙š, y[×øNI(JÖ¦−~p“Âp“k~2
°v, ¦9mU%0ž

fBž? (ä0¸ˇ:DC ∼ 75Hz ) ž²ÑÚPmU, ‚àéª[×Â, D>>ÿ%5Ú mU°¥p“k~2
&õð!!(, Ó0·¦ÿ%®i™ hexamethonium bromide (20 mg/kg) J Î>>ÿ%º4, p“Ìÿ%ÚmUvÍ$í*
ÆmM

43å¶J ñìPi ì, J(Fm‘øçN|ckÈ(J-
…T
A'˝×}´Ç '˝ Ð|7á*¶, ú”À˘Ú”J 20 éižp7¶, «¿ obex ‡(® 2.0 … 2 (˝¬® 2.0 … *¶[Þ¿ 0.5-2.5 mm 2í>>EU–
ÊJÚŒéíjvƒ

>>EU–(, kÇá ÚŒé5‡, lÓ0·¦ÿ%-w X¸í®i™ gallamine (50 mg/kg), β1-adrenoceptor í®i™ atenolol (1 mg/kg) £ angiotensin II converting

enzyme í{„™ captopril (10 mg/kg)
¤(©½
B
ü} Z^k·¦®“ÓJ,H

™¾íû}5ø
QOJ ÚŒé (Anapulse stimulator model 302-T, WPI Inc., New Haven, CT) ªW.°‘Kíjš0§Œé
Œé‘Kí¡bà-: 0§vÑ 500 ms, 0§#Ñ 10-50 mA, 0§ä0Ñ 50 Hz, c 0§ívÑ 500 ms
7s 0§È ívÑ 0.6 B 50 ”, ø_ ŒéívÈÑ 80-90 ”
s_ 5ÈíȽv†eÓ í¦9£>>ÿ%º4Sv+ ì7ì
k…õð2, ©øÁ43תêA 20 _ í Œé

õð!!(, ¦64”p“k~qí©_ ŒéF)5¦9£>>ÿ%5éªmU, ‚àé ª/bPž²Â BIOPAC (System, Inc., Goleta, CA) J 6 kHz ¦šä0ž²ÑbPm U
>>ÿ%íbPmUÇy‚à½0vÈqÑ 20 …”íbP }ÂT‡TÜ

3.2 ÌAÑŒé¶}

ýÁ Wistar 43 (350 B 450 s) J pentobarbital (50 mg/kg) ,}(, àø¶MFH êA$ Ó0ž£q‡>>ÿ%íp“Ú”
ÇÕJø.²•Â”Ú”â:5.p1%

‡JRd.íw , JZp“ãܬ˙2d.w íwÚP‰“
GXêH(, Ó&Ó)+

Bú4w(2¦?ßÞò<¥¦vZâÓ0žlJ pentobarbital (12.5 mg/kg) Tü™

¾,}, p“ÓÊ`~¬˙2, ¦9£>>ÿ%ímU
çÓ¢)+ƒ?ßÞò<¥¦(

ÿT¢p“1úÓªW>9¥¦ ÎGX
GXêA(, à,5p“˙åy½µøŸ
p“

F)5¦9£ÿ%mU©½ø} ¦ 64 ”ívÈÅ, ÄÑ©¨p“í,vÑù
üB ú
ü} , FJ©¨p“ª)ƒ 13 ’e

ÇÕýÁ Wistar 43 (½¾ 350 B 450 s) k$ Ó0ž£qd.wÚP¸‡>>ÿ

%º45p“Ú”GXêA(, &Ó)+Bú4w(2¦?ßÞò<¥¦v, âÓ0žl J pentobarbital (40 mg/kg) Tœ×™¾,}, p“ÓÊ`~¬˙2¦9£>>ÿ%í

mU
çÓ¢)+ò<¥¦(, T¢p“
p“F)5¦9£ÿ%mU©½ü} ¦ 64 ”í vÈÅ
ÄÑ©¨p“í,vÑþ
B
} , FJ©¨p“ª}A 15 í’e

Ê [2] ¥¹d³N|, ¦9ä$2 0.02 ƒ 1.7 Hz íä0¸ˇq, >>ÿ%x|wŠ 0×üí?‰
úkÔìä0í>>ÿ%º4‰}¨A°ä0í¦9‰
7/ÓOŒéä 0íÓ‹, >>ÿ%º4ú¨A¦9‰¾}ANb-±

ûı ä$}&DvÈå

ÊbPmUTÜ,, 'Ö$lj¶\àV,¿.°bPmUÈíóÉ4íì¾j¶
Ê

¥³, Bbbú¥<\àV,¿ä$íóÉ$lj¶, dªø¥íªWj„Dn

4.1 Cross-Correlation

I x(n), y(n) Ñs Ì&íbPmU , †…b˛¤Èí cross-correlation ÿuløw 2ø bPmU x(n) ì, Çø bPmU y(n) y%â‚àvÈôb (time delay) VD x(n) ªWq (y°w‚M, 6ÿu

rxy(m) = E{x(n)y(n + m)}

FJ

r_xx(m) = E{x(n)x(n + m)}

r_yy(m) = E{y(n)y(n + m)}

6âkvÈôb¥_ÄÖ, U)mUTÜ,í cross-correlation }.°k$l, íÀøbM

$, 7ZJø åí$æÊ
ԁVz, †Bb˚ r_xx ¸ r_yy Ñ autocorrelation

4.2 Cross-Covariance

ÇøéN cross-correlation í$l¾ÿu cross-covariance
løs mU, lÁ _

킁M(, yd cross-correlation «, 6ÿu

cxy(m) = E{[x(n) − µx][y(n + m) − µy]}

µ_x Ñ x(n) 킁M, µ_y Ñ y(n) 킁M, ÄÑ‚MÊ«vx(4É[FJ,H c_xy

¢ªJŸAJ-$

c_xy(m) = r_xy(m) − µ_xµ_y

4.3 CSD D PSD

7 cross-correlation í–1(, ÿªJªø¥«nä$}&íj¶
s mU x(n), y(n) í Cross Spectral Density (CSD), ÿu‚à x(n), y(n) í cross-correlation V½hì 2ø_híúiÖá

P_xy(ω) =X

m

r_xy(m)e^−iωm

7

P_xx(ω) =X

m

r_xx(m)e^−iωm P_yy(ω) =X

m

r_yy(m)e^−iωm

}u x(n) D y(n) í Power Spectral Density (PSD)

I

X(ω) =X

n

x(n)e^−iωn Y (ω) =X

n

y(n)e^−iωn

}Ñ x(n) D y(n) íúiÖá, y‹, r_xy ¥‘ä, ÿªJ½hZŸ,Hí CSD t

P_xy(ω) = X

m

r_xy(m)e^−iωm

= X

m

X

n

x(n)y(n + m)e^−iωm

= X

m

X

n

x(n)y(m)e^−iωme^iωn

=  X

n

x(n)e^iωn X

m

y(m)e^−iωm

= X(ω) · Y (ω)

°Ü

P_xx(ω) = X(ω) · X(ω) P_yy(ω) = Y (ω) · Y (ω)

4.4 Phase ƒbDž²ƒb (Transfer Function)

ílì2ž²ƒbt

P_xy(ω) = H(ω)P_xx D Phase ƒbt

Φ(ω) = tan⁻¹

nIm[P_xy(ω)]

Re[P_xy(ω)]

o

J-Bb%âøÔy8”Vzp Phase Dž²ƒb

I x(t), y(t) s mUÈíÉ[ÉÀÓívÈÏ τ , 6ÿu

x(t + τ ) = y(t)

,Þ¥‘äuJ y(t) dÑ™íVõí: J τ > 0 [ýÊv, x(t) äl y(t), J τ < 0 H[ x(t) r( y(t), τ = 0 †u x(t) = y(t)
¥älDr(íÉ[¦²kJ x(t) C y(t) dÑ™íVõí!‹

ú,H y(t) ªWZ sž², )ƒ

Y (ω) = Z

y(t)e^−iωtdt

= Z

x(t + τ )e^−iωtdt

= e^iωτ Z

x(t)e^−iωtdt

= e^iωτX(ω)

ø,ÞFRû|Víäp CSD t, )ƒ

P_xy(ω) = X(ω) · e^iωτX(ω) = e^iωτP_xx(ω)

Dž²ƒbTªœÿ}êÛ

H(ω) = e^iωτ

6ÿu x(t) D y(t) ÊÉvÈÏí8”-, wž²ƒbcuÄÑvÈÏF¨AíóPÏ e^iωτ 7˛

ÄÑ P_xx …™uø_õƒb, Pà Euler t, ÿ})ƒ

P_xy(ω) = P_xx(ω) cos(ωτ ) + iP_xx(ω) sin(ωτ )

I

Re[P_xy(ω)] = P_xx(ω) cos(ωτ ) Im[P_xy(ω)] = P_xx(ω) sin(ωτ ) }Ñ P_xy íõ¶D™¶
FJ

Im[P_xy(ω)]

Re[P_xy(ω)] = tan(ωτ ) I

Φ(ω) = tan⁻¹nIm[P_xy(ω)]

Re[P_xy(ω)]

o

= ωτ

Φ(ω) ÿuÊøOd³cí ”phase”
Ĥ‡ú.°íä0õ ω ÿªJv| x(t), y(t) óúívÈÏ, 6ÿu

τ = Φ(ω) ω b·<íu, ,b<2.â ω 6= 0 nW

FJ, úø_ ìíä0õ ω 7k, à‹vÈÏ τ = 0 † x(t) = y(t), X(ω) = Y (ω), P_xy(ω) = P_xx(ω), H(ω) = 1 7/ Φ(ω) = 0

4.5 Coherence

Coherence \àV,¿s mUÊä$íóN

C_xy(ω) = |P_xy(ω)|²

P_xx(ω)P_yy(ω) = P_xy(ω)P_xy(ω) P_xx(ω)P_yy(ω)

wM}k [0, 1] , MQ¡ 1, H[s mUÊä$íóN×
Jø‡Hí P_xx, P_xy, P_yy òQp, †

C_xy(ω) = X(ω) · Y (ω) · X(ω) · Y (ω) X(ω) · X(ω) · Y (ω) · Y (ω)

= X(ω) · Y (ω) · X(ω) · Y (ω) X(ω) · X(ω) · Y (ω) · Y (ω)

= 1

})ƒÌÑ 1 íbƒb, 1Ì¡?‰, 6.?TX’m
ÑU)Flí coherence M

¡?‰, J-zpsZªíj¶

4.6 Bartlett D Welch j¶

‡Hl coherence j¶, uòQ* P_xx, P_xy, P_yy )V
;d& N í x(n) u}0Êc_

v,, Í(‚àø_&D x(n) ó°í ”¢” z x(n) *c_v,¦-V
¥óòQ/

ÄÉí ”¢”, ÿ}U) P_xx, P_xy, P_yy '×í variance, ¥ší!‹, ÿuBbz x(n)

&Ó‹6ÌÈk9

ѱQ P_xx í variance, ø& N í x(n) HàA m ¨&ÌÑ ^N_m /.½Lí}Òm U
}ú©ÒmUªW P_xx «, )ƒ}ÒmU®Aí P_xx¹ , P_xx² , . . . , P_xx^m Í(¦ P_xx = (P_xx¹ + P_xx² + . . . + P_xx^m)/m V¦HŸlì2í P_xx
à¤, ÿ?±Q P_xx í variance
7 /ŸmU x(n) F?~’í}ÒmUbñÖ, w P_xx 5 variance M±Q^‹ÿpé
ç m Q¡ N v, P_xx íM¡k 0
°ší, 6ªJUàó°íj¶Vl P_xy ¸ P_yy

¥ømU}’Ñ.½Lí}ÒmUj¶, ÿ˚Ñ Bartlett j¶

Bartlett j¶˛%ªJ^±Q P_xx, P_xy ¸ P_yy í variance, OEÿÜ
Ì&ím U, F?}’í}ÒmU,buÌí, |Öÿ N ¨, ©¨}ÒmUÿɨÖø_bM, ¥ š ͪJ®ƒ|Öí}ÒmU, º6ÄÑ©¨}ÒmU¬s7„?¨Ö—D’m

Ñj² Bartlett j¶íÿÜ, Ê 1976  Welch TX7ø_Z¾íj¶
3bíZªÿu, o r}ÒmUªJóí½L
Žâ½LíÓ‹, ÿªJ)ƒ—DÖí}ÒmU, 7/©¨}

ÒmU·?DÖ—Dí’m
à¤, ÿ?y^±Ql P_xx, P_xy, P_yy vF¨A variance

MATLAB í Signal Processing Toolbox ³, TXø_Uà Welch j¶í cohere( ) ƒ

Vls mUí coherence, Êú Welch j¶š7jJ(, ÿ?Uà cohere( ) ¥_

ƒVl

J-u MATLAB 5 cohere( ) ƒ펵:

[coh, f]=cohere(x, y, nFFT, Fs, window, noverlap)

w2 x, y us ÄeàVl˛¤ coherence ímU, nFFT Ñ}ÒmUí&, Fs u¦

šä0
window †u6}ÒmU‹ ”¢”, Wà: hamming, hanning 
”¢” í&.

âD nFFT íMó°
‹7 ”¢” í}ÒmUʪW FFT vÿªJÁý leakage íßÞ

|(, noverlap Ñ}ÒmUÈ˛¤í½L
ç noverlap=0 v, ÿóçkUà Barlett j

¶

J-BbøJøPõðV‡ nFFT ¸ noverlap ¥s_¡b
BbàVdõðímUÑ allcos ¸ allsin, ì2à-: ÀPvÈqËÓä0Ñ k í cos D sin ƒb, }J 20 ÑȽ ªWÚ‹)ƒs_hƒb

f_cos(x) = X

k

cos(2πkx) k = 0, 20, 40 . . . , k_max

f_sin(x) =X

k

sin(2πkx) k = 0, 20, 40 . . . , k_max

k_max Ñ|×ËÓä0, / f_cos(x), f_sin(x) x ^π₂ íóPÏ
ÊvÈ x ∈ [0, t] í¸ˇq, ú f_cos(x) D f_sin(x) J f_s í¦šä0ªW¦š)ƒbPmU allcos D allsin
Ĥ allcos D allsin &ÌÑ N(=f_s× t)

I

nFFT = N ∗ (m/10), m = 1, 2, ..., 10 noverlap = floor(nFFT ∗ op), op = 0, 0.1, ..., 0.9

Í(‡ú.°í}ÒmU& (nFFT) D½Lì}ª op (overlap percentage) V}H p cohere( ) l allcos D allsin í coherence M, 1$lwM×k 0.5 F25ì}ª
- [uJ k_max = 100, f_s = 1024, t = 10 ªWlíWä, w2 - H[.x¡?‰íÕ”, ].8$l

op 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

m1 66 33 33 44 33 68 33 34 33 33

2 66 31 38 44 54 69 31 31 31 31

3 71 59 53 42 37 74 59 38 31 33

4 78 53 69 42 55 73 46 63 30 31

5 70 73 76 80 85 75 80 79 78 80

6 - - - - 56 82 65 48 71 34

7 - - - 56 63 69 86

8 - - - 68 59

9 - - - 80

10 - - - -

%â,[ªJõ|, Î7µ<ÄÑ nFFT &¬ÅF¨A allcos D allsin mU˛¤Ì¶}

< (coherence ·u 1, J - [ý) .8$lí8”Õ, Ê m=5 v, Ìœ×íM
ÖÍ op=90%, m=6 v 85.91 íM, OuUà&œÅí}ÒmU&Dœòí½Lì}ª, }ªwF ¯ÖyÖí}ÒmU, ¨Ay×íl¾, FJ op=40%, m=5 }uªœß í²Ï, 6ÿu nFFT=^N₂ / noverlap=^N₅

¤Õ, } à,[bM´ k_max ¥_ÄÖ
k_max qìí×, cñbMÿ}O-±, ÄÑ œ íä0¸ˇ, ÿóú}yÖ coherence ük 0.5 íä0A}
Ou¹UbM-±, óú

|×MßÞT´u.‰
J¤ZªTÑl coherence v, ²Ï¡bvíø_YW

J,Fní nFFT D noverlap s¡b, ø}Êüıl¦9D>>ÿ%º4mU˛¤

Èí coherence vUà

üı bW}&

5.1 }&j¶

Ê [2] d³Tƒ, Fbíä$}&j¶uløp“¦9¸>>ÿ%º4íbPmU}¨d×

àZ sž², ¦Ì, yªWä$}&
¥.%^òímU%%¨ÖrÖÆm, 7 àƒ ä$}&í¹”¸Äü4
ʤBbùª Wavelet í;¶, ı‚à Wavelet …™x_@

4í ”¢” íÔ4, V6ŒBbªWmUÊ×àZ sž²‡í‡0TÜ, J‚±Ql¾1 .Ü˙š^‹
ÊõT,, ‚à MATLAB VªWlD˙Çê
J-Êä$}&,, 6}

ø·<‰rÕ2Ê 0.02 ƒ 1.7 Hz íä0¸ˇqVªWhô

øæ¦9 (BP) ¸>>ÿ%º4 (SNA) DmU“vÈí’e-p, }ªW~š}j, ømU}jA.°µíQäDòä
Ê 0.02 ƒ 1.7 Hz ¸ˇq, >>ÿ%º4Œéä0}

ú¦9ßÞ°äËÓ
¤°äËÓ, æÊk¦9D>>ÿ%º4í®µQämU³

MATLAB í Wavelet Toolbox TXD~šƒbóÉíƒ
Wà: dwt( ) D wavedec( ) }TXøµ~š}jDÖµ~š}jŠ?
Éuú£>Íí~šƒb7k, Uà Wavelet Toolbox FTX탥7}mU%¬}j(, QäDòä}j[b&.£üí¥^‹

J j = 10 & 1024 ÑW:

n A_n D_n N/2ⁿ diff 1 520 520 512 8 2 268 268 256 12 3 142 142 128 14 4 79 79 64 15 5 46 48 32 16 6 32 32 16 16 7 24 24 8 16 8 20 20 4 16 9 18 18 2 16 10 17 17 1 16

¤Ïæ4, ¢}ú}j(íQä[bÊZ sž²v¨Aä$Rí.£ü!‹
ku, ŸJ Wavelet Toolbox FVíZ‚4, ½hªŸ~š}jí˙V^£mU}j(íQòäm U&.£üí½æ

5.2 ˙š[b¦)

‡ú Biothogonal 6.8 (bior6.8), …Ûbs Qò˙š[b, w2ø àVªW}j, Çø àVªW¯A, 6ÿuÛbû .°í˙š[b, Qä}j (Ld), Qä¯A (Lr), òä}j (Hd) Dòä¯A (Hr) û [b
Uà Wavelet Toolbox ³í wfilters( ) ÿªJ¦)¥û Qòä˙š[b, 9lJ .mat í$æ[, Ê.bv-p

5.3 ~š}jƒ

mydwt( ) uªW~š}jíƒ, 3bíÆj¶VA [1, Ch4]
Žµà-:

[A, D, L, tout]=mydwt(sig, step, tin)

w2 sig ukªW~š}jímU, step ubªW}jí ”Ÿb”, tin †umUFp“ív È
mydwt( ) ƒÉTÜ 2 ícbŸj&ímU, JpmU.¯, †S¦i mUíj

VTÜ
1øú@ívÈõf# tout
ÉbmU…™Fp“ívÈDÅ, i mUíT 1Ì.]

âk V_j = V_j−1L W_j−1, Ê£>~šƒb˛È1.A , kuUà A_n(Approximation)

¸ D_n (Detail) }[ýmUÊ n µ~š}jíQäDòä[b, 6ÿu

L D

Í(ø©Ÿ}jíQäDòämU}YÕ¯9ƒ A £ D ³Þ
â L p“®µ~š}jí QòämUí&
°v, L …™í&6H[7¤mUF}jíŸb

adcoefget( ) ª* A D D ³¦|®µ}jíQäDòä[b
Žµà-:

w=adcoefget(x, L, level)

w2 x ªJu A C D, L uæ[®µQäDòä&í²¾, level †uNìk¦|¨øµ QäCòämU, Í(f# w

¦9C>>ÿ%º4mUÊ%â mydwt( ) ªW}j(, ªJzFíQä[bDòä[b

½hõAs_.°éímU, N¬Z sž²ªWä$}&, ¥¶}Uà fftspec( ) VêA

[spec, freq]=fftspec(sig, time)

fftspec( ) ÛbpkªWZ sž²ímU (sig) £wFp“ívÈ (time), Í(fmU íä$ (spec) Dwú@íä0W (freq)

5.4 !‹Dn

J [2] qTXF‚ sti3 í BP ¸ SNA mUÑW (Figure 7), Œéä0 0.63 Hz í SNA ú BP F¨Aí°äPÓ%¬ FFT (Àʪc (Figure 1)
OuFl|Víä$Dw coherence 1.Ü; (Figure 3)
ŸÄÊkòQªW FFT í BP ¸ SNA mU…™Öí Æm, 7¥<ÆmZ‹27}&!‹
kuÊú BP ¸ SNA ªW FFT ‡, lømUªW~

š}j, ©%¬øŸ~š}j, í¶}ÿ}Ožƒòä, ÿóçuú©øµQämU ªW˙š

%¬~š}j(, BP ¸ SNA ÿªJ}„jAs .°ÍíQä[b (An) Dòä[b (Dn)
ªJêÛ, ׶}íòä[b·uÆm¸¶Mä0íž
7khôí 0.02 ƒ 1.7 ä 0¸ˇÿæÊk®µíQä[b³Þ
ÎÝ%¬ÝÖµí~š}jnœ}zBbbhôí ä0žƒòä
Í7¬Öí~š}j, F)ƒíQäDòäí&¬s, JyúwªW FFT 6̶y×)Bó’m

ú®µQä[b}ªW FFT, phase ¸ coherence íl, ¦|øµQäí}&!‹1 DòQªW FFT í!‹ªœÿ}êÛ, °äPÓy‹Àʪc (Figure 2), c_ä$6ÄÑ

~š}j^‹7‰)ªœ´, 6œßí coherence (Figure 4)
J./ú%-µíQ äTl, Ê. àhôíä0¸ˇq, ÖÍ˙ší^‹M/Ouw!‹·¸øµí!‹Ï æ1.× (Figure 5, Figure 6), ¥H[׶}íÆmÊøµ~š}jÿ˛%˙Î

©øµíQämU&·uw,øµíQämU&íøš, v,Fú@í Ó‹øI, ú@í¦šä0Áš, ª¿ä OÁš
Jz©µQä[bí FFT Fú@í|×ä0¦|

Võ, ÿ}õƒ|ת¿ä í‰“, ©%¬øµ}j, F)ª¿ä ÿúÁš
cì F Ñm Uª¿ä , † n µ}jíQä (A_n) ª¿ä ÿu F/2ⁿ
‹, F uÌí, FJ.ª?

}Ì¢`í}- 
I [f_min, f_max] Ñkhôíä0¸ˇ, † f_max ≤ F/2ⁿ

6ÿu

n ≤ log₂(F/f_max)

Êkhôíä0¸ˇq, |ÖÉbd n µí~š}jÿªJ7
Wচä0 6144 , ª¿ä  Ñ 3022 , cìbhôíä0¸ˇÊ [0,2] †

n ≤ log₂(3022/2) = 10.5613 6ÿuÉbd 10 µªJ

Î7 0.63 Hz í°äËÓä0Õ, ¦9mUä$Ê 0.05 Hz íä0_.üíPÙ
ÑõÀ

sti3 í¦9mU%¬ 13 µ~š}j(ªJõƒÊ 10 µJ‡·uòò,,í¯Aš, Ou ƒ7 10 µJ(ÿ7p鉓 (Figure 8)
Î7¦9…™í pulse Õ, yõƒ7Ê 10 µJ‡·õ.ƒíø_Ûï — ø_Qäíš
*¥_Qäší°Q¸ˇV²|ú@ä 0, ÿ}êۏßD¦9ä$2 0.05 Hz íä0ó¯
ƒ7 13 µ, BbFbhôí 0.63 Hz í°äËÓä06˛%žƒòä, FJ¥_Qäš6ÿêréÛ|V7

Z sž²ªJ)ømUx¨<.°ä0íA}, OuºÌ¶)øÖ¥<ä0ímUÊv

,íÅó
7~šž²ªJømU„jAÊv,Ö.°AMíQäDòämU, Ouº

³Ÿ¶püíN|mUFÖíä0
FJø~šž²»ºZ sž²ÿªJuèmUÊv

Dä0í[

Figure 1: òQªW FFT í BP D SNA ä$

Figure 2: øµ~š}jí BP D SNA Qää$

Figure 3: BP, SNA òQUà FFT í Phase, Coherence

Figure 4: BP, SNA øµ~š}jQäí Phase, Coherence

Figure 5:
µ~š}jí BP D SNA Qää$

Figure 6: BP, SNA
µ~š}jQäí Phase, Coherence

Figure 7: BP D SNA mU

Figure 8: %¬
µ~š}jí BP D SNA QämU