講義案内 前田研究室 maedalab Fourier

フ エ変換と地震 スペクト 分析

Spectrum Analysis of Earthquake and Fourier Transform

* ^{こ プ ント 新 地震動 ペ ト 解析入門 大崎順彦}, ^{鹿島出版会 を元 作成し} います ：The original of this document was from the above text book.

- ^{世界最初 強震記録： 強震計} Strong-Motion Acceleration: SMAC

Imperial Valley Earthquake at EL Centro(1940.5.18) 326gal^{： 人類初 強震 記録}

(Record of the first strong motion due to earthquake in the human history) Arvin-Tahachapi Earthquake at Taft(1952.7.12) 147gal

- ^{地震波 特長} Properties of Earthquake Wave

対象 す 地盤－構造物系 対し 与え 影響を考え ため 有力 手

最大振幅 maximum amplitude

時間 duration time

包絡曲線 envelope curve : ^主要動 principal shock 波数 numbers of wave

振動周期 _periods

エネ _Energy

Duration time

Principal shock

Maximum amplitude

Finite Fourier Approximation of Time History and Time Series

and its Formulations

1) Approximation of digital time history data with Tri-angle series

Discrete System:

_{t } 0

: data sampling interval

 ^t

⁰

^{, x}

⁰



^,

  ^t

¹

^{, x}

^{1 ,}

 ^t

²

^{, x}

²



^{, …,}

 ^t

^N¹

^{, x}

^N¹



^･･…N =(N-1) +1: data: N conditions

Duration Time:

T _ N _ t

(1.1)

Time:

_{t  m  t}

(m= 0, 1, 2, ^･_{…, N-1)} (1.2)

A data point:

x

_m

_ x _{ } m _ t

N=16(m=0 _{– 15) : x0}, x₁, _{…….., x15}. duration time: T=16_t

2) Approximation of digital time history data with infinite tri-angle series

 









































kt

B

t

B

t

B

B

kt

A

t

A

t

A

A

k k

sin

2

sin

sin

cos

2

cos

cos

2 1

0

2 1

0 (2.1)

What is the period Tp of cos(kt)or sin(kt)?

  ^{      } _ ^{  } _ ^

 ^{k t T kt k t} k

kt cos

_p

cos 2 _ cos ^{2 }

cos

T

_p

_ ² k ^



: As k increases, the period Tp decrease (the frequency f=1/T_p increases).

 



_k^_0

^A

^k

^{cos kt  B}

^k

^{sin kt}

^(2.2)

replace

t

by

t

T



2

or

t

t

N _



2



_k^_0

_ ^{ A}

^k

^{cos 2} _N ^ _ ^kt _t ^{ B}

^k

^{sin 2} _N ^ _ ^kt _t ^ _

^(2.3)

3) Approximation of digital time history data with finite triangle series

Set k to be from 0 to N/2





_

_ ^ _ ^ _{ } ^{ }

_

_ ^{ } _ ^



²

0 2

0

sin 2

cos 2

sin 2

cos 2

N

k

k k

N

k

k k

m

N

B km

N

A km

t

N

B kt

t

N

A kt

x ^{   }

(3.1)

 

 















2 2

1 0

2 2

1 0

,

,

,

,

,

,

,

,

,

,

,

,

N k

N k

B

B

B

B

B

A

A

A

A

A

Here, Number of unknown coefficients is 2(N/2+ 1) (3.2)

Number of unknown coefficient 2(N/2+1) =N+2 > number of conditions (data) N From partial consideration,

For the case of

k _ 0

, ₀

cos ² A

₀

1 A

₀

N

A ^ km _{  }

and ₀

sin ² _{ B}

₀

_ 0 _ 0

N

B ^ km

(3.3)

For the case of

k _ N 2

, ₂

sin ² _ B

₂

sin m _ 0

N

B

_N

^ km

_N

_

(3.4)

Consequently, Eq.(3.1) is reduced to

 

N

m

A N

N

B km

N

A km

A

x

_N

N

k

k k

m

2

cos 2

sin 2

cos 2

₂

1 2

1

0

 _ ^{ }  ^ _ ^  ^

 _

^



(3.5)

For convenience

 

N

m

A N

N

B km

N

A km

A

x

_N

N

k

k k

m

2

cos 2

2 2

2 sin

cos

2

₂

1 2

1 0





 _

 

  _



 _

^



(3.6)

 

 

















 1 2 2

1

2 1 2 2

1 0

,

,

,

,

,

,

,

,

,

,

,

,

N k

N N k

B

B

B

B

A

A

A

A

A

A

N/2+ 1+ N/2-1= N (3.7)

Therefore,

Numbers of unknown coefficient N = Condition Equation N (3.8)

4) Determination of A

_k

and B

_k

with orthogonal property of triangle functions

三角関数 ？/ Why do we use triangle functions for Fourier approximation?

三角関数系 直交性を利用す (We utilize orthogonal property for Triangle Functions)

 ^{ }   ^{ } 



  ^cos  ^cos   ^cos 

cos

2

(a)

       



  ^sin  ^cos   ^sin 

cos

2

(b)

 ^{ }   ^{ } 



  ^sin   ^cos   ^cos 

sin

2

(c)



 ^{1 cos 2}

cos

2

²

_{ }

(d)



 ^{1 cos 2}

sin

2

²

_{ }

(e)

       

sin 2

sin 2

2

cos 1

1

cos

2

cos

cos

cos



 

















N

N

N

 

 



 _ ^























(f)

       

sin 2

sin 2

2

sin 1

1

sin

2

sin

sin

sin



 

















N

N

N

 

 



 _ 























(g)

if

_{ } 0

, summarize the results.

sin 2

sin 2

2

cos 1

cos

1

0



 



N

N

m

N

m

 



_^



^(h)

sin 2

sin 2

2

sin 1

sin

1

0



 



N

N

m

N

m

 



_^



⁽ⁱ⁾

 





 









 





 

 





 



















1

0 1

0 1

0

2 0

2 cos

sin

0

2 2

2 sin

sin

0

2 2

2 cos

cos

N

m N

m N

m

N

km

N

lm

l

k

l

k

N

N

km

N

lm

l

k

l

k

N

N

km

N

lm













(j)

For Ak

たとえ _Akを求める / For example, we calculate the factor Ak,

 

N

m

A N

N

B lm

N

A lm

A

x

_N

N

l

l l

m

2

cos 2

2 2

2 sin

cos

2

₂

1 2

1

0

 _ ^{ }  ^ _ ^  ^

 _

^



(4.1)

1)^{上式の両辺に}

cos _ 2 _ km N _

^を掛ける / Multiplication of

cos _ 2 _ km N _

to Eq.(4.1).

 

N

km

N A

km

x

_m

_ cos ^{2 }

2 2

cos _

⁰



_^

_ ^{ } _ ^



²¹

1

cos 2

sin 2

cos 2

cos 2

N

l

l

l

N

km

N

B lm

N

km

N

A ^ lm ^{  }

 

N

km

N

m

A

_N

_ N _{2 }

2 cos

cos 2

2



2 ^(4.2)

2)

m _ 0

^から

m _{ N } 1

^{ま 総和を る} / Summation from m=0 to m=N-1 in Eq. (4.2)

  _



^N_^1₀

^{cos 2 } ₂

⁰ _m^N_^₀¹

^{cos 2}

m

m

N

km

N A

km

x _ ^

( -> 0 )

 

_^

_ ^

_^

_ ^



²¹

1 1

0

cos 2

cos 2

N

l

l N

m

N

km

N

A ^ lm ^

 

_^

_ ^

_^

_ ^



²¹

1 1

0

cos 2

sin 2

N

l

l N

m

N

km

N

B ^ lm ^

( -> 0 )

 

N

km

N

m

A

_N

_ N _{2 }

2 cos

cos 2

2



2 ( -> 0 ) (4.3)

1st term, 3rd term and 4th term in right formula =0 with account for the orthogonal

  _{ }



^N_^₀¹

^{cos 2 }

^N_l²₁^{1 l}

^ _

_m^N_^₀¹

^{cos 2 cos 2 } _

m

m

N

km

N

A lm

N

km

x _ ^{ }

(4.4)

2 0

0

2 0

2 cos

cos

_{1 2 2 1}

1 2

1

1

0



























 

 











 

 ^{k N}

N

l

N

m

l

^A

A N

A

N A

km

N

A ^ lm ^

(4.5)

N

x km

A N

N

m m

k

^

cos 2

2

¹



_^1



^(4.6)

 



 



































1

2

,

,

2

,

1

2

,

1

2

,

,

2

,

1

,

0

sin 2

2

cos 2

2

1

1 1

1

N

k

N

N

k

N

x km

B N

N

x km

A N

N

m m k

N

m m k





(4.7)



_^



¹

0

0

¹

2

N

m

x

m

N

A

: mean value (4.8)

3)^時間関数

x _{ } t

^の近似: Fourier Approximation

t

m

t _{ }

,

t

m t

 

^(4.9)

  ^{ }

t

N

t

A N

t

N

B kt

t

N

A kt

A

t

x

_N

N

k

k

k

^ _ ^{ } 

 



 

 

 _

^



2

cos 2

2 2

2 sin

cos

2

₂

1 2

1 0







_(4.10)

5) Spectrum Properties

  ^{ }

t

N

t

A N

t

N

B kt

t

N

A kt

A

t

x

_N

N

k

k

k

^ _ ^{ } 

 



 

 

 _

^



2

cos 2

2 2

2 sin

cos

2

₂

1 2

1 0







_(5.1)

1)^{周波数 周期} Frequency/Period

T

k k

t

k

N ^ 

 ²  ²

 

^(5.2)

k

T

N

k

T T

k k

 



  ^

2

,

T

N

k

T

k

f T

k

k

^{  } 

1

(5.3)

－

_{k  0}

_,

₀

0



 ^f

f

_k : ^直流成分(Cascade Component)



_^



¹

0

0

¹

2

N

m

x

m

N

A

全体 ゼ 点 _(5.4)

－

_{k  0}

_,

_{ 0}

f

k :

2 / 1 2 / 2

1

^{f f}

N

^f

N

f _{   }

_

_

,

T

₁

_ T

₂

_{  } T

_N/21

_ T

_N/2 (5.5)

－ 分解す 周波数 トビトビ Discontinuity of Decomposed Frequency

t

f N

f

f

_{k k}

 









1

1 ^(5.6)

2)^{基本振動数}(Fundamental Frequency)

T

N

T

f T

 



 ^{1 1 1}

1

1 ^(5.7)

3)^{ナイキスト振動数}(Nyquist Frequency) ^分解能：Resolving power

検出可能 高周波数 限界値 /: Limit value of detection possible high frequency

T

f T

N

N

^{  2} 

1

1

2

2 ^(5.8)

01

.

 0

t

^{(sec.) →}

^f

N

^{50 Hz}

01

.

0

2

1

2



 

^(5.9)

4)^{振幅 位相角}(Amplitude/Phase Angle) ^情報 2^つ

   ^t     ^t  ^X   ^t   

A

_k

cos B

_k

sin

_k

cos

(5.10)

2 2

k k

k

^{A B}

X _{ }

(5.11)

 

 



^

k k

A

1

B

 tan

^(5.12)

4)^{フ エ 振幅 スペクト} (Fourier Amplitude Spectrum/ Fourier Spectrum)

X

k

T

2

dimension:

_{  } X

_k

sec .

(5.13) 5)^{パワ スペクト} (Power Spectrum): Invariant Value





^N_^1₀ ²

^{ }

^{0 2}

^{ 2}

^N_k^/_²₁^{1 k 2}

^

^{N2 2}

m

m

^{t T C T C T C}

x C

_k: ^{複素数フ エ振幅} (5.14)

0 5 10 15 20 -800

-600 -400 -200 0 200 400 600 800

PI-83m, NS-component

Acc., (gal)

Time (sec.)

0 20 40 60 80 100

-250 -200 -150 -100 -50 0 50 100 150 200 250

Kushiro West port, NS-component

Acc., (gal)

Time (sec.)

0 20 40 60 80 100

-500 -400 -300 -200 -100 0 100 200 300 400 500

Sanriku Harukaoki, NS-component

Acc., (gal)

Time (sec.)

0.1 1 10

1 10 100

Kushirooki Harukaoki PI-83NS

Fourier Amp, (cm/sec)

Period, (sec.)

6) Finite Fourier Approximation with Complex Number

ib

a

c _{ }

^:

c

: complex number,

_a

: real part,

_b

:imaginary part,

i _{ } 1

^(6.1)

2

2

b

a

c _{ }

: absolute value (6.2)

* 2

c

c

c _{ }

,

c

^*

_ a _ ib

: conjugate complex number (6.3)

   



_{cos i sin}

e

^i

_{ }

: Euler’s Formula ^(6.4)

 

  ^ _ ^{ }



























i i

i i

e

e

e

e

2

sin 1

2

cos 1

(6.5)

Approximation with Complex Number

   

 

   

  ^ _ ^{ }















 N km i N km i

N km i N km i

e

e

N i

km

e

N e

km

/ 2 /

2

/ 2 /

2

2

1

sin 2

2

1

cos 2













(6.6)

Finite series



_^



¹

0 N 2

k

N i km k

m

^{C e}

x



,

m

= 0, 1, 2,..., N-1 (6.7)

2

k k k

iB

C _ A ^

,

k

= 0, 1, 2,..., N-1 N (6.8)

Determination of

C

_k



_^{ }



¹

0

1

^N 2 m

N i km m

k

^{x e}

C N



,

k

= 0, 1, 2,..., N-1 N (6.9)

k N

k

^C

C _

_ : folding frequency

f

_N

t

 

2

1

2 ^(6.10)

    _ ^{ }





k k

k k

C

B

C

al

A

Im

2

Re

2

,

k

= 0, 1, 2,..., N/2 (6.11)

7) Fast Fourier Transform FFT

C

0

C

₁

C

₂

C

₃

C

₄

C

₅

C

₆

C

₇

^1×8

一回分割

C

0

C

₂

C

₄

C

₆

^2×4

C

1

C

₃

C

₅

C

₇

2^回分割

C

0

C

₄

4×2

C

2

C

₆

C

1

C

₅

C

3

C

₇

3^回分割

C

0

^8×1

C

4

C

2

C

6

C

1

C

5

C

3

C

7

－ 計算時間(Time for Fourier Coefficient Calculations):

T

_{ca l} Fourier Transform(FT)

T

_{ca l}

_ N

²

Fast Fourier Transform(FFT)

T

_{ca l}

_ N log

₂

N

Comparison for Cal. Time

N Factor _{Ratio for}

T

ca l

4094 2×23×89 12.9

4095 3²×5×7×13 3.9

4096 2¹² 1

4097 17×241 28

4098 2×3×683 77

4099 - 460

4100 2²×5²×41 6.3

－ 後続 ゼロ(Trailing Zero):

N 2^{累乗 い ゼ を後 さ}

_N=3000 → N=3000+1096=4096=2¹² ン 効果(Link Effect)^を解消

8) Link effect

(a) Periodic Function: earthquake motion transformed by Fourier series (b) Non-periodic function: real earthquake motion

Link Effect in Fourier transform

9) Fourier Integral: Discrete system / Continuous System

Time

Frequency or Period

Spectrum

Time Domain Frequency Domain

Fourier Transfom

(Fourier Integral)

Fourier Inverse Transfom

  



_^_

^

_^_



k

T i kt k k

T i kt

k

^{e TC e} T

C

t

x ( ) ¹

2

2_{ }

: for Discrete System (9.1)



^

 

^

 ¹

^T_T²₂ ^i2Tkt

k

^{x t e dt}

C T



,

_{  } k _{ }

(9.2)

T

f

_k

_ k

,

f T

f

f _

_{k 1}

_

_k

_ ¹



 ^(9.3)

  

 



 





















function

continuos

f

F

discr ete

TC

T df

f

T f

k

T

k

^{: :}

1 0

(9.4)

 

     

     

 

 





 

 





































 

























dt

e

t

x

f

F

dt

e

t

T x

TC

df

e

f

F

t

T x

e

TC

t

x

function

continuos

f

F

discr ete

TC

T df

f

T f

k

T

ft T i

T

T i kt k

ft i k

T i kt k k

 

 

2 2 2

2

2 2

1

) 1

(

:

:

1 0

(9.5)

Fourier transform(Fourier integral):

_{     }









T

TC

k

f

F

t

x lim

(9.6)

Fourier inverse transform:

F _{ } f _ x _{ } t

(9.7) Fourier Spectrum:

T C

_k



^{k k}



k

^{A iB}

TC _ T _

2

^(9.8)

k k

k

k

^X

B T

T A

C

T 2 2

2

2

 



^(9.9)

10) Smoothing / Filters

a) Data Window b) Spectral Windouw

c) Lag Window

0.01 0.1 1 10

0

10

20

30

40

50

Number of Data=2048, Nyquist Frequency=1/(2*0.01) Spectrum Window, Parzen's Filter

Band=0.0Hz

Fo urie r Amp, (c m/se c)

Period, (sec.)

0.01 0.1 1 10

0

10

20

30

40

Number of Data=2048, Nyquist Frequency=1/(2*0.01) Spectrum Window, Parzen's Filter

Band=0.5Hz Band=1.0Hz Band=2.0Hz

Fo urie r Amp, (c m/se c)

Period, (sec.)

地震による建物の被害と地盤の関係 : Relation between the damage of the

building by the earthquake and ground condition: ^{おもしろジ テク 技報}

堂

Complete collapse

rate of buildings Wooden building Storehouse

Ocean, water front

Reclaimed Land Alluvial Land

Dilluvial Land

東京のウ タ フロントの経緯 埋立て事業の経緯 : Process of water-

front and Reclamation in Tokyo-bay

Tokyo Bay

Reclaimed Land Alluvial Land

Dilluvial Land

Recent time

1923

1855

推薦図書 Recommendation Text and Papers

1) ^{新 地震動 スペクト 解析入門 大崎順彦}, ^{鹿島出版会}

2) ^{スペクト 解析 日野幹雄}, ^朝倉書店

3) ^{フ エ解析 大石進一}, ^{岩波書店 理工系 数学入門コ ス}

For examples

1) “The Fourier Integral and Its Applications”, papoulis, A.(1962), McGraw-Hill

2) “Random Data: Analysis and measurement Procedures”, Bendat, J.S. and Piersol, A.G.(1971), John Wiley & Sons.