豊橋技科大（地盤地震工学） 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:

_{Δt >} 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 _Δ t

(1.1)

Time:

t ₌ m _Δ t

(m=0, 1, 2, ^･…, N-1) (1.2) A data point:

x

_m

₌ x _{( )} m _Δ t

N=16(m=0 – 15) : x0, x1, …….., x15. duration time: T=16_×Δt

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 _π cos ^{2 π}

cos

T

_p

₌ ² k ^π

∴

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

[ ]

∑

^∞₌₀

^{cos + sin}

k

k

k

^{kt B kt}

A

(2.2)

replace

t

by

t

T

π

2

or

t

t

N _Δ

π

2

∑

^∞₌₀

_⎢⎣ ^{⎡ cos 2} _Δ ^{+ sin 2} _{Δ ⎥⎦} ^⎤

k

k

k

N t

B kt

t

N

A ^π kt ^π

(2.3)

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

Set k to be from 0 to N/2

∑

∑

₌

_⎢⎣ ^⎡ _Δ ⁺ _{Δ ⎥⎦} ^{⎤ =}

₌

_⎢⎣ ^{⎡ +} _⎥⎦ ^⎤

=

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 ^{π π π π}

(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 ^π km _{= ⋅ =}

and ₀

sin ² _{= B}

₀

_⋅ 0 _≡ 0

N

B ^π km

(3.3) For the case of

k ₌ N 2

, ₂

sin ² ₌ B

₂

sin m _≡ 0

N

B

_N

^π km

_N

_π

(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

+ _⎢⎣ ^{⎡ π} + ^π _⎥⎦ ^⎤ + ^π

= _∑

⁻

=

(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

+ _⎢⎣ ^{⎡ π} + ^π _⎥⎦ ^⎤ + ^π

= _∑

⁻

=

(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)

( α β ) ( α β )

β

α ⋅ ^cos = ^cos + + ^cos −

cos

2

(a)

( ^{α β} ) ( ^{α β} )

β

α ⋅ ^sin = ^cos + − ^sin −

cos

2

(b)

( α β ) ( α β )

β

α ⋅ ^sin = − ^cos + + ^cos −

sin

2

(c)

α

α ^{1 cos 2}

cos

2

²

_{= +}

(d)

α

α ^{1 cos 2}

sin

2

²

_{= −}

(e)

( ) ( ) { ( ) }

sin 2

sin 2

2

cos 1

1

cos

2

cos

cos

cos

β

β β

α

β

α

β

α

β

α

α

N

N

N

⎟ ⎠

⎜ ⎞

⎝ ⎛ ₊ −

=

−

+

+

⋅⋅

⋅⋅

+

+

+

+

+

(f)

( ) ( ) { ( ) }

sin 2

sin 2

2

sin 1

1

sin

2

sin

sin

sin

β

β β

α

β

α

β

α

β

α

α

N

N

N

⎟ ⎠

⎜ ⎞

⎝

⎛ ₊ −

=

−

+

+

⋅⋅

⋅⋅

+

+

+

+

+

(g)

if

_{α =} 0

, summarize the results.

sin 2

sin 2

2

cos 1

cos

1

0

β

β β

β

N

N

m

N

m

− ⋅

∑

₌⁻

=

^(h)

sin 2

sin 2

2

sin 1

sin

1

0

β

β β

β

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

π

π

π

π

π

π

(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

π

π

π ₊

⎥⎦ ⎤

⎢⎣ ⎡ ₊

+

= _∑

⁻

=

(4.1)

1)^{上式の両辺に}

cos ₍ 2 _π km N ₎

^を掛ける / Multiplication of

cos ₍ 2 _π km N ₎

to Eq.(4.1).

( ^{km N} ) ^A _N ^km

x

_m

_π cos ^{2 π}

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 ^π lm ^{π π π}

( )

N

km

N

m

A

_N

_π N 2 _π

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₌⁻¹₀

^{cos 2}

m

m

N

A km

N

km

x _π ^π

( -> 0 )

∑ ∑

₌⁻

_⎢⎣ ^⎡

₌⁻

_⎥⎦ ^⎤

+

²¹

1 1

0

cos 2

cos 2

N

l l N

m

^N

km

N

A ^π lm ^π

∑ ∑

₌⁻

_⎢⎣ ^⎡

₌⁻

_⎥⎦ ^⎤

+

²¹

1 1

0

cos 2

sin 2

N

l l N

m

N

km

N

B ^π lm ^π

( -> 0 )

( )

N

km

N

m

A

_N

_π N 2 _π

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 _π ^{π π}

(4.4)

2 0

0

2 0

2 cos

cos

_{1 2 21}

1 2

1 1

0

⋅

+

⋅⋅

⋅

+

⋅

+

⋅⋅

⋅

+

⋅

+

⋅

⎥⎦ =

⎢⎣ ⎤

⎡

−

−

=

−

∑ ∑

= ^{k N}

N

l N

m

l

^A

A N

A

N A

km

N

A ^π lm ^π

(4.5)

N

x km

A N

N

m m k

π

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

π

π

(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 _{= Δ}

,

t

m t

= Δ

^(4.9)

( ) ^{t A A} _N ^kt _t ^B _N ^kt _t ^A _N ^{( ) N} _t ^t

x

_N

N

k

k

k

_⎥ + _Δ

⎦

⎢ ⎤

⎣

⎡

+ Δ

+ Δ

≈ _∑

⁻

=

2

cos 2

2 2

2 sin

cos

2

₂

1 2

1 0

π

π

π

_(4.10)

5) Spectrum Properties

( ) ^{t A A} _N ^kt _t ^B _N ^kt _t ^A _N ^{( ) N} _t ^t

x

_N

N

k

k

k

^⎥ _⎦ ^{⎤ +} Δ

⎢ ⎣

⎡

+ Δ

+ Δ

≈ _∑

⁻

=

2

cos 2

2 2

2 sin

cos

2

₂

1 2

1 0

π

π

π

_(5.1)

1)^{周波数 周期} Frequency/Period

T

k k

t

k

N ^π π

ω = ² _Δ = ²

^(5.2)

k

T

N

k

T T

k k

= Δ

=

= _ω ^{2 π}

^,

T

N

k

T

k

f T

k

k

= ¹ = = _Δ

^(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}

= Δ

−

≡

Δ

₊1

¹

^(5.6)

2)基本振動数(Fundamental Frequency)

T

N

T

f T

= Δ

=

= ^{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

= = _Δ

2

1

1

2

2 ^(5.8)

01

.

= 0

Δt

^{(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

( ) ω ^t + ( ) ω ^t = ^X ( ω ^t + φ )

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

φ 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

^{Δ =}

⁰²

^{+ 2}

^N_k^/₌²₁^{−1 k2}

⁺

^N22

m

m

t T C T C T C

x C

k: ^{複素数フ エ振幅} (5.14)

-8 - - - 8

PI-83m, NS-component

Acc., (gal)

Time (sec.)

8 -

- - - -

Kushiro West port, NS-component

Acc., (gal)

Time (sec.)

8 -

- - - -

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)

*

_c

2

c

c _{⋅ =}

,

c

^*

₌ a ₋ ib

: conjugate complex number (6.3)

( ^{θ θ} )

θ

_{cos i sin}

e

^±i

_{= ±}

: Euler’s Formula (6.4)

( )

( ) ^⎪ _⎭ ^{⎪ ⎬}

⎫

−

=

+

=

−

− θ θ

θ θ

θ

θ

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

π π

π π

π

π

(6.6)

Finite series

∑

₌⁻

=

¹

0 N 2

k N i km

k

m

^{C e}

x

π

,

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

π

,

k

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

k N

k

^C

C ₌

₋ : folding frequency

f

_N

t

= Δ

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

_cal Fourier Transform(FT)

T

_cal

_∝ N

²

Fast Fourier Transform(FFT)

T

_cal

_∝ N log

₂

N

Comparison for Cal. Time

N Factor _{Ratio for}

T

cal

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

2π π

: for Discrete System (9.1)

∫

⁻

( )

⁻

= ¹

^T_T²₂ ^i2Tkt

k

^{x t e dt}

C T

π

,

_{− ∞ ≤} k _{≤ ∞}

(9.2)

T

f

_k

₌ k

,

f T

f

f ₌

_k1

₋

_k

₌ ¹

Δ

₊ ^(9.3)

⎪ ( )

⎪ ⎪

⎩

⎪⎪ ⎪

⎨

⎧

→

≈

→

=

Δ

→

∞

→

function

continuos

f

F

discrete

TC

T df

f

T f

k

T

k

^{: :}

1 0

(9.4)

( )

( ) ( ) ( )

( ) ( ) ( )

⎪ ⎪

⎪ ⎪

⎪

⎩

⎪⎪

⎪ ⎪

⎪

⎨

⎧

=

→

=

=

→

=

→

≈

→

=

Δ

→

∞

→

∫

∫

∑ ∫

∞

∞

−

−

−

−

∞

∞

−

∞

−∞

=

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

discrete

TC

T df

f

T f

k

T

ft T i

T T i kt

k

ft i

k

T i kt

k k

π π π π

2 2 2

2

2 2

1

) 1

(

:

:

1 0

(9.5)

Fourier transform(Fourier integral):

_{( ) ( ) ( )}

∞

=

→

→

T ^k

TC

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

F o urier Amp, ( cm/sec )

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

F o urier Amp, ( cm/sec )

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.