地盤災害システム論 前田研究室 maedalab Fourier

G _G G

E_EOEO_OTT_TE_ECEC_CHH_HN_NINI_ICC_CA_AAL_LL

_E E _E

N_NGNG_GII_INN_NEE_EEE_ERR_RII_INN_NGG_G

_L L _L

A_AAB_BB._{. .} [email protected].

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^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/

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

)

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[ ]

∑

^∞

=

+

0

sin

cos

k

k

k

^{kt B kt}

A

(2.2)

replace

t

by

t

T

π

2

or

t

N∆ t

π

2

∑

^∞

=0

⎢⎣ ^{⎡ ∆ + ∆} ⎥⎦ ^⎤

sin 2

cos 2

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

∑

∑

=

⎢⎣ ^{⎡ ∆ + ∆} ⎥⎦ ^{⎤ =}

=

⎢⎣ ^{⎡ +} ⎥⎦ ^⎤

=

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)

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4) Determination of A

k

and B

k

with orthogonal property of triangle functions

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^EEE^OOO

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^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/ 3)

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^{G G} G

^EEE^OOO

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^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/

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

π

π

π

(5.1)

1)

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^CCIC^IIE^EEN^NNC^CC^EEE^&&&

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^EEE^OOO

E E _E

^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/

G G G

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S S _S

^CCIC^IIE^EEN^NNC^CC^EEE^&&&

^{G G} G

^EEE^OOO

E E _E

^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/

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)

( 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

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S S _S

^CCIC^IIE^EEN^NNC^CC^EEE^&&&

^{G G} G

^EEE^OOO

E E _E

^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/

7) Fast Fourier Transform

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^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/

8) Link effect

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

Link Effect in Fourier transform

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E E _E

^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/

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)

∫

−

( )

=

² − 2

1

^T 2 T

T i kt

k

^{x t e dt}

C T

π

,

− ∞ ≤ k ≤ ∞

(9.2)

T

f

_k

= k

,

f T

f

f =

_{k 1}

−

_k

= ¹

∆

₊ (9.3)

⎪ ( )

⎪ ⎪

⎩

⎪⎪

⎪

⎨

⎧

→

≈

→

=

∆

→

∞

→

function

continuos

f

F

discrete

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

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

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)

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E E _E

^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/

10) Smoothing / Filters

a) Data Window b) Spectral Windouw

c) Lag Window

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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

Fourier 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

Fourier Amp, (cm/sec)

Period, (sec.)

G G G

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S S _S

^CCIC^IIE^EEN^NNC^CC^EEE^&&&

^{G G} G

^EEE^OOO

E E _E

^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/

G G G

^EEE^OOO

S S _S

^CCIC^IIE^EEN^NNC^CC^EEE^&&&

^{G G} G

^EEE^OOO

E E _E

^NNGN^GGI^II^NNEN^EEE^EER^RRI^IIN^NN^GG G^LLL^AAA^BBB http://www.cm.nitech.ac.jp/maeda-lab/