Unit Step Dirac Delta du t t ( ) Function u t t () δ − = t t Function … · 2016. 12. 5. · 1...

1

G. AhmadiME 529-Stochastics G. AhmadiME 529-Stochastics

!! Special FunctionsSpecial Functions

!! Differential EquationsDifferential Equations

!! Fourier Series and TransformsFourier Series and Transforms

!! Probability and Random ProcessesProbability and Random Processes

!! Linear System AnalysisLinear System Analysis

G. AhmadiME 529-Stochastics

Unit Step Unit Step Function Function ( )

⎭⎬⎫

⎩⎨⎧

<≥

=−0

00 tt0

tt1ttu

u(t-to)

to


DiracDirac Delta Delta Function Function ( ) ( )

dtttdu

tt 00

−=−δ

δ(t-to)

to

2


( ) ( ) 1dtttdttt 0

0

t

t 00 =−δ=−δ ∫∫ε+

ε−

+∞

∞−

( ) ( ) ( ) ( ) ( )0

t

t 00 tfdttttfdttttf 0

0

=−δ=−δ ∫∫ε+

ε−

+∞

∞−

( ) ( ) ( ) ( )00

t

1011 ttutfdttttf −=−δ∫ ∞−

( )[ ] ( )00 tta1tta −δ=−δ


Error Error Function Function ( ) ∫ −

π=

x

0

t dte2xerf2

( ) ( ) ∫∞ −

π=−=

x

t dte2xerf1xerfc2

( ) ( )∞== erfc00erf ( ) ( )xerfxerf −=−

( ) ∫∞ −

=1

xt

n dtt

exEExponential Exponential IntegralsIntegrals ( ) ∫ ∞−

=x t

i dttexE


Linear FirstLinear First--Order Order ( ) ( )xQyxP

dxdy

=+

( ) ( )( )∫ ∫+∫=

−− x

0 11

dxxPdxxPdxxQecey

x

1x 21x

0 1

( )xQybdxdy

=+ ( )∫ −−=x

0 11)xx(b dxxQey 1


0bydxdya

dxyd2

2

=++

212 m,mforSolve0bamm →=++

alRemm 12 =≠ xm2

xm1

21 ececy +=

mmm 12 == )xcc(ey 21mx +=

3


qipm1 +=

qipm2 −=2ap −=

4

2abq −=

( )qxsincqxcoscey 21px +=


Particular SolutionsParticular Solutions ( )xRbydxdya

dxyd2

2

=++

( ) ( )∫∫ −−

−+

−= dxxRe

mmedxxRe

mmey xm

12

xmxm

21

xm

P2

21

1

( ) ( )∫∫ −− −= dxxRxeedxxRexey mxmxmxmxP

( ) ( )∫∫ −− −= dxqxsinxReq

qxcosedxqxcosxReq

qxsiney pxpx

pxpx

P


( )xsbydxdyax

dxydx 2

22 =++

mAxy = ( ) 01 =++− bammm

21 m2

m1 xAxAy +=

LetLet


⎟⎠⎞

⎜⎝⎛=

xyF

dxdy

xyv =

( ) cvvF

dvxln +−

= ∫

vdxdvx

dxdy

+=LetLet

4


( ) ( ) 0dyy,xNdxy,xM =+

( )yxy,x

xN

yM 2

∂∂ϕ∂

=∂∂

=∂∂

xM

∂ϕ∂

=y

N∂ϕ∂

=

( ) consty,x =ϕ

WithWith


( ) 0ynxdxdyx

dxydx 2222

22 =−β++

( ) ( )xYCxJCy n2n1 β+β=

( )xJn β ( )xYn β

SolutionsSolutions

Bessel Bessel FunctionsFunctions


Fourier Cosine Series Fourier Cosine Series

( ) ∑∞

=⎟⎠⎞

⎜⎝⎛ π

+π

+=1n

nn0

Lxnsinb

Lxncosa

2axf

( )∫−π

=L

Ln dxL

xncosxfL1a ( )∫−

π=

L

Ln dxL

xnsinxfL1b

( ) ( )xfxf −=

( ) ∑∞

=

π+=

1nn

0

Lxncosa

2axf ( )∫

π=

L

0n dxL

xncosxfL2a


Fourier Exponential Series Fourier Exponential Series

( ) ( )xfxf −−=

( ) ∑∞

=

π=

1nn L

xnsinbxf ( )∫π

=L

0n dxL

xnsinxfL2b

( ) ∑∞

−∞=

ω=n

xin

necxf ( )∫−ω−=

L

L

xin dxexf

L21c n

Ln

nπ

=ω

5


( ) ∑∞

−∞=

π

=n

Lxin

necxf LxL <<−

( )∫−π

−′=

L

LL

xin

n dxxfeL21c

( ) ( ) ( )∑∫+∞

∞−−

′−ω ′=L

L

xxi dxxfeL21xf n

Ln

nπ

=ωLπ

=ω∆

∞→L ∫∑ ω=ω∆ gdgn

FES FES FES

Replacing for cn


( ) ( ) ( )∫ ∫∞+

∞−

∞+

∞−

′−ω ω′′π

= dxdxfe21xf xxi

( ) ( )∫+∞

∞−

′ω− ′′=ω xdxfef xi

( ) ( )∫∞+

∞−

ω ωωπ

= dfe21xf xi

Fourier Integral RepresentationFourier Integral Representation

Fourier Transform (Exponential) Fourier Transform (Exponential)


( ) ( )∫∞

′′′ω=ω0c xdxfxcosf

( ) ( )∫∞

ωωωπ

=0 c dfxcos2xf

( ) ( )∫∞

′′′ω=ω0s xdxfxsinf

( ) ( )∫∞

ωωωπ

=0 s dfxsin2xf

Fourier Cosine Fourier Cosine TransformTransform

Fourier Sine Fourier Sine TransformTransform


( ) ( )ωω==⎭⎬⎫

⎩⎨⎧ℑ ∫

∞+

∞−

ω− fidxdx

xdfedxdf xi

( )ωω−=⎭⎬⎫

⎩⎨⎧

ℑ fdx

fd 22

2

( ) ( )ωω=⎭⎬⎫

⎩⎨⎧

ℑ fidx

fd nn

n

6


( )02

2

xxbfdxdfa

dxfd

−δ=++ +∞<<∞− x

( ) ( ) ( ) 0xi2 efbfaif ω−=ω+ωω+ωω−

Taking Fourier TransformTaking Fourier Transform

( )ω+ω−

=ωω−

iabef 2

xi 0

( )( )

∫∞+

∞−

−ω

ωω+ω−π

= diab

e21xf 2

xxi 0


( ) ( ) ( ) ( )∫+∞

∞−ξξ−ξ= dxffxf*xf 2121 ( ) ( )ωω 21 ff

( )0xx −δ 0xie ω−

xe α−22

2α+ωα

xcos 0ω ( ) ( )[ ]00 ω+ωδ+ω−ωδπ

xcose x βα− ( )( ) 222222

222

42

ωα+α−β−ω

β+α+ωα

⎥⎦

⎤⎢⎣

⎡β

βα

+βα− xsinxcose x ( )( ) 222222

22

44

ωα+α−β−ω

β+αα

( )xf ( )ωf

( )01 xxf + ( )ωω fe 0xi


xcose22 x βα− ( ) ( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

αβ−ω

−+⎭⎬⎫

⎩⎨⎧

αβ+ω

−απ

2

2

2

2

4exp

4exp

2

( )xf ( )ωf

22xe α−⎭⎬⎫

⎩⎨⎧

αω

−απ

2

2

4exp

( )xdxd

n

n

δ ( )niω( )xJ0

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ <ω

ω−elsewhere0

11

22


f(t) X(t)( )( )ωHth

( ) ( ) ( )∫ τττ−=t

0dfthtX

( ) ( ) ( ) ( ) ( )tf*thdfthtX =τττ−= ∫+∞

∞−

h(t)=Impulse Responseh(t)=Impulse Response H(H(ωω)=System Function)=System Function

( ) ( )∫+∞

∞−

ω−=ω dttheH ti

7


( ) ( ) ( )ωω=ω f~Hx~

( ) ( ) ( ) ( ) ( ) ( )ωω=ωω=ω=ω ff2222

xx SHfHT1x

T1S

( ) ( ) ( )ωω=ω ff2

xx SHS

Fourier Fourier TransformTransform

Spectral Spectral RelationshipRelationship


( )tfxx =α+& ( ) teth α−=

( )tfxx2x 200 =ω+ζω+ &&&

( ) tsine1th dt

d

0 ωω

= ζω− 20d 1 ζ−ω=ω


( )tnxx =α+& i1ii1i nx

txx

=α+∆−

++

)t(xx ii =ii1i n

t1tx

t11x

∆α+∆

+∆α+

=+

Finite difference

1tforntxx ii1i <<∆α∆+=+

1tforn1xt

1x ii1i >>∆αα

+∆α

=+


-3 -2.5

-2 -1.5

-1 -0.5

0 0.5

1 1.5

2 2.5

3

Whi

te N

oise

0 0.0001 0.0002 0.0003 0.0004 0.0005 Time (s)

8


-150000

-100000

-50000

0

50000

100000 B

row

nian

For

ce

0 0.0001 0.0002 0.0003 0.0004 0.0005 Time (s)


-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

Part

icle

Vel

ocity

0 0.0001 0.0002 0.0003 0.0004 0.0005 Time (s)


-2E-08 0

2E-08 4E-08 6E-08 8E-08 1E-07

1.2E-07 1.4E-07 1.6E-07

Part

icle

Pos

ition

0 0.0001 0.0002 0.0003 0.0004 0.0005 Time (s)


-4E-08

-2E-08

0

2E-08

4E-08

6E-08

8E-08

Part

icle

Pos

ition

0 0.0001 0.0002 0.0003 0.0004 0.0005 Time (s)

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Unit Step Dirac Delta du t t ( ) Function u t t () δ − = t t Function … · 2016. 12. 5. · 1...

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