r.v., Z-transform, Laplace transform

(1)

Cheng-Fu Chou, CMLab, CSIE, NTU

r.v., Z-transform,

Laplace transform

Cheng-Fu Chou 2005.10.04

P. 2

Cheng-Fu Chou, CMLAB, CSIE, NTU

Outline

2 or more r.v. z transform Laplace transform P. 3

PDF for 2 R.V.

Marginal Density Function 2

( , )

[

,

];

( , )

XY XY XY

F

x y

P X

x Y

y

d F

x y

f

x y

dxdy

d

2 2 1 1 1 2 1 2 | [ , ] ( )

Independent : ( ) ( ) ( ) Prob[X x]Prob[Y y] ( , ) ( | ) Prob[ | ] ( ) x y XY x y XY X Y XY X Y Y P x X x y Y y f xy dydx F xy F x F y F x y F x y X x Y y F y d d d d d d

³ ³

P. 4

Function of Random Variable

Func. of r.v.

One important r.v. is where Xi are independent

( ) ( ) [ ] [{ : ( ( )) }] Y Y g X F y P Y dy P w g X w dy 1 n i i

Y

¦

X

(2)

P. 5

Y = X

₁

+ X

₂ PDF pdf Convolution 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 2 2 2

( )

[

]

[

]

(

)

[

( )

]

( )

(

)

( )

Y y x X X y x X X X X

F y

P Y

y

P X

X

y

f

x x dx dx

f

x dx f

x dx

F

y

x

f

x dx

f f f f f f f f

d

³ ³

³

1 2 2 2 2

( )

(

)

( )

Y X X

f

y

f

y

x

f

x dx

f

³

1 2

( )

Y X X

f

y

f

y

f

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P. 6

Convolution Ex.

• f(n) • 2/3 as n = 1 • 1/3 as n = 2 • g(n) • 1/2 as n = 1 • 1/2 as n = 2 h(n) = f(n)g(n) = ? P. 7

Ex. (cont.)

h(0) = f(0)g(0) = 0 h(1) = f(1)g(0) + f(0)g(1) = 0 h(2) = f(2)g(0) + f(1)g(1) + f(0)g(2) = 1/3 h(3) = f(3)g(0)+f(2)g(1)+f(1)g(2)+f(0)g(3) = ½ h(4) = f(4)g(0)+f(3)g(1)+f(2)g(2)+f(1)g(3)+f(0)g(4) = 1/6 P. 8

z transform

Consider a function of discrete time fn s.t. – fn t 0 for n = 0, 1, 2, … – fn = 0 for n = -1, -2, … – 0

( ) where ( )

n n _n n

f

F z

¦

f

f z

(3)

P. 9

Examples

Ex1: Ex2: 0 0

0,1, 2,...

( )

(

)

n n n n n n n

f

A

for n

F z

A

z

A

Z

z

D

f f

?

$

¦

0

Convolution property

n n n n k k k

f

g

¦

f

g

P. 10

Convolution Property

0 0 0 0 0 0 0 0 [ ] [ ] [ ] ( ) ( n n n n n k k n k n n k k n k k n k k n k k n k k n k k n k k n k k n k k m k m k m f g f g z f g z z g z f z g z f z g z f z G z F f f f f f f f f

¦ ¦

¦

) z P. 11

Properties of z transform

0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1. ( ) 2. ( ) ( ) 3. ( ) 1 1 4. [ ( ) ] 5. ( ) 6. ( ) 7. ( ) ( ) n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n f F z f z af bg aF z bG z a f F az f f z f z F z f z z f f z z f z zF z dz d nf nf z f z z F z dz dz f g F z G z f f f f f f f

¦

P. 12

Properties (cont.)

1 0 0 0 0 0 0

8. ( )

( )

(1

) ( )

( )

9. ,

0,1, 2,...

1

10. (n is a parameter of )

( )

11. (1)

12. (0)

n n n n n k n k k k k k n k k n k n n n n

f

F z

zF z

z F z

F z

f n

f z

f z z

z

f

F z

a

F

f

F

f

f f f f

w

¦

¦¦

¦

(4)

P. 13

z Transform pair

0 0 2 2 0 1 0 1. ( ) 1 0 2. 1 3. 1 0,1,... 1 4. 1 5. 1 6. (1 ) 7. (1 ) 1 8. ( ) ! !

Read and derive Table I.1 and I.2

n k n k n n n k n k n n n n n n z n n U F z otherwise U z for n z z z z A A A z z z n z z n z z F z e n n G G D D D D D D f f f ® ¯ ¦ ¦ ¦ P. 14

z transform : difference equation

Ex: 1 2 0 1

1

6

5 6( )

2, 3, 4,...

5

6 0;

5

n n n n

g

n

g

P. 15

z-transform and moment

0 1 1 0 0 2 2 2 0 2 2 2 1 2 0 ( ) ( ) ( ) ( ) ( 1) ( ) ( ) n n n n n z n n n n n n z n n G z f z dG z dG z nf z nf X dz dz d G z n n f z dz d G z n n f X X dz f f f f f

¦

P. 16

Laplace transform

Def: Ex 1. Ex 2.

1. * ( )

( )

;

2. ( )

* ( )

st

F

s

f t e dt

f t

F

s

f f

³

0 ( )

0 otherwise

at

Ae

t

f t

t

®

¯

1

0 ( )

0

0 t

t

G

_®

t

¯

(5)

P. 17

Convolution

f(t) and g(t) take on non-zero values for tt0

( )

(

) ( )

f t

g t

f

f t

x g x dx

f

_³

P. 18

Properties

1 1. ( ) ( ) * ( ) * ( ) 2. ( ) * ( ) 3. ( ) * ( ) 4. ( ) * ( ) 5. ( ) * ( ) * ( ) 6. ( ) ( 1) ( ) 7. * ( 1) 1 as at n n n n s s af t bg t aF s bG s t f aF as a f t a e F s e f t F s a d tf t F s ds d F s t f t ds f t F s ds t f

_³

P. 19

Properties (cont.)

1 1 ( ) 8. ?( * ( ) (0 )) ( ) 9. * ( ) (0 ) ... (0 ) * ( ) 10. ( ) * ( ) 11. ... ( )( ) n n n n n t t t _n n df t sF s f dt d f t s F s s f f dt F s f t dt s F s f t dt s f f f

³

³ ³

P. 20

Differential eq.

Find f(t) 2 2 ( ) 6 ( ) 9 ( ) 2 ; (0 ) (0 ) 0; d f t f t f t t dt dt df f dt

(6)

P. 21

Random Sum

1 2 0 0 0 Given ... , find

are i.i.d. is a discrete non-negative r.v. * ( ) * ( | ) P[ ] * ( | ) [ * ( )] * ( ) [ * ( )] [ ] Note that ( ) [ ] Relate with [ * ( )] * ( ) N i n n n n n n n Y X X X Y X N Y s Y s N n N n Y s N n X s Y s X s P N n N z P N n z z X s Y s f f f

¦

[ * ( )] N X s P. 22

Ex.

1 2 ...

are i.i.d. exponentally distributed, N is geometrically distributed. (a) Find [ ]. (b) what is var[ ]. N i Y X X X X E Y Y

r.v., Z-transform, Laplace transform

r.v., Z-transform,

Laplace transform

Outline

PDF for 2 R.V.

( , )

[

,

];

( , )

( , )

F

x y

P X

x Y

y

d F

x y

f

x y

dxdy

d

d

³ ³

Function of Random Variable

Y

¦

X

Y = X

+ X

( )

[

]

[

]

(

)

[

( )

]

( )

(

)

( )

F y

P Y

y

P X

X

y

f

x x dx dx

f

x dx f

x dx

F

y

x

f

x dx

d



d



³ ³

³ ³

³

( )

(

)

( )

f

y

f

y

x

f

x dx



³