# r.v., Z-transform, Laplace transform

## 全文

(1)

Cheng-Fu Chou, CMLab, CSIE, NTU

Cheng-Fu Chou, CMLab, CSIE, NTU

### Laplace transform

Cheng-Fu Chou 2005.10.04

P. 2

Cheng-Fu Chou, CMLAB, CSIE, NTU

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

### 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

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

### X

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P. 5

Cheng-Fu Chou, CMLAB, CSIE, NTU

Cheng-Fu Chou, CMLAB, CSIE, NTU

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

### x dx

f  f f f  f f f f

1 2 2 2 2

Y X X

f

f

1 2

Y X X

### y

P. 6

Cheng-Fu Chou, CMLAB, CSIE, NTU

Cheng-Fu Chou, CMLAB, CSIE, NTU

### 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

n n n n

f

### f z

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P. 9

Cheng-Fu Chou, CMLAB, CSIE, NTU

Cheng-Fu Chou, CMLAB, CSIE, NTU

Ex1: Ex2: 0 0

n n n n n n n

f f

0

n n n n k k k



### g

P. 10

Cheng-Fu Chou, CMLAB, CSIE, NTU

Cheng-Fu Chou, CMLAB, CSIE, NTU

### 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

1 0 0 0 0 0 0

### 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 f  f

### ¦

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P. 13

Cheng-Fu Chou, CMLAB, CSIE, NTU

Cheng-Fu Chou, CMLAB, CSIE, NTU

### 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

Cheng-Fu Chou, CMLAB, CSIE, NTU

Cheng-Fu Chou, CMLAB, CSIE, NTU

Ex: 1 2 0 1

n n n n

 

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.

st

f  f

at



### ¯

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P. 17

Cheng-Fu Chou, CMLAB, CSIE, NTU

Cheng-Fu Chou, CMLAB, CSIE, NTU

### Convolution

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

f

f

## ³

### 

P. 18

Cheng-Fu Chou, CMLAB, CSIE, NTU

Cheng-Fu Chou, CMLAB, CSIE, NTU

### 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             

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        

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    

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P. 21

Cheng-Fu Chou, CMLAB, CSIE, NTU

Cheng-Fu Chou, CMLAB, CSIE, NTU

### 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

Cheng-Fu Chou, CMLAB, CSIE, NTU

Cheng-Fu Chou, CMLAB, CSIE, NTU

### 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   

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