(8-9)

Assoc.Prof.Piya Kovintavewat, Ph.D.

Data Storage Technology Research CenterData Storage Technology Research Center

Nakhon Pathom Rajabhat University

http://home.npru.ac.th/piya

http://home.npru.ac.th/piya

Outline

LTI

.. 2

(Fourier transform) (Fourier transform)

(spectrum)

..

3

CtFT: continuous-time Fourier transform

( )x t

t

( )px t

( )x t

t1T1T

t1T1T

.. 4

( )px t

( ) ( )0 0/2/2

1 T

jk t jk tp k k p

k T

x t a e a x t e dtT

=

= = 0 2 /T =

( )1 X jk( )0ka X jkT =

ak xp(t)

( ) ( ) ( )0 00 0 01 12jk t jk tp k kx t X jk e X jk eT

= = = =

.. 5

2k kT = =

.. 6

.. 7

.. 8

.. 9

Example 1p

( )x t ( )1

0 368

t0

0.368

1/ a

( )X( )X j1/ a

( )X j/ 4

/ 2

aa

12a

/ 4a

a

/ 2

.. 10

.. 11

Example 2p

.. 12

.. 13

Example 3p

.. 14

Exercise 1

.. 15

.. 16

.. 17

.. 18

Example 4p

.. 19

.. 20

Example 5p

( )x t

0 2TTT2T t1T

( )X j1T

( )X j

1T

1T

.. 210 0

1

1

0

.. 22

.. 23

CtFT ( ) ( )CtFTx t X j ( ) ( )CtFTy t Y j

( ) ( ) ( ) ( )CtFTax t by t aX j bY j + + ( ) ( )( )0j t CtFTt X j

( ) ( )00 j tCtFTx t t e X j

( ) ( )( )0 0j Cte x t X j ( ) ( )CtFTx t X j

( ) ( )j( ) ( ) ( )n nCtFTnd x t j X jdt dt

( ) ( ) ( )nn CtFT nd X jjt x t d

.. 24

( ) ( ) nd

( ) 1CtFT jx t X

( )X j2( ) ( )1/x t t T =

1 1 =2 =

2

1

t0 111 1

0.5 =

0 2

0.5

11 2 4 4

242 4

( ) ( ) ( ) ( )CtFT X Y ( ) ( ) ( ) ( )CtFTx t y t X j Y j

( ) ( ) ( ) ( )CtFTx t y t X j Y j

( ) ( )CtFTx t X j ( ) ( ) ( ) ( )x t y t X j Y j

..

25

( ) ( )j

( ) ( )CtFTx t X j ( ) ( )2CtFTX t x j

1t ( ) ( ) ( ) ( )1 0t CtFTx d X j Xj

+

( ) ( ) ( ) ( )1 0 CtFTx t x t X j djt

+

( ) ( )0x t dt X =

( ) ( )1 0X j d x = ( ) ( )2 j

.. 26

( ) ( )2 212

x t dt X j d

= 2

( ) ( )2 21x t dt E X j dE = ..

27

( ) ( ) 2

x t dt E X j dE ==

.. 28

.. 29

Example 5p

.. 30

Exercise 2

.. 31

Example 6p

.. 32

.. 33

LTI LTI

( ) ( )0 0

k kN M

k kk kk k

d y t d x ta b

dt dt= == N

.. 34

Exercise 3

.. 35

.. 36

LTI

( ) ( )dy t Kx t t= ( ) ( )dj tY j Ke X j =( ) ( ) ( ) ( ) K td

( ) ( )dj tY j Ke X j = LTI ( ) ( ) ( )d j H jj tH j Ke H j e = = LTI ( ) ( ) ( )djH j Ke H j e = =

.. 37

Example 7p

.. 38

.. 39

Exercise 4

.. 40

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Exercise 8

.. 42

.. 43

.. 44

LTI LTI

(pass band ( t b d) (stop band)

.. 45

1( )H j ( )H j

c c

1

c c

1

() ()

( )H j ( )H j

1

1

1()

1 22 1()

1 22

.. 46

.. 47

.. 48

C L ( )H j1

R( )x t ( )y t

( )i t f0.707

B

.. 49

Lf HfRf (Hz)

( )x t ( )y t( )i t

.. 50