สัญญาณและระบบการแปลงฟูเรียร์ที่ต่อเนื่องทางเวลา...
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สัญญาณและระบบการแปลงฟูเรียร์ที่ต่อเนื่องทางเวลาTRANSCRIPT
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(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
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(Fourier transform) (Fourier transform)
(spectrum)
..
3
CtFT: continuous-time Fourier transform
( )x t
t
( )px t
( )x t
t1T1T
t1T1T
.. 4
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( )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
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.. 7
.. 8
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.. 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
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.. 11
Example 2p
.. 12
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.. 13
Example 3p
.. 14
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Exercise 1
.. 15
.. 16
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.. 17
.. 18
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Example 4p
.. 19
.. 20
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Example 5p
( )x t
0 2TTT2T t1T
( )X j1T
( )X j
1T
1T
.. 210 0
1
1
0
.. 22
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.. 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
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( ) 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
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( ) ( )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
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.. 29
Example 5p
.. 30
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Exercise 2
.. 31
Example 6p
.. 32
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.. 33
LTI LTI
( ) ( )0 0
k kN M
k kk kk k
d y t d x ta b
dt dt= == N
.. 34
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Exercise 3
.. 35
.. 36
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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
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.. 39
Exercise 4
.. 40
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.. 41
Exercise 8
.. 42
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.. 43
.. 44
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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
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.. 47
.. 48
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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