Chapter 5 离散时间傅立叶变换
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Transcript of Chapter 5 离散时间傅立叶变换
![Page 1: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/1.jpg)
Chapter 5
离散时间傅立叶变换
![Page 2: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/2.jpg)
(2 )( 2 ) ( )
sF分析周期信号
[ ]x n kNx
k
12 1N 1周期的
0
2
N
1
0
sin[(2 1) ]2sin( )2
N
k
0
0n1N1N NN 2N2N NN
[ ]x n
n1N1N
1非周期的
分析非周期信号sF kNx
12 1N
22
1sin[(2 1) ]2sin( )2
N
( )jX e
0
<3.12>
5.0 引言3.6 周期信号的傅立叶级数展开
2
2
[ ]
1[ ]
kk N
kk N
jk nN
jk nN
x n x e
x x n eN
N 0 0
![Page 3: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/3.jpg)
[ ]x n
n1N1N
非周期信号
5.1 非周期信号的表示 : 离散时间傅里叶变换
A. 离散时间傅里叶变换的导出
N
[ ]x n
n1N1N NN 2N2N
周期信号 周期 : N 0
2
N
![Page 4: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/4.jpg)
0
( )j j nX e e
0
0
0
[ ]
1[ ]
kk N
kk N
jk n
jk n
x n x e
x x n eN
0
0
0
1[ ]
2
[ ]
kk N
kn
jk n
jk n
x n Nx e
Nx x n e
0( 1)2
N
0 00
1[ ] ( )
2 k N
jk jk nx n X e e
( ) [ ]j
n
j ndefine x e x n e
2
2
1[ ]
2j j nx n X e e d
1
0( ) |j j n
kX e e
0k
0 0( )jk j nX e e
面积
0 0
2
[ ] [ ]
( 2 )k N
x n x n
Nd N
0 0
3
0
0
0
0
( ) | ( )
1 1( ) | ( )
k
k
jkjk
jkjk
x e x e Nx
x x e x eN N
![Page 5: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/5.jpg)
于是得到非周期信号的离散傅里叶变换对 (DTFT) :
也得到一种新的傅里叶级数系数的表示方法 :
02
1( )k
jk k
Nx X e
N 3
非周期信号的离散傅里叶变换
周期信号的傅里叶级数系数 (DFS) 周期为 N
2
1[ ]
2
( ) [ ]n
j
j j n
j n
x n X e e d
X e x n e
2
1 反
( 频谱)正
复杂信号= 系数( )基本信号( )
系数( )
相似程度(的)
=复杂信号 (与)基本信号( )
![Page 6: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/6.jpg)
B. 例 <5.1>
[ ] [ ]nx n a u n | | 1a 1
1 jae 等比数列求和
| | 1a ( ) [ ]
n
j njX e x n e
| ( ) |
( )
j
j
X e
X e
00 0[ ( )]1
| ( ) |2
j X jj n eX ee
x[n] 的谱密度
( )X e j
0| ( ) |j
x e
| ( ) |jX e
22
x[n] 的频谱分布
0( )j
X e
0
0
0
n
n
j na e
![Page 7: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/7.jpg)
B. 例 <5.1>
![Page 8: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/8.jpg)
B. 例 <5.2>
| | 1a
![Page 9: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/9.jpg)
B. 例 <5.2>
![Page 10: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/10.jpg)
B. 例 <5.3>
![Page 11: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/11.jpg)
C. 收敛性问题
| [ ] |n
x n
[ ]x n 绝对可加
![Page 12: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/12.jpg)
B. 例 <5.4>
![Page 13: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/13.jpg)
5.2 周期信号的傅里叶变换
![Page 14: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/14.jpg)
5.2 周期信号的傅里叶变换
0( ) 2 ( )kk
jX e x k
2 / N
周期 N
F
2 / N
0[ ] jk nk
k Nx n x e
![Page 15: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/15.jpg)
5.2 周期信号的傅里叶变换
![Page 16: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/16.jpg)
![Page 17: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/17.jpg)
0( ) 2 ( )kk
jX e x k
F
0 2 / N 0[ ] jk n
kk N
x n x e
2 2 ( )jX e
2 /8 0
0
2
20N
<3.12> 1 2 40N N
[ ]x n
sF
F
2
8 16 24-8-16-24 0 40-40 k
1/8
kx
2 0 2
![Page 18: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/18.jpg)
5.3 DFT 性质
5.3.2 线性 kk
Fk
k
1 2[ ] [ ]x n x n 1 2( ) ( )j jX e X e F
1[ ]x n
2[ ]x n2 ( )jX e
1( )jX e F
F
![Page 19: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/19.jpg)
[ ]x n ( )jX e F
0[ ]x n n 0 ( )j n je X e F
5.3.3 时移特性与频移特性
5.3 DFT 性质
![Page 20: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/20.jpg)
5.3 DFT 性质
![Page 21: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/21.jpg)
5.3 DFT 性质
![Page 22: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/22.jpg)
5.3 DFT 性质
![Page 23: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/23.jpg)
5.3 DFT 性质
![Page 24: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/24.jpg)
5.3 DFT 性质
![Page 25: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/25.jpg)
5.3 DFT 性质
![Page 26: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/26.jpg)
5.3 DFT 性质
![Page 27: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/27.jpg)
5.3 DFT性质
![Page 28: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/28.jpg)
5.3 DFT 性质
![Page 29: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/29.jpg)
5.3 DFT 性质
![Page 30: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/30.jpg)
5.3 DFT 性质
![Page 31: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/31.jpg)
5.3 DFT性质
![Page 32: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/32.jpg)
5.4 卷积性质
[ ]x n ( )jX e
[ ]h n ( )jH e
F
F
( )jY e [ ]y n
?F
[ ] [ ]x n h n ( ) ( )j jX e H e F
F
[ ]x n LTI
[ ]h n[ ]y n
( ) ( ) ( )j j jX e H e Y e
<例 >
![Page 33: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/33.jpg)
5.4 卷积性质
![Page 34: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/34.jpg)
5.4 卷积性质
![Page 35: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/35.jpg)
5.4 卷积性质
![Page 36: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/36.jpg)
5.4 卷积性质
![Page 37: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/37.jpg)
5.4 卷积性质
有
![Page 38: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/38.jpg)
5.4 卷积性质
![Page 39: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/39.jpg)
5.4 卷积性质
![Page 40: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/40.jpg)
5.4 卷积性质
![Page 41: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/41.jpg)
[ ]g n
5.5 相乘性质
F
[ ]x n ( )jX e
[ ]p n ( )jP e
F
F
循环卷积 周期卷积 < DSP >
( )
2
1( ) ( ) ( )
2j j jG e X e P e d
?
2 [ ] [ ]x n p n
1( ) ( )
2j jX e P e
F
F 1
2
![Page 42: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/42.jpg)
5.5 相乘性质
![Page 43: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/43.jpg)
5.5 相乘性质
![Page 44: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/44.jpg)
5.3 DFT 性质
5.3.2 线性 kk
Fk
k
1 2[ ] [ ]x n x n 1 2( ) ( )j jX e X e F
1[ ]x n
2[ ]x n2 ( )jX e
1( )jX e F
F
![Page 45: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/45.jpg)
[ ]x n ( )jX e F
0[ ]x n n 0 ( )j n je X e F
5.3.3 时移特性与频移特性
5.3 DFT 性质
![Page 46: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/46.jpg)
5.3 DFT 性质
![Page 47: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/47.jpg)
5.3 DFT 性质
![Page 48: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/48.jpg)
5.3 DFT 性质
![Page 49: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/49.jpg)
5.3 DFT 性质
![Page 50: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/50.jpg)
5.3 DFT 性质
![Page 51: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/51.jpg)
5.3 DFT 性质
![Page 52: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/52.jpg)
5.3 DFT 性质
![Page 53: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/53.jpg)
5.3 DFT性质
![Page 54: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/54.jpg)
5.3 DFT 性质
![Page 55: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/55.jpg)
5.3 DFT 性质
![Page 56: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/56.jpg)
5.3 DFT 性质
![Page 57: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/57.jpg)
5.3 DFT性质
![Page 58: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/58.jpg)
5.4 卷积性质
[ ]x n ( )jX e
[ ]h n ( )jH e
F
F
( )jY e [ ]y n
?F
[ ] [ ]x n h n ( ) ( )j jX e H e F
F
[ ]x n LTI
[ ]h n[ ]y n
( ) ( ) ( )j j jX e H e Y e
<例 >
![Page 59: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/59.jpg)
5.4 卷积性质
![Page 60: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/60.jpg)
5.4 卷积性质
![Page 61: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/61.jpg)
5.4 卷积性质
![Page 62: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/62.jpg)
5.4 卷积性质
![Page 63: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/63.jpg)
5.4 卷积性质
有
![Page 64: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/64.jpg)
5.4 卷积性质
![Page 65: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/65.jpg)
5.4 卷积性质
![Page 66: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/66.jpg)
5.4 卷积性质
![Page 67: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/67.jpg)
[ ]g n
5.5 相乘性质
F
[ ]x n ( )jX e
[ ]p n ( )jP e
F
F
循环卷积 周期卷积 < DSP >
( )
2
1( ) ( ) ( )
2j j jG e X e P e d
?
2 [ ] [ ]x n p n
1( ) ( )
2j jX e P e
F
F 1
2
![Page 68: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/68.jpg)
5.5 相乘性质
![Page 69: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/69.jpg)
5.5 相乘性质
![Page 70: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/70.jpg)
5.5 相乘性质
![Page 71: Chapter 5 离散时间傅立叶变换](https://reader035.fdocuments.net/reader035/viewer/2022081416/56813625550346895d9d9bbc/html5/thumbnails/71.jpg)
(P277)
表达
( )jP e [ ]p n
① 知道变换对
② 性质 (无对偶性质)
解题
5.6 DTFT 变换对与性质列表
F? ?
[ ]ix n ( )jiX e F
√表达性质
[ ]x n ( )jX e F?
[ ]x n [ ]g n? ( )jX e ( )jG e ?
( & )F
( & )
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C. Examples
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