信号与系统第三章课件

1. Continuous-time system
yt est ht ht e st
h
e st d
e st h es d
est ht yt e stes
与时间无关
Defining
H s h t estdt ——Eigenvalue (特征值）
Eigenfunction 特征函数
4
2
3
xt 1 1 cos 2t cos4t 2 cos6t
2
3
Consider a real periodic signal xt xt
ak e jk0t
ak e jk0t
a e jk0t k
a e jk0t k
k
k
k
k
real periodic xt
e st
H sest
2
Chapter 3
Fourier Series
2. Discrete-time system
yn zn hn hn zn
z n hn yn
hkznk
k
zn hkzk k
与时间无关
Defining H z hnzn ——Eigenvalue (特征值） n
x t 1 e j4 t 1 e j4 t 1 e j7 t 1 e j7 t
2
2
2
2
1 2
e
j12e
j4t
1 2
e
j12e
j 4 t
1 2
e
j 21e
j 7 t
1 2
e
j 21e
j7t
5
Chapter 3
Fourier Series
§3.3 Fourier Series Representation（傅立叶级数）
Re
A e e jk jk0t k
k 1
2 ak Bk jCk
xt a0 2 Ak cosk0t k k 1
xt a0 2 Re Bk jCk e jk0t
k 1
xt a0 2
Bk
cos
k0t
Ck
sin
k0t
8
k 1
Chapter 3
Fourier Series
Example :
Consider an LTI system for which the input xt 1 1 cos2t
and
the impulse response
ht
etut
determine
the
2 output
yt
x t e j0t 1 e j2 t e j2 t 4
k 0,1,2,
xt ake jk0t ——Fourier Series k
ak ——Fourier Series Coefficients
Spectral Coefficients （频谱系数）
Baidu Nhomakorabeaa0
Constant Component
a1
Fundamental Component
a2
Second Harmonic Component 6
Chapter 3
Fourier Series
Example 3.2
3
x t
ak e jk 2 t
k 3
a0 1
, a1 1 / 4
a2 1 / 2 , a3 1 / 3
x t 1 1 e j2 t e j2 t 1 e j4 t e j4 t 1 e j6 t e j6 t
ak ak
7
Chapter 3
Fourier Series
xt
ak e jk0t a0
ak e jk0t ak e jk0t
k
k 1
x t a0 2 Re ake jk0t
a e jk0t k
k 1
1 ak Ake jk
x t a0 2
Chapter 3 Fourier Series Representations
of Periodic Signals
1
Chapter 3
Fourier Series
§3.2 The Response of LTI Systems to Complex Exponentials LTI 系统对复指数信号的响应
9
Chapter 3
Fourier Series
x t
ak e jk0t
k
yt ak H jk0 e jk0t k
§3.3.2 Determination of Fourier Series Representation
1 xt e j2 t y t e j2t3
H s h t estdt t 3 estdt e3s
S
x t e j2t H s
e j2t e j6e j2t e j2t3
s j2
2 xt cos 4t cos 7t yt cos 4t 3 cos 7t 3
Eigenfunction 特征函数
zn
H z zn
3
Chapter 3
Fourier Series
Continuous-time system
eskt H sk eskt
akeskt ak H sk eskt
k
k
xt
yt
Discrete-time system
Particularly
H j et u t e j tdt ete j tdt
0
H j
e j 1t
j 1
0
1
j 1
0
yt H j0e j0 t 1 H j2 e j2 t 1 H j2 e j2 t
4
4
1 e j2t
1 e j 2t
yt 1 4 4
1 j2 1 j2
of Continuous-time Periodic Signals
§3.3.1 Linear Combinations （线性组合）
of Harmonically Related Complex Exponentials
xt T0 xt
k t e jk0 t
0
2
T0
——Fundamental frequency
zkn H
zk
z
n k
xn
a
k
z
n k
k
yn ak H zk zkn
k
e j t H j e j t
Fourier Analysis
e j n H e j e j n
Fourier Analysis
4
Chapter 3
Fourier Series
Example 3.1
Consider an LTI system : ht t 3 yt xt 3