信号与系统 第三章课件

Suppose x(t) is real ,then x (t )  x (t )
     jk0t jk0t   jk0t  x (t )    ak e jk0t    ak e   a e  a e  k k k  k   k   k   *
0 0
T0
T0

  x(t )e
0 T0

0, T0 ,
kn kn
 jn0t
dt anT0
1 T0  jn0t a  x ( t ) e dt consequently n  0 T0 Notice : the integration can be over any interval of length T
could be represented by harmonically related sinusoids in 1807.
some story about Fourier
• Born in France in 1768 • Fourier claimed that any periodic signal could be represented by harmonically related sinusoids in 1807 • Due to Lagrange „s objection his paper never appeared • His paper appeared in “The Analytical Theory of Heat” in 1822 • Dirichlet provide precise conditions in 1829
1768—1830
傅里叶的两个最重要的贡献——
• “周期信号都可以表示为成谐波关系的正弦信 号的加权和”——傅里叶的第一个主要论点 • “非周期信号都可以用正弦信号的加权积分来 表示”——傅里叶的第二个主要论点
3.2 The Response of LTI Systems to Complex Exponentials(LTI系统对复指 数信号的响应)
 Periodic Singal can be represented as linear
combination of complex exponentials.
3.1 Historical Perspective (历史的回顾)
1、The concept of using trigonometric sums to
 
y(t)= 1/2e-j12ej4t + 1/2ej12e-j4t+ 1/2e-j21ej7t+ 1/2ej21e-j7t =cos[4(t-3)}+cos[7(t-3)]
3.3 Fourier Series Representation of Continuous-Time Periodic Signals(连 续时间周期信号的傅里叶级数表示)
describe periodic phenomena goes back to
Babylonians
2、Euler examined the motion of Vibrating string is a linear combination of a few normal mode in 1748.
 ak  a

 k
or
a  ak
* k
ak is expressed in polar form as
x(t ) 
k 
ak  Ak e jk


Ak e jk e jk0t a0 
k 

1
Ak e j ( k0t k )   Ak e j ( k0t k )
h( k ) z  k  H ( z ) z n
y(n) 
k 


z
( nk )
h(k )  z
n
k 


Eigenfunction

Eigenfunction in-> Same function out with gain Eigenvalue Gain is called “Eigenvalue” discrete time
Chapter 3 Fourier Series
Representation of Periodic Signals 第3章 周期信号的傅里叶级数表示
Main content ：
1. The Frequency Analysis of Periodic Siganl(周期信号的频域分析） 2. The Frequency Analysis of LTI（ LTI系统的
0
Each of these signals has a fundamental frequency that is 2 multiple of ω0, each is periodic with period T
Thus , x(t ) 
the form is referred to as the Fourier series representation

continuious time
e
st
h(t )
H (s)e
st
z
n
h( n)
Eigenfunction
H ( z )Z n
Eigenvalue
Conclusion:

complex exponential signal e 、z n are eigenfuction
H ( s)   h(t )e st dt
这表明用傅里叶级数可以表示连续时间周期信号，即: 连续时间 周期信号可以分解成无数多个复指数谐波分量。
k 
jk0t a e  k , k  0, 1, 2

is also periodic,
Example 2： x(t )  cos  t  1 e j0t  1 e  j0t 0
x(t )   ak e sk t
y (t )   ak H ( sk )e sk t
x(n)   ak Z kn
k
k
y(n)   ak H ( Z k )Z kn
k
k
qustion
How broad a class of signals could be represented as a linear combination of complex exponentials?
Some alternative form for the Fourier series (Cont)
ak is expressed in rectangular form as ak  Bk  jCk
x(t )  a0 

k 
 (B
1
k
 jCk )e
jk0t
  ( Bk  jCk )e
3.1 Historical Perspective (cont)
3、Largange criticized the use of trigonometric
series to examine vibrating string in 1759.
4、Fourier claimed that any periodic signal
3.3.1. Linear Combinations of Harmonically Related Complex exponentials
The set of harmonically related complex exponentials
k (t )  {e
jk0t
} k  0, 1, 2,
k 1

jk0t
jk0t  jk0t   a0    ( B  jC ) e  ( B  jC ) e k k k  k  k 1
Q a  a k
* k
 Bk  jCk  Bk  jCk
thus Bk  B k
Ck  Ck
Conclusion: the real part of ak is even , the imaginary part of ak is odd
1 a1  2
2
2
Example 3 ： x(t )  cos  t  2cos3 t 0 0 1 j0t  j0t j 30t  j 30t  [e  e ] e e 2 1 a3  1 a1  2
Some alternative form for the Fourier series
x (t ) 
k 
jk0t a e  k ,

2 0  T0
j ( k  n )0t
x(t )e
 jn0t

k 
ae
k


T0
0wenku.baidu.com
x(t )e
 jn0t
dt 
k 
a 
k

T0
0
e
j ( k  n )0t
dt

T0
0
e
j ( k  n )0t
dt   cos(k  n)0tdt  j  sin(k  n)0tdt

continuios time
st
e
st
n
discrete time
z
n
e
h(t )
y(t )   e
  s ( t  )
y(t )
z
h( n)

y ( n)
Using Time domain method，
h( )d  e
st


h( )e s d  H (s)est
k 1
Some alternative form for the Fourier series (CONT)
 a0  [ A k e jk0t e j k  Ak e jk0t e jk ]
k 1 
Q a  ak
* k
 Ak e
ak
 k 1

 jk
thus： Ak  A k Conclusion:
频域分析） 3. Properties of Fourier Series（傅立叶级数的 性质）
3.0 Introduction(引言)
The basis for time domain (chapter 2) 1) Signal can be represented as linear combination of shift impulses。 2) System is LTI。
k 1
k 1 
——trigonometric functions form (another form)
3.3.2. Determination of the Fourier Series Representation of a continuous-time Periodic Signal Assuming periodic signal x(t) can be represented with the Fourier series
k  k
 Ak e
j k
is even ,and
 jk0t
k is odd
 Ak e
jk0t
 x(t )  a0  [ A k e
k 1
e
j  k
e
j k
]
 a0  2 Ak cos(k0t   k )
——trigonometric functions form
Example 1 ( 3.1)： • a LTI systems y(t)=x(t-3) , now the input x(t)=cos(4t)+cos(7t), detemin y(t)? x(t)= 1/2ej4t +1/2e-j4t+1/2ej7t+1/2e-j7t
H (s)    (  3)es d  e3s
 
st
of LTI systems H ( s ) 、H ( z ) are eignevalue.
H ( z) 
k  n h ( n ) z  
The usefulness of decomposition in term of eigenfuction is important for the analysis of LTI systems . If ：
Some alternative form for the Fourier series (Cont)
jk0t  jk0t   x(t )  a0   ( Bk  jCk )e  ( Bk  jCk )e  
 a0  2  Bk cos k0t  Ck sin k0t 