X( j)
2 T1 1 T 2



2 T


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4.3 傅里叶变换的性质 Properties of the Fourier Transform
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Eg1:
x(t) e u(t 3) X(jω ) ?
at
at
解 :e
1 u (t ) j a
例3： x(t )
1 ak T
T

(t nT )
n
均匀冲激串
1 dt T

2 T 2
(t ) e

j
2 kt T

2 T 2
T
1 ( t ) dt T
2 X ( j ) T
x (t )
1
2 ( k ) 0 ( k0 ) T k k
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实信号的正负 • 若 x(t ) 是实信号，则 x(t ) x (t ) 频率成份互为 于是有: X ( j ) X * ( j ) 共轭对称。
Wang Zhengyong
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讨论傅里叶变换的性质，旨在通过这些性 质揭示信号时域特性与频域特性之间的关系， 同时掌握和运用这些性质可以简化傅里叶变 换对的求取。
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4.2 周期信号的傅里叶变换
The Fourier Transformation of Periodic Signals
X ( jω ) e j , | | 5
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Eg2:
≮X1( j) ≮X( j) a
2. 时移: Time Shifting
连续时间信号: 若 x(t ) X ( j ) 则 x(t t0 ) X ( j )e jt0 离散时间信号: 若 x ( n ) X ( e j ), 则
x ( n n 0 ) X ( e j ) e j n 0
这表明信号的时移只影响它的相频特性，其相频 特性会增加一个线性相移。
X( j) X( j)
≮ X ( j )
≮ X ( j )
即：模是偶函数，相位是奇函数
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离散时间信号:
ax1 (n) bx2 (n) aX1 (e j ) bX 2 (e j )
FT
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Wang Zhengyong
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考查
x (t ) e
j 0 t
FT
2 ( 0 )
表明: 周期性复指数信号的频谱是一个冲激。 若 x(t ) e jk0t 则 X ( j ) 2 ( k0 ) 当把周期信号表示为傅里叶级数时:
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3. 共轭及共轭对称性质: Conjugate and Symmetry 连续时间信号:
若 则 证明:

FT
x ( t ) e at u ( t 3) e 3 a e a ( t 3 ) u ( t 3)
Time shifting property
X ( j ) e
3a
1 j 3 e j a
X ( jω ) ?
?
sin 5( t 1) x (t ) ( t 1)

x (t ) X ( j )
x* (t ) X * ( j )
由 X ( j ) x (t )e j t dt
可得
X ( j )
所以 即
*
*

x * ( t ) e j t dt
X ( j ) x* (t )e jt dt

x*(t) X*( j)
a
a

/ 2
/ 4
a

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• 如果 x(t ) x( t ) ,信号是实偶函数。则
X ( j ) x (t )e jt dt
4. 微分与积分: Differentiation and Integration 时域微分性质: 若 x(t ) X ( j)
dx(t ) 则 j X ( j ) dt
j j [ ( 0 ) ( 0 )] 0
j
X ( j)
j

0
0
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*
若 X ( j) Re[ X ( j)] j Im[ X ( j)] 则可得
Re[ X ( j )] Re[ X ( j )]
Im[ X ( j )] Im[ X ( j )]
• 若 X ( j)
即实部是偶函数 虚部是奇函数
X ( j) e
j≮X ( j ) 则可得出
x(t ) e u(t ), a 0
1 X ( j ) | X ( j ) | e j ≮X ( j ) a j
X ( j ) 1 a
X ( j )
1/ a
1 2a
0
2 2
at
,
≮
X ( j ) tg
a
/4

周期信号的FT: X ( j )
2 a ( k )
k 0 k
表明：周期信号的傅里叶变换由一系列冲激组成，每 一个冲激分别位于信号的各次谐波频率处，其冲激强 度正比于对应的傅里叶级数系数 。
ak
1 j 0 t j 0 t x ( t ) sin t [ e e ] 例1： 0 2j X ( j ) [ ( 0 ) ( 0 )]
X ( j ) 是虚奇函数。
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连续时间信号: 若 x(t) X ( j)
则
x* (t) X * ( j)
X ( j )
2 T
t
2T T 0


2 2 X ( j) ( k ) T k T
2 0 2 T T
T
2T
x(t )
n
(t nT )
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j [≮X ( j ) a ] j≮X ( j ) ja X ( j ) | X ( j ) | e | X ( j ) | e e 1 x(t )和X ( j); 已知 ja ( ) X j e 设x1 (t )的FT为X1 ( j)且满足 | X1 ( j) | X ( j) ,相位如图, x(.t a) x1x((tt) 表示 试将x1 ( t )用
例4. 周期性矩形脉冲
T T1
(t ) x
1
t
0
T1 T
sin( k0T1 ) 2T1 2T1 ak Sa( k0T1 ) sin c( k0T1 / ) T k T

X ( j )
k

2 sin( k 0 T1 ) ( k 0 ) k
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1. 线性: Linearity 连续时间信号:
FT FT
若 x(t) X ( j), y(t) Y ( j) 则 ax(t ) by(t ) aX ( j ) bY ( j ) FT
?
设实信号x(t ) xe (t ) xo (t )且 X(j )=Re[X(j )]+jIm[X(j )] 则xe (t ) ?; xo (t ) ?
FT FT
正确答案:
xe (t ) Re[X(j )] xo (t ) jIm[X(j )]
FT
FT
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