第五章讲义——离散时间傅立叶变换

~
− jk ( 2π / N ) n
+∞
define the envelop of Nak is X (e jw ) X (e ) =
jw
n=−∞
∑ x [ n] e
)
n=−∞
n=−∞ − jk ( 2π / N ) n
1 = N
∑
+∞
x [ n] e−
− jk ( 2π / N ) n
∑
+∞
x [ n] e− jwn
jw
Example 5.3
1, | n |≤ N1 x[n] = ,determine X e jw 0, | n |> N1
( )
X (e ) =
jω
sinω ( N1 + 1 ) 2 sin(ω / 2)
5.0 Introduce
The difference of CFS and DFS
% (t) = x
CFS
ak
k =−∞
∑
T
+∞
ak e jkw0t
1 = T
∫
% ( t ) e − jkw0t dt x
% [ n] = x
DFS
k =< N >
∑
ak e
jkω0 n
1 ak = N
n=< N >
CFT
X ( jw ) = ∫ 1 x(t) = 2π
+∞ −∞
DFT
− jwt
x(t)e
dt
X (e
jω
)= 1 2π
n=−∞
∑
2
+∞
x[n]e − jωn
jω
∫
+∞
−∞
X ( jw ) e jwt dt
x[n] =
∫ π X (e
)e jωn dω
X (e jw ) is periodic, period is 2π w → 0, low fren] =
k= N
∑
1 X e jkw0 e jkw0 n N
(
1 = 2π
~
k= N
∑
X e jkw0 e jkw0n w0
(
)
N → ∞, x [ n] → x [ n] and ∑ →∫
1 x [ n] = 2π
∫π
2
X ( e jw ) e jwndw
The difference of CFT and DFT
∑
% [ n] e − jkω0n a = a x k k + rN
The development of CFT
T →∞⇒
CFT:
% ( t) → x ( t ) x 2π w0 = → 0, ∑ →∫ T
+∞ −∞
X ( jw ) = ∫ 1 x(t) = 2π
x(t)e
− jwt
dt
jwt
∫
+∞
−∞
X ( jw ) e dw
5.1 Representation of aperiodic signals: the discrete-time Fourier transform
DFT:
N →∞⇒ % [ n] → x [ n] x
1 ak = N
n=− N1
∑
N2
x [ n] e−
Nak =
离散时间傅立叶变换也可以看成是周期信号X(ejw)在频域内 展开为傅立叶级数，系数为x【n]
Example 5.1
x[n] = a u[ n] , | a |< 1,determine X e
n
( )
jw
Example 5.2
x[n] = a , | a |< 1,determine X e
n
( )
第五章讲义离散时间傅立叶变换离散傅立叶反变换离散傅立叶变换推导傅立叶变换傅立叶变换的性质傅立叶变换红外光谱常用傅立叶变换稀疏傅立叶变换傅立叶逆变换傅立叶反变换
Signals and Systems
Chapter 5 The Discrete-Time Fourier Transform
Liu Ke, School of Automation Engineering