《信号与系统》第五章课件(英文版)

1. Development of the Discrete-Time Fourier Transform (离散时间傅立叶变换 的导出);
2. Basic Fourier Transform Pairs (常用信 号的离散时间傅立叶变换对);
3. The Fourier Transform for Periodic Signals (离散时间周期信号的傅立叶变换);
X ( e jω
)
=
sin(
2N1
+
1
)
ω
2
ω
sin
2
ak
=
1 N
sin
π
N
k( 2N1
π
+1)
,
sin k
N
so
ak
=
1 N
X ( e jω ) ω = 2π k N
②Compare with the corresponding c-t aperiodic square wave signal
X( jω ) = 2T1 sinωT1 ωT1
② Are there convergence issues
associated with
∫ x [n] = 1 X ( e jω )e jωndω ?
2π 2π
NO!
Because the integral in this equation is
over a finite interval of integration.
3)Using the property of the Fourier transform to examine frequencydomain analysis of signals and LTI systems.
note that the similarities and differences between continuous-time and discrete-time Fourier transform analysis.
j 2π kn
ake N ,
k =< N >
∑ [ ] ak
=
1
x%
N n=<N>
n
− j 2π kn
eN
∑ [ ] ∑ [ ] so
Nak =
N /2
x%
− j 2π kn
ne N =
+∞
x
− j 2π kn
ne N
n=− N / 2
n = −∞
as N → ∞ , 2π k → ω
N
Define:
∫ ( ) We approximate x[n] by xˆ [n] =
1
+W
X
e jω
e jωndω
2π −W
δ பைடு நூலகம்n]
5.2 The Fourier Transform For Periodic Signals(p367)（周期信号的离散傅里叶变换）
9consider the Fourier transform of the sequence
5.0 Introduction
Analytical objects : aperiodic discretetime signals and systems Analytical methods: (similar to CTFT) 1)An aperiodic d-t signal can be viewed a periodic d-t signals with an infinite period. 2)As the period becomes infinite, the discrete-time Fourier series representation becomes the discrete-time Fourier transform .
X( e jω ) =
1
1+ a2 − 2acosω
Phase:
X ( e jω ) = − tg−1 a sinω 1 − a cosω
All of values are of the same sign
Low-frequency signal
Decaying
0<a<1
Decaying alternate in value
Consider a discrete-time periodic signal:
N → ∞ periodic
aperiodic
for a d-t periodic signal x% [n] , we have
the discrete-time Fourier series pair:
[ ] ∑ x% n =
Example 5.3(p365)
x
[n]
=
⎧1, ⎨⎩0,
n ≤ N1 n > N1
∑ X ( e jω
)=
N1
e − jω n
n=− N1
=
sin(
2
N
1
+
1
)
ω
2
sin ω
2
Real and even sequence
N1 = 2
Real and even function
①Compare with the corresponding periodic square wave signal
Example 5.1(p362) x [n] = anu[n] , a < 1
∑ [ ] ∑( ) ∞
X ( e jω ) = anu
n= −∞
∞
n e− jωn =
n=0
ae− jω
n
=
1−
1 ae− jω
Where X ( e jω ) is a complex function
Magnitude:
x[n] = e jω0n
In c-t time, we saw ( ) e jω0t ↔ 2πδ ω − ω0
(Note the d-t Fourier transform must be periodic
in ω with 2π )
∞
∑ Therefore, we expect e jω0n ↔ 2πδ（ω − ω0 − 2π k）
high frequencies are the values of ω near odd multiples of π .
Low-frequency signal
high-frequency signal
5.1.2 examples of discrete-time Fourier transform (p362)（离散傅里叶变换的例子）
[ ] ∫ x n = 1
−∞
X ( e jω )e jωndω
2π 2π
discrete-time Fourier transform pair
9Differences between the c-t and d-t Fourier transform :
1) periodicity of the discrete-time transform X ( e jω )
5.1 Representation of Aperiodic Signals: The Discrete-Time Fourier Transform (p359)（非周期信号的表达：离散傅里叶变换）
5.1.1 Development of the Discrete-Time Fourier Transform (离散傅里叶变换的导出)
Ch5 The Discrete-Time Fourier Transform 第5章 离散时间傅立叶变换
V Abbreviations（缩写）：
1. CFS :The Continuous-Time Fourier Series ——连续时间傅立叶级数
2. DFS :The Discrete-Time Fourier Series ——离散时间傅立叶级数
k = −∞
To check the validity of the expression above,
∫ ∫ 1
2π
X (e jω )e jωndω =
2π
2π
0
δ
(ω
- ω0 )e
jωndω
=
e jω0n
∞
∑ so e jω0n ↔ 2πδ（ω − ω0 − 2π k） k = −∞
consider an arbitrary periodic sequence x [n]
of as a linear combination of complex
exponentials infinitesimally close in
frequency and with amplitudes 1 X( e jω )dω 2π
结论：
∞
∑ [ ] X ( e jω ) = x n e− jωn
n=0
∞
∞
∑ ∑ = a ne jωn + a ne − jωn
n=1
n=0
=
ae jω 1 − ae jω
+
1
−
1 ae −
jω
=
1− a2
1 + a2 − 2a cos ω
Low-frequency signal
0<a<1
Real and even function
Real and even sequence
X(
e
jω
)
=
sin(
2N1 +
ω
1)ω
2
sin
2
Example 5.4(p367)
δ [n]
x [n] = δ [n]
1
n
0
∞
∑ [ ] X(e jω ) = x n e− jωn = 1
n=−∞
X(e jω )
1
−π 0
ω
π
5.1.3 Convergence Issues associated with the Discrete-Time Fourier Transform（p366) (离散 傅里叶变换的收敛问题)
¾Question:
①What is the conditions on x[n] that
guarantee the convergence of this ∞
∑ sum X ( e jω ) = [ ] x n e− jωn −∞ +∞
Answer 1: Condition: ∑ x[n] < ∞ n = −∞ ( i.e. x[n] is absolutely summable)
4. Properties of the Discrete-Time Fourier Transform (傅立叶变换的性质);
5. The frequency response and frequency-domain methods for discrete-time signals and systems (离 散系统的频率响应与频域分析方法);
lim Nak
N →∞
X（e jω）
∑ [ ] ∞
X（ e jω）=
x n e − jω n
n = −∞
discrete-time Fourier transform
Compare X（e jω）with ak , we have
ak
=
1 N
X (e jω ) ω=2π k N
=
1 N
X(e jkω0 )
high-frequency signal
−1 < a < 0
Example 5.2 (p364) x [n] = a n , a < 1
x[n] = [ a−nu −n−1] + anu[n]
−1
∞
∑ ∑ X ( e jω ) =
a − ne − jω n + a ne − jω n
n = −∞
3. CTFT :The Continuous-Time Fourier Transform ——连续时间傅立叶变换
4. DTFT :The Discrete-Time Fourier Transform ——离散时间傅立叶变换
Focus on it in this chapter
V Main content
Answer 2:
Condition: ∑+∞ x[n] 2 < ∞ n= −∞
( i.e. the sequence has finite energy)
The two conditions above can guarantee ∞
the convergence of ∑ [ ] X ( e jω ) = x n e− jωn −∞
X (e j(ω +2π ) ) = X (e jω )
2) the finite interval of integration in the synthesis equation.
¾In discrete-time ,
Low frequencies are the values of ω near even multiple of π ;
so
[ ] ∑ x% n = 1
X (e jkω0 ) ⋅ e jkω0n ,
N k=< N >
ω0
=
2π
N
∑ =
1
2π
k =< N >
X (e jkω0 ) ⋅ e jkω0n
⋅ω0
as N → ∞
∫ ∴ x [n ] = 1 X ( e jω )e jω nd ω
2π 2π
an aperiodic sequence can be thought