chapter 4 the continue-time fourier transformmin.sjtu.edu.cn/files/ss2019/min_ss_chap4.pdf4.1 the...
TRANSCRIPT
![Page 1: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/1.jpg)
Chapter 4 The Continue-Time Fourier
Transform
Instructor: Hongkai Xiong (熊红凯)
Distinguished Professor (特聘教授)
http://min.sjtu.edu.cn
TAs: Yuhui Xu,Qi Wang
Department of Electronic Engineering
Shanghai Jiao Tong University
2019-04
![Page 2: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/2.jpg)
Foreword of the Chapter
• By exploiting the properties of superposition and time invariance, if we know the response of an LTI system to some inputs, we actually know the response to many inputs
• If we can find sets of “basic” signals so that
▫ We can represent rich classes of signals as linear combinations of these building block signals.
▫ The response of LTI Systems to these basic signals are both simple and insightful.
![Page 3: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/3.jpg)
• Candidate sets of basic signal
▫ Unit impulse function and its delays
▫ Complex exponential signals (Eigenfunctions of all LTI systems)
• In this Chapter, we will focus on: why, how, what
▫ Can we represent aperiodic signals as “sums or integrals” of complex exponentials
▫ How to represent aperiodic signals as “sums or integrals” of complex exponentials
▫ What kinds of aperiodic signals can we represent as “sums or integrals” of complex exponentials? (how large types of such signals can benefit from the Fourier Transform?)
][/)( nt
sttj ee / nnj ze /
![Page 4: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/4.jpg)
Topic
4.0 Introduction
4.1 The Continuous-Time Fourier Transform
4.2 The Fourier Transform for Periodic Signals
4.3 Properties of the Continuous-Time Fourier Transform
4.4 The Convolution Property
4.5 The multiplication Property
4.6 System Characterized by Linear Constant-Coefficient
Differential Equations
![Page 5: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/5.jpg)
Topic
4.0 Introduction
4.1 The Continuous-Time Fourier Transform
4.2 The Fourier Transform for Periodic Signals
4.3 Properties of the Continuous-Time Fourier Transform
4.4 The Convolution Property
4.5 The multiplication Property
4.6 System Characterized by Linear Constant-Coefficient
Differential Equations
![Page 6: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/6.jpg)
4.0 Introduction
• Fourier Series Representation
▫ It decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The discrete-time Fourier transform is a periodic
• Fourier Transform
▫ A representation of aperiodic signals as linear combinations of complex exponentials
![Page 7: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/7.jpg)
T1 kept fixed T increases
Motivating Example
Discrete
frequency
points
become
denser in ω
as T
increases
![Page 8: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/8.jpg)
• Then for periodic square wave, the spectrum of x(t), i.e. {ak},
are , the spectrum space is
• Then for square pulse, the spectrum X(jω) are , the
spectrum space is , i.e. the complex exponentials
occur at a continuum of frequencies
Tk
Tkak
0
10 )sin(2
T
20
)sin(2 1T
02
0 T
![Page 9: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/9.jpg)
Topic
4.0 Introduction
4.1 The Continuous-Time Fourier Transform
4.2 The Fourier Transform for Periodic Signals
4.3 Properties of the Continuous-Time Fourier Transform
4.4 The Convolution Property
4.5 The multiplication Property
4.6 System Characterized by Linear Constant-Coefficient
Differential Equations
![Page 10: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/10.jpg)
4.1.1 Development
• To derive the spectrum for aperiodic signals x(t), we can
approximate it by a periodic signal with infinite period
T
)(~ tx
-T1 T1
…… ……
-T1 0 T1 T
)(tx
)(~ tx
)()(~lim txtxT
![Page 11: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/11.jpg)
Assuming (1) is converged, we define
![Page 12: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/12.jpg)
• Thus
𝑥 𝑡 =
𝑘
𝑎𝑘𝑒𝑗𝑘𝜔0𝑡 =
𝑘
1
𝑇𝑋(𝑗𝑘𝜔0)𝑒
𝑗𝑘𝜔0𝑡
=1
2𝜋
𝑘=−∞
∞
𝑋(𝑗𝑘𝜔0)𝑒𝑗𝑘𝜔0𝑡𝜔0
• When 𝑇 → ∞
𝑥 𝑡 =1
2𝜋න−∞
∞
𝑋(𝑗𝜔)𝑒𝑗𝜔𝑡𝑑𝜔
𝑥 𝑡 =1
2𝜋∞−∞𝑋(𝑗𝜔)𝑒𝑗𝜔𝑡𝑑𝜔
𝑋 𝑗𝜔 = ∞−∞𝑥(𝑡)𝑒−𝑗𝜔𝑡𝑑𝑡
Synthesis equation
Analysis equation
![Page 13: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/13.jpg)
4.1.2 Convergence
• What kinds of signals can be represented in Fourier Transform (satisfies one of the following 2 conditions)
▫ 1、Finite energy
Then we are guaranteed that:
𝑋(𝑗𝜔) is finite
∞−∞
𝑒(𝑡) 2𝑑𝑡 = 0
(𝑒 𝑡 = ො𝑥 𝑡 − 𝑥(𝑡) ො𝑥 𝑡 =1
2𝜋∞−∞𝑋(𝑗𝜔)𝑒𝑗𝜔𝑡𝑑𝜔)
![Page 14: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/14.jpg)
▫ 2、Dirichlet conditions, require that
𝑥(𝑡) be absolutely integrable
𝑥(𝑡) have a finite number of maxima and minima within any finite interval
𝑥(𝑡) have a finite number of discontinuities within any finite interval. Furthermore, each of these discontinuities must be finite
Then we guarantee that
ො𝑥 𝑡 is equal to 𝑥(𝑡) for any 𝑡 except at a discontinuity, where it is equal to the average of the values on either side of the discontinuity
𝑋(𝑗𝜔) is finite
![Page 15: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/15.jpg)
Examples • Exponential function
Magnitude Spectrum Phase Spectrum
Even symmetry Odd symmetry
If α is complex, x(t)
is absolutelty
integrable as long
as Re{α}>0
![Page 16: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/16.jpg)
22
2)(0,)(
jXetx
t
![Page 17: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/17.jpg)
Examples
• Unit impulse 𝑥 𝑡 = 𝛿(𝑡)
𝑋 𝑗𝜔 = න−∞
+∞
𝛿(𝑡)𝑒−𝑗𝜔𝑡𝑑𝑡 = 1
• DC Signal
)(2)(1)( jXtx
)(21
)(2
1
2
1)(
2
1
de tj
![Page 18: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/18.jpg)
Example
• Rectangle Pulse Signal
1
1
,0
,1)(
Tt
Tttx 1
1 1
sin( ) 2 =2 ( )a
TX j T S T
𝑆𝑎 𝑥 =𝑠𝑖𝑛𝑥
𝑥
𝑠𝑖𝑛𝑥 𝑥 =𝑠𝑖𝑛𝜋𝑥
𝜋𝑥
![Page 19: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/19.jpg)
Topic
4.0 Introduction
4.1 The Continuous-Time Fourier Transform
4.2 The Fourier Transform for Periodic Signals
4.3 Properties of the Continuous-Time Fourier Transform
4.4 The Convolution Property
4.5 The multiplication Property
4.6 System Characterized by Linear Constant-Coefficient
Differential Equations
![Page 20: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/20.jpg)
• For a periodic signal x(t) with fundamental frequency , what’s its FT?
T
20
k
tjk
keatx 0)(
k
tjk
k
k
tjk
k eaeatx ][][)]([ 00
?0 tjk
ethe question becomes:
![Page 21: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/21.jpg)
• Thanks to the impulse function, suppose𝑋 𝑗𝜔 = 𝛿 𝜔 − 𝜔0
𝑥 𝑡 =1
2𝜋න−∞
∞
𝛿 𝜔 − 𝜔0 𝑒𝑗𝜔𝑡𝑑𝜔 =1
2𝜋𝑒𝑗𝜔0𝑡
• That is 𝑒𝑗𝜔0𝑡 ↔ 2𝜋𝛿 𝜔 − 𝜔0
• So
𝑥 𝑡 =
𝑘
𝑎𝑘𝑒𝑗𝑘𝜔0𝑡 ↔ 𝑋 𝑗𝜔 =
𝑘
2𝜋𝑎𝑘 𝛿 𝜔 − 𝑘𝜔0
— All the energy is
concentrated in one
frequency — ωo,
![Page 22: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/22.jpg)
• So for a periodic signal x(t) with fundamental frequency , its FT is:
▫ Fourier Series Coefficient
▫ Fourier Transform
• The FT can be interpreted as a train of impulses occurring at the harmonically related frequencies and for which the area of the impulse at the kth harmonic frequency kω0 is 2π times the kth F.S. coefficient ak
T
20
2
2
0
0
)(1
)(
T
T
tjk
k
tjk
k
dtetxT
a
eatx
0 0
2( ) ( 2 ( )k
k
x t X j a kT
) ,
![Page 23: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/23.jpg)
• Example: )cos()( 0ttx
0 0
1 1
1 1( )
2 2
1
2
0 1
j t j t
k
x t e e
a a
a k
,
)]()([)( 00 jX
ka
k1-1
1/2
00
)( jX
Similarly:
)]()([sin 000 jt
![Page 24: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/24.jpg)
• Example:
k
kTttx )()(
……
ka
0 0( ) ( )k k
t kT k ……
( )X j
k
Same function in the frequency-domain!
Note: (period in t) T ⇔ (period in ω) 2π/T
![Page 25: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/25.jpg)
Topic
4.0 Introduction
4.1 The Continuous-Time Fourier Transform
4.2 The Fourier Transform for Periodic Signals
4.3 Properties of the Continuous-Time Fourier Transform
4.4 The Convolution Property
4.5 The multiplication Property
4.6 System Characterized by Linear Constant-Coefficient
Differential Equations
![Page 26: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/26.jpg)
• Linearity
• Time Shifting
)()( jXtx )()( jYty
)()()()( jbYjaXtbytax
)()( jXtx
)()( 0
0 jXettx
tj
![Page 27: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/27.jpg)
• Time and Frequency Scaling
)()( jXtx
)(||
1)(
a
jX
aatx
1a )()( jXtx for
compressed in time ⇔ stretched in frequency
( ) ?x at b
a
bj
ea
jX
abatx
)(||
1)(
![Page 28: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/28.jpg)
• Example: Determine the Fourier Transform of the following signals
2( ) ( )tx t e u t
2( 1)( ) ( )tx t e u t
2( ) ( 1)tx t e u t
1.
2.
3.
![Page 29: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/29.jpg)
• Differentiation
• The differentiation operation enhances high-frequency components in the effective frequency band of a signal
• Without any further information about the DC component of the original signal, we cannot completely recover it from its differentials
)()( jXtx
)()(
jXjdt
tdx
![Page 30: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/30.jpg)
• Integration
dxtg
t
)()( )()( jXtx
)()()()(*)()( jUjXjGtutxtg
0
1
u(t)
where u(t) is the unit step function, defined as
![Page 31: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/31.jpg)
• 𝑢 𝑡 =1+𝑠𝑔𝑛(𝑡)
2
• 𝑥 𝑡 = 𝑠𝑔𝑛 𝑡 ↔ 𝑋 𝑗𝜔 =2
𝑗𝜔
0
1
u(t)
0
1
-1
sgn(t)
0
1
DC
)(2)(1)( jXtx
)(1
)(
j
jU
Odd
function
![Page 32: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/32.jpg)
• 𝑢 𝑡 =1+𝑠𝑔𝑛(𝑡)
2
• 𝑥 𝑡 = 𝑠𝑔𝑛 𝑡 ↔ 𝑋 𝑗𝜔 =2
𝑗𝜔
0
1
u(t)
0
1
-1
sgn(t)
0
1
DC
)(2)(1)( jXtx
)(1
)(
j
jU
Odd
function
![Page 33: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/33.jpg)
)()()()(*)()( jUjXjGtutxtg
)()0()(1
)()()(
XjXj
jGdxtg
t
)(1
)(
j
jU
according to:
The integration operation diminishes high-frequency components in the effective frequency band of a signal
![Page 34: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/34.jpg)
• Example: triangle pulse
1
2
2
' ( )x t
2
2
2
2
)(tx
)2
||0
2||
||21
)(
t
tt
tx
,
,
'
1
11
'
1
( ) )
)( ) (0) ( )
(0)= ( ) 0
x t X j
X jX j X
j
X x t dt
(
(
又Since
)4
(2
)4
(sin8
)( 2
2
2
SajX
![Page 35: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/35.jpg)
• 2 approaches to calculate X(0) :
01. (0) ( ) |
2. (0) ( )
X X j
X x t dt
![Page 36: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/36.jpg)
• Example: sgn(t)
)0(1
)0(1)sgn()(
t
tttx
,
,
0
1
-1
sgn(t)
![Page 37: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/37.jpg)
• By defining the sgn function as a special exponential function
• By representing the sgn function in terms of unit step functions
0)0(
)0()(1
te
tetx
t
t
,
,
1
0
-1
)(1 tx
)(lim)sgn( 10
txt
jjXt
2)(lim)sgn( 1
0=
)()()sgn( tutut
jjjt
2]
1)([)
1)(()sgn(
![Page 38: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/38.jpg)
• By exploiting integration property
)()(2)(sgn1)(2)sgn( 1
' txtttut
1 1
( )( ) ( )
dx tx t X j
dt 设
)()()(1
xtxdttx
t
t
xdttxtx )()()( 1
Suppose
)()(2)()0()(
)( 11
xX
j
jXjX
2)(1 =jX
jj
t2
)()1(2)(22
)sgn(
When x(-∞)≠0
![Page 39: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/39.jpg)
• Duality
▫ Both time and frequency are continuous and in general aperiodic
▫ Suppose f() and g() are two functions related by
Same except for
these differences
)()( jXtx )(2)( jxtX
![Page 40: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/40.jpg)
• Examplesin
( )Wt
x tt
=
W
WjX
,0
,1)(
![Page 41: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/41.jpg)
• Example
• Example
1( )x t
t
00
0/2)()sgn()(
jjXttx
)sgn()( jjX
21
1)(
ttx
22
2)(0}Re{)(
jXetx
t,
ejX )(
![Page 42: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/42.jpg)
• Other duality properties
▫ (1) Frequency Shifting
)()( jXtx
))(()( 00
jXtxetj
![Page 43: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/43.jpg)
• Example:
)]()([
)]()([
][2
1sin
)]()([
][2
1cos
00
00
0
00
0
00
00
j
j
eej
t
eet
tjtj
tjtj
![Page 44: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/44.jpg)
▫ (2)Differentiation in frequency domain
▫ (3)Integration in frequency domain
d
jdXtjtx
)()(
d
jdXjttx
)()(
dXtxtxjt
)()()0()(1
t
dXjt
tx)(
)(when x(0)=0,
![Page 45: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/45.jpg)
• Example: 2( ) ( ) ?tx t te u t
2
2
2
1( )
2
1 1( ) ( )
2 (2 )
t
t
e u tj
dte u t j
d j j
2( ) ( 1) ?tx t te u t
![Page 46: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/46.jpg)
• Example :To determine x(t) according to X(jω)
1 1
)( jX
1
)(' jX
11
Hints: To exploit the
differentiation property in
frequency domain
'
1
11
'
1
) ( )
)( ) (0) ( )
(0)= ( ) 0
X j x t
x tx t x t
jt
x X j d
(
(
又
![Page 47: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/47.jpg)
• Conjugation and Conjugate Symmetry
)()( jXtx
)()( jXtx
If x(t) is real valued
)()( jXjX —Conjugate Symmetry
)]([)](Re[)( jXjIjXjX m
)](Re[)](Re[ jXjX
)](Im[)](Im[ jXjX
①
![Page 48: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/48.jpg)
)(|)(|)( jXjejXjX
|)(||)(| jXjX =
)()( jXjX =
②
evenandrealtx )( evenandrealjX )(
oddandrealtx )( oddandimaginarypurelyjX )(
)]([)]()([2
1)( jXRetxtxtxe
)]([)]()([2
1)(0 jXjItxtxtx m
③
![Page 49: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/49.jpg)
• Example:
)}({2)()()( || tueEtuetueetx t
v
ttt
jtue t
1
)(
222
2}
2
1Re{2)(
jtx
![Page 50: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/50.jpg)
• Parseval’s Relation
dffXdjXdttx
222 |)(||)(|2
1|)(|
2|)(| jX
d
T
jXdttx
T TT
T
2
2 |)(|lim
2
1|)(|
1lim
T
jX
T
2|)(|lim
——Energy-density Spectrum
——Power-density Spectrum
and:
2|)(| fX——Energy per unit frequency (Hz)
![Page 51: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/51.jpg)
• Example1:
• Example2:
)(tx
)( jX
2/
-1 -0.5 10.5
sin 2( )
tx t
t=
dttx 2|)(|
dttx 2|)(|
To determine
To determine
![Page 52: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/52.jpg)
• Example:
-1-2 1 2
1
2
)(tx
t
To use the FT of typical signals and FT
properties to determine the FT of the following
signals
![Page 53: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/53.jpg)
• Solution 1:
2( ) 2 ( ) ( 2) ( 2)x t g t u t u t
2
1
-2
1
1-1
2
+ +
gτ(t) is the rectangle pulse with width of τand unit magnitude
( )g t
1
2
-
2
t
![Page 54: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/54.jpg)
• Solution 2: '
1
1 1
( ) ( )
( ) ( )
x t x t
x t X j
设
11
( )( ) (0) ( ) 2 ( ) ( )
X jX j X x
j
则
-1
1
2
-2
'( )x t
t1 2
Assuming :
Then
![Page 55: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/55.jpg)
• Example: To determine the FC of the periodic signal by using FT
2
1T
2
1T1T
2
3 1T
)(tf
t
。。。 。。。
![Page 56: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/56.jpg)
• Let be the basic signal
•
)()( 00 Ftf
)(0 tf1
0 2/1T
)('
0 tf
0 2/1T1
2
T
)("
0 tf
0
1
2
T
1
2
T
" '1 10
1 1
2 2( ) ( ) ( ) ( )
2 2
T Tf t t t t
T T
1|)(
10
1
nn FT
F
![Page 57: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/57.jpg)
• Example :To determine x(t) according to )( jX
0 0
A
|)(| jX )( jX
0 0 w
A
|)(| jX )( jX
00
2/
2/
1.
2.
Notes: They
have different
phase
spectrum
![Page 58: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/58.jpg)
Topic
4.0 Introduction
4.1 The Continuous-Time Fourier Transform
4.2 The Fourier Transform for Periodic Signals
4.3 Properties of the Continuous-Time Fourier Transform
4.4 The Convolution Property
4.5 The multiplication Property
4.6 System Characterized by Linear Constant-Coefficient
Differential Equations
![Page 59: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/59.jpg)
4.4.1 Convolution Property
![Page 60: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/60.jpg)
• Example: the Triangle Impulse Signal
![Page 61: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/61.jpg)
4.4.2 Frequency Response
• Definition:
dtth )( -stable system
( )( ) ( ) ( ) / ( )
( )
Y jY j X j H j H j
X j
Conditioned on:
Then:
![Page 62: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/62.jpg)
4.4.2 Frequency Response
• The frequency response H(jω) can completely represent a stable LTI system (NOT all LTI systems)
H1(jω) H2(jω) H1(jω) · H2(jω)
Series interconnection of LTI systems (Cascaded system)
H1(jω)
H2(jω)
H1(jω) + H2(jω)
Parallel interconnection of LTI systems
![Page 63: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/63.jpg)
4.4.2 Frequency Response
• The frequency response is the F.T. of the impulse response, it captures the change in complex amplitude of the Fourier transform of the input at each frequency ω
▫ For a complex exponential input x(t), as a consequence of the eigenfunction property, the output y(t) can be expressed as:
▫ For a sinusoid input x(t), as a consequence of the eigenfunction property, the output y(t) can be expressed as:
jHjejHjH
Magnitude gain Phase shifting
tjtjejHtyetx 0
0
0 |)()()(
))(cos()()()cos()( 0000 jHtjHtyttx
![Page 64: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/64.jpg)
• Example: Consider an LTI system with If the input x(t)=sin(t), determine the
output y(t)
• Solution:
1( )
1H j
j
2
1
1( )
1
1( )
1
( ) tan ( )
H jj
H j
H j
4sin
2
1
))1(sin()1()(
t
jHtjHty
![Page 65: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/65.jpg)
• Example: Consider an LTI system with
for the input x(t)
Determine the output of the system
)()( tueth t
)()( 2 tuetx t
)(*)()( thtxty
![Page 66: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/66.jpg)
• Example: for a system with Gaussian response, i.e. the unit impulse response is Gaussian, consider the output of the system with a Gaussian input
Gaussian × Gaussian = Gaussian ⇒ Gaussian ∗ Gaussian = Gaussian
![Page 67: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/67.jpg)
Why: Log-Magnitude and Phase to illustrate the frequency
response
𝑌(𝑗𝜔) = 𝐻 𝑗𝜔 × 𝑋 𝑗𝜔
𝑙𝑜𝑔 𝑌(𝑗𝜔) = 𝑙𝑜𝑔 𝐻 𝑗𝜔 + 𝑙𝑜𝑔 𝑋 𝑗𝜔
∠𝐻 𝑗𝜔 = ∠𝐻1 𝑗𝜔 + ∠𝐻2(𝑗𝜔)
𝐻1(𝑗𝜔) 𝐻2(𝑗𝜔)
𝑙𝑜𝑔 𝐻(𝑗𝜔) = 𝑙𝑜𝑔 𝐻1 𝑗𝜔 + 𝑙𝑜𝑔 𝐻2 𝑗𝜔
∠𝑌 𝑗𝜔 = ∠𝐻 𝑗𝜔 + ∠𝑋(𝑗𝜔)
Easy to add
Easy to add
Cascading:
![Page 68: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/68.jpg)
How: Plotting Log-Magnitude and Phase
• a) For real-valued signals and systems
• b) For historical reasons, log-magnitude is usually plotted in units of decibels (dB):
Plot for ω ≥ 0, often with a
logarithmic scale for
frequency in CT
Why 20 log10(.)power magnitude
So… 20 dB or 2 bels:
= 10 amplitude gain
= 100 power gain
![Page 69: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/69.jpg)
• A Typical Bode plot for a second-order CT system
20 log10|H(jω)| and ∠ H(jω) vs. log10ω
![Page 70: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/70.jpg)
4.4.3 Filtering
-a process in which the relative complex magnitudes of the frequency components in a signal are changed or some frequency components are completely eliminated
• Frequency-Selective Filters—systems that are designed to pass some frequency components
undistorted, and diminish/eliminate others significantly
• Typical types of frequency-selective filters▫ LPF(Low-pass Filter)
▫ HPF(High-pass Filter)
▫ BPF(Band-pass Filter)
▫ BSF (Band-stop Filter
![Page 71: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/71.jpg)
)( jH
cc
1
c
cjH
,0
1)(
,LPF
HPF
BPF
passbandstopband
c
cjH
,0
1)(
,
21
21
,0
1)(
cc
ccjH
,,
cutoff frequency
upper cutoff frequencylower cutoff frequency
![Page 72: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/72.jpg)
• Example :
1/4-1/4
1
x(t)
1-1 t
。。。。。。
)( jH
3 3
To determine the response of the LPF to the signal x(t)
![Page 73: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/73.jpg)
• Some typical systems
▫ ① Delay
▫ ② Differentiator
0
0( ) ( )
( ) ( )j t
y t x t t
Y j X j e
0
)(
)()(
tje
jX
jYjH
)()(
)()(
jXjjY
dt
tdxty
j
jX
jYjH
)(
)()(
![Page 74: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/74.jpg)
▫ ③ Integrator
when
)()0()(
)(
)()(
Xj
jXjY
dxty
t
0)()0(
dttxX
jjX
jYjH
1
)(
)()(
![Page 75: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/75.jpg)
• Example: to determine outputs of the system with H(jω) in the figure with the following input signals
jtetx )(
)6)((
1)(
jjjX
-1 1
2j
-2j
)( jH
1、
2、
![Page 76: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/76.jpg)
• Example: for the following signal x(t) with period of 1
To determine the output of the system with frequency response H(jω) with the input x(t)
1)2
1(,0
)2
1(,2sin
)(
mtm
mtmttx
)33(3
)(
jjH
![Page 77: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/77.jpg)
• Solution:
∴ x(t) contains the frequency components:
Only the DC and the first order harmonic components are within
the passband of the LPF
k
k katx )(2)( 0
22
0 T
,4,2,0
)( jH Differe
ntiator
3
1
33
)(1 ty )(ty
and
![Page 78: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/78.jpg)
Filters
• Zero-phase shifting Ideal LPF
)( jH
cc
c
cjH
||,0
||,1)(
![Page 79: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/79.jpg)
sin( )
sin
c
c c
c
th t
t
t
t
)(1
2
1)( tSits c
dxx
xySi
sin)(
Unit impulse response Unit step response
where
The unit impulse response of the HPF is t
ttth c
sin)()(
![Page 80: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/80.jpg)
• Linear Phase Ideal LPF
c
tjejH
||,)( 0
|)(| jH
cc
)( jH
cc
c
cjH
||,0
||,1|)(| ctjH ||,)( 0
Result: Linear phase ⇔ simply a rigid shift in time, no distortion
Nonlinear phase ⇔ distortion as well as shift
𝑌 𝑗𝜔 = 𝑒−𝑗𝜔𝑡0𝑋 𝑗𝜔 𝑦 𝑡 = 𝑥(𝑡 − 𝑡0)time-shift
![Page 81: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/81.jpg)
▫ Unit impulse response:
▫ Unit step response:
)(
)(sin)(
0
0
tt
ttth c
)]([1
2
1)( 0ttSits c
![Page 82: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/82.jpg)
• How do we think about signal delay when the phase is nonlinear?
Concept of Group Delay
When the signal is narrow-band and concentrated near ω0, H
(jw) ~ linear with ω near ω0, then the differential of H (jw) at ω0
reflects the time delay.
For frequencies “near” ω0
For w “near” ω0
Τ(ω0)Time delay of
the original signal
![Page 83: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/83.jpg)
• Non-ideal LPF
R
C
1( ) ,H j
j RC
( )H j
2 2( )H j
1( ) tan ( )H j
)( jH
![Page 84: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/84.jpg)
)()( tueth t )()1()( tuets t
1e
)(th
1 t
1
11 e
)(ts
t
1
Unit impulse response Unit step response
• causal h(t <0) = 0, decaying
• s(t) non-oscillation and non-overshoot
![Page 85: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/85.jpg)
• Time domain and frequency domain aspects of non-ideal filter
▫ Trade-offs between time domain and frequency domain characteristics, i.e. the width of transition band ↔ the setting time of the step response
Definitions:
Passband ripple: δ1
Stopband ripple: δ2
Definitions:
Rise time: trSetting time: tsOvershoot: Δ
Ringing frequency ωr
Passband Transition Stopband
Rise time
Setting
time
Setting time: the time at which the step response settles to within δ (a specified tolerance) of its final value
![Page 86: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/86.jpg)
Topic
4.0 Introduction
4.1 The Continuous-Time Fourier Transform
4.2 The Fourier Transform for Periodic Signals
4.3 Properties of the Continuous-Time Fourier Transform
4.4 The Convolution Property
4.5 The multiplication Property
4.6 System Characterized by Linear Constant-Coefficient
Differential Equations
![Page 87: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/87.jpg)
4.5.1 Multiplication Property
)()()()( 2211 jXtxjXtx
)()()( 21 txtxtx
)]()([2
1)( 21
jXjXjX
![Page 88: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/88.jpg)
• Example :
E
0 1
2
T
)]2
()([sin)( 11
TtututEtx
![Page 89: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/89.jpg)
• Example :
t
t
sin
t
t
2
sin
]}2sin
[]sin
[{2
1)(
t
t
t
ttx
1
-1 1
1
-1/2 1/2
-3/2 -1/2 1/2 3/2
1/2
Then
2
sin sin2 ?
tt
dtt
![Page 90: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/90.jpg)
• Example :
𝑟 𝑡 = 𝑠 𝑡 × 𝑃 𝑡 ↔ 𝑅 𝑗𝜔 =1
2𝜋[𝑆 𝑗𝜔 ∗ 𝑃 𝑗𝜔 ]
𝐹𝑜𝑟 𝑝 𝑡 = 𝑐𝑜𝑠𝜔0𝑡 ↔ 𝑃 𝑗𝜔 = 𝜋[𝛿 𝜔 − 𝜔0 + 𝛿(𝜔 + 𝜔0)]
𝑅 𝑗𝜔 =1
2[𝑆(𝑗 𝜔 − 𝜔0 ) + 𝑆(𝑗 𝜔 + 𝜔0 )]
![Page 91: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/91.jpg)
4.5.2 Modulation
• Why?
▫ More efficient to transmit E&M signals at higher frequencies
▫ Transmitting multiple signals through the same medium using different carriers
▫ Transmitting through “channels” with limited passbands
▫ Others...
• How?
▫ Many methods
▫ Focus here for the most part on Amplitude Modulation (AM)
![Page 92: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/92.jpg)
Amplitude modulation
(AM)
Drawn assuming:
Modulating
signal
Carrier
signal
Modulated
signal
![Page 93: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/93.jpg)
• Synchronous Demodulation of Sinusoidal AM
𝑋 𝑗𝜔 =1
2𝜋𝑋𝑐 𝑗𝜔 ∗ 𝐶 𝑗𝜔
=1
2𝑋𝑐 𝑗 𝜔 − 𝜔0 + 𝑋𝑐 𝑗 𝜔 + 𝜔0
=1
2X jω +
1
4[𝑋𝑐(𝑗 𝜔 − 2𝜔0 ) +
𝑋𝑐(𝑗 𝜔 + 2𝜔0 )]
ො𝑥 𝑡 = 𝑥𝑐(𝑡) × 𝑐(𝑡)
If 𝜃 = 0
What if 𝜃 ≠ 0?
![Page 94: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/94.jpg)
• Synchronous Demodulation (with phase error) in the Frequency Domain
𝑐𝑜𝑠(𝜔𝑐𝑡 + 𝜃) ↔ 𝜋𝑒𝑗𝜃𝛿 𝜔 − 𝜔0 + 𝜋𝑒−𝑗𝜃𝛿(𝜔 + 𝜔0)
![Page 95: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/95.jpg)
• Asynchronous Demodulation
▫ Assume ωc>> ωM, so signal envelope looks like x(t)
▫ Add same carrier with amplitude A to signal
A = 0 ⇒ DSB/SC (Double Side Band, Suppressed Carrier)A > 0 ⇒ DSB/WC (Double Side Band, With Carrier)
![Page 96: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/96.jpg)
In order for it to function properly, the envelope function mustbe
positive for all time, i.e.A+ x(t) > 0 for all t.
Demo: Envelope detection for asynchronous demodulation.
Advantages of asynchronous demodulation:
— Simpler in design and implementation.
Disadvantages of asynchronous demodulation:
— Requires extra transmitting power [Acosωct]2to make sure A+
x(t) > 0 ⇒Maximum power efficiency = 1/3 (P8.27)
![Page 97: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/97.jpg)
• Example:
)(tx )(ty
)(1 ty
t5cos
)( jH
t
ttx
2sin)(
)( jH
5 5 For
To determine )(tySignal processing in
frequency domain
![Page 98: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/98.jpg)
• Double-Sideband (DSB) and Single-Sideband (SSB) AM
DSB, occupies 2ωMbandwidth in ω> 0.
Each sideband approach only occupies ωMbandwidth in ω> 0.
Since x(t) and y(t)
are real, from Conjugate symmetry both LSB and USB signals carry exactly the
same information.
![Page 99: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/99.jpg)
• Single-Sideband (SSB) AM
Can also get SSB/SC
or SSB/WC
![Page 100: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/100.jpg)
• An implementation of SSB modulation, p600,figure 8.21-22
Hilbert
Transform
tth
j
jjH
1)(
0
0)(
![Page 101: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/101.jpg)
• Frequency-Division Multiplexing (FDM)(Examples: Radio-station signals and analog cell phones)
All the channels
can share the
same medium.
![Page 102: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/102.jpg)
• FDM in the Frequency-Domain
![Page 103: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/103.jpg)
• Demultiplexing and Demodulation
▫ Channels must not overlap ⇒Bandwidth Allocation
▫ It is difficult (and expensive) to design a highly selective bandpass filter with a tunable center frequency
▫ Solution –Superheterodyne Receivers
ωa needs to be tunable
![Page 104: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/104.jpg)
• The Superheterodyne Receiver
▫ Operation principle:
— Down convert from ωc to ωIF, and use a coarse tunable BPF for the front end.
— Use a sharp-cutoff fixed BPF at ωIF to get rid of other signals.
![Page 105: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/105.jpg)
4.5.3 Sampling
• Most of the signals we encounter are CT signals, e.g. x(t). How do we convert them into DT signals x[n] to take advantages of the rapid progress and tools of digital signal processing
▫ — Sampling, taking snap shots of x(t) every T seconds
• T –sampling period, x[n] ≡x(nT), n= ..., -1, 0, 1, 2, ... —Regularly spaced samples
• Applications and Examples
▫ —Digital Processing of Signals
▫ —Images in Newspapers
▫ —Sampling Oscilloscope
▫ —…
How do we perform sampling?
![Page 106: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/106.jpg)
• Why/When Would a Set of Samples Be Adequate?▫ Observation: Lots of signals have the same samples
▫ By sampling we throw out lots of information –all values of x(t) between sampling points are lost.
▫ Key Question for Sampling:
Under what conditions can we reconstruct the
original CT signal x(t) from its samples?
![Page 107: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/107.jpg)
• Impulse Sampling—Multiplying x(t) by the sampling function
![Page 108: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/108.jpg)
• Analysis of Sampling in the Frequency Domain
Multiplication Property =>
=Sampling Frequency
![Page 109: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/109.jpg)
• Illustration of sampling in the frequency-domain for a band-limited (X(jω)=0 for |ω| > ωM) signal
![Page 110: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/110.jpg)
• Reconstruction of x(t) from sampled signals
If there is no overlap
between shifted
spectra, a LPF can
reproduce x(t) from xp(t)
![Page 111: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/111.jpg)
Suppose x(t) is band-limited, so that
X(jω)=0 for |ω| > ωM
Then x(t) is uniquely determined by its
samples {x(nT)} if
where ωs = 2π/T
![Page 112: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/112.jpg)
• Observations▫ (1) In practice, we obviously don’t
sample with
impulses or implement ideal lowpass filters
— One practical example: The Zero-Order Hold
▫ (2) Sampling is fundamentally a time varying operation, since we multiply x(t) with a time-varying function p(t). However, H(jω) is the identity system (which is TI) for band-limited x(t) satisfying the sampling theorem (ωs > 2ωM).
▫ (3) What if ωs <= 2ωM? Something different: more later.
![Page 113: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/113.jpg)
).(
.
.
)(,
2
2
,...,2,1,0),()(
.0)(
:
txequalexactlywillsignaloutputresultingthe
iffrequencycutoffandTgainwithfilterlowpassidealan
throughprocessedthenistrainimpulseThisvaluessamplesuccessive
arethatamplitudeshaveimpulsesuccessivewhichintrainimpulse
periodicageneratingbytxtreconstruccanwesamplestheseGiven
Twhere
ifnnTxsamplesitsbydetermineduniquelyistx
ThenforjXwithsignallimitedbandabeusLet
TheoremSampling
mcms
c
s
s
ms
s
m
![Page 114: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/114.jpg)
• Example:
:
.0)()(
signalsfollowingtheforrateNyquisttheDetermine
forjXwithtxsignallimitedbandaConsider m
dt
tdx
tx
tx
)()3(
)()2(
1)(2)1(
2
![Page 115: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/115.jpg)
• Time-Domain Interpretation of Reconstruction of Sampled Signals —Band-Limited Interpolation
The lowpass filter interpolates the samples assuming x(t) contains no
energy at frequencies >= ωc
![Page 116: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/116.jpg)
• Graphic Illustration of Time-Domain Interpolation
▫ Original CT signal
▫ After Sampling
▫ After passing the LPF
![Page 117: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/117.jpg)
• Interpolation Methods (1): Band-limited Interpolation: ideal LPF, i.e. sinc function in time domain
)]([)(
)()()()(
scs
n
c
cc
s
n
sr
nTtSanTx
SanTtnTxtx
![Page 118: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/118.jpg)
• Interpolation Methods (2): Zero-Order Hold
)(txr
0T
)(0 th
])2/sin(2
[)( 2/
0
T
ejH Tj
![Page 119: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/119.jpg)
)2/sin(2
)(
)(
)()(
2
0T
jHe
jH
jHjH
Tj
r
T
Reconstruct filter)(0 th)(0 tx
)()( jHth rr )(
)(
tx
tr)(tx p
)(tp
)(tx
H(jω) – Ideal interpolation filter (LPF)
![Page 120: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/120.jpg)
• Interpolation Methods (3): First-Order Hold —Linear interpolation
2
1 ]2/
)2/sin([
1)(
T
TjH
-T
)(1 th
T
)(txr
![Page 121: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/121.jpg)
• Under sampling and Aliasing
When ωs ≦ 2ωM => Under-sampling
![Page 122: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/122.jpg)
Xr (jω)≠X(jω) Distortion due to aliasing
— Higher frequencies of x(t) are “folded back” and take on the “aliases”of lower frequencies
— Note that at the sample times, xr(nT) = x(nT)
![Page 123: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/123.jpg)
• Example:
X(t) = cos(ω0t + Φ)
Sampling of cosω0t
Aliasing case:
Then with the ideal LPF with
cut off frequency of ωM<
ωc< ωs- ω0 , the
reconstructed signal is
cos((ωs-ω0)t)
Ref. Q7.38
![Page 124: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/124.jpg)
• Example: AM with an Arbitrary Periodic Carrier
C(t) – periodic with period T, carrier frequency ωc = 2π/T
![Page 125: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/125.jpg)
• Example: Modulating a (Periodic) Rectangular Pulse Train
In practice, we can use a (periodic) rectangular pulse train instead of impulses, since the later is impractical
![Page 126: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/126.jpg)
![Page 127: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/127.jpg)
• Discussions on modulating a (Periodic) Rectangular Pulse Train
▫ 1) We get a similar picture with any c(t) that is periodic with period T
▫ 2) As long as ωc= 2π/T > 2ωM, there is no overlap in the shifted and
scaled replicas of X(jω). Consequently, assuming a0≠0:
x(t) can be recovered by passing y(t) through a LPF
▫ 3) Pulse Train Modulation is the basis for Time-Division Multiplexing
▫ Assign time slots instead of frequency slots to different
channels, e.g. AT&T wireless phones
▫ 4) Really only need samples{x(nT)} when ωc> 2ωM⇒Pulse Amplitude
Modulation
![Page 128: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/128.jpg)
Topic
4.0 Introduction
4.1 The Continuous-Time Fourier Transform
4.2 The Fourier Transform for Periodic Signals
4.3 Properties of the Continuous-Time Fourier Transform
4.4 The Convolution Property
4.5 The multiplication Property
4.6 System Characterized by Linear Constant-Coefficient
Differential Equations
![Page 129: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/129.jpg)
LTI Systems Described by LCCDE’s
(Linear-constant-coefficient differential equations)
Using the Differentiation Property
Transform both sides of the equation
1) Rational, can usePFE to get h(t)
2) If X(jω) is rationale.g. x(t)=Σcie
-at u(t)
then Y(jω) is also rational
PFE: Partial-fraction expansion
![Page 130: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/130.jpg)
• Example:
)(2)(
)(3)(
4)(
2
2
txdt
tdxty
dt
tdy
dt
tyd
3)(4)(
2)(
2
jj
jjH
![Page 131: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/131.jpg)
• Zero-state response of LTI systems——Partial-fraction expansion method
• Example:
31)3)(1(
2
3)(4)(
2)( 21
2
j
A
j
A
jj
j
jj
jjH
vj
2
1|
3
2|)()1( 111
vv
v
vvHvA
2
1|
1
2|)()3( 332
vv
v
vvHvA
3
2
1
1
2
1
)(
jj
jH
)(1
tuej
t
)(2
1)(
2
1)( 3 tuetueth tt
Let
then
and
![Page 132: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/132.jpg)
• Example: To calculate the zero-state response of the system discussed in previous example
3)1(1)3()1(
2)()()( 2
2
1211
2
j
A
j
A
j
A
jj
jjXjHjY
4
1|
)3(
1|]
3
2[|)]()1[(
)!12(
11211
2
11
vvv
vv
v
dv
dvYv
dv
dA
2
1|)()1( 1
2
12 vvYvA
jtue t
1
)(
4
1|
)1(
2|)()3( 3232
vv
v
vvYvA
2
1 1 1
4 2 4( )1 ( 1) 3
Y jj j j
2)(
1]
1[)(
jjd
djtute t
)(]4
1
2
1
4
1[)( 3 tueteety ttt
high-order pole point
![Page 133: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/133.jpg)
• Homework BASIC PROBLEMS WITH ANSWER: 4.1, 4.4
BASIC PROBLEMS: 4.21, 4.22, 4.25, 4.32, 6.21, 6.22, 7.3, 7.4, 8.22, 8.30
![Page 134: Chapter 4 The Continue-Time Fourier Transformmin.sjtu.edu.cn/files/ss2019/MIN_SS_chap4.pdf4.1 The Continuous-Time Fourier Transform 4.2 The Fourier Transform for Periodic Signals 4.3](https://reader035.vdocuments.site/reader035/viewer/2022071502/61224e9c5f59fe12da7af85a/html5/thumbnails/134.jpg)
Many Thanks
Q & A
134