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Time and Frequency Characterization
of Signals and Systems
Chapter 6
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Chapter 6 Time and Frequency Characterization
§ 6.1 The Magnitude-Phase Representation
（ 幅度 -相位） of the Fourier Transform
jXjejXjX
dejXtx tj 2 1
jX —— Magnitude
2?jX —— Energy-Density
Xj? —— Phase
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tttx 6c o s2c o s41
tttx 6c o s42c o s2
Chapter 6 Time and Frequency Characterization
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Chapter 6 Time and Frequency Characterization
-1 0 1 ω
1jX1
t ttx?s in1?
-11 -10 –9 0 9 10 11 ω
1
jX 2
tt ttx 10c o ss i n22
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( ) c o s ( ) c o s ( ) c o s ( )1 2 3121 2 4 623x t t t t
0321
12,8,4 321
93.0,7.2,6 321
2.7,1.4,2.1 321
Chapter 6 Time and Frequency Characterization
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Chapter 6 Time and Frequency Characterization
2,A black-and-white picture
F1 2 1 2,,x t t X j j
22111 2 1 2 1 2,,jt jtX j j x t t e e d t d t
Figure 6.2
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12(,)X j j12| (,) |X j j
Chapter 6 Time and Frequency Characterization
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,| (,) |
,(,)
12
12 0
M ag ni tu de X j j
P hase X j j


,| (,) |
,(,)
12
12
1M a g nit ude X j j
P has e X j j


Chapter 6 Time and Frequency Characterization
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M a g ni tu d e
Chapter 6 Time and Frequency Characterization
,(,)12 P h a s e X j j
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Chapter 6 Time and Frequency Characterization
§ 6.2 The Magnitude-Phase Representation of the
Frequency Response of LTI Systems
Y j X j H j
Y j X j H j
Y j X j H j
Hj? —— Gain
Hj —— Phase Shift
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Chapter 6 Time and Frequency Characterization
ttttx 3c o sc o s3/c o s
c o s / 3 1 / 3 c o s 1 c o s 3 3y t t t t
§ 6.2.1 Linear and Nonlinear Phase
1,Linear Phase
0 tjejH 1jH 0 tjH
0 tjXjY
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Chapter 6 Time and Frequency Characterization
3/23c o s2/c o s3/3/c o s tttty
33/23c o s2/c o s3/3/c o s tttty
2,Nonlinear Phase
11 jjjH 1jH ωtg-jH -12
txty
t0
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Chapter 6 Time and Frequency Characterization
§ 6.2.2 Group delay （群延时）
Consider a narrowband signal
窄带信号
0?





2
0
2
0
0
0


jX jH
jj eejHjXjY α—— Group delayat ω=ω
0
The Group Delay at each frequency
jHdd
0
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0( ) ( )t s ig n a l d e la y
Example,0( ) ( )y t x t t
0()H j t
0() jtH j e
Chapter 6 Time and Frequency Characterization
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§ 6.2.3 Log-Magnitude and Bode Plots
对数模和波特图
Chapter 6 Time and Frequency Characterization
Bode Plots,Log-Frequency—— Log-Magnitude
s i n?jH
jHjHlo g
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Chapter 6 Time and Frequency Characterization
Example 1
txty
dt
tdy
22 /1/1lg20lg20jH
0lg20 /1 a jH
lg20lg20lg20 /1 b jH
It is approximately a linear function of lg(ω)
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Chapter 6 Time and Frequency Characterization
Example 2
2 22
2 2 n n n
d y t d y t y t x t
d t d t

2
2 22
n
nn
Hj
jj



n?
—— undamped natural frequency（无阻尼自然频率）
—— damping ratio（阻尼系数）
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Chapter 6 Time and Frequency Characterization
112c t c th t M e e u t
12 n tnh t te u t
012
2
sin 1
1
n t
n
n
eh t t u t


2 1
/ 2 / 1nn
Hj
jj


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Chapter 6 Time and Frequency Characterization
§ 6.4 Time-Domain and Frequency-Domain
Aspects of Nonideal filters
jX1
jX 2
overlap
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Chapter 6 Time and Frequency Characterization
Nonideal Filter
Passband ripple 12?
Stopband ripple 2?
ts
The unit
response
jH
Magnitude
2
overshoot
Rise time rt st tSetting time
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11
Transition
Passband
Stopband
2?
0?
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Chapter 6 Time and Frequency Characterization
Homework:
6.5 6.23