Fall 2008
Signals and Systems
Chapter SS-6Time & Frequency Characterization of
Signals and Systems
Feng-Li LianSDU-BME
Sep08 – Dec08
Figures and images used in these lecture notes are adopted from“Signals & Systems” by Alan V. Oppenheim and Alan S. Willsky, 1997
SDUBME-SS6-TimeFreq-2Shou shui Wei © 2008Introduction
Time- & Frequency-Domain Characterization:
SDUBME-SS6-TimeFreq-3Introduction
Time- & Frequency-Domain Characterization:
SDUBME-SS6-TimeFreq-4Introduction
Time- & Frequency-Domain Characterization:
LTI System
Shou shui Wei © 2008
Shou shui Wei © 2008
SDUBME-SS6-TimeFreq-5
Shou shui Wei©2008Introduction
Time-Domain Frequency-Domain
Convolution
Transformation
Differential or
Difference
Equations
Algebraic
Equations
Convolution Multiplication
System
DesignTime-Domain
Considerations
Frequency-Domain
Considerations
SystemModel
& Operations
TechniquesConvolutionMultiplication
SDUBME-SS6-TimeFreq-6Shou shui Wei©2008Outline
The Magnitude-Phase Representation of the Fourier Transform
The Magnitude-Phase Representation of Frequency Response of LTI Systems
Time-Domain Properties of Ideal Frequency-Selective Filters
Time-Domain and Frequency-Domain Aspects of Non-ideal Filters
1st-Order & 2nd-Order Continuous-Time Systems
1st-Order & 2nd-Order Discrete-Time Systems
Time- & Frequency-Domain Analysis of Systems
SDUBME-SS6-TimeFreq-7Shou shui Wei©2008Magnitude-Phase Representation of Fourier Transform
Magnitude & Phase Representation:
SDUBME-SS6-TimeFreq-8Shou shui Wei©2008Magnitude-Phase Representation of Fourier Transform
Magnitude & Phase Angle:
- 2 - 1 .5 - 1 - 0 .5 0 0 . 5 1 1 . 5 2
- 1
0
1 1
- 2 - 1 .5 - 1 - 0 .5 0 0 . 5 1 1 . 5 2
- 1
0
1 1 / 2c o s( 2π t + φ 1 )
- 2 - 1 .5 - 1 - 0 .5 0 0 . 5 1 1 . 5 2
- 1
0
1 c os ( 4π t + φ 2)
- 2 - 1 .5 - 1 - 0 .5 0 0 . 5 1 1 . 5 2
- 1
0
1 2 / 3c o s( 6π t + φ 3 )
- 2 - 1 .5 - 1 - 0 .5 0 0 . 5 1 1 . 5 2
0
2
4
SDUBME-SS6-TimeFreq-9Shou shui Wei©2008Magnitude-Phase Representation of Fourier Transform
Magnitude & Phase Angle:
-2 -1.5 -1 -0.5 0 0 .5 1 1.5 2
-1
0
1 1
-2 -1.5 -1 -0.5 0 0 .5 1 1.5 2
-1
0
1 1/2c os( 2π t + φ1 )
-2 -1.5 -1 -0.5 0 0 .5 1 1.5 2
-1
0
1 cos(4π t + φ 2)
-2 -1.5 -1 -0.5 0 0 .5 1 1.5 2
-1
0
1 2/3c os( 6π t + φ3 )
-2 -1.5 -1 -0.5 0 0 .5 1 1.5 2
0
2
4
SDUBME-SS6-TimeFreq-10Shou shui Wei©2008Magnitude-Phase Representation of Fourier Transform
Magnitude & Phase Angle:
SDUBME-SS6-TimeFreq-11Shou shui Wei©2008Magnitude-Phase Representation of Fourier Transform
Magnitude & Phase Angle in Images:
SDUBME-SS6-TimeFreq-12Shou shui Wei©2008Outline
The Magnitude-Phase Representation of the Fourier Transform
The Magnitude-Phase Representation of Frequency Response of LTI Systems
Time-Domain Properties of Ideal Frequency-Selective Filters
Time-Domain and Frequency-Domain Aspects of Non-ideal Filters
1st-Order & 2nd-Order Continuous-Time Systems
1st-Order & 2nd-Order Discrete-Time Systems
Time- & Frequency-Domain Analysis of Systems
SDUBME-SS6-TimeFreq-13Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
Magnitude & Phase Distortions:
LTI System
SDUBME-SS6-TimeFreq-14Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
Log-Magnitude & Bode Plots: LTI System
SDUBME-SS6-TimeFreq-15Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
Log-Magnitude & Bode Plots:
SDUBME-SS6-TimeFreq-16Shou shui Wei©2008Outline
The Magnitude-Phase Representation of the Fourier Transform
The Magnitude-Phase Representation of Frequency Response of LTI Systems
Time-Domain Properties of Ideal Frequency-Selective Filters
Time-Domain and Frequency-Domain Aspects of Non-ideal Filters
1st-Order & 2nd-Order Continuous-Time Systems
1st-Order & 2nd-Order Discrete-Time Systems
Time- & Frequency-Domain Analysis of Systems
SDUBME-SS6-TimeFreq-17Shou shui Wei©2008First-Order & Second-Order CT Systems
First-Order CT Systems:
LTI System
Time Constant
SDUBME-SS6-TimeFreq-18Shou shui Wei©2008First-Order & Second-Order CT Systems
First-Order CT Systems:
SDUBME-SS6-TimeFreq-19Shou shui Wei©2008First-Order & Second-Order CT Systems
First-Order CT Systems:
SDUBME-SS6-TimeFreq-20Shou shui Wei©2008First-Order & Second-Order CT Systems
First-Order CT Systems:
SDUBME-SS6-TimeFreq-21Shou shui Wei©2008First-Order & Second-Order CT Systems
Second-Order CT Systems:
SDUBME-SS6-TimeFreq-22Shou shui Wei©2008First-Order & Second-Order CT Systems
Second-Order CT Systems:
SDUBME-SS6-TimeFreq-23Shou shui Wei©2008First-Order & Second-Order CT Systems
Second-Order CT Systems:
SDUBME-SS6-TimeFreq-24Shou shui Wei©2008First-Order & Second-Order CT Systems
Second-Order CT Systems:
SDUBME-SS6-TimeFreq-25Shou shui Wei©2008First-Order & Second-Order CT Systems
Second-Order CT Systems:
SDUBME-SS6-TimeFreq-26Shou shui Wei©2008First-Order & Second-Order CT Systems
Second-Order CT Systems:
SDUBME-SS6-TimeFreq-27Shou shui Wei©2008First-Order & Second-Order CT Systems
SDUBME-SS6-TimeFreq-28Shou shui Wei©2008First-Order & Second-Order CT Systems
Second-Order CT Systems:
SDUBME-SS6-TimeFreq-29Shou shui Wei©2008First-Order & Second-Order CT Systems
Second-Order CT Systems:
NTUEE-SS6-TimeFreq-30Shou shui Wei©2008First-Order & Second-Order CT Systems
Second-Order CT Systems:
SDUBME-SS6-TimeFreq-31Shou shui Wei©2008First-Order & Second-Order CT Systems
Example 6.4:
NTUEE-SS6-TimeFreq-32Shou shui Wei©2008First-Order & Second-Order CT Systems
Example 6.5:
SDUBME-SS6-TimeFreq-33Shou shui Wei©2008First-Order & Second-Order CT Systems
Example 6.5:
NTUEE-SS6-TimeFreq-34Shou shui Wei©2008Outline
The Magnitude-Phase Representation of the Fourier Transform
The Magnitude-Phase Representation of Frequency Response of LTI Systems
Time-Domain Properties of Ideal Frequency-Selective Filters
Time-Domain and Frequency-Domain Aspects of Non-ideal Filters
1st-Order & 2nd-Order Continuous-Time Systems
1st-Order & 2nd-Order Discrete-Time Systems
Time- & Frequency-Domain Analysis of Systems
SDUBME-SS6-TimeFreq-35Shou shui Wei©2008First-Order & Second-Order DT Systems
First-Order DT Systems:
LTI System
NTUEE-SS6-TimeFreq-36Shou shui Wei©2008First-Order & Second-Order DT Systems
Impulse Response of First-Order DT Systems:
SDUBME-SS6-TimeFreq-37Shou shui Wei©2008First-Order & Second-Order DT Systems
Step Response of First-Order DT Systems:
NTUEE-SS6-TimeFreq-38Shou shui Wei©2008First-Order & Second-Order DT Systems
Magnitude & Phase of Frequency Response:
SDUBME-SS6-TimeFreq-39Shou shui Wei©2008First-Order & Second-Order DT Systems
Magnitude & Phase of Frequency Response:
NTUEE-SS6-TimeFreq-40Shou shui Wei©2008First-Order & Second-Order DT Systems
Second-Order DT Systems:
SDUBME-SS6-TimeFreq-41Shou shui Wei©2008First-Order & Second-Order DT Systems
Impulse Response of 2nd-Order DT Systems:
NTUEE-SS6-TimeFreq-42Shou shui Wei©2008First-Order & Second-Order DT Systems
SDUBME-SS6-TimeFreq-43Shou shui Wei©2008First-Order & Second-Order DT Systems
Step Response of 2nd-Order DT Systems:
NTUEE-SS6-TimeFreq-44Shou shui Wei©2008First-Order & Second-Order DT Systems
SDUBME-SS6-TimeFreq-45Shou shui Wei©2008First-Order & Second-Order DT Systems
Magnitude & Phase of Frequency Response:
NTUEE-SS6-TimeFreq-46Shou shui Wei©2008Outline
The Magnitude-Phase Representation of the Fourier Transform
The Magnitude-Phase Representation of Frequency Response of LTI Systems
Time-Domain Properties of Ideal Frequency-Selective Filters
Time-Domain and Frequency-Domain Aspects of Non-ideal Filters
1st-Order & 2nd-Order Continuous-Time Systems
1st-Order & 2nd-Order Discrete-Time Systems
Time- & Frequency-Domain Analysis of Systems
SDUBME-SS6-TimeFreq-47Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
Magnitude & Phase Distortions:
LTI System
NTUEE-SS6-TimeFreq-48Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
Linear Phase: H
SDUBME-SS6-TimeFreq-49Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
Linear Phase: H
NTUEE-SS6-TimeFreq-50Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
Group Delay & Phase:
• Linear Phase & Delay:
• Nonlinear Phase & Group Delay
SDUBME-SS6-TimeFreq-51Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
Example 6.1: H3H2H1
NTUEE-SS6-TimeFreq-52Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
H1
0 500 1000 1500 2000 2500-400
-300
-200
-100
0
rad/sec
deg
ree
0 500 1000 1500 2000 2500-200
-100
0
100
200
rad/sec
deg
ree
SDUBME-SS6-TimeFreq-53Shou shui Wei©2008
0 500 1000 1500 2000 2500-350
-300
-250
-200
-150
-100
-50
0
rad/sec
deg
ree
0 500 1000 1500 2000 2500-200
-100
0
100
200
rad/sec
deg
ree
Magnitude-Phase Representation of Frequency Response of LTI Systems
0 500 1000 1500 2000 2500-400
-300
-200
-100
0
rad/sec
deg
ree
0 500 1000 1500 2000 2500-200
-100
0
100
200
rad/sec
deg
ree
0 500 1000 1500 2000 2500-400
-300
-200
-100
0
rad/sec
deg
ree
0 500 1000 1500 2000 2500-200
-100
0
100
200
rad/sec
deg
ree
H3H2H1
NTUEE-SS6-TimeFreq-54Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
H3H2H1
0 500 1000 1500 2000 2500-200
0
200
rad/sec
deg
ree
0 500 1000 1500 2000 2500-200
0
200
rad/sec
deg
ree
0 500 1000 1500 2000 2500-200
0
200
rad/sec
deg
ree
0 500 1000 1500 2000 2500-200
0
200
rad/sec
deg
ree
SDUBME-SS6-TimeFreq-55Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
H3H2H1
0 500 1000 1500 2000 2500-400
-200
0
rad/sec
deg
ree
0 500 1000 1500 2000 2500-400
-200
0
rad/sec
deg
ree
0 500 1000 1500 2000 2500-400
-200
0
rad/sec
deg
ree
500 1000 1500 2000 2500-1000
-500
0
rad/sec
deg
ree
NTUEE-SS6-TimeFreq-56Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
SDUBME-SS6-TimeFreq-57Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
NTUEE-SS6-TimeFreq-58Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
H3H2H1
SDUBME-SS6-TimeFreq-59Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems
Example 6.2:Analog Transmission Performance on the Switched Telecommunication Networks (AT&T/Bell)
No
n-c
on
stan
t G
rou
p D
elay
(m
s)
Mag
nit
ud
e (2
0lo
g10
|H(j
w)|
, dB
)
Frequency (Hz)Frequency (Hz)
Short
Medium
Short
Medium
NTUEE-SS6-TimeFreq-60Shou shui Wei©2008Outline
The Magnitude-Phase Representation of the Fourier Transform
The Magnitude-Phase Representation of Frequency Response of LTI Systems
Time-Domain Properties of Ideal Frequency-Selective Filters
Time-Domain and Frequency-Domain Aspects of Non-ideal Filters
1st-Order & 2nd-Order Continuous-Time Systems
1st-Order & 2nd-Order Discrete-Time Systems
Time- & Frequency-Domain Analysis of Systems
SDUBME-SS6-TimeFreq-61Shou shui Wei©2008Time-Domain Properties of Ideal Frequency-Selective Filters
Ideal Lowpass Filters:
NTUEE-SS6-TimeFreq-62Shou shui Wei©2008Time-Domain Properties of Ideal Frequency-Selective Filters
Ideal Lowpass Filters:
SDUBME-SS6-TimeFreq-63Shou shui Wei©2007Time-Domain Properties of Ideal Frequency-Selective Filters
Ideal Lowpass Filters with Linear Phase:
NTUEE-SS6-TimeFreq-64Shou shui Wei©2007Time-Domain Properties of Ideal Frequency-Selective Filters
Step Response of Ideal Lowpass Filters:ringing
rise time
SDUBME-SS6-TimeFreq-65Shou shui Wei©2008Outline
The Magnitude-Phase Representation of the Fourier Transform
The Magnitude-Phase Representation of Frequency Response of LTI Systems
Time-Domain Properties of Ideal Frequency-Selective Filters
Time-Domain and Frequency-Domain Aspects of Non-ideal Filters
1st-Order & 2nd-Order Continuous-Time Systems
1st-Order & 2nd-Order Discrete-Time Systems
Time- & Frequency-Domain Analysis of Systems
NTUEE-SS6-TimeFreq-66Shou shui Wei©2008Time- & Frequency-Domain Aspects of Non-ideal Filters
Overlapping Spectra: H
Ideal Band-Pass Filter ?
SDUBME-SS6-TimeFreq-67Shou shui Wei©2008Time- & Frequency-Domain Aspects of Non-ideal Filters
Desired Filter Characteristics: H
NTUEE-SS6-TimeFreq-68Shou shui Wei©2008Time- & Frequency-Domain Aspects of Non-ideal Filters
Example 6.3: Two Frequently Used Filters:
• Butterworth, Chebyshev, Elliptic filters
SDUBME-SS6-TimeFreq-69Shou shui Wei©2008Time- & Frequency-Domain Aspects of Non-ideal Filters
Example 6.3: Two Frequently Used Filters:
• Butterworth filter
• Elliptic filter
Mag
nitu
de
Frequency (Hz) Time (ms)
Frequency Response Step Response
– Fifth-order rational frequency response
– Cutoff frequency = 500 Hz
NTUEE-SS6-TimeFreq-70Shou shui Wei©2008Outline
The Magnitude-Phase Representation of the Fourier Transform
The Magnitude-Phase Representation of Frequency Response of LTI Systems
Time-Domain Properties of Ideal Frequency-Selective Filters
Time-Domain and Frequency-Domain Aspects of Non-ideal Filters
1st-Order & 2nd-Order Continuous-Time Systems
1st-Order & 2nd-Order Discrete-Time Systems
Time- & Frequency-Domain Analysis of Systems
SDUBME-SS6-TimeFreq-71Shou shui Wei©2008Time-Domain & Frequency-Domain Analysis of Systems
DT Non-recursive Filters:
• Recursive or infinite impulse response (IIR) filters
> Impossible to design a causal, recursive filter
with exactly linear phase (related to time delay)
• Non-recursive or finite impulse response (FIR) filters
> Can have exactly linear phase (related to time delay)
NTUEE-SS6-TimeFreq-72Shou shui Wei©2008Time-Domain & Frequency-Domain Analysis of Systems
Log-Magnitude Plots:
SDUBME-SS6-TimeFreq-73Shou shui Wei©2008Time-Domain & Frequency-Domain Analysis of Systems
Lowpass Filtering
on Dow Jones Weekly
Stock Market Index:
NTUEE-SS6-TimeFreq-74Shou shui Wei©2008Time-Domain & Frequency-Domain Analysis of Systems
General Form of DT Non-recursive Filters:
SDUBME-SS6-TimeFreq-75Shou shui Wei©2008Time-Domain & Frequency-Domain Analysis of Systems
General Form of DT Non-recursive Filters:
NTUEE-SS6-TimeFreq-76Shou shui Wei©2008Time-Domain & Frequency-Domain Analysis of Systems
Comparison on a Linear Amplitude Scale:
Log Scale
Linear Scale
SDUBME-SS6-TimeFreq-77Shou shui Wei©2008Time-Domain & Frequency-Domain Analysis of Systems
Lowpass Non-recursive Filter with 251 Coefficients:
Coefficients determined by the Parks-McClellan algorithm
SDUBME-SS6-TimeFreq-78Shou shui Wei©2008Outline
Geometric Evaluation of the Fourier Transform
of The Laplace Transform & Z-Transform
SDUBME-SS6-TimeFreq-79Shou shui Wei©2008Magnitude-Phase Representation in Complex Plane (s-plane & z-plane)
In s-plane or z-plane: