Signals and Systems - Shandong UniversityFall 2008 Signals and Systems Chapter SS-6 Time & Frequency Characterization of Signals and Systems Feng-Li Lian SDU-BME Sep08 – Dec08 Figures

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


SDUBME-SS6-TimeFreq-4Introduction


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:


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



-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




Magnitude & Phase Angle in Images:









SDUBME-SS6-TimeFreq-13Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems

Magnitude & Phase Distortions:

LTI System


Log-Magnitude & Bode Plots: LTI System


Log-Magnitude & Bode Plots:









SDUBME-SS6-TimeFreq-17Shou shui Wei©2008First-Order & Second-Order CT Systems

First-Order CT Systems:

LTI System

Time Constant








Second-Order CT Systems:
















NTUEE-SS6-TimeFreq-30Shou 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:


Example 6.5:

NTUEE-SS6-TimeFreq-34Shou shui Wei©2008Outline








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:


Step Response of First-Order DT Systems:


Magnitude & Phase of Frequency Response:




Second-Order DT Systems:


Impulse Response of 2nd-Order DT Systems:



Step Response of 2nd-Order DT Systems:













Magnitude & Phase Distortions:

LTI System

NTUEE-SS6-TimeFreq-48Shou shui Wei©2008Magnitude-Phase Representation of Frequency Response of LTI Systems

Linear Phase: H


Linear Phase: H


Group Delay & Phase:

• Linear Phase & Delay:

• Nonlinear Phase & Group Delay


Example 6.1: H3H2H1


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


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


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




H3H2H1


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









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









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









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:


Lowpass Filtering

on Dow Jones Weekly

Stock Market Index:


General Form of DT Non-recursive Filters:


General Form of DT Non-recursive Filters:


Comparison on a Linear Amplitude Scale:

Log Scale

Linear Scale


Lowpass Non-recursive Filter with 251 Coefficients:

Coefficients determined by the Parks-McClellan algorithm


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:

Documents

Signals and Systems - Shandong UniversityFall 2008 Signals and Systems Chapter SS-6 Time & Frequency Characterization of Signals and Systems Feng-Li Lian SDU-BME Sep08 – Dec08 Figures