103-2 ss01 ss - 國立臺灣大學cc.ee.ntu.edu.tw/.../signalssystems/103-2_ss01_ss.pdf• the...
TRANSCRIPT
Spring 2015
信號與系統Signals and Systems
Chapter SS-1Signals and Systems
Feng-Li LianNTU-EE
Feb15 – Jun15
Figures and images used in these lecture notes are adopted from“Signals & Systems” by Alan V. Oppenheim and Alan S. Willsky, 1997
Feng-Li Lian © 2015NTUEE-SS1-SS-2Outline
Introduction
Continuous-Time & Discrete-Time Signals
Transformations of the Independent Variable
Exponential & Sinusoidal Signals
The Unit Impulse & Unit Step Functions
Continuous-Time & Discrete-Time Systems
Basic System Properties
Feng-Li Lian © 2015NTUEE-SS1-SS-3Signals & Systems
Signals & Systems:
Is about using mathematical techniques to help describe and analyze systemswhich process signals
• Signals are variables that carry information
• Systems process input signals to produce output signals
System
Output
Signal
Input
Signal
Feng-Li Lian © 2015NTUEE-SS1-SS-4Signals & Systems: Signals
Discrete-Time Signals:
The weekly Dow-Jones stock market index
Boy-One Dormitory, 2/13-2/21, 2013, every 30 min
http://big5.jrj.com.cn/
Feng-Li Lian © 2015NTUEE-SS1-SS-5Signals & Systems: Signals
Discrete-Time Signals:
A monochromatic picture
117 151 112 136 110 95 166 131 125 159 114 130 159 133
164 209 173 171 168 126 225 189 184 219 172 180 206 188
154 151 143 174 156 160 186 179 163 193 167 161 167 168
173 179 172 182 195 181 187 205 183 205 196 186 180 193
155 188 149 145 181 131 166 185 166 181 164 164 166 171
176 176 113 178 169 159 177 184 171 182 148 159 174 164
162 198 141 164 202 149 181 142 176 182 189 136 165 176
143 143 148 127 132 97 177 152 160 74 163 119 162 143
62 51 40 32 95 82 28 39 75 35 60 58 77 28
26 67 14 34 3 49 22 69 48 26 81 67 49 91
176 134 129 97 185 120 212 160 173 139 197 166 207 192
243 255 250 255 254 242 242 255 254 222 255 253 239 252
248 255 226 238 255 255 249 255 255 255 230 252 255 255
243 238 255 253 245 247 255 238 255 244 255 237 249 241
Feng-Li Lian © 2015NTUEE-SS1-SS-6Signals & Systems: Signals
Continuous-Time Signals: Source voltage & capacity voltage
in a simple RC circuit
Recording of a speech signal
Feng-Li Lian © 2015NTUEE-SS1-SS-7Signals & Systems: Signals
Graphical Representations of Signals:
Continuous-time signals x(t) or xc(t)
Discrete-time signals x[n] or xd[n]
Feng-Li Lian © 2015NTUEE-SS1-SS-8Signals & Systems: Signals
Energy & Power of a resistor:
Instantaneous power
Total energy over a finite time interval
Average power over a finite time interval
Feng-Li Lian © 2015NTUEE-SS1-SS-9Signals & Systems: Signals
Signal Energy & Power:
Total energy over a finite time interval
Time-averaged power over a finite time interval
Feng-Li Lian © 2015NTUEE-SS1-SS-10Signals & Systems: Signals
Signal Energy & Power:
Total energy over an infinite time interval
Time-averaged power over an infinite time interval
Feng-Li Lian © 2015NTUEE-SS1-SS-11Signals & Systems: Signals
Three Classes of Signals:
Finite total energy & zero average power
Finite average power & infinite total energy
Infinite average power & infinite total energy
Feng-Li Lian © 2015NTUEE-SS1-SS-12Outline
Introduction
Continuous-Time & Discrete-Time Signals
Transformations of the Independent Variable– Time Shift
– Time Reversal
– Time Scaling
– Periodic Signals
– Even & Odd Signals
Exponential & Sinusoidal Signals
The Unit Impulse & Unit Step Functions
Continuous-Time & Discrete-Time Systems
Basic System Properties
Feng-Li Lian © 2015NTUEE-SS1-SS-13Signals & Systems: Transformation of the Independent Variable
Time Shift:
Feng-Li Lian © 2015NTUEE-SS1-SS-14Signals & Systems: Transformation
Time Reversal:
Feng-Li Lian © 2015NTUEE-SS1-SS-15Signals & Systems: Transformation
Time Scaling:
Feng-Li Lian © 2015NTUEE-SS1-SS-16Signals & Systems: Transformation
x ( t ) → x ( - t + 1 )
Feng-Li Lian © 2015NTUEE-SS1-SS-17Signals & Systems: Transformation
x ( t ) → x ( - t + 1 )
Feng-Li Lian © 2015NTUEE-SS1-SS-18Signals & Systems: Transformation
x ( t ) → x ( 3/2 t + 1 )
Feng-Li Lian © 2015NTUEE-SS1-SS-19Signals & Systems: Transformation
x ( t ) → x ( a t - b )
– |a| < 1: linearly stretched
– |a| > 1: linearly compressed
– a < 0: time reversal
– b > 0: delayed time shift
– b < 0: advanced time shift
Problems:• P1.21 for CT
• P1.22 for DT
Feng-Li Lian © 2015NTUEE-SS1-SS-20Signals & Systems: Transformation
CT & DT Periodic Signals:
Feng-Li Lian © 2015NTUEE-SS1-SS-21Signals & Systems: Transformation
Periodic Signals:
• A periodic signal is unchanged by a time shift of T or N
• They are also periodic with period
– 2T, 3T, 4T, …
– 2N, 3N, 4N, …
• T or N is called the fundamental period
denoted as T0 or N0
Feng-Li Lian © 2015NTUEE-SS1-SS-22Signals & Systems: Transformation
Periodic signal ?
Problems:• P1.25 for CT
• P1.26 for DT
Feng-Li Lian © 2015NTUEE-SS1-SS-23Signals & Systems: Transformation
Even & odd signals:
Feng-Li Lian © 2015NTUEE-SS1-SS-24Signals & Systems: Transformation
Even-odd decomposition of a signal:
• Any signal can be broken into a sum of one even signal and one odd signal
Feng-Li Lian © 2015NTUEE-SS1-SS-25Signals & Systems: Transformation
Even-odd decomposition of a DT signal:
Problems:• P1.23 for CT
• P1.24 for DT
Feng-Li Lian © 2015NTUEE-SS1-SS-26Signals & Systems: Transformation
Uniqueness of even-odd decomposition:
Feng-Li Lian © 2015NTUEE-SS1-SS-27Outline
Introduction
Continuous-Time & Discrete-Time Signals
Transformations of the Independent Variable– Time Shift
– Time Reversal
– Time Scaling
– Periodic Signals
– Even & Odd Signals
Exponential & Sinusoidal Signals
The Unit Impulse & Unit Step Functions
Continuous-Time & Discrete-Time Systems
Basic System Properties
Feng-Li Lian © 2015NTUEE-SS1-SS-28Signals & Systems: Exponential & Sinusoidal Signals
Magnitude & Phase Representation:
Euler’s relation:
Feng-Li Lian © 2015NTUEE-SS1-SS-29Signals & Systems: Exponential & Sinusoidal Signals
CT Complex Exponential Signals:
• where C & a are, in general, complex numbers
Feng-Li Lian © 2015NTUEE-SS1-SS-30Signals & Systems: Exponential & Sinusoidal Signals
Real exponential signals:
• If C & a are real
Feng-Li Lian © 2015NTUEE-SS1-SS-31Signals & Systems: Exponential & Sinusoidal Signals
Periodic complex exponential signals:
• If a is purely imaginary
• It is periodic
– Because let
– Then
– Hence
Feng-Li Lian © 2015NTUEE-SS1-SS-32Signals & Systems: Exponential & Sinusoidal Signals
Periodic sinusoidal signals:
Feng-Li Lian © 2015NTUEE-SS1-SS-33Signals & Systems: Exponential & Sinusoidal Signals
Period & Frequency:
Feng-Li Lian © 2015NTUEE-SS1-SS-34Signals & Systems: Exponential & Sinusoidal Signals
Euler’s relation:
Feng-Li Lian © 2015NTUEE-SS1-SS-35Signals & Systems: Exponential & Sinusoidal Signals
Total energy & average power:
Problem:• P1.3
Feng-Li Lian © 2015NTUEE-SS1-SS-36Signals & Systems: Exponential & Sinusoidal Signals
Harmonically related periodic exponentials
Feng-Li Lian © 2015NTUEE-SS1-SS-37Signals & Systems: Exponential & Sinusoidal Signals
General complex exponential signals:
Feng-Li Lian © 2015NTUEE-SS1-SS-38Signals & Systems: Exponential & Sinusoidal Signals
General complex exponential signals:
Feng-Li Lian © 2015NTUEE-SS1-SS-39Signals & Systems: Exponential & Sinusoidal Signals
DT complex exponential signal or sequence:
• where C & a are, in general, complex numbers
Feng-Li Lian © 2015NTUEE-SS1-SS-40Signals & Systems: Exponential & Sinusoidal Signals
Real exponential signals:
• If C & a are real
Feng-Li Lian © 2015NTUEE-SS1-SS-41Signals & Systems: Exponential & Sinusoidal Signals
DT Complex Exponential & Sinusoidal Signals
• If b is purely imaginary (or |a| = 1)
Feng-Li Lian © 2015NTUEE-SS1-SS-42Signals & Systems: Exponential & Sinusoidal Signals
Euler’s relation:
And,
Feng-Li Lian © 2015NTUEE-SS1-SS-43Signals & Systems: Exponential & Sinusoidal Signals
General complex exponential signals:
Feng-Li Lian © 2015NTUEE-SS1-SS-44Signals & Systems: Exponential & Sinusoidal Signals
Periodicity properties of DT complex exponentials:
• The signal with frequency ω0 is identical to the signals with frequencies
• Only need to consider a frequency interval of length 2π– Usually use
• The low frequencies are located at The high frequencies are located at
Feng-Li Lian © 2015NTUEE-SS1-SS-45
-10 0 10
-1
0
1
n
cos(0*pi*t)
-10 0 10
-1
0
1
n
cos(1/8*pi*t)
-10 0 10
-1
0
1
n
cos(1/4*pi*t)
-10 0 10
-1
0
1
n
cos(1/2*pi*t)
-10 0 10
-1
0
1
n
cos(1*pi*t)
-10 0 10
-1
0
1
n
cos(3/2*pi*t)
-10 0 10
-1
0
1
n
cos(7/4*pi*t)
-10 0 10
-1
0
1
n
cos(15/8*pi*t)
-10 0 10
-1
0
1
n
cos(2*pi*t)
Signals & Systems: Exponential & Sinusoidal Signals
CT exponential signals
t t t
Feng-Li Lian © 2015NTUEE-SS1-SS-46
-10 0 10
-1
0
1
n
cos(2*pi*t)
-10 0 10
-1
0
1
n
cos(17/8*pi*t)
-10 0 10
-1
0
1
n
cos(9/4*pi*t)
-10 0 10
-1
0
1
n
cos(5/2*pi*t)
-10 0 10
-1
0
1
n
cos(3*pi*t)
-10 0 10
-1
0
1
n
cos(7/2*pi*t)
-10 0 10
-1
0
1
n
cos(15/4*pi*t)
-10 0 10
-1
0
1
n
cos(31/8*pi*t)
-10 0 10
-1
0
1
n
cos(4*pi*t)
Signals & Systems: Exponential & Sinusoidal Signals
CT exponential signals
t t t
Feng-Li Lian © 2015NTUEE-SS1-SS-47
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
t
cos(0*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(0*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1/8*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1/8*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1/4*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1/4*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1/2*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1/2*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(3/2*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(3/2*pi*n)
Signals & Systems: Exponential & Sinusoidal Signals
CT & DT exponential signals
t
t
t
t
t
t
n
n
n
n
n
n
Feng-Li Lian © 2015NTUEE-SS1-SS-48
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(7/4*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(7/4*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(15/8*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(15/8*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(2*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(2*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(17/8*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(17/8*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(9/4*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(9/4*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(5/2*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(5/2*pi*n)
Signals & Systems: Exponential & Sinusoidal Signals
CT & DT exponential signals
t
t
t
t
t
t
n
n
n
n
n
n
Feng-Li Lian © 2015NTUEE-SS1-SS-49
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(3*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(3*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(7/2*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(7/2*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(15/4*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(15/4*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(31/8*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(31/8*pi*n)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(4*pi*t)
-10 0 10-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(4*pi*n)
Signals & Systems: Exponential & Sinusoidal Signals
CT & DT exponential signals
t
t
t
t
t n
n
n
n
n
Feng-Li Lian © 2015NTUEE-SS1-SS-50
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(0*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1/8*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1/4*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1/2*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(1*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(3/2*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(7/4*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(15/8*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(2*pi*n)
Signals & Systems: Exponential & Sinusoidal Signals
Periodicity properties of DT exponential signals
Feng-Li Lian © 2015NTUEE-SS1-SS-51
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(2*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(17/8*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(9/4*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(5/2*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(3*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(7/2*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(15/4*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(31/8*pi*n)
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
n
cos(4*pi*n)
Signals & Systems: Exponential & Sinusoidal Signals
Periodicity properties of DT exponential signals
Feng-Li Lian © 2015NTUEE-SS1-SS-52Signals & Systems: Exponential & Sinusoidal Signals
Periodicity properties of DT exponential signals
Periodicity of N > 0
That is,
Hence,
Problem:• P1.35
Feng-Li Lian © 2015NTUEE-SS1-SS-53Signals & Systems: Exponential & Sinusoidal Signals
Periodicity properties of DT exponential signals
Feng-Li Lian © 2015NTUEE-SS1-SS-54Signals & Systems: Exponential & Sinusoidal Signals
Harmonically related periodic exponentials
• Only N distinct periodic exponentials in the set
Feng-Li Lian © 2015NTUEE-SS1-SS-55Signals & Systems: Exponential & Sinusoidal Signals
Comparison of CT & DT signals:
Problem:• P1.36
CT DT
CT
DT
Feng-Li Lian © 2015NTUEE-SS1-SS-56Outline
Introduction
Continuous-Time & Discrete-Time Signals
Transformations of the Independent Variable– Time Shift
– Time Reversal
– Time Scaling
– Periodic Signals
– Even & Odd Signals
Exponential & Sinusoidal Signals
The Unit Impulse & Unit Step Functions
Continuous-Time & Discrete-Time Systems
Basic System Properties
Feng-Li Lian © 2015NTUEE-SS1-SS-57Signals & Systems: Unit Impulse & Unit Step Functions
DT Unit Impulse & Unit Step Sequences
Unit impulse (or unit sample)
Unit step
Feng-Li Lian © 2015NTUEE-SS1-SS-58Signals & Systems: Unit Impulse & Unit Step Functions
Relationship Between Impulse & Step
First difference
Running sum
Feng-Li Lian © 2015NTUEE-SS1-SS-59Signals & Systems: Unit Impulse & Unit Step Functions
Relationship Between Impulse & Step
Alternatively,
Feng-Li Lian © 2015NTUEE-SS1-SS-60Signals & Systems: Unit Impulse & Unit Step Functions
Sample by Unit Impulse For x[n] More generally,
Feng-Li Lian © 2015NTUEE-SS1-SS-61Signals & Systems: Unit Impulse & Unit Step Functions
CT Unit Impulse & Unit Step Functions
Unit step function
Unit impulse function
Feng-Li Lian © 2015NTUEE-SS1-SS-62Signals & Systems: Unit Impulse & Unit Step Functions
Relationship Between Impulse & Step
Running integral
First derivative
• But, u(t) is discontinuous at t = 0, hence, not differentiable
• Use approximation
Feng-Li Lian © 2015NTUEE-SS1-SS-63Signals & Systems: Unit Impulse & Unit Step Functions
Relationship Between Impulse & Step
Use approximation
Feng-Li Lian © 2015NTUEE-SS1-SS-64Signals & Systems: Unit Impulse & Unit Step Functions
Relationship Between Impulse & Step
Feng-Li Lian © 2015NTUEE-SS1-SS-65Signals & Systems: Unit Impulse & Unit Step Functions
Sample by Unit Impulse Function
For x(t)
More generally,
Feng-Li Lian © 2015NTUEE-SS1-SS-66Signals & Systems: Unit Impulse & Unit Step Functions
Example 1.7:
Feng-Li Lian © 2015NTUEE-SS1-SS-67Signals & Systems
Exponential, Sinusoidal, Impulse, Step functionstime frequency
Feng-Li Lian © 2015NTUEE-SS1-SS-68Signals & Systems
Summation or Integration of Three Functions
Feng-Li Lian © 2015NTUEE-SS1-SS-69Outline
Introduction
Continuous-Time & Discrete-Time Signals
Transformations of the Independent Variable– Time Shift
– Time Reversal
– Time Scaling
– Periodic Signals
– Even & Odd Signals
Exponential & Sinusoidal Signals
The Unit Impulse & Unit Step Functions
Continuous-Time & Discrete-Time Systems
Basic System Properties
Feng-Li Lian © 2015NTUEE-SS1-SS-70Signals & Systems: CT & DT Systems
Physical Systems & Mathematical Descriptions
Examples of physical systems are
signal processing, communications,
electromechanical motors, automotive vehicles,
chemical-processing plants
A system can be viewed as a process
in which input signals are transformed by the system
or cause the system to respond in some way,
resulting in other signals or outputs
Feng-Li Lian © 2015NTUEE-SS1-SS-71Signals & Systems: CT & DT Systems
Simple examples of CT systems
RC circuit
Feng-Li Lian © 2015NTUEE-SS1-SS-72Signals & Systems: CT & DT Systems
Simple examples of CT systems
Automobile
Feng-Li Lian © 2015NTUEE-SS1-SS-73Signals & Systems: CT & DT Systems
Simple examples of DT systems
Balance in a bank account
Feng-Li Lian © 2015NTUEE-SS1-SS-74Signals & Systems: CT & DT Systems
Simple examples of DT systems
Digital simulation of differential equation
Feng-Li Lian © 2015NTUEE-SS1-SS-75Signals & Systems: CT & DT Systems
Interconnections of Systems:• Audio System:
Bass Treble EqualizerMicrophone
or TapeSpeaker
Feng-Li Lian © 2015NTUEE-SS1-SS-76Signals & Systems: CT & DT Systems
Interconnections of Systems
Series or cascade interconnection of 2 systems
> e.x. radio receiver + amplifier
Parallel interconnection of 2 systems
> e.x. audio system with several microphones or speakers
Feng-Li Lian © 2015NTUEE-SS1-SS-77Signals & Systems: CT & DT Systems
Interconnections of Systems
Series-parallel interconnection
Feedback interconnection
> e.x. cruise control, electrical circuit
Feng-Li Lian © 2015NTUEE-SS1-SS-78Chapter 1: Signals and Systems
Introduction
Continuous-Time & Discrete-Time Signals
Transformations of the Independent Variable– Time Shift
– Time Reversal
– Time Scaling
– Periodic Signals
– Even & Odd Signals
Exponential & Sinusoidal Signals
The Unit Impulse & Unit Step Functions
Continuous-Time & Discrete-Time Systems
Basic System Properties– Systems with or without memory
– Invertibility & Inverse Systems
– Causality
– Stability
– Time Invariance
– Linearity
Feng-Li Lian © 2015NTUEE-SS1-SS-79Signals & Systems: Basic System Properties
Systems with or without memory
Memoryless systems• Output depends only on the input at that same time
Systems with memory
(resistor)
(delay)
(accumulator)
Feng-Li Lian © 2015NTUEE-SS1-SS-80Signals & Systems: Basic System Properties
Invertibility & Inverse Systems
Invertible systems• Distinct inputs lead to distinct outputs
Feng-Li Lian © 2015NTUEE-SS1-SS-81Signals & Systems: Basic System Properties
Causality
Causal systems• Output depends only on input at present time & in the past
• Non-causal systems
Feng-Li Lian © 2015NTUEE-SS1-SS-82Signals & Systems: Basic System Properties
Stability
Stable systems• Small inputs lead to responses that do not diverge
• Every bounded input excites a bounded output– Bounded-input bounded-output stable (BIBO stable)
– For all |x(t)| < a, then |y(t)| < b, for all t
• Balance in a bank account?
Feng-Li Lian © 2015NTUEE-SS1-SS-83Signals & Systems: Basic System Properties
Example 1.13: Stability
Feng-Li Lian © 2015NTUEE-SS1-SS-84Signals & Systems: Basic System Properties
Time Invariance
Time-invariant systems• Behavior & characteristics of system are fixed over time
• A time shift in the input signal results in an identical time shift in the output signal
Feng-Li Lian © 2015NTUEE-SS1-SS-85Signals & Systems: Basic System Properties
Time Invariance
Example of time-invariant system (Example 1.14)
Feng-Li Lian © 2015NTUEE-SS1-SS-86Signals & Systems: Basic System Properties
Time Invariance
Example of time-varying system (Example 1.16)
Feng-Li Lian © 2015NTUEE-SS1-SS-87Signals & Systems: Basic System Properties
Linearity
Linear systems• If an input consists of the weighted sum of several signals,
then the output is the superposition of the responses of the system to each of those signals
Feng-Li Lian © 2015NTUEE-SS1-SS-88Signals & Systems: Basic System Properties
Linearity
Linear systems• In general,
• OR,
Feng-Li Lian © 2015NTUEE-SS1-SS-89Signals & Systems: Basic System Properties
Linearity
Example 1.17:
Feng-Li Lian © 2015NTUEE-SS1-SS-90Signals & Systems: Basic System Properties
Linearity
Example 1.18:
Feng-Li Lian © 2015NTUEE-SS1-SS-91Signals & Systems: Basic System Properties
Linearity
Example 1.20:
Feng-Li Lian © 2015NTUEE-SS1-SS-92Signals & Systems: Basic System Properties
Linearity
Example 1.20:
However,
Feng-Li Lian © 2015NTUEE-SS1-SS-93Chapter 1: Signals and Systems
Introduction
Continuous-Time & Discrete-Time Signals
Transformations of the Independent Variable– Time Shift
– Time Reversal
– Time Scaling
– Periodic Signals
– Even & Odd Signals
Exponential & Sinusoidal Signals
The Unit Impulse & Unit Step Functions
Continuous-Time & Discrete-Time Systems
Basic System Properties– Systems with or without memory
– Invertibility & Inverse Systems
– Causality
– Stability
– Time Invariance
– Linearity
Feng-Li Lian © 2015NTUEE-SS1-SS-94
Periodic Aperiodic
FS– CT
– DT(Chap 3)
FT– DT
– CT (Chap 4)
(Chap 5)
Bounded/Convergent
LT
zT – DT
Time-Frequency
(Chap 9)
(Chap 10)
Unbounded/Non-convergent
– CT
CT-DT
Communication
Control
(Chap 6)
(Chap 7)
(Chap 8)
(Chap 11)
Signals & Systems LTI & Convolution(Chap 1) (Chap 2)
Flowchart
Feng-Li Lian © 2015NTUEE-SS1-SS-95Signals & Systems: Exponential & Sinusoidal Signals
Problem 1.26 (Page 61)
-25 -20 -15 -10 -5 0 5 10 15 20 25-1.5
-1
-0.5
0
0.5
1
1.5
L = 25;n = -L:L;
x = cos( pi/8 * (n.^2) );
figure(1)stem( n, x, 'o' ); hold on;
axis([-L L -1.5 1.5])
Feng-Li Lian © 2015NTUEE-SS1-SS-96Signals & Systems: Exponential & Sinusoidal Signals
Problem 1.27 (Page 62)
Feng-Li Lian © 2015NTUEE-SS1-SS-97Signals & Systems: Exponential & Sinusoidal Signals
Problem 1.27 (Page 62)