text : a. v. oppenheim, a. s. willsky, and s. h. nawab ...contents.kocw.or.kr/document/signals and...
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Dept. of Electronics Eng. DH26029 Signals and Systems
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• Text : A. V. Oppenheim, A. S. Willsky, and S. H. Nawab,
Signals & Systems, 2nd Ed., Prentice-Hall, NJ, 1997
• Course Evaluation
• MATLAB-Programming Reports: 6%,
• Problem-Solving Reports: 4%
• Quiz: 30% (15% each)
• Midterm Exam: 40%
• Final exam 40%
Dept. of Electronics Eng. DH26029 Signals and Systems
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Communications Digital Signal
Processing Control
Circuits &
Systems
Fourier
Transform
DFT
Z-Transform
Laplace
Transform
Differential
Equations
Linear
Algebra
Complex
Variables
Dept. of Electronics Eng. DH26029 Signals and Systems
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1. Signals and Systems
• Signals
– Detectable physical quantities or variables by means of which messages or
information can be transmitted.
– Represented as functions of one or more independent variables
Dept. of Electronics Eng. DH26029 Signals and Systems
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1. Signals and Systems • Continuous-time signal: independent variable
– Voltage and Current in RC circuit
– Audio Signal, Audio Tape, Video Tape
– Analog TV Signal
• Discrete-time signal: independent variable – Stock market record
– Digital Audio Tape (DAT)
– Input to DAC
– Sampling of Continuous-time Signals
)( txt
][ nxn
010101010011·······
··
Dept. of Electronics Eng. DH26029 Signals and Systems
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• Continuous-time signal processing
- RC Circuit
RCj
CjR
CjjH
jVs
jVC
1
1
1
1
)()(
)(
22 )(1
1|)(|
RCjH
- Cut-off frequency : RC
1
?)()cos()( 1 tVtAtV CS
- Solution in Time-domain?
- Solution in Frequency domain?
Dept. of Electronics Eng. DH26029 Signals and Systems
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• Continuous-time and discrete-time signal
Dept. of Electronics Eng. DH26029 Signals and Systems
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• Continuous-time and discrete-time signal
Dept. of Electronics Eng. DH26029 Signals and Systems
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1.1.2 Signal energy and power
)(1
)()()( 2 tvR
titvtp
2
1
2
1
)(1
)( 2t
t
t
tdttv
RdttpE
• Average power P
2
1
2
1
)(11
)(1 2
1212
t
t
t
tdttv
Rttdttp
ttP
• Power and Energy in Electric Circuits
Dept. of Electronics Eng. DH26029 Signals and Systems
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• Continuous-time signal: energy
• Discrete-time signal : energy
2
1
2|)(|t
tdttx
2|][|2
1
n
nn
nx
• Energy over an infinite time interval
dttxdttx
T
TTE22 |)(||)(|lim
22 |][||][|lim
n
N
NnN
nxnxE
Dept. of Electronics Eng. DH26029 Signals and Systems
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• Average Power
T
TTdttx
TP
2|)(|2
1lim
2|][|12
1lim
N
NnN
nxNP
• Classification of Signals
))( :(ex , iii)
0, ii)
0 i)
ttxPE
EPP
PE
Dept. of Electronics Eng. DH26029 Signals and Systems
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1.2 Transformation of the Independent Variable
1.2.1 Example of transformations of the independent variables
Dept. of Electronics Eng. DH26029 Signals and Systems
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Dept. of Electronics Eng. DH26029 Signals and Systems
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Dept. of Electronics Eng. DH26029 Signals and Systems
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Example 1.1
)(tx
)1( tx
)1( tx
tx23
123 tx
Dept. of Electronics Eng. DH26029 Signals and Systems
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1.2.2 Periodic signals
• Fundamental Period : The smallest positive value among the periods
]2[][][ , period :
)2()()( , period :
NnxNnxnxN
TtxTtxtxT
][][
)()(
Nnxnx
Ttxtx
00 , NT
Dept. of Electronics Eng. DH26029 Signals and Systems
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Example 1.4 Check of periodicity
][][ Nnxnx
0 if)sin(
0 if)cos()(
tt
tttx
Dept. of Electronics Eng. DH26029 Signals and Systems
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1.2.3 Even and odd signals
• Even signal :
• Odd signal :
)()( txtx ][][ nxnx
)()( txtx ][][ nxnx
Dept. of Electronics Eng. DH26029 Signals and Systems
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• Even-odd decomposition of a signal
)()(2
1)( txtxtxEv
)()(2
1)( txtxtxOd
Dept. of Electronics Eng. DH26029 Signals and Systems
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1.3.1 Continuous-Time Complex Exponential and Sinusoidal Signal
1.3 Exponential and sinusoidal signal
atCetx )(
• Real Exponential Signals
Real :, aC
Dept. of Electronics Eng. DH26029 Signals and Systems
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• Periodic complex exponential and sinusoidal signals
tjetx 0)(
)(00 Ttjtjee
TjtjTtjeee 000 )(
10 Tj
e
TT of valuepositivesmallest the,2
0
0
if
imaginarypurely :a
Dept. of Electronics Eng. DH26029 Signals and Systems
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tjtetj
00 sincos0
tjjtjj eeA
eeA
tA 00
22)cos( 0
)(
00Re)cos(
tjeAtA
)(
00Im)sin(
tjeAtA
• Euler’s Relation:
11
0
periodperiod ET
P
000
2 000 1 TdtdteE
TTtj
period
12
1lim
20
dteT
PT
T
tj
T
• Periodic Signals: infinite total energy but finite average power