lecture 3
DESCRIPTION
signalsTRANSCRIPT
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ECEN3513 Signal Analysis
Reading: Section 1.1
Instructor: Dr. Guoliang FanSchool of Electrical and Computer Engineering
Oklahoma State University
Lecture #3
Continuous-time and Discrete-time Signals
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ECEN 3513 Signal Analysis 2
Wisdom for the day
A good start is half-way to success.
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ECEN 3513 Signal Analysis 3
Goals
To introduce the mathematical representations of both
continuous-time (CT) and discrete-time (DT) signals.
To compute the energy and power of a given signal,
and to classify signals into three basic classes
according to their energy and power.
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ECEN 3513 Signal Analysis 4
Definitions
Continuous-time (CT) signals: defined at every instant of time over a continuous domain, such as an interval; or a union of intervals.
Discrete-time (DT) signals: taking values only at a countable or finite set of points on the real line, and these time instants are equally-spaced.
)(tx
][nx
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ECEN 3513 Signal Analysis 5
CT Signal Examples
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ECEN 3513 Signal Analysis 6
DT Signal Examples
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ECEN 3513 Signal Analysis 7
Signal Energy and Power
In general, we can define the total energy for any continuous-time or discrete-time signal as:
The time average power is defined as follows
Note: The terms of “power” and “energy” are used here independently of whether the signal quantities are related to physical energy. It is for convenience.
(CT) )(2
1
2
t
tdttxE
(CT) )(1 2
1
2
12
t
tdttx
ttP
(DT) ][2
1
2
n
nn
nxE
(DT) ][1
1 2
1
2
12
n
nn
nxnn
P
Absolute value (real numbers) or Modulus (complex numbers)
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ECEN 3513 Signal Analysis 8
Signal Energy and Power (Cont’d)
Sometimes, it may be interesting in examining energy in signals over an infinite time interval as:
Similarly, we can define the time-averaged power over an infinite time interval as
)(lim2
T
TTdttxE
(CT) )(2
1lim
2
T
TTdttx
TP
N
NnN
nxE2
][lim
(DT) ][12
1lim
2
N
NnN
nxN
P
(CT) )(2
dttx
(DT) ][2
n
nx
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ECEN 3513 Signal Analysis 9
Three Classes of Signals
Class I: The class of signals with finite total energy. The average power will be
Case II: The class of signals with finite average power. The total energy
Case III: Otherwise The average power is not finite. The total energy is not finite.
E
.0P
P
.E
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ECEN 3513 Signal Analysis 10
Example (1)
Given signal ttx sin)( dttxE
2)(
dttT
dttxT
PT
TT
T
TT )(sin2
1lim)(
2
1lim 22
T
T
T
TTdt
t
Tdt
T 2
)2cos(
2
1
2
1
2
1lim
dtt
)(sin2
dtt
T
T
TT
2
)2cos(1
2
1lim
2
)2cos(1)(sin2 xx
2
1dt
T
T
TT
2
1
2
1lim
T
Tt
T )2sin(
8
1 ))2sin()2(sin(8
1TT
T 0 T
T
T
4
)2sin(
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ECEN 3513 Signal Analysis 11
Example (2)
)3()( tjetx
dttxE
2)( dte tj
2)3(
dt
T
TTdttx
TP
2)(
2
1lim 1
norm)unit (the
1)( batje
)sin()cos( :equationEular tjte jt
)(sin)(cos 22 tte jt 1
T
TTdt
T1
2
1lim
T
TT 2
2lim
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ECEN 3513 Signal Analysis 12
Example (3)
ttx )(
dttxE
2)( dtt
2
3
3t
dttxT
PT
TT 2)(
2
1lim
T
TT
tT
3
3
1
2
1lim
3
2
2
1lim
3T
TT
3lim
2TT
Or by inspection?
dttT
T
TT 2
2
1lim
dttx
)(2
33 )(3
1
3
1
2
1lim TT
TT
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ECEN 3513 Signal Analysis 13
Example (4)
)0,1][( ][3
1][
nnununx
n
N
NnN
nxE2
][lim
0
2
3
1
n
n
0 9
1
n
n
8
9
91
1
1
N
NnN
nxN
P2
][12
1lim 0
series) (geometric
)1(1
0
0
aa
aa
n
nn
n
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Example (5)
ECEN 3513 Signal Analysis 14
)1( 1
][ nn
nx
N
NnN
nxE2
][lim
1
21
n n
condition) econvergenc series-(
)1( 1
1
p
pnnp
N
NnN
nxN
P2
][12
1lim 0
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ECEN 3513 Signal Analysis 15
Four Sufficient Conditions
A finite signal of finite lengthCase I
A periodic & finite signal
A constant signal
A signal of infinite length that goes to infinity
)(tft
Case II
Case III
E
P
Ctf )(
)()(,)( tfTtfCtf
CtfTttf )( and ,0)(
OTW
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ECEN 3513 Signal Analysis 16
More Examples
To decide the class of signals given below.
ttx 4)(1
4)(2 tx
ttx
3cos)(3
nnx ][1
odd. is 1
even; is 1][2 n
nnx
tjetx 34 )(
njenx 33 ][
0,1
][4 nn
nx
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ECEN 3513 Signal Analysis 17
Homework #1
Hwk#1.a 1.1, 1.2, and 1.3 (CT and DT Signals)