Post on 31-Dec-2015
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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
ECEN 3513 Signal Analysis 2
Wisdom for the day
A good start is half-way to success.
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.
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
ECEN 3513 Signal Analysis 5
CT Signal Examples
ECEN 3513 Signal Analysis 6
DT Signal Examples
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)
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
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
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(
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
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
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
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
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
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
ECEN 3513 Signal Analysis 17
Homework #1
Hwk#1.a 1.1, 1.2, and 1.3 (CT and DT Signals)