Lecture 3

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.


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.


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


CT Signal Examples


DT Signal Examples


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)


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


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


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(


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


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


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)


)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


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


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


Homework #1

Hwk#1.a 1.1, 1.2, and 1.3 (CT and DT Signals)

Lecture 3

Documents

Transcript of Lecture 3