Leo Lam © 2010-2012

Signals and Systems

EE235

Leo Lam © 2010-2012

People types

There are 10 types of people in the world:

Those who know binary and those who don’t.

Leo Lam © 2010-2012

Today’s menu

• From Friday:– Manipulation of signals– To Do: Really memorize u(t), r(t), p(t)

• Even and odd signals• Dirac Delta Function

Leo Lam © 2010-2012

How to find LCM

• Factorize and group• Your turn: 225 and 270’s LCM

• Answer: 1350

Leo Lam © 2010-2012

Even and odd signals

An even signal is such that:

tSymmetrical across

the t=0 axis

tAsymmetrical across

the t=0 axis

An odd signal is such that:

( ) ( )e ex t x t

( ) ( )o ox t x t 0

( ) 2 ( )L L

e e

L

x t dt x t dt

( ) 0L

o

L

x t dt

Leo Lam © 2010-2012

Even and odd signals

1 1( ) ( ( ) ( )) ( ( ) ( ))

2 2x t x t x t x t x t

Every signal sum of an odd and even signal.

( ) ( )e ex t x t

Even signal is such that:

The even and odd parts of a signal

Odd signal is such that:

( ) ( )o ox t x t

1( ) ( ( ) ( ))

21

( ) ( ( ) ( ))2

e

o

x t x t x t

x t x t x t

Leo Lam © 2010-2012

Even and odd signals

1( ) ( ( ) ( ))

21

( ) ( ( ) ( ))2

e

o

x t x t x t

x t x t x t

Euler’s relation:

j te What are the even and odd parts of

)sin()(2

1

)cos()(2

1

)sin()cos(

tjee

tee

tjte

tjtj

tjtj

tj

Even part

Odd part

Leo Lam © 2010-2012

Summary:

• Even and odd signals• Breakdown of any signals to the even and odd

components

Leo Lam © 2010-2012

Delta function δ(t)

“a spike of signal at time 0”

0

The Dirac delta is: • The unit impulse or impulse• Very useful• Not a function, but a “generalized function”)

Leo Lam © 2010-2012

Delta function δ(t)

0lim

Each rectangle has area 1, shrinking width, growing height ---limit is (t)

1

1

Leo Lam © 2010-2012

Dirac Delta function δ(t)

“a spike of signal at time 0”

0

It has height = , width = 0, and area = 1

• δ(t) Rules1. δ(t)=0 for t≠02. Area:

3. If x(t) is continuous at t0, otherwise undefined

1)( dtt

)()()()()( 0000 txtttxtttx

0 t0

Shifted to time instant t0:

Leo Lam © 2010-2012

Dirac Delta example

• Evaluate

10

2

)( dtt

= 0. Because δ(t)=0 for all t≠0

Leo Lam © 2010-2012

Dirac Delta – Your turn

• Evaluate

= 1. Why?/ 4

sin( / 2) ( / 2)t dt

Change of variable: / 2t ( 1 )d

d dtdt

/2

/4 /4sin( / 2) sin( / 2)( / 2) ( )t dt d

1

1

Leo Lam © 2010-2011

Scaling the Dirac Delta

• Proof:

• Suppose a>0

• a<0

( )at dt

d dat a dt

dt a

/

/

1 1( ) ( ) ( )

t a

t a

dat dt d

a a a

/

/

1 1( ) ( ) ( )

a

a

d dd

a a a a

Leo Lam © 2010-2011

Scaling the Dirac Delta

• Proof:

• Generalizing the last result

( )at dt

1 1( ) ( )

t

tat dt d

a a

Leo Lam © 2010-2011

• Multiplication of a function that is continuous at t0 by δ(t) gives a scaled impulse.

• Sifting Properties

• Relation with u(t)

Summary: Dirac Delta Function

0 0 0( ) ( ) ( ) ( )x t t t x t t t

0 0( ) ( ) ( )x t t t dt x t

( ) ( )t

u t d

( ) ( )d

t u tdt