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Lecture 2: Thu Aug 23, 2018

Lecture

>Dirac impulse definition

>Dirac properties

> properties of signals: even and odd

> the even-odd decomposition

> properties of signals: power vs energy

1

The Difference Between the Unit Step and the Dirac Impulse

u( t ) d( t )

2

Dirac Impulse “Function”

An idealization of a incredibly short pulse that integrates to one:

0

s!0

tt0

t0

1/D

D

d( t ) = limD!0

pD( t ) = lims!0

fs( t )

pD( t ) fs( t )

!

3

Properties of Dirac Impulse

• zero for all t 6= 0, yet integrates to one: d( t )dt = 1

• Integrates to unit step: u( t ) = d( t )dt

• derivative of unit step: d( t ) = u( t )

• Sampling property: Multiplying anything by a delta function yields a scaled delta function:

x( t )d(t – t0) = x( t0)d(t – t0)

• Sifting property:

x( t )d(t – t0)dt = x(t0)

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ddt-----

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4

How to Plot

d(t) + 2d(t – 1)

t0 1

5

How to Plot

0

d(t) + 2d(t – 1)

t1

t0 1

6

Pop Quiz

Simplify x( t ) = t3cos(99pt)d(t + 1)

7

Time Shift

x( t – t0)

> shifts to right (delay) when t0 > 0

> shifts to left (advance) when t0 < 0

8

Time Scale

x(at)

> compresses when a > 1

> expands when 0 < a < 1

9

Time Reversal

-4 -2 2 40

x( t )

t

-4 -2 2 40

x(–t)

t

10

Example: Shift, then Time Scale

Sketch x(3t – 5), when x( t ) is defined below:

-4 -2 2 4 6 8 100

x( t )

t

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Example: Shift, then Time Scale

Sketch x(3t – 5), when x( t ) is defined below:

-4 -2 2 4 6 8 100

x( t )

-4 -2 2 4 6 8 100

x(t – 5)

-4 -2 2 4 6 8 100

x(3t – 5)

DELAY

TIME SCALE

t

t

t

12

Another Time Scale and Shift Example

Sketch x(2t – 1), when x( t ) is defined on the left-hand side below:

SHIF

T FIRST

SCALE FIRST

(Same answer either way)

13

Another Time Scale and Shift Example

Sketch x(2t – 1), when x( t ) is defined on the left-hand side below:

SHIF

T FIRST

SCALE FIRST

(Same answer either way)

14

Time Scaling an Impuse

Pop Quiz 1: What is d(– t)?

Pop Quiz 2: What is d(4t)?

15

Time Scaling an Impuse

Pop Quiz 1: What is d(– t)? d(– t) = d( t )

Pop Quiz 2: What is d(4t)?

Answer: d(4t) = d( t ), since it is the limit of:14---

t0

1/D

D/4

pD( 4 t )

(Height would need to be 4/D to have unit area)

16

Types of Symmetry

Even ) x(–t) = x( t ). Examples: t4, e–j t j, cos( t ), ...

Odd ) x(–t) = –x( t ). Example: t3, sin( t ), ...

Theorem: Any signal can be decomposed into even and odd parts:

x( t ) = xe( t ) + xo( t )

where xe( t ) = is even

and xo( t ) = is odd.

EVEN ODD NEITHER

x t( ) x t–( )+2

-------------------------------

x t( ) x t–( )–2

------------------------------

17

Example: Write as Even + Odd Signal

–1 0 1

1

t

x( t )

18

Example: Write as Even + Odd Signal

–1 0 1

1

t

–1 0 1t

1

x( t )

x( – t )

19

Example: Write as Even + Odd Signal

–1 0 1

1

t

–1 0 1t

ADD

SUBTRACT

1

0.5

1–1 t

0.5

–0.5

xe( t ) xo( t )

1

x( t )

x( – t )

( AN

D D

I VI D

E B

Y 2

)( A

ND

DI V

I DE

BY

2 )

1–1

20

Pop Quiz

Let:

• xe( t ) and ye( t ) be even

• xo( t ) and yo( t ) be odd

(a) the product xe( t )ye( t ) of even functions is [even ][odd ][ neither ]

(b) the product xo( t )yo( t ) of odd functions is [even ][odd ][ neither ]

(c) the product xe( t )xo( t ) of even by odd is [even ][odd ][ neither ]

21

Categorizing Signals

• odd, even, neither

• periodic, nonperiodic

• causal, noncausal, neither

• “energy” (finite energy, zero power)

• “power” (infinite energy, finite power)

22

Energy and Power

Energy:

E = x2( t )dt

Power:

P = x2( t )dt

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limt® ¥

12t-----

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