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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)
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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)?
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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
------------------------------
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Example: Write as Even + Odd Signal
–1 0 1
1
t
x( t )
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Example: Write as Even + Odd Signal
–1 0 1
1
t
–1 0 1t
1
x( t )
x( – t )
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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
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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 ]
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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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