discrete -time signals and systems continuous-time signals & systems...
TRANSCRIPT
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Yogananda Isukapalli
Discrete - Time Signals and Systems
Continuous-Time signals & systemsIntroduction
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Mathematical Representation of Signals
• Independent variable can be• ‘Time’, ex. Speech signal• ‘spatial co-ordinates’, ex. Image• ‘time and space’, ex. Video• ‘numerical index’
• Function can be,• Of many Dimensions• Continuous, Notation “( )”, as in x(t)• Discrete, Notation “[ ]”, as in x[n]
Function of an independent variable, in Mathematical Sense
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Examples of common Continuous signals
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Continued…. Examples
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Continued…. Examples
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Classification of Continuous-Time Signals
0 2
0 0
2
2
)
lim (
)
1l
)
)
1 ( )
im ( )2
;
L
T
L
L
L
L
L
Average Power
B
x t
A
x t dt
dt
C
x
L
t dtT
Periodic S
Average Pow
Energy Signals
ignals
r
E
e
P
P
®
®
-
-
¥
¥
é ùê úë û
=
=
= ò
ò
ò
)
( ) ( )
D
x t x t dt
Area¥
-¥é ù =ë ûA ò
-
Infinite length signals: Example 10
0 ) ( ) cos( ) 3 / , 10
( ) 10cos(3 )0
with rad s A an
a xd
A
t
t tt
x t
w fw p f
p=
-¥ <
-
Examples 2&3: Infinite length signals
0) ,
( ) (
)
3
b PeriodicsignalsSquare wave with fundamental frequen
tcy o
x t THf
x tz
-= ¥ <
-
Finite length signals
[ ]( ) ( 3( ) sin(4 ) )( ) ( 2) ( 4), ( )... ,
u ux t t t t
p t u t u t u t unit step sig next sectiona nlp -= -
= - - -Fig 16.7
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Unit Step Signal
1 00 0
( ) tt
u t ³ìí
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Finite continuous pulse
( )u t T-
T
( ) ( )u t u t T- -
T
Fig 16.9
Fig 16.10
Fig 16.11
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One Sided Signals: Application of Unit step signalUnit step signal operates on the ideal non-causalinfinite length signal into a causal one
Fig 16.13
Fig 16.12
( ) ( )x t u t
( ) ( )x t u t
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Rectangular Pulse
( )... 2 ( ) ( ) ( )x t is derived from unit step functionsx t A u t a u t aé ùë û= + - -
( ) ( / 2 )x t rect t a=
A
a- a t
Fig 16.15
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Unit Ramp Signal
( )r t t=
0 t1
1
( )...
(
) 1 , (
)
(
(
)
)t
u unit step signa
u
l
d dAlso note th
u
atdt dt
d
r t t
r t
t
t
t t-¥
= = =
= ò
Fig 16.16
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Unit Impulse Signal
(1)=
The area of the impulse is wrUnit impulse signal is symb
itten in paranthesses next toolized with an arro
the arrow dw
hea-
2
1
1 2
) ( ) 0 01 0
) ( )0
) ( ) 0) ( ) ( );
t
t
a t t
t tb t dt
otherwise
c t undefined at td t t even function
d
d
dd d
= ¹
<
-
0ˆ( ) lim ( )
ˆ( )ˆ:
( )
iu t u t
du tDefinition
s an infinite
t
ly small value
dtd
D®
=
D=
Unit Impulse & Unit Step Signals
ˆ( )u t
D t
1
0
( )u t
Fig 16.18
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Unit Impulse & Unit Step Signals, contd..
ˆˆ( )) (du tdt
td =
D t0
1D
( ); ( )
'
)( ) (t
Thus unit - impulse can't be defined as a single pointIt is an approximation of narrow pulses whose ar
u t
ea is '1
t ddu t
dtd t td
-¥== ò
Fig 16.19
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Unit Impulse: Approximation with pulses
'1'
Notice that the area of any pulse isImpulse results at the derivative of any discontinuous point
Fig 16.20
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Unit Impulse: Derivative at discontinuity
Notice the location and magnitude of impulse
( )( ) du ttdt
d =
Fig 16.21
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22
Unit Impulse- Derivative of a discontinuous function
2( 1)2( 1)
2( 1) 2
0
2( 1)
2
( )
1
1
( )
( ) ( 1) ( 1)
( 1) 2
( 1) 2 ( 1
( ) ( 1)
)
)
( 1
tt
t t
t
t
x t e u
dx t du t dee u t
d
e t
t dt dte
t
t
t
e
t e u
ud
d
- -- -
- - - -
-
-
-
-
-= +
= -
-
= -
= - -
- -
-
Fig 16.22 Fig 16.23
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Unit Impulse - Scaling property
0
0
0
0 0( ) ( ) ( ) ( )
( ) ( ) ( )f t t t dt f
f t t t t
t
f t t
d
d d¥
-¥-
-
=
- =
ò
The impulse isshown as pulse
-
( )
( )( )1 1 21 2 2
( )( ) ( ) ( )
1 1( ) ( );
( ) ( ) (
(
)
)
) (f t t t t tsignal multiplied by tf t
f t t t f t t
wo shifted un
bat b
it impulse
t a ta
s
b
t
b ta a
d
d d
d
d
d
d
d
= - + -
+ = + - =
-
-
- +
Unit Impulse - Scaling property
Fig 16.25
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Unit Impulse - Example , ( ) sin(20 ) ( 1 80)Evaluate the expression x t t tp d= -
( ) sin(20 ) ( 1 80) sin(20 ) ( 1 80) ( 1 80)
1 800.707
the continuous function multiplied by an impulse becomesan impulse with a size depending only on the value ofcontinuo
x t t tt
us function
t
ev
p dp dd
= -= -= -
( ) sin(20 0.707
0
) ( 1 80) ( 1 80)
.
707
aluated at the time location of impulse
x t dt t t dt tp d d¥ ¥ ¥
-¥ -¥ -¥= - = -
=
ò ò ò
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Unit Impulse - Examples0
0
3 32 2
3 33
3
( )
( 3) 3 ( 3)
3 ( 3)
( ) ( 2) (2 2 ) ( 2)
6 ( 2)
)
)
)
3
6
j t j tt t e dt
t t dt t dt
t dt
t t t dt t dt
t
a
b
c
dt
ew wd
d d
d
d d
d
¥-
-¥¥ ¥
-¥ -¥¥
-¥
- -
-
-- =
- = -
= - =
+ - = + -
= - =
ò
ò ò
ò
ò ò
ò
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Operations on continuous-time signals
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28
-
29
Operations on continuous-time signals, contd…
( )x t
(2 )x t
( 2)x t
-
1-
( )x t
t0 1 2
2-
( / 2)x t
t0 2 4
2-
(2 / 2)x t+
t0 24-
Operations on continuous-time signals, contd…
Fig 16.28
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James H. McClellan, Ronald W. Schafer and Mark A. Yoder, “ 9.1, and 9.2 “SignalProcessing First”, Prentice Hall, 2003
Reference