Microelectric Circuits by Meiling CHEN 1 Lecture 14 Feedback.
Lesson 3 Signals and systems Linear system. Meiling CHEN2 (1) Unit step function Shift a Linear...
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Lesson 3 Signals and systems Linear system
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Transcript of Lesson 3 Signals and systems Linear system. Meiling CHEN2 (1) Unit step function Shift a Linear...
Lesson 3
Signals and systems
Linear system
Meiling CHEN 2
(1) Unit step function
0,0
0,1)(
t
ttu
at
atatu
,0
,1)(
t
1
t
1
a
Shift a
Linear system
Meiling CHEN 3
(2) Unit impulse function
)]()([1
lim)(0
atutua
ta
ta
a
1
Area=1
)(tf
)()( tfdt
dt
t
)1()(t
Amplitude
width 0
t
)(k)( atk
a
Linear system
Meiling CHEN 4
(3) Unit doublet function )(t
t
)1()(' t
Linear system
Meiling CHEN 5
t
)(tf
t
)(* tf
…
n
nTttftf )()()(*
Sampling
Linear system
Meiling CHEN 6
(4) sign function
0
0
0
,1
,0
,1
)sgn(
t
t
t
t
(5) Unit ramp signal
0,0
0,)(
t
tttr
t
)(tr
tdutror
dt
tdrtu )()(
)()(
Linear system
Meiling CHEN 7
(6) parabolic signal
t
)(tf p
0,0
0,)(
2
t
tttf p
(7) sinc signal
t
)(tf p
t
ttc
sin
)(sin
Linear system
Meiling CHEN 8
Signal Classification
• Periodic and aperiodic
• Even and odd
• Real and complex
• Continuous-time and discrete-time
• Deterministic and stochastic (random)
• Causal and noncausal
Linear system
Meiling CHEN 9
Periodic signals )()( pTtftf
Even signals
odd signals
)()( tftf
)()( tftf
t
)(tf
t
)(tf
Linear system
Meiling CHEN 10
Causal signals
Anticausal signals
0,0)( tallfortf
0,0)( tallfortf
Linear system
Meiling CHEN 11
Causal and noncausal system
Example: distinguish between causal and noncausal systems in the following:
t
)(tu
1 2
(1) Case I )()( tuty
t
)(ty
12Noncausal system
0)(
0)(1
tybut
tutwhen
Linear system
Meiling CHEN 12
(2) Case II
causal system
)()( tuty
t
)(ty
1 2
Delay system
(3) Case III )2()()( tututy
causal systemAt present past
Linear system
Meiling CHEN 13
(4) Case IV )2()()( tututy
noncausal systemAt present future
(5) Case V
noncausal system
stepunitistuiftuty )()()( 2
t
)(ty
0)(
0)(0
tybut
tutwhen
Linear system
Meiling CHEN 14
Signal operations
• Simple operation : ＋、－• Convolution : *
Linear system
Meiling CHEN 15
simple operation
)(tf
)1()()()( trtrtutf
)(tu)1( tr
)(tr
Linear system
Meiling CHEN 16
Convolution Integral :)()()()(
)()()()(
tgtvdtgtv
dtgvtvtg
Linear system
Linear system …
)(t )(th
)(tu )()( thtu
Linear system
Meiling CHEN 17
Linear systemI.C.=0
)(tLinear system
)(th
Impulse response
)()]([ sHthL Transfer function of the system
Linear systemI.C.=0
)(tfAny input
)(ty zs
Zero state response
)()()( thtfty zs
Meiling CHEN 18
2/4)( tth )(tu
2 3 8
Example : Graphical convolution
(1) 2t
2 3
)(u)( th
tt
t8t
0)( ty
Linear system
Meiling CHEN 19
(2) 32 t
2 3
)(u)( th
t8t
td
tty
2)
24(2)(
(3) 63 t
2 3
)(u)( th
t8t
3
2)
24(2)(
dt
ty
Linear system
Meiling CHEN 20
(4) 116 t
2 3
)(u )( th
t8t
3
8)
24(2)(
td
tty
(5) t11
0)( ty2 3
)(u)( th
t8t
Linear system
Meiling CHEN 21
Linear system
0)( ty2t
32 t
t
dt
ty2
)2
4(2)(
63 t
3
2)
24(2)(
dt
ty
116 t
3
8)
24(2)(
td
tty
t11 0)( ty
Ans:
Meiling CHEN 22
)(th)(tu )()()( thtuty
)(sH)(sU )()()( sHsUsY
integral
Algebra operator
Laplace and convolution Linear system
Meiling CHEN 23
Example 2/4)( tth
8t
)(tu
2 3 t
Linear system
s
eesU
ussUeetuL
tttu
ss
ss
1)(
)0()()]([
)3()2()(
32
32
8 t
)(th
8
4
)(t
2
8
2
8
2
12)(
)0()0()(
2
1
2
1)]([(
)8(2
1)(
2
1)()(
s
essH
hshsHs
esthL
tttth
s
s
Meiling CHEN 24
Linear system
3
328
2
)1)(12()()()(
s
eeessHsUsY
sss
)()]()([ sfetutfL s Hint:
Meiling CHEN 25
Laplace transform
0)()}({)( dtetxtxLsX st
jswhere Complex frequency
For causal signals pass through linear time-invariant causal systems
f(t) F(s) f(t) F(s)
1
u(t)
r(t)s
1
)(t
21s
)(1
asate
ttu 0sin)(
ttu 0cos)(
20
20
s
20
2 ss
nttu )( 1!ns
n
31s
221)( ttf p
Meiling CHEN 26
Laplace transform properties
)()()]()([ sGsFtgtfL
)()]([ asFtfeL at
)()]()([ sfetutfL s
)0()()]([ fssFtfL
)0()0()0()()]([ 121)( nnnnn ffsfssFstfL
s
sFdfL
t )(])([
0
ds
sdFttfL
)()]([
n
nnn
ds
sFdtftL
)()1()]([
Linear system
