Linear system
Lesson 3
Signals and systems
Linear system
(1) Unit step function
1, t  0
u (t )  
0 , t  0
1
t
Shift a
1, t  a
u (t  a)  
0 , t  a
1
a
Meiling CHEN
t
2
Linear system
(2) Unit impulse function
1
a
1
 (t )  lim [u (t )  u (t  a)]
a 0 a
 (t )
 (t ) 
f (t )
Area=1
(1)
d
f (t )
dt
t
a
Amplitude

t
width  0
k (t  a)
(k )
a
t
Meiling CHEN
3
Linear system
(3) Unit doublet function
 (t )
 ' (t )
(1)
t
Meiling CHEN
4
Linear system
Sampling
f (t )
f * (t )
…
t
f * (t ) 
t

 f (t ) (t  nT )
n  
Meiling CHEN
5
Linear system
(4) sign function
 1, t  0

sgn( t )   0, t  0
 1, t  0

(5) Unit ramp signal
r (t )
t, t  0
r (t )  
0, t  0
dr (t )
u (t ) 
dt
or
t
t
r (t )   u ( )d

Meiling CHEN
6
Linear system
(6) parabolic signal
f p (t )
t 2 , t  0
f p (t )  
 0, t  0
t
(7) sinc signal
f p (t )
sin t
sin c(t ) 
t
t
Meiling CHEN
7
Linear system
Signal Classification
•
•
•
•
•
•
Periodic and aperiodic
Even and odd
Real and complex
Continuous-time and discrete-time
Deterministic and stochastic (random)
Causal and noncausal
Meiling CHEN
8
Linear system
Periodic signals f (t )  f (t  T p )
f (t )
Even signals
f (t )  f (t )
t
f (t )
odd signals
f (t )   f (t )
t
Meiling CHEN
9
Linear system
Causal signals
f (t )  0, for all
Anticausal signals f (t )  0, for all
Meiling CHEN
t 0
t 0
10
Linear system
Causal and noncausal system
Example: distinguish between causal and noncausal systems
in the following:
u (t )
1
2
t
(1) Case I y (t )  u (t )
y (t )
when
but
2
1
t
Meiling CHEN
t  1
u (t )  0
y (t )  0
Noncausal system
11
Linear system
(2) Case II y (t )  u (t   )
y (t )
Delay system
1  
(3) Case III
2 
t
causal system
y (t )  u (t )  u (t  2)
causal system
At present
past
Meiling CHEN
12
Linear system
(4) Case IV
y (t )  u (t )  u (t  2)
noncausal system
At present
(5) Case V
future
y(t )  u(t 2 ) if
y (t )
u(t ) is unit
when
but
t
Meiling CHEN
t0
step
u (t )  0
y (t )  0
noncausal system
13
Linear system
Signal operations
• Simple operation : ＋、－
• Convolution : *
Meiling CHEN
14
Linear system
simple operation
f (t )
f (t )  u (t )  r (t )  r (t  1)
u (t )
r (t  1)
r (t )
Meiling CHEN
15
Linear system

Convolution Integral :
g (t )  v(t )   v( ) g (t   )d


  v(t   ) g (t )d  v(t )  g (t )

 (t )
h(t )
Linear system
u (t )  h(t )
u (t )
Linear system
Meiling CHEN
…
16
Linear system
 (t )
Linear system
I.C.=0
h(t )
Impulse response
L[h(t )]  H ( s)
f (t )
Any input
Transfer function of the system
Linear system
I.C.=0
y zs (t )
Zero state response
y zs (t )  f (t )  h(t )
Meiling CHEN
17
Linear system
Example : Graphical convolution
h(t )  4  t / 2
u (t )
2
3
t
t
8
(1) t  2
u ( )
h(t   )
y (t )  0
t 8
t 2
3
Meiling CHEN
18
Linear system
(2)  2  t  3
u ( )
h(t   )
t 
y (t )   2(4 
)d
2
2
t
t 8
 2t
3
(3) 3  t  6
h(t   )
u ( )
t 
y (t )   2(4 
)d
2
2
3
t 8  2
3
t
Meiling CHEN
19
Linear system
(4) 6  t  11
u ( )
h(t   )
t 
y (t )   2(4 
)d
t 8
2
3
2
t 8 3
t
(5) 11  t
u ( )
2
h(t   )
3
t 8
t
Meiling CHEN
y (t )  0
20
Linear system
t  2
2 t 3
Ans:
3t 6
6  t  11
11  t
y (t )  0
t 
y (t )   2(4 
)d
2
2
3
t 
y (t )   2(4 
)d
2
2
3
t 
y (t )   2(4 
)d
t 8
2
y (t )  0
t
Meiling CHEN
21
Linear system
Laplace and convolution
u (t )
h(t )
y (t )  u (t )  h(t )
integral
U (s )
H (s )
Y (s)  U (s) H ( s)
Algebra operator
Meiling CHEN
22
Linear system
Example
h(t )  4  t / 2
u (t )
2
3
t
t
h(t )
u(t )   (t  2)   (t  3)
 (t )
L[u (t )]  e 2 s  e 3 s  sU ( s )  u (0  )
3 s
e e
U ( s) 
s
2s
8
8
1

4
8
t
1
1



h (t )   (t )   (t )   (t  8)
2
2
1 1
L[( h(t )]  s   e 8 s
2 2
 s 2 H ( s )  sh (0  )  h(0  )
2 s  1  e 8 s
H (s) 
2s 2
Meiling CHEN
23
Linear system
(2s  1  e 8 s )(e 2 s  e 3s  1)
Y ( s)  U ( s) H ( s) 
2s 3
Hint: L[ f (t   )u (t   )]  e s f ( s)
Meiling CHEN
24
Laplace transform
For causal signals pass through linear time-invariant
causal systems

X (s)  L{x(t )}   x(t )e dt
 st
0
f(t)
where s    j
Complex frequency
F(s)
f(t)
 (t )
1
u(t ) sin 0t
u(t)
1
s
u (t ) cos 0t
r(t)
1
s2
1
s3
u (t )t n
f p (t )  t
1 2
2
e
Meiling CHEN
 at
F(s)
0
s 2  02
s
s 2  02
n!
s n1
1
( s a )
25
Linear system
Laplace transform properties
L[f (t )  g (t )]  F ( s)  G ( s)
L[e  at f (t )]  F (s  a)
L[ f (t   )u (t   )]  e s f ( s)
L[ f (t )]  sF ( s)  f (0  )
L[ f ( n) (t )]  s n F (s)  s n1 f (0  )  s n2 f (0  )    f n1 (0  )
 dF ( s )
L[tf (t )] 
ds
t
F ( s)
L[  f ( )d ] 
s
0
n
d F ( s)
L[t f (t )]  (1)
ds n
n
n
Meiling CHEN
26