16.362 Signal and System I
• The representation of discrete-time signals in terms of impulse x [ 0 ]

[ n ]
 x [ 0 ] x [ 1 ]

[ n

1 ]
 x [ 1 ] x [ k ]

[ n
 k ]
 x [ k ] x [ n ]
 k



 x [ k ]

[ n
 k ]
Example u [ n ]
 k



 u [ k ]

[ n
 k ]
 k
 


0

[ n
 k ]

16.362 Signal and System I
• The representation of discrete-time signals in terms of impulse x [ n ]
 k



 x [ k ]

[ n
 k ]

[ n ] h [ n ] x [ n ]
 k



 x [ k ]

[ n
 k ] y [ n ]
 h [ n ] y [ n ]
 k



 x [ k ] h [ n
 k ]
 k




 h [ k ] x [ n
 k ] y [ n ]
 h [ n ]
 x [ n ] y [ n ]
 x [ n ]
 h [ n ]
Convolution

16.362 Signal and System I
• The representation of continuous-time signals in terms of impulse x ( t )
 


 x ( t ' )

( t
 t ' ) dt ' y ( t )
 


 x ( t ' ) h ( t
 t ' ) dt ' y ( t )
 h ( t )
 x ( t )
• Properties of LIT systems
Commutative property y ( t )
 h ( t )
 x ( t ) y ( t )
 x ( t )
 h ( t )
Distributive property y ( t )


 h
1
( t )
 h
1
( t )
 h
2 x ( t )
( t )


 x ( t ) h
2
( t )
 x ( t )

16.362 Signal and System I
• Properties of LIT systems
Associative property y ( t )

 h
1
 h
1
( t
( t
)
 h
1
( t
 h
2
)
)


 h
2 h
2
( t
( t
( t
)
)

)
 x


( t ) x ( t x ( t )

)
Causality h ( t )

0 , for t<0.
h [ n ]

0 , for n<0.
Stability



 h ( t ) dt
  n



 h [ n ]
 

16.362 Signal and System I
• The unit step response of an LTI system

[ n ] h [ n ] y [ n ] y [ n ]
 k





[ k ] h [ n
 k ] y [ n ]
 k



 h [ k ]

[ n
 k ]
 h [ n ] u [ n ] h [ n ] s [ n ] s [ n ]
 k



 h [ k ] u [ n
 k ]
 k
 n 
 h [ k ]

16.362 Signal and System I
• The unit step response of an LTI system u [ n ] h [ n ] s
1
[ n ] s
1
[ n ]
 k
 n 
 h [ k ] u [ n

1 ] h [ n ] s
2
[ n ] s
2
[ n ]

 k



 h [ k ] u [ n k n


1 
 h [ k ]

1
 k ] s
2
[ n ]
 k n


1 
 h [ k ]
 s
1
[ n

1 ] s
1
[ n ]
 s
1
[ n

1 ]
 h [ n ]

16.362 Signal and System I
• The unit step response of an LTI system

[ n ] h [ n ] h [ n ] u [ n ] h [ n ] s [ n ] s [ n ]
 s [ n

1 ]
 h [ n ]

16.362 Signal and System I
• Linear constant-coefficient difference equations x [ n ] + y [ n ] delay y [ n ]

1
2 y [ n

1 ]
 x [ n ]
1
2 y [ n ]

?
h [ n ]

?
h [ n ] y [ n ] depends on x[n]. We don’t know y[n] unless x[n] is given.
But h[n] doesn’t depend on x[n]. We should be able to obtain h[n] without x[n].
How?
• LTI system response properties, this chapter.
• Discrete Fourier transform, --- Ch. 5.

16.362 Signal and System I
• Linear constant-coefficient difference equations

[ n ]
+
1
2 h [ n ] h [ n ]

1
2 h [ n

1 ]
 
[ n ]
When n

1, h [ n ]

1
2 h [ n

1 ] h [ n ] delay y [ n ]

1
2 y [ n

1 ]
 x [ n ] h [ n ]

1
2 h [ n

1 ]
 
[ n ] h [ n ] h [ n

1 ]

1
2 h [ n ]


A

1
2 n h [ n ]


A

1
2 n u [ n ] Causality

16.362 Signal and System I
• Linear constant-coefficient difference equations

[ n ]
+ h [ n ] y [ n ]

1
2 y [ n

1 ]
 x [ n ] delay h [ n ]

1
2 h [ n

1 ]
 
[ n ] h [ n ]
1
2 h [ n ]


A

1
2 n u [ n ]
Determine A by initial condition:
When n = 0, h [ n ]

1
2 h [ n

1 ]
 
[ n ] h [ 0 ]
 
[ 0 ]

1 h [ 0 ]


A

1
2
0 u [ 0 ] A = 1

16.362 Signal and System I

• Linear constant-coefficient difference equations
[ n

1 ] h [ n ] delay y [ n ]

1
2 y [ n

1 ]
 x [ n ] h [ n ]

1
2 h [ n

1 ]
 
[ n ]
1
2 h [ n ] y [ n ]

?
+ h [ n ]

1
2 n u [ n ]
Two ways:
(1) Repeat the procedure
(2) y [ n ]
 x [ n ]
 h [ n ] y [ n ]
 
[ n

1 ]
 h [ n ]
 h [ n

1 ]

1
2 n

1 u [ n

1 ]

16.362 Signal and System I
• The unit step response of an LTI system, continuous time

( t ) h ( t ) y ( t ) y ( t )






 h ( t )
(

) h ( n
 
) d
 u ( t ) h ( t ) s ( t ) s ( t )
  
  h (

) u ( t
 
) d

 
 t
 h (

) d
 ds ( t )
 h ( t ) dt

x ( t )
16.362 Signal and System I
• Linear constant-coefficient difference equations
1
2
+ d dt y ( t ) y ( t )
 
1
2 dy dt

1
2 x ( t ) h ( t ) y ( t )

?

1
2 h ( t )

?
dy

2 y ( t )
 x ( t ) dt y ( t ) depends on x(t). We don’t know y(t) unless x(t) is given.
But h(t) doesn’t depend on x(t). We should be able to obtain h(t) without x(t).
How?
• LTI system response properties, this chapter.
• Continuous time Fourier transform.

16.362 Signal and System I

( t )
• Linear constant-coefficient difference equations
1
2
+ d dt y ( t ) y ( t )
 
1
2 dy dt

1
2 x ( t ) h ( t ) 
1
2 y ( t )
 
1
2 dy
 dt
1
2

( t )
When t>0, y ( t )
 
1
2 dy dt
Determine A by initial condition: y ( t )
 
1
2 dy dt

1
2

( t ) y ( t )

Ae

2 t h ( t )

Ae

2 t u ( t )
Causality

16.362 Signal and System I
• Linear constant-coefficient difference equations

( t )
1
2
+ y ( t ) y ( t )
 
1
2 dy dt

1
2 x ( t ) d dt h ( t ) 
1
2
Determine A by initial condition: y ( t )
 
1
2 dy
 dt
1
2

( t ) h ( t )

Ae

2 t u ( t )
Ae

2 t u ( t )
 
1
2
(

2 ) Ae

2 t u ( t )

(

1
2
) Ae

2 t

( t )

1
2

( t )
A = 1 h ( t )
 e

2 t u ( t )

16.362 Signal and System I
• Linear constant-coefficient difference equations x ( t )

Ke 3 t u ( t ) 1
2
+ y ( t ) d dt h ( t ) 
1
2 y ( t )
 
1
2 dy dt

1
2 x ( t ) h ( t )
 e

2 t u ( t ) y ( t )
 x ( t )
 h ( t )
 


 x (

) h ( t
 
) d

 



Ke
3
 u (

)[ e

2 ( t
 
) u ( t
 
)] d

  o t
Ke 3

[ e

2 ( t
 
) ] d


Ke

2 t
 o t e
5
 d


K
[ e 3 t
5
 e

2 t ] y ( t )

K
[ e
3 t
5
 e

2 t
] u ( t )

16.362 Signal and System I
• Singularity functions
Define: u
0
( t )
 
( t ) u
1
( t )
 d

( t ) dt u n
( t )
 d n

( t ) dt n u

1
( t )
 
 t


(

) d
  u ( t ) u

2
( t )
 u

1
( t )
 u

1
( t )
 


 u (

) u ( t
 
) d

 
 t
 u (

) d
 u
 n
( t )
 u ( t )
 u ( t )
   u ( t )
 
 t
 u

( n

1 )
(

) d


16.362 Signal and System I
• Singularity functions x ( t )
 u
0
( t )
 x ( t )
 
( t )
 x ( t ) x ( t )
 u
0
( t )
 x ( t ) x ( t )
 u
1
( t )


 
  u
1
(

) x ( t
 
) d




 x ( t
 
) du
0
(

)

 x ( t




  u
0
(

) u
0
)
(

) |

 
 


 dx ( t d ( t




)
) d
 u
0
(

) dx ( t
 
)
 dx
 u
0
( t ) dt
 dx dt x ( t )
 u
1
( t )
 dx ( t ) dt x ( t )
 u n
( t )
 d n x ( t ) dt n

16.362 Signal and System I
• Singularity functions x ( t )
 u
0
( t )
 x ( t ) u
1
( t )
 u
1
( t )
 du
1
( t )
 u
2
( t ) dt x ( t )
 u
1
( t )
 dx ( t ) dt x ( t )
 u n
( t )
 d n x ( t ) dt n u
1
( t )
 u
1
( t )
    u
1
( t )
 u k
( t ) k terms

16.362 Signal and System I
• Singularity functions x ( t )
 u

1
( t )

 


 x ( t )
 u ( t ) x (

) u ( t
 
) d

 
 t
 x (

) d
 x ( t )
 u
0
( t )
 x ( t ) x ( t )
 u

1
( t )
 
 t
 x (

) d
 x ( t )
 u n
( t )
 d n x ( t ) dt n u

2
( t )
 u ( t )
 u ( t ) x ( t )
 u

2
( t )
 x ( t )
 u ( t )
 u ( t )
 
 t
 x (

) d
  u ( t )

 t   
  x (

' ) d

' d


16.362 Signal and System I
• Singularity functions --- discrete time
Define: u
1
[ n ]
 
[ n ]
 
[ n

1 ] u k
[ n ]
 u k

1
[ n ]
 u k

1
[ n

1 ] x [ n ]
 u
1
[ n ]
 x [ n ]
 x [ n
 x [ n ]


1 ]

[ n ]
 x [ n ]
 
[ n

1 ] x [ n ]
 u
1
[ n ]
 x [ n ]
 x [ n

1 ] u
1
[ n ]
 u
1
[ n ]
 u
1
[ n ]
 u
1
[ n

1 ]

16.362 Signal and System I
• Singularity functions --- discrete time
Define: u

1
[ n ]
 u [ n ] u

2
[ n ]
 u

1
[ n ]
 u

1
[ n ]
 k



 u [ k ] u [ n
 k ]
 k
 n 
 u [ k ]
 n

1 x [ n ]
 u

1
[ n ]

 x [ n ]
 u [ n ]



 x [ k ] u [ n
 k ]

 n 
 x [ k ]