2 Linear Time-Invariant Systems
2.1 Discrete-time LTI system,The convolution sum
2.1.1 The Representation of Discrete-time Signals
in Terms of Impulses
2,Linear Time-Invariant Systems




k
knkx
nxnxnxnxnxnx
][][
]2[]2[]1[]1[][]0[]1[]1[]2[]2[][

If x[n]=u[n],then


0
][][
k
knnx?
2 Linear Time-Invariant Systems
2 Linear Time-Invariant Systems
2.1.2 The Discrete-time Unit Impulse Response
and the Convolution Sum Representation
of LTI Systems
(1) Unit Impulse(Sample) Response
LTIx[n]=?[n] y[n]=h[n]
Unit Impulse Response,h[n]
2 Linear Time-Invariant Systems
(2) Convolution Sum of LTI System
LTIx[n] y[n]=?
Solution:
Question,
[n] h[n]
[n-k] h[n-k]
x[k]?[n-k] x[k] h[n-k]





kk
knhkxnyknkxnx ][][][][][][?
2 Linear Time-Invariant Systems
2 Linear Time-Invariant Systems
2 Linear Time-Invariant Systems
( Convolution Sum )So



k
knhkxny ][][][
or y[n] = x[n] * h[n]
(3) Calculation of Convolution Sum
Time Inversal,h[k] h[-k]
Time Shift,h[-k] h[n-k]
Multiplication,x[k]h[n-k]
Summing,


k
knhkxny ][][][
Example 2.1 2.2 2.3 2.4 2.5
2 Linear Time-Invariant Systems
2.2 Continuous-time LTI system,
The convolution integral
2.2.1 The Representation of Continuous-time
Signals in Terms of Impulses



o t h e r w is e
tt
,0
0,1)(?Define
We have the expression,



k
ktkxtx )()()(
Therefore,



k
ktkxtx )()(lim)( 0?
2 Linear Time-Invariant Systems
2 Linear Time-Invariant Systems
or
dtxtx )()()(
2 Linear Time-Invariant Systems
2.2.2 The Continuous-time Unit impulse Response
and the convolution Integral Representation
of LTI Systems
(1) Unit Impulse Response
LTIx(t)=?(t) y(t)=h(t)
(2) The Convolution of LTI System
LTIx(t) y(t)=?
2 Linear Time-Invariant Systems
A,LTI?(t) h(t)
x(t) y(t)=?
dtxtx )()()(
Because of
dthxty )()()(
So,we can get
( Convolution Integral )
or y(t) = x(t) * h(t)
2 Linear Time-Invariant Systems
B,)(lim)(
0 tt
or y(t) = x(t) * h(t)
)(lim)( 0 thth
LTI?(t) h(t)?
(t) h?(t)
)()( tht
)()( kthkt?
)()()()( kthkxktkx?






kk
kthkxktkx )()(lim)()(lim
00
dtxtx )()()( dthxty )()()(
( Convolution Integral )
2 Linear Time-Invariant Systems
2 Linear Time-Invariant Systems
(3) Computation of Convolution Integral
Time Inversal,h(?) h(-?)
Time Shift,h(-?) h(t-?)
Multiplication,x(?)h(t-?)
Integrating,
dthxty )()()(
Example 2.6 2.8
2 Linear Time-Invariant Systems
2.3 Properties of Linear Time Invariant System
Convolution formula:
dthxthtxty )()()(*)()(



k
knhkxnhnxny ][][][*][][
h(t)
x(t) y(t)=x(t)*h(t)
h[n]
x[n] y[n]=x[n]*h[n]
2 Linear Time-Invariant Systems
2.3.1 The Commutative Property
Discrete time,x[n]*h[n]=h[n]*x[n]
Continuous time,x(t)*h(t)=h(t)*x(t)
h(t)
x(t) y(t)=x(t)*h(t)
x(t)
h(t) y(t)=h(t)*x(t)

2 Linear Time-Invariant Systems
2.3.2 The Distributive Property
Discrete time,
x[n]*{h1[n]+h2[n]}=x[n]*h1[n]+x[n]*h2[n]
Continuous time,
x(t)*{h1(t)+h2(t)}=x(t)*h1(t)+x(t)*h2(t)
h1(t)+h2(t)
x(t) y(t)=x(t)*{h1(t)+h2(t)}
h1(t)x(t) y(t)=x(t)*h1(t)+x(t)*h2(t)
h2(t)
Example 2.10
2 Linear Time-Invariant Systems
2.3.3 The Associative Property
Discrete time,
x[n]*{h1[n]*h2[n]}={x[n]*h1[n]}*h2[n]
Continuous time,
x(t)*{h1(t)*h2(t)}={x(t)*h1(t)}*h2(t)
h1(t)*h2(t)
x(t) y(t)=x(t)*{h1(t)*h2(t)}
h1(t)x(t)
y(t)=x(t)*h1(t)*h2(t)
h2(t)
2 Linear Time-Invariant Systems
2.3.4 LTI system with and without Memory
Memoryless system:
Discrete time,y[n]=kx[n],h[n]=k?[n]
Continuous time,y(t)=kx(t),h(t)=k?(t)
k?(t)
x(t) y(t)=kx(t)=x(t)*k?(t)
k?[n]
x[n] y[n]=kx[n]=x[n]*k?[n]
Imply that,x(t)*?(t)=x(t) and x[n]*?[n]=x[n]
2 Linear Time-Invariant Systems
2.3.5 Invertibility of LTI system
Original system,h(t)
Reverse system,h1(t)
(t)
x(t) x(t)*?(t)=x(t)
So,for the invertible system,
h(t)*h1(t)=?(t) or h[n]*h1[n]=?[n]
h(t)
x(t) x(t)
h1(t)
Example 2.11 2.12
2 Linear Time-Invariant Systems
2.3.6 Causality for LTI system
Discrete time system satisfy the condition:
h[n]=0 for n<0
Continuous time system satisfy the condition:
h(t)=0 for t<0
2 Linear Time-Invariant Systems
2.3.7 Stability for LTI system
Definition of stability:
Every bounded input produces a bounded
output,
Discrete time system:





kk
khknxknhkxny ][][][][][ 或
If |x[n]|<B,the condition for |y[n]|<A is


k
kh |][|
Anyt he nkhif
khBkhknxny
k
kk









|][|,|][|
|][||][||][||][|
2 Linear Time-Invariant Systems
Continuous time system:
If |x(t)|<B,the condition for |y(t)|<A is
dhtxdthxty )()()()()( 或
dh |)(|
Atyt h e ndhif
dhBdhtxty









|)(|,|)(|
|)(||)(||)(||)(|


Example 2.13
2 Linear Time-Invariant Systems
2.3.8 The Unit Step Response of LTI system
Discrete time system:
]1[][][][][][][ )1(

nhnhnsnhnhkhns n
k
或
h[n]
[n] h[n]
u[n] s[n]=u[n]*h[n]
Continuous time system:
h(t)
(t) h(t)
u(t) s(t)=u(t)*h(t)
)(')()()()( )1( tsththdhts t 或
2 Linear Time-Invariant Systems
2.4 Causal LTI Systems Described by
Differential and Difference Equation
Discrete time system,Differential Equation
Continuous time system,Difference Equation
2 Linear Time-Invariant Systems
2.4.1 Linear Constant-Coefficient Differential Equation
A general Nth-order linear constant-coefficient
differential equation:


M
k
k
k
k
N
k
k
k
k dt
tdxb
dt
tyda
00
)()(
or
)()(')()(
)()(')()(
01
)1(
1
)(
01
)1(
1
)(
txbtxbtxbtxb
tyatyatyatya
M
M
M
M
N
N
N
N


and initial condition:
y(t0),y’(t0),……,y (N-1)(t0) ( N values )
2 Linear Time-Invariant Systems
2.4.2 Linear Constant-Coefficient Difference Equation
A general Nth-order linear constant-coefficient
difference equation:



M
k
k
N
k
k knxbknya
00
][][
or
][]1[)]1([][
][]1[)]1([][
011
011
nxbnxbMnxbMnxb
nyanyaNnyaNnya
MM
NN


and initial condition:
y[0],y[-1],……,y[ -(N-1)] ( N values )
Example 2.15
2 Linear Time-Invariant Systems
2.4.3 Block Diagram Representations of First-order
Systems Described by Differential and Difference
Equation
(1) Dicrete time system
Basic elements:
A,An adder
B,Multiplication by a coefficient
C,An unit delay
2 Linear Time-Invariant Systems
Basic elements,
2 Linear Time-Invariant Systems
Example,y[n]+ay[n-1]=bx[n]
2 Linear Time-Invariant Systems
(2) Continuous time system
Basic elements:
A,An adder
B,Multiplication by a coefficient
C,An (differentiator) integrator
2 Linear Time-Invariant Systems
Basic elements,
2 Linear Time-Invariant Systems
Example,y’(t)+ay(t)=bx(t)
2 Linear Time-Invariant Systems
2.5 Singularity Functions
2.5.1 The unit impulse as idealized short pulse



o t h e r w is e
tt
,0
0,1)(?
)(lim)( 0 tt
(1)
1
(2) )(*)()( tttr

)(lim)( 0 trt
2 Linear Time-Invariant Systems




t
dxtxtutx
txttx
tttxttttx
ttxtttx
txttx
txttx


)()()(*)()6(
)(')('*)()5(
)()(*)()4(
)()(*)()3(
)()0()()()2(
)()(*)()1(
)1(
2121
00
Several important formula:
Problems:
2.1 2.3 2.5 2.7 2.10 2.11 2.12
2.18 2.19 2.20 2.23 2.24 2.40 2.47