Linear Time-Invariant Systems (LTI)

1
Linear Time-Invariant Systems (LTI)
2
Linear Time-Invariant Systems (LTI) Superpositio
n
3
Linear Time-Invariant Systems (LTI) Superpositio
n Convolution
4
Convolution is commutative, associative
and distributive
5
Convolution is commutative, associative
and distributive

6
Convolution is commutative, associative
and distributive

7
Convolution is distributive
(cascaded systems)
8
Convolution is commutative, associative
and distributive Proof (associative) --
exchange order of integration
9
Linear Time Invariant (LTI) Systems
Note that
is that is time
time reversed and shifted by t (as a function of
) The integral is evaluated for each t to
get y(t)
10

• Observe that if h(t) 0 for tlt0, then
• y(t)0 for tlt0 because h(t-t) 0, tlt0
• Maximum overlap of h(t-t) and x(t-t)
• gives t for the maximum value of y(t)
• 3. If h(t) 0 for tlt0 and tgtt0, and
• if x(t) 0 for tlt0 and tgtt1
• then y(t)0 for tlt0 and
• y(t)0 for tgtt0 t1

11
Example
12
(No Transcript)
13
y(t)
1
0 1 2 3
14
Example
1
1
h(t)
x(t)
0
0
y(t)x(t)?h(t)
t
15
Example
1
1
h(t)
x(t)
0
0
y(t)0 for tlt0
y(t)x(t)?h(t)
t
16
Example
1
1
h(t)
x(t)
0
0
y(t)0 for tlt0
y(t)x(t)?h(t)
t