M. J. Roberts - 8/16/04
Solutions 3-2
Causal
Memory:
The response at any time,
t
t
=
0
, depends only on the excitation at time,
t
t
=
0
and not on any
past values.
System has no memory.
Invertibility:
There are many value of the excitation that all cause a response of zero and there are many
values of the excitation that all cause a response of one.
Therefore the system is not
invertible.
2.
Show that a system with excitation, x
t
( )
, and response, y
t
( )
, described by
y
x
x
t
t
t
( )
=
−
(
)
−
−
(
)
5
3
is linear but not causal and not invertible.
Causality:
At time,
t
=
0, y
x
x
0
5
3
( )
=
−
(
)
−
( )
.
Therefore the response at time,
t
=
0, depends on the
excitation at a later time,
t
=
3.
Not Causal
Memory:
At time,
t
=
0, y
x
x
0
5
3
( )
=
−
(
)
−
( )
.
Therefore the response at time,
t
=
0, depends on the
excitation at a previous time,
t
= −
5.
System has memory.
Invertibility:
A counterexample will demonstrate that the system is not invertible.
Let the excitation be a
constant,
K
.
Then the response is y
t
K
K
( )
=
−
=
0.
This is the response, no matter what
K
is.
Therefore when the response is a constant zero, the excitation cannot be determined.
Not Invertible.
3.
Show that a system with excitation, x
t
( )
, and response, y
t
( )
, described by
y
x
t
t
( )
=
2
is linear, time variant and non-causal.
Time Invariance:
Let x
g
1
t
t
( )
=
( )
.
Then
y
g
1
2
t
t
( )
=
.
Let x
g
2
0
t
t
t
( )
=
−
(
)
.
Then
y
g
y
g
2
0
1
0
0
2
2
t
t
t
t
t
t
t
( )
=
−
≠
−
(
)
=
−
.
Time Variant
Causality: