Name:
ECE 301: Quiz 2
Purdue University, Summer 2009
1. Let the response
y
(
t
) to a continuoustime system be related to the
input
x
(
t
) by the following relationship:
y
(
t
) = 2
x
(
t
2
)
.
With yes/no answers (no proofs necessary) state whether the system
is:
(a) memoryless
No.
y
(2) = 2
x
(4) so that
y
(
t
) at time
t
= 2 depends
on future values of the input
x
(
t
).
(b) causal
No.
By the same argument as above.
(c) invertible
No.
The response
y
(
t
) depends on the input
x
(
t
) only
for positive values of
t
. Therefore consider two inputs
x
1
(
t
) and
x
2
(
t
) such that
x
1
(
t
) =
x
2
(
t
) for
t
≥
0 but
x
1
(
t
)
negationslash
=
x
2
(
t
) for
t <
0.
These two inputs will produce identical outputs, and therefore the
system is
not
invertible.
(d) stable
Yes.
If the input is bounded, there exists some bound
B <
∞
such that

x
(
t
)

< B
for all
t
. Therefore the magnitude of
the output

y
(
t
)

=

2
x
(
t
2
)

= 2

x
(
t
2
)
 ≤
2
B <
∞
.
(e) linear
Yes.
Let
x
1
(
t
) and
x
2
(
t
) are inputs with corresponding
outputs
y
1
(
t
) = 2
x
1
(
t
2
) and
y
2
(
t
) = 2
x
2
(
t
2
), respectively. Now let
a third input
x
3
(
t
) =
ax
1
(
t
) +
bx
2
(
t
) for some constants
a
and
b
.
We have that the system’s response to
x
3
(
t
) is
y
3
(
t
) = 2
x
3
(
t
2
) =
2(
ax
1
(
t
2
) +
bx
2
(
t
2
) =
ay
1
(
t
) +
by
2
(
t
).
(f) time invariant
No.
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 Spring '06
 V."Ragu"Balakrishnan
 Input/output, following relationship

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