ECE633 Signals and Systems I, Fall 2009 – Homework 3 Solutions
1
From:
Linear Systems and Signals, 2
nd
ed
.
B. P. Lathi, Oxford University Press, 2005.
1.6-1
Write the input-output relationship for an ideal integrator.
Determine the zero-input and
zero-state components of the response.
( )
( )
( )
( )
( )
N
( )
0
0
0
zero-input
zero-state
0
t
t
t
y t
x
d
x
d
x
d
y
x
d
τ
τ
τ
τ
τ
τ
τ
τ
−∞
−∞
=
=
+
=
+
∫
∫
∫
∫
±²³²´
1.7-1
For the systems described by the following equations, with the input
( )
x t
and the output
( )
y t
,
determine which of the systems are linear and which are nonlinear.
(a)
( )
( )
2
2
dy
y t
x
t
dt
+
=
(e)
( )
( )
2
2
dy
y t
x t
dt
⎛
⎞
+
=
⎜
⎟
⎝
⎠
(b)
( )
( )
2
3
dy
ty t
t x t
dt
+
=
(f)
(
)
( )
( )
sin
2
dy
dx
t
y t
x t
dt
dt
+
=
+
(c)
( )
( )
3
2
y t
x t
+
=
(g)
( )
( )
2
dy
dx
y t
x t
dt
dt
+
=
(d)
( )
( )
2
dy
y
t
x t
dt
+
=
(h)
( )
( )
t
y t
x
d
τ
τ
−∞
=
∫
(a)
( )
( )
( )
( )
2
2
1
2
1
2
1
2
2
2
dy
dy
y
t
y
t
x
t
x
t
dt
dt
+
+
+
=
+
[
]
( )
( )
( )
( )
2
1
2
1
2
1
2
2
d y
y
y
t
y
t
x
t
x
t
dt
+
+
+
≠
+
⎡
⎤
⎡
⎤
⎣
⎦
⎣
⎦
non-linear
(b)
( )
( )
( )
( )
2
2
1
2
1
2
1
2
3
3
dy
dy
ty
t
ty
t
t x
t
t x
t
dt
dt
+
+
+
=
+
[
]
( )
( )
( )
( )
1
2
2
1
2
1
2
3
d y
y
t
y
t
y
t
t
x
t
x
t
dt
+
+
+
=
+
⎡
⎤
⎡
⎤
⎣
⎦
⎣
⎦
linear
(c)
( )
( )
( )
( )
1
2
1
2
3
3
2
2
y
t
y
t
x
t
x
t
+
+
+
=
+
( )
( )
( )
( )
1
2
1
2
3
2
y
t
y
t
x
t
x
t
+
+
≠
+
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- Fall '09
- Staff
- Trigraph
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