ECE633F09_HW3solutions - ECE633 Signals and Systems I Fall...

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 + + ≠ +