Signals and Systems, Final Exam
Solutions
Spring 2005, Edited by bypeng
1.
[21 points] Consider a continuous-time system with impulse response
3
1
( )
( )
( )
2
( )
t
t
h t
t
e
u t
e u t
δ
−
−
=
+
+
a)
[3] Determine the transfer function of the
inverse
system of
h
1
.
b)
[3] Is the inverse system causal and stable?
c)
[3] Sketch the asymptotic approximation to the gain and phase of the Bode plot for the transfer
function of the inverse system.
d)
[3] Draw a direct-form representation of the inverse system using a minimum number of
integrators and multipliers.
e)
[3] Repeat Step d) and draw a parallel-form representation of the inverse system.
f)
[3] Repeat Step d) and draw a cascade-form representation of the inverse system.
g)
[3] Suppose the continuous-time system
1
( )
h t
is cascaded with a sampler, a converter that
converts an impulse train to a sequence, and an discrete-time LTI system
2
[ ]
h
n
, as shown in
the following block diagram. Determine the frequency response
2
(
)
j
H
e
ω
such that
[ ]
[ ]
w n
n
δ
=
when the input
( )
x t
is a unit impulse.
h
1
(
t
)
Conversion
of impulse
train to a
sequence
h
2
[
n
]
w
[
n
]
y
[
n
]
y
(
t
)
x
(
t
)
( )
(
)
n
p t
t
nT
δ
+∞
=−∞
=
−
∑
Solution:
a)
We need the transfer function of the inverse system
1
( )
I
H
s
satisfying
1
1
( )
( )
1
I
H
s H
s
=
, and we
have
2
1
1
2
(3
)(1
)
(1
)
2(3
)
10
7
(2
)(5
)
( )
1
3
1
(3
)(1
)
(3
)(1
)
(3
)(1
)
s
s
s
s
s
s
s
s
H
s
s
s
s
s
s
s
s
s
+
+
+
+
+
+
+
+
+
+
=
+
+
=
=
=
+
+
+
+
+
+
+
+
so
1
(3
)(1
)
( )
(2
)(5
)
s
s
H
s
s
s
+
+
=
+
+
; There are three possible ROCs
Re[ ]
5
s
< −
;
5
Re[ ]
2
s
−
<
< −
;
Re[ ]
2
s
> −
. But we need the ROC includes
Re[ ]
1
s
> −
, so the ROC is
Re[ ]
2
s
> −
.
b)
Since
1
( )
I
H
s
is rational and the ROC is right-hand-sided, the inverse system is causal.
Since
1
( )
I
H
s
has all poles in the left half side of the s-plane, the inverse system is stable.
c)
The sketch of the Bode plot is as the following figure. [Note that the dashed line is the actual
response while the solid line is the approximation using the technique in Section 6.5/9.4 in the
textbook. One should use the technique in Section 6.5/9.4 to get the full credit.]

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