ECE 493
HW #11 –
Version 1.2 February 11, 2011
Spring 2011
Univ. of Illinois
Due Tues, Feb 15
Prof. Allen
Topic of this homework:
Convergence of transforms, ROC, Classification of systems (polezero
placement).
Deliverable: Show your work.
1
Convergence
1. If
F
(
s
) = 1
/
(1 +
s
) and the ROC is
σ
0
>

1, find the inverse Laplace transform (
L

1
).
2. If
F
(
s
) = 1
/
(1 +
s
) and the ROC is
σ
0
<

1, find the inverse Laplace transform.
1. If
F
(
s
) = 1
/
(
s

1) and the ROC is
σ
0
>
1, find the inverse Laplace transform.
2. If
F
(
s
) = 1
/
(
s

1) and the ROC is
σ
0
<
1, find the inverse Laplace transform.
3. Find the inverse Laplace transform of
F
(
s
) =
1
(
s

1)(
s
+ 1)
(1)
(a) if the ROC is between the two poles.
(b) if the ROC is to the right of
σ
= 1.
(c) if the ROC is to the left of
σ
=

1.
2
Parseval’s Thm
What is Parseval’s Theorem for the
1. Fourier Transform?
2.
z
transform?
3. Fourier series?
4. DFT?
5. Laplace Transform?
6. Derive this result for the case of the FT, starting from the basic definition of the FT transform
and its inverse.
1
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3
Filter classes
Let
s
=
σ
+
iω
be the Laplace (complex) frequency.
A
causal
filter
h
(
t
)
↔
H
(
s
) is one that is zero for negative time. It necessarily has a Laplace
transform
L
.
An
finite impulse response
(FIR) filter has finite duration, namely if
f
(
t
) is FIR, then it is zero
for
t <
0 and for
t > T
where
T
is some time. FIR filters only have
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 Spring '11
 JontB.Allen
 Laplace

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