ECE600
Introduction to Digital Signal Processing
Autumn 2010
Homework #5
Oct. 29, 2010
HOMEWORK ASSIGNMENT #5
Due Fri. Nov. 5, 2010
(in class)
1. Say that we sampled a signal
x
(
t
) at the rate
1
T
= 990 MHz to produce
x
[
n
]. Now, however, we
wish that we had samples of
x
(
t
) at the rate 999 MHz, which we will refer to as
y
[
m
]. Describe
how
x
[
n
] can be resampled to get
y
[
m
].
2. After a recent lecture, a student asked, “What does the DTFT of a periodic signal look like?” This
problem is meant to address that question.
(a) Say that ˜
x
[
n
] is an
N
periodic signal (i.e., ˜
x
[
n
] = ˜
x
[
n
+
N
] for all
n
) and say that
{
x
[
n
]
}
N

1
n
=0
is a length
N
signal that contains one period of ˜
x
[
n
] (i.e.,
x
[
n
] = ˜
x
[
n
] for
n
= 0
,
1
,...,N
−
1).
Derive a simplified expression for
˜
X
(
e
jω
) in terms of the DFT
X
[
k
]. (
Hint
: In stating your
answer, remember that the DFT
X
[
k
] is valid only for
k
= 0
,
1
,...,N
−
1. Thus, the expression
X
[
(
k
)
N
] may come in useful.)
(b) Confirm your answer with the signal
x
[
n
] =
e
j
2
π
N
n
defined for
n
= 0
,
1
,...,N
−
1.
To do
this, compute the DFT
X
[
k
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 Fall '08
 Clymer,B
 Digital Signal Processing, Signal Processing, linear convolution, P. Schniter, 6circular convolution, 4circular convolution

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