ENEE630 ADSP
RECITATION 2 w/ solution
September 17/18, 2009
1.
Consider the structures shown in Fig. R2.1, with input transforms and filter responses as
indicated. Sketch the quantities
Y
0
(
e
jω
) and
Y
1
(
e
jω
).
Figure R2.1:
Solution:
Comment:
For a downsampled signal, it’s still possible to recover the original signal (not necessarily
lowpass) using filters and multirate building blocks as long as there is no aliasing (
H
1
in this
problem).
2.
For each case shown in the Fig. R2.2, prove or disprove whether the left system is equivalent
to the right system? Assume
M, L, K
are all integers larger than 1.
Solution:
Assume the input, output, intermediate signals are
x
(
n
)
, y
(
n
), and
u
(
n
), respectively.
(a). (FD)
U
(
z
) =
X
(
z
M
),
Y
2
(
z
) =
1
M
∑
M

1
k
=0
U
(
z
1
/M
W
k
M
) =
1
M
∑
M

1
k
=0
X
(
zW
kM
M
) =
X
(
z
).
(TD)
u
(
n
) =
x
(
n
M
) if
n
is a multiple of
M
, and
u
(
n
) = 0 otherwise. Then,
y
2
(
n
) =
u
(
Mn
) =
x
(
n
).
(b). (FD)
U
(
z
) =
1
M
∑
M

1
k
=0
X
(
z
1
/M
W
k
M
),
Y
2
(
z
) =
U
(
z
M
) =
1
M
∑
M

1
k
=0
X
(
zW
k
M
).
(TD)
u
(
n
) =
x
(
Mn
),
y
2
(
n
) =
u
(
n
M
) =
x
(
n
) only if
n
is a multiple of
M
, i.e.,
y
2
(
n
) =
(
x
(
n
)
n
is multiple of
M
0
otherwise.
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 Spring '10
 wu
 French Revolution, Rightwing politics, Leftwing politics, Political spectrum, 1 K, multirate building blocks

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