Digital Signal Processing Reference
In-Depth Information
(-
1
)
n
(-
1
)
n
X
H
lp
(e
j
ω
)
X
+
y(n)
x(n)
H
lp
(e
j
ω
)
FIGURE 3.6
Figure for problem (b).
We wish to derive new filters from this prototype by manipulation
of the impulse response
h
(
n
).
i.
Plot the frequency response
H
1
(
e
j
ω
) for the system whose impulse
response is
h
1
(
n
)
= h
(2
n
)
.
ii.
Plot the frequency response
H
2
(
e
jw
) for the system whose impulse
response is as follows:
h
2
(
n
) =
h
(
n
/2),
n
= 0, ±2, ±4, …
h
2
(
n
) = 0,
otherwise
iii. Plot the frequency response
H
3
(
e
j
ω
) for the system whose impulse
response is
h
3
(
n
)
= e
j
π
n
h
(
n
)
.
b.
Consider the system shown in Figure 3.6 with input
x
(
n
) and output
y
(
n
). The LTI systems shown with frequency response
H
lp
(
e
jw
)are
ideal low-pass filters with cutoff frequency
/
4 rad. and unity gain
in the passband. Show that the overall system acts as an
ideal bandstop
filter
, where the stopband is in the region
π
π
/
4
≤
ω
≤
3
π
/
4.
c.
Suppose we have two 4-point sequences
x
(
n
) and
h
(
n
) as follows:
x
(
n
) = cos(
π
n
/2),
n
= 0, 1, 2, 3
h
(
n
) = 2
n
,
n
= 0, 1, 2, 3
i. Calculate the 4-point DFT
X
(
k
).
ii. Calculate the 4-point DFT
H
(
k
).
iii. Calculate
y
(
n
)
= x
(
n
)
h
(
n
) by doing the circular convolution
directly.
iv. Calculate
y
(
n
) of part (iii) by multiplying the DFTs of
x
(
n
) and
h
(
n
) and performing an inverse DFT.
