Digital Signal Processing Reference
In-Depth Information
12.3.
Prove the linearity property of the DFT.
= j
dX(
Æ
)
dÆ
12.4.
Prove that
5
nx[n]
6
.
12.5.
Given a filter
h[n]
described by the difference equation,
y[n] = 3x[n] + 5x[n - 1] + 3x[n - 2]
(a)
Find and plot the impulse filter response of the filter.
(b)
Find the descrete-time Fourier transform DTFT
(c)
Does
h[n],
H(Æ).
H(Æ)
have linear phase?
12.6.
For each of the systems that follow, with filters with DTFT as given, determine
whether they have linear phase or nonlinear phase. If they have linear phase, deter-
mine the phase.
1
1 - .7e
- jÆ
(a)
H
1
(Æ) =
H
2
(Æ) = 1 + 3e
- jÆ
+ 3e
- 2jÆ
+ e
- 3jÆ
(b)
(c)
(d)
H
3
(Æ) = 2e
2jÆ
+ 3e
jÆ
+ 7 + 3e
- jÆ
+ 2e
- 2jÆ
H
4
(Æ) = 1 + 2e
- 4jÆ
+ 4e
- 6jÆ
12.7.
Given
x
0
[n] = d[n] + d[n - 2] + d[n - 4],
find
X
0
(Æ) and X(Æ)
(the DTFT of the
periodic version of
x
0
[n]
). Assume
N = 5.
y[n] = x[
3
],
12.8.
Given a discrete-time function
where
x[n]
has DTFT
X(Æ),
find
Y(Æ),
the DTFT of
y[n],
in terms of
X(Æ).
12.9.
Consider a discrete-time periodic function
x[n]
with DTFT
4
q
2p
2pk
4
2pk
4
X(Æ) =
X
0
¢
≤
d
¢
Æ-
≤
.
-
q
X
0
(
2pk
4
The values of
)
are
4,
k = 0
2pk
4
0,
k = 1
¢
≤
d
X
0
=
4,
k = 2
0,
k = 3.
Find
x
0
[n],
where
x
0
[n]
is one period of
x[n],
that is,
x[n],
0 … n … N - 1
b
x
0
[n] =
0,
otherwise.
12.10.
We wish to design a finite impulse response (FIR) filter
h[n].
We have the following
constraints on the DTFT
H(Æ)
of the filter:
