Digital Signal Processing Reference
In-Depth Information
3.10
Find the DTFT of the following two functions:
(a)
x
1
(n)
=
e
−
0
.
5
n
u(n)
=
x(
−
n
−
2
)
where
x(n)
=
5
−
n
u(n)
.
3.11
Given the DTFT of
x
1
(n)
(b)
x
2
(n)
x
as
X
1
(e
jω
)
, express the
={
−
}
10
10
in terms of
X
1
(e
jω
)
.
DTFT of
x
2
(n)
={
−
−
}
10
1010
10
Express the DTFT of
x
3
(n)
={
10
−
110
−
10
}
in terms
of
X
1
(e
jω
)
.
3.12
An LTI-DT system is described by the difference equation
y(n)
−
0
.
5
y(n
−
1
)
=
x(n)
−
bx(n
−
1
)
Determine the value of
b
(other than 0.5) such that the square of the mag-
nitude of its transfer function
H(e
jω
)
is a constant equal to
b
2
for all
frequencies.
z
−
N
)/N
.
Determine the frequency response of the filter in a closed-form expression
for
N
3.13
A comb filter is defined by its transfer function
H(z)
=
(
1
−
=
10.
3.14
Show that the magnitude response of an IIR filter with
2
+
0
.
2
z
−
1
−
0
.
2
z
−
3
−
2
z
−
4
H(z)
=
(
1
−
z
−
4
)
is a real function of
ω
and an even function of
ω
.
3.15
A discrete-time signal with a lowpass frequency response that is a con-
stant equal to 5 and has a bandwidth equal to 0
.
4
π
is the input to an ideal
bandpass filter with a passband between
ω
p
1
=
0
.
3
π
and
ω
p
2
=
0
.
6
π
and
a magnitude of 4. What is the bandwidth of the output signal?
3.16
A DT signal
x(n)
+
10 cos
(
0
.
9
πn)
is the
input to an allpass filter with a constant magnitude of 5 for all frequencies.
What is the output
y(n)
of the filter?
=
4cos
(
0
.
4
πn)
+
6cos
(
0
.
8
πn)
a
n
u(n)
and
h(n)
b
n
u(n)
,where0
<a<
1
,
0
<b<
1,
3.17
Given
x
(n)
=
=
a
n
+
1
b
n
+
1
/a
a
=
b
show that
y(n)
=
x(n)
∗
h(n)
≡
−
−
b
.
n
2
(
0
.
1
)
n
u(n)
.
3.18
Find the DTFT of
x(n)
=
3.19
Prove that
Nk
N
−
1
±
2
N,...
0 t r ie
=
0
,
±
N,
e
j(
2
π/N)kn
=
n
=
0
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