Digital Signal Processing Reference
In-Depth Information
The DTFT
Y
2
(
Ω
) of the output sequence is therefore given by
Y
2
(
Ω
)
=
3
π
−
π
8
+
π
8
δ
1
+ δ
1
Ω
Ω
−
π
4
+
π
4
+
5
π
δ
Ω
0
+ δ
Ω
0
−
π
8
+
π
8
=
3
π
δ
+ δ
.
Ω
Ω
Calculating the inverse DTFT, the output sequence is obtained as
π
8
k
y
2
[
k
]
=
3 cos
.
The LTID system
H
2
(
Ω
) acts like an ideal lowpass filter as the sinusoidal
component 3 cos(
π
k
/
8) with low fundamental frequency
Ω
1
= π/
8isnot
attenuated, while the sinusoidal component 3 cos(
π
k
/
4) with high fundamental
frequency
Ω
1
= π/
4 is blocked from the output.
(iii) For the transfer function in Eq. (11.67), the values of the transfer
function
H
3
(
Ω
) at frequencies
Ω
1
=π/
8 and
Ω
2
=π/
4 are given by
= π/
8
,
H
3
(
π/
8)
=
0
,
H
3
(
π/
8)
=
0
,<
H
3
(
π/
8)
=
0 radians;
Ω
=−π/
8
,
H
3
(
−π/
8)
=
0
,
H
3
(
−π/
8)
=
0
,<
H
3
(
−π/
8)
=
0 radians;
Ω
= π/
4
,
H
3
(
π/
4)
=
1
,
H
3
(
π/
4)
=
1
,<
H
3
(
π/
4)
=
0 radians;
Ω
=−π/
4
,
H
3
(
−π/
4)
=
1
,
H
3
(
−π/
4)
=
1
,<
H
3
(
−π/
4)
=
0 radians.
The DTFT
Y
3
(
Ω
) of the output sequence is therefore given by
Ω
−
π
8
+
π
8
Y
3
(
Ω
)
=
3
π
δ
0
+ δ
0
Ω
Ω
−
π
4
+
π
4
+
5
π
δ
1
+ δ
1
Ω
Ω
−
π
4
−
π
4
=
5
π
δ
+ δ
.
Ω
Ω
Calculating the inverse DTFT, the output sequence is obtained as
π
4
k
y
3
[
k
]
=
5 cos
.
The LTID system
H
3
(
Ω
) acts like an ideal highpass filter as the sinusoidal com-
ponent 3 cos(
π
k
/
8) with lower fundamental frequency
Ω
1
= π
/8 is blocked,
while the sinusoidal component 3 cos(
π
k
/
8) with higher fundamental frequency
Ω
1
= π/
4 is unattenuated in the output sequence.
11.8 Continuous- a
nd discrete-time Fourier transforms
In Chapters 4, 5, and 10, we derived frequency representations for CT and DT
waveforms. In particular, we considered the following four frequency represen-
tations:
(1) CTFT for CT periodic signals;
(2) CTFT for CT aperiodic signals;
(3) DTFT for DT periodic sequences;
(4) DTFT for DT aperiodic sequences.
Search WWH ::
Custom Search