Digital Signal Processing Reference
In-Depth Information
Ω
)
H(
Ω
, rad./sec.
0
(a)
H(e
j
ω
)
ω
,
rad.
π
π
π
π
-
4
-
2
0
2
4
(b)
FIGURE 3.1
(a) Analog frequency response and (b) digital frequency response.
Analog frequency response and digital frequency response
One of the most important differences between discrete-time systems and
analog systems is that discrete-time systems have a periodic frequency
response,
H
(
e
j
), while analog systems have a nonperiodic Fourier transform
ω
H
(
j
Ω
)
.
Figure 3.1 illustrates this difference in between
H
(
j
Ω
) and
H
(
e
j
).
ω
3.1.2
Discrete Fourier Transform
The Discrete Fourier Transform (DFT) is a practical extension of the DTFT,
which is discrete in both
time and the frequency domains
.
As discussed in
the previous section, the DTFT
X
(
ω
)
is a periodic function with period 2
π
radians. This property is used to the divide the frequency interval (0, 2
π
)
into
N
points, to yield the DFT of the discrete-time sequence
x
(
n
), 0
≤
n
≤
N
- 1 as follows:
−
∑
N
1
−
j
2
π
kN
(3.4)
Xk
()
=
X
( )
ω
ωπ
=
xne
()
,
0
≤
k
≤
N
−
1
=
2
kN
n
=
0
