Digital Signal Processing Reference
In-Depth Information
4
0.5
p
3
0.25
p
2
0
1
−0.25
p
0
−0.5
p
−
p
−0.75
p
−0.5
p
−0.25
p
0
0.25
p
0.5
p
0.75
p
p
−
p
−0.75
p
−0.5
p
−0.25
p
0
0.25
p
0.5
p
0.75
p
p
(a)
(b)
If instead the DTFT is to be plotted within the range
−π ≤
Ω
≤ π
, then the
DTFT coefficients can be rearranged as follows:
Fig. 12.8. Spectral estimation of
DT sequences using the DFT in
Example 12.10. (a) Estimated
magnitude spectrum;
(b) estimated phase spectrum.
The dashed lines show the
continuous spectrum obtained
from the DTFT.
X
(
Ω
r
)
=
[4
,
2
,
0
,
2]
for
Ω
r
=
[
−π, −
0
.
5
π,
0
,
0
.
5
π
] radians
/
s
.
The magnitude and phase spectra obtained from the DTFT coefficients are
sketched using stem plots in Figs. 12.8(a) and (b). For comparison, we use Eq.
(11.28b) to derive the DTFT for
x
[
k
]. The DTFT is given by
3
x
[
k
]e
−
j
Ω
k
=
2
+
e
−
j
Ω
+
e
−
j3
Ω
.
X
(
Ω
)
=
k
=
0
The actual magnitude and phase spectra based on the above DTFT expression
are plotted in Figs. 12.8(a) and (b) respectively (see dashed lines). Although
the DFT coefficients provide exact values of the DTFT at the discrete fre-
quencies
Ω
r
=
[0
,
0
.
5
π
,
π,
1
.
5
π
] radians
/
s, no information is available on
the characteristics of the magnitude and phase spectra for the intermediate
frequencies. This is a consequence of the low resolution used by the DFT
to discretize the DTFT frequency
Ω
. Section 12.3.1 introduces the concept
of zero padding, which allows us to improve the resolution used by the
DFT.
12.3.1 Zero padding
To improve the resolution of the frequency axis
Ω
in the DFT domain, a com-
monly used approach is to append the DT sequences with additional zero-valued
samples. This process is called
zero padding
, and for an aperiodic sequence
x
[
k
]
of length
N
is defined as follows:
x
[
k
]0
≤
k
≤
N
−
1
0
x
zp
[
k
]
=
N
≤
k
≤
M
−
1
.
The zero-padded sequence
x
zp
[
k
] has an increased length of
M
. The frequency
resolution
Ω
of the zero-padded sequence is improved from 2
π/
N
to 2
π/
M
.
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