Digital Signal Processing Reference
In-Depth Information
X
(
)
4
/3
2
/3
/3
0
/3
2
/3
Figure 12.7
6
/3
4
/3
2
/3
0
2
/3
4
/3
6
/3
X(Æ)
for Example 12.7.
1
3
2p
3
4p
3
B
¢
≤
e
j2p/3
¢
≤
e
j4p/3
R
x[1] =
X
0
(0) + X
0
+ X
0
1
3
[2 + (-1)
l
120°
+ (-1)
l
240°
]
=
1
3
[2 + 0.5 - j0.866 + 0.5 + j0.866] = 1;
=
1
3
2p
3
4p
3
B
¢
≤
e
j4p/3
¢
≤
e
j8p/3
R
x[2] =
X
0
(0) + X
0
+ X
0
1
3
[2 + 0.5 - j0.866 + 0.5 + j0.866] = 1.
=
These values are seen to be correct.
■
In the next section, a transform called the discrete Fourier transform will be
defined for the distinct values it is independent of the
derivations of this section. However, these derivations give us an interpretation of
the discrete Fourier transform for cases in which the values of
x
0
[0], x
0
[1],
Á , x
0
[N - 1],
x
0
[n]
are samples
from a continuous-time signal.
12.4
DISCRETE FOURIER TRANSFORM
In Chapters 5 and 6 and in earlier sections of this chapter, we have presented for-
mulas for calculation of Fourier transforms. However, for both the actual Fourier
transform and the discrete-time Fourier transform, the formulas produce continu-
ous functions of frequency. In this section, we develop an approximation of the
Fourier transform that can be calculated from a finite set of discrete-time samples of
