Digital Signal Processing Reference
In-Depth Information
TABLE 12.5
Fourier Transform and DFT Approximation
l
F
(
V
)
l
F
[
k
]
k
V
F
(
V
)
T
F
[
k
]
% Error
0
0
2.000
2.2000
10.0
0
0
1
1.9635
0.9411
0.8524
9.4
-1.9635
-1.9635
2
3.9270
0.3601
0.4828
34.1
-0.7854
-0.7854
3
5.8905
0.1299
0.0702
45.9
-2.7489
0.3927
4
7.8540
0.2546
0.2000
21.5
-1.5708
-1.5708
5
9.8175
0.0780
0.2359
202.6
-0.3927
-0.3927
6
11.7810
0.1200
0.0828
31.0
-2.3562
0.7854
7
13.7445
0.1344
0.1133
15.7
-1.1781
-1.1781
2.5
2
1.5
1
0.5
0.1
0
5
10
15
20
25
30
Figure 12.17
Comparison of the DFT with
the Fourier transform.
Frequency (rad/s)
represents one period of the periodic extension of the original signal. These are the
values used to calculate from (12.33). From this presentation, we see that the
DFT we calculate not only is based on sampled values of the analog signal, but also in-
volves a convolution with the Fourier transform of the windowing function.
This implied multiplication of the periodic extension of the sampled signal by the
windowing function results in the phenomenon called
spectrum-leakage distortion,
which arises from the spectrum spreading that develops from truncating a signal. This
phenomenon can be illustrated by the truncated cosine shown in Figure 12.19(a). This
signal is the product of a cosine wave and a rectangular pulse, as shown in Figure
12.19(b). From our previous study of the Fourier transform and Table 5.2, we know that
X[k]
cos(v
1
t)
Î
"
p[d(v-v
1
) + d(v + v
1
)]
and
rect(t/T)
Î
"
Tsinc(Tv/2)
,
as shown in Figure 12.19(c). From the Fourier transform properties (Table 5.1),
1
2p
F
1
(v)
*
F
2
(v).
f
1
(t)f
2
(t)
Î
"
