Cryptography Reference
In-Depth Information
4
2
50
100
150
200
250
-2
-4
-6
x
64
Figure 14.7: 256 points calculated from the equation
(2 +
)sin(4
×
2
πx/
256)
.
30
30
20
20
10
10
50
100
150
200
250
50
100
150
200
250
-10
-10
-20
-20
-30
-30
Figure 14.8: A graph of the real and imaginary parts of the Fourier
transform computed from the data in Figure 14.7.
Figure 14.7 shows a graph of the function
(2+
x
64
256)
.
If the 256 points from this example are fed into a Fourier transform,
the result has its largest values at the
)sin(4
×
2
πx/
y
251
position. The
second spike is caused by aliasing. The values in the first
n
2
y
4
and the
elements
report the Fourier transform of the function computed from left to
right and the second
n
2
elements carry the result of computing it from
right to left. The results are mirrors.
Figure 14.8 shows the real and imaginary parts of the Fourier
transform applied to the data in Figure 14.7. Many physicists and
electrical engineers who use these algorithms to analyze radio phe-
nomena like to say that most of the “energy” can be found in the
imaginary part at
y
251
. Some mathematicians talk about how
Figure 14.8 shows the “frequency space”, while Figure 14.7 shows
the “function space”. In both cases, the graphs are measuring the
amount that the data can be modeled by each element.
y
4
and
