Cryptography Reference
In-Depth Information
n
n
c j =
f
(
x i )cos(
jπx i /v
)
d j =
f
(
x i )sin(
jπx i /v
)
.
i=1
i=1
Discrete approaches are almost certainly going to be more inter-
esting to modern steganographers because so much data is stored
and transported in digital form.
14.6 The Fast Fourier Transform
The calculus may be beautiful, but digital data doesn't come in conti-
nous functions. Luckily, mathematicians have found versions of the
equations suitable for calculation. In fact, the version for discrete
dataknownastheFastFourierTransform(FFT)isthefoundation
for many of the digital electronics used for sound, radio, and images.
Almost all multimedia software today uses some form of the FFT to
analyze data, find the dominant harmonic characteristics, and then
use this information to enhance or perhaps compress the data. Mu-
sicians use FFT-based algorithms to add reverb, dampen annoying
echoes, or change the acoustics of the hall where the recording was
made. Record companies use FFTs to digitize music and store it on
CDs. Teenagers use FFTs again to convert the music into MP3 files.
The list goes on and on.
The details behind the FFT are beyond the scope of this topic.
The algorithm uses a clever numerical juggling routine often called
a “butterfly algorithm” to minimize the number of multiplications.
The end result is a long vector of numbers summarizing the strength
of various frequencies in the signal.
To be more precise, an FFT algorithm accepts a vector of
n
el-
ements,
{x 0 ,x 1 ...x n−1 , and returns another vector of
n
elements
{y 0 ,y 1 ...y n−1 ,where
n
1
n
x r e 2πi(r−1)(s−1)/n
y s =
.
r=1
This equation is used byMathematica and its Wavelet Explorer pack-
age, the program that created many of the pictures in this section.
Others use slight variations designed to solve particular problems.
The vector that emerges is essentially a measure of howwell each
function matches the underlying data. For instance, the fourth ele-
ment in the vector measures how much the graph has in common
with cos(4
×
2
πx
)+
i
sin(4
×
2
πx
) .
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