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141
CHAPTER
8
The Discrete Fourier Transform
Fourier analysis is a family of mathematical techniques, all based on decomposing signals into
sinusoids.
The discrete Fourier transform (DFT) is the family member used with
digitized
signals.
This is the first of four chapters on the
real DFT
, a version of the discrete Fourier
transform that uses real numbers to represent the input and output signals.
The
complex DFT
,
a more advanced technique that uses complex numbers, will be discussed in Chapter 31.
In this
chapter we look at the mathematics and algorithms of the Fourier decomposition, the heart of the
DFT.
The Family of Fourier Transform
Fourier analysis is named after
Jean Baptiste Joseph Fourier
(17681830),
a French mathematician and physicist.
(Fourier is pronounced:
, and is
for
@
¯e
@
¯a
always capitalized).
While many contributed to the field, Fourier is honored
for his mathematical discoveries and insight into the practical usefulness of the
techniques.
Fourier was interested in heat propagation, and presented a paper
in 1807 to the Institut de France on the use of sinusoids to represent
temperature distributions.
The paper contained the controversial claim that any
continuous periodic signal could be represented as the sum of properly chosen
sinusoidal waves.
Among the reviewers were two of history's most famous
mathematicians, Joseph Louis Lagrange (17361813), and Pierre Simon de
Laplace (17491827).
While Laplace and the other reviewers voted to publish the paper, Lagrange
adamantly protested.
For nearly 50 years, Lagrange had insisted that such an
approach could not be used to represent signals with
corners
, i.e.,
discontinuous slopes, such as in square waves.
The Institut de France bowed
to the prestige of Lagrange, and rejected Fourier's work.
It was only after
Lagrange died that the paper was finally published, some 15 years later.
Luckily, Fourier had other things to keep him busy, political activities,
expeditions to Egypt with Napoleon, and trying to avoid the guillotine after the
French Revolution (literally!).
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View Full DocumentThe Scientist and Engineer's Guide to Digital Signal Processing
142
Sample number
0
4
8
12
16
40
20
0
20
40
60
80
DECOMPOSE
SYNTHESIZE
FIGURE 81a
(see facing page)
Amplitude
Who was right?
It's a split decision.
Lagrange was correct in his assertion that
a summation of sinusoids cannot form a signal with a corner.
However, you
can get
very
close.
So close that the difference between the two has
zero
energy
.
In this sense, Fourier was right, although 18th century science knew
little about the concept of energy.
This phenomenon now goes by the name:
Gibbs Effect
, and will be discussed in Chapter 11.
Figure 81 illustrates how a signal can be decomposed into sine and cosine
waves.
Figure (a) shows an example signal, 16 points long, running from
sample number 0 to 15.
Figure (b) shows the Fourier decomposition of this
signal, nine cosine waves and nine sine waves, each with a different
frequency and amplitude.
Although far from obvious, these 18 sinusoids
add to produce the
waveform in (a).
It should be noted that the objection
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 Summer '09

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