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EEL 3135: DiscreteTime Signals and Systems
Dr. Fred J. Taylor, Professor
Lesson Title: Fourier Techniques
Lesson Number: 30
[Chap 111 to 11:4]
Background:
Brief Biography of Jean Baptiste Joseph Fourier (17681830)
Fourier was the son of a tailor and orphaned at the age of eight.
Educated at a nonsectarian military
college where he excelled in math, he considered entering the priesthood when the French Revolution
broke out.
Fourier joined the revolution which left France, in its aftermath, without a viable
intelligentsia.
Fourier himself had several close encounters with the guillotine.
The persecution of
scientists was finally abated through the intervention of Napoleon.
Napoleon, in fact, encouraged the
academic production scientists which benefited Fourier who, at the age of twentysix, was appointed
to the faculty of the new Ecole Normale.
Napoleon, in 1789, visited Egypt with a group of scientists including Fourier.
Fourier was appointed to
the new Institut d’Egypte which became famous for its work with the Rosetta stone.
Returning to
France in 1801, Fourier’s career advanced and retreated along with Napoleon’s exiles and along with
his several changes of political loyalties.
In 1807, while serving as the Prefect of Grenoble, Fourier
claimed the work that would make him famous.
His studies in heat transfer were collected in a paper
that introduced what we now would consider to be the foundations of orthogonal functions and the
Fourier series.
As was the custom of the time, papers were judged by the eminent scholars of the
day.
They were, in this case, Lagrange, Laplace, Legendre, and a fourth scholar.
They found the
paper innovative but lacking mathematical elegance and rigor.
Lagrange, however, did not believe
the results were correct and vigorously opposed its publication over the support of the other three
judges.
Fourier, unfortunately, could not provide a rigorous defense for his work (Dirichlet later
completed the proof).
The paper would wait fifteen more years to see print in the text
Théorie
analytique de la chaleur
.
In Chapter 11, Fourier transform techniques are introduced for continuoustime signals. The
continuoustime Fourier transform and its derivatives are intrinsic to the study of continuous
time signals.
The
continuoustime
Fourier transform
(CTFT) is defined in terms of the
analysis
equation
:
∫
∞
∞


=
dt
x(t)e
)
X(j
t
j
ϖ
1
and the
synthesis equation
:
∫
∞
∞

Ω
=
d
)e
X(j
x(t)
j
t
2
where
ϖ
is called the
analog frequency
and is measured in radians per second, and X(j
ϖ
) is
called the
spectrum
of x(t).
The Fourier transform, while be a fundamental scientific tool for
centuries, demonstrated a systemic weakness once we entered the digital computer era.
The
production the spectrum of x(t) is seen to require both a prior knowledge of the signal and its
integral over all time.
This is an unrealistic task to assign a digital computer from a
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 Spring '08
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