Welcome to Module 4 of Digital Signal
Processing.
This module is concerned with
understanding spectra and the Fourier
transform.
This is a fundamental tool that will be
used in, in this class ever after.
The idea of understanding spectra is as
old as science itself.
Ancient Greek mathematicians and
physicists, as well as Arab scientists,
were interested to understand why is
rainbow was actually having all its
beautiful colors.
But it took until the 14th century, until
the scientific experiment was done by a
Dominican monk Theodore from Freyberg who
held a bottle of water in the sun and
recognized the colors of rainbow,
Including the secondary rainbow.
This experiment was only an early step, it
took another 300 years until Descartes and
Newton gave a full explanation of how
white light can actually be decomposed
into the colors of the rainbow.
In particular, Newton proposed several
experiments to take a prism, decompose
white light into rainbow colors, take
another prism and recompose the colors of
the rainbow into white light.
And this is shown in a beautiful picture
here, that these are reenactment of
actually the Newton experiment.
Newton also proposed what is known as a
crucial experiment, which shows that pure
colors are actually icon functions of
optical elements, proving them once and
for all that white light is simply made up
of many, many fundamental frequencies.
And so this notion of spectrum, which is
very intuitive from the rainbow, is
actually the very notion that we are going
to study using the Fourier transform.
The main character of the current module
is Joseph Fourier, a 19th century
mathematician and physicist who lived in
France.
Joseph Fourier provided the mathematics to
really understand what Newton and
Descartes proposed as physicist.
And the way he did this is to propose to
the compose any well-behaved function into
a sum of harmonic sines and cosines.
How this could be done?
He wrote longhand in a famous book called,
Theorie analytique de La chaleur and you
can see it looks very, very complicated.
Don't be scared, by now we understand this
fully.
We have much more powerful notation and in
particular we have computers that will
compute the Fourier transform for us.
Nonetheless, the idea of Fourier is that
sines and cosines can be used to decompose
functions is still very fundamental in
many areas, and not only in signal
processing, but all applied sciences and
physics.
And, the reason Fourier was studying this
was a physical problem.
He wanted to understand the solution to
the heat equation.
And it turns out that sines and cosines
are eigenfunctions of the heat equation.
And therefore, represent the very natural
bases where to search a solution for this
partial differential equation.