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Welcome to Module 4 of Digital Signal
Processing.

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This module is concerned with
understanding spectra and the Fourier

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transform.
This is a fundamental tool that will be

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used in, in this class ever after.
The idea of understanding spectra is as

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old as science itself.
Ancient Greek mathematicians and

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physicists, as well as Arab scientists,
were interested to understand why is

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rainbow was actually having all its
beautiful colors.

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But it took until the 14th century, until
the scientific experiment was done by a

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Dominican monk Theodore from Freyberg who
held a bottle of water in the sun and

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recognized the colors of rainbow,
Including the secondary rainbow.

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This experiment was only an early step, it
took another 300 years until Descartes and

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Newton gave a full explanation of how
white light can actually be decomposed

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into the colors of the rainbow.
In particular, Newton proposed several

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experiments to take a prism, decompose
white light into rainbow colors, take

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another prism and recompose the colors of
the rainbow into white light.

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And this is shown in a beautiful picture
here, that these are reenactment of

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actually the Newton experiment.
Newton also proposed what is known as a

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crucial experiment, which shows that pure
colors are actually icon functions of

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optical elements, proving them once and
for all that white light is simply made up

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of many, many fundamental frequencies.
And so this notion of spectrum, which is

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very intuitive from the rainbow, is
actually the very notion that we are going

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to study using the Fourier transform.
The main character of the current module

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is Joseph Fourier, a 19th century
mathematician and physicist who lived in

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France.
Joseph Fourier provided the mathematics to

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really understand what Newton and
Descartes proposed as physicist.

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And the way he did this is to propose to
the compose any well-behaved function into

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a sum of harmonic sines and cosines.
How this could be done?

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He wrote longhand in a famous book called,
Theorie analytique de La chaleur and you

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can see it looks very, very complicated.
Don't be scared, by now we understand this

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fully.
We have much more powerful notation and in

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particular we have computers that will
compute the Fourier transform for us.

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Nonetheless, the idea of Fourier is that
sines and cosines can be used to decompose

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functions is still very fundamental in
many areas, and not only in signal

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processing, but all applied sciences and
physics.

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And, the reason Fourier was studying this
was a physical problem.

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He wanted to understand the solution to
the heat equation.

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And it turns out that sines and cosines
are eigenfunctions of the heat equation.

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And therefore, represent the very natural
bases where to search a solution for this

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partial differential equation.