Digital Signal Processing is the branch of engineering that, in the space of just a few decades, has enabled unprecedented levels of interpersonal communication and of on-demand entertainment. By reworking the principles of electronics, telecommunication and computer science into a unifying paradigm, DSP is a the heart of the digital revolution that brought us CDs, DVDs, MP3 players, mobile phones and countless other devices. The goal, for students of this course, will be to learn the fundamentals of Digital Signal Processing from the ground up. Starting from the basic definition of a discrete-time signal, we will work our way through Fourier analysis, filter design, sampling, interpolation and quantization to build a DSP toolset complete enough to analyze a practical communication system in detail. Hands-on examples and demonstration will be routinely used to close the gap between theory and practice. To make the best of this class, it is recommended that you are proficient in basic calculus and linear algebra; several programming examples will be provided in the form of Python notebooks but you can use your favorite programming language to test the algorithms described in the course.
Signal of the Day: Exoplanet hunting
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Из курса от партнера École Polytechnique Fédérale de Lausanne
Digital Signal Processing
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Module 2: Vector Spaces
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Paolo PrandoniLecturer
School of Computer and Communication Science
Martin VetterliProfessor
School of Computer and Communication Sciences
Hello and welcome to this new installment of The Signal of the Day,
where we will be talking about Exoplanet Hunting.
As you would see, this is yet another cool application of Digital Signal Processing.
But before we proceed to the actual topic, we would like to acknowledge the help of
the KeyOps team from the University of Bern in Switzerland.
In particular, we like to thank and
who provided us with the datasets and helped us this Signal of the Day.
Let us start exploration of the topic, with this image of the Milky Way.
Our galaxy observed from Earth.
It contains between 200 and 400 billion stars,
out of which the sun, our star, is just one.
If we zoom out and look now at an image of the universe taken by the space telescope
Hubble, the universe contains an estimated number of
100 billion galaxies, each containing hundreds of billions of stars.
Hence the likelihood is very high that another star planet system
comparable to our solar system exists.
But how do we look for these so called exoplanets?
Given the immensity of the universe, and the number of stars
it's sounds like it's worse than actually looking for a needle in a haystack.
In comparison,
the presence of a star is much easier to detect as a star emits its own light.
Also the characteristics of this slide allow estimating a characteristic
about the star like its ranges, distance, or temperature at its surface.
For exoplanet we have to go one step further and use digital signal processing.
In a minute, we are going to study one method to find exoplanets,
but let us first make a short historical discretion.
20 years ago, Michelle Mayo and
Deirdre Kelo, from the University of Geneva, discovered the first exoplanet.
Deirdre Kelo, pictured here on the left,
was then a PhD student and his aim was not to hunt for exoplanets but
to measure very precisely the radial velocity of stars.
Collecting measurements and processing them using a special algorithm,
he noticed that one of the stars he was solving behaved in rather unusual way.
Unlike other stars the radial velocity of this star was varying strongly.
He thought originally this was created by a bug in his software.
But after a couple of months of further measurements and
double checking, he came up to the conclusion that this variation in velocity
was caused by the presence of a planet.
This first discovered exoplanet, named Pegasi 51,
has the radius of Jupiter with a revolution period of only 4.2 days.
That is 1,000 times smaller in comparison with Jupiter.
This went against existing beliefs about planets.
However, a couple of months later, the discovery was confirmed by measurements
from another team and since then, hundreds of other exoplanets have been discovered.
The original radial velocity method is quite involved and
suddenly beyond the scope every Signal of the Day.
We'll instead explain another method called the transit method.
Consider the following simplified model, where a planet of radius rp
revolves around a star of radius rs, here in red.
We also call d the distance between the planet and its star.
An observer, here on the left,
is located far away and observes the system.
For the sake of simplicity, we'll also assume the observer is located
in the same plane as the revolution plane of the planet around its star.
This diagram is just a top view of this simplified model.
Let us now illustrate what happens when the planet passes in front of the star,
as seen from the observer.
The top figure represent our simplified model from the point of view
of the observer, with the star in red and the planet in black.
Again, below we are going to plot the evolution over time
of the flux of light measured by the observer.
That is, the amount of light emitted by the star reaching this observer.
As the planet transits in front of it's star, that is it passes in front of it,
the observer measures the reduction of the flux of the light.
Indeed, the planet is partially masking surface of the star
in effect similar to what happens during an eclipse
It can be shown that the relative change in the flux is equal to the ratio
between the radius of the planet and the radius of the star to the square.
Indeed,the masking effect of the planet is proportional to its surface,
Irp square, and the amount of light emitted by a star
is also proportional to its surface by rs square.
This relative change of flux is called transit depth.
What happens now if we adopt a point of view of an alien
that tries to detect the presence of planets in our solar system.
In this image from the NASA, we perceive the relative size of the planets in
our solar system compared with the size of the sun on the left.
In the case of Earth, applying the formula for
the transit depth given before, you obtain a reduction of 0.1%.
In the case of Jupiter, which is roughly ten times bigger than Earth,
the flux reduction is of the order of 1%.
The best available telescope today has a capacity to
detect transit depths of the order of 0.1%.
So it has more chances to detect a planet the size of Jupiter, but
virtually none to detect planet the size of Earth.
Let us now look at an example of an actual transit data.
Scientists use either a space based or Earth telescope to observe a distant star.
And use a CCD camera to measure the amount of light,
technically the amount of photons, from the star that reaches the observer.
Whether space based or Earth based observations, scientists needs to
constantly realign the telescope to track the movement of the star.
This is a complex procedure but let's assume this is properly taken care of.
What we obtain is a set of images over a period of observations
like the one on the left here.
The measurement process is obviously not perfect and
there are many sources of noise that corrupt this image.
For example, the CCD sensor introduced various sorts of noise but
knowing the characteristic of the sensor, it is possible to remove these effects.
This is where digital signal processing comes into play.
The actual procedure is statistical, so
it's not covered by the material you studied in class.
You will need to follow on another course on statistical signal processing
to fully comprehend these matters.
But the main lesson here is that in the seemingly unrelated field,
digital signal processing plays a central role.
After signal processing methods have been applied,
we obtain a de-noised image, like the one represented on the right here.
From this series of images, we can extract the light curve,
such as this example measured by the CoRoT mission.
Notice that in this case, the transit depth is a approximately equal to 3%.
Looking now into a near future, CHEOPS is a European mission
managed by the University of Bern to build a space based telescope.
It will launch in 2017 and will be able to detect
exoplanet with a transient depth of 0.1%.
Who knows?
One day following this class, you will be able to join the CHEOPS team as
a digital signal processing expert and help them detecting a new exoplanet.
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