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Welcome to Module 10. 
This is the end.

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This is the last model obviously and 
should show some perspective of where we

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can go with the stuff you have learned 
from this online class.

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So it has four parts. 
We're going to talk about courses that we

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give here at EPFL that follow up on the 
Basic Digital Signal Processing class.

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Then we'll talk about some research 
projects, where techniques that we have

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learned, are actually being used. 
We're also going to talk about a few

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start-ups that came out of research from 
the lab, and finish with

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acknowledgements. 
Module 10.1.

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What sort of classes can you take once 
you have mastered Digital Signal

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Processing basics? 
Well, there is a classical course called

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Statistical Signal Processing. 
We give, here, a class on Audio and

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Acoustic Signal Processing. 
We have a follow up course, which is more

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Mathematical on the Foundations of Signal 
Processing and we give a doctoral course

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on advanced topics. 
So the first classical class in on

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statistical signal processing. 
So why do we need statistical tools to

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process signals? 
Well, so far we have seen mostly

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deterministic signals. 
But most deterministic signals, if they

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are measured, will be hampered by noise. 
Second, signals change over time.

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So we need to adapt signal processing 
methods to changing conditions.

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And finally, to do estimation, so to find 
some information from a signal.

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So that's called optimal estimation, 
requires stochastic models and

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statistical techniques. 
What sort of problems can we address?

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So every communications problem that you 
can think of will need statistical signal

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processing. 
So you saw the Wi-Fi system in module 9.

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That's a typical example where noise is 
dominant and you need to have,

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statisfical methods to recover signals 
from noise.

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Then we have examples in biological 
signal processing, for example spikes

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working. 
And one classic other example is adaptive

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filtering for echo cancellation. 
In module 5.12 we saw reverberation and

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the inverse called dereverberation. 
And this is used in for example hands

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free telephone communications, and 
requires adaptive filtering.

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Let's look at just one image here. 
It's from biological signal processing.

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So there is a measurement here in the 
brain of a grasshopper, and it's to

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figure out when the olfactory system of 
the grasshopper is actually active.

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So there are some electrical probes here. 
Here are neural spikes coming out and you

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need to do two things which are 
statistical in nature.

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One is to classify the spikes, or if you 
blow up here's a signal you're going to

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see these neural spikes and they have 
different types.

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But if you classify correctly you can 
identify Certain characteristic.

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And the other one is that you have 
changing characteristic over time.

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You might have very low activity here, 
and some very high activity, so you would

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like to model this to figure out when a 
certain neuron is actually active.

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Okay, so they are two examples of 
statistical signal processing in the

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context of biological. 
So, the outline of the class, is that we

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start with basic models, then we look at 
these exemplary applications, for

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example, Wireless Transmission, Echo 
Cancellation and Spikes sorting.

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You can go to the website of the class 
here, and see all the details and the

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outline. 
A second class we're giving, which builds

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up on digital signal processing is one on 
signal processing for audio and

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acoustics. 
The objective is to understand acoustics,

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but also psychoacoustics, because signal 
processing for acoustics has to deal with

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a human perceptual system. 
So spatial hearing, for example, is very

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sophisticated and very important if you 
do multi-channel audio processing.

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Then, we want to understand manipulation 
processing of audio signals.

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Last but not least, understand 
state-of-the-art methods in audio signal

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processing, including on consumer-only 
audio.

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On the right side here we have a 
beautiful picture of a so-called

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spectrogram, a spectrogram is a local 
Fourier analysis over time.

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So you move, here is time, here is the 
spectrum, that changes over time.

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And here is a small piece of music, and 
you see all the harmonics and then we

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change the note. 
We have other harmonics, and so on.

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So this is the most basic signal 
processing for acoustics.

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But of course if you do it for 
multi-channels it's Becomes very

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complicated, and leads to sophisticated 
processing techniques.

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Here is an example done by people in the 
lab, on so called Auralization.

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So if a piece a music and you like to 
simulate its rendering, in different

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environments. 
So here's a rendering which simulate a

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concert hall, here's a rendering which 
simulate a classroom, here's a rendering

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would be for example, for multimedia 
system, and here's a rendering would be

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in an open public space. 
These spaces are all very, very

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different, and then if you want to 
predict how something sounds In a given

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environment then this is a perfect 
system.

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Okay the outline of the class, spatial 
hearing is the first important topic, and

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recording methodologies. 
Then multi-channel audio, in this class

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we always talk about the single signal 
but here in audio signal processing you

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can have dozens of channels or even 
hundreds of channels.

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Talking about spacial filtering, coding, 
which you're all familiar with MP3 which

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we discussed briefly as our much more 
sophisticated methods for multi channel

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audio. 
Last but not least auralization to do

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stimulation of acoustic environments like 
in the previous slide.

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Again, you have details on the website 
for the course.

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The next class we teach is called, 
Mathematical Foundations of Signal

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Processing. 
As we have seen in module six, the world

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is analog, but computation is on digital 
computers.

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So how do we go From the analog world, 
to, back to the analog world, using

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processing methodologies on computers. 
And the examples are audio as we've just

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seen, sensor networks that we are 
going to discuss in a minute, imaging,

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light on digital cameras, computer 
graphics, and so on.

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And the key mathematical concepts we have 
sort of Seen in this class at an

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elementary levels, so it's sampling and 
interpo-, sampling and interpolation or

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approximation and compression. 
Now, in this class we do this in much

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more detail. 
Okay, but the basic picture is the same

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as we saw in module six. 
You have an analog world here that we

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inhabit, and we have a digital world 
where we do the processing.

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So for example, questions are you want to 
build a sensor network.

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Here is EPFL campus. 
You want to measure temperature, how many

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sensors should you put on the campus to 
sense temperature accurately?

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And once you have sensed the temperature, 
how should you reconstruct?

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We're going to discuss this also in the 
research topic later on.

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But that's a basic question of signal 
processing.

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It's a sampling question. 
How many sensors?

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And an interpolation question, how do we 
reconstruct the spatial field of

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temperature over time? 
The course outline is, we do again,

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Hildred space geometry, but now in great 
detail, so if you didn't enjoy Module

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two, here in the current class. 
you may want to take this one but be

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ready for some much more difficult stuff, 
but beautiful stuff in my view.

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Then we discuss discrete-time systems and 
sequences, functions of continuous time

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and systems on continuous time, and then 
there is a big part on sampling,

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interpolation, and approximation. 
Finally we discuss some applications.

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This class is based on the textbook that 
has been mentioned in this class, also

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that is just coming out now, with 
Vetterli, Kovacevic and V.

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Goyal it's called Foundations of Signal 
Processing and is also available open

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access under this website. 
Last but not least, we give doctoral

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courses here and doctoral courses are 
really at the state of the art of what is

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being done in Signal Processing, what is 
being researched and published currently.

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And it's based on the fact that the 
classical approach as discuss in

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undergrad and masters level classes are 
sometimes limited and need to be

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extended. 
To do state-of-the-art signal processing

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research, this sometimes uses, quite 
sophisticated mathematical tools, that

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you also need to understand and maybe 
apply, or, modify for a signal processing

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problem. 
So here is an sample from a class like

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this it's a compressed sensing, it has 
been mentioned in the forum, compressed

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sensing is a very interesting technique 
where you try to acquire the analog world

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by taking very, very few samples in 
particle or meta and you reconstruct

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using regularization. 
So here's just a picture.

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I don't have time to really explain it. 
But it's a picture where you solve a

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linear system, essentially. 
But you solve it using different

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regularization. 
So we worked always with the l 2 norm.

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We measured the sum of squares of a 
sequence, for example.

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There are other norms that are possible. 
The l 1, that's the sum of absolute

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values, or the l infinity, that's the 
maximum value in sequence and if you

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regularize a problem you solve your 
linear system using these different norms

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you get very different solutions. 
And one of them happens to be very sparse

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so if you are looking for a solution that 
has very few non zero terms then L1

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regularization which is what hes using 
compressed sensing will give you an

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interesting solution. 
We'll come back to this in the research

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projects, because it is actually used 
currently in the lab to solve some very

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interesting problems. 
Okay, so this Advanced Topics class

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actually moves from topic to topic, so 
sometimes it's on Fourier and wavelet

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processing, sometimes its on mathematical 
principles, sometimes it's on Simply

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reading groups on advanced topics. 
Again, there is a website, and there is

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volume 2 of Fourier and wavelets sequence 
here, which is the basis for the first

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version of this class. 
Okay.

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So that was an overview of the classes 
you could take if you were, for example,

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at EPFL. 
Lots of that material is actually online,

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so if you are interested, you can 
actually also learn it online from our

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website.