Welcome to Module 10. This is the end. This is the last model obviously and should show some perspective of where we can go with the stuff you have learned from this online class. So it has four parts. We're going to talk about courses that we give here at EPFL that follow up on the Basic Digital Signal Processing class. Then we'll talk about some research projects, where techniques that we have learned, are actually being used. We're also going to talk about a few start-ups that came out of research from the lab, and finish with acknowledgements. Module 10.1. What sort of classes can you take once you have mastered Digital Signal Processing basics? Well, there is a classical course called Statistical Signal Processing. We give, here, a class on Audio and Acoustic Signal Processing. We have a follow up course, which is more Mathematical on the Foundations of Signal Processing and we give a doctoral course on advanced topics. So the first classical class in on statistical signal processing. So why do we need statistical tools to process signals? Well, so far we have seen mostly deterministic signals. But most deterministic signals, if they are measured, will be hampered by noise. Second, signals change over time. So we need to adapt signal processing methods to changing conditions. And finally, to do estimation, so to find some information from a signal. So that's called optimal estimation, requires stochastic models and statistical techniques. What sort of problems can we address? So every communications problem that you can think of will need statistical signal processing. So you saw the Wi-Fi system in module 9. That's a typical example where noise is dominant and you need to have, statisfical methods to recover signals from noise. Then we have examples in biological signal processing, for example spikes working. And one classic other example is adaptive filtering for echo cancellation. In module 5.12 we saw reverberation and the inverse called dereverberation. And this is used in for example hands free telephone communications, and requires adaptive filtering. Let's look at just one image here. It's from biological signal processing. So there is a measurement here in the brain of a grasshopper, and it's to figure out when the olfactory system of the grasshopper is actually active. So there are some electrical probes here. Here are neural spikes coming out and you need to do two things which are statistical in nature. One is to classify the spikes, or if you blow up here's a signal you're going to see these neural spikes and they have different types. But if you classify correctly you can identify Certain characteristic. And the other one is that you have changing characteristic over time. You might have very low activity here, and some very high activity, so you would like to model this to figure out when a certain neuron is actually active. Okay, so they are two examples of statistical signal processing in the context of biological. So, the outline of the class, is that we start with basic models, then we look at these exemplary applications, for example, Wireless Transmission, Echo Cancellation and Spikes sorting. You can go to the website of the class here, and see all the details and the outline. A second class we're giving, which builds up on digital signal processing is one on signal processing for audio and acoustics. The objective is to understand acoustics, but also psychoacoustics, because signal processing for acoustics has to deal with a human perceptual system. So spatial hearing, for example, is very sophisticated and very important if you do multi-channel audio processing. Then, we want to understand manipulation processing of audio signals. Last but not least, understand state-of-the-art methods in audio signal processing, including on consumer-only audio. On the right side here we have a beautiful picture of a so-called spectrogram, a spectrogram is a local Fourier analysis over time. So you move, here is time, here is the spectrum, that changes over time. And here is a small piece of music, and you see all the harmonics and then we change the note. We have other harmonics, and so on. So this is the most basic signal processing for acoustics. But of course if you do it for multi-channels it's Becomes very complicated, and leads to sophisticated processing techniques. Here is an example done by people in the lab, on so called Auralization. So if a piece a music and you like to simulate its rendering, in different environments. So here's a rendering which simulate a concert hall, here's a rendering which simulate a classroom, here's a rendering would be for example, for multimedia system, and here's a rendering would be in an open public space. These spaces are all very, very different, and then if you want to predict how something sounds In a given environment then this is a perfect system. Okay the outline of the class, spatial hearing is the first important topic, and recording methodologies. Then multi-channel audio, in this class we always talk about the single signal but here in audio signal processing you can have dozens of channels or even hundreds of channels. Talking about spacial filtering, coding, which you're all familiar with MP3 which we discussed briefly as our much more sophisticated methods for multi channel audio. Last but not least auralization to do stimulation of acoustic environments like in the previous slide. Again, you have details on the website for the course. The next class we teach is called, Mathematical Foundations of Signal Processing. As we have seen in module six, the world is analog, but computation is on digital computers. So how do we go From the analog world, to, back to the analog world, using processing methodologies on computers. And the examples are audio as we've just seen, sensor networks that we are going to discuss in a minute, imaging, light on digital cameras, computer graphics, and so on. And the key mathematical concepts we have sort of Seen in this class at an elementary levels, so it's sampling and interpo-, sampling and interpolation or approximation and compression. Now, in this class we do this in much more detail. Okay, but the basic picture is the same as we saw in module six. You have an analog world here that we inhabit, and we have a digital world where we do the processing. So for example, questions are you want to build a sensor network. Here is EPFL campus. You want to measure temperature, how many sensors should you put on the campus to sense temperature accurately? And once you have sensed the temperature, how should you reconstruct? We're going to discuss this also in the research topic later on. But that's a basic question of signal processing. It's a sampling question. How many sensors? And an interpolation question, how do we reconstruct the spatial field of temperature over time? The course outline is, we do again, Hildred space geometry, but now in great detail, so if you didn't enjoy Module two, here in the current class. you may want to take this one but be ready for some much more difficult stuff, but beautiful stuff in my view. Then we discuss discrete-time systems and sequences, functions of continuous time and systems on continuous time, and then there is a big part on sampling, interpolation, and approximation. Finally we discuss some applications. This class is based on the textbook that has been mentioned in this class, also that is just coming out now, with Vetterli, Kovacevic and V. Goyal it's called Foundations of Signal Processing and is also available open access under this website. Last but not least, we give doctoral courses here and doctoral courses are really at the state of the art of what is being done in Signal Processing, what is being researched and published currently. And it's based on the fact that the classical approach as discuss in undergrad and masters level classes are sometimes limited and need to be extended. To do state-of-the-art signal processing research, this sometimes uses, quite sophisticated mathematical tools, that you also need to understand and maybe apply, or, modify for a signal processing problem. So here is an sample from a class like this it's a compressed sensing, it has been mentioned in the forum, compressed sensing is a very interesting technique where you try to acquire the analog world by taking very, very few samples in particle or meta and you reconstruct using regularization. So here's just a picture. I don't have time to really explain it. But it's a picture where you solve a linear system, essentially. But you solve it using different regularization. So we worked always with the l 2 norm. We measured the sum of squares of a sequence, for example. There are other norms that are possible. The l 1, that's the sum of absolute values, or the l infinity, that's the maximum value in sequence and if you regularize a problem you solve your linear system using these different norms you get very different solutions. And one of them happens to be very sparse so if you are looking for a solution that has very few non zero terms then L1 regularization which is what hes using compressed sensing will give you an interesting solution. We'll come back to this in the research projects, because it is actually used currently in the lab to solve some very interesting problems. Okay, so this Advanced Topics class actually moves from topic to topic, so sometimes it's on Fourier and wavelet processing, sometimes its on mathematical principles, sometimes it's on Simply reading groups on advanced topics. Again, there is a website, and there is volume 2 of Fourier and wavelets sequence here, which is the basis for the first version of this class. Okay. So that was an overview of the classes you could take if you were, for example, at EPFL. Lots of that material is actually online, so if you are interested, you can actually also learn it online from our website.