Now before I talk about the mathematics of continuous wavelength transform, I would like to give you an example. I would like to actually use the transform on this specific signal, on this guy here. And show you the result so that you know what we are after. In a way, I would like to give you the intuition before we actually look at the internals. So, if I applied continuous wavelength transform to this signal, I will get a bunch of coefficients. And the best way to give you the intuition is to plot this coefficients on something called a scalogram. I need to plot that looks like this, and I'll try to explain it to you. It's kind of a heat map, right? But you will notice first that here we have a time component. If you remember in the Fourier transform, we only get frequency and signal strength. Here, we start with time. And if you go back to this signal, you will notice that it starts at 0, goes up to 10, and it's pretty much the same on the scalogram. It starts at 0 goes up to 1000 because of the sampling rate of 100. It shows this plot covers the entire signal. The coefficients of the transform cover the entire signal. Then up here we have the scale, and the scale is more very intuitive. And I'd like to think about the scale as inverse of the frequency. If it helps, you can think about the scale as a scale on a map, right? So if I use these screenshots from Google Maps, for example, this is a scale of 20 meters, right? And it shows you, for example, this is the IBM office at South Bank where I spent a lot of time, right here. This building here is the Waterloo Campus of Kings College where I also spent a lot of time. And this thing here is, well, I'm not exactly sure what it is. It could be a river, it could be a lake, I don't know. Now, if I zoom out a bit, and I set my scale to 500 meters, so my scale is bigger. I don't see the fine details anymore. You know the IBM office is somewhere around here. This is the Waterloo Campus. I can't see the details of the buildings, but I can definitely see that this here is actually the river Thames. Right, so, pretty much the same as on a map, the scale on the scaleogram shows you different details in different scales. And the lower the scale is, the higher frequencies you see on the plotted this region. So we have the time, we have the scale and the color represents the strength, the magnitude of the signal. So it goes up or down. And what can this plot tell us? Well, I see here that, in the low frequencies area, I have a signal that starts around 0. And then it goes up because of the color here. Then it comes back to 0, then it goes down, becomes negative. Then up again, down again, up again, down again and so on. And this kind of continues for the whole duration of the signal. And this information is actually this component in the signal. This guy, right? That goes up and down and it continues from the beginning of the signal to the end of the signal. If you look again at the scalogram, you see these guys here. And because they are in the top of the plot, this kind of means that they represent high frequencies because the scale here is low. And they appear at around 200 which on the other plot is 2. Right here, right. And what that represent, is these guys, these bursts in the signal. These spikes here, right? And we can see this by the intensity of the color. We have one sharp jump here. Then go back to negative. Then even higher jump in the amplitude. Then back down and so on. And the bottom line is the scalogram plugs the coefficients. But you can immediately see that now we're not confined to either the frequency or the time domain. We can have both. We can look at the frequencies and at the same time we can look at the strength of the signal in time. There is a catch, the catch is that the continuous where the transform gives us different resolutions and different scales. But I'm not going to go into these details. The idea is that we can have all the information in one place and this is very, very useful. And I want to actually show you another variant of this plot. I'm using the same coefficients but I will do this in 3D and you will see that it's even more interesting to kind of look at the signal from this perspective. Okay, so this is our signal in a 3D plot. Again, we have the Amplitude. We have the Scale. And we have the Time. And you can look at the plot now. And you can see that in the higher scale region, we have this continuous wave that goes through time. And it's always here. And then in the lower scales where we have high frequencies, we have this sharp berth, here. If we rotate this way, you will see them better, right. Okay, so these are our bursts and here we have the discomponent that goes through the entire signal that's always there. Or you can take a look from this perspective and that's the time, right? And you see something very similar to actually the time domain, right? You have this constant components on the background and then you have the sharp burst at the foreground. And these are the same coefficients that I used for the scalogram. So you can plot the coefficients in very interesting and exciting ways. Okay, so let's look at the continuous wavelength transform.