Welcome to this session of Fourier Transform. Let's start with simple definition. We say that the Fourier transform decomposes a signal into the frequencies that make it up. What does this mean? So imagine that you have a signal, let's take an example, we have a signal s1, right? And on the y axis here, you have the amplitude of the signal, on the x axis you have the time. So if you want to measure the strength of the signal, let's say at some specific time, you take a point here. And you look at the distance between the 0 on the y axis and that's your strength, that's the amplitude of the signal. If you look the same for another signal s2, let me pick another color. We select the same moment in time and we measure this distance. So in this case, this distance is 2 whatever it means. I haven't defined the scale of my axis is just the amplitude of the signal. And then for the 1 on the top, the distance is 1. Now what happens if you emit these two signals of the same time, right? So what happens if you get a signal s1 + s2 is for the same point in time, let me pick another color, at the same point in time. Your distance here will be the sum of these two distances. So because these two signals are being added together. So you have one from s1, which is here. And then you will have two from s2, which is here. So you end up with a total distance of three. So when you add signals together, you are just kind of summing these of waves. And the question now is, if I give you signal s1 + s2 only, can you recover the original signals s1 and s2? That are being joined up together to make the new signal, as s1 + s2. And that's what Fourier transform does. It takes a complex signal and it decomposes it to the frequencies that made it up, so these two signals, right? Now this example is very, very simple, but I think that I have the perfect tool to show you complex signals and Fourier transform. To do this, we would have to leave my PC for a moment and you have to join me in my living room. So here we are in front of my piano and we're here because the piano is essentially a signal generator. Each key on the piano generates a unique sound wave, because when the hammer hits the strings based on the thickness of the strings and also the tension. This inpart generates air pressure which is essentially a sound wave. And each key on the piano has its unique frequency. So the range of the piano goes from about I think 27 hertz here in the low end, and it goes all the way up to about 4200 hertz here in the high end. So what I like to do now, is I would like to pick a core which consist of different notes. So we have different frequencies mix together and see if we can actually recover the region of frequencies using Fourier transform. So if I pick a chord maybe this one, [SOUND] so this chord is comprised of three individual notes. So we have E flat, E natural, [SOUND] we have G sharp [SOUND] and we have C sharp. [SOUND] And when we press the keys together we get this lovely sound, this very nice chord. So what we would like to do is if we have the data only for this complex signal if we have the data just for the signal of the chord. [SOUND] Can we actually apply Fourier transform and recover the original notes in the chord which are A natural, G sharp, and C sharp? So what I will do is will record the chord. [SOUND] Okay, now save it, upload it to my PC, and then use Fourier transform and see if we can actually uncover these three individual nodes based on the complex signal all along. So let's walk back upstairs. So we're back at my PC. I have uploaded the file, and the first thing I will do is I will play it to you. So the file is here. It's called output wave and I will double-click it. So listen carefully. [SOUND] Okay, this is my recording. Now, how can I recover the original signals? What will the Fourier transform do for me? I want to show you this plot, and I'll try to do some explaining. We have three different original signals. Let's say this is your signal 1, s1. This is s2, and this is s3. And these signals here on the chart don't reflect the frequencies of the piano notes. This is just for illustration, but the idea is that you have three different signals. Each one generated by a piano key, and when we merge them together you get something like this. That's the signal of s1 + s2 + s3 and on the, sorry, s2 + s3, and on the y axis here you have amplitude, on the x axis you have the time, right? And the duration of my clip are something like 4 seconds, so we have 4 seconds here. What Fourier transform does, is it kind of moves us from the time domain to the frequency domain. So we will get a plot like this one where we have on the x axis different frequencies, right? And we said that the piano varies between something like 20 hertz and let's say I'm interested to look up to 4200 hertz. And then on the y axis, we will get the strength of the signal. So in this case, you can see we have three distinct signals. This is the strength, the amplitude of the first one. That's the one, With the lowest frequency, so that's why it comes first in the chart. That's s2, this signal is the in the middle between these two, in terms of frequency. So it come second on the chart because the frequency increases in this direction. And then s3 comes in the height of the bars, shows the amplitude of the signal. So this is what Fourier transform does, it helps us transition between the time and frequency domain. And actually, there is also the inverse Fourier transform which I'm not going to cover in this session, but just to let you know, it exist. So if you want to transition from the frequency to the time domain to something like this, what you need is IFT inverse Fourier transform.
