What is the Fourier Transform? ("Brilliant explanation!")

So

what

is the Fourier

transform

? I'm just going to give an intuitive

explanation

. There is another video on the channel that explains the actual Fourier

transform

equation, so I encourage you to check out the details in the link below this video. but intuitively, if you have a signal and it can be any signal, maybe my voice signal in time and we have measured it and we can have this shape, then you can represent it in terms of the time domain like this image here or you can represent it in another way. way and this is

what

is called a transform, so the Fourier transform is just an equivalent representation of the signal which happens to be, if it is in the time domain, then the representation is in the frequency domain and we use capital . for the signal in the frequency domain and a lower case in the time domain, that's a convention, so I'm going to draw some examples of these here and discuss some of them, give you a little more intuition, so what?

What do we mean by represent? in the frequency domain, well it turns out that any signal can be composed of sinusoidal components and that is the essence of the Fourier transform and what do I mean by that, let's take a look, let's say this signal that could be My voice would have a low pass characteristic, what do I mean by that? Well, I can only make sounds with my voice up to a certain frequency, so maybe I'll call that frequency bandwidth, that's the highest frequency component I can. I do it by trying to make the pitch of my voice higher and higher, so all the frequency components of my signals that I can output are at frequencies lower than that value.

Now here I am drawing a negative value and what do I mean by that? I'll talk about that in a minute and again there are more videos on the channel about what negative frequency is if that confuses you, but these two see that they are equivalent representations and we put here the Fourier transform that you can write this way with an equation that generate this signal or you can write one with an equation in terms of frequency and they are equivalent and that's why it's called transformation, you can transform in one direction, you can transform backwards, let's try and intuitively. understand a little more what we mean by this frequency domain, let's look at some examples, so here's an example of a cause, a waveform, okay, I'm going to take a cos wave, let's say yt is equal to cos of 2 pi f1 of t, so that's the waveform.

I can write a mathematical equation and I can plot this waveform. The cos waveform starts when time equals zero. It starts at one and oscillates like that. I think we are familiar with this and of course time is negative. Also, and this is the form of the cause, so it has a frequency component. I think we're familiar with the fact that it causes oscillations at a certain frequency and I've called this frequency one which is what I've drawn here, so in the frequency domain, hopefully, it's intuitive to think about that and think well, will only have a single particular waveform, will only have a single frequency component, let's say I draw here I can represent that by a delta function because a delta function is something that only exists at that exact value, so it only exists exactly at f1 and again I'll come back to explain why we have negative in a minute, but this would be y of capital f and of f and I think I hope it's It's intuitive to think that if you have a cos waveform that only has one frequency like it's at a single frequency, then in the frequency domain there would be no other frequency values, only the value at f1 and therefore, because there are infinitely close frequencies available to us in this continuous range of frequencies, this must be a delta function , okay, and if the height of this is one, then the height of these two, because we have positives and negatives, will be half and again i' We'll come back to more of that in a minute, but intuitively a cosway form with a frequency has one component plus its corresponding negative, what if we had another, say another shape of roadway, but with a higher frequency, well, this is just a The waveform looks like this in the time domain.

Of course, all of these peaks should be the same height. I'm just drawing it quickly here, but let's say this was a zt again. I can write it as a mathematical expression here in terms of cos of 2. pi f 2 t again mathematically especially in the time domain or I can represent it in the frequency domain with a math with a graph in ft, this is a higher frequency , so it's going to be higher than f1, so this is where f2 is and then of course, a match with a negative, which I'll get to in a minute f2 okay, now this is going to be z of f okay, so this is the transform of Fourier that we're transforming from the time domain to the frequency domain um okay, so uh Let's think now about what we would have if we had a continuum of all of these at different frequencies, so hopefully you can see that if we had different amounts of each of these different frequencies and many, many more at all the different frequencies, so all these delta functions. they could be infinitely close together, you would have the entire range of frequencies available and different amounts of all those frequencies, if you add them together in the right way, they will give you this signal here, so that is the essence of the Fourier transform. is that a signal can be made up of sinusoids and cause at different frequencies, if you add them up with the correct weights, you end up with the signal you have here, change the weighting slightly, you get a different signal and we can see it in the time domain or we can see this in the frequency domain with the amount of each of these frequencies being represented here.

If you add them all up, they will all come on one axis and you will have contributions. of multiple uh sinusoids with different weightings going together, they give you an overall picture in the frequency domain that matches, corresponds and is equivalent to this frequency, the main picture here, so let's understand these negatives and get more insight into how they all add up and so on, let's think about another frequency or sorry, another signal that's at the same frequency as this one here, so it's still at f2, but let's say it's a sine waveform, so let's think about that for a minute, so this is sinusoidal. of 2pi f2t, so each sine wave starts at 0 and goes up to 1 and has the same.

I'm trying to draw this with the same number of oscillations as the previous one, but it started at 0, so it has the same frequency as this one. one here, but it has a different phase and so what does that mean in our Fourier transform domain? Well, it's still going to be two peaks because it has the same frequency component and that's really what I've talked about so far. What am I doing? is missing in this image because these two things cannot be the same it cannot be that these two are the same image this is a Fourier transform it is a unique transformation so if you have gone from here from here to here if you go back here you would have to go back to this waveform, but if you can see here these two match, then something is missing and the answer is and that's why we need the negative, the answer is that at each frequency there are There are two different signals that can be orthogonal and they can be , are the sign and the cause, okay, and therefore we need two components in the frequency domain and this is really one of the main reasons why we need complex numbers.

Explore that a little bit more here just for a minute, so let's remember that and we'll use this as an example so sine of 2 pi ft 2 pi f 2 t can be written as in complex numbers 1 divided by 2 j from e to j 2 pi f 2 t minus e at least j 2 pi f 2 t so I hope that's something you remember from the basic complex numbers and their relationship to sinuses, so now let's think about this one, so here we have a positive result. a complex number with a positive frequency of f2, so it relates to this delta function here and this is a complex number with a negative frequency, so j 2 pi t, but if you plug in the negative frequency that gives us this component here, so these two complex numbers, this one relates to that delta function, this one relates to that delta function and so far we've been graphing, we hope you can realize that so far we've been graphing the magnitudes, but as it is a complex number, we also need to plot the phases, so let's look at that in this example just for a minute, so one over two j which is equal to negative half of j, just multiply the top and bottom uh by um for j and y and you get that result because the j gives you a j squared which is the negative uh and this is the same in polar coordinates so it's a good idea to think about that for a minute or this complex exponential is half times e to the power a minus j of pi in 2 because e a the minus j prime 2 is equal to j because this is equal to cos plus j sine and the cos of negative pi in 2 is 0 and the sine of negative pi in 2 is negative 1 and that gives you gives that minus in front, okay, so this is the number that's in front here, so the number that multiplies the positive frequency is this number here that has a phase of minus pi in two, so there is an important additional element here in our image, so now we realize that this was the magnitude and we should We put magnitude lines here now that we recognize this, uh, that there is a complex requirement, so this is the magnitude and now we also have to plot the phase, so I'm going to plot this for the phase of w f as a function. of f now for the positive frequency, the positive f2, we had this phase of minus pi in two, so there is a phase here of minus pi in two and for the negative frequency, well, this number here goes in with this negative and this negative comes to the front. of this this negative uh multiplying by half with the negative that rotates the phase in pi, so now we have positive pi in two for the corresponding to the negative frequency, okay, this is the important thing for a complex number, which is what that we have in the frequency domain because we are dealing with sinusoids and causes why two different sinusoids, two different waveforms occupy the same frequencies or exist at the same frequencies, therefore, we need their amplitudes and their phases, so what we need the complex representation of amplitudes and phases. and here we can see that there are now two graphs that correspond to each of these waveforms.

Here I have not drawn the phase for this one, but for the because, you can look at it and you can realize that the phase is zero for both, so if I were to draw the phase graph for z, it would be zero everywhere, you can confirm this using the same type of logic as here and the same for this frequency, there will also be a corresponding sinusoid and therefore there will be an amplitude and a phase for Those who go back to the Fourier transform now, hopefully, we understand this picture that they understand about the negative frequency, which is actually just this phase component to represent the phases also because of the cos and sine at each frequency and the transform is one way to represent. any time domain waveform, you could think of it as a sum of all the different sinusoids at different frequencies; that's a way of thinking that actually corresponds to the inverse Fourier transform equation or you could think of it as the sum of all the different frequency components that make up the signal and that correspond again to the Fourier transform equation.

I won't write the equations here. You can find the equations or look at the links in the details below this video to find the video about the

explanation

. of the Fourier transform equation. I hope this helped you understand more about the Fourier transform. If you liked it, like the video, like it. Help others call the video. Subscribe to the channel to watch more videos and see the link the website link in the information below where there is a complete categorized list of all the videos on the channel along with summary sheets summary sheets in pdf