How do Complex Numbers relate to Real Signals?

Well, let's explore how a

complex

number

relate

s to

real

signals

and to do this we're going to consider the circle, so here's a circle and we have. I'm drawing two horizontal axes and one vertical axes and what we can do is label. The point on the circle has a width distance and a radius. Well, for this example, let's call it 1, convert it to a unit of radius and then there will be an angle and now we're going to put the angle relative to the horizontal axis. If you can imagine yourself as this point, this point can move around the circle and imagine looking at this circle if you are on the paper in the plane of the paper and this was your eye and you were looking along that axis. and so if you're looking from there to here as the dot moves around the circle you have no concept of whether it's near or far from you, if you're just looking from that side of the view you don't know.

If you have any depth of field, all you're seeing is this point going up and down and you can probably think that if you're looking from this side here, you can imagine that as it moves, let's say from this point here to here, it's going to going from a negative value to a positive value and if this point was moving at a constant rate around the circle, then it's in the time it takes to go from here to there from this angle here since you're looking here. We're going to see it go from a pretty big negative number to a

real

ly big positive number now in the same period of time, as it moves from here to here from the point of view of your eyes, it just goes up a little bit and then goes down a little bit. little so that as it moves from here to here all you can see is it goes up and down in quantity, whereas when it made the same arc when it extended the same arc between down here and up here what we saw was that it changed quite a bit and It turns out that this is exactly a sine wave, so if I plot it as a function of theta, if we just looked with our side eye, then it will start, let's say it starts here, so this is our starting point, so when it starts here it starts with the value of 0 and as it moves around here if you look from the side all you see is it goes up, you can't see the depth you can only see it goes up then it reaches the top and then it goes back down as it moves, all you see it do is go up and down, up and down, and it goes up and down according to this sine wave, so it's actually sine. theta that we are familiar with in mathematics and that is the waveform and many waveforms in real life, the waveforms that I am generating as I speak now, sine waves, if we also imagine looking from the north Augen or direction from below, let's say if we were looking from below, we would see something very similar except it would start like this is the positive direction here, it would start as a big number, so when we are Looking from below or from above we would see it as a big number there and then as it moves around the circle we can't see the depth of field in the vertical direction, all we can see is it goes from here.

In the shot here it goes from a big number like who is down there, we only see it move there because we can't see the depth and then it will as it moves over here, we see it move there and then move back. on and so on so what we're seeing there is this waveform here and this waveform if this is the time dimension down here plotting with time as it moves and this waveform here I put it on its side this waveform here is a form that starts at the maximum value and then goes down, it is also a sine waveform, but it is a sine waveform that at 0 at a time is equal to 0 or a thick theta, sorry , it should say theta is equal to 0, it's starting at the maximum starting at 1 and we know that that waveform is cos theta, so if I'm plotting it with respect to theta, this is cos theta and we have sine theta, so that if we look in two orthogonal directions while this point moves around a circle, we would see a sine wave from one direction and from a 90 degree orthogonal direction we would see a cos wave.

Well, what is the mathematical expression for

complex

numbers

? So if we just think about conflict, these are real signs, let's think about complex

numbers

, if you just think about complex numbers for us a minute, the full complex number notation says we can write an equation for this point if this is the real one, so if we label this as real, we label this as imaginary, so this point here is the amplitude which in this case is 1 multiplied by e to the power of J, which is the complex number J e to the power of J theta, so this point is the point e to the power of J theta and has a magnitude of 1.

So how do we write a relationship between these real

signals

? that we know in the real world and we generate with anything that oscillates the signals in the power supply sinusoidal signals the signals that we generate by vibrating our voice are sinusoidal signals and many of them mixed and so on, but if we only have one sinusoid, what do we do? How do we

relate

it to this here? It's pretty simple, it's a complex number and we can find the real component, which is the distance here, and the imaginary component, which is the height up here, so the real component and the imaginary component. component okay, so e al J theta is equal to this is the real component here because this one here where we look from below shows us how far it is along this real axis, so this is cos theta plus on the imaginary axis that is this over here is J because it's on the imaginary sine theta axis, that's like the complex number that can sometimes seem pretty arbitrary, especially to engineers and engineering students and people who think about the real world and think about what things are like. mathematics of complex numbers.

It relates to the real world and the signs that we know, so we must go one step further because from the bottle you think and obtain another equation if we moved in the other direction instead of doing it this way, we moved from this way. then we would have e to negative J theta theta then he to negative J theta and what would that equal? Well, if we look from below, we are still starting here and we are only because we are coming towards us. or moving away from this direction we can't know the depth of field so as we move in the negative direction we will still see a cos waveform starting here and going down here with this waveform wouldn't change if this point moves down first here of course it would be the opposite because this way if this moves down we see it move in negative instead of positive so this is minus J sine theta so here are two equations to figure out what does this mean. are the complex numbers in terms of real things, these real waveforms, so let's go the other way, so if we add the left side to each other and we have the right side to each other, we could have e raised to J theta plus e Sorry , e at least J theta, so we add those two and now go ahead with these two, so we have 2 cos theta and these would cancel because this is the plus sign J and this is the minus sign J, so they would cancel. we have 2 cos so let's put the 2 on the other side and make that 1/2 and then we have cos theta so here is a real waveform that we know in the real world a cos waveform and this is how to write it in terms of complex numbers so that those complex numbers are useful, very useful in real world signals.

One more step to make it even more real is to think about what happens if this point, as we move, moves at exactly a constant speed. in time, so I've shown it here as a function of theta, so for any value of theta we can find what the value is here, but let's relate it to real time, that is, to time, and if we were to move at a constant speed. angular rate over here so theta we could replace theta with Omega T so T is time and Omega is the angular rate so for a large value of Omega you would be moving fast a small value of Omega you would be moving slow and then we can Plot our waveforms as a function of time and we would have different rates of change of our waveform, so this would be a one by one value of Omega.

For a higher value of Omega you could change more quickly and this is a higher frequency sine wave here and then for different values ​​of Omega you get different sine waves and therefore when you sir a person speaking in a low voice would have a low frequency, a person with a higher voice would have a higher frequency to show that image there more and so we have an equation now where we are replacing theta with Omega T and that is actually a real signal that you would say and if you put, for example, 50 Hertz or 60 Hertz in some countries for the power supply, if it were a power signal, a voltage signal in In PowerPoint, then we would have cos of Omega and there is a relationship of Omega with the actual frequencies that It is related by 2 pi.

F just goes to the angular frequency, these are real frequencies, so you could have a waveform here to represent the power waveform. like COS of 2pi multiplied by 60 if it is 60 hertz multiplied and that would be the waveform that is really the waveform of the power in the power supply and now simply replacing theta with 2pi ft here we have a way to write If it we reduce in complex numbers and in complex numbers we can use all the machinery and mathematical methods of using complex numbers to solve all kinds of problems and questions about what would happen to that power supply if we passed it through a circuit or through a capacitor or through of an inductor and solve all kinds of doubts that you would like to know about real electrical circuits with sinusoidal inputs