Complex exponentials spin

In the last video, we did a quick review of the exponential and what it means. Then we looked and figured out what the magnitude of an exponential is. The magnitude is equal to one. Now we’re going to look closely at this complex exponential as it represents a part of a cosine.

Now we're going to keep combining some of our ideas from the last couple of videos. You remember, one of the things we did was we used Euler's formula and turned it inside out. We developed an expression for cosine. So, if I say cosine of theta, I can say that equals 1/2 times e to the J theta plus e to the minus J theta. This is cosine of theta expressed as two separate exponentials.

Now we're going to take a really special step. I'm going to put in an argument right here of time. I'm going to say cosine of Omega T. T is time, and this shape here, this symbol here is the lowercase Omega from the Greek alphabet, and that's the frequency. So, time is in units of seconds, and Omega is a frequency, so it's in units of one/seconds. That's the units of frequency or per seconds, and when these multiply together, we get a dimensionless number right here.

We can take the cosine of a dimensionless number. So, what is this equal to? This equals 1/2 * e to the J Omega T plus e to the minus J Omega T. When we make T the argument of the cosine here, T is the stuff that keeps going up and up and up. The number T gets just gets bigger all the time, and so we ended up with a cosine waveform.

I'll just make a bad cosine looking thing here; that’s what a cosine looks like, and it keeps going and going and going. So, we have an idea of what a cosine wave looks like. The frequency determines how fast this goes up and down or how often it goes up and down. But now what I want to do is I want to look at a really special thing.

I want to look at what this thing right here, what is this thing e to the J Omega T? What we see is this cosine here is made of two of these things. So, whatever these things are, I can make a cosine out of them. Now we're going to look really carefully at e to the J Omega T.

What we just reviewed was that this is a complex number. Let's draw that complex number. So we're going to put a number out here; we know it falls on the unit circle. We know its angle is whatever is multiplying the J up in this exponent. Whatever is up in the exponent is the angle of this thing. So, this angle right here is Omega T, and we know the magnitude of this is, as we decided before, the magnitude is one. That's why it falls on the unit circle.

Okay, so now look at this. Here's this number T that's determining the angle, and that means what that means is that the angle is increasing with time. If time is equal to zero, the point is right here at time equals zero because the angle is zero. As time proceeds, the angle keeps starting to grow and grow, and it basically keeps growing. It keeps going; it comes back to here after Omega T equals 2 pi. Then what happens? It goes and keeps going around it again, and this basically goes along for as long as time goes along.

So here's this complex number moving along the unit circle in time over and over and over again. This is a number that is rotating. The number is rotating, so I can write here e to the J Omega T, and I know that because time's up here, I know it's rotating in time.

Now I'm going to put a different number on there. Let's put it over here, say, well, let's actually start this number at zero, and I'm going to call this number e to the minus J Omega T. What does that number look like? That's this guy here, that's this one here; we’ll make him orange. So, at time equals zero, it's e to the zero or one, just as we would expect.

Now, as time gets bigger, the angle, the thing multiplying J is minus Omega T, and so the angle is becoming more and more negative. So after a little bit of time, it's here, and after a little bit more time, it's here. What we notice is it keeps rotating this way. This is what happens when you have e to the minus J Omega T; you rotate in this direction, and it keeps going and going and going.

So these two numbers are pretty similar in behavior, except one rotates counterclockwise and the other rotates clockwise in our coordinate system, which is the complex plane. So in summary, if you see either of these shapes, e to the plus J Omega T or e to the minus J Omega T, what pops into your head is a number that spins.

For me, the simple idea is I have a number here, and I have a number here. They both spin in a complex space. To represent those in mathematical notation, I need this kind of notation here, which is a little bit awkward, but as I get used to it, e to the J Omega T is a spinning number, e to the minus J Omega T is a spinning number. This is an amazingly powerful idea, and we'll be able to describe every signal that happens using these kinds of terms.