Negative frequency

I want to talk a little bit about one of the quirkier ideas in signal processing, and that's the idea of negative frequency. This is a phrase that may not initially make any sense at all. What does it mean to be a negative frequency? Could there be a sine wave that goes up and down at a rate of minus 10 cycles per second? What on earth does that mean? Or could there be a radio station of minus 680 kHz? What is that? That doesn't sound like it means anything.

There is a sense in which negative frequency is understandable, and we're going to just quickly talk about that here. You remember we can describe a cosine like this: cosine of Omega t equal 12 e+ J Omega t plus e to the minus J Omega t. Each of these components, each of these complex exponentials here, we can draw as a rotating complex number. So for this first one, we could actually draw it if we want to. We could draw a complex number out here in space and think of it as a rotating vector. That would be e to the J Omega T; this one has a plus sign.

Now the other thing, this term over here would look like a similar thing. It would be some vector in space—there's that number right there, e to the minus J Omega T—and this one rotating in the negative direction. So that means this is rotating this way.

So the idea of a negative frequency, when we talk about rotating vectors, makes good sense. If we are rotating in the positive direction like this, you could say that's a positive frequency. If we're rotating in the negative direction like this, you could say that's a negative frequency. So Omega T gives us the speed, and this sign right here and this sign right here give us the direction. The frequency here is plus Omega T, and the frequency here is minus Omega T. So in this sense of rotating vectors, negative frequency seems like a pretty simple idea.

Okay, so let's go where it's not a simple idea, and that's when we do this thing we did before, where we projected these vectors onto a cosine wave and spread out the time axis linearly, like this, going down the page for the cosine. Remember we did this: we drew a line here on this side; we're going to do plus Omega T, so we're going to plot e to the plus J Omega T here. This will be the real axis, this will be the imaginary axis, and down here, this will be the voltage axis, and this will be the time axis.

So when we start out at time equals zero, we have—we projected down here—and we got that value right there. As time goes on, if we tip our arrow up like that to that point, then we project down to the cosine curve right here. If we let our arrow go all the way to the other side, it projects like this down; it projects down to this point on the cosine curve.

As we go further and further—let's go straight down for a second—that one projects to right there, and then as we come over here, it projects to that point right there. Eventually, when we get back to home again, we get back to zero; the projection is right to this point here, and so we've, with one rotation, carved out one cycle of the cosine. So that seems pretty clear. Omega T is there; Omega T is down here.

Okay, so let's do it again, but this time, we'll go over on this side and we'll plot e to the minus J Omega T. All right, and so this again is the real axis, and this is the imaginary, and this is the voltage axis, and this is the time axis. So let's start out again; we're going straight sideways at time equals zero.

So let's project that down, and okay, that's pretty good; that's the same as before; we got the same V here. Now let's tip it down; let's say we rotate this way a little bit, and here's our new position of our vector. That projection, after a little bit of time, goes to right here, and then if we go over this way, eventually rotate some more, we'll project to this point here.

When we're straight sideways, it'll project to this peak right here, and you can see what's happening is basically the exact same thing is happening as happened on the left, which is these points carve out a cosine wave just as we'd expect. When this vector gets all the way back around to zero, we've done one cycle and we're back home, and now we're projecting to this point here.

What happened here is we took two different vectors rotating in opposite directions. This one clearly had a positive frequency because it was going counterclockwise; this one had a negative frequency because it was going clockwise. Both of them carved out the exact same cosine wave when we got done.

So in the vector world, where we're spinning vectors around, it seems very natural to talk about plus and minus frequency. But when we cast this back into, say, a real world V of T, notice that the idea of negative frequency just sort of melts away. It evaporates; it's not really there anymore, and it's been removed by this process of projection.

So when the idea of negative frequency comes up and it seems like it doesn't make sense in this time domain view of the signals, just remember that when we go back up here and we look at the rotating vectors, that it just means which way the vector is spinning.