Negative frequency

I want to talk a little bit about one of the quirkier ideas in 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's what what does it mean to be a negative frequency could there be a sine wave that goes up and down at a at a rate of minus 10 cycles per second what what on earth does that mean or could there be a radio station of minus 680 kilohertz 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 equals 1/2 e to the plus J Omega T plus e to the minus J Omega T and each of these components each of these complex Exponential's here we can draw as a rotating complex number so for this first one we could actually draw it if we wanted to we could draw it like this we can draw a complex number out here in space and think of it as a rotating vector and 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 makes good sense if we are rotating in the positive direction like this you could say that's a positive frequency and if we're rotating in the negative direction like this you can 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 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 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 and then 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 don't it projects down to this point on the cosine curve and as we go farther and farther let's go straight down for a second that one projects two right there and as we come over here it projects to that point right there and 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 we've 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 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 and that projection after a little bit of time it goes to right here and then if we go over this way eventually rotate some more we'll project to this point here and 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 is happened on the left which is these points carve out a cosine wave just as we'd expect and 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 of rotating in opposite directions this one clearly had a positive frequency because it was going counterclockwise this one had a negative frequency because it's going clockwise and both of them carved out the exact same cosine wave when we got done so in the 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 and say a real-world V of T notice that the idea of negative frequency just sort of it melts away it evaporates it's not 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 of the signals but then just remember that when we go back up here and we look at the rotating vectors that it just means which way the vectors spinning