In this video, we're going to talk about the Signals that are the most important for this course. we're going to see these signals over, and over, and over again. one you've already seen already, is the Sinusoid, and signals that are basically related to the sinusoid in some way. Well, today we're going to talk about a very different class of signals, they're pulse-like signals. These, you haven't seen in your calculus course. These are going to be interesting signals as you'll see. But probably the most important thing in this video, is the segment on how to think about signals. How to build them and how to think about them in a simpler Parts. That's going to be a very very important thing we're going to talk about. So here's our sinusoid, and we've seen it before, and as we've said it has an amplitude A which determines how big it is, so this negative Right down here as a size of -A, goes down to -A. The frequency that turns how often this reappears. This is a periodic signal. Which means it repeats, and it's certainly clear And if you hop over to the next peak, your going to get the same signal back again. What occurred here, is going to re-occur here, etc. Well the period we call capital T. And just a symbol that we'll use often for, what the period of a signal is. How is it related to f naut? And if you think about it for a second, the sine wave repeats, whenever f naut times t, is an integer. Which means that t has to be 1 over f0. So f0 again is the frequency in Hertz, and this formula gives t in seconds. So, hertz has units of inverse seconds if you will. And then finally there is the phase, which basically defines how this starts at the origin. So that value right there. Is A sine of T. So if T is 0, this thing starts at 0 of the origin and, therefore there's no, what's called, phase offset. Now I should point out, that if you have cosine, 2pie f naut t It's another sinusoid. That's equal to a phase shifted sine wave. When it's shifted by 90 degrees, pi over 2. You should verify that formula is correct. I hope I didn't make an error. I think it's right. and therefore, a sine wave can be written in terms of a phase shifted cosine, also. So. There's really no difference between this cosine and a sine, they're phase-shifted versions of each other. So this is what phase does, it moves things around in time. Now the next signal we're going to talk about is the complex exponential Yes, complex valued signals. So j here is the square root of -1 and this signal is incredibly important for us. It's incredibly important for all electrical engineers. This is another periodic signal. Its period is the same as the sinusoid we saw before and its period = 1 / f not or 1 / frequency. Now you can plot this in the complex plane. So this is the real part. This is the imaginary part. And it turns out, that it goes around in a circle. So, if you, The angle at any one time, is 2pi f0t, for everytime. So, as time increases, this angle increases, And this, complex exponential, looks like an arrow that goes around, and around, and around. Unit circle. And the time it takes it, to go, around once, is the period. That makes it periodic. Again I think it's pretty clear, again, that that period has to be such F nought times T is an integer. That makes the period one over F nought. Now I hope you all know Oiler's relationship Which says that a complex exponential actually consists of a cosine and a sine, so the cosine is the real part of the complex exponential. The sine is the imaginary part of the complex exponential. I cannot stress how important these signals are in our course. So in particular, the complex value nature of it is very very important to understand. So, if you're a little unfamiliar with complex numbers Oiler's formula. I suggest you look at the complex numbers video, which I've made. I hope it will help you, develop a better mathematical appreciation of complex numbers and complex valued signals. Okay. Let's go on to our pulses. pulses are going to be a little different that what you've seen in Calculus courses, because most of them are discountinuous. You probably shied away from discontinuous functions in your calculus course but pulse signals occur all the time. And perhaps the simplest signal is what we call the unit step = 0 for t less than 0, and then = 1 for t greater than 0. So it looks like a step, a step up, where it gets its name. Now you probably noticed right away, I didn't define it at the origin. And that's just the way it is. Its value at the origin is undefined. Turns out it doesn't matter what its value at the origin is. And you may not like that, but get used to it. It turns out it's going to occur all the time, and it turns out we don't really care what the value is at the origin. It just plain doesn't matter. And so another signal that has discontinuities is what we call the Unit pulse. By the way, the word unit here refers to the fact that the amplitude of this pulse, that now it jumps. The size of its continuity is 1. So the amplitude of this pulse, is 1, because it's a unit pulse. So, As the formula says, it's 0, for t less than 0. It's then 1 in between 0 and delta, and delta is called the pulse width. And, then it's zero again afterwards. So, it it pops up, stays constant then goes down and continues on, so that's a single pulse. We're going to discover the pulses used very frequently in digital communications. It's used to represent. It's, that's what makes it extremely important. And the next signal we're going to talk about is the square wave. So, a square wave is a periodic signal, it has a period of capital T as I've labeled it here, because you can see it repeats every capital T. Goes on forever. the amplitude of a square wave is, there is no convention for that, there is no, we don't really talk about the unit square wave. You explicitly have to give the amplitude, which I've just labeled here as A. So it has discontinuities and The origin T over 2, T, 3 T over 2. So it's discontinuous ever T over 2. And, what it's value is at those points, it doesn't matter, it's discontinuous. Just have to appreciate that as we go through the course. Alright, here comes the good stuff. Here comes the part that's very, very important. We have to build signals, what do we mean by that? The thing we have to first talk about, before we get into building signals, is what's called a signal delay. So, here, the delay, is tao, and I claim this signal, is delayed in time, by tao, so lets see how that works. The example I like to use is a unit step. Because this point of discontinuity gives us an handle, to latch on to where the origin is, in the signal. So, I'm going to pretend that my signal s is a unit step. So here it is, just u(t), just like we've seen in, earlier in this video. If I delay it, I claim it looks like this. So, let's see if we can figure this out. The discontinuity occurs where the argument of u is zero. So argument u is still a function. It's now 0 when T equals tau. That's where the discontinuity is. So this means that this formula right now, corresponds to a, to a step that's been delayed, shifted over in time, to tau. So U. Of, t plus tao, is called a signal advance. That means, this discontinuity, now occurs at -tao, wherever this argument is 0. So, we usually think of tao as being a positive number. So that the sign in the argument here tells you if it's the signal delay or an advance. Signal delay is going to be a very important thing in building signals as you're about to see. So let's have a little game. Let's see if you can tell me what this signal is. So we have here a unit step and a delayed unit step. It's delayed by delta. And I subtracted the two signals. What does it look like? Okay, well, let's try to plot it. We'll plot the pieces. So the unit step course looks like that. The delayed unit step is just as I shown you already. It looks like that, occurring at delta. But we have a minus sign over here, so what does that do? Well that inverts it, flips it over, so now it looks like that. So what's the sum of the red curve, and the, and the green curve? It's zero here, and all we get is the red guy up here, and then when they subtract, the red and the green cancel each other, and we get that. Ta-da. What we have, is the unit pulse. So, actually, you can think of the pulse As being a sum of the unit step and a delayed unit step by sum as the same as the differece here, there's a difference between a unit step and a delayed unit step and now decompose, The signal into what I claim are simpler pieces. I actually look at this pulse, and I see those two components when I look at it. Let's try an example here to see how this works for a little bit more complicated case. So This thinking about signals as a sum or difference of simpler signals is called signal superposition. This word superposition is going to occur again and again here, so superposition means just to add them up. 'Kay? Here's our example. How would you build this? How would you deconstruct it? How would you think? What are the pieces that you see in this signal? Okay. Well, what I see, is I see that big discontinuity. Right away, the first thing that jumps out is that it's discontinue. That tells me that's gotta be a unit step in here somewhere. And if we put in a unit step, negative unit step because it, the step is going down. If the origin, of course that's not the right place. We need to shift it over to 1, well, we've seen already. That's the play. So we just move it over. And now we get a discontinuity at the right place. Alright, that's good. But now, we want to, talk about this earlier part of the wave form, it looks like it goes up. Linearly, with a slope of one. How do we describe that? Well, we can describe that by what's called the ramp function. that's why I called it r(t) and I want to point out it's the integral of the unit step. So the ramp function is 0, 4t less than 0, and then it goes up with a slope of 1 forever and that's the same as this interval. Okay. So, now, what do we got? If we add together the green curve and the red curve? let's see we got this, in this range, always going to be 0 back there. There's no, red curve 0, so we get the ramp, and then you get a discontinuity, because of the, In steps which jumps down. The ramp is continuous, the unit step is discontinuous that causes the discontinuity when you add'em up. But it keeps on going forever because of the ramp. I now that's not quite what we want, we want something that cancels This ramp part, in this area, only in that area, what would you do? Think about that for a second. What signal could you add out there, or subtract, to make that part disappear? And I think you know what the answer is Hence it is a delayed ramp. So we now have R, loose signal is that delayed ramp and when you add them all together you get the black curve. And so our final expression for S of T. Is, that it's a sum, a superposition of L and 3g signals. It has a ramp, a ramp collade, and it also has a unit step. This is what I mean by signal decomposition. Signal super position is building signals, decomposition is thinking about the pieces. I would also point out that this ramp, is built from a unit step. It's the integral of it. So basically this signal s, although it looks a little screwy Turns out it's actually built, the pieces that bfuilt it are integrals and the, of the unit steps and the unit step itself. So the unit step is the key thing because in this signal, even though this early part of the signal doesn't look anything like the unit step and it turns out it lurch underneath inside and. That's kind of cool. We're going to use this over and over, and over again as we go through this course. Now okay, so let's review the important signals, are sinusoids and more importantly is the complex exponential it turns out. And the real and imaginary parts of the complex exponential, corresponding to the sinusoid. We've talked about the unit step. Discontinuous signals, are going to occur over and over again, both theoretically, and in practice. Pulses are used, to communicate bits, and as we've seen, a pulse can be considered as a superposition of unit step. Of unit step functions. So this construction in deconstructing signals is a sum of simpler signals. Oh, is this important. This is going to make your life easy in doing calculations. And I can't stress, it also helps you understand what is going on. We'll see this in the succeeding videos.