1 00:00:00,012 --> 00:00:04,835 In this video, we're going to talk about the Signals that are the most important 2 00:00:04,835 --> 00:00:08,497 for this course. we're going to see these signals over, 3 00:00:08,497 --> 00:00:12,969 and over, and over again. one you've already seen already, is the 4 00:00:12,969 --> 00:00:17,962 Sinusoid, and signals that are basically related to the sinusoid in some way. 5 00:00:17,962 --> 00:00:22,382 Well, today we're going to talk about a very different class of signals, they're 6 00:00:22,382 --> 00:00:25,777 pulse-like signals. These, you haven't seen in your calculus 7 00:00:25,777 --> 00:00:28,522 course. These are going to be interesting signals 8 00:00:28,522 --> 00:00:31,557 as you'll see. But probably the most important thing in 9 00:00:31,557 --> 00:00:34,702 this video, is the segment on how to think about signals. 10 00:00:34,702 --> 00:00:38,468 How to build them and how to think about them in a simpler Parts. 11 00:00:38,468 --> 00:00:44,676 That's going to be a very very important thing we're going to talk about. 12 00:00:44,676 --> 00:00:51,388 So here's our sinusoid, and we've seen it before, and as we've said it has an 13 00:00:51,388 --> 00:00:59,091 amplitude A which determines how big it is, so this negative Right down here as a 14 00:00:59,091 --> 00:01:07,136 size of -A, goes down to -A. The frequency that turns how often this 15 00:01:07,136 --> 00:01:11,784 reappears. This is a periodic signal. 16 00:01:11,784 --> 00:01:20,065 Which means it repeats, and it's certainly clear And if you hop over to 17 00:01:20,065 --> 00:01:26,736 the next peak, your going to get the same signal back again. 18 00:01:26,736 --> 00:01:32,583 What occurred here, is going to re-occur here, etc. 19 00:01:32,583 --> 00:01:40,442 Well the period we call capital T. And just a symbol that we'll use often 20 00:01:40,442 --> 00:01:47,837 for, what the period of a signal is. How is it related to f naut? And if you 21 00:01:47,837 --> 00:01:55,352 think about it for a second, the sine wave repeats, whenever f naut times t, is 22 00:01:55,352 --> 00:02:02,957 an integer. Which means that t has to be 1 over f0. 23 00:02:02,957 --> 00:02:15,582 So f0 again is the frequency in Hertz, and this formula gives t in seconds. 24 00:02:15,582 --> 00:02:21,594 So, hertz has units of inverse seconds if you will. 25 00:02:21,594 --> 00:02:31,106 And then finally there is the phase, which basically defines how this starts 26 00:02:31,106 --> 00:02:35,982 at the origin. So that value right there. 27 00:02:35,982 --> 00:02:43,739 Is A sine of T. So if T is 0, this thing starts at 0 of 28 00:02:43,739 --> 00:02:51,100 the origin and, therefore there's no, what's called, phase offset. 29 00:02:51,100 --> 00:02:59,745 Now I should point out, that if you have cosine, 2pie f naut t It's another 30 00:02:59,745 --> 00:03:07,107 sinusoid. That's equal to a phase shifted sine 31 00:03:07,107 --> 00:03:14,336 wave. When it's shifted by 90 degrees, pi over 32 00:03:14,336 --> 00:03:17,922 2. You should verify that formula is 33 00:03:17,922 --> 00:03:21,532 correct. I hope I didn't make an error. 34 00:03:21,532 --> 00:03:27,497 I think it's right. and therefore, a sine wave can be written 35 00:03:27,497 --> 00:03:31,802 in terms of a phase shifted cosine, also. So. 36 00:03:31,802 --> 00:03:38,517 There's really no difference between this cosine and a sine, they're phase-shifted 37 00:03:38,517 --> 00:03:43,277 versions of each other. So this is what phase does, it moves 38 00:03:43,277 --> 00:03:47,987 things around in time. Now the next signal we're going to talk 39 00:03:47,987 --> 00:03:56,407 about is the complex exponential Yes, complex valued signals. 40 00:03:56,407 --> 00:04:11,443 So j here is the square root of -1 and this signal is incredibly important for 41 00:04:11,443 --> 00:04:15,342 us. It's incredibly important for all 42 00:04:15,342 --> 00:04:20,607 electrical engineers. This is another periodic signal. 43 00:04:20,607 --> 00:04:27,722 Its period is the same as the sinusoid we saw before and its period = 1 / f not or 44 00:04:27,722 --> 00:04:32,327 1 / frequency. Now you can plot this in the complex 45 00:04:32,327 --> 00:04:37,057 plane. So this is the real part. 46 00:04:37,057 --> 00:04:47,742 This is the imaginary part. And it turns out, that it goes around in 47 00:04:47,742 --> 00:04:51,742 a circle. So, if you, 48 00:04:51,742 --> 00:05:00,624 The angle at any one time, is 2pi f0t, for everytime. 49 00:05:00,624 --> 00:05:11,467 So, as time increases, this angle increases, And this, complex exponential, 50 00:05:11,467 --> 00:05:17,172 looks like an arrow that goes around, and around, and around. 51 00:05:17,172 --> 00:05:21,932 Unit circle. And the time it takes it, to go, around 52 00:05:21,932 --> 00:05:26,102 once, is the period. That makes it periodic. 53 00:05:26,102 --> 00:05:34,417 Again I think it's pretty clear, again, that that period has to be such F nought 54 00:05:34,417 --> 00:05:40,963 times T is an integer. That makes the period one over F nought. 55 00:05:40,963 --> 00:05:48,291 Now I hope you all know Oiler's relationship Which says that a complex 56 00:05:48,291 --> 00:05:56,101 exponential actually consists of a cosine and a sine, so the cosine is the real 57 00:05:56,101 --> 00:06:03,063 part of the complex exponential. The sine is the imaginary part of the 58 00:06:03,063 --> 00:06:08,272 complex exponential. I cannot stress how important these 59 00:06:08,272 --> 00:06:13,225 signals are in our course. So in particular, the complex value 60 00:06:13,225 --> 00:06:17,530 nature of it is very very important to understand. 61 00:06:17,530 --> 00:06:25,187 So, if you're a little unfamiliar with complex numbers Oiler's formula. 62 00:06:25,187 --> 00:06:30,289 I suggest you look at the complex numbers video, which I've made. 63 00:06:30,289 --> 00:06:36,447 I hope it will help you, develop a better mathematical appreciation of complex 64 00:06:36,447 --> 00:06:39,815 numbers and complex valued signals. Okay. 65 00:06:39,815 --> 00:06:45,152 Let's go on to our pulses. pulses are going to be a little different 66 00:06:45,152 --> 00:06:50,237 that what you've seen in Calculus courses, because most of them are 67 00:06:50,237 --> 00:06:53,727 discountinuous. You probably shied away from 68 00:06:53,727 --> 00:06:59,807 discontinuous functions in your calculus course but pulse signals occur all the 69 00:06:59,807 --> 00:07:04,058 time. And perhaps the simplest signal is what 70 00:07:04,058 --> 00:07:10,946 we call the unit step = 0 for t less than 0, and then = 1 for t greater than 0. 71 00:07:10,946 --> 00:07:16,293 So it looks like a step, a step up, where it gets its name. 72 00:07:16,293 --> 00:07:22,912 Now you probably noticed right away, I didn't define it at the origin. 73 00:07:22,912 --> 00:07:27,932 And that's just the way it is. Its value at the origin is undefined. 74 00:07:27,932 --> 00:07:32,373 Turns out it doesn't matter what its value at the origin is. 75 00:07:32,373 --> 00:07:35,722 And you may not like that, but get used to it. 76 00:07:35,722 --> 00:07:41,271 It turns out it's going to occur all the time, and it turns out we don't really 77 00:07:41,271 --> 00:07:46,462 care what the value is at the origin. It just plain doesn't matter. 78 00:07:46,462 --> 00:07:54,280 And so another signal that has discontinuities is what we call the Unit 79 00:07:54,280 --> 00:07:59,406 pulse. By the way, the word unit here refers to 80 00:07:59,406 --> 00:08:05,904 the fact that the amplitude of this pulse, that now it jumps. 81 00:08:05,904 --> 00:08:13,102 The size of its continuity is 1. So the amplitude of this pulse, is 1, 82 00:08:13,102 --> 00:08:19,324 because it's a unit pulse. So, As the formula says, it's 0, for t 83 00:08:19,324 --> 00:08:24,225 less than 0. It's then 1 in between 0 and delta, and 84 00:08:24,225 --> 00:08:31,209 delta is called the pulse width. And, then it's zero again afterwards. 85 00:08:31,209 --> 00:08:38,173 So, it it pops up, stays constant then goes down and continues on, so that's a 86 00:08:38,173 --> 00:08:42,827 single pulse. We're going to discover the pulses used 87 00:08:42,827 --> 00:08:46,891 very frequently in digital communications. 88 00:08:46,891 --> 00:08:52,262 It's used to represent. It's, that's what makes it extremely 89 00:08:52,262 --> 00:08:56,502 important. And the next signal we're going to talk 90 00:08:56,502 --> 00:09:01,937 about is the square wave. So, a square wave is a periodic signal, 91 00:09:01,937 --> 00:09:07,707 it has a period of capital T as I've labeled it here, because you can see it 92 00:09:07,707 --> 00:09:11,328 repeats every capital T. Goes on forever. 93 00:09:11,328 --> 00:09:17,905 the amplitude of a square wave is, there is no convention for that, there is no, 94 00:09:17,905 --> 00:09:21,715 we don't really talk about the unit square wave. 95 00:09:21,715 --> 00:09:27,380 You explicitly have to give the amplitude, which I've just labeled here 96 00:09:27,380 --> 00:09:31,200 as A. So it has discontinuities and The origin 97 00:09:31,200 --> 00:09:36,469 T over 2, T, 3 T over 2. So it's discontinuous ever T over 2. 98 00:09:36,469 --> 00:09:43,728 And, what it's value is at those points, it doesn't matter, it's discontinuous. 99 00:09:43,728 --> 00:09:48,962 Just have to appreciate that as we go through the course. 100 00:09:48,962 --> 00:09:54,120 Alright, here comes the good stuff. Here comes the part that's very, very 101 00:09:54,120 --> 00:09:57,774 important. We have to build signals, what do we mean 102 00:09:57,774 --> 00:10:02,957 by that? The thing we have to first talk about, before we get into building 103 00:10:02,957 --> 00:10:10,451 signals, is what's called a signal delay. So, here, the delay, is tao, and I claim 104 00:10:10,451 --> 00:10:17,886 this signal, is delayed in time, by tao, so lets see how that works. 105 00:10:17,886 --> 00:10:25,502 The example I like to use is a unit step. Because this point of discontinuity gives 106 00:10:25,502 --> 00:10:30,567 us an handle, to latch on to where the origin is, in the signal. 107 00:10:30,567 --> 00:10:34,572 So, I'm going to pretend that my signal s is a unit step. 108 00:10:34,572 --> 00:10:40,152 So here it is, just u(t), just like we've seen in, earlier in this video. 109 00:10:40,152 --> 00:10:44,973 If I delay it, I claim it looks like this. 110 00:10:44,973 --> 00:10:53,867 So, let's see if we can figure this out. The discontinuity occurs where the 111 00:10:53,867 --> 00:11:00,522 argument of u is zero. So argument u is still a function. 112 00:11:00,522 --> 00:11:07,444 It's now 0 when T equals tau. That's where the discontinuity is. 113 00:11:07,444 --> 00:11:15,681 So this means that this formula right now, corresponds to a, to a step that's 114 00:11:15,681 --> 00:11:20,420 been delayed, shifted over in time, to tau. 115 00:11:20,420 --> 00:11:24,632 So U. Of, t plus tao, is called a signal 116 00:11:24,632 --> 00:11:29,537 advance. That means, this discontinuity, now 117 00:11:29,537 --> 00:11:34,227 occurs at -tao, wherever this argument is 0. 118 00:11:34,227 --> 00:11:40,062 So, we usually think of tao as being a positive number. 119 00:11:40,062 --> 00:11:46,253 So that the sign in the argument here tells you if it's the signal delay or an 120 00:11:46,253 --> 00:11:49,758 advance. Signal delay is going to be a very 121 00:11:49,758 --> 00:11:54,683 important thing in building signals as you're about to see. 122 00:11:54,683 --> 00:12:00,012 So let's have a little game. Let's see if you can tell me what this 123 00:12:00,012 --> 00:12:04,928 signal is. So we have here a unit step and a delayed 124 00:12:04,928 --> 00:12:08,292 unit step. It's delayed by delta. 125 00:12:08,292 --> 00:12:15,748 And I subtracted the two signals. What does it look like? Okay, well, let's 126 00:12:15,748 --> 00:12:19,542 try to plot it. We'll plot the pieces. 127 00:12:19,542 --> 00:12:30,717 So the unit step course looks like that. The delayed unit step is just as I shown 128 00:12:30,717 --> 00:12:38,342 you already. It looks like that, occurring at delta. 129 00:12:38,342 --> 00:12:47,095 But we have a minus sign over here, so what does that do? Well that inverts it, 130 00:12:47,095 --> 00:12:56,004 flips it over, so now it looks like that. So what's the sum of the red curve, and 131 00:12:56,004 --> 00:13:03,901 the, and the green curve? It's zero here, and all we get is the red guy up here, 132 00:13:03,901 --> 00:13:10,721 and then when they subtract, the red and the green cancel each other, and we get 133 00:13:10,721 --> 00:13:11,899 that. Ta-da. 134 00:13:11,899 --> 00:13:18,232 What we have, is the unit pulse. So, actually, you can think of the pulse 135 00:13:18,232 --> 00:13:27,085 As being a sum of the unit step and a delayed unit step by sum as the same as 136 00:13:27,085 --> 00:13:36,832 the differece here, there's a difference between a unit step and a delayed unit 137 00:13:36,832 --> 00:13:43,562 step and now decompose, The signal into what I claim are simpler pieces. 138 00:13:43,562 --> 00:13:49,477 I actually look at this pulse, and I see those two components when I look at it. 139 00:13:49,477 --> 00:13:55,487 Let's try an example here to see how this works for a little bit more complicated 140 00:13:55,487 --> 00:13:59,307 case. So This thinking about signals as a sum 141 00:13:59,307 --> 00:14:04,877 or difference of simpler signals is called signal superposition. 142 00:14:04,877 --> 00:14:11,472 This word superposition is going to occur again and again here, so superposition 143 00:14:11,472 --> 00:14:16,062 means just to add them up. 'Kay? Here's our example. 144 00:14:16,062 --> 00:14:23,681 How would you build this? How would you deconstruct it? How would you think? What 145 00:14:23,681 --> 00:14:28,261 are the pieces that you see in this signal? Okay. 146 00:14:28,261 --> 00:14:33,032 Well, what I see, is I see that big discontinuity. 147 00:14:33,032 --> 00:14:38,284 Right away, the first thing that jumps out is that it's discontinue. 148 00:14:38,284 --> 00:14:42,823 That tells me that's gotta be a unit step in here somewhere. 149 00:14:42,823 --> 00:14:48,594 And if we put in a unit step, negative unit step because it, the step is going 150 00:14:48,594 --> 00:14:52,051 down. If the origin, of course that's not the 151 00:14:52,051 --> 00:14:55,755 right place. We need to shift it over to 1, well, 152 00:14:55,755 --> 00:14:58,247 we've seen already. That's the play. 153 00:14:59,447 --> 00:15:05,212 So we just move it over. And now we get a discontinuity at the 154 00:15:05,212 --> 00:15:08,712 right place. Alright, that's good. 155 00:15:08,712 --> 00:15:16,277 But now, we want to, talk about this earlier part of the wave form, it looks 156 00:15:16,277 --> 00:15:22,542 like it goes up. Linearly, with a slope of one. 157 00:15:22,542 --> 00:15:34,181 How do we describe that? Well, we can describe that by what's called the ramp 158 00:15:34,181 --> 00:15:39,897 function. that's why I called it r(t) and I want to 159 00:15:39,897 --> 00:15:44,032 point out it's the integral of the unit step. 160 00:15:44,032 --> 00:15:50,987 So the ramp function is 0, 4t less than 0, and then it goes up with a slope of 1 161 00:15:50,987 --> 00:15:55,332 forever and that's the same as this interval. 162 00:15:55,332 --> 00:15:58,720 Okay. So, now, what do we got? If we add 163 00:15:58,720 --> 00:16:05,025 together the green curve and the red curve? let's see we got this, in this 164 00:16:05,025 --> 00:16:11,970 range, always going to be 0 back there. There's no, red curve 0, so we get the 165 00:16:11,970 --> 00:16:18,973 ramp, and then you get a discontinuity, because of the, In steps which jumps 166 00:16:18,973 --> 00:16:22,782 down. The ramp is continuous, the unit step is 167 00:16:22,782 --> 00:16:28,147 discontinuous that causes the discontinuity when you add'em up. 168 00:16:28,147 --> 00:16:32,235 But it keeps on going forever because of the ramp. 169 00:16:32,235 --> 00:16:38,722 I now that's not quite what we want, we want something that cancels This ramp 170 00:16:38,722 --> 00:16:46,607 part, in this area, only in that area, what would you do? Think about that for a 171 00:16:46,607 --> 00:16:51,367 second. What signal could you add out there, or 172 00:16:51,367 --> 00:16:59,192 subtract, to make that part disappear? And I think you know what the answer is 173 00:16:59,192 --> 00:17:06,377 Hence it is a delayed ramp. So we now have R, loose signal is that 174 00:17:06,377 --> 00:17:14,422 delayed ramp and when you add them all together you get the black curve. 175 00:17:14,422 --> 00:17:24,419 And so our final expression for S of T. Is, that it's a sum, a superposition of L 176 00:17:24,419 --> 00:17:31,717 and 3g signals. It has a ramp, a ramp collade, and it 177 00:17:31,717 --> 00:17:38,482 also has a unit step. This is what I mean by signal 178 00:17:38,482 --> 00:17:43,509 decomposition. Signal super position is building 179 00:17:43,509 --> 00:17:47,782 signals, decomposition is thinking about the pieces. 180 00:17:47,782 --> 00:17:52,758 I would also point out that this ramp, is built from a unit step. 181 00:17:52,758 --> 00:17:57,772 It's the integral of it. So basically this signal s, although it 182 00:17:57,772 --> 00:18:02,953 looks a little screwy Turns out it's actually built, the pieces that bfuilt it 183 00:18:02,953 --> 00:18:07,000 are integrals and the, of the unit steps and the unit step itself. 184 00:18:07,000 --> 00:18:11,573 So the unit step is the key thing because in this signal, even though this early 185 00:18:11,573 --> 00:18:16,325 part of the signal doesn't look anything like the unit step and it turns out it 186 00:18:16,325 --> 00:18:20,068 lurch underneath inside and. That's kind of cool. 187 00:18:20,068 --> 00:18:26,326 We're going to use this over and over, and over again as we go through this 188 00:18:26,326 --> 00:18:30,616 course. Now okay, so let's review the important 189 00:18:30,616 --> 00:18:37,279 signals, are sinusoids and more importantly is the complex exponential it 190 00:18:37,279 --> 00:18:40,942 turns out. And the real and imaginary parts of the 191 00:18:40,942 --> 00:18:44,592 complex exponential, corresponding to the sinusoid. 192 00:18:44,592 --> 00:18:49,567 We've talked about the unit step. Discontinuous signals, are going to occur 193 00:18:49,567 --> 00:18:53,542 over and over again, both theoretically, and in practice. 194 00:18:53,542 --> 00:18:59,092 Pulses are used, to communicate bits, and as we've seen, a pulse can be considered 195 00:18:59,092 --> 00:19:03,107 as a superposition of unit step. Of unit step functions. 196 00:19:03,107 --> 00:19:08,552 So this construction in deconstructing signals is a sum of simpler signals. 197 00:19:08,552 --> 00:19:12,742 Oh, is this important. This is going to make your life easy in 198 00:19:12,742 --> 00:19:16,882 doing calculations. And I can't stress, it also helps you 199 00:19:16,882 --> 00:19:21,755 understand what is going on. We'll see this in the succeeding videos.