1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,280 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,280 --> 00:00:18,450 at ocw.mit.edu. 8 00:00:21,970 --> 00:00:24,790 DENNIS FREEMAN: So today the idea 9 00:00:24,790 --> 00:00:27,880 is to think about CT systems in exactly the same way 10 00:00:27,880 --> 00:00:31,852 that we thought about DT systems last time. 11 00:00:31,852 --> 00:00:33,310 For the past couple of weeks, we've 12 00:00:33,310 --> 00:00:36,320 been looking at many representations of both CT 13 00:00:36,320 --> 00:00:37,360 and DT systems. 14 00:00:37,360 --> 00:00:42,250 Last lecture we made a picture of a bunch of different ways 15 00:00:42,250 --> 00:00:43,960 to think about DT systems and we tried 16 00:00:43,960 --> 00:00:47,610 to think about relations between them. 17 00:00:47,610 --> 00:00:49,690 The new thing from last time was this idea 18 00:00:49,690 --> 00:00:53,200 of a z transform, which formally is the link between a system 19 00:00:53,200 --> 00:00:56,380 function, a function of z, and a unit sample 20 00:00:56,380 --> 00:01:00,849 response, a function of n. 21 00:01:00,849 --> 00:01:02,640 We showed last time that because we already 22 00:01:02,640 --> 00:01:04,800 understand a whole lot of these other connections 23 00:01:04,800 --> 00:01:08,292 that connection is very straightforward. 24 00:01:08,292 --> 00:01:10,000 What we're going to do today is precisely 25 00:01:10,000 --> 00:01:13,150 the same thing for CT. 26 00:01:13,150 --> 00:01:15,559 The boxes barely changed. 27 00:01:15,559 --> 00:01:17,350 You should see the relationship immediately 28 00:01:17,350 --> 00:01:19,600 between the former set of boxes and this set of boxes. 29 00:01:23,200 --> 00:01:25,750 And we'll do a very similar thing. 30 00:01:25,750 --> 00:01:27,790 We'll look at a relationship between the system 31 00:01:27,790 --> 00:01:30,280 function, which is a function of s, 32 00:01:30,280 --> 00:01:33,820 and the impulse response, which is a function of t. 33 00:01:33,820 --> 00:01:35,980 We'll look at that similar function for CT, 34 00:01:35,980 --> 00:01:39,100 the thing that's analogous to the z transform in DT. 35 00:01:39,100 --> 00:01:42,790 So Laplace transform, just like in DT 36 00:01:42,790 --> 00:01:45,360 where it maps a function of time to a function of z, 37 00:01:45,360 --> 00:01:48,790 here it maps a function of time, which in CT we'll 38 00:01:48,790 --> 00:01:49,840 write that way. 39 00:01:49,840 --> 00:01:54,100 It maps that to a function of s. 40 00:01:54,100 --> 00:02:01,564 So s is going to be something like x of t 41 00:02:01,564 --> 00:02:04,500 e to the minus st dt. 42 00:02:04,500 --> 00:02:07,000 So the idea is going to be that this was a function of time. 43 00:02:07,000 --> 00:02:08,465 This is only a function of s. 44 00:02:08,465 --> 00:02:10,090 We get the function of s by integrating 45 00:02:10,090 --> 00:02:12,610 some function of time, but we integrate time out 46 00:02:12,610 --> 00:02:14,390 so time disappears. 47 00:02:14,390 --> 00:02:15,820 So the idea then is that we end up 48 00:02:15,820 --> 00:02:21,940 with a map linking a function of time to a function of s. 49 00:02:21,940 --> 00:02:24,080 Presumably you've seen this before. 50 00:02:24,080 --> 00:02:27,860 This is a topic in 1803. 51 00:02:27,860 --> 00:02:29,060 Smile. 52 00:02:29,060 --> 00:02:29,900 Nod your head. 53 00:02:29,900 --> 00:02:32,090 Make me feel like I'm connecting to you, right. 54 00:02:32,090 --> 00:02:33,960 Right, so you've all seen this before. 55 00:02:33,960 --> 00:02:35,180 Nothing's new. 56 00:02:35,180 --> 00:02:37,640 Actually there is one tiny thing that's new. 57 00:02:37,640 --> 00:02:39,770 In 1803 they use a variant of the Laplace 58 00:02:39,770 --> 00:02:42,800 transform that we will call the unilateral Laplace 59 00:02:42,800 --> 00:02:45,485 transform, which means they started their integrals at 0. 60 00:02:47,990 --> 00:02:50,930 For reasons that will be clear at the very end of the course, 61 00:02:50,930 --> 00:02:55,200 when we do Fourier transforms to make 62 00:02:55,200 --> 00:02:57,840 the transition between this kind of a transform and a Fourier 63 00:02:57,840 --> 00:03:01,200 transform easier we will do something 64 00:03:01,200 --> 00:03:03,930 called a bilateral transform. 65 00:03:03,930 --> 00:03:06,540 The only difference being that we will start our integrals 66 00:03:06,540 --> 00:03:09,040 at minus infinity. 67 00:03:09,040 --> 00:03:13,650 There are subtle differences between those two transforms. 68 00:03:13,650 --> 00:03:18,360 Rest assured, that the big picture is identical. 69 00:03:18,360 --> 00:03:20,880 And when there is an important difference 70 00:03:20,880 --> 00:03:23,740 we will point that out. 71 00:03:23,740 --> 00:03:27,030 So for the rest of today and for the rest of two weeks 72 00:03:27,030 --> 00:03:29,910 I will only talk about bilateral transforms, the kind 73 00:03:29,910 --> 00:03:31,710 that we integrate over all of time. 74 00:03:34,602 --> 00:03:36,310 And the easiest way I know to get started 75 00:03:36,310 --> 00:03:38,050 is to just try some, right. 76 00:03:38,050 --> 00:03:40,570 You could study properties of the math. 77 00:03:40,570 --> 00:03:42,750 That's probably the way it was studied in 1803. 78 00:03:42,750 --> 00:03:47,700 I find it easier to develop some intuition for what's going on 79 00:03:47,700 --> 00:03:50,760 here by just doing some. 80 00:03:50,760 --> 00:03:52,500 So here's probably the simplest Laplace 81 00:03:52,500 --> 00:03:55,140 transform we'll look at. 82 00:03:55,140 --> 00:03:59,220 What's the Laplace transform of a function of time 83 00:03:59,220 --> 00:04:02,370 that exponentially decays with positive time in a 0 84 00:04:02,370 --> 00:04:06,000 for time less than 0? 85 00:04:06,000 --> 00:04:09,000 As you might expect, the first transform we'll do 86 00:04:09,000 --> 00:04:12,420 involve simply plugging that mathematical formula 87 00:04:12,420 --> 00:04:14,890 into the definition. 88 00:04:14,890 --> 00:04:18,240 So the x of s, the Laplace transform, 89 00:04:18,240 --> 00:04:20,910 is always the function of time integrated against e 90 00:04:20,910 --> 00:04:23,370 to the minus st. 91 00:04:23,370 --> 00:04:25,680 We're only doing bilateral but notice 92 00:04:25,680 --> 00:04:29,460 that the 0 for t less than 0 cuts off the bottom. 93 00:04:29,460 --> 00:04:31,230 So in fact, you would get the same answer 94 00:04:31,230 --> 00:04:35,760 if you did bilateral or unilateral. 95 00:04:35,760 --> 00:04:40,170 And then we're left with running this integral 96 00:04:40,170 --> 00:04:43,950 over infinite time this kind of a function, which comes out 97 00:04:43,950 --> 00:04:45,880 looking like this. 98 00:04:45,880 --> 00:04:49,650 The only tricky thing is thinking about the implications 99 00:04:49,650 --> 00:04:53,080 of that infinity thing. 100 00:04:53,080 --> 00:04:56,550 There's going to be certain values of s for which 101 00:04:56,550 --> 00:05:00,180 that integral diverges. 102 00:05:00,180 --> 00:05:03,140 So to think about that, think about the thing 103 00:05:03,140 --> 00:05:04,480 that we're integrating. 104 00:05:04,480 --> 00:05:09,480 We're integrating something that looks like e to the pt. 105 00:05:09,480 --> 00:05:12,620 We're integrating against some function. 106 00:05:12,620 --> 00:05:16,240 So define the original function of time is e to the minus t, 107 00:05:16,240 --> 00:05:17,990 here we have e to the minus st. Let's just 108 00:05:17,990 --> 00:05:22,970 focus on that one for a moment, the e to the st thing. 109 00:05:22,970 --> 00:05:25,280 We're going to generally be worried about values of s, 110 00:05:25,280 --> 00:05:28,550 or here, values of p, that have complex values. 111 00:05:28,550 --> 00:05:30,800 We're going to be looking at the s plane just the same 112 00:05:30,800 --> 00:05:32,665 as last time we looked at the z plane. 113 00:05:32,665 --> 00:05:34,790 So we're going to think about this thing might have 114 00:05:34,790 --> 00:05:36,155 a real and imaginary component. 115 00:05:36,155 --> 00:05:38,155 So we might have something like looks like this. 116 00:05:41,860 --> 00:05:45,280 Sigma would be the real part of p. 117 00:05:45,280 --> 00:05:48,790 Omega would be the imaginary part of p. 118 00:05:48,790 --> 00:05:52,420 And somebody want to hazard a guess 119 00:05:52,420 --> 00:05:55,720 at what that might look like if I tried to expand it 120 00:05:55,720 --> 00:05:58,880 as a real and imaginary part? 121 00:05:58,880 --> 00:06:01,360 Someone wanted to hazard a guess at the name 122 00:06:01,360 --> 00:06:02,920 of the equation I might use? 123 00:06:05,430 --> 00:06:06,760 AUDIENCE: Euler. 124 00:06:06,760 --> 00:06:08,700 DENNIS FREEMAN: Euler's equation, sure. 125 00:06:08,700 --> 00:06:10,500 So I would expand this by Euler's equation 126 00:06:10,500 --> 00:06:12,840 and I get something of the form e to the sigma t, 127 00:06:12,840 --> 00:06:17,460 cos omega t plus j sine omega t. 128 00:06:20,547 --> 00:06:22,880 The point of writing that out is that I 129 00:06:22,880 --> 00:06:25,220 hope it's clear that convergence of the integral 130 00:06:25,220 --> 00:06:29,765 is going to depend critically on the real or the imaginary part? 131 00:06:29,765 --> 00:06:32,890 Real, right? 132 00:06:32,890 --> 00:06:35,169 So the convergence is going to be determined by this. 133 00:06:35,169 --> 00:06:37,210 This is the thing that's affecting the magnitude. 134 00:06:37,210 --> 00:06:39,251 That's the thing that's getting smaller or bigger 135 00:06:39,251 --> 00:06:42,980 as I go to infinity or minus infinity. 136 00:06:42,980 --> 00:06:48,330 The cosine term, the sine term, those are oscillating. 137 00:06:48,330 --> 00:06:53,460 They don't make the function be more or less convergent 138 00:06:53,460 --> 00:06:55,867 than it was or originally. 139 00:06:55,867 --> 00:06:57,450 So when I'm thinking about convergence 140 00:06:57,450 --> 00:07:00,240 I'm going to be thinking about the real part. 141 00:07:00,240 --> 00:07:04,320 So over here if I want to make this integral converge 142 00:07:04,320 --> 00:07:07,305 I want to be thinking about t going from 0 to infinity. 143 00:07:07,305 --> 00:07:08,730 Well, 0 is not a problem. 144 00:07:08,730 --> 00:07:11,530 e to the 0 is 1, that's going to work. 145 00:07:11,530 --> 00:07:13,420 It's the infinity part that's a problem. 146 00:07:13,420 --> 00:07:17,160 So if I have infinity plugged in for t, 147 00:07:17,160 --> 00:07:21,060 what's the range of values for s for plus 1 that makes sense? 148 00:07:21,060 --> 00:07:24,300 That's the important question, right. 149 00:07:24,300 --> 00:07:26,010 So I'm going to want to constrain 150 00:07:26,010 --> 00:07:31,770 the real part of that number, s plus 1, to be bigger than 0. 151 00:07:31,770 --> 00:07:34,320 If the real part of that number is bigger than 0, 152 00:07:34,320 --> 00:07:36,300 then this thing will converge to 0. 153 00:07:39,560 --> 00:07:42,180 If that converges to 0 then I have a simple answer, 154 00:07:42,180 --> 00:07:43,940 it's just the answer in the bottom. 155 00:07:43,940 --> 00:07:46,190 It's in the bottom so I have to put a minus sign, that 156 00:07:46,190 --> 00:07:47,340 kills that minus sign. 157 00:07:47,340 --> 00:07:50,000 So my answer is 1 over s plus 1. 158 00:07:50,000 --> 00:07:51,230 Easy, right? 159 00:07:51,230 --> 00:07:54,680 And we will say that the Laplace transform has a functional 160 00:07:54,680 --> 00:07:56,322 form, 1 over s plus 1. 161 00:07:56,322 --> 00:07:58,280 And it has region of convergence, the real part 162 00:07:58,280 --> 00:07:59,485 of s bigger then minus 1. 163 00:07:59,485 --> 00:08:02,030 Just solving that inequality for the real part of s so 164 00:08:02,030 --> 00:08:04,190 I can draw it on an s plane. 165 00:08:04,190 --> 00:08:06,590 So we'll associate then with this 166 00:08:06,590 --> 00:08:11,480 s transform a picture which we will commonly call the pole 0 167 00:08:11,480 --> 00:08:12,630 pattern. 168 00:08:12,630 --> 00:08:16,350 Is just a picture of the s plane which 169 00:08:16,350 --> 00:08:20,890 shows me the singularities, the poles and zeroes of the Laplace 170 00:08:20,890 --> 00:08:23,469 transform and the region of convergence. 171 00:08:23,469 --> 00:08:24,510 So there's a single pole. 172 00:08:24,510 --> 00:08:27,110 There's a pole at s equals a minus 1, 173 00:08:27,110 --> 00:08:29,940 so I indicate that by the x. 174 00:08:29,940 --> 00:08:32,880 And I have region of convergence which I'll illustrate here 175 00:08:32,880 --> 00:08:35,650 with the gray area. 176 00:08:35,650 --> 00:08:39,480 So it converges for all values of s that are in the gray area. 177 00:08:39,480 --> 00:08:41,010 OK, easy right? 178 00:08:41,010 --> 00:08:43,140 Trivial. 179 00:08:43,140 --> 00:08:46,400 OK, so now you get to talk to your neighbor. 180 00:08:46,400 --> 00:08:47,700 OK, turn to your neighbor. 181 00:08:47,700 --> 00:08:49,110 Say, hi. 182 00:08:49,110 --> 00:08:51,030 [INTERPOSING VOICES] 183 00:08:51,030 --> 00:08:53,190 And figure out the Laplace transform 184 00:08:53,190 --> 00:08:56,090 of a slightly more, but not very complicated, function. 185 00:09:01,080 --> 00:09:04,573 [INTERPOSING VOICES] 186 00:10:53,060 --> 00:10:54,999 DENNIS FREEMAN: So what's the answer? 187 00:10:54,999 --> 00:10:57,290 Everybody raise your hands, show some number of fingers 188 00:10:57,290 --> 00:11:01,700 equal to the number of the correct solution. 189 00:11:01,700 --> 00:11:04,620 Wonderful. 190 00:11:04,620 --> 00:11:08,000 The universal answer-- I think it was universal-- was 1. 191 00:11:08,000 --> 00:11:09,210 So how do you get 1? 192 00:11:09,210 --> 00:11:12,057 What do I do? 193 00:11:12,057 --> 00:11:14,390 How do I find the Laplace transform of the top function? 194 00:11:14,390 --> 00:11:17,680 [INAUDIBLE] 195 00:11:21,450 --> 00:11:22,900 DENNIS FREEMAN: Precisely. 196 00:11:22,900 --> 00:11:27,060 So what I do is just stick it in the formula, do it twice. 197 00:11:27,060 --> 00:11:30,030 Laplace transform, like the z transform, is linear. 198 00:11:30,030 --> 00:11:32,400 It will turn out that the Laplace of a sum 199 00:11:32,400 --> 00:11:33,585 is the sum of the Laplaces. 200 00:11:39,190 --> 00:11:44,142 If I simply stick the expression for x2 into the definition, 201 00:11:44,142 --> 00:11:45,725 you can see it splits into two pieces. 202 00:11:48,320 --> 00:11:50,090 And I get two parts. 203 00:11:50,090 --> 00:11:52,520 1 over s plus 1 just like I got-- 204 00:11:52,520 --> 00:11:56,820 so the first part looks just like x1, the first example. 205 00:11:56,820 --> 00:11:58,340 The second one looks almost the same 206 00:11:58,340 --> 00:12:00,980 except there's a 2 where there used to be 1. 207 00:12:00,980 --> 00:12:06,050 Shockingly, it changes one of the ones into a 2. 208 00:12:06,050 --> 00:12:09,570 And then the only issue is where is the region of convergence. 209 00:12:09,570 --> 00:12:13,160 So this part converges if the real part of s 210 00:12:13,160 --> 00:12:15,758 is bigger than minus 1. 211 00:12:15,758 --> 00:12:21,030 This converges if the real part of s is bigger than minus 2. 212 00:12:21,030 --> 00:12:22,950 So they both converge if the real part of s 213 00:12:22,950 --> 00:12:24,740 is bigger than minus? 214 00:12:24,740 --> 00:12:26,000 AUDIENCE: 1. 215 00:12:26,000 --> 00:12:29,110 DENNIS FREEMAN: 1, right. 216 00:12:29,110 --> 00:12:32,680 If I'm in the region of s where both the first part 217 00:12:32,680 --> 00:12:37,449 and the second part converges then I'm fine. 218 00:12:37,449 --> 00:12:38,990 So that was all very straightforward. 219 00:12:38,990 --> 00:12:44,020 I get an answer that was number one, OK. 220 00:12:44,020 --> 00:12:48,050 So the transform is just the sum of those two pieces. 221 00:12:48,050 --> 00:12:49,750 And the region is the region of overlap. 222 00:12:54,450 --> 00:12:55,980 A more interesting case, and where 223 00:12:55,980 --> 00:12:57,390 things are a little bit different when 224 00:12:57,390 --> 00:12:59,306 we think about the bilateral Laplace transform 225 00:12:59,306 --> 00:13:01,890 than unilateral, is that the regions 226 00:13:01,890 --> 00:13:05,850 can be more complicated. 227 00:13:05,850 --> 00:13:12,020 So here's an example where I've got a backwards traveling 228 00:13:12,020 --> 00:13:13,710 function. 229 00:13:13,710 --> 00:13:15,980 So now the function only exists for t less than 0. 230 00:13:20,059 --> 00:13:22,100 As you might expect, just stick it in the formula 231 00:13:22,100 --> 00:13:24,240 and see what happens. 232 00:13:24,240 --> 00:13:27,800 So stick this expression into here. 233 00:13:27,800 --> 00:13:32,580 Now it lips off the top of the integral. 234 00:13:32,580 --> 00:13:34,330 But when you're grinding through integrals 235 00:13:34,330 --> 00:13:35,880 it's hard to tell the difference. 236 00:13:35,880 --> 00:13:39,090 You get something that looks very much the same. 237 00:13:39,090 --> 00:13:41,730 The thing that is different, notice that the function 238 00:13:41,730 --> 00:13:42,960 was upside down. 239 00:13:42,960 --> 00:13:45,720 It was minus something rather than plus something 240 00:13:45,720 --> 00:13:48,600 like the first example was. 241 00:13:48,600 --> 00:13:52,110 So the first example was e to the minus t, t bigger than 0. 242 00:13:52,110 --> 00:13:55,470 This example is minus e to the minus t, t less than 0. 243 00:13:58,540 --> 00:14:05,700 That gave me a minus sign here but the nontrivial limit 244 00:14:05,700 --> 00:14:08,001 is at the top rather than at the bottom. 245 00:14:08,001 --> 00:14:09,750 That's the reason I put the minus sign in. 246 00:14:09,750 --> 00:14:11,333 So it would kill the other minus sign. 247 00:14:14,180 --> 00:14:20,530 Also, since I need convergence at t equals minus infinity, 248 00:14:20,530 --> 00:14:23,870 the region flips. 249 00:14:23,870 --> 00:14:27,830 I still need to have that sort of thing decay 250 00:14:27,830 --> 00:14:30,920 so that the integral exists. 251 00:14:30,920 --> 00:14:34,620 But now the functions are flipped in time. 252 00:14:34,620 --> 00:14:40,920 So the important range of s is now flipped. 253 00:14:40,920 --> 00:14:46,980 So now I get the same functional form exactly, but a region 254 00:14:46,980 --> 00:14:50,119 that's on the other side. 255 00:14:50,119 --> 00:14:51,660 That's one of the central differences 256 00:14:51,660 --> 00:14:57,560 between the bilateral and the unilateral transform. 257 00:14:57,560 --> 00:14:59,540 I need to tell you the region of convergence 258 00:14:59,540 --> 00:15:02,572 for you to know which of those two functions 259 00:15:02,572 --> 00:15:03,530 that I'm talking about. 260 00:15:07,400 --> 00:15:10,260 OK, so the important thing from this example 261 00:15:10,260 --> 00:15:13,920 is the functional form looks exactly the same. 262 00:15:13,920 --> 00:15:16,290 I had a functional form in time, e to the minus t, 263 00:15:16,290 --> 00:15:18,390 e to the minus t. 264 00:15:18,390 --> 00:15:21,000 It was that functional form of time 265 00:15:21,000 --> 00:15:24,132 that I crank through the integration for, 266 00:15:24,132 --> 00:15:25,590 and it's not surprising then that I 267 00:15:25,590 --> 00:15:29,762 get a functional form for the s transform that looks the same. 268 00:15:29,762 --> 00:15:31,720 If you start with functional forms in time that 269 00:15:31,720 --> 00:15:34,875 look the same, you get functional forms in s 270 00:15:34,875 --> 00:15:37,160 that look the same. 271 00:15:37,160 --> 00:15:40,420 However, since the region of t was 272 00:15:40,420 --> 00:15:45,990 different for the two functions the region of s is different. 273 00:15:45,990 --> 00:15:47,400 There's also the negative sign. 274 00:15:47,400 --> 00:15:52,190 There's the fact that this flipped this way, 275 00:15:52,190 --> 00:15:55,220 and that's related to the fact that my region of integration 276 00:15:55,220 --> 00:15:59,060 extends from minus infinity to 0 versus 0 to infinity. 277 00:15:59,060 --> 00:16:04,160 The important limit flipped from being the bottom limit 278 00:16:04,160 --> 00:16:06,020 to the top limit. 279 00:16:06,020 --> 00:16:07,380 OK, that's roughly it. 280 00:16:10,610 --> 00:16:12,960 So just to make sure that everybody is with me, 281 00:16:12,960 --> 00:16:15,112 what's the Laplace transform of this symmetric 282 00:16:15,112 --> 00:16:15,820 looking function? 283 00:17:50,670 --> 00:17:53,147 So which of those functions is the Laplace transform 284 00:17:53,147 --> 00:17:54,230 of the symmetric function? 285 00:17:57,390 --> 00:18:00,750 It's about 3 calories to raise your hand, right? 286 00:18:00,750 --> 00:18:03,420 Good exercise, makes you breathe. 287 00:18:03,420 --> 00:18:06,130 The answer is? 288 00:18:06,130 --> 00:18:07,110 Oh, come on. 289 00:18:10,540 --> 00:18:14,960 OK, it's more like 90% correct. 290 00:18:14,960 --> 00:18:17,540 So not everybody has it right. 291 00:18:17,540 --> 00:18:19,330 What am I going to do? 292 00:18:19,330 --> 00:18:21,330 It's just like the last one, right? 293 00:18:21,330 --> 00:18:22,640 Just stick it in. 294 00:18:25,850 --> 00:18:29,810 If you think about writing the integral of this 295 00:18:29,810 --> 00:18:33,560 the complicated part is this absolute value sine thing. 296 00:18:33,560 --> 00:18:36,050 So split it into two parts, that way the absolute value 297 00:18:36,050 --> 00:18:37,790 is one thing when t is bigger than 0 298 00:18:37,790 --> 00:18:39,831 and it's a different thing when t is less than 0. 299 00:18:43,120 --> 00:18:45,910 Now looks like just the sum of two functions that 300 00:18:45,910 --> 00:18:47,080 are both trivial. 301 00:18:47,080 --> 00:18:49,689 One is right sided, the other is left sided, 302 00:18:49,689 --> 00:18:51,730 so we might be expecting something a little funky 303 00:18:51,730 --> 00:18:52,790 compared to the last one. 304 00:18:57,230 --> 00:19:01,640 So this guy that is left sided in time 305 00:19:01,640 --> 00:19:06,420 ends up with a left-sided transform, 306 00:19:06,420 --> 00:19:09,040 just like the previous examples. 307 00:19:09,040 --> 00:19:12,390 So we see that this part, which looks 308 00:19:12,390 --> 00:19:17,680 like a pole at 1, so we're going to get 309 00:19:17,680 --> 00:19:20,940 a part of the transform that has a pole at 1. 310 00:19:20,940 --> 00:19:24,360 And it's going to be valid for regions in s space 311 00:19:24,360 --> 00:19:28,450 for s magnitude less than 1, OK. 312 00:19:28,450 --> 00:19:33,510 Then we're going to get this other piece, that 313 00:19:33,510 --> 00:19:37,215 corresponds to a pole minus 1. 314 00:19:37,215 --> 00:19:40,260 And it's right sided, so like the right sided examples 315 00:19:40,260 --> 00:19:42,060 that we saw before there's a right sided 316 00:19:42,060 --> 00:19:44,500 region of convergence here. 317 00:19:44,500 --> 00:19:49,240 So it's going to be right of its pole, but it's pole is minus 1. 318 00:19:49,240 --> 00:19:51,100 So in sum, the region is going to be 319 00:19:51,100 --> 00:19:54,450 to the right of the pole at minus 1, 320 00:19:54,450 --> 00:19:58,270 and to the left of the pole at 1 because we 321 00:19:58,270 --> 00:19:59,650 have to be in the part of s space 322 00:19:59,650 --> 00:20:04,520 where both of those two integrals converged. 323 00:20:04,520 --> 00:20:09,870 So we end up with the ROC being the intersection, OK. 324 00:20:09,870 --> 00:20:12,540 So we get a pole 0 pattern that looks like this. 325 00:20:12,540 --> 00:20:14,580 There are two poles now. 326 00:20:14,580 --> 00:20:16,329 When you add together a part that 327 00:20:16,329 --> 00:20:18,120 looks like a pole a minus 1 and a part that 328 00:20:18,120 --> 00:20:20,285 looks like a pole at 1, you get two poles. 329 00:20:23,500 --> 00:20:27,600 And the region becomes the band in between. 330 00:20:27,600 --> 00:20:30,570 Again, because the regions of s transforms 331 00:20:30,570 --> 00:20:32,640 because of Euler's expression is determined 332 00:20:32,640 --> 00:20:35,750 by the real part of s. 333 00:20:35,750 --> 00:20:39,926 So that we're always going to get some kind of vertical band. 334 00:20:39,926 --> 00:20:42,050 The band is always going to be delimited by a pole, 335 00:20:42,050 --> 00:20:45,810 just like it was in a z transform. 336 00:20:45,810 --> 00:20:46,740 There's two poles. 337 00:20:46,740 --> 00:20:48,240 So all we really need to worry about 338 00:20:48,240 --> 00:20:51,330 is which region bounded by two poles 339 00:20:51,330 --> 00:20:54,110 we're going to correspond to. 340 00:20:54,110 --> 00:20:59,244 OK, so the answer was two. 341 00:20:59,244 --> 00:21:00,910 Written in a little bit of a funky form, 342 00:21:00,910 --> 00:21:03,160 but if factor this you can see that there's two poles, 343 00:21:03,160 --> 00:21:04,881 one at 1, and one at minus 1. 344 00:21:08,570 --> 00:21:11,930 So hopefully that's all trivial, that's all stuff 345 00:21:11,930 --> 00:21:14,655 you can do in your sleep. 346 00:21:14,655 --> 00:21:16,280 What I want to do is think a little bit 347 00:21:16,280 --> 00:21:18,050 about what the implications are. 348 00:21:18,050 --> 00:21:19,520 What are we doing? 349 00:21:19,520 --> 00:21:24,440 What is a region of convergence? 350 00:21:24,440 --> 00:21:28,330 So I want to think now about four different functions, 351 00:21:28,330 --> 00:21:30,100 in four different poles zero patterns, 352 00:21:30,100 --> 00:21:32,721 in four different regions. 353 00:21:32,721 --> 00:21:34,720 And I want to think about the defining integral, 354 00:21:34,720 --> 00:21:36,430 that one up there. 355 00:21:36,430 --> 00:21:42,530 I want to think about x of s is the integral over all of time 356 00:21:42,530 --> 00:21:46,210 of x of te to the minus st dt. 357 00:21:46,210 --> 00:21:48,959 What are we doing when we take a Laplace transform? 358 00:21:48,959 --> 00:21:50,500 And what are we doing when we specify 359 00:21:50,500 --> 00:21:53,260 a region of convergence? 360 00:21:53,260 --> 00:21:55,030 Because I can help you think about 361 00:21:55,030 --> 00:21:57,280 what should the answer be, which is helpful 362 00:21:57,280 --> 00:21:58,690 when you're trying to think about 363 00:21:58,690 --> 00:22:00,160 does my answer make any sense? 364 00:22:00,160 --> 00:22:03,520 Or what am I doing anyway? 365 00:22:03,520 --> 00:22:07,870 So x1, this was e to the minus t. 366 00:22:07,870 --> 00:22:10,292 So it had a pole at minus 1 in the region to the right. 367 00:22:10,292 --> 00:22:12,000 Right-sided function, right-sided region. 368 00:22:14,840 --> 00:22:17,370 Here we have a more complicated function, 369 00:22:17,370 --> 00:22:20,600 which can be the sum of two poles, one 370 00:22:20,600 --> 00:22:23,962 at minus 1 and 1 at minus 2. 371 00:22:23,962 --> 00:22:25,670 And we got a region that was to the right 372 00:22:25,670 --> 00:22:27,770 of the rightmost pole. 373 00:22:27,770 --> 00:22:29,540 Right-sided function, right-sided region. 374 00:22:32,620 --> 00:22:36,080 Here was x3, which was the negative part of that one. 375 00:22:36,080 --> 00:22:40,340 We got the same pole minus 1 one the left-sided region. 376 00:22:40,340 --> 00:22:42,619 And here was the symmetric one where we found 377 00:22:42,619 --> 00:22:43,910 that the region was in between. 378 00:22:43,910 --> 00:22:45,950 So what I want to think now is what 379 00:22:45,950 --> 00:22:49,400 happens when we have a particular value of s that 380 00:22:49,400 --> 00:22:51,110 is inside or outside those regions. 381 00:22:51,110 --> 00:22:53,439 What's really going on? 382 00:22:53,439 --> 00:22:54,980 So let's start by thinking about what 383 00:22:54,980 --> 00:22:59,455 would happen if I considered an s depicted by this red x. 384 00:22:59,455 --> 00:23:00,830 I probably shouldn't have used x. 385 00:23:00,830 --> 00:23:02,900 I should have probably used a star. 386 00:23:02,900 --> 00:23:04,550 That's not a pole. 387 00:23:04,550 --> 00:23:09,050 That's the value of s in this integral. 388 00:23:09,050 --> 00:23:12,970 What happens if I choose to integrate against the function 389 00:23:12,970 --> 00:23:15,705 e to the minus st where s is 0? 390 00:23:19,272 --> 00:23:21,230 If I choose to integrate against that function, 391 00:23:21,230 --> 00:23:25,120 that function as depicted by red here. 392 00:23:25,120 --> 00:23:26,660 So the convergence of the integral, 393 00:23:26,660 --> 00:23:27,740 the convergence of the thing that I'm 394 00:23:27,740 --> 00:23:29,690 calling a Laplace transform depends entirely 395 00:23:29,690 --> 00:23:32,500 on the convergence of x. 396 00:23:32,500 --> 00:23:34,030 That's what s equals 0 means. 397 00:23:36,760 --> 00:23:40,820 s equals 0 says, the converge of the transform 398 00:23:40,820 --> 00:23:44,810 depends entirely on the convergence of the function. 399 00:23:44,810 --> 00:23:46,900 So that means that this function converges, 400 00:23:46,900 --> 00:23:51,070 it's in the region of convergence. 401 00:23:51,070 --> 00:23:55,310 This one converges, it's in the region of convergence. 402 00:23:55,310 --> 00:23:59,750 This one does not converge, this one diverges. 403 00:23:59,750 --> 00:24:03,890 As you go to minus infinity the function is unbounded. 404 00:24:03,890 --> 00:24:06,320 The transform doesn't exist. 405 00:24:06,320 --> 00:24:09,261 s equals 0 is not in the region. 406 00:24:09,261 --> 00:24:10,135 That's what it means. 407 00:24:12,920 --> 00:24:18,680 Here, if I integrate against e to the 0 both sides converge, 408 00:24:18,680 --> 00:24:20,640 there is no problem. 409 00:24:20,640 --> 00:24:21,440 I'm in the region. 410 00:24:25,120 --> 00:24:27,960 So now if I think about a different s, what if I move 411 00:24:27,960 --> 00:24:30,220 s a little bit to the left? 412 00:24:30,220 --> 00:24:31,315 Say, minus 1/2. 413 00:24:34,150 --> 00:24:38,000 If I'm thinking about integrating against e 414 00:24:38,000 --> 00:24:40,410 to the minus st, and if s is minus 1/2, 415 00:24:40,410 --> 00:24:44,120 that's e to the 1/2 t. 416 00:24:44,120 --> 00:24:48,020 e to the 1/2 t is a function that for all time 417 00:24:48,020 --> 00:24:50,730 becomes greater as I go toward positive infinity, 418 00:24:50,730 --> 00:24:52,850 exponentially greater. 419 00:24:52,850 --> 00:24:57,080 So if I think about what happens if I multiplied this x of t 420 00:24:57,080 --> 00:25:01,500 by that weighting function, does the product 421 00:25:01,500 --> 00:25:02,460 converge or diverge? 422 00:25:05,990 --> 00:25:10,860 It converges because the convergence of the blue line 423 00:25:10,860 --> 00:25:15,250 is faster than the divergence of the red line. 424 00:25:15,250 --> 00:25:18,400 Even though the red line is diverging, 425 00:25:18,400 --> 00:25:22,460 the product is overall convergent. 426 00:25:22,460 --> 00:25:24,570 OK that's because of the relative positions 427 00:25:24,570 --> 00:25:27,470 of the red x and the blue x. 428 00:25:27,470 --> 00:25:30,860 The blue function is converging faster than the red function is 429 00:25:30,860 --> 00:25:31,640 diverging. 430 00:25:31,640 --> 00:25:34,010 The product converges. 431 00:25:34,010 --> 00:25:36,050 So this x is in the region of convergence. 432 00:25:36,050 --> 00:25:38,670 That's what it means. 433 00:25:38,670 --> 00:25:41,880 Similarly here, this is diverging. 434 00:25:41,880 --> 00:25:43,379 The red curve is still diverging. 435 00:25:43,379 --> 00:25:45,420 It's the same red curves are all these red curves 436 00:25:45,420 --> 00:25:47,700 are diverging to the right. 437 00:25:47,700 --> 00:25:49,960 This one is converging. 438 00:25:49,960 --> 00:25:53,200 It's ultimately converging as the sum of two things, e 439 00:25:53,200 --> 00:25:55,740 to the minus t and e to the minus 2t, both of which 440 00:25:55,740 --> 00:26:01,670 are fast compared to the explosion e to the 1/2 t. 441 00:26:01,670 --> 00:26:04,880 So I'm in the region. 442 00:26:04,880 --> 00:26:08,286 Here it's exploding in a region of time 443 00:26:08,286 --> 00:26:09,285 that I don't care about. 444 00:26:12,230 --> 00:26:14,320 And it's convergent for this region of time 445 00:26:14,320 --> 00:26:17,005 that I do care about, but it's not convergent enough. 446 00:26:20,620 --> 00:26:25,720 So the integral still diverges, I'm not in the region. 447 00:26:25,720 --> 00:26:28,540 And finally, this one is more complicated. 448 00:26:28,540 --> 00:26:32,950 The integrand, this thing that I'm integrating against, 449 00:26:32,950 --> 00:26:38,380 it's always becoming greater as I go toward positive infinity. 450 00:26:38,380 --> 00:26:42,460 That tends to be convergent on this side 451 00:26:42,460 --> 00:26:44,410 and divergent on that side, but it's not 452 00:26:44,410 --> 00:26:49,390 divergent enough to make the function not converge. 453 00:26:49,390 --> 00:26:52,430 So I'm in the region. 454 00:26:52,430 --> 00:26:55,220 If I make an even bigger pole. 455 00:26:55,220 --> 00:27:03,890 So say I put e to the st, if I put that at minus 1 1/2, 456 00:27:03,890 --> 00:27:08,000 then that is fast enough that it breaks the product. 457 00:27:08,000 --> 00:27:11,520 The product is no longer integrable. 458 00:27:11,520 --> 00:27:15,990 So this divergent trend is enough to make that diverge. 459 00:27:15,990 --> 00:27:19,080 I'm not in the region anymore. 460 00:27:19,080 --> 00:27:22,080 Similarly here, I'm not in the region 461 00:27:22,080 --> 00:27:26,775 because I've made it fast enough that one of the parts diverges. 462 00:27:29,991 --> 00:27:31,490 One of the parts coverages, but they 463 00:27:31,490 --> 00:27:37,190 have to both converge in order for the interval to converge. 464 00:27:37,190 --> 00:27:40,070 Here, I have finally made the left-hand side 465 00:27:40,070 --> 00:27:44,750 convergent enough to make the product converge. 466 00:27:44,750 --> 00:27:46,880 So that's fine. 467 00:27:46,880 --> 00:27:50,030 And here, this is accelerating so fast 468 00:27:50,030 --> 00:27:52,820 that although it was stabilizing here 469 00:27:52,820 --> 00:27:57,050 it became non-convergent over here. 470 00:27:57,050 --> 00:27:59,090 What I want you to get from this is the idea 471 00:27:59,090 --> 00:28:00,770 that you can think physically about what 472 00:28:00,770 --> 00:28:02,230 the region of convergence is. 473 00:28:02,230 --> 00:28:04,670 The region of convergence is a function 474 00:28:04,670 --> 00:28:10,010 that I stick into the integral to make the integral converge. 475 00:28:10,010 --> 00:28:11,810 So the region of convergence corresponds 476 00:28:11,810 --> 00:28:15,770 to those exponents that makes sense for which 477 00:28:15,770 --> 00:28:17,990 the integral will converge. 478 00:28:17,990 --> 00:28:22,465 OK, with that vast new insight this problem is now trivial. 479 00:28:25,080 --> 00:28:31,950 Enumerate all possible functions x of t 480 00:28:31,950 --> 00:28:33,648 for which that's the transform. 481 00:30:12,077 --> 00:30:13,410 So how many functions are there? 482 00:30:19,492 --> 00:30:20,450 Keep going, keep going. 483 00:30:20,450 --> 00:30:24,140 I'm still looking for a right answer. 484 00:30:24,140 --> 00:30:25,587 That's a clue. 485 00:30:25,587 --> 00:30:27,170 Right, I don't see a right answer yet. 486 00:30:27,170 --> 00:30:27,990 So keep going. 487 00:30:27,990 --> 00:30:28,790 Keep going. 488 00:30:28,790 --> 00:30:31,030 Actually, I do now. 489 00:30:31,030 --> 00:30:32,470 I see two right answers. 490 00:30:32,470 --> 00:30:34,000 Can we make it three? 491 00:30:34,000 --> 00:30:34,650 Going once. 492 00:30:34,650 --> 00:30:36,390 Going twice. 493 00:30:36,390 --> 00:30:37,150 Two right answers? 494 00:30:40,790 --> 00:30:43,650 OK, the right answer is three. 495 00:30:43,650 --> 00:30:45,480 So now that I've told you the right answer, 496 00:30:45,480 --> 00:30:47,900 you rationalize for me why is the right answer three? 497 00:30:53,060 --> 00:30:54,230 Talk to your neighbor. 498 00:30:54,230 --> 00:30:56,968 Why is the right answer three? 499 00:30:56,968 --> 00:31:00,447 [INTERPOSING VOICES] 500 00:32:03,640 --> 00:32:08,060 OK, who can volunteer a concise statement 501 00:32:08,060 --> 00:32:10,520 for why the right answer is three and not four? 502 00:32:13,570 --> 00:32:14,524 Yes? 503 00:32:14,524 --> 00:32:17,863 [INAUDIBLE] 504 00:32:19,300 --> 00:32:20,460 That's exactly right. 505 00:32:20,460 --> 00:32:21,920 There are two poles, and there are 506 00:32:21,920 --> 00:32:27,920 three ways you can divide up the s plane by two poles. 507 00:32:27,920 --> 00:32:33,390 So there's four functions here, let's think about four s 508 00:32:33,390 --> 00:32:33,890 planes. 509 00:32:38,680 --> 00:32:42,940 So each of these functions-- so here's a pole at minus 2. 510 00:32:42,940 --> 00:32:44,740 Here's a pole at 2. 511 00:32:44,740 --> 00:32:46,630 Minus 2, 2, minus 2, 2, minus 2, 2. 512 00:32:46,630 --> 00:32:48,175 They all have the same poles. 513 00:32:54,390 --> 00:32:58,700 OK, where's the region that corresponds to function one? 514 00:33:03,660 --> 00:33:07,280 So here's pole at minus 2. e to the minus 515 00:33:07,280 --> 00:33:11,510 2t corresponds to a pole at minus 2. 516 00:33:11,510 --> 00:33:13,490 u of t is right-sided function. 517 00:33:13,490 --> 00:33:15,110 Where's the region that corresponds 518 00:33:15,110 --> 00:33:17,450 to the pole minus 2? 519 00:33:17,450 --> 00:33:18,980 To the right of minus 2. 520 00:33:21,650 --> 00:33:27,140 So this first function converges to the right of minus 2. 521 00:33:27,140 --> 00:33:36,970 This function is a pole at 2, e to the 2t and it's right sided. 522 00:33:36,970 --> 00:33:41,700 So it's right sided with regard to a pole at 2. 523 00:33:41,700 --> 00:33:43,380 So what's the region that corresponds 524 00:33:43,380 --> 00:33:46,140 to a right-sided time function that 525 00:33:46,140 --> 00:33:48,060 corresponds to a pole at 2, or that's right 526 00:33:48,060 --> 00:33:49,365 sided with regard to here. 527 00:33:51,870 --> 00:33:54,540 So the net region of convergence is that region. 528 00:33:57,160 --> 00:33:57,710 Make sense? 529 00:34:00,700 --> 00:34:05,520 So then I have a pole at minus 2 with the right sided. 530 00:34:05,520 --> 00:34:06,620 OK, so that's this. 531 00:34:09,139 --> 00:34:13,650 And a pole as 2, which is left sided. 532 00:34:13,650 --> 00:34:16,820 So that's this. 533 00:34:16,820 --> 00:34:19,250 So that corresponds to a region here. 534 00:34:23,929 --> 00:34:27,980 Then the third one I have a pole at minus 2, which is left 535 00:34:27,980 --> 00:34:30,350 sided. 536 00:34:30,350 --> 00:34:32,905 So that means I want to have convergence here. 537 00:34:36,199 --> 00:34:38,480 And I have a pole at 2 which is right sided, 538 00:34:38,480 --> 00:34:40,280 and I need convergence here. 539 00:34:40,280 --> 00:34:43,610 There's no way to make that convergent. 540 00:34:43,610 --> 00:34:46,010 If I choose my s to make the first part convergent then 541 00:34:46,010 --> 00:34:49,330 my second part is not convergent, and vice versa. 542 00:34:49,330 --> 00:34:52,330 So there's no way to do that one. 543 00:34:52,330 --> 00:34:53,130 So this is no. 544 00:34:53,130 --> 00:34:53,820 This is yes. 545 00:34:53,820 --> 00:34:56,460 This is yes. 546 00:34:56,460 --> 00:35:00,825 And then finally, this is a pole at minus 2 to the left. 547 00:35:03,670 --> 00:35:08,920 And a pole at 2 to the left. 548 00:35:08,920 --> 00:35:10,590 So that's that one. 549 00:35:10,590 --> 00:35:11,970 So that's OK. 550 00:35:11,970 --> 00:35:15,890 So the idea is that there's only three ways to chop up 551 00:35:15,890 --> 00:35:19,000 a space delimited by two poles. 552 00:35:19,000 --> 00:35:21,372 OK, so it looks like you to have four 553 00:35:21,372 --> 00:35:23,830 because we're used to thinking about the two by two matrix. 554 00:35:23,830 --> 00:35:24,996 That's not the way it works. 555 00:35:29,460 --> 00:35:32,880 OK, two last things. 556 00:35:32,880 --> 00:35:34,230 First last thing. 557 00:35:34,230 --> 00:35:35,850 Probably the most important thing 558 00:35:35,850 --> 00:35:38,250 about the Laplace transform, probably the only reason 559 00:35:38,250 --> 00:35:41,190 we bother with that whatever is that you can use it to solve 560 00:35:41,190 --> 00:35:42,810 differential equations. 561 00:35:42,810 --> 00:35:45,420 That's probably the most important reason 562 00:35:45,420 --> 00:35:47,010 for even talking about it. 563 00:35:47,010 --> 00:35:49,600 The fact that we can take it is of zero consequence. 564 00:35:49,600 --> 00:35:52,052 We can take lots of integrals. 565 00:35:52,052 --> 00:35:53,760 The interesting thing about this integral 566 00:35:53,760 --> 00:35:57,360 is that it helps us to solve differential equations. 567 00:35:57,360 --> 00:36:01,020 And the trick is that if we start with the differential 568 00:36:01,020 --> 00:36:03,504 equation we can take the Laplace transform 569 00:36:03,504 --> 00:36:04,920 of the whole differential equation 570 00:36:04,920 --> 00:36:07,260 and that will end up making sense. 571 00:36:07,260 --> 00:36:12,050 Just like the z transform, the Laplace transform is linear. 572 00:36:12,050 --> 00:36:13,710 The Laplace transform of a sum is 573 00:36:13,710 --> 00:36:16,020 sum of the Laplace transforms. 574 00:36:16,020 --> 00:36:18,660 That's pretty simple just by looking at the definition 575 00:36:18,660 --> 00:36:22,380 because we can distribute the e to the minus st over a sum, 576 00:36:22,380 --> 00:36:26,280 if x were a sum it distributes. 577 00:36:26,280 --> 00:36:27,790 So that's easy. 578 00:36:27,790 --> 00:36:30,750 The other important thing is the derivative. 579 00:36:30,750 --> 00:36:33,210 What's the Laplace transform of a derivative? 580 00:36:33,210 --> 00:36:35,250 That turns out to be easy. 581 00:36:35,250 --> 00:36:38,160 If that weren't easy, we wouldn't even bother with it. 582 00:36:38,160 --> 00:36:40,390 But it is easy. 583 00:36:40,390 --> 00:36:43,020 You can see that if we make the assumption that x 584 00:36:43,020 --> 00:36:48,190 is the Laplace transform of x, and then if we define 585 00:36:48,190 --> 00:36:50,170 y is the derivative of x, we can ask 586 00:36:50,170 --> 00:36:51,944 what's the Laplace transform of y 587 00:36:51,944 --> 00:36:53,610 and that will tell us what's the Laplace 588 00:36:53,610 --> 00:36:57,420 transform of the derivative. 589 00:36:57,420 --> 00:36:58,950 Take the definition of transform, 590 00:36:58,950 --> 00:37:04,910 stick in y equals x dot and integrate by parts. 591 00:37:04,910 --> 00:37:07,400 You all can integrate by parts much better 592 00:37:07,400 --> 00:37:12,650 than I can because I took it 35 years ago. 593 00:37:12,650 --> 00:37:18,195 But it's very easy to see that if we call this u and this v 594 00:37:18,195 --> 00:37:23,240 dot, integral u dv is uv minus the integral v du. 595 00:37:26,240 --> 00:37:29,030 The trick is that taking integrals and derivatives 596 00:37:29,030 --> 00:37:34,180 of this part is easy, it's an exponential. 597 00:37:34,180 --> 00:37:37,150 Exponentials are the one function 598 00:37:37,150 --> 00:37:40,510 whose derivative has the same shape as the function. 599 00:37:40,510 --> 00:37:43,800 That's the reason it's easy. 600 00:37:43,800 --> 00:37:50,300 So that means that I can easily integrate that part. 601 00:37:50,300 --> 00:37:51,960 I can easily differentiate that part 602 00:37:51,960 --> 00:37:54,626 to get something that looks just like this, except as multiplied 603 00:37:54,626 --> 00:37:55,370 by minus s. 604 00:37:58,250 --> 00:37:59,810 And that minus s is the only thing 605 00:37:59,810 --> 00:38:02,570 that ends up being important. 606 00:38:02,570 --> 00:38:05,450 The Laplace transform of the derivative 607 00:38:05,450 --> 00:38:10,980 is s times the Laplace transform of the original function. 608 00:38:10,980 --> 00:38:14,090 OK, differentiating time is the same 609 00:38:14,090 --> 00:38:18,350 as multiplying by s in a Laplace transform. 610 00:38:18,350 --> 00:38:20,900 Because of that it's trivial to think 611 00:38:20,900 --> 00:38:26,050 about the Laplace transform of a differential equation. 612 00:38:26,050 --> 00:38:28,720 Linearity says you can do it term wise. 613 00:38:28,720 --> 00:38:30,610 And a differential equation is something 614 00:38:30,610 --> 00:38:32,500 that has a bunch of derivatives in it. 615 00:38:32,500 --> 00:38:35,170 Those all turn into multiply by s. 616 00:38:35,170 --> 00:38:37,990 So you end up then with this differential equation 617 00:38:37,990 --> 00:38:39,175 being replaced by this. 618 00:38:45,360 --> 00:38:49,810 So the Laplace transform of the derivative is s times y. 619 00:38:49,810 --> 00:38:52,240 The Laplace transform of y is y. 620 00:38:52,240 --> 00:38:55,530 Now I have to figure out the Laplace transform of delta. 621 00:38:55,530 --> 00:38:59,037 OK, turns out that's easy. 622 00:38:59,037 --> 00:39:00,620 What's the Laplace transform of delta? 623 00:39:00,620 --> 00:39:02,630 That's why we like delta functions. 624 00:39:02,630 --> 00:39:04,580 Delta functions seem mathematically bizarre 625 00:39:04,580 --> 00:39:06,530 but they're so easy to work with, 626 00:39:06,530 --> 00:39:08,990 that's the only reason we use them. 627 00:39:08,990 --> 00:39:11,180 If you think about the Laplace transform of delta 628 00:39:11,180 --> 00:39:13,760 just stick it in the formula. 629 00:39:13,760 --> 00:39:15,830 The interesting thing about the delta function 630 00:39:15,830 --> 00:39:20,950 is that it's 0 almost everywhere. 631 00:39:20,950 --> 00:39:23,040 So we don't need to worry about the product 632 00:39:23,040 --> 00:39:25,080 except where the delta function is not 0. 633 00:39:28,250 --> 00:39:32,740 Because the delta function makes the time axis mostly 0. 634 00:39:32,740 --> 00:39:36,540 The only place that's not 0 is 0. 635 00:39:36,540 --> 00:39:39,690 So the only value of this function that matters 636 00:39:39,690 --> 00:39:44,437 in the least is its value at 0. 637 00:39:44,437 --> 00:39:46,770 If you think about what that looks like is as a picture. 638 00:39:50,340 --> 00:39:55,730 So if we have something that looks like e to the minus st 639 00:39:55,730 --> 00:39:59,480 and if we multiply by an impulse. 640 00:39:59,480 --> 00:40:01,260 The impulse is 0 except at 0. 641 00:40:01,260 --> 00:40:03,260 The only thing that survives that multiplication 642 00:40:03,260 --> 00:40:08,360 is the value of e to the minus st at t equals 0. 643 00:40:08,360 --> 00:40:10,280 We can think about that as a limit. 644 00:40:10,280 --> 00:40:12,290 We thought about the delta function 645 00:40:12,290 --> 00:40:16,010 as a limit of a function that looks like this, minus epsilon, 646 00:40:16,010 --> 00:40:18,770 epsilon t, 1 over 2 epsilon. 647 00:40:18,770 --> 00:40:22,820 So that the limit as epsilon goes to 0 becomes a delta. 648 00:40:22,820 --> 00:40:26,060 That's how we defined the impulse function. 649 00:40:26,060 --> 00:40:28,430 If we think about that definition in terms 650 00:40:28,430 --> 00:40:32,300 of this product you can see that you 651 00:40:32,300 --> 00:40:36,270 get exactly this expression. 652 00:40:36,270 --> 00:40:41,510 So if you multiply this times this all you pick out 653 00:40:41,510 --> 00:40:42,330 is this part. 654 00:40:45,800 --> 00:40:48,410 As you make epsilon smaller, and smaller, and smaller, 655 00:40:48,410 --> 00:40:52,262 you focus just on that point right at 0. 656 00:40:52,262 --> 00:40:53,470 The area of the impulse is 1. 657 00:40:53,470 --> 00:40:58,880 Therefore, the integral of this thing is everywhere 1. 658 00:40:58,880 --> 00:41:00,530 So this sifts out. 659 00:41:00,530 --> 00:41:01,710 That's what we'll call it. 660 00:41:01,710 --> 00:41:03,335 So this is called the sifting property. 661 00:41:08,130 --> 00:41:11,370 It's the nice part about the delta function. 662 00:41:11,370 --> 00:41:13,500 If you integrate a function with regard 663 00:41:13,500 --> 00:41:18,110 to time where the function is multiplied by delta, 664 00:41:18,110 --> 00:41:20,600 that integral is the value of the function at 0. 665 00:41:20,600 --> 00:41:23,840 It sifts out the 0 value. 666 00:41:23,840 --> 00:41:27,260 That means that the Laplace transform of the delta function 667 00:41:27,260 --> 00:41:29,370 is 1. 668 00:41:29,370 --> 00:41:31,830 It's the most simple function you can imagine. 669 00:41:31,830 --> 00:41:35,330 So the net effect of that is that this differential 670 00:41:35,330 --> 00:41:38,690 equation, the Laplace transform of the differential equation 671 00:41:38,690 --> 00:41:40,565 becomes an algebraic equation. 672 00:41:45,400 --> 00:41:49,050 That means that we can solve it by doing algebra. 673 00:41:49,050 --> 00:41:52,500 So in particular, we can readily solve the equation 674 00:41:52,500 --> 00:41:54,830 to find an expression for y. 675 00:41:54,830 --> 00:41:57,530 We saw that previously. 676 00:41:57,530 --> 00:41:59,720 We recognize ys. 677 00:41:59,720 --> 00:42:01,580 It's a recognition thing. 678 00:42:01,580 --> 00:42:03,950 It's a table look up sort of thing. 679 00:42:03,950 --> 00:42:09,110 We know the form 1 over s plus 1. 680 00:42:09,110 --> 00:42:11,150 So we know the time function that is the answer. 681 00:42:13,670 --> 00:42:15,440 So notice that when we do this just 682 00:42:15,440 --> 00:42:17,390 the way that happened with the z transform. 683 00:42:17,390 --> 00:42:18,950 When we solve a differential equation 684 00:42:18,950 --> 00:42:22,190 by using Laplace transform we basically don't use calculus. 685 00:42:24,662 --> 00:42:26,870 We use the Laplace transform to turn the differential 686 00:42:26,870 --> 00:42:30,620 equation which is calculus in 1803 and that kind of stuff, 687 00:42:30,620 --> 00:42:33,200 into algebra, which is high school. 688 00:42:33,200 --> 00:42:35,060 We do everything with algebra. 689 00:42:35,060 --> 00:42:39,014 The only annoying thing is this table look up thing. 690 00:42:39,014 --> 00:42:40,430 It's a little annoying that we had 691 00:42:40,430 --> 00:42:46,975 to do the inverse Laplace transform by table look up. 692 00:42:46,975 --> 00:42:50,410 It's a little dissatisfying. 693 00:42:50,410 --> 00:42:55,630 So I'll just mention that there is a formal way of not 694 00:42:55,630 --> 00:42:57,220 doing table look up. 695 00:42:59,656 --> 00:43:01,030 For the kinds of problems we will 696 00:43:01,030 --> 00:43:03,910 look at using this formula is so much more 697 00:43:03,910 --> 00:43:07,880 difficult than table look up that we will never 698 00:43:07,880 --> 00:43:11,240 use that formula. 699 00:43:11,240 --> 00:43:13,160 It will be useful for proving things, 700 00:43:13,160 --> 00:43:16,496 it will not be useful for inverting things. 701 00:43:16,496 --> 00:43:17,870 Because of the form of the things 702 00:43:17,870 --> 00:43:19,640 we do, linear differential equations 703 00:43:19,640 --> 00:43:22,010 with constant coefficients, it will always 704 00:43:22,010 --> 00:43:23,960 be easier to do table look up then 705 00:43:23,960 --> 00:43:25,970 it will be to run this integral. 706 00:43:25,970 --> 00:43:29,247 If you're interested in that integral, fine, take 1804. 707 00:43:29,247 --> 00:43:30,830 There's a bunch of people who know all 708 00:43:30,830 --> 00:43:33,000 about how that integral works. 709 00:43:33,000 --> 00:43:37,320 It's spectacularly interesting but we won't use it here. 710 00:43:37,320 --> 00:43:42,020 So the upshot then is that the Laplace transform, 711 00:43:42,020 --> 00:43:47,840 we learn about it because it's very useful 712 00:43:47,840 --> 00:43:53,110 and its utility comes from a bunch of properties. 713 00:43:53,110 --> 00:43:55,600 The ones we illustrated today, we use the fact 714 00:43:55,600 --> 00:43:58,990 that it was linear and we use the fact 715 00:43:58,990 --> 00:44:00,550 that there's a simple relationship 716 00:44:00,550 --> 00:44:02,590 with differentiation. 717 00:44:02,590 --> 00:44:05,470 Using just those facts it turns out easy 718 00:44:05,470 --> 00:44:09,550 to do differential equations using the Laplace transform. 719 00:44:09,550 --> 00:44:11,290 And there's many other things I've 720 00:44:11,290 --> 00:44:12,850 illustrated in the last two slides 721 00:44:12,850 --> 00:44:16,360 but I won't go over them right now. 722 00:44:16,360 --> 00:44:20,860 The idea of taking limits using Laplace transforms. 723 00:44:20,860 --> 00:44:22,990 It turns out that the Laplace transform is what 724 00:44:22,990 --> 00:44:25,880 we call a reciprocal function. 725 00:44:25,880 --> 00:44:29,695 Since the integrand depends and the product of s and t, 726 00:44:29,695 --> 00:44:33,950 s gets large, corresponds to t gets small. 727 00:44:33,950 --> 00:44:37,570 So you can take the limit for things 728 00:44:37,570 --> 00:44:41,900 with time getting small by looking at s getting big. 729 00:44:41,900 --> 00:44:44,530 Similarly, you can do the reverse. 730 00:44:44,530 --> 00:44:48,490 You can look at time gets big by looking at s gets small. 731 00:44:48,490 --> 00:44:50,876 That's just properties of the Laplace transform. 732 00:44:50,876 --> 00:44:52,000 And there's lots of others. 733 00:44:52,000 --> 00:44:54,670 Besides the table there's lots of properties 734 00:44:54,670 --> 00:44:56,920 we won't have time to go over but I 735 00:44:56,920 --> 00:44:58,060 will hint at some of them. 736 00:44:58,060 --> 00:45:00,880 And in particular, there's a very useful transformation 737 00:45:00,880 --> 00:45:02,990 where you take the Laplace transform of a circuit. 738 00:45:02,990 --> 00:45:05,720 And that's in the homework. 739 00:45:05,720 --> 00:45:08,540 So that's extremely useful. 740 00:45:08,540 --> 00:45:11,099 The major point then is just that there's 741 00:45:11,099 --> 00:45:12,890 lots of relationships, there's lots of ways 742 00:45:12,890 --> 00:45:14,370 we think about CT systems. 743 00:45:14,370 --> 00:45:17,480 This Laplace transform is a new one 744 00:45:17,480 --> 00:45:22,360 and it's very useful because of the properties in that table. 745 00:45:22,360 --> 00:45:24,470 OK, thanks a lot.