1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high-quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation, or view additional materials from 6 00:00:13,470 --> 00:00:19,290 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:19,290 --> 00:00:21,518 ocw.mit.edu. 8 00:00:21,518 --> 00:00:56,170 [MUSIC PLAYING] 9 00:00:56,170 --> 00:00:58,350 PROFESSOR: Over the last series of lectures, in 10 00:00:58,350 --> 00:01:00,940 discussing filtering, modulation, and sampling, 11 00:01:00,940 --> 00:01:03,720 we've seen how powerful and useful the 12 00:01:03,720 --> 00:01:06,490 Fourier transform is. 13 00:01:06,490 --> 00:01:09,210 Beginning with this lecture, and over the next several 14 00:01:09,210 --> 00:01:14,400 lectures, I'd like to develop and exploit a generalization 15 00:01:14,400 --> 00:01:18,600 of the Fourier transform, which will not only lead to 16 00:01:18,600 --> 00:01:22,240 some important new insights about signals and systems, but 17 00:01:22,240 --> 00:01:27,110 also will remove some of the restrictions that we've had 18 00:01:27,110 --> 00:01:30,300 with the Fourier transform. 19 00:01:30,300 --> 00:01:34,510 The generalization that we'll be talking about in the 20 00:01:34,510 --> 00:01:38,330 continuous time case is referred to as the Laplace 21 00:01:38,330 --> 00:01:42,060 transform, and in the discrete time case, is referred to as 22 00:01:42,060 --> 00:01:43,680 the z transform. 23 00:01:43,680 --> 00:01:46,810 What I'd like to do in today's lecture is begin on the 24 00:01:46,810 --> 00:01:49,110 continuous time case, namely a discussion 25 00:01:49,110 --> 00:01:51,010 of the Laplace transform. 26 00:01:51,010 --> 00:01:55,130 Continue that into the next lecture, and following that 27 00:01:55,130 --> 00:01:58,690 develop the z transform for discrete time. 28 00:01:58,690 --> 00:02:02,150 And also, as we go along, exploit 29 00:02:02,150 --> 00:02:05,080 the two notions together. 30 00:02:05,080 --> 00:02:09,630 Now, to introduce the notion of the Laplace transform, let 31 00:02:09,630 --> 00:02:12,250 me remind you again of what led us 32 00:02:12,250 --> 00:02:15,380 into the Fourier transform. 33 00:02:15,380 --> 00:02:19,910 We developed the Fourier transform by considering the 34 00:02:19,910 --> 00:02:26,210 idea of representing signals as linear combinations of 35 00:02:26,210 --> 00:02:27,660 basic signals. 36 00:02:27,660 --> 00:02:33,060 And in the Fourier transform, in the continuous time case, 37 00:02:33,060 --> 00:02:36,190 the basic signals that we picked in the representation 38 00:02:36,190 --> 00:02:38,800 were complex exponentials. 39 00:02:38,800 --> 00:02:43,650 And in what we had referred to as the synthesis equation, the 40 00:02:43,650 --> 00:02:48,020 synthesis equation corresponded to, in effect, a 41 00:02:48,020 --> 00:02:52,230 decomposition as a linear combination, a decomposition 42 00:02:52,230 --> 00:02:55,350 of x of t as a linear combination of complex 43 00:02:55,350 --> 00:02:56,980 exponentials. 44 00:02:56,980 --> 00:03:00,440 And of course, associated with this was the corresponding 45 00:03:00,440 --> 00:03:05,010 analysis equation that, in effect, gave us the amplitudes 46 00:03:05,010 --> 00:03:08,850 associated with the complex exponentials. 47 00:03:08,850 --> 00:03:11,550 Now, why did we pick complex exponentials? 48 00:03:11,550 --> 00:03:16,760 Well, recall that the reason was that complex exponentials 49 00:03:16,760 --> 00:03:20,250 are eigenfunctions of linear time-invariant systems, and 50 00:03:20,250 --> 00:03:21,900 that was very convenient. 51 00:03:21,900 --> 00:03:27,900 Specifically, if we have a linear time-invariant system 52 00:03:27,900 --> 00:03:34,080 with an impulse response h of t, what we had shown is that 53 00:03:34,080 --> 00:03:37,750 that class of systems has the property that if we put in a 54 00:03:37,750 --> 00:03:43,070 complex exponential, we get out a complex exponential at 55 00:03:43,070 --> 00:03:47,100 the same frequency and with a change in amplitude. 56 00:03:47,100 --> 00:03:51,790 And this change in amplitude, in fact, corresponded as we 57 00:03:51,790 --> 00:03:55,180 showed as the discussion went along, to the Fourier 58 00:03:55,180 --> 00:04:00,380 transform of the system impulse response. 59 00:04:00,380 --> 00:04:06,630 So the notion of decomposing signals into complex 60 00:04:06,630 --> 00:04:09,710 exponentials was very intimately connected, and the 61 00:04:09,710 --> 00:04:13,470 Fourier transform was very intimately connected, with the 62 00:04:13,470 --> 00:04:16,810 eigenfunction property of complex exponentials for 63 00:04:16,810 --> 00:04:20,070 linear time-invariant systems. 64 00:04:20,070 --> 00:04:24,370 Well, complex exponentials of that type are not the only 65 00:04:24,370 --> 00:04:28,470 eigenfunctions for linear time-invariant systems. 66 00:04:28,470 --> 00:04:33,210 In fact, what you've seen previously is that if we took 67 00:04:33,210 --> 00:04:39,290 a more general exponential, e to the st, where s is a more 68 00:04:39,290 --> 00:04:40,760 general complex number. 69 00:04:40,760 --> 00:04:45,670 Not just j omega, but in fact sigma plus j omega. 70 00:04:45,670 --> 00:04:50,110 For any value of s, the complex exponential is an 71 00:04:50,110 --> 00:04:51,370 eigenfunction. 72 00:04:51,370 --> 00:04:55,350 And we can justify that simply by substitution into the 73 00:04:55,350 --> 00:04:57,040 convolution integral. 74 00:04:57,040 --> 00:05:01,060 In other words, the response to this complex exponential is 75 00:05:01,060 --> 00:05:05,570 the convolution of the impulse response with the excitation. 76 00:05:05,570 --> 00:05:12,070 And notice that we can break this term into a product, e to 77 00:05:12,070 --> 00:05:16,030 the st e to the minus s tau. 78 00:05:16,030 --> 00:05:21,300 And the e to the st term can come outside the integration. 79 00:05:21,300 --> 00:05:25,160 And consequently, just carrying through that algebra, 80 00:05:25,160 --> 00:05:32,760 would reduce this integral to an integral with an e to the 81 00:05:32,760 --> 00:05:35,040 st factor outside. 82 00:05:35,040 --> 00:05:38,130 So just simply carrying through the algebra, what we 83 00:05:38,130 --> 00:05:42,360 would conclude is that a complex exponential with any 84 00:05:42,360 --> 00:05:47,110 complex number s would generate, as an output, a 85 00:05:47,110 --> 00:05:51,960 complex exponential of the same form multiplied by 86 00:05:51,960 --> 00:05:53,460 whatever this integral is. 87 00:05:53,460 --> 00:05:56,520 And this integral, of course, will depend on what 88 00:05:56,520 --> 00:05:58,110 the value of s is. 89 00:05:58,110 --> 00:06:00,030 But that's all that it will depend on. 90 00:06:00,030 --> 00:06:06,340 Or said another way, what this all can be denoted as is some 91 00:06:06,340 --> 00:06:11,560 function h of s that depends on the value of s. 92 00:06:11,560 --> 00:06:16,370 So finally then, e to the st as an excitation to a linear 93 00:06:16,370 --> 00:06:20,430 time-invariant system generates a response, which is 94 00:06:20,430 --> 00:06:26,840 a complex constant depending on s, multiplying the same 95 00:06:26,840 --> 00:06:30,340 function that excited the system. 96 00:06:30,340 --> 00:06:35,840 So what we have then is the eigenfunction property, more 97 00:06:35,840 --> 00:06:41,300 generally, in terms of a more general complex exponential 98 00:06:41,300 --> 00:06:46,660 where the complex factor is given by this integral. 99 00:06:46,660 --> 00:06:51,830 Well, in fact, what that integral corresponds to is 100 00:06:51,830 --> 00:06:57,120 what we will define as the Laplace transform of the 101 00:06:57,120 --> 00:06:58,860 impulse response. 102 00:06:58,860 --> 00:07:04,710 And in fact, we can apply this transformation to a more 103 00:07:04,710 --> 00:07:08,240 general time function that may or may not be the impulse 104 00:07:08,240 --> 00:07:12,370 response of a linear time-invariant system. 105 00:07:12,370 --> 00:07:17,130 And so, in general, it is this transformation on a time 106 00:07:17,130 --> 00:07:22,020 function which is the Laplace transform of that time 107 00:07:22,020 --> 00:07:25,050 function, and it's a function of s. 108 00:07:25,050 --> 00:07:31,120 So the definition of the Laplace transform is that the 109 00:07:31,120 --> 00:07:37,120 Laplace transform of a time function x of t is the result 110 00:07:37,120 --> 00:07:39,990 of this transformation on x of t. 111 00:07:39,990 --> 00:07:45,280 It's denoted as x of s, and as a shorthand notation as we had 112 00:07:45,280 --> 00:07:49,040 with the Fourier transform, then we have in the time 113 00:07:49,040 --> 00:07:52,640 domain, the time function x of t, and in the Laplace 114 00:07:52,640 --> 00:07:57,170 transform domain, the function x of s. 115 00:07:57,170 --> 00:08:01,560 And these then represent a transform pair. 116 00:08:01,560 --> 00:08:07,190 Now, let me remind you that the development of that 117 00:08:07,190 --> 00:08:11,170 mapping is exactly the process the we went through initially 118 00:08:11,170 --> 00:08:14,670 in developing a mapping that ended up giving 119 00:08:14,670 --> 00:08:17,260 us the Fourier transform. 120 00:08:17,260 --> 00:08:20,710 Essentially, what we've done is just broadened our horizon 121 00:08:20,710 --> 00:08:24,140 somewhat, or our notation somewhat. 122 00:08:24,140 --> 00:08:27,750 And rather than pushing just a complex exponential through 123 00:08:27,750 --> 00:08:31,830 the system, we've pushed a more general time function e 124 00:08:31,830 --> 00:08:35,760 to the st, where s is a complex number with both a 125 00:08:35,760 --> 00:08:39,080 real part and an imaginary part. 126 00:08:39,080 --> 00:08:41,440 Well, the discussion that we've gone through so far, of 127 00:08:41,440 --> 00:08:45,440 course, is very closely related to what we went 128 00:08:45,440 --> 00:08:48,020 through for the Fourier transform. 129 00:08:48,020 --> 00:08:49,860 The mapping that we've ended up with is 130 00:08:49,860 --> 00:08:51,850 called the Laplace transform. 131 00:08:51,850 --> 00:08:54,490 And as you can well imagine and perhaps, may have 132 00:08:54,490 --> 00:08:59,200 recognized already, there's a very close connection between 133 00:08:59,200 --> 00:09:03,500 the Laplace transform and the Fourier transform. 134 00:09:03,500 --> 00:09:06,440 Well, to see one of the connections, what we can 135 00:09:06,440 --> 00:09:11,240 observe is that if we look at the Fourier transform 136 00:09:11,240 --> 00:09:16,020 expression and if we look at the Laplace transform 137 00:09:16,020 --> 00:09:21,720 expression, where s is now a general complex number sigma 138 00:09:21,720 --> 00:09:25,720 plus j omega, these two expressions, in fact, are 139 00:09:25,720 --> 00:09:31,730 identical if, in fact, sigma is equal to 0. 140 00:09:31,730 --> 00:09:36,530 If sigma is equal to 0 so that s is just j omega, then all 141 00:09:36,530 --> 00:09:40,550 that this transformation is, is the same as that. 142 00:09:40,550 --> 00:09:44,490 Substitute in s equals j omega and this is what we get. 143 00:09:47,530 --> 00:09:51,190 What this then tells us is that if we have the Laplace 144 00:09:51,190 --> 00:09:57,390 transform, and if we look at the Laplace transform at s 145 00:09:57,390 --> 00:10:05,350 equals j omega, then that, in fact, corresponds to the 146 00:10:05,350 --> 00:10:09,420 Fourier transform of x of t. 147 00:10:09,420 --> 00:10:13,910 Now, there is a slight notational issue that this 148 00:10:13,910 --> 00:10:15,650 raises, and it's very 149 00:10:15,650 --> 00:10:17,530 straightforward to clean it up. 150 00:10:17,530 --> 00:10:19,520 But it's something that it's-- 151 00:10:19,520 --> 00:10:23,980 you have to just kind of focus on for a second to understand 152 00:10:23,980 --> 00:10:25,470 what the issue is. 153 00:10:25,470 --> 00:10:30,510 Notice that on the left-hand side of this equation, x of s 154 00:10:30,510 --> 00:10:33,550 representing the Laplace transform. 155 00:10:33,550 --> 00:10:36,690 When we look at that with sigma equal to 0 or s equal to 156 00:10:36,690 --> 00:10:41,100 j omega, our natural inclination is to write that 157 00:10:41,100 --> 00:10:44,140 as x of j omega, of course. 158 00:10:44,140 --> 00:10:46,470 On the other hand, the right-hand side of the 159 00:10:46,470 --> 00:10:49,970 equation, namely the Fourier transform of x of t, we've 160 00:10:49,970 --> 00:10:55,500 typically written as x of omega. 161 00:10:55,500 --> 00:10:58,210 Focusing on the fact that it's a function of 162 00:10:58,210 --> 00:11:00,580 this variable omega. 163 00:11:00,580 --> 00:11:03,250 Well, there's a slight awkwardness here because here 164 00:11:03,250 --> 00:11:05,740 we're talking about an argument j omega, here we're 165 00:11:05,740 --> 00:11:07,920 talking about an argument omega. 166 00:11:07,920 --> 00:11:11,370 And a very straightforward way of dealing with that is to 167 00:11:11,370 --> 00:11:16,760 simply change our notation for the Fourier transform, 168 00:11:16,760 --> 00:11:19,850 recognizing that the Fourier transform, of course, is a 169 00:11:19,850 --> 00:11:22,740 function of omega, but it's also, in fact, a 170 00:11:22,740 --> 00:11:24,470 function of j omega. 171 00:11:24,470 --> 00:11:28,060 And if we write it that way, then the two 172 00:11:28,060 --> 00:11:29,440 notations come together. 173 00:11:29,440 --> 00:11:33,520 In other words, the Laplace transform at s equals j omega 174 00:11:33,520 --> 00:11:36,260 just simply reduces both mathematically and 175 00:11:36,260 --> 00:11:39,100 notationally to the Fourier transform. 176 00:11:39,100 --> 00:11:43,400 So the notation that we'll now be adopting for the Fourier 177 00:11:43,400 --> 00:11:48,110 transform is the notation whereby we express the Fourier 178 00:11:48,110 --> 00:11:53,190 transform no longer simply as x of omega, but choosing as 179 00:11:53,190 --> 00:11:55,760 the argument j omega. 180 00:11:55,760 --> 00:11:59,620 Simple notational change. 181 00:11:59,620 --> 00:12:02,360 Now, here we see one relationship between the 182 00:12:02,360 --> 00:12:06,560 Fourier transform and the Laplace transform. 183 00:12:06,560 --> 00:12:10,180 Namely that the Laplace transform for s equals j omega 184 00:12:10,180 --> 00:12:12,770 reduces to the Fourier transform. 185 00:12:12,770 --> 00:12:17,340 We also have another important relationship. 186 00:12:17,340 --> 00:12:22,880 In particular, the fact that the Laplace transform can be 187 00:12:22,880 --> 00:12:28,160 interpreted as the Fourier transform of a modified 188 00:12:28,160 --> 00:12:29,710 version of x of t. 189 00:12:29,710 --> 00:12:32,820 Let me show you what I mean. 190 00:12:32,820 --> 00:12:35,000 Here, of course, we have the relationship 191 00:12:35,000 --> 00:12:36,190 that we just developed. 192 00:12:36,190 --> 00:12:38,225 Namely that s equals j omega. 193 00:12:38,225 --> 00:12:42,540 The Laplace transform reduces to the Fourier transform. 194 00:12:42,540 --> 00:12:45,810 But now let's look at the more general Laplace transform 195 00:12:45,810 --> 00:12:47,440 expression. 196 00:12:47,440 --> 00:12:51,800 And if we substitute in s equals sigma plus j omega, 197 00:12:51,800 --> 00:12:57,500 which is the general form for this complex variable s, and 198 00:12:57,500 --> 00:13:04,820 we carry through some of the algebra, breaking this into 199 00:13:04,820 --> 00:13:06,840 the product of two exponentials, z to the minus 200 00:13:06,840 --> 00:13:10,360 sigma t times z to the minus j omega t. 201 00:13:10,360 --> 00:13:15,530 We now have this expression where, of course, in both of 202 00:13:15,530 --> 00:13:20,140 these there is a dt. 203 00:13:20,140 --> 00:13:26,380 And now when we look at this, what we observe is that this, 204 00:13:26,380 --> 00:13:30,930 in fact, is the Fourier transform of something. 205 00:13:30,930 --> 00:13:32,160 What's the something? 206 00:13:32,160 --> 00:13:35,830 It's not x of t anymore, it's the Fourier transform of x of 207 00:13:35,830 --> 00:13:40,730 t multiplied by e to the minus sigma t. 208 00:13:40,730 --> 00:13:44,790 So if we think of these two terms together, this integral 209 00:13:44,790 --> 00:13:46,930 is just the Fourier transform. 210 00:13:46,930 --> 00:13:51,070 It's the Fourier transform of x of t multiplied by an 211 00:13:51,070 --> 00:13:52,170 exponential. 212 00:13:52,170 --> 00:13:57,600 If sigma is greater than 0, it's an exponential that 213 00:13:57,600 --> 00:13:59,720 decays with time. 214 00:13:59,720 --> 00:14:02,540 If sigma is less than 0, it's an exponential 215 00:14:02,540 --> 00:14:05,380 that grows with time. 216 00:14:05,380 --> 00:14:09,720 So we have then this additional relationship, which 217 00:14:09,720 --> 00:14:13,010 tells us that the Laplace transform is the Fourier 218 00:14:13,010 --> 00:14:19,630 transform of an exponentially weighted time function. 219 00:14:19,630 --> 00:14:24,290 Now, this exponential weighting has some important 220 00:14:24,290 --> 00:14:26,000 significance. 221 00:14:26,000 --> 00:14:29,800 In particular, recall that there were issues of 222 00:14:29,800 --> 00:14:31,840 convergence with the Fourier transform. 223 00:14:31,840 --> 00:14:35,590 In particular, the Fourier transform 224 00:14:35,590 --> 00:14:36,940 may or may not converge. 225 00:14:36,940 --> 00:14:40,700 And for convergence, in fact, what's required is that the 226 00:14:40,700 --> 00:14:43,250 time function that we're transforming be absolutely 227 00:14:43,250 --> 00:14:45,140 integrable. 228 00:14:45,140 --> 00:14:49,920 Now, we can have a time function that isn't absolutely 229 00:14:49,920 --> 00:14:52,930 integrable because, let's say, it grows 230 00:14:52,930 --> 00:14:56,760 exponentially as time increases. 231 00:14:56,760 --> 00:15:00,940 But when we multiply it by this exponential factor that's 232 00:15:00,940 --> 00:15:05,050 embodied in the Laplace transform, in fact that brings 233 00:15:05,050 --> 00:15:07,970 the function back down for positive time. 234 00:15:07,970 --> 00:15:13,430 And we'll impose absolute integrability on the product 235 00:15:13,430 --> 00:15:17,190 of x of t times e to the minus sigma t. 236 00:15:17,190 --> 00:15:22,300 And so the conclusion, an important point is that the 237 00:15:22,300 --> 00:15:25,610 Laplace transform, the Fourier transform of this product may 238 00:15:25,610 --> 00:15:27,490 converge, even though the Fourier 239 00:15:27,490 --> 00:15:29,530 transform of x of t doesn't. 240 00:15:29,530 --> 00:15:33,460 In other words, the Laplace transform may converge even 241 00:15:33,460 --> 00:15:35,970 when the Fourier transform doesn't converge. 242 00:15:35,970 --> 00:15:38,030 And we'll see that and we'll see examples of it as the 243 00:15:38,030 --> 00:15:40,830 discussion goes along. 244 00:15:40,830 --> 00:15:43,500 Now let me also draw your attention to the fact, 245 00:15:43,500 --> 00:15:48,720 although we won't be working through this in detail. 246 00:15:48,720 --> 00:15:53,520 To the fact that this equation, in effect, provides 247 00:15:53,520 --> 00:16:00,990 the basis for us to figure out how to express x of t in terms 248 00:16:00,990 --> 00:16:03,100 of the Laplace transform. 249 00:16:03,100 --> 00:16:06,350 In effect, we can apply the inverse Fourier transform to 250 00:16:06,350 --> 00:16:10,870 this, thereby to this, account for the exponential factor by 251 00:16:10,870 --> 00:16:12,980 bringing it over to the other side. 252 00:16:12,980 --> 00:16:17,260 And if you go through this, and in fact, you'll have an 253 00:16:17,260 --> 00:16:19,860 opportunity to go through this both in the video course 254 00:16:19,860 --> 00:16:26,180 manual and also it's carried through in the text, what you 255 00:16:26,180 --> 00:16:30,600 end up with is a synthesis equation, an expression for x 256 00:16:30,600 --> 00:16:34,070 of t in terms of x of s which corresponds 257 00:16:34,070 --> 00:16:35,730 to a synthesis equation. 258 00:16:35,730 --> 00:16:40,070 And which now builds x of t out of a linear combination of 259 00:16:40,070 --> 00:16:44,570 not necessarily functions of the form e to the j omega t, 260 00:16:44,570 --> 00:16:50,930 but in terms of functions or basic signals which are more 261 00:16:50,930 --> 00:16:56,160 general exponentials e to the st. 262 00:16:56,160 --> 00:17:01,720 OK, well, let's just look at some examples of the Laplace 263 00:17:01,720 --> 00:17:03,320 transform of some time functions. 264 00:17:03,320 --> 00:17:07,980 And these examples that I'll go through are all examples 265 00:17:07,980 --> 00:17:10,410 that are worked out in the text. 266 00:17:10,410 --> 00:17:12,790 And so I don't want to focus on the algebra. 267 00:17:12,790 --> 00:17:15,990 What I'd like to focus on are some of the issues and the 268 00:17:15,990 --> 00:17:18,450 interpretation. 269 00:17:18,450 --> 00:17:24,010 Let's first of all, look at the example in the text, which 270 00:17:24,010 --> 00:17:26,329 is Example 9.1. 271 00:17:26,329 --> 00:17:31,720 If we take the Fourier transform of this exponential, 272 00:17:31,720 --> 00:17:35,745 then, as you well know, the result we have is 1 over j 273 00:17:35,745 --> 00:17:37,500 omega plus a. 274 00:17:37,500 --> 00:17:41,010 And that can't converge for any a. 275 00:17:41,010 --> 00:17:44,680 In particular, it's only for a greater than 0. 276 00:17:44,680 --> 00:17:48,090 What that really means is that for convergence of the Fourier 277 00:17:48,090 --> 00:17:52,840 transform, this has to be a decaying exponential. 278 00:17:52,840 --> 00:17:56,620 It can't be an increasing exponential. 279 00:17:56,620 --> 00:18:01,620 If instead we apply the Laplace transform to this, 280 00:18:01,620 --> 00:18:06,030 applying the Laplace transform is the same as taking the 281 00:18:06,030 --> 00:18:11,430 Fourier transform of x of t times an exponential, and the 282 00:18:11,430 --> 00:18:14,160 exponent that we would multiply by is e to 283 00:18:14,160 --> 00:18:15,710 the minus sigma t. 284 00:18:15,710 --> 00:18:21,100 So in effect, taking the Laplace transform of this is 285 00:18:21,100 --> 00:18:26,032 like taking the Fourier transform of e to the minus at 286 00:18:26,032 --> 00:18:29,860 e to the minus sigma t. 287 00:18:29,860 --> 00:18:34,490 And if we carry that through, just working through the 288 00:18:34,490 --> 00:18:38,750 integral, we end up with a Laplace transform, which is 1 289 00:18:38,750 --> 00:18:42,200 over s plus a. 290 00:18:42,200 --> 00:18:47,710 But just as in the Fourier transform, the Fourier 291 00:18:47,710 --> 00:18:50,980 transform won't converge for any a. 292 00:18:50,980 --> 00:18:56,110 Now what happens is that the Laplace transform will only 293 00:18:56,110 --> 00:19:00,530 converge when the Fourier transform of this converges. 294 00:19:00,530 --> 00:19:04,430 Said another way, it's when the combination of a plus 295 00:19:04,430 --> 00:19:07,490 sigma is greater than 0. 296 00:19:07,490 --> 00:19:11,860 So we would require that, if I write it over here, a plus 297 00:19:11,860 --> 00:19:14,000 sigma is greater than 0. 298 00:19:14,000 --> 00:19:19,510 Or that sigma is greater than minus a. 299 00:19:19,510 --> 00:19:24,460 So in fact, in the Laplace transform of this, we have an 300 00:19:24,460 --> 00:19:26,880 expression 1 over s plus a. 301 00:19:26,880 --> 00:19:32,260 But we also require, in interpreting that, that the 302 00:19:32,260 --> 00:19:37,430 real part of s be greater than minus a. 303 00:19:37,430 --> 00:19:42,120 So that, essentially, the Fourier transform of x of t 304 00:19:42,120 --> 00:19:45,300 times e to the minus sigma t converges. 305 00:19:45,300 --> 00:19:49,960 So it's important to recognize that the algebraic expression 306 00:19:49,960 --> 00:19:55,040 that we get is only valid for certain values of the 307 00:19:55,040 --> 00:19:56,580 real part of s. 308 00:19:56,580 --> 00:20:01,180 And so, for this example, we can summarize it as this 309 00:20:01,180 --> 00:20:05,110 exponential has a Laplace transform, which is 1 over s 310 00:20:05,110 --> 00:20:09,560 plus a, where s is restricted to the range the real part of 311 00:20:09,560 --> 00:20:11,430 s greater than minus a. 312 00:20:14,020 --> 00:20:18,900 Now, we haven't had this issue before of restrictions on what 313 00:20:18,900 --> 00:20:20,210 the value of s is. 314 00:20:20,210 --> 00:20:22,600 With the Fourier transform, either it converged or it 315 00:20:22,600 --> 00:20:23,880 didn't converge. 316 00:20:23,880 --> 00:20:27,760 With the Laplace transform, there are certain values of s. 317 00:20:27,760 --> 00:20:30,620 We now have more flexibility, and so there's certain values 318 00:20:30,620 --> 00:20:33,690 of the real part of s for which it converges and certain 319 00:20:33,690 --> 00:20:36,360 values for which it doesn't. 320 00:20:36,360 --> 00:20:39,110 The values of s for which the Laplace 321 00:20:39,110 --> 00:20:42,880 transform converges is-- 322 00:20:42,880 --> 00:20:47,570 the values are referred to as the region of convergence of 323 00:20:47,570 --> 00:20:50,580 the Laplace transform. 324 00:20:50,580 --> 00:20:56,130 And it's important to recognize that in specifying 325 00:20:56,130 --> 00:20:59,370 the Laplace transform, what's required is not only the 326 00:20:59,370 --> 00:21:05,370 algebraic expression, but also the domain or set of values of 327 00:21:05,370 --> 00:21:12,210 s for which that algebraic expression is valid. 328 00:21:12,210 --> 00:21:16,670 Just to underscore that point, let me draw your attention to 329 00:21:16,670 --> 00:21:21,890 another example in the text, which is Example 9.2. 330 00:21:21,890 --> 00:21:27,445 In Example 9.2, we have an exponential for negative time, 331 00:21:27,445 --> 00:21:30,040 0 for positive time. 332 00:21:30,040 --> 00:21:33,380 And if you carry through the algebra there, you end up with 333 00:21:33,380 --> 00:21:35,950 a Laplace transform expression, which is again 1 334 00:21:35,950 --> 00:21:38,660 over s plus a. 335 00:21:38,660 --> 00:21:42,280 Exactly the same algebraic expression as we had for the 336 00:21:42,280 --> 00:21:44,490 previous example. 337 00:21:44,490 --> 00:21:49,660 The important distinction is that now the real part of s is 338 00:21:49,660 --> 00:21:52,170 restricted to be less than minus a. 339 00:21:52,170 --> 00:21:56,750 And so, in fact, if you compare this example with the 340 00:21:56,750 --> 00:22:00,180 one above it, and let's just look back at the answer that 341 00:22:00,180 --> 00:22:03,350 we had there. 342 00:22:03,350 --> 00:22:07,240 If you compare those two examples, here the algebraic 343 00:22:07,240 --> 00:22:11,820 expression is 1 over s plus a with a certain region of 344 00:22:11,820 --> 00:22:13,130 convergence. 345 00:22:13,130 --> 00:22:18,260 Here the algebraic expression is 1 over s plus a. 346 00:22:18,260 --> 00:22:21,260 And the only difference between those two is the 347 00:22:21,260 --> 00:22:23,180 domain or region of convergence. 348 00:22:23,180 --> 00:22:27,840 So there is another complication, or twist, now. 349 00:22:27,840 --> 00:22:32,040 Not only do we need to generate the algebraic 350 00:22:32,040 --> 00:22:37,200 expression, but we also have to be careful to specify the 351 00:22:37,200 --> 00:22:41,500 region of convergence over which that algebraic 352 00:22:41,500 --> 00:22:44,280 expression is valid. 353 00:22:44,280 --> 00:22:49,450 Now, later on in this lecture, and actually also as the 354 00:22:49,450 --> 00:22:52,830 discussion of the Laplace transform goes on, we'll begin 355 00:22:52,830 --> 00:22:56,350 to see and understand more about how the region of 356 00:22:56,350 --> 00:22:59,360 convergence relates to various 357 00:22:59,360 --> 00:23:03,420 properties of the time function. 358 00:23:03,420 --> 00:23:07,070 Well, let's finally look at one additional example from 359 00:23:07,070 --> 00:23:10,550 the text, And this is Example 9.3. 360 00:23:10,550 --> 00:23:15,420 And what it consists of is the time function, which is the 361 00:23:15,420 --> 00:23:18,620 sum of two exponentials. 362 00:23:18,620 --> 00:23:22,190 And although we haven't formally talked about 363 00:23:22,190 --> 00:23:24,920 properties of the Laplace transform yet, one of the 364 00:23:24,920 --> 00:23:26,210 properties that we'll see-- and it's 365 00:23:26,210 --> 00:23:28,050 relatively easy to develop-- 366 00:23:28,050 --> 00:23:32,260 is the fact that the Laplace transform of a sum is the sum 367 00:23:32,260 --> 00:23:33,430 of the Laplace transform. 368 00:23:33,430 --> 00:23:38,330 So, in fact, we can get the Laplace transform of the sum 369 00:23:38,330 --> 00:23:44,910 of these two terms as the sum of the Laplace transforms. 370 00:23:44,910 --> 00:23:49,390 So for this one, we know from the example that we looked at 371 00:23:49,390 --> 00:23:53,810 previously, Example 9.1, that this is of the form 1 over s 372 00:23:53,810 --> 00:23:57,940 plus 1 with a region of convergence, which is the real 373 00:23:57,940 --> 00:24:01,180 part of s greater than minus 1. 374 00:24:01,180 --> 00:24:04,030 For this one, we have a Laplace transform which is 1 375 00:24:04,030 --> 00:24:08,810 over s plus 2 with a region of convergence which is the real 376 00:24:08,810 --> 00:24:12,270 part of s greater than minus 2. 377 00:24:12,270 --> 00:24:16,010 So for the two of them together, we have to take the 378 00:24:16,010 --> 00:24:18,140 overlap of those two regions. 379 00:24:18,140 --> 00:24:21,540 In other words, we have to take the region that 380 00:24:21,540 --> 00:24:24,810 encompasses both the real part of s greater than minus 1 and 381 00:24:24,810 --> 00:24:27,230 the real part of s greater than minus 2. 382 00:24:27,230 --> 00:24:31,870 And if we put those together, then we have a combined region 383 00:24:31,870 --> 00:24:34,140 of convergence, which is the real part of s 384 00:24:34,140 --> 00:24:36,610 greater than minus 1. 385 00:24:36,610 --> 00:24:39,140 So this is the expression. 386 00:24:39,140 --> 00:24:44,270 And for this particular example, what we have is a 387 00:24:44,270 --> 00:24:46,450 ratio of polynomials. 388 00:24:46,450 --> 00:24:49,830 The ratio of polynomials, there's a numerator polynomial 389 00:24:49,830 --> 00:24:52,080 and a denominator polynomial. 390 00:24:52,080 --> 00:24:59,020 And it's convenient to summarize these by plotting 391 00:24:59,020 --> 00:25:02,180 the roots of the numerator polynomial and the roots of 392 00:25:02,180 --> 00:25:06,110 the denominator polynomial in the complex plane. 393 00:25:06,110 --> 00:25:10,650 And the complex plane which they're plotted is referred to 394 00:25:10,650 --> 00:25:13,350 the s-plane. 395 00:25:13,350 --> 00:25:18,590 So we can, for example, take the denominator polynomial and 396 00:25:18,590 --> 00:25:24,230 summarize it by specifying the fact, or by representing the 397 00:25:24,230 --> 00:25:28,710 fact that it has roots at s equals minus 1 and at s 398 00:25:28,710 --> 00:25:30,190 equals minus 2. 399 00:25:30,190 --> 00:25:33,950 And I've done that in this picture by putting an x where 400 00:25:33,950 --> 00:25:37,850 the roots of the denominator polynomial are. 401 00:25:37,850 --> 00:25:42,720 The numerator polynomial has a root at s equals minus 3/2, 402 00:25:42,720 --> 00:25:45,070 and I've represented that by a circle. 403 00:25:45,070 --> 00:25:48,510 So these are the roots of the denominator polynomial and 404 00:25:48,510 --> 00:25:50,070 this is the root of the numerator 405 00:25:50,070 --> 00:25:52,220 polynomial for this example. 406 00:25:52,220 --> 00:25:56,540 And also, for this example, we can represent the region of 407 00:25:56,540 --> 00:26:00,300 convergence, which is the real part of s 408 00:26:00,300 --> 00:26:01,600 greater than minus 1. 409 00:26:01,600 --> 00:26:06,720 And so that's, in fact, the region over here. 410 00:26:06,720 --> 00:26:09,850 There is also, if I draw these, just the roots of the 411 00:26:09,850 --> 00:26:13,180 numerator and denominator of polynomials, I would need an 412 00:26:13,180 --> 00:26:16,820 additional piece of information to specify the 413 00:26:16,820 --> 00:26:18,460 algebraic expression completely. 414 00:26:18,460 --> 00:26:21,250 Namely, a multiplying constant out in front 415 00:26:21,250 --> 00:26:24,190 of the whole thing. 416 00:26:24,190 --> 00:26:31,220 Well, this particular example, has the Laplace transform as a 417 00:26:31,220 --> 00:26:32,600 rational function. 418 00:26:32,600 --> 00:26:35,950 Namely, it's one polynomial in the numerator and another 419 00:26:35,950 --> 00:26:38,040 polynomial in the denominator. 420 00:26:38,040 --> 00:26:41,610 And in fact, as we'll see, Laplace transforms, which are 421 00:26:41,610 --> 00:26:45,170 ratios of polynomials, form a very important class. 422 00:26:45,170 --> 00:26:48,430 They, in fact, represent systems that are describable 423 00:26:48,430 --> 00:26:51,810 by linear constant coefficient differential equations. 424 00:26:51,810 --> 00:26:53,510 You shouldn't necessarily-- 425 00:26:53,510 --> 00:26:55,420 in fact, for sure you shouldn't see 426 00:26:55,420 --> 00:26:56,850 why that's true now. 427 00:26:56,850 --> 00:26:59,050 We'll see that later. 428 00:26:59,050 --> 00:27:06,060 But that means that Laplace transforms that are rational 429 00:27:06,060 --> 00:27:10,280 functions, namely, the ratio of a numerator polynomial 430 00:27:10,280 --> 00:27:14,180 divided by the denominator polynomial, become very 431 00:27:14,180 --> 00:27:18,640 important in the discussion that follows. 432 00:27:18,640 --> 00:27:22,760 And in fact, we have some terminology for this. 433 00:27:22,760 --> 00:27:27,030 The roots of the numerator polynomial are referred to as 434 00:27:27,030 --> 00:27:30,750 the zeroes of the Laplace transform. 435 00:27:30,750 --> 00:27:34,165 Because, of course, those are the values of s at which x of 436 00:27:34,165 --> 00:27:36,790 s becomes 0. 437 00:27:36,790 --> 00:27:40,130 And the roots of the denominator polynomial are 438 00:27:40,130 --> 00:27:44,900 referred to as the poles of the Laplace transform. 439 00:27:44,900 --> 00:27:49,840 And those are the values of s at which the Laplace 440 00:27:49,840 --> 00:27:52,090 transform blows up. 441 00:27:52,090 --> 00:27:53,600 Namely, becomes infinite. 442 00:27:53,600 --> 00:27:57,430 If you think of setting s equal to a value where this 443 00:27:57,430 --> 00:28:00,780 denominator polynomial goes to 0, of course, x 444 00:28:00,780 --> 00:28:02,880 of s becomes infinite. 445 00:28:02,880 --> 00:28:06,860 And what we would expect and, of course, we'll see 446 00:28:06,860 --> 00:28:09,360 that this is true. 447 00:28:09,360 --> 00:28:12,670 What we would expect is that wherever that happens, there 448 00:28:12,670 --> 00:28:14,750 must be some problem with convergence 449 00:28:14,750 --> 00:28:15,930 of the Laplace transform. 450 00:28:15,930 --> 00:28:18,160 And indeed, the Laplace transform doesn't 451 00:28:18,160 --> 00:28:20,960 converge at the poles. 452 00:28:20,960 --> 00:28:24,390 Namely, at the roots of the denominator polynomial. 453 00:28:24,390 --> 00:28:28,120 So, in fact, let's focus in on that a little further. 454 00:28:28,120 --> 00:28:33,980 Let's examine and talk about the region of convergence of 455 00:28:33,980 --> 00:28:38,200 the Laplace transform, and how it's associated both with 456 00:28:38,200 --> 00:28:41,970 properties of the time function, and also with the 457 00:28:41,970 --> 00:28:46,720 location of the poles of the Laplace transform. 458 00:28:46,720 --> 00:28:50,320 And as we'll see, there are some very specific and 459 00:28:50,320 --> 00:28:54,500 important relationships and conclusions that we can draw 460 00:28:54,500 --> 00:28:58,790 about how the region of convergence is constrained and 461 00:28:58,790 --> 00:29:05,790 associated with the locations of the poles in the s-plane. 462 00:29:05,790 --> 00:29:11,310 Well, to begin with, we can, of course, make the statement 463 00:29:11,310 --> 00:29:14,350 as I've just made that the region of 464 00:29:14,350 --> 00:29:17,260 convergence contains no poles. 465 00:29:17,260 --> 00:29:22,600 In particular, if I think of this general rational 466 00:29:22,600 --> 00:29:29,310 function, the poles of x of s are the values of s at which 467 00:29:29,310 --> 00:29:30,870 the denominator is 0. 468 00:29:30,870 --> 00:29:34,560 Or equivalently, x of s blows up. 469 00:29:34,560 --> 00:29:38,160 And of course then, that implies that the expression 470 00:29:38,160 --> 00:29:41,460 has no longer converged. 471 00:29:41,460 --> 00:29:43,370 Well, that's one statement that we can make. 472 00:29:43,370 --> 00:29:46,160 Now, there are some others. 473 00:29:46,160 --> 00:29:51,170 And one, for example, is the statement that if I have a 474 00:29:51,170 --> 00:29:55,820 point in the s-plane that corresponds to convergence, 475 00:29:55,820 --> 00:30:00,680 then in fact any line in the s-plane with that same real 476 00:30:00,680 --> 00:30:05,100 part will also be a set of values for which the Laplace 477 00:30:05,100 --> 00:30:07,110 transform converges. 478 00:30:07,110 --> 00:30:09,450 And what's the reason for that? 479 00:30:09,450 --> 00:30:15,910 The reason for that is that s is sigma plus j omega and 480 00:30:15,910 --> 00:30:20,840 convergence of the Laplace transform is associated with 481 00:30:20,840 --> 00:30:25,580 convergence of the Fourier transform of e to the minus 482 00:30:25,580 --> 00:30:28,390 sigma t times x of t. 483 00:30:28,390 --> 00:30:31,950 And so the convergence only depends on sigma. 484 00:30:31,950 --> 00:30:37,010 If it only depends on sigma, then if it converges for one 485 00:30:37,010 --> 00:30:39,200 value of sigma-- 486 00:30:39,200 --> 00:30:43,980 I'm sorry, for a value of sigma for some value of omega, 487 00:30:43,980 --> 00:30:47,840 then it will converge for that same sigma for 488 00:30:47,840 --> 00:30:50,130 any value of omega. 489 00:30:50,130 --> 00:30:55,650 The conclusion then is that the region of convergence, if 490 00:30:55,650 --> 00:30:58,110 I have a point, then I also have a line. 491 00:30:58,110 --> 00:31:01,080 And so what that suggests is that as we look at the region 492 00:31:01,080 --> 00:31:05,930 of convergence, it in fact corresponds to strips in the 493 00:31:05,930 --> 00:31:09,360 complex plane. 494 00:31:09,360 --> 00:31:12,930 Now, finally we can tie together the region of 495 00:31:12,930 --> 00:31:17,980 convergence to the convergence of the Fourier transform. 496 00:31:17,980 --> 00:31:23,220 In particular, since we know that the Laplace transform 497 00:31:23,220 --> 00:31:28,480 reduces to the Fourier transform when the complex 498 00:31:28,480 --> 00:31:34,770 variable s is equal to j omega, the implication is that 499 00:31:34,770 --> 00:31:40,000 if we have the Laplace transform and if the Laplace 500 00:31:40,000 --> 00:31:43,300 transform reduces to the Fourier transform when 501 00:31:43,300 --> 00:31:44,440 sigma equals 0. 502 00:31:44,440 --> 00:31:48,140 In other words, when s is equal to j omega, then the 503 00:31:48,140 --> 00:31:52,600 Fourier transform of x of t converging is equivalent to 504 00:31:52,600 --> 00:31:57,530 the statement that the Laplace transform converges for sigma 505 00:31:57,530 --> 00:31:58,930 equal to 0. 506 00:31:58,930 --> 00:32:00,310 In other words, that the region of 507 00:32:00,310 --> 00:32:04,340 convergence includes what? 508 00:32:04,340 --> 00:32:06,690 The j omega axis in the s-plane. 509 00:32:10,880 --> 00:32:14,230 So we have then some statements that kind of tie 510 00:32:14,230 --> 00:32:17,680 together the location of the poles and the region of 511 00:32:17,680 --> 00:32:18,270 convergence. 512 00:32:18,270 --> 00:32:21,340 Let me make one other statement, which is a much 513 00:32:21,340 --> 00:32:22,800 harder statement to justify. 514 00:32:22,800 --> 00:32:25,240 And I won't try to, I'll just simply state it. 515 00:32:25,240 --> 00:32:27,810 And that is that the region of convergence of the Laplace 516 00:32:27,810 --> 00:32:31,320 transform is a connected region. 517 00:32:31,320 --> 00:32:37,330 In other words, if the entire region consists of a single 518 00:32:37,330 --> 00:32:42,480 strip in the s-plane, it can't consist of a strip over here, 519 00:32:42,480 --> 00:32:44,650 for example, and a strip over there. 520 00:32:44,650 --> 00:32:50,420 Well, let me emphasize some of those points a little further. 521 00:32:50,420 --> 00:32:59,060 Let's suppose that I have a Laplace transform, and the 522 00:32:59,060 --> 00:33:03,280 Laplace transform that I'm talking about is a rational 523 00:33:03,280 --> 00:33:09,610 function, which is 1 over s plus 1 times s plus 2. 524 00:33:09,610 --> 00:33:14,460 Then the pole-zero pattern, as it's referred to, in the 525 00:33:14,460 --> 00:33:18,160 s-plane, the location of the roots of the numerator and 526 00:33:18,160 --> 00:33:19,490 denominator polynomials. 527 00:33:19,490 --> 00:33:22,230 Of course, there is no numerator polynomial. 528 00:33:22,230 --> 00:33:25,850 The denominator polynomial roots, which I've represented 529 00:33:25,850 --> 00:33:28,440 by these x's, are shown here. 530 00:33:28,440 --> 00:33:33,430 And so this is the pole-zero pattern. 531 00:33:33,430 --> 00:33:37,690 And from what I've said, the region of convergence can't 532 00:33:37,690 --> 00:33:42,590 include any poles and it must correspond to 533 00:33:42,590 --> 00:33:44,860 strips in the s-plane. 534 00:33:44,860 --> 00:33:49,160 And furthermore, it must be just one connected region 535 00:33:49,160 --> 00:33:51,640 rather than multiple regions. 536 00:33:51,640 --> 00:33:57,090 And so with this algebraic expression then, the possible 537 00:33:57,090 --> 00:34:00,250 choices for the region of convergence consistent with 538 00:34:00,250 --> 00:34:02,550 those properties are the following. 539 00:34:02,550 --> 00:34:05,830 One of them would be a region of convergence to the 540 00:34:05,830 --> 00:34:09,489 right of this pole. 541 00:34:09,489 --> 00:34:16,190 A second would be a region of convergence which lies between 542 00:34:16,190 --> 00:34:21,570 the two poles as I show here. 543 00:34:21,570 --> 00:34:27,350 And a third is a region of convergence which is to the 544 00:34:27,350 --> 00:34:31,199 left of this pole. 545 00:34:31,199 --> 00:34:36,010 And because of the fact that I said without proof that the 546 00:34:36,010 --> 00:34:39,870 region of convergence must be a single strip, it can't be 547 00:34:39,870 --> 00:34:41,159 multiple strips. 548 00:34:41,159 --> 00:34:44,460 In fact, we could not consider, as a possible region 549 00:34:44,460 --> 00:34:48,060 of convergence, what I show here. 550 00:34:48,060 --> 00:34:52,810 So, in fact, this is not a valid region of convergence. 551 00:34:52,810 --> 00:34:56,750 There are only three possibilities associated with 552 00:34:56,750 --> 00:34:58,900 this pole-zero pattern. 553 00:34:58,900 --> 00:35:02,270 Namely, to the right of this pole, between the two poles, 554 00:35:02,270 --> 00:35:06,500 and to the left of this pole. 555 00:35:06,500 --> 00:35:11,250 Now, to carry the discussion further, we can, in fact, 556 00:35:11,250 --> 00:35:16,490 associate the region of convergence of the Laplace 557 00:35:16,490 --> 00:35:21,180 transform with some very specific characteristics of 558 00:35:21,180 --> 00:35:22,540 the time function. 559 00:35:22,540 --> 00:35:28,790 And what this will do is to help us understand how for 560 00:35:28,790 --> 00:35:32,320 various choices of the region of convergence, the 561 00:35:32,320 --> 00:35:35,740 interpretation that we can impose on the 562 00:35:35,740 --> 00:35:37,310 related time function. 563 00:35:37,310 --> 00:35:40,360 Let me show you what I mean. 564 00:35:40,360 --> 00:35:46,860 Suppose that we start with a time function as I indicate 565 00:35:46,860 --> 00:35:53,820 here, which is a finite duration time function. 566 00:35:53,820 --> 00:35:58,230 In other words, it's 0 except in some time interval. 567 00:35:58,230 --> 00:36:03,200 Now, recall that the Fourier transform converges if the 568 00:36:03,200 --> 00:36:06,580 time function has the property that it's absolutely 569 00:36:06,580 --> 00:36:07,450 integrable. 570 00:36:07,450 --> 00:36:10,870 And as long as everything's stays finite in terms of 571 00:36:10,870 --> 00:36:14,130 amplitudes in a finite duration signal, there's no 572 00:36:14,130 --> 00:36:17,830 difficulty that we're going to run into here. 573 00:36:17,830 --> 00:36:21,040 Now, here the Fourier transform will converge. 574 00:36:21,040 --> 00:36:25,920 And now the question is, what can we say about the region of 575 00:36:25,920 --> 00:36:29,770 convergence of the Laplace transform? 576 00:36:29,770 --> 00:36:34,700 Well, the Laplace transform is the Fourier transform of the 577 00:36:34,700 --> 00:36:37,810 time function multiplied by an exponential. 578 00:36:37,810 --> 00:36:41,820 And so we can ask about whether we can destroy the 579 00:36:41,820 --> 00:36:45,560 absolute integrability of this by multiplying by an 580 00:36:45,560 --> 00:36:49,510 exponential that grows to fast or decays 581 00:36:49,510 --> 00:36:50,490 too fast, or whatever. 582 00:36:50,490 --> 00:36:53,130 And let's take a look at that. 583 00:36:53,130 --> 00:36:57,310 Suppose that this time function is absolutely 584 00:36:57,310 --> 00:36:59,230 integrable. 585 00:36:59,230 --> 00:37:05,550 And let's multiply it by a decaying exponential. 586 00:37:05,550 --> 00:37:09,540 So this is now x of t times z to the minus sigma t if I 587 00:37:09,540 --> 00:37:11,960 think of multiplying these two together. 588 00:37:11,960 --> 00:37:16,260 And what you can see is that for positive time, sort of 589 00:37:16,260 --> 00:37:20,140 thinking informally, I'm helping the integrability of 590 00:37:20,140 --> 00:37:23,680 the product because I'm pushing this part down. 591 00:37:23,680 --> 00:37:25,960 For negative time, unfortunately, I'm making 592 00:37:25,960 --> 00:37:28,030 things grow. 593 00:37:28,030 --> 00:37:31,710 But I don't let them grow indefinitely because there's 594 00:37:31,710 --> 00:37:36,180 some time before which this is equal to 0. 595 00:37:36,180 --> 00:37:44,660 Likewise, if I had a growing exponential, then for a 596 00:37:44,660 --> 00:37:48,960 growing exponential for negative time, or for this 597 00:37:48,960 --> 00:37:52,720 part, I'm making things smaller. 598 00:37:52,720 --> 00:37:55,580 For positive time, eventually this exponential is growing 599 00:37:55,580 --> 00:37:56,930 without bound. 600 00:37:56,930 --> 00:38:01,370 But the time function stops at some point. 601 00:38:01,370 --> 00:38:08,580 So the idea then kind of is that for a finite duration 602 00:38:08,580 --> 00:38:12,760 time function, no matter what kind of exponential I multiply 603 00:38:12,760 --> 00:38:16,720 by, whether it's going this way or going this way, because 604 00:38:16,720 --> 00:38:19,870 of the fact that essentially the limits on the integral are 605 00:38:19,870 --> 00:38:24,940 finite, I'm guaranteed that I'll always maintain absolute 606 00:38:24,940 --> 00:38:26,370 integrability. 607 00:38:26,370 --> 00:38:30,890 And so, in fact then, for a finite duration time function, 608 00:38:30,890 --> 00:38:33,800 the region of convergence is the entire s-plane. 609 00:38:37,100 --> 00:38:41,890 Now, we can also make statements about other kinds 610 00:38:41,890 --> 00:38:43,470 of time functions. 611 00:38:43,470 --> 00:38:52,280 And let's look at a time function which I define as a 612 00:38:52,280 --> 00:38:55,470 right-sided time function. 613 00:38:55,470 --> 00:39:01,720 And a right-sided time function is one which is 0 up 614 00:39:01,720 --> 00:39:06,200 until some time, and then it goes on after that, presumably 615 00:39:06,200 --> 00:39:08,050 off to infinity. 616 00:39:08,050 --> 00:39:12,850 Now, let me remind you that the whole issue here with the 617 00:39:12,850 --> 00:39:18,540 region of convergence has to do with exponentials that we 618 00:39:18,540 --> 00:39:23,850 can multiply a time function by and have the product end up 619 00:39:23,850 --> 00:39:26,430 being absolutely integrable. 620 00:39:26,430 --> 00:39:32,020 Well, suppose that when I multiply this time function by 621 00:39:32,020 --> 00:39:35,400 an exponential which, let's say decays. 622 00:39:35,400 --> 00:39:39,860 But an exponential e to the minus sigma 0 t, what you can 623 00:39:39,860 --> 00:39:43,730 see sort of intuitively is that if this product is 624 00:39:43,730 --> 00:39:49,030 absolutely integrable, if I were to increase sigma 0, then 625 00:39:49,030 --> 00:39:52,090 I'm making things even better for positive time because I'm 626 00:39:52,090 --> 00:39:53,640 pushing them down. 627 00:39:53,640 --> 00:39:57,730 And whereas they might be worse for negative time, that 628 00:39:57,730 --> 00:40:01,170 doesn't matter because before some time the 629 00:40:01,170 --> 00:40:02,540 product is equal to 0. 630 00:40:02,540 --> 00:40:08,200 So if this product is absolutely integrable, then if 631 00:40:08,200 --> 00:40:12,840 I chose an exponential e to the minus sigma 1t where sigma 632 00:40:12,840 --> 00:40:17,370 1 is greater than sigma 0, then that product will also be 633 00:40:17,370 --> 00:40:19,470 absolutely integrable. 634 00:40:19,470 --> 00:40:23,270 And we can draw an important conclusion about that, about 635 00:40:23,270 --> 00:40:25,420 the region of convergence from that. 636 00:40:25,420 --> 00:40:30,700 In particular, we can make the statement that if the time 637 00:40:30,700 --> 00:40:38,310 function is right-sided and if convergence occurs for some 638 00:40:38,310 --> 00:40:44,250 value sigma 0, then in fact, we will have convergence of 639 00:40:44,250 --> 00:40:49,700 the Laplace transform for all values of the real part of s 640 00:40:49,700 --> 00:40:51,770 greater than sigma 0. 641 00:40:51,770 --> 00:40:56,610 The reason, of course, being that if sigma 0 increases, 642 00:40:56,610 --> 00:41:03,210 then the exponential decays even faster for positive time. 643 00:41:03,210 --> 00:41:06,070 Now what that says then thinking another way, in terms 644 00:41:06,070 --> 00:41:09,700 of the region of convergence as we might draw it in the 645 00:41:09,700 --> 00:41:13,080 s-plane, is that if we have a point that's in the region of 646 00:41:13,080 --> 00:41:17,450 convergence corresponding to some value sigma 0, then all 647 00:41:17,450 --> 00:41:21,580 values of s to the right of that in the s-plane will also 648 00:41:21,580 --> 00:41:24,330 be in the region of convergence. 649 00:41:24,330 --> 00:41:26,800 We can also combine that with the statement that for 650 00:41:26,800 --> 00:41:31,490 rational functions we know that there can't be any poles 651 00:41:31,490 --> 00:41:33,190 in the region of convergence. 652 00:41:33,190 --> 00:41:36,190 If you put those two statements together, then we 653 00:41:36,190 --> 00:41:41,830 end up with a statement that if x of t is right-sided and 654 00:41:41,830 --> 00:41:46,060 if its Laplace transform is rational, then the region of 655 00:41:46,060 --> 00:41:51,280 convergence is to the right of the rightmost pole. 656 00:41:51,280 --> 00:41:55,130 So we have here a very important insight, which tells 657 00:41:55,130 --> 00:41:59,980 us that we can infer some property about the time 658 00:41:59,980 --> 00:42:02,640 function from the region of convergence. 659 00:42:02,640 --> 00:42:05,880 Or conversely, if we know something about the time 660 00:42:05,880 --> 00:42:09,440 function, namely being right-sided, then we can infer 661 00:42:09,440 --> 00:42:12,570 something about the region of convergence. 662 00:42:12,570 --> 00:42:17,090 Well, in addition to right-sided signals, we can 663 00:42:17,090 --> 00:42:19,510 also have left-sided signals. 664 00:42:19,510 --> 00:42:22,370 And a left-sided signal is essentially a right-sided 665 00:42:22,370 --> 00:42:23,910 signal turned around. 666 00:42:23,910 --> 00:42:27,480 In other words, a left-sided signal is one that is 667 00:42:27,480 --> 00:42:30,460 0 after some time. 668 00:42:30,460 --> 00:42:32,850 Well, we can carry out exactly the same 669 00:42:32,850 --> 00:42:34,580 kind of argument there. 670 00:42:34,580 --> 00:42:37,800 Namely, if the signal goes off to infinity in the negative 671 00:42:37,800 --> 00:42:42,690 time direction and stops some place for positive time, if I 672 00:42:42,690 --> 00:42:46,110 have an exponential that I can multiply it by and have that 673 00:42:46,110 --> 00:42:48,620 product be absolutely integrable. 674 00:42:48,620 --> 00:42:51,590 And if I choose an exponential that decays even faster for 675 00:42:51,590 --> 00:42:54,560 negative time so that I'm pushing the stuff way out 676 00:42:54,560 --> 00:42:58,830 there down even further, then I enhance the 677 00:42:58,830 --> 00:43:00,850 integrability even more. 678 00:43:00,850 --> 00:43:03,830 And you might have to think through that a little bit, but 679 00:43:03,830 --> 00:43:07,170 it's exactly the flip side of the argument 680 00:43:07,170 --> 00:43:08,890 for right-sided signals. 681 00:43:08,890 --> 00:43:14,470 And the conclusion then is that if we have a left-sided 682 00:43:14,470 --> 00:43:20,010 signal and we have a point, a value of the real part of s 683 00:43:20,010 --> 00:43:23,400 which is in the region of convergence, then in fact, all 684 00:43:23,400 --> 00:43:29,140 values to the left of that point in the s-plane will also 685 00:43:29,140 --> 00:43:32,500 be in the region of convergence. 686 00:43:32,500 --> 00:43:35,330 Now, similar to the statement that we made for right-sided 687 00:43:35,330 --> 00:43:39,230 signals, if x of t is left-sided and, in fact, we're 688 00:43:39,230 --> 00:43:41,720 talking about a rational Laplace transform, which we 689 00:43:41,720 --> 00:43:43,830 most typically will. 690 00:43:43,830 --> 00:43:47,730 Then, in fact, we can make the statement that the region of 691 00:43:47,730 --> 00:43:53,290 convergence is to the left of the leftmost pole because we 692 00:43:53,290 --> 00:43:56,730 know if we find a point that's in the region of convergence, 693 00:43:56,730 --> 00:43:58,930 everything to the left of that has to be in the region of 694 00:43:58,930 --> 00:43:59,870 convergence. 695 00:43:59,870 --> 00:44:02,480 We can't have any poles in the region of convergence. 696 00:44:02,480 --> 00:44:04,960 You put those two statements together and it says it's to 697 00:44:04,960 --> 00:44:07,710 the left of the leftmost pole. 698 00:44:07,710 --> 00:44:12,340 Now the final situation is the situation where we have a 699 00:44:12,340 --> 00:44:14,680 signal which is neither right-sided nor left-sided. 700 00:44:14,680 --> 00:44:18,010 It goes off to infinity for positive time and it goes off 701 00:44:18,010 --> 00:44:20,100 to infinity for negative time. 702 00:44:20,100 --> 00:44:23,460 And there the thing to kind of recognize is that if you 703 00:44:23,460 --> 00:44:28,370 multiply by an exponential, and it's decaying very fast 704 00:44:28,370 --> 00:44:30,750 for positive time, it's going to be growing very fast for 705 00:44:30,750 --> 00:44:32,060 negative time. 706 00:44:32,060 --> 00:44:35,060 Conversely, if it's decaying very fast for negative time, 707 00:44:35,060 --> 00:44:37,530 it's growing very fast for positive time. 708 00:44:37,530 --> 00:44:39,840 And there's this notion of trying to balance 709 00:44:39,840 --> 00:44:41,710 the value of sigma. 710 00:44:41,710 --> 00:44:44,620 And in effect, what that says is that the region of 711 00:44:44,620 --> 00:44:47,810 convergence can't extent too far to the left or 712 00:44:47,810 --> 00:44:49,450 too far to the right. 713 00:44:49,450 --> 00:44:55,790 Said another way for a two-sided signal, if we have a 714 00:44:55,790 --> 00:44:59,560 point which is in the region of convergence, then that 715 00:44:59,560 --> 00:45:06,530 point defines a strip in the s-plane that takes that point 716 00:45:06,530 --> 00:45:09,960 and extends it to the left until you bump into a pole, 717 00:45:09,960 --> 00:45:12,990 and extends it to the right until you bump it into a pole. 718 00:45:15,845 --> 00:45:20,460 So you begin to then see that we can tie together some 719 00:45:20,460 --> 00:45:24,040 properties of the region of convergence and the 720 00:45:24,040 --> 00:45:25,940 right-sidedness, or left-sidedness, or 721 00:45:25,940 --> 00:45:29,030 two-sidedness of the time function. 722 00:45:29,030 --> 00:45:32,750 And you'll have a chance to examine that in more detail in 723 00:45:32,750 --> 00:45:35,010 the video course manual. 724 00:45:35,010 --> 00:45:41,440 Let's conclude this lecture by talking about how we might get 725 00:45:41,440 --> 00:45:46,040 the time function given the appliance transform. 726 00:45:46,040 --> 00:45:49,830 Well, if we have a Laplace transform, we can, in 727 00:45:49,830 --> 00:45:53,010 principle, get the time function back again by 728 00:45:53,010 --> 00:45:56,230 recognizing this relationship between the Laplace transform 729 00:45:56,230 --> 00:45:59,360 and the Fourier transform, and using the formal Fourier 730 00:45:59,360 --> 00:46:00,650 transform expression. 731 00:46:00,650 --> 00:46:04,510 Or equivalently, the formal inverse Laplace transform 732 00:46:04,510 --> 00:46:07,190 expression, which is in the text. 733 00:46:07,190 --> 00:46:11,040 But more typically what we would do is what we've done 734 00:46:11,040 --> 00:46:15,910 also with the Fourier transform, which is to use 735 00:46:15,910 --> 00:46:21,580 simple Laplace transform pairs together with the notion of 736 00:46:21,580 --> 00:46:23,490 the partial fraction expansion. 737 00:46:23,490 --> 00:46:28,020 And let's just go through that with an example. 738 00:46:28,020 --> 00:46:34,350 Let's suppose that I have a Laplace transform as I 739 00:46:34,350 --> 00:46:38,380 indicated here in its pole-zero plot and a region of 740 00:46:38,380 --> 00:46:42,060 convergence which is to the right of this pole. 741 00:46:42,060 --> 00:46:45,140 And what we can identify from the region of convergence, in 742 00:46:45,140 --> 00:46:47,980 fact, is that we're talking about a 743 00:46:47,980 --> 00:46:50,660 right-sided time function. 744 00:46:50,660 --> 00:46:54,300 So the region of convergence is the real part of s greater 745 00:46:54,300 --> 00:46:55,900 than minus 1. 746 00:46:55,900 --> 00:47:01,050 And now looking down at the algebraic expression, we have 747 00:47:01,050 --> 00:47:04,610 the algebraic expression for this, as I indicated here, 748 00:47:04,610 --> 00:47:09,560 equivalently expanded in a partial fraction expansion, as 749 00:47:09,560 --> 00:47:10,250 I show below. 750 00:47:10,250 --> 00:47:13,590 So if you just simply combine these together, that's the 751 00:47:13,590 --> 00:47:15,030 same as this. 752 00:47:15,030 --> 00:47:19,360 And the region of convergence is the real part of s greater 753 00:47:19,360 --> 00:47:21,410 than minus 1. 754 00:47:21,410 --> 00:47:24,830 Now, the region of convergence of-- 755 00:47:24,830 --> 00:47:28,360 this is the sum of two terms, so the time function is the 756 00:47:28,360 --> 00:47:30,320 sum of two time functions. 757 00:47:30,320 --> 00:47:37,760 And the region of convergence of the combination must be the 758 00:47:37,760 --> 00:47:39,850 intersection of the region of convergence 759 00:47:39,850 --> 00:47:41,960 associated with each one. 760 00:47:41,960 --> 00:47:46,990 Recognizing that this is to the right of the poles, that 761 00:47:46,990 --> 00:47:51,290 tells us immediately that each of these two then would 762 00:47:51,290 --> 00:47:55,110 correspond to the Laplace transform of a right-sided 763 00:47:55,110 --> 00:47:57,180 time function. 764 00:47:57,180 --> 00:48:00,910 Well, let's look at it term by term. 765 00:48:00,910 --> 00:48:06,620 The first term is the factor 1 over s plus 1 with a region of 766 00:48:06,620 --> 00:48:10,620 convergence to the right of this pole. 767 00:48:10,620 --> 00:48:15,070 And this algebraically corresponds 768 00:48:15,070 --> 00:48:16,510 to what I've indicated. 769 00:48:16,510 --> 00:48:21,610 And this, in fact, is similar to, or a special case of the 770 00:48:21,610 --> 00:48:24,940 example that we pointed to at the beginning of the lecture. 771 00:48:24,940 --> 00:48:27,010 Namely, Example 9.1. 772 00:48:27,010 --> 00:48:29,370 And so we can just simply use that result. 773 00:48:29,370 --> 00:48:32,120 If you think back to that example or refer to your 774 00:48:32,120 --> 00:48:37,570 notes, we know that time function of the form e to the 775 00:48:37,570 --> 00:48:41,360 minus a t gives us the Laplace transform, which is 1 over s 776 00:48:41,360 --> 00:48:47,530 plus a with the real part of s greater than minus a. 777 00:48:47,530 --> 00:48:50,620 And so this is the Laplace transform of the first. 778 00:48:50,620 --> 00:48:53,080 Or, I'm sorry, this is the inverse Laplace transform of 779 00:48:53,080 --> 00:48:55,310 the first term. 780 00:48:55,310 --> 00:49:03,610 If we now consider the pole at s equals minus 2, and here is 781 00:49:03,610 --> 00:49:06,790 the region of convergence that we originally began with. 782 00:49:06,790 --> 00:49:12,370 In fact, we can having removed the pole at minus 1, extend 783 00:49:12,370 --> 00:49:16,370 this region of convergence to this pole. 784 00:49:16,370 --> 00:49:21,820 And we now have an algebraic expression, which is minus 1 785 00:49:21,820 --> 00:49:26,370 over s plus 2, the real part of s greater than minus 1. 786 00:49:26,370 --> 00:49:29,190 Although, in fact, we can extend the region of 787 00:49:29,190 --> 00:49:31,870 convergence up to the pole. 788 00:49:31,870 --> 00:49:37,250 And the inverse transform of this is now, again, referring 789 00:49:37,250 --> 00:49:40,840 to the same example, minus e to the minus 2t 790 00:49:40,840 --> 00:49:43,210 times the unit step. 791 00:49:43,210 --> 00:49:49,530 And if we simply put the two terms together then, adding 792 00:49:49,530 --> 00:49:54,100 the one that we have here to what we had before, we have a 793 00:49:54,100 --> 00:50:00,860 total inverse Laplace transform, which is that. 794 00:50:00,860 --> 00:50:05,890 So essentially, what's happened is that each of the 795 00:50:05,890 --> 00:50:09,540 poles has contributed an exponential factor. 796 00:50:09,540 --> 00:50:13,030 And because of the region of convergence being to the right 797 00:50:13,030 --> 00:50:18,000 of all those poles, that is consistent with the notion 798 00:50:18,000 --> 00:50:20,080 that both of those terms correspond to 799 00:50:20,080 --> 00:50:22,720 right-sided time functions. 800 00:50:22,720 --> 00:50:29,260 Well, let's just focus for a second or two on the same 801 00:50:29,260 --> 00:50:31,260 pole-zero pattern. 802 00:50:31,260 --> 00:50:36,620 But instead of a region of convergence which is to the 803 00:50:36,620 --> 00:50:40,030 right of the poles as we had before, we'll now take a 804 00:50:40,030 --> 00:50:43,310 region of convergence which is between the two poles. 805 00:50:43,310 --> 00:50:47,700 And I'll let you work through this more leisurely in the 806 00:50:47,700 --> 00:50:49,510 video course manual. 807 00:50:49,510 --> 00:50:53,200 But when we carry out the partial fraction expansion, as 808 00:50:53,200 --> 00:51:00,060 I've done below, we would now associate with this pole a 809 00:51:00,060 --> 00:51:02,490 region of convergence to the right. 810 00:51:02,490 --> 00:51:07,200 With this pole, a region of convergence to the left. 811 00:51:07,200 --> 00:51:13,610 And so what we would have is the sum of a right-sided time 812 00:51:13,610 --> 00:51:15,520 function due to this pole. 813 00:51:15,520 --> 00:51:20,160 And in fact it's of the form e to the minus t for t positive. 814 00:51:20,160 --> 00:51:26,330 And a left-sided time function due to this pole. 815 00:51:26,330 --> 00:51:28,590 And in fact, that's of the form e to the 816 00:51:28,590 --> 00:51:30,970 minus 2t for t negative. 817 00:51:30,970 --> 00:51:35,280 And so, in fact, the answer that we will get when we 818 00:51:35,280 --> 00:51:39,120 decompose this, use the partial fraction expansion, 819 00:51:39,120 --> 00:51:42,870 being very careful about associating the region of 820 00:51:42,870 --> 00:51:46,230 convergence of this pole to the right and of this pole to 821 00:51:46,230 --> 00:51:50,530 the left, we'll have then, when we're all done, a time 822 00:51:50,530 --> 00:51:55,570 function which will be of the form e to the minus t times 823 00:51:55,570 --> 00:51:58,660 the unit step for t positive. 824 00:51:58,660 --> 00:52:03,380 And then we'll have a term of the form e to the-- 825 00:52:03,380 --> 00:52:06,750 I'm sorry, this would be e to the minus 2t since this is at 826 00:52:06,750 --> 00:52:09,150 minus 2 and this is at minus 1. 827 00:52:09,150 --> 00:52:11,960 This would be a plus sign and this would be minus e to the 828 00:52:11,960 --> 00:52:15,300 minus t for t negative. 829 00:52:15,300 --> 00:52:20,120 And you'll look at that a little more carefully when you 830 00:52:20,120 --> 00:52:21,970 sit down with the video course manual. 831 00:52:24,950 --> 00:52:28,920 OK, well, what we've gone through, rather quickly, is an 832 00:52:28,920 --> 00:52:32,510 introduction to the Laplace transform. 833 00:52:32,510 --> 00:52:36,970 And a couple of points to underscore again, is the fact 834 00:52:36,970 --> 00:52:41,190 that the Laplace transform is very closely associated with 835 00:52:41,190 --> 00:52:42,690 the Fourier transform. 836 00:52:42,690 --> 00:52:47,280 And in fact, the Laplace transform for s equals j omega 837 00:52:47,280 --> 00:52:49,570 reduces to the Fourier transform. 838 00:52:49,570 --> 00:52:52,790 But more generally, the Laplace transform is the 839 00:52:52,790 --> 00:52:59,760 Fourier transform of x of t with an exponential weighting. 840 00:52:59,760 --> 00:53:02,870 And there are some exponentials for which that 841 00:53:02,870 --> 00:53:05,010 product converges. 842 00:53:05,010 --> 00:53:08,430 There are other exponentials for which that product has a 843 00:53:08,430 --> 00:53:10,780 Fourier transform that doesn't converge. 844 00:53:10,780 --> 00:53:14,370 That then imposes on the discussion of the Laplace 845 00:53:14,370 --> 00:53:18,860 transform what we refer to as the region of convergence. 846 00:53:18,860 --> 00:53:22,670 And it's very important to understand that in specifying 847 00:53:22,670 --> 00:53:28,670 a Laplace transform, it's important to identify not only 848 00:53:28,670 --> 00:53:34,870 the algebraic expression, but also the values of s for which 849 00:53:34,870 --> 00:53:35,620 it's valid. 850 00:53:35,620 --> 00:53:37,740 Namely, the region of convergence 851 00:53:37,740 --> 00:53:39,490 of the Laplace transform. 852 00:53:39,490 --> 00:53:44,350 Finally what we did was to tie together some properties of a 853 00:53:44,350 --> 00:53:48,200 time function with things that we can say about the region of 854 00:53:48,200 --> 00:53:52,060 convergence of its Laplace transform. 855 00:53:52,060 --> 00:53:54,910 Now, just as with the Fourier transform, the Laplace 856 00:53:54,910 --> 00:53:58,670 transform has some very important properties. 857 00:53:58,670 --> 00:54:04,140 And out of these properties, both are some mechanisms for 858 00:54:04,140 --> 00:54:09,760 using the Laplace transform for such systems as those 859 00:54:09,760 --> 00:54:12,280 described by linear constant coefficient 860 00:54:12,280 --> 00:54:14,080 differential equations. 861 00:54:14,080 --> 00:54:17,900 But more importantly, the properties will help us. 862 00:54:17,900 --> 00:54:23,550 As we understand them further, will help us in using and 863 00:54:23,550 --> 00:54:27,890 exploiting the Laplace transform to study and 864 00:54:27,890 --> 00:54:31,700 understand linear time-invariant systems. 865 00:54:31,700 --> 00:54:34,970 And that's what we'll go on to next time. 866 00:54:34,970 --> 00:54:38,530 In particular, talking about properties, and then 867 00:54:38,530 --> 00:54:42,610 associating with linear time-invariant systems much of 868 00:54:42,610 --> 00:54:46,610 the discussion that we've had today relating 869 00:54:46,610 --> 00:54:48,040 to the Laplace transform. 870 00:54:48,040 --> 00:54:49,290 Thank you.