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 00:00:20,320 --> 00:00:22,320 DENNIS FREEMAN: So for the last couple of times, 9 00:00:22,320 --> 00:00:25,740 we've been looking at Fourier series 10 00:00:25,740 --> 00:00:29,190 as a way of looking at signals as a way of being composed out 11 00:00:29,190 --> 00:00:31,140 of sinusoids, much the way we had previously 12 00:00:31,140 --> 00:00:36,990 looked at frequency responses as a way of thinking about systems 13 00:00:36,990 --> 00:00:38,970 characterized by sinusoids. 14 00:00:38,970 --> 00:00:42,600 And for the past two sessions, we've looked at Fourier series 15 00:00:42,600 --> 00:00:44,220 not because they were terribly useful, 16 00:00:44,220 --> 00:00:47,170 but because they were terribly simple. 17 00:00:47,170 --> 00:00:49,770 Today, I want to do the much more difficult task, 18 00:00:49,770 --> 00:00:51,810 but much more interesting task of thinking 19 00:00:51,810 --> 00:00:54,180 about the general case for thinking 20 00:00:54,180 --> 00:00:58,560 about a sinusoidal decomposition of an arbitrary signal, one 21 00:00:58,560 --> 00:01:00,870 that is not necessarily periodic. 22 00:01:00,870 --> 00:01:03,180 So I should say upfront, what I'm going 23 00:01:03,180 --> 00:01:05,790 to talk about is motivational. 24 00:01:05,790 --> 00:01:06,990 It's not a proof. 25 00:01:06,990 --> 00:01:09,450 Proving Fourier series convergence 26 00:01:09,450 --> 00:01:11,247 is actually very complicated. 27 00:01:11,247 --> 00:01:13,080 It's something that mathematicians worked on 28 00:01:13,080 --> 00:01:14,950 for about 100 years. 29 00:01:14,950 --> 00:01:17,730 So I am not going to try to prove things 30 00:01:17,730 --> 00:01:19,200 in any rigorous fashion, but I am 31 00:01:19,200 --> 00:01:22,290 going to try to motivate things so 32 00:01:22,290 --> 00:01:26,400 that you should at least expect that such a thing should exist. 33 00:01:26,400 --> 00:01:29,580 So the idea, the motivation is going to be 34 00:01:29,580 --> 00:01:33,810 how can I think about an aperiodic signal 35 00:01:33,810 --> 00:01:36,030 within a periodic framework because I already have 36 00:01:36,030 --> 00:01:38,400 worked out all the details. 37 00:01:38,400 --> 00:01:42,090 The details for Fourier series are relatively simple, well, 38 00:01:42,090 --> 00:01:44,700 at least compared to a Fourier transform, which is harder. 39 00:01:44,700 --> 00:01:48,370 Fourier series themselves are not that easy. 40 00:01:48,370 --> 00:01:50,820 But if I believe the Fourier series idea, 41 00:01:50,820 --> 00:01:52,710 is there a way to leverage that to think 42 00:01:52,710 --> 00:01:54,300 about aperiodic signals? 43 00:01:54,300 --> 00:01:57,840 And the idea is going to be let's take an aperiodic signal. 44 00:01:57,840 --> 00:02:00,130 I've tried to choose something terribly simple. 45 00:02:00,130 --> 00:02:04,230 It's the simplest thing that I could think of it isn't zero. 46 00:02:04,230 --> 00:02:08,600 So it's one for a while and zero most of the time. 47 00:02:08,600 --> 00:02:10,110 But I could make that signal, which 48 00:02:10,110 --> 00:02:13,950 is clearly not periodic, by thinking about periodically 49 00:02:13,950 --> 00:02:15,330 extending it. 50 00:02:15,330 --> 00:02:16,170 Copy it. 51 00:02:16,170 --> 00:02:19,800 Add it to itself many times, each time 52 00:02:19,800 --> 00:02:21,720 shifted by a capital T. 53 00:02:21,720 --> 00:02:24,770 This signal is obviously periodic. 54 00:02:24,770 --> 00:02:26,640 This transformation is obviously going 55 00:02:26,640 --> 00:02:29,130 to take any signal regardless and turn it 56 00:02:29,130 --> 00:02:32,550 into something that is periodic in cap T. 57 00:02:32,550 --> 00:02:36,270 So if I did that, then I could kind of trivially say, 58 00:02:36,270 --> 00:02:37,950 well, the aperiodic thing is just 59 00:02:37,950 --> 00:02:41,790 the limit when capital T goes to infinity of the periodic thing. 60 00:02:41,790 --> 00:02:43,560 OK, that's pretty trivial. 61 00:02:43,560 --> 00:02:46,080 OK, that's obviously true. 62 00:02:46,080 --> 00:02:51,000 The trick is, what if I took a Fourier series in the middle? 63 00:02:51,000 --> 00:02:53,760 What if I periodically extended this thing 64 00:02:53,760 --> 00:02:55,770 to get something that is periodic, 65 00:02:55,770 --> 00:02:58,567 take a Fourier series of this thing, 66 00:02:58,567 --> 00:03:00,150 and then take the limit of the series? 67 00:03:02,670 --> 00:03:06,270 So that's the thing I'm going to do over the next three slides. 68 00:03:06,270 --> 00:03:10,770 So think about a general, aperiodic signal, periodically 69 00:03:10,770 --> 00:03:15,605 extended so it's now periodic in cap T, take a Fourier series-- 70 00:03:19,132 --> 00:03:21,090 just to motivate the kind of math that happens, 71 00:03:21,090 --> 00:03:25,020 I've written out the math for this particularly simple signal 72 00:03:25,020 --> 00:03:28,416 that is one for a while and zero most of the time. 73 00:03:28,416 --> 00:03:31,190 The Fourier series coefficient, a sub k 74 00:03:31,190 --> 00:03:34,227 is obviously 1 over t, the period, 75 00:03:34,227 --> 00:03:35,310 integral over the period-- 76 00:03:35,310 --> 00:03:39,590 I took the symmetric period because it's the easiest one-- 77 00:03:39,590 --> 00:03:42,965 signal of interest, basis function integrated over time. 78 00:03:45,730 --> 00:03:48,140 And that's pretty trivially-- 79 00:03:48,140 --> 00:03:49,310 those integrals are easy. 80 00:03:49,310 --> 00:03:50,750 That was chosen that way. 81 00:03:50,750 --> 00:03:53,050 And so I get an answer that looks like that. 82 00:03:53,050 --> 00:03:57,170 The thing I want you to see about the answer 83 00:03:57,170 --> 00:04:01,760 is that I can think about it as a function of omega or k. 84 00:04:01,760 --> 00:04:04,610 And that's what I've plotted here. 85 00:04:04,610 --> 00:04:08,630 In particular, if I multiply a sub k by capital T, 86 00:04:08,630 --> 00:04:11,720 so as to kill this 1 over t thing, 87 00:04:11,720 --> 00:04:19,750 and if I plot t times a sub k, I get a relationship 2 sine omega 88 00:04:19,750 --> 00:04:27,790 S over omega, omega being k 2pi over t. 89 00:04:27,790 --> 00:04:32,830 But for the Fourier series, that only exists for k and integer. 90 00:04:32,830 --> 00:04:36,800 So that's what's represented by the blue bars. 91 00:04:36,800 --> 00:04:39,670 But what I want you to see is just from the math, 92 00:04:39,670 --> 00:04:45,250 the envelope doesn't depend on t. 93 00:04:45,250 --> 00:04:46,880 OK, that's the trick. 94 00:04:46,880 --> 00:04:49,750 So the idea is I'm plotting the Fourier coefficients a sub 95 00:04:49,750 --> 00:04:52,570 k as a function of k. 96 00:04:52,570 --> 00:04:58,990 So k equals 0, 1, 2, 3, 4, 5, et cetera. 97 00:04:58,990 --> 00:05:02,050 But I notice that the envelope can be written strictly 98 00:05:02,050 --> 00:05:05,320 as a function of omega where there is a simple relationship 99 00:05:05,320 --> 00:05:06,880 between omega and k. 100 00:05:06,880 --> 00:05:09,760 But omega is defined across the entire axis, 101 00:05:09,760 --> 00:05:12,640 and it's represented by this light black curve. 102 00:05:12,640 --> 00:05:16,990 That's more apparent if I think about increasing capital 103 00:05:16,990 --> 00:05:21,300 T. What if I were to keep the base waveform the same, 104 00:05:21,300 --> 00:05:23,000 but change capital T? 105 00:05:23,000 --> 00:05:27,850 Say I double capital T. The thing that happens 106 00:05:27,850 --> 00:05:31,930 is the envelope stays the same, but the spacing 107 00:05:31,930 --> 00:05:35,740 of the k's becomes condensed. 108 00:05:35,740 --> 00:05:40,140 There's more k's in a given number of hertzes, 109 00:05:40,140 --> 00:05:42,100 in frequency, than there was before. 110 00:05:42,100 --> 00:05:44,890 And if I double it again, it doubles again. 111 00:05:44,890 --> 00:05:46,480 The envelope didn't change. 112 00:05:46,480 --> 00:05:48,220 The k's did. 113 00:05:50,706 --> 00:05:52,580 The interesting thing about that construction 114 00:05:52,580 --> 00:05:56,330 is that it has separated out the part that depends on capital T 115 00:05:56,330 --> 00:05:59,180 from the part that doesn't depend on capital T. Capital 116 00:05:59,180 --> 00:06:02,390 T was this arbitrary thing that I 117 00:06:02,390 --> 00:06:04,640 used to take an aperiodic signal and turn it 118 00:06:04,640 --> 00:06:06,260 into a periodic signal. 119 00:06:06,260 --> 00:06:09,900 And it has an effect on the answer 120 00:06:09,900 --> 00:06:14,420 that can be separated from the other part of the answer. 121 00:06:14,420 --> 00:06:19,070 Some part of the answer depends on the base waveform. 122 00:06:19,070 --> 00:06:23,510 Some other part of the answer depends on capital T. Well, 123 00:06:23,510 --> 00:06:26,000 that's nice because now if I think about taking the limit 124 00:06:26,000 --> 00:06:29,060 as capital T goes to infinity, I have a prayer of interpreting 125 00:06:29,060 --> 00:06:32,540 things because part of my answer is changing with t and part 126 00:06:32,540 --> 00:06:33,470 of it isn't. 127 00:06:33,470 --> 00:06:36,590 So all I need to do now is focus on the part that is changing 128 00:06:36,590 --> 00:06:38,420 with t and separate it from the part that's 129 00:06:38,420 --> 00:06:41,160 not changing with t. 130 00:06:41,160 --> 00:06:43,100 So now, I can think about taking-- 131 00:06:43,100 --> 00:06:47,390 so I just plug in this expression 132 00:06:47,390 --> 00:06:51,950 here for this integral. 133 00:06:51,950 --> 00:06:53,960 And what I get then is something that 134 00:06:53,960 --> 00:07:00,180 looks a lot like a Fourier series or even 135 00:07:00,180 --> 00:07:01,820 a Laplace transform. 136 00:07:01,820 --> 00:07:05,570 I get an integral-- ignore the limit part for a moment. 137 00:07:05,570 --> 00:07:08,030 I get an integral of something times 138 00:07:08,030 --> 00:07:11,000 some sort of a weighting function. 139 00:07:11,000 --> 00:07:13,760 And I get something over here where the integral 140 00:07:13,760 --> 00:07:17,980 was over time, but the function over here 141 00:07:17,980 --> 00:07:19,690 doesn't have time in it. 142 00:07:19,690 --> 00:07:22,000 It only has omega in it. 143 00:07:22,000 --> 00:07:23,500 That's the sense in which it sort of 144 00:07:23,500 --> 00:07:30,100 looks like the analysis formula for either Fourier 145 00:07:30,100 --> 00:07:31,984 series or Laplace transforms. 146 00:07:31,984 --> 00:07:33,400 It looks like the analysis formula 147 00:07:33,400 --> 00:07:40,020 because I'm calculating a T ak, the components of the series, 148 00:07:40,020 --> 00:07:43,530 or this new thing, E of omega, that doesn't depend-- 149 00:07:43,530 --> 00:07:45,180 neither of those depended on t. 150 00:07:47,970 --> 00:07:50,340 OK, so the idea is that when I do 151 00:07:50,340 --> 00:07:52,200 this kind of a limiting operation 152 00:07:52,200 --> 00:07:54,750 on the periodic extension, I get something 153 00:07:54,750 --> 00:08:01,590 that ends up looking like a transform relationship. 154 00:08:01,590 --> 00:08:03,990 And if I think about going the other way, 155 00:08:03,990 --> 00:08:06,450 doing a synthesis operation, I can 156 00:08:06,450 --> 00:08:08,010 think about how I would construct 157 00:08:08,010 --> 00:08:11,077 x of t out of the Fourier coefficients 158 00:08:11,077 --> 00:08:12,660 But now, there's a simple relationship 159 00:08:12,660 --> 00:08:15,630 between the Fourier series coefficients, 160 00:08:15,630 --> 00:08:24,230 a sub k, and this thing Ew, which I've represented here. 161 00:08:24,230 --> 00:08:26,320 And I don't like the t. 162 00:08:26,320 --> 00:08:29,620 So I'll do a substitution from here. 163 00:08:29,620 --> 00:08:32,740 t can be written as omega 0 over 2pi. 164 00:08:32,740 --> 00:08:34,690 1 over t can be written as omega 0 ever 2pi. 165 00:08:37,872 --> 00:08:39,330 And now, I've got everything I need 166 00:08:39,330 --> 00:08:44,760 to think about how that sum approaches an integral 167 00:08:44,760 --> 00:08:47,070 in a Riemann sum kind of sense. 168 00:08:47,070 --> 00:08:49,770 Think about as I add more and more-- 169 00:08:49,770 --> 00:08:52,920 as I make capital T get bigger and bigger, 170 00:08:52,920 --> 00:08:54,630 omega 0 gets smaller and smaller. 171 00:08:57,840 --> 00:09:01,020 As I make the capital T bigger and bigger, 172 00:09:01,020 --> 00:09:03,720 the spacing gets smaller and smaller. 173 00:09:03,720 --> 00:09:08,640 Increasingly, I can think about this function E of omega 174 00:09:08,640 --> 00:09:12,630 as being smooth and increasingly constant 175 00:09:12,630 --> 00:09:16,360 over the small interval between the bars. 176 00:09:16,360 --> 00:09:20,820 So I can think about the sum as a Riemann sum passed 177 00:09:20,820 --> 00:09:23,970 to the integral as the limit. 178 00:09:23,970 --> 00:09:28,590 So when I do that, omega 0 is the spacing 179 00:09:28,590 --> 00:09:29,705 between the two adjacents. 180 00:09:29,705 --> 00:09:31,080 It's the region over which I want 181 00:09:31,080 --> 00:09:35,730 to think about that integrand being constant. 182 00:09:35,730 --> 00:09:41,730 And so that in the limit, this omega 0 passes to d omega. 183 00:09:41,730 --> 00:09:44,760 And I'm left with something that looks like a synthesis 184 00:09:44,760 --> 00:09:47,190 equation. 185 00:09:47,190 --> 00:09:51,410 So if I just write those equations here and think 186 00:09:51,410 --> 00:09:54,800 about this Ew thing being some kind of a transform, which I'll 187 00:09:54,800 --> 00:10:00,000 mysteriously write as x of j omega, 188 00:10:00,000 --> 00:10:04,430 then the result for the aperiodic case 189 00:10:04,430 --> 00:10:07,250 has a structure that looks very much like the Fourier series, 190 00:10:07,250 --> 00:10:10,140 or for that matter like the Laplace transform. 191 00:10:10,140 --> 00:10:16,460 What it says is that I can synthesize an arbitrary x of t 192 00:10:16,460 --> 00:10:19,970 by adding together a whole bunch of components 193 00:10:19,970 --> 00:10:25,250 that already depend on omega weighted by some weighting 194 00:10:25,250 --> 00:10:25,750 function. 195 00:10:25,750 --> 00:10:27,910 s this looks like a synthesis equation 196 00:10:27,910 --> 00:10:31,160 very much like the synthesis equation for Fourier series 197 00:10:31,160 --> 00:10:33,700 or for Laplace transforms. 198 00:10:33,700 --> 00:10:36,900 And I get an analysis equation that similarly 199 00:10:36,900 --> 00:10:38,410 has the same form again. 200 00:10:38,410 --> 00:10:42,430 I take the x of t and figure out the component 201 00:10:42,430 --> 00:10:46,450 that should be at omega by multiplying 202 00:10:46,450 --> 00:10:50,980 by a complex exponential and integrating. 203 00:10:50,980 --> 00:10:54,270 OK, I have to emphasize this is not a proof. 204 00:10:54,270 --> 00:10:57,040 All I wanted to do was kind of motivate 205 00:10:57,040 --> 00:11:00,100 the way you can think about an aperiodic signal 206 00:11:00,100 --> 00:11:02,620 as being periodic in some time interval 207 00:11:02,620 --> 00:11:04,240 and passed to the limit. 208 00:11:04,240 --> 00:11:05,890 And if you do that, you can sort of 209 00:11:05,890 --> 00:11:09,900 see where the equations are coming from. 210 00:11:09,900 --> 00:11:11,540 OK? 211 00:11:11,540 --> 00:11:13,821 So the idea then-- 212 00:11:13,821 --> 00:11:14,320 whoops. 213 00:11:18,000 --> 00:11:20,570 So the idea then is that we will use 214 00:11:20,570 --> 00:11:24,410 these relationships to define an analysis and synthesis 215 00:11:24,410 --> 00:11:26,810 of aperiodic signals. 216 00:11:26,810 --> 00:11:29,960 And we'll refer to that as a Fourier transform. 217 00:11:29,960 --> 00:11:32,540 The Fourier transform will let us 218 00:11:32,540 --> 00:11:36,690 have insights that are completely 219 00:11:36,690 --> 00:11:39,030 analogous to the Fourier series, except they now 220 00:11:39,030 --> 00:11:40,450 apply for aperiodic signals. 221 00:11:40,450 --> 00:11:42,120 So in particular, we'll be able to think 222 00:11:42,120 --> 00:11:44,160 about a signal being composed of a bunch 223 00:11:44,160 --> 00:11:45,240 of sinusoidal components. 224 00:11:45,240 --> 00:11:49,240 And we'll be able to think about systems as filters. 225 00:11:52,020 --> 00:11:54,390 OK, so I've already alluded to the fact 226 00:11:54,390 --> 00:11:58,350 that the Fourier transform relations 227 00:11:58,350 --> 00:12:03,840 look very similar in form to the Laplace transform relations. 228 00:12:03,840 --> 00:12:08,010 And so I've illustrated the analysis equations here just 229 00:12:08,010 --> 00:12:12,130 to emphasize the similarity. 230 00:12:12,130 --> 00:12:14,750 The Laplace transform, you'll remember, had the-- 231 00:12:14,750 --> 00:12:17,610 we integrated some signal that was 232 00:12:17,610 --> 00:12:20,670 a function of time times a complex exponential integrated 233 00:12:20,670 --> 00:12:26,010 over time to get a Laplace transform that was 234 00:12:26,010 --> 00:12:31,860 a function of s, not time. 235 00:12:31,860 --> 00:12:35,670 It was a way of having an alternative representation 236 00:12:35,670 --> 00:12:36,930 for the signal. 237 00:12:36,930 --> 00:12:38,214 There was no new information. 238 00:12:38,214 --> 00:12:39,630 The same information was contained 239 00:12:39,630 --> 00:12:41,910 in s of x as was contained in x of t. 240 00:12:41,910 --> 00:12:44,880 Except now, where it was organized by time, 241 00:12:44,880 --> 00:12:47,959 now, it's organized by s. 242 00:12:47,959 --> 00:12:50,250 We get the same sort of thing with a Fourier transform. 243 00:12:50,250 --> 00:12:54,470 And in fact, this gives away the mysterious reason 244 00:12:54,470 --> 00:12:58,882 for calling it x of j omega in the previous slide. 245 00:12:58,882 --> 00:13:01,340 You can see that a different way to think about the Fourier 246 00:13:01,340 --> 00:13:04,490 transform is that it's simply-- 247 00:13:04,490 --> 00:13:07,100 a trivial way to think about it, it's the value-- 248 00:13:07,100 --> 00:13:12,010 the Fourier transform is the value of the Laplace transform 249 00:13:12,010 --> 00:13:14,690 evaluated s equals j omega. 250 00:13:14,690 --> 00:13:20,270 All you do is you take this expression for the Laplace 251 00:13:20,270 --> 00:13:24,230 transform, and every place there was an s, make s equal j omega. 252 00:13:24,230 --> 00:13:27,300 And you get this equation. 253 00:13:27,300 --> 00:13:29,670 So that's the reason we like the notation. 254 00:13:29,670 --> 00:13:33,350 The Fourier transform is x of j omega. 255 00:13:33,350 --> 00:13:35,120 There are confusions that arise by that. 256 00:13:35,120 --> 00:13:36,839 And I'll talk about those in a moment. 257 00:13:36,839 --> 00:13:38,630 But for the time being, the important thing 258 00:13:38,630 --> 00:13:41,750 is that the Fourier transform can 259 00:13:41,750 --> 00:13:46,820 be viewed as a special case looking at the j omega 260 00:13:46,820 --> 00:13:51,040 axis of the Laplace transform. 261 00:13:51,040 --> 00:13:53,140 OK? 262 00:13:53,140 --> 00:13:57,250 So that view points out two things. 263 00:13:57,250 --> 00:13:59,900 There's a lot of similarities, and there are some differences. 264 00:13:59,900 --> 00:14:02,620 First, the similarities-- because you 265 00:14:02,620 --> 00:14:06,490 can regard the Fourier transform as kind of a special case-- 266 00:14:06,490 --> 00:14:07,524 that's not really true. 267 00:14:07,524 --> 00:14:09,940 And I will say something about that by the end of the hour 268 00:14:09,940 --> 00:14:10,820 as well. 269 00:14:10,820 --> 00:14:13,420 But because it's kind of a special case of the Laplace 270 00:14:13,420 --> 00:14:18,004 transform, the Fourier transform inherits 271 00:14:18,004 --> 00:14:19,920 a lot of the important properties of a Laplace 272 00:14:19,920 --> 00:14:21,270 transform. 273 00:14:21,270 --> 00:14:24,950 In particular, the two things that we looked at most 274 00:14:24,950 --> 00:14:27,750 has been linearity. 275 00:14:27,750 --> 00:14:30,000 Because the Laplace transform is linear, 276 00:14:30,000 --> 00:14:34,120 we can do all manner of things with it. 277 00:14:34,120 --> 00:14:38,240 The same as we use the properties of linear systems 278 00:14:38,240 --> 00:14:41,480 to simplify our view of how to think about a system, 279 00:14:41,480 --> 00:14:44,480 we could, for example, because systems are linear, 280 00:14:44,480 --> 00:14:47,450 we can look at the response of a system 281 00:14:47,450 --> 00:14:50,690 to a sum of inputs as the sum of the responses 282 00:14:50,690 --> 00:14:51,710 to the individuals. 283 00:14:51,710 --> 00:14:54,830 That's a very important property that we used of systems 284 00:14:54,830 --> 00:14:56,300 as a result of linearity. 285 00:14:56,300 --> 00:14:59,870 We did the same thing with Laplace transforms. 286 00:14:59,870 --> 00:15:01,785 The Laplace transform of a sum is the sum 287 00:15:01,785 --> 00:15:04,340 of a Laplace transforms. 288 00:15:04,340 --> 00:15:07,676 And in conjunction with the differentiation roll 289 00:15:07,676 --> 00:15:09,050 by which we knew that the Laplace 290 00:15:09,050 --> 00:15:12,610 transform of a derivative is s times the Laplace transform 291 00:15:12,610 --> 00:15:17,090 the function, the combination of linearity 292 00:15:17,090 --> 00:15:19,040 and the differentiation role allowed 293 00:15:19,040 --> 00:15:23,360 us to apply Laplace transforms to turn differential equations 294 00:15:23,360 --> 00:15:26,870 into algebraic equations. 295 00:15:26,870 --> 00:15:32,580 Precisely the same thing will work with Fourier transforms. 296 00:15:32,580 --> 00:15:34,620 For reasons that should be clear, 297 00:15:34,620 --> 00:15:37,330 if the Laplace transform has the property of linearity, 298 00:15:37,330 --> 00:15:40,530 so does the Fourier. 299 00:15:40,530 --> 00:15:43,560 And if the Laplace transform is simply 300 00:15:43,560 --> 00:15:44,964 related to the Fourier transform, 301 00:15:44,964 --> 00:15:46,380 then there's a simple relationship 302 00:15:46,380 --> 00:15:48,690 between the Fourier transform of a derivative 303 00:15:48,690 --> 00:15:52,480 and the Fourier transform of the underlying function. 304 00:15:52,480 --> 00:15:55,600 So in the Laplace transform, you multiply by s. 305 00:15:55,600 --> 00:15:57,600 Not very surprisingly, in the Fourier transform, 306 00:15:57,600 --> 00:16:00,300 you multiply by j omega. 307 00:16:00,300 --> 00:16:02,850 So there's enormous similarity. 308 00:16:02,850 --> 00:16:05,670 And in fact, most of what you know about Laplace, 309 00:16:05,670 --> 00:16:09,420 you can immediately carry over into Fourier. 310 00:16:09,420 --> 00:16:12,496 There are some differences. 311 00:16:12,496 --> 00:16:13,870 And if there weren't differences, 312 00:16:13,870 --> 00:16:16,090 we probably wouldn't bother with talking about both of them. 313 00:16:16,090 --> 00:16:16,589 Right? 314 00:16:16,589 --> 00:16:19,030 There are some things that will be easy to think about 315 00:16:19,030 --> 00:16:20,110 with Fourier transforms. 316 00:16:20,110 --> 00:16:21,627 And that's the reason we do it. 317 00:16:21,627 --> 00:16:23,710 There are some things that are easy to think about 318 00:16:23,710 --> 00:16:24,760 with Laplace transforms. 319 00:16:24,760 --> 00:16:28,670 Otherwise, we would have just skipped straight to Fourier. 320 00:16:28,670 --> 00:16:34,012 So there are some things that Fourier and Laplace share. 321 00:16:34,012 --> 00:16:35,720 There are some things that are different. 322 00:16:35,720 --> 00:16:39,744 One of the biggest differences is the domain. 323 00:16:39,744 --> 00:16:41,410 When we think about a Laplace transform, 324 00:16:41,410 --> 00:16:44,860 we think about x of s. 325 00:16:44,860 --> 00:16:49,300 The domain or the Laplace transform is the domain of s. 326 00:16:49,300 --> 00:16:52,510 The domain of s, s is a complex number. 327 00:16:52,510 --> 00:16:55,060 For that reason, when we thought out Laplace transforms, 328 00:16:55,060 --> 00:16:59,560 we always talked about what does the Laplace transform 329 00:16:59,560 --> 00:17:02,590 look like in the s plane. 330 00:17:02,590 --> 00:17:04,770 And we thought about the real part of the s 331 00:17:04,770 --> 00:17:08,390 and the imaginary part of s. 332 00:17:08,390 --> 00:17:11,950 When we think about Fourier transforms, 333 00:17:11,950 --> 00:17:16,720 we're thinking about a transform with real domain. 334 00:17:16,720 --> 00:17:21,369 Rather than thinking about x of s as a complex number, 335 00:17:21,369 --> 00:17:23,770 we're going to think of x of j omega, omega 336 00:17:23,770 --> 00:17:30,550 a real number that's a little confusing, right? 337 00:17:30,550 --> 00:17:35,620 Just sort of to confuse you, we rewrite the one 338 00:17:35,620 --> 00:17:40,500 that is a complex number as s-- 339 00:17:40,500 --> 00:17:43,474 no indication whatever that it's complex. 340 00:17:43,474 --> 00:17:44,890 And the one that is a real number, 341 00:17:44,890 --> 00:17:49,070 we put a j in front of it to remind you that it's real. 342 00:17:49,070 --> 00:17:52,930 I apologize, I don't know why we do this. 343 00:17:52,930 --> 00:17:56,280 So just remember that s, which looks kind of real isn't. 344 00:17:56,280 --> 00:17:58,830 It's complex. 345 00:17:58,830 --> 00:18:01,016 And j omega, which looks kind of complex, 346 00:18:01,016 --> 00:18:02,640 well, it's the omega part that matters. 347 00:18:02,640 --> 00:18:03,620 It's real. 348 00:18:03,620 --> 00:18:06,630 OK, so the important thing is the Laplace transform, 349 00:18:06,630 --> 00:18:09,680 the domain of a Laplace transform 350 00:18:09,680 --> 00:18:13,190 is complex number s, real and imaginary parts, 351 00:18:13,190 --> 00:18:16,330 characterized by a plane. 352 00:18:16,330 --> 00:18:19,880 The domain of the Fourier transform is real. 353 00:18:19,880 --> 00:18:21,880 That's enormously important. 354 00:18:21,880 --> 00:18:25,110 And we'll come back to that over and over again. 355 00:18:25,110 --> 00:18:27,430 But just to drive home the point, 356 00:18:27,430 --> 00:18:29,110 one of the things we thought about 357 00:18:29,110 --> 00:18:33,280 with the Laplace transform was this idea of eigenfunctions 358 00:18:33,280 --> 00:18:35,590 and eigenvalues. 359 00:18:35,590 --> 00:18:37,360 It was an idea of linearity. 360 00:18:37,360 --> 00:18:41,820 It was the idea that we can think about a system 361 00:18:41,820 --> 00:18:45,190 by how you put in a function, like E to the st, 362 00:18:45,190 --> 00:18:46,580 and calculate the output. 363 00:18:46,580 --> 00:18:50,350 Well, if the output, if the system is linear time invariant 364 00:18:50,350 --> 00:18:54,970 and can be characterized by a Laplace transform h of s, 365 00:18:54,970 --> 00:18:58,970 what's the output of that system when the input is e to the st? 366 00:19:03,950 --> 00:19:04,630 Everybody shout. 367 00:19:04,630 --> 00:19:07,765 It will make me feel much better. 368 00:19:07,765 --> 00:19:09,140 If you all shout at once, I won't 369 00:19:09,140 --> 00:19:10,520 be able to understand a word you said, 370 00:19:10,520 --> 00:19:12,228 and I'll assume you said the right thing. 371 00:19:16,474 --> 00:19:18,140 OK I didn't understand a thing you said. 372 00:19:18,140 --> 00:19:25,950 So I assume you all said h of s e to the st. Right, 373 00:19:25,950 --> 00:19:31,240 e to the st is an eigenfunction of a linear time invariant 374 00:19:31,240 --> 00:19:33,010 system. 375 00:19:33,010 --> 00:19:35,140 Eigenfunction means the function in 376 00:19:35,140 --> 00:19:37,420 is the same form as the function out 377 00:19:37,420 --> 00:19:39,740 except it could be multiplied by a constant. 378 00:19:39,740 --> 00:19:41,500 The constant is the eigenvalue. 379 00:19:41,500 --> 00:19:45,640 The eigenvalue is h of s. 380 00:19:45,640 --> 00:19:49,180 If we wanted to know, for example, if we wanted 381 00:19:49,180 --> 00:19:52,690 to characterize a very simple system, 382 00:19:52,690 --> 00:19:54,910 we might have a system of the form 1 over 1 plus s. 383 00:19:54,910 --> 00:19:57,444 We might have a signal of the form 1 over 1 plus s. 384 00:19:57,444 --> 00:19:58,860 So let's say we have a system now. 385 00:19:58,860 --> 00:20:01,930 Let's say that x represents some kind of a system. 386 00:20:01,930 --> 00:20:04,270 Then we would have said that that's a pole. 387 00:20:04,270 --> 00:20:07,490 Where's the pole? 388 00:20:07,490 --> 00:20:10,410 Minus 1-- we would have said we have a system 389 00:20:10,410 --> 00:20:13,518 with a single pole at minus 1. 390 00:20:13,518 --> 00:20:16,440 I would never have drawn this complicated picture 391 00:20:16,440 --> 00:20:18,780 at the bottom because it would be frightening. 392 00:20:18,780 --> 00:20:20,820 I would always draw something friendly 393 00:20:20,820 --> 00:20:22,830 like the picture over here. 394 00:20:22,830 --> 00:20:28,110 Right, the entire system can be understood by a single x. 395 00:20:28,110 --> 00:20:31,650 OK well, if you were computing eigenfunctions and eigenvalues, 396 00:20:31,650 --> 00:20:34,185 you would like to know what's the magnitude and phase. 397 00:20:36,920 --> 00:20:40,970 Sorry, the x of s is a complex valued 398 00:20:40,970 --> 00:20:44,100 function of complex domain. 399 00:20:44,100 --> 00:20:48,360 s is a complex number, and the answer is a complex number. 400 00:20:48,360 --> 00:20:51,260 So we'd like to know the real and imaginary parts 401 00:20:51,260 --> 00:20:56,037 of h of s or the magnitude and phase equivalently. 402 00:20:56,037 --> 00:20:58,370 If we want to know the magnitude and phase, for example, 403 00:20:58,370 --> 00:21:02,180 of h of s, in principle, we need to know what magnitude could it 404 00:21:02,180 --> 00:21:04,280 be for all the different s's. 405 00:21:04,280 --> 00:21:06,740 So what's plotted here is a picture 406 00:21:06,740 --> 00:21:11,987 of the magnitude of this function 407 00:21:11,987 --> 00:21:13,570 as a function of all the different s's 408 00:21:13,570 --> 00:21:17,860 that can be an eigenfunction. 409 00:21:17,860 --> 00:21:20,470 Right, so for all of the-- so any 410 00:21:20,470 --> 00:21:24,420 s is an eigenfunction of the system. 411 00:21:24,420 --> 00:21:27,000 And that plot plots the magnitude 412 00:21:27,000 --> 00:21:30,690 of the associated eigenvalue. 413 00:21:30,690 --> 00:21:35,460 The point is that I have to tell you a complex plane 414 00:21:35,460 --> 00:21:38,301 number of values. 415 00:21:38,301 --> 00:21:38,800 Right? 416 00:21:38,800 --> 00:21:42,900 There a value for s equals 1, s equals minus 1, s equals 2, 417 00:21:42,900 --> 00:21:49,127 s equals minus 2, s equals j, s equals 2j, s equals 17 plus 5j. 418 00:21:49,127 --> 00:21:51,210 All the different values, all the different points 419 00:21:51,210 --> 00:21:54,120 in the s plane have a different associated eigenvalue. 420 00:21:54,120 --> 00:21:55,920 And to completely characterize this system, 421 00:21:55,920 --> 00:21:58,050 I have to tell you all of those. 422 00:21:58,050 --> 00:22:01,130 By contrast, if I think about the Fourier transform, 423 00:22:01,130 --> 00:22:02,910 the Fourier transform maps a function 424 00:22:02,910 --> 00:22:08,870 of time to a function of omega. 425 00:22:08,870 --> 00:22:11,280 The complete characterization of the Fourier transform 426 00:22:11,280 --> 00:22:13,220 is showed here. 427 00:22:13,220 --> 00:22:15,230 All I need to worry about is what 428 00:22:15,230 --> 00:22:17,630 are all the possible values of omega. 429 00:22:17,630 --> 00:22:19,520 I'm thinking now instead of thinking s, 430 00:22:19,520 --> 00:22:24,050 I'm thinking how would I compose x of t 431 00:22:24,050 --> 00:22:26,949 by summing together a bunch of sine waves. 432 00:22:26,949 --> 00:22:28,490 The reason I want to think about that 433 00:22:28,490 --> 00:22:31,280 is because I want to think about systems in terms 434 00:22:31,280 --> 00:22:33,420 of frequency responses. 435 00:22:33,420 --> 00:22:36,200 So I want to know which frequencies are amplified, 436 00:22:36,200 --> 00:22:38,780 which ones are attenuated, which ones are phase delayed, 437 00:22:38,780 --> 00:22:41,690 which ones are phase advanced. 438 00:22:41,690 --> 00:22:44,060 And in order to do that kind of construction, 439 00:22:44,060 --> 00:22:47,030 all I need to know is what's the magnitude 440 00:22:47,030 --> 00:22:52,220 and angle of the system function for all possible values 441 00:22:52,220 --> 00:22:52,720 of omega. 442 00:22:55,540 --> 00:22:57,150 So that's an enormous difference. 443 00:22:57,150 --> 00:23:00,340 Instead of having, in the previous case, 444 00:23:00,340 --> 00:23:05,640 I had a function of time turning into a function of two space. 445 00:23:05,640 --> 00:23:09,030 Function of one space turned into a function of two space. 446 00:23:09,030 --> 00:23:10,890 Here I have a function of one space turning 447 00:23:10,890 --> 00:23:13,990 into a function of one space. 448 00:23:13,990 --> 00:23:17,670 So that is conceptually a whole lot simpler. 449 00:23:17,670 --> 00:23:19,890 Even more importantly, it is going 450 00:23:19,890 --> 00:23:21,430 to give rise to something that we'll 451 00:23:21,430 --> 00:23:24,330 spend most of the time for the rest of the term on-- 452 00:23:24,330 --> 00:23:29,190 the notion of signal processing where we can alternatively 453 00:23:29,190 --> 00:23:35,010 represent a signal x not by its time samples, 454 00:23:35,010 --> 00:23:38,930 but instead by its frequency samples. 455 00:23:38,930 --> 00:23:42,130 It would be very difficult to use that technique. 456 00:23:42,130 --> 00:23:46,560 Although it would work perfectly, 457 00:23:46,560 --> 00:23:48,330 there would be an explosion of information 458 00:23:48,330 --> 00:23:53,070 if we tried to use a signal processing technique with this 459 00:23:53,070 --> 00:23:55,140 where we represent this one-dimensional signal 460 00:23:55,140 --> 00:24:00,510 by a two-dimensional transform because we would be exploding 461 00:24:00,510 --> 00:24:01,950 the amount of information. 462 00:24:01,950 --> 00:24:04,140 We would be increasing substantially 463 00:24:04,140 --> 00:24:08,310 the amount of information required to specify the signal. 464 00:24:08,310 --> 00:24:13,120 When we do the Fourier, there is no such explosion. 465 00:24:13,120 --> 00:24:15,280 It was a one-dimensional function of time. 466 00:24:15,280 --> 00:24:17,810 It is a one-dimensional function of omega. 467 00:24:21,520 --> 00:24:26,067 OK, OK, I've been talking too much. 468 00:24:26,067 --> 00:24:27,650 I would like you to make sure that you 469 00:24:27,650 --> 00:24:30,270 understand the mechanics of what I've just said. 470 00:24:30,270 --> 00:24:33,590 So here's a signal, x1 of t. 471 00:24:33,590 --> 00:24:36,050 Which of these, if any, represents 472 00:24:36,050 --> 00:24:39,544 the Fourier transform? 473 00:24:39,544 --> 00:24:40,460 You're all very quiet. 474 00:24:40,460 --> 00:24:41,660 Look at your neighbor. 475 00:24:41,660 --> 00:24:42,590 Don't be quiet. 476 00:24:42,590 --> 00:24:43,430 And then start. 477 00:24:43,430 --> 00:24:45,195 And then you can go back to being quiet. 478 00:24:45,195 --> 00:24:47,020 [SIDE CONVERSATIONS] 479 00:26:05,020 --> 00:26:05,640 So it's quiet. 480 00:26:05,640 --> 00:26:07,220 So I assume that means convergence. 481 00:26:07,220 --> 00:26:12,747 So which function represents the Fourier transform of x1 of t? 482 00:26:12,747 --> 00:26:13,830 Everybody raise your hand. 483 00:26:13,830 --> 00:26:15,960 Indicate by a number of fingers. 484 00:26:15,960 --> 00:26:18,900 And it's overwhelmingly correct, which is wonderful. 485 00:26:18,900 --> 00:26:20,230 That's the point. 486 00:26:20,230 --> 00:26:22,170 The point is Fourier transforms are easy. 487 00:26:22,170 --> 00:26:23,820 And you've all got it. 488 00:26:23,820 --> 00:26:29,100 So it's trivial to run this kind of an integral. 489 00:26:29,100 --> 00:26:31,590 It's not very different from doing a Laplace transform. 490 00:26:31,590 --> 00:26:33,760 Here I've indicated the Laplace transform. 491 00:26:33,760 --> 00:26:34,260 Right? 492 00:26:34,260 --> 00:26:37,050 We do e to the minus t. x of t is 1 or 0. 493 00:26:37,050 --> 00:26:40,860 We change the limits to indicate the 1 or zeroness. 494 00:26:40,860 --> 00:26:42,990 Very trivial here, we get a slightly different 495 00:26:42,990 --> 00:26:45,330 looking answer because instead of e to the st, 496 00:26:45,330 --> 00:26:48,090 we have e to the j omega t. 497 00:26:48,090 --> 00:26:51,700 But otherwise, it's pretty much the same. 498 00:26:51,700 --> 00:26:55,810 The big difference, though, is again the domain. 499 00:26:55,810 --> 00:26:57,890 So if you think about the answer-- 500 00:26:57,890 --> 00:27:00,234 so the answer is four like all of you said-- 501 00:27:00,234 --> 00:27:02,400 if you think about the answer from the point of view 502 00:27:02,400 --> 00:27:07,830 of eigenfunctions and eigenvalues, 503 00:27:07,830 --> 00:27:10,390 you have to think about a two space. 504 00:27:10,390 --> 00:27:13,380 The two space for even that simple function, 505 00:27:13,380 --> 00:27:18,990 sort of the least complicated function I could think of, 506 00:27:18,990 --> 00:27:21,460 is illustrated here. 507 00:27:21,460 --> 00:27:23,070 And what you're supposed to see there 508 00:27:23,070 --> 00:27:27,970 is if I were to integrate x of t e to the minus st 509 00:27:27,970 --> 00:27:36,670 dt to get x of s, if I think about s as sigma plus j omega-- 510 00:27:36,670 --> 00:27:39,480 it has a real part and an imaginary part-- 511 00:27:39,480 --> 00:27:43,000 the real part, as I make the real part big, e to the st 512 00:27:43,000 --> 00:27:45,780 becomes something that explodes. 513 00:27:45,780 --> 00:27:49,020 And you can see that manifest here over in this region. 514 00:27:49,020 --> 00:27:51,030 So this is the real axis this way. 515 00:27:51,030 --> 00:27:53,550 This is the imaginary axis that way. 516 00:27:53,550 --> 00:27:56,220 You can see that as you go to bigger 517 00:27:56,220 --> 00:27:59,310 numbers in the positive real direction, 518 00:27:59,310 --> 00:28:02,480 the magnitude explodes. 519 00:28:02,480 --> 00:28:05,000 If you go in the negative direction 520 00:28:05,000 --> 00:28:09,242 because there was a sum here, the magnitude explodes again. 521 00:28:09,242 --> 00:28:10,700 You get this horrible function that 522 00:28:10,700 --> 00:28:12,990 spends a lot of its time near infinity. 523 00:28:12,990 --> 00:28:15,020 Right? 524 00:28:15,020 --> 00:28:18,170 So that's a complicated picture by comparison to the picture 525 00:28:18,170 --> 00:28:20,314 that you get if you look at Fourier transform. 526 00:28:20,314 --> 00:28:21,980 So if you look at the Fourier transform, 527 00:28:21,980 --> 00:28:24,530 you get something that's relatively simpler. 528 00:28:24,530 --> 00:28:27,980 We're only looking along the imaginary axis now. 529 00:28:27,980 --> 00:28:30,770 Furthermore, there's an easy way to interpret this. 530 00:28:30,770 --> 00:28:33,830 This is explicitly telling us if you put a certain frequency 531 00:28:33,830 --> 00:28:36,800 into the system, say this represented a system function, 532 00:28:36,800 --> 00:28:38,420 if this represented a system function, 533 00:28:38,420 --> 00:28:41,570 it's telling you that there's a simple way of thinking 534 00:28:41,570 --> 00:28:44,150 about how it amplifies or attenuates frequencies. 535 00:28:44,150 --> 00:28:44,780 Right? 536 00:28:44,780 --> 00:28:47,240 It likes frequencies near the middle. 537 00:28:47,240 --> 00:28:49,020 There's a lot of frequencies-- 538 00:28:49,020 --> 00:28:52,240 so if this represented a system function, 539 00:28:52,240 --> 00:28:56,834 it would pass with a gain of two frequencies near 0. 540 00:28:56,834 --> 00:28:59,000 And the magnitude would be smaller for these others. 541 00:28:59,000 --> 00:29:00,583 And there is a phase relationship too. 542 00:29:00,583 --> 00:29:04,770 So there's insights that you can get from this Fourier 543 00:29:04,770 --> 00:29:07,920 representation that are less easy to get from the Laplace. 544 00:29:07,920 --> 00:29:10,860 I mean the Laplace was a complete specification 545 00:29:10,860 --> 00:29:14,357 of a signal or a system, either. 546 00:29:14,357 --> 00:29:15,690 So all the information is there. 547 00:29:15,690 --> 00:29:18,326 It's just that it's more apparent-- 548 00:29:18,326 --> 00:29:20,700 some of the information is more apparent-- in the Fourier 549 00:29:20,700 --> 00:29:23,910 representation. 550 00:29:23,910 --> 00:29:29,960 OK, second question, what if I stretched the time axis? 551 00:29:29,960 --> 00:29:35,060 x1 was 1 between minus 1 and 1. x2 is 1 between minus 2 and 2. 552 00:29:35,060 --> 00:29:37,460 So all I'm doing, stretching the axis. 553 00:29:37,460 --> 00:29:40,760 What happens to the Fourier transform? 554 00:29:40,760 --> 00:29:41,720 Look at your neighbor. 555 00:29:41,720 --> 00:29:43,500 Choose a number. 556 00:29:43,500 --> 00:29:45,480 [SIDE CONVERSATIONS] 557 00:30:34,520 --> 00:30:39,194 OK, which answer tells me what happens when I stretch time? 558 00:30:39,194 --> 00:30:40,860 So everybody raise your hand and tell me 559 00:30:40,860 --> 00:30:46,140 some number between 0 and 5, 1 and 5 actually. 560 00:30:46,140 --> 00:30:50,190 OK, 20% correct. 561 00:30:50,190 --> 00:30:52,890 S 562 00:30:52,890 --> 00:30:55,180 So what's going to happen? 563 00:30:55,180 --> 00:31:01,440 Well, it's pretty easy to simply do out the integral again. 564 00:31:01,440 --> 00:31:03,450 Right, so that's the sort of most primitive way 565 00:31:03,450 --> 00:31:05,100 you can think about it. 566 00:31:05,100 --> 00:31:08,810 If you simply run the integral, I've 567 00:31:08,810 --> 00:31:10,610 written it in a kind of funny way. 568 00:31:10,610 --> 00:31:11,750 Right? 569 00:31:11,750 --> 00:31:15,430 So a lot of you said one for the answer. 570 00:31:18,804 --> 00:31:19,970 This kind of looks like one. 571 00:31:19,970 --> 00:31:22,314 Why is that not one? 572 00:31:22,314 --> 00:31:23,230 That's actually three. 573 00:31:26,840 --> 00:31:30,200 Why do I like to write it as-- 574 00:31:30,200 --> 00:31:32,180 instead of writing 2 since 2 omega over omega 575 00:31:32,180 --> 00:31:34,460 I like to write 4 signed 2 omega over 2 omega. 576 00:31:34,460 --> 00:31:37,070 Why do I like that? 577 00:31:37,070 --> 00:31:38,495 Because I'm completely random. 578 00:31:41,169 --> 00:31:42,710 AUDIENCE: Omega is the same that way? 579 00:31:42,710 --> 00:31:43,205 DENNIS FREEMAN: Excuse me. 580 00:31:43,205 --> 00:31:45,185 AUDIENCE: Omega can be the same-- like you 581 00:31:45,185 --> 00:31:47,189 can have omega absorb the 2. 582 00:31:47,189 --> 00:31:48,730 DENNIS FREEMAN: That's kind of right. 583 00:31:48,730 --> 00:31:51,050 So can you unscramble the sentence slightly? 584 00:31:53,850 --> 00:31:54,900 What is more-- yes. 585 00:31:54,900 --> 00:31:57,440 AUDIENCE: Aren't they the same form that we use? 586 00:31:57,440 --> 00:31:59,700 DENNIS FREEMAN: It's the same form in what sense? 587 00:31:59,700 --> 00:32:01,140 I mean what's the same about it? 588 00:32:01,140 --> 00:32:02,330 Yes. 589 00:32:02,330 --> 00:32:03,152 Yes. 590 00:32:03,152 --> 00:32:06,104 AUDIENCE: Like I thought I was going to say omega 591 00:32:06,104 --> 00:32:08,056 is near 0 when number two is 4. 592 00:32:08,056 --> 00:32:09,430 DENNIS FREEMAN: Correct, correct. 593 00:32:09,430 --> 00:32:13,830 If you think about what happens for omega near 0, 594 00:32:13,830 --> 00:32:16,530 I've got the sine of 2 omega, which is-- 595 00:32:19,450 --> 00:32:22,540 what's the sine of 2 omega when you make it 0? 596 00:32:22,540 --> 00:32:23,140 0. 597 00:32:23,140 --> 00:32:24,580 So I have 0 over 0. 598 00:32:24,580 --> 00:32:25,984 Bad. 599 00:32:25,984 --> 00:32:26,650 So what do I do? 600 00:32:29,980 --> 00:32:31,050 L'Hospital. 601 00:32:31,050 --> 00:32:33,120 So if I do L'Hospital's rule, then I 602 00:32:33,120 --> 00:32:35,490 can make this thing look like one. 603 00:32:35,490 --> 00:32:39,510 And if I write it in the form sine 2 omega over 2 omega, 604 00:32:39,510 --> 00:32:43,350 that has a value near 0 that approaches 1. 605 00:32:43,350 --> 00:32:45,516 So the amplitude is 4. 606 00:32:45,516 --> 00:32:46,890 So that's a way of separating out 607 00:32:46,890 --> 00:32:50,400 the part that's unity amplitude from the part that 608 00:32:50,400 --> 00:32:52,750 is the constant that multiplies the amplitude. 609 00:32:52,750 --> 00:32:55,230 So the amplitude is 4. 610 00:32:55,230 --> 00:32:59,790 And frequency, which had been pi, moves to pi over 2. 611 00:32:59,790 --> 00:33:02,660 So the point is that-- 612 00:33:02,660 --> 00:33:05,210 so the answer is number three. 613 00:33:05,210 --> 00:33:11,750 The peak increases, and the frequency spacing decreases. 614 00:33:11,750 --> 00:33:17,650 But more generally, the point is that if I stretched time, 615 00:33:17,650 --> 00:33:19,782 I compress frequency. 616 00:33:19,782 --> 00:33:21,740 But I compress frequency in a very special way. 617 00:33:21,740 --> 00:33:28,360 I compress frequency in an area-preserving way. 618 00:33:28,360 --> 00:33:32,080 That's why the peak popped up. 619 00:33:32,080 --> 00:33:34,980 So what I'd like to do is think about a general scaling rule. 620 00:33:34,980 --> 00:33:40,050 If I wanted to think about scaling x1 into x2, 621 00:33:40,050 --> 00:33:45,340 such that x2 is a scaled version of time compared to x1, 622 00:33:45,340 --> 00:33:49,460 so if I wanted x2 of t to be x1 of at, 623 00:33:49,460 --> 00:33:53,160 and if I wanted to stretch x1 to turn it into x2, 624 00:33:53,160 --> 00:33:54,570 should I make a1-- 625 00:33:54,570 --> 00:33:56,810 should I make a bigger or less than 1? 626 00:34:01,634 --> 00:34:03,800 I'm trying to generalize the result that I just did. 627 00:34:03,800 --> 00:34:04,880 Right? 628 00:34:04,880 --> 00:34:06,800 So I stretched x1 into x2. 629 00:34:06,800 --> 00:34:12,587 And what I saw is that frequency shrunk, and amplitude went up. 630 00:34:12,587 --> 00:34:14,420 So now, I'm thinking about what would happen 631 00:34:14,420 --> 00:34:16,230 if I did that in general. 632 00:34:16,230 --> 00:34:19,850 If I took x1, and I stretched it by setting 633 00:34:19,850 --> 00:34:23,750 x2 equal to x1 of at, would I want 634 00:34:23,750 --> 00:34:28,219 a to be bigger or less than 1 if I want to stretch x1 635 00:34:28,219 --> 00:34:30,324 to turn it into x2? 636 00:34:30,324 --> 00:34:31,710 AUDIENCE: Less than 1. 637 00:34:31,710 --> 00:34:34,530 DENNIS FREEMAN: Less than 1 because then the logic 638 00:34:34,530 --> 00:34:40,050 is that if I wanted, for example, x2 of 2 to be x1 of 1, 639 00:34:40,050 --> 00:34:42,199 stretch x2-- 640 00:34:42,199 --> 00:34:48,690 stretch x1, sorry, so that it's value x2 at the position 2 641 00:34:48,690 --> 00:34:53,230 is the same as the original function x1 at 1. 642 00:34:53,230 --> 00:34:55,000 If I stretched it, then I clearly 643 00:34:55,000 --> 00:34:57,070 have to have a equal to a half in that case. 644 00:34:57,070 --> 00:35:00,890 And in general, stretching would correspond to a less than 1. 645 00:35:00,890 --> 00:35:03,010 And now, I can think about where that fits 646 00:35:03,010 --> 00:35:07,300 in the transform relationship. 647 00:35:07,300 --> 00:35:11,990 Think about finding the Fourier transform of x2, 648 00:35:11,990 --> 00:35:18,350 and substituting x1 of at for x2, 649 00:35:18,350 --> 00:35:21,350 and then making this relationship look 650 00:35:21,350 --> 00:35:24,450 more like a Fourier transform. 651 00:35:24,450 --> 00:35:26,330 So I don't want the at to be here. 652 00:35:26,330 --> 00:35:28,730 I want function of t. 653 00:35:28,730 --> 00:35:31,535 So I can rewrite at as tau. 654 00:35:34,450 --> 00:35:37,930 Now, this looks like a Fourier transform except that 655 00:35:37,930 --> 00:35:40,720 I've changed all my t's to tau's. 656 00:35:40,720 --> 00:35:43,660 And the point is that that transformation of tau 657 00:35:43,660 --> 00:35:47,200 equals at shows up in two places. 658 00:35:47,200 --> 00:35:50,230 There's an explicit time here. 659 00:35:50,230 --> 00:35:52,075 And there's a time dependence in the dt. 660 00:35:54,830 --> 00:35:57,710 So the dt one is the one that gives me the shrinking 661 00:35:57,710 --> 00:36:01,280 and swelling of the axis. 662 00:36:01,280 --> 00:36:06,320 And the 1 over a from here is the one that gives me 663 00:36:06,320 --> 00:36:09,410 the changing amplitude. 664 00:36:09,410 --> 00:36:12,000 That's how you get the area-preserving property. 665 00:36:12,000 --> 00:36:14,990 However much it got compressed-- however 666 00:36:14,990 --> 00:36:18,110 much it stretched in time so that it became compressed 667 00:36:18,110 --> 00:36:21,110 in frequency, whatever the factor is that compressed it 668 00:36:21,110 --> 00:36:23,694 in frequency, it also makes, by the same factor, 669 00:36:23,694 --> 00:36:24,485 it makes it taller. 670 00:36:26,829 --> 00:36:29,370 We'd like to build up intuition for how the Fourier transform 671 00:36:29,370 --> 00:36:29,580 works. 672 00:36:29,580 --> 00:36:31,829 That's the reason for doing these kinds of properties. 673 00:36:34,820 --> 00:36:38,680 So now, there's another way of thinking about that same thing 674 00:36:38,680 --> 00:36:41,475 by thinking about what we call the moment theorems. 675 00:36:45,730 --> 00:36:47,800 Here what we think about is what would 676 00:36:47,800 --> 00:36:50,270 happen if we evaluated the Fourier transform at omega 677 00:36:50,270 --> 00:36:52,610 equals 0. 678 00:36:52,610 --> 00:36:56,400 Well, omega equals 0 is associated with a particularly 679 00:36:56,400 --> 00:37:00,780 simple complex exponential. 680 00:37:00,780 --> 00:37:03,840 If the frequency is 0, e to the j0 t is 1. 681 00:37:06,560 --> 00:37:09,260 So what you see is that the value 682 00:37:09,260 --> 00:37:10,960 of the Fourier transform at omega 683 00:37:10,960 --> 00:37:13,335 equals 0 is the area under the curve. 684 00:37:17,200 --> 00:37:20,640 So the idea then is that if I took 685 00:37:20,640 --> 00:37:25,270 an x of t, which was x1, which was 1 between minus 1 and 1, 686 00:37:25,270 --> 00:37:26,550 there's an area of 2. 687 00:37:26,550 --> 00:37:28,300 And that's a way of directly saying, well, 688 00:37:28,300 --> 00:37:30,240 the Fourier transform at 0 better be 2. 689 00:37:32,572 --> 00:37:34,030 And the intuitive thing that you're 690 00:37:34,030 --> 00:37:35,590 supposed to take away from that is 691 00:37:35,590 --> 00:37:38,620 when you look at a Fourier transform, the value at 0 692 00:37:38,620 --> 00:37:40,450 is the dc. 693 00:37:40,450 --> 00:37:46,500 How much constant is there in that signal? 694 00:37:46,500 --> 00:37:48,660 So there's a very explicit representation 695 00:37:48,660 --> 00:37:50,520 for the frequency content. 696 00:37:50,520 --> 00:37:52,770 I mean that's what the Fourier transform is all about. 697 00:37:52,770 --> 00:37:55,200 And in particular, the zero frequency is dc. 698 00:37:55,200 --> 00:37:56,240 It's the average value. 699 00:37:58,770 --> 00:38:01,870 That kind of a relationship works both ways. 700 00:38:01,870 --> 00:38:06,150 If you were to use the synthesis formula 701 00:38:06,150 --> 00:38:11,565 and think about how do you synthesize x of 0, 702 00:38:11,565 --> 00:38:12,940 well, it's the same sort of thing 703 00:38:12,940 --> 00:38:17,050 except now the t is 0 instead of the omega being 0. 704 00:38:17,050 --> 00:38:19,660 And what we get is 1 over 2pi times 705 00:38:19,660 --> 00:38:25,170 the area under the transform. 706 00:38:25,170 --> 00:38:28,310 So what that says is whatever is going on 707 00:38:28,310 --> 00:38:33,800 over in this wiggly thing, the net area, the average value, 708 00:38:33,800 --> 00:38:38,690 divided by 2pi has to equal the value at x1 of 0. 709 00:38:38,690 --> 00:38:41,540 x1 of 0 is clearly 1. 710 00:38:41,540 --> 00:38:46,370 So that means the area under this thing must be 2pi. 711 00:38:46,370 --> 00:38:48,026 That wasn't particularly clear. 712 00:38:48,026 --> 00:38:49,400 I mean I don't know automatically 713 00:38:49,400 --> 00:38:51,140 the area under that curve. 714 00:38:51,140 --> 00:38:53,840 But that's such a frequently recurring thing 715 00:38:53,840 --> 00:38:59,240 that it's useful to notice that the area under this funny curve 716 00:38:59,240 --> 00:39:02,285 happens to be precisely the area of this inscribed triangle. 717 00:39:04,870 --> 00:39:06,910 So the height of the triangle is 2. 718 00:39:06,910 --> 00:39:10,300 Half the base is pi. 719 00:39:10,300 --> 00:39:11,440 So the area is 2pi. 720 00:39:14,080 --> 00:39:20,400 I'm sure some Greek knew this. 721 00:39:20,400 --> 00:39:21,390 But I don't. 722 00:39:21,390 --> 00:39:22,800 So if somebody can think of a way 723 00:39:22,800 --> 00:39:25,760 to derive that answer without using Fourier transforms-- 724 00:39:25,760 --> 00:39:27,630 I can do it with Fourier transforms. 725 00:39:27,630 --> 00:39:31,350 And I can look it up in books where the authors also 726 00:39:31,350 --> 00:39:32,940 use Fourier transforms. 727 00:39:32,940 --> 00:39:36,030 But I'm sure some ancient Greek can do this. 728 00:39:36,030 --> 00:39:38,160 So the question is if anybody can figure out 729 00:39:38,160 --> 00:39:41,320 how the ancient Greeks would have come to that conclusion, 730 00:39:41,320 --> 00:39:43,430 I would be very interested to know. 731 00:39:43,430 --> 00:39:47,494 Does everybody get-- so areas of inscribed whatevers, 732 00:39:47,494 --> 00:39:48,660 right, that's what they did. 733 00:39:48,660 --> 00:39:49,170 Right? 734 00:39:49,170 --> 00:39:51,780 So I would like to know how to get 735 00:39:51,780 --> 00:39:55,620 the fact that the area under this wiggly function 2 sine 736 00:39:55,620 --> 00:40:00,450 omega over omega is 2pi without knowing Fourier transforms. 737 00:40:00,450 --> 00:40:01,790 So that's an open challenge. 738 00:40:01,790 --> 00:40:04,620 So try to figure out how to prove that without using 739 00:40:04,620 --> 00:40:08,310 Fourier transforms. 740 00:40:08,310 --> 00:40:11,910 So if we use the moment idea, and we think about this scaling 741 00:40:11,910 --> 00:40:16,560 thing, we come up with a very interesting result 742 00:40:16,560 --> 00:40:19,110 that if you were to stretch x1, which 743 00:40:19,110 --> 00:40:21,330 had been 1 just between minus 1 and 1, 744 00:40:21,330 --> 00:40:24,960 to turn it into x2, which was 1 between minus 2 and 2, 745 00:40:24,960 --> 00:40:30,222 and just keep stretching, what would happen? 746 00:40:30,222 --> 00:40:31,680 Well, it gets skinnier and skinnier 747 00:40:31,680 --> 00:40:34,150 and skinnier and skinnier. 748 00:40:34,150 --> 00:40:37,312 But in a very special way, the area is the same. 749 00:40:37,312 --> 00:40:39,520 Even though it got skinnier and skinnier and skinnier 750 00:40:39,520 --> 00:40:43,030 and skinnier, the area is the same. 751 00:40:43,030 --> 00:40:47,720 If you keep doing that, it turns into an impulse. 752 00:40:47,720 --> 00:40:50,460 Well, that's pretty interesting. 753 00:40:50,460 --> 00:40:52,970 That's an alternative way of deriving an impulse. 754 00:40:52,970 --> 00:40:55,935 An impulse, we think about an impulse 755 00:40:55,935 --> 00:40:58,120 as a generalized function. 756 00:40:58,120 --> 00:41:00,250 Any function that has the property 757 00:41:00,250 --> 00:41:06,650 that in some kind of a limit, the area shrinks towards 0, 758 00:41:06,650 --> 00:41:10,030 but the area doesn't change, that 759 00:41:10,030 --> 00:41:11,339 turns into a delta function. 760 00:41:11,339 --> 00:41:13,630 That's a different way of thinking about the definition 761 00:41:13,630 --> 00:41:15,140 of a delta function. 762 00:41:15,140 --> 00:41:17,560 And so we just found something very interesting. 763 00:41:17,560 --> 00:41:23,880 The Fourier transform of the constant 1 764 00:41:23,880 --> 00:41:28,680 seems to be a delta function as 0 of area 2pi. 765 00:41:28,680 --> 00:41:30,030 Well, that's pretty interesting. 766 00:41:30,030 --> 00:41:31,446 What's the Laplace transform of 1? 767 00:41:38,450 --> 00:41:39,950 Too shocking. 768 00:41:39,950 --> 00:41:41,900 What's the Laplace transform of 1? 769 00:41:44,738 --> 00:41:46,160 AUDIENCE: It's a delta function. 770 00:41:46,160 --> 00:41:48,844 DENNIS FREEMAN: Delta function. 771 00:41:48,844 --> 00:41:50,260 What's the Laplace transform of 1? 772 00:41:50,260 --> 00:41:54,980 And So Laplace transform, right, so x of s, 773 00:41:54,980 --> 00:42:00,240 integral 1e to the minus st dt. 774 00:42:00,240 --> 00:42:02,070 What's the Laplace transform of 1? 775 00:42:12,300 --> 00:42:13,139 AUDIENCE: 1 over s. 776 00:42:13,139 --> 00:42:14,180 DENNIS FREEMAN: 1 over s. 777 00:42:14,180 --> 00:42:15,240 How about 1 over s? 778 00:42:18,990 --> 00:42:21,415 Yes? 779 00:42:21,415 --> 00:42:21,915 No? 780 00:42:25,880 --> 00:42:28,000 1 over s-- yes. 781 00:42:28,000 --> 00:42:28,800 1 over s-- no. 782 00:42:31,380 --> 00:42:32,710 Me. 783 00:42:32,710 --> 00:42:34,570 1 over s, no. 784 00:42:34,570 --> 00:42:37,345 Why not? 785 00:42:37,345 --> 00:42:41,170 AUDIENCE: You would just use the su of 1 u of t. 786 00:42:41,170 --> 00:42:42,500 DENNIS FREEMAN: It's 1, yes. 787 00:42:42,500 --> 00:42:48,501 So the Laplace transform of u of t is 1 over s. 788 00:42:48,501 --> 00:42:49,001 Right? 789 00:42:52,005 --> 00:42:54,380 You remember there was a region of convergence associated 790 00:42:54,380 --> 00:42:57,080 with Laplace transforms. 791 00:42:57,080 --> 00:42:59,180 The region of convergence, we thought about 792 00:42:59,180 --> 00:43:06,399 like if you had a time function like u of t, 793 00:43:06,399 --> 00:43:07,940 then it would converge as long as you 794 00:43:07,940 --> 00:43:15,020 multiplied by some factor that generally attenuated. 795 00:43:15,020 --> 00:43:18,050 So that bounded what kinds of s's worked. 796 00:43:18,050 --> 00:43:22,340 We needed the real part of s bigger than 0. 797 00:43:22,340 --> 00:43:25,790 Because if the real part of s was the other way, 798 00:43:25,790 --> 00:43:27,450 the interval would diverge-- 799 00:43:27,450 --> 00:43:29,180 bad. 800 00:43:29,180 --> 00:43:33,900 Right so we could find the Laplace transform of u of t-- 801 00:43:38,630 --> 00:43:41,600 real part of s bigger than 0. 802 00:43:41,600 --> 00:43:45,260 Or we could find the Laplace transform of a backward step. 803 00:43:48,130 --> 00:43:52,000 The region of convergence would flip. 804 00:43:52,000 --> 00:43:54,240 And we got a sign change. 805 00:43:56,770 --> 00:43:59,430 But the Laplace transform of 1 doesn't exist. 806 00:44:02,500 --> 00:44:06,940 There is no region of convergence for the function 1. 807 00:44:06,940 --> 00:44:10,030 That's a big difference between Fourier and Laplace as well. 808 00:44:10,030 --> 00:44:16,300 Even though Fourier, is in some sense, a subset of Laplace, 809 00:44:16,300 --> 00:44:18,730 there are some signals that have Fourier transforms 810 00:44:18,730 --> 00:44:21,980 and not Laplace transforms, and so in that sense, 811 00:44:21,980 --> 00:44:23,951 Laplace is a subset of Fourier. 812 00:44:23,951 --> 00:44:25,450 So in fact, you better think of them 813 00:44:25,450 --> 00:44:28,720 as Venn diagrams that overlap. 814 00:44:28,720 --> 00:44:32,470 So there are some signals that have both, 815 00:44:32,470 --> 00:44:36,880 but there are some signals that have one and not the other. 816 00:44:36,880 --> 00:44:45,370 OK, so the final and maybe most important property 817 00:44:45,370 --> 00:44:48,010 of Laplace transforms is that they have 818 00:44:48,010 --> 00:44:51,620 a simple, inverse relationship. 819 00:44:51,620 --> 00:44:53,370 You may remember that I talked about there 820 00:44:53,370 --> 00:44:55,120 being an inverse relationship for Laplace. 821 00:44:59,100 --> 00:45:01,930 So you can think about x of t being 822 00:45:01,930 --> 00:45:06,860 1 over j 2pi, the integral over some sort of a contour of x 823 00:45:06,860 --> 00:45:11,400 of s e to the st ds. 824 00:45:11,400 --> 00:45:13,240 And I told you don't ever try to do that 825 00:45:13,240 --> 00:45:17,710 without going over to math and talking to those folks first. 826 00:45:17,710 --> 00:45:20,170 That's complicated. 827 00:45:20,170 --> 00:45:22,210 The interesting thing about this relationship 828 00:45:22,210 --> 00:45:25,690 is that it's really simple. 829 00:45:25,690 --> 00:45:29,560 So there's a very simple relationship between a Fourier 830 00:45:29,560 --> 00:45:33,860 transform and its inverse. 831 00:45:33,860 --> 00:45:38,380 OK, so I think I'll defer talking about that 832 00:45:38,380 --> 00:45:44,150 until next time, the reason being that I want 833 00:45:44,150 --> 00:45:45,690 to end a little earlier today. 834 00:45:45,690 --> 00:45:48,890 So I'll finish talking about the rest of the slides 835 00:45:48,890 --> 00:45:51,100 on the next lecture.