1 00:00:00,120 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:04,059 Commons license. 3 00:00:04,059 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,750 continue to offer high quality educational resources for free. 5 00:00:10,750 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,310 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,310 --> 00:00:18,480 at ocw.mit.edu. 8 00:00:21,780 --> 00:00:24,505 DENNIS FREEMAN: Hello. 9 00:00:24,505 --> 00:00:29,890 So there's almost certainly no need to show this, 10 00:00:29,890 --> 00:00:33,160 but the good news is this is the last time I'll ever 11 00:00:33,160 --> 00:00:35,420 show a slide like this. 12 00:00:35,420 --> 00:00:36,626 It'll never happen again. 13 00:00:36,626 --> 00:00:37,750 At least not to this crowd. 14 00:00:37,750 --> 00:00:41,110 So you can all rest assured this is the last time you'll 15 00:00:41,110 --> 00:00:43,780 see something like this. 16 00:00:43,780 --> 00:00:46,675 Any questions on this, everybody's happy? 17 00:00:49,180 --> 00:00:52,210 Those are two separate questions, I guess. 18 00:00:52,210 --> 00:00:53,590 Any questions on this? 19 00:00:56,120 --> 00:00:57,940 OK, good. 20 00:00:57,940 --> 00:01:01,390 So today I want to finish up talking 21 00:01:01,390 --> 00:01:04,450 about Fourier transforms by comparing a lot of stuff we've 22 00:01:04,450 --> 00:01:05,129 already done. 23 00:01:05,129 --> 00:01:07,840 So there's nothing new today. 24 00:01:07,840 --> 00:01:09,370 This is intended, if anything, to be 25 00:01:09,370 --> 00:01:12,010 a bit of a review in light of things 26 00:01:12,010 --> 00:01:14,740 that might happen tomorrow. 27 00:01:14,740 --> 00:01:18,250 And also just to make things look simpler. 28 00:01:18,250 --> 00:01:20,860 Now, the best way to make things look simpler 29 00:01:20,860 --> 00:01:24,220 is to start with something that looks incredibly complicated. 30 00:01:24,220 --> 00:01:27,020 OK, so this is my thing that looks very complicated. 31 00:01:27,020 --> 00:01:31,581 This is simply a summary of the transforms 32 00:01:31,581 --> 00:01:32,830 that we've talked about today. 33 00:01:32,830 --> 00:01:35,420 We've talked about four of them. 34 00:01:35,420 --> 00:01:38,980 And the point of the slide is first, 35 00:01:38,980 --> 00:01:41,020 it looks a little complicated. 36 00:01:41,020 --> 00:01:43,930 But much more importantly, by the end of the hour, 37 00:01:43,930 --> 00:01:45,500 it's supposed to look very simple. 38 00:01:45,500 --> 00:01:46,000 Yes? 39 00:01:46,000 --> 00:01:48,445 AUDIENCE: Could we increase the volume? 40 00:01:48,445 --> 00:01:50,570 DENNIS FREEMAN: Could we increase the volume a bit? 41 00:01:50,570 --> 00:01:53,860 Well, I know that I can shout more. 42 00:01:53,860 --> 00:01:57,580 And if they notice me, they'll turn the volume up, too. 43 00:01:57,580 --> 00:01:58,870 So I will try to shout more. 44 00:01:58,870 --> 00:02:04,190 So if I don't, please tell me to do so. 45 00:02:04,190 --> 00:02:09,639 So the idea, then, is to try to look for some structure. 46 00:02:09,639 --> 00:02:11,530 Not only within one of the transforms, 47 00:02:11,530 --> 00:02:16,660 but also across all of the transforms. 48 00:02:16,660 --> 00:02:20,470 And to start this off, let's just think about it 49 00:02:20,470 --> 00:02:22,120 at the most basic level. 50 00:02:22,120 --> 00:02:26,480 What were the kinds of signals that we thought about? 51 00:02:26,480 --> 00:02:27,980 The kinds of signals that we thought 52 00:02:27,980 --> 00:02:30,850 about when we did DT Fourier series, DT Fourier 53 00:02:30,850 --> 00:02:34,940 transforms, CT Fourier series, and CT Fourier transforms? 54 00:02:34,940 --> 00:02:40,570 The kinds of signals look different, right? 55 00:02:40,570 --> 00:02:44,290 So in the top ones, we had discrete time. 56 00:02:44,290 --> 00:02:47,470 In the bottom ones, we have continuous time. 57 00:02:47,470 --> 00:02:51,070 In the left we have things that are periodic. 58 00:02:51,070 --> 00:02:54,047 In the right we had things that were not periodic. 59 00:02:54,047 --> 00:02:55,380 Four different kinds of signals. 60 00:02:55,380 --> 00:02:57,463 Not surprising there would be four different kinds 61 00:02:57,463 --> 00:02:59,590 of transforms. 62 00:02:59,590 --> 00:03:04,210 An interesting thing happens if you look at the Fourier domain. 63 00:03:04,210 --> 00:03:05,770 I haven't switched around the order. 64 00:03:05,770 --> 00:03:09,720 It was DT Fourier, DT Fourier transform, CT series, 65 00:03:09,720 --> 00:03:10,974 CT transform. 66 00:03:10,974 --> 00:03:11,640 It's still that. 67 00:03:14,530 --> 00:03:16,750 I've drawn a sort of iconic view of how 68 00:03:16,750 --> 00:03:22,510 we would draw a picture of the Fourier transform of interest. 69 00:03:22,510 --> 00:03:24,150 Where is discrete? 70 00:03:28,080 --> 00:03:29,760 So discrete in the previous picture-- 71 00:03:29,760 --> 00:03:32,220 which was a picture of time-- 72 00:03:32,220 --> 00:03:33,762 NNTT. 73 00:03:33,762 --> 00:03:35,220 Where was discrete in this picture? 74 00:03:39,070 --> 00:03:40,300 Up. 75 00:03:40,300 --> 00:03:44,290 Where is discrete in this picture? 76 00:03:44,290 --> 00:03:45,910 Left, that's weird. 77 00:03:48,970 --> 00:03:50,695 Where is periodic in this picture? 78 00:03:53,770 --> 00:03:56,130 Left. 79 00:03:56,130 --> 00:03:57,800 Where is periodic in this picture? 80 00:04:02,110 --> 00:04:05,531 Is there a periodic in this picture? 81 00:04:05,531 --> 00:04:06,030 Top. 82 00:04:06,030 --> 00:04:08,760 Why do you say top? 83 00:04:08,760 --> 00:04:10,950 Ah, unit circle! 84 00:04:10,950 --> 00:04:13,050 Unit circle is periodic. 85 00:04:13,050 --> 00:04:17,490 So there's discreteness and periodicity in both pictures, 86 00:04:17,490 --> 00:04:20,890 but they're not at the same place. 87 00:04:20,890 --> 00:04:23,050 Well, that's kind of interesting, right? 88 00:04:23,050 --> 00:04:25,049 Somehow, discrete and periodic map 89 00:04:25,049 --> 00:04:26,590 to things like discrete and periodic, 90 00:04:26,590 --> 00:04:29,110 but they don't map directly. 91 00:04:29,110 --> 00:04:30,380 That's kind of interesting. 92 00:04:30,380 --> 00:04:31,960 So part of the goal today is to try 93 00:04:31,960 --> 00:04:33,168 to get to the bottom of that. 94 00:04:33,168 --> 00:04:35,380 To understand why that's true. 95 00:04:35,380 --> 00:04:39,670 You already know a lot about how different transformations are 96 00:04:39,670 --> 00:04:41,210 related. 97 00:04:41,210 --> 00:04:43,000 So let's think of a simple example 98 00:04:43,000 --> 00:04:45,690 of the relation between Fourier series and Fourier transforms. 99 00:04:45,690 --> 00:04:48,148 Imagine that we have one of those signals that was periodic 100 00:04:48,148 --> 00:04:51,400 in time, continuous in time. 101 00:04:51,400 --> 00:04:52,870 Then there'd be a Fourier series-- 102 00:04:52,870 --> 00:04:55,360 represented this way. 103 00:04:55,360 --> 00:04:57,220 So I'd end up with a Fourier series that has 104 00:04:57,220 --> 00:04:59,620 a bunch of coefficients, a, k. 105 00:04:59,620 --> 00:05:01,870 Can somebody tell me, what would the Fourier transform 106 00:05:01,870 --> 00:05:02,370 look like? 107 00:05:05,322 --> 00:05:06,780 Fourier transform-- are you allowed 108 00:05:06,780 --> 00:05:08,863 to take the Fourier transform of a periodic thing? 109 00:05:14,054 --> 00:05:14,554 Yeah? 110 00:05:14,554 --> 00:05:17,637 AUDIENCE: Could we take the same thing with impulses? 111 00:05:17,637 --> 00:05:19,720 DENNIS FREEMAN: Take the same thing with impulses. 112 00:05:19,720 --> 00:05:21,650 OK, I like lots of those words. 113 00:05:21,650 --> 00:05:25,182 Could you put them together in slightly a different order? 114 00:05:25,182 --> 00:05:26,515 Where should I put the impulses? 115 00:05:29,776 --> 00:05:30,758 Yeah? 116 00:05:30,758 --> 00:05:33,166 AUDIENCE: At the k omega not. 117 00:05:33,166 --> 00:05:35,247 DENNIS FREEMAN: At the k omega not. 118 00:05:35,247 --> 00:05:36,080 How do you get that? 119 00:05:36,080 --> 00:05:36,800 That's exactly right. 120 00:05:36,800 --> 00:05:37,591 How'd you get that? 121 00:05:37,591 --> 00:05:41,324 AUDIENCE: Because the [INAUDIBLE].. 122 00:05:53,670 --> 00:05:55,170 DENNIS FREEMAN: So there's some kind 123 00:05:55,170 --> 00:06:00,999 of a relationship between the k's, in the ak's, and frequency 124 00:06:00,999 --> 00:06:02,790 that ought to be represented as an impulse, 125 00:06:02,790 --> 00:06:05,250 is what you're saying? 126 00:06:05,250 --> 00:06:07,410 One way you can see that is to take the Fourier 127 00:06:07,410 --> 00:06:08,710 transform of this expression. 128 00:06:08,710 --> 00:06:10,876 What would you get if you took the Fourier transform 129 00:06:10,876 --> 00:06:12,420 of x of t, x of j omega? 130 00:06:12,420 --> 00:06:13,500 That's pretty easy. 131 00:06:13,500 --> 00:06:16,083 What would you get if you took the Fourier transform of a sum? 132 00:06:18,675 --> 00:06:20,175 What do you always get when you take 133 00:06:20,175 --> 00:06:22,387 the Fourier transfer of a sum? 134 00:06:22,387 --> 00:06:24,095 The sum of the Fourier transforms, right? 135 00:06:24,095 --> 00:06:25,850 It's within your operator. 136 00:06:25,850 --> 00:06:28,880 That's one of the most powerful reasons we like it. 137 00:06:28,880 --> 00:06:32,780 So sum of the functions, take the transform of each one. 138 00:06:32,780 --> 00:06:34,160 Add them back up. 139 00:06:34,160 --> 00:06:36,980 How do you take the Fourier transform of e to the j omega 140 00:06:36,980 --> 00:06:38,510 not t? 141 00:06:38,510 --> 00:06:41,170 What's the Fourier transform? e to the j omega 142 00:06:41,170 --> 00:06:44,570 not t transforms to what? 143 00:06:44,570 --> 00:06:50,840 Fourier transform-- impulse. 144 00:06:50,840 --> 00:06:53,000 So how do I know that? 145 00:06:53,000 --> 00:06:57,700 So impulse-- what else, anything? 146 00:06:57,700 --> 00:06:58,230 Am I done? 147 00:07:00,589 --> 00:07:01,880 Where should I put the impulse? 148 00:07:07,140 --> 00:07:08,260 What should I put-- 149 00:07:08,260 --> 00:07:11,800 how should I indicate where I put it? 150 00:07:11,800 --> 00:07:15,640 So I need to say omega minus omega not, or something 151 00:07:15,640 --> 00:07:17,510 like that, right? 152 00:07:17,510 --> 00:07:20,940 So how do I know that? 153 00:07:20,940 --> 00:07:23,970 And am I done? 154 00:07:23,970 --> 00:07:24,606 Yeah? 155 00:07:24,606 --> 00:07:25,500 AUDIENCE: Multiply by 2 pi. 156 00:07:25,500 --> 00:07:26,910 DENNIS FREEMAN: Multiply by 2 pi. 157 00:07:26,910 --> 00:07:28,285 And a good way to do that is just 158 00:07:28,285 --> 00:07:31,684 to memorize an absurd number of equations, right? 159 00:07:31,684 --> 00:07:33,600 Let's think about how you would remember that. 160 00:07:33,600 --> 00:07:36,700 So if you wanted to take the Fourier transform of anything, 161 00:07:36,700 --> 00:07:37,200 right? 162 00:07:37,200 --> 00:07:43,111 You would say, h of j omega is some integral of anything e 163 00:07:43,111 --> 00:07:47,550 to the minus j omega t dt, right? 164 00:07:47,550 --> 00:07:50,620 What happens if I plug x of t equals e to the j omega 165 00:07:50,620 --> 00:07:52,433 not t into this expression? 166 00:07:55,540 --> 00:07:58,554 Can you close the integral? 167 00:07:58,554 --> 00:08:00,470 I mean, that's the way you would do it, right? 168 00:08:00,470 --> 00:08:02,803 I want to take the Fourier transform of e to the j omega 169 00:08:02,803 --> 00:08:04,340 not t. 170 00:08:04,340 --> 00:08:06,800 You stick it in the formula, and it should come out, right? 171 00:08:10,862 --> 00:08:12,820 So how many of you can integrate this function? 172 00:08:18,000 --> 00:08:19,080 No one, good. 173 00:08:19,080 --> 00:08:22,137 That's good, because I can't either. 174 00:08:22,137 --> 00:08:23,970 What's hard about integrating that function? 175 00:08:29,675 --> 00:08:31,300 That was a much easier question, right? 176 00:08:31,300 --> 00:08:33,880 Can you integrate was a hard question. 177 00:08:33,880 --> 00:08:34,870 Well, maybe not. 178 00:08:34,870 --> 00:08:36,760 It's binary. 179 00:08:36,760 --> 00:08:38,409 What's difficult about integrating 180 00:08:38,409 --> 00:08:40,330 that function of time? 181 00:08:40,330 --> 00:08:44,169 What if I try to integrate e to the j omega not 182 00:08:44,169 --> 00:08:47,339 t e to the minus j omega t dt? 183 00:08:47,339 --> 00:08:49,380 What's difficult about integrating that function? 184 00:08:55,200 --> 00:08:58,460 What do we do when we see these sorts of things-- 185 00:08:58,460 --> 00:08:59,870 e to the j omega not t? 186 00:08:59,870 --> 00:09:03,050 How do you think about e to the-- 187 00:09:03,050 --> 00:09:05,680 complex exponential. 188 00:09:05,680 --> 00:09:08,696 You can name out things like Euler's equation, right? 189 00:09:08,696 --> 00:09:09,820 So e to the j omega not t-- 190 00:09:09,820 --> 00:09:11,903 you might want to say, well, that's the same thing 191 00:09:11,903 --> 00:09:15,190 as cosine omega not t plus j sine omega not t. 192 00:09:20,190 --> 00:09:22,550 If I took just the first term-- cosine omega not t, 193 00:09:22,550 --> 00:09:24,081 does it have a Laplace transform? 194 00:09:26,790 --> 00:09:28,790 Doesn't have a Laplace transform, hmm. 195 00:09:28,790 --> 00:09:32,770 How about a Fourier transform? 196 00:09:32,770 --> 00:09:35,020 Why does it have a Fourier transform and not a Laplace 197 00:09:35,020 --> 00:09:35,856 transform? 198 00:09:41,320 --> 00:09:44,600 So Laplace transforms-- there's only a Laplace transform 199 00:09:44,600 --> 00:09:47,600 if there's some kind of a multiplicative kernel-- 200 00:09:47,600 --> 00:09:50,760 e to the minus st-- 201 00:09:50,760 --> 00:09:54,920 that has some values that make the integral converge. 202 00:09:54,920 --> 00:09:57,960 There's no value of s for which e to the j omega 203 00:09:57,960 --> 00:10:00,380 not t will converge. 204 00:10:00,380 --> 00:10:03,417 The amplitude of e to the j omega not t-- 205 00:10:03,417 --> 00:10:05,750 the real part has an amplitude that fluctuates like this 206 00:10:05,750 --> 00:10:07,700 and never decays. 207 00:10:07,700 --> 00:10:09,980 Goes on forever and ever in both directions. 208 00:10:09,980 --> 00:10:12,230 There's no exponential function that you 209 00:10:12,230 --> 00:10:17,510 can multiply times that function to make it converge. 210 00:10:17,510 --> 00:10:20,635 There is no Laplace transform. 211 00:10:20,635 --> 00:10:22,120 Is there a Fourier transform? 212 00:10:22,120 --> 00:10:23,882 Yeah. 213 00:10:23,882 --> 00:10:24,590 What's different? 214 00:10:24,590 --> 00:10:26,780 What makes us be able to do a Fourier transform 215 00:10:26,780 --> 00:10:30,700 and not a Laplace transform? 216 00:10:30,700 --> 00:10:33,660 Delta functions. 217 00:10:33,660 --> 00:10:35,675 We never write a Laplace transform 218 00:10:35,675 --> 00:10:37,830 with a delta function, ever. 219 00:10:37,830 --> 00:10:47,450 If you write on your exam h of s is some function of delta of s, 220 00:10:47,450 --> 00:10:49,390 that's definitely wrong, right? 221 00:10:49,390 --> 00:10:50,490 We just never do that. 222 00:10:53,130 --> 00:10:55,800 We're much more liberal about using delta functions 223 00:10:55,800 --> 00:10:58,320 in Fourier transforms. 224 00:10:58,320 --> 00:11:02,130 That's one of the reasons they're powerful. 225 00:11:02,130 --> 00:11:05,280 And one of the ways you can think about that 226 00:11:05,280 --> 00:11:07,930 is that delta functions are easy to integrate. 227 00:11:07,930 --> 00:11:09,129 That's not quite enough. 228 00:11:09,129 --> 00:11:10,920 The other thing that's important to realize 229 00:11:10,920 --> 00:11:14,580 is that we've got this integral here-- 230 00:11:14,580 --> 00:11:18,150 it's either going to be zero or infinity. 231 00:11:18,150 --> 00:11:22,230 There's going to be certain values of omega 232 00:11:22,230 --> 00:11:23,730 that when we multiply this together, 233 00:11:23,730 --> 00:11:26,063 we're going to get something that goes on and on forever 234 00:11:26,063 --> 00:11:28,156 and ever and never decays. 235 00:11:28,156 --> 00:11:29,530 That's going to give us infinity. 236 00:11:29,530 --> 00:11:32,260 And there's going to be other values 237 00:11:32,260 --> 00:11:35,970 that the long-term integral's going to go to zero. 238 00:11:35,970 --> 00:11:40,540 And that information is summarized by the impulses, 239 00:11:40,540 --> 00:11:41,340 right? 240 00:11:41,340 --> 00:11:44,310 So if we choose omega not-- 241 00:11:44,310 --> 00:11:48,220 if we choose omega to be omega not-- 242 00:11:48,220 --> 00:11:50,470 then when we do the integral, which comes out 243 00:11:50,470 --> 00:11:52,645 a function of omega, we're going to get something 244 00:11:52,645 --> 00:11:54,074 that goes to infinity. 245 00:11:56,780 --> 00:11:58,129 OK? 246 00:11:58,129 --> 00:12:00,670 The easy way to think about that is to do the inverse Fourier 247 00:12:00,670 --> 00:12:02,950 transform, right? 248 00:12:02,950 --> 00:12:08,660 The inverse Fourier transform is x of t 249 00:12:08,660 --> 00:12:14,440 is 1 over 2 pi in the integral x of j omega e to the j omega t 250 00:12:14,440 --> 00:12:17,500 d omega. 251 00:12:17,500 --> 00:12:22,470 Now, if we put the delta function in this, it's easy. 252 00:12:22,470 --> 00:12:24,900 Because the delta function over here-- 253 00:12:24,900 --> 00:12:27,222 if we say that-- 254 00:12:27,222 --> 00:12:28,997 it's x over here and it's h over there. 255 00:12:28,997 --> 00:12:29,580 But who cares? 256 00:12:29,580 --> 00:12:30,580 That should have been x. 257 00:12:38,790 --> 00:12:41,160 If we put over there that the x of j 258 00:12:41,160 --> 00:12:44,700 omega is 2 pi delta omega minus omega not, 259 00:12:44,700 --> 00:12:46,980 then we can see that the function of the delta 260 00:12:46,980 --> 00:12:49,095 is just to sift out a particular value of e 261 00:12:49,095 --> 00:12:53,490 to the j omega t, the one at omega not. 262 00:12:53,490 --> 00:12:55,860 And the 2 pi's obvious, too. 263 00:12:55,860 --> 00:12:58,039 The 2 pi came from the 1 over 2 pi, right? 264 00:12:58,039 --> 00:13:00,330 We had to have a 2 pi out front so that the 1 over 2 pi 265 00:13:00,330 --> 00:13:01,620 killed it. 266 00:13:01,620 --> 00:13:03,630 OK, anyway-- so the idea, then, is 267 00:13:03,630 --> 00:13:06,390 that simply by taking the Fourier transform of this, 268 00:13:06,390 --> 00:13:09,300 we can get an expression for the Fourier transform. 269 00:13:09,300 --> 00:13:12,270 And that tells us something about the great utility 270 00:13:12,270 --> 00:13:14,730 of using impulses, right? 271 00:13:14,730 --> 00:13:16,980 Because we use impulses in the Fourier transform, 272 00:13:16,980 --> 00:13:19,080 we can represent a lot more signals 273 00:13:19,080 --> 00:13:21,810 than we could have represented in the Laplace transform. 274 00:13:21,810 --> 00:13:24,535 That's one of the utilities of a Fourier. 275 00:13:24,535 --> 00:13:26,910 And there's a big relationship between the Fourier series 276 00:13:26,910 --> 00:13:29,980 and the Fourier transform. 277 00:13:29,980 --> 00:13:33,040 So in particular, if we have a signal that's periodic in time, 278 00:13:33,040 --> 00:13:36,084 it'll have both a series and a transform. 279 00:13:36,084 --> 00:13:38,250 The transform is a bunch of delta functions weighted 280 00:13:38,250 --> 00:13:43,110 by the Fourier series coefficients and 2 pi. 281 00:13:43,110 --> 00:13:45,600 And what that says is, had we been 282 00:13:45,600 --> 00:13:48,600 willing to put in impulses to begin with, 283 00:13:48,600 --> 00:13:51,030 we would never have had to define the series. 284 00:13:51,030 --> 00:13:53,940 We did the series just because it was easy. 285 00:13:53,940 --> 00:13:57,500 Convergence issues were more clear. 286 00:13:57,500 --> 00:14:00,690 But if we had been willing to think about delta functions 287 00:14:00,690 --> 00:14:03,060 in the transforms-- which would have been an extremely 288 00:14:03,060 --> 00:14:06,780 foreign concept coming straight from Laplace, where 289 00:14:06,780 --> 00:14:09,090 you would never see that. 290 00:14:09,090 --> 00:14:11,190 Had we been willing to accept delta functions 291 00:14:11,190 --> 00:14:12,780 in the transforms from the outset, 292 00:14:12,780 --> 00:14:14,969 we never would have needed series. 293 00:14:14,969 --> 00:14:17,010 So the relationship between series and transforms 294 00:14:17,010 --> 00:14:21,510 has to do with the fact that the series represents 295 00:14:21,510 --> 00:14:22,440 periodic signals. 296 00:14:22,440 --> 00:14:25,500 Periodic signals go on forever. 297 00:14:25,500 --> 00:14:29,160 There's never a Laplace transform, 298 00:14:29,160 --> 00:14:30,210 and that means that-- 299 00:14:30,210 --> 00:14:31,830 there's never a Laplace transform. 300 00:14:31,830 --> 00:14:34,455 That means the Fourier transform has a bunch of delta functions 301 00:14:34,455 --> 00:14:38,600 in it, OK? 302 00:14:38,600 --> 00:14:40,340 OK, so what I want to do, then, is 303 00:14:40,340 --> 00:14:41,840 think through other relationships 304 00:14:41,840 --> 00:14:43,145 that are on this table. 305 00:14:43,145 --> 00:14:45,020 And in particular, I want to think about them 306 00:14:45,020 --> 00:14:48,780 by thinking about the underlying structure of the signals. 307 00:14:48,780 --> 00:15:05,890 So think about having DTFS, DTFT, CTFS, and CTFT. 308 00:15:05,890 --> 00:15:08,470 So what I want to do is think about the relationships 309 00:15:08,470 --> 00:15:12,640 among those four transforms by thinking about how 310 00:15:12,640 --> 00:15:15,300 they differ in time, right? 311 00:15:15,300 --> 00:15:18,790 So the top guys are discrete in time. 312 00:15:18,790 --> 00:15:20,590 The bottom guys are continuous in time. 313 00:15:20,590 --> 00:15:24,670 The left guys are periodic in time. 314 00:15:24,670 --> 00:15:26,830 The right ones are aperiodic in time. 315 00:15:26,830 --> 00:15:28,480 So what I want to do to think about how 316 00:15:28,480 --> 00:15:32,650 they relate to each other, is think about a base case. 317 00:15:32,650 --> 00:15:36,927 Think about a signal that's down here-- aperiodic CT. 318 00:15:36,927 --> 00:15:39,260 And think about what you would have to do to the signal, 319 00:15:39,260 --> 00:15:43,540 and therefore what would you have to do to the transform. 320 00:15:43,540 --> 00:15:45,370 If you wanted to move this way-- 321 00:15:45,370 --> 00:15:48,470 I'll do that one first. 322 00:15:48,470 --> 00:15:50,090 If you move up-- 323 00:15:50,090 --> 00:15:52,430 I'll do that one second. 324 00:15:52,430 --> 00:15:53,810 Or if you move that way-- 325 00:15:53,810 --> 00:15:55,070 I'll do that one third. 326 00:15:55,070 --> 00:15:58,150 So really, the rest of the hour is just doing three examples. 327 00:15:58,150 --> 00:16:00,320 Well, four examples. 328 00:16:00,320 --> 00:16:03,290 What's the transform of a base case? 329 00:16:03,290 --> 00:16:06,730 Then what would happen as I move through this table? 330 00:16:06,730 --> 00:16:09,530 OK, everybody's with the game plan? 331 00:16:09,530 --> 00:16:10,917 So here's my base case. 332 00:16:10,917 --> 00:16:13,250 I want to think about the Fourier transform of a signal, 333 00:16:13,250 --> 00:16:16,760 x of t, that's a triangle. 334 00:16:16,760 --> 00:16:18,657 We could calculate the Fourier transform just 335 00:16:18,657 --> 00:16:19,490 from the definition. 336 00:16:19,490 --> 00:16:21,730 That's trivial, right? 337 00:16:21,730 --> 00:16:23,360 Good way, bad way. 338 00:16:23,360 --> 00:16:24,135 Good way. 339 00:16:27,860 --> 00:16:29,780 Smile. 340 00:16:29,780 --> 00:16:30,320 Better. 341 00:16:30,320 --> 00:16:33,140 OK, bad way-- 342 00:16:33,140 --> 00:16:35,352 I can think of a better way to do it. 343 00:16:35,352 --> 00:16:36,560 What's a better way to do it? 344 00:16:40,560 --> 00:16:41,060 Yeah? 345 00:16:41,060 --> 00:16:42,060 AUDIENCE: [INAUDIBLE]. 346 00:16:59,837 --> 00:17:01,420 DENNIS FREEMAN: So one way is to think 347 00:17:01,420 --> 00:17:06,614 about turning straight lines into constants via derivatives 348 00:17:06,614 --> 00:17:08,030 and integrals and stuff like that. 349 00:17:08,030 --> 00:17:08,988 That's a very good way. 350 00:17:08,988 --> 00:17:10,994 Can somebody think of a different way? 351 00:17:10,994 --> 00:17:12,160 That'll work, it'll be fine. 352 00:17:12,160 --> 00:17:12,400 Yes? 353 00:17:12,400 --> 00:17:14,585 AUDIENCE: Is that the convolution of two square 354 00:17:14,585 --> 00:17:15,460 pulses? 355 00:17:15,460 --> 00:17:17,501 DENNIS FREEMAN: Convolution of two square pulses, 356 00:17:17,501 --> 00:17:18,880 that's wonderful. 357 00:17:18,880 --> 00:17:23,530 So a different way we can do it is to convolve a square pulse 358 00:17:23,530 --> 00:17:24,510 with a square pulse. 359 00:17:24,510 --> 00:17:26,560 That's particularly good for this example, 360 00:17:26,560 --> 00:17:30,886 because a square pulse is one of our canonical forms. 361 00:17:30,886 --> 00:17:32,510 So if we think about what's the Fourier 362 00:17:32,510 --> 00:17:33,676 transform of a square pulse? 363 00:17:33,676 --> 00:17:36,200 We all know that it's something that has the form sine 364 00:17:36,200 --> 00:17:38,030 omega over omega. 365 00:17:38,030 --> 00:17:41,660 So all we really need to do is fill in the constants. 366 00:17:41,660 --> 00:17:43,040 Filling in the constants is easy, 367 00:17:43,040 --> 00:17:45,410 because we know the moment theorems. 368 00:17:45,410 --> 00:17:47,060 The moment theorem says that whatever 369 00:17:47,060 --> 00:17:49,670 the area is under this curve, it has 370 00:17:49,670 --> 00:17:54,810 to be the height of that curve at 0. 371 00:17:54,810 --> 00:17:56,400 Whatever is the area under this curve, 372 00:17:56,400 --> 00:17:57,900 it has to be the height of that one. 373 00:17:57,900 --> 00:17:59,524 Except that you have to divide by 2 pi, 374 00:17:59,524 --> 00:18:02,260 because the annoying 2 pi factor. 375 00:18:02,260 --> 00:18:04,950 The annoying 2 pi factor is something 376 00:18:04,950 --> 00:18:06,690 that's in the inverse transform and not 377 00:18:06,690 --> 00:18:07,800 in the Fourier transform. 378 00:18:07,800 --> 00:18:10,830 That makes it asymmetric. 379 00:18:10,830 --> 00:18:14,910 So in order to choose the constants-- the area under this 380 00:18:14,910 --> 00:18:19,610 is 1, so that height must be 1. 381 00:18:19,610 --> 00:18:21,740 The area under this, by magic, turns 382 00:18:21,740 --> 00:18:24,980 into the area of that triangle, which is 2 pi. 383 00:18:24,980 --> 00:18:26,750 Divide by 2 pi is 1. 384 00:18:26,750 --> 00:18:29,480 So that means that this frequency must be 2 pi 385 00:18:29,480 --> 00:18:32,900 in order to make that constant be 1. 386 00:18:32,900 --> 00:18:34,160 So we're done. 387 00:18:34,160 --> 00:18:38,250 So we know, then, that the transform of this signal, 388 00:18:38,250 --> 00:18:42,400 which is the convolution of this signal with itself, 389 00:18:42,400 --> 00:18:46,570 is just the product of this function with itself. 390 00:18:46,570 --> 00:18:50,830 Convolution in time maps to multiplication in frequency. 391 00:18:50,830 --> 00:18:52,550 OK? 392 00:18:52,550 --> 00:18:53,550 So that's the base case. 393 00:18:53,550 --> 00:18:55,260 Now what I want to think about is 394 00:18:55,260 --> 00:18:57,000 how I would march through the table. 395 00:18:57,000 --> 00:18:58,580 That's my case down here. 396 00:18:58,580 --> 00:19:00,330 So what I want to do first is think about, 397 00:19:00,330 --> 00:19:03,180 how would I convert that this way? 398 00:19:03,180 --> 00:19:05,430 And what's the implication for that conversion 399 00:19:05,430 --> 00:19:11,260 in terms of time, and in terms of frequency? 400 00:19:11,260 --> 00:19:14,520 So one way to think about the transformation from a transform 401 00:19:14,520 --> 00:19:15,690 to a series-- 402 00:19:15,690 --> 00:19:17,891 you can only series with things that are periodic. 403 00:19:17,891 --> 00:19:19,890 The transformation that we talked about in class 404 00:19:19,890 --> 00:19:22,170 was periodic extension. 405 00:19:22,170 --> 00:19:24,300 So you think about taking the base case, which 406 00:19:24,300 --> 00:19:27,250 was a single triangle wave, and turning it 407 00:19:27,250 --> 00:19:29,670 into a sequence of triangle waves. 408 00:19:29,670 --> 00:19:31,800 There's lots of ways to think about that. 409 00:19:31,800 --> 00:19:34,870 But the way I want to motivate is to think about it-- 410 00:19:34,870 --> 00:19:37,260 well, no, backing up. 411 00:19:37,260 --> 00:19:39,480 So if we think about the triangle 412 00:19:39,480 --> 00:19:42,000 wave and transformation and make it into a periodic triangle 413 00:19:42,000 --> 00:19:43,770 wave, we could just substitute this 414 00:19:43,770 --> 00:19:45,660 into the definition of transform. 415 00:19:45,660 --> 00:19:46,760 That's ugly, why? 416 00:19:52,510 --> 00:19:55,270 For the same reason that one was ugly. 417 00:19:55,270 --> 00:19:56,310 What's ugly about this? 418 00:19:59,630 --> 00:20:02,916 The signal goes on and on and on and on and on. 419 00:20:02,916 --> 00:20:04,290 There will be certain frequencies 420 00:20:04,290 --> 00:20:06,330 where it'll blow up. 421 00:20:06,330 --> 00:20:08,160 Those will be infinity. 422 00:20:08,160 --> 00:20:11,190 There will be other frequencies where it is 0. 423 00:20:11,190 --> 00:20:13,470 There could be frequencies where it's minus infinity, 424 00:20:13,470 --> 00:20:17,047 and those are the only three options. 425 00:20:17,047 --> 00:20:18,630 Since it goes on and on and on and on, 426 00:20:18,630 --> 00:20:20,046 if it's anything different from 0, 427 00:20:20,046 --> 00:20:24,057 it has to be either infinity or minus infinity. 428 00:20:24,057 --> 00:20:26,390 It's going to be a train of impulses for the same reason 429 00:20:26,390 --> 00:20:28,670 that thing was some kind of a train of impulses. 430 00:20:28,670 --> 00:20:30,795 So we want to have a good way of thinking about it. 431 00:20:30,795 --> 00:20:32,900 I told you one way to think about this-- 432 00:20:32,900 --> 00:20:34,650 by taking the inverse transform. 433 00:20:34,650 --> 00:20:40,440 This is a little harder to guess the transforms, 434 00:20:40,440 --> 00:20:44,040 so you can verify it with the inverse. 435 00:20:44,040 --> 00:20:46,950 But you can form this by thinking 436 00:20:46,950 --> 00:20:55,350 about convolving the base waveform with an impulse train. 437 00:20:55,350 --> 00:20:57,225 So what if I had an impulse train? 438 00:21:00,420 --> 00:21:02,130 So I'm-- oh, it's on the next slide. 439 00:21:02,130 --> 00:21:05,910 What if I had an impulse train like so? 440 00:21:05,910 --> 00:21:07,860 If I think about having an impulse train, 441 00:21:07,860 --> 00:21:14,400 if I had the base waveform and an appropriate impulse train, 442 00:21:14,400 --> 00:21:20,550 how could I manufacture the signal of interest, which 443 00:21:20,550 --> 00:21:22,440 is this periodic extension? 444 00:21:22,440 --> 00:21:23,940 So if I have these two signals, what 445 00:21:23,940 --> 00:21:27,520 would I do to get that signal? 446 00:21:27,520 --> 00:21:29,270 Convolve. 447 00:21:29,270 --> 00:21:33,000 So if I knew the transform of this, 448 00:21:33,000 --> 00:21:36,902 which I do because this is the base case which we just did. 449 00:21:36,902 --> 00:21:38,610 If I knew the transform of the base case, 450 00:21:38,610 --> 00:21:41,550 and if I knew the transform of the impulse train-- 451 00:21:41,550 --> 00:21:44,360 impulse train is something that goes to function of time. 452 00:21:44,360 --> 00:21:46,390 0-- let's say that they're separated by t. 453 00:21:46,390 --> 00:21:48,450 Let's say each one has a height of 1. 454 00:21:48,450 --> 00:21:51,620 What's the Fourier transform of an impulse train? 455 00:22:00,232 --> 00:22:02,190 I can see you're all on the edge of your seats. 456 00:22:06,000 --> 00:22:09,640 How do you find the Fourier transform of an impulse train? 457 00:22:12,414 --> 00:22:13,330 OK, ask your neighbor. 458 00:24:07,010 --> 00:24:10,310 So how do you find the Fourier transform of an impulse train? 459 00:24:14,230 --> 00:24:17,430 Look it up in the tables! 460 00:24:17,430 --> 00:24:18,045 Boo, hiss. 461 00:24:21,280 --> 00:24:24,420 How do you find the Fourier transform of an impulse train? 462 00:24:28,630 --> 00:24:30,970 Is there anything easy about an impulse train? 463 00:24:30,970 --> 00:24:33,700 What a horrendous thing, right? 464 00:24:33,700 --> 00:24:36,010 It has an infinite number of impulses, yuck. 465 00:24:36,010 --> 00:24:41,790 Each impulse is horribly misbehaved. 466 00:24:41,790 --> 00:24:43,652 How would I find the Fourier transform 467 00:24:43,652 --> 00:24:44,610 of a train of impulses? 468 00:24:44,610 --> 00:24:44,950 Yes? 469 00:24:44,950 --> 00:24:46,200 AUDIENCE: So if you actually just plug 470 00:24:46,200 --> 00:24:47,950 that in to the formula, then you get 471 00:24:47,950 --> 00:24:51,100 different exponents and like, [INAUDIBLE],, 472 00:24:51,100 --> 00:24:53,780 and then, since there's negative infinity to infinity, 473 00:24:53,780 --> 00:24:56,750 they add up to the [INAUDIBLE]. 474 00:24:56,750 --> 00:24:58,980 DENNIS FREEMAN: So you're saying that somehow I 475 00:24:58,980 --> 00:25:00,510 should stick it into-- 476 00:25:00,510 --> 00:25:05,520 so I'm thinking about finding the Fourier transform by going 477 00:25:05,520 --> 00:25:08,860 back to the direct formula. 478 00:25:08,860 --> 00:25:13,910 So if I put an infinite train of impulses here, what would I do? 479 00:25:13,910 --> 00:25:19,740 So I'd have a sum of delta of t minus, 480 00:25:19,740 --> 00:25:27,020 say, n times t sum over n e to the minus j omega t dt. 481 00:25:34,780 --> 00:25:35,590 That's a good step. 482 00:25:38,630 --> 00:25:40,850 So now, what I might do is interchange 483 00:25:40,850 --> 00:25:43,780 the order, because that's what everybody does, right? 484 00:25:43,780 --> 00:25:45,990 And it's a little dicey doing that, right? 485 00:25:45,990 --> 00:25:48,680 You're always safe to do that if the integrand is absolutely 486 00:25:48,680 --> 00:25:49,400 summable. 487 00:25:49,400 --> 00:25:53,430 That's not precisely absolutely summable. 488 00:25:53,430 --> 00:25:56,310 Everybody knows what absolutely summable means? 489 00:25:59,260 --> 00:26:01,200 So it means if you took the absolute value 490 00:26:01,200 --> 00:26:02,730 and you summed it over all of time, 491 00:26:02,730 --> 00:26:04,521 you would get something less than infinity. 492 00:26:04,521 --> 00:26:06,720 That's not quite the truth here, right? 493 00:26:06,720 --> 00:26:09,810 So some of the sums in Fourier series and Fourier transforms 494 00:26:09,810 --> 00:26:11,210 can be hard, OK? 495 00:26:11,210 --> 00:26:13,830 But throwing caution to the wind, 496 00:26:13,830 --> 00:26:16,390 we would probably get something like that. 497 00:26:21,940 --> 00:26:25,270 And then we could say, well, that just samples it. 498 00:26:25,270 --> 00:26:27,815 We could sample this thing at t equals nt. 499 00:26:27,815 --> 00:26:33,210 So that would be e to the minus j omega nt. 500 00:26:35,860 --> 00:26:38,650 Then I end up summing a large number 501 00:26:38,650 --> 00:26:41,320 of complex exponentials-- 502 00:26:41,320 --> 00:26:43,948 large being infinite. 503 00:26:43,948 --> 00:26:46,235 It's kind of ugly. 504 00:26:46,235 --> 00:26:46,735 Yes? 505 00:26:46,735 --> 00:26:49,705 AUDIENCE: Could you now [INAUDIBLE].. 506 00:26:56,150 --> 00:26:58,310 So I could split this into two sums-- 507 00:26:58,310 --> 00:27:02,170 n less than 0, and a similar sum-- n bigger than 0. 508 00:27:02,170 --> 00:27:06,414 And then throw in the 0 term, just for good measure. 509 00:27:06,414 --> 00:27:09,495 AUDIENCE: And then we can see you have a bunch of sines-- 510 00:27:09,495 --> 00:27:11,870 actually, in this case, you have a bunch of cosine terms. 511 00:27:11,870 --> 00:27:15,838 So you know that-- by having the cosine terms, [INAUDIBLE].. 512 00:27:18,350 --> 00:27:20,600 DENNIS FREEMAN: So you just said something that seemed 513 00:27:20,600 --> 00:27:21,725 to take a left turn, there. 514 00:27:21,725 --> 00:27:23,990 What was the thing about series? 515 00:27:23,990 --> 00:27:25,752 AUDIENCE: So if you take your Fourier 516 00:27:25,752 --> 00:27:30,570 series of a sum of cosines, it's just the coefficient 517 00:27:30,570 --> 00:27:32,030 of the same sign? 518 00:27:32,030 --> 00:27:35,300 DENNIS FREEMAN: So, that sounds good. 519 00:27:35,300 --> 00:27:37,610 Could we use that idea of taking a Fourier series 520 00:27:37,610 --> 00:27:38,990 a little earlier in the proof? 521 00:27:42,200 --> 00:27:45,810 How would I take the Fourier transform of that? 522 00:27:45,810 --> 00:27:49,900 How would I take the Fourier series of that? 523 00:27:49,900 --> 00:27:51,358 Yeah? 524 00:27:51,358 --> 00:27:54,760 AUDIENCE: All 1's. 525 00:27:54,760 --> 00:27:56,260 All 1's. 526 00:27:56,260 --> 00:27:59,020 DENNIS FREEMAN: All the 1's, yes, yes. 527 00:27:59,020 --> 00:28:00,010 This is periodic. 528 00:28:00,010 --> 00:28:02,170 It has a series. 529 00:28:02,170 --> 00:28:03,840 Trick? 530 00:28:03,840 --> 00:28:08,590 No, no, no-- clever observation, not a trick. 531 00:28:08,590 --> 00:28:10,450 So it's a periodic waveform. 532 00:28:10,450 --> 00:28:12,730 It has a series. 533 00:28:12,730 --> 00:28:14,470 OK, why is that good? 534 00:28:14,470 --> 00:28:17,680 Periodic waveform series, I only need to look at one period. 535 00:28:17,680 --> 00:28:21,850 I reduce an infinite number of impulses to one. 536 00:28:21,850 --> 00:28:24,890 That's a good move. 537 00:28:24,890 --> 00:28:26,880 So all I need to do is-- 538 00:28:26,880 --> 00:28:28,430 to find the series, all I need to do 539 00:28:28,430 --> 00:28:32,600 is think about the Fourier series sum. 540 00:28:32,600 --> 00:28:35,420 Which is almost the same thing, except that now I've 541 00:28:35,420 --> 00:28:38,450 got something that has k's in it, right? 542 00:28:38,450 --> 00:28:43,024 Because a series only has harmonic frequencies. 543 00:28:43,024 --> 00:28:44,690 So what I can do is, instead of thinking 544 00:28:44,690 --> 00:28:47,630 about a Fourier series-- 545 00:28:47,630 --> 00:28:50,620 so now I can say, instead of finding the Fourier transform, 546 00:28:50,620 --> 00:28:52,610 let's find the Fourier series. 547 00:28:52,610 --> 00:28:56,050 That's find a sub k, which would be 1 over t 548 00:28:56,050 --> 00:29:01,630 to the integral on t of x of t e to the minus j 2 pi 549 00:29:01,630 --> 00:29:05,210 kt by t on t, OK? 550 00:29:05,210 --> 00:29:08,990 So now I only need to find the series. 551 00:29:08,990 --> 00:29:10,850 And by integrating over t-- 552 00:29:10,850 --> 00:29:12,469 OK, well I'll do the easy one, right? 553 00:29:12,469 --> 00:29:13,760 Never do something that's hard. 554 00:29:13,760 --> 00:29:15,380 I can choose any interval of t. 555 00:29:15,380 --> 00:29:18,140 Let me take that one. 556 00:29:18,140 --> 00:29:20,240 So the only thing that persists, then-- 557 00:29:20,240 --> 00:29:21,770 so if I do minus-- 558 00:29:21,770 --> 00:29:27,190 whatever, t by 2 to t by 2, then this becomes delta of t. 559 00:29:29,860 --> 00:29:35,815 And delta of t sifts out the 0 value of this guy. 560 00:29:35,815 --> 00:29:38,760 But e to 0 is 1. 561 00:29:38,760 --> 00:29:43,800 So this is 1 over t for all k. 562 00:29:43,800 --> 00:29:45,090 We knew that, right? 563 00:29:45,090 --> 00:29:49,950 The Fourier series for a unit impulse is 1 everywhere. 564 00:29:49,950 --> 00:29:52,320 All the harmonics have the same size-- 565 00:29:52,320 --> 00:29:56,580 same size, same magnitude, same phase everywhere. 566 00:29:56,580 --> 00:29:59,010 Then, since we know the series, we 567 00:29:59,010 --> 00:30:02,490 can trivially convert it into a Fourier transform, right? 568 00:30:02,490 --> 00:30:06,540 Using the deep insight from a few slides ago, right? 569 00:30:06,540 --> 00:30:10,680 So how do you take a series and turn it into a transform? 570 00:30:10,680 --> 00:30:12,090 Substitute for every k. 571 00:30:14,725 --> 00:30:18,000 AUDIENCE: [INAUDIBLE]. 572 00:30:18,000 --> 00:30:20,970 DENNIS FREEMAN: Put an impulse in place of every k 573 00:30:20,970 --> 00:30:23,070 at the frequency-- omega-- 574 00:30:23,070 --> 00:30:26,560 that corresponds to that k, OK? 575 00:30:26,560 --> 00:30:30,242 The k's-- 0, 1, 2, 3, are the 0, 1, 2, 3 harmonics. 576 00:30:30,242 --> 00:30:32,200 The harmonics are harmonics of the fundamental. 577 00:30:32,200 --> 00:30:36,460 The fundamental is 2 pi over t. 578 00:30:36,460 --> 00:30:39,890 So you put one impulse at every multiple of 2 pi 579 00:30:39,890 --> 00:30:45,730 over capital T. And you multiply them all by 2 pi. 580 00:30:45,730 --> 00:30:47,226 So here's your answer. 581 00:30:47,226 --> 00:30:48,850 You start with this thing you recognize 582 00:30:48,850 --> 00:30:50,500 as a periodic waveform. 583 00:30:50,500 --> 00:30:53,400 You turn it into a series. 584 00:30:53,400 --> 00:30:56,310 Then you convert every one of the harmonics 585 00:30:56,310 --> 00:30:59,340 into a continuous frequency representation 586 00:30:59,340 --> 00:31:05,250 so that the spacing is the base period 2 pi over cap T. 587 00:31:05,250 --> 00:31:07,020 The height of these guys was 1 over T, 588 00:31:07,020 --> 00:31:09,030 because of the annoying 1 over T factor in front 589 00:31:09,030 --> 00:31:12,590 of the Fourier series formula. 590 00:31:12,590 --> 00:31:15,380 And you get an extra 2 pi, because the annoying 1 over 2 591 00:31:15,380 --> 00:31:21,390 pi in the inverse Fourier formula, OK? 592 00:31:21,390 --> 00:31:23,625 So what we see, then, is that the Fourier transform 593 00:31:23,625 --> 00:31:27,190 of an impulse train is an impulse train. 594 00:31:27,190 --> 00:31:29,820 And so we can use that, then, to find the Fourier transform 595 00:31:29,820 --> 00:31:31,260 of this mass, remember? 596 00:31:31,260 --> 00:31:32,760 Oh, I've got-- oh, here it is. 597 00:31:32,760 --> 00:31:34,320 I was taking my base waveform, which 598 00:31:34,320 --> 00:31:37,290 was a triangle, trying to figure out 599 00:31:37,290 --> 00:31:39,780 what would happen if I did an infinite extension of that-- 600 00:31:39,780 --> 00:31:41,970 an infinite periodic extension. 601 00:31:41,970 --> 00:31:45,390 I think about the extension as being convolving the base 602 00:31:45,390 --> 00:31:47,520 waveform. 603 00:31:47,520 --> 00:31:48,460 Cover that up. 604 00:31:48,460 --> 00:31:52,684 If I think about convolving the base wave, form-- 605 00:31:52,684 --> 00:31:54,450 oh, no, it's gone. 606 00:31:54,450 --> 00:31:56,700 If I think about convolving the base waveform 607 00:31:56,700 --> 00:32:00,750 with periodic train of impulses, that 608 00:32:00,750 --> 00:32:02,700 means that-- so the time waveform is 609 00:32:02,700 --> 00:32:04,390 the convolution of two things. 610 00:32:04,390 --> 00:32:06,900 That means that the Fourier transform is 611 00:32:06,900 --> 00:32:10,420 the multiplying of two things. 612 00:32:10,420 --> 00:32:14,620 So I write down the transform of the base waveform-- 613 00:32:14,620 --> 00:32:16,690 thing that we did a long time ago. 614 00:32:16,690 --> 00:32:19,170 I write down the Fourier transform of the impulse train. 615 00:32:25,804 --> 00:32:27,220 The period of the impulse train is 616 00:32:27,220 --> 00:32:29,511 the same as the period of the periodic extension, which 617 00:32:29,511 --> 00:32:30,560 is capital T. 618 00:32:30,560 --> 00:32:32,350 It was 4. 619 00:32:32,350 --> 00:32:35,050 So that means the impulses are separated 620 00:32:35,050 --> 00:32:39,330 by 2 pi over 4 pi over 2. 621 00:32:39,330 --> 00:32:42,480 And the height of the impulse is always 622 00:32:42,480 --> 00:32:43,600 equal to their separation. 623 00:32:43,600 --> 00:32:47,200 So they're also pi over 2. 624 00:32:47,200 --> 00:32:49,600 So then all I do is multiply this times this, 625 00:32:49,600 --> 00:32:51,400 and that gives me the transform. 626 00:32:51,400 --> 00:32:54,700 Convolving time, multiplying frequency. 627 00:32:54,700 --> 00:32:56,980 The important thing that I found is that the effect 628 00:32:56,980 --> 00:32:58,540 of periodic extension-- 629 00:33:01,100 --> 00:33:03,230 I took a base waveform and turned it 630 00:33:03,230 --> 00:33:04,820 into a string of waveforms, right? 631 00:33:04,820 --> 00:33:08,930 So over here, I took a base waveform 632 00:33:08,930 --> 00:33:16,940 and turned it into a repeated thing, like so, in time. 633 00:33:16,940 --> 00:33:19,610 The effect of the periodic extension, 634 00:33:19,610 --> 00:33:25,720 which we understand trivially in time, is to sample frequency. 635 00:33:25,720 --> 00:33:30,640 What I get in the frequency representation is samples. 636 00:33:30,640 --> 00:33:33,720 The samples being controlled by how long 637 00:33:33,720 --> 00:33:37,160 is the periodic extension. 638 00:33:37,160 --> 00:33:39,050 So what I get is samples. 639 00:33:39,050 --> 00:33:41,270 Instead of getting all of these, I just get part. 640 00:33:41,270 --> 00:33:45,340 I just get a uniformly-spaced sample set separated by pi 641 00:33:45,340 --> 00:33:48,320 over 2. 642 00:33:48,320 --> 00:33:52,260 So then the effect of the periodic extension 643 00:33:52,260 --> 00:33:58,940 was to sample the Fourier transform of the base waveform. 644 00:33:58,940 --> 00:34:03,380 And that's the reason we can think about the Fourier series 645 00:34:03,380 --> 00:34:05,070 as being just a sequence of numbers. 646 00:34:05,070 --> 00:34:07,930 In fact, those numbers are the same. 647 00:34:07,930 --> 00:34:10,310 Or, they differ by 2 pi, right? 648 00:34:10,310 --> 00:34:13,370 So I have to multiply these guys by 2 pi 649 00:34:13,370 --> 00:34:16,150 to get the Fourier transform coefficients, 650 00:34:16,150 --> 00:34:19,310 because the Fourier transform of cosine of e to the j omega 651 00:34:19,310 --> 00:34:25,290 not t, I should say, is 2 pi delta, OK? 652 00:34:25,290 --> 00:34:27,989 So the reason the Fourier series coefficients 653 00:34:27,989 --> 00:34:33,270 are related the way they are, the reason they sample 654 00:34:33,270 --> 00:34:35,280 the Fourier transform, is because we 655 00:34:35,280 --> 00:34:37,940 can regard the periodic extension, which 656 00:34:37,940 --> 00:34:39,420 was convolved by an impulse train 657 00:34:39,420 --> 00:34:41,840 as being equivalently multiplying by an impulse 658 00:34:41,840 --> 00:34:44,433 train and frequency. 659 00:34:44,433 --> 00:34:45,906 OK? 660 00:34:45,906 --> 00:34:47,239 That's what I wanted you to see. 661 00:34:47,239 --> 00:34:50,020 I wanted you to see that if you think 662 00:34:50,020 --> 00:34:54,639 about periodic extension-- 663 00:34:54,639 --> 00:34:56,437 which is moving left in this diagram. 664 00:34:56,437 --> 00:34:58,770 If you think about moving left, what you're really doing 665 00:34:58,770 --> 00:35:01,750 is sampling in frequency. 666 00:35:01,750 --> 00:35:05,490 So that's the reason it came out having a sampled representation 667 00:35:05,490 --> 00:35:06,030 on the left. 668 00:35:08,750 --> 00:35:11,610 The periodic extension convolving with an impulse 669 00:35:11,610 --> 00:35:13,610 train was the same as multiplying by the impulse 670 00:35:13,610 --> 00:35:17,690 train and frequency, which has the effect of sampling 671 00:35:17,690 --> 00:35:19,990 the frequency response. 672 00:35:19,990 --> 00:35:24,440 OK, so now the effect of moving this way 673 00:35:24,440 --> 00:35:26,090 was the same sampling in frequency. 674 00:35:26,090 --> 00:35:27,506 Well, what happens if you actually 675 00:35:27,506 --> 00:35:30,211 sample in time, which is what you do when you go up? 676 00:35:33,182 --> 00:35:35,140 It's got to be an [INAUDIBLE] like this, right? 677 00:35:37,770 --> 00:35:40,885 So, the most useful thing about the Fourier transform 678 00:35:40,885 --> 00:35:42,760 is that the Fourier transform and the inverse 679 00:35:42,760 --> 00:35:45,490 transform look almost the same. 680 00:35:45,490 --> 00:35:48,230 That's the most useful property of the transform. 681 00:35:48,230 --> 00:35:51,610 Now, the Fourier transform has a simple inverse relationship, 682 00:35:51,610 --> 00:35:56,520 and is almost precisely the same except for the annoying 2 pi 683 00:35:56,520 --> 00:35:59,470 and except for a change of sign-- 684 00:35:59,470 --> 00:36:03,790 e to the j omega t, compared to e to the minus j omega t. 685 00:36:03,790 --> 00:36:05,420 Except for those trivial differences, 686 00:36:05,420 --> 00:36:06,760 they're the same thing. 687 00:36:06,760 --> 00:36:10,270 So we might expect that if we understood the effect of moving 688 00:36:10,270 --> 00:36:16,210 that way, which is convolve in time, multiply in frequency, 689 00:36:16,210 --> 00:36:19,330 you might expect an analogous thing to happen this way. 690 00:36:19,330 --> 00:36:21,730 And of course, it does. 691 00:36:21,730 --> 00:36:24,760 So here, let's think about the base waveform-- 692 00:36:24,760 --> 00:36:26,575 same base waveform. 693 00:36:26,575 --> 00:36:28,450 But now what we're doing is sampling in time. 694 00:36:28,450 --> 00:36:30,100 And I'll just arbitrarily, for the sake 695 00:36:30,100 --> 00:36:34,480 of illustration, sample it with capital T equals a half. 696 00:36:34,480 --> 00:36:38,380 Question is-- what would that sampling in time 697 00:36:38,380 --> 00:36:40,180 do to the Fourier transform? 698 00:36:40,180 --> 00:36:41,200 OK, sampling is time. 699 00:36:41,200 --> 00:36:43,480 That's something we understand completely in the time domain. 700 00:36:43,480 --> 00:36:45,146 What would happen in the Fourier domain? 701 00:36:47,610 --> 00:36:49,730 OK, well that's a little hard to compare, 702 00:36:49,730 --> 00:36:52,810 because the domain of these two signals-- 703 00:36:52,810 --> 00:36:56,920 this is continuous domain, T. This is discrete domain. 704 00:36:56,920 --> 00:36:58,917 And that's kind of annoying. 705 00:36:58,917 --> 00:37:01,000 So it's kind of hard, because they're not on equal 706 00:37:01,000 --> 00:37:02,890 footing to compare them. 707 00:37:02,890 --> 00:37:05,170 So what I can do is map the same information 708 00:37:05,170 --> 00:37:09,670 that was here into a representation over here that 709 00:37:09,670 --> 00:37:12,630 is continuous in time. 710 00:37:12,630 --> 00:37:15,970 The information over here was 0 everywhere, 711 00:37:15,970 --> 00:37:18,790 except one half-and-half. 712 00:37:18,790 --> 00:37:22,240 Here it's 0 everywhere except one half-and-half. 713 00:37:22,240 --> 00:37:25,810 Anyone know why I used impulses instead of just 714 00:37:25,810 --> 00:37:28,150 making the function different? 715 00:37:28,150 --> 00:37:30,850 Why didn't I, instead of using the complicated-- wouldn't it 716 00:37:30,850 --> 00:37:34,760 be much easier if I didn't use the impulses? 717 00:37:34,760 --> 00:37:42,490 And if instead, I just said, OK, my xp of t-- 718 00:37:42,490 --> 00:37:46,300 why didn't I say xp of t is almost everywhere 719 00:37:46,300 --> 00:37:53,565 0, except at that point, at that point, and at that point? 720 00:37:53,565 --> 00:37:54,440 Why didn't I do that? 721 00:38:03,136 --> 00:38:03,636 Yes? 722 00:38:03,636 --> 00:38:05,088 AUDIENCE: [INAUDIBLE]. 723 00:38:08,970 --> 00:38:11,480 DENNIS FREEMAN: Right, let's assume that it is continuous. 724 00:38:11,480 --> 00:38:19,240 And that there's exactly three points that differ 725 00:38:19,240 --> 00:38:21,690 from the straight line at 0. 726 00:38:21,690 --> 00:38:23,150 Yes? 727 00:38:23,150 --> 00:38:25,320 The integral's 0. 728 00:38:25,320 --> 00:38:28,610 That never works. 729 00:38:28,610 --> 00:38:30,590 In fact, the most trivial representation 730 00:38:30,590 --> 00:38:35,830 that only has energy at three places is three impulses. 731 00:38:35,830 --> 00:38:39,390 That's the only way to get non-trivial energy. 732 00:38:39,390 --> 00:38:41,230 OK, so I can't do that. 733 00:38:41,230 --> 00:38:43,190 I'm forced to do something like this. 734 00:38:43,190 --> 00:38:44,312 So now the idea is-- 735 00:38:44,312 --> 00:38:46,020 I switched to this kind of representation 736 00:38:46,020 --> 00:38:50,260 that has the same information, but has the same domain-- 737 00:38:50,260 --> 00:38:52,887 CT, continuous time. 738 00:38:52,887 --> 00:38:54,970 So that now I can think about-- what's the Fourier 739 00:38:54,970 --> 00:38:57,140 transform of this guy? 740 00:38:57,140 --> 00:38:59,350 Well, it's the same idea. 741 00:38:59,350 --> 00:39:05,620 The way I can think about computing 742 00:39:05,620 --> 00:39:10,147 this thing from that thing is to multiply in time by an impulse 743 00:39:10,147 --> 00:39:10,794 train. 744 00:39:13,540 --> 00:39:16,300 What's the effect of multiplying time by an impulse train? 745 00:39:16,300 --> 00:39:19,570 What's the effect in frequency? 746 00:39:19,570 --> 00:39:21,670 Multiply in time by an impulse train, 747 00:39:21,670 --> 00:39:23,600 what are you doing to frequency? 748 00:39:23,600 --> 00:39:26,130 Convolving frequency. 749 00:39:26,130 --> 00:39:28,320 So we take the original base waveform, 750 00:39:28,320 --> 00:39:29,380 but now we convolve it. 751 00:39:31,890 --> 00:39:34,970 And lo and behold, we convolve this signal-- 752 00:39:34,970 --> 00:39:37,470 which could be anything-- with this signal-- which is always 753 00:39:37,470 --> 00:39:38,310 periodic. 754 00:39:38,310 --> 00:39:42,630 If you convolve in anything with a periodic, what do you get? 755 00:39:42,630 --> 00:39:46,240 Periodic, inevitable. 756 00:39:46,240 --> 00:39:49,380 So the fact that I sampled in time 757 00:39:49,380 --> 00:39:51,630 leads to periodic in frequency. 758 00:39:51,630 --> 00:39:53,980 That's the point. 759 00:39:53,980 --> 00:39:57,460 Sampling in time leads to periodic in frequency. 760 00:39:57,460 --> 00:39:59,750 That's always going to be true. 761 00:39:59,750 --> 00:40:01,870 So we end up with a relationship here. 762 00:40:01,870 --> 00:40:04,600 The sampled waveform is going to be periodic. 763 00:40:04,600 --> 00:40:07,300 In here it's periodic, and 4 pi. 764 00:40:07,300 --> 00:40:09,670 That's a little different from what we've seen before. 765 00:40:09,670 --> 00:40:12,130 If we had looked at the x of n, we 766 00:40:12,130 --> 00:40:14,830 would have seen something that was periodic in 2 pi, 767 00:40:14,830 --> 00:40:19,250 but that's because the frequencies are different. 768 00:40:19,250 --> 00:40:20,320 One 769 00:40:20,320 --> 00:40:22,580 The first one I showed is a function 770 00:40:22,580 --> 00:40:27,420 of little omega, which has the units radians per second. 771 00:40:27,420 --> 00:40:31,130 This one is a function of cap Omega, which has the units 772 00:40:31,130 --> 00:40:32,750 radians, they're different. 773 00:40:32,750 --> 00:40:34,610 One's frequency in discrete time. 774 00:40:34,610 --> 00:40:39,350 One's frequency in continuous time. 775 00:40:39,350 --> 00:40:41,660 And there's a relationship, and that relationship 776 00:40:41,660 --> 00:40:44,510 is always this. 777 00:40:44,510 --> 00:40:48,420 That's the reason I write the two frequencies differently. 778 00:40:48,420 --> 00:40:52,100 One way to relate the two frequencies is via sampling. 779 00:40:52,100 --> 00:40:54,470 That's a way of taking a continuous time signal 780 00:40:54,470 --> 00:40:56,300 and turning it into a discrete time signal. 781 00:40:56,300 --> 00:40:59,480 When you do that, if you sample by the sampling 782 00:40:59,480 --> 00:41:02,120 interval capital T, that dictates 783 00:41:02,120 --> 00:41:04,350 that this relationship's going to be like that. 784 00:41:04,350 --> 00:41:06,400 And it's very general. 785 00:41:06,400 --> 00:41:09,350 All you need to do is look at the definitions. 786 00:41:09,350 --> 00:41:16,220 If you look at the definition of the DT Fourier transform-- 787 00:41:16,220 --> 00:41:22,465 that's this-- the CT Fourier transform-- that's this-- 788 00:41:25,190 --> 00:41:28,370 and just remember that we form this thing 789 00:41:28,370 --> 00:41:30,710 by samples of that thing. 790 00:41:34,260 --> 00:41:36,750 So now, if you think about flipping the order-- 791 00:41:36,750 --> 00:41:40,272 and again, not worrying about whether or not 792 00:41:40,272 --> 00:41:40,980 it will converge. 793 00:41:40,980 --> 00:41:43,380 You just assume it will for the time being. 794 00:41:43,380 --> 00:41:47,340 Just flip the order of the summation and the integration. 795 00:41:47,340 --> 00:41:55,340 This then sifts out the value of little t equals n cap T, 796 00:41:55,340 --> 00:41:58,550 and we get a formula for the continuous time Fourier 797 00:41:58,550 --> 00:42:02,000 transform that looks just like the formula 798 00:42:02,000 --> 00:42:04,100 for the discrete time Fourier transform. 799 00:42:06,720 --> 00:42:10,120 The only difference in the two is that one 800 00:42:10,120 --> 00:42:11,800 has discrete domain, n. 801 00:42:11,800 --> 00:42:14,200 The other has continuous domain, time. 802 00:42:14,200 --> 00:42:17,020 And by sampling, we convert continuous domain, time, 803 00:42:17,020 --> 00:42:19,840 into discrete domain, n. 804 00:42:19,840 --> 00:42:22,930 So we're always going to end up with this relationship that 805 00:42:22,930 --> 00:42:26,770 the discrete frequency-- that's generated by sampling-- 806 00:42:26,770 --> 00:42:28,180 the discrete frequency is related 807 00:42:28,180 --> 00:42:31,110 to the continuous frequency multiplied by capital 808 00:42:31,110 --> 00:42:33,970 T. Capital T has the units of seconds. 809 00:42:33,970 --> 00:42:39,080 So that takes radians per second and turns it into radians. 810 00:42:39,080 --> 00:42:43,840 OK, almost done, one more. 811 00:42:43,840 --> 00:42:47,860 So I've gone from here to here. 812 00:42:47,860 --> 00:42:49,960 What happens if I do a periodic extension? 813 00:42:49,960 --> 00:42:53,740 The answer was that you convolve in time, 814 00:42:53,740 --> 00:42:56,380 which is the same as multiply in frequency 815 00:42:56,380 --> 00:42:59,230 by an impulse train, which is the same as sampling 816 00:42:59,230 --> 00:43:01,740 in frequency. 817 00:43:01,740 --> 00:43:04,200 Convolving time, sampling frequency. 818 00:43:04,200 --> 00:43:06,450 That's what happened when I did this one. 819 00:43:06,450 --> 00:43:09,270 I just now did this one. 820 00:43:09,270 --> 00:43:12,910 What happens if you sample in time? 821 00:43:12,910 --> 00:43:16,976 Sample in time is multiplied by an impulse train. 822 00:43:16,976 --> 00:43:19,225 Multiplied by an impulse train is the same as convolve 823 00:43:19,225 --> 00:43:21,880 in frequency. 824 00:43:21,880 --> 00:43:25,640 Convolve in frequency makes it periodic. 825 00:43:25,640 --> 00:43:28,290 Sample in time, periodic in frequency. 826 00:43:28,290 --> 00:43:32,620 The last example is just to do this direction. 827 00:43:32,620 --> 00:43:36,210 What happens if I take the DTFT and try 828 00:43:36,210 --> 00:43:39,252 to turn that into a DTFS? 829 00:43:39,252 --> 00:43:40,710 Not surprisingly, what I have to do 830 00:43:40,710 --> 00:43:43,110 is the same thing I did in continuous time. 831 00:43:43,110 --> 00:43:45,270 If I want to change this DT signal-- which 832 00:43:45,270 --> 00:43:48,210 is aperiodic-- into a periodic DT signal, 833 00:43:48,210 --> 00:43:51,360 I do periodic extension. 834 00:43:51,360 --> 00:43:54,090 I can think about the periodically-extended waveform, 835 00:43:54,090 --> 00:43:58,850 here extended by multiples of 8, as being 836 00:43:58,850 --> 00:44:03,180 the original waveform convolved with that unit sample train. 837 00:44:07,120 --> 00:44:10,420 And so if I convolve by the unit sample train, what 838 00:44:10,420 --> 00:44:11,410 do I do in frequency? 839 00:44:15,320 --> 00:44:19,520 Convolve by unit sample train in time, 840 00:44:19,520 --> 00:44:22,590 gives [INAUDIBLE] and frequency. 841 00:44:22,590 --> 00:44:26,220 Multiply-- so I'm expecting that the answer will be sampled, 842 00:44:26,220 --> 00:44:28,290 and that's what happens. 843 00:44:28,290 --> 00:44:34,880 So I take the Fourier transform of this case. 844 00:44:34,880 --> 00:44:36,800 Which, like all Fourier transforms, 845 00:44:36,800 --> 00:44:41,860 is periodic in 2 pi. 846 00:44:41,860 --> 00:44:44,800 And now I think about convolving it 847 00:44:44,800 --> 00:44:49,850 with this unit sample train, which 848 00:44:49,850 --> 00:44:52,300 has the same relationship as the impulse train. 849 00:44:52,300 --> 00:44:54,950 And the impulse train-- the train 850 00:44:54,950 --> 00:44:57,710 of impulses separated by capital T, turned 851 00:44:57,710 --> 00:45:01,040 into a train of impulses separated by 2 pi over t. 852 00:45:01,040 --> 00:45:06,080 Here separated by capital N, turns into impulse trains 853 00:45:06,080 --> 00:45:08,090 separated by 2 pi over n-- 854 00:45:08,090 --> 00:45:10,230 same thing. 855 00:45:10,230 --> 00:45:13,820 So if this is 8 over here, I do 2 pi 856 00:45:13,820 --> 00:45:16,850 over 8, which is pi over 4. 857 00:45:16,850 --> 00:45:20,780 So convolving in discrete time is the same 858 00:45:20,780 --> 00:45:24,335 as multiplying by this impulse train, which gives me, 859 00:45:24,335 --> 00:45:29,410 then, a string of samples from that waveform. 860 00:45:29,410 --> 00:45:31,410 So the effect of going this way, which 861 00:45:31,410 --> 00:45:34,860 is periodic extension, which is the same as convolving in time, 862 00:45:34,860 --> 00:45:38,410 is sampling in frequency. 863 00:45:38,410 --> 00:45:40,820 OK, so the idea, then, is that that's 864 00:45:40,820 --> 00:45:43,860 a way of rationalizing how the Fourier series would 865 00:45:43,860 --> 00:45:44,360 have worked. 866 00:45:46,970 --> 00:45:49,400 So the Fourier series was a discrete set of numbers 867 00:45:49,400 --> 00:45:51,475 exactly for that reason. 868 00:45:51,475 --> 00:45:52,850 I started out with something that 869 00:45:52,850 --> 00:45:56,030 was continuous in frequency, but I sampled it 870 00:45:56,030 --> 00:46:02,440 because I was multiplying by an impulse train. 871 00:46:02,440 --> 00:46:07,257 So what I hope this did, OK, was dragged you 872 00:46:07,257 --> 00:46:09,340 through a couple of examples that will make things 873 00:46:09,340 --> 00:46:11,542 a little bit easier tomorrow. 874 00:46:11,542 --> 00:46:13,000 But I also hope that you understand 875 00:46:13,000 --> 00:46:15,490 how we went from a diagram that looks kind 876 00:46:15,490 --> 00:46:18,310 of complicated an arbitrary-- 877 00:46:18,310 --> 00:46:21,490 here we had DT on the top and CT on the bottom, 878 00:46:21,490 --> 00:46:25,640 periodic on the left and not periodic on the right. 879 00:46:25,640 --> 00:46:28,930 The transforms had all that things turn 90 degrees. 880 00:46:28,930 --> 00:46:33,760 We ended up with discrete on the left instead of on the top, 881 00:46:33,760 --> 00:46:36,790 and we ended up with periodic in the top 882 00:46:36,790 --> 00:46:39,260 rather than on the left. 883 00:46:39,260 --> 00:46:46,180 And that's entirely because of the relationships 884 00:46:46,180 --> 00:46:48,460 between the various Fourier transforms. 885 00:46:48,460 --> 00:46:49,689 So the idea is-- 886 00:46:49,689 --> 00:46:51,730 and you can understand all of those relationships 887 00:46:51,730 --> 00:46:54,880 by simply thinking about impulse trains. 888 00:46:54,880 --> 00:46:57,810 We take an aperiodic signal, turn it into a periodic 889 00:46:57,810 --> 00:47:00,190 by convolution in time, which is the same as multiplying 890 00:47:00,190 --> 00:47:03,570 frequency, which is the same as sampling. 891 00:47:03,570 --> 00:47:06,570 We sample in time by multiplying by an impulse train, which 892 00:47:06,570 --> 00:47:08,340 is the same as convolving in frequency, 893 00:47:08,340 --> 00:47:10,530 which makes it periodic. 894 00:47:10,530 --> 00:47:15,360 So the hope is that this looks like more than just-- 895 00:47:15,360 --> 00:47:18,450 1, 2, 3, 4, 5, 6, 7, 8-- eight independent equations, 896 00:47:18,450 --> 00:47:20,000 they're not. 897 00:47:20,000 --> 00:47:23,730 It's really two equations-- the transform and the inverse 898 00:47:23,730 --> 00:47:27,710 transform-- and they're almost the same equation, 899 00:47:27,710 --> 00:47:30,250 except for taking into account the special properties 900 00:47:30,250 --> 00:47:31,950 of the different domains. 901 00:47:31,950 --> 00:47:35,716 OK, have a good time, and see you tomorrow.