1 00:00:00,000 --> 00:00:00,030 2 00:00:00,030 --> 00:00:02,285 The following content is provided under a Creative 3 00:00:02,285 --> 00:00:03,610 Commons license. 4 00:00:03,610 --> 00:00:05,460 Your support will help MIT OpenCourseWare 5 00:00:05,460 --> 00:00:09,940 continue to offer high-quality educational resources for free. 6 00:00:09,940 --> 00:00:12,530 To make a donation, or to view additional materials 7 00:00:12,530 --> 00:00:15,840 from hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15,840 --> 00:00:20,150 at ocw.mit.edu. 9 00:00:20,150 --> 00:00:21,840 PROFESSOR STRANG: OK, hi. 10 00:00:21,840 --> 00:00:25,810 So I've got homework nine for you. 11 00:00:25,810 --> 00:00:28,020 Ready to return at the end. 12 00:00:28,020 --> 00:00:31,440 Also, the department asked me to do evaluations, 13 00:00:31,440 --> 00:00:33,490 but that's the end of the lecture. 14 00:00:33,490 --> 00:00:39,250 Then, so everybody knows there's a quiz tomorrow night. 15 00:00:39,250 --> 00:00:45,030 And shall I just remember the four questions on the quiz? 16 00:00:45,030 --> 00:00:49,580 I mean, not the details but the general idea of the questions. 17 00:00:49,580 --> 00:00:51,110 Details OK too. 18 00:00:51,110 --> 00:00:53,070 Yes. 19 00:00:53,070 --> 00:00:58,270 Yeah, so there'll be one question on a Fourier series. 20 00:00:58,270 --> 00:01:03,870 And you should know the energy equality 21 00:01:03,870 --> 00:01:09,460 for all of these possibilities, connecting the function squared 22 00:01:09,460 --> 00:01:11,740 with the coefficient squared. 23 00:01:11,740 --> 00:01:17,630 A second one on the discrete Fourier transform. 24 00:01:17,630 --> 00:01:19,200 Cyclic stuff. 25 00:01:19,200 --> 00:01:25,060 The third one on the Fourier integral. 26 00:01:25,060 --> 00:01:35,490 And have a look at the applications to solving an ODE. 27 00:01:35,490 --> 00:01:38,920 I did one in class. 28 00:01:38,920 --> 00:01:42,310 The one in class was the one in the book 29 00:01:42,310 --> 00:01:47,800 -u''+a^2*u=f(x), so this will be. 30 00:01:47,800 --> 00:01:50,610 So have a look at that application. 31 00:01:50,610 --> 00:01:53,770 This is, of course, on minus infinity to infinity. 32 00:01:53,770 --> 00:02:00,310 and then a fourth question on convolution. 33 00:02:00,310 --> 00:02:01,000 OK. 34 00:02:01,000 --> 00:02:03,610 And this afternoon, of course, I'll 35 00:02:03,610 --> 00:02:07,950 be here to answer any questions from the homework, 36 00:02:07,950 --> 00:02:13,270 from any source, for these topics. 37 00:02:13,270 --> 00:02:15,060 Are there any questions just now, though? 38 00:02:15,060 --> 00:02:19,950 I'm OK to take questions. 39 00:02:19,950 --> 00:02:24,660 I thought I'd discussed today a topic that 40 00:02:24,660 --> 00:02:27,870 involves both Fourier series and Fourier integrals. 41 00:02:27,870 --> 00:02:32,650 It's a kind of cool connection and it's 42 00:02:32,650 --> 00:02:36,300 linked to the name of Claude Shannon who 43 00:02:36,300 --> 00:02:41,440 created information theory, who was a Bell Labs guy and then 44 00:02:41,440 --> 00:02:43,790 an MIT professor. 45 00:02:43,790 --> 00:02:49,560 So I should put his name in. 46 00:02:49,560 --> 00:02:50,060 Shannon. 47 00:02:50,060 --> 00:02:54,400 OK, so this is, yeah. 48 00:02:54,400 --> 00:02:57,160 You'll see. 49 00:02:57,160 --> 00:02:59,680 So it's not on the quiz but it gives me 50 00:02:59,680 --> 00:03:03,790 a chance to say something important, and at the same time 51 00:03:03,790 --> 00:03:07,140 review Fourier series and Fourier integrals. 52 00:03:07,140 --> 00:03:11,130 So let me start with the problem. 53 00:03:11,130 --> 00:03:17,250 The problem comes for an A to D converter. 54 00:03:17,250 --> 00:03:19,620 So what does that mean? 55 00:03:19,620 --> 00:03:22,550 That means, this A is analog. 56 00:03:22,550 --> 00:03:28,080 That means we have a function, A for analog. 57 00:03:28,080 --> 00:03:30,590 And D for digital. 58 00:03:30,590 --> 00:03:37,270 So we have a function, like-- So f(x), say, 59 00:03:37,270 --> 00:03:39,650 all the way minus infinity to infinity, 60 00:03:39,650 --> 00:03:43,300 so we'll be doing, that's where the Fourier 61 00:03:43,300 --> 00:03:45,190 integral's going to come up. 62 00:03:45,190 --> 00:03:47,380 So that's analog. 63 00:03:47,380 --> 00:03:51,210 All x, it's some curve. 64 00:03:51,210 --> 00:03:55,940 And people build, and you can buy, 65 00:03:55,940 --> 00:04:01,530 and they're sold in large quantities, something that 66 00:04:01,530 --> 00:04:03,900 just samples that function. 67 00:04:03,900 --> 00:04:05,750 Say, at the integers. 68 00:04:05,750 --> 00:04:11,300 So now I'll sample that function and let 69 00:04:11,300 --> 00:04:14,880 me take the period of the sample to be one, so that I'm 70 00:04:14,880 --> 00:04:20,010 going to take the values f(n). 71 00:04:20,010 --> 00:04:22,350 So now I've got something digital 72 00:04:22,350 --> 00:04:26,510 that I can work with, that I can compute with. 73 00:04:26,510 --> 00:04:36,370 And, so the sampling theorem-- Well, I mean, the question is-- 74 00:04:36,370 --> 00:04:40,450 Yeah, the sampling theorem is about this question, 75 00:04:40,450 --> 00:04:45,410 and it seems a crazy question, when do these numbers -- 76 00:04:45,410 --> 00:04:47,420 That's just a sequence of numbers. 77 00:04:47,420 --> 00:04:52,690 This x was all the way from minus infinity to infinity. 78 00:04:52,690 --> 00:04:58,520 And similarly, n is numbers all the way from minus infinity 79 00:04:58,520 --> 00:05:01,520 to infinity, they're just samples. 80 00:05:01,520 --> 00:05:06,800 When does that tell me the function? 81 00:05:06,800 --> 00:05:10,260 When can I learn from those samples, 82 00:05:10,260 --> 00:05:13,950 when do I have total information about the function? 83 00:05:13,950 --> 00:05:16,170 Now, you'll say impossible. 84 00:05:16,170 --> 00:05:16,990 Right? 85 00:05:16,990 --> 00:05:24,410 So suppose I draw a function f(x), OK? 86 00:05:24,410 --> 00:05:31,560 And I'm going to sample it at these points. 87 00:05:31,560 --> 00:05:37,150 All the way, so these are the numbers, these are my f(n). 88 00:05:37,150 --> 00:05:42,780 Sampling at equal intervals, because if we 89 00:05:42,780 --> 00:05:45,510 want to use Fourier ideas, equal spacing 90 00:05:45,510 --> 00:05:47,220 is the right thing to have. 91 00:05:47,220 --> 00:05:50,340 So when could I recover the function in between? 92 00:05:50,340 --> 00:05:52,580 Well, you'd say, never. 93 00:05:52,580 --> 00:05:55,640 Because how do I know what that function 94 00:05:55,640 --> 00:05:57,530 could be doing in between. 95 00:05:57,530 --> 00:06:04,640 So let me take the case when all the samples are zero. 96 00:06:04,640 --> 00:06:06,540 And let's think about that case. 97 00:06:06,540 --> 00:06:12,700 Suppose, what could the function be if all the samples, 98 00:06:12,700 --> 00:06:18,500 if these are all zeroes, forever. 99 00:06:18,500 --> 00:06:21,900 OK, well there's one leading candidate 100 00:06:21,900 --> 00:06:26,830 for the function, the zero function. 101 00:06:26,830 --> 00:06:31,370 Now, you'll see the whole point of the sampling theorem 102 00:06:31,370 --> 00:06:36,140 if you think about other, what other functions? 103 00:06:36,140 --> 00:06:37,460 Familiar functions, yeah. 104 00:06:37,460 --> 00:06:42,030 I mean, we could, obviously, any, all sorts of things. 105 00:06:42,030 --> 00:06:45,670 But since we're doing Fourier, we 106 00:06:45,670 --> 00:06:51,110 like to pick on the sines, cosines, the special functions, 107 00:06:51,110 --> 00:06:53,440 and think about those in particular. 108 00:06:53,440 --> 00:06:57,100 So, somebody said sines. 109 00:06:57,100 --> 00:07:02,850 Now, what function, so a sine function certainly, 110 00:07:02,850 --> 00:07:07,300 the sine function hits zero infinitely often. 111 00:07:07,300 --> 00:07:14,160 What frequency, so sine of what would give me the same answer? 112 00:07:14,160 --> 00:07:16,440 The same samples. 113 00:07:16,440 --> 00:07:19,930 If I put this sine function that you're going to tell me, 114 00:07:19,930 --> 00:07:24,450 so you're going to tell me it's sine of something 115 00:07:24,450 --> 00:07:27,540 will have these zero values at all 116 00:07:27,540 --> 00:07:32,390 the integers, at zero, one, two, minus one, minus two, and so 117 00:07:32,390 --> 00:07:32,940 on. 118 00:07:32,940 --> 00:07:36,780 So what would do the job? 119 00:07:36,780 --> 00:07:37,730 Sine of? 120 00:07:37,730 --> 00:07:45,390 Of what will hit zero. 121 00:07:45,390 --> 00:07:48,790 So I'm looking for a sine function, 122 00:07:48,790 --> 00:07:51,620 I guess I'm looking first for the function that 123 00:07:51,620 --> 00:07:54,130 just does that. 124 00:07:54,130 --> 00:07:56,880 And what is it? sin(pi*x). 125 00:07:56,880 --> 00:08:01,160 And now tell me some more. 126 00:08:01,160 --> 00:08:03,150 Tell me another function. 127 00:08:03,150 --> 00:08:07,360 Which will also, it won't be that graph. 128 00:08:07,360 --> 00:08:09,620 sin(2pi*x). 129 00:08:09,620 --> 00:08:14,800 And all the rest, OK? 130 00:08:14,800 --> 00:08:18,500 Let me just use a word that's kind of a handy word. 131 00:08:18,500 --> 00:08:22,300 Of course, let's put zero on the list here. 132 00:08:22,300 --> 00:08:24,210 OK. 133 00:08:24,210 --> 00:08:29,000 So this is where k, the frequency, usually appears. 134 00:08:29,000 --> 00:08:33,790 This is where k-- 135 00:08:33,790 --> 00:08:37,390 The word I want to introduce is alias. 136 00:08:37,390 --> 00:08:44,670 This frequency, pi, is an alias for this at frequency zero. 137 00:08:44,670 --> 00:08:49,110 Here's the, it's a different function 138 00:08:49,110 --> 00:08:51,370 but yet the samples are the same. 139 00:08:51,370 --> 00:08:55,200 So if you're only looking at the samples 140 00:08:55,200 --> 00:08:59,360 you're getting the same answer but somehow 141 00:08:59,360 --> 00:09:01,070 the function has a different name. 142 00:09:01,070 --> 00:09:05,360 So that frequency and this frequency, and all those others 143 00:09:05,360 --> 00:09:06,880 would be alias. 144 00:09:06,880 --> 00:09:11,010 Can I just write that word down, because you see it often. 145 00:09:11,010 --> 00:09:13,760 Alias. 146 00:09:13,760 --> 00:09:16,850 That means two frequencies, like pi and 2pi, 147 00:09:16,850 --> 00:09:20,580 and zero or whatever, that give you the same samples. 148 00:09:20,580 --> 00:09:27,980 OK, so now comes Shannon's question. 149 00:09:27,980 --> 00:09:32,810 So we have to make some assumption on the function. 150 00:09:32,810 --> 00:09:38,730 To knock out those possibilities. 151 00:09:38,730 --> 00:09:44,160 We want to know a limited class of functions. 152 00:09:44,160 --> 00:09:47,370 Which don't include these guys. 153 00:09:47,370 --> 00:09:50,290 So that within this limited class of functions, 154 00:09:50,290 --> 00:09:54,900 this is the only candidate and we have this possibility 155 00:09:54,900 --> 00:09:57,110 of doing the impossible. 156 00:09:57,110 --> 00:10:02,340 Of determining that if I know zeroes here, 157 00:10:02,340 --> 00:10:05,670 the function has to be zero everywhere. 158 00:10:05,670 --> 00:10:08,620 OK, now the question is what class of functions? 159 00:10:08,620 --> 00:10:13,140 We want to eliminate these guys, and sort of, your instinct 160 00:10:13,140 --> 00:10:19,260 is, you want to eliminate functions that, you know, 161 00:10:19,260 --> 00:10:21,770 if it's not zero then it's got to get up and back 162 00:10:21,770 --> 00:10:24,400 down in every thing. 163 00:10:24,400 --> 00:10:26,980 It could do different things in different intervals. 164 00:10:26,980 --> 00:10:33,160 But somehow it's got to have some of these frequencies. 165 00:10:33,160 --> 00:10:36,370 Pi or higher. 166 00:10:36,370 --> 00:10:37,810 Would have to be in there. 167 00:10:37,810 --> 00:10:40,450 So this is the instinct. 168 00:10:40,450 --> 00:10:48,870 That if I limit the frequency band, so I'm going to say f(x) 169 00:10:48,870 --> 00:10:53,560 is band-limited, can I introduce that word? 170 00:10:53,560 --> 00:10:55,740 I'll maybe take a moment just ask you 171 00:10:55,740 --> 00:10:59,480 if you've seen that word before. 172 00:10:59,480 --> 00:11:01,850 How many have seen this word, band-limited? 173 00:11:01,850 --> 00:11:04,410 Quite a few but not half. 174 00:11:04,410 --> 00:11:05,070 OK. 175 00:11:05,070 --> 00:11:10,600 Band-limited means the band is a band of frequencies. 176 00:11:10,600 --> 00:11:14,290 So the function's band-limited when 177 00:11:14,290 --> 00:11:19,270 its transform, this tells me how much of each frequency 178 00:11:19,270 --> 00:11:20,190 there is. 179 00:11:20,190 --> 00:11:23,420 If this is zero, in some band. 180 00:11:23,420 --> 00:11:32,480 In some band, let's say, all frequencies below something. 181 00:11:32,480 --> 00:11:36,110 And let's not even put equal in there. 182 00:11:36,110 --> 00:11:37,580 OK. 183 00:11:37,580 --> 00:11:40,190 But that's not critical. 184 00:11:40,190 --> 00:11:44,120 Band-limited, I have to tell you the size of the band. 185 00:11:44,120 --> 00:11:48,590 And the size of the band, the limiting frequency 186 00:11:48,590 --> 00:11:51,130 is this famous Nyquist frequency, 187 00:11:51,130 --> 00:11:53,270 so Nyquist is a guy's name. 188 00:11:53,270 --> 00:11:57,420 And the Nyquist frequency in our problem here is pi. 189 00:11:57,420 --> 00:12:00,090 This is the Nyquist frequency. 190 00:12:00,090 --> 00:12:08,780 If we let that frequency, that's the borderline frequency. 191 00:12:08,780 --> 00:12:12,510 And there would be a similar Nyquist sampling rate. 192 00:12:12,510 --> 00:12:16,680 So Nyquist is the guy who studied 193 00:12:16,680 --> 00:12:20,220 the sort of borderline case. 194 00:12:20,220 --> 00:12:26,940 So the point is that if our, say, band-limited by pi, 195 00:12:26,940 --> 00:12:29,710 I have to tell you, so band-limited means there's 196 00:12:29,710 --> 00:12:31,550 some limit on the band. 197 00:12:31,550 --> 00:12:37,030 And our interest is when that limit is the Nyquist frequency. 198 00:12:37,030 --> 00:12:43,310 The one we don't want to allow, so we-- This is the point. 199 00:12:43,310 --> 00:12:45,320 So this will be the idea. 200 00:12:45,320 --> 00:12:52,570 That if we take this class of function, 201 00:12:52,570 --> 00:12:56,600 that band-limited-- Those are called band-limited functions, 202 00:12:56,600 --> 00:12:58,770 and they're band-limited specifically 203 00:12:58,770 --> 00:13:01,940 by the Nyquist limit. 204 00:13:01,940 --> 00:13:05,050 If we take those, then the idea is 205 00:13:05,050 --> 00:13:13,990 that then we can reconstruct from the samples. 206 00:13:13,990 --> 00:13:17,980 Because the only function that has zero samples in that 207 00:13:17,980 --> 00:13:20,560 class is the zero function. 208 00:13:20,560 --> 00:13:22,490 You see that class has knocked out, 209 00:13:22,490 --> 00:13:26,130 is not allowing these guys. 210 00:13:26,130 --> 00:13:29,270 Of course, haven't proved anything yet. 211 00:13:29,270 --> 00:13:31,920 And I haven't shown how to reconstruct. 212 00:13:31,920 --> 00:13:35,790 Well, of course, we quickly reconstructed the zero function 213 00:13:35,790 --> 00:13:41,200 out of those zeroes, but now let me take another obviously 214 00:13:41,200 --> 00:13:43,570 important possible sample. 215 00:13:43,570 --> 00:13:48,560 Suppose I get zero samples except at that point, where 216 00:13:48,560 --> 00:13:51,910 it's one. 217 00:13:51,910 --> 00:14:00,270 OK, now the question is what function, f(x)-- 218 00:14:00,270 --> 00:14:04,490 Can I fill in, in between zero, zero, zero, zero, one, zero, 219 00:14:04,490 --> 00:14:11,010 zero, zero, can I fill in exactly one function that comes 220 00:14:11,010 --> 00:14:14,040 from this class? 221 00:14:14,040 --> 00:14:19,030 So that I have now the answer for this highly important 222 00:14:19,030 --> 00:14:19,930 sample? 223 00:14:19,930 --> 00:14:23,310 The sample that's all zeroes except for the delta sample, 224 00:14:23,310 --> 00:14:24,420 you could say. 225 00:14:24,420 --> 00:14:27,130 OK, so I'm looking for the function now 226 00:14:27,130 --> 00:14:30,760 which is one at that point. 227 00:14:30,760 --> 00:14:32,780 And zero at the others. 228 00:14:32,780 --> 00:14:36,150 So here's a key function. 229 00:14:36,150 --> 00:14:39,370 And I'll show you what it is. 230 00:14:39,370 --> 00:14:43,980 So the function, this function that I'm going to mention, 231 00:14:43,980 --> 00:14:49,860 will get down here, it'll oscillate, it'll go forever. 232 00:14:49,860 --> 00:14:52,450 It's not like a spline. 233 00:14:52,450 --> 00:14:58,030 Splines made it to zero and stayed there, right? 234 00:14:58,030 --> 00:15:00,140 The cubic spline, for example. 235 00:15:00,140 --> 00:15:08,480 OK, but I guess, yeah, that somehow that function, 236 00:15:08,480 --> 00:15:11,160 we're not in that league. 237 00:15:11,160 --> 00:15:13,650 We're in this band-limited league. 238 00:15:13,650 --> 00:15:20,080 In a way you could say that, I mean, 239 00:15:20,080 --> 00:15:24,140 what's the key connection between dropoff of the Fourier 240 00:15:24,140 --> 00:15:25,280 transform? 241 00:15:25,280 --> 00:15:28,790 So if the Fourier transform drops off fast, 242 00:15:28,790 --> 00:15:31,640 what does that tell me about the function? 243 00:15:31,640 --> 00:15:32,850 It's smooth, thanks. 244 00:15:32,850 --> 00:15:34,260 That's exactly the right word. 245 00:15:34,260 --> 00:15:36,280 If the Fourier transform drops off fast, 246 00:15:36,280 --> 00:15:38,080 the function is smooth. 247 00:15:38,080 --> 00:15:42,060 OK, this is really an extreme case. 248 00:15:42,060 --> 00:15:46,470 That it's dropped off totally. 249 00:15:46,470 --> 00:15:50,770 You know, it's not just decay rate, it's just zonk, out. 250 00:15:50,770 --> 00:15:53,870 Beyond this band of frequencies. 251 00:15:53,870 --> 00:15:57,700 And so that gives, you could say, sort 252 00:15:57,700 --> 00:15:59,620 of a hyper-smooth function. 253 00:15:59,620 --> 00:16:01,890 I mean, so smooth that you know everything 254 00:16:01,890 --> 00:16:03,130 by knowing the sample. 255 00:16:03,130 --> 00:16:07,710 OK, now I'm ready to write down the key function, 256 00:16:07,710 --> 00:16:11,420 a famous function that has those samples. 257 00:16:11,420 --> 00:16:17,960 And that function is sin(pi*x)/(pi*x). 258 00:16:17,960 --> 00:16:21,820 259 00:16:21,820 --> 00:16:24,450 I don't know if you ever thought about this function, 260 00:16:24,450 --> 00:16:25,640 and it has a name. 261 00:16:25,640 --> 00:16:27,700 Do you know its name? 262 00:16:27,700 --> 00:16:28,220 Sinc. 263 00:16:28,220 --> 00:16:29,980 It's the sinc function. 264 00:16:29,980 --> 00:16:32,060 Which is a little-- you know, the name's 265 00:16:32,060 --> 00:16:34,390 a little unfortunate. 266 00:16:34,390 --> 00:16:37,680 Mainly because you know, you're using those same letters 267 00:16:37,680 --> 00:16:41,550 S I N but it's a c that turns it, 268 00:16:41,550 --> 00:16:46,120 that gives it-- So this is called the sinc function. 269 00:16:46,120 --> 00:16:47,660 sinc(x). 270 00:16:47,660 --> 00:16:50,530 But the main thing is its formula. 271 00:16:50,530 --> 00:16:56,410 OK, well everybody sees that at x=1, the sin(pi) is zero, 272 00:16:56,410 --> 00:17:00,600 at x=2, the sin(pi) is zero, all these ones we've seen already. 273 00:17:00,600 --> 00:17:03,230 And now what happens at equals zero? 274 00:17:03,230 --> 00:17:07,460 Do you recognize that this function, as x goes to zero, 275 00:17:07,460 --> 00:17:10,230 is a perfectly good function? 276 00:17:10,230 --> 00:17:14,750 I mean, it becomes 0/0 at x=0. 277 00:17:14,750 --> 00:17:17,860 But the limit, we know to be one. 278 00:17:17,860 --> 00:17:18,360 Right? 279 00:17:18,360 --> 00:17:25,550 This sin(theta)/theta is one as theta approaches zero, right? 280 00:17:25,550 --> 00:17:28,630 So that does have that correct sample. 281 00:17:28,630 --> 00:17:32,130 And now what's the, I claim that that 282 00:17:32,130 --> 00:17:33,690 is a band-limited function. 283 00:17:33,690 --> 00:17:40,650 And you'll see that that function pushes the limit. 284 00:17:40,650 --> 00:17:46,620 It's right-- Nyquist barely lets it in. 285 00:17:46,620 --> 00:17:48,190 Now here's a calculation. 286 00:17:48,190 --> 00:17:50,470 So this is our practice. 287 00:17:50,470 --> 00:17:59,360 What is f hat of k for that function? 288 00:17:59,360 --> 00:18:01,450 Now, let me think how to do this one. 289 00:18:01,450 --> 00:18:07,720 So just to understand this better, 290 00:18:07,720 --> 00:18:10,390 I want to see that that is a band-limited function 291 00:18:10,390 --> 00:18:14,130 and what is its Fourier transform. 292 00:18:14,130 --> 00:18:18,980 OK, now once again here we have a function where if I want to 293 00:18:18,980 --> 00:18:25,220 do-- How best to do this? 294 00:18:25,220 --> 00:18:33,010 You could say well, just do it. 295 00:18:33,010 --> 00:18:35,830 As I would say on the quiz, just go ahead and do it. 296 00:18:35,830 --> 00:18:41,470 But you'll see I'm going to have a problem, I think. 297 00:18:41,470 --> 00:18:44,040 But this gives us a chance to remember the formula. 298 00:18:44,040 --> 00:18:47,450 So what's the formula for the Fourier integral transform 299 00:18:47,450 --> 00:18:48,680 of this particular? 300 00:18:48,680 --> 00:18:51,640 So my function is the sinc function, 301 00:18:51,640 --> 00:18:55,400 sin(pi*x) over sin(pi*x). 302 00:18:55,400 --> 00:19:03,000 So how do I get its, I do a what here? e^(-ikx), 303 00:19:03,000 --> 00:19:05,650 and am I doing dx or dk? 304 00:19:05,650 --> 00:19:07,290 dx, right? 305 00:19:07,290 --> 00:19:10,790 And I'm going from minus infinity to infinity. 306 00:19:10,790 --> 00:19:14,640 And am I, do I have a 2pi? 307 00:19:14,640 --> 00:19:16,160 Yes or no? 308 00:19:16,160 --> 00:19:17,850 Who knows, anyway? 309 00:19:17,850 --> 00:19:21,150 Right. 310 00:19:21,150 --> 00:19:22,960 In the book I didn't? 311 00:19:22,960 --> 00:19:25,750 OK. 312 00:19:25,750 --> 00:19:33,100 Now, well, I don't know the answer. 313 00:19:33,100 --> 00:19:37,640 But so let's, it's much better to start with the answer, 314 00:19:37,640 --> 00:19:42,350 right, and check that-- So let me 315 00:19:42,350 --> 00:19:44,470 say what I think the answer is. 316 00:19:44,470 --> 00:19:51,340 I think the answer is, it's a function that's 317 00:19:51,340 --> 00:19:59,060 exactly as I say, it pushes the limit from, this is k. 318 00:19:59,060 --> 00:20:03,770 It's the square wave, it's zero, the height is one. 319 00:20:03,770 --> 00:20:08,010 It's the function that's zero all the way here, 320 00:20:08,010 --> 00:20:09,660 all the way there. 321 00:20:09,660 --> 00:20:13,740 I think that that's the Fourier transform of that function. 322 00:20:13,740 --> 00:20:20,370 And just before we check it, see how is Nyquist got really 323 00:20:20,370 --> 00:20:22,550 pushed up to the wall, right? 324 00:20:22,550 --> 00:20:26,590 Because the frequency is non-zero right, 325 00:20:26,590 --> 00:20:30,170 all the way through pi. 326 00:20:30,170 --> 00:20:36,020 But pi is just one point there. 327 00:20:36,020 --> 00:20:41,460 And anyway, I think we won't get into philosophical discussion 328 00:20:41,460 --> 00:20:45,460 about whether, you know, is that limited to pi? 329 00:20:45,460 --> 00:20:49,440 I don't know whether to put, you saw me chicken out here. 330 00:20:49,440 --> 00:20:52,620 I didn't know whether to put less or equal or not, 331 00:20:52,620 --> 00:20:55,640 and I still don't. 332 00:20:55,640 --> 00:21:00,770 But this is making the point that that's the key frequency. 333 00:21:00,770 --> 00:21:03,260 So this particular function, what I'm saying 334 00:21:03,260 --> 00:21:07,080 is this particular function has all the frequencies 335 00:21:07,080 --> 00:21:12,480 in equal amounts over a band, and nothing outside that band. 336 00:21:12,480 --> 00:21:14,260 And that's the Nyquist band. 337 00:21:14,260 --> 00:21:18,250 OK, now why is this the correct answer here? 338 00:21:18,250 --> 00:21:25,080 I guess the smart way would be, this is a good function, 339 00:21:25,080 --> 00:21:26,140 easy function. 340 00:21:26,140 --> 00:21:32,270 So let's take the transform in the other direction. 341 00:21:32,270 --> 00:21:34,710 Start from here and get to here. 342 00:21:34,710 --> 00:21:35,210 Right? 343 00:21:35,210 --> 00:21:36,500 That would be convincing. 344 00:21:36,500 --> 00:21:41,470 Because we do know that that pair of formulas 345 00:21:41,470 --> 00:21:46,220 for f connected to f hat connected to f, 346 00:21:46,220 --> 00:21:47,620 they go together. 347 00:21:47,620 --> 00:21:54,750 So if I can show that I go from here, that that Fourier 348 00:21:54,750 --> 00:21:56,710 integral takes me from here to there, 349 00:21:56,710 --> 00:21:59,080 then this guy will take me back. 350 00:21:59,080 --> 00:22:00,550 So let me just do that. 351 00:22:00,550 --> 00:22:05,120 Because that's a very very important one that you 352 00:22:05,120 --> 00:22:07,470 should be prepared for. 353 00:22:07,470 --> 00:22:07,970 Right. 354 00:22:07,970 --> 00:22:10,580 So now what do I want to do? 355 00:22:10,580 --> 00:22:23,780 Here's my function of k, and I'm hoping that I recall-- Now, 356 00:22:23,780 --> 00:22:27,240 what do I do when I want to do the transform 357 00:22:27,240 --> 00:22:29,010 in the opposite direction? 358 00:22:29,010 --> 00:22:37,120 It'll be an e^(+ikx), right? d what? dk, now. 359 00:22:37,120 --> 00:22:39,700 From k equal minus infinity to to infinity. 360 00:22:39,700 --> 00:22:43,800 And now I think I do put in the 2pi, is that right? 361 00:22:43,800 --> 00:22:50,740 And the question is, does that bring back the sinc function? 362 00:22:50,740 --> 00:22:55,640 If it does, then this was OK in the other direction. 363 00:22:55,640 --> 00:22:57,790 If the transform's correct in one direction then 364 00:22:57,790 --> 00:22:59,430 the inverse transform will be correct. 365 00:22:59,430 --> 00:23:03,370 So I just plan to do this integral. f 366 00:23:03,370 --> 00:23:07,380 hat of k, of course, is an easy integral now. f hat of k 367 00:23:07,380 --> 00:23:13,320 is one over, between minus pi and pi, 368 00:23:13,320 --> 00:23:17,040 so I only have to do over that range where f hat of k 369 00:23:17,040 --> 00:23:20,480 is just a one. 370 00:23:20,480 --> 00:23:23,660 And now that's an integral we can certainly do. 371 00:23:23,660 --> 00:23:30,490 So I have 1/(2pi), integrating e^(ikx) will give me e^(ikx) 372 00:23:30,490 --> 00:23:33,100 over ix. 373 00:23:33,100 --> 00:23:35,680 Now, remember I'm integrating dk. 374 00:23:35,680 --> 00:23:36,510 Oh, look. 375 00:23:36,510 --> 00:23:40,690 See, we're showing this x now in the denominator. 376 00:23:40,690 --> 00:23:44,340 That we're hoping for. 377 00:23:44,340 --> 00:23:50,360 And now I have to do that between k is minus pi and pi. 378 00:23:50,360 --> 00:23:56,550 So this is like, so I get 1/(2pi), 379 00:23:56,550 --> 00:24:06,390 e^(i*pi*k) minus e^(-i*pi*x), right? 380 00:24:06,390 --> 00:24:07,790 Over the ix. 381 00:24:07,790 --> 00:24:10,350 382 00:24:10,350 --> 00:24:13,560 OK so far? 383 00:24:13,560 --> 00:24:17,710 I was doing a k integral and I get an x answer. 384 00:24:17,710 --> 00:24:21,350 And I want to be sure that this x answer is the x answer I 385 00:24:21,350 --> 00:24:23,890 want, it's the sinc function. 386 00:24:23,890 --> 00:24:24,890 OK, it is. 387 00:24:24,890 --> 00:24:25,900 Right? 388 00:24:25,900 --> 00:24:31,290 I recognize the sine, e^(i*theta)-e^(-i*theta), 389 00:24:31,290 --> 00:24:36,530 divided by two, I guess. 390 00:24:36,530 --> 00:24:38,260 Is the sine, right? 391 00:24:38,260 --> 00:24:40,700 So I have 1/(2pi). 392 00:24:40,700 --> 00:24:46,740 And here is ix-- Well, no, the i is part of that sine. 393 00:24:46,740 --> 00:24:54,930 So I'm just using the fact that e^(i*theta)-e^(-i*theta) is, 394 00:24:54,930 --> 00:24:59,650 this is cosine plus i sine, subtract cosine. 395 00:24:59,650 --> 00:25:05,630 But subtract minus i sine, so that will be 2i*sin(theta), 396 00:25:05,630 --> 00:25:07,080 right? 397 00:25:07,080 --> 00:25:11,530 We all know, and now theta is pi*x here. 398 00:25:11,530 --> 00:25:17,780 So I have two-- let me keep the i there, and 2i*sin(pi*x). 399 00:25:17,780 --> 00:25:20,780 You see it works. 400 00:25:20,780 --> 00:25:25,170 Just using this, replacing this by the sine, 401 00:25:25,170 --> 00:25:27,440 the two cancels the two. 402 00:25:27,440 --> 00:25:32,060 The i cancels the i, and I have sin(pi*x) over pi*x, 403 00:25:32,060 --> 00:25:34,010 that's the sinc function. 404 00:25:34,010 --> 00:25:42,380 OK, so that's the function we've now checked. 405 00:25:42,380 --> 00:25:44,860 We've checked two things about this function. 406 00:25:44,860 --> 00:25:48,190 It has the right samples, zero, zero, zero, zero, 407 00:25:48,190 --> 00:25:51,790 one at x=0 and then back to zero. 408 00:25:51,790 --> 00:25:55,650 It is band-limited, so it's the guy, 409 00:25:55,650 --> 00:26:00,570 if this was my sample, if this was my f(n), zero, zero, zero, 410 00:26:00,570 --> 00:26:05,820 one, zero, zero, zero, then I've got it. 411 00:26:05,820 --> 00:26:10,950 It's the right function. 412 00:26:10,950 --> 00:26:15,840 OK, we can create Shannon's sampling formula. 413 00:26:15,840 --> 00:26:20,830 Shannon's sampling formula gives me the f(x) for any f(n). 414 00:26:20,830 --> 00:26:23,380 415 00:26:23,380 --> 00:26:25,370 Maybe you can spot that, now. 416 00:26:25,370 --> 00:26:28,200 So this is going to use the shift invariance. 417 00:26:28,200 --> 00:26:34,120 Oh yeah, let's-- Tell me what the, let's take one step here. 418 00:26:34,120 --> 00:26:39,300 Suppose my f(n)'s were zero, zero, zero, zero, and a one 419 00:26:39,300 --> 00:26:40,780 there. 420 00:26:40,780 --> 00:26:46,720 So suppose-- This is, for the exam too, this idea of shifting 421 00:26:46,720 --> 00:26:49,890 is simple and basic. 422 00:26:49,890 --> 00:26:52,270 And it's a great thing to be able to do. 423 00:26:52,270 --> 00:26:56,050 So this wouldn't be the right answer. 424 00:26:56,050 --> 00:26:59,230 That function produced the one at zero. 425 00:26:59,230 --> 00:27:04,500 Tell me what function, copying this idea, 426 00:27:04,500 --> 00:27:11,570 will produce all zeroes except for a one at this point. 427 00:27:11,570 --> 00:27:17,400 So I'll put this question over here. 428 00:27:17,400 --> 00:27:24,220 Suppose I'm all zeroes at that point but at this point I'm up 429 00:27:24,220 --> 00:27:26,400 and then I go back to zeroes. 430 00:27:26,400 --> 00:27:33,440 What function is giving me that? 431 00:27:33,440 --> 00:27:36,820 You see it does decay because of the x in the denominator, 432 00:27:36,820 --> 00:27:40,580 it kind of goes to zero but not very fast. 433 00:27:40,580 --> 00:27:45,370 OK, what's that function? 434 00:27:45,370 --> 00:27:48,810 I replace x by x-1, right. 435 00:27:48,810 --> 00:27:53,940 So the function here is sin(pi(x-1)), 436 00:27:53,940 --> 00:27:56,360 divided by pi(x-1). 437 00:27:56,360 --> 00:27:59,230 I just change the x to x-1. 438 00:27:59,230 --> 00:28:05,940 Now again, at x=0, this is sin(pi), sin(-pi), 439 00:28:05,940 --> 00:28:10,500 it's safely zero, but at x=1 that's now the point where 440 00:28:10,500 --> 00:28:13,660 I'm getting 0/0. 441 00:28:13,660 --> 00:28:18,080 And the numbers are right to give me the exact answer, one. 442 00:28:18,080 --> 00:28:24,110 OK, so now we see what to do if the sampling turned out 443 00:28:24,110 --> 00:28:25,790 to give us this answer. 444 00:28:25,790 --> 00:28:29,440 And now can you tell me the whole formula? 445 00:28:29,440 --> 00:28:31,660 Can you tell me the whole formula, 446 00:28:31,660 --> 00:28:35,210 so now I'm ready for Shannon's sampling theorem. 447 00:28:35,210 --> 00:28:41,200 Is that f(x), if f(x) is band-limited, 448 00:28:41,200 --> 00:28:45,040 then I can tell you what it is at all x. 449 00:28:45,040 --> 00:28:48,730 So I'm going back to the beginning of this lecture. 450 00:28:48,730 --> 00:28:52,880 It's a miracle that this is possible. 451 00:28:52,880 --> 00:28:57,690 That we can write down a formula for f(x) at all x, 452 00:28:57,690 --> 00:29:01,530 only using f(n)'s. 453 00:29:01,530 --> 00:29:05,550 OK, now it's going to be a sum. 454 00:29:05,550 --> 00:29:09,250 From n equal minus infinity to infinity, 455 00:29:09,250 --> 00:29:12,970 because I'm going to use all the f(n)'s to produce the f(x). 456 00:29:12,970 --> 00:29:17,910 And now what do I put in there? 457 00:29:17,910 --> 00:29:23,500 So I want my formula to be correct. 458 00:29:23,500 --> 00:29:28,810 I want my formula to be correct in this case, 459 00:29:28,810 --> 00:29:34,130 so that if all the f's were zero except for the middle one, 460 00:29:34,130 --> 00:29:37,200 then I want to put sin(pi*x)/(pi*x) in there. 461 00:29:37,200 --> 00:29:39,120 The sinc function. 462 00:29:39,120 --> 00:29:41,500 And I also want to get this one right. 463 00:29:41,500 --> 00:29:45,650 If all the f's are zero, so there'll only be one term. 464 00:29:45,650 --> 00:29:48,920 If this is the term, I want that to show up. 465 00:29:48,920 --> 00:29:53,710 OK, what do I-- Yeah, you can tell me. 466 00:29:53,710 --> 00:29:57,740 Suppose this is hitting at n. 467 00:29:57,740 --> 00:30:00,640 We'll just fix this and then you'll see it. 468 00:30:00,640 --> 00:30:06,460 Suppose that all the others, n, n-1, n+1, all those, it's zero, 469 00:30:06,460 --> 00:30:09,630 but at x=n, it's one. 470 00:30:09,630 --> 00:30:11,730 Now what should I have chosen? 471 00:30:11,730 --> 00:30:14,420 What's the correct-- I'm just going 472 00:30:14,420 --> 00:30:19,190 to make it easy for all of us to, yes. 473 00:30:19,190 --> 00:30:25,460 What's the good sinc function which peaks at a point n? 474 00:30:25,460 --> 00:30:27,790 Again, I'm just shifting it over. 475 00:30:27,790 --> 00:30:30,810 So what do I do? 476 00:30:30,810 --> 00:30:34,920 Put in, what do I write here? n. 477 00:30:34,920 --> 00:30:36,500 I shift the whole thing by n. 478 00:30:36,500 --> 00:30:39,660 So that's the right answer when this hits at n. 479 00:30:39,660 --> 00:30:42,150 So now maybe you see that this is 480 00:30:42,150 --> 00:30:51,970 going to be the right answer for all of them. 481 00:30:51,970 --> 00:30:56,130 You see that, we're using linearity and shift invariance. 482 00:30:56,130 --> 00:31:00,240 The shift invariance is telling us this answer 483 00:31:00,240 --> 00:31:03,620 for wherever the one hits. 484 00:31:03,620 --> 00:31:05,350 That's what we need. 485 00:31:05,350 --> 00:31:08,740 And then by linearity, I put together 486 00:31:08,740 --> 00:31:12,200 whatever the f is at that point, that 487 00:31:12,200 --> 00:31:14,320 would just amplify the sinc. 488 00:31:14,320 --> 00:31:17,800 And then I have to put them in for all the other values. 489 00:31:17,800 --> 00:31:20,410 That's the Shannon formula. 490 00:31:20,410 --> 00:31:25,940 That's the Shannon formula, and this function is band-limited, 491 00:31:25,940 --> 00:31:27,160 let's see. 492 00:31:27,160 --> 00:31:29,630 What's the-- Oh, yeah. 493 00:31:29,630 --> 00:31:34,120 What's the, do you see that this one, 494 00:31:34,120 --> 00:31:36,180 that this guy is band-limited? 495 00:31:36,180 --> 00:31:37,970 We checked, right? 496 00:31:37,970 --> 00:31:42,640 We checked that this one, sin(pi*x)/(pi*x), 497 00:31:42,640 --> 00:31:47,190 that was band-limited because we actually found the band. 498 00:31:47,190 --> 00:31:50,510 Now, that just gives us another chance to think. 499 00:31:50,510 --> 00:31:51,560 Allowed. 500 00:31:51,560 --> 00:31:55,280 What's the Fourier transform of this guy? 501 00:31:55,280 --> 00:31:58,560 My claim is that it's also in this band. 502 00:31:58,560 --> 00:32:04,200 Non-zero only in the band, and zero outside the Nyquist band. 503 00:32:04,200 --> 00:32:08,440 What is the transform of that? 504 00:32:08,440 --> 00:32:11,360 What happens if you shift a function, what 505 00:32:11,360 --> 00:32:14,280 happens to its transform? 506 00:32:14,280 --> 00:32:23,670 So that's one of the key rules that makes Fourier so special. 507 00:32:23,670 --> 00:32:28,300 If I took this sine, let me write this guy again. 508 00:32:28,300 --> 00:32:31,710 This was the un-shifted one. 509 00:32:31,710 --> 00:32:35,180 That connected to the, what am I going 510 00:32:35,180 --> 00:32:40,440 to call that, the box function. 511 00:32:40,440 --> 00:32:42,950 The box function, the square wave. 512 00:32:42,950 --> 00:32:45,380 Well, box is good. 513 00:32:45,380 --> 00:32:49,040 Now, what if I shift the function? 514 00:32:49,040 --> 00:32:50,670 If I shift a function, what happens 515 00:32:50,670 --> 00:32:53,120 to its Fourier transform? 516 00:32:53,120 --> 00:32:55,300 Anybody remember? 517 00:32:55,300 --> 00:33:02,090 You multiply it by, so if I shift the function, 518 00:33:02,090 --> 00:33:04,410 I just multiply this box function, 519 00:33:04,410 --> 00:33:07,790 this is a box function in the k, times something, 520 00:33:07,790 --> 00:33:19,140 e to the i shift, and the shift was one, right? 521 00:33:19,140 --> 00:33:24,210 Is it just e to the ik, d being the shift distance. 522 00:33:24,210 --> 00:33:26,340 Oh, the shift distance was n. 523 00:33:26,340 --> 00:33:27,970 Right. 524 00:33:27,970 --> 00:33:30,430 And possibly minus, who knows. 525 00:33:30,430 --> 00:33:33,500 OK, but what's the point here? 526 00:33:33,500 --> 00:33:38,100 The point is that it's still zero outside the box. 527 00:33:38,100 --> 00:33:43,700 Inside the box, instead of being one, it's this complex guy. 528 00:33:43,700 --> 00:33:45,660 But no change. 529 00:33:45,660 --> 00:33:47,810 It's still zero outside the box. 530 00:33:47,810 --> 00:33:49,370 It's still band-limited. 531 00:33:49,370 --> 00:33:54,010 So this is the transform of this guy. 532 00:33:54,010 --> 00:33:56,820 And then the transform of this combination 533 00:33:56,820 --> 00:34:00,960 would be still in the box, multiplied 534 00:34:00,960 --> 00:34:05,130 by some messy expression. 535 00:34:05,130 --> 00:34:07,000 So what was I doing there? 536 00:34:07,000 --> 00:34:10,000 I was just checking that, sure enough, this guy 537 00:34:10,000 --> 00:34:11,420 is band-limited. 538 00:34:11,420 --> 00:34:20,770 And it's band-limited, it gives us the right f(n)'s, of course. 539 00:34:20,770 --> 00:34:29,180 Everybody sees that at x=n, let's just have a look now. 540 00:34:29,180 --> 00:34:30,900 We've got this great formula. 541 00:34:30,900 --> 00:34:33,330 Plug in x=n. 542 00:34:33,330 --> 00:34:37,660 What happens when you plug in at one of the samples, 543 00:34:37,660 --> 00:34:43,710 you look to see what this A to D converter produced at time n, 544 00:34:43,710 --> 00:34:46,210 and let's just see. 545 00:34:46,210 --> 00:34:49,200 So at x=n, the left side is f(n). 546 00:34:49,200 --> 00:34:52,130 Why is the right side f of that n? 547 00:34:52,130 --> 00:34:53,440 That particular n? 548 00:34:53,440 --> 00:34:59,070 Maybe I should give a specific letter to that n. 549 00:34:59,070 --> 00:35:04,840 So at that particular sample, this left side is f(N), 550 00:35:04,840 --> 00:35:06,940 and I hope that the right side gives me 551 00:35:06,940 --> 00:35:10,670 f at that capital N. That particular one. 552 00:35:10,670 --> 00:35:12,400 Why does it? 553 00:35:12,400 --> 00:35:14,510 You're all seeing that. 554 00:35:14,510 --> 00:35:19,170 At x equal capital N, these guys are all zero, 555 00:35:19,170 --> 00:35:21,920 except for one of them. 556 00:35:21,920 --> 00:35:27,700 Except for the one when little n and capital N are the same. 557 00:35:27,700 --> 00:35:29,880 Then that becomes the one. 558 00:35:29,880 --> 00:35:34,440 And I'm getting f at capital N. So it will give me that, 559 00:35:34,440 --> 00:35:39,030 for the n=1, the one term, yeah. 560 00:35:39,030 --> 00:35:41,900 I don't know if it was necessary to say that. 561 00:35:41,900 --> 00:35:48,390 You've got the idea of the sampling formula. 562 00:35:48,390 --> 00:35:51,770 I could say more about the sampling, 563 00:35:51,770 --> 00:36:03,570 just to realize that the technology, communication 564 00:36:03,570 --> 00:36:07,340 theory is always trying to, like, 565 00:36:07,340 --> 00:36:10,480 to have a greater bandwidth. 566 00:36:10,480 --> 00:36:12,510 You always want a greater bandwidth. 567 00:36:12,510 --> 00:36:17,220 But if the bandwidth, which is this, is increased, 568 00:36:17,220 --> 00:36:19,240 well by the way, what does happen? 569 00:36:19,240 --> 00:36:30,190 Suppose it's band-limited by pi, oh, by pi/T. 570 00:36:30,190 --> 00:36:35,610 Let's just, I normalize things to choose samples 571 00:36:35,610 --> 00:36:37,400 every integer. 572 00:36:37,400 --> 00:36:39,380 Zero, one, two, three. 573 00:36:39,380 --> 00:36:43,290 And that turned out that the Nyquist frequency was pi. 574 00:36:43,290 --> 00:36:50,260 Now, what sampling rate would correspond 575 00:36:50,260 --> 00:36:54,980 to this band, which could be-- Well, let 576 00:36:54,980 --> 00:36:56,860 me just say what it is. 577 00:36:56,860 --> 00:36:59,730 That would be the Nyquist frequency 578 00:36:59,730 --> 00:37:10,710 for sampling every T. Instead of a sampling interval of one, 579 00:37:10,710 --> 00:37:17,920 if I sample every T, 2T, 3T, -T, my sampling rate is T, 580 00:37:17,920 --> 00:37:21,870 so if T is small, I'm sampling much more. 581 00:37:21,870 --> 00:37:24,910 Suppose T is 1/4. 582 00:37:24,910 --> 00:37:29,430 If T is 1/4, then I'm doing four samplings. 583 00:37:29,430 --> 00:37:33,040 I'm taking four samples, I'm paying more for this A 584 00:37:33,040 --> 00:37:35,720 to D converter, because it's taking four samples where 585 00:37:35,720 --> 00:37:39,780 previously it took one. 586 00:37:39,780 --> 00:37:41,420 How do I get paid back? 587 00:37:41,420 --> 00:37:43,240 What's the reward? 588 00:37:43,240 --> 00:37:51,850 The reward is if T is 1/4, then the Nyquist limit is 4pi. 589 00:37:51,850 --> 00:37:55,390 I can get a broader band of signals 590 00:37:55,390 --> 00:37:58,390 by sampling them more often. 591 00:37:58,390 --> 00:38:00,490 Let me just say that again, because that's 592 00:38:00,490 --> 00:38:03,070 the fundamental idea behind it. 593 00:38:03,070 --> 00:38:08,350 If I sample more often, say, so fast sampling would be small-- 594 00:38:08,350 --> 00:38:20,850 Fast samples would be small t and then a higher Nyquist. 595 00:38:20,850 --> 00:38:23,140 A higher band limit. 596 00:38:23,140 --> 00:38:24,390 More functions allowed. 597 00:38:24,390 --> 00:38:29,270 If I sample more often I'm able to catch on to more functions. 598 00:38:29,270 --> 00:38:33,190 If I sample-- And that's what, I mean, 599 00:38:33,190 --> 00:38:37,460 now, communications want wide bands. 600 00:38:37,460 --> 00:38:41,280 And this is where they get limited. 601 00:38:41,280 --> 00:38:46,060 I mean, this is, you could say, the fundamental, 602 00:38:46,060 --> 00:38:48,830 I don't know whether to say physical limit, sort 603 00:38:48,830 --> 00:38:53,610 of maybe Fourier limit on sampling theory. 604 00:38:53,610 --> 00:38:56,810 Is exactly this Nyquist frequency. 605 00:38:56,810 --> 00:39:00,160 OK, questions or discussion about that. 606 00:39:00,160 --> 00:39:04,110 OK. 607 00:39:04,110 --> 00:39:07,350 So that's an example that allowed 608 00:39:07,350 --> 00:39:13,110 us to do a lot of things. 609 00:39:13,110 --> 00:39:16,380 I did want to ask for your help doing these evaluations. 610 00:39:16,380 --> 00:39:19,840 Let me say what I'm going to do this afternoon. 611 00:39:19,840 --> 00:39:24,260 I'm going to answer all the questions I can, 612 00:39:24,260 --> 00:39:27,180 and I planned, when there is a pause, 613 00:39:27,180 --> 00:39:34,990 and nobody else asks, I plan to compute the Fourier integral 614 00:39:34,990 --> 00:39:38,580 and Fourier series, say, Fourier series, 615 00:39:38,580 --> 00:39:48,510 for a function that has, it's going to be like the one today 616 00:39:48,510 --> 00:39:53,820 except this is going to have a height of 1/h, 617 00:39:53,820 --> 00:39:56,490 and a width of h. 618 00:39:56,490 --> 00:40:03,630 So that's, in case you're not able to be here this afternoon, 619 00:40:03,630 --> 00:40:06,520 I thought I'd just say in advance what calculations 620 00:40:06,520 --> 00:40:07,740 I thought I would do. 621 00:40:07,740 --> 00:40:11,570 So there's a particular function f(x), 622 00:40:11,570 --> 00:40:13,880 it happens to be an even function. 623 00:40:13,880 --> 00:40:16,520 We'll compute its Fourier coefficients, in 624 00:40:16,520 --> 00:40:20,870 and we'll let h go to zero. 625 00:40:20,870 --> 00:40:21,930 To see what happens. 626 00:40:21,930 --> 00:40:23,540 It's just a good example that you 627 00:40:23,540 --> 00:40:27,180 may have seen on older exams. 628 00:40:27,180 --> 00:40:32,740 OK, well can I just say a personal word 629 00:40:32,740 --> 00:40:36,210 before I pass out-- So evaluations, 630 00:40:36,210 --> 00:40:39,640 if you're willing to help, and just leave them on the table, 631 00:40:39,640 --> 00:40:41,930 would be much appreciated. 632 00:40:41,930 --> 00:40:45,630 I'll stretch out the homeworks. 633 00:40:45,630 --> 00:40:48,570 I just want to say I've enjoyed teaching you guys. 634 00:40:48,570 --> 00:40:49,960 Very much. 635 00:40:49,960 --> 00:40:52,730 Thank you all, and-- Thanks. 636 00:40:52,730 --> 00:40:56,990