1 00:00:00,000 --> 00:00:02,360 The following content is provided under a Creative 2 00:00:02,360 --> 00:00:03,640 Commons license. 3 00:00:03,640 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue to 4 00:00:06,540 --> 00:00:09,515 offer high quality educational resources for free. 5 00:00:09,515 --> 00:00:12,810 To make a donation or to view additional materials from 6 00:00:12,810 --> 00:00:16,870 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,870 --> 00:00:18,120 ocw.mit.edu. 8 00:00:22,980 --> 00:00:27,580 PROFESSOR: --And go on with lectures 8 to 10. 9 00:00:30,530 --> 00:00:33,090 First I want to briefly review what we said 10 00:00:33,090 --> 00:00:36,320 about measurable functions. 11 00:00:36,320 --> 00:00:40,580 Again, I encourage you if you hate this material and you 12 00:00:40,580 --> 00:00:43,700 think it's only for mathematicians, 13 00:00:43,700 --> 00:00:45,950 please let me know. 14 00:00:45,950 --> 00:00:48,190 I don't know whether it's appropriate to cover it in 15 00:00:48,190 --> 00:00:50,530 this class either. 16 00:00:50,530 --> 00:00:54,810 I think it probably is because you need to know a little more 17 00:00:54,810 --> 00:01:00,340 mathematics when you deal with these Fourier transforms and 18 00:01:00,340 --> 00:01:01,900 Fourier series. 19 00:01:04,490 --> 00:01:08,510 When you're dealing with communication theory, then you 20 00:01:08,510 --> 00:01:10,330 need to know for signal processing 21 00:01:10,330 --> 00:01:12,100 and things like that. 22 00:01:12,100 --> 00:01:15,750 When you get into learning a little more, my sense at this 23 00:01:15,750 --> 00:01:21,070 point is it's much easier to learn a good deal more than it 24 00:01:21,070 --> 00:01:23,660 is to learn just a little bit more, because a little bit 25 00:01:23,660 --> 00:01:27,280 more you're always faced with all of these questions of what 26 00:01:27,280 --> 00:01:29,610 does this really mean. 27 00:01:29,610 --> 00:01:33,250 It turns out that if you put just a little bit of measure 28 00:01:33,250 --> 00:01:38,000 theory into your thinking, it makes all these problems very 29 00:01:38,000 --> 00:01:39,710 much simpler. 30 00:01:39,710 --> 00:01:42,540 So you remember we talked about what a measurable set 31 00:01:42,540 --> 00:01:45,400 was last time. 32 00:01:45,400 --> 00:01:50,380 What a measurable set is, is a set which you can essentially 33 00:01:50,380 --> 00:01:55,150 break up into a countable union of intervals, and 34 00:01:55,150 --> 00:02:03,030 something which is bounded by a countble set of intervals 35 00:02:03,030 --> 00:02:05,340 which have arbitrarily small measure. 36 00:02:05,340 --> 00:02:09,310 So you can take all of these sets of zero measure -- 37 00:02:09,310 --> 00:02:12,970 countable sets, cantor sets, all of these things, and you 38 00:02:12,970 --> 00:02:16,140 can just throw all of those out from consideration. 39 00:02:16,140 --> 00:02:19,960 That's what makes measure theory useful and interesting 40 00:02:19,960 --> 00:02:22,530 and what simplifies things. 41 00:02:22,530 --> 00:02:26,750 So we then said that a function is measurable if each 42 00:02:26,750 --> 00:02:31,930 of these sets here -- if the set of times for which a is 43 00:02:31,930 --> 00:02:34,860 less than or equal to u of t is less than or equal to b -- 44 00:02:34,860 --> 00:02:37,500 these things are these level sets here. 45 00:02:37,500 --> 00:02:42,920 It's a set of values, t, along this axis in which the 46 00:02:42,920 --> 00:02:47,470 function is nailed between any two of these points here. 47 00:02:52,230 --> 00:02:55,370 For a function to be measurable, all of these sets 48 00:02:55,370 --> 00:02:57,910 in here have to be measurable. 49 00:02:57,910 --> 00:03:01,390 We haven't given you any examples of sets which are not 50 00:03:01,390 --> 00:03:04,990 measurable, and therefore, we haven't given you any examples 51 00:03:04,990 --> 00:03:07,720 of functions which are not measurable. 52 00:03:07,720 --> 00:03:15,360 It's very, very difficult to construct such functions, and 53 00:03:15,360 --> 00:03:18,480 since it's so difficult to construct them, it's easier to 54 00:03:18,480 --> 00:03:23,510 just say that all of the functions we can think of are 55 00:03:23,510 --> 00:03:24,540 measurable. 56 00:03:24,540 --> 00:03:27,550 That includes an awful lot more functions than you ever 57 00:03:27,550 --> 00:03:30,640 deal with in engineering. 58 00:03:30,640 --> 00:03:37,410 If you ever make a serious mistake in your engineering 59 00:03:37,410 --> 00:03:41,670 work because of thinking that a function is measurable when 60 00:03:41,670 --> 00:03:45,750 it's not measurable, I will give you $1,000. 61 00:03:45,750 --> 00:03:48,610 I make that promise that you because I don't think it's 62 00:03:48,610 --> 00:03:51,290 ever happened in history, I don't think it ever will 63 00:03:51,290 --> 00:03:58,440 happen in history, and that's just how sure I am of it. 64 00:03:58,440 --> 00:04:01,580 Unless you try to deliberately make such a mistake in order 65 00:04:01,580 --> 00:04:03,080 to collect $1,000. 66 00:04:03,080 --> 00:04:08,090 Of course, that's not fair. 67 00:04:08,090 --> 00:04:13,900 Then we said the way we define this approximation to the 68 00:04:13,900 --> 00:04:17,650 integral, namely, we always defined it from underneath, 69 00:04:17,650 --> 00:04:21,810 and therefore, if we started to take these intervals here 70 00:04:21,810 --> 00:04:25,960 with epsilon and split them in half, the thing that happens 71 00:04:25,960 --> 00:04:29,210 is you always get extra components put in. 72 00:04:29,210 --> 00:04:33,560 So that as you make the scaling finer and finer, what 73 00:04:33,560 --> 00:04:36,970 happens is the approximation into the integral gets larger 74 00:04:36,970 --> 00:04:38,080 and larger. 75 00:04:38,080 --> 00:04:40,970 Because of that, if you're dealing with a non-negative 76 00:04:40,970 --> 00:04:45,800 function, the only two things that can happen, as you go to 77 00:04:45,800 --> 00:04:50,350 the limit of finer and finer scaling, is one, you come to a 78 00:04:50,350 --> 00:04:53,830 finite limit, and two, you come to an infinite limit. 79 00:04:53,830 --> 00:04:55,900 Nothing else can happen. 80 00:04:55,900 --> 00:05:00,440 You see that's remarkably simple as mathematics go. 81 00:05:00,440 --> 00:05:04,340 This is dealing with all of these functions you can ever 82 00:05:04,340 --> 00:05:06,980 think of, and those are the only two things that can 83 00:05:06,980 --> 00:05:08,230 happen here. 84 00:05:12,660 --> 00:05:17,370 So then we went on to say that a function as L1, if it's 85 00:05:17,370 --> 00:05:21,820 measurable, and if the integral of its magnitude is 86 00:05:21,820 --> 00:05:23,830 less than infinity, and so far we're still 87 00:05:23,830 --> 00:05:27,710 talking about real functions. 88 00:05:27,710 --> 00:05:38,520 If u of t is L1, then you can take the integral of u of t 89 00:05:38,520 --> 00:05:41,210 and you can split it up into two things. 90 00:05:41,210 --> 00:05:45,790 You can split it up into the set of times over which u of t 91 00:05:45,790 --> 00:05:48,630 is positive, and the set of times over 92 00:05:48,630 --> 00:05:51,270 which u of t is negative. 93 00:05:51,270 --> 00:05:54,880 The set of times over which u of t is equal to zero doesn't 94 00:05:54,880 --> 00:05:56,900 contribute to the integral at all so you can 95 00:05:56,900 --> 00:05:59,410 forget about that. 96 00:05:59,410 --> 00:06:03,040 This is well-defined as the function is measurable. 97 00:06:03,040 --> 00:06:06,540 What's now happening is that if the integral of this 98 00:06:06,540 --> 00:06:11,060 magnitude is less than infinity, then both this has 99 00:06:11,060 --> 00:06:14,890 to be less than infinity, and this has to be less than 100 00:06:14,890 --> 00:06:18,860 infinity, which says that so long as you're dealing with L1 101 00:06:18,860 --> 00:06:24,540 functions, you never have to worry about this nasty 102 00:06:24,540 --> 00:06:28,320 situation where the negative part of the function is 103 00:06:28,320 --> 00:06:32,060 infinite, the positive part of the function is infinite, and 104 00:06:32,060 --> 00:06:35,610 therefore, the difference, plus infinity, minus infinity 105 00:06:35,610 --> 00:06:38,090 doesn't make any sense at all. 106 00:06:38,090 --> 00:06:41,360 And instead of having an integral which is infinite or 107 00:06:41,360 --> 00:06:45,030 finite, you have a function which might just be undefined 108 00:06:45,030 --> 00:06:46,040 completely. 109 00:06:46,040 --> 00:06:50,090 Well this says that if the function is L1, you can't ever 110 00:06:50,090 --> 00:06:54,990 have any problem with integrals of this sort thing 111 00:06:54,990 --> 00:06:55,710 being undefined. 112 00:06:55,710 --> 00:06:59,800 So, and in fact, u of t is always integral and has a 113 00:06:59,800 --> 00:07:02,560 finite value in this case. 114 00:07:02,560 --> 00:07:06,340 Now, we say that a complex function is measurable if both 115 00:07:06,340 --> 00:07:09,210 the real part is measurable and the imaginary part is 116 00:07:09,210 --> 00:07:10,350 measurable. 117 00:07:10,350 --> 00:07:13,470 Therefore, you don't have to worry about complex functions 118 00:07:13,470 --> 00:07:18,000 as really being any more than twice as complicated as real 119 00:07:18,000 --> 00:07:20,530 functions, and conceptually they aren't any more 120 00:07:20,530 --> 00:07:24,340 complicated at all because you just treat them as two 121 00:07:24,340 --> 00:07:27,100 separate functions, and everything you know about real 122 00:07:27,100 --> 00:07:32,170 functions you now know about complex functions. 123 00:07:32,170 --> 00:07:34,570 Now, as far as Fourier theory is concerned -- 124 00:07:34,570 --> 00:07:39,490 Fourier series, Fourier integrals, discrete time, 125 00:07:39,490 --> 00:07:45,200 Fourier transforms -- all of these things, if u of t is an 126 00:07:45,200 --> 00:07:51,880 L1 function, then u of t times e to the 2 pi ft has to be L1, 127 00:07:51,880 --> 00:07:55,350 and this has to be true for all possible values of f. 128 00:07:55,350 --> 00:07:56,860 Why is that? 129 00:07:56,860 --> 00:08:00,330 Well, it's just that the absolute value of u of t for 130 00:08:00,330 --> 00:08:04,650 any given t, is exactly the same as the absolute value of 131 00:08:04,650 --> 00:08:08,510 u of t times e to the 2 pi i ft. 132 00:08:08,510 --> 00:08:13,170 Namely, this is a quantity whose magnitude is always 1, 133 00:08:13,170 --> 00:08:18,870 and therefore, this quantity, this magnitude is equal to 134 00:08:18,870 --> 00:08:20,850 this magnitude. 135 00:08:20,850 --> 00:08:24,150 Therefore, when you integrate this magnitude, you get the 136 00:08:24,150 --> 00:08:28,460 same thing as when you integrate this magnitude. 137 00:08:28,460 --> 00:08:30,830 So, there's no problem here. 138 00:08:33,370 --> 00:08:37,640 So what that says, using this idea, too, of positive and 139 00:08:37,640 --> 00:08:41,420 negative parts, it says that the integral of u of t times e 140 00:08:41,420 --> 00:08:47,710 to the 2 pi i ft dt has to exist for all real f. 141 00:08:47,710 --> 00:08:51,300 That covers Fourier series and Fourier integral. 142 00:08:51,300 --> 00:08:54,980 We haven't talked about the Fourier integral yet, but it 143 00:08:54,980 --> 00:09:00,510 says that just by defining u of t to be L1, you avoid all 144 00:09:00,510 --> 00:09:04,570 of these problems of when these Fourier integrals exist 145 00:09:04,570 --> 00:09:05,700 and when they don't exist. 146 00:09:05,700 --> 00:09:06,950 They always exist. 147 00:09:11,580 --> 00:09:14,100 So let's talk about the Fourier transform then since 148 00:09:14,100 --> 00:09:18,140 we're backing into this slowly anyway. 149 00:09:18,140 --> 00:09:21,540 What you've learned in probably two or three classes 150 00:09:21,540 --> 00:09:25,490 by now is that the Fourier transform is 151 00:09:25,490 --> 00:09:27,630 defined in this way. 152 00:09:27,630 --> 00:09:31,740 Namely, the transform, a function of frequency, u hat 153 00:09:31,740 --> 00:09:37,120 of f, we use hats here instead of capital letters because we 154 00:09:37,120 --> 00:09:41,160 like to use capital letters for random variables. 155 00:09:41,160 --> 00:09:44,500 So that this transform here is the integral from minus 156 00:09:44,500 --> 00:09:49,140 infinity to infinity of the original function, we start 157 00:09:49,140 --> 00:09:53,950 with u of t times e to the minus 2 pi ift dt. 158 00:09:53,950 --> 00:09:55,950 Now, what's the nice part about this? 159 00:09:55,950 --> 00:10:02,600 The nice part about this is that if u of t is L1, then in 160 00:10:02,600 --> 00:10:08,590 fact this Fourier transform has to exist everywhere. 161 00:10:08,590 --> 00:10:12,710 Namely, what you've learned before in all of these classes 162 00:10:12,710 --> 00:10:17,600 says essentially that if these functions are well-behaved 163 00:10:17,600 --> 00:10:20,140 then these transforms exist. 164 00:10:20,140 --> 00:10:23,750 What you've learned about well-behaved is completely 165 00:10:23,750 --> 00:10:27,060 circular, namely, a function is well-behaved if the 166 00:10:27,060 --> 00:10:31,040 transform exists, and the transform exists if the 167 00:10:31,040 --> 00:10:32,690 function is well-behaved. 168 00:10:32,690 --> 00:10:38,670 You have no idea of what kinds of functions the Fourier 169 00:10:38,670 --> 00:10:40,740 transform exists and what kinds of 170 00:10:40,740 --> 00:10:42,270 functions it doesn't exist. 171 00:10:42,270 --> 00:10:45,650 There's another thing buried in here, in this transform 172 00:10:45,650 --> 00:10:46,980 relationship. 173 00:10:46,980 --> 00:10:50,660 Namely, the first thing is we're trying to define the 174 00:10:50,660 --> 00:10:54,390 transform function of frequency in terms of the 175 00:10:54,390 --> 00:10:56,020 function of time. 176 00:10:56,020 --> 00:10:57,860 Well and good. 177 00:10:57,860 --> 00:11:01,810 Then we try to define a function of time in terms of a 178 00:11:01,810 --> 00:11:04,700 function of frequency. 179 00:11:04,700 --> 00:11:10,130 What's hidden here is if you start with u of t, you then 180 00:11:10,130 --> 00:11:13,770 get a function of frequency, and this sort of implicitly 181 00:11:13,770 --> 00:11:16,040 says that you get back again the same 182 00:11:16,040 --> 00:11:18,680 thing you started with. 183 00:11:18,680 --> 00:11:21,730 That, in fact, is the most ticklish part of all of this. 184 00:11:21,730 --> 00:11:27,270 So that if we start with an L1 function u of t, we wind up 185 00:11:27,270 --> 00:11:30,590 with a Fourier transform, u hat of f. 186 00:11:30,590 --> 00:11:32,670 Then u hat of f goes in here. 187 00:11:32,670 --> 00:11:37,620 We hope we might get back the sum transform here. 188 00:11:37,620 --> 00:11:40,420 The nasty thing, and one of the reasons we're going to get 189 00:11:40,420 --> 00:11:47,080 away from L1 in just a minute, is that if a function is L1, 190 00:11:47,080 --> 00:11:51,200 it's Fourier transform is not necessarily L1. 191 00:11:51,200 --> 00:11:54,720 What that means is that you have to learn all of this 192 00:11:54,720 --> 00:11:58,130 stuff about L1 functions, and then as soon as you take the 193 00:11:58,130 --> 00:12:03,810 Fourier transform, bingo, it's all gone up in a lot of smoke, 194 00:12:03,810 --> 00:12:07,730 and you have to start all over again saying something about 195 00:12:07,730 --> 00:12:10,820 what properties the transform might have. 196 00:12:10,820 --> 00:12:14,790 But anyway, it's nice to start with because when u of t is 197 00:12:14,790 --> 00:12:18,745 L1, we know that this function actually exists. 198 00:12:18,745 --> 00:12:21,120 It actually exists as a complex number. 199 00:12:21,120 --> 00:12:23,750 It exists as a complex number for every 200 00:12:23,750 --> 00:12:26,160 possible real f here. 201 00:12:26,160 --> 00:12:29,840 Namely, there aren't any if, and's, but's or maybe's here, 202 00:12:29,840 --> 00:12:32,290 there's nothing like L2 convergence that we were 203 00:12:32,290 --> 00:12:35,260 talking about a little bit before. 204 00:12:35,260 --> 00:12:37,160 This just exists, period. 205 00:12:40,340 --> 00:12:45,070 There's something more here, that if this is L1, this not 206 00:12:45,070 --> 00:12:48,945 only exists but it's also a continuous function, and those 207 00:12:48,945 --> 00:12:51,030 don't prove that. 208 00:12:51,030 --> 00:12:54,860 If you've taken a course in analysis and you know what a 209 00:12:54,860 --> 00:12:58,850 complex function is and you're quite patient, you can sit 210 00:12:58,850 --> 00:13:02,090 down and actually show this yourselves, but 211 00:13:02,090 --> 00:13:03,340 it's a bit of a pain. 212 00:13:05,750 --> 00:13:07,890 So for a well-behaved function, the first integral 213 00:13:07,890 --> 00:13:11,780 exists for all f, the second exists for all t, and results 214 00:13:11,780 --> 00:13:13,670 in the original u of t. 215 00:13:13,670 --> 00:13:17,570 But then more specifically, if u of t is L1, the first 216 00:13:17,570 --> 00:13:21,740 integral exists for all f and it's a continuous function. 217 00:13:21,740 --> 00:13:24,950 It's also a bounded function, as we'll see in a little bit. 218 00:13:24,950 --> 00:13:28,510 If u hat of f is L1, the second integral exists for all 219 00:13:28,510 --> 00:13:31,410 t, and u of t is continuous. 220 00:13:31,410 --> 00:13:35,900 Therefore, if you assume at the output at the onset of 221 00:13:35,900 --> 00:13:40,180 things that both this is L1 and this is L1, then 222 00:13:40,180 --> 00:13:43,400 everything is also continuous and you have a very nice 223 00:13:43,400 --> 00:13:46,840 theory, which doesn't apply to too many of the things that 224 00:13:46,840 --> 00:13:48,730 you're interested in. 225 00:13:48,730 --> 00:13:50,720 And I'll explain why in a little bit. 226 00:13:54,230 --> 00:13:58,510 Anyway, for these well-behaved functions, we have all of 227 00:13:58,510 --> 00:14:02,640 these relationships that I'm sure you've learned in 228 00:14:02,640 --> 00:14:04,820 whatever linear systems course you've taken. 229 00:14:07,400 --> 00:14:09,980 Since you should know all of these things, I just want to 230 00:14:09,980 --> 00:14:14,640 briefly talk about them I mean this linearity idea is 231 00:14:14,640 --> 00:14:16,380 something you would just use without 232 00:14:16,380 --> 00:14:19,210 thinking about it, I think. 233 00:14:19,210 --> 00:14:21,730 In other words, if you'd never learned this and you were 234 00:14:21,730 --> 00:14:25,830 trying to work out a problem you would just use it anyway, 235 00:14:25,830 --> 00:14:30,380 because anything respectable has to have -- well, anything 236 00:14:30,380 --> 00:14:32,820 respectable, again, means anything 237 00:14:32,820 --> 00:14:34,070 which has this property. 238 00:14:36,620 --> 00:14:39,590 This conjugate property you can derive that easily from 239 00:14:39,590 --> 00:14:43,050 the Fourier transform relationships also, if you 240 00:14:43,050 --> 00:14:45,010 have a well-behaved function. 241 00:14:45,010 --> 00:14:49,760 This quantity here, this duality, is particularly 242 00:14:49,760 --> 00:14:57,890 interesting because it isn't really duality, it's something 243 00:14:57,890 --> 00:15:00,820 called hermitian duality. 244 00:15:00,820 --> 00:15:04,650 You start out with this formula here to go to there 245 00:15:04,650 --> 00:15:08,650 and you use almost the same formula to get back again. 246 00:15:08,650 --> 00:15:13,710 The only difference is instead of a minus 2 pi ift, you have 247 00:15:13,710 --> 00:15:15,960 a plus 2 pi ift. 248 00:15:15,960 --> 00:15:19,400 In other words, this is the conjugate of this, which is 249 00:15:19,400 --> 00:15:22,480 why this is called hermitian duality. 250 00:15:22,480 --> 00:15:27,100 But aside from that, everything you learn about the 251 00:15:27,100 --> 00:15:31,480 Fourier transform, you also automatically know about the 252 00:15:31,480 --> 00:15:35,740 inverse Fourier transform for these well-behaved functions. 253 00:15:38,360 --> 00:15:40,980 Otherwise you don't know whether the other one exists 254 00:15:40,980 --> 00:15:45,170 or not, and we'll certainly get into that. 255 00:15:45,170 --> 00:15:49,290 So, the duality is expressed this way, the Fourier 256 00:15:49,290 --> 00:15:54,520 transform of, bleah. 257 00:15:54,520 --> 00:15:58,900 If you take a function, u of t, and regard that as a 258 00:15:58,900 --> 00:16:04,410 Fourier transform, then -- 259 00:16:04,410 --> 00:16:07,800 I always have trouble saying this. 260 00:16:07,800 --> 00:16:11,660 If you take a function, u of t, and then you regard that as 261 00:16:11,660 --> 00:16:13,480 a function of frequency -- 262 00:16:13,480 --> 00:16:16,840 OK, that's this -- and then you regard it as a function of 263 00:16:16,840 --> 00:16:21,170 minus frequency, namely, you substitute minus f for t in 264 00:16:21,170 --> 00:16:23,330 whatever time function you start with. 265 00:16:23,330 --> 00:16:27,170 You start with a time function u of t, you substitute minus f 266 00:16:27,170 --> 00:16:31,300 for t, which gives you a function of frequency. 267 00:16:31,300 --> 00:16:35,780 The inverse Fourier transform of that is what you get by 268 00:16:35,780 --> 00:16:40,660 taking the Fourier transform of u of t, and then 269 00:16:40,660 --> 00:16:44,360 substituting t for f in it. 270 00:16:44,360 --> 00:16:47,990 It's much harder to say it than to do. 271 00:16:47,990 --> 00:16:50,270 This time shift, you've all seen that. 272 00:16:50,270 --> 00:16:53,880 If you shift a function in time, the only thing that 273 00:16:53,880 --> 00:16:58,150 happens is you get this rotating term in it. 274 00:16:58,150 --> 00:17:02,410 Same thing for a frequency shift. 275 00:17:02,410 --> 00:17:05,200 You want to have an interesting exercise, take 276 00:17:05,200 --> 00:17:09,320 time shift plus duality and derive the scaling and 277 00:17:09,320 --> 00:17:13,380 frequency from it -- it has to work, and of course, it does. 278 00:17:13,380 --> 00:17:14,640 Scaling -- 279 00:17:14,640 --> 00:17:17,360 there's this relationship here. 280 00:17:17,360 --> 00:17:20,440 This one is always a funny one. 281 00:17:20,440 --> 00:17:25,480 It's a little strange because when you scale here, it's not 282 00:17:25,480 --> 00:17:29,250 too surprising that when you take a function and you squash 283 00:17:29,250 --> 00:17:32,120 it down that the Fourier transform 284 00:17:32,120 --> 00:17:35,160 gets squashed upwards. 285 00:17:35,160 --> 00:17:38,840 Because in a sense what you're doing when you squash a 286 00:17:38,840 --> 00:17:42,040 function down you're making everything happen faster than 287 00:17:42,040 --> 00:17:45,080 it did before, which means that all the frequencies in it 288 00:17:45,080 --> 00:17:47,420 are going to get higher than they are before. 289 00:17:47,420 --> 00:17:50,780 But also when you squash it down, the amount of energy in 290 00:17:50,780 --> 00:17:53,570 the function is going to go down. 291 00:17:53,570 --> 00:17:56,600 One of the most important properties that we're going to 292 00:17:56,600 --> 00:18:00,470 find and what you ought to know already is that when you 293 00:18:00,470 --> 00:18:05,790 take the energy in a function, you get the same answer as you 294 00:18:05,790 --> 00:18:09,190 get when you take the energy in the Fourier transform. 295 00:18:09,190 --> 00:18:13,940 Namely, you integrate u of f squared over frequency and you 296 00:18:13,940 --> 00:18:16,620 get the same as if you integrate u of t 297 00:18:16,620 --> 00:18:18,450 squared over time. 298 00:18:18,450 --> 00:18:20,710 That's an important check that you use on 299 00:18:20,710 --> 00:18:22,560 all sorts of things. 300 00:18:22,560 --> 00:18:25,490 The thing that happens now then when you're scaling is 301 00:18:25,490 --> 00:18:28,550 when you scale a function, u of t, you bring it down and 302 00:18:28,550 --> 00:18:30,950 you spread it out when you bring it down. 303 00:18:30,950 --> 00:18:36,070 The frequency function goes up so the energy in the time 304 00:18:36,070 --> 00:18:39,590 function goes down, the energy in the frequency function goes 305 00:18:39,590 --> 00:18:44,400 up, and you need something in order to keep the energy 306 00:18:44,400 --> 00:18:47,090 relationship working properly. 307 00:18:47,090 --> 00:18:48,480 This is what you need. 308 00:18:48,480 --> 00:18:50,260 Actually, if you derive this, the t 309 00:18:50,260 --> 00:18:53,360 just falls out naturally. 310 00:18:53,360 --> 00:19:00,210 So we get the same thing if we do scaling and frequency. 311 00:19:00,210 --> 00:19:03,010 I don't think I put that down but it's the same 312 00:19:03,010 --> 00:19:04,090 relationship. 313 00:19:04,090 --> 00:19:06,110 There's differentiation. 314 00:19:06,110 --> 00:19:10,230 Differentiation we won't talk about or use it a whole lot. 315 00:19:10,230 --> 00:19:14,120 All of these things turn out to be remarkably robust. 316 00:19:14,120 --> 00:19:18,820 When you're dealing with L1 functions or L2 functions and 317 00:19:18,820 --> 00:19:21,300 you scale them or you shift them or do any of those 318 00:19:21,300 --> 00:19:25,980 things, if they're L1, they're still L1 after you're through, 319 00:19:25,980 --> 00:19:28,910 if they're L2, they're L2 after you're through. 320 00:19:28,910 --> 00:19:31,760 If you differentiate a function, all 321 00:19:31,760 --> 00:19:33,000 those properties change. 322 00:19:33,000 --> 00:19:35,950 You can't be sure of anything anymore. 323 00:19:35,950 --> 00:19:39,380 There's convolution, which I'm sure you've derived many 324 00:19:39,380 --> 00:19:44,290 times, and which one of the exercises derives again. 325 00:19:44,290 --> 00:19:52,370 There's correlation, and what this sort of relationship says 326 00:19:52,370 --> 00:19:55,790 is taking products at the frequency domain is the same 327 00:19:55,790 --> 00:20:00,220 as going through this convolution relationship and 328 00:20:00,220 --> 00:20:01,470 the time domain. 329 00:20:01,470 --> 00:20:04,930 Of course, there's a dual relation to that, which you 330 00:20:04,930 --> 00:20:07,680 don't use very often but it still exists. 331 00:20:07,680 --> 00:20:16,280 Correlation is you actually get correlation by using one 332 00:20:16,280 --> 00:20:20,420 of the conjugate properties on the convolution and I'm sure 333 00:20:20,420 --> 00:20:22,060 you've all seen that. 334 00:20:25,640 --> 00:20:29,700 That's something you should all have been using and 335 00:20:29,700 --> 00:20:31,910 familiar with for a long time. 336 00:20:31,910 --> 00:20:38,100 Two special cases of the Fourier transform is that u of 337 00:20:38,100 --> 00:20:42,990 zero is what happens when you take the Fourier transform and 338 00:20:42,990 --> 00:20:45,440 you evaluate it at t equals zero. 339 00:20:45,440 --> 00:20:50,570 You get u of zero is just the integral of u hat of f. 340 00:20:50,570 --> 00:20:55,090 u hat of zero is just the integral of u of t. 341 00:20:55,090 --> 00:20:57,560 What do you use these for? 342 00:20:57,560 --> 00:21:01,350 Well the thing I use them for is this half the time when 343 00:21:01,350 --> 00:21:06,630 you're working out a problem it's obvious by inspection 344 00:21:06,630 --> 00:21:10,240 what this integral is or it's obvious by inspection what 345 00:21:10,240 --> 00:21:16,210 this integral is, and by doing that you can check whether you 346 00:21:16,210 --> 00:21:18,500 have all the constants right in the 347 00:21:18,500 --> 00:21:20,880 transform that you've taken. 348 00:21:20,880 --> 00:21:24,270 I don't know anybody who can take Fourier transforms 349 00:21:24,270 --> 00:21:27,480 without getting at least one constant wrong at least half 350 00:21:27,480 --> 00:21:29,440 the time they do it. 351 00:21:29,440 --> 00:21:32,210 I probably get one constant wrong about three-quarters of 352 00:21:32,210 --> 00:21:35,500 the time that I do it, and I'm sure I'll do it here in class 353 00:21:35,500 --> 00:21:36,330 a number of times. 354 00:21:36,330 --> 00:21:43,720 I hope you find it, but it's one of these things we all do. 355 00:21:43,720 --> 00:21:46,610 This is one of the best ways of finding out what you've 356 00:21:46,610 --> 00:21:49,700 done and going back and checking it. 357 00:21:49,700 --> 00:21:55,460 Parseval's Theorem, Parseval's Theorem is really just this 358 00:21:55,460 --> 00:21:59,730 convolution equation which we're applying 359 00:21:59,730 --> 00:22:02,520 at tau equals zero. 360 00:22:02,520 --> 00:22:05,490 You take the convolution, you apply it at tau equals zero 361 00:22:05,490 --> 00:22:06,350 and what do you get? 362 00:22:06,350 --> 00:22:12,620 You get the integral of u of t times some other conjugate -- 363 00:22:12,620 --> 00:22:15,610 the conjugate of the other function is equal to the 364 00:22:15,610 --> 00:22:21,560 integral of u hat of f times the complex 365 00:22:21,560 --> 00:22:24,190 conjugate of v of f. 366 00:22:26,840 --> 00:22:31,050 Much more important than this is what happens if v happens 367 00:22:31,050 --> 00:22:34,980 to be the same as u, and that gives you the energy equation 368 00:22:34,980 --> 00:22:36,640 here, which is what I was talking about. 369 00:22:36,640 --> 00:22:42,720 It says the integral in a function you can find it two 370 00:22:42,720 --> 00:22:45,580 ways, either by looking at it in time or by 371 00:22:45,580 --> 00:22:47,810 looking at it in frequency. 372 00:22:47,810 --> 00:22:50,450 I urge you to always think about doing this whenever 373 00:22:50,450 --> 00:22:55,000 you're working problems, because often the Fourier 374 00:22:55,000 --> 00:22:59,520 integral it's very easy to find the integral and the and 375 00:22:59,520 --> 00:23:02,300 the time function is very difficult. 376 00:23:02,300 --> 00:23:05,570 A good example of this is sinc functions 377 00:23:05,570 --> 00:23:08,220 and rectangular functions. 378 00:23:08,220 --> 00:23:12,800 Anybody can take a rectangular function, square it and 379 00:23:12,800 --> 00:23:14,470 integrate it. 380 00:23:14,470 --> 00:23:17,770 It takes a good deal of skill if you don't use this 381 00:23:17,770 --> 00:23:21,890 relationship to take a sinc function, to square it and to 382 00:23:21,890 --> 00:23:24,760 integrate it. 383 00:23:24,760 --> 00:23:28,630 You can do it if you're skillful at integration, you 384 00:23:28,630 --> 00:23:31,170 might regard it as a challenge, but after you get 385 00:23:31,170 --> 00:23:34,550 done you realize that you've really wasted a lot of time 386 00:23:34,550 --> 00:23:37,170 because this is the right way of doing it here. 387 00:23:43,980 --> 00:23:48,480 Now, as I mentioned before, it's starting to look like 388 00:23:48,480 --> 00:23:52,750 Fourier series and Fourier integrals are much nicer when 389 00:23:52,750 --> 00:23:58,450 you have L1 functions, and they are, but L1 functions are 390 00:23:58,450 --> 00:24:01,880 not terribly useful as far as most 391 00:24:01,880 --> 00:24:04,710 communication functions go. 392 00:24:04,710 --> 00:24:08,690 In other words, not enough functions are L1 to provide 393 00:24:08,690 --> 00:24:11,860 suitable models for the communication systems that we 394 00:24:11,860 --> 00:24:13,110 want to look at. 395 00:24:16,710 --> 00:24:19,820 In fact, most of the models that we're going to look at, 396 00:24:19,820 --> 00:24:23,810 the functions that we're dealing with are L2 functions. 397 00:24:23,810 --> 00:24:28,330 One of the reasons for this is this sinc function, sine x 398 00:24:28,330 --> 00:24:31,120 over x function is not L1. 399 00:24:34,500 --> 00:24:38,660 A sinc function just goes down as 1 over t, and since it goes 400 00:24:38,660 --> 00:24:42,350 down as 1 over t, you take the absolute value of it and you 401 00:24:42,350 --> 00:24:45,810 integrate 1 over t and what do you get when you integrate it 402 00:24:45,810 --> 00:24:48,450 from minus infinity to infinity. 403 00:24:48,450 --> 00:24:53,340 A function that's 1 over t, well, if you really want to go 404 00:24:53,340 --> 00:24:56,650 through the trouble of integrating it, you can 405 00:24:56,650 --> 00:25:00,650 integrate it over limits and you get limits where you have 406 00:25:00,650 --> 00:25:04,190 to evaluate the limits on a logarithmic function. 407 00:25:04,190 --> 00:25:05,840 When you get all done with that you 408 00:25:05,840 --> 00:25:06,950 get an infinite value. 409 00:25:06,950 --> 00:25:07,830 And you can see this. 410 00:25:07,830 --> 00:25:11,610 You could take 1 over t and you just look at it, as you go 411 00:25:11,610 --> 00:25:14,050 further and further out it just gets bigger and bigger 412 00:25:14,050 --> 00:25:15,570 without limit. 413 00:25:15,570 --> 00:25:18,220 So, sinc t is not L1. 414 00:25:18,220 --> 00:25:21,230 Sinc function is a function we'd like to use. 415 00:25:21,230 --> 00:25:26,000 Any function with a discontinuity can't be the 416 00:25:26,000 --> 00:25:30,390 Fourier transform of any L1 function. 417 00:25:30,390 --> 00:25:32,620 In other words, we said that if you take the Fourier 418 00:25:32,620 --> 00:25:36,920 transform of an L1 function, one of the nice things about 419 00:25:36,920 --> 00:25:40,830 it is you get a continuous function. 420 00:25:40,830 --> 00:25:43,520 One of the nasty things about it is you get 421 00:25:43,520 --> 00:25:45,900 a continuous function. 422 00:25:45,900 --> 00:25:50,080 Since you get a continuous function it says that any time 423 00:25:50,080 --> 00:25:52,970 you want to deal with transforms which are not 424 00:25:52,970 --> 00:26:02,050 continuous, you can't be talking about time functions 425 00:26:02,050 --> 00:26:03,420 which are L1. 426 00:26:03,420 --> 00:26:06,440 One of the frequency functions we want to look at a great 427 00:26:06,440 --> 00:26:10,030 deal is a frequency function corresponding to a 428 00:26:10,030 --> 00:26:12,020 band-limited function. 429 00:26:12,020 --> 00:26:15,370 When you take a band-limited function you just chop it off 430 00:26:15,370 --> 00:26:19,000 at some frequency, and usually when you chop it off, you chop 431 00:26:19,000 --> 00:26:22,000 it off and get a discontinuity. 432 00:26:22,000 --> 00:26:26,300 When you chop it off and get a discountinuity, bingo, the 433 00:26:26,300 --> 00:26:30,090 time function you're dealing with cannot be L1. 434 00:26:30,090 --> 00:26:34,340 It has to dribble away much too slowly as 435 00:26:34,340 --> 00:26:35,700 time goes to infinity. 436 00:26:35,700 --> 00:26:39,760 That's an extraordinarily important thing to remember. 437 00:26:39,760 --> 00:26:44,210 Any time you get a function which is discontinuous in the 438 00:26:44,210 --> 00:26:49,840 frequency domain, the function cannot go to zero any faster 439 00:26:49,840 --> 00:26:53,660 in a time domain than 1 over t and vice versa in the 440 00:26:53,660 --> 00:26:55,860 frequency domain. 441 00:26:55,860 --> 00:26:59,460 L1 functions sometimes have infinite energy. 442 00:26:59,460 --> 00:27:03,080 In other words, sinc t is not L1 -- 443 00:27:03,080 --> 00:27:07,120 well, that's not a good example because that's not L1, 444 00:27:07,120 --> 00:27:11,370 and it also has infinite energy, but you can just as 445 00:27:11,370 --> 00:27:14,720 easily find functions which drop off a little more slowly 446 00:27:14,720 --> 00:27:18,170 than sinc t, and which have infinite 447 00:27:18,170 --> 00:27:21,300 energy because they--. 448 00:27:21,300 --> 00:27:22,980 Excuse me. 449 00:27:22,980 --> 00:27:27,720 Sometimes you have functions which go off to infinity too 450 00:27:27,720 --> 00:27:32,760 fast as you approach time equals zero, things which are 451 00:27:32,760 --> 00:27:34,960 a little bit like impulses but not really. 452 00:27:34,960 --> 00:27:38,740 Impulses are awful and we'll talk about them in a minute, 453 00:27:38,740 --> 00:27:42,490 because they don't have finite energy as we said before. 454 00:27:42,490 --> 00:27:46,590 We have functions which just slowly go off to infinity and 455 00:27:46,590 --> 00:27:50,540 they are L2 but they aren't L1. 456 00:27:50,540 --> 00:27:52,800 Why do we care about those weird functions? 457 00:27:52,800 --> 00:27:56,020 We care about them, as I said before, because we would like 458 00:27:56,020 --> 00:27:59,680 to be able to make statements which are simple which we can 459 00:27:59,680 --> 00:28:02,260 believe in. 460 00:28:02,260 --> 00:28:05,800 In other words, you don't want to go through a course like 461 00:28:05,800 --> 00:28:10,550 this with a whole bunch of things that you have to leave. 462 00:28:10,550 --> 00:28:12,930 It's very nice to have some things that you really 463 00:28:12,930 --> 00:28:17,670 believe, and whether you believe them or not, it's nice 464 00:28:17,670 --> 00:28:19,490 to have theorems about them. 465 00:28:19,490 --> 00:28:22,210 Even if you don't believe the theorems, at least you have 466 00:28:22,210 --> 00:28:25,590 theorems so you can fool other people about 467 00:28:25,590 --> 00:28:27,920 it, if nothing else. 468 00:28:27,920 --> 00:28:31,350 Well, it turns out that L2 functions are really the right 469 00:28:31,350 --> 00:28:33,530 class to look at here. 470 00:28:45,440 --> 00:28:47,480 Oh, I think I left out one of the most 471 00:28:47,480 --> 00:28:48,540 important things here. 472 00:28:48,540 --> 00:28:51,360 Maybe I put it down here. 473 00:28:51,360 --> 00:28:53,130 No, probably not. 474 00:28:53,130 --> 00:28:57,480 One of the reasons we want to deal with L2 functions is if 475 00:28:57,480 --> 00:29:04,460 you're dealing with compression, for example, and 476 00:29:04,460 --> 00:29:09,220 you take a function, if the function has infinite energy 477 00:29:09,220 --> 00:29:15,300 in it, then one of the things that happens is that any time 478 00:29:15,300 --> 00:29:19,060 you expand it into any kind of orthonormal expansion or 479 00:29:19,060 --> 00:29:22,260 orthongonal expansion, which we'll talk about later, you 480 00:29:22,260 --> 00:29:26,000 have coefficients, which have infinite energy. 481 00:29:26,000 --> 00:29:30,860 In other words, they have infinite values, or the sum 482 00:29:30,860 --> 00:29:34,370 squared of the coefficients is equal to infinity. 483 00:29:34,370 --> 00:29:37,850 When we try to compress those we're going to find that no 484 00:29:37,850 --> 00:29:42,840 matter how we do it our mean square error 485 00:29:42,840 --> 00:29:45,100 is going to be infinite. 486 00:29:45,100 --> 00:29:50,020 Yes, we will talk about that later when we get to talking 487 00:29:50,020 --> 00:29:52,130 about expansions. 488 00:29:52,130 --> 00:29:56,220 So for all those reasons L1 isn't the right thing, L2 is 489 00:29:56,220 --> 00:29:58,800 the right thing. 490 00:29:58,800 --> 00:30:03,510 A function going from the reals into the complexes, in 491 00:30:03,510 --> 00:30:07,350 other words, a complex valued function is L2 if it's 492 00:30:07,350 --> 00:30:12,530 measurable, and if this integral is less than 493 00:30:12,530 --> 00:30:15,950 infinity, in other words, if that has finite energy. 494 00:30:15,950 --> 00:30:18,620 Primarily it means it has finite energy because all the 495 00:30:18,620 --> 00:30:21,390 functions you can think of are measurable. 496 00:30:21,390 --> 00:30:23,890 So it really says you're dealing with functions that 497 00:30:23,890 --> 00:30:25,620 have finite energy here. 498 00:30:31,510 --> 00:30:34,040 So, let's go on to Fourier transforms then. 499 00:30:37,670 --> 00:30:39,950 Interesting simple theorem. 500 00:30:39,950 --> 00:30:42,850 I think I stated this last time, also. 501 00:30:42,850 --> 00:30:50,980 If a function is L2 and its time limited, it's also L1. 502 00:30:50,980 --> 00:30:55,130 So we've already found that if functions are L1, they have 503 00:30:55,130 --> 00:30:58,360 Fourier transforms that exist. 504 00:30:58,360 --> 00:31:03,070 The reason for this is if you square u of t, take the 505 00:31:03,070 --> 00:31:07,320 magnitude squared of u of t for any given t, it has to be 506 00:31:07,320 --> 00:31:11,550 less than or equal to the sum of u of t plus 1. 507 00:31:11,550 --> 00:31:13,080 In fact, it has to be less than that. 508 00:31:13,080 --> 00:31:14,330 Why? 509 00:31:19,530 --> 00:31:22,520 How would you prove this if you had to prove it? 510 00:31:25,980 --> 00:31:29,770 Well, you say gee, this is two separate terms here. 511 00:31:29,770 --> 00:31:32,730 Why don't I look at two separate cases. 512 00:31:32,730 --> 00:31:38,320 The two separate cases are first, suppose u of t itself 513 00:31:38,320 --> 00:31:40,930 as a magnitude less than 1. 514 00:31:40,930 --> 00:31:45,530 If u of t has and magnitude less than 1 and you square it, 515 00:31:45,530 --> 00:31:47,630 you get something even smaller. 516 00:31:47,630 --> 00:31:51,770 So, any time u of t has a magnitude less than 1, u 517 00:31:51,770 --> 00:31:57,190 squared of t is less than or equal to u of t. 518 00:31:57,190 --> 00:31:59,530 Blah blah blah blah blah blah blah blah. 519 00:31:59,530 --> 00:32:00,930 If u of t -- 520 00:32:10,020 --> 00:32:11,270 what did I do here? 521 00:32:19,320 --> 00:32:22,620 No wonder I couldn't explain this to you. 522 00:32:27,330 --> 00:32:30,620 Let's try it that way and see if it works. 523 00:32:30,620 --> 00:32:33,118 If you can prove something, turn it around and see if you 524 00:32:33,118 --> 00:32:36,350 can prove it then. 525 00:32:36,350 --> 00:32:38,160 Now, two cases. 526 00:32:38,160 --> 00:32:41,940 First one, let's suppose that u of t is less 527 00:32:41,940 --> 00:32:43,440 than or equal to 1. 528 00:32:43,440 --> 00:32:46,430 Well then, u of t is less than or equal to 1. 529 00:32:46,430 --> 00:32:50,160 And this is positive so this is less than or equal to that. 530 00:32:50,160 --> 00:32:53,420 Let's look at the other case. u of t is greater than or 531 00:32:53,420 --> 00:32:55,560 equal to 1, magnitude. 532 00:32:55,560 --> 00:32:57,840 Well then, it's less than or equal to u 533 00:32:57,840 --> 00:32:59,540 of t magnitude squared. 534 00:32:59,540 --> 00:33:02,530 So either way this is less than or equal to that. 535 00:33:02,530 --> 00:33:07,390 What that says is the integral over any finite limits of u of 536 00:33:07,390 --> 00:33:12,700 t dt is less than or equal to the integral of this. 537 00:33:12,700 --> 00:33:15,870 Well, the integral of this splits up into the integral 538 00:33:15,870 --> 00:33:22,940 magnitude u of t squared dt plus the integral of 1. 539 00:33:22,940 --> 00:33:25,900 Now, the integral of 1 over any finite limits is 540 00:33:25,900 --> 00:33:28,300 just b minus a. 541 00:33:28,300 --> 00:33:30,860 That's where the finite limits come in. 542 00:33:30,860 --> 00:33:34,120 Finite limits say you don't have to worry about this term 543 00:33:34,120 --> 00:33:36,050 because it's finite. 544 00:33:36,050 --> 00:33:42,650 So that says at any time you have an L2 function over a 545 00:33:42,650 --> 00:33:46,710 finite range, that function is also L1 546 00:33:46,710 --> 00:33:48,960 over that finite range. 547 00:33:48,960 --> 00:33:52,830 Which says that any time you take a Fourier transform of an 548 00:33:52,830 --> 00:33:57,610 L2 function, which is only non-zero over a finite range, 549 00:33:57,610 --> 00:34:00,910 bingo, it's L1 and you get all these nice properties. 550 00:34:00,910 --> 00:34:03,910 It has to exist, it has to be continuous, it has to be 551 00:34:03,910 --> 00:34:06,120 bounded, and all of that neat stuff. 552 00:34:09,040 --> 00:34:14,060 So for any L2 function u of t, what I'm going to try to do 553 00:34:14,060 --> 00:34:17,850 now, and I'm just copying what a guy by the name of 554 00:34:17,850 --> 00:34:20,810 Plancherel did a long time ago. 555 00:34:20,810 --> 00:34:24,530 The thing that Plancherel did was he said how do I know when 556 00:34:24,530 --> 00:34:28,260 a Fourier transform exists or not. 557 00:34:28,260 --> 00:34:31,040 I would like to make it exist for L2 558 00:34:31,040 --> 00:34:33,900 functions, how do I do it? 559 00:34:33,900 --> 00:34:37,000 Well, he said OK, the thing I'm going to do is to take the 560 00:34:37,000 --> 00:34:42,290 function u of t and I'm first going to truncate it. 561 00:34:42,290 --> 00:34:45,290 In fact, if you think in terms of Reimann integration and 562 00:34:45,290 --> 00:34:48,940 things like that, any time you take an integral from minus 563 00:34:48,940 --> 00:34:53,140 infinity to plus infinity, what do you mean by it? 564 00:34:53,140 --> 00:34:56,250 You mean the limit as you integrate the function over 565 00:34:56,250 --> 00:35:00,510 finite limits and then you let the limits go to infinity. 566 00:35:00,510 --> 00:35:04,230 So all we're doing is the same trick here. 567 00:35:04,230 --> 00:35:09,180 So we're going to take u of t, we're going to truncate it to 568 00:35:09,180 --> 00:35:13,900 some minus a to plus a over some finite range, no matter 569 00:35:13,900 --> 00:35:15,720 how big a happens to be. 570 00:35:15,720 --> 00:35:20,570 We're going to call the function b sub a of t, u of t 571 00:35:20,570 --> 00:35:22,580 truncated to these limits. 572 00:35:22,580 --> 00:35:27,690 In other words, u of t times a rectangular function evaluated 573 00:35:27,690 --> 00:35:30,400 at t over 2a. 574 00:35:30,400 --> 00:35:34,650 Now, can all of you look at this function and see that it 575 00:35:34,650 --> 00:35:38,560 just means truncate from minus a to plus a? 576 00:35:38,560 --> 00:35:39,430 No. 577 00:35:39,430 --> 00:35:44,260 Well, you should learn to do this. 578 00:35:44,260 --> 00:35:47,170 One of the ways to do it is to say OK, the rectangular 579 00:35:47,170 --> 00:35:52,330 function is defined as having the value 1 between minus 1/2 580 00:35:52,330 --> 00:35:55,330 and plus 1/2 and it's zero everywhere else. 581 00:35:57,860 --> 00:36:00,570 I think I said that before in class, didn't I? 582 00:36:00,570 --> 00:36:03,650 Certainly it's in the notes. 583 00:36:03,650 --> 00:36:16,490 Rectangle ft equals 1 for t less than or equal 584 00:36:16,490 --> 00:36:19,960 to 1/2, zero else. 585 00:36:23,980 --> 00:36:27,360 So with this definition you just evaluate what happens 586 00:36:27,360 --> 00:36:32,870 when t is equal to minus a, you get rectangle of minus a 587 00:36:32,870 --> 00:36:36,150 over 2a, which is minus 1/2. 588 00:36:36,150 --> 00:36:39,910 When t is equal to a, you're up to the other limit, so this 589 00:36:39,910 --> 00:36:44,650 function is 1 for t between minus a and plus a and zero 590 00:36:44,650 --> 00:36:46,230 everywhere else. 591 00:36:46,230 --> 00:36:50,940 Please get used to using this and become a little facile at 592 00:36:50,940 --> 00:36:55,800 sorting out what it means because it's a very handy way 593 00:36:55,800 --> 00:36:59,300 to avoid writing awkward things like this. 594 00:36:59,300 --> 00:37:05,870 So va of t by what we've said is both L2 and its L1. 595 00:37:05,870 --> 00:37:10,770 We started out with a function which is L2 and we truncated. 596 00:37:10,770 --> 00:37:16,360 Then according to this theorem here, this function va of t is 597 00:37:16,360 --> 00:37:18,670 now time limited. 598 00:37:18,670 --> 00:37:24,300 It's also L2, and therefore, by the theorem it's also L1. 599 00:37:24,300 --> 00:37:29,660 Therefore, it's continue and you can take the Fourier 600 00:37:29,660 --> 00:37:35,700 transform of it -- v hat a of f exists for all f and it's 601 00:37:35,700 --> 00:37:36,590 continuous. 602 00:37:36,590 --> 00:37:42,990 So this function is just the Fourier transform that you get 603 00:37:42,990 --> 00:37:46,400 when you truncate the function, which is what you 604 00:37:46,400 --> 00:37:48,780 would think of as a way to find the 605 00:37:48,780 --> 00:37:51,840 Fourier transform anyway. 606 00:37:51,840 --> 00:37:54,810 I mean if this is not a reasonable approximation to 607 00:37:54,810 --> 00:37:58,650 the Fourier transform a function, you haven't modeled 608 00:37:58,650 --> 00:38:01,530 the function very well. 609 00:38:01,530 --> 00:38:05,470 Because when a gets extraordinarily large, if 610 00:38:05,470 --> 00:38:08,320 there's anything of significance that happens 611 00:38:08,320 --> 00:38:13,300 before year ten to the minus 6 or which happens after year 612 00:38:13,300 --> 00:38:17,120 ten to the plus 6, and you're dealing with electronic 613 00:38:17,120 --> 00:38:21,530 speeds, your models don't make any sense. 614 00:38:21,530 --> 00:38:26,050 So for anything of any interest, these functions here 615 00:38:26,050 --> 00:38:29,550 are going to start approximating u of t, and 616 00:38:29,550 --> 00:38:32,590 therefore we hope this will start approximating the 617 00:38:32,590 --> 00:38:35,520 Fourier transform of u of t. 618 00:38:35,520 --> 00:38:37,500 Who can make that more precise for me? 619 00:38:40,440 --> 00:38:44,500 What happens when u of t has finite energy? 620 00:38:47,210 --> 00:38:51,900 If it has finite energy it means that the integral of u 621 00:38:51,900 --> 00:38:54,410 of t magnitude squared over the 622 00:38:54,410 --> 00:38:58,460 infinite interval is finite. 623 00:38:58,460 --> 00:39:01,650 So you start integrating it over bigger and bigger 624 00:39:01,650 --> 00:39:04,580 minus a to plus a. 625 00:39:04,580 --> 00:39:08,630 What happens is as the minus a to plus a gets bigger and 626 00:39:08,630 --> 00:39:11,180 bigger and bigger, you're including more and more of the 627 00:39:11,180 --> 00:39:16,760 function, so that the integral of u of t squared over that 628 00:39:16,760 --> 00:39:19,460 bigger and bigger interval has to be getting closer and 629 00:39:19,460 --> 00:39:23,780 closer to the overall integral of u of t. 630 00:39:23,780 --> 00:39:29,800 Which says that the energy in u of t minus the a of t has to 631 00:39:29,800 --> 00:39:33,760 get very, very small as a gets large. 632 00:39:33,760 --> 00:39:36,560 That's one of the reasons why we like to deal with finite 633 00:39:36,560 --> 00:39:38,200 energy functions. 634 00:39:38,200 --> 00:39:42,650 By definition they cannot have a appreciable energy outside 635 00:39:42,650 --> 00:39:43,990 of very large limits. 636 00:39:43,990 --> 00:39:46,950 How large those large limits have to be depends on the 637 00:39:46,950 --> 00:39:50,600 function, but if you make them large enough you will always 638 00:39:50,600 --> 00:39:55,550 get negligible energy outside of those limits. 639 00:39:55,550 --> 00:39:59,580 So then we can take the Fourier transform of this 640 00:39:59,580 --> 00:40:03,750 function within those limits and we get something which we 641 00:40:03,750 --> 00:40:07,500 hope is going to be a reasonable approximation of 642 00:40:07,500 --> 00:40:10,300 the Fourier transform of u of t. 643 00:40:16,400 --> 00:40:19,820 That's what Plancherel said. 644 00:40:19,820 --> 00:40:26,610 Plancherel said if we have an L2 function, u of t, then 645 00:40:26,610 --> 00:40:30,900 there is an L2 function u hat of f, which is really the 646 00:40:30,900 --> 00:40:33,270 Fourier transform of u of t. 647 00:40:33,270 --> 00:40:39,060 Some people call this the Plancherel transform of u of t 648 00:40:39,060 --> 00:40:42,990 and say that indeed Plancherel was the one that invented 649 00:40:42,990 --> 00:40:44,790 Fourier transforms or Plancherel 650 00:40:44,790 --> 00:40:47,110 transforms for L2 functions. 651 00:40:47,110 --> 00:40:50,190 That's probably giving him a little too much credit, and 652 00:40:50,190 --> 00:40:54,360 Fourier somewhat less than due credit. 653 00:40:54,360 --> 00:40:57,590 But it was a neat theorem. 654 00:40:57,590 --> 00:41:03,140 What he said is that there is a function u hat of f, which 655 00:41:03,140 --> 00:41:06,240 we'll call the Fourier transform, which has the 656 00:41:06,240 --> 00:41:13,660 property that when you take the integral of the difference 657 00:41:13,660 --> 00:41:20,830 between u hat of f and the transform of b sub a of t, 658 00:41:20,830 --> 00:41:22,800 when you take the integral of this dt -- 659 00:41:22,800 --> 00:41:26,080 in other words, when you evaluate the energy in the 660 00:41:26,080 --> 00:41:34,880 difference between u hat of f and v sub a of f, 661 00:41:34,880 --> 00:41:37,570 that goes to zero. 662 00:41:37,570 --> 00:41:39,960 Well this isn't a big deal. 663 00:41:39,960 --> 00:41:43,090 In other words, this is plausible, since this integral 664 00:41:43,090 --> 00:41:46,610 has to go to zero for an L2 function, that's what we just 665 00:41:46,610 --> 00:41:52,760 said, then therefore, using the energy relation, this also 666 00:41:52,760 --> 00:41:55,570 has to go to zero. 667 00:41:55,570 --> 00:41:58,830 So is this another example where Plancherel just came 668 00:41:58,830 --> 00:42:03,510 along at the right time and he said something totally trivial 669 00:42:03,510 --> 00:42:05,960 and became famous because of it? 670 00:42:05,960 --> 00:42:09,320 I mean as I've urged all of you, work on problems that 671 00:42:09,320 --> 00:42:10,680 other people haven't worked on. 672 00:42:10,680 --> 00:42:12,940 If you're lucky, you will do something 673 00:42:12,940 --> 00:42:15,700 trivial and become famous. 674 00:42:15,700 --> 00:42:18,130 As another piece of philosophy, you become far 675 00:42:18,130 --> 00:42:20,670 more famous for doing something trivial than for 676 00:42:20,670 --> 00:42:23,990 doing something difficult, because everybody remembers 677 00:42:23,990 --> 00:42:25,870 something trivial. 678 00:42:25,870 --> 00:42:28,390 And if you do something difficult nobody even 679 00:42:28,390 --> 00:42:32,030 understands it. 680 00:42:32,030 --> 00:42:38,150 But no, it wasn't that because there's something hidden here. 681 00:42:38,150 --> 00:42:42,500 He says a function like this exists. 682 00:42:42,500 --> 00:42:47,740 In other words, the problem is these functions get closer and 683 00:42:47,740 --> 00:42:50,690 closer to something. 684 00:42:50,690 --> 00:42:53,440 They get closer and closer to each other as a gets bigger 685 00:42:53,440 --> 00:42:55,490 and bigger. 686 00:42:55,490 --> 00:42:57,030 You can show that because you have a 687 00:42:57,030 --> 00:42:59,920 handle on these functions. 688 00:42:59,920 --> 00:43:03,000 Whether they get closer and closer to a real bonafide 689 00:43:03,000 --> 00:43:05,900 function or not is another question. 690 00:43:09,950 --> 00:43:15,320 Back when you studied arithmetic, if you were in any 691 00:43:15,320 --> 00:43:18,730 kind of advanced class studying arithmetic, you 692 00:43:18,730 --> 00:43:20,980 studied the rational numbers and the 693 00:43:20,980 --> 00:43:23,440 real numbers you remember. 694 00:43:23,440 --> 00:43:28,500 And you remember the problem of what happens if you take a 695 00:43:28,500 --> 00:43:34,760 sequence of rational numbers which is approaching a limit. 696 00:43:34,760 --> 00:43:37,700 There's a big problem there because when you take a 697 00:43:37,700 --> 00:43:40,900 sequence of rational numbers that approaches a limit, the 698 00:43:40,900 --> 00:43:43,680 limit might not be rational. 699 00:43:43,680 --> 00:43:47,250 In other words, when you take sequences of things you can 700 00:43:47,250 --> 00:43:50,790 get out of the domain of the things you're working with. 701 00:43:50,790 --> 00:43:54,030 Now, we can't get out of the domain of being L2, but we 702 00:43:54,030 --> 00:43:56,770 might get out of domain of measurable functions, we might 703 00:43:56,770 --> 00:43:59,820 get out of the domain of functions at all. 704 00:43:59,820 --> 00:44:03,480 We can have all sorts of strange things happen. 705 00:44:03,480 --> 00:44:07,640 The nice thing here, which was really a theorem by 706 00:44:07,640 --> 00:44:08,490 [? Resenage ?] 707 00:44:08,490 --> 00:44:10,470 a long time ago. 708 00:44:10,470 --> 00:44:13,740 It says that when you take cosine sequences of L2 709 00:44:13,740 --> 00:44:17,950 functions, they converge to an L2 function. 710 00:44:17,950 --> 00:44:21,200 So that's really what's involved in here. 711 00:44:21,200 --> 00:44:24,420 So maybe this should be called the [? Resenage ?] 712 00:44:24,420 --> 00:44:26,880 transform, I don't know. 713 00:44:26,880 --> 00:44:31,220 But anyway, whatever this says, the theorem says, the 714 00:44:31,220 --> 00:44:34,080 first part of Plancherel's theorem says that this 715 00:44:34,080 --> 00:44:39,050 function exists and you get a handle on it by taking this 716 00:44:39,050 --> 00:44:43,260 transform, making a bigger and bigger, and it says it will 717 00:44:43,260 --> 00:44:47,010 converge to something in this energy sense. 718 00:44:47,010 --> 00:44:50,480 Bingo, when you're all done this goes to zero. 719 00:44:50,480 --> 00:44:54,750 We're going to denote this function as a limit and a mean 720 00:44:54,750 --> 00:44:57,230 of the Fourier transform. 721 00:44:57,230 --> 00:45:00,970 In other words, we do have a Fourier transform in the same 722 00:45:00,970 --> 00:45:03,630 sense that we had a Fourier series before. 723 00:45:03,630 --> 00:45:06,275 We didn't know weather the Fourier series would converge 724 00:45:06,275 --> 00:45:09,600 at every point, but we knew that it converged at enough 725 00:45:09,600 --> 00:45:13,190 points, namely, almost everywhere, everywhere but on 726 00:45:13,190 --> 00:45:14,710 a set of measure zero. 727 00:45:14,710 --> 00:45:18,910 It converges so that, in fact, you get this kind of 728 00:45:18,910 --> 00:45:21,400 relationship. 729 00:45:21,400 --> 00:45:23,890 Now, do you have to worry about that? 730 00:45:23,890 --> 00:45:24,270 No. 731 00:45:24,270 --> 00:45:28,340 Again, this is one of these very nice things that says 732 00:45:28,340 --> 00:45:32,700 there is a Fourier transform, you don't have to worry about 733 00:45:32,700 --> 00:45:36,200 what goes on at these oddball sets where the function has 734 00:45:36,200 --> 00:45:38,400 discountinuities and things like that. 735 00:45:38,400 --> 00:45:40,010 You can forget all of that. 736 00:45:40,010 --> 00:45:43,560 You can be as careless as you've ever been, and now you 737 00:45:43,560 --> 00:45:46,640 know that it all works out mathematically, so long as you 738 00:45:46,640 --> 00:45:48,600 stick to L2 functions. 739 00:45:48,600 --> 00:45:53,010 So, sticking to L2 functions says you can be a careless 740 00:45:53,010 --> 00:45:56,850 engineer, you can use lousy mathematics and you'll always 741 00:45:56,850 --> 00:45:58,510 get the right answer. 742 00:45:58,510 --> 00:46:02,600 So, it's nice for engineers, it's nice for me. 743 00:46:02,600 --> 00:46:05,730 I don't like to be careful all the time. 744 00:46:05,730 --> 00:46:11,130 I like to be careful once and then solve that and go on. 745 00:46:11,130 --> 00:46:13,850 Well, because of time frequency duality, you can do 746 00:46:13,850 --> 00:46:17,860 exactly the same thing in the frequency domain. 747 00:46:17,860 --> 00:46:22,170 So, you start out defining some b, which is bigger than 748 00:46:22,170 --> 00:46:26,470 zero which is arbitrarily large, you define a finite 749 00:46:26,470 --> 00:46:34,850 bandwidth approximation as w hat sub b of f is u hat of f. 750 00:46:34,850 --> 00:46:36,820 We now know that u hat of f exists 751 00:46:36,820 --> 00:46:38,900 and it's an L2 function. 752 00:46:38,900 --> 00:46:42,220 u hat of f times this rectangular function, 753 00:46:42,220 --> 00:46:43,240 that's f over 2b. 754 00:46:43,240 --> 00:46:48,510 In other words, it's u hat of f truncated to a big bandwidth 755 00:46:48,510 --> 00:46:49,810 minus b to plus b. 756 00:46:52,860 --> 00:46:56,510 Since w sub b of f is L1, as well as 757 00:46:56,510 --> 00:47:00,140 L2, this always exists. 758 00:47:00,140 --> 00:47:02,800 So long as you deal with a finite bandwidth, this 759 00:47:02,800 --> 00:47:04,520 quantity exists. 760 00:47:04,520 --> 00:47:06,950 It exists for all t and r. 761 00:47:06,950 --> 00:47:08,980 It's continuous. 762 00:47:08,980 --> 00:47:11,980 The second part of Plancherel's theorem then says 763 00:47:11,980 --> 00:47:16,550 that the limit as b goes to infinity of the integral of u 764 00:47:16,550 --> 00:47:21,130 of t minus this truncated function, magnitude squared 765 00:47:21,130 --> 00:47:25,420 the energy in that, goes to zero. 766 00:47:25,420 --> 00:47:29,060 This now is a little different than what we did before. 767 00:47:29,060 --> 00:47:31,720 It's easier in the sense that we don't have to worry about 768 00:47:31,720 --> 00:47:35,120 the existence of this function because we started out with 769 00:47:35,120 --> 00:47:37,530 this to start with. 770 00:47:37,530 --> 00:47:40,710 It's a little harder because we know that a function 771 00:47:40,710 --> 00:47:44,020 exists, but we don't know that it's u of t. 772 00:47:44,020 --> 00:47:47,150 So, in fact, poor old Plancherel had to do something 773 00:47:47,150 --> 00:47:52,040 other than just this very simple argument that says all 774 00:47:52,040 --> 00:47:53,430 the energy works out right. 775 00:47:53,430 --> 00:47:56,720 He had to also show that you really wind up with the right 776 00:47:56,720 --> 00:47:58,660 function when you get all through with it. 777 00:47:58,660 --> 00:48:01,650 But again, this is the same kind of energy convergence 778 00:48:01,650 --> 00:48:02,470 that we had before. 779 00:48:02,470 --> 00:48:02,840 Yeah? 780 00:48:02,840 --> 00:48:05,900 AUDIENCE: Could you discuss non-uniqueness? 781 00:48:05,900 --> 00:48:07,940 Clearly, [INAUDIBLE] 782 00:48:07,940 --> 00:48:12,260 u hat f to satisfy Plancherel 1 and Plancherel 2. 783 00:48:12,260 --> 00:48:16,710 PROFESSOR: Yeah, in fact, any two functions which are L2 784 00:48:16,710 --> 00:48:18,980 equivalent. 785 00:48:18,980 --> 00:48:23,270 But you see the nice thing is when you take this finite 786 00:48:23,270 --> 00:48:26,370 bandwidth approximation there's only one. 787 00:48:26,370 --> 00:48:28,370 It's only when you get to the limit that all 788 00:48:28,370 --> 00:48:31,720 of this mess occurs. 789 00:48:31,720 --> 00:48:34,390 If you take these different possible functions, u hat of 790 00:48:34,390 --> 00:48:40,880 f, which just differ in these negligible sets of measure 791 00:48:40,880 --> 00:48:44,720 zero, those don't affect this integral. 792 00:48:44,720 --> 00:48:48,190 Sets of measure zero don't affect integrals at all. 793 00:48:48,190 --> 00:48:51,640 So the mathematicians deal with L2 theory by talking 794 00:48:51,640 --> 00:48:55,280 about equivalence classes of functions. 795 00:48:55,280 --> 00:48:57,920 I find it hard to think about equivalence classes of 796 00:48:57,920 --> 00:49:01,560 functions and partitioning the set of all functions into a 797 00:49:01,560 --> 00:49:04,120 bunch of equivalence classes. 798 00:49:04,120 --> 00:49:07,375 So I just sort of remember in the back of my mind that there 799 00:49:07,375 --> 00:49:11,630 are all these functions which differ in a strange way. 800 00:49:11,630 --> 00:49:13,990 We'll talk about that more when we get to the sampling 801 00:49:13,990 --> 00:49:16,580 theorem later today, because there it 802 00:49:16,580 --> 00:49:18,300 happens to be important. 803 00:49:18,300 --> 00:49:20,470 Here it's not really important, here we don't have 804 00:49:20,470 --> 00:49:23,410 to worry about it. 805 00:49:23,410 --> 00:49:26,570 Anyway, we can always get back to the u of t that we started 806 00:49:26,570 --> 00:49:29,670 with in this way. 807 00:49:29,670 --> 00:49:33,920 Now, this says that all L2 functions have Fourier 808 00:49:33,920 --> 00:49:37,470 transforms in this very nice sense. 809 00:49:37,470 --> 00:49:40,720 In other words, at this point you don't have to worry about 810 00:49:40,720 --> 00:49:43,750 continuity, you don't have to worry about how fast things 811 00:49:43,750 --> 00:49:47,070 drop off, you don't have to worry about anything. 812 00:49:47,070 --> 00:49:50,660 So long as you have finite energy functions, this 813 00:49:50,660 --> 00:49:53,840 beautiful result always holds true. 814 00:49:53,840 --> 00:49:58,390 There always is a Fourier transform, it always has this 815 00:49:58,390 --> 00:50:02,350 nice property that it has the same energy as the function 816 00:50:02,350 --> 00:50:04,240 you started with. 817 00:50:04,240 --> 00:50:09,820 The only nasty thing, as Dave pointed out, is that, in fact, 818 00:50:09,820 --> 00:50:12,270 it might not be a unique function, but it's close 819 00:50:12,270 --> 00:50:13,490 enough to unique. 820 00:50:13,490 --> 00:50:15,790 It's unique in an engineering sense. 821 00:50:25,020 --> 00:50:29,570 The other thing is that L2 wave forms don't include some 822 00:50:29,570 --> 00:50:32,880 of your favorite wave forms. 823 00:50:32,880 --> 00:50:37,510 They don't include constants, they don't include sine waves, 824 00:50:37,510 --> 00:50:40,610 and they don't include Dirac impulse functions. 825 00:50:40,610 --> 00:50:42,640 All of them have infinite energy. 826 00:50:42,640 --> 00:50:45,790 I pointed out in class, spent quite a bit of time explaining 827 00:50:45,790 --> 00:50:49,580 why an impulse function had infinite energy by looking at 828 00:50:49,580 --> 00:50:54,040 it as a very narrow pulse of width epsilon and a height 1 829 00:50:54,040 --> 00:50:59,330 over epsilon, and showing that the energy in that is 1 over 830 00:50:59,330 --> 00:51:02,710 epsilon, and as epsilon goes to zero and the pulse gets 831 00:51:02,710 --> 00:51:07,270 narrower and narrower, bingo, the energy goes to infinity. 832 00:51:07,270 --> 00:51:11,060 Constants are the same way, they extend on and on forever. 833 00:51:11,060 --> 00:51:14,870 Therefore, they have infinite energy, except if the constant 834 00:51:14,870 --> 00:51:17,220 happens to be zero. 835 00:51:17,220 --> 00:51:20,780 Sine waves are the same way, they dribble on forever. 836 00:51:23,400 --> 00:51:28,560 So the question is are these good models of reality? 837 00:51:28,560 --> 00:51:31,450 The answer is they're good for some things and they're very 838 00:51:31,450 --> 00:51:33,810 bad for other things. 839 00:51:33,810 --> 00:51:37,420 The point in this course is that if you're looking for 840 00:51:37,420 --> 00:51:43,140 wave forms that are good models for either the kinds of 841 00:51:43,140 --> 00:51:46,930 functions that we're going to quantize, namely, source wave 842 00:51:46,930 --> 00:51:50,320 forms, or if you're looking for the kinds of things that 843 00:51:50,320 --> 00:51:54,170 we're going to transmit on channels, these 844 00:51:54,170 --> 00:51:56,720 are very lousy functions. 845 00:51:56,720 --> 00:51:59,860 They don't make any sense in a communication context. 846 00:52:02,660 --> 00:52:06,370 But anyway, where did these things come from? 847 00:52:06,370 --> 00:52:09,660 Constants and sine waves result from refusing to 848 00:52:09,660 --> 00:52:11,720 explicitly model when very 849 00:52:11,720 --> 00:52:14,920 long-lasting functions terminate. 850 00:52:14,920 --> 00:52:17,260 In other words, if you're looking at a carrier function 851 00:52:17,260 --> 00:52:26,640 in a communication's system, sine of 2 pi, f carrier times 852 00:52:26,640 --> 00:52:31,620 t, it just keeps on wiggling around forever. 853 00:52:31,620 --> 00:52:35,030 Since you want to talk about that over the complete time of 854 00:52:35,030 --> 00:52:39,680 interest, you don't want to say what the time of interest 855 00:52:39,680 --> 00:52:44,190 is, you don't want to admit to your employer that this thing 856 00:52:44,190 --> 00:52:47,560 is going to stop working after one month because you want to 857 00:52:47,560 --> 00:52:50,320 let him think that he's going to make a profit off of this 858 00:52:50,320 --> 00:52:55,300 forever, and you don't want to commit to putting it into use 859 00:52:55,300 --> 00:52:57,740 in one month when you know it's going to get delayed for 860 00:52:57,740 --> 00:52:58,670 a whole year. 861 00:52:58,670 --> 00:52:59,940 So you want to think of this as 862 00:52:59,940 --> 00:53:01,400 something which is permanent. 863 00:53:01,400 --> 00:53:04,770 You don't want to answer the question at what time does it 864 00:53:04,770 --> 00:53:08,450 start and at what time does it end, because for many of the 865 00:53:08,450 --> 00:53:11,120 questions you ask, you can just regard 866 00:53:11,120 --> 00:53:15,110 it as going on forever. 867 00:53:15,110 --> 00:53:17,440 You have the same thing with impulses. 868 00:53:17,440 --> 00:53:25,420 Impulses are always models of short pulses. 869 00:53:25,420 --> 00:53:28,300 If you put these short pulses through a filter, the only 870 00:53:28,300 --> 00:53:30,680 thing which is of interest in them is what 871 00:53:30,680 --> 00:53:32,670 their integral is. 872 00:53:32,670 --> 00:53:35,040 And since the only thing of interest is their integral, 873 00:53:35,040 --> 00:53:39,120 you call it an impulse and you don't worry about just how 874 00:53:39,120 --> 00:53:43,800 narrow it is, except that it has infinite energy and, 875 00:53:43,800 --> 00:53:46,980 therefore, whenever you start to deal with a situation in 876 00:53:46,980 --> 00:53:51,870 which energy is important, these becomes lousy models. 877 00:53:51,870 --> 00:53:54,320 So we can't use these when we're 878 00:53:54,320 --> 00:53:56,070 talking about L2 functions. 879 00:53:56,070 --> 00:53:59,540 That's the price we pay for dealing with L2 functions. 880 00:53:59,540 --> 00:54:03,320 But it's a small price because for almost all the things 881 00:54:03,320 --> 00:54:06,570 we'll be dealing with, it's the energy of the functions 882 00:54:06,570 --> 00:54:09,810 that are really important. 883 00:54:09,810 --> 00:54:12,790 So as communication wave forms, infinite energy wave 884 00:54:12,790 --> 00:54:16,090 forms make mean square error quantization results 885 00:54:16,090 --> 00:54:17,860 meaningless. 886 00:54:17,860 --> 00:54:20,980 In other words, when you sample these infinite energy 887 00:54:20,980 --> 00:54:24,985 wave forms you get results that don't make any sense, and 888 00:54:24,985 --> 00:54:28,250 they make those channel results meaningless. 889 00:54:28,250 --> 00:54:30,940 Therefore, from now on, whether I remember to say it 890 00:54:30,940 --> 00:54:34,620 or not, everything we deal with, unless we're looking at 891 00:54:34,620 --> 00:54:38,740 counter examples to something, is going to be an L2 function. 892 00:54:45,630 --> 00:54:48,310 Let's go on. 893 00:54:48,310 --> 00:54:52,430 I'm starting to feel like I'm back in our signals and 894 00:54:52,430 --> 00:54:56,570 systems course, because at this point I'm defining my 895 00:54:56,570 --> 00:55:00,100 third different kind of transform. 896 00:55:00,100 --> 00:55:02,990 Fortunately, this is the last transform we will have to talk 897 00:55:02,990 --> 00:55:06,050 about, so we're all done with this. 898 00:55:06,050 --> 00:55:09,860 The other nice thing is that the discrete time Fourier 899 00:55:09,860 --> 00:55:15,020 transform happens to be just the time frequency dual of the 900 00:55:15,020 --> 00:55:17,110 Fourier series. 901 00:55:17,110 --> 00:55:24,480 So that whether you've ever studied the dtft or not, you 902 00:55:24,480 --> 00:55:27,460 already know everything there is to know about it, because 903 00:55:27,460 --> 00:55:30,550 the only things there are to know about it are the results 904 00:55:30,550 --> 00:55:33,050 about Fourier series. 905 00:55:33,050 --> 00:55:36,280 So the theorem is really the same theorem that we had for 906 00:55:36,280 --> 00:55:38,570 Fourier series. 907 00:55:38,570 --> 00:55:43,360 Assume that you have a function of frequency, u hat 908 00:55:43,360 --> 00:55:47,020 of f -- before we had a function of time, now we have 909 00:55:47,020 --> 00:55:49,590 a function of frequency. 910 00:55:49,590 --> 00:55:52,680 Suppose it's defined over the interval minus w 911 00:55:52,680 --> 00:55:56,260 to plus w into c. 912 00:55:56,260 --> 00:55:59,260 In other words, a way we often say that a function is 913 00:55:59,260 --> 00:56:03,410 truncated is to say it goes from some interval into c. 914 00:56:03,410 --> 00:56:08,510 This is a complex function which is non-zero only for f 915 00:56:08,510 --> 00:56:12,290 in this finite bandwidth range. 916 00:56:12,290 --> 00:56:16,240 We want to assume that this is L2 and thus, we 917 00:56:16,240 --> 00:56:19,510 know it's also L1. 918 00:56:19,510 --> 00:56:23,380 Then we're going to take the Fourier coefficients. 919 00:56:23,380 --> 00:56:26,710 Before we thought of the Fourier coefficients as 920 00:56:26,710 --> 00:56:30,020 corresponding to what goes on at different frequencies, now 921 00:56:30,020 --> 00:56:33,180 we're going to regard them as time quantities, and we'll see 922 00:56:33,180 --> 00:56:35,360 exactly why later. 923 00:56:35,360 --> 00:56:37,240 So we'll define these as the Fourier 924 00:56:37,240 --> 00:56:39,690 coefficients of this function. 925 00:56:39,690 --> 00:56:43,280 So they're 1 over 2w times this integral here. 926 00:56:43,280 --> 00:56:46,650 You remember before when we dealt with the Fourier series, 927 00:56:46,650 --> 00:56:50,220 we went from minus t over 2 to plus t over 2. 928 00:56:50,220 --> 00:56:53,250 Now we're going from minus w to plus w. 929 00:56:53,250 --> 00:56:53,850 Why? 930 00:56:53,850 --> 00:56:57,150 It's just convention, there's no real reason. 931 00:56:57,150 --> 00:57:02,790 So that what's happening here is that this is a Fourier 932 00:57:02,790 --> 00:57:06,310 series formula for a coefficient where we're 933 00:57:06,310 --> 00:57:11,710 substituting w for t over 2. 934 00:57:11,710 --> 00:57:15,130 We're putting a plus sign in the exponent instead of a 935 00:57:15,130 --> 00:57:15,990 minus sign. 936 00:57:15,990 --> 00:57:21,500 In other words, we're doing this hermitian duality bit. 937 00:57:21,500 --> 00:57:23,600 What's the other difference? 938 00:57:23,600 --> 00:57:26,090 We're interchanging time and frequency. 939 00:57:26,090 --> 00:57:29,470 Aside from that it's exactly the same theorem that we 940 00:57:29,470 --> 00:57:32,180 established -- well, that we stated before. 941 00:57:32,180 --> 00:57:37,880 So we know that this quantity here, since u hat of f is L1, 942 00:57:37,880 --> 00:57:41,690 this is finite and it exists -- that's just a finite 943 00:57:41,690 --> 00:57:48,320 complex number and nothing more -- for all integer k. 944 00:57:48,320 --> 00:57:54,580 Also, the convergence result when we go back, this is the 945 00:57:54,580 --> 00:57:57,790 formula we try to use to go back to the function we 946 00:57:57,790 --> 00:57:59,670 started with. 947 00:57:59,670 --> 00:58:02,020 It's just a finite approximation 948 00:58:02,020 --> 00:58:04,300 to the Fourier series. 949 00:58:04,300 --> 00:58:08,350 We're saying that as k zero gets larger and larger, this 950 00:58:08,350 --> 00:58:12,190 finite approximation approaches the function in 951 00:58:12,190 --> 00:58:14,450 energy sense. 952 00:58:14,450 --> 00:58:19,420 So this is exactly what you should mean by a Fourier 953 00:58:19,420 --> 00:58:20,430 series anyway. 954 00:58:20,430 --> 00:58:22,175 It's exactly what you should mean by a 955 00:58:22,175 --> 00:58:24,110 Fourier transform anyway. 956 00:58:24,110 --> 00:58:28,610 As you go to the limit with more and more terms you get 957 00:58:28,610 --> 00:58:34,010 something which is equal to what you started with, except 958 00:58:34,010 --> 00:58:37,070 on the set of measure zero. 959 00:58:37,070 --> 00:58:40,240 In other words, it converges everywhere where it matters. 960 00:58:40,240 --> 00:58:43,270 It converges to something in the sense that the 961 00:58:43,270 --> 00:58:47,160 energy is the same. 962 00:58:47,160 --> 00:58:50,860 I said that in such a way that makes it a little simpler than 963 00:58:50,860 --> 00:58:53,660 it really is. 964 00:58:53,660 --> 00:59:01,940 You can't always say that this converges in any nice way. 965 00:59:01,940 --> 00:59:06,280 Next time I'm going to show you a truly awful function 966 00:59:06,280 --> 00:59:10,910 which we'll use in the Fourier series instead of dtft, which 967 00:59:10,910 --> 00:59:16,360 is time limited and which is just incredibly messy and 968 00:59:16,360 --> 00:59:19,460 it'll show you why you have to be a little bit careful about 969 00:59:19,460 --> 00:59:22,250 stating these results. 970 00:59:22,250 --> 00:59:26,250 But you don't have to worry about it most of the time, 971 00:59:26,250 --> 00:59:29,560 because this theorem is still true. 972 00:59:29,560 --> 00:59:31,960 It's just that you have to be a little careful about how to 973 00:59:31,960 --> 00:59:35,230 interpret it because you don't know whether this is going to 974 00:59:35,230 --> 00:59:37,340 reach a limit or not. 975 00:59:37,340 --> 00:59:39,720 All you know is that this will be true, this energy 976 00:59:39,720 --> 00:59:42,960 difference goes to zero. 977 00:59:42,960 --> 00:59:48,060 Also, the energy in the function is equal to the 978 00:59:48,060 --> 00:59:49,800 energy in the coefficients. 979 00:59:49,800 --> 00:59:52,610 This was the thing that we found so useful with the 980 00:59:52,610 --> 00:59:54,810 Fourier series. 981 00:59:54,810 --> 01:00:00,420 It's why we can play this game that we play with mean square 982 01:00:00,420 --> 01:00:06,550 quantization error of taking a function and then turning it 983 01:00:06,550 --> 01:00:11,690 into a sequence of samples, trying to quantize the samples 984 01:00:11,690 --> 01:00:16,050 for minimum mean square error and associate the mean square 985 01:00:16,050 --> 01:00:18,590 error in the samples with the mean square 986 01:00:18,590 --> 01:00:20,120 error on the function. 987 01:00:20,120 --> 01:00:24,280 You can't do that with anything that I know of other 988 01:00:24,280 --> 01:00:25,830 than mean square error. 989 01:00:25,830 --> 01:00:28,790 If you want to deal with other kinds of quantization errors, 990 01:00:28,790 --> 01:00:31,320 you have a real problem going from 991 01:00:31,320 --> 01:00:34,170 coefficients to functions. 992 01:00:34,170 --> 01:00:43,310 And finally, for any set of numbers u sub k, if the sum is 993 01:00:43,310 --> 01:00:46,470 less than infinity, in other words, if you're dealing with 994 01:00:46,470 --> 01:00:52,770 a sequence of finite energy, there always is such a 995 01:00:52,770 --> 01:00:54,540 frequency function. 996 01:00:54,540 --> 01:00:57,340 Many people when they use the discrete time Fourier 997 01:00:57,340 --> 01:01:01,860 transform think of starting with the sequence and taking a 998 01:01:01,860 --> 01:01:04,920 sequence and saying well it's nice to think of this sequence 999 01:01:04,920 --> 01:01:09,270 in the frequency domain, and then they say a function f 1000 01:01:09,270 --> 01:01:13,100 exists such that this is true, or they just say that this is 1001 01:01:13,100 --> 01:01:16,300 equal to that without worrying about the convergence at all, 1002 01:01:16,300 --> 01:01:18,150 which is more common. 1003 01:01:18,150 --> 01:01:20,620 But since we've already gone through all of this for the 1004 01:01:20,620 --> 01:01:25,760 Fourier series, we might as well say it right here also. 1005 01:01:25,760 --> 01:01:27,970 So there's really nothing different here. 1006 01:01:27,970 --> 01:01:34,090 But now the question is why do these u of k's -- 1007 01:01:34,090 --> 01:01:38,060 I mean why do we think of those as time coefficients? 1008 01:01:38,060 --> 01:01:40,420 I mean what's really going on in this discrete 1009 01:01:40,420 --> 01:01:42,660 time Fourier transform. 1010 01:01:42,660 --> 01:01:47,050 At this point it just looks like a lot of mathematics and 1011 01:01:47,050 --> 01:01:50,930 it's hard to interpret what any of these things mean. 1012 01:01:50,930 --> 01:01:53,040 Well, the next thing I want to do is to go into 1013 01:01:53,040 --> 01:01:55,760 the sampling theorem. 1014 01:01:55,760 --> 01:01:59,480 The sampling theorem, in fact, is going to interpret for you 1015 01:01:59,480 --> 01:02:02,920 what this discrete time Fourier transform is, because 1016 01:02:02,920 --> 01:02:05,760 the sampling theorem and the discrete time Fourier 1017 01:02:05,760 --> 01:02:10,270 transform are just intimately related, they're hand and 1018 01:02:10,270 --> 01:02:21,280 glove with each other, and that's the next 1019 01:02:21,280 --> 01:02:22,800 thing we want to do. 1020 01:02:22,800 --> 01:02:27,230 But first we have to re-write this a little bit. 1021 01:02:27,230 --> 01:02:32,180 We're going to say that this frequency function is the 1022 01:02:32,180 --> 01:02:35,810 limit in the mean of this -- 1023 01:02:35,810 --> 01:02:40,630 this rectangular function is what we use just to make sure 1024 01:02:40,630 --> 01:02:43,190 we're only talking about frequency between 1025 01:02:43,190 --> 01:02:47,250 minus w and plus w. 1026 01:02:47,250 --> 01:02:50,355 The limit in the mean, there's a little notational trick that 1027 01:02:50,355 --> 01:02:55,510 we use so that we can think of this as just a limit instead 1028 01:02:55,510 --> 01:02:59,310 of thinking of it as this crazy thing that we just 1029 01:02:59,310 --> 01:03:02,120 derive, which is really not so crazy. 1030 01:03:04,900 --> 01:03:08,540 That means we can talk about this transform without always 1031 01:03:08,540 --> 01:03:14,060 rubbing our noses in all of this mess here. 1032 01:03:14,060 --> 01:03:16,900 It just means that once in awhile we go back and think 1033 01:03:16,900 --> 01:03:18,630 what does this really mean. 1034 01:03:18,630 --> 01:03:22,110 It means convergence in energy rather than convergence 1035 01:03:22,110 --> 01:03:26,640 point-wise, because we might not have convergence 1036 01:03:26,640 --> 01:03:27,890 point-wise. 1037 01:03:29,650 --> 01:03:34,660 So we're going to write this also as the limit in the mean 1038 01:03:34,660 --> 01:03:37,580 of the sum over k of u of k. 1039 01:03:37,580 --> 01:03:40,480 We're going to glop all of this together. 1040 01:03:40,480 --> 01:03:45,940 This is just some function of k and a frequency. 1041 01:03:45,940 --> 01:03:50,920 So we're going to call that phi sub k of f at some wave 1042 01:03:50,920 --> 01:03:56,150 form, and the wave form is this. 1043 01:03:56,150 --> 01:04:01,940 What happens if you look at the relationship between to 1044 01:04:01,940 --> 01:04:06,030 phi k of f and phi k prime of f? 1045 01:04:06,030 --> 01:04:08,200 Namely, if you look at two different functions. 1046 01:04:11,220 --> 01:04:14,490 These two functions are orthongonal to each other for 1047 01:04:14,490 --> 01:04:21,570 the same reason that the functions that the sinusoid 1048 01:04:21,570 --> 01:04:27,040 you used in the Fourier series are orthongonal. 1049 01:04:27,040 --> 01:04:31,790 Namely, you take this function, you multiply it by e 1050 01:04:31,790 --> 01:04:35,450 to the minus 2 pi ik prime f over 2w times 1051 01:04:35,450 --> 01:04:37,070 this rectangular function. 1052 01:04:37,070 --> 01:04:40,880 You integrate from minus w to w and what do you get? 1053 01:04:40,880 --> 01:04:43,490 You're just integrating a sinusoid -- 1054 01:04:43,490 --> 01:04:45,880 the whole thing is one big sinusoid -- 1055 01:04:45,880 --> 01:04:49,520 over one period of that sinusoid or multiple periods 1056 01:04:49,520 --> 01:04:50,630 of the sinusoid. 1057 01:04:50,630 --> 01:04:54,650 Actually, k minus k prime periods of the sinusoid. 1058 01:04:54,650 --> 01:04:57,360 And when you integrate a sinusoid over a period, what 1059 01:04:57,360 --> 01:04:58,140 do you get? 1060 01:04:58,140 --> 01:05:00,200 You get zero. 1061 01:05:00,200 --> 01:05:03,330 So it says that these functions are all orthongonal 1062 01:05:03,330 --> 01:05:04,460 to each other. 1063 01:05:04,460 --> 01:05:08,460 So, presto, we have another orthongonal expansion just 1064 01:05:08,460 --> 01:05:11,850 like the Fourier series gave us an orthongonal expansion. 1065 01:05:11,850 --> 01:05:14,020 And in fact, it's the same orthongonal expansion. 1066 01:05:31,050 --> 01:05:36,400 Now the next thing to observe is that we have done the 1067 01:05:36,400 --> 01:05:41,220 Fourier transform and we've also done the discrete time 1068 01:05:41,220 --> 01:05:43,070 Fourier transform. 1069 01:05:43,070 --> 01:05:48,010 In both of them we're dealing with some frequency function. 1070 01:05:48,010 --> 01:05:50,720 Now we're dealing with some frequency function which is 1071 01:05:50,720 --> 01:05:54,560 limited to minus w to plus w, but we have two 1072 01:05:54,560 --> 01:05:57,570 expansions for it. 1073 01:05:57,570 --> 01:06:00,740 We have the Fourier transform, so we can go to a function u 1074 01:06:00,740 --> 01:06:03,710 of t, and we also have this discrete 1075 01:06:03,710 --> 01:06:07,070 time Fourier transform. 1076 01:06:07,070 --> 01:06:13,070 So u of t is equal to this Fourier transform here. 1077 01:06:13,070 --> 01:06:17,580 Again, I should write limit in the mean here, but then I 1078 01:06:17,580 --> 01:06:19,390 think about and I say do I have to write 1079 01:06:19,390 --> 01:06:21,220 limit in the mean? 1080 01:06:21,220 --> 01:06:22,090 No. 1081 01:06:22,090 --> 01:06:24,520 I don't need a limit in the mean here because I'm 1082 01:06:24,520 --> 01:06:26,970 integrating this over finite limits. 1083 01:06:26,970 --> 01:06:31,050 Since I'm taking a function over finite limits, u hat of f 1084 01:06:31,050 --> 01:06:36,150 is over these limits, is an L1 function, therefore, this 1085 01:06:36,150 --> 01:06:37,010 integral exists. 1086 01:06:37,010 --> 01:06:41,510 This function is a continuous function. 1087 01:06:41,510 --> 01:06:45,070 There aren't any sets of measure zero involved here. 1088 01:06:45,070 --> 01:06:49,740 This is one specific function which is always the same. 1089 01:06:49,740 --> 01:06:52,460 You know what it is exactly at every point. 1090 01:06:52,460 --> 01:06:56,370 At every point t, this converges. 1091 01:06:56,370 --> 01:06:58,890 So then what we're going to do, what the sampling theorem 1092 01:06:58,890 --> 01:07:05,030 does is it relates this to what you get with a dtft. 1093 01:07:05,030 --> 01:07:09,090 So the sampling theorem says let u hat of f be an L2 1094 01:07:09,090 --> 01:07:20,050 function which goes from minus ww to c, and let u of t be 1095 01:07:20,050 --> 01:07:23,730 this, namely, that, which we now know exists and is 1096 01:07:23,730 --> 01:07:25,110 continuous. 1097 01:07:25,110 --> 01:07:28,750 Define capital T as 1 over 2w. 1098 01:07:28,750 --> 01:07:32,270 You don't have to do that if you don't want to, but it's a 1099 01:07:32,270 --> 01:07:35,250 little easier to think in terms of some increment of 1100 01:07:35,250 --> 01:07:37,880 time, T, here. 1101 01:07:37,880 --> 01:07:42,120 Then u of t is continuous, L2 and bounded. 1102 01:07:42,120 --> 01:07:44,570 It's bounded by u of t less than or equal to this. 1103 01:07:44,570 --> 01:07:46,030 Why is that? 1104 01:07:53,295 --> 01:07:57,670 It doesn't make any sense as I stated it. 1105 01:07:57,670 --> 01:07:58,940 Now it makes sense. 1106 01:07:58,940 --> 01:08:01,720 OK, its magnitude is bounded. 1107 01:08:01,720 --> 01:08:04,880 Its magnitude is bounded because if you take u of t 1108 01:08:04,880 --> 01:08:09,310 magnitude, it's equal to the magnitude of this which is 1109 01:08:09,310 --> 01:08:12,740 less than or equal to the integral of the magnitude of 1110 01:08:12,740 --> 01:08:18,130 this, which is equal to the integral of the magnitude of 1111 01:08:18,130 --> 01:08:22,370 just u hat of f, which is what we have here. 1112 01:08:22,370 --> 01:08:25,470 So all of that works nicely. 1113 01:08:25,470 --> 01:08:29,520 So, u of t is a nice, well-defined function. 1114 01:08:29,520 --> 01:08:33,150 Then the other part of it is that u of t is equal to the 1115 01:08:33,150 --> 01:08:38,800 sum if its values at these sample points times sinc of t 1116 01:08:38,800 --> 01:08:40,810 minus kt over t. 1117 01:08:40,810 --> 01:08:43,490 Now you've probably seen this sampling theorem before. 1118 01:08:43,490 --> 01:08:46,430 How many people haven't seen this before? 1119 01:08:46,430 --> 01:08:50,130 I mean aside from the question of trying to do it -- 1120 01:08:50,130 --> 01:08:52,870 do it in a way that makes sense. 1121 01:08:52,870 --> 01:08:53,890 OK, you've all seen it. 1122 01:08:53,890 --> 01:08:55,080 So good. 1123 01:08:55,080 --> 01:09:00,160 What it's saying is you can represent a function in terms 1124 01:09:00,160 --> 01:09:03,310 of just knowing what its samples are. 1125 01:09:03,310 --> 01:09:06,890 Or you can take the function, you can sample it, and when 1126 01:09:06,890 --> 01:09:14,340 you sample it if you put these little sinc hats around all 1127 01:09:14,340 --> 01:09:17,180 the samples, you get back to the function again. 1128 01:09:17,180 --> 01:09:20,490 So you take all the samples, you then put these sincs 1129 01:09:20,490 --> 01:09:22,630 around them, add them all up, and 1130 01:09:22,630 --> 01:09:27,680 bingo, you got the function. 1131 01:09:27,680 --> 01:09:29,880 Let's see why that's true. 1132 01:09:35,820 --> 01:09:38,290 Here's the sinc function here. 1133 01:09:38,290 --> 01:09:43,290 The important thing about sinc t, sine pi t over pi t, which 1134 01:09:43,290 --> 01:09:47,410 you can see by just looking at the sine function, is it has 1135 01:09:47,410 --> 01:09:51,040 the value 1 when t is equal to zero. 1136 01:09:51,040 --> 01:09:53,780 I mean to get that value 1, you really have to go through 1137 01:09:53,780 --> 01:09:57,830 a limiting operation here to think of sine pi t when t is 1138 01:09:57,830 --> 01:10:01,580 very small as being approximately equal to pi t. 1139 01:10:01,580 --> 01:10:05,580 When you divide pi t by pi t you get 1, so that's its value 1140 01:10:05,580 --> 01:10:08,490 there and value around there. 1141 01:10:08,490 --> 01:10:13,580 At every other sample point, namely, at t equals 1, sine of 1142 01:10:13,580 --> 01:10:15,220 pi t is zero. 1143 01:10:15,220 --> 01:10:19,200 At t equals 2, sine of pi t is equal to zero. 1144 01:10:19,200 --> 01:10:25,030 So the sinc function is 1 at zero and is zero at every 1145 01:10:25,030 --> 01:10:27,660 other integer point. 1146 01:10:27,660 --> 01:10:32,240 Now to see why it's true and to understand what the dtft is 1147 01:10:32,240 --> 01:10:39,270 all about, note that we have said that a frequency 1148 01:10:39,270 --> 01:10:44,150 function, u hat of f, can be expressed as the sum over k -- 1149 01:10:44,150 --> 01:10:47,530 and I should use a limit in the mean here but I'm not -- 1150 01:10:47,530 --> 01:10:51,970 of uk times this transform relationship here. 1151 01:10:54,560 --> 01:10:59,450 Well, these are these functions that we talked about 1152 01:10:59,450 --> 01:11:00,700 awhile ago. 1153 01:11:09,010 --> 01:11:09,980 They're these functions. 1154 01:11:09,980 --> 01:11:15,840 They're the sinusoids, periodic sinusoids in k 1155 01:11:15,840 --> 01:11:18,510 truncated in frequency. 1156 01:11:24,430 --> 01:11:31,880 So we know that u hat of f is equal to that dtft expansion. 1157 01:11:31,880 --> 01:11:35,360 If I take the inverse Fourier transform of that, I can take 1158 01:11:35,360 --> 01:11:38,490 the inverse Fourier transform of all these functions and 1159 01:11:38,490 --> 01:11:43,590 I'll get u of t is equal to the sum over k, of uk pk of t. 1160 01:11:43,590 --> 01:11:45,710 I'm being careless about the mathematics here. 1161 01:11:45,710 --> 01:11:48,510 I've been careful about it all along. 1162 01:11:48,510 --> 01:11:52,450 The notes does it carefully, particularly in the appendix. 1163 01:11:52,450 --> 01:11:54,750 I'm not going to worry about that here. 1164 01:11:54,750 --> 01:11:58,520 If I take the function pk of f, which is this truncated 1165 01:11:58,520 --> 01:12:03,340 sinusoid, and I take the inverse transform of that -- 1166 01:12:03,340 --> 01:12:07,350 take the transform of this, I get this. 1167 01:12:07,350 --> 01:12:10,870 Can you see that just by inspection? 1168 01:12:10,870 --> 01:12:13,090 If you were really hot on these things, if you just 1169 01:12:13,090 --> 01:12:16,830 finished taking 6.003, you could probably see that by 1170 01:12:16,830 --> 01:12:18,630 inspection. 1171 01:12:18,630 --> 01:12:23,490 If you remember all of those relationships that we went 1172 01:12:23,490 --> 01:12:26,650 through before, you can see it by inspection. 1173 01:12:26,650 --> 01:12:31,260 The Fourier transform of erect function is a sinc function. 1174 01:12:31,260 --> 01:12:35,520 This exponential here when you go into the time domain 1175 01:12:35,520 --> 01:12:39,010 corresponds to a time shift, so that gives rise to this 1176 01:12:39,010 --> 01:12:41,350 time shift here. 1177 01:12:41,350 --> 01:12:45,150 The 1 over t is just one of these constants you have to 1178 01:12:45,150 --> 01:12:48,250 keep straight, and which I would do just by integrating 1179 01:12:48,250 --> 01:12:51,110 the things to see what I get. 1180 01:12:51,110 --> 01:13:00,970 Finally, u of kt, if I look at this, is just 1 over t times u 1181 01:13:00,970 --> 01:13:05,220 of k, because these functions here are these sinc functions, 1182 01:13:05,220 --> 01:13:08,350 which are zero everywhere but on their own point. 1183 01:13:08,350 --> 01:13:17,240 So if I look at u of kt, it's a sum over k of ck of kt, and 1184 01:13:17,240 --> 01:13:26,160 ck of kt is only 1 when little t is equal to k times capital 1185 01:13:26,160 --> 01:13:30,690 T, and therefore, I get that point there. 1186 01:13:30,690 --> 01:13:35,740 Therefore, u of kt is just 1 over t times u of k. 1187 01:13:35,740 --> 01:13:39,950 That finishes the sampling theorem except for really 1188 01:13:39,950 --> 01:13:47,110 tracing through all of these things about convergence, but 1189 01:13:47,110 --> 01:13:50,350 it also tells you what the discrete time Fourier 1190 01:13:50,350 --> 01:13:52,380 transform is. 1191 01:13:52,380 --> 01:13:56,420 Because the discrete time Fourier transform is just 1192 01:13:56,420 --> 01:13:59,560 scaled samples of u of t. 1193 01:13:59,560 --> 01:14:02,380 In other words, you start out with this frequency function, 1194 01:14:02,380 --> 01:14:05,110 you take the inverse Fourier transform of it, you get a 1195 01:14:05,110 --> 01:14:06,280 time function. 1196 01:14:06,280 --> 01:14:10,050 You take the samples of that, you scale them, and those are 1197 01:14:10,050 --> 01:14:12,800 the coefficients in the discrete time Fourier 1198 01:14:12,800 --> 01:14:15,410 transform, which is what you use discrete time Fourier 1199 01:14:15,410 --> 01:14:17,100 transforms for. 1200 01:14:17,100 --> 01:14:20,520 You think of sampling of function, then you represent 1201 01:14:20,520 --> 01:14:23,450 the function in terms of those samples and then you want to 1202 01:14:23,450 --> 01:14:26,820 go into the frequency domain as a way of dealing with the 1203 01:14:26,820 --> 01:14:30,110 properties of those samples, and all you're doing is just 1204 01:14:30,110 --> 01:14:34,690 going into the Fourier transform of u of t, and all 1205 01:14:34,690 --> 01:14:37,550 of that works out. 1206 01:14:42,780 --> 01:14:46,370 There's one bizarre thing here, and I'm going to talk 1207 01:14:46,370 --> 01:14:50,170 about that more next time. 1208 01:14:50,170 --> 01:14:54,340 That is that when you look at this time frequency limited 1209 01:14:54,340 --> 01:15:03,080 function, u hat of f, u hat of f can be very badly behaved. 1210 01:15:03,080 --> 01:15:06,340 You can have a frequency limited function which does 1211 01:15:06,340 --> 01:15:08,330 all sorts of crazy things. 1212 01:15:08,330 --> 01:15:11,770 Since it's frequency limited as inverse transform, it's 1213 01:15:11,770 --> 01:15:12,820 beautifully behaved. 1214 01:15:12,820 --> 01:15:15,460 It's just the sum of sinc functions -- it's bounded, 1215 01:15:15,460 --> 01:15:18,580 it's continuous and everything else. 1216 01:15:18,580 --> 01:15:22,180 When we go back into the frequency domain it is just as 1217 01:15:22,180 --> 01:15:23,980 ugly as can be. 1218 01:15:23,980 --> 01:15:29,050 So what we have in the sampling theorem, it comes out 1219 01:15:29,050 --> 01:15:31,570 particularly clearly. 1220 01:15:31,570 --> 01:15:36,240 Is that L1 functions in one domain are nice, continuous, 1221 01:15:36,240 --> 01:15:39,910 beautiful functions which are not L1 in the other domain. 1222 01:15:39,910 --> 01:15:44,260 So you sort of go from L1 to continuous. 1223 01:15:44,260 --> 01:15:47,360 Now when we're dealing with L2 functions, they're not 1224 01:15:47,360 --> 01:15:50,630 continuous, they're not anything else, but you always 1225 01:15:50,630 --> 01:15:52,380 go from L2 to L2. 1226 01:15:52,380 --> 01:15:56,240 In other words, you can't get out of the L2 domain, and 1227 01:15:56,240 --> 01:16:01,140 therefore, when you're dealing with L2 functions, all you 1228 01:16:01,140 --> 01:16:04,700 have to worry about is L2 functions, because you always 1229 01:16:04,700 --> 01:16:06,890 stay there no matter what. 1230 01:16:06,890 --> 01:16:10,120 When we start talking about stochastic processes, we'll 1231 01:16:10,120 --> 01:16:12,510 find out you always stay there also. 1232 01:16:12,510 --> 01:16:15,450 In other words, you only have to know about one thing. 1233 01:16:15,450 --> 01:16:18,380 We've seen here that to interpret what these Fourier 1234 01:16:18,380 --> 01:16:23,230 transforms mean, it's nice to have a little idea about L1 1235 01:16:23,230 --> 01:16:28,400 functions also because when we think of going to the limit, 1236 01:16:28,400 --> 01:16:33,450 when we go to the limit we get something which is badly 1237 01:16:33,450 --> 01:16:37,390 behaved, and for these finite time and finite frequency 1238 01:16:37,390 --> 01:16:41,000 approximations, we have things which are beautifully behaved. 1239 01:16:41,000 --> 01:16:43,280 I'm going to stop now so we can pass out the quizzes.