1 00:00:00,000 --> 00:00:02,360 The following content is provided under a Creative 2 00:00:02,360 --> 00:00:03,630 Commons license. 3 00:00:03,630 --> 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 --> 00:00:16,780 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,780 --> 00:00:19,260 ocw.mit.edu. 8 00:00:19,260 --> 00:00:23,510 PROFESSOR: We started to talk about the discrete time 9 00:00:23,510 --> 00:00:27,080 Fourier transform last time. 10 00:00:27,080 --> 00:00:31,120 This is probably something you've been exposed to before, 11 00:00:31,120 --> 00:00:35,270 and as in many cases, we're looking at it in a very 12 00:00:35,270 --> 00:00:38,830 different way than what you probably looked at it before. 13 00:00:38,830 --> 00:00:42,900 The discrete time Fourier transform is simply the time 14 00:00:42,900 --> 00:00:46,900 frequency dual of the Fourier series. 15 00:00:46,900 --> 00:00:49,420 Nothing more than that. 16 00:00:49,420 --> 00:00:51,450 For any L2 function -- 17 00:00:54,070 --> 00:00:56,290 now we're talking about a function of frequency, but a 18 00:00:56,290 --> 00:00:57,660 function is a function. 19 00:00:57,660 --> 00:01:02,690 So this is a complex valued function of frequency. 20 00:01:02,690 --> 00:01:09,710 If it's limited, if it's truncated to the band from 21 00:01:09,710 --> 00:01:17,480 minus w to plus w, then, in fact, it's inverse Fourier 22 00:01:17,480 --> 00:01:21,460 transform is given by this limit in the mean of the sum 23 00:01:21,460 --> 00:01:24,520 of the coefficients times v sub k of f. 24 00:01:24,520 --> 00:01:27,640 This is exactly the same as the Fourier transform 25 00:01:27,640 --> 00:01:34,210 replacing times with frequencies, replacing w for t 26 00:01:34,210 --> 00:01:48,340 over 2, and in the complex frequency you're replacing a 27 00:01:48,340 --> 00:01:53,910 to the plus 2 pi i here with e to the minus 2 pi i in the 28 00:01:53,910 --> 00:01:54,940 Fourier series. 29 00:01:54,940 --> 00:01:57,420 So those are the only differences, it's just 30 00:01:57,420 --> 00:02:00,900 notational differences, and aside from that, it works 31 00:02:00,900 --> 00:02:02,300 exactly the same way. 32 00:02:02,300 --> 00:02:05,380 The coefficients are given by this. 33 00:02:05,380 --> 00:02:07,560 We showed, when we were talking about the Fourier 34 00:02:07,560 --> 00:02:11,980 series, that these coefficients exist as complex 35 00:02:11,980 --> 00:02:14,670 numbers, they're always finite. 36 00:02:14,670 --> 00:02:17,900 You can calculate them if you want to. 37 00:02:17,900 --> 00:02:21,900 This quantity here can be rather fishy. 38 00:02:21,900 --> 00:02:25,730 This is this limit in the mean which says that you have to 39 00:02:25,730 --> 00:02:30,060 calculate this by looking at the sum here over a finite 40 00:02:30,060 --> 00:02:33,600 sum, over a finite sum this is well-defined and 41 00:02:33,600 --> 00:02:35,140 behaves very nicely. 42 00:02:35,140 --> 00:02:38,860 As you go to the limit funny things can happen, but the 43 00:02:38,860 --> 00:02:41,940 thing that we showed is in terms of energy, nothing funny 44 00:02:41,940 --> 00:02:43,760 can happen. 45 00:02:43,760 --> 00:02:46,170 I'm going to give you an example of this kind of funny 46 00:02:46,170 --> 00:02:50,340 business as we talk about the sampling theorem in just a 47 00:02:50,340 --> 00:02:52,410 couple of minutes. 48 00:02:52,410 --> 00:02:57,080 So the u hat of f has to be L1 since it is 49 00:02:57,080 --> 00:02:59,030 limited in this way. 50 00:02:59,030 --> 00:03:02,280 It has a continuous inverse transform, which is this. 51 00:03:02,280 --> 00:03:08,430 OK, so you can go from -- 52 00:03:08,430 --> 00:03:10,080 blah blah blah blah blah. 53 00:03:10,080 --> 00:03:13,500 The discrete Fourier transform is simply a transform between 54 00:03:13,500 --> 00:03:17,260 a function and a sequence of terms. 55 00:03:17,260 --> 00:03:20,940 Now this bit here is something we haven't 56 00:03:20,940 --> 00:03:23,690 talked about before. 57 00:03:23,690 --> 00:03:27,560 Because now we've talked about Fourier transforms also, and 58 00:03:27,560 --> 00:03:31,900 if you have an L2 function, u hat of f, that L2 function has 59 00:03:31,900 --> 00:03:36,700 a Fourier transform and an inverse Fourier transform in 60 00:03:36,700 --> 00:03:41,320 this case, which is u of t, which is given by this 61 00:03:41,320 --> 00:03:43,270 expression here, the usual expression 62 00:03:43,270 --> 00:03:45,050 for the Fourier transform. 63 00:03:45,050 --> 00:03:49,240 Now, the thing that's peculiar about this Fourier transform 64 00:03:49,240 --> 00:03:56,200 is that since u hat of f is limited, is truncated, this 65 00:03:56,200 --> 00:03:58,820 transform here exists everywhere. 66 00:03:58,820 --> 00:04:01,250 We don't need a limit in the mean here. 67 00:04:01,250 --> 00:04:04,280 You can calculate this for every t this exists. 68 00:04:04,280 --> 00:04:07,380 This is a point-wise convergent thing. 69 00:04:07,380 --> 00:04:10,770 This is the thing that you're used to when you think of 70 00:04:10,770 --> 00:04:14,230 functions, because the function is defined everywhere 71 00:04:14,230 --> 00:04:15,770 in this case. 72 00:04:15,770 --> 00:04:18,380 And since we're going to go into the sampling theorem, it 73 00:04:18,380 --> 00:04:21,260 had damned well better be defined everywhere because 74 00:04:21,260 --> 00:04:23,210 otherwise the sampling theorem wouldn't make 75 00:04:23,210 --> 00:04:24,460 any sense at all. 76 00:04:30,490 --> 00:04:33,830 So, the inverse Fourier transform of the discrete time 77 00:04:33,830 --> 00:04:36,750 Fourier transform, as we said, is this, it's a 78 00:04:36,750 --> 00:04:38,450 limit in the mean. 79 00:04:38,450 --> 00:04:40,790 Namely, you take more and more terms here. 80 00:04:40,790 --> 00:04:45,250 You get closer and closer to this in terms of energy. 81 00:04:45,250 --> 00:04:48,000 It doesn't say anything about what happens for particular 82 00:04:48,000 --> 00:04:50,330 values of f. 83 00:04:50,330 --> 00:04:57,480 This is a sampling expansion with t equals 1 over 2w. 84 00:05:14,080 --> 00:05:19,770 OK, let's go back here. 85 00:05:19,770 --> 00:05:23,330 We're talking about some function of frequency which 86 00:05:23,330 --> 00:05:29,150 has a Fourier series, it also has a Fourier transform. 87 00:05:29,150 --> 00:05:32,750 What we're interested in now is what is the relationship 88 00:05:32,750 --> 00:05:36,190 between these coefficients here and a discrete time 89 00:05:36,190 --> 00:05:39,910 Fourier transform and this function here. 90 00:05:39,910 --> 00:05:42,300 What I want to show you is that, in fact, these two 91 00:05:42,300 --> 00:05:45,140 things are very closely related. 92 00:05:45,140 --> 00:05:48,760 You can now go back to the Fourier series itself and 93 00:05:48,760 --> 00:05:53,090 relate the Fourier series coefficients to the Fourier 94 00:05:53,090 --> 00:05:55,810 transform and you'll get the same kind of sampling 95 00:05:55,810 --> 00:05:58,660 representation that we're getting right now. 96 00:06:03,550 --> 00:06:32,360 So, if we -- there's something missing in what I'm 97 00:06:32,360 --> 00:06:33,610 trying to say here. 98 00:06:37,860 --> 00:06:40,770 Oh, I see what I'm trying to do, sorry. 99 00:06:40,770 --> 00:06:43,740 What I'm trying to do is to take the inverse Fourier 100 00:06:43,740 --> 00:06:46,480 transform of u hat of f, which is given by 101 00:06:46,480 --> 00:06:48,030 this expression here. 102 00:06:48,030 --> 00:06:50,330 Temporarily I'm going to forget about the fact that 103 00:06:50,330 --> 00:06:53,540 this is a limit in the mean, throw mathematics to the 104 00:06:53,540 --> 00:06:57,030 winds, and simply take this inverse transform. 105 00:06:57,030 --> 00:06:59,730 Take the inverse transform of this also. 106 00:06:59,730 --> 00:07:03,230 So, when I take the inverse transform of the sum, what I'm 107 00:07:03,230 --> 00:07:08,370 going to get is the sum over k of u sub k, and then in place 108 00:07:08,370 --> 00:07:13,780 of this frequency function -- these are orthogonal functions 109 00:07:13,780 --> 00:07:16,730 here, the things I listed on the previous page -- 110 00:07:16,730 --> 00:07:20,270 I got u of t is the sum of u sub k times these time 111 00:07:20,270 --> 00:07:21,980 functions now. 112 00:07:21,980 --> 00:07:28,950 These time functions are just the Fourier transforms of 113 00:07:28,950 --> 00:07:32,120 these frequency functions here. 114 00:07:32,120 --> 00:07:36,190 This is a set of orthogonal wave forms here, which are 115 00:07:36,190 --> 00:07:38,230 truncated sinusoids. 116 00:07:38,230 --> 00:07:42,280 I want to take the Fourier transform of these and the 117 00:07:42,280 --> 00:07:45,240 Fourier transforms of these -- 118 00:07:45,240 --> 00:07:47,810 I should have put them both on the same slide so you could 119 00:07:47,810 --> 00:07:49,810 see what they are -- 120 00:07:49,810 --> 00:07:53,140 but in fact, in your homework you're going to take the 121 00:07:53,140 --> 00:07:57,190 inverse Fourier transform of that and show that it is, in 122 00:07:57,190 --> 00:07:59,040 fact, this. 123 00:07:59,040 --> 00:08:03,710 That's just a nice exercise in taking rectangular functions 124 00:08:03,710 --> 00:08:09,420 and sinc functions and applying shifts in both time 125 00:08:09,420 --> 00:08:10,890 and frequency. 126 00:08:10,890 --> 00:08:16,240 When you do this, this turns out to have the inverse 127 00:08:16,240 --> 00:08:17,590 transform of this. 128 00:08:17,590 --> 00:08:21,530 So this is u of t is just this, where this comes from 129 00:08:21,530 --> 00:08:26,580 here, and then this turns out to be the inverse transform of 130 00:08:26,580 --> 00:08:31,220 this function here, which, in fact, is just this. 131 00:08:34,960 --> 00:08:36,210 Sorry for all of that. 132 00:08:40,400 --> 00:08:43,690 So now if we want to try to understand what this means, 133 00:08:43,690 --> 00:08:48,020 suppose that you take a function u of t, which is, in 134 00:08:48,020 --> 00:08:52,190 fact, the inverse transform of this u hat of f, which is 135 00:08:52,190 --> 00:08:56,810 truncated to a finite band limit. 136 00:08:56,810 --> 00:09:03,010 If we then take v of t equal to u of t everywhere except on 137 00:09:03,010 --> 00:09:06,920 the sample points, and on every sample point we simply 138 00:09:06,920 --> 00:09:11,060 add 1 to v -- in other words, v of kt is equal to 139 00:09:11,060 --> 00:09:14,190 u of kT plus 1. 140 00:09:14,190 --> 00:09:18,150 So we take this nice smooth function that we have here, 141 00:09:18,150 --> 00:09:28,800 and at every sample point we simply add 1. 142 00:09:35,490 --> 00:09:39,610 So now the question is, is this new function I have still 143 00:09:39,610 --> 00:09:41,620 baseband limited or isn't it? 144 00:09:46,620 --> 00:09:49,750 You see you can't answer that question because we weren't 145 00:09:49,750 --> 00:09:52,790 careful enough to say what we meant by a 146 00:09:52,790 --> 00:09:55,760 baseband limited function. 147 00:09:55,760 --> 00:09:59,050 Usually when you talk about a baseband limited function, 148 00:09:59,050 --> 00:10:01,540 you're talking about a function whose Fourier 149 00:10:01,540 --> 00:10:07,830 transform is zero, except in the range minus w to plus w. 150 00:10:07,830 --> 00:10:12,750 Well, this new function v of t here has the property that its 151 00:10:12,750 --> 00:10:17,770 Fourier transform is zero outside of minus w to plus w. 152 00:10:20,550 --> 00:10:25,980 And therefore if you define baseband limited as functions 153 00:10:25,980 --> 00:10:30,770 whose Fourier transform is zero outside of limits, then 154 00:10:30,770 --> 00:10:34,790 the sampling theorem doesn't hold. 155 00:10:34,790 --> 00:10:37,140 So what do we do about this? 156 00:10:37,140 --> 00:10:40,940 Well, the easiest thing to do about it is to change what we 157 00:10:40,940 --> 00:10:45,580 mean by baseband limited to what you would have meant if 158 00:10:45,580 --> 00:10:49,410 we hadn't going through all of this mathematics. 159 00:10:49,410 --> 00:10:52,500 In other words, the Fourier transform cuts both ways -- u 160 00:10:52,500 --> 00:10:57,010 of t has a Fourier transform, u hat of f, u hat of f has an 161 00:10:57,010 --> 00:10:59,590 inverse transform, u of t. 162 00:10:59,590 --> 00:11:05,000 What we mean now by baseband limited is that u of t is the 163 00:11:05,000 --> 00:11:09,020 inverse transform of a frequency function, which is 164 00:11:09,020 --> 00:11:11,830 limited to minus w over plus w. 165 00:11:11,830 --> 00:11:16,640 In other words, we will regard u of t here as being baseband 166 00:11:16,640 --> 00:11:19,120 limited, we will not regard v of t as 167 00:11:19,120 --> 00:11:21,830 being baseband limited. 168 00:11:21,830 --> 00:11:25,490 Because we have these two functions, u of t and v of t, 169 00:11:25,490 --> 00:11:29,380 which both have the same Fourier transforms, but which 170 00:11:29,380 --> 00:11:33,050 are not equal to each other. 171 00:11:33,050 --> 00:11:37,050 We want to, at this point, say that the only one which is 172 00:11:37,050 --> 00:11:41,630 really baseband limited is the function which is the inverse 173 00:11:41,630 --> 00:11:43,160 transform of u of t. 174 00:11:43,160 --> 00:11:45,860 If you read the sampling theorem as it's written in the 175 00:11:45,860 --> 00:11:49,430 notes, that's exactly what it says. 176 00:11:49,430 --> 00:11:53,200 It defines this function u of t to which the sampling 177 00:11:53,200 --> 00:11:55,290 theorem applies in that way. 178 00:11:55,290 --> 00:11:57,100 Well anyway, this is the sampling 179 00:11:57,100 --> 00:11:58,620 theorem at this point. 180 00:11:58,620 --> 00:12:01,310 It says you can take u of t, you can 181 00:12:01,310 --> 00:12:03,330 express it in this way. 182 00:12:03,330 --> 00:12:10,870 If I go back to the previous slide, what we did here by 183 00:12:10,870 --> 00:12:19,390 comparing u of t with this expression for u sub k. u sub 184 00:12:19,390 --> 00:12:22,660 k are the coefficients in the discrete time Fourier 185 00:12:22,660 --> 00:12:31,510 transform. u of t is this inverse transform. 186 00:12:31,510 --> 00:12:34,620 These quantities here are almost the same. 187 00:12:34,620 --> 00:12:39,530 This quantity here is just evaluating this at particular 188 00:12:39,530 --> 00:12:40,300 frequencies. 189 00:12:40,300 --> 00:12:45,060 Namely, if for frequency f I substitute in, if we're time 190 00:12:45,060 --> 00:12:54,930 t, I substitute in k over 2w, then this formula just becomes 191 00:12:54,930 --> 00:12:55,820 that formula. 192 00:12:55,820 --> 00:13:01,720 In other words, 2w times u sub k is equal to u 193 00:13:01,720 --> 00:13:03,450 evaluated at k over 2w. 194 00:13:07,310 --> 00:13:10,130 You already saw that when you looked at the Fourier series. 195 00:13:10,130 --> 00:13:12,630 When you looked at the Fourier series, you saw these 196 00:13:12,630 --> 00:13:15,590 coefficients, which, in fact, look like the Fourier 197 00:13:15,590 --> 00:13:17,930 transform terms. 198 00:13:17,930 --> 00:13:20,870 And which, in fact, were the same as the Fourier transform 199 00:13:20,870 --> 00:13:27,760 except for a scale factor at some particular frequency. 200 00:13:27,760 --> 00:13:31,970 Here we have these coefficients, which are, in 201 00:13:31,970 --> 00:13:35,980 fact, scaled versions of the inverse transform of 202 00:13:35,980 --> 00:13:37,750 particular times. 203 00:13:37,750 --> 00:13:41,970 So the conclusion from that is that, in fact, what the 204 00:13:41,970 --> 00:13:46,780 discrete time Fourier transform is is it's simply 205 00:13:46,780 --> 00:13:50,530 the Fourier transform of the sampling theorem expansion. 206 00:13:50,530 --> 00:13:55,940 The two of them are duals in a very different way than the 207 00:13:55,940 --> 00:14:02,830 Fourier series and the dtft are duals. 208 00:14:02,830 --> 00:14:07,650 The dtft and the Fourier series are duals in the sense 209 00:14:07,650 --> 00:14:12,300 that if you take the expressions for the Fourier 210 00:14:12,300 --> 00:14:16,560 series and change frequency for time and time for 211 00:14:16,560 --> 00:14:19,810 frequency, you get to the dtft. 212 00:14:19,810 --> 00:14:23,670 Here what we're doing is taking the dtft and simply 213 00:14:23,670 --> 00:14:28,500 taking the inverse Fourier transform of it, so that the 214 00:14:28,500 --> 00:14:31,700 sampling theorem is, in fact, the Fourier 215 00:14:31,700 --> 00:14:34,710 transform of the dtft. 216 00:14:34,710 --> 00:14:37,440 It's not the dual, it's the Fourier transform itself. 217 00:14:45,350 --> 00:14:50,830 Well, the discrete time Fourier transform generalizes 218 00:14:50,830 --> 00:14:54,820 to arbitrary frequency intervals just as well as to a 219 00:14:54,820 --> 00:14:56,170 baseband interval. 220 00:14:56,170 --> 00:14:59,660 Namely, you can do exactly the same thing as what we've just 221 00:14:59,660 --> 00:15:05,060 done if you're looking not at the frequency at the range of 222 00:15:05,060 --> 00:15:09,000 frequencies from minus w to plus w, but you shift that up 223 00:15:09,000 --> 00:15:13,040 to any old place you want to and look at delta minus w to 224 00:15:13,040 --> 00:15:14,840 delta plus w. 225 00:15:14,840 --> 00:15:17,990 And the fact the discrete time Fourier transform, if you 226 00:15:17,990 --> 00:15:20,560 don't put the rectangular function in it is going to be 227 00:15:20,560 --> 00:15:22,070 periodic anyway. 228 00:15:22,070 --> 00:15:24,140 It's exactly the same thing that we had 229 00:15:24,140 --> 00:15:26,200 with a Fourier series. 230 00:15:26,200 --> 00:15:29,270 With a Fourier series, we could find the Fourier series 231 00:15:29,270 --> 00:15:33,230 for a function limited between minus t over 2 232 00:15:33,230 --> 00:15:35,030 to plus t over 2. 233 00:15:35,030 --> 00:15:39,420 Now by duality, we can find the dtft for a function 234 00:15:39,420 --> 00:15:44,960 limited between minus w plus delta and plus w plus delta. 235 00:15:44,960 --> 00:15:47,150 That's what we're doing here. 236 00:15:47,150 --> 00:15:54,980 So the dtft in generalized form is now this, and v sub k 237 00:15:54,980 --> 00:16:00,050 is now the integral from delta minus w to delta plus w of the 238 00:16:00,050 --> 00:16:02,780 same old thing as before. 239 00:16:02,780 --> 00:16:06,860 This is equal to this, which is the same old thing as 240 00:16:06,860 --> 00:16:15,140 before, except we now have this shifted frequency in the 241 00:16:15,140 --> 00:16:17,830 rectangular function. 242 00:16:17,830 --> 00:16:22,450 So if we take the inverse Fourier transform of this, 243 00:16:22,450 --> 00:16:27,500 again, using the same duality we had before, we get v of t 244 00:16:27,500 --> 00:16:31,510 is equal to the sum times the sinc function. 245 00:16:35,230 --> 00:16:36,780 And the only difference now that we're expanding this 246 00:16:36,780 --> 00:16:40,240 given frequency band not centered around zero but 247 00:16:40,240 --> 00:16:43,460 centered around something else, the only difference in 248 00:16:43,460 --> 00:16:47,380 the sampling theorem is now we have this rotating term 249 00:16:47,380 --> 00:16:55,260 gyrating around up at this frequency, k over 2w. 250 00:16:55,260 --> 00:17:00,080 That's the only way in which this is different from the 251 00:17:00,080 --> 00:17:02,430 sampling theorem that we had before. 252 00:17:06,270 --> 00:17:08,810 I wish I could put more things on one slide but you wouldn't 253 00:17:08,810 --> 00:17:10,740 see them if I could. 254 00:17:10,740 --> 00:17:16,450 So here's a Fourier transform of the sum 1 over t, u of kt 255 00:17:16,450 --> 00:17:18,090 time the sinc function. 256 00:17:18,090 --> 00:17:20,500 Here it's the same thing. 257 00:17:23,190 --> 00:17:26,410 That's one reason for comparing 258 00:17:26,410 --> 00:17:27,660 these things sometimes. 259 00:17:36,600 --> 00:17:37,970 1 over t in here. 260 00:17:43,970 --> 00:17:49,600 OK, times, so it's the same thing with this rotating term 261 00:17:49,600 --> 00:17:51,290 which is the only difference. 262 00:17:51,290 --> 00:17:56,210 Now, how many of you can see that the Fourier transform of 263 00:17:56,210 --> 00:18:00,910 this quantity is equal to this sinc function? 264 00:18:00,910 --> 00:18:01,780 I can't do that. 265 00:18:01,780 --> 00:18:05,080 It's one of the things you're going to do in your homework. 266 00:18:05,080 --> 00:18:08,410 I get confused every time I do it, and I got confused enough 267 00:18:08,410 --> 00:18:12,800 to leave out the 1 over t this time when I did it. 268 00:18:12,800 --> 00:18:18,750 You just have to be patient with that and do it a few 269 00:18:18,750 --> 00:18:23,360 times and you'll find that it's not -- 270 00:18:23,360 --> 00:18:26,720 well, it becomes automatic after awhile. 271 00:18:26,720 --> 00:18:28,840 It's one place where you need plug and chug. 272 00:18:31,560 --> 00:18:36,170 So now that we've generalized the dtft to look at any old 273 00:18:36,170 --> 00:18:39,250 frequency band instead of just the frequency band around 274 00:18:39,250 --> 00:18:42,290 zero, we can do the same thing that we 275 00:18:42,290 --> 00:18:44,170 did with time functions. 276 00:18:44,170 --> 00:18:46,000 Namely, with a time function -- 277 00:18:46,000 --> 00:18:46,210 Yes? 278 00:18:46,210 --> 00:18:49,600 AUDIENCE: [INAUDIBLE]. 279 00:18:49,600 --> 00:18:51,510 PROFESSOR: You don't think it should be a 1 over t? 280 00:18:51,510 --> 00:18:58,030 Well, you very well might be right because I didn't think 281 00:18:58,030 --> 00:19:00,820 it should have been either when I wrote it down. 282 00:19:00,820 --> 00:19:03,280 But I don't see -- 283 00:19:03,280 --> 00:19:04,530 AUDIENCE: [INAUDIBLE]. 284 00:19:09,560 --> 00:19:10,550 PROFESSOR: It was the--. 285 00:19:10,550 --> 00:19:11,800 AUDIENCE: [INAUDIBLE]. 286 00:19:22,630 --> 00:19:25,080 PROFESSOR: Oh, that's the difference, yes, of course. 287 00:19:27,940 --> 00:19:29,950 It's not the coefficient that I have here, 288 00:19:29,950 --> 00:19:32,400 it's the actual -- 289 00:19:32,400 --> 00:19:35,180 yeah. 290 00:19:35,180 --> 00:19:36,430 Yes. 291 00:19:40,780 --> 00:19:42,340 So it should be the same as the-- 292 00:19:54,610 --> 00:19:55,620 Oh, I see the problem. 293 00:19:55,620 --> 00:19:57,940 I shouldn't have had the 1 over t here, should I? 294 00:20:00,860 --> 00:20:02,110 No. 295 00:20:06,580 --> 00:20:09,900 I know one of them couldn't be right. 296 00:20:09,900 --> 00:20:15,500 It is right in the notes, so you can sort it out there. 297 00:20:20,100 --> 00:20:25,480 So, the thing we did when we were dealing with a Fourier 298 00:20:25,480 --> 00:20:30,140 series is we took an arbitrary function of time, we segmented 299 00:20:30,140 --> 00:20:33,450 it into time intervals and then we expanded each one of 300 00:20:33,450 --> 00:20:36,870 those time intervals into a Fourier series. 301 00:20:36,870 --> 00:20:40,740 By doing that we could take an arbitrary L2 function and 302 00:20:40,740 --> 00:20:46,320 represent it as an orthogonal expansion over this double sum 303 00:20:46,320 --> 00:20:50,720 of time shifts and frequency terms. 304 00:20:50,720 --> 00:20:57,070 We call that the truncated sinusoidal expansion, the t 305 00:20:57,070 --> 00:21:01,270 spaced truncated sinusoids and we made an expansion out of 306 00:21:01,270 --> 00:21:05,100 that that would allow us to express any old L2 function in 307 00:21:05,100 --> 00:21:06,870 terms of that. 308 00:21:06,870 --> 00:21:08,700 Here we're going to do the same thing. 309 00:21:08,700 --> 00:21:12,390 We can take an arbitrary frequency function, separate 310 00:21:12,390 --> 00:21:14,260 into bands of frequencies. 311 00:21:14,260 --> 00:21:18,610 You often want to do this in digital communication when 312 00:21:18,610 --> 00:21:21,750 you're looking at transmitting information in different 313 00:21:21,750 --> 00:21:24,830 bands, which you do all the time in radio. 314 00:21:24,830 --> 00:21:27,850 Somebody has a certain part of the spectrum, they transmit a 315 00:21:27,850 --> 00:21:30,430 signal there, somebody else has another part of the 316 00:21:30,430 --> 00:21:33,970 spectrum, they transmit a signal there. 317 00:21:33,970 --> 00:21:37,060 You can look at those different signals which are in 318 00:21:37,060 --> 00:21:39,960 different frequency bands, they're all orthogonal to each 319 00:21:39,960 --> 00:21:43,770 other, they don't interfere with each other at all. 320 00:21:43,770 --> 00:21:45,340 We're doing the same thing here. 321 00:21:45,340 --> 00:21:48,780 We're just saying an arbitrary function can be split into 322 00:21:48,780 --> 00:21:52,300 different frequency bands, each one of those frequency 323 00:21:52,300 --> 00:21:58,660 bands can be represented both by a dtft, which is the thing 324 00:21:58,660 --> 00:22:02,040 we just did on the last slide, and by sampling theorem 325 00:22:02,040 --> 00:22:05,940 expression, which is what we get when we take the inverse 326 00:22:05,940 --> 00:22:10,240 Fourier transform of the dtft. 327 00:22:10,240 --> 00:22:14,600 So when we do that what we get is a perfectly arbitrary 328 00:22:14,600 --> 00:22:20,140 frequency function which exists from minus infinity to 329 00:22:20,140 --> 00:22:21,720 plus infinity. 330 00:22:21,720 --> 00:22:26,440 We can represent it as the sum of all these separate 331 00:22:26,440 --> 00:22:28,000 frequency functions. 332 00:22:28,000 --> 00:22:31,420 I just threw a limit in the mean here because I'm not 333 00:22:31,420 --> 00:22:36,730 being careful about what happens where we separate from 334 00:22:36,730 --> 00:22:39,000 one frequency function to the next. 335 00:22:39,000 --> 00:22:43,860 Namely, at frequency w do I use one term or do I use the 336 00:22:43,860 --> 00:22:45,840 other term or do I use the sum. 337 00:22:45,840 --> 00:22:47,480 But we don't want to worry about that. 338 00:22:47,480 --> 00:22:49,440 We don't want to even think about it. 339 00:22:49,440 --> 00:22:51,730 So we put a limit in the mean here. 340 00:22:51,730 --> 00:22:59,990 So the v sub hat m of f then is going to be the part of u 341 00:22:59,990 --> 00:23:05,140 of f which is in this particular frequency range. 342 00:23:05,140 --> 00:23:08,540 That's completely the analogy of taking a time function, 343 00:23:08,540 --> 00:23:11,810 looking at that time function over a particular range of 344 00:23:11,810 --> 00:23:16,300 time, and here what we're doing is taking a frequency 345 00:23:16,300 --> 00:23:20,920 function, segementing it into different frequency intervals 346 00:23:20,920 --> 00:23:27,330 so that the end frequency interval is then just this 347 00:23:27,330 --> 00:23:30,820 with a rectangular function to truncate it. 348 00:23:30,820 --> 00:23:35,570 If I take the inverse Fourier transform of this what I'm 349 00:23:35,570 --> 00:23:41,440 going to get is u of t, take the inverse transform of all 350 00:23:41,440 --> 00:23:45,750 of these terms, so I'll get the sum of vm of t. 351 00:23:45,750 --> 00:23:53,140 Now, vm of t is the inverse Fourier transform of vm of f. 352 00:23:53,140 --> 00:23:57,210 You take that quantity, take the inverse transform of it, 353 00:23:57,210 --> 00:24:01,080 and sure enough you get this kind of expression here. 354 00:24:01,080 --> 00:24:09,350 It's a sampling theorem in v sub m of f with this rotating 355 00:24:09,350 --> 00:24:11,240 frequency term here, which is just the 356 00:24:11,240 --> 00:24:13,940 thing that we had before. 357 00:24:13,940 --> 00:24:17,280 So, all we're doing here is starting out with some 358 00:24:17,280 --> 00:24:21,030 arbitrary frequency function, we're segementing it in 359 00:24:21,030 --> 00:24:22,610 frequencies. 360 00:24:22,610 --> 00:24:28,920 Each frequency band then has a dtft associated with it. 361 00:24:28,920 --> 00:24:33,070 When we take the inverse Fourier transform of that 362 00:24:33,070 --> 00:24:38,480 dtft, what we get is a sampling expansion for that 363 00:24:38,480 --> 00:24:40,270 particular frequency band. 364 00:24:40,270 --> 00:24:46,180 So what this is doing here, finally when we get all done 365 00:24:46,180 --> 00:24:50,120 with this, is I'm just combining the sampling theorem 366 00:24:50,120 --> 00:24:53,830 expansion in each frequency range, which 367 00:24:53,830 --> 00:24:56,790 is what u of t is. 368 00:24:56,790 --> 00:25:00,390 So I've taken u of t, I split it up into different frequency 369 00:25:00,390 --> 00:25:05,110 ranges, I've expressed what's in each frequency range in 370 00:25:05,110 --> 00:25:07,850 terms of a sampling theorem. 371 00:25:07,850 --> 00:25:11,330 The sampling theorem terms are these with these rotating 372 00:25:11,330 --> 00:25:17,990 terms in them corresponding to the mth frequency range. 373 00:25:17,990 --> 00:25:24,680 This is completely analogous then to the truncated sinc 374 00:25:24,680 --> 00:25:27,180 function expansion we had before. 375 00:25:30,470 --> 00:25:36,090 This becomes a little more sensible if we substitute a 376 00:25:36,090 --> 00:25:40,020 sampling time, t, for 1 over 2w. 377 00:25:40,020 --> 00:25:43,590 Namely, all of these expressions here are talking 378 00:25:43,590 --> 00:25:48,600 about what happens when you sample these individual 379 00:25:48,600 --> 00:25:52,530 frequency bands at intervals 1 over 2w. 380 00:25:52,530 --> 00:25:56,130 So we'll just call that capital T to make the formula 381 00:25:56,130 --> 00:25:58,140 look a little simpler. 382 00:25:58,140 --> 00:26:02,620 Then we get u of t is this limit in the mean of this 383 00:26:02,620 --> 00:26:05,690 whole expression there. 384 00:26:05,690 --> 00:26:10,260 So it's a double sum, it's a sum over time, over the 385 00:26:10,260 --> 00:26:16,870 samples, so there's one term for each time, kT, and there's 386 00:26:16,870 --> 00:26:20,610 one term for each frequency interval. 387 00:26:20,610 --> 00:26:26,550 Frequencies are indexed by m, time is indexed by k. 388 00:26:26,550 --> 00:26:30,170 So the thing we have here is an expansion now in terms of 389 00:26:30,170 --> 00:26:35,120 coefficients -- these are just called coefficients again. 390 00:26:35,120 --> 00:26:44,780 This expansion which looks suspiciously like the t space 391 00:26:44,780 --> 00:26:47,720 truncated sinusoids that we had before. 392 00:26:47,720 --> 00:26:52,990 The only difference is that, the terms were truncated in 393 00:26:52,990 --> 00:26:58,810 time; here, the terms are truncated in frequency. 394 00:26:58,810 --> 00:27:03,110 So the different terms making up this expansion, these 395 00:27:03,110 --> 00:27:08,900 orthogonal terms here, in one case what we have is a sinc 396 00:27:08,900 --> 00:27:12,710 function which is translated in time and then 397 00:27:12,710 --> 00:27:15,200 translated in frequency. 398 00:27:15,200 --> 00:27:19,070 In another case we have the rectangular function, which is 399 00:27:19,070 --> 00:27:23,320 translated in time and then translated in frequency. 400 00:27:23,320 --> 00:27:25,970 So in that sense, these two expansions are almost the 401 00:27:25,970 --> 00:27:31,220 same, and you can think of doing expansions perhaps in 402 00:27:31,220 --> 00:27:36,500 other things also and we'll talk more about that later. 403 00:27:36,500 --> 00:27:47,330 So, this then is just this thing we're going to call a t 404 00:27:47,330 --> 00:27:51,600 spaced sinc weighted sinusoid expansion. 405 00:27:51,600 --> 00:27:54,930 So the only thing we have is this one sinc function which 406 00:27:54,930 --> 00:27:58,160 is this hat sort of function. 407 00:27:58,160 --> 00:28:03,120 The terms in here are those functions shifted in time by 408 00:28:03,120 --> 00:28:08,740 some number of sampling intervals, t, and then shifted 409 00:28:08,740 --> 00:28:14,830 in frequency by some number of frequency bands, 2w. 410 00:28:14,830 --> 00:28:17,560 See, the original frequency bands that we had went from 411 00:28:17,560 --> 00:28:19,910 minus w to plus w. 412 00:28:19,910 --> 00:28:23,130 The next one goes from w to 3w, the next one 413 00:28:23,130 --> 00:28:25,080 goes from 3w to 5w. 414 00:28:25,080 --> 00:28:29,360 So the frequency bands we're talking about here 415 00:28:29,360 --> 00:28:32,260 are of width 2w. 416 00:28:32,260 --> 00:28:38,140 The time intervals we're talking about are of width t. 417 00:28:38,140 --> 00:28:40,030 Why do people confuse you that way? 418 00:28:42,550 --> 00:28:46,160 Well, because all of this happened a long time before 419 00:28:46,160 --> 00:28:52,440 people realized how closely the duality relationship 420 00:28:52,440 --> 00:28:54,730 between time and frequency was. 421 00:28:54,730 --> 00:28:58,420 So people wanted to talk about frequencies, baseband 422 00:28:58,420 --> 00:28:59,020 frequencies. 423 00:28:59,020 --> 00:29:03,240 You talk about a baseband limited to w and you're 424 00:29:03,240 --> 00:29:06,750 talking about positive frequencies, because engineers 425 00:29:06,750 --> 00:29:11,740 used to deal with cosines and sine, and there weren't any 426 00:29:11,740 --> 00:29:14,010 such thing as negative frequencies. 427 00:29:14,010 --> 00:29:17,170 Then they decided everything was easier when they dealt 428 00:29:17,170 --> 00:29:21,230 with complex sinusoids, negative frequencies reared 429 00:29:21,230 --> 00:29:24,990 their ugly head, but people didn't want to change their 430 00:29:24,990 --> 00:29:28,490 notation for what a frequency band was, 431 00:29:28,490 --> 00:29:30,600 which is probably good. 432 00:29:30,600 --> 00:29:34,130 So we're simply stuck with this incompatibility of 433 00:29:34,130 --> 00:29:38,240 dealing with frequencies one way and dealing with 434 00:29:38,240 --> 00:29:39,490 time the other way. 435 00:29:45,980 --> 00:29:49,900 We can look at that as increments of time t, and 436 00:29:49,900 --> 00:29:54,170 increments of frequency, 1 over t, but, in fact, 1 over t 437 00:29:54,170 --> 00:29:58,100 is 2w, so the increments in time we're using in both of 438 00:29:58,100 --> 00:30:02,140 these expansions, are t, the increments in frequency we're 439 00:30:02,140 --> 00:30:05,510 using are 2w. 440 00:30:05,510 --> 00:30:10,140 Now, there's a relatively long section in the notes talking 441 00:30:10,140 --> 00:30:13,030 about degrees of freedom, which is a 442 00:30:13,030 --> 00:30:15,400 pretty important topic. 443 00:30:15,400 --> 00:30:20,940 It's a little bit fishy mathematically, but it really 444 00:30:20,940 --> 00:30:22,550 makes good engineering sense. 445 00:30:26,310 --> 00:30:30,910 It's an idea which is important both in terms of 446 00:30:30,910 --> 00:30:35,870 taking source wave forms and representing them in 447 00:30:35,870 --> 00:30:42,130 orthogonal expansions, and in taking frequency functions and 448 00:30:42,130 --> 00:30:45,400 representing them -- 449 00:30:45,400 --> 00:30:51,040 well, it is also important in terms of taking things that we 450 00:30:51,040 --> 00:30:57,950 transmit where you have bits coming into an encoder. 451 00:30:57,950 --> 00:31:01,570 We're going to turn those bits into signals, we're going to 452 00:31:01,570 --> 00:31:04,170 turn those signals into wave forms. 453 00:31:04,170 --> 00:31:07,480 Those wave forms will usually be thought of as things that 454 00:31:07,480 --> 00:31:10,650 we transmit in time, and we also transmit them in 455 00:31:10,650 --> 00:31:16,310 frequency, because we often use some kind of multiplexing 456 00:31:16,310 --> 00:31:18,120 between different frequency bands. 457 00:31:18,120 --> 00:31:20,990 We want to have a common way of thinking about all of these 458 00:31:20,990 --> 00:31:26,120 things, and this is the way that we're going to do it. 459 00:31:26,120 --> 00:31:29,370 Namely, if we're thinking in terms of a particular sampling 460 00:31:29,370 --> 00:31:32,610 time, t, and we want to look at a very, very large 461 00:31:32,610 --> 00:31:36,040 frequency band, and therefore, look at many multiplex 462 00:31:36,040 --> 00:31:40,380 frequency bands, we can say how many coefficients can we 463 00:31:40,380 --> 00:31:43,500 send on this channel? 464 00:31:43,500 --> 00:31:46,990 When we look at it in these terms of different frequency 465 00:31:46,990 --> 00:31:51,470 bands, the number of coefficients we can send is 466 00:31:51,470 --> 00:31:54,210 over a period of time, t zero. 467 00:31:54,210 --> 00:31:57,670 We can send t zero over t different 468 00:31:57,670 --> 00:32:00,520 coefficients in time. 469 00:32:00,520 --> 00:32:02,220 We now look at frequency. 470 00:32:02,220 --> 00:32:07,070 We have some very broad frequency band, w zero. 471 00:32:07,070 --> 00:32:10,950 The number of different bands that we have 472 00:32:10,950 --> 00:32:18,130 is w zero over 2w. 473 00:32:23,430 --> 00:32:26,110 As a result of all of this when you add everything up, 474 00:32:26,110 --> 00:32:31,720 you get 2t zero w zero degrees of freedom over this overall 475 00:32:31,720 --> 00:32:34,270 bandwidth of w zero. 476 00:32:34,270 --> 00:32:37,710 Now remember, an overall bandwidth of w zero in terms 477 00:32:37,710 --> 00:32:45,200 of these complex frequencies goes from minus w to plus w. 478 00:32:45,200 --> 00:32:47,790 Minus w zero to plus w zero. 479 00:32:47,790 --> 00:32:51,620 The time interval goes from minus t zero over 2 to plus t 480 00:32:51,620 --> 00:32:52,520 zero over 2. 481 00:32:52,520 --> 00:32:56,710 So this factor 2 here, which we always talk about in terms 482 00:32:56,710 --> 00:32:59,310 of number of degrees of freedom, is really a 483 00:32:59,310 --> 00:33:02,380 consequence of the fact that we measure time intervals and 484 00:33:02,380 --> 00:33:05,840 frequency intervals in a slightly different way. 485 00:33:05,840 --> 00:33:09,620 But anyway, whether we look at it in terms of one expansion 486 00:33:09,620 --> 00:33:12,850 or the other expansion the answer we get is the same. 487 00:33:12,850 --> 00:33:16,210 If you take some large time interval, some large frequency 488 00:33:16,210 --> 00:33:20,650 interval, tuck as many numbers as you can in that interval, 489 00:33:20,650 --> 00:33:22,820 this is what you come up with. 490 00:33:22,820 --> 00:33:27,540 Now, why did I say that this is just slightly fishy? 491 00:33:27,540 --> 00:33:31,660 Well, it's slightly fishy because if you take a function 492 00:33:31,660 --> 00:33:34,880 and you truncate it in time -- 493 00:33:34,880 --> 00:33:38,110 if we take this function and truncate it to minus t zero 494 00:33:38,110 --> 00:33:42,640 over 2 and plus t zero over 2, even though that might be ten 495 00:33:42,640 --> 00:33:47,920 years, how can we limit the frequency? 496 00:33:47,920 --> 00:33:48,640 Well, we can't. 497 00:33:48,640 --> 00:33:51,490 Because when we take the Fourier transform of a time 498 00:33:51,490 --> 00:33:56,410 limited function, it exists for all frequencies. 499 00:33:56,410 --> 00:34:00,540 The same thing happens if we try to limit it in frequency, 500 00:34:00,540 --> 00:34:03,650 it squirts out forever in time. 501 00:34:03,650 --> 00:34:05,370 So you can't get around that. 502 00:34:05,370 --> 00:34:10,420 The thing that saves us is that if t zero and w zero are 503 00:34:10,420 --> 00:34:14,270 both large enough, these functions all dribble away 504 00:34:14,270 --> 00:34:18,180 quickly enough that it doesn't make any difference. 505 00:34:18,180 --> 00:34:19,540 You know it has to. 506 00:34:19,540 --> 00:34:24,920 If you think in terms of the sampling theorem, and you try 507 00:34:24,920 --> 00:34:30,930 to think about it carefully in mathematics, what does it say? 508 00:34:30,930 --> 00:34:34,830 If you want to transmit a function by putting these 509 00:34:34,830 --> 00:34:38,550 little sine x over x hats around each of the 510 00:34:38,550 --> 00:34:42,140 coefficients in the function, when do you have to start 511 00:34:42,140 --> 00:34:45,830 transmitting those wave forms? 512 00:34:45,830 --> 00:34:49,670 You have to start transmitting them at minus infinity. 513 00:34:49,670 --> 00:34:52,650 I mean we turn on our transmitter and we somehow 514 00:34:52,650 --> 00:34:55,530 have to have been transmitting for an infinite amount of time 515 00:34:55,530 --> 00:34:59,020 before we send the first symbol. 516 00:34:59,020 --> 00:35:01,560 Well that's ridiculous, of course. 517 00:35:01,560 --> 00:35:06,450 So that we always approximate these sinusoids by sinusoids 518 00:35:06,450 --> 00:35:10,560 which are truncated, and we always have some engineering 519 00:35:10,560 --> 00:35:13,900 faith that what we're throwing away is not important. 520 00:35:13,900 --> 00:35:17,170 The only place that it's important is when you start 521 00:35:17,170 --> 00:35:20,740 talking about things which are really zero everywhere and 522 00:35:20,740 --> 00:35:22,100 then it becomes important. 523 00:35:22,100 --> 00:35:26,660 But the idea of degrees of freedom is a very sensible 524 00:35:26,660 --> 00:35:30,070 idea until you try to express it precisely. 525 00:35:33,570 --> 00:35:38,670 Let's get on to something called aliasing. 526 00:35:42,640 --> 00:35:46,360 We're going to spend most of the rest of today talking 527 00:35:46,360 --> 00:35:48,790 about aliasing. 528 00:35:48,790 --> 00:35:53,560 I want to try to explain why it is that we want to spend 529 00:35:53,560 --> 00:35:57,220 time on this, because there are really two 530 00:35:57,220 --> 00:35:59,530 things going on here. 531 00:35:59,530 --> 00:36:02,640 One of the things that are going on is that if you want 532 00:36:02,640 --> 00:36:07,070 to look at a wave form and do some processing on it, the 533 00:36:07,070 --> 00:36:11,190 usual way to do it with the digital technology we have 534 00:36:11,190 --> 00:36:15,130 today, is to take that wave form and sample it very, very 535 00:36:15,130 --> 00:36:17,710 rapidly in time. 536 00:36:17,710 --> 00:36:22,710 Then process the hell out of all those samples. 537 00:36:22,710 --> 00:36:25,580 We do that hardly caring whether it's band limited, 538 00:36:25,580 --> 00:36:28,530 hardly caring about the information in it, hardly 539 00:36:28,530 --> 00:36:33,050 caring about anything, we just want to sample it so fast that 540 00:36:33,050 --> 00:36:37,620 we essentially approximate the function very well. 541 00:36:37,620 --> 00:36:42,610 Now when we do that something's going to get lost, 542 00:36:42,610 --> 00:36:47,030 because when we take those samples we're ignoring what 543 00:36:47,030 --> 00:36:49,700 happens between the samples. 544 00:36:49,700 --> 00:36:54,020 In some approximate sense, wave forms are always smooth 545 00:36:54,020 --> 00:36:57,210 because they always get filtered by something before 546 00:36:57,210 --> 00:36:59,950 anybody looks at them. 547 00:36:59,950 --> 00:37:04,220 And because they're smooth, if you'd sample it fast enough, 548 00:37:04,220 --> 00:37:07,230 your fast enough samples are going to look 549 00:37:07,230 --> 00:37:08,220 like the wave form. 550 00:37:08,220 --> 00:37:10,820 If you just connect them with straight lines you're going to 551 00:37:10,820 --> 00:37:14,340 get a very good approximation of what the wave form is. 552 00:37:14,340 --> 00:37:18,140 But then you stop and ask, and when you stop and ask you're 553 00:37:18,140 --> 00:37:21,570 in trouble because you say well, how fast do I have to 554 00:37:21,570 --> 00:37:25,730 sample, and if I sample that fast, how much error am I 555 00:37:25,730 --> 00:37:28,040 going to make? 556 00:37:28,040 --> 00:37:34,280 That's the question for which aliasing gives you the answer. 557 00:37:34,280 --> 00:37:37,560 So we want to explore it for that reason. 558 00:37:37,560 --> 00:37:40,850 The other reason that we want to explore it is when we start 559 00:37:40,850 --> 00:37:44,500 talking about modulation, we'll start talking about 560 00:37:44,500 --> 00:37:48,650 something called Nyquist criterion, and that's best 561 00:37:48,650 --> 00:37:52,690 looked at in terms of aliasing again, and we'll see why that 562 00:37:52,690 --> 00:37:54,540 is when we get there. 563 00:37:54,540 --> 00:37:59,140 So for both of those reasons, and also for the reason of 564 00:37:59,140 --> 00:38:04,580 trying to understand these expansions in terms of source 565 00:38:04,580 --> 00:38:07,870 wave forms, we want to understand what this 566 00:38:07,870 --> 00:38:11,960 relationship is between the samples of a function that 567 00:38:11,960 --> 00:38:16,530 isn't quite band limited and the function itself. 568 00:38:16,530 --> 00:38:19,670 So the thing we're going to do to try to understand that is 569 00:38:19,670 --> 00:38:22,600 instead of studying just the samples, which is what you 570 00:38:22,600 --> 00:38:28,220 usually do, if you're just looking at the samples and 571 00:38:28,220 --> 00:38:31,240 you're trying to say how much error do I incur by doing 572 00:38:31,240 --> 00:38:35,350 that, and we're looking at mean square error, somehow or 573 00:38:35,350 --> 00:38:38,660 other we have to get back to wave forms and compare the 574 00:38:38,660 --> 00:38:42,560 resulting wave form with the wave form we started with. 575 00:38:42,560 --> 00:38:48,250 So the thing that we're going to do is to take our function 576 00:38:48,250 --> 00:38:55,440 u of t, we're going to sample it at some very rapid speed, 577 00:38:55,440 --> 00:38:59,925 and then we're going to take those samples and recreate a 578 00:38:59,925 --> 00:39:02,220 wave form by the sampling theorem. 579 00:39:02,220 --> 00:39:06,110 So we're going to call the approximation to u of t some 580 00:39:06,110 --> 00:39:11,210 approximation s of t -- s of t is, in fact, baseband limited 581 00:39:11,210 --> 00:39:18,490 at this point to a frequency w where t is equal to 1 over 2w, 582 00:39:18,490 --> 00:39:23,415 and we simply have this sampling theorem applied to u 583 00:39:23,415 --> 00:39:28,480 of kt, as if u of kt came from a band limited wave form. 584 00:39:28,480 --> 00:39:31,820 So this is, in a sense, an interpolation formula. 585 00:39:31,820 --> 00:39:34,040 It's a little better than taking these samples and 586 00:39:34,040 --> 00:39:37,770 joining them by straight lines, because we're, in fact, 587 00:39:37,770 --> 00:39:42,480 joining them by these smoother sinc functions. 588 00:39:42,480 --> 00:39:48,180 The question is how close is s of t to u of t and how do we 589 00:39:48,180 --> 00:39:50,540 look at that question? 590 00:39:50,540 --> 00:39:53,740 Well, we now have a nice way of looking at it because after 591 00:39:53,740 --> 00:40:05,640 going through this t spaced sinc weighted sinusoidal 592 00:40:05,640 --> 00:40:12,660 expansion, we have an exact expression for u of t, which 593 00:40:12,660 --> 00:40:19,970 is the limit in the mean of these frequency 594 00:40:19,970 --> 00:40:22,110 terms in u of t. 595 00:40:22,110 --> 00:40:25,840 Namely, to get this expression, remember the thing 596 00:40:25,840 --> 00:40:30,340 we did was to take arbitrary u of t, split it up into little 597 00:40:30,340 --> 00:40:34,480 frequency bands, each of width w, then apply the sampling 598 00:40:34,480 --> 00:40:37,690 theorem to each of those frequency terms, which we can 599 00:40:37,690 --> 00:40:39,920 do exactly. 600 00:40:39,920 --> 00:40:44,490 So this, in fact, is an exact expansion -- don't worry about 601 00:40:44,490 --> 00:40:47,210 the limit in the mean right now, we're going to get rid of 602 00:40:47,210 --> 00:40:49,640 that in a while. 603 00:40:49,640 --> 00:40:52,200 So we have s of t, which is this. 604 00:40:52,200 --> 00:40:56,340 We have u of t, which is this. 605 00:40:56,340 --> 00:41:03,580 Now, we look at this and we say OK, s of kt, namely, the 606 00:41:03,580 --> 00:41:10,740 case sample of s of t, is simply u of kt. 607 00:41:10,740 --> 00:41:12,590 That's what happens here. 608 00:41:12,590 --> 00:41:21,130 You put in any old time which is some integer j times 609 00:41:21,130 --> 00:41:25,540 capital T and you look at this expression here, and the sinc 610 00:41:25,540 --> 00:41:35,020 function is only non-zero when j capital T over t, when if 611 00:41:35,020 --> 00:41:39,080 you take the sinc function of an integer it's zero unless 612 00:41:39,080 --> 00:41:40,950 the integer is zero. 613 00:41:40,950 --> 00:41:47,050 The sinc function goes through zero at every integer point 614 00:41:47,050 --> 00:41:49,740 except for zero itself where it's equal to 1. 615 00:41:52,530 --> 00:41:54,750 That's the thing which makes the sync function nice. 616 00:42:00,380 --> 00:42:02,770 It dribbles away -- 617 00:42:02,770 --> 00:42:11,380 2, 3 and so forth minus 1. 618 00:42:11,380 --> 00:42:21,340 So it's zero at all those sample times, so that the case 619 00:42:21,340 --> 00:42:26,710 sample of s of ts of kt is just equal to u of kt. 620 00:42:26,710 --> 00:42:30,750 Namely, what we're doing in this approximation, which is 621 00:42:30,750 --> 00:42:34,870 what one usually does if one samples something, is we're 622 00:42:34,870 --> 00:42:39,840 assuming that the approximation is correct at 623 00:42:39,840 --> 00:42:42,300 the sample points, and we're arranging it so it's correct 624 00:42:42,300 --> 00:42:43,490 at the sample points. 625 00:42:43,490 --> 00:42:49,400 So, s of kt then is equal to u of kt from this. 626 00:42:49,400 --> 00:42:52,260 u of kt from this -- 627 00:42:52,260 --> 00:42:59,560 OK, now take t and substitute k times capital T in here. 628 00:43:03,630 --> 00:43:07,990 And if we substitute k times capital T in here -- let's 629 00:43:07,990 --> 00:43:13,560 not, let's substitute j times capital T to avoid a conflict 630 00:43:13,560 --> 00:43:15,560 with notation here. 631 00:43:15,560 --> 00:43:23,400 Then this T here becomes jT and what we have is 632 00:43:23,400 --> 00:43:30,140 sinc of j minus k. 633 00:43:30,140 --> 00:43:36,650 So we have a sum here over all k of sync of j minus k. 634 00:43:36,650 --> 00:43:42,190 Sinc of j minus k is zero for all integers k, except for j, 635 00:43:42,190 --> 00:43:47,770 therefore, this quantity is zero every time k is 636 00:43:47,770 --> 00:43:49,000 not equal to j. 637 00:43:49,000 --> 00:43:51,140 Therefore, we just have the sum over m. 638 00:43:51,140 --> 00:43:56,610 So, s of kt is equal to u of kt, which is equal to the sum 639 00:43:56,610 --> 00:44:03,170 over all frequency bands, m, of v sub m of kt. 640 00:44:03,170 --> 00:44:04,420 Now what is this saying? 641 00:44:06,980 --> 00:44:11,390 The thing this is saying is if you take this function u of t, 642 00:44:11,390 --> 00:44:16,630 which has a bunch of different frequency bands in it, each of 643 00:44:16,630 --> 00:44:19,170 those frequency bands has a sampling theorem 644 00:44:19,170 --> 00:44:20,630 associated with it. 645 00:44:20,630 --> 00:44:24,480 Each one of those frequency bands is represented by its 646 00:44:24,480 --> 00:44:28,150 samples at periods of time, k. 647 00:44:28,150 --> 00:44:33,400 But as soon as we look at u of kt, if we only have the 648 00:44:33,400 --> 00:44:40,040 samples u of kt, there's no way to tell which frequency 649 00:44:40,040 --> 00:44:42,650 band it's coming from. 650 00:44:42,650 --> 00:44:45,230 So all of these different frequency bands 651 00:44:45,230 --> 00:44:48,840 all get alias together. 652 00:44:48,840 --> 00:44:53,960 If I just look at u of kt, what it is is the sum over all 653 00:44:53,960 --> 00:44:58,760 these frequency bands of the samples of the individual 654 00:44:58,760 --> 00:45:00,010 frequency functions. 655 00:45:02,750 --> 00:45:07,120 So that if you tell me what u of kt is, I can't tell you 656 00:45:07,120 --> 00:45:09,310 what these samples are. 657 00:45:09,310 --> 00:45:13,380 All I know is what the sum of them is. 658 00:45:13,380 --> 00:45:17,190 So, in fact, if you start out with a function which instead 659 00:45:17,190 --> 00:45:23,810 of being baseband limited to w, in fact, is sitting between 660 00:45:23,810 --> 00:45:30,050 w and 3w, and there's nothing in this baseband, and I look 661 00:45:30,050 --> 00:45:34,300 at these samples and then I recreate things this way, what 662 00:45:34,300 --> 00:45:35,590 am I doing? 663 00:45:35,590 --> 00:45:41,160 I'm just taking that function at w to 3w and translating 664 00:45:41,160 --> 00:45:46,410 down in frequency to minus w to w. 665 00:45:46,410 --> 00:45:48,340 You can't tell. 666 00:45:48,340 --> 00:45:51,080 There's no way to tell just from the samples which 667 00:45:51,080 --> 00:45:53,430 frequency band we're looking at. 668 00:45:53,430 --> 00:45:57,850 So all these things get mixed together. 669 00:45:57,850 --> 00:46:05,170 So, s of t then, since s of t is this times these sinc 670 00:46:05,170 --> 00:46:11,480 functions, and this is the sum of all the vm's, s of t is 671 00:46:11,480 --> 00:46:16,700 just this double sum now where all of these are now down at 672 00:46:16,700 --> 00:46:19,090 this baseband frequency interval. 673 00:46:19,090 --> 00:46:21,880 There's no way to tell them apart. 674 00:46:21,880 --> 00:46:24,900 We have this double sum here so I'm adding up all of these 675 00:46:24,900 --> 00:46:29,430 different coefficients and they're all mixed together all 676 00:46:29,430 --> 00:46:31,360 down in this one frequency band. 677 00:46:40,560 --> 00:46:48,300 So, u of t is represented this way, double sum, vm of kt, vm 678 00:46:48,300 --> 00:46:55,250 of kt, sinc of t over t minus k, sinc of t over t minus k. 679 00:46:55,250 --> 00:47:00,980 This rotating term up here, an s of t, what I've done 680 00:47:00,980 --> 00:47:04,930 effectively is to get rid of all these rotating terms. 681 00:47:04,930 --> 00:47:11,020 It simplifies it enormously, but I changed the function. 682 00:47:11,020 --> 00:47:16,440 In fact, this function is low path limited to w and this 683 00:47:16,440 --> 00:47:19,810 function has all of the glory of an arbitrary set of 684 00:47:19,810 --> 00:47:22,030 frequencies in it. 685 00:47:22,030 --> 00:47:25,870 By just looking at these samples, I've lost all of this 686 00:47:25,870 --> 00:47:30,840 stuff and it's just back down to a baseband limited function 687 00:47:30,840 --> 00:47:33,880 at this point. 688 00:47:33,880 --> 00:47:36,650 So, if I look at the difference between u of t and 689 00:47:36,650 --> 00:47:42,100 s of t, what I'm going to get -- and this is expressed a 690 00:47:42,100 --> 00:47:44,210 little differently than the way it is in the notes but 691 00:47:44,210 --> 00:47:54,170 it's the same thing -- just the sum over k and m, and it 692 00:47:54,170 --> 00:47:57,380 has these sinc functions in it, which both 693 00:47:57,380 --> 00:47:59,580 of these terms have. 694 00:47:59,580 --> 00:48:05,930 Then u of t has these rotating terms and s of t doesn't, so s 695 00:48:05,930 --> 00:48:09,020 of t just has one in place of the rotating term. 696 00:48:09,020 --> 00:48:15,610 So the difference between u of t and s of t is just this big 697 00:48:15,610 --> 00:48:18,950 monster sum here, which is looking at all the different 698 00:48:18,950 --> 00:48:24,920 frequency bands at all the different sampling times. 699 00:48:24,920 --> 00:48:28,430 If I now try to look at the energy difference between u of 700 00:48:28,430 --> 00:48:31,330 t and s of t, because that's what I'm interested in -- how 701 00:48:31,330 --> 00:48:35,200 much error have I accumulated, how much mean square error 702 00:48:35,200 --> 00:48:36,570 have I gotten? 703 00:48:36,570 --> 00:48:42,480 By taking u of t and sampling it and then viewing it as a 704 00:48:42,480 --> 00:48:45,090 low pass function. 705 00:48:45,090 --> 00:48:49,050 This energy difference, well, we have a set of coefficients 706 00:48:49,050 --> 00:48:52,390 here, we have a set of functions here. 707 00:48:52,390 --> 00:48:56,260 These functions are all orthogonal to each other. 708 00:48:56,260 --> 00:48:57,750 Why are they orthogonal? 709 00:48:57,750 --> 00:49:02,180 Well, because sinc functions are orthogonal, which I think 710 00:49:02,180 --> 00:49:09,210 we've shown on our homework, and space time functions are 711 00:49:09,210 --> 00:49:10,180 orthogonal. 712 00:49:10,180 --> 00:49:14,220 So, all of these functions are orthogonal to -- 713 00:49:14,220 --> 00:49:15,620 excuse me -- 714 00:49:15,620 --> 00:49:19,330 spaced frequency functions are orthogonal to each other, and 715 00:49:19,330 --> 00:49:24,730 the sinc weighted functions are orthogonal to each other. 716 00:49:24,730 --> 00:49:29,760 So all I'm doing here is expanding all of this in terms 717 00:49:29,760 --> 00:49:32,350 of these orthogonal functions. 718 00:49:32,350 --> 00:49:41,290 I have to do this separately for these terms -- 719 00:49:44,760 --> 00:49:47,960 I know what happens. 720 00:49:47,960 --> 00:49:52,410 This works when it's cold and it doesn't work when it's hot. 721 00:49:52,410 --> 00:49:54,030 Interesting. 722 00:49:54,030 --> 00:49:57,590 So I'm separating this into this term and this term. 723 00:49:57,590 --> 00:50:01,360 Now, look at what this is when m is equal to zero. 724 00:50:01,360 --> 00:50:05,680 When m is equal to zero, e to the 2 pi i, zero t over t 725 00:50:05,680 --> 00:50:08,180 minus 1 is zero. 726 00:50:08,180 --> 00:50:16,240 So, all of the m terms here are collapsed into zero, so 727 00:50:16,240 --> 00:50:21,190 there isn't any error down at baseband in a sense. 728 00:50:21,190 --> 00:50:27,870 All of the error occurs due to these frequency terms larger 729 00:50:27,870 --> 00:50:29,970 than minus w to w. 730 00:50:29,970 --> 00:50:32,890 Well, that's the way it should be. 731 00:50:32,890 --> 00:50:38,090 Because we know that if u of t didn't have any terms outside 732 00:50:38,090 --> 00:50:41,920 of minus w to plus w, the sampling theorem would be 733 00:50:41,920 --> 00:50:46,510 absolutely rock solid with no error. 734 00:50:46,510 --> 00:50:50,260 So the errors are due to two kinds of terms. 735 00:50:50,260 --> 00:50:55,500 One, they're due to these terms, which is this thing. 736 00:50:55,500 --> 00:51:01,250 Two, they're due to these terms, which is this. 737 00:51:01,250 --> 00:51:08,830 The only difference between this is that this is a square 738 00:51:08,830 --> 00:51:13,530 of a sum and this is a sum of squares. 739 00:51:13,530 --> 00:51:16,180 Sum of squares are nicer, we can deal with them more 740 00:51:16,180 --> 00:51:22,020 nicely, and we understand what that is better. 741 00:51:22,020 --> 00:51:26,340 A sum and then taking the square of it after we take the 742 00:51:26,340 --> 00:51:27,720 sum is a good deal dirtier. 743 00:51:30,450 --> 00:51:34,030 The trouble is that this difference 744 00:51:34,030 --> 00:51:36,890 doesn't have to be L2. 745 00:51:36,890 --> 00:51:40,310 In other words, the mean square error we got from doing 746 00:51:40,310 --> 00:51:43,200 this can be infinite. 747 00:51:43,200 --> 00:51:47,620 One of the problems in the current problem set is a nice 748 00:51:47,620 --> 00:51:51,640 simple example of this. 749 00:51:51,640 --> 00:51:56,240 So, s of t need not have finite energy. 750 00:51:56,240 --> 00:52:02,490 That's a real kicker, because all of this theory is very 751 00:52:02,490 --> 00:52:04,770 nice until this point. 752 00:52:04,770 --> 00:52:08,270 I mean the sampling theorem works perfectly because when 753 00:52:08,270 --> 00:52:11,710 you're dealing with baseband limited functions -- 754 00:52:21,860 --> 00:52:28,460 for up baseband limited function, the sinc functions 755 00:52:28,460 --> 00:52:30,590 give you a perfect approximation. 756 00:52:30,590 --> 00:52:34,550 Namely, the sampling theorem works with no error at all. 757 00:52:34,550 --> 00:52:38,790 As soon as you get a function which spills out into higher 758 00:52:38,790 --> 00:52:46,030 frequencies, you can, in fact, have these samples coming up 759 00:52:46,030 --> 00:52:51,330 with infinite energy in them, and at that point this 760 00:52:51,330 --> 00:52:55,040 difference here can have infinite energy. 761 00:52:55,040 --> 00:52:58,510 In fact, you would do much, much better if you simply 762 00:52:58,510 --> 00:53:01,450 represented u of t by zero. 763 00:53:01,450 --> 00:53:05,350 You'd do infinitely better in terms of mean square error 764 00:53:05,350 --> 00:53:07,590 just by throwing it away and saying I'm not going even 765 00:53:07,590 --> 00:53:09,230 bother to approximate it. 766 00:53:09,230 --> 00:53:12,020 I'll just call it zero and nothing else. 767 00:53:12,020 --> 00:53:15,180 If you did that, you would only have the error in u of t 768 00:53:15,180 --> 00:53:18,940 and you wouldn't have all this generated stuff from all these 769 00:53:18,940 --> 00:53:22,150 terms that you're throwing away. 770 00:53:22,150 --> 00:53:26,540 When we start looking at random processes, and really 771 00:53:26,540 --> 00:53:28,990 the only thing that we're interested in here is random 772 00:53:28,990 --> 00:53:32,010 processes because noise is a random 773 00:53:32,010 --> 00:53:35,340 process, signals are random. 774 00:53:35,340 --> 00:53:38,240 So when we start looking at random processes, the thing 775 00:53:38,240 --> 00:53:44,380 that we're going to find is that this square of a sum in 776 00:53:44,380 --> 00:53:47,560 terms of expected value is going to be approximately the 777 00:53:47,560 --> 00:53:51,770 same as this sum of squares. 778 00:53:51,770 --> 00:53:55,090 So that this term and this term are going to be roughly 779 00:53:55,090 --> 00:53:58,830 equal for most of the stochastic processes that we 780 00:53:58,830 --> 00:54:03,100 deal with, so this problem doesn't occur of s of t having 781 00:54:03,100 --> 00:54:06,230 infinite energy. 782 00:54:06,230 --> 00:54:10,730 What that means is that these two terms are of 783 00:54:10,730 --> 00:54:13,160 roughly equal magnitude. 784 00:54:13,160 --> 00:54:16,010 Now, we can understand something more 785 00:54:16,010 --> 00:54:18,320 about these two terms. 786 00:54:18,320 --> 00:54:26,690 The term here is what you get, namely, these are the parts of 787 00:54:26,690 --> 00:54:30,650 u of t which are outside of the frequency range 788 00:54:30,650 --> 00:54:33,530 from minus w to w. 789 00:54:33,530 --> 00:54:37,670 A reasonable alternative to simply sampling this function 790 00:54:37,670 --> 00:54:41,200 would be to filter the function first. 791 00:54:41,200 --> 00:54:45,760 If you filter the function first what's going to happen? 792 00:54:45,760 --> 00:54:49,170 These terms will go away. 793 00:54:49,170 --> 00:54:51,260 These terms will stay. 794 00:54:51,260 --> 00:54:52,620 You still have the aliasing -- 795 00:54:52,620 --> 00:54:55,680 you can't get rid of that. 796 00:54:55,680 --> 00:54:59,640 But you can get rid of these terms. 797 00:54:59,640 --> 00:55:00,950 Excuse me. 798 00:55:00,950 --> 00:55:02,710 I take that back. 799 00:55:06,640 --> 00:55:09,610 I seem to have a binary problem today, whatever. 800 00:55:09,610 --> 00:55:13,180 I seem to be mixing up proof and falsehood. 801 00:55:13,180 --> 00:55:18,190 If you filter a function you're clearly going to get 802 00:55:18,190 --> 00:55:21,160 rid of all of the aliasing, because after you filtered the 803 00:55:21,160 --> 00:55:23,230 function you will have a band limited 804 00:55:23,230 --> 00:55:25,320 function sitting there. 805 00:55:25,320 --> 00:55:28,500 So the only error you're going to get is the error in what 806 00:55:28,500 --> 00:55:30,300 you've thrown away. 807 00:55:30,300 --> 00:55:34,080 What you've thrown away is, in fact, this quantity here. 808 00:55:43,600 --> 00:55:46,280 Slow down, say it right. 809 00:55:46,280 --> 00:55:48,440 What you have thrown away is all of the 810 00:55:48,440 --> 00:55:50,410 extra frequency terms. 811 00:55:50,410 --> 00:55:53,620 So the error is all of those frequency terms that you've 812 00:55:53,620 --> 00:55:55,270 thrown away. 813 00:55:55,270 --> 00:55:58,860 These are the frequency terms that you've thrown away. 814 00:55:58,860 --> 00:56:03,290 This is the error that you wind up with after you filter. 815 00:56:03,290 --> 00:56:06,300 These terms here are the aliasing terms. 816 00:56:06,300 --> 00:56:09,030 These are the things which, in fact, could be infinite if 817 00:56:09,030 --> 00:56:11,940 you're unlucky enough. 818 00:56:11,940 --> 00:56:16,220 These you can get rid of by filtering. 819 00:56:16,220 --> 00:56:19,570 So now the question is should you filter first or should you 820 00:56:19,570 --> 00:56:21,290 just sample and say to hell with it? 821 00:56:24,270 --> 00:56:27,830 Well, if you think about why you want to sample and use 822 00:56:27,830 --> 00:56:32,910 digital signal processing, what you're trying to do is to 823 00:56:32,910 --> 00:56:36,500 avoid building very complex analog filters. 824 00:56:36,500 --> 00:56:40,230 So if you try to build a very, very sharp filter which is 825 00:56:40,230 --> 00:56:44,200 getting rid of all of those out of band terms, you're sort 826 00:56:44,200 --> 00:56:47,980 of throwing away a lot of the reason for trying to sample to 827 00:56:47,980 --> 00:56:49,320 start with. 828 00:56:49,320 --> 00:56:52,550 So the usual conclusion is no, we're not going to filter 829 00:56:52,550 --> 00:56:55,220 first, or we're only going to have a very crude filtering 830 00:56:55,220 --> 00:56:59,370 operation first, and we'll just sample a little faster if 831 00:56:59,370 --> 00:57:03,050 we have to, because sampling faster is easier. 832 00:57:03,050 --> 00:57:07,690 So we usually just sample faster and avoid these terms. 833 00:57:11,340 --> 00:57:15,550 Now we want to look at aliasing viewed in frequency 834 00:57:15,550 --> 00:57:17,820 terms, and we've almost been doing this. 835 00:57:17,820 --> 00:57:21,240 We've been sort of talking around it a little bit. 836 00:57:21,240 --> 00:57:28,360 We're viewing an arbitrary function, u of t, in terms of 837 00:57:28,360 --> 00:57:33,770 a sum of these frequency limited functions. 838 00:57:33,770 --> 00:57:37,570 We're just arbitrarily taking a function, splitting it up 839 00:57:37,570 --> 00:57:39,610 into these frequency bands. 840 00:57:39,610 --> 00:57:42,100 In terms of frequency what we're doing is just 841 00:57:42,100 --> 00:57:47,780 segementing the Fourier transform u of f. 842 00:57:47,780 --> 00:57:52,150 As far as the samples are concerned, uf of kt is just 843 00:57:52,150 --> 00:57:55,600 equal to the sum of the samples in each of these 844 00:57:55,600 --> 00:57:58,100 frequency bands -- that's what we've been saying. 845 00:57:58,100 --> 00:58:01,660 That's really what aliasing is at a fundamental level. 846 00:58:01,660 --> 00:58:06,470 So when you sample you can't tell which frequency band each 847 00:58:06,470 --> 00:58:09,580 of these samples come from, and they usually come a little 848 00:58:09,580 --> 00:58:12,770 bit from each one. 849 00:58:12,770 --> 00:58:17,720 So, s of t now, I'd like to split up s of t in the same 850 00:58:17,720 --> 00:58:19,490 way that I split up u of t. 851 00:58:19,490 --> 00:58:23,770 I'd like to split up s of t, even though I can't do this 852 00:58:23,770 --> 00:58:27,090 from looking at s of t, I can do it mathematically. 853 00:58:27,090 --> 00:58:32,280 I want to split up s of t into the contributions to s of t 854 00:58:32,280 --> 00:58:34,380 from all of these different bands. 855 00:58:34,380 --> 00:58:39,800 So I want to view the mth frequency band as the sum of 856 00:58:39,800 --> 00:58:46,710 vm of kt times sinc of t over t minus k. 857 00:58:46,710 --> 00:58:52,530 Namely, look at what s of t was to start with, somewhere 858 00:58:52,530 --> 00:58:54,660 back here a long time ago. 859 00:59:07,360 --> 00:59:11,070 What we found was that s of t is equal to this quantity 860 00:59:11,070 --> 00:59:15,130 here, which is the sum over time terms, the sampling 861 00:59:15,130 --> 00:59:17,920 terms, and a sum over frequency term. 862 00:59:17,920 --> 00:59:23,140 What I'm doing now is I'm just defining s sub m of t to be 863 00:59:23,140 --> 00:59:26,590 this sum in here for one particular value of m. 864 00:59:26,590 --> 00:59:30,260 I'm just splitting this double sum into a 865 00:59:30,260 --> 00:59:31,500 number of separate terms. 866 00:59:31,500 --> 00:59:35,690 So this is the contribution to s of t from the 867 00:59:35,690 --> 00:59:37,870 mth frequency band. 868 00:59:37,870 --> 00:59:41,350 So, s of t is the sum of all of these different frequency 869 00:59:41,350 --> 00:59:46,870 contributions in this quantity. vm of t, which came 870 00:59:46,870 --> 00:59:51,170 from u of t, same quantity here except for the complex 871 00:59:51,170 --> 00:59:57,110 exponential here. vm of t is the same quantity with the 872 00:59:57,110 --> 00:59:59,100 rotating term at the end of it. 873 00:59:59,100 --> 01:00:05,830 Namely, this is the actual signal at this 874 01:00:05,830 --> 01:00:08,020 mth frequency band. 875 01:00:08,020 --> 01:00:12,980 So the contribution to it and the sampling approximation is 876 01:00:12,980 --> 01:00:17,350 this, the actual term is this. 877 01:00:17,350 --> 01:00:25,640 So we look at these two and we say gee, this looks like -- if 878 01:00:25,640 --> 01:00:28,640 we look at this in the frequency domain this looks 879 01:00:28,640 --> 01:00:32,170 like just a frequency shift. 880 01:00:32,170 --> 01:00:38,120 So the thing we can then say is that v sub m of f, namely 881 01:00:38,120 --> 01:00:44,010 the Fourier transform of this, differs from the Fourier 882 01:00:44,010 --> 01:00:48,890 transform of this just by a frequency shift. 883 01:00:48,890 --> 01:00:52,810 If we take the frequency shift formula and frequency and look 884 01:00:52,810 --> 01:00:58,080 at it in time it just becomes that rotating term there. 885 01:00:58,080 --> 01:01:01,960 Namely, frequency shifts look like complex exponential 886 01:01:01,960 --> 01:01:03,210 multipliers. 887 01:01:05,360 --> 01:01:09,660 We already know that d sub m of f is the mth frequency band 888 01:01:09,660 --> 01:01:10,810 and u of t. 889 01:01:10,810 --> 01:01:16,370 So in fact, it's u of f truncated to the 890 01:01:16,370 --> 01:01:18,210 mth frequency band. 891 01:01:18,210 --> 01:01:19,460 That's what this said . 892 01:01:19,460 --> 01:01:23,460 So, u hat of f rect ft minus m is just equal 893 01:01:23,460 --> 01:01:25,290 to this term here. 894 01:01:25,290 --> 01:01:29,180 So this says that s hat of f, which is the sum of all these 895 01:01:29,180 --> 01:01:34,130 terms, is just the sum over m, sum over all the frequency 896 01:01:34,130 --> 01:01:42,440 terms of u hat of f plus m over t times a rectangular 897 01:01:42,440 --> 01:01:45,130 function of ft. 898 01:01:45,130 --> 01:01:52,960 All this is saying is that s of f in frequency, it's all 899 01:01:52,960 --> 01:01:55,450 down at baseband now. 900 01:01:55,450 --> 01:02:02,710 Each of these frequency terms in u of t, up at some band and 901 01:02:02,710 --> 01:02:08,840 frequency, when we sample it, we're effectively bringing it 902 01:02:08,840 --> 01:02:09,530 down to base then. 903 01:02:09,530 --> 01:02:12,990 When we sample it and multiply by the sinc functions we're 904 01:02:12,990 --> 01:02:14,590 bringing it down to baseband. 905 01:02:14,590 --> 01:02:18,110 So this is saying when you look at this approximation, 906 01:02:18,110 --> 01:02:23,290 the space band approximation, what we wind up with is this 907 01:02:23,290 --> 01:02:28,805 frequency function evaluated at all of these different -- 908 01:02:28,805 --> 01:02:33,100 well, just evaluated at different times. 909 01:02:33,100 --> 01:02:39,160 So, s hat of f is, in fact, frequency limited to minus w 910 01:02:39,160 --> 01:02:43,890 to plus w, and it has all of these different terms 911 01:02:43,890 --> 01:02:44,970 contributing to it. 912 01:02:44,970 --> 01:02:48,070 It's just looking at aliasing and frequency instead of 913 01:02:48,070 --> 01:02:49,980 looking at aliasing and time. 914 01:02:49,980 --> 01:02:52,090 In both cases it's the same thing. 915 01:02:52,090 --> 01:02:55,850 In one case, all of the samples get mixed together. 916 01:02:55,850 --> 01:02:57,610 In the other case, all of the frequency 917 01:02:57,610 --> 01:02:58,920 bands get mixed together. 918 01:03:02,470 --> 01:03:07,990 I hope this will be clearer in terms of this. 919 01:03:22,120 --> 01:03:31,280 Let's take some arbitrary function here -- a most 920 01:03:31,280 --> 01:03:37,650 amazing thing, these things all get. 921 01:03:47,800 --> 01:03:52,210 It seems as if latex -- a thing for drawing pictures is 922 01:03:52,210 --> 01:03:56,510 fine on my screen but not fine when I print things. 923 01:03:56,510 --> 01:03:58,130 But anyway, this is fine as it is. 924 01:03:58,130 --> 01:04:01,260 It's slightly different from the one in the notes, which is 925 01:04:01,260 --> 01:04:03,780 also goofed up in the same way. 926 01:04:03,780 --> 01:04:07,420 Let's suppose we start out with a frequency function 927 01:04:07,420 --> 01:04:09,310 which looks like this. 928 01:04:09,310 --> 01:04:11,620 Actually, it should have something else 929 01:04:11,620 --> 01:04:14,640 there but that's OK. 930 01:04:14,640 --> 01:04:20,630 This is minus 1 over 2t, this is plus 1 over 2t. 931 01:04:20,630 --> 01:04:22,730 In other words, minus w to w. 932 01:04:22,730 --> 01:04:28,380 Here we have another frequency band from w to 3w, another 933 01:04:28,380 --> 01:04:31,800 frequency band from 3w to 5w. 934 01:04:31,800 --> 01:04:36,250 What that formula said is that each frequency band gets 935 01:04:36,250 --> 01:04:40,920 picked up and stuck down in the baseband area. 936 01:04:40,920 --> 01:04:44,820 So, in fact, what's happening is that this part of the 937 01:04:44,820 --> 01:04:47,430 frequency function is going to be picked 938 01:04:47,430 --> 01:04:50,030 up, stuck down there. 939 01:04:50,030 --> 01:04:53,370 So this goes over there. 940 01:04:58,080 --> 01:05:03,870 This quantity here is in the band from 3w to 5w. 941 01:05:03,870 --> 01:05:07,410 It gets picked up and put in here also. 942 01:05:07,410 --> 01:05:12,560 It's gotten picked up and put down there. 943 01:05:12,560 --> 01:05:16,870 So I have this part, this part, and this part just stays 944 01:05:16,870 --> 01:05:18,320 where it is -- this is the part that's 945 01:05:18,320 --> 01:05:20,580 actually band limited. 946 01:05:20,580 --> 01:05:26,100 So that s hat of f now is going to be the sum of this 947 01:05:26,100 --> 01:05:32,130 and this and this, and one disappeared term which was 948 01:05:32,130 --> 01:05:39,150 supposed to be here, and that's going to come in there. 949 01:05:39,150 --> 01:05:45,650 When you add up all of these -- this goes over to there. 950 01:05:45,650 --> 01:05:48,500 When you add up all of these you get the total frequency 951 01:05:48,500 --> 01:05:50,220 function s hat of f. 952 01:05:53,930 --> 01:05:57,130 All of you understand the mechanics of this? 953 01:05:57,130 --> 01:06:02,000 I mean graphically you can just find s hat of f from 954 01:06:02,000 --> 01:06:05,830 taking all of these things and folding them all into this 955 01:06:05,830 --> 01:06:09,810 baseband approximation. 956 01:06:09,810 --> 01:06:19,240 So that you can look at the error that you get in terms of 957 01:06:19,240 --> 01:06:23,160 sampling a function which is not quite band limited in 958 01:06:23,160 --> 01:06:25,150 these terms, also. 959 01:06:25,150 --> 01:06:29,610 The aliasing then looks like the translation from something 960 01:06:29,610 --> 01:06:34,540 which is in this band stuck into the main band between 961 01:06:34,540 --> 01:06:36,350 minus w and w. 962 01:06:36,350 --> 01:06:40,290 Translation of this thing stuck into the main band. 963 01:06:40,290 --> 01:06:43,450 Translation of this stuck into the main band. 964 01:06:43,450 --> 01:06:46,150 They're all added together in such a way that you can't tell 965 01:06:46,150 --> 01:06:48,600 which one came from where. 966 01:06:48,600 --> 01:06:54,680 So once you look at this, you can't go back to here, which 967 01:06:54,680 --> 01:06:56,680 is why people call it aliasing. 968 01:06:56,680 --> 01:07:00,470 These things are all just mixed together and there's no 969 01:07:00,470 --> 01:07:03,170 way to get back. 970 01:07:03,170 --> 01:07:08,100 The theorem that corresponds to all of this, and the 971 01:07:08,100 --> 01:07:10,950 theorem is pretty much proven in the appendix if you 972 01:07:10,950 --> 01:07:12,850 want to read it. 973 01:07:12,850 --> 01:07:16,480 If you're not interested in those mathematical details 974 01:07:16,480 --> 01:07:18,830 you're certainly welcome not to read it. 975 01:07:18,830 --> 01:07:24,290 I'm not going to have any problems on it. 976 01:07:24,290 --> 01:07:26,340 I'm not going to have any quiz problems on it, we might have 977 01:07:26,340 --> 01:07:29,120 a problem on it. 978 01:07:29,120 --> 01:07:34,440 It turns out that just L1 and L2 is not enough when you're 979 01:07:34,440 --> 01:07:37,730 dealing with aliasing. 980 01:07:37,730 --> 01:07:41,860 Aliasing, since you're both sampling and looking at things 981 01:07:41,860 --> 01:07:46,640 at arbitrarily large frequencies, the mathematics 982 01:07:46,640 --> 01:07:48,380 just gets messy. 983 01:07:48,380 --> 01:07:52,260 So the condition that you need is that the frequency function 984 01:07:52,260 --> 01:07:55,150 you're dealing with in order for all of these aliasing 985 01:07:55,150 --> 01:08:01,190 results to hold true, is that the limit as f goes to 986 01:08:01,190 --> 01:08:08,160 infinity of u hat of f times some function, f to the 1 plus 987 01:08:08,160 --> 01:08:10,190 epsilon has to go to zero. 988 01:08:10,190 --> 01:08:13,770 In other words, this is saying that u hat of f has to go to 989 01:08:13,770 --> 01:08:18,630 zero with increasing frequency fast enough. 990 01:08:18,630 --> 01:08:22,470 It has to go to zero a little faster than 1 over f. 991 01:08:22,470 --> 01:08:26,240 If it went to zero, it was 1 over f, that would be 992 01:08:26,240 --> 01:08:29,040 guaranteed by a thing, L2. 993 01:08:29,040 --> 01:08:32,870 That's not enough here. 994 01:08:32,870 --> 01:08:35,850 You need the stronger condition. 995 01:08:35,850 --> 01:08:40,070 So if it goes to zero fast enough as f gets very, very 996 01:08:40,070 --> 01:08:45,350 large -- you know any function you're going to deal with you 997 01:08:45,350 --> 01:08:48,990 can always model it so that it does this. 998 01:08:48,990 --> 01:08:53,320 Because you simply can't transmit wave forms that have 999 01:08:53,320 --> 01:08:55,920 arbitrarily high frequencies in them. 1000 01:08:55,920 --> 01:08:59,060 I mean no matter what kind of antenna you use to transmit 1001 01:08:59,060 --> 01:09:01,870 them, if it's an optical antenna you can get to much 1002 01:09:01,870 --> 01:09:05,750 higher frequencies, but no matter what kind of antenna 1003 01:09:05,750 --> 01:09:08,595 you use you're going to be limited somehow in frequency, 1004 01:09:08,595 --> 01:09:13,320 and it's going to drop off much faster with increasing 1005 01:09:13,320 --> 01:09:14,700 frequency than this. 1006 01:09:14,700 --> 01:09:17,670 So this isn't any sort of practical limitation on 1007 01:09:17,670 --> 01:09:21,520 aliasing, it's just that it's there and it limits the models 1008 01:09:21,520 --> 01:09:24,900 you can create somewhat. 1009 01:09:24,900 --> 01:09:29,570 If you have this condition then it says that the Fourier 1010 01:09:29,570 --> 01:09:36,470 transform of u of t has to be an L1 function. 1011 01:09:36,470 --> 01:09:40,380 The inverse transform, u of t, has to be 1012 01:09:40,380 --> 01:09:41,740 continuous and bounded. 1013 01:09:41,740 --> 01:09:46,560 In other words, when we go from u hat of f to u of t we 1014 01:09:46,560 --> 01:09:48,990 get a bonafide function there. 1015 01:09:48,990 --> 01:09:54,520 It's not something which is a limit in the mean. 1016 01:09:54,520 --> 01:09:56,690 Because as soon as we're dealing with samples of 1017 01:09:56,690 --> 01:09:59,550 something you really need a function to talk about. 1018 01:09:59,550 --> 01:10:03,750 You can't have something which is -- 1019 01:10:03,750 --> 01:10:07,830 well, you can't have something which has these little extra 1020 01:10:07,830 --> 01:10:09,300 things on it. 1021 01:10:09,300 --> 01:10:10,640 You can't live with that. 1022 01:10:10,640 --> 01:10:13,810 So the theorem says that this frequency 1023 01:10:13,810 --> 01:10:15,510 function has to be L1. 1024 01:10:15,510 --> 01:10:18,540 It says the inverse transform has to be 1025 01:10:18,540 --> 01:10:20,120 continuous and bounded. 1026 01:10:20,120 --> 01:10:23,510 We've already seen that L1 functions have continuous and 1027 01:10:23,510 --> 01:10:26,900 bounded Fourier transforms. 1028 01:10:26,900 --> 01:10:31,140 For any t greater than zero, the sampling approximation, 1029 01:10:31,140 --> 01:10:36,240 namely, s of t, which is this, is going to be bounded and 1030 01:10:36,240 --> 01:10:37,660 continuous. 1031 01:10:37,660 --> 01:10:42,390 And s hat of f satisfies this relationship here. 1032 01:10:42,390 --> 01:10:45,200 So this is the frequency version of 1033 01:10:45,200 --> 01:10:47,740 the aliasing formula. 1034 01:10:47,740 --> 01:10:49,780 Here you still have the limit in the mean here. 1035 01:10:49,780 --> 01:10:52,450 As soon as you go back to frequencies you can't say 1036 01:10:52,450 --> 01:10:54,300 anything precise anymore. 1037 01:10:54,300 --> 01:10:57,650 In time, everything is precise. 1038 01:10:57,650 --> 01:11:00,240 These functions are bonafide functions. 1039 01:11:00,240 --> 01:11:03,630 In terms of frequencies they're not quite. 1040 01:11:12,010 --> 01:11:17,160 I want to just start talking about these L2 functions. 1041 01:11:19,760 --> 01:11:22,350 As you realize, we've not only been talking about L2 1042 01:11:22,350 --> 01:11:26,760 functions, we've been talking about L1 functions also, and 1043 01:11:26,760 --> 01:11:28,660 now these crazy functions which go off 1044 01:11:28,660 --> 01:11:32,010 to zero fast enough. 1045 01:11:32,010 --> 01:11:34,330 The thing we're basically interested in, aside from 1046 01:11:34,330 --> 01:11:38,360 sampling, is the L2 functions because they're the ones where 1047 01:11:38,360 --> 01:11:41,160 you can go from function to Fourier 1048 01:11:41,160 --> 01:11:43,330 transform and back again. 1049 01:11:43,330 --> 01:11:47,710 They're the ones which work or these orthogonal expansions, 1050 01:11:47,710 --> 01:11:49,170 and they're the main things we're going to 1051 01:11:49,170 --> 01:11:50,420 be interested in. 1052 01:11:53,330 --> 01:11:58,670 When we try to go further talking about these functions, 1053 01:11:58,670 --> 01:12:01,820 it turns out that it's much easier to think about them in 1054 01:12:01,820 --> 01:12:05,380 terms of vector spaces. 1055 01:12:05,380 --> 01:12:09,020 Some of you probably have thought about wave forms 1056 01:12:09,020 --> 01:12:13,450 before as vectors, some of you probably haven't. 1057 01:12:17,060 --> 01:12:20,140 I'm sure all of you are familiar with vectors at least 1058 01:12:20,140 --> 01:12:24,390 in terms of a notational convenience as a way of 1059 01:12:24,390 --> 01:12:27,560 representing n tuples of numbers. 1060 01:12:27,560 --> 01:12:30,280 You can always take an n tuple of numbers and instead of 1061 01:12:30,280 --> 01:12:37,010 writing out u1, u2, u3, up to u sub n, you can say vector u, 1062 01:12:37,010 --> 01:12:40,050 and you can manipulate those vectors, you can add vectors, 1063 01:12:40,050 --> 01:12:43,170 you can multiply them by scalers, you can do all of the 1064 01:12:43,170 --> 01:12:47,720 neat things that people do, even without knowing anything 1065 01:12:47,720 --> 01:12:50,390 about vector spaces. 1066 01:12:50,390 --> 01:12:54,530 It's not much of an extension on that to take accountably 1067 01:12:54,530 --> 01:13:01,520 infinite sequence of numbers and represent it as a vector, 1068 01:13:01,520 --> 01:13:05,320 especially if you've never done it before. 1069 01:13:05,320 --> 01:13:07,960 If you've done it before and if you've thought about it 1070 01:13:07,960 --> 01:13:12,180 before, you will realize there are some problems there. 1071 01:13:12,180 --> 01:13:14,010 But at least conceptually there's no problem. 1072 01:13:14,010 --> 01:13:20,120 You can think about a sequence of numbers as being a vector, 1073 01:13:20,120 --> 01:13:22,960 you can add those sequences, you can multiply them by 1074 01:13:22,960 --> 01:13:26,680 scalers, you can do all sorts of neat things with them. 1075 01:13:26,680 --> 01:13:33,610 Since we've shown how to represent wave forms as 1076 01:13:33,610 --> 01:13:37,820 sequences of coefficients in orthogonal expansions, it then 1077 01:13:37,820 --> 01:13:41,220 isn't much of an extension to think about wave forms as 1078 01:13:41,220 --> 01:13:43,280 being vectors. 1079 01:13:43,280 --> 01:13:48,250 But the nice thing about doing this is that you're not really 1080 01:13:48,250 --> 01:13:50,340 stuck to a particular orthogonal 1081 01:13:50,340 --> 01:13:51,690 expansion when you do this. 1082 01:13:51,690 --> 01:13:56,140 You can think of wave forms as being vectors in just as nice 1083 01:13:56,140 --> 01:14:00,910 a sense as you can think about these n tuples as being 1084 01:14:00,910 --> 01:14:04,820 vectors, and that's what we're trying to get at. 1085 01:14:04,820 --> 01:14:09,620 These orthogonal expansions we're going to be looking at 1086 01:14:09,620 --> 01:14:13,570 are really viewed most easily in terms of vector space. 1087 01:14:13,570 --> 01:14:16,720 All of the questions about convergence of orthogonal 1088 01:14:16,720 --> 01:14:21,190 expansions and things like that, limits and all of that, 1089 01:14:21,190 --> 01:14:24,530 all of these are very natural in vector spaces. 1090 01:14:24,530 --> 01:14:27,340 They're so natural that people often forget about the fact 1091 01:14:27,340 --> 01:14:29,750 that they're taking limits even. 1092 01:14:29,750 --> 01:14:32,200 So it just looks nice there. 1093 01:14:32,200 --> 01:14:40,340 But as soon as we do this, what we're going to be doing 1094 01:14:40,340 --> 01:14:48,390 constantly from now on is trying to say what we know 1095 01:14:48,390 --> 01:14:51,610 when we're looking at two-dimensional vectors, we 1096 01:14:51,610 --> 01:14:53,210 have all sorts of pretty pictures 1097 01:14:53,210 --> 01:14:55,600 about how they behave. 1098 01:14:55,600 --> 01:14:58,420 We are going to be trying to use those pretty pictures in 1099 01:14:58,420 --> 01:15:01,760 two-dimensional space to understand what happens with 1100 01:15:01,760 --> 01:15:05,440 wave forms, and in order to do that we have to know a little 1101 01:15:05,440 --> 01:15:09,850 bit more about vectors than just vectors are things you 1102 01:15:09,850 --> 01:15:14,710 can add and multiply scalers by. 1103 01:15:14,710 --> 01:15:19,010 So I will bore you for two minutes by quickly going 1104 01:15:19,010 --> 01:15:22,230 through the axioms of a vector space. 1105 01:15:22,230 --> 01:15:25,540 My reason for going through this is when you define a 1106 01:15:25,540 --> 01:15:30,650 vector space axiomatically, then you can, in fact, prove 1107 01:15:30,650 --> 01:15:34,630 that wave forms satisfy all those axioms. 1108 01:15:34,630 --> 01:15:38,300 Once you prove that wave forms satisfy all those axioms, then 1109 01:15:38,300 --> 01:15:40,180 everything you know about two-dimensional and 1110 01:15:40,180 --> 01:15:45,670 three-dimensional vectors and all of that stuff all applies. 1111 01:15:45,670 --> 01:15:48,540 Namely, everything which is general for vector spaces 1112 01:15:48,540 --> 01:15:50,730 applies to these vectors. 1113 01:15:50,730 --> 01:15:53,050 So the axioms are the following. 1114 01:15:53,050 --> 01:15:55,230 You start out with a set of elements 1115 01:15:55,230 --> 01:15:57,270 which you call vectors. 1116 01:15:57,270 --> 01:16:01,010 In terms of n tuples, the set of elements is just the set of 1117 01:16:01,010 --> 01:16:03,470 different n tuples. 1118 01:16:03,470 --> 01:16:07,440 You also start out -- well, for here we don't have to 1119 01:16:07,440 --> 01:16:09,040 worry about scalers. 1120 01:16:09,040 --> 01:16:13,220 So we have the set of n tuples now, or perhaps it might be a 1121 01:16:13,220 --> 01:16:17,450 set of sequences or a set of wave forms. 1122 01:16:17,450 --> 01:16:20,830 The things we insist on before we will call this set a vector 1123 01:16:20,830 --> 01:16:25,430 space are that we have an operation 1124 01:16:25,430 --> 01:16:27,760 which we call addition. 1125 01:16:27,760 --> 01:16:30,380 Fortunately for n tuples, addition is what you would 1126 01:16:30,380 --> 01:16:31,240 think it would be. 1127 01:16:31,240 --> 01:16:33,810 It's element-wise addition. 1128 01:16:33,810 --> 01:16:38,400 For sequences, it's element-wise addition again. 1129 01:16:38,400 --> 01:16:42,550 For wave forms, it's function addition. 1130 01:16:42,550 --> 01:16:44,870 You have communitivity -- 1131 01:16:44,870 --> 01:16:48,380 you can add things in either order. 1132 01:16:48,380 --> 01:16:51,690 I mean look, if nobody pointed this out to you, you'd do it 1133 01:16:51,690 --> 01:16:53,580 anyway, right? 1134 01:16:53,580 --> 01:16:56,370 So that it looks like nothing. 1135 01:16:56,370 --> 01:17:00,920 But in fact, you can invent things, mathematical objects, 1136 01:17:00,920 --> 01:17:03,660 which you can deal with which are not communitive. 1137 01:17:03,660 --> 01:17:06,670 So this is saying to be a vector space it has to have 1138 01:17:06,670 --> 01:17:08,680 this communitivity operation. 1139 01:17:08,680 --> 01:17:10,020 Associativity -- 1140 01:17:10,020 --> 01:17:12,580 again, it's hard to see why you would say something like 1141 01:17:12,580 --> 01:17:16,630 this, but all we're defining is this addition operation, 1142 01:17:16,630 --> 01:17:22,000 namely, there's a definition of addition, and we know by 1143 01:17:22,000 --> 01:17:27,170 axiom that the sum has to be in this vector space. 1144 01:17:27,170 --> 01:17:32,800 As soon as we assume this associativity axiom, it says 1145 01:17:32,800 --> 01:17:37,430 OK, this is a vector in the space, this and this are both 1146 01:17:37,430 --> 01:17:38,930 vectors in the space. 1147 01:17:38,930 --> 01:17:41,390 Therefore, this sum has to be in the space. 1148 01:17:41,390 --> 01:17:46,530 Therefore this plus the sum has to be in the space. 1149 01:17:46,530 --> 01:17:48,600 What associativity says is that that's the 1150 01:17:48,600 --> 01:17:51,820 same element as this. 1151 01:17:51,820 --> 01:17:54,520 Once you see this you leave out the parentheses because 1152 01:17:54,520 --> 01:17:57,290 you can add as many things as you want to. 1153 01:17:57,290 --> 01:18:01,100 Same thing -- well, here there has to be a unique 1154 01:18:01,100 --> 01:18:03,690 vector, zero -- 1155 01:18:03,690 --> 01:18:07,730 v plus zero is equal to v for all v, and there's only one 1156 01:18:07,730 --> 01:18:09,920 vector that has that property. 1157 01:18:09,920 --> 01:18:12,590 We're going to see some problems there in a little bit 1158 01:18:12,590 --> 01:18:14,990 when we start studying these L2 functions, but we'll do 1159 01:18:14,990 --> 01:18:17,120 that next time. 1160 01:18:17,120 --> 01:18:21,160 Finally, for every v there's a unique negative vector so that 1161 01:18:21,160 --> 01:18:22,740 the sum is equal to zero. 1162 01:18:22,740 --> 01:18:24,300 All the operations that you're used to. 1163 01:18:24,300 --> 01:18:26,130 We'll go through the other things next time.