1 00:00:00,135 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high-quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:15,780 from hundreds of MIT courses, visit 7 00:00:15,780 --> 00:00:18,060 MITOpenCourseWare at ocw.mit.edu. 8 00:00:23,080 --> 00:00:26,438 [MUSIC PLAYING] 9 00:00:55,790 --> 00:00:57,410 ALAN OPPENHEIM: Hi. 10 00:00:57,410 --> 00:00:59,420 So it turns out that you guys missed out 11 00:00:59,420 --> 00:01:02,210 on some of the best parts of these lectures, which 12 00:01:02,210 --> 00:01:04,940 is all the kibitzing that goes on in back of the cameras 13 00:01:04,940 --> 00:01:08,090 just before they get started. 14 00:01:08,090 --> 00:01:13,490 Well, in the last lecture, we stressed the equivalence 15 00:01:13,490 --> 00:01:17,060 of finite length and periodic sequences. 16 00:01:17,060 --> 00:01:22,790 In particular, as hopefully you recall, 17 00:01:22,790 --> 00:01:28,400 we talked about a finite length sequence, x of n, 18 00:01:28,400 --> 00:01:32,040 which was of length capital N. In other words, 19 00:01:32,040 --> 00:01:39,380 it's equal to 0, except in the range 0 to capital N minus 1. 20 00:01:39,380 --> 00:01:44,240 We stressed also the fact that if the sequence is considered 21 00:01:44,240 --> 00:01:47,360 a finite length capital N, then it 22 00:01:47,360 --> 00:01:50,990 can also be considered of longer length than that, 23 00:01:50,990 --> 00:01:54,530 or equivalently the statement that it's a finite length 24 00:01:54,530 --> 00:01:58,220 doesn't imply that some of the values in the interval 0 to N 25 00:01:58,220 --> 00:02:02,720 minus 1 can't also be 0. 26 00:02:02,720 --> 00:02:04,910 In relating a finite length sequence 27 00:02:04,910 --> 00:02:09,740 to a periodic sequence, we generated a periodic sequence, 28 00:02:09,740 --> 00:02:14,090 x tilde of n, by simply repeating the finite length 29 00:02:14,090 --> 00:02:18,770 sequence over and over again with a period of capital 30 00:02:18,770 --> 00:02:24,500 N. That is x of n was added to itself, shifted by capital N, 31 00:02:24,500 --> 00:02:28,190 shifted by 2 capital N, shifted by 3 capital N. 32 00:02:28,190 --> 00:02:31,960 And because of the fact that it's a finite length, 33 00:02:31,960 --> 00:02:34,120 there is no interference between each 34 00:02:34,120 --> 00:02:37,450 of these individual replicas of x of n. 35 00:02:37,450 --> 00:02:40,330 The result then is a periodic sequence, 36 00:02:40,330 --> 00:02:45,770 one period of which corresponds to the finite length sequence. 37 00:02:45,770 --> 00:02:48,190 Well, we'll have occasion frequently 38 00:02:48,190 --> 00:02:51,430 to refer to this periodic sequence 39 00:02:51,430 --> 00:02:54,420 in describing the finite length sequence. 40 00:02:54,420 --> 00:02:57,130 And consequently it's convenient to use 41 00:02:57,130 --> 00:03:00,640 a slightly different notation than the summation 42 00:03:00,640 --> 00:03:02,440 that we've been using. 43 00:03:02,440 --> 00:03:05,590 In particular, we can rewrite this 44 00:03:05,590 --> 00:03:08,600 as the periodic sequence, x tilde 45 00:03:08,600 --> 00:03:15,370 of n, equal to x of n modulo capital N. 46 00:03:15,370 --> 00:03:22,410 That is x of n modulo capital N, in other words, the argument 47 00:03:22,410 --> 00:03:28,860 of x of n is little n taken modulo capital N. Another way 48 00:03:28,860 --> 00:03:32,760 of writing that notationally which is convenient, 49 00:03:32,760 --> 00:03:37,230 is just to simply express that as a double parentheses 50 00:03:37,230 --> 00:03:40,260 with a subscript capital N, so that this 51 00:03:40,260 --> 00:03:46,530 means x of n, the argument n, taken modulo capital N. 52 00:03:46,530 --> 00:03:51,570 So, you can see then that as little n varies 53 00:03:51,570 --> 00:03:55,260 over the interval 0 to capital N minus 1, 54 00:03:55,260 --> 00:04:01,650 we see the finite length sequence x of n as little n 55 00:04:01,650 --> 00:04:08,340 is equal to capital N. Capital N taken modulo, capital N is 0. 56 00:04:08,340 --> 00:04:11,760 So we'll see x of 0 back again. 57 00:04:11,760 --> 00:04:14,840 And as little n varies in the range capital N 58 00:04:14,840 --> 00:04:17,850 to 2 capital N minus 1. 59 00:04:17,850 --> 00:04:20,730 That argument taken modulo N again 60 00:04:20,730 --> 00:04:25,830 repeats the values of x of n the range n equals 0 61 00:04:25,830 --> 00:04:28,410 to capital N minus 1. 62 00:04:28,410 --> 00:04:31,680 Expressing it this way or equivalently expressing it 63 00:04:31,680 --> 00:04:35,910 this way, as I stressed before in the previous lecture, 64 00:04:35,910 --> 00:04:40,610 corresponds to taking the finite length sequence, x of n, 65 00:04:40,610 --> 00:04:42,660 and simply wrapping it around the circumference 66 00:04:42,660 --> 00:04:44,310 of a cylinder. 67 00:04:44,310 --> 00:04:47,230 As we run around the circumference of the cylinder, 68 00:04:47,230 --> 00:04:52,610 we see x of n repeated over and over again. 69 00:04:52,610 --> 00:04:56,660 Likewise, we were able to get back to the finite length 70 00:04:56,660 --> 00:04:59,930 sequence from the periodic sequence 71 00:04:59,930 --> 00:05:03,410 by simply multiplying by a rectangular sequence, 72 00:05:03,410 --> 00:05:06,860 in other words extracting a single period. 73 00:05:06,860 --> 00:05:09,740 A single period of the periodic sequence 74 00:05:09,740 --> 00:05:14,610 corresponds to the finite length sequence that we began with. 75 00:05:14,610 --> 00:05:17,730 In the last lecture we then introduced the discrete Fourier 76 00:05:17,730 --> 00:05:22,890 series of x tilde of n, and the discrete Fourier 77 00:05:22,890 --> 00:05:27,360 series coefficients we denoted as capital X tilde of k, 78 00:05:27,360 --> 00:05:30,090 and chose to interpret them, as you recall, 79 00:05:30,090 --> 00:05:36,420 as a periodic sequence, periodic in k with a period of capital 80 00:05:36,420 --> 00:05:41,760 N. The discrete Fourier series representation, just 81 00:05:41,760 --> 00:05:45,210 to summarize it again for you, we've 82 00:05:45,210 --> 00:05:49,530 rewritten here that is a sum of complex exponentials 83 00:05:49,530 --> 00:05:54,000 harmonically related generate x tilde of n. 84 00:05:54,000 --> 00:05:56,840 And the Fourier series coefficients, capital X tilde 85 00:05:56,840 --> 00:06:01,450 of k were given by this expression. 86 00:06:01,450 --> 00:06:04,500 And I stress again the fact that we 87 00:06:04,500 --> 00:06:08,760 chose to interpret the Fourier series coefficients 88 00:06:08,760 --> 00:06:10,620 as a periodic sequence. 89 00:06:10,620 --> 00:06:14,890 And it didn't particularly matter, because in any case 90 00:06:14,890 --> 00:06:17,670 we only use a single period of that 91 00:06:17,670 --> 00:06:23,370 in constructing the periodic sequence, x tilde of n. 92 00:06:23,370 --> 00:06:25,650 Now in this lecture, what we'd like to do 93 00:06:25,650 --> 00:06:29,640 is apply the idea of the discrete Fourier 94 00:06:29,640 --> 00:06:33,960 series and the similarity or basically the equivalence 95 00:06:33,960 --> 00:06:36,930 of finite length and periodic sequences 96 00:06:36,930 --> 00:06:42,240 to a Fourier series representation of finite length 97 00:06:42,240 --> 00:06:43,800 sequences. 98 00:06:43,800 --> 00:06:48,450 And basically the idea is that we 99 00:06:48,450 --> 00:06:51,120 can convert the finite length sequence 100 00:06:51,120 --> 00:06:52,560 to a periodic sequence. 101 00:06:52,560 --> 00:06:55,340 We've seen how to do that, and go back and forth. 102 00:06:55,340 --> 00:06:59,730 Generate a Fourier series representation for that, 103 00:06:59,730 --> 00:07:03,210 and then simply interpret the Fourier series coefficients 104 00:07:03,210 --> 00:07:06,870 as what we'll call the discrete Fourier transform 105 00:07:06,870 --> 00:07:09,780 of the finite length sequence. 106 00:07:09,780 --> 00:07:13,590 So, the discrete Fourier transform then 107 00:07:13,590 --> 00:07:18,080 is given by the sum from n equals 0 108 00:07:18,080 --> 00:07:22,140 to capital N minus 1 of x of n times 109 00:07:22,140 --> 00:07:28,260 w sub capital N to the nk, corresponding to the fact 110 00:07:28,260 --> 00:07:36,060 that if we generate a periodic sequence from x of n, 111 00:07:36,060 --> 00:07:40,290 we can recognize that in this summation for the Fourier 112 00:07:40,290 --> 00:07:43,410 series coefficients, in fact, we only 113 00:07:43,410 --> 00:07:47,940 use one period of x tilde of n anyway. 114 00:07:47,940 --> 00:07:53,484 So that we could simply replace in here x tilde of n by x of n. 115 00:07:53,484 --> 00:07:54,900 And of course, since the summation 116 00:07:54,900 --> 00:07:57,570 runs from 0 to capital N minus 1, 117 00:07:57,570 --> 00:08:00,610 we can't tell the difference. 118 00:08:00,610 --> 00:08:07,080 This then is by definition the discrete Fourier transform 119 00:08:07,080 --> 00:08:12,030 of the finite length sequence, x of n. 120 00:08:12,030 --> 00:08:18,220 And it is denoted by x of k. 121 00:08:18,220 --> 00:08:21,460 It corresponds to the Fourier series coefficients 122 00:08:21,460 --> 00:08:26,170 of the periodic equivalent sequence for x of n, 123 00:08:26,170 --> 00:08:29,530 except that it is useful to maintain 124 00:08:29,530 --> 00:08:33,580 the duality between the time domain and frequency domain 125 00:08:33,580 --> 00:08:37,169 by interpreting the Fourier transform, 126 00:08:37,169 --> 00:08:40,510 the discrete Fourier transform to be a finite length 127 00:08:40,510 --> 00:08:44,275 sequence, since the sequence that we're computing 128 00:08:44,275 --> 00:08:47,500 the Fourier transform from likewise 129 00:08:47,500 --> 00:08:50,630 starts as a finite length sequence. 130 00:08:50,630 --> 00:08:52,490 Well, that's straightforward. 131 00:08:52,490 --> 00:08:57,700 It just simply involves taking the Fourier series coefficients 132 00:08:57,700 --> 00:09:03,210 for k in the range 0 to capital N minus 1, 133 00:09:03,210 --> 00:09:08,450 and simply setting them to 0 outside that range. 134 00:09:08,450 --> 00:09:12,320 In other words, the discrete Fourier transform, 135 00:09:12,320 --> 00:09:18,020 x of k of a finite length sequence x of n, 136 00:09:18,020 --> 00:09:23,500 is equal to the Fourier series coefficients 137 00:09:23,500 --> 00:09:28,950 of the periodic equivalent of x of n 138 00:09:28,950 --> 00:09:32,400 multiplied by a rectangular sequence 139 00:09:32,400 --> 00:09:34,150 to extract a single period. 140 00:09:36,780 --> 00:09:40,800 We can likewise then get the Fourier series coefficients 141 00:09:40,800 --> 00:09:44,910 from the discrete Fourier transform, x of k, 142 00:09:44,910 --> 00:09:48,240 by simply periodically repeating x 143 00:09:48,240 --> 00:09:53,220 of k with a period of capital N. Or equivalently, we 144 00:09:53,220 --> 00:09:58,410 can write that the Fourier series coefficients, x tilde 145 00:09:58,410 --> 00:10:04,110 of k, are equal to the discrete Fourier transform values 146 00:10:04,110 --> 00:10:10,890 with the argument taken modulo capital N. This corresponds 147 00:10:10,890 --> 00:10:15,150 to simply periodically repeating capital X of k, 148 00:10:15,150 --> 00:10:19,020 like wrapping capital X of k around the circumference 149 00:10:19,020 --> 00:10:22,290 of a cylinder, running around the circumference again 150 00:10:22,290 --> 00:10:23,590 and again. 151 00:10:23,590 --> 00:10:28,200 And what we see are the Fourier series coefficients. 152 00:10:28,200 --> 00:10:32,640 Now perhaps this sounds a little involved or intricate. 153 00:10:32,640 --> 00:10:35,290 The idea is really very simple. 154 00:10:35,290 --> 00:10:40,590 It just simply is that the discrete Fourier transform 155 00:10:40,590 --> 00:10:43,380 corresponds to the Fourier series 156 00:10:43,380 --> 00:10:48,270 coefficients for the periodic equivalent of the finite length 157 00:10:48,270 --> 00:10:49,530 sequence. 158 00:10:49,530 --> 00:10:52,450 And to maintain the duality, in this case, 159 00:10:52,450 --> 00:10:55,710 we choose to simply extract one period of the Fourier series 160 00:10:55,710 --> 00:11:00,840 coefficients so that now we have a finite length sequence. 161 00:11:00,840 --> 00:11:04,345 Its discrete Fourier transform is likewise a finite length 162 00:11:04,345 --> 00:11:04,845 sequence. 163 00:11:07,770 --> 00:11:11,930 And the properties of the discrete Fourier transform, 164 00:11:11,930 --> 00:11:15,420 as we'll see, the differences in the properties 165 00:11:15,420 --> 00:11:17,430 between this and the Fourier transforms 166 00:11:17,430 --> 00:11:21,690 we've talked about in previous lectures are related very 167 00:11:21,690 --> 00:11:26,100 closely to the implied periodicity in the sequence 168 00:11:26,100 --> 00:11:30,690 or in the discrete Fourier transform. 169 00:11:30,690 --> 00:11:32,250 Before discussing the properties, 170 00:11:32,250 --> 00:11:37,800 however, it's useful to relate the discrete Fourier transform 171 00:11:37,800 --> 00:11:41,820 to the Z transform, as we've talked about it previously. 172 00:11:41,820 --> 00:11:47,740 In particular, here we have the Fourier transform, 173 00:11:47,740 --> 00:11:54,330 the discrete Fourier transform relationship, capital X of k 174 00:11:54,330 --> 00:11:58,650 is the sum from n equals capital N minus 1 of x of n, 175 00:11:58,650 --> 00:12:01,290 w sub N to the nk. 176 00:12:01,290 --> 00:12:08,610 Although that has to be multiplied by R sub n of k 177 00:12:08,610 --> 00:12:11,490 to extract a single period. 178 00:12:11,490 --> 00:12:15,400 The inverse discrete Fourier transform relationship 179 00:12:15,400 --> 00:12:20,660 then is the inverse Fourier series relationship 180 00:12:20,660 --> 00:12:25,500 1 over capital N the sum x sub k w sub of capital 181 00:12:25,500 --> 00:12:27,930 N to the minus nk. 182 00:12:27,930 --> 00:12:31,500 And again, since we're no longer talking 183 00:12:31,500 --> 00:12:35,280 about a periodic sequence, it's necessary to extract 184 00:12:35,280 --> 00:12:44,190 one period, this multiplied by R sub capital N of n. 185 00:12:44,190 --> 00:12:46,470 On the other hand, the Z transform 186 00:12:46,470 --> 00:12:51,580 of the finite length sequence is the sum of x of n times 187 00:12:51,580 --> 00:12:54,090 z to the minus n. 188 00:12:54,090 --> 00:12:56,010 In general, of course, the sum running 189 00:12:56,010 --> 00:12:58,560 from minus infinity to plus infinity, 190 00:12:58,560 --> 00:13:01,680 but because this is a finite length sequence, 191 00:13:01,680 --> 00:13:08,175 this is a sum from n equals 0 to capital N minus 1, 192 00:13:08,175 --> 00:13:11,430 because of the fact that the sequence x of n 193 00:13:11,430 --> 00:13:17,530 is 0 outside the range n equals zero to capital N minus 1. 194 00:13:17,530 --> 00:13:22,710 Well, to relate the Z transform to the discrete Fourier 195 00:13:22,710 --> 00:13:29,100 transform, we simply need to compare this expression 196 00:13:29,100 --> 00:13:31,360 with this expression. 197 00:13:31,360 --> 00:13:34,530 And what we see, ignoring for a moment 198 00:13:34,530 --> 00:13:40,440 the capital R sub N of k, that these two expressions are 199 00:13:40,440 --> 00:13:48,900 equal if we pick z equal to w sub capital N to the minus k. 200 00:13:48,900 --> 00:13:54,120 So the discrete Fourier transform coefficients 201 00:13:54,120 --> 00:13:58,500 are equal to the Z transform, if we 202 00:13:58,500 --> 00:14:08,100 choose z equal to w sub capital N to the minus k, 203 00:14:08,100 --> 00:14:13,340 and look at this for values of k equal to 0, 204 00:14:13,340 --> 00:14:18,116 1, up through capital N minus 1. 205 00:14:18,116 --> 00:14:22,970 What that says then, is that the discrete Fourier transform 206 00:14:22,970 --> 00:14:26,270 corresponds to samples of the Z transform; 207 00:14:26,270 --> 00:14:27,740 and where are those samples? 208 00:14:27,740 --> 00:14:30,950 Well, those samples are on the unit circle. 209 00:14:30,950 --> 00:14:35,930 Because the magnitude of w is equal to 1. 210 00:14:35,930 --> 00:14:39,260 And for example, if we chose the case were capital 211 00:14:39,260 --> 00:14:44,660 N was equal to 8, then we would find that the discrete Fourier 212 00:14:44,660 --> 00:14:49,370 transform coefficients fall at points on the unit circle, 213 00:14:49,370 --> 00:14:55,550 equally spaced in angle with eight points around the unit 214 00:14:55,550 --> 00:15:01,080 circle; 1, 2, 3, 4, 5, 6, 7, 8. 215 00:15:01,080 --> 00:15:02,430 I did it right. 216 00:15:02,430 --> 00:15:07,560 So these are equally-spaced points around the unit circle. 217 00:15:07,560 --> 00:15:12,540 For capital N equal to 8, then the discrete Fourier transform 218 00:15:12,540 --> 00:15:17,040 corresponds to sampling the Z transform at these eight 219 00:15:17,040 --> 00:15:18,960 points. 220 00:15:18,960 --> 00:15:22,650 And then the only effect that this capital R sub N of k 221 00:15:22,650 --> 00:15:27,090 has is that it requires that we interpret this 222 00:15:27,090 --> 00:15:31,052 as a finite length sequence, the set of eight values, 223 00:15:31,052 --> 00:15:33,510 rather than running around, and around, and around the unit 224 00:15:33,510 --> 00:15:37,100 circle over and over again. 225 00:15:37,100 --> 00:15:41,260 So, we have then the discrete Fourier transform. 226 00:15:41,260 --> 00:15:44,830 We have the notion that the discrete Fourier transform 227 00:15:44,830 --> 00:15:46,720 really is nothing different. 228 00:15:46,720 --> 00:15:50,020 It's exactly the same as the discrete Fourier series, 229 00:15:50,020 --> 00:15:52,180 except that it's one period of a discrete Fourier 230 00:15:52,180 --> 00:15:54,220 series extracted. 231 00:15:54,220 --> 00:15:58,240 And furthermore, it corresponds to sampling 232 00:15:58,240 --> 00:16:01,810 the Z transform at equally-spaced points 233 00:16:01,810 --> 00:16:04,000 around the unit circle. 234 00:16:04,000 --> 00:16:06,220 Apparently because of the fact that the sequence 235 00:16:06,220 --> 00:16:09,580 is finite length, from that set of samples 236 00:16:09,580 --> 00:16:14,350 we're able to get back exactly the sequence x of n. 237 00:16:14,350 --> 00:16:16,780 That isn't obviously true, in general. 238 00:16:16,780 --> 00:16:19,390 If we had an arbitrary x of n, and we 239 00:16:19,390 --> 00:16:22,420 sampled a Z transform around the unit circle, 240 00:16:22,420 --> 00:16:26,530 in general we wouldn't be able to get x of n back again. 241 00:16:26,530 --> 00:16:28,840 But if x of n is a finite length, 242 00:16:28,840 --> 00:16:32,500 then we are able to obtain x of n 243 00:16:32,500 --> 00:16:36,890 from the samples of its Z transform. 244 00:16:36,890 --> 00:16:37,390 OK. 245 00:16:37,390 --> 00:16:40,000 Well, I'd like now to look at some 246 00:16:40,000 --> 00:16:44,190 of the properties of the discrete Fourier transform. 247 00:16:44,190 --> 00:16:46,870 Again, there are lots of properties 248 00:16:46,870 --> 00:16:48,020 that we can talk about. 249 00:16:48,020 --> 00:16:50,860 I'll focus on a few, and primarily 250 00:16:50,860 --> 00:16:54,070 what I want to focus on is the difference 251 00:16:54,070 --> 00:16:58,000 between the properties for the discrete Fourier transform 252 00:16:58,000 --> 00:17:00,940 and the properties for the Fourier transform 253 00:17:00,940 --> 00:17:05,079 or the Z transform as we have been talking about them 254 00:17:05,079 --> 00:17:07,660 over the last several lectures. 255 00:17:07,660 --> 00:17:14,560 Well, let's look first of all at the shifting property. 256 00:17:14,560 --> 00:17:19,119 And let me stress that all of the properties, 257 00:17:19,119 --> 00:17:21,880 as we go through them, as we present them 258 00:17:21,880 --> 00:17:24,220 for the discrete Fourier transform, 259 00:17:24,220 --> 00:17:29,440 will relate to the corresponding property 260 00:17:29,440 --> 00:17:33,100 for the periodic equivalent of the finite length sequence. 261 00:17:33,100 --> 00:17:36,700 So when we want to ask what happens to the discrete Fourier 262 00:17:36,700 --> 00:17:40,780 transform when we do such and such to the sequence, 263 00:17:40,780 --> 00:17:44,440 the basic idea that basically what we want to ask 264 00:17:44,440 --> 00:17:47,740 is what happens to the Fourier series coefficients 265 00:17:47,740 --> 00:17:52,310 for the periodic equivalent of that finite length sequence. 266 00:17:52,310 --> 00:17:52,810 All right. 267 00:17:52,810 --> 00:17:56,500 Let's talk about the shifting property. 268 00:17:56,500 --> 00:17:59,560 We have a sequence, x of n. 269 00:17:59,560 --> 00:18:05,020 It has a discrete Fourier transform capital X of k. 270 00:18:05,020 --> 00:18:07,960 And then we have the periodic equivalent 271 00:18:07,960 --> 00:18:10,630 of x of n, x tilde of n. 272 00:18:10,630 --> 00:18:15,070 And it has a discrete Fourier series coefficients, 273 00:18:15,070 --> 00:18:19,870 which are the periodic extension of the discrete Fourier 274 00:18:19,870 --> 00:18:22,780 transform. 275 00:18:22,780 --> 00:18:26,140 Well, the shifting property that we want to introduce then 276 00:18:26,140 --> 00:18:29,350 corresponds to the finite length sequence 277 00:18:29,350 --> 00:18:34,460 which is implied by shifting the periodic sequence, x tilde 278 00:18:34,460 --> 00:18:36,770 of n. 279 00:18:36,770 --> 00:18:41,650 We know that if we shift the periodic sequence, x tilde 280 00:18:41,650 --> 00:18:47,850 of n plus m, the Fourier series coefficients 281 00:18:47,850 --> 00:18:52,680 that we get are x tilde of k multiplied 282 00:18:52,680 --> 00:18:57,330 by a complex exponential w sub capital N to the minus km. 283 00:19:00,270 --> 00:19:03,720 So the shifting property that we want to introduce then 284 00:19:03,720 --> 00:19:07,410 for the discrete Fourier transform, 285 00:19:07,410 --> 00:19:12,450 corresponds to extracting shifting x of n in such a way 286 00:19:12,450 --> 00:19:14,820 that it corresponds to extracting 287 00:19:14,820 --> 00:19:19,890 one period of the shifted periodic sequence, x 288 00:19:19,890 --> 00:19:21,780 tilde of n. 289 00:19:21,780 --> 00:19:28,140 If we do that, if we extract one period of x1 tilde of n 290 00:19:28,140 --> 00:19:30,960 to get a finite length sequence back, 291 00:19:30,960 --> 00:19:36,130 then we get a discrete Fourier transform, 292 00:19:36,130 --> 00:19:38,370 which corresponds to extracting one period 293 00:19:38,370 --> 00:19:40,930 of this periodic sequence. 294 00:19:40,930 --> 00:19:44,880 And that then is simply the original discrete Fourier 295 00:19:44,880 --> 00:19:49,560 transform capital X of k multiplied again by w sub 296 00:19:49,560 --> 00:19:52,800 capital N to the minus km. 297 00:19:52,800 --> 00:19:55,080 Now this shifting that we're talking about 298 00:19:55,080 --> 00:19:57,660 is a little different than the kind 299 00:19:57,660 --> 00:20:01,530 of shifting that we've been doing normally in the Fourier 300 00:20:01,530 --> 00:20:03,540 transform and Z transform. 301 00:20:03,540 --> 00:20:04,890 And it has to be. 302 00:20:04,890 --> 00:20:07,140 Because we're talking about a sequence 303 00:20:07,140 --> 00:20:09,330 that's a finite length. 304 00:20:09,330 --> 00:20:14,050 That is it's 0 outside the range, 0 to capital N minus 1. 305 00:20:14,050 --> 00:20:18,780 And if we just simply shift that sequence linearly, 306 00:20:18,780 --> 00:20:22,800 then we're going to end up with a sequence that no longer is 0 307 00:20:22,800 --> 00:20:25,780 outside the range 0 to capital N minus 1. 308 00:20:25,780 --> 00:20:27,420 So there's a slightly different kind 309 00:20:27,420 --> 00:20:29,910 of shifting that we're talking about. 310 00:20:29,910 --> 00:20:32,710 It's referred to as a circular shift. 311 00:20:32,710 --> 00:20:37,800 and I'd like to illustrate what that shifting corresponds to 312 00:20:37,800 --> 00:20:39,460 by referring to one of the view graphs. 313 00:20:45,900 --> 00:20:51,240 Here I have a sequence, x of n, which is a finite length. 314 00:20:51,240 --> 00:20:57,870 So it's nonzero only in the region of the blue values. 315 00:20:57,870 --> 00:21:02,760 And to generate the shifted, quote, "shifted" version 316 00:21:02,760 --> 00:21:07,410 of x of n, we first generated the periodic sequence, 317 00:21:07,410 --> 00:21:09,750 x tilde of n. 318 00:21:09,750 --> 00:21:12,990 So this is the periodic replica or equivalent 319 00:21:12,990 --> 00:21:15,270 of this finite length sequence. 320 00:21:15,270 --> 00:21:17,820 So we simply have this period repeated over here, 321 00:21:17,820 --> 00:21:20,700 repeated here, repeated there, et cetera. 322 00:21:20,700 --> 00:21:24,540 It was this sequence that we shift. 323 00:21:24,540 --> 00:21:31,000 So for example, if we shifted to the left by two values, 324 00:21:31,000 --> 00:21:35,370 we then have this periodic sequence shifted to the left 325 00:21:35,370 --> 00:21:37,530 by two values. 326 00:21:37,530 --> 00:21:43,230 But to recover, to extract one period again in the range 0 327 00:21:43,230 --> 00:21:49,200 to capital N minus 1, we want to extract this period 328 00:21:49,200 --> 00:21:51,570 of this periodic sequence. 329 00:21:51,570 --> 00:21:56,220 That's what the function capital R sub capital N of n 330 00:21:56,220 --> 00:21:57,480 does for us. 331 00:21:57,480 --> 00:22:04,410 And the result then is this one period extracted, and the rest 332 00:22:04,410 --> 00:22:06,550 of the sequence, 0. 333 00:22:06,550 --> 00:22:10,350 Now, we could of course, when we go through this shifting, 334 00:22:10,350 --> 00:22:13,350 carry ourselves through all of these steps. 335 00:22:13,350 --> 00:22:16,020 But there's a more straightforward way 336 00:22:16,020 --> 00:22:19,080 to look at how you get this sequence directly 337 00:22:19,080 --> 00:22:21,030 from this one. 338 00:22:21,030 --> 00:22:25,620 You can see that we could have interpreted 339 00:22:25,620 --> 00:22:31,140 the change from here to here, either by linearly shifting 340 00:22:31,140 --> 00:22:34,560 this sequence or by taking each one 341 00:22:34,560 --> 00:22:39,600 of the periods in this sequence and circularly shifting. 342 00:22:39,600 --> 00:22:42,450 As this shifts to the left, taking points 343 00:22:42,450 --> 00:22:45,900 as they drop out of this period and recirculating them 344 00:22:45,900 --> 00:22:47,790 around to the beginning of the period, 345 00:22:47,790 --> 00:22:50,170 or to the side of the period. 346 00:22:50,170 --> 00:22:54,780 Well, you can see that that's in essence, what we got here. 347 00:22:54,780 --> 00:22:57,550 And in fact, it's in essence, what we get here. 348 00:22:57,550 --> 00:23:03,450 This corresponds not to a linear shift of x of n, 349 00:23:03,450 --> 00:23:08,670 but to a circular shift of x of n. 350 00:23:08,670 --> 00:23:11,970 As we shift this sequence to the left by two 351 00:23:11,970 --> 00:23:17,970 values, instead of just simply having these points drop off 352 00:23:17,970 --> 00:23:21,930 the end of this period from 0 to n minus 1, 353 00:23:21,930 --> 00:23:24,670 we wrap them around to the other side 354 00:23:24,670 --> 00:23:27,960 so that they come in over here instead 355 00:23:27,960 --> 00:23:30,840 of wrapping around there. 356 00:23:30,840 --> 00:23:35,490 Now one way to look at this is to think of x of n, 357 00:23:35,490 --> 00:23:37,530 again as we've been trying to do, 358 00:23:37,530 --> 00:23:41,280 wrapped around the circumference of a cylinder. 359 00:23:41,280 --> 00:23:43,650 And what we've done in the kind of shift 360 00:23:43,650 --> 00:23:46,260 that we're talking about here, is 361 00:23:46,260 --> 00:23:51,060 just simply rotated the cylinder by two points, rather than 362 00:23:51,060 --> 00:23:53,280 the usual kind of linear shift where 363 00:23:53,280 --> 00:23:56,220 we have the sequence stretched out in a straight line 364 00:23:56,220 --> 00:23:58,980 and we shift the sequence linearly. 365 00:23:58,980 --> 00:24:00,960 Here we have a circular shift. 366 00:24:00,960 --> 00:24:03,360 Think of the sequence wrapped around the circumference 367 00:24:03,360 --> 00:24:08,100 of a cylinder, and the cylinder just rotated by two points 368 00:24:08,100 --> 00:24:13,290 so that the result is that the two points that are shifted out 369 00:24:13,290 --> 00:24:15,855 of this window between 0 and n minus 1 370 00:24:15,855 --> 00:24:19,290 get shifted in from the other side. 371 00:24:19,290 --> 00:24:24,480 Notationally a way to say that is as I've done here, really 372 00:24:24,480 --> 00:24:27,450 what this expression corresponds to is 373 00:24:27,450 --> 00:24:31,500 that we started with x of n. 374 00:24:31,500 --> 00:24:33,960 We formed the periodic replica of that 375 00:24:33,960 --> 00:24:35,790 using the modular notation that we 376 00:24:35,790 --> 00:24:38,400 introduced at the beginning of the lecture. 377 00:24:38,400 --> 00:24:43,960 Shift that by two points, and then extract a single period. 378 00:24:43,960 --> 00:24:46,020 So this is a long way-- 379 00:24:46,020 --> 00:24:48,180 well I guess it's really a short way of going 380 00:24:48,180 --> 00:24:49,860 through all these steps. 381 00:24:49,860 --> 00:24:55,290 But it's the idea and not the mathematical formalism 382 00:24:55,290 --> 00:24:56,940 that's important. 383 00:24:56,940 --> 00:25:01,560 Think of this as a circular shift of this finite length 384 00:25:01,560 --> 00:25:07,390 sequence inside the interval 0 to capital N minus 1. 385 00:25:07,390 --> 00:25:07,890 OK. 386 00:25:07,890 --> 00:25:15,000 So this then is our shifting property. 387 00:25:15,000 --> 00:25:20,740 I've rewritten it in a slightly different way here, 388 00:25:20,740 --> 00:25:24,490 corresponding to using the modular sort of notation. 389 00:25:24,490 --> 00:25:28,907 That is the statement is that a circular shift of x of n-- 390 00:25:28,907 --> 00:25:31,240 and that's what all this means is just simply a circular 391 00:25:31,240 --> 00:25:33,130 shift-- 392 00:25:33,130 --> 00:25:36,910 has as its discrete Fourier transform 393 00:25:36,910 --> 00:25:40,390 the original discrete Fourier transform multiplied 394 00:25:40,390 --> 00:25:43,390 by this complex exponential. 395 00:25:43,390 --> 00:25:45,430 Because of the fact that we've interpreted 396 00:25:45,430 --> 00:25:50,770 the discrete Fourier transform as a finite length sequence, 397 00:25:50,770 --> 00:25:53,440 we have again a duality between the time domain 398 00:25:53,440 --> 00:25:55,330 and the frequency domain. 399 00:25:55,330 --> 00:25:59,260 In particular, if we apply a circular shift 400 00:25:59,260 --> 00:26:03,930 to the discrete Fourier transform values, 401 00:26:03,930 --> 00:26:06,240 the resulting finite length sequence 402 00:26:06,240 --> 00:26:09,540 is the original finite length sequence multiplied 403 00:26:09,540 --> 00:26:12,310 by a complex exponential. 404 00:26:12,310 --> 00:26:16,140 Well, the details of the algebra involved in working this out 405 00:26:16,140 --> 00:26:17,760 aren't important right now. 406 00:26:17,760 --> 00:26:19,950 You'll have plenty of opportunity 407 00:26:19,950 --> 00:26:24,780 to look at those at your leisure in the text, 408 00:26:24,780 --> 00:26:28,300 also to work some of this in the study guide. 409 00:26:28,300 --> 00:26:33,180 The important concept right now, the important thing to cement, 410 00:26:33,180 --> 00:26:37,530 is the notion of a circular shift of a finite length 411 00:26:37,530 --> 00:26:38,910 sequence. 412 00:26:38,910 --> 00:26:41,640 And think of it as the finite length sequence 413 00:26:41,640 --> 00:26:45,570 as it shifts, points that come outside the interval 0 414 00:26:45,570 --> 00:26:48,030 to capital N minus 1, wrap around 415 00:26:48,030 --> 00:26:49,320 and come in on the other side. 416 00:26:51,990 --> 00:26:55,080 Now we also have symmetry properties 417 00:26:55,080 --> 00:26:58,830 for the discrete Fourier transform, which are basically 418 00:26:58,830 --> 00:27:01,800 a consequence of the symmetry properties 419 00:27:01,800 --> 00:27:05,100 for the discrete Fourier series. 420 00:27:05,100 --> 00:27:07,230 I'll remind you that for the discrete Fourier 421 00:27:07,230 --> 00:27:12,570 series, for a periodic sequence that's real, 422 00:27:12,570 --> 00:27:15,390 we had the symmetry property that 423 00:27:15,390 --> 00:27:18,840 said that the real part of the Fourier series coefficients 424 00:27:18,840 --> 00:27:20,280 are even. 425 00:27:20,280 --> 00:27:22,540 And we wrote that in a couple of ways. 426 00:27:22,540 --> 00:27:26,070 One of which was to state that x sub R of k 427 00:27:26,070 --> 00:27:31,320 is equal to x sub R of capital N minus k. 428 00:27:31,320 --> 00:27:35,370 And the imaginary part of the Fourier series coefficients 429 00:27:35,370 --> 00:27:36,300 are odd. 430 00:27:36,300 --> 00:27:42,780 That is they're equal to minus x sub I of capital N minus k. 431 00:27:42,780 --> 00:27:48,265 Well, the discrete Fourier transform 432 00:27:48,265 --> 00:27:51,240 is one period of the discrete Fourier series. 433 00:27:51,240 --> 00:27:54,210 And consequently, this implies the same type 434 00:27:54,210 --> 00:27:58,740 of symmetry properties for the discrete Fourier transform, 435 00:27:58,740 --> 00:28:01,790 keeping in mind again that we want to relate these properties 436 00:28:01,790 --> 00:28:04,510 just to a single period. 437 00:28:04,510 --> 00:28:07,320 So we have the discrete Fourier transform. 438 00:28:07,320 --> 00:28:11,050 Again, we'll assume that x of n is real. 439 00:28:11,050 --> 00:28:16,090 And the statement then, implied by simply extracting one period 440 00:28:16,090 --> 00:28:18,640 out of the discrete Fourier series, 441 00:28:18,640 --> 00:28:22,660 is that the real part of the discrete Fourier transform 442 00:28:22,660 --> 00:28:28,420 coefficients are equal to one period extracted out 443 00:28:28,420 --> 00:28:35,860 of the periodic sequence, which is x sub R of N minus k, 444 00:28:35,860 --> 00:28:38,170 repeated over and over again. 445 00:28:38,170 --> 00:28:40,480 Periodically that's what the modular notation 446 00:28:40,480 --> 00:28:43,300 did for us, one period extracted out of that 447 00:28:43,300 --> 00:28:46,750 that's what this rectangular window does for us. 448 00:28:46,750 --> 00:28:49,840 And we have a similar type of statement 449 00:28:49,840 --> 00:28:51,880 about the imaginary part. 450 00:28:51,880 --> 00:28:53,470 This is basically a statement that 451 00:28:53,470 --> 00:29:00,640 says that the real part is even, if we interpret even correctly. 452 00:29:00,640 --> 00:29:07,850 And the imaginary part is odd, if we interpret odd correctly. 453 00:29:07,850 --> 00:29:10,880 And in particular, the interpretation 454 00:29:10,880 --> 00:29:13,040 is a little different than the way 455 00:29:13,040 --> 00:29:16,850 we've been talking about even and odd sequences. 456 00:29:16,850 --> 00:29:19,730 Well, to see what this expression means, 457 00:29:19,730 --> 00:29:21,050 it looks a little involved. 458 00:29:21,050 --> 00:29:22,940 It's got some modular things in it, 459 00:29:22,940 --> 00:29:25,790 and multiplying by a rectangular window. 460 00:29:25,790 --> 00:29:29,570 But that's mainly notational formalism. 461 00:29:29,570 --> 00:29:31,670 And what these two expressions are saying 462 00:29:31,670 --> 00:29:34,570 are very straightforward. 463 00:29:34,570 --> 00:29:40,290 Let's look, for example, at what this implies for k 464 00:29:40,290 --> 00:29:45,930 equal to 0 or for k equal to 1. 465 00:29:45,930 --> 00:29:52,390 For k equal to 0, we have x sub R of capital N modulo capital 466 00:29:52,390 --> 00:29:58,140 N, or equivalently up here we have capital X sub R of 0 467 00:29:58,140 --> 00:30:06,310 is X sub R of N minus 0 modulo capital N times R sub N of 0. 468 00:30:06,310 --> 00:30:09,720 Well, R sub N of 0, according to the way 469 00:30:09,720 --> 00:30:14,490 that we've defined this rectangular window, is unity. 470 00:30:14,490 --> 00:30:18,400 And what is all this equal to? 471 00:30:18,400 --> 00:30:24,470 Well, it's capital N modulo capital N, which is 0. 472 00:30:24,470 --> 00:30:26,530 In other words, this argument at 0, 473 00:30:26,530 --> 00:30:29,680 this simply is a trivial statement 474 00:30:29,680 --> 00:30:34,180 that says that X Sub R of 0 is equal to X sub R of 0. 475 00:30:34,180 --> 00:30:38,770 And that really doesn't tell us too much. 476 00:30:38,770 --> 00:30:41,500 For k equal to 1, we add that X sub R of 1 477 00:30:41,500 --> 00:30:45,810 is X sub R of capital N minus 1 modulo capital 478 00:30:45,810 --> 00:30:49,340 N times R sub capital N of 1. 479 00:30:49,340 --> 00:30:53,200 This again is unity. 480 00:30:53,200 --> 00:30:56,200 And since capital N minus 1 is in the range 0 481 00:30:56,200 --> 00:31:00,220 to capital N minus 1 that taken, modulo N 482 00:31:00,220 --> 00:31:08,480 is just simply X sub R of capital N minus 1. 483 00:31:08,480 --> 00:31:13,350 So what this says is X sub R of 0 is equal to X sub R of 0. 484 00:31:13,350 --> 00:31:17,780 So as X sub R of 1 is equal to X sub R of N minus 1. 485 00:31:17,780 --> 00:31:21,160 And if we repeated this, we'd find X sub R of 2 equal to X 486 00:31:21,160 --> 00:31:23,140 sub R of N minus 2. 487 00:31:23,140 --> 00:31:25,630 And that's in the context of the discrete Fourier 488 00:31:25,630 --> 00:31:31,750 transform, what we mean by the Fourier transform being even. 489 00:31:31,750 --> 00:31:34,480 So as an example, here we have what 490 00:31:34,480 --> 00:31:39,250 can correspond to a real part. 491 00:31:39,250 --> 00:31:44,400 And we have X sub R of k. 492 00:31:44,400 --> 00:31:50,665 We have X sub R of 1 equal to X sub R of capital N minus 1. 493 00:31:50,665 --> 00:31:59,110 Well, that says that this value is equal to that value. 494 00:31:59,110 --> 00:32:03,280 X sub R of 2 equal to X sub R of N minus 2. 495 00:32:03,280 --> 00:32:08,290 That says that this value is equal to that value. 496 00:32:11,090 --> 00:32:14,480 And likewise this value is equal to this one. 497 00:32:14,480 --> 00:32:17,720 This one is equal to that one, et cetera. 498 00:32:17,720 --> 00:32:21,680 In a similar way, we have a statement that tells us 499 00:32:21,680 --> 00:32:27,390 that the imaginary part is odd in exactly the same sense. 500 00:32:27,390 --> 00:32:30,140 In other words that there's a symmetry 501 00:32:30,140 --> 00:32:35,330 as we look at the imaginary part coming in from both sides 502 00:32:35,330 --> 00:32:36,770 of the sequence. 503 00:32:36,770 --> 00:32:38,600 Incidentally, I've indicated here 504 00:32:38,600 --> 00:32:42,950 by a dotted line what would correspond 505 00:32:42,950 --> 00:32:48,360 to the periodic extension of this set of DFT values. 506 00:32:48,360 --> 00:32:51,500 In other words, this would be this one repeated over again. 507 00:32:51,500 --> 00:32:54,710 And if we were thinking of the discrete Fourier series, 508 00:32:54,710 --> 00:32:58,280 this would just be periodic replicas. 509 00:32:58,280 --> 00:32:59,930 Although in fact, the discrete Fourier 510 00:32:59,930 --> 00:33:02,570 transform, since we've interpreted it 511 00:33:02,570 --> 00:33:06,530 as a finite length sequence, the discrete Fourier transform 512 00:33:06,530 --> 00:33:11,010 would have 0 values on both ends of this. 513 00:33:11,010 --> 00:33:11,510 All right. 514 00:33:11,510 --> 00:33:15,440 So the corresponding symmetry for the imaginary part 515 00:33:15,440 --> 00:33:25,180 says that if we look at the imaginary part at n equals 1, 516 00:33:25,180 --> 00:33:29,170 then that's equal to minus the imaginary part at capital 517 00:33:29,170 --> 00:33:31,480 N minus 1. 518 00:33:31,480 --> 00:33:34,810 If we look at the imaginary part at n equals 2, 519 00:33:34,810 --> 00:33:38,170 then that's equal to minus the imaginary part 520 00:33:38,170 --> 00:33:41,600 at capital N minus 2. 521 00:33:41,600 --> 00:33:45,130 In other words, there is an antisymmetry 522 00:33:45,130 --> 00:33:47,680 as we look at the imaginary part coming 523 00:33:47,680 --> 00:33:49,510 in from both ends of the period. 524 00:33:49,510 --> 00:33:52,510 So that's the kind of evenness and oddness 525 00:33:52,510 --> 00:33:55,240 that we're talking about when we talk 526 00:33:55,240 --> 00:33:58,270 about the symmetry properties for the discrete Fourier 527 00:33:58,270 --> 00:33:59,950 transform. 528 00:33:59,950 --> 00:34:02,920 In other words, it's an even function in the sense 529 00:34:02,920 --> 00:34:08,110 that as we come in from the two ends, we see a symmetry. 530 00:34:08,110 --> 00:34:09,100 The real part is even. 531 00:34:09,100 --> 00:34:10,810 The imaginary part is odd, in the sense 532 00:34:10,810 --> 00:34:12,670 that we see an antisymmetry. 533 00:34:12,670 --> 00:34:14,710 As we come in from the two ends of the period. 534 00:34:18,270 --> 00:34:19,949 OK. 535 00:34:19,949 --> 00:34:23,190 That is just a simple introduction 536 00:34:23,190 --> 00:34:27,550 to the symmetry properties. 537 00:34:27,550 --> 00:34:31,060 We have also as we've talked about with 538 00:34:31,060 --> 00:34:35,080 our other transforms, the convolution property. 539 00:34:35,080 --> 00:34:40,150 And the convolution property relates very heavily 540 00:34:40,150 --> 00:34:44,690 to the type of convolution that we 541 00:34:44,690 --> 00:34:49,380 were talking about for the discrete Fourier series. 542 00:34:49,380 --> 00:34:56,449 What we want to ask is, what is the sequence, finite length 543 00:34:56,449 --> 00:35:02,840 sequence x3 of n, whose discrete Fourier transform is 544 00:35:02,840 --> 00:35:07,790 the product of the discrete Fourier transform of x1 of n, 545 00:35:07,790 --> 00:35:13,110 and the discrete Fourier transform of x2 of n. 546 00:35:13,110 --> 00:35:19,350 Well, we know what the sequence is, the periodic sequence is, 547 00:35:19,350 --> 00:35:22,380 whose discrete Fourier series is the product 548 00:35:22,380 --> 00:35:26,010 of the discrete Fourier series of the periodic equivalent 549 00:35:26,010 --> 00:35:31,820 of x1 of n and the periodic equivalent of x2 of n. 550 00:35:31,820 --> 00:35:34,520 And this sequence must just simply 551 00:35:34,520 --> 00:35:40,280 be one period extracted from x3 tilde of n. 552 00:35:40,280 --> 00:35:45,740 In other words, x3 of n is equal to x3 tilde of n multiplied 553 00:35:45,740 --> 00:35:49,530 by a rectangular sequence. 554 00:35:49,530 --> 00:35:55,760 Well, we know how to form from the last lecture, 555 00:35:55,760 --> 00:35:59,660 the convolution that corresponds to multiplying 556 00:35:59,660 --> 00:36:02,180 the discrete Fourier series together. 557 00:36:02,180 --> 00:36:07,560 And that convolution was the periodic sequences x1 tilde 558 00:36:07,560 --> 00:36:13,940 of m and x2 tilde of n minus m, summed over 559 00:36:13,940 --> 00:36:19,915 one period that is from m equals 0 to capital N minus 1. 560 00:36:19,915 --> 00:36:22,380 Well, the finite length sequence, 561 00:36:22,380 --> 00:36:25,520 x3 of n that corresponds to the product of these two 562 00:36:25,520 --> 00:36:30,020 discrete Fourier transforms, corresponds then 563 00:36:30,020 --> 00:36:36,470 to extracting a single period from this periodic sequence, 564 00:36:36,470 --> 00:36:42,140 in other words, multiplying this by R sub capital N of n. 565 00:36:44,930 --> 00:36:48,140 So this then is the kind of convolution 566 00:36:48,140 --> 00:36:51,710 that results from multiplying the discrete Fourier 567 00:36:51,710 --> 00:36:56,720 transforms together, or we could write this in a different way. 568 00:36:56,720 --> 00:37:00,320 We can express this periodic sequence 569 00:37:00,320 --> 00:37:04,790 using our modular notation in terms of x1 of m, 570 00:37:04,790 --> 00:37:10,760 as x1 of m taken modulo capital N. 571 00:37:10,760 --> 00:37:15,800 And this periodic sequence is x2 of n minus m 572 00:37:15,800 --> 00:37:19,550 that argument taken modulo capital N. 573 00:37:19,550 --> 00:37:23,490 Well, we can take one additional step here, 574 00:37:23,490 --> 00:37:26,330 which may or may not be useful. 575 00:37:26,330 --> 00:37:29,900 Observe that we only use values of m 576 00:37:29,900 --> 00:37:33,290 in this expression in the interval from 0 577 00:37:33,290 --> 00:37:35,550 to capital N minus 1. 578 00:37:35,550 --> 00:37:38,330 So in fact, although this is a periodic sequence, 579 00:37:38,330 --> 00:37:40,580 we're only using one period of it. 580 00:37:40,580 --> 00:37:49,970 So this piece is equivalently x1 of m. 581 00:37:49,970 --> 00:37:51,530 And what is this? 582 00:37:51,530 --> 00:37:56,000 Well, think back to the kind of circular shifting 583 00:37:56,000 --> 00:37:59,630 that we were talking about doing with finite length sequences. 584 00:37:59,630 --> 00:38:02,750 What this basically corresponds to doing 585 00:38:02,750 --> 00:38:07,250 is applying not a linear shift to x2 of n, 586 00:38:07,250 --> 00:38:10,610 but a circular shift, flipping x2 and applying 587 00:38:10,610 --> 00:38:14,180 a circular shift to x2 of n. 588 00:38:14,180 --> 00:38:16,190 And then of course, this is just in here 589 00:38:16,190 --> 00:38:19,670 to allow us to extract a single period. 590 00:38:19,670 --> 00:38:22,670 This then corresponds to what we'll 591 00:38:22,670 --> 00:38:28,670 refer to as a circular convolution of this sequence, 592 00:38:28,670 --> 00:38:33,710 or rather the sequence x1 of n, and the finite length sequence 593 00:38:33,710 --> 00:38:35,860 x2 of n. 594 00:38:35,860 --> 00:38:39,110 It's a circular convolution. 595 00:38:39,110 --> 00:38:42,960 And notationally we'll denote that as x1 596 00:38:42,960 --> 00:38:45,590 of n, circularly convolved, keeping 597 00:38:45,590 --> 00:38:49,100 track of what the circumference of this cylinder 598 00:38:49,100 --> 00:38:53,270 is that we're wrapping the finite-length sequences around, 599 00:38:53,270 --> 00:38:56,660 circularly convolved, an endpoint circular convolution 600 00:38:56,660 --> 00:38:59,300 with x2 of n. 601 00:38:59,300 --> 00:39:02,180 Now, I wouldn't expect at this point 602 00:39:02,180 --> 00:39:06,440 that what this circular convolution is in detail, 603 00:39:06,440 --> 00:39:09,170 I wouldn't expect that to be very clear. 604 00:39:09,170 --> 00:39:11,930 We have some modular notation in there, 605 00:39:11,930 --> 00:39:13,500 and we're multiplying by this. 606 00:39:13,500 --> 00:39:15,830 And there was some periodic something somewhere, 607 00:39:15,830 --> 00:39:19,550 and where did we extract the period. 608 00:39:19,550 --> 00:39:21,750 A lot of our attention at this point, 609 00:39:21,750 --> 00:39:24,710 and in fact through most of the next lecture, 610 00:39:24,710 --> 00:39:28,940 is going to be focused on interpreting and getting 611 00:39:28,940 --> 00:39:33,290 a feeling for what the circular convolution means. 612 00:39:33,290 --> 00:39:35,990 There are lots of pictures that I'll 613 00:39:35,990 --> 00:39:39,800 try to suggest to help in forming 614 00:39:39,800 --> 00:39:42,980 an idea of circular convolution. 615 00:39:42,980 --> 00:39:46,790 Let me just offer one now and show you 616 00:39:46,790 --> 00:39:52,280 a simple example of circular convolution, with the notion 617 00:39:52,280 --> 00:39:55,580 that in the next lecture we'll be focusing almost entirely 618 00:39:55,580 --> 00:39:57,410 on what circular convolution means 619 00:39:57,410 --> 00:40:01,141 in relation to linear convolution, et cetera. 620 00:40:01,141 --> 00:40:01,640 But look. 621 00:40:01,640 --> 00:40:03,800 Here's what this means. 622 00:40:03,800 --> 00:40:05,870 First of all, linear convolution, what do we do? 623 00:40:05,870 --> 00:40:07,970 We have two sequences. 624 00:40:07,970 --> 00:40:09,890 They're laid out flat. 625 00:40:09,890 --> 00:40:14,120 We do a linear convolution by flipping one of the sequences, 626 00:40:14,120 --> 00:40:16,820 and then shifting the two sequences linearly 627 00:40:16,820 --> 00:40:19,820 with relation to each other, multiplying 628 00:40:19,820 --> 00:40:22,700 and adding up the values from minus infinity 629 00:40:22,700 --> 00:40:25,100 to plus infinity. 630 00:40:25,100 --> 00:40:27,500 The circular convolution basically 631 00:40:27,500 --> 00:40:32,180 corresponds to wrapping one of the sequences 632 00:40:32,180 --> 00:40:35,180 around the circumference of a cylinder, 633 00:40:35,180 --> 00:40:38,360 taking the other sequence, flipping it, wrapping 634 00:40:38,360 --> 00:40:42,050 that around the circumference of another cylinder, 635 00:40:42,050 --> 00:40:45,020 putting the two cylinders inside each other, 636 00:40:45,020 --> 00:40:48,920 and then the shifting as we change the value of n 637 00:40:48,920 --> 00:40:51,980 can be thought of as rotating one of the cylinders 638 00:40:51,980 --> 00:40:57,590 inside the other, multiplying the sequence values together, 639 00:40:57,590 --> 00:41:00,860 and then adding that product around the cylinder. 640 00:41:00,860 --> 00:41:05,850 That is adding from m equals 0 to capital N minus 1. 641 00:41:05,850 --> 00:41:08,480 And what this thing does basically, 642 00:41:08,480 --> 00:41:11,840 since it extracts one period of a periodic sequence, 643 00:41:11,840 --> 00:41:14,990 it corresponds essentially to just simply 644 00:41:14,990 --> 00:41:19,550 cutting the cylinder and laying the answer around flat again. 645 00:41:19,550 --> 00:41:22,130 That's the notion of circular convolution, 646 00:41:22,130 --> 00:41:24,590 and it results from the fact that we're 647 00:41:24,590 --> 00:41:28,550 thinking of the arguments in the convolution taken 648 00:41:28,550 --> 00:41:31,130 modulo capital N. 649 00:41:31,130 --> 00:41:35,360 Well, let's just look at a simple example with the idea 650 00:41:35,360 --> 00:41:39,540 that as we go on to the next lecture, 651 00:41:39,540 --> 00:41:43,460 we'll be concentrating some more on the interpretation of what 652 00:41:43,460 --> 00:41:44,990 circular convolution means. 653 00:41:50,880 --> 00:41:51,480 All right. 654 00:41:51,480 --> 00:41:56,340 Here we have a sequence x1 of m. 655 00:41:56,340 --> 00:42:00,900 Remember that in the circular convolution, 656 00:42:00,900 --> 00:42:04,320 we're summing over an index m. 657 00:42:04,320 --> 00:42:07,680 So here we have x1 of m. 658 00:42:07,680 --> 00:42:10,230 Here we have x2 of m. 659 00:42:10,230 --> 00:42:18,430 And what we want to form is x2 of n minus m taken 660 00:42:18,430 --> 00:42:23,110 modulo capital N. So first of all, 661 00:42:23,110 --> 00:42:30,090 here we have x2 of minus m, modulo capital N. 662 00:42:30,090 --> 00:42:33,970 And what that corresponds to is flipping x2 of m, 663 00:42:33,970 --> 00:42:40,690 flipping it over, and then repeating it periodically. 664 00:42:40,690 --> 00:42:44,740 And the periodic replicas I've designated here 665 00:42:44,740 --> 00:42:48,130 with dashed lines, so that we focus completely 666 00:42:48,130 --> 00:42:53,240 on the interval 0 to capital N minus 1. 667 00:42:53,240 --> 00:42:56,050 So in fact, notice that we could have thought 668 00:42:56,050 --> 00:43:02,230 of flipping this modulo capital N by first flipping 669 00:43:02,230 --> 00:43:04,600 it, and then taking the values over here, 670 00:43:04,600 --> 00:43:08,300 and just simply replacing them on the right-hand side. 671 00:43:08,300 --> 00:43:12,610 In other words, it's these values that ended up over here. 672 00:43:12,610 --> 00:43:15,310 When we flip this sequence, of course, the value at 0 673 00:43:15,310 --> 00:43:20,560 stays where it is, and the rest of those values end 674 00:43:20,560 --> 00:43:23,905 up again in the range 0 to capital N minus 1, 675 00:43:23,905 --> 00:43:26,800 but flipped over. 676 00:43:26,800 --> 00:43:29,590 If we shift this periodic sequence 677 00:43:29,590 --> 00:43:36,545 by one value to the left, then what we have is-- 678 00:43:39,290 --> 00:43:42,230 sorry, shift it by one value to the right, 679 00:43:42,230 --> 00:43:46,460 then we end up with this periodic sequence. 680 00:43:46,460 --> 00:43:51,032 Again, concentrate on the range 0 to capital N minus 1, 681 00:43:51,032 --> 00:43:55,580 and you can see that that is a circular shift 682 00:43:55,580 --> 00:44:02,180 of one point of one period of this periodic sequence. 683 00:44:02,180 --> 00:44:04,010 In other words, if you circularly 684 00:44:04,010 --> 00:44:07,100 shift this to the right, that point 685 00:44:07,100 --> 00:44:10,460 comes back in on the other side. 686 00:44:10,460 --> 00:44:13,490 Here is the same thing shifted. 687 00:44:13,490 --> 00:44:16,400 Again, another point, so we can think 688 00:44:16,400 --> 00:44:19,970 of having taken this one period, shifting it to the right, 689 00:44:19,970 --> 00:44:23,390 and this point comes back in again on the other side. 690 00:44:23,390 --> 00:44:26,120 Of course, the way we're really getting it is-- 691 00:44:26,120 --> 00:44:27,680 or another way to think of it is-- 692 00:44:27,680 --> 00:44:31,400 it's the corresponding value from the next period over that 693 00:44:31,400 --> 00:44:32,720 gets shifted in. 694 00:44:32,720 --> 00:44:35,210 But I'd prefer now that you begin 695 00:44:35,210 --> 00:44:40,500 thinking of it as a circular shift of just that one period. 696 00:44:40,500 --> 00:44:43,040 So, you can see that as we continue 697 00:44:43,040 --> 00:44:47,990 changing the value of n, then this one period 698 00:44:47,990 --> 00:44:49,910 gets circularly shifted. 699 00:44:49,910 --> 00:44:51,590 This value will end up over here, 700 00:44:51,590 --> 00:44:55,070 then this value will end up over here, et cetera. 701 00:44:55,070 --> 00:44:59,570 Well, if we carry out the convolution of x1 of m and x2 702 00:44:59,570 --> 00:45:03,340 of m, for example for little n equals 1, 703 00:45:03,340 --> 00:45:06,830 we take this, multiply it by that, 704 00:45:06,830 --> 00:45:12,390 and sum up the values in the range 0 to capital N minus 1. 705 00:45:12,390 --> 00:45:14,480 Similarly for little n equals 2, we 706 00:45:14,480 --> 00:45:17,810 take this and multiply it by that, 707 00:45:17,810 --> 00:45:20,120 sum up the values in the product in the range 708 00:45:20,120 --> 00:45:23,594 0 to capital N minus 1. 709 00:45:23,594 --> 00:45:27,320 One thing that you should be able to see in this example 710 00:45:27,320 --> 00:45:31,610 is that the answer will look quite a bit different 711 00:45:31,610 --> 00:45:34,550 than a linear convolution of these two, 712 00:45:34,550 --> 00:45:40,640 because in fact, notice that we're multiplying always by-- 713 00:45:40,640 --> 00:45:42,260 let's call this amplitude 1. 714 00:45:42,260 --> 00:45:44,930 We're multiplying always by 1. 715 00:45:44,930 --> 00:45:49,820 And no matter how we circularly shifted this one period, 716 00:45:49,820 --> 00:45:53,820 each value would still get multiplied by the same amount. 717 00:45:53,820 --> 00:45:57,440 And consequently, the resulting values in the convolution 718 00:45:57,440 --> 00:46:01,940 will be the sum of these values no matter how much circular 719 00:46:01,940 --> 00:46:03,710 shifting that we do. 720 00:46:03,710 --> 00:46:07,670 So that for this particular example, the convolution 721 00:46:07,670 --> 00:46:10,460 comes out to look like this. 722 00:46:10,460 --> 00:46:12,950 That is the values in the convolution 723 00:46:12,950 --> 00:46:17,580 are constant over the range 0 to capital N minus 1. 724 00:46:17,580 --> 00:46:20,630 Then of course, we had this either cutting the cylinder, 725 00:46:20,630 --> 00:46:22,520 if you want to look at it that way, 726 00:46:22,520 --> 00:46:26,390 or multiplying it by R sub capital N of n, 727 00:46:26,390 --> 00:46:30,930 and consequently we get 0 outside that range. 728 00:46:30,930 --> 00:46:31,430 All right. 729 00:46:31,430 --> 00:46:34,330 Well, what I would hope at this point, 730 00:46:34,330 --> 00:46:37,790 I would certainly expect that the idea 731 00:46:37,790 --> 00:46:41,420 of circular convolution isn't totally clear. 732 00:46:41,420 --> 00:46:44,090 I would expect that a little bit of the notion 733 00:46:44,090 --> 00:46:45,590 is coming out of the haze. 734 00:46:45,590 --> 00:46:50,840 That is the notion that things are being shifted circularly 735 00:46:50,840 --> 00:46:52,760 to implement the convolution. 736 00:46:52,760 --> 00:46:55,190 That's the kind of idea that I would hope 737 00:46:55,190 --> 00:46:57,920 is becoming somewhat clearer. 738 00:46:57,920 --> 00:47:05,270 And in the next lecture, what I intend to focus on entirely 739 00:47:05,270 --> 00:47:09,980 is the additional interpretation of circular convolution, 740 00:47:09,980 --> 00:47:13,910 with the hope that by the end of the next lecture 741 00:47:13,910 --> 00:47:17,900 the notions of the circular shifting and the difference 742 00:47:17,900 --> 00:47:21,770 between circular convolution and linear convolution 743 00:47:21,770 --> 00:47:22,680 will be clear. 744 00:47:22,680 --> 00:47:24,400 Thank you.