1 00:00:00,090 --> 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 --> 00:00:17,280 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,280 --> 00:00:18,450 at ocw.mit.edu. 8 00:00:53,220 --> 00:00:54,450 ALAN V. OPPENHEIM: OK. 9 00:00:54,450 --> 00:00:56,820 In the last several lectures, we've 10 00:00:56,820 --> 00:01:01,940 been discussing the Fourier and Z-transforms. 11 00:01:01,940 --> 00:01:03,880 And we've seen in particular that the Fourier 12 00:01:03,880 --> 00:01:07,320 and Z-transforms provide us with a set 13 00:01:07,320 --> 00:01:12,030 of important analytical tools for representing discrete time 14 00:01:12,030 --> 00:01:17,980 signals, and also for dealing with discrete time systems. 15 00:01:17,980 --> 00:01:21,630 We saw, for example, that through the use of the Fourier 16 00:01:21,630 --> 00:01:26,410 transform or the Z-transform, we could convert convolution 17 00:01:26,410 --> 00:01:29,260 in the time domain to multiplication 18 00:01:29,260 --> 00:01:32,260 in either the frequency domain in the Fourier transform 19 00:01:32,260 --> 00:01:35,170 case, or more generally, in the Z 20 00:01:35,170 --> 00:01:39,740 domain in the Z-transform case. 21 00:01:39,740 --> 00:01:43,790 Now one of the things that it's important to recognize 22 00:01:43,790 --> 00:01:47,420 is that for the most part, the Fourier transform 23 00:01:47,420 --> 00:01:52,010 and the Z-transform are primarily analytical tools. 24 00:01:52,010 --> 00:01:55,700 That is, it would be hard to imagine 25 00:01:55,700 --> 00:01:58,580 implementing, for example, a discrete time 26 00:01:58,580 --> 00:02:02,630 system by first computing the Fourier 27 00:02:02,630 --> 00:02:05,810 transform of the sequences, multiplying 28 00:02:05,810 --> 00:02:08,810 the Fourier transforms together, and then computing 29 00:02:08,810 --> 00:02:10,789 the inverse transform. 30 00:02:10,789 --> 00:02:14,450 One of the reasons that that's obviously a difficult thing 31 00:02:14,450 --> 00:02:18,230 to do computationally is that we saw, 32 00:02:18,230 --> 00:02:23,240 as we discussed, for example, the Fourier transform, 33 00:02:23,240 --> 00:02:25,790 that the Fourier transform is a function 34 00:02:25,790 --> 00:02:27,980 of a continuous variable. 35 00:02:27,980 --> 00:02:31,340 That is, omega in the Fourier transform 36 00:02:31,340 --> 00:02:33,472 is a continuous variable. 37 00:02:33,472 --> 00:02:35,180 So that, in fact, if we wanted to compute 38 00:02:35,180 --> 00:02:38,030 the Fourier transform explicitly, 39 00:02:38,030 --> 00:02:40,880 we would have to compute it at an infinite number 40 00:02:40,880 --> 00:02:42,970 of frequencies. 41 00:02:42,970 --> 00:02:48,990 Similarly, we have a situation like that for the Z-transform. 42 00:02:48,990 --> 00:02:51,060 Well, today I'd like to introduce 43 00:02:51,060 --> 00:02:55,890 a third transform, which I'll refer to as 44 00:02:55,890 --> 00:02:59,570 the discrete Fourier transform. 45 00:02:59,570 --> 00:03:03,930 The discrete Fourier transform is similar in style 46 00:03:03,930 --> 00:03:06,600 to the Fourier transform and the Z-transform, 47 00:03:06,600 --> 00:03:11,010 as we've been talking about, in the sense that more or less, 48 00:03:11,010 --> 00:03:13,590 the discrete Fourier transform maps 49 00:03:13,590 --> 00:03:17,640 convolution to multiplication. 50 00:03:17,640 --> 00:03:20,050 Also, there are properties of Fourier 51 00:03:20,050 --> 00:03:22,620 transform-- the discrete Fourier transform-- 52 00:03:22,620 --> 00:03:25,380 that are similar to the properties of the Fourier 53 00:03:25,380 --> 00:03:29,820 transform and Z-transform, as we've been talking about them. 54 00:03:29,820 --> 00:03:33,330 But the discrete Fourier transform 55 00:03:33,330 --> 00:03:36,600 is different than the other transforms 56 00:03:36,600 --> 00:03:41,520 that we've been discussing in a number of important respects. 57 00:03:41,520 --> 00:03:47,930 One of the respects, one of the reasons that it's different 58 00:03:47,930 --> 00:03:54,380 is that it is a transform that can be explicitly evaluated. 59 00:03:54,380 --> 00:03:57,890 And consequently, it is important, 60 00:03:57,890 --> 00:04:01,580 not only in the analysis of discrete time systems, 61 00:04:01,580 --> 00:04:06,200 but also, as we'll see, in the implementation of discrete time 62 00:04:06,200 --> 00:04:07,130 systems. 63 00:04:07,130 --> 00:04:10,970 That is, many digital signal processing algorithms, 64 00:04:10,970 --> 00:04:16,339 as we'll see, actually involve the explicit computation 65 00:04:16,339 --> 00:04:19,399 of the discrete Fourier transform, which 66 00:04:19,399 --> 00:04:23,720 is the Fourier transform that we're about to introduce. 67 00:04:23,720 --> 00:04:27,290 Now let me try to explain a little about what 68 00:04:27,290 --> 00:04:29,030 the discrete Fourier transform is 69 00:04:29,030 --> 00:04:33,040 before we look at it in detail. 70 00:04:33,040 --> 00:04:37,060 Basically, the discrete Fourier transform 71 00:04:37,060 --> 00:04:40,480 is related to the Fourier transform 72 00:04:40,480 --> 00:04:43,000 that we've been discussing in the sense 73 00:04:43,000 --> 00:04:47,590 that it corresponds to samples of the Fourier transform, 74 00:04:47,590 --> 00:04:52,540 or more generally, samples of the Z-transform. 75 00:04:52,540 --> 00:04:58,000 Now that requires that we impose some restrictions 76 00:04:58,000 --> 00:05:00,820 on the sequences that we're representing 77 00:05:00,820 --> 00:05:02,440 through that transform. 78 00:05:02,440 --> 00:05:07,420 And as we'll see, it turns out that we can represent sequences 79 00:05:07,420 --> 00:05:09,730 that are of finite length. 80 00:05:09,730 --> 00:05:14,290 That is, only have a finite number of non-zero samples. 81 00:05:14,290 --> 00:05:17,920 We can represent those sequences by samples of their Fourier 82 00:05:17,920 --> 00:05:19,120 transform. 83 00:05:19,120 --> 00:05:24,960 Those samples then correspond to the discrete Fourier transform. 84 00:05:24,960 --> 00:05:28,380 Well, there are some issues that arise 85 00:05:28,380 --> 00:05:30,780 in looking at the discrete Fourier transform, 86 00:05:30,780 --> 00:05:34,830 or DFT, as I'll refer to it. 87 00:05:34,830 --> 00:05:39,720 And a number of these issues relate to the fact 88 00:05:39,720 --> 00:05:44,220 that the discrete Fourier transform 89 00:05:44,220 --> 00:05:49,750 has properties that are somewhat different, and also somewhat 90 00:05:49,750 --> 00:05:52,570 similar, to the Fourier transform properties 91 00:05:52,570 --> 00:05:54,820 that we've been discussing. 92 00:05:54,820 --> 00:05:56,890 There are lots of ways of introducing 93 00:05:56,890 --> 00:05:59,830 the discrete Fourier transform. 94 00:05:59,830 --> 00:06:04,990 And the one that I guess I find the most interesting, 95 00:06:04,990 --> 00:06:09,190 and the most satisfactory, is to relate the discrete Fourier 96 00:06:09,190 --> 00:06:12,850 transform to the discrete Fourier 97 00:06:12,850 --> 00:06:16,690 series for periodic sequences. 98 00:06:16,690 --> 00:06:19,960 And in particular, relate the notion 99 00:06:19,960 --> 00:06:25,060 of finite length sequences to periodic sequences. 100 00:06:25,060 --> 00:06:27,280 Well, let me explain in a little more detail what 101 00:06:27,280 --> 00:06:30,280 I mean by that. 102 00:06:30,280 --> 00:06:33,720 Let's consider, first of all, a sequence 103 00:06:33,720 --> 00:06:40,950 x of n, which I'll restrict to be a finite length sequence. 104 00:06:40,950 --> 00:06:46,090 The sequence, for example, one example indicated here, 105 00:06:46,090 --> 00:06:53,290 has a set of non-zero values only over a finite range 106 00:06:53,290 --> 00:06:56,010 of the argument n. 107 00:06:56,010 --> 00:06:59,790 Here is a sequence that is of finite length. 108 00:06:59,790 --> 00:07:03,370 And I've chosen to refer to it as a sequence of finite length 109 00:07:03,370 --> 00:07:08,540 capital N. 0 outside the range from little n 110 00:07:08,540 --> 00:07:12,260 equals 0 to little n equals capital N minus 1. 111 00:07:12,260 --> 00:07:15,730 It's 0 outside that range. 112 00:07:15,730 --> 00:07:19,150 It's interesting, just as an aside, 113 00:07:19,150 --> 00:07:22,180 to note that for this particular sequence, 114 00:07:22,180 --> 00:07:25,690 it's also actually 0 outside the range from little n 115 00:07:25,690 --> 00:07:29,410 equals 1 to capital N minus 1. 116 00:07:29,410 --> 00:07:33,970 And in general, obviously, if I talk about a finite length 117 00:07:33,970 --> 00:07:37,780 sequence of length capital N, I can also 118 00:07:37,780 --> 00:07:40,810 refer to it as a finite length sequence of length 119 00:07:40,810 --> 00:07:44,530 greater than capital N. That is, the important statement 120 00:07:44,530 --> 00:07:46,990 about the sequence being finite length, 121 00:07:46,990 --> 00:07:51,010 of finite length capital N, is that the sequence values are 122 00:07:51,010 --> 00:07:56,740 0 outside the range 0 to capital N minus 1. 123 00:07:56,740 --> 00:07:59,200 Although obviously, the sequence could also 124 00:07:59,200 --> 00:08:02,065 be 0 inside that range for some of the values. 125 00:08:04,630 --> 00:08:10,430 Now the basic notion that leads to the discrete Fourier 126 00:08:10,430 --> 00:08:13,600 transform, or one way of looking at the discrete Fourier 127 00:08:13,600 --> 00:08:18,040 transform, is to recognize that if I have a finite length 128 00:08:18,040 --> 00:08:21,510 sequence, as I have here, I could 129 00:08:21,510 --> 00:08:25,950 construct from that sequence a periodic sequence. 130 00:08:25,950 --> 00:08:31,200 And let me denote the periodic sequence by x tilde of n. 131 00:08:31,200 --> 00:08:34,169 In general, by the way, when I refer to a sequence 132 00:08:34,169 --> 00:08:36,030 with a tilde on it, that will always 133 00:08:36,030 --> 00:08:38,760 correspond to a periodic sequence. 134 00:08:38,760 --> 00:08:42,030 And let me construct this periodic sequence 135 00:08:42,030 --> 00:08:45,180 by simply taking the finite length sequence 136 00:08:45,180 --> 00:08:49,770 and repeating it over and over again with a period of capital 137 00:08:49,770 --> 00:08:52,930 N. In other words, I can construct 138 00:08:52,930 --> 00:08:58,890 the periodic sequence, x tilde of n equal to x of n 139 00:08:58,890 --> 00:09:02,370 plus x of n shifted to the left by capital 140 00:09:02,370 --> 00:09:07,560 N and x of n shifted to the right by capital N and 2n, 2 141 00:09:07,560 --> 00:09:10,620 capital N, and 3 capital N, et cetera. 142 00:09:10,620 --> 00:09:14,370 In other words, just simply taking this sequence 143 00:09:14,370 --> 00:09:17,580 and repeating it over and over again 144 00:09:17,580 --> 00:09:20,100 with a period of capital N. 145 00:09:20,100 --> 00:09:24,660 So obviously I can generate, from a finite length 146 00:09:24,660 --> 00:09:29,050 sequence, a periodic sequence. 147 00:09:29,050 --> 00:09:32,160 And in fact, I could get the finite length sequence 148 00:09:32,160 --> 00:09:38,010 back from the periodic sequence by simply extracting one period 149 00:09:38,010 --> 00:09:40,140 of this periodic sequence. 150 00:09:40,140 --> 00:09:46,680 In other words, I can get x of n back from x tilde of n simply 151 00:09:46,680 --> 00:09:53,100 by multiplying by unity for n between 0 and capital N 152 00:09:53,100 --> 00:10:00,650 minus 1, and 0 outside that range. 153 00:10:00,650 --> 00:10:02,200 Now the important point here-- 154 00:10:02,200 --> 00:10:04,270 there are a couple of important points. 155 00:10:04,270 --> 00:10:08,260 One of them is that if I have a finite length sequence, 156 00:10:08,260 --> 00:10:11,230 I can turn it into a periodic sequence. 157 00:10:11,230 --> 00:10:13,540 If I have a periodic sequence, I can 158 00:10:13,540 --> 00:10:15,460 get back to the finite length sequence 159 00:10:15,460 --> 00:10:18,280 simply by extracting one period. 160 00:10:18,280 --> 00:10:22,450 Or what that essentially says is that there really 161 00:10:22,450 --> 00:10:25,900 isn't much difference between a finite length sequence 162 00:10:25,900 --> 00:10:28,030 and a periodic sequence. 163 00:10:28,030 --> 00:10:30,430 That is, a finite length sequence 164 00:10:30,430 --> 00:10:34,440 is defined by capital N values. 165 00:10:34,440 --> 00:10:37,950 A periodic sequence is also defined by capital N values 166 00:10:37,950 --> 00:10:41,490 because once I specify a single period, 167 00:10:41,490 --> 00:10:44,460 then I don't have any more degrees 168 00:10:44,460 --> 00:10:48,060 of freedom in specifying the rest of the sequence. 169 00:10:48,060 --> 00:10:51,300 And that's an important point to keep in mind, particularly 170 00:10:51,300 --> 00:10:53,440 as we go through the next several lectures. 171 00:10:53,440 --> 00:10:58,610 A finite length sequence is very similar to a periodic sequence 172 00:10:58,610 --> 00:11:03,690 in that both of them are simply defined by capital N values. 173 00:11:03,690 --> 00:11:05,670 One way to think of that, by the way, 174 00:11:05,670 --> 00:11:11,190 is to think of taking a finite length sequence of capital N 175 00:11:11,190 --> 00:11:13,040 values. 176 00:11:13,040 --> 00:11:15,440 Instead of drawing it along a straight line 177 00:11:15,440 --> 00:11:19,130 as I've done here, imagine taking this finite length 178 00:11:19,130 --> 00:11:21,440 sequence and wrapping it around the circumference 179 00:11:21,440 --> 00:11:23,380 of a cylinder. 180 00:11:23,380 --> 00:11:27,400 So we start with the finite length sequence and just simply 181 00:11:27,400 --> 00:11:31,330 display it wrapped around the circumference of a cylinder. 182 00:11:31,330 --> 00:11:34,060 As we run around the cylinder, over and over 183 00:11:34,060 --> 00:11:40,780 again, what we see is the periodic sequence x tilde of n. 184 00:11:40,780 --> 00:11:43,810 So in a sense, the periodic sequence 185 00:11:43,810 --> 00:11:46,240 is just simply like the finite length sequence, 186 00:11:46,240 --> 00:11:48,730 but wrapped on a cylinder instead of laid out 187 00:11:48,730 --> 00:11:50,260 in a straight line. 188 00:11:50,260 --> 00:11:52,810 And that's also a picture that will 189 00:11:52,810 --> 00:11:55,420 recur several times as we go through the next several 190 00:11:55,420 --> 00:11:56,530 lectures. 191 00:11:56,530 --> 00:12:00,130 So we can generate the periodic sequence from the finite length 192 00:12:00,130 --> 00:12:01,690 sequence. 193 00:12:01,690 --> 00:12:07,700 We can also recover the sequence x of n 194 00:12:07,700 --> 00:12:12,440 from the periodic sequence by extracting just one period, 195 00:12:12,440 --> 00:12:18,020 as I've indicated here, x of n is x tilde of n in the range, 196 00:12:18,020 --> 00:12:21,530 little n between 0 and capital N minus 1. 197 00:12:21,530 --> 00:12:24,200 And it's equal to 0, otherwise. 198 00:12:24,200 --> 00:12:27,770 Or what we'll use as a convenient form of notation 199 00:12:27,770 --> 00:12:33,560 for that is to express x of n, the finite length sequence, 200 00:12:33,560 --> 00:12:38,480 as the periodic sequence, x tilde of n multiplied 201 00:12:38,480 --> 00:12:44,750 by another sequence, which I'll denote as R sub capital N of n, 202 00:12:44,750 --> 00:12:46,880 where R sub capital N of n is just 203 00:12:46,880 --> 00:12:49,810 simply a rectangular sequence. 204 00:12:49,810 --> 00:12:54,530 So R sub capital N of n is 1 for little n between 0 205 00:12:54,530 --> 00:13:00,010 and capital N minus 1, and it's equal to 0, otherwise. 206 00:13:00,010 --> 00:13:00,510 All right. 207 00:13:00,510 --> 00:13:03,490 So it's important to begin, at this point, 208 00:13:03,490 --> 00:13:08,150 to think of finite length and periodic sequences 209 00:13:08,150 --> 00:13:10,940 as more or less the same type of thing in the sense 210 00:13:10,940 --> 00:13:15,010 that it's easy to go back and forth from one to the other. 211 00:13:15,010 --> 00:13:18,240 Now, why is this point of view important? 212 00:13:18,240 --> 00:13:22,140 Well, we know certainly in the continuous time case 213 00:13:22,140 --> 00:13:26,250 that a periodic time function can be represented by a Fourier 214 00:13:26,250 --> 00:13:27,730 series. 215 00:13:27,730 --> 00:13:30,760 In the discrete time case, a periodic sequence, 216 00:13:30,760 --> 00:13:34,960 likewise, can be represented by a Fourier series. 217 00:13:34,960 --> 00:13:38,440 And the idea, that is, the key point 218 00:13:38,440 --> 00:13:41,410 behind the discrete Fourier transform 219 00:13:41,410 --> 00:13:46,450 is that we can use the Fourier series representation 220 00:13:46,450 --> 00:13:50,590 of the periodic sequence to represent the finite length 221 00:13:50,590 --> 00:13:51,670 sequence. 222 00:13:51,670 --> 00:13:53,960 That is, that, in essence, provides 223 00:13:53,960 --> 00:13:58,720 a Fourier kind of representation for a finite length sequence. 224 00:13:58,720 --> 00:14:04,930 So we have the notion then that the periodic sequence x 225 00:14:04,930 --> 00:14:10,940 tilde of n has a Fourier series representation. 226 00:14:10,940 --> 00:14:15,660 We can compute the discrete Fourier series 227 00:14:15,660 --> 00:14:18,780 of this periodic sequence. 228 00:14:18,780 --> 00:14:20,190 And as we'll see-- 229 00:14:20,190 --> 00:14:21,840 not in this lecture, but as we'll 230 00:14:21,840 --> 00:14:24,690 see in more detail in the next lecture-- 231 00:14:24,690 --> 00:14:29,160 it is discrete Fourier series representation 232 00:14:29,160 --> 00:14:33,120 of this periodic sequence that is 233 00:14:33,120 --> 00:14:38,820 what we'll call the discrete Fourier transform of x of n. 234 00:14:38,820 --> 00:14:42,240 So let's begin then with a discussion 235 00:14:42,240 --> 00:14:46,740 of the discrete Fourier series of periodic sequences 236 00:14:46,740 --> 00:14:51,630 with the idea that we'll be applying that representation 237 00:14:51,630 --> 00:14:54,810 to the representation of finite length sequence. 238 00:14:54,810 --> 00:14:57,180 And that representation is what will 239 00:14:57,180 --> 00:15:01,490 correspond to the discrete Fourier transform. 240 00:15:01,490 --> 00:15:01,990 OK. 241 00:15:01,990 --> 00:15:06,930 So we want to talk about the discrete Fourier series. 242 00:15:06,930 --> 00:15:13,430 We're considering a sequence x tilde of n, which is periodic, 243 00:15:13,430 --> 00:15:19,160 and it's period we'll call capital N. 244 00:15:19,160 --> 00:15:21,860 In the continuous time case, we know 245 00:15:21,860 --> 00:15:24,470 that if we have a periodic time function, 246 00:15:24,470 --> 00:15:28,910 we can represent it as a linear combination of harmonically 247 00:15:28,910 --> 00:15:31,070 related complex exponentials. 248 00:15:31,070 --> 00:15:33,920 That's the Fourier series representation 249 00:15:33,920 --> 00:15:36,150 in the continuous time case. 250 00:15:36,150 --> 00:15:39,500 And I'll just simply state without proof 251 00:15:39,500 --> 00:15:42,500 that the same kind of relationship 252 00:15:42,500 --> 00:15:44,750 holds in the discrete time case. 253 00:15:44,750 --> 00:15:50,900 That is, we can represent a periodic sequence x tilde of n 254 00:15:50,900 --> 00:15:56,540 as a linear combination of complex exponentials, which 255 00:15:56,540 --> 00:16:00,260 are harmonically related to the frequency. 256 00:16:00,260 --> 00:16:04,100 Or equivalently, the reciprocal of the period, 257 00:16:04,100 --> 00:16:08,360 so that this forms a general relationship 258 00:16:08,360 --> 00:16:13,510 for a discrete Fourier series of a periodic sequence x 259 00:16:13,510 --> 00:16:15,440 tilde of n. 260 00:16:15,440 --> 00:16:17,510 That is, these are the harmonically related 261 00:16:17,510 --> 00:16:22,310 complex exponentials, just as we form linear combinations 262 00:16:22,310 --> 00:16:26,120 of harmonically related complex exponentials for the Fourier 263 00:16:26,120 --> 00:16:29,560 series in a continuous time case. 264 00:16:29,560 --> 00:16:31,190 What are the Fourier coefficients? 265 00:16:31,190 --> 00:16:34,840 Well, it's, of course, these capital X tildes 266 00:16:34,840 --> 00:16:41,030 of k, which we'll have a little more to say about in a minute. 267 00:16:41,030 --> 00:16:46,520 Well, notice that I haven't specified as of yet any limits 268 00:16:46,520 --> 00:16:48,560 on this summation. 269 00:16:48,560 --> 00:16:52,910 And in particular, what we need to examine 270 00:16:52,910 --> 00:16:57,140 is how many distinct periodically or harmonically 271 00:16:57,140 --> 00:17:01,590 related complex exponentials there are. 272 00:17:01,590 --> 00:17:06,819 Well, let's take a look at the complex exponential, 273 00:17:06,819 --> 00:17:10,020 the set of complex exponentials e to the j, 2 274 00:17:10,020 --> 00:17:13,869 pi over capital N, little nk. 275 00:17:13,869 --> 00:17:16,000 And the statement that I want to make 276 00:17:16,000 --> 00:17:20,290 is that these complex exponentials are periodic. 277 00:17:20,290 --> 00:17:23,060 Of course, we know that they're periodic in n. 278 00:17:23,060 --> 00:17:25,429 But they're are also periodic in k. 279 00:17:28,300 --> 00:17:34,366 As we vary k from 0 to capital N minus 1, 280 00:17:34,366 --> 00:17:38,240 we generate all of the possible harmonically related 281 00:17:38,240 --> 00:17:41,900 complex exponentials with this fundamental frequency, 2 282 00:17:41,900 --> 00:17:44,810 pi over capital N. Well, we can see 283 00:17:44,810 --> 00:17:52,800 that very simply by substituting in for k, k plus capital N. 284 00:17:52,800 --> 00:17:56,310 And then recognizing that we can break 285 00:17:56,310 --> 00:17:59,430 this complex exponential into the product of two 286 00:17:59,430 --> 00:18:06,420 complex exponentials e to the j, 2 pi over capital N, nk, 287 00:18:06,420 --> 00:18:11,970 and e to the j, 2 pi over capital N times 288 00:18:11,970 --> 00:18:15,600 little n times capital N. 289 00:18:15,600 --> 00:18:19,370 Well, these capital N's cancel each other out. 290 00:18:19,370 --> 00:18:23,450 This factor is then e to the j, 2 pi times little n. 291 00:18:23,450 --> 00:18:27,650 Well, any integer multiple of 2 pi up here then just 292 00:18:27,650 --> 00:18:31,250 simply reduces this to unity. 293 00:18:31,250 --> 00:18:36,790 So that, in fact, e to the j, 2 pi over capital N, little n 294 00:18:36,790 --> 00:18:40,095 times k plus capital N is the same as 295 00:18:40,095 --> 00:18:45,310 e to the j, 2 pi over capital N, little n times k. 296 00:18:45,310 --> 00:18:47,620 Well, that shouldn't be surprising, actually, 297 00:18:47,620 --> 00:18:50,740 because that's a point that's come up time 298 00:18:50,740 --> 00:18:53,170 and again as we've been going through these lectures. 299 00:18:53,170 --> 00:18:59,950 The point being that sinusoids in the discrete time 300 00:18:59,950 --> 00:19:04,750 domain, as we vary sinusoids in frequency-- 301 00:19:04,750 --> 00:19:10,400 we've seen time and again, in the range 0 to 2 pi-- 302 00:19:10,400 --> 00:19:14,230 in fact, those are all the sinusoids that we can generate. 303 00:19:14,230 --> 00:19:16,180 And if we keep going in frequency, 304 00:19:16,180 --> 00:19:19,070 we just simply see the same ones over again. 305 00:19:19,070 --> 00:19:22,210 So this is a consequence of that. 306 00:19:22,210 --> 00:19:24,420 But for the discrete Fourier series, 307 00:19:24,420 --> 00:19:25,900 it says an important thing. 308 00:19:25,900 --> 00:19:29,710 It says that in forming the discrete Fourier 309 00:19:29,710 --> 00:19:36,410 series, once we've used the complex exponentials for k 310 00:19:36,410 --> 00:19:39,890 between 0 and capital N minus 1, we've 311 00:19:39,890 --> 00:19:42,440 used all the complex exponentials 312 00:19:42,440 --> 00:19:45,680 with this fundamental frequency that we have. 313 00:19:45,680 --> 00:19:48,080 And if we keep going with k, we're 314 00:19:48,080 --> 00:19:51,800 just simply going to see the same complex exponentials over 315 00:19:51,800 --> 00:19:53,900 and over again. 316 00:19:53,900 --> 00:19:56,480 What does that say about the discrete Fourier series? 317 00:19:56,480 --> 00:19:59,900 It says that in the representation 318 00:19:59,900 --> 00:20:06,590 of the discrete Fourier series, the limits on this sum 319 00:20:06,590 --> 00:20:10,400 don't range from minus infinity to plus infinity. 320 00:20:10,400 --> 00:20:16,265 They run simply from 0 to capital N minus 1. 321 00:20:16,265 --> 00:20:21,500 So once I've looked at these linear combination 322 00:20:21,500 --> 00:20:24,500 of these complex exponentials for k between 0 323 00:20:24,500 --> 00:20:28,880 and capital N minus 1, there are no new complex exponentials 324 00:20:28,880 --> 00:20:32,790 with that fundamental frequency that I'm able to find. 325 00:20:32,790 --> 00:20:37,980 So this then is the form of the discrete Fourier series. 326 00:20:37,980 --> 00:20:42,150 There's one additional insertion that I'd like to make. 327 00:20:42,150 --> 00:20:47,190 And this is just simply a normalization factor. 328 00:20:47,190 --> 00:20:51,090 I want to multiply this by 1 over capital N. Obviously, that 329 00:20:51,090 --> 00:20:53,200 doesn't make any essential difference. 330 00:20:53,200 --> 00:20:55,770 It's just a factor for normalization. 331 00:20:55,770 --> 00:21:00,900 And it plays a role which is similar in the continuous time 332 00:21:00,900 --> 00:21:07,540 case to the role that 2 pi or 1 over 2 pi plays. 333 00:21:07,540 --> 00:21:08,050 All right. 334 00:21:08,050 --> 00:21:12,470 So here we have the discrete Fourier series. 335 00:21:12,470 --> 00:21:16,370 It's relatively straightforward to show 336 00:21:16,370 --> 00:21:21,110 that you can obtain the Fourier series coefficients x 337 00:21:21,110 --> 00:21:25,430 tilde of k from x tilde of n through 338 00:21:25,430 --> 00:21:29,180 an inverse relationship, which is the relationship 339 00:21:29,180 --> 00:21:31,920 that I've indicated here. 340 00:21:31,920 --> 00:21:36,540 So this is then, in essence, the inverse discrete Fourier 341 00:21:36,540 --> 00:21:40,140 series, or equivalently, the relationship 342 00:21:40,140 --> 00:21:43,320 for obtaining the discrete Fourier 343 00:21:43,320 --> 00:21:49,160 series coefficients from the periodic sequence x tilde of n. 344 00:21:49,160 --> 00:21:53,750 Now notice that I've happened to write the Fourier series 345 00:21:53,750 --> 00:21:57,710 coefficients with a tilde over them, 346 00:21:57,710 --> 00:22:04,310 implying that those coefficients are themselves periodic. 347 00:22:04,310 --> 00:22:08,490 Well, are they periodic, or aren't they periodic? 348 00:22:08,490 --> 00:22:11,520 What I've been saying here and what I've been saying here-- 349 00:22:11,520 --> 00:22:14,250 and I've spent a long time saying this-- 350 00:22:14,250 --> 00:22:18,330 that there are only a finite number of complex exponentials. 351 00:22:18,330 --> 00:22:21,915 Once I've looked at them in the range 0 to capital N minus 1, 352 00:22:21,915 --> 00:22:24,930 I've seen all the ones I can see. 353 00:22:24,930 --> 00:22:29,080 And in that sense, the Fourier series coefficients, 354 00:22:29,080 --> 00:22:32,730 x tilde of k, are finite. 355 00:22:32,730 --> 00:22:35,731 That is, there are only a finite number of them. 356 00:22:35,731 --> 00:22:37,730 Well, the important point is that there are only 357 00:22:37,730 --> 00:22:42,530 a finite number of distinct coefficients. 358 00:22:42,530 --> 00:22:45,470 Also in this relationship, whether I 359 00:22:45,470 --> 00:22:48,995 consider these as periodic or not periodic, 360 00:22:48,995 --> 00:22:51,530 I still only use a finite number of them. 361 00:22:51,530 --> 00:22:55,970 That is, I only use them in the range k equals 0 to capital 362 00:22:55,970 --> 00:22:57,620 N minus 1. 363 00:22:57,620 --> 00:23:02,390 And how I choose to interpret capital X tilde of k 364 00:23:02,390 --> 00:23:05,720 outside the range 0 to capital N minus 1 365 00:23:05,720 --> 00:23:09,290 is going to have absolutely no effect on the evaluation 366 00:23:09,290 --> 00:23:11,850 of this summation. 367 00:23:11,850 --> 00:23:15,970 Well, it turns out to be convenient to interpret 368 00:23:15,970 --> 00:23:21,690 the Fourier series coefficients as being periodic in k. 369 00:23:21,690 --> 00:23:25,380 Well, in fact, the relationship as I've written it here 370 00:23:25,380 --> 00:23:28,920 makes it evident, in this particular relationship, 371 00:23:28,920 --> 00:23:31,170 that the coefficients are periodic. 372 00:23:31,170 --> 00:23:39,940 In other words, if I substitute in for k, k plus capital N, 373 00:23:39,940 --> 00:23:45,130 then I'll get back exactly the same relation that I have here. 374 00:23:45,130 --> 00:23:49,450 Because of the fact that e to the minus j 2 pi over capital 375 00:23:49,450 --> 00:23:53,710 N times little n times k plus capital 376 00:23:53,710 --> 00:24:00,740 N is equal to e the minus j 2 pi over capital 377 00:24:00,740 --> 00:24:04,490 N times little n times k. 378 00:24:04,490 --> 00:24:06,430 So that if I simply examined-- 379 00:24:06,430 --> 00:24:09,330 I asked from this relationship-- 380 00:24:09,330 --> 00:24:14,000 what does capital X tilde of k plus capital N come out to be? 381 00:24:14,000 --> 00:24:17,450 If I simply substitute that in, then because of the fact 382 00:24:17,450 --> 00:24:20,620 that these two complex exponentials are the same, 383 00:24:20,620 --> 00:24:22,370 I'll find that x-- 384 00:24:22,370 --> 00:24:26,740 capital X tilde of k is equal to capital X 385 00:24:26,740 --> 00:24:30,920 tilde of k plus capital N, k plus 2 capital N, et cetera. 386 00:24:30,920 --> 00:24:35,000 That is, it's a periodic sequence, although I only 387 00:24:35,000 --> 00:24:38,300 use one period of it in reconstructing 388 00:24:38,300 --> 00:24:42,130 the periodic sequence x tilde of n. 389 00:24:42,130 --> 00:24:45,880 I choose to do that mainly because of duality. 390 00:24:45,880 --> 00:24:50,080 That is, mainly because it is convenient to think 391 00:24:50,080 --> 00:24:54,400 of these as a periodic sequence so that I 392 00:24:54,400 --> 00:24:57,490 have one periodic sequence representing 393 00:24:57,490 --> 00:25:03,460 another periodic sequence, this being a periodic sequence in k 394 00:25:03,460 --> 00:25:12,080 with a period of capital N, and this being a periodic sequence 395 00:25:12,080 --> 00:25:16,340 in n with a period also of capital N. 396 00:25:16,340 --> 00:25:19,040 So now with a discrete Fourier series, 397 00:25:19,040 --> 00:25:22,160 there's a duality in the, what we 398 00:25:22,160 --> 00:25:25,280 could call the time domain and the frequency domain. 399 00:25:25,280 --> 00:25:27,410 And the duality is there in part because we've 400 00:25:27,410 --> 00:25:32,240 chosen to represent, or think of the Fourier series coefficients 401 00:25:32,240 --> 00:25:35,550 as periodic, as a periodic sequence, 402 00:25:35,550 --> 00:25:38,930 although we only use a finite number of those values 403 00:25:38,930 --> 00:25:43,640 in actually explicitly evaluating the Fourier 404 00:25:43,640 --> 00:25:46,800 series for x tilde of n. 405 00:25:46,800 --> 00:25:50,040 Well, this then is the Fourier series. 406 00:25:50,040 --> 00:25:53,400 Let me finally rewrite it one other way, 407 00:25:53,400 --> 00:25:58,050 which just introduces some notation that's convenient. 408 00:25:58,050 --> 00:26:03,130 Let me define w sub capital N as e 409 00:26:03,130 --> 00:26:08,350 to the minus j 2 pi over capital N. In that case, 410 00:26:08,350 --> 00:26:12,510 then just simply rewriting the Fourier series, 411 00:26:12,510 --> 00:26:15,480 we have x tilde of n is 1 over capital 412 00:26:15,480 --> 00:26:21,240 N, the sum of capital X tilde of k, w sub capital 413 00:26:21,240 --> 00:26:24,180 N to the minus nk. 414 00:26:24,180 --> 00:26:27,240 And the Fourier series coefficients 415 00:26:27,240 --> 00:26:31,240 expressed in terms of capital W sub n 416 00:26:31,240 --> 00:26:39,110 is the sum of x tilde of n, w sub capital N to the nk. 417 00:26:39,110 --> 00:26:42,740 Well, the discrete Fourier series 418 00:26:42,740 --> 00:26:46,370 has properties, just as the Fourier 419 00:26:46,370 --> 00:26:50,730 transform and the Z-transform has had a number of properties. 420 00:26:50,730 --> 00:26:55,850 And again, as we've done with the other transforms, 421 00:26:55,850 --> 00:26:58,940 I won't spend a lot of time on the details 422 00:26:58,940 --> 00:27:03,440 of either enumerating the properties or proving them. 423 00:27:03,440 --> 00:27:06,320 But let me just illustrate one or two 424 00:27:06,320 --> 00:27:11,880 to give you some idea as to what these properties involve. 425 00:27:11,880 --> 00:27:16,560 Well, first of all, as we've talked about in the Fourier 426 00:27:16,560 --> 00:27:19,530 transform and Z-transform cases, there 427 00:27:19,530 --> 00:27:25,620 is a shifting property that tells us how the Fourier series 428 00:27:25,620 --> 00:27:28,440 coefficients of a periodic sequence 429 00:27:28,440 --> 00:27:31,260 are related to the Fourier series coefficients 430 00:27:31,260 --> 00:27:33,120 of that sequence shifted. 431 00:27:33,120 --> 00:27:39,240 And in particular, it turns out that if capital X of k, capital 432 00:27:39,240 --> 00:27:41,730 X tilde of k are the Fourier series 433 00:27:41,730 --> 00:27:45,420 coefficients for little x tilde of n, 434 00:27:45,420 --> 00:27:47,970 then the Fourier series coefficients 435 00:27:47,970 --> 00:27:53,590 for that sequence shifted, that is, little x tilde of n plus m, 436 00:27:53,590 --> 00:27:56,260 corresponds to multiplying the Fourier series 437 00:27:56,260 --> 00:28:04,940 coefficients by w sub capital N to the minus km. 438 00:28:04,940 --> 00:28:10,430 Shifting the sequence involves multiplying the Fourier series 439 00:28:10,430 --> 00:28:14,150 coefficients by a complex exponential, which 440 00:28:14,150 --> 00:28:16,170 is what this is. 441 00:28:16,170 --> 00:28:20,930 And that is similar to what we've seen with the Fourier 442 00:28:20,930 --> 00:28:25,490 transform, shifting a sequence, multiply the Fourier transform 443 00:28:25,490 --> 00:28:28,060 by a complex exponential. 444 00:28:28,060 --> 00:28:32,520 And the Z-transform, we had exactly the same situation. 445 00:28:32,520 --> 00:28:36,480 We have a dual relationship, which 446 00:28:36,480 --> 00:28:40,440 expresses the result of shifting the Fourier series 447 00:28:40,440 --> 00:28:42,130 coefficients. 448 00:28:42,130 --> 00:28:45,120 If we shift the Fourier series coefficients, 449 00:28:45,120 --> 00:28:49,320 the result is multiplication by a complex exponential 450 00:28:49,320 --> 00:28:54,310 of the original sequence, little x tilde of n. 451 00:28:54,310 --> 00:28:56,020 In fact, one of the things that's 452 00:28:56,020 --> 00:28:59,650 true with the discrete Fourier series that hasn't been true 453 00:28:59,650 --> 00:29:02,910 with the Fourier transform or the Z-transform 454 00:29:02,910 --> 00:29:07,990 is there is a strong duality between the time 455 00:29:07,990 --> 00:29:12,610 domain, the discrete time domain, and the Fourier 456 00:29:12,610 --> 00:29:15,370 or frequency domain. 457 00:29:15,370 --> 00:29:21,690 In particular, we begin with a discrete periodic sequence, 458 00:29:21,690 --> 00:29:24,540 and we end up in the Fourier domain with, again, 459 00:29:24,540 --> 00:29:27,310 a discrete periodic sequence. 460 00:29:27,310 --> 00:29:29,590 In fact, something to think about 461 00:29:29,590 --> 00:29:34,670 is the fact that the Fourier series coefficients we've said, 462 00:29:34,670 --> 00:29:38,920 or we've chosen to interpret them, as being periodic. 463 00:29:38,920 --> 00:29:42,040 That implies that they themselves have a Fourier 464 00:29:42,040 --> 00:29:43,850 series representation. 465 00:29:43,850 --> 00:29:46,120 And something to think about is, what 466 00:29:46,120 --> 00:29:48,730 are the Fourier series coefficients 467 00:29:48,730 --> 00:29:49,970 of that periodic sequence? 468 00:29:54,340 --> 00:29:54,910 All right. 469 00:29:54,910 --> 00:29:59,110 So one important point then is that we have this duality 470 00:29:59,110 --> 00:30:00,820 between the two domains. 471 00:30:00,820 --> 00:30:05,110 And essentially, any property that we have in the time domain 472 00:30:05,110 --> 00:30:07,420 will find the dual property in the frequency 473 00:30:07,420 --> 00:30:10,150 domain for the discrete Fourier series. 474 00:30:10,150 --> 00:30:12,910 And that will the hold true when, 475 00:30:12,910 --> 00:30:16,340 later on in the next lecture, we talk about the discrete Fourier 476 00:30:16,340 --> 00:30:18,470 transform. 477 00:30:18,470 --> 00:30:21,890 Another useful property, which we've also 478 00:30:21,890 --> 00:30:26,780 talked about for the Fourier transform, 479 00:30:26,780 --> 00:30:29,540 are the set of symmetry properties. 480 00:30:29,540 --> 00:30:34,160 And in particular, if we consider x tilde of n 481 00:30:34,160 --> 00:30:38,260 to be a real a periodic sequence, 482 00:30:38,260 --> 00:30:40,960 then there are symmetry relationships 483 00:30:40,960 --> 00:30:45,430 between the real part of the Fourier series, 484 00:30:45,430 --> 00:30:48,940 and symmetry relations for the imaginary part of the Fourier 485 00:30:48,940 --> 00:30:50,080 series. 486 00:30:50,080 --> 00:30:54,280 In particular, we can think of expressing the Fourier series 487 00:30:54,280 --> 00:30:59,840 coefficients in terms of their real part plus j 488 00:30:59,840 --> 00:31:03,010 times the imaginary part. 489 00:31:03,010 --> 00:31:06,940 And the symmetry relationships that result 490 00:31:06,940 --> 00:31:10,660 are that if the periodic sequence is real, 491 00:31:10,660 --> 00:31:14,530 then the real part of the Fourier series coefficients 492 00:31:14,530 --> 00:31:21,490 are even, x sub R of k is equal to x sub R of minus k. 493 00:31:21,490 --> 00:31:26,140 And that's what we refer to as the property 494 00:31:26,140 --> 00:31:28,690 of, in the case of the Fourier transform, the Fourier 495 00:31:28,690 --> 00:31:31,300 transform being even. 496 00:31:31,300 --> 00:31:34,270 We can also write this in another way that 497 00:31:34,270 --> 00:31:36,190 will become important when we talk about 498 00:31:36,190 --> 00:31:38,960 the discrete Fourier transform. 499 00:31:38,960 --> 00:31:42,490 In particular, since this is a periodic sequence, 500 00:31:42,490 --> 00:31:44,440 since s sub R of-- 501 00:31:44,440 --> 00:31:47,110 since x of k is periodic, obviously 502 00:31:47,110 --> 00:31:49,370 it's real part is also periodic-- 503 00:31:49,370 --> 00:31:54,760 we can replace k by k plus capital N, or k 504 00:31:54,760 --> 00:31:58,630 minus capital N. And we can rewrite this 505 00:31:58,630 --> 00:32:04,750 as a statement that says that x sub R tilde of k 506 00:32:04,750 --> 00:32:12,390 is equal to x sub R tilde of capital N minus k. 507 00:32:12,390 --> 00:32:14,190 It shouldn't be particularly evident 508 00:32:14,190 --> 00:32:20,060 why we want to do that here, although it becomes important 509 00:32:20,060 --> 00:32:22,340 when we apply some of these notions 510 00:32:22,340 --> 00:32:24,890 to the representation of finite length sequence 511 00:32:24,890 --> 00:32:28,880 and sequences in the discrete Fourier transform. 512 00:32:28,880 --> 00:32:31,910 Likewise, for the imaginary part of the Fourier series 513 00:32:31,910 --> 00:32:36,040 coefficients, we end up with a symmetry property 514 00:32:36,040 --> 00:32:39,670 that says that the imaginary part of the Fourier series 515 00:32:39,670 --> 00:32:41,275 coefficients are odd. 516 00:32:44,010 --> 00:32:47,940 And again, as we did with the real part of the Fourier series 517 00:32:47,940 --> 00:32:51,000 coefficients, we can rewrite this 518 00:32:51,000 --> 00:32:58,690 to say that x sub i tilde of k is equal to minus x sub i 519 00:32:58,690 --> 00:33:03,300 tilde of N minus k. 520 00:33:06,800 --> 00:33:10,460 Finally, we can, in a similar manner, 521 00:33:10,460 --> 00:33:14,270 look at the magnitude of the Fourier series coefficients 522 00:33:14,270 --> 00:33:17,630 and the angle of the Fourier series coefficients. 523 00:33:17,630 --> 00:33:21,620 And just as we've seen with the Fourier transform, 524 00:33:21,620 --> 00:33:24,710 the result that we'll find is that the magnitude 525 00:33:24,710 --> 00:33:27,350 of the Fourier series coefficients 526 00:33:27,350 --> 00:33:31,610 are even, an even function of k. 527 00:33:31,610 --> 00:33:36,080 And the angle of the Fourier series coefficients 528 00:33:36,080 --> 00:33:39,880 are an odd function of k. 529 00:33:39,880 --> 00:33:42,370 So we have symmetry properties like we 530 00:33:42,370 --> 00:33:44,890 had with the Fourier transform. 531 00:33:44,890 --> 00:33:49,990 An important thing to keep track of at this point, because we'll 532 00:33:49,990 --> 00:33:53,410 want to refer back to this when we talk 533 00:33:53,410 --> 00:33:55,930 about the discrete Fourier transform, 534 00:33:55,930 --> 00:33:59,590 is that in talking about the property of even or odd, 535 00:33:59,590 --> 00:34:03,190 we can either talk about it as we do here, 536 00:34:03,190 --> 00:34:05,890 or in terms of a shift-- 537 00:34:05,890 --> 00:34:09,070 that is, in terms of a statement that x sub bar of k 538 00:34:09,070 --> 00:34:13,510 is x sub R of capital N minus k, which essentially relates 539 00:34:13,510 --> 00:34:18,580 the evenness of this periodic sequence 540 00:34:18,580 --> 00:34:24,770 to the relationship between values within one period. 541 00:34:24,770 --> 00:34:30,120 Finally, we have another very important property, which 542 00:34:30,120 --> 00:34:32,569 is the convolution property. 543 00:34:32,569 --> 00:34:34,110 This is a property that, again, we've 544 00:34:34,110 --> 00:34:38,020 had for the Fourier transform and for the Z-transform. 545 00:34:38,020 --> 00:34:40,530 It's a property that states, essentially, 546 00:34:40,530 --> 00:34:45,810 that the convolution of two periodic sequences 547 00:34:45,810 --> 00:34:50,550 results in the multiplication of the discrete Fourier series 548 00:34:50,550 --> 00:34:54,570 coefficients, with just one minor twist, which again will 549 00:34:54,570 --> 00:34:58,140 become important when we relate this to the discrete Fourier 550 00:34:58,140 --> 00:34:59,490 transform. 551 00:34:59,490 --> 00:35:03,630 In particular, we have a periodic sequence, 552 00:35:03,630 --> 00:35:05,640 x1 tilde of n. 553 00:35:05,640 --> 00:35:09,120 And here its Fourier series coefficients. 554 00:35:09,120 --> 00:35:13,110 A second periodic sequence, x2 tilde of n, 555 00:35:13,110 --> 00:35:16,200 with its Fourier series coefficients. 556 00:35:16,200 --> 00:35:20,220 And what we'd like to ask is, what 557 00:35:20,220 --> 00:35:26,010 is the periodic sequence, x3 tilde of n, 558 00:35:26,010 --> 00:35:28,740 whose Fourier series coefficients are 559 00:35:28,740 --> 00:35:34,290 the product of x1 tilde of k and x2 tilde of k? 560 00:35:34,290 --> 00:35:37,680 In other words, if we multiply the Fourier series coefficients 561 00:35:37,680 --> 00:35:41,910 together, what is the sequence that that corresponds to? 562 00:35:41,910 --> 00:35:45,390 The answer, which involves just simply a little bit of algebra 563 00:35:45,390 --> 00:35:50,280 to verify, is that the resulting sequence 564 00:35:50,280 --> 00:35:57,690 is a sum of x1 tilde of m, x2 tilde of n minus m. 565 00:35:57,690 --> 00:36:03,000 Well, that looks exactly like a convolution 566 00:36:03,000 --> 00:36:05,700 with one important difference. 567 00:36:05,700 --> 00:36:09,150 And that is that the limits on the summation 568 00:36:09,150 --> 00:36:12,240 don't go from minus infinity to plus infinity 569 00:36:12,240 --> 00:36:14,370 as they did in the case of the Fourier 570 00:36:14,370 --> 00:36:16,770 transform and Z-transform. 571 00:36:16,770 --> 00:36:19,020 Ordinary linear convolution, as we've always 572 00:36:19,020 --> 00:36:21,300 been talking about it, involved the sum 573 00:36:21,300 --> 00:36:23,460 from minus infinity to plus infinity, 574 00:36:23,460 --> 00:36:27,780 of x1 of m, x2 of n minus m. 575 00:36:27,780 --> 00:36:31,780 Here we have, as the limits on the sum, 0 576 00:36:31,780 --> 00:36:33,960 to capital N minus 1. 577 00:36:33,960 --> 00:36:38,670 The summation is carried out only over one period. 578 00:36:38,670 --> 00:36:45,610 We have also a dual property, in other words, the property that 579 00:36:45,610 --> 00:36:50,710 relates the Fourier series coefficients of the product 580 00:36:50,710 --> 00:36:54,310 of two sequences to the Fourier series coefficients 581 00:36:54,310 --> 00:36:56,620 of the individual sequences. 582 00:36:56,620 --> 00:37:00,460 And because of the duality now that we have between the time 583 00:37:00,460 --> 00:37:04,280 domain and the frequency domain for the Fourier series, 584 00:37:04,280 --> 00:37:06,850 these Fourier series coefficients 585 00:37:06,850 --> 00:37:11,890 are, again, the convolution of the two Fourier series 586 00:37:11,890 --> 00:37:15,470 coefficients for x1 of n and x2 of n. 587 00:37:15,470 --> 00:37:18,880 There's a normalization factor, 1 over capital N. 588 00:37:18,880 --> 00:37:25,878 But again, the limits on the sum involve 0 to capital N minus 1. 589 00:37:25,878 --> 00:37:29,870 So this is slightly different than the convolution 590 00:37:29,870 --> 00:37:31,860 as we've been talking about. 591 00:37:31,860 --> 00:37:34,820 It is referred to generally, and we'll 592 00:37:34,820 --> 00:37:38,420 be referring to it as a periodic convolution. 593 00:37:38,420 --> 00:37:43,850 The important distinction being that for this convolution, 594 00:37:43,850 --> 00:37:46,670 it involves a summation, not for minus infinity 595 00:37:46,670 --> 00:37:51,500 to plus infinity, but simply a summation over only one period 596 00:37:51,500 --> 00:37:54,270 of the periodic sequences. 597 00:37:54,270 --> 00:37:58,050 Well, to drive this home, let me just 598 00:37:58,050 --> 00:38:05,580 show you with a simple example what these sequences look like, 599 00:38:05,580 --> 00:38:08,610 and in particular, what the shifting involves, 600 00:38:08,610 --> 00:38:11,490 and apply an interpretation to that. 601 00:38:11,490 --> 00:38:13,972 And so let's return to the view graph. 602 00:38:20,860 --> 00:38:27,350 What I'm indicating here is a sequence x2 tilde 603 00:38:27,350 --> 00:38:36,390 of m, a sequence x1 tilde of m, and to form the convolution 604 00:38:36,390 --> 00:38:38,790 of these two sequences. 605 00:38:38,790 --> 00:38:45,530 What I would like to do is generate x2 tilde, in general, 606 00:38:45,530 --> 00:38:54,740 of n minus m, multiply that by this, by x1 tilde of m, 607 00:38:54,740 --> 00:38:58,860 and then sum up the result over one period. 608 00:38:58,860 --> 00:39:02,200 Well, I'm indicating, by the way, 609 00:39:02,200 --> 00:39:06,320 in blue, just one period of this periodic sequence. 610 00:39:06,320 --> 00:39:08,660 And similarly in blue, one period 611 00:39:08,660 --> 00:39:10,890 of this periodic sequence. 612 00:39:10,890 --> 00:39:14,720 So let's examine, first of all, the sequence 613 00:39:14,720 --> 00:39:22,370 which corresponds to replacing this by x2 tilde of n minus m. 614 00:39:22,370 --> 00:39:25,610 And let's do it for the specific case 615 00:39:25,610 --> 00:39:27,710 where little n is equal to 2. 616 00:39:27,710 --> 00:39:35,120 So I've illustrated here the sequence x2 tilde of 2 minus m. 617 00:39:35,120 --> 00:39:38,870 And to generate this sequence from this one 618 00:39:38,870 --> 00:39:42,300 involves essentially two steps. 619 00:39:42,300 --> 00:39:46,550 The first step is to flip this sequence over, 620 00:39:46,550 --> 00:39:50,720 that's replacing m by minus m. 621 00:39:50,720 --> 00:39:55,790 And then the second step is to shift the sequence by an amount 622 00:39:55,790 --> 00:39:58,610 little n, depending on the argument that we 623 00:39:58,610 --> 00:40:00,440 want to stick in here. 624 00:40:00,440 --> 00:40:04,490 Now this is the result of doing that. 625 00:40:04,490 --> 00:40:09,230 We can think of this as having flipped this sequence over, 626 00:40:09,230 --> 00:40:17,020 and then shifted it by two points to the right. 627 00:40:17,020 --> 00:40:22,040 And the result is then this periodic sequence. 628 00:40:22,040 --> 00:40:26,860 An important thing to keep in mind, 629 00:40:26,860 --> 00:40:29,920 or to look at-- and again, this is a point that we'll be 630 00:40:29,920 --> 00:40:34,090 emphasizing in much more detail in the next lecture-- 631 00:40:34,090 --> 00:40:38,080 is that as we examine these blue points, 632 00:40:38,080 --> 00:40:40,510 we could think of having gotten those 633 00:40:40,510 --> 00:40:44,740 by simply flipping one period of this 634 00:40:44,740 --> 00:40:48,430 and circularly shifting that, circularly shifting 635 00:40:48,430 --> 00:40:53,820 those points, simply in the range 0 to capital N minus 1. 636 00:40:53,820 --> 00:40:56,320 That's an interpretation that will become much more 637 00:40:56,320 --> 00:41:00,190 evident in the next lecture. 638 00:41:00,190 --> 00:41:03,580 But then to form the convolution of this sequence 639 00:41:03,580 --> 00:41:09,420 with this sequence, we then, after having constructed 640 00:41:09,420 --> 00:41:14,350 x2 tilde of n minus m, carry out the multiplication 641 00:41:14,350 --> 00:41:16,810 of these two sequences. 642 00:41:16,810 --> 00:41:18,720 The result of that multiplication, 643 00:41:18,720 --> 00:41:23,440 I've indicated here, so that for this particular example, 644 00:41:23,440 --> 00:41:27,650 these three points get multiplied by these three. 645 00:41:27,650 --> 00:41:31,340 These four points, rather, get multiplied 646 00:41:31,340 --> 00:41:32,990 by these four values. 647 00:41:32,990 --> 00:41:37,610 The rest of the points in that one period get multiplied by 0. 648 00:41:37,610 --> 00:41:40,250 And then finally, to evaluate the result 649 00:41:40,250 --> 00:41:44,530 of the periodic convolution for little n equal to 2, 650 00:41:44,530 --> 00:41:50,391 we sum up those values in the range 0 to capital N minus 1. 651 00:41:50,391 --> 00:41:55,480 So it operates very much like a linear or ordinary convolution, 652 00:41:55,480 --> 00:41:59,020 as we've been talking about over the last several lectures. 653 00:41:59,020 --> 00:42:03,640 The important difference is that the summation is carried out 654 00:42:03,640 --> 00:42:07,241 just simply over one period. 655 00:42:07,241 --> 00:42:07,740 All right. 656 00:42:07,740 --> 00:42:12,780 Well, that completes the discussion 657 00:42:12,780 --> 00:42:18,540 as we want to present it, of the discrete Fourier series. 658 00:42:18,540 --> 00:42:20,850 As I indicated at the beginning of the lecture, 659 00:42:20,850 --> 00:42:27,510 our objective was eventually to develop a Fourier 660 00:42:27,510 --> 00:42:30,390 representation for finite length sequences, that is, 661 00:42:30,390 --> 00:42:32,940 the discrete Fourier transform. 662 00:42:32,940 --> 00:42:37,530 And in the next lecture, what we'll want to do 663 00:42:37,530 --> 00:42:40,770 is take the ideas of the discrete Fourier 664 00:42:40,770 --> 00:42:44,070 series, as we've talked about them here, 665 00:42:44,070 --> 00:42:48,510 and apply them to the representation of finite length 666 00:42:48,510 --> 00:42:51,630 sequences, resulting in what we'll call 667 00:42:51,630 --> 00:42:54,320 the discrete Fourier transform.