1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high-quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation, or view additional materials from 6 00:00:13,470 --> 00:00:19,290 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:19,290 --> 00:00:20,525 ocw.mit.edu. 8 00:00:20,525 --> 00:00:55,510 [MUSIC PLAYING] 9 00:00:55,510 --> 00:00:57,900 PROFESSOR: Over the last several lectures, we developed 10 00:00:57,900 --> 00:01:03,320 the Fourier representation for continuous-time signals. 11 00:01:03,320 --> 00:01:06,760 What I'd now like to do is develop a similar 12 00:01:06,760 --> 00:01:09,230 representation for discrete-time. 13 00:01:09,230 --> 00:01:12,940 And let me begin the discussion by reminding you of 14 00:01:12,940 --> 00:01:15,280 what our basic motivation was. 15 00:01:15,280 --> 00:01:19,780 The idea is that what we wanted to do was exploit the 16 00:01:19,780 --> 00:01:23,640 properties of linearity and time invariance for linear 17 00:01:23,640 --> 00:01:26,330 time-invariant systems. 18 00:01:26,330 --> 00:01:31,860 So in the case of linear time-invariant systems, the 19 00:01:31,860 --> 00:01:38,650 basic idea was to consider decomposing the input as a 20 00:01:38,650 --> 00:01:42,730 linear combination of basic inputs. 21 00:01:42,730 --> 00:01:46,620 And then, because of linearity, the output could be 22 00:01:46,620 --> 00:01:49,960 expressed as a linear combination of corresponding 23 00:01:49,960 --> 00:01:59,610 outputs where psi sub i is the output due to phi sub i. 24 00:01:59,610 --> 00:02:03,440 So basically, what we attempted to do was decompose 25 00:02:03,440 --> 00:02:07,710 the input, and then reconstruct the output through 26 00:02:07,710 --> 00:02:13,620 a linear combination of the outputs to those basic inputs. 27 00:02:13,620 --> 00:02:19,020 We then focused on the notion of choosing the basic inputs 28 00:02:19,020 --> 00:02:20,890 with two criteria in mind. 29 00:02:20,890 --> 00:02:25,900 One was to choose them so that a broad class of signals could 30 00:02:25,900 --> 00:02:30,020 be constructed out of those basic inputs. 31 00:02:30,020 --> 00:02:35,000 And the second was to choose the basic inputs, so that the 32 00:02:35,000 --> 00:02:38,800 response to those was easy to compute. 33 00:02:38,800 --> 00:02:41,570 And as you recall, one representation that we ended 34 00:02:41,570 --> 00:02:47,050 up with, with those basic criteria in mind, was the 35 00:02:47,050 --> 00:02:49,500 representation through convolution. 36 00:02:49,500 --> 00:02:52,610 And then in beginning the discussion of the Fourier 37 00:02:52,610 --> 00:02:55,980 representation of continuous-time signals, we 38 00:02:55,980 --> 00:03:01,880 chose as another set of basic inputs complex exponentials. 39 00:03:01,880 --> 00:03:07,860 So for continuous-time, we chose a set of basic inputs 40 00:03:07,860 --> 00:03:10,380 which were complex exponentials. 41 00:03:10,380 --> 00:03:15,470 The motivation there was the fact that the complex 42 00:03:15,470 --> 00:03:18,950 exponentials have what we refer to as the 43 00:03:18,950 --> 00:03:20,940 eigenfunction property. 44 00:03:20,940 --> 00:03:24,190 Namely, if we put complex exponentials into our 45 00:03:24,190 --> 00:03:29,540 continuous-time systems, then the output is a complex 46 00:03:29,540 --> 00:03:34,830 exponential of the same form with only a 47 00:03:34,830 --> 00:03:37,980 change in complex amplitude. 48 00:03:37,980 --> 00:03:42,450 And that change in complex amplitude is what we referred 49 00:03:42,450 --> 00:03:48,930 to as the frequency response of the system. 50 00:03:48,930 --> 00:03:52,040 And also, by the way, as it developed later, that 51 00:03:52,040 --> 00:03:55,470 frequency response, as you should now recognize from this 52 00:03:55,470 --> 00:03:59,380 expression, is in fact the Fourier transform, the 53 00:03:59,380 --> 00:04:03,120 continuous-time Fourier transform of the system 54 00:04:03,120 --> 00:04:06,170 impulse response. 55 00:04:06,170 --> 00:04:10,790 And the notion of decomposing a signal as a linear 56 00:04:10,790 --> 00:04:15,960 combination of these complex exponentials is what, first 57 00:04:15,960 --> 00:04:19,260 the Fourier series representation, and then later 58 00:04:19,260 --> 00:04:21,130 the Fourier transform representation 59 00:04:21,130 --> 00:04:22,680 corresponded to. 60 00:04:22,680 --> 00:04:28,390 And finally, to remind you of one additional point, the fact 61 00:04:28,390 --> 00:04:30,370 is that because of the eigenfunction 62 00:04:30,370 --> 00:04:34,640 property, the response-- 63 00:04:34,640 --> 00:04:37,580 once we have decomposed the input as a linear combination 64 00:04:37,580 --> 00:04:40,920 of complex exponentials, the response to that linear 65 00:04:40,920 --> 00:04:44,820 combination is straightforward to compute once we know the 66 00:04:44,820 --> 00:04:50,110 frequency response because of the eigenfunction property. 67 00:04:50,110 --> 00:04:54,790 Now, basically the same strategy and many of the same 68 00:04:54,790 --> 00:05:00,120 ideas work in discrete-time, paralleling almost exactly 69 00:05:00,120 --> 00:05:02,390 what happened in continuous-time. 70 00:05:02,390 --> 00:05:05,180 So the similarities between discrete-time and 71 00:05:05,180 --> 00:05:07,690 continuous-time are very strong. 72 00:05:07,690 --> 00:05:11,710 Although as we'll see, there are a number of differences. 73 00:05:11,710 --> 00:05:15,460 And it's important as we go through the discussion to 74 00:05:15,460 --> 00:05:20,210 illuminate not only the similarities, but obviously 75 00:05:20,210 --> 00:05:22,160 also the differences. 76 00:05:22,160 --> 00:05:26,800 Well, let's begin with the eigenfunction property, and 77 00:05:26,800 --> 00:05:31,900 let me just state that just as in continuous-time, if we 78 00:05:31,900 --> 00:05:36,580 consider a set of basic signals, which are complex 79 00:05:36,580 --> 00:05:40,770 exponential rules, then discrete-time linear time-m 80 00:05:40,770 --> 00:05:45,390 invariant systems have the eigenfunction property. 81 00:05:45,390 --> 00:05:50,200 Namely, if we put a complex exponential into the system, 82 00:05:50,200 --> 00:05:53,970 the response is a complex exponential at the same 83 00:05:53,970 --> 00:06:00,270 complex frequency, and simply multiplied by an appropriate 84 00:06:00,270 --> 00:06:03,620 complex factor, or constant. 85 00:06:03,620 --> 00:06:11,510 And just as we did in continuous-time, we will be 86 00:06:11,510 --> 00:06:15,980 referring to this complex constant, which is a function, 87 00:06:15,980 --> 00:06:19,910 of course, the frequency of the complex exponential input. 88 00:06:19,910 --> 00:06:24,400 We'll be referring to this as the frequency response. 89 00:06:24,400 --> 00:06:27,980 And although it's not particularly evident at this 90 00:06:27,980 --> 00:06:32,020 point, as the discussion develops through this lecture, 91 00:06:32,020 --> 00:06:36,620 what in fact will happen is very much paralleling 92 00:06:36,620 --> 00:06:38,340 continuous-time. 93 00:06:38,340 --> 00:06:43,990 This particular expression, in fact, will correspond to what 94 00:06:43,990 --> 00:06:48,550 we'll refer to as the Fourier transform, the discrete-time 95 00:06:48,550 --> 00:06:52,030 Fourier transform of the system impulse response. 96 00:06:52,030 --> 00:06:55,190 So there, of course, there's a very strong parallel between 97 00:06:55,190 --> 00:06:58,330 continuous time and discrete time. 98 00:06:58,330 --> 00:07:02,750 Now, just as we did in continuous-time, let's begin 99 00:07:02,750 --> 00:07:07,720 the discussion by first concentrating on periodic-- 100 00:07:07,720 --> 00:07:11,010 the representation through complex exponentials of 101 00:07:11,010 --> 00:07:16,030 periodic sequences, and then we'll generalize that 102 00:07:16,030 --> 00:07:18,270 discussion to the representation 103 00:07:18,270 --> 00:07:20,570 of aperiodic signals. 104 00:07:20,570 --> 00:07:26,190 So let's consider first a periodic signal, or in 105 00:07:26,190 --> 00:07:28,900 general, signals which are periodic. 106 00:07:28,900 --> 00:07:33,530 Period denoted by capital N. And then, of course, the 107 00:07:33,530 --> 00:07:39,610 fundamental frequency is 2 pi divided by capital N. 108 00:07:39,610 --> 00:07:45,200 Now, we can consider exponentials which have this 109 00:07:45,200 --> 00:07:50,390 as a fundamental frequency, or which are harmonics of that, 110 00:07:50,390 --> 00:07:53,880 and that would correspond to the class of complex 111 00:07:53,880 --> 00:07:59,290 exponentials of the form e to the jk omega 0 n. 112 00:07:59,290 --> 00:08:06,610 So these complex exponentials then, as k varies, are complex 113 00:08:06,610 --> 00:08:11,930 exponentials that are harmonically related, all of 114 00:08:11,930 --> 00:08:16,730 which are periodic with the same period capital N. 115 00:08:16,730 --> 00:08:19,680 Although the fundamental period is 116 00:08:19,680 --> 00:08:20,960 different for each of these. 117 00:08:20,960 --> 00:08:26,150 Each of them being related by an integer amount. 118 00:08:26,150 --> 00:08:29,830 Now, again, just as we did in continuous time, we can 119 00:08:29,830 --> 00:08:34,390 consider attempting to build our periodic signal out of a 120 00:08:34,390 --> 00:08:36,780 linear combination of these. 121 00:08:36,780 --> 00:08:43,280 And so we consider a periodic signal, which is a weighted 122 00:08:43,280 --> 00:08:48,060 sum of these complex exponentials. 123 00:08:48,060 --> 00:08:50,980 And, of course, this periodic signal-- 124 00:08:50,980 --> 00:08:52,420 this is a periodic signal. 125 00:08:52,420 --> 00:08:56,460 This can be verified, more or less, in a straightforward way 126 00:08:56,460 --> 00:08:58,150 by substitution. 127 00:08:58,150 --> 00:09:00,950 And, of course, one of the things that we'll want to 128 00:09:00,950 --> 00:09:04,930 address shortly is how broad a class of signals, again, can 129 00:09:04,930 --> 00:09:07,310 be represented by this sum? 130 00:09:07,310 --> 00:09:10,730 And another question obviously will be, how do we determine 131 00:09:10,730 --> 00:09:13,860 the coefficients a sub k? 132 00:09:13,860 --> 00:09:20,010 However, before we do that, let me focus on an important 133 00:09:20,010 --> 00:09:23,210 distinction between continuous-time and 134 00:09:23,210 --> 00:09:26,950 discrete-time in the context of these complex exponentials 135 00:09:26,950 --> 00:09:30,420 and this representation. 136 00:09:30,420 --> 00:09:34,140 When we talked about complex exponentials and sinusoids 137 00:09:34,140 --> 00:09:38,100 early in the course, one of the differences that we saw 138 00:09:38,100 --> 00:09:42,380 between continuous-time and discrete-time is that in 139 00:09:42,380 --> 00:09:46,990 continuous-time, as we vary the frequency variable, we see 140 00:09:46,990 --> 00:09:52,100 different complex exponentials as omega varies. 141 00:09:52,100 --> 00:09:56,580 Whereas, in discrete-time, we saw, in fact, that there was a 142 00:09:56,580 --> 00:09:57,750 periodicity. 143 00:09:57,750 --> 00:10:03,670 Or said another way, it's straightforward to verify that 144 00:10:03,670 --> 00:10:08,880 if we think of this class of complex exponentials. 145 00:10:08,880 --> 00:10:17,490 That, in fact, if we consider varying k by adding to it 146 00:10:17,490 --> 00:10:23,670 capital N, where capital N is the period of the fundamental 147 00:10:23,670 --> 00:10:25,690 complex exponential. 148 00:10:25,690 --> 00:10:33,700 Then in fact, if we replace k by k plus capital N, we'll see 149 00:10:33,700 --> 00:10:38,810 exactly the same complex exponentials over again. 150 00:10:38,810 --> 00:10:39,690 Now, what does that say? 151 00:10:39,690 --> 00:10:43,430 What it says is that if I consider this class of complex 152 00:10:43,430 --> 00:10:52,790 exponentials, as k varies from 0 through capital N minus 1, 153 00:10:52,790 --> 00:10:56,050 we will see all of the ones that there are to see. 154 00:10:56,050 --> 00:10:58,060 There aren't anymore. 155 00:10:58,060 --> 00:11:03,160 And so, in fact, if we can build x of n out of this 156 00:11:03,160 --> 00:11:08,600 linear combination, then we better be able to do it as k 157 00:11:08,600 --> 00:11:16,230 varies from 0 up to N minus 1. 158 00:11:16,230 --> 00:11:20,990 Because beyond that, we'll simply see the same complex 159 00:11:20,990 --> 00:11:22,870 exponentials over again. 160 00:11:22,870 --> 00:11:27,670 So, for example, if k takes on the value capital N, that will 161 00:11:27,670 --> 00:11:31,630 be exactly the same complex exponential as if 162 00:11:31,630 --> 00:11:34,300 k is equal to 0. 163 00:11:34,300 --> 00:11:41,070 So in fact, this sum ranges only over capital N of the 164 00:11:41,070 --> 00:11:43,430 distinct complex exponentials. 165 00:11:43,430 --> 00:11:47,100 Let's say, for example, from 0 to capital N minus 1 . 166 00:11:47,100 --> 00:11:52,130 Although, in fact, since these complex exponentials repeat in 167 00:11:52,130 --> 00:11:57,450 k, I could actually consider instead of from 0 to N minus 168 00:11:57,450 --> 00:12:01,240 1, I could consider from 1 to N, or from 2 to 169 00:12:01,240 --> 00:12:03,800 N plus 1, or whatever. 170 00:12:03,800 --> 00:12:09,260 Or said another way, in this representation, I could 171 00:12:09,260 --> 00:12:16,310 alternatively choose k outside this range, thinking of these 172 00:12:16,310 --> 00:12:20,430 coefficients simply as periodically repeating in k 173 00:12:20,430 --> 00:12:24,660 because of the fact that these complex exponentials 174 00:12:24,660 --> 00:12:27,580 periodically repeat in k. 175 00:12:27,580 --> 00:12:31,130 So, in fact, in place of this expression, it will be common 176 00:12:31,130 --> 00:12:35,080 in writing the Fourier series expression to write it as I've 177 00:12:35,080 --> 00:12:38,910 indicated here, where the implication is that these 178 00:12:38,910 --> 00:12:44,260 Fourier coefficients periodically repeat as k 179 00:12:44,260 --> 00:12:47,350 continues to repeat outside the interval from 180 00:12:47,350 --> 00:12:49,370 0 to N minus 1. 181 00:12:49,370 --> 00:12:54,520 And so this notation, in fact, says that what we're going to 182 00:12:54,520 --> 00:13:02,270 use is k ranging over one period of this periodic 183 00:13:02,270 --> 00:13:04,590 sequence, which is the Fourier series coefficients. 184 00:13:07,730 --> 00:13:13,330 So the expression that we have then for the Fourier series 185 00:13:13,330 --> 00:13:15,280 I've repeated here. 186 00:13:15,280 --> 00:13:18,360 And the implication now is that the 187 00:13:18,360 --> 00:13:20,650 a sub k's are periodic. 188 00:13:20,650 --> 00:13:22,820 They periodically repeat because, of course, these 189 00:13:22,820 --> 00:13:25,150 exponentials periodically repeat. 190 00:13:25,150 --> 00:13:29,490 This indicates that we only use them over one period. 191 00:13:29,490 --> 00:13:33,930 And now we can inquire as to how we determine the 192 00:13:33,930 --> 00:13:36,120 coefficients a sub k. 193 00:13:36,120 --> 00:13:39,480 Well, we can formally go through this much as we did in 194 00:13:39,480 --> 00:13:41,550 the continuous-time case. 195 00:13:41,550 --> 00:13:45,070 And we do, in fact, do that in the text, which involves 196 00:13:45,070 --> 00:13:48,310 substituting some sums and interchanging the orders of 197 00:13:48,310 --> 00:13:49,790 summation, et cetera. 198 00:13:49,790 --> 00:13:54,020 But let me draw your attention to the fact that this, in 199 00:13:54,020 --> 00:13:59,290 fact, can be thought of as capital N equations and 200 00:13:59,290 --> 00:14:02,210 capital N unknowns. 201 00:14:02,210 --> 00:14:08,100 In other words, we know x of n over a period, and so we know 202 00:14:08,100 --> 00:14:12,280 what the left-hand side of this is for capital N values. 203 00:14:12,280 --> 00:14:16,070 And we'd like to determine these constants a sub k. 204 00:14:16,070 --> 00:14:18,860 Well, it turns out that there is a nice convenient 205 00:14:18,860 --> 00:14:21,050 closed-form expression for that. 206 00:14:21,050 --> 00:14:26,060 And, in fact, if we evaluate the closed-form expression 207 00:14:26,060 --> 00:14:29,860 through any of a variety of algebraic manipulations, we 208 00:14:29,860 --> 00:14:35,290 end up then with the analysis equation. 209 00:14:35,290 --> 00:14:39,150 And the analysis equation, which tells us how to get the 210 00:14:39,150 --> 00:14:47,080 coefficients a sub k from x of n is what I've indicated here. 211 00:14:47,080 --> 00:14:53,020 And so this tells us how from x of n to get the a sub k's. 212 00:14:53,020 --> 00:14:58,140 And, of course, the first equation tells us how x of n 213 00:14:58,140 --> 00:15:01,120 is built up out of the a sub k. 214 00:15:01,120 --> 00:15:05,520 Notice incidentally that there is a strong duality between 215 00:15:05,520 --> 00:15:07,050 these two equations. 216 00:15:07,050 --> 00:15:11,590 And that's a duality that we'll return to, actually 217 00:15:11,590 --> 00:15:13,750 toward the end of the next lecture. 218 00:15:13,750 --> 00:15:16,340 Now, there is a real difference between the way 219 00:15:16,340 --> 00:15:19,630 those equations look and the way the continuous-time 220 00:15:19,630 --> 00:15:21,540 Fourier series looked. 221 00:15:21,540 --> 00:15:26,650 In the continuous-time case, let me remind you that it 222 00:15:26,650 --> 00:15:31,630 required an infinite number of coefficients to build up this 223 00:15:31,630 --> 00:15:33,800 continuous-time function. 224 00:15:33,800 --> 00:15:37,720 And so this was not simply a matter of identifying how to 225 00:15:37,720 --> 00:15:41,180 invert capital N or a finite number of equations and a 226 00:15:41,180 --> 00:15:44,280 finite number of unknowns. 227 00:15:44,280 --> 00:15:51,210 And the analysis equation was an integration as opposed to 228 00:15:51,210 --> 00:15:55,510 the synthesis equation, which is a summation. 229 00:15:55,510 --> 00:15:59,400 So there is a real difference there between the 230 00:15:59,400 --> 00:16:01,960 continuous-time and discrete-time cases. 231 00:16:01,960 --> 00:16:06,560 And the difference arises, to a large extent, because of 232 00:16:06,560 --> 00:16:09,480 this notion that in discrete-time, the complex 233 00:16:09,480 --> 00:16:16,050 exponentials are periodic in their frequency. 234 00:16:16,050 --> 00:16:21,470 So we have then to summarize the synthesis equation and the 235 00:16:21,470 --> 00:16:27,210 analysis equation for the discrete-time Fourier series. 236 00:16:27,210 --> 00:16:32,250 Again, x of n, our original signal is periodic. 237 00:16:32,250 --> 00:16:35,400 And, of course, the complex exponentials 238 00:16:35,400 --> 00:16:38,230 involved are periodic. 239 00:16:38,230 --> 00:16:40,680 They're periodic obviously in n. 240 00:16:40,680 --> 00:16:43,530 But in contrast to continuous-time, 241 00:16:43,530 --> 00:16:46,370 these repeat in k. 242 00:16:46,370 --> 00:16:51,070 In other words, as k omega 0 goes outside a range that 243 00:16:51,070 --> 00:16:53,500 covers a 2 pi interval. 244 00:16:53,500 --> 00:16:57,570 And because of that, we're imposing, in a sense, the 245 00:16:57,570 --> 00:17:00,450 interpretation that the a sub k's are 246 00:17:00,450 --> 00:17:02,820 likewise a periodic sequence. 247 00:17:02,820 --> 00:17:07,890 And in fact, if we look at the analysis equation, as we let k 248 00:17:07,890 --> 00:17:14,010 vary outside the range from 0 to N minus 1, what you can 249 00:17:14,010 --> 00:17:18,680 easily verify by substitution in here is that this sequence 250 00:17:18,680 --> 00:17:22,030 will, in fact, periodically repeat. 251 00:17:22,030 --> 00:17:26,300 So to underscore the difference between the 252 00:17:26,300 --> 00:17:30,320 continuous-time and discrete-time cases, we have 253 00:17:30,320 --> 00:17:33,730 this periodicity in the time domain, and that's a 254 00:17:33,730 --> 00:17:38,200 periodicity that is, of course, true in discrete-time 255 00:17:38,200 --> 00:17:43,520 and it's also true in continuous-time if we replace 256 00:17:43,520 --> 00:17:48,820 the integer variable by the discrete-time time variable. 257 00:17:48,820 --> 00:17:53,620 And we also, in discrete-time, have this periodicity in k, or 258 00:17:53,620 --> 00:17:55,430 in k omega 0. 259 00:17:55,430 --> 00:17:58,700 And correspondingly, a periodicity in the Fourier 260 00:17:58,700 --> 00:18:00,240 coefficients. 261 00:18:00,240 --> 00:18:06,980 And that is a set of properties that does not 262 00:18:06,980 --> 00:18:10,300 happen in continuous-time. 263 00:18:10,300 --> 00:18:15,520 And it is that that essentially leads to all of 264 00:18:15,520 --> 00:18:18,650 the important differences between discrete-time Fourier 265 00:18:18,650 --> 00:18:21,390 representations and continuous-time Fourier 266 00:18:21,390 --> 00:18:24,250 representations. 267 00:18:24,250 --> 00:18:29,430 Now, just quickly, let me draw your attention to the issue of 268 00:18:29,430 --> 00:18:33,400 convergence and when a sequence can and can't be 269 00:18:33,400 --> 00:18:35,370 represented, et cetera. 270 00:18:35,370 --> 00:18:39,620 And recall that in the continuous-time case, we 271 00:18:39,620 --> 00:18:44,530 focused on convergence in the context either of conditions, 272 00:18:44,530 --> 00:18:48,310 which I referred to as square integrability, or another set 273 00:18:48,310 --> 00:18:51,510 of conditions, which were the Dirichlet conditions. 274 00:18:51,510 --> 00:18:55,310 And there was this issue about when the signal does and 275 00:18:55,310 --> 00:18:58,480 doesn't converge at discontinuities, et cetera. 276 00:18:58,480 --> 00:19:02,330 Let me just simply draw your attention to the fact that in 277 00:19:02,330 --> 00:19:07,470 the discrete-time case, what we have is the representation 278 00:19:07,470 --> 00:19:12,620 of the periodic signal as a sum of a 279 00:19:12,620 --> 00:19:14,750 finite number of terms. 280 00:19:14,750 --> 00:19:17,350 This represents capital N equations 281 00:19:17,350 --> 00:19:19,850 and capital N unknowns. 282 00:19:19,850 --> 00:19:24,980 If we consider earth the partial sum, namely taking a 283 00:19:24,980 --> 00:19:31,640 smaller number of terms, then simply what happens is as the 284 00:19:31,640 --> 00:19:36,480 number of terms increases to the finite number required to 285 00:19:36,480 --> 00:19:41,620 represent x of n, we simply end up with the partial sum 286 00:19:41,620 --> 00:19:44,160 representing the finite [? length ?] sequence. 287 00:19:44,160 --> 00:19:48,600 What all that boils down to is the statement that in 288 00:19:48,600 --> 00:19:53,090 discrete-time there really are no convergence issues as there 289 00:19:53,090 --> 00:19:55,670 were in continuous-time. 290 00:19:55,670 --> 00:20:00,260 OK, well let's look at an example of the Fourier series 291 00:20:00,260 --> 00:20:03,390 representation for a particular signal. 292 00:20:03,390 --> 00:20:06,420 And the one that I've picked here is a simple one. 293 00:20:06,420 --> 00:20:14,430 Namely, a constant, a sine term, and a cosine term. 294 00:20:14,430 --> 00:20:17,860 Now, for this particular example, we can expand this 295 00:20:17,860 --> 00:20:22,720 out directly in terms of complex exponentials and 296 00:20:22,720 --> 00:20:26,710 essentially recognize this as a sum of complex exponentials. 297 00:20:26,710 --> 00:20:33,370 It's examined in more detail in Example 5.2 in the text. 298 00:20:33,370 --> 00:20:37,920 And if we look at the Fourier series coefficients, we can 299 00:20:37,920 --> 00:20:41,430 either look at it in terms of real and imaginary parts or 300 00:20:41,430 --> 00:20:44,160 magnitude and angle. 301 00:20:44,160 --> 00:20:48,260 On the left side here, I have the real part of the Fourier 302 00:20:48,260 --> 00:20:50,150 coefficients. 303 00:20:50,150 --> 00:20:55,480 And let me draw your attention to the fact that I've drawn 304 00:20:55,480 --> 00:21:01,330 this to specifically illuminate the periodicity of 305 00:21:01,330 --> 00:21:04,360 the Fourier series coefficients with a period of 306 00:21:04,360 --> 00:21:05,570 capital N. 307 00:21:05,570 --> 00:21:08,740 So here are the Fourier coefficients. 308 00:21:08,740 --> 00:21:12,920 And, in fact, it's this line that represents the DC, or 309 00:21:12,920 --> 00:21:17,510 constant term, and these two lines that represent the 310 00:21:17,510 --> 00:21:19,160 cosine term. 311 00:21:19,160 --> 00:21:23,540 And of course, these are the three terms that are required. 312 00:21:23,540 --> 00:21:27,500 Or equivalently, this one, this one, and this one. 313 00:21:27,500 --> 00:21:29,960 And then because of the periodicity of the Fourier 314 00:21:29,960 --> 00:21:34,140 series coefficients, this simply periodically repeats. 315 00:21:34,140 --> 00:21:39,150 So here is the real part and below it I show 316 00:21:39,150 --> 00:21:41,390 the imaginary part. 317 00:21:41,390 --> 00:21:45,560 And in the imaginary part, incidentally let me draw your 318 00:21:45,560 --> 00:21:53,090 attention to the fact that it's this term and this term 319 00:21:53,090 --> 00:21:56,760 in the imaginary part that represent the sinusoid. 320 00:21:56,760 --> 00:22:00,280 Whereas it's the symmetric terms in the real part the 321 00:22:00,280 --> 00:22:03,110 represent the cosine. 322 00:22:03,110 --> 00:22:07,580 OK, let's look at another example. 323 00:22:07,580 --> 00:22:12,420 This is another example from the text, and one that we'll 324 00:22:12,420 --> 00:22:18,000 be making frequent reference to in this particular lecture. 325 00:22:18,000 --> 00:22:24,090 And what it is, is a square wave. 326 00:22:24,090 --> 00:22:27,940 And I've expressed the Fourier series coefficients, which are 327 00:22:27,940 --> 00:22:30,340 algebraically developed in the text. 328 00:22:30,340 --> 00:22:35,000 I've expressed the Fourier series coefficients as samples 329 00:22:35,000 --> 00:22:38,616 of an envelope function. 330 00:22:38,616 --> 00:22:43,120 And so I've expressed it as samples of this particular 331 00:22:43,120 --> 00:22:46,120 function, which is referred to as a sin 332 00:22:46,120 --> 00:22:49,020 nx over sin x function. 333 00:22:49,020 --> 00:22:52,850 And let me just compare it to a continuous-time example, 334 00:22:52,850 --> 00:22:57,980 which is the continuous-time square wave, where with the 335 00:22:57,980 --> 00:23:02,840 continuous-times square wave the form of the Fourier series 336 00:23:02,840 --> 00:23:08,490 coefficients was as samples of what we refer to as a sin x 337 00:23:08,490 --> 00:23:11,050 over x function. 338 00:23:11,050 --> 00:23:15,820 Now, the sin nx over sin x function, which is the 339 00:23:15,820 --> 00:23:19,000 envelope of the Fourier series coefficients for the 340 00:23:19,000 --> 00:23:24,260 discrete-time periodic square wave plays the role-- 341 00:23:24,260 --> 00:23:26,490 and we'll see it very often in discrete-time-- 342 00:23:26,490 --> 00:23:31,080 that sin x over x does in continuous-time. 343 00:23:31,080 --> 00:23:33,180 And, in fact, we should understand right from the 344 00:23:33,180 --> 00:23:37,500 beginning that the sin x over x envelope couldn't possibly 345 00:23:37,500 --> 00:23:40,820 be the envelope of the discrete-time Fourier series 346 00:23:40,820 --> 00:23:42,220 coefficients. 347 00:23:42,220 --> 00:23:47,600 And one obvious reason is that it is not periodic. 348 00:23:47,600 --> 00:23:50,540 What we require, of course, from the discussion that I've 349 00:23:50,540 --> 00:23:53,530 just gone through is periodicity of the 350 00:23:53,530 --> 00:23:54,340 coefficients. 351 00:23:54,340 --> 00:23:58,150 And then consequently, also periodicity of the envelope in 352 00:23:58,150 --> 00:23:59,710 the discrete-time case. 353 00:23:59,710 --> 00:24:03,910 So once again, if we look back at the algebraic expression 354 00:24:03,910 --> 00:24:08,440 that I have, it's samples of the sin nx over sine x 355 00:24:08,440 --> 00:24:13,130 function that represent the Fourier series coefficients of 356 00:24:13,130 --> 00:24:16,160 this periodic square wave. 357 00:24:16,160 --> 00:24:24,650 Now, in the representation in the continuous-time case, we 358 00:24:24,650 --> 00:24:29,710 essentially had used the concept of an envelope to 359 00:24:29,710 --> 00:24:33,320 represent the Fourier series coefficients, and the notion 360 00:24:33,320 --> 00:24:35,990 that the Fourier series coefficients were samples of 361 00:24:35,990 --> 00:24:36,710 an envelope. 362 00:24:36,710 --> 00:24:41,500 And that is the same notion that we'll be using in 363 00:24:41,500 --> 00:24:43,110 discrete-time. 364 00:24:43,110 --> 00:24:49,560 So again for this square wave example, then what we have is 365 00:24:49,560 --> 00:24:54,520 an envelope function, the sin nx over sin x envelope 366 00:24:54,520 --> 00:24:59,720 function for a particular value of the period. 367 00:24:59,720 --> 00:25:04,280 Here indicated with a period of 10 samples. 368 00:25:04,280 --> 00:25:08,530 These samples of this envelope function would then represent 369 00:25:08,530 --> 00:25:12,340 the Fourier series coefficients. 370 00:25:12,340 --> 00:25:19,280 If we increased the period, then we would simply have a 371 00:25:19,280 --> 00:25:23,910 finer spacing on the samples of the envelope function to 372 00:25:23,910 --> 00:25:26,500 get the Fourier series coefficients. 373 00:25:26,500 --> 00:25:32,380 And likewise, if we increase the period still further, what 374 00:25:32,380 --> 00:25:37,260 we would have is an even finer spacing. 375 00:25:37,260 --> 00:25:40,970 So actually, as the period increases, and recall we used 376 00:25:40,970 --> 00:25:43,810 this in continuous-time also. 377 00:25:43,810 --> 00:25:47,700 As the period increases, we can view the Fourier series 378 00:25:47,700 --> 00:25:50,540 coefficients as samples of an envelope. 379 00:25:50,540 --> 00:25:54,970 And as the period increases, the sample spacing 380 00:25:54,970 --> 00:25:56,860 gets finer and finer. 381 00:25:56,860 --> 00:26:00,500 And in fact, as the period goes off essentially to 382 00:26:00,500 --> 00:26:04,380 infinity, the samples of the envelope, in effect, become 383 00:26:04,380 --> 00:26:06,140 the envelope. 384 00:26:06,140 --> 00:26:10,460 And recall also that this was essentially the trick that we 385 00:26:10,460 --> 00:26:16,850 used in continuous-time to allow us to develop or utilize 386 00:26:16,850 --> 00:26:21,590 the Fourier series to provide a representation of aperiodic 387 00:26:21,590 --> 00:26:24,410 signals as a linear combination of complex 388 00:26:24,410 --> 00:26:26,490 exponentials. 389 00:26:26,490 --> 00:26:35,220 In particular, what we did in the continuous-time case when 390 00:26:35,220 --> 00:26:41,280 we had an aperiodic signal was to consider constructing a 391 00:26:41,280 --> 00:26:44,500 periodic signal for which the aperiodic 392 00:26:44,500 --> 00:26:47,110 signal was one period. 393 00:26:47,110 --> 00:26:52,200 And then we developed the notion that since the periodic 394 00:26:52,200 --> 00:26:58,310 signal has a Fourier series, and since as the period of the 395 00:26:58,310 --> 00:27:02,760 periodic signal increases and goes to infinity, the periodic 396 00:27:02,760 --> 00:27:06,520 signal represents the aperiodic signal. 397 00:27:06,520 --> 00:27:09,930 Then, essentially, the Fourier series provides us with a 398 00:27:09,930 --> 00:27:11,740 representation. 399 00:27:11,740 --> 00:27:14,770 Now, we can do exactly the same thing in the 400 00:27:14,770 --> 00:27:16,420 discrete-time case. 401 00:27:16,420 --> 00:27:19,780 The statement is exactly the same, except that in the 402 00:27:19,780 --> 00:27:24,530 discrete-time case, instead of t as the independent variable, 403 00:27:24,530 --> 00:27:29,450 we simply make exactly the same statement, but with our 404 00:27:29,450 --> 00:27:31,730 discrete-time variable n. 405 00:27:31,730 --> 00:27:37,560 So the basic notion then in representing a discrete-time 406 00:27:37,560 --> 00:27:46,960 aperiodic signal is to first construct a periodic signal. 407 00:27:46,960 --> 00:27:50,450 Here we have the aperiodic signal. 408 00:27:50,450 --> 00:27:54,340 We construct a periodic signal by simply periodically 409 00:27:54,340 --> 00:27:58,840 replicating the aperiodic signal. 410 00:27:58,840 --> 00:28:04,890 The periodic signal and the aperiodic signal are identical 411 00:28:04,890 --> 00:28:07,310 for one period. 412 00:28:07,310 --> 00:28:12,120 And as the period goes off to infinity, it's the Fourier 413 00:28:12,120 --> 00:28:16,810 series representation of the periodic signal that provides 414 00:28:16,810 --> 00:28:21,050 a representation of the aperiodic signal. 415 00:28:21,050 --> 00:28:30,450 Again, to return to the example that we have been kind 416 00:28:30,450 --> 00:28:31,780 of working through this lecture. 417 00:28:31,780 --> 00:28:34,730 Namely, the periodic square wave. 418 00:28:34,730 --> 00:28:38,970 If we have an aperiodic signal, which is a rectangle, 419 00:28:38,970 --> 00:28:42,330 and we construct a periodic signal. 420 00:28:42,330 --> 00:28:44,130 And now we consider letting this 421 00:28:44,130 --> 00:28:46,570 period increase to infinity. 422 00:28:46,570 --> 00:28:51,320 We would first have this set of samples of the envelope. 423 00:28:51,320 --> 00:28:55,820 As the period increases, we would decrease the sample 424 00:28:55,820 --> 00:28:58,660 spacing to this set of samples. 425 00:28:58,660 --> 00:29:03,140 As the period increases further, it would be this set 426 00:29:03,140 --> 00:29:04,970 of samples. 427 00:29:04,970 --> 00:29:08,850 And as the period goes off to infinity, it's every point on 428 00:29:08,850 --> 00:29:10,050 the envelope. 429 00:29:10,050 --> 00:29:13,440 In fact, what the representation of the 430 00:29:13,440 --> 00:29:18,380 aperiodic signal is, is the envelope. 431 00:29:18,380 --> 00:29:20,670 OK, well, so that's the basic notion. 432 00:29:20,670 --> 00:29:23,650 It's no different than what we did in the 433 00:29:23,650 --> 00:29:25,580 continuous-time case. 434 00:29:25,580 --> 00:29:30,770 And mathematically, it develops in very much the same 435 00:29:30,770 --> 00:29:35,590 way as in the continuous-time case. 436 00:29:35,590 --> 00:29:42,130 Specifically, here is our representation through the 437 00:29:42,130 --> 00:29:47,550 Fourier series of the-- 438 00:29:47,550 --> 00:29:51,760 here is a representation through the envelope function. 439 00:29:51,760 --> 00:29:56,500 And this is the Fourier series synthesis equation where the 440 00:29:56,500 --> 00:30:02,660 equation below tells us how we get these Fourier coefficients 441 00:30:02,660 --> 00:30:05,970 or the envelope from x of n. 442 00:30:05,970 --> 00:30:08,860 Now, x tilde of n is the periodic signal. 443 00:30:08,860 --> 00:30:11,990 And we know that over one period, which is the only 444 00:30:11,990 --> 00:30:16,080 interval over which we use it, in fact, this is identical to 445 00:30:16,080 --> 00:30:17,980 the aperiodic signal. 446 00:30:17,980 --> 00:30:22,610 And so, in fact, we can rewrite this equation simply 447 00:30:22,610 --> 00:30:27,630 by substituting in instead of x tilde, the original 448 00:30:27,630 --> 00:30:29,540 aperiodic signal. 449 00:30:29,540 --> 00:30:33,950 And now we can use infinite limits on this sum. 450 00:30:33,950 --> 00:30:38,180 And what we would want to examine, mathematically, is 451 00:30:38,180 --> 00:30:46,950 what happens to the top equation as we let the period 452 00:30:46,950 --> 00:30:49,250 go off to infinity? 453 00:30:49,250 --> 00:30:53,090 And what happens is exactly identical, mathematically, to 454 00:30:53,090 --> 00:30:54,580 continuous-time. 455 00:30:54,580 --> 00:30:56,970 I won't belabor the details. 456 00:30:56,970 --> 00:31:01,980 Essentially it's this sum that goes to an integral. 457 00:31:01,980 --> 00:31:05,590 Omega 0, which is the fundamental frequency, is 458 00:31:05,590 --> 00:31:06,940 going towards 0. 459 00:31:06,940 --> 00:31:10,100 In fact, becomes the differential in the integral. 460 00:31:10,100 --> 00:31:13,200 And in the second equation, of course, this then 461 00:31:13,200 --> 00:31:15,210 becomes x of omega. 462 00:31:15,210 --> 00:31:19,980 And as N goes to infinity then, what the Fourier series 463 00:31:19,980 --> 00:31:25,020 becomes is the Fourier transform as summarized by the 464 00:31:25,020 --> 00:31:27,610 bottom two equations. 465 00:31:27,610 --> 00:31:33,490 So although there is a little bit of mathematical trickery. 466 00:31:33,490 --> 00:31:36,900 Or let's not call it trickery, but subtlety, to be tracked 467 00:31:36,900 --> 00:31:38,330 through in detail. 468 00:31:38,330 --> 00:31:43,300 The important conceptual thing to think about is this notion 469 00:31:43,300 --> 00:31:46,760 that we take the aperiodic signal, form a periodic 470 00:31:46,760 --> 00:31:49,230 signal, let the period go off to infinity. 471 00:31:49,230 --> 00:31:52,920 In which case, the Fourier series coefficients become 472 00:31:52,920 --> 00:31:54,630 these envelopes functions. 473 00:31:54,630 --> 00:31:56,910 And incidentally, mathematically, one of the 474 00:31:56,910 --> 00:31:59,890 sums ends up going to an integral. 475 00:31:59,890 --> 00:32:05,690 So what we have then is the discrete-time Fourier 476 00:32:05,690 --> 00:32:11,210 transform, which is a representation of 477 00:32:11,210 --> 00:32:13,080 an aperiodic signal. 478 00:32:13,080 --> 00:32:17,920 And we have the synthesis equation, which I show as the 479 00:32:17,920 --> 00:32:21,470 top equation on this transparency. 480 00:32:21,470 --> 00:32:27,940 And this is the integral that the Fourier series synthesis 481 00:32:27,940 --> 00:32:32,800 equation went to as the period went off to infinity. 482 00:32:32,800 --> 00:32:36,730 And we have the corresponding analysis equation, which is 483 00:32:36,730 --> 00:32:41,180 shown below, where this tells us the Fourier transform. 484 00:32:41,180 --> 00:32:44,890 In effect, the envelope or the Fourier series coefficients of 485 00:32:44,890 --> 00:32:47,250 that periodic signal. 486 00:32:47,250 --> 00:32:53,550 And here represented in terms of the aperiodic signal. 487 00:32:53,550 --> 00:32:56,215 So we have the analysis equation 488 00:32:56,215 --> 00:32:58,710 and synthesis equation. 489 00:32:58,710 --> 00:33:03,160 There are a number of things to focus on 490 00:33:03,160 --> 00:33:03,980 as you look at this. 491 00:33:03,980 --> 00:33:06,450 And we'll talk about some of its properties actually in the 492 00:33:06,450 --> 00:33:07,150 next lecture. 493 00:33:07,150 --> 00:33:11,120 But some of the points that I'd like you to think about 494 00:33:11,120 --> 00:33:16,700 and focus on is the fact that now there is somewhat of an 495 00:33:16,700 --> 00:33:20,420 imbalance or lack of duality between the time domain and 496 00:33:20,420 --> 00:33:21,890 frequency domain. 497 00:33:21,890 --> 00:33:24,530 x of n, which is our aperiodic signal, 498 00:33:24,530 --> 00:33:27,350 is of course, discrete. 499 00:33:27,350 --> 00:33:32,100 It's Fourier transform, x of omega, is a function of a 500 00:33:32,100 --> 00:33:33,320 continuous variable. 501 00:33:33,320 --> 00:33:36,870 Omega is a continuous variable. 502 00:33:36,870 --> 00:33:41,420 That is essentially what represents the envelope. 503 00:33:41,420 --> 00:33:46,060 Also, in the time domain x of n is aperiodic. 504 00:33:46,060 --> 00:33:49,270 It's not a periodic function. 505 00:33:49,270 --> 00:33:52,440 However, in the frequency domain, remember that the 506 00:33:52,440 --> 00:33:55,960 Fourier series coefficients were always periodic. 507 00:33:55,960 --> 00:34:00,090 Well, this envelope function then is also periodic with a 508 00:34:00,090 --> 00:34:03,820 period in omega of 2 pi. 509 00:34:03,820 --> 00:34:07,820 Once again, the reason for the periodicity, it all stems back 510 00:34:07,820 --> 00:34:12,449 to the fact that when we talk about complex exponentials-- 511 00:34:12,449 --> 00:34:15,199 and recall back to the early lectures. 512 00:34:15,199 --> 00:34:19,300 In discrete-time, as the frequency variable covers a 513 00:34:19,300 --> 00:34:25,980 range of 2 pi, when you proceed past that range, you 514 00:34:25,980 --> 00:34:28,610 simply see the same complex exponentials 515 00:34:28,610 --> 00:34:30,350 over and over again. 516 00:34:30,350 --> 00:34:33,300 And so obviously, anything that we do with them would 517 00:34:33,300 --> 00:34:37,480 have to be periodic in that frequency variable. 518 00:34:37,480 --> 00:34:42,480 All right, notationally, we'll, again, represent the 519 00:34:42,480 --> 00:34:44,520 discrete-time Fourier transform pair 520 00:34:44,520 --> 00:34:46,179 as I indicated here. 521 00:34:46,179 --> 00:34:49,980 And since it's a complex function of frequency may, on 522 00:34:49,980 --> 00:34:54,530 occasion, want to either represent it in rectangular 523 00:34:54,530 --> 00:34:59,180 form as I indicate in this equation, or in polar form as 524 00:34:59,180 --> 00:35:02,040 I indicate in this equation. 525 00:35:02,040 --> 00:35:04,930 Let's look at an example. 526 00:35:04,930 --> 00:35:10,590 And, of course, one example that we can look at is the one 527 00:35:10,590 --> 00:35:14,480 that has kind of been tracking us through this lecture, which 528 00:35:14,480 --> 00:35:18,260 is the example of a rectangle. 529 00:35:18,260 --> 00:35:24,130 Now, the rectangle, if we refer back to our argument of 530 00:35:24,130 --> 00:35:26,720 how we get a Fourier representation for an 531 00:35:26,720 --> 00:35:30,550 aperiodic signal, we would form a periodic signal where 532 00:35:30,550 --> 00:35:31,620 this is repeated. 533 00:35:31,620 --> 00:35:34,110 And that's our square wave example. 534 00:35:34,110 --> 00:35:36,740 As the period goes to infinity, the Fourier 535 00:35:36,740 --> 00:35:40,370 transform of this is represented by the envelope of 536 00:35:40,370 --> 00:35:44,140 those Fourier series coefficients, and that was our 537 00:35:44,140 --> 00:35:48,210 sin nx over sin x function, which in this particular case, 538 00:35:48,210 --> 00:35:54,320 for these particular numbers, is sin 5 omega over 2 divided 539 00:35:54,320 --> 00:35:57,890 by sin omega over 2. 540 00:35:57,890 --> 00:36:02,490 And notice, of course, as we would expect-- 541 00:36:02,490 --> 00:36:07,800 notice that this is a periodic function of the frequency 542 00:36:07,800 --> 00:36:13,830 variable omega repeating, of course, with a period of 2 pi. 543 00:36:13,830 --> 00:36:17,680 Whereas, in the time domain, the function was not a 544 00:36:17,680 --> 00:36:19,020 periodic function, it's aperiodic. 545 00:36:22,750 --> 00:36:26,170 Now, let's look at another example. 546 00:36:26,170 --> 00:36:31,880 Let's look at an example which is another signal that has 547 00:36:31,880 --> 00:36:34,780 kind of popped its head up from time to time as the 548 00:36:34,780 --> 00:36:36,720 lectures have gone along. 549 00:36:36,720 --> 00:36:40,110 A signal which is another aperiodic signal, which is a 550 00:36:40,110 --> 00:36:47,010 decaying exponential of this form with the factor a chosen 551 00:36:47,010 --> 00:36:49,890 between 0 and 1. 552 00:36:49,890 --> 00:36:54,670 And you can work out the algebra at your leisure. 553 00:36:54,670 --> 00:36:58,220 Basically, if we substitute into the Fourier transform 554 00:36:58,220 --> 00:37:02,830 analysis equation, it's this sum that we evaluate. 555 00:37:02,830 --> 00:37:07,665 Because we have a unit step here which shuts this off for 556 00:37:07,665 --> 00:37:11,250 n less than 0, we can change the limits on the sum. 557 00:37:11,250 --> 00:37:16,790 This then corresponds to the sum over an infinite number of 558 00:37:16,790 --> 00:37:19,160 terms of a geometric series. 559 00:37:19,160 --> 00:37:23,690 And that, as we've seen before, is 1 divided by 1 560 00:37:23,690 --> 00:37:27,340 minus a e to the minus j omega. 561 00:37:27,340 --> 00:37:32,610 So let's look at what that looks like. 562 00:37:32,610 --> 00:37:40,100 Here then we have, again, the expression in the time domain 563 00:37:40,100 --> 00:37:43,330 and the expression in the frequency domain. 564 00:37:43,330 --> 00:37:47,580 And let's, in particular, focus on what the magnitude of 565 00:37:47,580 --> 00:37:49,960 the Fourier transform looks like. 566 00:37:49,960 --> 00:37:54,790 It's as we show here. 567 00:37:54,790 --> 00:37:59,380 And for the particular values of a that I pick, namely 568 00:37:59,380 --> 00:38:03,460 between 0 and 1, it's larger at the origin 569 00:38:03,460 --> 00:38:05,340 than it is at pi. 570 00:38:05,340 --> 00:38:09,090 And then, of course, it is periodic. 571 00:38:09,090 --> 00:38:12,080 And the periodicity is inherent in the Fourier 572 00:38:12,080 --> 00:38:16,150 transform in discrete-time, so we really might only need to 573 00:38:16,150 --> 00:38:21,160 look at this either from minus pi to pi, or from 0 to 2 pi. 574 00:38:21,160 --> 00:38:24,710 The periodicity, of course, would imply what the rest of 575 00:38:24,710 --> 00:38:27,900 this is for other values of omega. 576 00:38:27,900 --> 00:38:31,290 Let me also draw your attention while we're on it to 577 00:38:31,290 --> 00:38:33,900 the fact that-- 578 00:38:33,900 --> 00:38:38,960 observe that if a were, in fact, negative, then this 579 00:38:38,960 --> 00:38:42,700 value would be less than this value. 580 00:38:42,700 --> 00:38:47,090 And in fact, for a negative, the magnitude of the frequency 581 00:38:47,090 --> 00:38:51,630 response would look like this except shifted by an amount in 582 00:38:51,630 --> 00:38:54,380 omega equal to pi. 583 00:38:54,380 --> 00:38:59,050 And this example will come up and play an important role in 584 00:38:59,050 --> 00:39:03,650 our discussion next time, so try to keep it in mind. 585 00:39:03,650 --> 00:39:06,710 And in fact, work it out more carefully between 586 00:39:06,710 --> 00:39:08,220 now and next time. 587 00:39:08,220 --> 00:39:12,800 And also, if you have a chance, focus on this issue of 588 00:39:12,800 --> 00:39:18,500 how it looks with a positive as compared with a negative. 589 00:39:18,500 --> 00:39:23,860 Now, we developed the Fourier transform by beginning with 590 00:39:23,860 --> 00:39:25,370 the Fourier series. 591 00:39:25,370 --> 00:39:29,350 We did that in continuous-time also. 592 00:39:29,350 --> 00:39:32,920 What I'd like to do now, just as we did in continuous-time, 593 00:39:32,920 --> 00:39:38,800 is now absorb the Fourier series within the broader 594 00:39:38,800 --> 00:39:41,540 framework of the Fourier transform. 595 00:39:41,540 --> 00:39:44,360 And there are two relationships between the 596 00:39:44,360 --> 00:39:48,000 Fourier series and the Fourier transform, which are identical 597 00:39:48,000 --> 00:39:54,250 to relationships that we had in the continuous-time case. 598 00:39:54,250 --> 00:40:00,980 Let me remind you that in continuous-time we had the 599 00:40:00,980 --> 00:40:09,290 statement that if we have a periodic signal, that in fact 600 00:40:09,290 --> 00:40:12,040 the Fourier series coefficients of that periodic 601 00:40:12,040 --> 00:40:19,340 signal is proportional to samples of the Fourier 602 00:40:19,340 --> 00:40:23,140 transform of one period. 603 00:40:23,140 --> 00:40:28,510 Well, in fact, let me remind you flows easily from all the 604 00:40:28,510 --> 00:40:33,210 things that we built up so far, because of the fact that 605 00:40:33,210 --> 00:40:36,670 the Fourier transform essentially, by definition, of 606 00:40:36,670 --> 00:40:44,600 the way we developed it, is what we get as the Fourier 607 00:40:44,600 --> 00:40:48,780 series coefficients, as we focus on one period, and then 608 00:40:48,780 --> 00:40:51,050 let the period go off to infinity. 609 00:40:51,050 --> 00:40:55,510 Well, looking at one period, the Fourier transform of that 610 00:40:55,510 --> 00:40:59,490 then is the envelope of the Fourier series coefficients. 611 00:40:59,490 --> 00:41:05,610 And so in continuous-time, we have this relationship. 612 00:41:05,610 --> 00:41:10,500 And in discrete-time, we have precisely the same 613 00:41:10,500 --> 00:41:16,240 relationship, except that here we're talking about an integer 614 00:41:16,240 --> 00:41:20,090 variable as opposed to the continuous variable, and a 615 00:41:20,090 --> 00:41:28,440 period of capital N as opposed to a period of t0. 616 00:41:28,440 --> 00:41:38,720 OK, so once again, if we return to our example, or if 617 00:41:38,720 --> 00:41:40,790 we return to a periodic signal. 618 00:41:40,790 --> 00:41:46,240 If we have a periodic signal and we consider the Fourier 619 00:41:46,240 --> 00:41:52,100 transform of one period, the Fourier series coefficients of 620 00:41:52,100 --> 00:41:56,880 this periodic signal are, in fact, samples-- 621 00:41:56,880 --> 00:42:01,800 as stated mathematically in the bottom equation, samples 622 00:42:01,800 --> 00:42:07,340 of the Fourier transform of one period. 623 00:42:07,340 --> 00:42:11,180 So x of omega is the Fourier transform of one period. 624 00:42:11,180 --> 00:42:13,760 a sub k's are the Fourier series coefficients of the 625 00:42:13,760 --> 00:42:15,350 periodic signal. 626 00:42:15,350 --> 00:42:19,190 And this relationship simply says they're related except 627 00:42:19,190 --> 00:42:23,030 for scale factor through samples along 628 00:42:23,030 --> 00:42:25,820 the frequency axis. 629 00:42:25,820 --> 00:42:33,370 And, of course, we saw this in the context of 630 00:42:33,370 --> 00:42:36,350 our square wave example. 631 00:42:36,350 --> 00:42:42,080 In the square wave example, we have a periodic signal, which 632 00:42:42,080 --> 00:42:45,690 is a periodic square wave. 633 00:42:45,690 --> 00:42:51,800 And the Fourier transform of one period, in fact, 634 00:42:51,800 --> 00:42:54,790 represents the envelope. 635 00:42:54,790 --> 00:42:57,180 And here we have the envelope function. 636 00:42:57,180 --> 00:43:02,520 Represents the envelope of the Fourier series coefficients. 637 00:43:02,520 --> 00:43:05,710 And the Fourier series coefficients are samples. 638 00:43:08,390 --> 00:43:14,460 So what we have then is a relationship back to the 639 00:43:14,460 --> 00:43:18,120 Fourier series coefficients from the Fourier transform 640 00:43:18,120 --> 00:43:21,750 that tells us that for a periodic signal now, the 641 00:43:21,750 --> 00:43:23,420 periodic signal-- 642 00:43:23,420 --> 00:43:27,170 the Fourier series coefficients are related, are 643 00:43:27,170 --> 00:43:31,920 samples of the Fourier transform of one period. 644 00:43:31,920 --> 00:43:36,830 Now, finally, to kind of bring things back in a circle and 645 00:43:36,830 --> 00:43:40,100 exactly identical to what we did in the continuous-time 646 00:43:40,100 --> 00:43:46,400 case, we can finally absorb the Fourier series in 647 00:43:46,400 --> 00:43:47,310 discrete-time. 648 00:43:47,310 --> 00:43:50,480 We can absorb it into the framework 649 00:43:50,480 --> 00:43:53,300 of the Fourier transform. 650 00:43:53,300 --> 00:44:00,800 Now, remember or recall how we did that when we tried to do a 651 00:44:00,800 --> 00:44:04,970 similar sort of thing in continuous-time. 652 00:44:04,970 --> 00:44:11,430 In continuous-time, what we essentially did is to develop 653 00:44:11,430 --> 00:44:15,040 that, more or less, by definition. 654 00:44:15,040 --> 00:44:18,150 We have a periodic signal. 655 00:44:18,150 --> 00:44:20,800 The periodic signal is represented through a Fourier 656 00:44:20,800 --> 00:44:24,360 series and Fourier series coefficients. 657 00:44:24,360 --> 00:44:27,570 Essentially what I pointed out at that time 658 00:44:27,570 --> 00:44:30,180 was that if we define-- 659 00:44:30,180 --> 00:44:33,780 take it as a definition, the Fourier transform of the 660 00:44:33,780 --> 00:44:39,220 periodic signal as an impulse train where the amplitudes of 661 00:44:39,220 --> 00:44:42,640 the impulses are proportional to the Fourier series 662 00:44:42,640 --> 00:44:44,680 coefficients. 663 00:44:44,680 --> 00:44:50,430 If we take that impulse train representation and simply plug 664 00:44:50,430 --> 00:44:58,460 it into the Fourier transform synthesis equation, what we 665 00:44:58,460 --> 00:45:04,860 end up with is the Fourier series synthesis equation. 666 00:45:04,860 --> 00:45:17,320 So in continuous-time, we had used this definition of the 667 00:45:17,320 --> 00:45:20,060 continuous-time Fourier transform 668 00:45:20,060 --> 00:45:22,610 of a periodic signal. 669 00:45:22,610 --> 00:45:27,670 And again, in discrete-time, it's simply a matter of using 670 00:45:27,670 --> 00:45:30,110 exactly the same expression. 671 00:45:30,110 --> 00:45:35,400 And using, instead, the appropriate variables related 672 00:45:35,400 --> 00:45:38,430 to discrete-time rather than the variables related to 673 00:45:38,430 --> 00:45:39,810 continuous-time. 674 00:45:39,810 --> 00:45:43,120 So in discrete-time, if we have a periodic signal, the 675 00:45:43,120 --> 00:45:47,150 Fourier transform of that periodic signal is defined as 676 00:45:47,150 --> 00:45:53,290 an impulse train where the amplitudes of the impulses are 677 00:45:53,290 --> 00:45:57,500 proportional to the Fourier series coefficients. 678 00:45:57,500 --> 00:46:00,660 If this expression is substituted into the synthesis 679 00:46:00,660 --> 00:46:05,340 equation for the Fourier transform, that will simply 680 00:46:05,340 --> 00:46:08,440 then reduce to the synthesis equation 681 00:46:08,440 --> 00:46:11,240 for the Fourier series. 682 00:46:11,240 --> 00:46:17,310 So once more returning to our example, which is the square 683 00:46:17,310 --> 00:46:19,170 wave example that we've carried through these 684 00:46:19,170 --> 00:46:23,780 lectures, or through this lecture, we can see that 685 00:46:23,780 --> 00:46:25,870 really what we're talking about really is 686 00:46:25,870 --> 00:46:27,120 a notational change. 687 00:46:30,740 --> 00:46:35,290 Here is the periodic signal and below it are the Fourier 688 00:46:35,290 --> 00:46:39,330 series coefficients, where I've removed the envelope 689 00:46:39,330 --> 00:46:43,420 function and just indicate the amplitudes of the coefficients 690 00:46:43,420 --> 00:46:46,610 indexed, of course, on the coefficient number. 691 00:46:46,610 --> 00:46:49,550 And so this represents a bar graph. 692 00:46:49,550 --> 00:46:52,760 And if instead of talking about the Fourier series 693 00:46:52,760 --> 00:46:59,040 coefficients, what I want to talk about is the Fourier 694 00:46:59,040 --> 00:47:03,300 transform, the Fourier transform, in essence, 695 00:47:03,300 --> 00:47:08,450 corresponds to simply redrawing that using impulses 696 00:47:08,450 --> 00:47:14,420 and using an axis that is essentially indexed on the 697 00:47:14,420 --> 00:47:18,910 fundamental frequency omega 0, rather than on the Fourier 698 00:47:18,910 --> 00:47:20,433 series coefficient number k. 699 00:47:23,884 --> 00:47:31,540 OK, so to summarize, what we've done is 700 00:47:31,540 --> 00:47:34,190 to pretty much parallel-- 701 00:47:34,190 --> 00:47:38,290 somewhat more quickly, the kind of development that we 702 00:47:38,290 --> 00:47:41,970 went through for continuous-time representation 703 00:47:41,970 --> 00:47:45,070 through complex exponentials, paralleled that for the 704 00:47:45,070 --> 00:47:47,390 discrete-time case. 705 00:47:47,390 --> 00:47:51,340 And pretty much the conceptual underpinnings of the 706 00:47:51,340 --> 00:47:55,230 development are identical in discrete-time and in 707 00:47:55,230 --> 00:47:56,480 continuous-time. 708 00:47:58,750 --> 00:48:04,510 We saw that there are some major differences, or 709 00:48:04,510 --> 00:48:06,700 important differences between continuous-time and 710 00:48:06,700 --> 00:48:08,560 discrete-time. 711 00:48:08,560 --> 00:48:11,240 And the difference, essentially 712 00:48:11,240 --> 00:48:14,080 relates to two aspects. 713 00:48:14,080 --> 00:48:18,790 One aspect is the fact that in discrete-time, we have a 714 00:48:18,790 --> 00:48:22,690 discrete representation in the time domain, whereas the 715 00:48:22,690 --> 00:48:25,020 independent variable in the frequency domain is a 716 00:48:25,020 --> 00:48:26,750 continuous variable. 717 00:48:26,750 --> 00:48:30,860 Whereas in continuous-time for the Fourier transform, we had 718 00:48:30,860 --> 00:48:35,120 a duality between the time domain and frequency domain. 719 00:48:35,120 --> 00:48:38,170 The other very important difference tied back to the 720 00:48:38,170 --> 00:48:41,310 difference between complex exponentials, continuous-time 721 00:48:41,310 --> 00:48:43,050 and discrete-time. 722 00:48:43,050 --> 00:48:45,980 In continuous-time, complex exponentials, as you vary the 723 00:48:45,980 --> 00:48:52,940 frequency, generate distinct time functions. 724 00:48:52,940 --> 00:48:56,500 In discrete-time, as you vary the frequency, once you've 725 00:48:56,500 --> 00:49:01,120 covered a frequency interval of 2 pi, then you've seen all 726 00:49:01,120 --> 00:49:02,240 the ones there are to see. 727 00:49:02,240 --> 00:49:03,680 There are no more. 728 00:49:03,680 --> 00:49:09,610 And this, in effect, imposes a periodicity on the Fourier 729 00:49:09,610 --> 00:49:13,390 domain representation of discrete-time signals. 730 00:49:13,390 --> 00:49:15,710 And some of those differences and, of course, lots of the 731 00:49:15,710 --> 00:49:20,380 similarities will surface, both as we use this 732 00:49:20,380 --> 00:49:26,120 representation and as we develop further properties. 733 00:49:26,120 --> 00:49:30,290 In the next lecture, what we'll do is to focus in, 734 00:49:30,290 --> 00:49:33,540 again, on the Fourier transform, the discrete-time 735 00:49:33,540 --> 00:49:39,050 Fourier transform, develop or illuminate some of the 736 00:49:39,050 --> 00:49:42,770 properties of the Fourier transform, and then see how 737 00:49:42,770 --> 00:49:46,850 these properties can be used for a number of things. 738 00:49:46,850 --> 00:49:49,360 For example, how the properties as they were in 739 00:49:49,360 --> 00:49:55,280 continuous-time can be used to efficiently generate the 740 00:49:55,280 --> 00:49:58,430 solution and analyze linear constant coefficient 741 00:49:58,430 --> 00:49:59,950 difference equations. 742 00:49:59,950 --> 00:50:03,680 And then beyond that, the concepts of filtering and 743 00:50:03,680 --> 00:50:05,090 modulation. 744 00:50:05,090 --> 00:50:09,220 And both the properties and interpretation, which will 745 00:50:09,220 --> 00:50:12,750 very strongly parallel the kinds of developments along 746 00:50:12,750 --> 00:50:14,650 those lines that we did in the last lecture. 747 00:50:14,650 --> 00:50:15,900 Thank you.