1 00:00:00,120 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:04,059 Commons license. 3 00:00:04,059 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,750 continue to offer high quality educational resources for free. 5 00:00:10,750 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,310 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,310 --> 00:00:18,480 at ocw.mit.edu. 8 00:00:26,200 --> 00:00:29,440 DENNIS FREEMAN: Hello, welcome. 9 00:00:29,440 --> 00:00:34,030 So as you might expect, there is yet another announcement. 10 00:00:34,030 --> 00:00:40,990 So this time, Exam 3, the last of the midterms, same rules-- 11 00:00:40,990 --> 00:00:43,120 everything is roughly the same. 12 00:00:43,120 --> 00:00:48,300 Walker, next Wednesday, 7:30 to 9:30. 13 00:00:48,300 --> 00:00:51,010 No recitations on the day of the exam. 14 00:00:51,010 --> 00:00:54,880 Coverage through Lecture 18, which is next lecture, coverage 15 00:00:54,880 --> 00:00:56,150 through Recitation 16-- 16 00:00:56,150 --> 00:00:59,890 that's tomorrow-- homeworks 1 to 10. 17 00:00:59,890 --> 00:01:01,300 Homework 10 won't be graded-- 18 00:01:01,300 --> 00:01:03,410 there will be solutions posted. 19 00:01:03,410 --> 00:01:07,240 3 pages, your old 2 plus 1. 20 00:01:07,240 --> 00:01:10,680 No calculators, blah, blah, blah-- same as last time. 21 00:01:10,680 --> 00:01:12,190 Designed to be one hour, two hours 22 00:01:12,190 --> 00:01:14,320 to complete-- same as last time. 23 00:01:14,320 --> 00:01:16,450 There'll be a review session-- 24 00:01:16,450 --> 00:01:21,171 there'll be one on Monday at 3 PM in 36-112 25 00:01:21,171 --> 00:01:23,170 and, of course, you can ask any review questions 26 00:01:23,170 --> 00:01:25,480 you'd like at any of the open office hours, 27 00:01:25,480 --> 00:01:31,960 but there'll be a formal review in 36-112 on Monday at 3:00. 28 00:01:31,960 --> 00:01:33,850 Prior term exams have already been posted 29 00:01:33,850 --> 00:01:35,620 and if you have a conflict, please tell me 30 00:01:35,620 --> 00:01:38,640 because I have to find rooms and a proctor. 31 00:01:38,640 --> 00:01:43,780 OK, questions, comments-- completely routine 32 00:01:43,780 --> 00:01:44,920 at this point, right? 33 00:01:44,920 --> 00:01:47,770 No problems-- everybody knows what they're doing? 34 00:01:47,770 --> 00:01:50,606 It'll be fun, it'll be enjoyable-- 35 00:01:50,606 --> 00:01:51,105 smile. 36 00:01:54,090 --> 00:01:56,630 Questions, comments? 37 00:01:56,630 --> 00:02:02,350 OK, so what I want to talk today about is DT, signal-processing. 38 00:02:02,350 --> 00:02:06,120 So formerly, I'm going to talk about things like DT Fourier 39 00:02:06,120 --> 00:02:06,690 series. 40 00:02:06,690 --> 00:02:09,530 Next time we'll talk about DT Fourier Transform, 41 00:02:09,530 --> 00:02:12,300 but the context for that-- the reason we do this-- 42 00:02:12,300 --> 00:02:15,540 is because the discrete time approaches are so 43 00:02:15,540 --> 00:02:20,430 useful for processing signals. 44 00:02:20,430 --> 00:02:22,740 And I want to motivate that by thinking 45 00:02:22,740 --> 00:02:27,510 about how signal processing has been viewed historically. 46 00:02:27,510 --> 00:02:29,790 For over the past 20 or 30 years, 47 00:02:29,790 --> 00:02:34,410 there's been enormous interest in signal processing 48 00:02:34,410 --> 00:02:38,760 and most of that interest evolved out of CT applications 49 00:02:38,760 --> 00:02:40,800 that we wanted to make better. 50 00:02:40,800 --> 00:02:42,836 We wanted to make our radio reception better, 51 00:02:42,836 --> 00:02:44,710 we wanted to make telephone reception better, 52 00:02:44,710 --> 00:02:46,922 we wanted to make hi-fis work better, 53 00:02:46,922 --> 00:02:48,630 we wanted to make television work better, 54 00:02:48,630 --> 00:02:51,600 photography, x-rays, blah, blah, blah. 55 00:02:51,600 --> 00:02:53,220 There were lots of reasons why we 56 00:02:53,220 --> 00:02:57,780 wanted to be able to alter the signals that 57 00:02:57,780 --> 00:02:59,620 were available to us. 58 00:02:59,620 --> 00:03:01,140 So for example, in radio-- 59 00:03:01,140 --> 00:03:04,020 how do you alter the signal so as to reduce static? 60 00:03:07,027 --> 00:03:08,860 In the early radios, there was a big problem 61 00:03:08,860 --> 00:03:10,120 with automatic gain control. 62 00:03:10,120 --> 00:03:13,180 So as you went over the hills, the line 63 00:03:13,180 --> 00:03:15,520 of sight to the radio station changed 64 00:03:15,520 --> 00:03:18,505 and that changed the gain so there was an interest 65 00:03:18,505 --> 00:03:20,170 in controlling gain. 66 00:03:20,170 --> 00:03:22,630 So there were lots of applications 67 00:03:22,630 --> 00:03:27,610 where we wanted to account for deficiencies in the hardware 68 00:03:27,610 --> 00:03:29,860 and that gave rise to the notion of signal processing. 69 00:03:29,860 --> 00:03:31,360 Those signal processing applications 70 00:03:31,360 --> 00:03:36,370 were largely in continuous time, because the signals were 71 00:03:36,370 --> 00:03:38,500 largely in continuous time-- 72 00:03:38,500 --> 00:03:40,780 all that's changed. 73 00:03:40,780 --> 00:03:42,580 Now when you say signal processing, 74 00:03:42,580 --> 00:03:45,840 you're almost always mean discrete. 75 00:03:45,840 --> 00:03:47,473 And there's a very good reason for it 76 00:03:47,473 --> 00:03:49,260 and it's digital electronics. 77 00:03:49,260 --> 00:03:51,720 Digital electronics are very inexpensive 78 00:03:51,720 --> 00:03:54,030 and they work very well. 79 00:03:54,030 --> 00:03:58,920 I want to give just one example chosen from my history. 80 00:03:58,920 --> 00:04:01,540 So like most teenage people-- 81 00:04:01,540 --> 00:04:03,000 I was teenage once, right? 82 00:04:03,000 --> 00:04:06,330 It was a long time ago, but I was one once. 83 00:04:06,330 --> 00:04:09,840 And like most teenage types, even then 84 00:04:09,840 --> 00:04:13,360 we got afflicted with the audio hi-fi 85 00:04:13,360 --> 00:04:17,010 listen to music virus problem. 86 00:04:17,010 --> 00:04:18,260 The good news is I recovered-- 87 00:04:18,260 --> 00:04:20,640 I'm fully functional, it's gone, right? 88 00:04:20,640 --> 00:04:23,250 So you have every reason to think that you'll recover, 89 00:04:23,250 --> 00:04:24,960 but that was a problem. 90 00:04:24,960 --> 00:04:29,970 I was interested in hi-fi and that was a hard problem. 91 00:04:33,050 --> 00:04:37,789 So if you think about how do you make a speaker system, 92 00:04:37,789 --> 00:04:39,330 one of the problems that you get when 93 00:04:39,330 --> 00:04:41,070 you try to design a speaker system 94 00:04:41,070 --> 00:04:43,761 is reproducing low frequencies. 95 00:04:43,761 --> 00:04:46,260 And my guess is that you're as interested in low frequencies 96 00:04:46,260 --> 00:04:47,772 as we were back then. 97 00:04:47,772 --> 00:04:49,230 The problem with low frequencies is 98 00:04:49,230 --> 00:04:56,100 that if you have just a speaker, the speaker 99 00:04:56,100 --> 00:04:57,090 has an electromagnet. 100 00:04:57,090 --> 00:04:59,610 The electromagnet moves so that it pushes the cone. 101 00:04:59,610 --> 00:05:04,080 The cone pushes air this way, but if cone is coming out, 102 00:05:04,080 --> 00:05:07,130 it also pulls air that way. 103 00:05:07,130 --> 00:05:10,880 So that's a problem because that means it can do that 104 00:05:10,880 --> 00:05:14,984 and the person who is over here hears nothing. 105 00:05:14,984 --> 00:05:17,150 That's especially bad at low frequencies for reasons 106 00:05:17,150 --> 00:05:21,010 I won't go into because the wavelengths are long, 107 00:05:21,010 --> 00:05:23,040 but that's the problem. 108 00:05:23,040 --> 00:05:24,330 So what do you do? 109 00:05:24,330 --> 00:05:25,710 Well, you put it in a box. 110 00:05:25,710 --> 00:05:28,820 So instead of having your speaker in space, 111 00:05:28,820 --> 00:05:31,790 you put the speaker in a box. 112 00:05:31,790 --> 00:05:33,710 Now when the cone goes that way, the air 113 00:05:33,710 --> 00:05:36,110 gets compressed toward the person-- 114 00:05:36,110 --> 00:05:40,300 that's our second person. 115 00:05:40,300 --> 00:05:44,541 And now the air that goes this way is constrained 116 00:05:44,541 --> 00:05:46,040 to the box, so it doesn't interfere, 117 00:05:46,040 --> 00:05:47,180 but what's the new problem? 118 00:05:51,937 --> 00:05:53,770 So putting it in the box solves one problem, 119 00:05:53,770 --> 00:05:55,103 but it introduces a new problem. 120 00:05:55,103 --> 00:05:57,606 What's the new problem? 121 00:05:57,606 --> 00:05:59,090 AUDIENCE: Box vibrates. 122 00:05:59,090 --> 00:06:00,820 DENNIS FREEMAN: The box will vibrate-- 123 00:06:00,820 --> 00:06:01,570 even worse. 124 00:06:06,564 --> 00:06:07,730 AUDIENCE: Negative pressure. 125 00:06:07,730 --> 00:06:09,063 DENNIS FREEMAN: Native pressure. 126 00:06:09,063 --> 00:06:11,270 So how does that show up as a performance problem? 127 00:06:11,270 --> 00:06:12,770 That's exactly right. 128 00:06:12,770 --> 00:06:15,800 What's the problem that's generated 129 00:06:15,800 --> 00:06:19,338 because of the negative pressure in the box? 130 00:06:19,338 --> 00:06:20,282 Yes? 131 00:06:20,282 --> 00:06:22,790 AUDIENCE: [INAUDIBLE] 132 00:06:22,790 --> 00:06:25,040 DENNIS FREEMAN: So there's something to do with gain-- 133 00:06:25,040 --> 00:06:26,060 that's exactly right. 134 00:06:26,060 --> 00:06:27,140 Because of the negative pressure, 135 00:06:27,140 --> 00:06:28,550 what happens to the speaker cone? 136 00:06:31,376 --> 00:06:32,670 AUDIENCE: Not be able to push. 137 00:06:32,670 --> 00:06:33,850 DENNIS FREEMAN: Yes, it can't move-- 138 00:06:33,850 --> 00:06:34,750 you put a load on it. 139 00:06:37,790 --> 00:06:39,804 If in order to come out, you have 140 00:06:39,804 --> 00:06:42,220 to generate negative pressure, but the negative pressure's 141 00:06:42,220 --> 00:06:44,240 going to pull it back in. 142 00:06:44,240 --> 00:06:48,110 So the problem is that you put the speaker in the box, 143 00:06:48,110 --> 00:06:53,140 now you no longer have the short circuit of the acoustic path. 144 00:06:53,140 --> 00:06:55,120 However, you've now introduced a load, 145 00:06:55,120 --> 00:06:58,660 which makes the speaker much harder to move. 146 00:06:58,660 --> 00:07:00,580 Everyone with me? 147 00:07:00,580 --> 00:07:02,680 So the problem was you put the speaker in the box 148 00:07:02,680 --> 00:07:07,300 and now the speaker, which was making a lot of air motion, 149 00:07:07,300 --> 00:07:10,330 makes very little air motion because of the box. 150 00:07:10,330 --> 00:07:12,280 So what do you do next? 151 00:07:12,280 --> 00:07:14,320 Make the box big. 152 00:07:14,320 --> 00:07:16,330 OK, well that works for a while. 153 00:07:16,330 --> 00:07:19,120 Ideally if you were trying to build a speaker for this room, 154 00:07:19,120 --> 00:07:22,870 you would make a box this size. 155 00:07:22,870 --> 00:07:25,030 Well, that's kind of an architectural waste, 156 00:07:25,030 --> 00:07:28,250 so that's not really the right solution. 157 00:07:28,250 --> 00:07:31,990 So here's an acoustic solution that was very popular 158 00:07:31,990 --> 00:07:34,510 when I was your age. 159 00:07:34,510 --> 00:07:36,670 So this is called a reflex port. 160 00:07:36,670 --> 00:07:39,730 So the idea is make a hole in the box-- 161 00:07:42,940 --> 00:07:46,986 so if you make a hole in the box, 162 00:07:46,986 --> 00:07:48,610 OK, now you're back to the same problem 163 00:07:48,610 --> 00:07:50,610 that if the speaker comes out this way, then 164 00:07:50,610 --> 00:07:52,930 it sucks air that way-- 165 00:07:52,930 --> 00:07:54,550 that's bad. 166 00:07:54,550 --> 00:07:57,464 So what you do instead is you put a tube 167 00:07:57,464 --> 00:07:59,380 and if you make the length of the tube exactly 168 00:07:59,380 --> 00:08:06,010 the right size, there is mass entrained in the tube. 169 00:08:06,010 --> 00:08:12,520 And then you have a system where the air is springy, 170 00:08:12,520 --> 00:08:17,540 but there's mass in here and you get a mass spring dashpot. 171 00:08:17,540 --> 00:08:20,390 So now when this is going in and out, 172 00:08:20,390 --> 00:08:23,130 the air goes in and out of this tube, 173 00:08:23,130 --> 00:08:24,920 and if you adjust the length of the tube 174 00:08:24,920 --> 00:08:29,690 so you get the mass right, you can get them going in phase. 175 00:08:29,690 --> 00:08:32,870 Anyone have a clue what I just talked about? 176 00:08:32,870 --> 00:08:38,210 The idea was to use mass spring dashpot resonance theory 177 00:08:38,210 --> 00:08:41,120 to change the acoustics of a speaker box 178 00:08:41,120 --> 00:08:44,110 so you get more low frequencies out. 179 00:08:44,110 --> 00:08:47,120 That was the way we thought about speaker design back then. 180 00:08:47,120 --> 00:08:49,620 The speakers that I bought were the Electric Voice Interface 181 00:08:49,620 --> 00:08:50,800 A's, right? 182 00:08:50,800 --> 00:08:53,200 At that time, they cost about $1,500, 183 00:08:53,200 --> 00:08:55,300 I was making $600 a month-- 184 00:08:55,300 --> 00:08:57,970 so times have changed slightly. 185 00:08:57,970 --> 00:09:03,400 So $1,500 was then a fortune, but I had to have them, right? 186 00:09:03,400 --> 00:09:04,470 There was any question. 187 00:09:04,470 --> 00:09:09,590 And the way these worked was like the reflex design. 188 00:09:09,590 --> 00:09:11,320 So in the reflex design, we had the port 189 00:09:11,320 --> 00:09:17,380 that was tuned so the mass resonated against the stiffness 190 00:09:17,380 --> 00:09:18,680 of the air. 191 00:09:18,680 --> 00:09:20,740 Here what you did-- 192 00:09:20,740 --> 00:09:23,920 this was not a speaker. 193 00:09:23,920 --> 00:09:25,660 This is the speaker-- 194 00:09:25,660 --> 00:09:26,554 AUDIENCE: [INAUDIBLE] 195 00:09:26,554 --> 00:09:27,970 DENNIS FREEMAN: --and the idea was 196 00:09:27,970 --> 00:09:30,790 that you adjust the mass and stiffness 197 00:09:30,790 --> 00:09:33,414 of this big passive thing. 198 00:09:33,414 --> 00:09:35,080 There were no electrical wires hooked up 199 00:09:35,080 --> 00:09:38,200 to that thing-- that was called a passive resonator. 200 00:09:38,200 --> 00:09:44,110 It was tuned so that when the 8-inch speaker up here was 201 00:09:44,110 --> 00:09:49,580 trying to move, it would move in resonance so that it 202 00:09:49,580 --> 00:09:53,160 was out of phase with the 8-inch speaker up on the top 203 00:09:53,160 --> 00:09:55,520 so they were both going in and out the same time. 204 00:09:55,520 --> 00:10:00,110 Same idea as the reflex, but you could get a bigger motion 205 00:10:00,110 --> 00:10:02,120 from this passive resonator than you 206 00:10:02,120 --> 00:10:03,755 could with the acoustic reflex. 207 00:10:10,510 --> 00:10:14,420 So the next big advance was by Amar Bose. 208 00:10:14,420 --> 00:10:16,120 Bose was here at the time-- he was 209 00:10:16,120 --> 00:10:20,000 a professor in electroengineering at that time 210 00:10:20,000 --> 00:10:21,920 and he invented the Bose 901s, which 211 00:10:21,920 --> 00:10:24,800 is something everybody had to have. 212 00:10:24,800 --> 00:10:27,920 What Bose did that was extraordinarily clever was 213 00:10:27,920 --> 00:10:30,870 he used really chintzy speakers-- 214 00:10:30,870 --> 00:10:32,705 CTS speakers, 4-inch diameter. 215 00:10:35,390 --> 00:10:39,560 They had absolutely crummy frequency response-- 216 00:10:39,560 --> 00:10:45,120 flat up to a couple of hundred Hertz and then their response 217 00:10:45,120 --> 00:10:47,670 fell roughly with one pole-- 218 00:10:47,670 --> 00:10:50,250 so that by the time you get out to the highest frequencies 219 00:10:50,250 --> 00:10:52,830 that we can hear, the response is greatly 220 00:10:52,830 --> 00:10:55,810 attenuated-- like 40 decibels. 221 00:10:55,810 --> 00:10:59,600 40 decibels is a factor of 100 in pressure. 222 00:10:59,600 --> 00:11:04,479 So the problem is that these speakers 223 00:11:04,479 --> 00:11:06,020 had exactly the same problems that we 224 00:11:06,020 --> 00:11:09,154 had in the acoustic reflex problem, 225 00:11:09,154 --> 00:11:10,320 except they were even worse. 226 00:11:10,320 --> 00:11:13,520 This was an 8-inch speaker, this is a 4-inch speaker, 227 00:11:13,520 --> 00:11:14,520 so they were even worse. 228 00:11:14,520 --> 00:11:15,800 But what Bose did-- 229 00:11:15,800 --> 00:11:19,160 he took advantage of electronics. 230 00:11:19,160 --> 00:11:22,520 He put an electronic signal processing box 231 00:11:22,520 --> 00:11:31,680 to put the inverse filter, so the box preprocessed the sound 232 00:11:31,680 --> 00:11:33,690 to boost the frequencies that he knew 233 00:11:33,690 --> 00:11:36,750 the speaker would attenuate. 234 00:11:36,750 --> 00:11:41,580 So this box was filled with op-amps, gains, buffers-- 235 00:11:41,580 --> 00:11:43,650 all the kinds of things you know about, 236 00:11:43,650 --> 00:11:45,390 which were then just barely available. 237 00:11:45,390 --> 00:11:50,400 That box cost $1,000 back then because that was very 238 00:11:50,400 --> 00:11:51,240 state-of-the-art. 239 00:11:51,240 --> 00:11:54,600 So the idea was that he switched the problem 240 00:11:54,600 --> 00:11:58,020 from being acoustics problem to an electronics problem-- 241 00:11:58,020 --> 00:11:58,770 very clever. 242 00:12:01,720 --> 00:12:07,750 Today we go one more step and we do the signal processing 243 00:12:07,750 --> 00:12:09,640 digitally. 244 00:12:09,640 --> 00:12:16,450 So any modern stereo system has a discrete time filter in it. 245 00:12:16,450 --> 00:12:19,680 The idea is that you take the signal that's coming off 246 00:12:19,680 --> 00:12:23,920 the source, like an MP3-- 247 00:12:23,920 --> 00:12:28,150 you take an MP3-type source, you run it through a digital 248 00:12:28,150 --> 00:12:28,940 filter-- 249 00:12:28,940 --> 00:12:33,740 this is a filter that works on numbers, not on volts-- 250 00:12:33,740 --> 00:12:39,580 and so you convert the source from the MP3 251 00:12:39,580 --> 00:12:41,680 into a stream of numbers that gets 252 00:12:41,680 --> 00:12:45,485 crunched digitally, giving rise to a different stream 253 00:12:45,485 --> 00:12:47,110 of numbers, which you then convert back 254 00:12:47,110 --> 00:12:48,860 into an analog signal. 255 00:12:48,860 --> 00:12:53,960 This is the approach you find in every modern stereo system. 256 00:12:53,960 --> 00:12:56,620 The reason being that it works so well. 257 00:12:56,620 --> 00:13:00,940 Here's a chip, the Texas Instruments TAS3004, 258 00:13:00,940 --> 00:13:04,510 which was specifically designed for car stereos. 259 00:13:04,510 --> 00:13:06,340 It's got amazing specs-- 260 00:13:06,340 --> 00:13:09,220 it has two channels, state-of-the-art converting 261 00:13:09,220 --> 00:13:13,870 rates, state-of-the-art digitisation. 262 00:13:13,870 --> 00:13:16,870 It has a processor internally that 263 00:13:16,870 --> 00:13:20,920 runs at 100 mega instructions per second. 264 00:13:20,920 --> 00:13:24,160 It implements all the common things-- treble, volume, 265 00:13:24,160 --> 00:13:27,190 loudness, everything you can imagine-- 266 00:13:27,190 --> 00:13:33,130 and it costs $9.63 off Digi-Key in units of one. 267 00:13:33,130 --> 00:13:38,425 And if you buy units of 500, they are $5.20 apiece-- 268 00:13:38,425 --> 00:13:42,400 a trivial amount of money by any standard. 269 00:13:42,400 --> 00:13:46,570 So the idea then is that it's just very effective 270 00:13:46,570 --> 00:13:49,052 to do number-crunching digitally. 271 00:13:49,052 --> 00:13:51,010 So that's where we want to talk about-- we want 272 00:13:51,010 --> 00:13:54,490 to talk about the same kinds of Fourier transforms 273 00:13:54,490 --> 00:13:58,330 and so forth that we've talked about with CT signals, 274 00:13:58,330 --> 00:14:02,320 but we want to know the new set of rules 275 00:14:02,320 --> 00:14:05,846 that apply when the signals are discrete in time. 276 00:14:05,846 --> 00:14:07,060 OK, everybody with me? 277 00:14:09,950 --> 00:14:15,290 So this is just a little bit of review. 278 00:14:15,290 --> 00:14:17,480 You'll remember that we started with discrete time, 279 00:14:17,480 --> 00:14:20,030 we did examples in discrete time when 280 00:14:20,030 --> 00:14:24,254 we were thinking about feedback, then we went on to the CT 281 00:14:24,254 --> 00:14:25,670 and thought about feedback systems 282 00:14:25,670 --> 00:14:28,310 and so forth, then we did CT Fourier 283 00:14:28,310 --> 00:14:32,600 because I think the Fourier stuff is easier to see in CT. 284 00:14:32,600 --> 00:14:34,640 Now we're folding back to do DT. 285 00:14:34,640 --> 00:14:39,080 So it's been about six weeks since we talked about DT, 286 00:14:39,080 --> 00:14:42,170 so a little bit of a refresher. 287 00:14:42,170 --> 00:14:44,690 We're going to be thinking about frequency responses, right? 288 00:14:44,690 --> 00:14:47,579 That was the basis of the Fourier transform, that's 289 00:14:47,579 --> 00:14:49,370 going to be the basis of the Fourier signal 290 00:14:49,370 --> 00:14:51,660 processing in discrete time. 291 00:14:51,660 --> 00:14:53,990 And the idea of frequency analysis 292 00:14:53,990 --> 00:14:57,080 starts with the idea of an eigenfunction. 293 00:14:57,080 --> 00:14:59,930 The eigenfunctions for a discrete time system 294 00:14:59,930 --> 00:15:01,700 are very similar to the eigenfunctions 295 00:15:01,700 --> 00:15:05,450 for a continuous time, but they're different, right? 296 00:15:05,450 --> 00:15:08,870 The eigenfunctions for a CT system 297 00:15:08,870 --> 00:15:14,050 are complex exponentials, e to the st. Nod your head-- 298 00:15:14,050 --> 00:15:17,090 it'll make me feel good to think that you can remember that we 299 00:15:17,090 --> 00:15:19,430 did e to the st at one point. 300 00:15:19,430 --> 00:15:21,890 So the eigenfunctions for CT systems 301 00:15:21,890 --> 00:15:25,880 are complex exponentials, e to the st. Eigenfunctions for DT 302 00:15:25,880 --> 00:15:29,870 systems are complex geometrics. 303 00:15:29,870 --> 00:15:32,600 Complex geometrics in DT play the same role 304 00:15:32,600 --> 00:15:35,990 as complex exponentials in CT, and they play that role 305 00:15:35,990 --> 00:15:39,000 for exactly the same reason. 306 00:15:39,000 --> 00:15:42,200 So when we wanted to prove that the complex exponentials were 307 00:15:42,200 --> 00:15:45,620 eigenfunctions of LTI systems, we took the complex exponential 308 00:15:45,620 --> 00:15:50,600 and we convolved it with the impulse response of an LTI 309 00:15:50,600 --> 00:15:51,680 system. 310 00:15:51,680 --> 00:15:54,200 All LTI systems can be represented by their impulse 311 00:15:54,200 --> 00:15:56,910 response. 312 00:15:56,910 --> 00:16:00,570 If you convolve any real-valued function 313 00:16:00,570 --> 00:16:03,590 with a complex exponential, you get back 314 00:16:03,590 --> 00:16:06,550 a complex exponential of the same shape, 315 00:16:06,550 --> 00:16:09,540 but possibly different complex amplitude. 316 00:16:09,540 --> 00:16:11,910 The same thing happens in DT. 317 00:16:11,910 --> 00:16:13,980 So in order to see that that's true, 318 00:16:13,980 --> 00:16:17,350 we think about the input to an LTI system being 319 00:16:17,350 --> 00:16:19,350 z to the n-- complex exponentials, 320 00:16:19,350 --> 00:16:22,290 so z is some complex number, z to the n 321 00:16:22,290 --> 00:16:26,520 is the evolution of the complex geometric 322 00:16:26,520 --> 00:16:29,750 as a function of time. 323 00:16:29,750 --> 00:16:33,680 We let the system be represented by the unit sample 324 00:16:33,680 --> 00:16:38,960 response, which is analogous to the unit impulse response. 325 00:16:38,960 --> 00:16:42,620 And we simply convolve the unit sample response 326 00:16:42,620 --> 00:16:46,640 with the complex geometric, and if you 327 00:16:46,640 --> 00:16:48,970 go through the convolution's thumb-- 328 00:16:48,970 --> 00:16:51,130 it's a thumb now because it's DT-- 329 00:16:51,130 --> 00:16:57,500 you find that the output has the same base, z, 330 00:16:57,500 --> 00:17:00,830 that the input did but now the amplitude 331 00:17:00,830 --> 00:17:03,710 has been modified by the value of the system 332 00:17:03,710 --> 00:17:09,880 function evaluated at the location, z, OK? 333 00:17:09,880 --> 00:17:11,460 We did this before-- this is review. 334 00:17:14,190 --> 00:17:19,200 Similarly, if you think about representing a discrete time 335 00:17:19,200 --> 00:17:22,589 system in terms of a linear difference 336 00:17:22,589 --> 00:17:24,540 equation with constant co-efficients, 337 00:17:24,540 --> 00:17:27,450 you can see that type of system can always 338 00:17:27,450 --> 00:17:31,360 be represented by the ratio of polynomials in z. 339 00:17:34,040 --> 00:17:35,620 The same as with CT-- 340 00:17:35,620 --> 00:17:38,470 we had the ratio of polynomials in s. 341 00:17:38,470 --> 00:17:41,740 Here it's the polynomials in z. 342 00:17:41,740 --> 00:17:46,450 And if we want to evaluate that h of z thing, the system 343 00:17:46,450 --> 00:17:48,460 function, there's a very easy way 344 00:17:48,460 --> 00:17:50,100 to do it since it's a polynomial. 345 00:17:50,100 --> 00:17:52,600 There's nothing different about polynomials in z 346 00:17:52,600 --> 00:17:56,074 and polynomials in s in far as their being polynomials. 347 00:17:56,074 --> 00:17:57,490 The fact that they're a polynomial 348 00:17:57,490 --> 00:17:58,948 means you can factor the numerator, 349 00:17:58,948 --> 00:18:00,640 you can factor the denominator, you 350 00:18:00,640 --> 00:18:03,520 can think about poles and zeros, and you can evaluate the system 351 00:18:03,520 --> 00:18:05,170 function. 352 00:18:05,170 --> 00:18:07,690 After you've factored the numerator and the denominator, 353 00:18:07,690 --> 00:18:10,780 think about each contribution. 354 00:18:10,780 --> 00:18:13,360 So z0 minus q0-- 355 00:18:13,360 --> 00:18:16,420 so if I have a zero at q0 and I want 356 00:18:16,420 --> 00:18:20,560 to know how big is the eigenvalue at z0, 357 00:18:20,560 --> 00:18:22,330 I think about the arrow that connects 358 00:18:22,330 --> 00:18:25,789 the 0 to the point of interest. 359 00:18:25,789 --> 00:18:27,580 And I can say something about the magnitude 360 00:18:27,580 --> 00:18:31,410 by the length of the arrow and the phase 361 00:18:31,410 --> 00:18:33,420 by the angle of the arrow-- 362 00:18:33,420 --> 00:18:34,320 same as CT. 363 00:18:37,600 --> 00:18:41,590 And I get the same divide and conquer idea. 364 00:18:41,590 --> 00:18:44,860 If I know how one 0 works and if I know how one pole works, 365 00:18:44,860 --> 00:18:48,520 I can combine the responses to the numerous poles 366 00:18:48,520 --> 00:18:52,000 in a complicated system in order to figure out how 367 00:18:52,000 --> 00:18:53,570 the complicated system works. 368 00:18:53,570 --> 00:18:56,320 I can think about the poles and zeros one at a times, 369 00:18:56,320 --> 00:18:58,210 just like I did for CT. 370 00:18:58,210 --> 00:19:04,230 So the magnitude is determined by the product and division 371 00:19:04,230 --> 00:19:07,580 of all the magnitude factors and the angle 372 00:19:07,580 --> 00:19:12,210 is determined by the sum and difference of all the angle 373 00:19:12,210 --> 00:19:13,029 factors. 374 00:19:13,029 --> 00:19:14,820 OK, I'm going over this quickly, but that's 375 00:19:14,820 --> 00:19:16,590 because I'm expecting that you know this, 376 00:19:16,590 --> 00:19:17,798 you just need to be reminded. 377 00:19:22,660 --> 00:19:25,300 If we want to think about the frequency response, 378 00:19:25,300 --> 00:19:27,010 we do the same thing we did in CT. 379 00:19:27,010 --> 00:19:30,970 We like complex geometrics, so we 380 00:19:30,970 --> 00:19:35,620 try to think about the eternal cosine waves 381 00:19:35,620 --> 00:19:39,310 as being expressed in terms of complex geometrics. 382 00:19:39,310 --> 00:19:44,930 The difference is that now we think about this e 383 00:19:44,930 --> 00:19:46,890 to the j omega 0 end-term. 384 00:19:52,690 --> 00:19:59,074 So we had the same form-- we had e to the j omega 0 t. 385 00:19:59,074 --> 00:20:00,990 We thought about that as a complex exponential 386 00:20:00,990 --> 00:20:03,180 with a frequency omega 0. 387 00:20:03,180 --> 00:20:09,660 Now we're thinking about e to the j omega 0 n as z 388 00:20:09,660 --> 00:20:12,690 being this e to the j omega 0 thing, OK? 389 00:20:12,690 --> 00:20:15,540 It's the same thing-- it's still Euler's equation, 390 00:20:15,540 --> 00:20:17,680 we're just parsing it differently. 391 00:20:17,680 --> 00:20:20,370 We used to think of it as a complex exponential, now 392 00:20:20,370 --> 00:20:22,740 we think of it as a complex geometric-- 393 00:20:22,740 --> 00:20:25,300 that's the only difference. 394 00:20:25,300 --> 00:20:27,225 So now what we need to do if we're 395 00:20:27,225 --> 00:20:30,240 thinking about the response to the cosine of omega 0 n, 396 00:20:30,240 --> 00:20:33,290 we have to think about adding two complex exponentials, one 397 00:20:33,290 --> 00:20:36,780 at z0 and one at z1, where z0 is e to the j omega 398 00:20:36,780 --> 00:20:41,480 0 an z1 is e to the minus j omega 0, OK? 399 00:20:45,310 --> 00:20:52,570 Then the response, since the z0 to the n is an eigenfunction, 400 00:20:52,570 --> 00:20:55,090 the answer is just that same eigenfunction 401 00:20:55,090 --> 00:20:58,780 premultiplied by the system function evaluated at z0 402 00:20:58,780 --> 00:21:01,470 and similarly here for z1. 403 00:21:01,470 --> 00:21:04,300 So we get an expression here that looks for all the world, 404 00:21:04,300 --> 00:21:10,840 except that this is the base of a geometric sequence. 405 00:21:10,840 --> 00:21:15,790 Except for that, it looks just the same as the CT example 406 00:21:15,790 --> 00:21:18,220 from before. 407 00:21:18,220 --> 00:21:21,190 Just like CT, the system function 408 00:21:21,190 --> 00:21:23,290 has conjugate symmetry and, in fact, 409 00:21:23,290 --> 00:21:26,260 that's for the same reason too. 410 00:21:26,260 --> 00:21:28,190 So you can see that there's complex symmetry 411 00:21:28,190 --> 00:21:31,810 just by thinking about the expansion in terms 412 00:21:31,810 --> 00:21:33,760 of the z transform. 413 00:21:33,760 --> 00:21:39,700 And if you think about the fact that the h of n's are all real, 414 00:21:39,700 --> 00:21:41,890 taking the complex conjugate of this side 415 00:21:41,890 --> 00:21:44,530 is the same as taking the complex conjugate of that. 416 00:21:44,530 --> 00:21:46,270 The complex conjugate of a sum is the sum 417 00:21:46,270 --> 00:21:47,380 of the complex conjugates. 418 00:21:47,380 --> 00:21:49,970 The only thing that's imaginary is that. 419 00:21:49,970 --> 00:21:53,430 So the only thing that happens is that the sign changes. 420 00:21:53,430 --> 00:21:59,240 So it's those complex conjugates' relationship in DT, 421 00:21:59,240 --> 00:22:02,180 just the same as there is in CT for exactly the same reason. 422 00:22:05,570 --> 00:22:07,620 And finally, we get our final answer, 423 00:22:07,620 --> 00:22:11,360 which is that if we want to know about the frequency response, 424 00:22:11,360 --> 00:22:15,410 we therefore think about what was the magnitude and angle 425 00:22:15,410 --> 00:22:17,120 of the system function. 426 00:22:17,120 --> 00:22:19,220 We only need to look at one of the frequencies 427 00:22:19,220 --> 00:22:20,930 since we know the frequency response has 428 00:22:20,930 --> 00:22:23,870 conjugate symmetry, and we can figure out 429 00:22:23,870 --> 00:22:27,380 the response to cos omega n. 430 00:22:27,380 --> 00:22:32,120 It's still cos omega n, except the magnitude 431 00:22:32,120 --> 00:22:34,370 is different by the magnitude of the system function 432 00:22:34,370 --> 00:22:37,934 and phase delayed by the angle of the system function-- 433 00:22:37,934 --> 00:22:39,350 everything looks exactly the same. 434 00:22:42,020 --> 00:22:44,660 The one thing that's different-- now we're getting something 435 00:22:44,660 --> 00:22:46,520 a little more interesting-- 436 00:22:46,520 --> 00:22:48,860 the one thing that's different is 437 00:22:48,860 --> 00:22:51,680 that we evaluate the system function 438 00:22:51,680 --> 00:22:54,279 when we're interested to find the frequency response. 439 00:22:54,279 --> 00:22:56,570 The thing we need to do is evaluate the system function 440 00:22:56,570 --> 00:22:58,910 on the unit circle. 441 00:22:58,910 --> 00:23:02,660 If we wanted to know the frequency response for a CT 442 00:23:02,660 --> 00:23:10,714 system, the frequency response for a CT system lives where? 443 00:23:10,714 --> 00:23:11,690 AUDIENCE: [INAUDIBLE] 444 00:23:11,690 --> 00:23:15,020 DENNIS FREEMAN: J omega-axis, right? 445 00:23:15,020 --> 00:23:18,810 Here, the frequency response lives on the unit circle. 446 00:23:18,810 --> 00:23:23,790 That's the only difference and that 447 00:23:23,790 --> 00:23:27,630 means that the vector diagrams work precisely 448 00:23:27,630 --> 00:23:31,680 the same way, except that we don't connect them up 449 00:23:31,680 --> 00:23:32,730 to the j omega-axis. 450 00:23:32,730 --> 00:23:36,560 We connect them up to the unit circle. 451 00:23:36,560 --> 00:23:37,370 So where is DC? 452 00:23:40,325 --> 00:23:43,170 AUDIENCE: [INAUDIBLE] 453 00:23:43,170 --> 00:23:49,770 DENNIS FREEMAN: DC means cos omega n where omega is 0. 454 00:23:49,770 --> 00:23:50,625 Where is that? 455 00:23:54,030 --> 00:23:57,095 Where was DC in CT? 456 00:23:57,095 --> 00:23:58,970 Frequency response lives on the j omega-axis. 457 00:23:58,970 --> 00:23:59,967 Where is DC? 458 00:23:59,967 --> 00:24:01,220 AUDIENCE: 0. 459 00:24:01,220 --> 00:24:02,200 DENNIS FREEMAN: 0. 460 00:24:02,200 --> 00:24:04,255 Where is DC in a DT system? 461 00:24:04,255 --> 00:24:05,470 AUDIENCE: 1. 462 00:24:05,470 --> 00:24:07,234 DENNIS FREEMAN: 1, OK? 463 00:24:07,234 --> 00:24:08,900 It's because we've shifted from thinking 464 00:24:08,900 --> 00:24:10,670 about complex exponentials to thinking 465 00:24:10,670 --> 00:24:13,700 about complex geometrics. 466 00:24:13,700 --> 00:24:19,580 So now DC lives on the unit circle at the place 467 00:24:19,580 --> 00:24:22,910 where the angle is 0, right? 468 00:24:22,910 --> 00:24:26,330 So now we have to think about things of this form. 469 00:24:26,330 --> 00:24:29,210 If we want omega 0 to be 0, we need 470 00:24:29,210 --> 00:24:35,460 to have this number be e to the 0, which is 1. 471 00:24:35,460 --> 00:24:37,180 So the DC response-- 472 00:24:37,180 --> 00:24:38,204 the thing that we plot-- 473 00:24:38,204 --> 00:24:40,370 we're going to plot frequency response just the same 474 00:24:40,370 --> 00:24:42,560 as we did in CT. 475 00:24:42,560 --> 00:24:45,350 They're going to have the same trick they had in CT. 476 00:24:45,350 --> 00:24:50,420 In CT, the system function has a non-zero answer 477 00:24:50,420 --> 00:24:54,410 throughout the entire s-plane. 478 00:24:54,410 --> 00:24:58,220 In DT, the system function has a non-zero response 479 00:24:58,220 --> 00:24:59,240 in the entire z-plane. 480 00:25:01,759 --> 00:25:04,050 The frequency response in CT lives on the j omega-axis, 481 00:25:04,050 --> 00:25:08,070 the frequency response in DT lives on the unit circle, 482 00:25:08,070 --> 00:25:10,390 right? 483 00:25:10,390 --> 00:25:12,590 But the frequency response itself 484 00:25:12,590 --> 00:25:15,800 is simply a plot of the magnitude and angle 485 00:25:15,800 --> 00:25:18,080 as a function of omega. 486 00:25:18,080 --> 00:25:21,530 I should mention that I sneakily use 487 00:25:21,530 --> 00:25:24,230 a different symbol for omega. 488 00:25:24,230 --> 00:25:30,050 I like to use a little omega for a CT and capital Omega for DT. 489 00:25:30,050 --> 00:25:32,540 The reason I like to do that is that the dimensions 490 00:25:32,540 --> 00:25:34,500 are different, right? 491 00:25:34,500 --> 00:25:36,390 When we think about little omega 0, 492 00:25:36,390 --> 00:25:39,222 we're thinking about cosine of omega 0 t. 493 00:25:39,222 --> 00:25:40,680 When we think about this one, we're 494 00:25:40,680 --> 00:25:45,394 thinking about cosine of omega 0 n. 495 00:25:45,394 --> 00:25:47,310 Can somebody tell me something different about 496 00:25:47,310 --> 00:25:49,000 little and capital Omega 0? 497 00:25:52,540 --> 00:25:53,710 What's different about them? 498 00:25:53,710 --> 00:25:56,100 Anything? 499 00:25:56,100 --> 00:25:56,703 Yes? 500 00:25:56,703 --> 00:25:58,870 AUDIENCE: The little one goes to infinity. 501 00:25:58,870 --> 00:26:00,328 DENNIS FREEMAN: The little one goes 502 00:26:00,328 --> 00:26:02,017 to infinity, that's correct. 503 00:26:02,017 --> 00:26:03,717 Yes? 504 00:26:03,717 --> 00:26:05,300 AUDIENCE: The little one [INAUDIBLE].. 505 00:26:08,120 --> 00:26:12,580 DENNIS FREEMAN: So you can think about omega 0-- 506 00:26:12,580 --> 00:26:19,670 omega 0 has dimensions like radian per second, 507 00:26:19,670 --> 00:26:26,630 where omega 0 has the dimensions of radians, right? 508 00:26:26,630 --> 00:26:28,860 So the dimension will be completely different-- 509 00:26:28,860 --> 00:26:31,550 that will be very important when we think about systems. 510 00:26:31,550 --> 00:26:33,700 So the last two weeks of the course, 511 00:26:33,700 --> 00:26:38,245 we'll think about systems that convert CT to DT and back. 512 00:26:38,245 --> 00:26:39,620 When we're doing that conversion, 513 00:26:39,620 --> 00:26:41,150 it's very important to keep track 514 00:26:41,150 --> 00:26:45,320 of how the frequencies change and this dimensional difference 515 00:26:45,320 --> 00:26:48,980 will be very handy to help us remember how to convert one 516 00:26:48,980 --> 00:26:50,760 into the other. 517 00:26:50,760 --> 00:26:52,820 So that's the reason for keeping them separate-- 518 00:26:52,820 --> 00:26:54,320 so they're dimensionally distinct. 519 00:26:54,320 --> 00:26:56,570 So all we need to do then to think about the frequency 520 00:26:56,570 --> 00:26:58,910 response-- in CT, the frequency response 521 00:26:58,910 --> 00:27:03,260 would be the magnitude and angle plotted versus little omega 0. 522 00:27:03,260 --> 00:27:07,650 All we do that's different is we plot against capital Omega 0. 523 00:27:07,650 --> 00:27:14,250 Capital Omega 0 specifies the angle on the unit circle. 524 00:27:14,250 --> 00:27:18,260 So if the angle is 0, we pick out 0 frequency. 525 00:27:18,260 --> 00:27:20,950 We think about that as having two pieces-- 526 00:27:20,950 --> 00:27:22,910 there's the piece associated with the pole, 527 00:27:22,910 --> 00:27:25,410 there's the piece that's associated with the 0. 528 00:27:25,410 --> 00:27:27,770 The 0 is in the top, so we take the length 529 00:27:27,770 --> 00:27:29,837 of the top divided by the length in the bottom, 530 00:27:29,837 --> 00:27:32,420 and that tells us the magnitude is 0, which is a number bigger 531 00:27:32,420 --> 00:27:34,290 than 1, right? 532 00:27:34,290 --> 00:27:36,630 Because the pole's in the bottom-- 533 00:27:36,630 --> 00:27:40,380 pole's the short one, pole's in the bottom. 534 00:27:40,380 --> 00:27:44,160 And as you increase capital Omega, 535 00:27:44,160 --> 00:27:47,680 you sweep out the frequency response. 536 00:27:47,680 --> 00:27:52,090 So by the time you get over to pi, 537 00:27:52,090 --> 00:27:55,184 the length of the vector associated with the pole 538 00:27:55,184 --> 00:27:57,600 is bigger than the length of the vector associated with 0, 539 00:27:57,600 --> 00:28:00,240 so now you're less than 1, OK? 540 00:28:00,240 --> 00:28:02,037 So everything's exactly the same as CT 541 00:28:02,037 --> 00:28:03,870 except that we're looking at the unit circle 542 00:28:03,870 --> 00:28:05,580 rather than looking at the j omega-axis. 543 00:28:08,890 --> 00:28:11,640 The same sort of thing happens for negative frequency, 544 00:28:11,640 --> 00:28:14,140 just like it did in CT, the only difference 545 00:28:14,140 --> 00:28:23,650 being that in CT, we were computing the frequency axis 546 00:28:23,650 --> 00:28:26,110 by looking at the j omega-axis. 547 00:28:26,110 --> 00:28:28,570 In DT, we're computing the frequency axis 548 00:28:28,570 --> 00:28:32,330 by looking at the unit circle. 549 00:28:32,330 --> 00:28:35,000 There is one huge difference. 550 00:28:35,000 --> 00:28:39,710 OK, so summary, everything's the same 551 00:28:39,710 --> 00:28:43,090 except we look at the unit circle-- 552 00:28:43,090 --> 00:28:46,960 except frequency responses are now periodic. 553 00:28:46,960 --> 00:28:49,330 Someone previously said that one of the differences 554 00:28:49,330 --> 00:28:52,060 between little omega big Omega is that little omega goes 555 00:28:52,060 --> 00:28:52,700 to infinity. 556 00:28:52,700 --> 00:28:54,340 That's absolutely true. 557 00:28:54,340 --> 00:28:56,788 What happens when big Omega goes to infinity? 558 00:28:59,600 --> 00:29:01,480 Well, angles wrap. 559 00:29:01,480 --> 00:29:04,030 So if you think about this diagram-- 560 00:29:04,030 --> 00:29:06,130 if you start with capital Omega being 561 00:29:06,130 --> 00:29:09,070 0, by the time you get over here capital Omega is pi. 562 00:29:09,070 --> 00:29:11,680 If you keep increasing, you'll come back here-- 563 00:29:11,680 --> 00:29:14,870 if you keep increasing, you go again. 564 00:29:14,870 --> 00:29:17,830 The big difference between CT and DT 565 00:29:17,830 --> 00:29:22,360 is that the DT complex exponentials 566 00:29:22,360 --> 00:29:26,380 are periodic in 2 pi-- 567 00:29:26,380 --> 00:29:28,990 every 2 pi, they repeat themselves. 568 00:29:28,990 --> 00:29:36,890 Because of that DT system functions repeat themselves. 569 00:29:36,890 --> 00:29:38,980 So if you think about evaluating the system 570 00:29:38,980 --> 00:29:41,200 function at some frequency omega 2, 571 00:29:41,200 --> 00:29:46,560 which happens to be omega 1 plus 2 pi k, 572 00:29:46,560 --> 00:29:49,500 you get the same thing as if you had evaluated the system 573 00:29:49,500 --> 00:29:56,250 function and omega 1 because the frequency e to the j omega 574 00:29:56,250 --> 00:30:00,210 2 is the same as the frequency e to the j omega 575 00:30:00,210 --> 00:30:04,400 1 because of the periodicity of angles. 576 00:30:04,400 --> 00:30:07,710 Is that clear? 577 00:30:07,710 --> 00:30:12,690 So we get something that's very different in DT. 578 00:30:12,690 --> 00:30:15,300 In DT, the frequency response we only 579 00:30:15,300 --> 00:30:18,955 ever plotted up to pi because above pi it repeats itself. 580 00:30:21,530 --> 00:30:23,370 By plotting it between minus pi and pi, 581 00:30:23,370 --> 00:30:27,320 I've showed a two pi range and the frequency response 582 00:30:27,320 --> 00:30:31,550 is always periodic in 2 pi because discrete frequencies 583 00:30:31,550 --> 00:30:32,990 are always periodic in 2 pi. 584 00:30:32,990 --> 00:30:33,694 Yes? 585 00:30:33,694 --> 00:30:35,444 AUDIENCE: How does that affect the filters 586 00:30:35,444 --> 00:30:37,920 if the number repeats itself? 587 00:30:37,920 --> 00:30:39,980 DENNIS FREEMAN: How does that affect the filters? 588 00:30:39,980 --> 00:30:44,130 That's an excellent question and not a trivial one to answer. 589 00:30:44,130 --> 00:30:47,390 So one of the big important differences, 590 00:30:47,390 --> 00:30:50,390 which you will under-appreciate right now, 591 00:30:50,390 --> 00:30:52,370 is that because of this repetition, 592 00:30:52,370 --> 00:30:55,180 there's no equivalent thing to bode. 593 00:30:55,180 --> 00:31:01,120 Now I know you all love bode and because you like it so much-- 594 00:31:01,120 --> 00:31:03,580 I'm being slightly sarcastic. 595 00:31:03,580 --> 00:31:06,220 I like it a lot because it's easy, right? 596 00:31:06,220 --> 00:31:08,200 You haven't quite got over the barrier yet 597 00:31:08,200 --> 00:31:11,180 and so I've heard some negative feedback about bode. 598 00:31:11,180 --> 00:31:13,630 Bode's wonderful, but in case you haven't got 599 00:31:13,630 --> 00:31:16,840 that message, one of the biggest problems in DT-- 600 00:31:16,840 --> 00:31:19,810 which comes about exactly because of the thing you said-- 601 00:31:19,810 --> 00:31:22,700 that the DT frequencies are periodic. 602 00:31:22,700 --> 00:31:27,400 We don't have an analog of bode, which 603 00:31:27,400 --> 00:31:29,380 means that when we try to sketch the magnitude 604 00:31:29,380 --> 00:31:32,140 and phase for a discrete time filter, 605 00:31:32,140 --> 00:31:34,180 it's a much harder problem. 606 00:31:34,180 --> 00:31:36,220 One for which we almost always use a computer 607 00:31:36,220 --> 00:31:39,100 because we don't know any good rules to let us 608 00:31:39,100 --> 00:31:41,354 think about it in our heads. 609 00:31:41,354 --> 00:31:42,770 So that's one of the implications, 610 00:31:42,770 --> 00:31:44,770 but there's a lot of others and, in fact, that's 611 00:31:44,770 --> 00:31:48,690 one of the main themes for the rest of the course. 612 00:31:48,690 --> 00:31:53,219 So the important difference then is two important differences, 613 00:31:53,219 --> 00:31:55,010 frequency response lives on the unit circle 614 00:31:55,010 --> 00:31:59,030 and discrete frequencies are periodic. 615 00:31:59,030 --> 00:32:03,330 OK, I've talked way too much, so finally I'm 616 00:32:03,330 --> 00:32:06,740 going to ask you to do something. 617 00:32:06,740 --> 00:32:13,350 I want you to think about three CT frequencies-- 618 00:32:13,350 --> 00:32:16,980 cos 3,000t, cos 4,000t, cos 5,000t-- 619 00:32:16,980 --> 00:32:20,400 and think about sampling them with a sampling 620 00:32:20,400 --> 00:32:23,550 interval capital T who is at 0.001 621 00:32:23,550 --> 00:32:29,380 and put them in order from lowest to highest DT frequency. 622 00:32:29,380 --> 00:32:33,478 So first, look at your neighbor, say hello. 623 00:32:33,478 --> 00:32:34,394 AUDIENCE: Hello. 624 00:32:37,395 --> 00:32:39,770 DENNIS FREEMAN: And now choose an answer between 0 and 5. 625 00:35:33,370 --> 00:35:38,074 OK, so which list goes from lowest to highest DT frequency? 626 00:35:38,074 --> 00:35:39,490 So raise your hand with the number 627 00:35:39,490 --> 00:35:42,440 of fingers equal to the answer between 0 and 5. 628 00:35:45,680 --> 00:35:46,605 High, high. 629 00:35:49,210 --> 00:35:52,320 OK, so I see people disagreeing with their partner. 630 00:35:52,320 --> 00:35:55,410 I think that's always good. 631 00:35:55,410 --> 00:35:58,290 OK, the correct answer is in the minority. 632 00:36:01,510 --> 00:36:06,600 So let's ask, according to the theory of lectures, 633 00:36:06,600 --> 00:36:07,600 what's the right answer? 634 00:36:12,960 --> 00:36:17,520 If you were the lecturer asking this question, 635 00:36:17,520 --> 00:36:20,328 what would you make the answer be? 636 00:36:20,328 --> 00:36:22,817 AUDIENCE: Probably none of them. 637 00:36:22,817 --> 00:36:25,150 DENNIS FREEMAN: None of them, except I don't have none-- 638 00:36:25,150 --> 00:36:29,570 but that would be the kind of thing I would do, I guess. 639 00:36:29,570 --> 00:36:31,610 If you were the lecturer and you wanted 640 00:36:31,610 --> 00:36:38,030 to make a point about x1, x2, x3, 3,000, 4,000, 5,000-- 641 00:36:38,030 --> 00:36:40,838 what would you want the answer to be? 642 00:36:40,838 --> 00:36:42,630 AUDIENCE: Number 5. 643 00:36:42,630 --> 00:36:47,600 DENNIS FREEMAN: Sure, you'd like to be number 5, completely 644 00:36:47,600 --> 00:36:49,420 backwards. 645 00:36:49,420 --> 00:36:51,830 OK, talk to your neighbor-- figure out 646 00:36:51,830 --> 00:36:53,947 why it's completely backwards. 647 00:36:53,947 --> 00:36:56,030 That is the right answer-- the theory of lecturers 648 00:36:56,030 --> 00:36:56,780 always works. 649 00:38:04,960 --> 00:38:07,340 So why is 5 the right answer? 650 00:38:13,470 --> 00:38:16,730 Is 5 the right answer? 651 00:38:16,730 --> 00:38:18,610 5 is the right answer-- 652 00:38:18,610 --> 00:38:19,780 why is 5 the right answer? 653 00:38:23,350 --> 00:38:26,500 OK, the corollary to the theory of lecturers-- 654 00:38:26,500 --> 00:38:27,890 always look at the last line. 655 00:38:27,890 --> 00:38:28,890 What was the last point? 656 00:38:31,818 --> 00:38:34,260 AUDIENCE: [INAUDIBLE] wrap [INAUDIBLE] 657 00:38:34,260 --> 00:38:37,040 DENNIS FREEMAN: Things wrap. 658 00:38:37,040 --> 00:38:40,520 So obviously these things must be wrapping, right, 659 00:38:40,520 --> 00:38:42,170 because according to the corollary 660 00:38:42,170 --> 00:38:45,560 to the theory of lecturers, it has to do with wrapping. 661 00:38:45,560 --> 00:38:49,146 According to the theory of lecturers, it's number 5. 662 00:38:49,146 --> 00:38:50,270 Why would it wrap that way? 663 00:38:53,070 --> 00:38:57,390 Well think about after you do the sampling, 664 00:38:57,390 --> 00:39:02,520 if you substitute capital T equals 0.001, 665 00:39:02,520 --> 00:39:04,820 you look up x1 of nT. 666 00:39:04,820 --> 00:39:09,420 So x1 of T is this, so substitute nT every place there 667 00:39:09,420 --> 00:39:14,220 was a T. So you get 3,000 nT. 668 00:39:14,220 --> 00:39:18,900 3,000 times capital T is 3. 669 00:39:18,900 --> 00:39:25,510 So x1 of n is cos of 3n, right, and that's written here. 670 00:39:25,510 --> 00:39:30,060 Similarly, the second one is 4n and the third one is 5n. 671 00:39:30,060 --> 00:39:38,210 So the discrete frequency, capital Omega, is 3, 4, or 5. 672 00:39:38,210 --> 00:39:45,320 Think about where 3, 4, or 5 are on the unit circle, OK? 673 00:39:45,320 --> 00:39:50,530 3 is a number just less than pi, so that puts 3 here. 674 00:39:54,800 --> 00:39:57,920 4 is a number that's bigger than pi, so you go around, 675 00:39:57,920 --> 00:40:01,370 you pass pi, and you go a little further. 676 00:40:01,370 --> 00:40:05,210 And 5 is a number that is slightly bigger than 3/4 677 00:40:05,210 --> 00:40:09,760 of 2 pi, OK? 678 00:40:09,760 --> 00:40:14,380 So by that logic, right, this is low frequencies-- 679 00:40:18,660 --> 00:40:21,150 positive and negatives are both necessary because 680 00:40:21,150 --> 00:40:22,560 of Euler's rule. 681 00:40:22,560 --> 00:40:25,020 In order to make a real value cosine function, 682 00:40:25,020 --> 00:40:27,900 right, we need e to the plus and we need e to the minus. 683 00:40:27,900 --> 00:40:29,790 So we start at 0 frequency. 684 00:40:29,790 --> 00:40:31,530 As we increase frequency, we're thinking 685 00:40:31,530 --> 00:40:34,600 about conjugate frequencies, like so. 686 00:40:34,600 --> 00:40:39,075 So this is the highest frequency, then they cross-- 687 00:40:42,140 --> 00:40:45,130 so which is the highest frequency among 3, 4, and 5? 688 00:40:47,830 --> 00:40:49,050 Oh, doesn't make any sense. 689 00:40:49,050 --> 00:40:50,966 According to the rules I just said, the answer 690 00:40:50,966 --> 00:40:52,830 must be 3, right? 691 00:40:52,830 --> 00:40:54,970 3 is closest to the high frequency, 692 00:40:54,970 --> 00:41:00,240 then the next closest is 4, and the lowest frequency must be 5. 693 00:41:00,240 --> 00:41:03,240 Why does that make any sense? 694 00:41:03,240 --> 00:41:05,910 The intuition comes from it if you think it about what's 695 00:41:05,910 --> 00:41:08,310 happening when you're sampling. 696 00:41:08,310 --> 00:41:13,390 Think about what would happen if capital Omega were a quarter. 697 00:41:13,390 --> 00:41:20,080 Then if we plotted cos omega t and compared the samples, 698 00:41:20,080 --> 00:41:28,020 cos 0.25n, you would see that at the discrete frequency, 699 00:41:28,020 --> 00:41:32,410 the samples are a very good representation of the unsampled 700 00:41:32,410 --> 00:41:34,100 signal. 701 00:41:34,100 --> 00:41:37,280 I'm trying to compare the CT signal to the DT sampled 702 00:41:37,280 --> 00:41:39,230 signal, OK? 703 00:41:39,230 --> 00:41:45,590 If you changed the discrete frequency-- if you double it-- 704 00:41:45,590 --> 00:41:47,390 you still get a good representation 705 00:41:47,390 --> 00:41:49,340 of the CT signal. 706 00:41:49,340 --> 00:41:52,940 If you double it again, you can still see the CT signal. 707 00:41:52,940 --> 00:41:55,450 I'm up to 1-- 708 00:41:55,450 --> 00:41:59,330 double it to 2, you can still sort of see it. 709 00:41:59,330 --> 00:42:00,770 Increase it to 3-- 710 00:42:00,770 --> 00:42:02,760 OK, now it's getting a little harder to see. 711 00:42:06,440 --> 00:42:07,840 Go a little higher to 4-- 712 00:42:12,160 --> 00:42:14,320 in this plot, the red line is showing 713 00:42:14,320 --> 00:42:17,710 the CT signal that was sampled. 714 00:42:17,710 --> 00:42:20,440 Here, the red curve is still showing the CT signal that 715 00:42:20,440 --> 00:42:24,970 was sampled, but the green curve is showing an alternative way 716 00:42:24,970 --> 00:42:27,340 to think about that set of samples. 717 00:42:27,340 --> 00:42:29,710 That set of samples could alternatively 718 00:42:29,710 --> 00:42:32,030 have come from the frequency 2 pi minus 4n. 719 00:42:34,680 --> 00:42:37,590 If you go to an even higher frequency-- 720 00:42:37,590 --> 00:42:41,530 if you go to 5, the problem is even worse. 721 00:42:41,530 --> 00:42:43,960 There's an alternative way of thinking about it 722 00:42:43,960 --> 00:42:47,100 that's a much lower frequency. 723 00:42:47,100 --> 00:42:52,260 If you go to 6, it's absurd to think about 6 724 00:42:52,260 --> 00:42:54,630 as being the frequency, right? 725 00:42:54,630 --> 00:42:57,260 The thing that your eyes sees is this low frequency thing. 726 00:42:57,260 --> 00:43:01,391 You would never guess that I had sampled the high frequency 727 00:43:01,391 --> 00:43:01,890 thing. 728 00:43:01,890 --> 00:43:05,780 That's what we mean by the frequencies wrapping. 729 00:43:05,780 --> 00:43:08,690 There are two different waveforms 730 00:43:08,690 --> 00:43:12,050 that you can sample and get the same blue circles-- 731 00:43:12,050 --> 00:43:13,340 the same samples. 732 00:43:13,340 --> 00:43:19,460 You could sample of 6n or 2 pi minus 6n 733 00:43:19,460 --> 00:43:21,170 and you would get the same thing. 734 00:43:21,170 --> 00:43:27,020 That's the reason we say that the discrete frequencies wrap, 735 00:43:27,020 --> 00:43:27,860 OK? 736 00:43:27,860 --> 00:43:30,900 So that's the reasoning behind thinking about this. 737 00:43:30,900 --> 00:43:35,480 So there's a way of thinking about 5 as this whole distance 738 00:43:35,480 --> 00:43:38,240 around like that, but there's another way 739 00:43:38,240 --> 00:43:41,180 of thinking about it as just the negative frequency coming 740 00:43:41,180 --> 00:43:43,770 this way. 741 00:43:43,770 --> 00:43:46,236 And since they come in pairs, regardless 742 00:43:46,236 --> 00:43:47,610 of the way I think about it, I've 743 00:43:47,610 --> 00:43:51,240 always got the pair there to make sense out 744 00:43:51,240 --> 00:43:54,010 of either interpretation. 745 00:43:54,010 --> 00:43:56,750 That's the hardest part in DT-- it's not hard, 746 00:43:56,750 --> 00:43:58,690 it's just the hardest part, OK? 747 00:43:58,690 --> 00:44:01,450 If you get that, that's the only trick in DT. 748 00:44:04,210 --> 00:44:06,826 So the answer then is number 5. 749 00:44:06,826 --> 00:44:08,200 So what kind of a filter is this? 750 00:44:12,102 --> 00:44:14,060 You remember, we want to have an intuition just 751 00:44:14,060 --> 00:44:16,670 like if we look at CT, right? 752 00:44:16,670 --> 00:44:22,880 If I told you that I had a CT and I looked at the s-plane, 753 00:44:22,880 --> 00:44:24,800 and I told you that there was a pole there-- 754 00:44:24,800 --> 00:44:26,320 what kind of a system is that? 755 00:44:26,320 --> 00:44:28,660 High-pass, low-pass, band-pass, stop-band. 756 00:44:32,060 --> 00:44:34,030 High-pass, low-pass, band-pass, what? 757 00:44:34,030 --> 00:44:37,380 CT system, single pole. 758 00:44:37,380 --> 00:44:41,080 Low pass-- it's a low pass because of bode, right? 759 00:44:41,080 --> 00:44:42,850 Right, you like bode, right? 760 00:44:42,850 --> 00:44:46,870 It's because if we think about frequencies near 0, 761 00:44:46,870 --> 00:44:49,120 we get one gain and if we think about frequencies 762 00:44:49,120 --> 00:44:54,130 becoming very large, the gain falls off linearly after that. 763 00:44:54,130 --> 00:44:56,470 What kind of a filter is this one? 764 00:45:00,630 --> 00:45:04,010 1, 2, 3, 4, 5? 765 00:45:04,010 --> 00:45:07,290 Yes, it's a high-pass filter, right? 766 00:45:07,290 --> 00:45:10,590 It's a high-pass filter because this pole 767 00:45:10,590 --> 00:45:13,700 is in the denominator, this vector is large 768 00:45:13,700 --> 00:45:15,327 when we're at DC, and it's short where 769 00:45:15,327 --> 00:45:17,285 we're into high frequencies in the denominator, 770 00:45:17,285 --> 00:45:20,610 so that makes the filter be big for high frequencies compared 771 00:45:20,610 --> 00:45:24,600 to low frequencies, OK? 772 00:45:24,600 --> 00:45:28,820 So that's the only thing that's different about DT. 773 00:45:28,820 --> 00:45:31,230 OK, in the last 10 minutes what I want to do 774 00:45:31,230 --> 00:45:33,954 is introduce the idea of a Fourier series 775 00:45:33,954 --> 00:45:36,120 because that's where we're going with all this junk. 776 00:45:36,120 --> 00:45:39,540 We want to understand DT frequency responses so that we 777 00:45:39,540 --> 00:45:42,340 can generate a Fourier series. 778 00:45:42,340 --> 00:45:47,010 So the idea is exactly the same as what we do in CT. 779 00:45:47,010 --> 00:45:55,000 We want to represent a DT signal as a sum of complex geometrics, 780 00:45:55,000 --> 00:45:55,500 right? 781 00:45:55,500 --> 00:45:59,370 We want to think of it as a sum of frequencies and the trick 782 00:45:59,370 --> 00:46:03,300 is to find out what that sum is. 783 00:46:03,300 --> 00:46:08,630 The thing that's different is that frequencies wrap. 784 00:46:08,630 --> 00:46:11,940 Because the frequencies wrap-- 785 00:46:11,940 --> 00:46:13,620 in CT the frequencies don't wrap. 786 00:46:13,620 --> 00:46:15,910 In CT, how many harmonics do we have to think about 787 00:46:15,910 --> 00:46:18,690 for an arbitrary CT signal? 788 00:46:18,690 --> 00:46:20,370 Infinite, right? 789 00:46:20,370 --> 00:46:25,092 When we think about a CT Fourier series, 790 00:46:25,092 --> 00:46:27,300 you think about the fundamental, the second harmonic, 791 00:46:27,300 --> 00:46:29,924 the third harmonic, the fourth, fifth, sixth, seventh they find 792 00:46:29,924 --> 00:46:31,530 10th, through infinity. 793 00:46:31,530 --> 00:46:34,180 In principle, they all matter. 794 00:46:34,180 --> 00:46:37,890 In DT, because of the wrapping, there 795 00:46:37,890 --> 00:46:40,610 is a finite number of them. 796 00:46:40,610 --> 00:46:43,610 It's pretty easy to see that-- 797 00:46:43,610 --> 00:46:49,280 choose any frequency, assume that that's periodic in capital 798 00:46:49,280 --> 00:46:49,780 N-- 799 00:46:49,780 --> 00:46:52,550 I'm just choosing the period to be capital N just like we would 800 00:46:52,550 --> 00:46:55,460 say the [INAUDIBLE] NCT is capital T. 801 00:46:55,460 --> 00:46:59,550 If the signal is periodic in capital N-- 802 00:46:59,550 --> 00:47:05,420 if it was originally a complex geometric, then-- 803 00:47:05,420 --> 00:47:07,880 because of the factoring properties of the complex 804 00:47:07,880 --> 00:47:09,320 geometric-- 805 00:47:09,320 --> 00:47:11,870 it must be the case that this number, 806 00:47:11,870 --> 00:47:14,315 in order to make it equal to that-- this number must be 1. 807 00:47:17,310 --> 00:47:18,900 The only way that number could be 1 808 00:47:18,900 --> 00:47:24,556 would be if capital Omega is a submultiple of 2 pi. 809 00:47:24,556 --> 00:47:25,930 That means that if we're thinking 810 00:47:25,930 --> 00:47:29,260 about periodic signals in 8, there 811 00:47:29,260 --> 00:47:31,360 are exactly eight frequencies we need 812 00:47:31,360 --> 00:47:34,920 to worry about and no more. 813 00:47:34,920 --> 00:47:37,270 The ninth one aliases back to the first, 814 00:47:37,270 --> 00:47:40,580 the 10th one aliases back to the second, et cetera. 815 00:47:40,580 --> 00:47:43,210 There is only eight of them. 816 00:47:43,210 --> 00:47:45,960 That's the big difference between CT and DT. 817 00:47:45,960 --> 00:47:51,400 In DT, if we were thinking about sequences of length 3, 818 00:47:51,400 --> 00:47:55,390 there would be three frequencies we'd have to worry about. 819 00:47:55,390 --> 00:48:00,070 If we were thinking of sequences of length 4, there's 4. 820 00:48:00,070 --> 00:48:03,250 The number of frequencies and the number of samples matches. 821 00:48:03,250 --> 00:48:05,389 In fact, that's not too surprising 822 00:48:05,389 --> 00:48:07,180 because if you just think about it in terms 823 00:48:07,180 --> 00:48:10,210 of knowns and unknowns, if we want to represent an arbitrary 824 00:48:10,210 --> 00:48:13,810 signal of length 4 in terms of frequencies, 825 00:48:13,810 --> 00:48:15,580 we better have four of them-- 826 00:48:15,580 --> 00:48:17,590 here they are. 827 00:48:17,590 --> 00:48:19,930 The idea is precisely the same as CT 828 00:48:19,930 --> 00:48:24,580 except that there is a finite number of harmonics 829 00:48:24,580 --> 00:48:27,340 that we have to worry about. 830 00:48:27,340 --> 00:48:30,070 The rules for figuring out what the harmonics are-- 831 00:48:30,070 --> 00:48:33,690 and there is also a convenient way to think about-- 832 00:48:33,690 --> 00:48:38,820 DT Fourier series as matrices. 833 00:48:38,820 --> 00:48:40,680 Since we have a finite number of frequencies 834 00:48:40,680 --> 00:48:41,880 and a finite number of times, there 835 00:48:41,880 --> 00:48:43,470 is some relation between the two-- 836 00:48:43,470 --> 00:48:45,540 you could have an arbitrary, linear relationship 837 00:48:45,540 --> 00:48:49,650 between those sets of numbers expressed by a matrix. 838 00:48:49,650 --> 00:48:52,290 So matrix turns out to be a very convenient way of thinking 839 00:48:52,290 --> 00:48:55,140 about DT Fourier series. 840 00:48:55,140 --> 00:48:57,720 And there's an orthogonality principle exactly 841 00:48:57,720 --> 00:49:01,050 like there was in CT. 842 00:49:01,050 --> 00:49:03,960 The orthogonality looks just like an inner product, 843 00:49:03,960 --> 00:49:06,350 just like it did in CT. 844 00:49:06,350 --> 00:49:10,140 In CT, it was the integral over any time of length capital 845 00:49:10,140 --> 00:49:14,190 T of the product of two frequencies 846 00:49:14,190 --> 00:49:17,070 is either 0 or capital T depending 847 00:49:17,070 --> 00:49:19,320 on whether the frequencies are the same or different-- 848 00:49:19,320 --> 00:49:21,910 same thing happens here. 849 00:49:21,910 --> 00:49:24,270 And because of that, there is a very simple way 850 00:49:24,270 --> 00:49:29,250 to sift out the k-th element of the series, 851 00:49:29,250 --> 00:49:31,710 just like there was an analogously simple way 852 00:49:31,710 --> 00:49:34,440 to sift out the k-th element of a CT series. 853 00:49:34,440 --> 00:49:40,080 All of this is precisely the same as what we did in CT. 854 00:49:40,080 --> 00:49:43,080 The result is analysis and synthesis formulas 855 00:49:43,080 --> 00:49:48,120 that look similar, except there are sums instead of integrals. 856 00:49:48,120 --> 00:49:49,740 They look just like the CT-- 857 00:49:49,740 --> 00:49:53,580 the interesting thing is that they both have a finite length. 858 00:49:53,580 --> 00:49:58,230 In CT, there was an infinite sum over k. 859 00:49:58,230 --> 00:50:04,200 In DT, there's a finite sum over k. 860 00:50:04,200 --> 00:50:07,650 And so that means that there is a very convenient matrix 861 00:50:07,650 --> 00:50:11,910 way of looking at how would you construct an arbitrary 862 00:50:11,910 --> 00:50:15,390 signal in time from frequency components 863 00:50:15,390 --> 00:50:18,750 and how would you compute the frequency components 864 00:50:18,750 --> 00:50:21,420 from the time components? 865 00:50:21,420 --> 00:50:24,120 It's just that in DT, the convenient way to express that 866 00:50:24,120 --> 00:50:26,730 relationship-- you can still express it as an analysis 867 00:50:26,730 --> 00:50:27,840 and synthesis formula-- 868 00:50:27,840 --> 00:50:30,030 that's what these are. 869 00:50:30,030 --> 00:50:32,230 But a much more friendly way of thinking about it 870 00:50:32,230 --> 00:50:33,800 is as a matrix. 871 00:50:33,800 --> 00:50:37,350 Those two representations are absolutely identical. 872 00:50:37,350 --> 00:50:42,555 So the idea then for today was generalize CT to DT. 873 00:50:42,555 --> 00:50:44,510 DT is really useful when you're trying 874 00:50:44,510 --> 00:50:47,060 to do signal processing with digital electronics. 875 00:50:47,060 --> 00:50:49,250 Digital electronics is the way to go because they're 876 00:50:49,250 --> 00:50:50,750 so inexpensive and powerful. 877 00:50:54,230 --> 00:51:00,050 So unit circle wrapped, finite length-- 878 00:51:00,050 --> 00:51:01,880 and that finite length one is really 879 00:51:01,880 --> 00:51:05,240 the key to signal processing because we can represent 880 00:51:05,240 --> 00:51:08,270 a finite length signal in time with a finite number 881 00:51:08,270 --> 00:51:08,970 of samples. 882 00:51:08,970 --> 00:51:11,200 There is a match and that's the reason that's 883 00:51:11,200 --> 00:51:14,840 the method of choice when we do digital signal processing, 884 00:51:14,840 --> 00:51:17,780 and we'll talk more about that next time.