WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.720 continue to offer high quality educational resources for free. 00:00:10.720 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:15.780 from hundreds of MIT courses, visit 00:00:15.780 --> 00:00:17.670 MITOpenCourseWare@OCW.MIT.edu. 00:00:27.210 --> 00:00:28.680 PROFESSOR: Welcome. 00:00:28.680 --> 00:00:31.140 One quick announcement-- if you have not yet 00:00:31.140 --> 00:00:34.680 picked up your graded exams, you can do so 00:00:34.680 --> 00:00:38.338 by seeing the TAs after the hour. 00:00:38.338 --> 00:00:41.170 OK? 00:00:41.170 --> 00:00:43.650 So today I want to continue to think 00:00:43.650 --> 00:00:48.660 about what we started last week, thinking about Fourier series. 00:00:48.660 --> 00:00:54.780 The idea is to develop a theory that 00:00:54.780 --> 00:00:59.610 lets us look at signals on the basis of frequency content, 00:00:59.610 --> 00:01:02.820 much as we looked at frequency responses 00:01:02.820 --> 00:01:05.850 as a characterization of systems, 00:01:05.850 --> 00:01:08.610 according to the way they process frequencies. 00:01:08.610 --> 00:01:11.850 And we saw last time that there were a number of kinds 00:01:11.850 --> 00:01:15.360 of signals, for example, musical signals, 00:01:15.360 --> 00:01:17.520 where that kind of an on approach-- 00:01:17.520 --> 00:01:20.940 thinking about the signal according to the frequencies 00:01:20.940 --> 00:01:21.870 that are in it-- 00:01:21.870 --> 00:01:25.290 makes a lot of sense and can lead to insight. 00:01:25.290 --> 00:01:27.600 We also developed some formalism. 00:01:27.600 --> 00:01:32.310 We figured out how you can break a signal into components 00:01:32.310 --> 00:01:37.090 and then assemble the components to generate the signal. 00:01:37.090 --> 00:01:39.810 And what I want to mention at the beginning of the hour today 00:01:39.810 --> 00:01:42.870 is just how to think about this operation 00:01:42.870 --> 00:01:45.810 in a more familiar way. 00:01:45.810 --> 00:01:50.430 We do this kind of a thing, breaking something 00:01:50.430 --> 00:01:53.826 into components all the time. 00:01:53.826 --> 00:01:55.200 One of the more familiar examples 00:01:55.200 --> 00:01:58.380 might be thinking about 3-space, right? 00:01:58.380 --> 00:02:02.220 The Cartesian analysis of 3-space is based on the idea 00:02:02.220 --> 00:02:05.070 that you can think about a vector location in 3-space 00:02:05.070 --> 00:02:06.090 as having components. 00:02:06.090 --> 00:02:07.965 There's a component in the x direction, the y 00:02:07.965 --> 00:02:09.180 direction, the z direction. 00:02:09.180 --> 00:02:10.919 That's completely analogous to the way 00:02:10.919 --> 00:02:15.330 we're thinking about Fourier representations for signals. 00:02:15.330 --> 00:02:20.070 So just like we would think about synthesizing 00:02:20.070 --> 00:02:24.570 the location of a point by adding together three pieces, 00:02:24.570 --> 00:02:28.290 and we would think about analyzing a point to figure out 00:02:28.290 --> 00:02:30.930 how big the components are in each of those directions, 00:02:30.930 --> 00:02:34.770 it's exactly the same when we think about Fourier series. 00:02:34.770 --> 00:02:38.860 We think about representing a signal as a sum of things. 00:02:38.860 --> 00:02:41.970 So the sum is precisely the same. 00:02:41.970 --> 00:02:44.750 This one happens to have an infinite number of terms, 00:02:44.750 --> 00:02:49.680 [? eh. ?] The top one has three terms, [? eh. ?] 00:02:49.680 --> 00:02:52.230 The principles are very similar. 00:02:52.230 --> 00:02:55.830 So we think about representing a signal as a sum of components. 00:02:55.830 --> 00:02:58.380 We think about representing a point in 3-space 00:02:58.380 --> 00:03:00.180 as a sum of components, and we think 00:03:00.180 --> 00:03:06.470 about analyzing the signal or the vector in 3-space, 00:03:06.470 --> 00:03:11.040 so that we figure out what each of those components are. 00:03:11.040 --> 00:03:13.980 And we do it in an operation where it's actually 00:03:13.980 --> 00:03:18.240 very convenient to think about the decomposition 00:03:18.240 --> 00:03:21.270 of the Fourier components using precisely the same language 00:03:21.270 --> 00:03:24.240 that we would use for thinking about vector spaces. 00:03:24.240 --> 00:03:26.701 So we would think about-- 00:03:26.701 --> 00:03:28.200 in the case of the Fourier, we think 00:03:28.200 --> 00:03:32.112 about integrating over the period sifts out a component. 00:03:32.112 --> 00:03:33.570 The analogous operation for 3-space 00:03:33.570 --> 00:03:35.700 is to think about a dot product. 00:03:35.700 --> 00:03:37.606 The way you take a vector and figure out 00:03:37.606 --> 00:03:39.480 the component in the x direction is to dot it 00:03:39.480 --> 00:03:42.720 with it, the dot product. 00:03:42.720 --> 00:03:44.670 In the Fourier case, we think about it 00:03:44.670 --> 00:03:45.810 as being an inner product. 00:03:48.550 --> 00:03:50.880 The idea is completely analogous. 00:03:50.880 --> 00:03:54.720 So we think about having the inner product of two things-- 00:03:54.720 --> 00:03:59.880 the reference direction and the vector. 00:03:59.880 --> 00:04:02.460 So reference direction and vector-- 00:04:02.460 --> 00:04:04.830 we think about it exactly the same way, except now it's 00:04:04.830 --> 00:04:07.316 an inner product, which means that after we've multiplied, 00:04:07.316 --> 00:04:08.190 we have to integrate. 00:04:08.190 --> 00:04:11.070 That's the only difference between inner product. 00:04:11.070 --> 00:04:13.830 Inner product implies some are [? integrated ?] after you've 00:04:13.830 --> 00:04:17.029 done the multiplication. 00:04:17.029 --> 00:04:19.880 So we do exactly the same thing, except that now we 00:04:19.880 --> 00:04:24.730 think about the inner product of a and b. 00:04:24.730 --> 00:04:29.530 That's just the integral, where we take the complex conjugate 00:04:29.530 --> 00:04:34.180 of one of the signals only because by defining it 00:04:34.180 --> 00:04:38.480 with a complex conjugate there, we set up the inner product 00:04:38.480 --> 00:04:41.320 so that the answer is zero unless we 00:04:41.320 --> 00:04:45.930 take the inner product of two things in the same direction. 00:04:45.930 --> 00:04:47.360 OK? 00:04:47.360 --> 00:04:50.300 By putting the minus sign there, if the two reference 00:04:50.300 --> 00:04:53.090 directions, that is to say the one 00:04:53.090 --> 00:04:59.180 characterized by k and m, the ones characterized by k and m, 00:04:59.180 --> 00:05:00.800 the inner product will be zero if we 00:05:00.800 --> 00:05:05.180 take the complex conjugate as long as k is not equal to m. 00:05:05.180 --> 00:05:08.050 k equals m is the only not zero component. 00:05:08.050 --> 00:05:11.270 OK, is that all clear? 00:05:11.270 --> 00:05:14.420 So to make sure that it's clear, here's a question. 00:05:17.507 --> 00:05:19.340 How many of the following pairs of functions 00:05:19.340 --> 00:05:23.015 are orthogonal in T equals 3? 00:05:25.412 --> 00:05:26.870 Part of the goal of the exercise is 00:05:26.870 --> 00:05:30.020 to figure out what the little caveat in T equals three means. 00:05:30.020 --> 00:05:32.810 So look at your neighbor, say hello, figure out 00:05:32.810 --> 00:05:35.636 a number between 0 and 4. 00:05:35.636 --> 00:05:38.624 [SIDE CONVERSATIONS] 00:07:48.340 --> 00:07:50.780 OK, so how many signals, how many of the pairs 00:07:50.780 --> 00:07:54.470 are orthogonal to each other? 00:07:54.470 --> 00:07:56.960 Raise your hand with some number between 0 and 4, 00:07:56.960 --> 00:07:59.990 unless you're completely bizarre and raise five, I mean. 00:07:59.990 --> 00:08:01.200 OK, come on, come on. 00:08:01.200 --> 00:08:02.760 Higher, so I can see them. 00:08:02.760 --> 00:08:04.080 Remember if you're wrong. 00:08:04.080 --> 00:08:08.710 It's your partner's fault, it's not your fault. 00:08:08.710 --> 00:08:11.530 OK, not quite. 00:08:11.530 --> 00:08:15.230 A lot of bad partners, no, no. 00:08:15.230 --> 00:08:16.460 Let's do the first one. 00:08:16.460 --> 00:08:21.290 Is the cos of 2 pi t orthogonal to the sine 00:08:21.290 --> 00:08:25.910 of 2 pi t over the interval capital T equals 3? 00:08:29.100 --> 00:08:31.770 Yes? 00:08:31.770 --> 00:08:34.400 No. 00:08:34.400 --> 00:08:37.049 I haven't a clue. 00:08:37.049 --> 00:08:38.695 I don't care. 00:08:38.695 --> 00:08:39.320 No, no, no, no. 00:08:39.320 --> 00:08:42.039 You all care, no. 00:08:42.039 --> 00:08:42.919 Are they orthogonal? 00:08:42.919 --> 00:08:43.460 So what do you-- 00:08:43.460 --> 00:08:45.043 how do I formally ask the question are 00:08:45.043 --> 00:08:46.015 they are orthogonal? 00:08:49.580 --> 00:08:52.820 OK, so it's either the last slide or the next slide. 00:08:52.820 --> 00:08:54.059 So go back to the last slide. 00:08:54.059 --> 00:08:55.600 What's it mean if they're orthogonal? 00:08:59.560 --> 00:09:00.191 Yeah? 00:09:00.191 --> 00:09:05.010 AUDIENCE: [INAUDIBLE] 00:09:05.010 --> 00:09:07.110 PROFESSOR: So how do I take the dot product? 00:09:07.110 --> 00:09:08.490 What do I do? 00:09:08.490 --> 00:09:11.152 AUDIENCE: [INAUDIBLE] conjugate. 00:09:11.152 --> 00:09:12.110 PROFESSOR: Conjugate 1. 00:09:12.110 --> 00:09:12.693 So I want to-- 00:09:12.693 --> 00:09:18.300 I'm thinking about 1 over T, the integral over T, a star of T, 00:09:18.300 --> 00:09:20.260 b of t, dt. 00:09:20.260 --> 00:09:21.990 Right? 00:09:21.990 --> 00:09:24.480 So the t comes in here, right, I'm 00:09:24.480 --> 00:09:26.640 integrating over a period t. 00:09:26.640 --> 00:09:31.180 So I take the two functions and I multiply them together. 00:09:31.180 --> 00:09:35.370 So I have this function, and I have that function. 00:09:35.370 --> 00:09:38.010 I multiply them together. 00:09:38.010 --> 00:09:41.700 If you multiply two sinusoids of the same frequency 00:09:41.700 --> 00:09:45.470 but different phase, what do you get? 00:09:45.470 --> 00:09:47.206 Another sinusoid, right? 00:09:47.206 --> 00:09:49.580 So you all know all these complicated trig relationships, 00:09:49.580 --> 00:09:50.079 right? 00:09:50.079 --> 00:09:50.920 Here's one of them. 00:09:50.920 --> 00:09:53.230 If you multiply cos of 2 pi T times the sine of pi T 00:09:53.230 --> 00:09:57.940 you get half the sign of double the frequency. 00:09:57.940 --> 00:09:58.480 OK? 00:09:58.480 --> 00:09:59.813 You don't need to memorize that. 00:09:59.813 --> 00:10:03.130 You just look at this picture, you look at that picture. 00:10:03.130 --> 00:10:07.490 This one over the interval 3 has 3 periods. 00:10:07.490 --> 00:10:08.620 Right? 00:10:08.620 --> 00:10:12.640 There are 3 periods of that waveform 00:10:12.640 --> 00:10:15.340 over the period of capital T. There's 00:10:15.340 --> 00:10:18.920 an integer number 3, same here. 00:10:18.920 --> 00:10:21.190 Here-- how many periods? 00:10:21.190 --> 00:10:22.630 Twice that. 00:10:22.630 --> 00:10:24.960 But it's exactly six. 00:10:24.960 --> 00:10:28.840 So you get a pure sinusoid. 00:10:28.840 --> 00:10:30.490 You get an integer number of periods. 00:10:30.490 --> 00:10:33.670 You integrate over an integer number of periods, you get 0. 00:10:33.670 --> 00:10:35.440 They're orthogonal. 00:10:35.440 --> 00:10:37.414 Had I chosen the period differently, 00:10:37.414 --> 00:10:38.830 they may not have been orthogonal. 00:10:38.830 --> 00:10:40.611 It depends on the period. 00:10:40.611 --> 00:10:41.110 OK? 00:10:41.110 --> 00:10:42.790 So the inner product depends on the period, 00:10:42.790 --> 00:10:44.415 because the inner product has something 00:10:44.415 --> 00:10:45.520 to do with integrator sum. 00:10:45.520 --> 00:10:50.550 And so the range over which you sum or integrate matters. 00:10:50.550 --> 00:10:54.570 How about cos 2 pi T cos 4 pi T? 00:10:54.570 --> 00:10:55.320 Orthogonal? 00:10:58.680 --> 00:10:59.640 Yeah? 00:10:59.640 --> 00:11:01.080 AUDIENCE: Yes. 00:11:01.080 --> 00:11:03.558 PROFESSOR: And the reason is? 00:11:03.558 --> 00:11:07.446 AUDIENCE: So think if you wrapped [INAUDIBLE] together, 00:11:07.446 --> 00:11:10.119 then there's a lot of symmetry that 00:11:10.119 --> 00:11:14.260 goes on [INAUDIBLE] is going to be 0. 00:11:14.260 --> 00:11:18.260 PROFESSOR: So now we've got two different frequencies. 00:11:18.260 --> 00:11:20.651 But we still get these funny cosine relationships 00:11:20.651 --> 00:11:22.400 that have to do with sums and differences. 00:11:22.400 --> 00:11:23.810 And the sums and differences both 00:11:23.810 --> 00:11:27.840 happen to be periodic over 3, over the interval capital 00:11:27.840 --> 00:11:29.690 T equals 3, right? 00:11:29.690 --> 00:11:33.390 So we still get the property that the average value here, 00:11:33.390 --> 00:11:36.470 which is what the interval was pulling out, the average is 0. 00:11:36.470 --> 00:11:38.500 So they're also orthogonal. 00:11:38.500 --> 00:11:41.390 How about cos 2 pi T sine pi T? 00:11:43.980 --> 00:11:46.735 OK, I've asked two questions, and they were both yes. 00:11:46.735 --> 00:11:48.340 So I'm getting bored at this point, 00:11:48.340 --> 00:11:52.336 so by the theory of questions in lecture, the answer is? 00:11:52.336 --> 00:11:54.779 [LAUGHTER] 00:11:54.779 --> 00:11:55.570 Now, wait a minute. 00:11:55.570 --> 00:11:57.100 I'm not that boring. 00:11:57.100 --> 00:11:59.730 Well, maybe. 00:11:59.730 --> 00:12:04.200 So is this periodic over a capital T equals 3? 00:12:07.610 --> 00:12:08.676 Ah, excuse me. 00:12:08.676 --> 00:12:10.300 I didn't say the right question, sorry. 00:12:10.300 --> 00:12:16.540 Is this function have an integer number of periods in the time 00:12:16.540 --> 00:12:19.319 interval capital T equals 3? 00:12:19.319 --> 00:12:21.360 What's the period-- what's the fundamental period 00:12:21.360 --> 00:12:22.800 of this waveform? 00:12:22.800 --> 00:12:23.580 AUDIENCE: 1. 00:12:23.580 --> 00:12:24.710 PROFESSOR: 1. 00:12:24.710 --> 00:12:29.037 So it has 3 periods over the interval cap T equals 3. 00:12:29.037 --> 00:12:29.870 What about this one? 00:12:33.502 --> 00:12:34.410 AUDIENCE: 2. 00:12:34.410 --> 00:12:35.610 PROFESSOR: Period is 2. 00:12:35.610 --> 00:12:39.290 How many periods are there in the time interval capital T 00:12:39.290 --> 00:12:40.297 equals 3? 00:12:40.297 --> 00:12:43.159 AUDIENCE: [INAUDIBLE] 00:12:43.159 --> 00:12:44.430 PROFESSOR: A period is 2. 00:12:44.430 --> 00:12:46.290 How many periods are there-- 00:12:46.290 --> 00:12:48.290 1 and 1/2, not an integer. 00:12:48.290 --> 00:12:49.760 Bad news, right? 00:12:49.760 --> 00:12:54.080 So integer number three, not integer number. 00:12:54.080 --> 00:12:57.170 If you were to integrate this over the period t 00:12:57.170 --> 00:13:00.440 equals 3, if I didn't multiply them 00:13:00.440 --> 00:13:03.170 if I just did that, if I just thought about that integral, 00:13:03.170 --> 00:13:05.600 I wouldn't get 0, right? 00:13:05.600 --> 00:13:08.600 There's more positives than there are negatives. 00:13:08.600 --> 00:13:11.720 And when I multiply them, the same sort of thing happens. 00:13:11.720 --> 00:13:14.340 I get two big peaks down and only one big peak up. 00:13:14.340 --> 00:13:18.980 It's because the resulting waveform no longer 00:13:18.980 --> 00:13:22.570 has an integer number of periods in the interval capital T 00:13:22.570 --> 00:13:23.270 equals 3. 00:13:23.270 --> 00:13:23.770 OK? 00:13:27.510 --> 00:13:30.900 Last one-- cos 2 pi T e to the-- 00:13:30.900 --> 00:13:33.735 whoops. 00:13:33.735 --> 00:13:37.410 Is that what I actually said? 00:13:37.410 --> 00:13:39.600 Good, I forgot the j. 00:13:39.600 --> 00:13:42.240 Because without the j, they would obviously not 00:13:42.240 --> 00:13:43.290 be orthogonal. 00:13:43.290 --> 00:13:44.550 Obviously, right? 00:13:44.550 --> 00:13:45.450 OK. 00:13:45.450 --> 00:13:48.400 I didn't mean to ask something quite that obvious. 00:13:48.400 --> 00:13:54.720 So what about cos 2 pi T and e to the j 2 pi T? 00:13:57.350 --> 00:14:00.870 Orthogonal. 00:14:00.870 --> 00:14:09.731 Not, I'm as clueless as I was on Part A. No, no, no, no, 00:14:09.731 --> 00:14:10.230 you're not. 00:14:10.230 --> 00:14:10.855 No, you're not. 00:14:10.855 --> 00:14:12.230 No, you're not. 00:14:12.230 --> 00:14:15.790 So how do you think about that? 00:14:15.790 --> 00:14:17.370 You can use Euler's expression. 00:14:17.370 --> 00:14:18.990 And if there had been a j there, this 00:14:18.990 --> 00:14:21.350 would have been a correct expression. 00:14:21.350 --> 00:14:22.050 OK? 00:14:22.050 --> 00:14:23.580 It's not quite a correct expression 00:14:23.580 --> 00:14:25.477 because I forgot to put the j there. 00:14:25.477 --> 00:14:27.060 But had there been a j there, it would 00:14:27.060 --> 00:14:30.600 have been cos 2 pi T plus j sine 2 pi T. 00:14:30.600 --> 00:14:33.330 And the awkward thing is that the cos and the cos 00:14:33.330 --> 00:14:35.820 are obviously not orthogonal with each other. 00:14:35.820 --> 00:14:40.650 A signal is not orthogonal with itself, OK? 00:14:40.650 --> 00:14:45.150 So because part of this signal is that signal, 00:14:45.150 --> 00:14:49.650 those two signals are not orthogonal. 00:14:49.650 --> 00:14:50.580 OK? 00:14:50.580 --> 00:14:52.940 Yes? 00:14:52.940 --> 00:14:54.620 OK, so that's kind of-- so that's 00:14:54.620 --> 00:14:56.180 the idea of orthogonality. 00:14:56.180 --> 00:14:59.810 It's a very good way to think about decompositions. 00:15:02.182 --> 00:15:04.640 And even though we only spent about half an hour last time, 00:15:04.640 --> 00:15:06.380 and only about 15 minutes this time, 00:15:06.380 --> 00:15:09.660 that is the whole theory of Fourier series. 00:15:09.660 --> 00:15:11.736 That doesn't mean we can't ask hard questions. 00:15:11.736 --> 00:15:13.110 There were a couple of questions. 00:15:13.110 --> 00:15:14.200 Yes, you were first. 00:15:14.200 --> 00:15:16.540 AUDIENCE: Is there a way to think about orthogonality 00:15:16.540 --> 00:15:20.260 using the Fourier [INAUDIBLE]. 00:15:20.260 --> 00:15:22.610 PROFESSOR: Well, the Fourier coefficients 00:15:22.610 --> 00:15:25.947 are the result of orthogonality. 00:15:25.947 --> 00:15:27.530 I don't think you can tell-- if I just 00:15:27.530 --> 00:15:29.060 told you a bunch of Fourier coefficients, 00:15:29.060 --> 00:15:30.920 I don't know if you can tell me something 00:15:30.920 --> 00:15:34.070 about the orthogonality of the underlying signals or not. 00:15:34.070 --> 00:15:36.975 AUDIENCE: What if [INAUDIBLE]. 00:15:36.975 --> 00:15:37.850 PROFESSOR: Excuse me? 00:15:37.850 --> 00:15:39.433 AUDIENCE: [INAUDIBLE] [? the period ?] 00:15:39.433 --> 00:15:41.570 and the Fourier [INAUDIBLE]. 00:15:44.370 --> 00:15:46.529 PROFESSOR: Let's see, so I'm not completely 00:15:46.529 --> 00:15:47.820 sure I know what you're asking. 00:15:47.820 --> 00:15:51.410 Certainly if you tell me that the Fourier's coefficients are 00:15:51.410 --> 00:15:55.030 blah, blah, blah, 3, 2 7, and 16. 00:15:55.030 --> 00:15:58.050 And if you tell me that you're working with a simple Fourier 00:15:58.050 --> 00:16:03.594 series periodic in 3, then you've told me everything. 00:16:03.594 --> 00:16:05.260 And so there's a way for me to backtrack 00:16:05.260 --> 00:16:06.760 that it was orthogonal. 00:16:06.760 --> 00:16:09.610 I am not sure if I'm connecting with you, so if I'm not, 00:16:09.610 --> 00:16:11.320 ask me after lecture to make sure that-- 00:16:11.320 --> 00:16:12.234 AUDIENCE: [INAUDIBLE] 00:16:12.234 --> 00:16:12.900 PROFESSOR: Sure. 00:16:12.900 --> 00:16:14.399 AUDIENCE: I think he's saying if you 00:16:14.399 --> 00:16:17.337 have two signals [INAUDIBLE] coefficients [INAUDIBLE] two 00:16:17.337 --> 00:16:21.281 signals, can I tell if those two signals are orthogonal 00:16:21.281 --> 00:16:27.220 [INAUDIBLE] the coefficients are orthogonal [INAUDIBLE].. 00:16:27.220 --> 00:16:31.150 PROFESSOR: If they have components in common, 00:16:31.150 --> 00:16:34.060 they couldn't possibly be orthogonal. 00:16:34.060 --> 00:16:38.750 So I would answer yes to that question. 00:16:38.750 --> 00:16:41.560 So if that's what you were-- so I think that's probably right. 00:16:41.560 --> 00:16:43.140 Does that sound right? 00:16:43.140 --> 00:16:44.430 Yeah, OK. 00:16:44.430 --> 00:16:45.630 Yes? 00:16:45.630 --> 00:16:48.130 AUDIENCE: I'm awfully confused by the complex conjugate, 00:16:48.130 --> 00:16:49.130 the [INAUDIBLE]. 00:16:49.130 --> 00:16:50.762 PROFESSOR: Yes, yes, yes. 00:16:50.762 --> 00:16:52.928 AUDIENCE: So does that mean we're taking the complex 00:16:52.928 --> 00:16:55.835 conjugate [? of a ?] and we're [? applying it ?] to b? 00:16:55.835 --> 00:16:57.710 PROFESSOR: We're taking the complex conjugate 00:16:57.710 --> 00:16:59.570 of the entire function. 00:16:59.570 --> 00:17:03.200 At every point in time, we take the complex conjugate of it. 00:17:03.200 --> 00:17:06.230 And it's especially useful to think about 00:17:06.230 --> 00:17:08.099 if you're doing something of the form-- 00:17:08.099 --> 00:17:17.480 if a of t were e to the j 2 pi mt and if b of t 00:17:17.480 --> 00:17:22.940 were e to the j 2 pi lt. 00:17:22.940 --> 00:17:25.190 The only thing we're trying to do-- 00:17:25.190 --> 00:17:27.030 but this comes up quite frequently-- 00:17:27.030 --> 00:17:28.490 the only thing we're trying to do 00:17:28.490 --> 00:17:32.420 is when you conjugate one of these, 00:17:32.420 --> 00:17:35.850 you rig it so that when you add the exponents, 00:17:35.850 --> 00:17:39.920 the result goes to 0 by putting the minus up there. 00:17:39.920 --> 00:17:40.653 That's all. 00:17:40.653 --> 00:17:42.028 AUDIENCE: It doesn't seem like we 00:17:42.028 --> 00:17:44.944 had to do any of that for the example we just worked on. 00:17:44.944 --> 00:17:47.860 It seems like there were just like [? signals ?] [INAUDIBLE].. 00:17:47.860 --> 00:17:49.290 PROFESSOR: Oh, interesting. 00:17:52.299 --> 00:17:53.340 That's a very good point. 00:17:53.340 --> 00:17:54.590 That's interesting. 00:17:54.590 --> 00:17:57.530 So I didn't intend to throw you a ringer. 00:17:57.530 --> 00:18:00.590 These were signals, all of these except that one, 00:18:00.590 --> 00:18:04.160 are real functions of time. 00:18:04.160 --> 00:18:06.980 That's why the complex conjugate didn't come up. 00:18:06.980 --> 00:18:07.790 So I apologize. 00:18:07.790 --> 00:18:10.970 I wasn't trying to make it seem tricky. 00:18:10.970 --> 00:18:14.480 OK, so it's because this function of time 00:18:14.480 --> 00:18:19.280 is everywhere real that we didn't need to rehearse this. 00:18:19.280 --> 00:18:20.920 We did have to do it in that one. 00:18:23.300 --> 00:18:23.800 OK? 00:18:26.800 --> 00:18:28.940 OK, so the point is that we've already covered, 00:18:28.940 --> 00:18:31.450 even though we've only done a little bit of work in lecture, 00:18:31.450 --> 00:18:34.150 we've already covered all of the theory. 00:18:34.150 --> 00:18:36.310 What remains though is to do some practice. 00:18:36.310 --> 00:18:39.760 And also what remains is to understand how this is useful. 00:18:39.760 --> 00:18:43.180 So it's not just music. 00:18:43.180 --> 00:18:47.800 The example that I want to talk about today is speech. 00:18:47.800 --> 00:18:50.350 The same sort of thing that we could do with music last time, 00:18:50.350 --> 00:18:51.610 we can do with speech. 00:18:51.610 --> 00:18:53.192 And here are some utterances. 00:18:53.192 --> 00:18:53.858 [AUDIO PLAYBACK] 00:18:53.858 --> 00:19:00.797 - Bat, bait, bet, beet, bit, bite, bought, boat, but, boot. 00:19:00.797 --> 00:19:01.380 [END PLAYBACK] 00:19:01.380 --> 00:19:02.755 PROFESSOR: All right, it was just 00:19:02.755 --> 00:19:05.650 intended to be a bunch of sounds that we can analyze 00:19:05.650 --> 00:19:08.890 with Fourier analysis to get some insight into how 00:19:08.890 --> 00:19:14.050 to think about, in particular, speech recognition and speech 00:19:14.050 --> 00:19:15.970 synthesis. 00:19:15.970 --> 00:19:18.580 So we can take those utterances, and all 00:19:18.580 --> 00:19:21.340 I did was write a little Python program to do the decomposition 00:19:21.340 --> 00:19:24.490 that I showed on the previous slides, 00:19:24.490 --> 00:19:27.850 so that I could break these time waveforms. 00:19:27.850 --> 00:19:31.670 Here I'm illustrating one, two, three, four, five, six periods. 00:19:31.670 --> 00:19:37.090 So I took one period of that sound 00:19:37.090 --> 00:19:41.680 and ran it through that kind of an integral 00:19:41.680 --> 00:19:43.910 to break it into Fourier components, which 00:19:43.910 --> 00:19:45.400 are showed here. 00:19:45.400 --> 00:19:47.500 And what I want you to see is just 00:19:47.500 --> 00:19:50.770 like you could have recognized a pattern here, 00:19:50.770 --> 00:19:55.060 and you might try to recognize which vowel is which 00:19:55.060 --> 00:19:58.780 by the signature in time. 00:19:58.780 --> 00:20:01.570 An alternative, and far more useful way 00:20:01.570 --> 00:20:04.150 of thinking about it, is to try to recognize 00:20:04.150 --> 00:20:07.480 the pattern in frequency. 00:20:07.480 --> 00:20:10.030 So there are characteristic differences in the sounds, 00:20:10.030 --> 00:20:12.701 and we'll look at the basis for why there are. 00:20:12.701 --> 00:20:14.200 There are characteristic differences 00:20:14.200 --> 00:20:18.520 in the sound that can help us to identify automatically, 00:20:18.520 --> 00:20:22.610 by a machine, what was being said. 00:20:22.610 --> 00:20:24.730 And so what we want to do is learn 00:20:24.730 --> 00:20:29.410 to think about a pattern that characterizes ah, 00:20:29.410 --> 00:20:34.151 ee, oo in the frequency domain, as opposed to the time domain. 00:20:34.151 --> 00:20:34.817 [AUDIO PLAYBACK] 00:20:34.817 --> 00:20:36.397 - Bat beet, boot. 00:20:36.397 --> 00:20:36.980 [END PLAYBACK] 00:20:36.980 --> 00:20:39.480 PROFESSOR: So there's something different about those sounds 00:20:39.480 --> 00:20:44.870 that manifests a difference in this Fourier signature. 00:20:44.870 --> 00:20:48.350 So that's one of the useful applications of this. 00:20:48.350 --> 00:20:50.060 And we'd like to understand that better. 00:20:50.060 --> 00:20:54.440 There's a really good physical reason why that happens. 00:20:54.440 --> 00:20:57.860 And it has to do with the way we produce speech. 00:20:57.860 --> 00:21:03.080 So you can think about speech as being generated by some source. 00:21:03.080 --> 00:21:05.180 Ultimately, the source of my speech 00:21:05.180 --> 00:21:10.070 is somewhere down here, which always amuses me 00:21:10.070 --> 00:21:14.270 when I see the cut off heads talking, like at Halloween 00:21:14.270 --> 00:21:16.310 and like on some cartoon shows. 00:21:16.310 --> 00:21:17.980 Because you can't do that, right? 00:21:17.980 --> 00:21:20.510 Because the source has to do with down here someplace, 00:21:20.510 --> 00:21:21.080 right? 00:21:21.080 --> 00:21:23.060 My lungs push in. 00:21:23.060 --> 00:21:24.800 That pushes air through something 00:21:24.800 --> 00:21:27.860 and starts making noise somehow. 00:21:27.860 --> 00:21:33.230 I'm going to focus today on things that we call voiced-- 00:21:33.230 --> 00:21:35.560 in a voiced sound. 00:21:35.560 --> 00:21:39.800 Ah, in a voiced sound, it's caused by vibrations 00:21:39.800 --> 00:21:40.715 of the vocal chords. 00:21:40.715 --> 00:21:44.910 So if you were to stick a camera down someone's throat, 00:21:44.910 --> 00:21:47.060 this is the sort of thing that you would see. 00:21:47.060 --> 00:21:52.100 It's an enormously complex structure 00:21:52.100 --> 00:21:57.020 whose mechanics are extremely difficult to understand. 00:21:57.020 --> 00:22:03.050 Because what happens is when you want to make a high sound, 00:22:03.050 --> 00:22:05.210 you tense the structure. 00:22:05.210 --> 00:22:07.970 You pull on some muscles that pull the cords. 00:22:07.970 --> 00:22:13.170 The cords are normally rattling pretty fast. 00:22:13.170 --> 00:22:16.170 And what you do is, you pull on a muscle 00:22:16.170 --> 00:22:20.507 that tenses them to make it go higher. 00:22:20.507 --> 00:22:22.590 But your intuition should say, now wait a minute-- 00:22:22.590 --> 00:22:25.290 you're making it longer to make it higher? 00:22:25.290 --> 00:22:28.140 And your intuition would be right. 00:22:28.140 --> 00:22:33.670 Normally, long organ pipes are higher or lower in frequency? 00:22:33.670 --> 00:22:34.640 AUDIENCE: Lower. 00:22:34.640 --> 00:22:37.300 PROFESSOR: Lower. 00:22:37.300 --> 00:22:40.720 So you have to do a lot of mental calculations 00:22:40.720 --> 00:22:44.090 in order to move these muscles correctly. 00:22:44.090 --> 00:22:46.570 So that the resulting frequency of the vibration 00:22:46.570 --> 00:22:47.590 comes out right. 00:22:47.590 --> 00:22:49.780 It's not obvious, because two things 00:22:49.780 --> 00:22:51.510 happen as you tense the muscle. 00:22:51.510 --> 00:22:55.750 The folds-- the vocal cords get longer which you would think 00:22:55.750 --> 00:22:58.930 would make the frequency lower, but they 00:22:58.930 --> 00:23:03.480 get tighter, which, of course, goes the other direction, 00:23:03.480 --> 00:23:03.980 right? 00:23:03.980 --> 00:23:05.310 So it's a very complicated thing. 00:23:05.310 --> 00:23:06.851 And in fact, it's something that goes 00:23:06.851 --> 00:23:08.270 bad with professional speakers. 00:23:08.270 --> 00:23:11.720 But even more, professional singers 00:23:11.720 --> 00:23:15.230 often have a lot of trouble with the enormous stress 00:23:15.230 --> 00:23:19.040 that happens on this structure with repeated use and repeated 00:23:19.040 --> 00:23:20.860 overuse. 00:23:20.860 --> 00:23:23.870 Anyway, this takes a real beating. 00:23:23.870 --> 00:23:27.764 But that's ultimately the source of speech. 00:23:27.764 --> 00:23:29.930 But if that were all you had, it wouldn't sound much 00:23:29.930 --> 00:23:31.790 like speech. 00:23:31.790 --> 00:23:35.960 A lot of the interesting stuff comes from these cavities 00:23:35.960 --> 00:23:37.760 that you intentionally manipulate 00:23:37.760 --> 00:23:41.000 as you're speaking to make the different characteristic 00:23:41.000 --> 00:23:43.880 sounds. 00:23:43.880 --> 00:23:51.020 So the idea then is that you have a source that contains 00:23:51.020 --> 00:23:53.030 information like frequency. 00:23:53.030 --> 00:23:57.320 What's the pitch of the utterance? 00:23:57.320 --> 00:24:02.240 But you have this other thing, which is acting like a filter. 00:24:02.240 --> 00:24:04.940 If you think about the whole thing as a system, 00:24:04.940 --> 00:24:09.620 we have a block which represents a filter, which 00:24:09.620 --> 00:24:12.039 is the thing that has a frequency response. 00:24:12.039 --> 00:24:13.580 The frequency response depends on how 00:24:13.580 --> 00:24:16.460 I've put my tongue in my mouth and how I've opened my lips 00:24:16.460 --> 00:24:17.356 and stuff like that. 00:24:17.356 --> 00:24:18.480 We'll see that in a minute. 00:24:18.480 --> 00:24:21.410 But it also depends on how the vocal folds-- 00:24:21.410 --> 00:24:25.710 it has an input, which is the vocal folds. 00:24:25.710 --> 00:24:29.330 So the idea then is the same kind of a source filter idea 00:24:29.330 --> 00:24:34.320 that we motivated last time by way of the RC filter example. 00:24:34.320 --> 00:24:37.070 If you put a resistor and a capacitor 00:24:37.070 --> 00:24:40.130 together with a source, a convenient way 00:24:40.130 --> 00:24:42.830 to think about that is as a low pass filter. 00:24:42.830 --> 00:24:45.170 We think about it having a frequency response. 00:24:45.170 --> 00:24:50.330 So the system, just the RC part, has a frequency response 00:24:50.330 --> 00:24:53.480 which we can characterize by a Bode diagram. 00:24:53.480 --> 00:24:54.860 So we can think about-- 00:24:54.860 --> 00:24:56.510 we did this last time-- 00:24:56.510 --> 00:24:59.270 so we can think about the low frequencies go through 00:24:59.270 --> 00:25:00.410 without attenuation. 00:25:00.410 --> 00:25:04.190 The gain is 1, and the phase is 0. 00:25:04.190 --> 00:25:07.820 So basically low frequencies go through the filter 00:25:07.820 --> 00:25:10.310 without any change. 00:25:10.310 --> 00:25:11.684 High frequencies are attenuated. 00:25:11.684 --> 00:25:13.100 The higher the frequency, the more 00:25:13.100 --> 00:25:20.030 the attenuation, and phase shifted by lagging pi over 2. 00:25:20.030 --> 00:25:23.090 So that's a way of thinking about the RC circuit 00:25:23.090 --> 00:25:24.980 as a low pass filter. 00:25:24.980 --> 00:25:26.510 And it gives us insight in the kinds 00:25:26.510 --> 00:25:28.790 of signals that go through and don't go through. 00:25:28.790 --> 00:25:33.710 So that, if we think about a signal like a square wave 00:25:33.710 --> 00:25:37.380 having a Fourier series decomposition, 00:25:37.380 --> 00:25:41.550 it only has odd components and the odd components fall with k. 00:25:41.550 --> 00:25:44.690 The magnitude of the component is inverse with k. 00:25:44.690 --> 00:25:49.640 So we get components that, if I plot on a log scale, 00:25:49.640 --> 00:25:53.625 the reciprocal relationship of the weight of the components, 00:25:53.625 --> 00:25:56.000 the magnitude of the components, makes it a straight line 00:25:56.000 --> 00:25:58.756 with a slope of minus 1. 00:25:58.756 --> 00:26:02.450 And now we can think about putting this signal into the RC 00:26:02.450 --> 00:26:07.040 filter and thinking about what the output should look like. 00:26:07.040 --> 00:26:09.080 If the frequency of the square wave, 00:26:09.080 --> 00:26:12.440 if the fundamental frequency, 2 pi 00:26:12.440 --> 00:26:17.250 over the period, if 2 pi over capital T, 00:26:17.250 --> 00:26:21.650 if 2 pi over capital T is some frequency that's low compared 00:26:21.650 --> 00:26:25.370 to the corner frequency of the low pass filter, 00:26:25.370 --> 00:26:28.440 basically the output of the filter, which 00:26:28.440 --> 00:26:32.390 is showed in green, overlaps the input, which is showed in red. 00:26:32.390 --> 00:26:35.780 You can't tell the difference because all the components 00:26:35.780 --> 00:26:39.750 have the same magnitude and phase as the input. 00:26:39.750 --> 00:26:42.420 But if you change the frequency of the square wave 00:26:42.420 --> 00:26:45.630 so that the fundamental is higher, 00:26:45.630 --> 00:26:48.870 some of the higher frequencies are attenuated 00:26:48.870 --> 00:26:51.700 and phase shifted. 00:26:51.700 --> 00:26:54.280 The shape of the waveform, showed in green, 00:26:54.280 --> 00:26:56.267 starts to deviate. 00:26:56.267 --> 00:26:57.850 If you go to still higher frequencies, 00:26:57.850 --> 00:26:59.080 the deviation's even greater. 00:26:59.080 --> 00:27:01.690 And if you go to high enough frequencies, 00:27:01.690 --> 00:27:04.810 they're all in the region where the magnitude is 00:27:04.810 --> 00:27:09.070 being attenuated by whatever frequency. 00:27:09.070 --> 00:27:12.730 So my dependence of 1 over k becomes 1 over k squared, 00:27:12.730 --> 00:27:17.560 and it goes from being a square wave to a triangle wave. 00:27:17.560 --> 00:27:20.430 So that's a way of thinking about the signal transformation 00:27:20.430 --> 00:27:21.450 in terms of a filter. 00:27:21.450 --> 00:27:22.590 We did that last time. 00:27:22.590 --> 00:27:25.880 What's going on with speech is exactly the same thing. 00:27:25.880 --> 00:27:27.510 What we want to do is think about-- 00:27:27.510 --> 00:27:32.310 the glottis makes some kind of a sound that goes into a filter. 00:27:32.310 --> 00:27:34.860 The filter is this thing that is controlled 00:27:34.860 --> 00:27:36.830 by my tongue's position and my jaw 00:27:36.830 --> 00:27:39.210 position and my lip position and stuff like that. 00:27:39.210 --> 00:27:41.790 And what comes out is speech. 00:27:41.790 --> 00:27:45.000 To demonstrate that, here's a film 00:27:45.000 --> 00:27:46.350 that was made by Ken Stevens. 00:27:46.350 --> 00:27:48.308 Ken Stevens was a professor in this department. 00:27:48.308 --> 00:27:50.790 He just recently retired. 00:27:50.790 --> 00:27:53.370 This was done when he was a graduate student. 00:27:53.370 --> 00:27:57.630 It's very hard to see because the contrast is not great. 00:27:57.630 --> 00:27:59.310 But you have to take into consideration 00:27:59.310 --> 00:28:01.050 this was made with X-rays. 00:28:01.050 --> 00:28:04.030 OK, we probably wouldn't do this today. 00:28:04.030 --> 00:28:07.530 It was a relatively large exposure 00:28:07.530 --> 00:28:10.620 to x-rays, which we sort of frown on these days. 00:28:10.620 --> 00:28:14.340 Just so you're not too worried, Ken Stevens, when he retired, 00:28:14.340 --> 00:28:17.140 had the longest teaching career in our history. 00:28:17.140 --> 00:28:21.380 He was a lecturer who actively lectured for 50 years. 00:28:21.380 --> 00:28:22.800 So he seemed to have done OK. 00:28:22.800 --> 00:28:23.910 He survived. 00:28:23.910 --> 00:28:26.280 So you don't need to worry about what happened to him. 00:28:26.280 --> 00:28:29.340 But we probably wouldn't repeat this. 00:28:29.340 --> 00:28:32.100 It's a little hard to see. 00:28:32.100 --> 00:28:35.400 The bone is easy, right, because x-rays don't 00:28:35.400 --> 00:28:37.500 go through bones very well. 00:28:37.500 --> 00:28:40.560 What you can just barely see is his lips. 00:28:40.560 --> 00:28:45.180 And it's important to watch the lips, too. 00:28:45.180 --> 00:28:50.250 It's also important that his chin is on a chin rest 00:28:50.250 --> 00:28:51.880 to simplify analysis. 00:28:51.880 --> 00:28:56.580 The idea of this was to get quantitative measurements 00:28:56.580 --> 00:29:00.450 to fit the source filter idea. 00:29:00.450 --> 00:29:06.074 OK, so now I'm going to play a film, a recording of him. 00:29:06.074 --> 00:29:10.058 [VIDEO PLAYBACK] 00:29:10.058 --> 00:29:11.552 - Test. 00:29:11.552 --> 00:29:13.046 Test. 00:29:13.046 --> 00:29:14.042 Test. 00:29:14.042 --> 00:29:17.030 [? The tongue. ?] [? The tongue. ?] 00:29:17.030 --> 00:29:18.524 The [INAUDIBLE]. 00:29:18.524 --> 00:29:22.508 The [INAUDIBLE].. [? The neck. ?] [INAUDIBLE].. 00:29:22.508 --> 00:29:24.002 [INAUDIBLE] 00:29:24.002 --> 00:29:25.496 [INAUDIBLE] 00:29:25.496 --> 00:29:26.990 [INAUDIBLE] 00:29:26.990 --> 00:29:28.982 [INAUDIBLE] 00:29:28.982 --> 00:29:30.476 [INAUDIBLE] 00:29:30.476 --> 00:29:33.962 [? Fox. ?] [? Clock. ?] The [INAUDIBLE].. 00:29:33.962 --> 00:29:35.456 The [INAUDIBLE]. 00:29:35.456 --> 00:29:36.950 [INAUDIBLE] 00:29:36.950 --> 00:29:41.432 [? Took. ?] [? Two. ?] [INAUDIBLE].. 00:29:41.432 --> 00:29:44.918 [? Tot. ?] [? Tech. ?] [INAUDIBLE].. 00:29:44.918 --> 00:29:46.910 [INAUDIBLE] 00:29:46.910 --> 00:29:48.404 [INAUDIBLE] 00:29:48.404 --> 00:29:49.898 [INAUDIBLE] 00:29:49.898 --> 00:29:51.392 [INAUDIBLE] 00:29:51.392 --> 00:29:53.384 [INAUDIBLE] 00:29:53.384 --> 00:29:58.025 Why did [INAUDIBLE] set the [INAUDIBLE] on top of his desk? 00:29:58.025 --> 00:29:59.858 I have put [? blood under ?] [? two clean ?] 00:29:59.858 --> 00:30:03.507 [? yellow shoes. ?] 00:30:03.507 --> 00:30:04.340 [END VIDEO PLAYBACK] 00:30:04.340 --> 00:30:05.834 [LAUGHTER] 00:30:05.834 --> 00:30:11.050 PROFESSOR: OK, so what you were supposed to see 00:30:11.050 --> 00:30:14.680 is that the thing that we associate with speech 00:30:14.680 --> 00:30:16.390 is only a small part of it. 00:30:16.390 --> 00:30:17.890 His lips were obviously moving. 00:30:17.890 --> 00:30:20.350 That's what we see. 00:30:20.350 --> 00:30:21.880 But if you were paying attention, 00:30:21.880 --> 00:30:27.389 his tongue was going up and down not a little bit, but a lot. 00:30:27.389 --> 00:30:29.680 So the gap between his tongue and the roof of his mouth 00:30:29.680 --> 00:30:31.385 was going from 0 to about that far. 00:30:34.810 --> 00:30:40.360 The velum back here was opening very broadly on occasion. 00:30:40.360 --> 00:30:42.580 So there was a significant variation 00:30:42.580 --> 00:30:48.860 in the shape of the structure through which the glottis wave 00:30:48.860 --> 00:30:50.470 form was passing. 00:30:50.470 --> 00:30:55.090 And that's the basis of the filtering that gives rise 00:30:55.090 --> 00:30:57.740 to the different speech sounds. 00:30:57.740 --> 00:31:01.030 So to convince you of that, here I 00:31:01.030 --> 00:31:04.936 have a carefully machined item. 00:31:04.936 --> 00:31:06.084 Let's see. 00:31:06.084 --> 00:31:07.000 I don't want this one. 00:31:07.000 --> 00:31:08.000 I want this one. 00:31:08.000 --> 00:31:10.930 So this is a Japanese oo. 00:31:10.930 --> 00:31:13.570 OK, now I don't know Japanese, so I have to just sort of trust 00:31:13.570 --> 00:31:15.153 the guy who made this that it actually 00:31:15.153 --> 00:31:17.110 sounds like a Japanese oo. 00:31:17.110 --> 00:31:20.380 The second one I'll show is a Japanese ee, which actually 00:31:20.380 --> 00:31:22.220 sounds more like an ee to me. 00:31:22.220 --> 00:31:26.260 But anyway, this model was made from measurements of the type 00:31:26.260 --> 00:31:28.300 that I just showed with Ken. 00:31:28.300 --> 00:31:31.420 So the idea was to estimate the size of those cavities 00:31:31.420 --> 00:31:33.520 through which the air was passing, 00:31:33.520 --> 00:31:37.020 and then make, by machining in Plexiglas, 00:31:37.020 --> 00:31:40.520 a structure that has that shape. 00:31:40.520 --> 00:31:43.670 So this was an early test of whether the source filter 00:31:43.670 --> 00:31:44.750 idea works. 00:31:44.750 --> 00:31:48.500 So if that is the explanation for how speech is generated, 00:31:48.500 --> 00:31:51.021 then I ought to be able to take a boring sound 00:31:51.021 --> 00:31:52.895 of the type that's generated by the glottis-- 00:31:52.895 --> 00:31:54.650 [BUZZING SOUND] 00:31:57.545 --> 00:32:00.500 And put it through this, and it should sound more like a vowel. 00:32:00.500 --> 00:32:01.730 OK? 00:32:01.730 --> 00:32:02.290 Got it? 00:32:02.290 --> 00:32:04.200 Know what I'm talking about? 00:32:04.200 --> 00:32:08.952 So this is a Japanese oo. 00:32:08.952 --> 00:32:11.850 [BUZZING SOUND] 00:32:11.850 --> 00:32:14.605 [COMBINES BUZZING SOUND WITH 'OO' SOUND] 00:32:14.605 --> 00:32:16.230 I don't know if anybody knows Japanese. 00:32:16.230 --> 00:32:17.856 I don't know if that sounds like an oo. 00:32:17.856 --> 00:32:19.563 Does anybody know Japanese, and does that 00:32:19.563 --> 00:32:20.580 sound like an oo or not? 00:32:23.410 --> 00:32:24.696 OK, I'll pass. 00:32:24.696 --> 00:32:28.590 [BUZZING SOUND] 00:32:28.590 --> 00:32:30.115 This is an ee. 00:32:30.115 --> 00:32:33.390 OK, now notice that the ee looks very different. 00:32:33.390 --> 00:32:34.770 Right? 00:32:34.770 --> 00:32:37.470 The question is whether that's a big enough difference 00:32:37.470 --> 00:32:44.130 to make the difference between an oo and an ee. 00:32:44.130 --> 00:32:47.270 I'm pressing the same button, nothing up my sleeve, nothing-- 00:32:47.270 --> 00:32:50.048 OK, so same button. 00:32:50.048 --> 00:32:53.520 [COMBINES BUZZING SOUND WITH 'EE' SOUND] 00:33:00.640 --> 00:33:02.570 OK, so what you're supposed to be convinced of 00:33:02.570 --> 00:33:07.780 is there is enough information in the shape 00:33:07.780 --> 00:33:12.440 of the vocal structures to account 00:33:12.440 --> 00:33:14.755 for the difference in the sounds. 00:33:17.380 --> 00:33:23.650 Now of course, we don't really care about the acoustics 00:33:23.650 --> 00:33:28.600 if we're trying to, for example, synthesize or analyze speech. 00:33:28.600 --> 00:33:30.670 We don't particularly care about that. 00:33:30.670 --> 00:33:32.650 We do like to know that there is a theory that underlies it, 00:33:32.650 --> 00:33:33.149 right? 00:33:33.149 --> 00:33:36.070 And there's a very sound physical basis 00:33:36.070 --> 00:33:39.610 for why we should think about the source filter idea. 00:33:39.610 --> 00:33:43.690 When I say source filter, source filter-- 00:33:43.690 --> 00:33:47.350 so everybody calls it the source filter model of speech. 00:33:47.350 --> 00:33:48.850 So is there any good physical reason 00:33:48.850 --> 00:33:50.980 for why that should be true? 00:33:50.980 --> 00:33:56.740 Of course, what we care about is the frequency response. 00:33:56.740 --> 00:33:58.920 So here what's showed is measurements 00:33:58.920 --> 00:34:01.860 of frequency responses taken from speakers. 00:34:01.860 --> 00:34:03.780 So now we don't do the x-ray thing. 00:34:03.780 --> 00:34:09.179 All we do is record somebody saying heed, had, hood, 00:34:09.179 --> 00:34:12.380 haw'd, who'd. 00:34:12.380 --> 00:34:16.070 And we look at men, women, and children, 00:34:16.070 --> 00:34:22.100 and we characterize how their frequency responses change 00:34:22.100 --> 00:34:24.210 when they make those different sounds. 00:34:24.210 --> 00:34:30.889 So what's showed here is that you get a relatively good fit 00:34:30.889 --> 00:34:36.409 by thinking about the frequency response having three formants. 00:34:36.409 --> 00:34:39.770 The formants are the peak frequencies. 00:34:39.770 --> 00:34:42.886 There's a theory, which I won't go into, 00:34:42.886 --> 00:34:44.510 for how you take this shape and turn it 00:34:44.510 --> 00:34:47.540 into a formant frequency. 00:34:47.540 --> 00:34:51.949 And given just the formant frequencies, or given 00:34:51.949 --> 00:34:54.650 the frequency response measured at uniform spacing 00:34:54.650 --> 00:34:57.260 across frequencies, there is a theory 00:34:57.260 --> 00:35:02.470 for how you can generate the smooth line, which is really-- 00:35:02.470 --> 00:35:05.050 this is an 11th order [? fit, ?] which 00:35:05.050 --> 00:35:09.550 means that there are 11 poles and no zeros. 00:35:09.550 --> 00:35:12.310 So what you do to get this shape then 00:35:12.310 --> 00:35:14.970 is take the locations and amplitudes 00:35:14.970 --> 00:35:20.020 of the formant frequencies and do a fit using poles. 00:35:20.020 --> 00:35:24.270 And so here's a table showing measured formant frequencies, 00:35:24.270 --> 00:35:29.440 F1, F2, and F3, for whatever, six different 00:35:29.440 --> 00:35:34.720 sounds for three different categories of speakers. 00:35:34.720 --> 00:35:35.980 OK? 00:35:35.980 --> 00:35:39.070 And that's kind of a complete analysis then in terms 00:35:39.070 --> 00:35:41.560 of the source filter idea. 00:35:41.560 --> 00:35:45.610 So this figure summarizes the idea. 00:35:45.610 --> 00:35:47.280 We think about source filters. 00:35:47.280 --> 00:35:49.450 So the source is the glottis. 00:35:49.450 --> 00:35:55.290 The filter is the formants created by the throat. 00:35:55.290 --> 00:35:58.900 And speech is the thing that comes out of the source filter. 00:35:58.900 --> 00:36:02.140 The source is some periodic waveform 00:36:02.140 --> 00:36:07.330 caused by the banging together of the vocal folds. 00:36:07.330 --> 00:36:15.430 The filter is the frequency response of the throat. 00:36:15.430 --> 00:36:18.310 And the result, then, is just passing this glottis wave 00:36:18.310 --> 00:36:21.160 form-- so this is a measured-- 00:36:21.160 --> 00:36:23.660 by sticking a microphone in somebody's throat, 00:36:23.660 --> 00:36:26.740 this is a measurement of what the glottis acoustics looks 00:36:26.740 --> 00:36:28.300 like. 00:36:28.300 --> 00:36:33.910 This is a Fourier decomposition of that periodic waveform. 00:36:33.910 --> 00:36:37.360 Then this is the frequency response of that thing. 00:36:37.360 --> 00:36:41.560 And this is the Fourier coefficients 00:36:41.560 --> 00:36:44.400 of the output for different sounds. 00:36:44.400 --> 00:36:45.510 So here is the frequency. 00:36:45.510 --> 00:36:50.230 So the same glottis signal underlies and ee sound and ah 00:36:50.230 --> 00:36:54.670 sound and generates two different spectra. 00:36:54.670 --> 00:36:57.100 We call that combination of magnitudes and angles. 00:36:57.100 --> 00:36:59.505 We call that the Fourier spectrum. 00:36:59.505 --> 00:37:00.880 So you get two different spectra, 00:37:00.880 --> 00:37:04.550 depending on the filter shape. 00:37:04.550 --> 00:37:07.140 OK? 00:37:07.140 --> 00:37:09.330 And that's the basis-- 00:37:09.330 --> 00:37:11.490 this theory, this source filter idea-- 00:37:11.490 --> 00:37:13.320 is the basis of the current technology 00:37:13.320 --> 00:37:16.690 for speech recognition and speech production. 00:37:16.690 --> 00:37:18.300 So I actually cheated. 00:37:18.300 --> 00:37:19.860 Those sounds that I played earlier-- 00:37:19.860 --> 00:37:22.410 bit, bat, bought, beat, all those things-- 00:37:22.410 --> 00:37:25.090 those were actually synthetic speech. 00:37:25.090 --> 00:37:27.510 OK, all I did is I ran a speech synthesizer, 00:37:27.510 --> 00:37:30.180 and I said, synthesize bit. 00:37:30.180 --> 00:37:32.850 So that was really a synthesized thing. 00:37:32.850 --> 00:37:34.920 That was not a real person. 00:37:34.920 --> 00:37:41.250 And so the synthesizer used this theory 00:37:41.250 --> 00:37:43.710 in order to generate this synthetic speech. 00:37:43.710 --> 00:37:47.650 We also use this theory in order to recognize speech. 00:37:47.650 --> 00:37:49.650 And you'll do a homework problem in Homework 10, 00:37:49.650 --> 00:37:53.700 I think it is, in which you'll build the primitive front 00:37:53.700 --> 00:37:59.360 end of a speech recognizer using this theory. 00:37:59.360 --> 00:38:01.940 And I'll give you a couple of utterances of different vowels 00:38:01.940 --> 00:38:05.180 and you'll have to classify which vowel is being said, 00:38:05.180 --> 00:38:08.000 according to some automatic speech recognizer 00:38:08.000 --> 00:38:10.310 based on this theory. 00:38:10.310 --> 00:38:13.430 The theory is also just fun, because a theory 00:38:13.430 --> 00:38:16.132 lets us figure out anomalies. 00:38:16.132 --> 00:38:17.840 So when somebody has a speech impediment, 00:38:17.840 --> 00:38:20.131 for example, when I did, when I was a little kid I did. 00:38:20.131 --> 00:38:21.560 And I was sent to speech school. 00:38:21.560 --> 00:38:23.310 Now they do a much better job because they 00:38:23.310 --> 00:38:27.320 do analysis to figure out what I'm doing wrong, 00:38:27.320 --> 00:38:29.120 using this sort of source filter idea. 00:38:29.120 --> 00:38:31.250 We can also use the source filter idea 00:38:31.250 --> 00:38:35.390 to understand paradoxes. 00:38:35.390 --> 00:38:38.990 So for example, I've told you before I work on hearing aids. 00:38:38.990 --> 00:38:40.820 I tried to make hearing aids hear. 00:38:40.820 --> 00:38:45.410 And so people with hearing deficiencies like mine, 00:38:45.410 --> 00:38:47.480 where I have sort of progressive age-related 00:38:47.480 --> 00:38:50.120 because I'm old, right, that's what happens. 00:38:50.120 --> 00:38:52.940 I have age-related hearing loss, which 00:38:52.940 --> 00:38:55.220 means that I'm losing high frequencies. 00:38:55.220 --> 00:38:56.900 I'm less sensitive to high frequencies. 00:38:56.900 --> 00:39:00.770 People like me, which is the vast majority of people 00:39:00.770 --> 00:39:03.620 my age-- 00:39:03.620 --> 00:39:07.580 it's easier to understand male speech than female speech. 00:39:07.580 --> 00:39:08.080 Why? 00:39:10.990 --> 00:39:12.080 Higher frequencies. 00:39:12.080 --> 00:39:16.480 So those higher frequencies shift 00:39:16.480 --> 00:39:18.066 some of the important stuff that I 00:39:18.066 --> 00:39:19.690 should be listening to into frequencies 00:39:19.690 --> 00:39:21.640 I don't hear anymore. 00:39:21.640 --> 00:39:22.630 Right? 00:39:22.630 --> 00:39:24.640 So that's a way of using this theory 00:39:24.640 --> 00:39:27.640 to try to understand what's wrong with me. 00:39:27.640 --> 00:39:30.310 But there's also things-- it's not just me. 00:39:30.310 --> 00:39:33.700 Normal people have trouble distinguishing female speech, 00:39:33.700 --> 00:39:36.621 especially in taxing environments, and one of those 00:39:36.621 --> 00:39:37.120 is singing. 00:39:40.170 --> 00:39:43.030 So if you consider altos and sopranos, 00:39:43.030 --> 00:39:46.350 sopranos are like, the worst, right? 00:39:46.350 --> 00:39:49.230 Because they are not only female, but they're at the high 00:39:49.230 --> 00:39:51.490 end the females. 00:39:51.490 --> 00:39:54.180 And there are those who complain about not 00:39:54.180 --> 00:39:57.360 being able to understand female singers. 00:39:57.360 --> 00:39:59.640 OK, so here's a demo that will help 00:39:59.640 --> 00:40:04.120 us to understand whether that's a valid kind of a criticism 00:40:04.120 --> 00:40:04.620 or not. 00:40:04.620 --> 00:40:10.110 So what I've got is a professional singer 00:40:10.110 --> 00:40:13.080 singing, la, la, la, la-- 00:40:13.080 --> 00:40:14.210 on a scale. 00:40:14.210 --> 00:40:16.430 So from low frequency to high frequency, 00:40:16.430 --> 00:40:18.180 then a different sound, a different sound, 00:40:18.180 --> 00:40:20.877 a different sound, a different sound. 00:40:20.877 --> 00:40:22.460 So the first thing that I want to do-- 00:40:22.460 --> 00:40:24.376 I want you to listen to those different sounds 00:40:24.376 --> 00:40:27.439 as she goes across the scale. 00:40:27.439 --> 00:40:29.480 Then I'm going to play just the low ones and just 00:40:29.480 --> 00:40:31.990 the high ones, just the low frequency ones and just 00:40:31.990 --> 00:40:32.990 the high frequency ones. 00:40:32.990 --> 00:40:36.080 So first, the different scales-- 00:40:36.080 --> 00:40:39.834 la, la, lore, loo, lee, OK? 00:40:39.834 --> 00:40:40.500 [AUDIO PLAYBACK] 00:40:40.500 --> 00:40:47.500 - La, la, la, la, la, la, la, la, la, la, la, la, la, la, la, 00:40:47.500 --> 00:40:48.500 la. 00:40:48.500 --> 00:40:50.500 La. 00:40:50.500 --> 00:40:55.000 Lore, lore, lore, lore, lore, lore, lore, lore, lore, lore, 00:40:55.000 --> 00:40:58.500 lore, lore, lore, lore, lore, lore, lore. 00:40:58.500 --> 00:41:00.000 Lore. 00:41:00.000 --> 00:41:05.750 Loo, loo, loo, loo, loo, loo, loo, loo, loo, loo, loo, loo, 00:41:05.750 --> 00:41:07.500 loo, loo, loo, loo. 00:41:07.500 --> 00:41:09.500 Loo. 00:41:09.500 --> 00:41:15.250 Ler, ler, ler, ler, ler, ler, ler, ler, ler, ler, ler, ler, 00:41:15.250 --> 00:41:17.000 ler, ler, ler, ler. 00:41:17.000 --> 00:41:19.000 Ler. 00:41:19.000 --> 00:41:24.750 Lee, lee, lee, lee, lee, lee, lee, lee, lee, lee, lee, lee, 00:41:24.750 --> 00:41:26.500 lee, lee, lee, lee. 00:41:26.500 --> 00:41:28.500 Lee. 00:41:28.500 --> 00:41:30.000 [END PLAYBACK] 00:41:30.000 --> 00:41:32.770 PROFESSOR: OK, so now what I've done 00:41:32.770 --> 00:41:37.420 is I've sliced out the lowest frequency, the very first 00:41:37.420 --> 00:41:40.810 of the scale from each of the sounds 00:41:40.810 --> 00:41:45.490 and pasted them together to get the low frequency run. 00:41:45.490 --> 00:41:51.470 And then I took out the high ones and pasted them together. 00:41:51.470 --> 00:41:52.110 OK? 00:41:52.110 --> 00:41:56.990 Exactly the same sounds, just played in a different order. 00:41:56.990 --> 00:41:59.540 So first the low frequency ones. 00:41:59.540 --> 00:42:01.022 SINGER: La. 00:42:01.022 --> 00:42:02.998 Lore. 00:42:02.998 --> 00:42:04.974 Loo. 00:42:04.974 --> 00:42:06.950 Ler. 00:42:06.950 --> 00:42:09.914 Lee. 00:42:09.914 --> 00:42:13.026 And the high frequency ones. 00:42:13.026 --> 00:42:15.022 La. 00:42:15.022 --> 00:42:16.519 Lore. 00:42:16.519 --> 00:42:18.515 Loo. 00:42:18.515 --> 00:42:20.511 Ler. 00:42:20.511 --> 00:42:21.509 Lee. 00:42:21.509 --> 00:42:26.000 [LAUGHTER] 00:42:26.000 --> 00:42:32.570 PROFESSOR: It's not her fault. She's doing everything right. 00:42:32.570 --> 00:42:34.030 And you can see that. 00:42:34.030 --> 00:42:40.010 Here is, again, a Python program analyzing those same segments. 00:42:40.010 --> 00:42:46.040 So what's shown here is the ee, the filter derived from the ee, 00:42:46.040 --> 00:42:50.360 by thinking about the lee, lee, lee, lee, lee, lee-- 00:42:50.360 --> 00:42:53.990 by looking at that sequence, and averaging 00:42:53.990 --> 00:42:57.620 across the frequencies. 00:42:57.620 --> 00:43:00.890 So here's the filter. 00:43:00.890 --> 00:43:08.180 Here's the filtered glottis spectrum for a low frequency 00:43:08.180 --> 00:43:10.880 and intermediate frequency and high frequency. 00:43:10.880 --> 00:43:14.340 What's the difference between the low, middle, and high? 00:43:17.330 --> 00:43:19.870 What's characteristically different at low and high? 00:43:26.110 --> 00:43:31.880 AUDIENCE: [INAUDIBLE] frequency, like high amplitude. 00:43:31.880 --> 00:43:37.230 PROFESSOR: So if you look at the low frequency, the low pitch, 00:43:37.230 --> 00:43:42.560 there are more frequency components in a given range. 00:43:42.560 --> 00:43:45.400 So if I say analyze the frequencies between 0 00:43:45.400 --> 00:43:50.110 and 1,000 hertz, 1,000 cycles per second, 00:43:50.110 --> 00:43:55.210 there are more lines when you have a low frequency. 00:43:55.210 --> 00:43:57.760 And so you get the density of the lines 00:43:57.760 --> 00:44:00.340 is greater for the low frequency utterance 00:44:00.340 --> 00:44:04.200 than it is for the high frequency utterance. 00:44:04.200 --> 00:44:09.450 The low frequency utterance is spaced close enough 00:44:09.450 --> 00:44:17.240 that you can clearly figure out this pattern from that spacing, 00:44:17.240 --> 00:44:20.660 because there are multiple lines per peak. 00:44:20.660 --> 00:44:23.430 The problem is that the speech waveforms 00:44:23.430 --> 00:44:26.870 have very sharp resonances. 00:44:26.870 --> 00:44:29.510 The peaks are narrow. 00:44:29.510 --> 00:44:32.720 So that as you go to a higher frequency, 00:44:32.720 --> 00:44:36.380 now it's very hard to see. 00:44:36.380 --> 00:44:39.140 So where there was two lines characterizing this guy, 00:44:39.140 --> 00:44:41.860 now there's one. 00:44:41.860 --> 00:44:44.460 And at the highest frequency, there's nothing there. 00:44:47.680 --> 00:44:51.340 Similarly with these peaks, again, several lines 00:44:51.340 --> 00:44:52.450 representing each peak. 00:44:52.450 --> 00:44:56.950 One line representing-- nothing representing this peak, nothing 00:44:56.950 --> 00:44:58.630 representing that peak. 00:44:58.630 --> 00:45:05.270 There is nothing about ee in that signal. 00:45:05.270 --> 00:45:08.520 And if you do the same analysis for ah, 00:45:08.520 --> 00:45:12.030 you get the same result. There's nothing about ee, 00:45:12.030 --> 00:45:13.740 and there's nothing about ah. 00:45:13.740 --> 00:45:15.940 There's just nothing there. 00:45:15.940 --> 00:45:19.470 There's no way anybody is going to tell those two sounds apart. 00:45:19.470 --> 00:45:24.840 So if the singer put her voice, her vocal tract, 00:45:24.840 --> 00:45:29.750 in precisely the right location, there 00:45:29.750 --> 00:45:32.330 would be no difference between those sounds, 00:45:32.330 --> 00:45:35.330 OK, regardless of what the director said. 00:45:35.330 --> 00:45:37.620 OK, so that's the problem. 00:45:37.620 --> 00:45:39.950 So that's a way of using the Fourier 00:45:39.950 --> 00:45:44.570 analysis to gain some insight into some anomalous situations. 00:45:44.570 --> 00:45:45.319 Yeah? 00:45:45.319 --> 00:45:46.777 AUDIENCE: Does this have more to do 00:45:46.777 --> 00:45:51.070 with the rate at which you're sampling? 00:45:51.070 --> 00:45:52.600 PROFESSOR: No. 00:45:52.600 --> 00:45:56.620 It has to do only with the frequency content 00:45:56.620 --> 00:45:59.800 of the glottis waveform. 00:45:59.800 --> 00:46:01.450 You can think about it as sampling. 00:46:01.450 --> 00:46:09.460 And that's a good insight, because the Fourier series only 00:46:09.460 --> 00:46:13.990 has components at integer multiples of a base frequency. 00:46:13.990 --> 00:46:18.680 So that means we're sampling in frequency, not in time. 00:46:18.680 --> 00:46:23.860 So we have this potentially continuous frequency response, 00:46:23.860 --> 00:46:26.230 which is characterizing this. 00:46:26.230 --> 00:46:27.250 That is continuous. 00:46:27.250 --> 00:46:29.680 I could excite this at any frequency that I want to. 00:46:29.680 --> 00:46:32.920 But the glottis waveform of the singer 00:46:32.920 --> 00:46:37.150 is only sampling that at particular frequencies-- 00:46:37.150 --> 00:46:45.010 C, C prime, C double prime, B, B prime, B double prime, right? 00:46:45.010 --> 00:46:47.410 So there's only certain frequencies at which 00:46:47.410 --> 00:46:51.625 the singer is sampling this. 00:46:51.625 --> 00:46:53.750 So there is a way of thinking about it as sampling. 00:46:53.750 --> 00:46:55.904 But it's not sampling due to my ADD converter 00:46:55.904 --> 00:46:56.820 or anything like that. 00:46:56.820 --> 00:46:59.450 It's sampling in frequency. 00:46:59.450 --> 00:47:03.790 So the point is that this kind of a source filter idea, 00:47:03.790 --> 00:47:05.830 and more generally, the filter idea, 00:47:05.830 --> 00:47:08.770 is such a powerful representation 00:47:08.770 --> 00:47:10.270 that next time we'll think about how 00:47:10.270 --> 00:47:15.020 to do the same sort of thing for non-periodic [? stimuli. ?] 00:47:15.020 --> 00:47:16.860 See you then.