WEBVTT 00:00:00.040 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:03.880 Commons license. 00:00:03.880 --> 00:00:06.920 Your support will help MIT OpenCourseWare continue to 00:00:06.920 --> 00:00:10.570 offer high quality educational resources for free. 00:00:10.570 --> 00:00:13.470 To make a donation or view additional materials from 00:00:13.470 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.515 ocw.mit.edu. 00:00:20.515 --> 00:00:56.490 [MUSIC PLAYING] 00:00:56.490 --> 00:00:58.660 PROFESSOR: Last time, we introduced the Laplace 00:00:58.660 --> 00:01:03.310 transform as a generalization of the Fourier transform, and, 00:01:03.310 --> 00:01:07.530 just as a reminder, the Laplace transform expression 00:01:07.530 --> 00:01:13.720 as we developed it is this integral, very much similar to 00:01:13.720 --> 00:01:16.870 the Fourier transform integral, except with a more 00:01:16.870 --> 00:01:19.210 general complex variable. 00:01:19.210 --> 00:01:22.940 And, in fact, we developed and talked about the relationship 00:01:22.940 --> 00:01:26.260 between the Laplace transform and the Fourier transform. 00:01:26.260 --> 00:01:30.880 In particular, the Laplace transform with the Laplace 00:01:30.880 --> 00:01:36.590 transform variable s, purely imaginary, in fact, reduces to 00:01:36.590 --> 00:01:38.970 the Fourier transform. 00:01:38.970 --> 00:01:43.320 Or, more generally, with the Laplace transform variable as 00:01:43.320 --> 00:01:48.550 a complex number, the Laplace transform is the Fourier 00:01:48.550 --> 00:01:53.430 transform of the corresponding time function with an 00:01:53.430 --> 00:01:55.280 exponential weighting. 00:01:55.280 --> 00:01:59.730 And, also, as you should recall, the exponential 00:01:59.730 --> 00:02:04.870 waiting introduced the notion that the Laplace transform may 00:02:04.870 --> 00:02:08.820 converge for some values of sigma and perhaps not for 00:02:08.820 --> 00:02:10.360 other values of sigma. 00:02:10.360 --> 00:02:15.890 So associated with the Laplace transform was what we refer to 00:02:15.890 --> 00:02:17.765 as the region of convergence. 00:02:20.400 --> 00:02:25.120 Now just as with the Fourier transform, there are a number 00:02:25.120 --> 00:02:28.470 of properties of the Laplace transform that are extremely 00:02:28.470 --> 00:02:35.250 useful in describing and analyzing signals and systems. 00:02:35.250 --> 00:02:39.200 For example, one of the properties that we, in fact, 00:02:39.200 --> 00:02:42.740 took advantage of in our discussion last time was the 00:02:42.740 --> 00:02:46.250 linearly the linearity property, which says, in 00:02:46.250 --> 00:02:49.980 essence, that the Laplace transform of the linear 00:02:49.980 --> 00:02:54.830 combination of two time functions is the same linear 00:02:54.830 --> 00:03:00.260 combination of the associated Laplace transforms. 00:03:00.260 --> 00:03:03.640 Also, there is a very important and useful property, 00:03:03.640 --> 00:03:05.980 which tells us how the 00:03:05.980 --> 00:03:09.650 derivative of a time function-- 00:03:09.650 --> 00:03:12.090 rather, the Laplace transform of the derivative-- 00:03:12.090 --> 00:03:14.710 is related to the Laplace transform. 00:03:14.710 --> 00:03:19.350 In particular, the Laplace transform of the derivative is 00:03:19.350 --> 00:03:23.320 the Laplace transform x of t multiplied by s. 00:03:23.320 --> 00:03:27.200 And, as you can see by just setting s equal to j omega, in 00:03:27.200 --> 00:03:30.230 fact, this reduces to the corresponding Fourier 00:03:30.230 --> 00:03:32.740 transform property. 00:03:32.740 --> 00:03:37.830 And a third property that we'll make frequent use of is 00:03:37.830 --> 00:03:40.390 referred to as the convolution property. 00:03:40.390 --> 00:03:43.910 Again, a generalization of the convolution property for 00:03:43.910 --> 00:03:45.700 Fourier transforms. 00:03:45.700 --> 00:03:49.270 Here the convolution property says that the Laplace 00:03:49.270 --> 00:03:53.390 transform of the convolution of two time functions is the 00:03:53.390 --> 00:03:58.620 product of the associated Laplace transforms. 00:03:58.620 --> 00:04:04.610 Now it's important at some point to think carefully about 00:04:04.610 --> 00:04:09.400 the region of convergence as we discuss these properties. 00:04:09.400 --> 00:04:13.430 And let me just draw your attention to the fact that in 00:04:13.430 --> 00:04:17.970 discussing properties fully and in detail, one has to pay 00:04:17.970 --> 00:04:25.090 attention not just to how the algebraic expression changes, 00:04:25.090 --> 00:04:27.990 but also what the consequences are for the region of 00:04:27.990 --> 00:04:30.990 convergence, and that's discussed in somewhat more 00:04:30.990 --> 00:04:35.130 detail in the text and I won't do that here. 00:04:35.130 --> 00:04:40.750 Now the convolution property leads to, of course, a very 00:04:40.750 --> 00:04:46.300 important and useful mechanism for dealing with linear time 00:04:46.300 --> 00:04:50.330 invariant systems, very much as the Fourier transform did. 00:04:50.330 --> 00:04:54.990 In particular, the convolution property tells us that if we 00:04:54.990 --> 00:04:59.280 have a linear time invariant system, the output in the time 00:04:59.280 --> 00:05:02.520 domain is the convolution of the input 00:05:02.520 --> 00:05:04.390 and the impulse response. 00:05:04.390 --> 00:05:08.450 In the Laplace transform domain, the Laplace transform 00:05:08.450 --> 00:05:12.530 of the output is the Laplace transform of the impulse 00:05:12.530 --> 00:05:16.760 response times the Laplace transform of the input. 00:05:16.760 --> 00:05:20.700 And again, this is a generalization of the 00:05:20.700 --> 00:05:24.220 corresponding property for Fourier transforms. 00:05:24.220 --> 00:05:28.020 In the case of the Fourier transform, the Fourier 00:05:28.020 --> 00:05:30.720 transform [? of ?] the impulse response we refer to as the 00:05:30.720 --> 00:05:32.430 frequency response. 00:05:32.430 --> 00:05:36.290 In the more general case with Laplace transforms, it's 00:05:36.290 --> 00:05:38.950 typical to refer to the Laplace transform of the 00:05:38.950 --> 00:05:43.530 impulse response as the system function. 00:05:43.530 --> 00:05:52.500 Now in talking about the system function, some issues 00:05:52.500 --> 00:05:54.120 of the region of convergence-- 00:05:54.120 --> 00:05:55.920 and for that matter, location of poles 00:05:55.920 --> 00:05:57.940 of the system function-- 00:05:57.940 --> 00:06:03.010 are closely tied in and related to issues of whether 00:06:03.010 --> 00:06:05.940 the system is stable and causal. 00:06:05.940 --> 00:06:08.570 And in fact, there's some useful statements that can be 00:06:08.570 --> 00:06:12.610 made that play an important role throughout the further 00:06:12.610 --> 00:06:13.640 discussion. 00:06:13.640 --> 00:06:19.030 For example, we know from previous discussions that 00:06:19.030 --> 00:06:21.380 there's a condition for stability of a system, which 00:06:21.380 --> 00:06:25.390 is absolute integrability of the impulse response. 00:06:25.390 --> 00:06:29.650 And that, in fact, is the same condition for convergence of 00:06:29.650 --> 00:06:33.180 the Fourier transform of the impulse response. 00:06:33.180 --> 00:06:39.350 What that says, really, is that if a system is stable, 00:06:39.350 --> 00:06:43.530 then the region of convergence of the system function must 00:06:43.530 --> 00:06:45.710 include the j omega axis. 00:06:45.710 --> 00:06:49.390 Which, of course, is where the Laplace transform reduces to 00:06:49.390 --> 00:06:52.380 the Fourier transform. 00:06:52.380 --> 00:06:56.540 So that relates the region of convergence and stability. 00:06:56.540 --> 00:07:00.790 Also, you recall from last time that we talked about the 00:07:00.790 --> 00:07:03.740 region of convergence associated with right sided 00:07:03.740 --> 00:07:05.050 time functions. 00:07:05.050 --> 00:07:07.300 In particular for a right sided time function, the 00:07:07.300 --> 00:07:10.030 region of convergence must be to the right of 00:07:10.030 --> 00:07:12.330 the rightmost pole. 00:07:12.330 --> 00:07:17.970 Well, if, in fact, we have a system that's causal, then 00:07:17.970 --> 00:07:22.040 that causality imposes the condition that the impulse 00:07:22.040 --> 00:07:23.820 response be right sided. 00:07:23.820 --> 00:07:27.510 And so, in fact, for causality, we would have a 00:07:27.510 --> 00:07:29.960 region of convergence associated with the system 00:07:29.960 --> 00:07:33.780 function, which is to the right of the rightmost pole. 00:07:33.780 --> 00:07:37.710 Now interestingly and very important is the consequence, 00:07:37.710 --> 00:07:40.380 if you put those two statements together, in 00:07:40.380 --> 00:07:45.620 particular, you're led to the conclusion that for stable 00:07:45.620 --> 00:07:51.770 causal systems, all the poles must be in the left half of 00:07:51.770 --> 00:07:53.700 the s-plane. 00:07:53.700 --> 00:07:54.870 What's the reason? 00:07:54.870 --> 00:07:59.010 The reason, of course, is that if the system is stable and 00:07:59.010 --> 00:08:01.720 causal, the region of convergence must be to the 00:08:01.720 --> 00:08:03.520 right of the rightmost pole. 00:08:03.520 --> 00:08:06.460 It must include the j omega axis. 00:08:06.460 --> 00:08:09.170 Obviously then, all the poles must be in the left half of 00:08:09.170 --> 00:08:10.660 the s-plane. 00:08:10.660 --> 00:08:14.400 And again, that's an issue that is discussed somewhat 00:08:14.400 --> 00:08:18.650 more carefully and in more detail in the text. 00:08:18.650 --> 00:08:23.840 Now, the properties that we're talking about here are not the 00:08:23.840 --> 00:08:26.620 only properties, there are many others. 00:08:26.620 --> 00:08:30.200 But these properties, in particular, provide the 00:08:30.200 --> 00:08:30.880 mechanism-- 00:08:30.880 --> 00:08:33.409 as they did with Fourier transforms-- 00:08:33.409 --> 00:08:37.280 for turning linear constant coefficient differential 00:08:37.280 --> 00:08:41.840 equations into algebraic equations and, corresponding, 00:08:41.840 --> 00:08:46.250 lead to a mechanism for dealing with and solving 00:08:46.250 --> 00:08:48.890 linear constant coefficient differential equations. 00:08:48.890 --> 00:08:53.280 And I'd like to illustrate that by looking at both first 00:08:53.280 --> 00:08:56.010 order and second order differential equations. 00:08:56.010 --> 00:08:59.400 Let's begin, first of all, with a first order 00:08:59.400 --> 00:09:01.940 differential equation. 00:09:01.940 --> 00:09:06.500 So what we're talking about is a first order system. 00:09:06.500 --> 00:09:11.930 What I mean by that is a system that's characterized by 00:09:11.930 --> 00:09:14.870 a first order differential equation. 00:09:14.870 --> 00:09:19.460 And if we apply to this equation the differentiation 00:09:19.460 --> 00:09:22.580 property, then the derivative-- 00:09:22.580 --> 00:09:25.910 the Laplace transform of the derivative is s times the 00:09:25.910 --> 00:09:28.540 Laplace transform of the time function. 00:09:28.540 --> 00:09:32.580 The linearity property allows us to combine these together. 00:09:32.580 --> 00:09:36.520 And so, consequently, applying the Laplace transform to this 00:09:36.520 --> 00:09:40.780 equation leads us to this algebraic equation, and 00:09:40.780 --> 00:09:43.990 following that through, leads us to the statement that the 00:09:43.990 --> 00:09:47.430 Laplace transform of the output is one over s plus a 00:09:47.430 --> 00:09:50.620 times the Laplace transform of the input. 00:09:50.620 --> 00:09:54.530 We know from the convolution property that this Laplace 00:09:54.530 --> 00:09:58.020 transform is the system function times x of s. 00:09:58.020 --> 00:10:02.530 And so, one over s plus a is the system function or 00:10:02.530 --> 00:10:04.670 equivalently, the Laplace transform 00:10:04.670 --> 00:10:07.420 of the impulse response. 00:10:07.420 --> 00:10:12.400 So, we can determine the impulse response by taking the 00:10:12.400 --> 00:10:17.560 inverse Laplace transform of h of s given by 00:10:17.560 --> 00:10:20.890 one over s plus a. 00:10:20.890 --> 00:10:24.600 Well, we can do that using the inspection method, which is 00:10:24.600 --> 00:10:28.620 one way that we have of doing inverse Laplace transforms. 00:10:28.620 --> 00:10:32.380 The question is then, what time function has a Laplace 00:10:32.380 --> 00:10:35.640 transform which is one over s plus a? 00:10:35.640 --> 00:10:38.280 The problem that we run into is that there are 00:10:38.280 --> 00:10:40.910 two answers to that. 00:10:40.910 --> 00:10:46.020 one over s plus a is the Laplace transform of an 00:10:46.020 --> 00:10:52.250 exponential for positive time, but one over s plus a is also 00:10:52.250 --> 00:10:57.550 the Laplace transform of an exponential for negative time. 00:10:57.550 --> 00:11:01.340 Which one of these do we end up picking? 00:11:01.340 --> 00:11:05.200 Well, recall that the difference between these was 00:11:05.200 --> 00:11:06.810 in their region of convergence. 00:11:06.810 --> 00:11:11.080 And in fact, in this case, this corresponded to a region 00:11:11.080 --> 00:11:14.280 of convergence, which was the real part of s 00:11:14.280 --> 00:11:16.740 greater than minus a. 00:11:16.740 --> 00:11:19.850 In this case, this was the corresponding Laplace 00:11:19.850 --> 00:11:24.100 transform, provided that the real part of s is 00:11:24.100 --> 00:11:26.970 less than minus a. 00:11:26.970 --> 00:11:30.400 So we have to decide which region of convergence that we 00:11:30.400 --> 00:11:33.980 pick and it's not the differential equation that 00:11:33.980 --> 00:11:37.650 will tell us that, it's something else that has to 00:11:37.650 --> 00:11:39.640 give us that information. 00:11:39.640 --> 00:11:40.520 What could it be? 00:11:40.520 --> 00:11:45.310 Well, what it might be is the additional information that 00:11:45.310 --> 00:11:49.210 the system is either stable or causal. 00:11:49.210 --> 00:11:52.560 So for example, if the system was causal, we would know that 00:11:52.560 --> 00:11:55.950 the region of convergence is to the right of the pole and 00:11:55.950 --> 00:11:59.320 that would correspond, then, to this 00:11:59.320 --> 00:12:02.630 being the impulse response. 00:12:02.630 --> 00:12:06.270 Whereas, with a negative-- 00:12:06.270 --> 00:12:08.320 I'm sorry with a positive-- 00:12:08.320 --> 00:12:12.820 if we knew that the system, let's say, was non-causal, 00:12:12.820 --> 00:12:17.280 then we would associate with this region of convergence and 00:12:17.280 --> 00:12:20.600 we would know then that this is the impulse response. 00:12:20.600 --> 00:12:26.060 So a very important point is that what we see is that the 00:12:26.060 --> 00:12:30.850 linear constant coefficient differential equation gives us 00:12:30.850 --> 00:12:36.770 the algebraic expression for the system function, but does 00:12:36.770 --> 00:12:39.190 not tell us about the region of convergence. 00:12:39.190 --> 00:12:41.690 We get the reach of convergence from some 00:12:41.690 --> 00:12:43.540 auxiliary information. 00:12:43.540 --> 00:12:44.860 What is that information? 00:12:44.860 --> 00:12:49.090 Well, it might, for example, be knowledge that the system 00:12:49.090 --> 00:12:52.930 is perhaps stable, which tells us that the region of 00:12:52.930 --> 00:12:57.290 convergence includes the j omega axis, or perhaps causal, 00:12:57.290 --> 00:13:00.920 which tells us that the region of convergence is to the right 00:13:00.920 --> 00:13:02.340 of the rightmost pole. 00:13:02.340 --> 00:13:06.190 So it's the auxiliary information that specifies for 00:13:06.190 --> 00:13:08.010 us the region of convergence. 00:13:08.010 --> 00:13:09.340 Very important point. 00:13:09.340 --> 00:13:11.980 The differential equation by itself does not completely 00:13:11.980 --> 00:13:16.250 specify the system, it only essentially tells us what the 00:13:16.250 --> 00:13:20.310 algebraic expression is for the system function. 00:13:20.310 --> 00:13:21.950 Alright that's a first order example. 00:13:21.950 --> 00:13:26.900 Let's now look at a second order system and the 00:13:26.900 --> 00:13:30.080 differential equation that I picked in this case. 00:13:30.080 --> 00:13:32.860 I've parameterized in a certain way, which we'll see 00:13:32.860 --> 00:13:34.150 will be useful. 00:13:34.150 --> 00:13:37.160 In particular, it's a second order differential equation 00:13:37.160 --> 00:13:40.800 and I chosen, just for simplicity, to not include any 00:13:40.800 --> 00:13:44.030 derivatives on the right hand side, although we could have. 00:13:44.030 --> 00:13:47.860 In fact, if we did, that would insert zeros into the system 00:13:47.860 --> 00:13:50.600 function, as well as the poles inserted by 00:13:50.600 --> 00:13:52.950 the left hand side. 00:13:52.950 --> 00:13:56.750 We can determine the system function in exactly the same 00:13:56.750 --> 00:14:00.840 way, namely, apply the Laplace transform to this equation. 00:14:00.840 --> 00:14:04.200 That would convert this differential equation to an 00:14:04.200 --> 00:14:06.290 algebraic equation. 00:14:06.290 --> 00:14:11.590 And now when we solve this algebraic equation for y of s, 00:14:11.590 --> 00:14:15.590 in terms of x of s, it will come out in the form of y of 00:14:15.590 --> 00:14:19.890 s, equal to h of s, times x of s. 00:14:19.890 --> 00:14:23.430 And h of s, in that case, we would get simply by dividing 00:14:23.430 --> 00:14:27.240 out by this polynomial [? in ?] s, and so the system 00:14:27.240 --> 00:14:32.120 function then is the expression that I have here. 00:14:32.120 --> 00:14:37.030 So this is the form for a second order system where 00:14:37.030 --> 00:14:38.240 there are two poles. 00:14:38.240 --> 00:14:42.980 Since this is a second order polynomial, there are no zeros 00:14:42.980 --> 00:14:47.090 associated with the fact that I had no derivatives of the 00:14:47.090 --> 00:14:51.120 input on the right hand side of the equation. 00:14:51.120 --> 00:14:54.110 Well, let's look at this example-- 00:14:54.110 --> 00:14:55.620 namely the second order system-- 00:14:55.620 --> 00:14:57.190 in a little more detail. 00:14:57.190 --> 00:15:01.100 And what we'll want to look at is the location of the poles 00:15:01.100 --> 00:15:02.880 and some issues such as, for example, 00:15:02.880 --> 00:15:05.720 the frequency response. 00:15:05.720 --> 00:15:11.170 So here again I have the algebraic expression for the 00:15:11.170 --> 00:15:13.020 system function. 00:15:13.020 --> 00:15:16.820 And as I indicated, this is a second order polynomial, which 00:15:16.820 --> 00:15:21.830 means that we can factor it into two roots. 00:15:21.830 --> 00:15:27.320 So c1 and c2 represent the poles of the system function. 00:15:27.320 --> 00:15:31.970 And in particular, in relation to the two parameters zeta and 00:15:31.970 --> 00:15:33.500 omega sub n-- 00:15:33.500 --> 00:15:39.160 if we look at what these roots are, then what we get are the 00:15:39.160 --> 00:15:42.670 two expressions that I have below. 00:15:42.670 --> 00:15:50.270 And notice, incidentally, that if zeta is less than one, then 00:15:50.270 --> 00:15:52.410 what's under the square root is negative. 00:15:52.410 --> 00:15:54.830 And so this, in fact, corresponds to 00:15:54.830 --> 00:15:56.600 an imaginary part-- 00:15:56.600 --> 00:15:59.410 an imaginary term for zeta less than one. 00:15:59.410 --> 00:16:04.680 And so the two roots, then, have a real part which is 00:16:04.680 --> 00:16:09.390 given by minus zeta omega sub n, and an imaginary part-- if 00:16:09.390 --> 00:16:14.020 I were to rewrite this and then express it in terms of j 00:16:14.020 --> 00:16:16.270 or the square root of minus one. 00:16:16.270 --> 00:16:21.200 Looking below, we'll have a real part which is minus zeta 00:16:21.200 --> 00:16:22.560 omega sub n-- 00:16:22.560 --> 00:16:25.140 an imaginary part which is omega sub n 00:16:25.140 --> 00:16:27.290 times this square root. 00:16:27.290 --> 00:16:31.700 So that's for zeta less than one and for zeta greater than 00:16:31.700 --> 00:16:35.400 one, the two roots, of course, will be real. 00:16:35.400 --> 00:16:40.260 Alright, so let's examine this for the case where zeta is 00:16:40.260 --> 00:16:41.720 less than one. 00:16:41.720 --> 00:16:46.690 And what that corresponds to, then, are two poles in the 00:16:46.690 --> 00:16:49.310 complex plane. 00:16:49.310 --> 00:16:54.670 And they have a real part and an imaginary part. 00:16:54.670 --> 00:16:58.570 And you can explore this in somewhat more detail on your 00:16:58.570 --> 00:17:02.220 own, but, essentially what happens is that as you keep 00:17:02.220 --> 00:17:07.980 the parameter omega sub n fixed and vary zeta, these 00:17:07.980 --> 00:17:11.619 poles trace out a circle. 00:17:11.619 --> 00:17:16.910 And, for example, where zeta equal to zero, the poles are 00:17:16.910 --> 00:17:21.810 on the j omega axis at omega sub n. 00:17:21.810 --> 00:17:31.370 As zeta increases and gets closer to one, the poles 00:17:31.370 --> 00:17:37.430 converge toward the real axis and then, in particular, for 00:17:37.430 --> 00:17:42.060 zeta greater than one, what we end up with are two poles on 00:17:42.060 --> 00:17:44.430 the real axis. 00:17:44.430 --> 00:17:48.310 Well, actually, the case that we want to look at a little 00:17:48.310 --> 00:17:51.460 more carefully is when the poles are complex. 00:17:51.460 --> 00:17:55.410 And what this becomes is a second order system, which as 00:17:55.410 --> 00:17:58.580 we'll see as the discussion goes on, has an impulse 00:17:58.580 --> 00:18:02.450 response which oscillates with time and correspondingly a 00:18:02.450 --> 00:18:06.380 frequency response that has a resonance. 00:18:06.380 --> 00:18:09.640 Well let's examine the frequency response a little 00:18:09.640 --> 00:18:11.250 more carefully. 00:18:11.250 --> 00:18:14.640 And what I'm assuming in the discussion is that, first of 00:18:14.640 --> 00:18:20.110 all, the poles are in the left half plane corresponding to 00:18:20.110 --> 00:18:22.640 zeta omega sub n being positive-- 00:18:22.640 --> 00:18:25.490 and so this is-- minus that is negative. 00:18:25.490 --> 00:18:29.710 And furthermore, I'm assuming that the poles are complex. 00:18:29.710 --> 00:18:33.770 And in that case, the algebraic expression for the 00:18:33.770 --> 00:18:38.110 system function is omega sub n squared in the numerator and 00:18:38.110 --> 00:18:42.700 two poles in the denominator, which are complex conjugates. 00:18:42.700 --> 00:18:48.360 Now, what we want to look at is the frequency response of 00:18:48.360 --> 00:18:49.720 the system. 00:18:49.720 --> 00:18:50.620 And 00:18:50.620 --> 00:18:55.370 that corresponds to looking at the Fourier transform of the 00:18:55.370 --> 00:18:59.450 impulse response, which is the Laplace transform on the j 00:18:59.450 --> 00:19:00.930 omega axis. 00:19:00.930 --> 00:19:05.190 So we want to examine what h of s is as we move along the j 00:19:05.190 --> 00:19:06.680 omega axis. 00:19:06.680 --> 00:19:11.810 And notice, that to do that, in this algebraic expression, 00:19:11.810 --> 00:19:15.920 we want to set s equal to j omega and then evaluate-- 00:19:15.920 --> 00:19:17.830 for example, if we want to look at the magnitude of the 00:19:17.830 --> 00:19:19.470 frequency response-- 00:19:19.470 --> 00:19:24.740 evaluate the magnitude of the complex number. 00:19:24.740 --> 00:19:27.100 Well, there's a very convenient way of doing that 00:19:27.100 --> 00:19:33.420 geometrically by recognizing that in the complex plane, 00:19:33.420 --> 00:19:36.000 this complex number minus that complex number 00:19:36.000 --> 00:19:37.870 represents a vector. 00:19:37.870 --> 00:19:41.360 And essentially, to look at the magnitude of this complex 00:19:41.360 --> 00:19:46.060 number corresponds to taking omega sub n squared and 00:19:46.060 --> 00:19:51.720 dividing it by the product of the lengths of these vectors. 00:19:51.720 --> 00:19:57.820 So let's look, for example, at the vector s minus c1, where s 00:19:57.820 --> 00:20:00.980 is on the j omega axis. 00:20:00.980 --> 00:20:08.920 And doing that, here is the vector c1, and here is the 00:20:08.920 --> 00:20:12.530 vector s-- which is j omega if we're looking, let's say, at 00:20:12.530 --> 00:20:14.650 this value of frequency-- 00:20:14.650 --> 00:20:18.120 and this vector, then, is the vector which is 00:20:18.120 --> 00:20:21.100 j omega minus c1. 00:20:21.100 --> 00:20:24.630 So in fact, it's the length of this vector that we want to 00:20:24.630 --> 00:20:28.240 observe as we change omega-- 00:20:28.240 --> 00:20:31.540 namely as we move along the j omega axis. 00:20:31.540 --> 00:20:35.470 We want to take this vector and this vector, take the 00:20:35.470 --> 00:20:39.140 lengths of those vectors, multiply them together, divide 00:20:39.140 --> 00:20:42.400 that into omega sub n squared, and that will give us the 00:20:42.400 --> 00:20:43.590 frequency response. 00:20:43.590 --> 00:20:46.920 Now that's a little hard to see how the frequency response 00:20:46.920 --> 00:20:48.750 will work out just looking at one point. 00:20:48.750 --> 00:20:53.610 Although notice that as we move along the j omega axis, 00:20:53.610 --> 00:20:57.230 as we get closer to this pole, this vector, in fact, gets 00:20:57.230 --> 00:21:01.160 shorter, and so we might expect , that 00:21:01.160 --> 00:21:02.490 the frequency response-- 00:21:02.490 --> 00:21:05.120 as we're moving along the j omega axis in the vicinity of 00:21:05.120 --> 00:21:06.160 that pole-- 00:21:06.160 --> 00:21:08.010 would start to peak. 00:21:08.010 --> 00:21:11.320 Well, I think that all of this is much better seen 00:21:11.320 --> 00:21:14.850 dynamically on the computer display, so let's go to the 00:21:14.850 --> 00:21:17.630 computer display and what we'll look at is a second 00:21:17.630 --> 00:21:19.180 order system-- 00:21:19.180 --> 00:21:21.720 the frequency response of it-- as we move 00:21:21.720 --> 00:21:25.340 along the j omega axis. 00:21:25.340 --> 00:21:31.220 So here we see the pole pair in the complex plane and to 00:21:31.220 --> 00:21:33.890 generate the frequency response, we want to look at 00:21:33.890 --> 00:21:37.660 the behavior of the pole vectors as we move vertically 00:21:37.660 --> 00:21:39.660 along the j omega axis. 00:21:39.660 --> 00:21:45.620 So we'll show the pole vectors and let's begin at omega 00:21:45.620 --> 00:21:46.800 equals zero. 00:21:46.800 --> 00:21:49.510 So here we have the pole vectors from the poles to the 00:21:49.510 --> 00:21:52.130 point omega equal to zero. 00:21:52.130 --> 00:21:56.480 And, as we move vertically along the j omega axis, we'll 00:21:56.480 --> 00:22:01.190 see how those pole vectors change in length. 00:22:01.190 --> 00:22:04.310 The magnitude of the frequency response is the reciprocal of 00:22:04.310 --> 00:22:07.440 the product of the lengths of those vectors. 00:22:07.440 --> 00:22:12.090 Shown below is the frequency response where we've begun 00:22:12.090 --> 00:22:14.800 just at omega equal to zero. 00:22:14.800 --> 00:22:22.800 And as we move vertically along the j omega axis and the 00:22:22.800 --> 00:22:27.000 pole vector lengths change, that will, then, influence 00:22:27.000 --> 00:22:28.980 what the frequency response looks like. 00:22:28.980 --> 00:22:34.400 We've started here to move a little bit away from omega 00:22:34.400 --> 00:22:39.640 equal to zero and notice that in the upper half plane the 00:22:39.640 --> 00:22:41.840 pole vector has gotten shorter. 00:22:41.840 --> 00:22:44.450 The pole vector for the pole in the lower half plane has 00:22:44.450 --> 00:22:45.830 gotten longer. 00:22:45.830 --> 00:22:50.070 And now, as omega increases further, that 00:22:50.070 --> 00:22:51.910 process will continue. 00:22:51.910 --> 00:22:55.690 And in particular, the pole vector associated with the 00:22:55.690 --> 00:22:59.530 pole in the upper half plane will be its 00:22:59.530 --> 00:23:02.200 shortest in the vicinity-- 00:23:02.200 --> 00:23:04.810 at a frequency in the vicinity of that pole-- 00:23:04.810 --> 00:23:08.830 and so, for that frequency, then, the frequency response 00:23:08.830 --> 00:23:13.450 will peak and we see that here. 00:23:13.450 --> 00:23:17.640 From this point as the frequency increases, 00:23:17.640 --> 00:23:20.655 corresponding to moving further vertically along the j 00:23:20.655 --> 00:23:25.280 omega axis, both pole vectors will increase in length. 00:23:25.280 --> 00:23:29.530 And that means, then, that the magnitude of the frequency 00:23:29.530 --> 00:23:31.530 response will decrease. 00:23:31.530 --> 00:23:35.250 For this specific example, the magnitude of the frequency 00:23:35.250 --> 00:23:39.900 response will asymptotically go to zero. 00:23:39.900 --> 00:23:44.490 So what we see here is that the frequency response has a 00:23:44.490 --> 00:23:49.510 resonance and as we see geometrically from the way the 00:23:49.510 --> 00:23:53.940 vectors behaved, that resonance in frequency is very 00:23:53.940 --> 00:23:58.630 clearly associated with the position of the poles. 00:23:58.630 --> 00:24:03.240 And so, in fact, to illustrate that further and dramatize it 00:24:03.240 --> 00:24:07.070 as long as we're focused on it, let's now look at the 00:24:07.070 --> 00:24:12.310 frequency response for the second order example as we 00:24:12.310 --> 00:24:14.440 change the pole positions. 00:24:14.440 --> 00:24:19.260 And first, what we'll do is let the polls move vertically 00:24:19.260 --> 00:24:22.610 parallel to the j omega axis and see how the frequency 00:24:22.610 --> 00:24:26.120 response changes, and then we'll have the polls move 00:24:26.120 --> 00:24:29.260 horizontally parallel to the real axis and see how the 00:24:29.260 --> 00:24:31.950 frequency response changes. 00:24:31.950 --> 00:24:35.560 To display the behavior of the frequency response as the 00:24:35.560 --> 00:24:40.020 poles move, we've changed the vertical scale on the 00:24:40.020 --> 00:24:42.890 frequency response somewhat. 00:24:42.890 --> 00:24:47.950 And now what we want to do is move the poles, first, 00:24:47.950 --> 00:24:51.030 parallel to the j omega axis, and then 00:24:51.030 --> 00:24:53.380 parallel to the real axis. 00:24:53.380 --> 00:24:57.210 Here we see the effect of moving the poles parallel to 00:24:57.210 --> 00:24:59.030 the j omega axis. 00:24:59.030 --> 00:25:02.880 And what we observe is that, in fact, the frequency 00:25:02.880 --> 00:25:07.390 location of the resonance shifts, basically tracking the 00:25:07.390 --> 00:25:10.520 location of the pole. 00:25:10.520 --> 00:25:16.770 If we now move the pole back down closer to the real axis, 00:25:16.770 --> 00:25:21.430 then this resonance will shift back toward its original 00:25:21.430 --> 00:25:25.100 location and so let's now see that. 00:25:39.060 --> 00:25:43.480 And here we are back at the frequency that we started at. 00:25:43.480 --> 00:25:47.790 Now we'll move the poles even closer to the real axis. 00:25:47.790 --> 00:25:52.760 The frequency location of the resonance will continue to 00:25:52.760 --> 00:25:55.340 shift toward lower frequencies. 00:25:55.340 --> 00:25:59.500 And also in the process, incidentally, the height over 00:25:59.500 --> 00:26:03.350 the resonant peak will increase because, of course, 00:26:03.350 --> 00:26:08.780 the lengths of the pole vectors are getting shorter. 00:26:08.780 --> 00:26:12.350 And so, we see now the resonance shifting down toward 00:26:12.350 --> 00:26:14.600 lower and lower frequency. 00:26:14.600 --> 00:26:20.960 And, finally, what we'll now do is move the poles back to 00:26:20.960 --> 00:26:25.400 their original position and the resonant peak will, of 00:26:25.400 --> 00:26:27.970 course, shift back up. 00:26:27.970 --> 00:26:33.000 And correspondingly the height or amplitude of the resonance 00:26:33.000 --> 00:26:34.250 will decrease. 00:26:39.480 --> 00:26:43.000 And now we're back at the frequency response that we had 00:26:43.000 --> 00:26:45.180 generated previously. 00:26:45.180 --> 00:26:49.160 Next we'd like to look at the behavior as the polls move 00:26:49.160 --> 00:26:50.740 parallel to the real axis. 00:26:50.740 --> 00:26:54.530 First closer to the j omega axis and then further away. 00:26:54.530 --> 00:26:58.710 As they move closer to the j omega axis, the resonance 00:26:58.710 --> 00:27:03.810 sharpens because of the fact that the pole vector gets 00:27:03.810 --> 00:27:06.060 shorter and responds-- 00:27:06.060 --> 00:27:10.070 or changes in length more quickly as we move past it 00:27:10.070 --> 00:27:13.470 moving along the j omega axis. 00:27:13.470 --> 00:27:18.360 So here we see the effect of moving the poles closer to the 00:27:18.360 --> 00:27:20.020 j omega axis. 00:27:20.020 --> 00:27:24.000 The resonance has gotten narrower in frequency and 00:27:24.000 --> 00:27:28.380 higher in amplitude, associated with the fact that 00:27:28.380 --> 00:27:29.715 the pole vector gets shorter. 00:27:33.330 --> 00:27:38.320 Next as we move back to the original location, the 00:27:38.320 --> 00:27:40.760 resonance will broaden once again and the 00:27:40.760 --> 00:27:43.035 amplitude will decrease. 00:27:54.160 --> 00:27:57.810 And then, if we continue to move the poles even further 00:27:57.810 --> 00:28:02.770 away from the real axis, the resonance will broaden even 00:28:02.770 --> 00:28:05.580 further and the amplitude of the peak 00:28:05.580 --> 00:28:07.305 will become even smaller. 00:28:13.830 --> 00:28:18.330 And finally, let's now look just move the poles back to 00:28:18.330 --> 00:28:23.310 their original position and we'll see the resonance narrow 00:28:23.310 --> 00:28:24.610 again and become higher. 00:28:34.540 --> 00:28:38.800 And so what we see then is that for a second order 00:28:38.800 --> 00:28:42.530 system, the behavior of the resonance basically is 00:28:42.530 --> 00:28:46.310 associated with the pole locations, the frequency of 00:28:46.310 --> 00:28:48.760 the resonance associated with the vertical position of the 00:28:48.760 --> 00:28:54.900 poles, and the sharpness of the resonance associated with 00:28:54.900 --> 00:28:59.020 the real part of the poles-- in other words, their position 00:28:59.020 --> 00:29:02.365 closer or further away from the j omega axis. 00:29:05.240 --> 00:29:10.460 OK, so for complex poles, then, for the second order 00:29:10.460 --> 00:29:15.690 system, what we see is that we get a resonant kind of 00:29:15.690 --> 00:29:21.020 behavior, and, in particular, then that resonate behavior 00:29:21.020 --> 00:29:26.010 tends to peak, or get peakier, as the value 00:29:26.010 --> 00:29:28.510 of zeta gets smaller. 00:29:28.510 --> 00:29:32.960 And here, just to remind you of what you saw, here is the 00:29:32.960 --> 00:29:38.080 frequency response with one particular choice of values-- 00:29:38.080 --> 00:29:40.320 well, this is normalized so that omega sub n is one-- one 00:29:40.320 --> 00:29:43.910 particular choice for zeta, namely 0.4. 00:29:43.910 --> 00:29:52.390 Here is what we have with zeta smaller, and, finally, here is 00:29:52.390 --> 00:29:56.200 an example where zeta has gotten even smaller than that. 00:29:56.200 --> 00:29:59.960 And what that corresponds to is the poles moving closer to 00:29:59.960 --> 00:30:02.980 the j omega axis, the corresponding frequency 00:30:02.980 --> 00:30:06.290 response getting peakier. 00:30:06.290 --> 00:30:12.620 Now in the time domain what happens is that we have, of 00:30:12.620 --> 00:30:17.660 course, these complex roots, which I indicated previously, 00:30:17.660 --> 00:30:21.180 where this represents the imaginary part because zeta is 00:30:21.180 --> 00:30:23.230 less than one. 00:30:23.230 --> 00:30:25.900 And in the time domain, we will have a form for the 00:30:25.900 --> 00:30:33.960 behavior, which is a e to the c one t, plus a conjugate, e 00:30:33.960 --> 00:30:38.670 to the c one conjugate t. 00:30:38.670 --> 00:30:43.270 And so, in fact, as the poles get closer 00:30:43.270 --> 00:30:44.740 to the j omega axis-- 00:30:44.740 --> 00:30:47.450 corresponding to zeta getting smaller-- 00:30:47.450 --> 00:30:53.110 as the polls get closer to the j omega axis, in the frequency 00:30:53.110 --> 00:30:55.840 domain the resonances get sharper. 00:30:55.840 --> 00:30:59.650 In the time domain, the real part of the poles has gotten 00:30:59.650 --> 00:31:04.320 smaller, and that means, in fact, that in the time domain, 00:31:04.320 --> 00:31:08.330 the behavior will be more oscillatory and less damped. 00:31:08.330 --> 00:31:12.850 And so just looking at that again. 00:31:12.850 --> 00:31:17.280 Here is, in the time domain, what happens. 00:31:17.280 --> 00:31:24.170 First of all, with the parameter zeta equal to 0.4, 00:31:24.170 --> 00:31:29.130 and it oscillates and exponentially dies out. 00:31:29.130 --> 00:31:35.890 Here is the second order system where zeta is now 0.2 00:31:35.890 --> 00:31:37.730 instead of 0.4. 00:31:37.730 --> 00:31:44.250 And, finally, the second order system where zeta is 0.1. 00:31:44.250 --> 00:31:49.930 And what we see as zeta gets smaller and smaller is that 00:31:49.930 --> 00:31:53.810 the oscillations are basically the same, but the exponential 00:31:53.810 --> 00:31:58.610 damping becomes less and less. 00:31:58.610 --> 00:32:02.150 Alright, now, this is a somewhat more detailed look at 00:32:02.150 --> 00:32:03.950 second order systems. 00:32:03.950 --> 00:32:07.010 And second order systems-- and for that 00:32:07.010 --> 00:32:08.390 matter, first order systems-- 00:32:08.390 --> 00:32:13.100 are systems that are important in their own right, but they 00:32:13.100 --> 00:32:18.600 also are important as basic building blocks for more 00:32:18.600 --> 00:32:21.750 general, in particular, for higher order systems. 00:32:21.750 --> 00:32:25.850 And the way in which that's done typically is by combining 00:32:25.850 --> 00:32:29.400 first and second order systems together in such a way that 00:32:29.400 --> 00:32:31.750 they implement higher order systems. 00:32:31.750 --> 00:32:35.330 And two very common connections are connections 00:32:35.330 --> 00:32:39.470 which are cascade connections, and connections which are 00:32:39.470 --> 00:32:42.170 parallel connections. 00:32:42.170 --> 00:32:47.980 In a cascade connection, we would think of combining the 00:32:47.980 --> 00:32:52.060 individual systems together as I indicate here in series. 00:32:52.060 --> 00:32:55.150 And, of course, from the convolution property, the 00:32:55.150 --> 00:32:59.430 overall system function is the product of the individual 00:32:59.430 --> 00:33:01.270 system functions. 00:33:01.270 --> 00:33:06.840 So, for example, if these were all second order systems, and 00:33:06.840 --> 00:33:11.500 I combine n of them together in cascade, the overall system 00:33:11.500 --> 00:33:15.010 would be a system that would have to n poles-- in other 00:33:15.010 --> 00:33:18.190 words, it would be a two n order system. 00:33:18.190 --> 00:33:21.140 That's one very common kind of connection. 00:33:21.140 --> 00:33:24.010 Another very common kind of connection for first and 00:33:24.010 --> 00:33:28.180 second order systems is a parallel connection, where, in 00:33:28.180 --> 00:33:31.360 that case, we connect the systems together 00:33:31.360 --> 00:33:33.990 as I indicate here. 00:33:33.990 --> 00:33:38.900 The overall system function is just simply the sum of these, 00:33:38.900 --> 00:33:41.710 and that follows from the linearity property. 00:33:41.710 --> 00:33:44.720 And so the overall system function would be as I 00:33:44.720 --> 00:33:47.160 indicate algebraically here. 00:33:47.160 --> 00:33:51.130 And notice that if each of these are second order 00:33:51.130 --> 00:33:56.020 systems, and I had capital N of them in parallel, when you 00:33:56.020 --> 00:33:59.080 think of putting the overall system function over one 00:33:59.080 --> 00:34:02.070 common denominator, that common denominator, in 00:34:02.070 --> 00:34:08.050 general, is going to be of order two N. So either the 00:34:08.050 --> 00:34:11.639 parallel connection or the cascade connection could be 00:34:11.639 --> 00:34:16.370 used to implement higher order systems. 00:34:16.370 --> 00:34:20.870 One very common context in which second order systems are 00:34:20.870 --> 00:34:25.739 combined together, either in parallel or in cascade, to 00:34:25.739 --> 00:34:30.219 form a more interesting system is, in 00:34:30.219 --> 00:34:31.929 fact, in speech synthesis. 00:34:31.929 --> 00:34:35.639 And what I'd like to do is demonstrate a speech 00:34:35.639 --> 00:34:40.780 synthesizer, which I have here, which in fact is a 00:34:40.780 --> 00:34:46.460 parallel combination of four second order systems, very 00:34:46.460 --> 00:34:49.960 much of the type that we've just talked about. 00:34:49.960 --> 00:34:53.170 I'll return to the synthesizer in a minute. 00:34:53.170 --> 00:34:56.949 Let me first just indicate what the basic idea is. 00:34:56.949 --> 00:35:00.930 In speech synthesis, what we're trying to represent or 00:35:00.930 --> 00:35:04.140 implement is something that corresponds 00:35:04.140 --> 00:35:06.450 to the vocal tract. 00:35:06.450 --> 00:35:09.570 The vocal tract is characterized by a set of 00:35:09.570 --> 00:35:10.760 resonances. 00:35:10.760 --> 00:35:12.700 And we can think of representing each of those 00:35:12.700 --> 00:35:15.160 resonances by a second order system. 00:35:15.160 --> 00:35:17.700 And then the higher order system corresponding to the 00:35:17.700 --> 00:35:21.350 vocal tract is built by, in this case, a parallel 00:35:21.350 --> 00:35:25.210 combination of those second order systems. 00:35:25.210 --> 00:35:30.470 So for the synthesizer, what we have connected together in 00:35:30.470 --> 00:35:35.710 parallel is four second order systems. 00:35:35.710 --> 00:35:41.130 And a control on each one of them that controls the center 00:35:41.130 --> 00:35:45.850 frequency or the resonant frequency of each of the 00:35:45.850 --> 00:35:48.930 second order systems. 00:35:48.930 --> 00:35:52.880 The excitation is an excitation that would 00:35:52.880 --> 00:35:56.270 represent the air flow through the vocal cords. 00:35:56.270 --> 00:35:59.950 The vocal cords vibrate and there are puffs of air through 00:35:59.950 --> 00:36:02.630 the vocal cords as they open and close. 00:36:02.630 --> 00:36:08.430 And so the excitation for the synthesizer corresponds to a 00:36:08.430 --> 00:36:12.640 pulse train representing the air flow 00:36:12.640 --> 00:36:14.270 through the vocal cords. 00:36:14.270 --> 00:36:17.850 The fundamental frequency of this representing the 00:36:17.850 --> 00:36:21.740 fundamental frequency of the synthesized voice. 00:36:21.740 --> 00:36:25.890 So that's the basic structure of the synthesizer 00:36:25.890 --> 00:36:30.380 And what we have in this analog synthesizer are 00:36:30.380 --> 00:36:35.600 separate controls on the individual center frequencies. 00:36:35.600 --> 00:36:38.250 There is a control representing the center 00:36:38.250 --> 00:36:40.930 frequency of the third resonator and the fourth 00:36:40.930 --> 00:36:42.660 resonator, and those are represented 00:36:42.660 --> 00:36:44.570 by these two knobs. 00:36:44.570 --> 00:36:48.860 And then the first and second resonators are controlled by 00:36:48.860 --> 00:36:50.650 moving this joystick. 00:36:50.650 --> 00:36:56.280 The first resonator by moving the joystick along this axis 00:36:56.280 --> 00:36:58.220 and the second resonator by moving the 00:36:58.220 --> 00:37:01.030 joystick along this axis. 00:37:01.030 --> 00:37:05.570 And then, in addition to controls on the four 00:37:05.570 --> 00:37:09.630 resonators, we can control the fundamental frequency of the 00:37:09.630 --> 00:37:13.360 excitation, and we do that with this knob. 00:37:13.360 --> 00:37:17.070 So let's, first of all, just listen to one of the 00:37:17.070 --> 00:37:20.490 resonators, and the resonator that I'll play 00:37:20.490 --> 00:37:22.310 is the fourth resonator. 00:37:22.310 --> 00:37:26.840 And what you'll hear first is the output as I vary the 00:37:26.840 --> 00:37:28.730 center frequency of that resonator. 00:37:28.730 --> 00:37:30.320 [BUZZING] 00:37:30.320 --> 00:37:32.190 So I'm lowering the center frequency. 00:37:35.310 --> 00:37:37.700 And then, bringing the center frequency back up. 00:37:40.790 --> 00:37:44.200 And then, as I indicated, I can also control the 00:37:44.200 --> 00:37:47.400 fundamental frequency of the excitation by 00:37:47.400 --> 00:37:48.541 turning this knob. 00:37:48.541 --> 00:37:51.310 [BUZZING] 00:37:51.310 --> 00:37:52.720 Lowering the fundamental frequency. 00:37:55.510 --> 00:37:58.486 And then, increasing the fundamental frequency. 00:38:01.960 --> 00:38:07.010 Alright, now, if the four resonators in parallel are an 00:38:07.010 --> 00:38:11.640 implementation of the vocal cavity, then, presumably, what 00:38:11.640 --> 00:38:15.600 we can synthesize when we put them all in are vowel sounds 00:38:15.600 --> 00:38:17.360 and let's do that. 00:38:17.360 --> 00:38:22.500 I'll now switch in the other resonators. 00:38:22.500 --> 00:38:27.530 When we do that, then, depending on what choice we 00:38:27.530 --> 00:38:30.920 have for the individual resonant frequencies, we 00:38:30.920 --> 00:38:32.990 should be able to synthesize vowel sounds. 00:38:32.990 --> 00:38:35.210 So here, for example, is the vowel e. 00:38:35.210 --> 00:38:37.640 [BUZZING "E"]. 00:38:37.640 --> 00:38:38.280 Here is 00:38:38.280 --> 00:38:39.025 [BUZZING "AH"] 00:38:39.025 --> 00:38:40.275 --ah. 00:38:41.875 --> 00:38:42.662 A. 00:38:42.662 --> 00:38:44.390 [BUZZING FLAT A] 00:38:44.390 --> 00:38:45.360 And, of course, we can-- 00:38:45.360 --> 00:38:46.662 [BUZZING OO] 00:38:46.662 --> 00:38:47.526 --generate 00:38:47.526 --> 00:38:47.902 [BUZZING "I"] 00:38:47.902 --> 00:38:50.160 --lots of other vowel sounds. 00:38:50.160 --> 00:38:51.170 [BUZZING "AH"] 00:38:51.170 --> 00:38:53.910 --and change the fundamental frequency at the same time. 00:38:53.910 --> 00:38:55.160 [CHANGES FREQUENCY UP AND DOWN] 00:39:01.600 --> 00:39:05.380 Now, if we want to synthesize speech it's not enough to just 00:39:05.380 --> 00:39:08.100 synthesize steady state vowels-- that gets boring 00:39:08.100 --> 00:39:08.920 after a while. 00:39:08.920 --> 00:39:13.360 Of course what happens with the vocal cavity is that it 00:39:13.360 --> 00:39:19.330 moves as a function of time and that's what generates the 00:39:19.330 --> 00:39:21.580 speech that we want to generate. 00:39:21.580 --> 00:39:26.520 And so, presumably then, if we change these resonant 00:39:26.520 --> 00:39:30.330 frequencies as a function of time appropriately, then we 00:39:30.330 --> 00:39:32.500 should be able to synthesize speech. 00:39:32.500 --> 00:39:36.650 And so by moving these resonances around, we can 00:39:36.650 --> 00:39:38.800 generate synthesized speech. 00:39:38.800 --> 00:39:43.600 And let's try it with some phrase. 00:39:43.600 --> 00:39:46.380 And I'll do that by simply adjusting the center 00:39:46.380 --> 00:39:47.630 frequencies appropriately. 00:39:50.310 --> 00:39:57.030 [BUZZING "HOW ARE YOU"] 00:39:57.030 --> 00:39:59.740 Well, hopefully you understood that. 00:39:59.740 --> 00:40:03.910 As you could imagine, I spent at least a few minutes before 00:40:03.910 --> 00:40:06.820 the lecture trying to practice that so that it would come out 00:40:06.820 --> 00:40:10.090 to be more or less intelligible. 00:40:10.090 --> 00:40:14.370 Now the system as I've just demonstrated it is, of course, 00:40:14.370 --> 00:40:19.490 a continuous time system or an analog speech synthesizer. 00:40:19.490 --> 00:40:23.540 There are many versions of digital or discrete time 00:40:23.540 --> 00:40:24.980 synthesizers. 00:40:24.980 --> 00:40:30.000 One of the first, in fact, being a device that many of 00:40:30.000 --> 00:40:33.310 you are very likely familiar with, which is the Texas 00:40:33.310 --> 00:40:36.680 Instruments Speak and Spell, which I show here. 00:40:36.680 --> 00:40:41.290 And what's very interesting and rather dramatic about this 00:40:41.290 --> 00:40:44.740 device is the fact that it implements the speech 00:40:44.740 --> 00:40:49.500 synthesis in very much the same way as I've demonstrated 00:40:49.500 --> 00:40:51.620 with the analog synthesizer. 00:40:51.620 --> 00:40:55.670 In this case, it's five second order filters in a 00:40:55.670 --> 00:40:58.170 configuration that's slightly different than a parallel 00:40:58.170 --> 00:41:02.130 configuration but conceptually very closely related. 00:41:02.130 --> 00:41:05.510 And let's take a look inside the box. 00:41:05.510 --> 00:41:09.520 And what we see there, with a slide that was kindly supplied 00:41:09.520 --> 00:41:13.720 by Texas Instruments, is the fact that there really are 00:41:13.720 --> 00:41:15.470 only four chips in there-- 00:41:15.470 --> 00:41:17.600 a controller chip, some storage. 00:41:17.600 --> 00:41:21.370 And the important point is the chip that's labeled as the 00:41:21.370 --> 00:41:25.860 speech synthesis chip, in fact, is what embodies or 00:41:25.860 --> 00:41:30.540 implements the five second order filters and, in 00:41:30.540 --> 00:41:34.470 addition, incorporates some other things-- some memory and 00:41:34.470 --> 00:41:36.310 also the [? DDA ?] 00:41:36.310 --> 00:41:37.270 converters. 00:41:37.270 --> 00:41:40.260 So, in fact, the implementation of the 00:41:40.260 --> 00:41:45.040 synthesizer is pretty much done on a single chip. 00:41:45.040 --> 00:41:48.940 Well that's a discrete time system. 00:41:48.940 --> 00:41:53.600 We've been talking for the last several lectures about 00:41:53.600 --> 00:41:57.230 continuous time systems and the Laplace transform. 00:41:57.230 --> 00:41:59.950 Hopefully what you've seen in this lecture and the previous 00:41:59.950 --> 00:42:07.110 lecture is the powerful tool that the Laplace transform 00:42:07.110 --> 00:42:11.830 affords us in analyzing and understanding system behavior. 00:42:14.440 --> 00:42:18.560 In the next lecture what I'd like to do is parallel the 00:42:18.560 --> 00:42:21.530 discussion for discrete time, turn our attention to the z 00:42:21.530 --> 00:42:26.330 transform, and, as you can imagine simply by virtue of 00:42:26.330 --> 00:42:31.670 the fact that I have shown you a digital and analog version 00:42:31.670 --> 00:42:35.040 of very much the same kind of system, the discussions 00:42:35.040 --> 00:42:38.960 parallel themselves very strongly and the z transform 00:42:38.960 --> 00:42:42.560 will play very much the same role in discrete time that the 00:42:42.560 --> 00:42:45.090 Laplace transform does in continuous time. 00:42:45.090 --> 00:42:46.340 Thank you.