1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation or view additional materials from 6 00:00:13,470 --> 00:00:19,290 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:19,290 --> 00:00:20,515 ocw.mit.edu. 8 00:00:20,515 --> 00:00:56,490 [MUSIC PLAYING] 9 00:00:56,490 --> 00:00:58,660 PROFESSOR: Last time, we introduced the Laplace 10 00:00:58,660 --> 00:01:03,310 transform as a generalization of the Fourier transform, and, 11 00:01:03,310 --> 00:01:07,530 just as a reminder, the Laplace transform expression 12 00:01:07,530 --> 00:01:13,720 as we developed it is this integral, very much similar to 13 00:01:13,720 --> 00:01:16,870 the Fourier transform integral, except with a more 14 00:01:16,870 --> 00:01:19,210 general complex variable. 15 00:01:19,210 --> 00:01:22,940 And, in fact, we developed and talked about the relationship 16 00:01:22,940 --> 00:01:26,260 between the Laplace transform and the Fourier transform. 17 00:01:26,260 --> 00:01:30,880 In particular, the Laplace transform with the Laplace 18 00:01:30,880 --> 00:01:36,590 transform variable s, purely imaginary, in fact, reduces to 19 00:01:36,590 --> 00:01:38,970 the Fourier transform. 20 00:01:38,970 --> 00:01:43,320 Or, more generally, with the Laplace transform variable as 21 00:01:43,320 --> 00:01:48,550 a complex number, the Laplace transform is the Fourier 22 00:01:48,550 --> 00:01:53,430 transform of the corresponding time function with an 23 00:01:53,430 --> 00:01:55,280 exponential weighting. 24 00:01:55,280 --> 00:01:59,730 And, also, as you should recall, the exponential 25 00:01:59,730 --> 00:02:04,870 waiting introduced the notion that the Laplace transform may 26 00:02:04,870 --> 00:02:08,820 converge for some values of sigma and perhaps not for 27 00:02:08,820 --> 00:02:10,360 other values of sigma. 28 00:02:10,360 --> 00:02:15,890 So associated with the Laplace transform was what we refer to 29 00:02:15,890 --> 00:02:17,765 as the region of convergence. 30 00:02:20,400 --> 00:02:25,120 Now just as with the Fourier transform, there are a number 31 00:02:25,120 --> 00:02:28,470 of properties of the Laplace transform that are extremely 32 00:02:28,470 --> 00:02:35,250 useful in describing and analyzing signals and systems. 33 00:02:35,250 --> 00:02:39,200 For example, one of the properties that we, in fact, 34 00:02:39,200 --> 00:02:42,740 took advantage of in our discussion last time was the 35 00:02:42,740 --> 00:02:46,250 linearly the linearity property, which says, in 36 00:02:46,250 --> 00:02:49,980 essence, that the Laplace transform of the linear 37 00:02:49,980 --> 00:02:54,830 combination of two time functions is the same linear 38 00:02:54,830 --> 00:03:00,260 combination of the associated Laplace transforms. 39 00:03:00,260 --> 00:03:03,640 Also, there is a very important and useful property, 40 00:03:03,640 --> 00:03:05,980 which tells us how the 41 00:03:05,980 --> 00:03:09,650 derivative of a time function-- 42 00:03:09,650 --> 00:03:12,090 rather, the Laplace transform of the derivative-- 43 00:03:12,090 --> 00:03:14,710 is related to the Laplace transform. 44 00:03:14,710 --> 00:03:19,350 In particular, the Laplace transform of the derivative is 45 00:03:19,350 --> 00:03:23,320 the Laplace transform x of t multiplied by s. 46 00:03:23,320 --> 00:03:27,200 And, as you can see by just setting s equal to j omega, in 47 00:03:27,200 --> 00:03:30,230 fact, this reduces to the corresponding Fourier 48 00:03:30,230 --> 00:03:32,740 transform property. 49 00:03:32,740 --> 00:03:37,830 And a third property that we'll make frequent use of is 50 00:03:37,830 --> 00:03:40,390 referred to as the convolution property. 51 00:03:40,390 --> 00:03:43,910 Again, a generalization of the convolution property for 52 00:03:43,910 --> 00:03:45,700 Fourier transforms. 53 00:03:45,700 --> 00:03:49,270 Here the convolution property says that the Laplace 54 00:03:49,270 --> 00:03:53,390 transform of the convolution of two time functions is the 55 00:03:53,390 --> 00:03:58,620 product of the associated Laplace transforms. 56 00:03:58,620 --> 00:04:04,610 Now it's important at some point to think carefully about 57 00:04:04,610 --> 00:04:09,400 the region of convergence as we discuss these properties. 58 00:04:09,400 --> 00:04:13,430 And let me just draw your attention to the fact that in 59 00:04:13,430 --> 00:04:17,970 discussing properties fully and in detail, one has to pay 60 00:04:17,970 --> 00:04:25,090 attention not just to how the algebraic expression changes, 61 00:04:25,090 --> 00:04:27,990 but also what the consequences are for the region of 62 00:04:27,990 --> 00:04:30,990 convergence, and that's discussed in somewhat more 63 00:04:30,990 --> 00:04:35,130 detail in the text and I won't do that here. 64 00:04:35,130 --> 00:04:40,750 Now the convolution property leads to, of course, a very 65 00:04:40,750 --> 00:04:46,300 important and useful mechanism for dealing with linear time 66 00:04:46,300 --> 00:04:50,330 invariant systems, very much as the Fourier transform did. 67 00:04:50,330 --> 00:04:54,990 In particular, the convolution property tells us that if we 68 00:04:54,990 --> 00:04:59,280 have a linear time invariant system, the output in the time 69 00:04:59,280 --> 00:05:02,520 domain is the convolution of the input 70 00:05:02,520 --> 00:05:04,390 and the impulse response. 71 00:05:04,390 --> 00:05:08,450 In the Laplace transform domain, the Laplace transform 72 00:05:08,450 --> 00:05:12,530 of the output is the Laplace transform of the impulse 73 00:05:12,530 --> 00:05:16,760 response times the Laplace transform of the input. 74 00:05:16,760 --> 00:05:20,700 And again, this is a generalization of the 75 00:05:20,700 --> 00:05:24,220 corresponding property for Fourier transforms. 76 00:05:24,220 --> 00:05:28,020 In the case of the Fourier transform, the Fourier 77 00:05:28,020 --> 00:05:30,720 transform [? of ?] the impulse response we refer to as the 78 00:05:30,720 --> 00:05:32,430 frequency response. 79 00:05:32,430 --> 00:05:36,290 In the more general case with Laplace transforms, it's 80 00:05:36,290 --> 00:05:38,950 typical to refer to the Laplace transform of the 81 00:05:38,950 --> 00:05:43,530 impulse response as the system function. 82 00:05:43,530 --> 00:05:52,500 Now in talking about the system function, some issues 83 00:05:52,500 --> 00:05:54,120 of the region of convergence-- 84 00:05:54,120 --> 00:05:55,920 and for that matter, location of poles 85 00:05:55,920 --> 00:05:57,940 of the system function-- 86 00:05:57,940 --> 00:06:03,010 are closely tied in and related to issues of whether 87 00:06:03,010 --> 00:06:05,940 the system is stable and causal. 88 00:06:05,940 --> 00:06:08,570 And in fact, there's some useful statements that can be 89 00:06:08,570 --> 00:06:12,610 made that play an important role throughout the further 90 00:06:12,610 --> 00:06:13,640 discussion. 91 00:06:13,640 --> 00:06:19,030 For example, we know from previous discussions that 92 00:06:19,030 --> 00:06:21,380 there's a condition for stability of a system, which 93 00:06:21,380 --> 00:06:25,390 is absolute integrability of the impulse response. 94 00:06:25,390 --> 00:06:29,650 And that, in fact, is the same condition for convergence of 95 00:06:29,650 --> 00:06:33,180 the Fourier transform of the impulse response. 96 00:06:33,180 --> 00:06:39,350 What that says, really, is that if a system is stable, 97 00:06:39,350 --> 00:06:43,530 then the region of convergence of the system function must 98 00:06:43,530 --> 00:06:45,710 include the j omega axis. 99 00:06:45,710 --> 00:06:49,390 Which, of course, is where the Laplace transform reduces to 100 00:06:49,390 --> 00:06:52,380 the Fourier transform. 101 00:06:52,380 --> 00:06:56,540 So that relates the region of convergence and stability. 102 00:06:56,540 --> 00:07:00,790 Also, you recall from last time that we talked about the 103 00:07:00,790 --> 00:07:03,740 region of convergence associated with right sided 104 00:07:03,740 --> 00:07:05,050 time functions. 105 00:07:05,050 --> 00:07:07,300 In particular for a right sided time function, the 106 00:07:07,300 --> 00:07:10,030 region of convergence must be to the right of 107 00:07:10,030 --> 00:07:12,330 the rightmost pole. 108 00:07:12,330 --> 00:07:17,970 Well, if, in fact, we have a system that's causal, then 109 00:07:17,970 --> 00:07:22,040 that causality imposes the condition that the impulse 110 00:07:22,040 --> 00:07:23,820 response be right sided. 111 00:07:23,820 --> 00:07:27,510 And so, in fact, for causality, we would have a 112 00:07:27,510 --> 00:07:29,960 region of convergence associated with the system 113 00:07:29,960 --> 00:07:33,780 function, which is to the right of the rightmost pole. 114 00:07:33,780 --> 00:07:37,710 Now interestingly and very important is the consequence, 115 00:07:37,710 --> 00:07:40,380 if you put those two statements together, in 116 00:07:40,380 --> 00:07:45,620 particular, you're led to the conclusion that for stable 117 00:07:45,620 --> 00:07:51,770 causal systems, all the poles must be in the left half of 118 00:07:51,770 --> 00:07:53,700 the s-plane. 119 00:07:53,700 --> 00:07:54,870 What's the reason? 120 00:07:54,870 --> 00:07:59,010 The reason, of course, is that if the system is stable and 121 00:07:59,010 --> 00:08:01,720 causal, the region of convergence must be to the 122 00:08:01,720 --> 00:08:03,520 right of the rightmost pole. 123 00:08:03,520 --> 00:08:06,460 It must include the j omega axis. 124 00:08:06,460 --> 00:08:09,170 Obviously then, all the poles must be in the left half of 125 00:08:09,170 --> 00:08:10,660 the s-plane. 126 00:08:10,660 --> 00:08:14,400 And again, that's an issue that is discussed somewhat 127 00:08:14,400 --> 00:08:18,650 more carefully and in more detail in the text. 128 00:08:18,650 --> 00:08:23,840 Now, the properties that we're talking about here are not the 129 00:08:23,840 --> 00:08:26,620 only properties, there are many others. 130 00:08:26,620 --> 00:08:30,200 But these properties, in particular, provide the 131 00:08:30,200 --> 00:08:30,880 mechanism-- 132 00:08:30,880 --> 00:08:33,409 as they did with Fourier transforms-- 133 00:08:33,409 --> 00:08:37,280 for turning linear constant coefficient differential 134 00:08:37,280 --> 00:08:41,840 equations into algebraic equations and, corresponding, 135 00:08:41,840 --> 00:08:46,250 lead to a mechanism for dealing with and solving 136 00:08:46,250 --> 00:08:48,890 linear constant coefficient differential equations. 137 00:08:48,890 --> 00:08:53,280 And I'd like to illustrate that by looking at both first 138 00:08:53,280 --> 00:08:56,010 order and second order differential equations. 139 00:08:56,010 --> 00:08:59,400 Let's begin, first of all, with a first order 140 00:08:59,400 --> 00:09:01,940 differential equation. 141 00:09:01,940 --> 00:09:06,500 So what we're talking about is a first order system. 142 00:09:06,500 --> 00:09:11,930 What I mean by that is a system that's characterized by 143 00:09:11,930 --> 00:09:14,870 a first order differential equation. 144 00:09:14,870 --> 00:09:19,460 And if we apply to this equation the differentiation 145 00:09:19,460 --> 00:09:22,580 property, then the derivative-- 146 00:09:22,580 --> 00:09:25,910 the Laplace transform of the derivative is s times the 147 00:09:25,910 --> 00:09:28,540 Laplace transform of the time function. 148 00:09:28,540 --> 00:09:32,580 The linearity property allows us to combine these together. 149 00:09:32,580 --> 00:09:36,520 And so, consequently, applying the Laplace transform to this 150 00:09:36,520 --> 00:09:40,780 equation leads us to this algebraic equation, and 151 00:09:40,780 --> 00:09:43,990 following that through, leads us to the statement that the 152 00:09:43,990 --> 00:09:47,430 Laplace transform of the output is one over s plus a 153 00:09:47,430 --> 00:09:50,620 times the Laplace transform of the input. 154 00:09:50,620 --> 00:09:54,530 We know from the convolution property that this Laplace 155 00:09:54,530 --> 00:09:58,020 transform is the system function times x of s. 156 00:09:58,020 --> 00:10:02,530 And so, one over s plus a is the system function or 157 00:10:02,530 --> 00:10:04,670 equivalently, the Laplace transform 158 00:10:04,670 --> 00:10:07,420 of the impulse response. 159 00:10:07,420 --> 00:10:12,400 So, we can determine the impulse response by taking the 160 00:10:12,400 --> 00:10:17,560 inverse Laplace transform of h of s given by 161 00:10:17,560 --> 00:10:20,890 one over s plus a. 162 00:10:20,890 --> 00:10:24,600 Well, we can do that using the inspection method, which is 163 00:10:24,600 --> 00:10:28,620 one way that we have of doing inverse Laplace transforms. 164 00:10:28,620 --> 00:10:32,380 The question is then, what time function has a Laplace 165 00:10:32,380 --> 00:10:35,640 transform which is one over s plus a? 166 00:10:35,640 --> 00:10:38,280 The problem that we run into is that there are 167 00:10:38,280 --> 00:10:40,910 two answers to that. 168 00:10:40,910 --> 00:10:46,020 one over s plus a is the Laplace transform of an 169 00:10:46,020 --> 00:10:52,250 exponential for positive time, but one over s plus a is also 170 00:10:52,250 --> 00:10:57,550 the Laplace transform of an exponential for negative time. 171 00:10:57,550 --> 00:11:01,340 Which one of these do we end up picking? 172 00:11:01,340 --> 00:11:05,200 Well, recall that the difference between these was 173 00:11:05,200 --> 00:11:06,810 in their region of convergence. 174 00:11:06,810 --> 00:11:11,080 And in fact, in this case, this corresponded to a region 175 00:11:11,080 --> 00:11:14,280 of convergence, which was the real part of s 176 00:11:14,280 --> 00:11:16,740 greater than minus a. 177 00:11:16,740 --> 00:11:19,850 In this case, this was the corresponding Laplace 178 00:11:19,850 --> 00:11:24,100 transform, provided that the real part of s is 179 00:11:24,100 --> 00:11:26,970 less than minus a. 180 00:11:26,970 --> 00:11:30,400 So we have to decide which region of convergence that we 181 00:11:30,400 --> 00:11:33,980 pick and it's not the differential equation that 182 00:11:33,980 --> 00:11:37,650 will tell us that, it's something else that has to 183 00:11:37,650 --> 00:11:39,640 give us that information. 184 00:11:39,640 --> 00:11:40,520 What could it be? 185 00:11:40,520 --> 00:11:45,310 Well, what it might be is the additional information that 186 00:11:45,310 --> 00:11:49,210 the system is either stable or causal. 187 00:11:49,210 --> 00:11:52,560 So for example, if the system was causal, we would know that 188 00:11:52,560 --> 00:11:55,950 the region of convergence is to the right of the pole and 189 00:11:55,950 --> 00:11:59,320 that would correspond, then, to this 190 00:11:59,320 --> 00:12:02,630 being the impulse response. 191 00:12:02,630 --> 00:12:06,270 Whereas, with a negative-- 192 00:12:06,270 --> 00:12:08,320 I'm sorry with a positive-- 193 00:12:08,320 --> 00:12:12,820 if we knew that the system, let's say, was non-causal, 194 00:12:12,820 --> 00:12:17,280 then we would associate with this region of convergence and 195 00:12:17,280 --> 00:12:20,600 we would know then that this is the impulse response. 196 00:12:20,600 --> 00:12:26,060 So a very important point is that what we see is that the 197 00:12:26,060 --> 00:12:30,850 linear constant coefficient differential equation gives us 198 00:12:30,850 --> 00:12:36,770 the algebraic expression for the system function, but does 199 00:12:36,770 --> 00:12:39,190 not tell us about the region of convergence. 200 00:12:39,190 --> 00:12:41,690 We get the reach of convergence from some 201 00:12:41,690 --> 00:12:43,540 auxiliary information. 202 00:12:43,540 --> 00:12:44,860 What is that information? 203 00:12:44,860 --> 00:12:49,090 Well, it might, for example, be knowledge that the system 204 00:12:49,090 --> 00:12:52,930 is perhaps stable, which tells us that the region of 205 00:12:52,930 --> 00:12:57,290 convergence includes the j omega axis, or perhaps causal, 206 00:12:57,290 --> 00:13:00,920 which tells us that the region of convergence is to the right 207 00:13:00,920 --> 00:13:02,340 of the rightmost pole. 208 00:13:02,340 --> 00:13:06,190 So it's the auxiliary information that specifies for 209 00:13:06,190 --> 00:13:08,010 us the region of convergence. 210 00:13:08,010 --> 00:13:09,340 Very important point. 211 00:13:09,340 --> 00:13:11,980 The differential equation by itself does not completely 212 00:13:11,980 --> 00:13:16,250 specify the system, it only essentially tells us what the 213 00:13:16,250 --> 00:13:20,310 algebraic expression is for the system function. 214 00:13:20,310 --> 00:13:21,950 Alright that's a first order example. 215 00:13:21,950 --> 00:13:26,900 Let's now look at a second order system and the 216 00:13:26,900 --> 00:13:30,080 differential equation that I picked in this case. 217 00:13:30,080 --> 00:13:32,860 I've parameterized in a certain way, which we'll see 218 00:13:32,860 --> 00:13:34,150 will be useful. 219 00:13:34,150 --> 00:13:37,160 In particular, it's a second order differential equation 220 00:13:37,160 --> 00:13:40,800 and I chosen, just for simplicity, to not include any 221 00:13:40,800 --> 00:13:44,030 derivatives on the right hand side, although we could have. 222 00:13:44,030 --> 00:13:47,860 In fact, if we did, that would insert zeros into the system 223 00:13:47,860 --> 00:13:50,600 function, as well as the poles inserted by 224 00:13:50,600 --> 00:13:52,950 the left hand side. 225 00:13:52,950 --> 00:13:56,750 We can determine the system function in exactly the same 226 00:13:56,750 --> 00:14:00,840 way, namely, apply the Laplace transform to this equation. 227 00:14:00,840 --> 00:14:04,200 That would convert this differential equation to an 228 00:14:04,200 --> 00:14:06,290 algebraic equation. 229 00:14:06,290 --> 00:14:11,590 And now when we solve this algebraic equation for y of s, 230 00:14:11,590 --> 00:14:15,590 in terms of x of s, it will come out in the form of y of 231 00:14:15,590 --> 00:14:19,890 s, equal to h of s, times x of s. 232 00:14:19,890 --> 00:14:23,430 And h of s, in that case, we would get simply by dividing 233 00:14:23,430 --> 00:14:27,240 out by this polynomial [? in ?] s, and so the system 234 00:14:27,240 --> 00:14:32,120 function then is the expression that I have here. 235 00:14:32,120 --> 00:14:37,030 So this is the form for a second order system where 236 00:14:37,030 --> 00:14:38,240 there are two poles. 237 00:14:38,240 --> 00:14:42,980 Since this is a second order polynomial, there are no zeros 238 00:14:42,980 --> 00:14:47,090 associated with the fact that I had no derivatives of the 239 00:14:47,090 --> 00:14:51,120 input on the right hand side of the equation. 240 00:14:51,120 --> 00:14:54,110 Well, let's look at this example-- 241 00:14:54,110 --> 00:14:55,620 namely the second order system-- 242 00:14:55,620 --> 00:14:57,190 in a little more detail. 243 00:14:57,190 --> 00:15:01,100 And what we'll want to look at is the location of the poles 244 00:15:01,100 --> 00:15:02,880 and some issues such as, for example, 245 00:15:02,880 --> 00:15:05,720 the frequency response. 246 00:15:05,720 --> 00:15:11,170 So here again I have the algebraic expression for the 247 00:15:11,170 --> 00:15:13,020 system function. 248 00:15:13,020 --> 00:15:16,820 And as I indicated, this is a second order polynomial, which 249 00:15:16,820 --> 00:15:21,830 means that we can factor it into two roots. 250 00:15:21,830 --> 00:15:27,320 So c1 and c2 represent the poles of the system function. 251 00:15:27,320 --> 00:15:31,970 And in particular, in relation to the two parameters zeta and 252 00:15:31,970 --> 00:15:33,500 omega sub n-- 253 00:15:33,500 --> 00:15:39,160 if we look at what these roots are, then what we get are the 254 00:15:39,160 --> 00:15:42,670 two expressions that I have below. 255 00:15:42,670 --> 00:15:50,270 And notice, incidentally, that if zeta is less than one, then 256 00:15:50,270 --> 00:15:52,410 what's under the square root is negative. 257 00:15:52,410 --> 00:15:54,830 And so this, in fact, corresponds to 258 00:15:54,830 --> 00:15:56,600 an imaginary part-- 259 00:15:56,600 --> 00:15:59,410 an imaginary term for zeta less than one. 260 00:15:59,410 --> 00:16:04,680 And so the two roots, then, have a real part which is 261 00:16:04,680 --> 00:16:09,390 given by minus zeta omega sub n, and an imaginary part-- if 262 00:16:09,390 --> 00:16:14,020 I were to rewrite this and then express it in terms of j 263 00:16:14,020 --> 00:16:16,270 or the square root of minus one. 264 00:16:16,270 --> 00:16:21,200 Looking below, we'll have a real part which is minus zeta 265 00:16:21,200 --> 00:16:22,560 omega sub n-- 266 00:16:22,560 --> 00:16:25,140 an imaginary part which is omega sub n 267 00:16:25,140 --> 00:16:27,290 times this square root. 268 00:16:27,290 --> 00:16:31,700 So that's for zeta less than one and for zeta greater than 269 00:16:31,700 --> 00:16:35,400 one, the two roots, of course, will be real. 270 00:16:35,400 --> 00:16:40,260 Alright, so let's examine this for the case where zeta is 271 00:16:40,260 --> 00:16:41,720 less than one. 272 00:16:41,720 --> 00:16:46,690 And what that corresponds to, then, are two poles in the 273 00:16:46,690 --> 00:16:49,310 complex plane. 274 00:16:49,310 --> 00:16:54,670 And they have a real part and an imaginary part. 275 00:16:54,670 --> 00:16:58,570 And you can explore this in somewhat more detail on your 276 00:16:58,570 --> 00:17:02,220 own, but, essentially what happens is that as you keep 277 00:17:02,220 --> 00:17:07,980 the parameter omega sub n fixed and vary zeta, these 278 00:17:07,980 --> 00:17:11,619 poles trace out a circle. 279 00:17:11,619 --> 00:17:16,910 And, for example, where zeta equal to zero, the poles are 280 00:17:16,910 --> 00:17:21,810 on the j omega axis at omega sub n. 281 00:17:21,810 --> 00:17:31,370 As zeta increases and gets closer to one, the poles 282 00:17:31,370 --> 00:17:37,430 converge toward the real axis and then, in particular, for 283 00:17:37,430 --> 00:17:42,060 zeta greater than one, what we end up with are two poles on 284 00:17:42,060 --> 00:17:44,430 the real axis. 285 00:17:44,430 --> 00:17:48,310 Well, actually, the case that we want to look at a little 286 00:17:48,310 --> 00:17:51,460 more carefully is when the poles are complex. 287 00:17:51,460 --> 00:17:55,410 And what this becomes is a second order system, which as 288 00:17:55,410 --> 00:17:58,580 we'll see as the discussion goes on, has an impulse 289 00:17:58,580 --> 00:18:02,450 response which oscillates with time and correspondingly a 290 00:18:02,450 --> 00:18:06,380 frequency response that has a resonance. 291 00:18:06,380 --> 00:18:09,640 Well let's examine the frequency response a little 292 00:18:09,640 --> 00:18:11,250 more carefully. 293 00:18:11,250 --> 00:18:14,640 And what I'm assuming in the discussion is that, first of 294 00:18:14,640 --> 00:18:20,110 all, the poles are in the left half plane corresponding to 295 00:18:20,110 --> 00:18:22,640 zeta omega sub n being positive-- 296 00:18:22,640 --> 00:18:25,490 and so this is-- minus that is negative. 297 00:18:25,490 --> 00:18:29,710 And furthermore, I'm assuming that the poles are complex. 298 00:18:29,710 --> 00:18:33,770 And in that case, the algebraic expression for the 299 00:18:33,770 --> 00:18:38,110 system function is omega sub n squared in the numerator and 300 00:18:38,110 --> 00:18:42,700 two poles in the denominator, which are complex conjugates. 301 00:18:42,700 --> 00:18:48,360 Now, what we want to look at is the frequency response of 302 00:18:48,360 --> 00:18:49,720 the system. 303 00:18:49,720 --> 00:18:50,620 And 304 00:18:50,620 --> 00:18:55,370 that corresponds to looking at the Fourier transform of the 305 00:18:55,370 --> 00:18:59,450 impulse response, which is the Laplace transform on the j 306 00:18:59,450 --> 00:19:00,930 omega axis. 307 00:19:00,930 --> 00:19:05,190 So we want to examine what h of s is as we move along the j 308 00:19:05,190 --> 00:19:06,680 omega axis. 309 00:19:06,680 --> 00:19:11,810 And notice, that to do that, in this algebraic expression, 310 00:19:11,810 --> 00:19:15,920 we want to set s equal to j omega and then evaluate-- 311 00:19:15,920 --> 00:19:17,830 for example, if we want to look at the magnitude of the 312 00:19:17,830 --> 00:19:19,470 frequency response-- 313 00:19:19,470 --> 00:19:24,740 evaluate the magnitude of the complex number. 314 00:19:24,740 --> 00:19:27,100 Well, there's a very convenient way of doing that 315 00:19:27,100 --> 00:19:33,420 geometrically by recognizing that in the complex plane, 316 00:19:33,420 --> 00:19:36,000 this complex number minus that complex number 317 00:19:36,000 --> 00:19:37,870 represents a vector. 318 00:19:37,870 --> 00:19:41,360 And essentially, to look at the magnitude of this complex 319 00:19:41,360 --> 00:19:46,060 number corresponds to taking omega sub n squared and 320 00:19:46,060 --> 00:19:51,720 dividing it by the product of the lengths of these vectors. 321 00:19:51,720 --> 00:19:57,820 So let's look, for example, at the vector s minus c1, where s 322 00:19:57,820 --> 00:20:00,980 is on the j omega axis. 323 00:20:00,980 --> 00:20:08,920 And doing that, here is the vector c1, and here is the 324 00:20:08,920 --> 00:20:12,530 vector s-- which is j omega if we're looking, let's say, at 325 00:20:12,530 --> 00:20:14,650 this value of frequency-- 326 00:20:14,650 --> 00:20:18,120 and this vector, then, is the vector which is 327 00:20:18,120 --> 00:20:21,100 j omega minus c1. 328 00:20:21,100 --> 00:20:24,630 So in fact, it's the length of this vector that we want to 329 00:20:24,630 --> 00:20:28,240 observe as we change omega-- 330 00:20:28,240 --> 00:20:31,540 namely as we move along the j omega axis. 331 00:20:31,540 --> 00:20:35,470 We want to take this vector and this vector, take the 332 00:20:35,470 --> 00:20:39,140 lengths of those vectors, multiply them together, divide 333 00:20:39,140 --> 00:20:42,400 that into omega sub n squared, and that will give us the 334 00:20:42,400 --> 00:20:43,590 frequency response. 335 00:20:43,590 --> 00:20:46,920 Now that's a little hard to see how the frequency response 336 00:20:46,920 --> 00:20:48,750 will work out just looking at one point. 337 00:20:48,750 --> 00:20:53,610 Although notice that as we move along the j omega axis, 338 00:20:53,610 --> 00:20:57,230 as we get closer to this pole, this vector, in fact, gets 339 00:20:57,230 --> 00:21:01,160 shorter, and so we might expect , that 340 00:21:01,160 --> 00:21:02,490 the frequency response-- 341 00:21:02,490 --> 00:21:05,120 as we're moving along the j omega axis in the vicinity of 342 00:21:05,120 --> 00:21:06,160 that pole-- 343 00:21:06,160 --> 00:21:08,010 would start to peak. 344 00:21:08,010 --> 00:21:11,320 Well, I think that all of this is much better seen 345 00:21:11,320 --> 00:21:14,850 dynamically on the computer display, so let's go to the 346 00:21:14,850 --> 00:21:17,630 computer display and what we'll look at is a second 347 00:21:17,630 --> 00:21:19,180 order system-- 348 00:21:19,180 --> 00:21:21,720 the frequency response of it-- as we move 349 00:21:21,720 --> 00:21:25,340 along the j omega axis. 350 00:21:25,340 --> 00:21:31,220 So here we see the pole pair in the complex plane and to 351 00:21:31,220 --> 00:21:33,890 generate the frequency response, we want to look at 352 00:21:33,890 --> 00:21:37,660 the behavior of the pole vectors as we move vertically 353 00:21:37,660 --> 00:21:39,660 along the j omega axis. 354 00:21:39,660 --> 00:21:45,620 So we'll show the pole vectors and let's begin at omega 355 00:21:45,620 --> 00:21:46,800 equals zero. 356 00:21:46,800 --> 00:21:49,510 So here we have the pole vectors from the poles to the 357 00:21:49,510 --> 00:21:52,130 point omega equal to zero. 358 00:21:52,130 --> 00:21:56,480 And, as we move vertically along the j omega axis, we'll 359 00:21:56,480 --> 00:22:01,190 see how those pole vectors change in length. 360 00:22:01,190 --> 00:22:04,310 The magnitude of the frequency response is the reciprocal of 361 00:22:04,310 --> 00:22:07,440 the product of the lengths of those vectors. 362 00:22:07,440 --> 00:22:12,090 Shown below is the frequency response where we've begun 363 00:22:12,090 --> 00:22:14,800 just at omega equal to zero. 364 00:22:14,800 --> 00:22:22,800 And as we move vertically along the j omega axis and the 365 00:22:22,800 --> 00:22:27,000 pole vector lengths change, that will, then, influence 366 00:22:27,000 --> 00:22:28,980 what the frequency response looks like. 367 00:22:28,980 --> 00:22:34,400 We've started here to move a little bit away from omega 368 00:22:34,400 --> 00:22:39,640 equal to zero and notice that in the upper half plane the 369 00:22:39,640 --> 00:22:41,840 pole vector has gotten shorter. 370 00:22:41,840 --> 00:22:44,450 The pole vector for the pole in the lower half plane has 371 00:22:44,450 --> 00:22:45,830 gotten longer. 372 00:22:45,830 --> 00:22:50,070 And now, as omega increases further, that 373 00:22:50,070 --> 00:22:51,910 process will continue. 374 00:22:51,910 --> 00:22:55,690 And in particular, the pole vector associated with the 375 00:22:55,690 --> 00:22:59,530 pole in the upper half plane will be its 376 00:22:59,530 --> 00:23:02,200 shortest in the vicinity-- 377 00:23:02,200 --> 00:23:04,810 at a frequency in the vicinity of that pole-- 378 00:23:04,810 --> 00:23:08,830 and so, for that frequency, then, the frequency response 379 00:23:08,830 --> 00:23:13,450 will peak and we see that here. 380 00:23:13,450 --> 00:23:17,640 From this point as the frequency increases, 381 00:23:17,640 --> 00:23:20,655 corresponding to moving further vertically along the j 382 00:23:20,655 --> 00:23:25,280 omega axis, both pole vectors will increase in length. 383 00:23:25,280 --> 00:23:29,530 And that means, then, that the magnitude of the frequency 384 00:23:29,530 --> 00:23:31,530 response will decrease. 385 00:23:31,530 --> 00:23:35,250 For this specific example, the magnitude of the frequency 386 00:23:35,250 --> 00:23:39,900 response will asymptotically go to zero. 387 00:23:39,900 --> 00:23:44,490 So what we see here is that the frequency response has a 388 00:23:44,490 --> 00:23:49,510 resonance and as we see geometrically from the way the 389 00:23:49,510 --> 00:23:53,940 vectors behaved, that resonance in frequency is very 390 00:23:53,940 --> 00:23:58,630 clearly associated with the position of the poles. 391 00:23:58,630 --> 00:24:03,240 And so, in fact, to illustrate that further and dramatize it 392 00:24:03,240 --> 00:24:07,070 as long as we're focused on it, let's now look at the 393 00:24:07,070 --> 00:24:12,310 frequency response for the second order example as we 394 00:24:12,310 --> 00:24:14,440 change the pole positions. 395 00:24:14,440 --> 00:24:19,260 And first, what we'll do is let the polls move vertically 396 00:24:19,260 --> 00:24:22,610 parallel to the j omega axis and see how the frequency 397 00:24:22,610 --> 00:24:26,120 response changes, and then we'll have the polls move 398 00:24:26,120 --> 00:24:29,260 horizontally parallel to the real axis and see how the 399 00:24:29,260 --> 00:24:31,950 frequency response changes. 400 00:24:31,950 --> 00:24:35,560 To display the behavior of the frequency response as the 401 00:24:35,560 --> 00:24:40,020 poles move, we've changed the vertical scale on the 402 00:24:40,020 --> 00:24:42,890 frequency response somewhat. 403 00:24:42,890 --> 00:24:47,950 And now what we want to do is move the poles, first, 404 00:24:47,950 --> 00:24:51,030 parallel to the j omega axis, and then 405 00:24:51,030 --> 00:24:53,380 parallel to the real axis. 406 00:24:53,380 --> 00:24:57,210 Here we see the effect of moving the poles parallel to 407 00:24:57,210 --> 00:24:59,030 the j omega axis. 408 00:24:59,030 --> 00:25:02,880 And what we observe is that, in fact, the frequency 409 00:25:02,880 --> 00:25:07,390 location of the resonance shifts, basically tracking the 410 00:25:07,390 --> 00:25:10,520 location of the pole. 411 00:25:10,520 --> 00:25:16,770 If we now move the pole back down closer to the real axis, 412 00:25:16,770 --> 00:25:21,430 then this resonance will shift back toward its original 413 00:25:21,430 --> 00:25:25,100 location and so let's now see that. 414 00:25:39,060 --> 00:25:43,480 And here we are back at the frequency that we started at. 415 00:25:43,480 --> 00:25:47,790 Now we'll move the poles even closer to the real axis. 416 00:25:47,790 --> 00:25:52,760 The frequency location of the resonance will continue to 417 00:25:52,760 --> 00:25:55,340 shift toward lower frequencies. 418 00:25:55,340 --> 00:25:59,500 And also in the process, incidentally, the height over 419 00:25:59,500 --> 00:26:03,350 the resonant peak will increase because, of course, 420 00:26:03,350 --> 00:26:08,780 the lengths of the pole vectors are getting shorter. 421 00:26:08,780 --> 00:26:12,350 And so, we see now the resonance shifting down toward 422 00:26:12,350 --> 00:26:14,600 lower and lower frequency. 423 00:26:14,600 --> 00:26:20,960 And, finally, what we'll now do is move the poles back to 424 00:26:20,960 --> 00:26:25,400 their original position and the resonant peak will, of 425 00:26:25,400 --> 00:26:27,970 course, shift back up. 426 00:26:27,970 --> 00:26:33,000 And correspondingly the height or amplitude of the resonance 427 00:26:33,000 --> 00:26:34,250 will decrease. 428 00:26:39,480 --> 00:26:43,000 And now we're back at the frequency response that we had 429 00:26:43,000 --> 00:26:45,180 generated previously. 430 00:26:45,180 --> 00:26:49,160 Next we'd like to look at the behavior as the polls move 431 00:26:49,160 --> 00:26:50,740 parallel to the real axis. 432 00:26:50,740 --> 00:26:54,530 First closer to the j omega axis and then further away. 433 00:26:54,530 --> 00:26:58,710 As they move closer to the j omega axis, the resonance 434 00:26:58,710 --> 00:27:03,810 sharpens because of the fact that the pole vector gets 435 00:27:03,810 --> 00:27:06,060 shorter and responds-- 436 00:27:06,060 --> 00:27:10,070 or changes in length more quickly as we move past it 437 00:27:10,070 --> 00:27:13,470 moving along the j omega axis. 438 00:27:13,470 --> 00:27:18,360 So here we see the effect of moving the poles closer to the 439 00:27:18,360 --> 00:27:20,020 j omega axis. 440 00:27:20,020 --> 00:27:24,000 The resonance has gotten narrower in frequency and 441 00:27:24,000 --> 00:27:28,380 higher in amplitude, associated with the fact that 442 00:27:28,380 --> 00:27:29,715 the pole vector gets shorter. 443 00:27:33,330 --> 00:27:38,320 Next as we move back to the original location, the 444 00:27:38,320 --> 00:27:40,760 resonance will broaden once again and the 445 00:27:40,760 --> 00:27:43,035 amplitude will decrease. 446 00:27:54,160 --> 00:27:57,810 And then, if we continue to move the poles even further 447 00:27:57,810 --> 00:28:02,770 away from the real axis, the resonance will broaden even 448 00:28:02,770 --> 00:28:05,580 further and the amplitude of the peak 449 00:28:05,580 --> 00:28:07,305 will become even smaller. 450 00:28:13,830 --> 00:28:18,330 And finally, let's now look just move the poles back to 451 00:28:18,330 --> 00:28:23,310 their original position and we'll see the resonance narrow 452 00:28:23,310 --> 00:28:24,610 again and become higher. 453 00:28:34,540 --> 00:28:38,800 And so what we see then is that for a second order 454 00:28:38,800 --> 00:28:42,530 system, the behavior of the resonance basically is 455 00:28:42,530 --> 00:28:46,310 associated with the pole locations, the frequency of 456 00:28:46,310 --> 00:28:48,760 the resonance associated with the vertical position of the 457 00:28:48,760 --> 00:28:54,900 poles, and the sharpness of the resonance associated with 458 00:28:54,900 --> 00:28:59,020 the real part of the poles-- in other words, their position 459 00:28:59,020 --> 00:29:02,365 closer or further away from the j omega axis. 460 00:29:05,240 --> 00:29:10,460 OK, so for complex poles, then, for the second order 461 00:29:10,460 --> 00:29:15,690 system, what we see is that we get a resonant kind of 462 00:29:15,690 --> 00:29:21,020 behavior, and, in particular, then that resonate behavior 463 00:29:21,020 --> 00:29:26,010 tends to peak, or get peakier, as the value 464 00:29:26,010 --> 00:29:28,510 of zeta gets smaller. 465 00:29:28,510 --> 00:29:32,960 And here, just to remind you of what you saw, here is the 466 00:29:32,960 --> 00:29:38,080 frequency response with one particular choice of values-- 467 00:29:38,080 --> 00:29:40,320 well, this is normalized so that omega sub n is one-- one 468 00:29:40,320 --> 00:29:43,910 particular choice for zeta, namely 0.4. 469 00:29:43,910 --> 00:29:52,390 Here is what we have with zeta smaller, and, finally, here is 470 00:29:52,390 --> 00:29:56,200 an example where zeta has gotten even smaller than that. 471 00:29:56,200 --> 00:29:59,960 And what that corresponds to is the poles moving closer to 472 00:29:59,960 --> 00:30:02,980 the j omega axis, the corresponding frequency 473 00:30:02,980 --> 00:30:06,290 response getting peakier. 474 00:30:06,290 --> 00:30:12,620 Now in the time domain what happens is that we have, of 475 00:30:12,620 --> 00:30:17,660 course, these complex roots, which I indicated previously, 476 00:30:17,660 --> 00:30:21,180 where this represents the imaginary part because zeta is 477 00:30:21,180 --> 00:30:23,230 less than one. 478 00:30:23,230 --> 00:30:25,900 And in the time domain, we will have a form for the 479 00:30:25,900 --> 00:30:33,960 behavior, which is a e to the c one t, plus a conjugate, e 480 00:30:33,960 --> 00:30:38,670 to the c one conjugate t. 481 00:30:38,670 --> 00:30:43,270 And so, in fact, as the poles get closer 482 00:30:43,270 --> 00:30:44,740 to the j omega axis-- 483 00:30:44,740 --> 00:30:47,450 corresponding to zeta getting smaller-- 484 00:30:47,450 --> 00:30:53,110 as the polls get closer to the j omega axis, in the frequency 485 00:30:53,110 --> 00:30:55,840 domain the resonances get sharper. 486 00:30:55,840 --> 00:30:59,650 In the time domain, the real part of the poles has gotten 487 00:30:59,650 --> 00:31:04,320 smaller, and that means, in fact, that in the time domain, 488 00:31:04,320 --> 00:31:08,330 the behavior will be more oscillatory and less damped. 489 00:31:08,330 --> 00:31:12,850 And so just looking at that again. 490 00:31:12,850 --> 00:31:17,280 Here is, in the time domain, what happens. 491 00:31:17,280 --> 00:31:24,170 First of all, with the parameter zeta equal to 0.4, 492 00:31:24,170 --> 00:31:29,130 and it oscillates and exponentially dies out. 493 00:31:29,130 --> 00:31:35,890 Here is the second order system where zeta is now 0.2 494 00:31:35,890 --> 00:31:37,730 instead of 0.4. 495 00:31:37,730 --> 00:31:44,250 And, finally, the second order system where zeta is 0.1. 496 00:31:44,250 --> 00:31:49,930 And what we see as zeta gets smaller and smaller is that 497 00:31:49,930 --> 00:31:53,810 the oscillations are basically the same, but the exponential 498 00:31:53,810 --> 00:31:58,610 damping becomes less and less. 499 00:31:58,610 --> 00:32:02,150 Alright, now, this is a somewhat more detailed look at 500 00:32:02,150 --> 00:32:03,950 second order systems. 501 00:32:03,950 --> 00:32:07,010 And second order systems-- and for that 502 00:32:07,010 --> 00:32:08,390 matter, first order systems-- 503 00:32:08,390 --> 00:32:13,100 are systems that are important in their own right, but they 504 00:32:13,100 --> 00:32:18,600 also are important as basic building blocks for more 505 00:32:18,600 --> 00:32:21,750 general, in particular, for higher order systems. 506 00:32:21,750 --> 00:32:25,850 And the way in which that's done typically is by combining 507 00:32:25,850 --> 00:32:29,400 first and second order systems together in such a way that 508 00:32:29,400 --> 00:32:31,750 they implement higher order systems. 509 00:32:31,750 --> 00:32:35,330 And two very common connections are connections 510 00:32:35,330 --> 00:32:39,470 which are cascade connections, and connections which are 511 00:32:39,470 --> 00:32:42,170 parallel connections. 512 00:32:42,170 --> 00:32:47,980 In a cascade connection, we would think of combining the 513 00:32:47,980 --> 00:32:52,060 individual systems together as I indicate here in series. 514 00:32:52,060 --> 00:32:55,150 And, of course, from the convolution property, the 515 00:32:55,150 --> 00:32:59,430 overall system function is the product of the individual 516 00:32:59,430 --> 00:33:01,270 system functions. 517 00:33:01,270 --> 00:33:06,840 So, for example, if these were all second order systems, and 518 00:33:06,840 --> 00:33:11,500 I combine n of them together in cascade, the overall system 519 00:33:11,500 --> 00:33:15,010 would be a system that would have to n poles-- in other 520 00:33:15,010 --> 00:33:18,190 words, it would be a two n order system. 521 00:33:18,190 --> 00:33:21,140 That's one very common kind of connection. 522 00:33:21,140 --> 00:33:24,010 Another very common kind of connection for first and 523 00:33:24,010 --> 00:33:28,180 second order systems is a parallel connection, where, in 524 00:33:28,180 --> 00:33:31,360 that case, we connect the systems together 525 00:33:31,360 --> 00:33:33,990 as I indicate here. 526 00:33:33,990 --> 00:33:38,900 The overall system function is just simply the sum of these, 527 00:33:38,900 --> 00:33:41,710 and that follows from the linearity property. 528 00:33:41,710 --> 00:33:44,720 And so the overall system function would be as I 529 00:33:44,720 --> 00:33:47,160 indicate algebraically here. 530 00:33:47,160 --> 00:33:51,130 And notice that if each of these are second order 531 00:33:51,130 --> 00:33:56,020 systems, and I had capital N of them in parallel, when you 532 00:33:56,020 --> 00:33:59,080 think of putting the overall system function over one 533 00:33:59,080 --> 00:34:02,070 common denominator, that common denominator, in 534 00:34:02,070 --> 00:34:08,050 general, is going to be of order two N. So either the 535 00:34:08,050 --> 00:34:11,639 parallel connection or the cascade connection could be 536 00:34:11,639 --> 00:34:16,370 used to implement higher order systems. 537 00:34:16,370 --> 00:34:20,870 One very common context in which second order systems are 538 00:34:20,870 --> 00:34:25,739 combined together, either in parallel or in cascade, to 539 00:34:25,739 --> 00:34:30,219 form a more interesting system is, in 540 00:34:30,219 --> 00:34:31,929 fact, in speech synthesis. 541 00:34:31,929 --> 00:34:35,639 And what I'd like to do is demonstrate a speech 542 00:34:35,639 --> 00:34:40,780 synthesizer, which I have here, which in fact is a 543 00:34:40,780 --> 00:34:46,460 parallel combination of four second order systems, very 544 00:34:46,460 --> 00:34:49,960 much of the type that we've just talked about. 545 00:34:49,960 --> 00:34:53,170 I'll return to the synthesizer in a minute. 546 00:34:53,170 --> 00:34:56,949 Let me first just indicate what the basic idea is. 547 00:34:56,949 --> 00:35:00,930 In speech synthesis, what we're trying to represent or 548 00:35:00,930 --> 00:35:04,140 implement is something that corresponds 549 00:35:04,140 --> 00:35:06,450 to the vocal tract. 550 00:35:06,450 --> 00:35:09,570 The vocal tract is characterized by a set of 551 00:35:09,570 --> 00:35:10,760 resonances. 552 00:35:10,760 --> 00:35:12,700 And we can think of representing each of those 553 00:35:12,700 --> 00:35:15,160 resonances by a second order system. 554 00:35:15,160 --> 00:35:17,700 And then the higher order system corresponding to the 555 00:35:17,700 --> 00:35:21,350 vocal tract is built by, in this case, a parallel 556 00:35:21,350 --> 00:35:25,210 combination of those second order systems. 557 00:35:25,210 --> 00:35:30,470 So for the synthesizer, what we have connected together in 558 00:35:30,470 --> 00:35:35,710 parallel is four second order systems. 559 00:35:35,710 --> 00:35:41,130 And a control on each one of them that controls the center 560 00:35:41,130 --> 00:35:45,850 frequency or the resonant frequency of each of the 561 00:35:45,850 --> 00:35:48,930 second order systems. 562 00:35:48,930 --> 00:35:52,880 The excitation is an excitation that would 563 00:35:52,880 --> 00:35:56,270 represent the air flow through the vocal cords. 564 00:35:56,270 --> 00:35:59,950 The vocal cords vibrate and there are puffs of air through 565 00:35:59,950 --> 00:36:02,630 the vocal cords as they open and close. 566 00:36:02,630 --> 00:36:08,430 And so the excitation for the synthesizer corresponds to a 567 00:36:08,430 --> 00:36:12,640 pulse train representing the air flow 568 00:36:12,640 --> 00:36:14,270 through the vocal cords. 569 00:36:14,270 --> 00:36:17,850 The fundamental frequency of this representing the 570 00:36:17,850 --> 00:36:21,740 fundamental frequency of the synthesized voice. 571 00:36:21,740 --> 00:36:25,890 So that's the basic structure of the synthesizer 572 00:36:25,890 --> 00:36:30,380 And what we have in this analog synthesizer are 573 00:36:30,380 --> 00:36:35,600 separate controls on the individual center frequencies. 574 00:36:35,600 --> 00:36:38,250 There is a control representing the center 575 00:36:38,250 --> 00:36:40,930 frequency of the third resonator and the fourth 576 00:36:40,930 --> 00:36:42,660 resonator, and those are represented 577 00:36:42,660 --> 00:36:44,570 by these two knobs. 578 00:36:44,570 --> 00:36:48,860 And then the first and second resonators are controlled by 579 00:36:48,860 --> 00:36:50,650 moving this joystick. 580 00:36:50,650 --> 00:36:56,280 The first resonator by moving the joystick along this axis 581 00:36:56,280 --> 00:36:58,220 and the second resonator by moving the 582 00:36:58,220 --> 00:37:01,030 joystick along this axis. 583 00:37:01,030 --> 00:37:05,570 And then, in addition to controls on the four 584 00:37:05,570 --> 00:37:09,630 resonators, we can control the fundamental frequency of the 585 00:37:09,630 --> 00:37:13,360 excitation, and we do that with this knob. 586 00:37:13,360 --> 00:37:17,070 So let's, first of all, just listen to one of the 587 00:37:17,070 --> 00:37:20,490 resonators, and the resonator that I'll play 588 00:37:20,490 --> 00:37:22,310 is the fourth resonator. 589 00:37:22,310 --> 00:37:26,840 And what you'll hear first is the output as I vary the 590 00:37:26,840 --> 00:37:28,730 center frequency of that resonator. 591 00:37:28,730 --> 00:37:30,320 [BUZZING] 592 00:37:30,320 --> 00:37:32,190 So I'm lowering the center frequency. 593 00:37:35,310 --> 00:37:37,700 And then, bringing the center frequency back up. 594 00:37:40,790 --> 00:37:44,200 And then, as I indicated, I can also control the 595 00:37:44,200 --> 00:37:47,400 fundamental frequency of the excitation by 596 00:37:47,400 --> 00:37:48,541 turning this knob. 597 00:37:48,541 --> 00:37:51,310 [BUZZING] 598 00:37:51,310 --> 00:37:52,720 Lowering the fundamental frequency. 599 00:37:55,510 --> 00:37:58,486 And then, increasing the fundamental frequency. 600 00:38:01,960 --> 00:38:07,010 Alright, now, if the four resonators in parallel are an 601 00:38:07,010 --> 00:38:11,640 implementation of the vocal cavity, then, presumably, what 602 00:38:11,640 --> 00:38:15,600 we can synthesize when we put them all in are vowel sounds 603 00:38:15,600 --> 00:38:17,360 and let's do that. 604 00:38:17,360 --> 00:38:22,500 I'll now switch in the other resonators. 605 00:38:22,500 --> 00:38:27,530 When we do that, then, depending on what choice we 606 00:38:27,530 --> 00:38:30,920 have for the individual resonant frequencies, we 607 00:38:30,920 --> 00:38:32,990 should be able to synthesize vowel sounds. 608 00:38:32,990 --> 00:38:35,210 So here, for example, is the vowel e. 609 00:38:35,210 --> 00:38:37,640 [BUZZING "E"]. 610 00:38:37,640 --> 00:38:38,280 Here is 611 00:38:38,280 --> 00:38:39,025 [BUZZING "AH"] 612 00:38:39,025 --> 00:38:40,275 --ah. 613 00:38:41,875 --> 00:38:42,662 A. 614 00:38:42,662 --> 00:38:44,390 [BUZZING FLAT A] 615 00:38:44,390 --> 00:38:45,360 And, of course, we can-- 616 00:38:45,360 --> 00:38:46,662 [BUZZING OO] 617 00:38:46,662 --> 00:38:47,526 --generate 618 00:38:47,526 --> 00:38:47,902 [BUZZING "I"] 619 00:38:47,902 --> 00:38:50,160 --lots of other vowel sounds. 620 00:38:50,160 --> 00:38:51,170 [BUZZING "AH"] 621 00:38:51,170 --> 00:38:53,910 --and change the fundamental frequency at the same time. 622 00:38:53,910 --> 00:38:55,160 [CHANGES FREQUENCY UP AND DOWN] 623 00:39:01,600 --> 00:39:05,380 Now, if we want to synthesize speech it's not enough to just 624 00:39:05,380 --> 00:39:08,100 synthesize steady state vowels-- that gets boring 625 00:39:08,100 --> 00:39:08,920 after a while. 626 00:39:08,920 --> 00:39:13,360 Of course what happens with the vocal cavity is that it 627 00:39:13,360 --> 00:39:19,330 moves as a function of time and that's what generates the 628 00:39:19,330 --> 00:39:21,580 speech that we want to generate. 629 00:39:21,580 --> 00:39:26,520 And so, presumably then, if we change these resonant 630 00:39:26,520 --> 00:39:30,330 frequencies as a function of time appropriately, then we 631 00:39:30,330 --> 00:39:32,500 should be able to synthesize speech. 632 00:39:32,500 --> 00:39:36,650 And so by moving these resonances around, we can 633 00:39:36,650 --> 00:39:38,800 generate synthesized speech. 634 00:39:38,800 --> 00:39:43,600 And let's try it with some phrase. 635 00:39:43,600 --> 00:39:46,380 And I'll do that by simply adjusting the center 636 00:39:46,380 --> 00:39:47,630 frequencies appropriately. 637 00:39:50,310 --> 00:39:57,030 [BUZZING "HOW ARE YOU"] 638 00:39:57,030 --> 00:39:59,740 Well, hopefully you understood that. 639 00:39:59,740 --> 00:40:03,910 As you could imagine, I spent at least a few minutes before 640 00:40:03,910 --> 00:40:06,820 the lecture trying to practice that so that it would come out 641 00:40:06,820 --> 00:40:10,090 to be more or less intelligible. 642 00:40:10,090 --> 00:40:14,370 Now the system as I've just demonstrated it is, of course, 643 00:40:14,370 --> 00:40:19,490 a continuous time system or an analog speech synthesizer. 644 00:40:19,490 --> 00:40:23,540 There are many versions of digital or discrete time 645 00:40:23,540 --> 00:40:24,980 synthesizers. 646 00:40:24,980 --> 00:40:30,000 One of the first, in fact, being a device that many of 647 00:40:30,000 --> 00:40:33,310 you are very likely familiar with, which is the Texas 648 00:40:33,310 --> 00:40:36,680 Instruments Speak and Spell, which I show here. 649 00:40:36,680 --> 00:40:41,290 And what's very interesting and rather dramatic about this 650 00:40:41,290 --> 00:40:44,740 device is the fact that it implements the speech 651 00:40:44,740 --> 00:40:49,500 synthesis in very much the same way as I've demonstrated 652 00:40:49,500 --> 00:40:51,620 with the analog synthesizer. 653 00:40:51,620 --> 00:40:55,670 In this case, it's five second order filters in a 654 00:40:55,670 --> 00:40:58,170 configuration that's slightly different than a parallel 655 00:40:58,170 --> 00:41:02,130 configuration but conceptually very closely related. 656 00:41:02,130 --> 00:41:05,510 And let's take a look inside the box. 657 00:41:05,510 --> 00:41:09,520 And what we see there, with a slide that was kindly supplied 658 00:41:09,520 --> 00:41:13,720 by Texas Instruments, is the fact that there really are 659 00:41:13,720 --> 00:41:15,470 only four chips in there-- 660 00:41:15,470 --> 00:41:17,600 a controller chip, some storage. 661 00:41:17,600 --> 00:41:21,370 And the important point is the chip that's labeled as the 662 00:41:21,370 --> 00:41:25,860 speech synthesis chip, in fact, is what embodies or 663 00:41:25,860 --> 00:41:30,540 implements the five second order filters and, in 664 00:41:30,540 --> 00:41:34,470 addition, incorporates some other things-- some memory and 665 00:41:34,470 --> 00:41:36,310 also the [? DDA ?] 666 00:41:36,310 --> 00:41:37,270 converters. 667 00:41:37,270 --> 00:41:40,260 So, in fact, the implementation of the 668 00:41:40,260 --> 00:41:45,040 synthesizer is pretty much done on a single chip. 669 00:41:45,040 --> 00:41:48,940 Well that's a discrete time system. 670 00:41:48,940 --> 00:41:53,600 We've been talking for the last several lectures about 671 00:41:53,600 --> 00:41:57,230 continuous time systems and the Laplace transform. 672 00:41:57,230 --> 00:41:59,950 Hopefully what you've seen in this lecture and the previous 673 00:41:59,950 --> 00:42:07,110 lecture is the powerful tool that the Laplace transform 674 00:42:07,110 --> 00:42:11,830 affords us in analyzing and understanding system behavior. 675 00:42:14,440 --> 00:42:18,560 In the next lecture what I'd like to do is parallel the 676 00:42:18,560 --> 00:42:21,530 discussion for discrete time, turn our attention to the z 677 00:42:21,530 --> 00:42:26,330 transform, and, as you can imagine simply by virtue of 678 00:42:26,330 --> 00:42:31,670 the fact that I have shown you a digital and analog version 679 00:42:31,670 --> 00:42:35,040 of very much the same kind of system, the discussions 680 00:42:35,040 --> 00:42:38,960 parallel themselves very strongly and the z transform 681 00:42:38,960 --> 00:42:42,560 will play very much the same role in discrete time that the 682 00:42:42,560 --> 00:42:45,090 Laplace transform does in continuous time. 683 00:42:45,090 --> 00:42:46,340 Thank you.