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 --> 00:00:20,600 ocw.mit.edu. 8 00:00:20,600 --> 00:00:55,444 [MUSIC PLAYING] 9 00:00:55,444 --> 00:00:58,610 PROFESSOR: Last time, we began the discussion of the 10 00:00:58,610 --> 00:01:01,240 z-transform. 11 00:01:01,240 --> 00:01:04,110 As with the Laplace transform in continuous time, we 12 00:01:04,110 --> 00:01:06,190 developed it as a generalization 13 00:01:06,190 --> 00:01:09,000 of the Fourier transform. 14 00:01:09,000 --> 00:01:14,230 The expression that we got for the z-transform is the sum 15 00:01:14,230 --> 00:01:17,050 that I indicate here. 16 00:01:17,050 --> 00:01:20,750 We also briefly talked about an inverse z-transform 17 00:01:20,750 --> 00:01:25,320 integral and some other informal methods of computing 18 00:01:25,320 --> 00:01:27,660 the inverse z-transform. 19 00:01:27,660 --> 00:01:32,220 But we focused in particular on the relationship between 20 00:01:32,220 --> 00:01:36,090 the z-transform and the Fourier transform, pointing 21 00:01:36,090 --> 00:01:41,800 out first of all that the z-transform, when we choose 22 00:01:41,800 --> 00:01:44,590 the magnitude of z equal to 1-- 23 00:01:44,590 --> 00:01:48,440 so the magnitude of z of the form e to the j omega-- 24 00:01:48,440 --> 00:01:52,150 just simply reduces to the Fourier 25 00:01:52,150 --> 00:01:55,210 transform of the sequence. 26 00:01:55,210 --> 00:02:00,160 Then, in addition, we explored the z-transform for z, a more 27 00:02:00,160 --> 00:02:02,640 general complex number. 28 00:02:02,640 --> 00:02:06,390 In the discrete time z-transform case, we expressed 29 00:02:06,390 --> 00:02:12,130 that complex number in polar form as r e to the j omega, 30 00:02:12,130 --> 00:02:17,800 and recognize that the z-transform expression in fact 31 00:02:17,800 --> 00:02:22,900 corresponds to the Fourier transform of the sequence 32 00:02:22,900 --> 00:02:25,400 exponentially weighted. 33 00:02:25,400 --> 00:02:30,610 Because of the exponential weighting, the z-transform 34 00:02:30,610 --> 00:02:35,100 converges for some values of r corresponding to some 35 00:02:35,100 --> 00:02:39,320 exponential waiting, and perhaps not for others, and 36 00:02:39,320 --> 00:02:44,890 that led to a notion which corresponded to the region of 37 00:02:44,890 --> 00:02:49,180 convergence associated with the z-transform. 38 00:02:49,180 --> 00:02:51,680 We talked some about properties of the region of 39 00:02:51,680 --> 00:02:55,230 convergence, particularly in relation to 40 00:02:55,230 --> 00:02:57,900 the pole-zero pattern. 41 00:02:57,900 --> 00:03:03,090 Now, they z-transform has a number of important and useful 42 00:03:03,090 --> 00:03:06,780 properties, just as the Laplace transform does. 43 00:03:06,780 --> 00:03:10,580 As one part of this lecture, what we'll want to do is 44 00:03:10,580 --> 00:03:14,440 exploit some of these properties in the context of 45 00:03:14,440 --> 00:03:16,930 systems described by linear constant coefficient 46 00:03:16,930 --> 00:03:19,360 difference equations. 47 00:03:19,360 --> 00:03:23,660 These particular properties that play an important role in 48 00:03:23,660 --> 00:03:25,280 that context are the properties 49 00:03:25,280 --> 00:03:27,570 that I indicate here. 50 00:03:27,570 --> 00:03:29,500 In particular, there is-- 51 00:03:29,500 --> 00:03:31,780 as with continuous time-- 52 00:03:31,780 --> 00:03:36,500 a linearity property that tells us that the z-transform 53 00:03:36,500 --> 00:03:40,870 of a linear combination of sequences is the same linear 54 00:03:40,870 --> 00:03:45,140 combination of the z-transforms, a shifting 55 00:03:45,140 --> 00:03:50,580 property that indicates that the z-transform of x of n 56 00:03:50,580 --> 00:03:56,740 shifted is the z transform of x of n multiplied by a factor 57 00:03:56,740 --> 00:03:59,780 z to the minus n 0. 58 00:03:59,780 --> 00:04:06,120 Then the convolution property for which the z-transform of a 59 00:04:06,120 --> 00:04:11,500 convolution of sequences is the product of the associated 60 00:04:11,500 --> 00:04:14,250 z-transforms. 61 00:04:14,250 --> 00:04:17,769 With all of these properties, of course, there is again the 62 00:04:17,769 --> 00:04:21,760 issue of what the associated region of convergence is in 63 00:04:21,760 --> 00:04:23,840 comparison with the region of convergence of 64 00:04:23,840 --> 00:04:26,500 the original sequences. 65 00:04:26,500 --> 00:04:30,040 That is an issue that's addressed somewhat more in the 66 00:04:30,040 --> 00:04:31,730 text and let's not go into it here. 67 00:04:34,240 --> 00:04:40,890 With the convolution property, the convolution property as in 68 00:04:40,890 --> 00:04:45,220 continuous time, of course, provides a mechanism for 69 00:04:45,220 --> 00:04:48,750 dealing with linear time invariant systems. 70 00:04:48,750 --> 00:04:52,640 In particular, in the time domain a linear time invariant 71 00:04:52,640 --> 00:04:55,980 system is described through convolution-- 72 00:04:55,980 --> 00:05:00,610 namely, the output is the convolution of the input and 73 00:05:00,610 --> 00:05:02,770 the impulse response. 74 00:05:02,770 --> 00:05:06,810 Because of the convolution property associated with the 75 00:05:06,810 --> 00:05:11,590 z-transform, the z-transform of the output is the 76 00:05:11,590 --> 00:05:15,650 z-transform of the input times the z-transform 77 00:05:15,650 --> 00:05:17,580 of the impulse response. 78 00:05:17,580 --> 00:05:22,410 Again, very much the same as what we had in continuous time 79 00:05:22,410 --> 00:05:26,590 and also what we had in the context of the discussion with 80 00:05:26,590 --> 00:05:29,410 the Fourier transform. 81 00:05:29,410 --> 00:05:32,290 In fact, because of the relationship between the 82 00:05:32,290 --> 00:05:37,280 z-transform and the Fourier transform, the z-transform of 83 00:05:37,280 --> 00:05:41,690 the impulse response evaluated on the unit circle-- 84 00:05:41,690 --> 00:05:44,570 in other words, for the magnitude of z equal to 1-- 85 00:05:44,570 --> 00:05:47,620 in fact corresponds to the frequency 86 00:05:47,620 --> 00:05:50,060 response of the system. 87 00:05:50,060 --> 00:05:53,400 More generally, when we talk about the z-transform of the 88 00:05:53,400 --> 00:05:58,300 impulse response, we will refer to it as the system 89 00:05:58,300 --> 00:06:00,225 function associated with the system. 90 00:06:03,130 --> 00:06:06,030 Now, the convolution property and these other properties, as 91 00:06:06,030 --> 00:06:09,740 I indicated, we will find useful in talking about 92 00:06:09,740 --> 00:06:13,260 systems which are described by linear constant coefficient 93 00:06:13,260 --> 00:06:18,460 difference equations, and in fact, we'll do that shortly. 94 00:06:18,460 --> 00:06:25,320 But first what I'd like to do is to continue to focus on the 95 00:06:25,320 --> 00:06:29,270 system function for linear time invariant systems, and 96 00:06:29,270 --> 00:06:33,350 make a couple of comments that tie back to some things that 97 00:06:33,350 --> 00:06:38,280 we said in the last lecture relating to the relationship 98 00:06:38,280 --> 00:06:43,510 between the region of convergence of a system, or-- 99 00:06:43,510 --> 00:06:47,080 I'm sorry, the region of convergence of a z-transform-- 100 00:06:47,080 --> 00:06:54,040 and the issue of where that is in relation to the poles of 101 00:06:54,040 --> 00:06:55,290 the z-transform. 102 00:06:56,980 --> 00:07:00,260 In particular, we can draw some conclusions tying back to 103 00:07:00,260 --> 00:07:05,780 that discussion about the pole locations of the system 104 00:07:05,780 --> 00:07:11,810 function in relation to whether the system is stable 105 00:07:11,810 --> 00:07:15,510 and whether the system is causal. 106 00:07:15,510 --> 00:07:19,960 In particular, recall from one of the early lectures way back 107 00:07:19,960 --> 00:07:25,830 when that stability for a system corresponded to the 108 00:07:25,830 --> 00:07:28,500 statement that the impulse response 109 00:07:28,500 --> 00:07:31,450 is absolutely summable. 110 00:07:31,450 --> 00:07:35,640 Furthermore, when we talked about the Fourier transform, 111 00:07:35,640 --> 00:07:42,220 the Fourier transform of a sequence converges if the 112 00:07:42,220 --> 00:07:44,500 sequence is absolutely summable. 113 00:07:44,500 --> 00:07:50,630 So in fact, the condition for stability of a system and the 114 00:07:50,630 --> 00:07:53,540 condition for convergence of the Fourier transform of its 115 00:07:53,540 --> 00:07:58,820 impulse response are the same condition, namely absolute 116 00:07:58,820 --> 00:08:00,900 summability. 117 00:08:00,900 --> 00:08:02,750 Now what does this mean? 118 00:08:02,750 --> 00:08:07,870 What it means is that if the Fourier transform converges, 119 00:08:07,870 --> 00:08:11,140 that means that the z-transform converges on the 120 00:08:11,140 --> 00:08:13,640 unit circle. 121 00:08:13,640 --> 00:08:18,200 Consequently, if the system is stable, then the system 122 00:08:18,200 --> 00:08:22,500 function, the z-transform or the impulse response, must 123 00:08:22,500 --> 00:08:25,210 also converge on the unit circle. 124 00:08:25,210 --> 00:08:28,610 In other words, the impulse response must have a Fourier 125 00:08:28,610 --> 00:08:31,160 transform that converges. 126 00:08:31,160 --> 00:08:36,549 So for a stable system, then, the region of convergence of 127 00:08:36,549 --> 00:08:39,450 the system function must include the unit 128 00:08:39,450 --> 00:08:42,620 circle in the z-plane. 129 00:08:42,620 --> 00:08:50,080 So we see how stability relates to the location of the 130 00:08:50,080 --> 00:08:54,450 region of convergence, and we can also relate causality to 131 00:08:54,450 --> 00:08:55,670 the region of convergence. 132 00:08:55,670 --> 00:09:01,850 In particular, we know that if a system is causal, then the 133 00:09:01,850 --> 00:09:05,980 impulse response is right-sided. 134 00:09:05,980 --> 00:09:08,250 For a sequence that's right-sided, the region of 135 00:09:08,250 --> 00:09:12,060 convergence of its z-transform must be outside 136 00:09:12,060 --> 00:09:15,110 the outermost pole. 137 00:09:15,110 --> 00:09:18,800 So for causality, the region of convergence of the system 138 00:09:18,800 --> 00:09:21,710 function must be outside the outermost pole. 139 00:09:21,710 --> 00:09:25,350 For stability, the region of convergence must include the 140 00:09:25,350 --> 00:09:30,150 unit circle, and we can also then draw from that the 141 00:09:30,150 --> 00:09:34,840 conclusion that if we have a system that's causal unstable, 142 00:09:34,840 --> 00:09:39,430 then all poles must be inside the unit circle because of the 143 00:09:39,430 --> 00:09:41,130 fact that the poles must-- 144 00:09:41,130 --> 00:09:44,420 because of the fact that the region of convergence must be 145 00:09:44,420 --> 00:09:47,480 outside the outermost pole and has to also 146 00:09:47,480 --> 00:09:49,910 include the unit circle. 147 00:09:49,910 --> 00:09:55,780 So for example, if we had let's say a system with a 148 00:09:55,780 --> 00:10:05,840 system function as I indicate here with the algebraic 149 00:10:05,840 --> 00:10:09,930 expression for the system function being the expression 150 00:10:09,930 --> 00:10:13,810 that I indicate here with the pole at z equals a third and 151 00:10:13,810 --> 00:10:18,290 another pole at z equals 2 and a zero at the origin. 152 00:10:18,290 --> 00:10:24,210 If, in fact, the system was causal, corresponding to an 153 00:10:24,210 --> 00:10:28,050 impulse response that's right-sided, this then would 154 00:10:28,050 --> 00:10:34,050 be the region of convergence of the system function. 155 00:10:34,050 --> 00:10:38,220 Alternatively, if I knew that the system was stable, then I 156 00:10:38,220 --> 00:10:40,910 know that the region of convergence must include the 157 00:10:40,910 --> 00:10:45,080 unit circle, and so this would be then the region of 158 00:10:45,080 --> 00:10:46,640 convergence. 159 00:10:46,640 --> 00:10:53,690 And what you might now want to ask yourselves is if instead 160 00:10:53,690 --> 00:10:57,980 the region of convergence for the system function is this, 161 00:10:57,980 --> 00:11:00,570 then is the system causal? 162 00:11:00,570 --> 00:11:02,390 That's the first question. 163 00:11:02,390 --> 00:11:06,340 The second question is is the system stable? 164 00:11:06,340 --> 00:11:08,840 Remembering that for causality, the region of 165 00:11:08,840 --> 00:11:12,150 convergence must be outside the outermost pole, and for 166 00:11:12,150 --> 00:11:16,540 stability it must include the unit circle. 167 00:11:16,540 --> 00:11:25,200 Now what I'd like to do is look at the properties of the 168 00:11:25,200 --> 00:11:29,430 z-transform, and in particular exploit these properties in 169 00:11:29,430 --> 00:11:33,850 the context of systems that are described by linear 170 00:11:33,850 --> 00:11:36,930 constant coefficient difference equations. 171 00:11:36,930 --> 00:11:40,230 The three basic properties that play a key role in that 172 00:11:40,230 --> 00:11:44,980 discussion are the linearity property, the shifting 173 00:11:44,980 --> 00:11:49,860 property, and the convolution property. 174 00:11:49,860 --> 00:11:52,430 These, then, are the properties 175 00:11:52,430 --> 00:11:55,380 that I want to exploit. 176 00:11:55,380 --> 00:11:59,740 So let's do that by first looking at a first order 177 00:11:59,740 --> 00:12:02,480 difference equation. 178 00:12:02,480 --> 00:12:06,310 In the case of a first order difference equation, which 179 00:12:06,310 --> 00:12:09,170 I've written as I indicate here-- 180 00:12:09,170 --> 00:12:11,930 no terms on the right hand side, but we could have terms 181 00:12:11,930 --> 00:12:13,860 of course, in general-- 182 00:12:13,860 --> 00:12:17,900 y of n minus ay of n minus 1 equals x of n. 183 00:12:17,900 --> 00:12:24,630 We can use the linearity property, so that if we take 184 00:12:24,630 --> 00:12:28,570 the z-transform of both sides of this expression, that will 185 00:12:28,570 --> 00:12:32,110 then be the z-transform of this term plus the z-transform 186 00:12:32,110 --> 00:12:34,180 of this term. 187 00:12:34,180 --> 00:12:39,790 So using those properties and together with the shifting 188 00:12:39,790 --> 00:12:44,010 property, the property that tells us that the z-transform 189 00:12:44,010 --> 00:12:49,110 of y of n minus 1 is z to the minus 1 times the z-transform 190 00:12:49,110 --> 00:12:53,520 of y of n, we then convert the difference equation to an 191 00:12:53,520 --> 00:12:55,400 algebraic expression. 192 00:12:55,400 --> 00:12:58,940 And we can solve this algebraic expression for the 193 00:12:58,940 --> 00:13:02,600 z-transform of the output in terms of the 194 00:13:02,600 --> 00:13:06,100 z-transform of the input. 195 00:13:06,100 --> 00:13:11,820 Now what we know from the convolution property is that 196 00:13:11,820 --> 00:13:15,670 for a system, the z-transform of the output is the system 197 00:13:15,670 --> 00:13:18,440 function times the z-transform of the input. 198 00:13:18,440 --> 00:13:24,720 So this factor that we have, then, must correspond to the 199 00:13:24,720 --> 00:13:29,260 system function, or equivalently the z-transform 200 00:13:29,260 --> 00:13:32,360 of the impulse response of the system. 201 00:13:32,360 --> 00:13:36,200 In fact, then, if we have this z-transform, we could figure 202 00:13:36,200 --> 00:13:41,560 out what the impulse response of the system is by computing 203 00:13:41,560 --> 00:13:45,730 or determining what the inverse z-transform, except 204 00:13:45,730 --> 00:13:52,590 for the fact that expression is an algebraic expression and 205 00:13:52,590 --> 00:13:55,940 doesn't yet totally specify the z-transform because we 206 00:13:55,940 --> 00:14:00,210 don't yet know what the region of convergence is. 207 00:14:00,210 --> 00:14:02,670 How do we get the region of convergence? 208 00:14:02,670 --> 00:14:05,320 Well, we have the same issue here as we had with the 209 00:14:05,320 --> 00:14:06,980 Laplace transform-- 210 00:14:06,980 --> 00:14:13,440 namely, the point that the difference equation tells us, 211 00:14:13,440 --> 00:14:17,340 in essence, what the algebraic expression is for the system 212 00:14:17,340 --> 00:14:22,080 function, but doesn't specify the region of convergence. 213 00:14:22,080 --> 00:14:26,910 That is specified by either explicitly, because one way or 214 00:14:26,910 --> 00:14:30,590 another we know what the impulse response is, or 215 00:14:30,590 --> 00:14:33,940 implicitly, because we know certain properties of the 216 00:14:33,940 --> 00:14:40,130 system, such as causality and or stability. 217 00:14:40,130 --> 00:14:46,030 If I, let's say, imposed on this system, in addition to 218 00:14:46,030 --> 00:14:55,580 the difference equation, the condition of causality, then 219 00:14:55,580 --> 00:14:59,530 what that requires is that the impulse response be 220 00:14:59,530 --> 00:15:03,700 right-sided or the region of convergence be outside the 221 00:15:03,700 --> 00:15:05,330 outermost pole. 222 00:15:05,330 --> 00:15:09,350 So, for this example that would require, then, that the 223 00:15:09,350 --> 00:15:12,940 region of convergence correspond to the magnitude of 224 00:15:12,940 --> 00:15:18,190 z greater than the magnitude of a. 225 00:15:18,190 --> 00:15:21,670 What you might think about is whether I would get the same 226 00:15:21,670 --> 00:15:25,960 condition if I required instead that 227 00:15:25,960 --> 00:15:29,310 the system be stable. 228 00:15:29,310 --> 00:15:31,500 Furthermore, you could think about the question of whether 229 00:15:31,500 --> 00:15:35,480 I could specify or impose on this system that it be both 230 00:15:35,480 --> 00:15:37,090 stable and causal. 231 00:15:37,090 --> 00:15:39,760 The real issue-- and let me just kind of point to it-- 232 00:15:39,760 --> 00:15:43,970 is that the answer to those questions relate to whether 233 00:15:43,970 --> 00:15:46,730 the magnitude of a is less than 1 or the magnitude of a 234 00:15:46,730 --> 00:15:48,160 is greater than 1. 235 00:15:48,160 --> 00:15:50,810 If the magnitude of a is less than 1 and I specify 236 00:15:50,810 --> 00:15:57,320 causality, that will also mean that the system is stable. 237 00:15:57,320 --> 00:16:02,170 In any case, given this region of convergence, then the 238 00:16:02,170 --> 00:16:06,340 impulse response is the inverse transform of that 239 00:16:06,340 --> 00:16:08,780 z-transform, which is a to the n times u of n. 240 00:16:11,490 --> 00:16:15,680 Now let's look at a second order equation, and there's a 241 00:16:15,680 --> 00:16:17,470 very similar strategy. 242 00:16:17,470 --> 00:16:21,430 For the second order equation, the one that I've picked is of 243 00:16:21,430 --> 00:16:24,700 this particular form, and I've written the coefficients 244 00:16:24,700 --> 00:16:28,270 parametrically as I indicate here for a specific reason, 245 00:16:28,270 --> 00:16:29,880 which we'll see shortly. 246 00:16:29,880 --> 00:16:32,870 Again, I can apply the z-transform to this 247 00:16:32,870 --> 00:16:38,580 expression, and I've skipped an algebraic step or two here. 248 00:16:38,580 --> 00:16:42,970 When I do this, then, again I use the linearity property and 249 00:16:42,970 --> 00:16:46,790 the shifting property, and I end up with this algebraic 250 00:16:46,790 --> 00:16:47,890 expression. 251 00:16:47,890 --> 00:16:54,240 If I now solve that to express y of z in terms of x of z and 252 00:16:54,240 --> 00:16:58,790 a function of z, then this is what I get 253 00:16:58,790 --> 00:17:00,040 for the system function. 254 00:17:02,430 --> 00:17:06,970 So this is now a second order system function, and we'll 255 00:17:06,970 --> 00:17:12,960 have two zeroes at the origin and it will have two poles. 256 00:17:12,960 --> 00:17:17,339 Again, there's the question of what we assume about the 257 00:17:17,339 --> 00:17:18,349 region of convergence-- 258 00:17:18,349 --> 00:17:20,240 I haven't specified that yet. 259 00:17:20,240 --> 00:17:25,990 But if we, let's say, assume that the system is causal, 260 00:17:25,990 --> 00:17:29,050 which I will tend to do, then that means that the region of 261 00:17:29,050 --> 00:17:32,830 convergence is outside the outermost pole. 262 00:17:32,830 --> 00:17:34,080 Now, where are the poles? 263 00:17:34,080 --> 00:17:38,920 Well, let me just kind of indicate that if-- 264 00:17:38,920 --> 00:17:40,010 and you can verify this 265 00:17:40,010 --> 00:17:43,250 algebraically at your leisure-- 266 00:17:43,250 --> 00:17:52,680 that if the cosine theta term is less than 1, then the roots 267 00:17:52,680 --> 00:17:56,110 of this polynomial will be complex. 268 00:17:56,110 --> 00:18:00,570 And in fact, the poles are at r e to the 269 00:18:00,570 --> 00:18:02,740 plus or minus j theta. 270 00:18:02,740 --> 00:18:08,970 So for cosine theta less than 1, then the poles are complex, 271 00:18:08,970 --> 00:18:13,960 and the complex poles at an angle theta and with a 272 00:18:13,960 --> 00:18:18,070 distance from the origin equal to the parameter r. 273 00:18:18,070 --> 00:18:21,550 In fact, let's just look at that. 274 00:18:21,550 --> 00:18:27,640 What I show here is the pole zero pattern. 275 00:18:27,640 --> 00:18:34,490 Assuming that r is less than 1, and that cosine theta is 276 00:18:34,490 --> 00:18:40,300 less than 1, and we have a complex pole pair shown here-- 277 00:18:40,300 --> 00:18:44,210 now if we assume that the system was causal, that means 278 00:18:44,210 --> 00:18:48,240 that the region of convergence is outside these poles. 279 00:18:48,240 --> 00:18:51,300 That would then include the unit circle, which means that 280 00:18:51,300 --> 00:18:54,290 the system is also stable. 281 00:18:54,290 --> 00:18:57,370 In fact, as long as the reach of convergence includes the 282 00:18:57,370 --> 00:19:02,850 unit circle, we can also talk about the frequency response 283 00:19:02,850 --> 00:19:03,680 of the system-- 284 00:19:03,680 --> 00:19:05,240 namely, we can evaluate the system 285 00:19:05,240 --> 00:19:08,500 function on the unit circle. 286 00:19:08,500 --> 00:19:13,270 We, in fact, evaluated the Fourier transform associated 287 00:19:13,270 --> 00:19:16,660 with this pole zero pattern last time. 288 00:19:16,660 --> 00:19:21,210 Recall that the frequency response, then, is one that 289 00:19:21,210 --> 00:19:26,620 has a resonant character with the resonant peak being 290 00:19:26,620 --> 00:19:33,010 roughly in the vicinity of the angle of the pole location. 291 00:19:33,010 --> 00:19:36,490 As the parameter r varies-- let's say, as r gets smaller-- 292 00:19:36,490 --> 00:19:38,710 this peak tends to broaden. 293 00:19:38,710 --> 00:19:41,980 As r gets closer to 1, the resonance 294 00:19:41,980 --> 00:19:43,230 tends to get sharper. 295 00:19:49,020 --> 00:19:55,100 This is now a look at the z-transform, and we see very 296 00:19:55,100 --> 00:19:59,000 strong parallels to the Laplace transform. 297 00:19:59,000 --> 00:20:02,920 In fact, throughout the course, I've tried to 298 00:20:02,920 --> 00:20:03,700 emphasize-- 299 00:20:03,700 --> 00:20:05,590 and it just naturally happens-- 300 00:20:05,590 --> 00:20:10,370 that there are very strong relationships and parallels 301 00:20:10,370 --> 00:20:14,890 between continuous time and discrete time. 302 00:20:14,890 --> 00:20:19,940 In fact, at one point we specifically mapped from 303 00:20:19,940 --> 00:20:22,960 continuous time to discrete time when we talked about 304 00:20:22,960 --> 00:20:27,460 discrete time processing of continuous time signals. 305 00:20:27,460 --> 00:20:32,760 What I'd like to do now is turn our attention to another 306 00:20:32,760 --> 00:20:38,000 very important reason for mapping from continuous time 307 00:20:38,000 --> 00:20:41,600 to discrete time, and in the process of doing this, what 308 00:20:41,600 --> 00:20:46,670 we'll need to do is exploit fairly heavily the insight, 309 00:20:46,670 --> 00:20:51,210 intuition, and procedures that we've developed for the 310 00:20:51,210 --> 00:20:55,150 Laplace transform and the z-transform. 311 00:20:55,150 --> 00:20:59,540 Specifically, what I would like to begin is a discussion 312 00:20:59,540 --> 00:21:06,530 relating to mapping continuous time filters to discrete time 313 00:21:06,530 --> 00:21:12,260 filters, or continuous time system functions to discrete 314 00:21:12,260 --> 00:21:14,410 time system functions. 315 00:21:14,410 --> 00:21:17,170 Now, why would we want to do that? 316 00:21:17,170 --> 00:21:21,130 Well, there are at least several reasons for wanting to 317 00:21:21,130 --> 00:21:25,140 map continuous time filters to discrete time filters. 318 00:21:25,140 --> 00:21:29,940 One, of course, is the fact that in some situations what 319 00:21:29,940 --> 00:21:34,140 we're interested in doing is processing continuous time 320 00:21:34,140 --> 00:21:38,220 signals with discrete time systems-- 321 00:21:38,220 --> 00:21:45,050 or, said another way, simulate continuous time systems with 322 00:21:45,050 --> 00:21:46,690 discrete time systems. 323 00:21:46,690 --> 00:21:49,530 So it would be natural in a setting like that to think of 324 00:21:49,530 --> 00:21:53,500 mapping the desired continuous time filter to a 325 00:21:53,500 --> 00:21:55,840 discrete time filter. 326 00:21:55,840 --> 00:21:58,480 So that's one very important context. 327 00:21:58,480 --> 00:22:02,880 There's another very important context in which this is done, 328 00:22:02,880 --> 00:22:11,650 and that is in the context or for the purpose of exploiting 329 00:22:11,650 --> 00:22:16,180 established design procedures for continuous time filters. 330 00:22:16,180 --> 00:22:18,440 The point is the following. 331 00:22:18,440 --> 00:22:22,240 We may or may not be processing a sample continuous 332 00:22:22,240 --> 00:22:25,260 time signal with our discrete time filter-- it may just be 333 00:22:25,260 --> 00:22:28,630 discrete time signals that we're working with. 334 00:22:28,630 --> 00:22:34,460 But in that situation, still, we need to design the 335 00:22:34,460 --> 00:22:38,280 appropriate discrete time filter. 336 00:22:38,280 --> 00:22:46,300 Historically, there is a very rich history associated with 337 00:22:46,300 --> 00:22:50,290 design of continuous time filters. 338 00:22:50,290 --> 00:22:54,860 In many cases, it's possible and very worthwhile and 339 00:22:54,860 --> 00:23:00,790 efficient to take those designs and map them to 340 00:23:00,790 --> 00:23:04,880 discrete time designs to use them as discrete time filters. 341 00:23:04,880 --> 00:23:08,650 So, another very important reason for talking about the 342 00:23:08,650 --> 00:23:13,120 kinds of mappings that we will be going into is to simply 343 00:23:13,120 --> 00:23:16,800 take advantage of what has been done historically in the 344 00:23:16,800 --> 00:23:18,050 continuous time case. 345 00:23:21,960 --> 00:23:26,160 Now, if we want to map continuous time filters to 346 00:23:26,160 --> 00:23:31,330 discrete time filters, then in continuous time, we're talking 347 00:23:31,330 --> 00:23:34,820 about a system function and an associated 348 00:23:34,820 --> 00:23:37,840 differential equation. 349 00:23:37,840 --> 00:23:43,230 In discrete time, there is the corresponding system function 350 00:23:43,230 --> 00:23:46,105 and the corresponding difference equation. 351 00:23:49,170 --> 00:23:53,960 Basically what we want to do is generate from a continuous 352 00:23:53,960 --> 00:23:59,640 time system in some way a discrete time system that 353 00:23:59,640 --> 00:24:05,650 meets an associated set of desired specifications. 354 00:24:05,650 --> 00:24:09,530 Now, there are certain constraints that it's 355 00:24:09,530 --> 00:24:13,410 reasonable and important to impose on whatever kinds of 356 00:24:13,410 --> 00:24:15,940 mappings we use. 357 00:24:15,940 --> 00:24:19,230 Obviously, we want a mapping that will take our continuous 358 00:24:19,230 --> 00:24:21,900 time system function and map it to a 359 00:24:21,900 --> 00:24:24,780 discrete time system function. 360 00:24:24,780 --> 00:24:28,210 Correspondingly in the time domain, there is a continuous 361 00:24:28,210 --> 00:24:32,520 time impulse response that maps to the associated 362 00:24:32,520 --> 00:24:35,610 discrete time impulse response. 363 00:24:35,610 --> 00:24:37,710 These are more or less natural. 364 00:24:37,710 --> 00:24:41,310 The two that are important and sometimes easy to lose sight 365 00:24:41,310 --> 00:24:45,450 of are the two that I indicate here. 366 00:24:45,450 --> 00:24:50,970 In particular, if we are mapping a continuous time 367 00:24:50,970 --> 00:24:55,130 filter with, let's say, a desired or desirable frequency 368 00:24:55,130 --> 00:24:59,680 response to a discrete time filter and we would like to 369 00:24:59,680 --> 00:25:03,360 preserve the good qualities of that frequency response as we 370 00:25:03,360 --> 00:25:09,460 look at the discrete time frequency response, then it's 371 00:25:09,460 --> 00:25:14,680 important what happens in the s-plane for the continuous 372 00:25:14,680 --> 00:25:19,860 time filter along the j omega axis relate in a nice way to 373 00:25:19,860 --> 00:25:24,310 what happens in the z-plane around the unit circle, 374 00:25:24,310 --> 00:25:29,170 because it's this over here that represents the frequency 375 00:25:29,170 --> 00:25:33,440 response in continuous time and this contour over here 376 00:25:33,440 --> 00:25:36,570 that represents the frequency response in discrete time. 377 00:25:36,570 --> 00:25:38,090 So that's an important property. 378 00:25:38,090 --> 00:25:41,100 We want to kind of the j omega axis to 379 00:25:41,100 --> 00:25:44,370 map to the unit circle. 380 00:25:44,370 --> 00:25:49,160 Another more or less natural condition to impose is a 381 00:25:49,160 --> 00:25:53,960 condition that if we are assured in some way that our 382 00:25:53,960 --> 00:25:59,400 continuous time filter is stable, then we would like to 383 00:25:59,400 --> 00:26:02,510 concentrate on design procedures that more or less 384 00:26:02,510 --> 00:26:07,550 preserve that and will give us stable digital filters. 385 00:26:07,550 --> 00:26:11,150 So these are kind of reasonable conditions to 386 00:26:11,150 --> 00:26:13,150 impose on the procedure. 387 00:26:13,150 --> 00:26:16,960 What I'd like to do in the remainder of this lecture is 388 00:26:16,960 --> 00:26:22,040 look at two common procedures for mapping continuous time 389 00:26:22,040 --> 00:26:25,370 filters to discrete time filters. 390 00:26:25,370 --> 00:26:28,020 The first one that I want to talk about is one that, in 391 00:26:28,020 --> 00:26:35,210 fact, is very frequently used, and also one that, as we'll 392 00:26:35,210 --> 00:26:40,990 see for a variety of reasons, is highly undesirable. 393 00:26:40,990 --> 00:26:44,930 The second is one that is also frequently used, and as we'll 394 00:26:44,930 --> 00:26:50,280 see is, in certain situations, very desirable. 395 00:26:50,280 --> 00:26:53,830 The first one that I want to talk about is the more or less 396 00:26:53,830 --> 00:27:00,260 intuitive simple procedure of mapping a differential 397 00:27:00,260 --> 00:27:04,790 equation to a difference equation by simply replacing 398 00:27:04,790 --> 00:27:07,265 derivatives by differences. 399 00:27:09,890 --> 00:27:13,870 The idea is that a derivative is more or less a difference, 400 00:27:13,870 --> 00:27:16,390 and there's some dummy parameter capital T that I've 401 00:27:16,390 --> 00:27:19,720 thrown in here, which I won't focus too much on. 402 00:27:19,720 --> 00:27:24,870 But in any case, this seems to have some plausibility. 403 00:27:24,870 --> 00:27:27,730 If we take the differential equation and do this with all 404 00:27:27,730 --> 00:27:32,410 the derivatives, both in terms of y of t and x of t, what 405 00:27:32,410 --> 00:27:36,560 we'll end up with is a difference equation. 406 00:27:36,560 --> 00:27:42,540 Now what we can use are the properties of the Laplace 407 00:27:42,540 --> 00:27:45,920 transform and the z-transform to see what this means in 408 00:27:45,920 --> 00:27:47,440 terms of a mapping-- 409 00:27:47,440 --> 00:27:50,940 in particular, using the differentiation property for 410 00:27:50,940 --> 00:27:52,390 Laplace transforms. 411 00:27:52,390 --> 00:27:56,970 In the Laplace transform domain, we would have this. 412 00:27:56,970 --> 00:28:02,400 Using the properties for the z-transform, the z-transform 413 00:28:02,400 --> 00:28:04,740 of this expression would be this. 414 00:28:04,740 --> 00:28:09,950 So, in effect, what it says is that every place in the system 415 00:28:09,950 --> 00:28:12,370 function or in the differential equation that we 416 00:28:12,370 --> 00:28:16,460 would be multiplying by s when Laplace transformed. 417 00:28:16,460 --> 00:28:18,000 In the difference equation, we would be 418 00:28:18,000 --> 00:28:21,070 multiplying by this factor. 419 00:28:21,070 --> 00:28:26,840 In fact, what this means is that the mapping from 420 00:28:26,840 --> 00:28:32,000 continuous time to discrete time corresponds to taking the 421 00:28:32,000 --> 00:28:38,200 system function and replacing s wherever we see it by 1 422 00:28:38,200 --> 00:28:43,110 minus z to the minus 1 over capital T. So if we have a 423 00:28:43,110 --> 00:28:47,400 system function in continuous time and we map it to a 424 00:28:47,400 --> 00:28:51,220 discrete time system function this way by replacing 425 00:28:51,220 --> 00:28:55,700 derivatives by differences, then that corresponds to 426 00:28:55,700 --> 00:29:01,390 replacing s by 1 minus z to the minus 1 over capital T. 427 00:29:01,390 --> 00:29:06,190 Now we'll see shortly what this mapping actually means 428 00:29:06,190 --> 00:29:09,980 more specifically in relating the s-plane to the z-plane. 429 00:29:09,980 --> 00:29:13,370 Let me just quickly, because I want to refer to this, also 430 00:29:13,370 --> 00:29:16,670 point to another procedure very much like backward 431 00:29:16,670 --> 00:29:20,910 differences which corresponds to replacing derivatives not 432 00:29:20,910 --> 00:29:23,770 by the backward differences that I just showed, but by 433 00:29:23,770 --> 00:29:25,640 forward differences. 434 00:29:25,640 --> 00:29:29,600 In that case, then, the mapping corresponds to 435 00:29:29,600 --> 00:29:36,030 replacing s by z to the minus 1 over capital T. It looks 436 00:29:36,030 --> 00:29:39,270 very similar to the previous case. 437 00:29:39,270 --> 00:29:43,330 So there the relationship between these system functions 438 00:29:43,330 --> 00:29:44,860 is what I indicate here. 439 00:29:47,800 --> 00:29:53,590 Let's just take a look at what those mappings correspond to 440 00:29:53,590 --> 00:29:56,730 when we look at this specifically in the s-plane 441 00:29:56,730 --> 00:29:59,250 and in the z-plane. 442 00:29:59,250 --> 00:30:04,920 What I show here is the s-plane, and of course it's 443 00:30:04,920 --> 00:30:07,350 things on the left half of the s-plane, poles on the left 444 00:30:07,350 --> 00:30:10,350 half of the s-plane that would guarantee stability. 445 00:30:10,350 --> 00:30:14,340 It's the j omega axis the tells us about the frequency 446 00:30:14,340 --> 00:30:21,630 response, and in the z-plane it's the unit circle that 447 00:30:21,630 --> 00:30:24,490 tells us about the frequency response. 448 00:30:24,490 --> 00:30:27,560 Things inside the unit circle, or poles inside the unit 449 00:30:27,560 --> 00:30:30,170 circle, that guarantee stability. 450 00:30:30,170 --> 00:30:35,310 Now the mapping from s-plane to the z-plane corresponding 451 00:30:35,310 --> 00:30:40,660 to replacing derivatives by backward differences in fact 452 00:30:40,660 --> 00:30:48,140 can be shown to correspond to mapping the j omega axis not 453 00:30:48,140 --> 00:30:51,780 to the unit circle, but to the little circle that I show 454 00:30:51,780 --> 00:30:55,230 here, which is inside the unit circle. 455 00:30:55,230 --> 00:30:58,550 The left half of the s-plane maps to the 456 00:30:58,550 --> 00:31:00,430 inside of that circle. 457 00:31:00,430 --> 00:31:01,210 What does that mean? 458 00:31:01,210 --> 00:31:06,500 That means that if we have a really good frequency response 459 00:31:06,500 --> 00:31:11,150 characteristic along this contour in the s-plane, we'll 460 00:31:11,150 --> 00:31:13,270 see that same frequency response 461 00:31:13,270 --> 00:31:14,910 along this little circle. 462 00:31:14,910 --> 00:31:17,030 That's not the one that we want, though-- 463 00:31:17,030 --> 00:31:20,420 we would like to see that same frequency response around the 464 00:31:20,420 --> 00:31:22,080 unit circle. 465 00:31:22,080 --> 00:31:26,960 To emphasize this point even more-- suppose, for example, 466 00:31:26,960 --> 00:31:33,230 that we had a pair of poles in our continuous time system 467 00:31:33,230 --> 00:31:36,000 function that looked like this. 468 00:31:36,000 --> 00:31:42,610 Then, where they're likely to end up in the z-plane is 469 00:31:42,610 --> 00:31:45,050 inside the unit circle, of course. 470 00:31:45,050 --> 00:31:50,080 But if the poles here are close to the j omega axis, 471 00:31:50,080 --> 00:31:53,930 that means that these poles will be close to this circle, 472 00:31:53,930 --> 00:31:58,320 but in fact might be very far away from the unit circle. 473 00:31:58,320 --> 00:32:01,530 What would happen, then, is that if we saw in the 474 00:32:01,530 --> 00:32:06,220 continuous time filter a very sharp resonance, the discrete 475 00:32:06,220 --> 00:32:08,900 time filter in fact might very well have that resonance 476 00:32:08,900 --> 00:32:12,660 broadened considerably because the poles are so far away from 477 00:32:12,660 --> 00:32:14,500 the unit circle. 478 00:32:14,500 --> 00:32:19,530 Now, one plus with this method, and it's about the 479 00:32:19,530 --> 00:32:25,570 only one, is the fact that the left half of the s-plane maps 480 00:32:25,570 --> 00:32:30,110 inside the unit circle-- in fact, inside a circle inside 481 00:32:30,110 --> 00:32:32,250 the unit circle, and so stability is always 482 00:32:32,250 --> 00:32:33,870 guaranteed. 483 00:32:33,870 --> 00:32:36,170 Let me just quickly mention, and you'll have a chance to 484 00:32:36,170 --> 00:32:38,910 look at this a little more carefully in the video course 485 00:32:38,910 --> 00:32:42,960 manual, that for forward differences instead of 486 00:32:42,960 --> 00:32:51,120 backward differences, this contour in the s-plane maps to 487 00:32:51,120 --> 00:32:56,380 a line in the z-plane, which is a line tangent to the unit 488 00:32:56,380 --> 00:33:01,250 circle, and in fact is the line that I showed there. 489 00:33:01,250 --> 00:33:05,890 So not only are forward differences equally bad in 490 00:33:05,890 --> 00:33:09,270 terms of the issue of whether they map from the j omega axis 491 00:33:09,270 --> 00:33:13,440 to the unit circle, but they have a further difficulty 492 00:33:13,440 --> 00:33:19,360 associated with them, which is the difficulty they may not 493 00:33:19,360 --> 00:33:21,505 and generally don't guarantee stability. 494 00:33:25,070 --> 00:33:29,840 Now, that's one method, and one, as I indicated, that's 495 00:33:29,840 --> 00:33:35,600 often used partly because it seems 496 00:33:35,600 --> 00:33:37,510 so intuitively plausible. 497 00:33:37,510 --> 00:33:40,960 What you can see is that by understanding carefully the 498 00:33:40,960 --> 00:33:44,500 issues and the techniques of Laplace and z-transforms, you 499 00:33:44,500 --> 00:33:48,280 can begin to see what some of the difficulties with those 500 00:33:48,280 --> 00:33:50,450 methods are. 501 00:33:50,450 --> 00:33:54,080 The next method that I'd like to talk about is a method 502 00:33:54,080 --> 00:33:57,000 that, in fact, is very commonly used. 503 00:33:57,000 --> 00:34:01,360 It's a very important, useful method, which kind of can be 504 00:34:01,360 --> 00:34:06,330 motivated by thinking along the lines of mapping the 505 00:34:06,330 --> 00:34:11,630 continuous time filter to a discrete time filter in such a 506 00:34:11,630 --> 00:34:15,580 way that the shape of the impulse response is 507 00:34:15,580 --> 00:34:16,620 preserved-- 508 00:34:16,620 --> 00:34:20,989 and, in fact, more specifically so that the 509 00:34:20,989 --> 00:34:26,590 discrete time impulse response corresponds to samples of the 510 00:34:26,590 --> 00:34:28,960 continuous time impulse response. 511 00:34:28,960 --> 00:34:32,800 And this is a method that's referred to as impulse 512 00:34:32,800 --> 00:34:35,260 invariance. 513 00:34:35,260 --> 00:34:40,790 So what impulse invariance corresponds to is designing 514 00:34:40,790 --> 00:34:46,100 the filter in such a way that the discrete time filter 515 00:34:46,100 --> 00:34:51,550 impulse response is simply a sample version of the 516 00:34:51,550 --> 00:34:57,200 continuous time filter impulse response with a sampling 517 00:34:57,200 --> 00:35:01,450 period which I denote here as capital T. That will turn into 518 00:35:01,450 --> 00:35:06,350 a slightly confusing parameter shortly, and perhaps carried 519 00:35:06,350 --> 00:35:08,270 over into the next lecture. 520 00:35:08,270 --> 00:35:11,900 Hopefully, we'll get that straighted out, though, within 521 00:35:11,900 --> 00:35:14,030 those two lectures. 522 00:35:14,030 --> 00:35:18,680 Remembering the issues of sampling, the discrete time 523 00:35:18,680 --> 00:35:22,970 frequency response, then since the frequency responses the 524 00:35:22,970 --> 00:35:27,180 Fourier transform of the impulse response is related to 525 00:35:27,180 --> 00:35:31,950 the continuous time impulse response as I indicate here, 526 00:35:31,950 --> 00:35:41,130 what this says is that it is the superposition of 527 00:35:41,130 --> 00:35:46,600 replications of the continuous time frequency response, 528 00:35:46,600 --> 00:35:51,440 linearly scaled in frequency and shifted and 529 00:35:51,440 --> 00:35:53,120 added to each other. 530 00:35:53,120 --> 00:35:56,830 It's the same old sort of shifting, adding, or aliasing 531 00:35:56,830 --> 00:35:58,430 issue-- the same sampling issues that 532 00:35:58,430 --> 00:36:01,740 we've addressed before. 533 00:36:01,740 --> 00:36:04,690 This equation will help us understand what the frequency 534 00:36:04,690 --> 00:36:06,290 response looks like. 535 00:36:06,290 --> 00:36:11,560 But in terms of an analytical procedure for mapping the 536 00:36:11,560 --> 00:36:14,540 continuous time system function to a discrete time 537 00:36:14,540 --> 00:36:20,020 system function, we can see that and develop it in the 538 00:36:20,020 --> 00:36:22,200 following way. 539 00:36:22,200 --> 00:36:27,000 Let's consider the continuous time system function expanded 540 00:36:27,000 --> 00:36:30,250 in a partial fraction expansion. 541 00:36:30,250 --> 00:36:33,320 And just for convenience, I'm picking first order poles-- 542 00:36:33,320 --> 00:36:44,540 this can be generalized multiple order poles, and we 543 00:36:44,540 --> 00:36:45,730 won't do that here. 544 00:36:45,730 --> 00:36:48,200 The same basic strategy applies. 545 00:36:48,200 --> 00:36:54,120 If we expand this in a partial fraction expansion, and we 546 00:36:54,120 --> 00:36:57,210 look at the impulse response associated with this-- we know 547 00:36:57,210 --> 00:36:59,940 how to take the inverse of Laplace transform of this, 548 00:36:59,940 --> 00:37:03,280 where I'm just naturally assuming causality throughout 549 00:37:03,280 --> 00:37:05,490 the discussion-- 550 00:37:05,490 --> 00:37:09,070 the continuous time impulse response, then, is the sum of 551 00:37:09,070 --> 00:37:14,610 exponentials with these amplitudes and at these 552 00:37:14,610 --> 00:37:17,910 complex exponential frequencies. 553 00:37:17,910 --> 00:37:21,570 Now, impulse invariance corresponds to sampling this, 554 00:37:21,570 --> 00:37:26,090 and so the discrete time impulse response is simply a 555 00:37:26,090 --> 00:37:29,700 sampled version of this. 556 00:37:29,700 --> 00:37:33,720 The A sub k, of course, carries down. 557 00:37:33,720 --> 00:37:35,440 We have the exponential-- 558 00:37:35,440 --> 00:37:39,340 we're sampling at t equals n capital T, and so we've 559 00:37:39,340 --> 00:37:43,550 replaced that here, and then the unit step to truncate 560 00:37:43,550 --> 00:37:46,550 things for negative time. 561 00:37:46,550 --> 00:37:50,360 Let's manipulate this further, and eventually what we want to 562 00:37:50,360 --> 00:37:52,530 get is a relationship-- 563 00:37:52,530 --> 00:37:53,230 a mapping-- 564 00:37:53,230 --> 00:37:58,360 from the continuous time to the discrete time filter. 565 00:37:58,360 --> 00:38:04,380 We have this step, and we can rewrite that now as I show 566 00:38:04,380 --> 00:38:08,860 here, just simply taking this n outside, and we have e to 567 00:38:08,860 --> 00:38:13,770 the s sub k capital T. Now this is of the form the sum of 568 00:38:13,770 --> 00:38:19,100 terms like A sub k times is beta to the n. 569 00:38:19,100 --> 00:38:23,750 We can compute the z-transform of this, and the z-transform 570 00:38:23,750 --> 00:38:26,840 that we get I show here-- 571 00:38:26,840 --> 00:38:30,680 it's simply A sub k over 1 minus e to the s sub k capital 572 00:38:30,680 --> 00:38:36,700 T z to the minus 1, simply carrying this term or this 573 00:38:36,700 --> 00:38:39,720 parameter down. 574 00:38:39,720 --> 00:38:42,750 So we started with a continuous time system 575 00:38:42,750 --> 00:38:46,800 function, which was a sum of terms like A sub k over s 576 00:38:46,800 --> 00:38:49,730 minus s sub k-- the poles were at s sub k. 577 00:38:49,730 --> 00:38:54,200 We now have the discrete time filter in this form. 578 00:38:54,200 --> 00:38:57,110 Consequently, then, this procedure of impulse 579 00:38:57,110 --> 00:39:03,400 invariance corresponds to mapping the continuous time 580 00:39:03,400 --> 00:39:07,840 filter to a discrete time filter by mapping the poles in 581 00:39:07,840 --> 00:39:10,080 the continuous time filter. 582 00:39:10,080 --> 00:39:15,870 According to this mapping, the continuous time filter pole at 583 00:39:15,870 --> 00:39:20,530 s sub k gets mapped to a pole e to the s sub k capital T, 584 00:39:20,530 --> 00:39:27,280 and the coefficients A sub k are preserved. 585 00:39:27,280 --> 00:39:31,210 That, then, algebraically, is what the procedure of impulse 586 00:39:31,210 --> 00:39:34,960 invariance corresponds to. 587 00:39:34,960 --> 00:39:39,630 Let's look at how we interpret some of this in 588 00:39:39,630 --> 00:39:42,180 the frequency domain. 589 00:39:42,180 --> 00:39:49,020 In particular, we have the expression that tells us how 590 00:39:49,020 --> 00:39:54,700 the discrete time frequency response is related to the 591 00:39:54,700 --> 00:39:57,470 continuous time frequency response. 592 00:39:57,470 --> 00:40:02,570 This is the expression that we had previously when we had 593 00:40:02,570 --> 00:40:05,490 talked about issues of sampling. 594 00:40:05,490 --> 00:40:07,910 So that means that we would form the discrete time 595 00:40:07,910 --> 00:40:12,460 frequency response by taking the continuous time 1, scaling 596 00:40:12,460 --> 00:40:17,100 it in frequency according to this parameter capital T, and 597 00:40:17,100 --> 00:40:21,190 then adding replications of that together. 598 00:40:21,190 --> 00:40:26,890 So if this is the continuous time frequency response, just 599 00:40:26,890 --> 00:40:30,460 simply an ideal low-pass filter with a cutoff frequency 600 00:40:30,460 --> 00:40:36,400 of omega sub c, then the frequency scaling operation 601 00:40:36,400 --> 00:40:42,130 would keep the same basic shape but linearly scale the 602 00:40:42,130 --> 00:40:47,600 frequency axis so that we now have omega sub c times T. Then 603 00:40:47,600 --> 00:40:52,060 the discrete time frequency response would be a 604 00:40:52,060 --> 00:40:57,400 superposition of these added together at multiples of 2 pi 605 00:40:57,400 --> 00:41:00,190 in discrete time frequency. 606 00:41:00,190 --> 00:41:03,350 So that's what we have here-- 607 00:41:03,350 --> 00:41:06,245 so this is the discrete time frequency response. 608 00:41:08,920 --> 00:41:11,810 This looks very nice-- it looks like impulse invariance 609 00:41:11,810 --> 00:41:15,610 will take the continuous time frequency response, just 610 00:41:15,610 --> 00:41:18,470 simply linearly scale the frequency axis, and 611 00:41:18,470 --> 00:41:21,210 incidentally periodically repeat it. 612 00:41:21,210 --> 00:41:24,360 We know that for an ideal low-pass filter, 613 00:41:24,360 --> 00:41:25,670 that looks just fine. 614 00:41:25,670 --> 00:41:28,940 In fact, for a band-limited frequency response, 615 00:41:28,940 --> 00:41:31,300 that looks just fine. 616 00:41:31,300 --> 00:41:35,355 But we know also that any time that we're sampling-- 617 00:41:35,355 --> 00:41:38,210 and here we're sampling the impulse response-- 618 00:41:38,210 --> 00:41:42,210 we have an effect in the frequency domain or the 619 00:41:42,210 --> 00:41:44,440 potential for an affect an effect that 620 00:41:44,440 --> 00:41:46,710 we refer to as aliasing. 621 00:41:46,710 --> 00:41:52,810 So in fact, if instead of the ideal low-pass filter we had 622 00:41:52,810 --> 00:41:56,030 taken a filter that was an approximation to a low-pass 623 00:41:56,030 --> 00:42:01,510 filter, then the corresponding frequency scale version would 624 00:42:01,510 --> 00:42:03,590 look as I've shown here. 625 00:42:03,590 --> 00:42:08,390 And now as we add these together, then what we will 626 00:42:08,390 --> 00:42:13,070 have is some potential for distortion corresponding to 627 00:42:13,070 --> 00:42:17,530 the fact that these replications overlap, and what 628 00:42:17,530 --> 00:42:19,660 that will lead to is aliasing. 629 00:42:22,330 --> 00:42:26,720 So some things that we can say about impulse invariance is 630 00:42:26,720 --> 00:42:29,270 that we have an algebraic procedure-- 631 00:42:29,270 --> 00:42:32,370 and I'll illustrate with another example shortly-- 632 00:42:32,370 --> 00:42:35,200 for taking a continuous time system function and mapping it 633 00:42:35,200 --> 00:42:37,680 to a discrete time system function. 634 00:42:37,680 --> 00:42:44,290 It has a very nice property in terms of mapping, the mapping 635 00:42:44,290 --> 00:42:48,910 from the frequency axis in continuous time due to the 636 00:42:48,910 --> 00:42:50,190 unit circle-- 637 00:42:50,190 --> 00:42:53,750 namely, to a first approximation. 638 00:42:53,750 --> 00:42:57,900 As long as there's no aliasing, the mapping is just 639 00:42:57,900 --> 00:43:01,200 simply a linear scaling of the frequency axis, although there 640 00:43:01,200 --> 00:43:02,910 may be some aliasing. 641 00:43:02,910 --> 00:43:06,640 That means, of course, that this method can only be used 642 00:43:06,640 --> 00:43:09,810 if the frequency response that's being mapped, or if the 643 00:43:09,810 --> 00:43:13,300 system that's being mapped, has a frequency response 644 00:43:13,300 --> 00:43:15,300 that's approximately low-pass-- 645 00:43:15,300 --> 00:43:16,940 it has to be approximately band-limited. 646 00:43:20,520 --> 00:43:23,970 Then what we have is some of potential distortion, which 647 00:43:23,970 --> 00:43:26,540 comes about because of aliasing. 648 00:43:26,540 --> 00:43:30,660 Also because of the mapping, the fact that poles at s sub k 649 00:43:30,660 --> 00:43:36,290 get mapped to poles at e to the s sub k capital T, if the 650 00:43:36,290 --> 00:43:39,960 analog or continuous time filter is stable-- 651 00:43:39,960 --> 00:43:43,050 meaning that the real part of s sub k is negative-- 652 00:43:43,050 --> 00:43:47,240 then the discrete time filter is guaranteed to be stable. 653 00:43:47,240 --> 00:43:51,220 In other words, the magnitude of z sub k will be guaranteed 654 00:43:51,220 --> 00:43:53,060 to be less than 1. 655 00:43:53,060 --> 00:43:56,300 I'm assuming, of course, in that discussion that we are 656 00:43:56,300 --> 00:43:58,730 always imposing causality on the systems. 657 00:44:01,700 --> 00:44:06,030 To just look at the algebraic mapping a little more 658 00:44:06,030 --> 00:44:09,650 carefully, let's take a simple example. 659 00:44:09,650 --> 00:44:15,130 Here is an example of a system, a continuous time 660 00:44:15,130 --> 00:44:21,690 system, where I simply have a resident pole pair with an 661 00:44:21,690 --> 00:44:26,000 imaginary part along the imaginary axis of omega sub r 662 00:44:26,000 --> 00:44:28,660 and a real part of minus alpha. 663 00:44:28,660 --> 00:44:34,650 And so the associated system function then is just the 664 00:44:34,650 --> 00:44:39,110 expression which incorporates the two poles, and I've put in 665 00:44:39,110 --> 00:44:42,850 a scale factor of 2 omega sub r. 666 00:44:42,850 --> 00:44:47,160 And now to design the discrete time filter using impulse 667 00:44:47,160 --> 00:44:49,990 invariance, you would carry out a partial fraction 668 00:44:49,990 --> 00:44:53,690 expansion of this, and that partial fraction expansion is 669 00:44:53,690 --> 00:44:55,990 shown below. 670 00:44:55,990 --> 00:44:59,550 We have a pole at minus alpha minus j omega r and at minus 671 00:44:59,550 --> 00:45:02,130 alpha plus j omega r. 672 00:45:02,130 --> 00:45:07,430 And to determine the discrete time filter based on impulse 673 00:45:07,430 --> 00:45:10,640 invariance, we would map the poles and preserve the 674 00:45:10,640 --> 00:45:14,480 coefficients a sub k, referred to as the residues. 675 00:45:14,480 --> 00:45:18,740 And so the discrete time filter that we would end up 676 00:45:18,740 --> 00:45:25,560 with as a system function, which I indicate here-- 677 00:45:25,560 --> 00:45:30,870 and we have, as I said, preserved the residue, and the 678 00:45:30,870 --> 00:45:34,820 pole is now at a to the minus alpha T, e to the minus j 679 00:45:34,820 --> 00:45:40,020 omega sub r T. That's one term, and the other term in 680 00:45:40,020 --> 00:45:44,950 the sum has a pole at the complex conjugate location. 681 00:45:44,950 --> 00:45:50,090 If we were to add these two factors together, then what we 682 00:45:50,090 --> 00:45:57,420 would get is both poles and a 0 at the origin. 683 00:45:57,420 --> 00:46:04,270 In fact, then, the pole is defined by its angle, and this 684 00:46:04,270 --> 00:46:15,050 angle is e to the j omega sub r capital T, and by its 685 00:46:15,050 --> 00:46:24,030 radius, and this radius is e to the minus alpha capital T. 686 00:46:24,030 --> 00:46:28,310 Now we can look at the frequency response associated 687 00:46:28,310 --> 00:46:31,600 with that, and let's just do that. 688 00:46:31,600 --> 00:46:40,730 For the original continuous time frequency response, what 689 00:46:40,730 --> 00:46:45,900 we have is simply a resonant character, as I've shown here. 690 00:46:45,900 --> 00:46:50,730 And if we map this using impulse invariance, which we 691 00:46:50,730 --> 00:46:56,920 just did, the resulting frequency response that we get 692 00:46:56,920 --> 00:46:59,670 is the frequency response which I indicate. 693 00:46:59,670 --> 00:47:03,920 We see that that's basically identical to the continuous 694 00:47:03,920 --> 00:47:09,430 time frequency response, except for a linear scaling in 695 00:47:09,430 --> 00:47:13,510 the frequency axis, if you just compare the dimensions 696 00:47:13,510 --> 00:47:21,210 along which the frequency axis is shown except for one minor 697 00:47:21,210 --> 00:47:25,500 issue, which is particularly highlighted when we look at 698 00:47:25,500 --> 00:47:30,180 the frequency response at higher frequencies. 699 00:47:30,180 --> 00:47:35,140 What's the reason why those two curves don't quite follow 700 00:47:35,140 --> 00:47:38,980 each other at higher frequencies? 701 00:47:38,980 --> 00:47:41,790 Well, the reason is aliasing. 702 00:47:41,790 --> 00:47:44,980 In other words, what's happened is that in the 703 00:47:44,980 --> 00:47:50,320 process so applying impulse invariance, the frequency 704 00:47:50,320 --> 00:47:53,770 response of the original continuous time filter is 705 00:47:53,770 --> 00:47:57,540 approximately preserved, except for some distortion, 706 00:47:57,540 --> 00:48:01,420 that distortion corresponding to aliasing. 707 00:48:01,420 --> 00:48:06,000 Well, just for comparison, let's look at what would 708 00:48:06,000 --> 00:48:08,490 happen if we took the same example-- 709 00:48:08,490 --> 00:48:10,710 and we're not going to work it through here carefully. 710 00:48:10,710 --> 00:48:15,100 We're not work it through it all, not even not carefully. 711 00:48:15,100 --> 00:48:19,560 If we took the same example and mapped it to a discrete 712 00:48:19,560 --> 00:48:22,720 time filter by replacing derivatives by backward 713 00:48:22,720 --> 00:48:28,200 differences, what happens in that case is that we get a 714 00:48:28,200 --> 00:48:32,030 frequency response that I indicate here. 715 00:48:32,030 --> 00:48:36,720 Notice that the resonance in the original continuous time 716 00:48:36,720 --> 00:48:41,020 filter is totally lost. 717 00:48:41,020 --> 00:48:44,950 In fact, basically the character of the continuous 718 00:48:44,950 --> 00:48:47,510 time frequency response is lost. 719 00:48:47,510 --> 00:48:48,410 What's the reason? 720 00:48:48,410 --> 00:48:49,920 Well, the reason goes back to the 721 00:48:49,920 --> 00:48:51,810 picture that I drew before. 722 00:48:51,810 --> 00:48:58,920 The continuous time filter had a pair of resident poles close 723 00:48:58,920 --> 00:49:00,830 to the j omega axis. 724 00:49:00,830 --> 00:49:03,850 When those get mapped using backward differences, they end 725 00:49:03,850 --> 00:49:07,850 up close to this little circle that's inside the unit circle, 726 00:49:07,850 --> 00:49:10,700 but in fact for this example, are far away 727 00:49:10,700 --> 00:49:11,950 from the unit circle. 728 00:49:16,350 --> 00:49:20,320 So far we have one useful technique for mapping 729 00:49:20,320 --> 00:49:23,600 continuous time filters to discrete time filters. 730 00:49:23,600 --> 00:49:27,160 In part to highlight some of the issues, I focused 731 00:49:27,160 --> 00:49:31,780 attention also on some not so useful methods-- 732 00:49:31,780 --> 00:49:35,410 namely, mapping derivatives to forward or backward 733 00:49:35,410 --> 00:49:37,630 differences. 734 00:49:37,630 --> 00:49:43,360 Next time what I would like to do is look at impulse 735 00:49:43,360 --> 00:49:46,470 invariance for another example-- 736 00:49:46,470 --> 00:49:51,360 namely, the design of a Butterworth filter, and I'll 737 00:49:51,360 --> 00:49:54,970 talk more specifically about what Butterworth filters are 738 00:49:54,970 --> 00:49:57,550 at the beginning of that lecture. 739 00:49:57,550 --> 00:50:01,170 Then, in addition, what we'll introduce is another very 740 00:50:01,170 --> 00:50:05,600 useful technique, which has some difficulties which 741 00:50:05,600 --> 00:50:11,360 impulse invariance doesn't have, but avoids the principal 742 00:50:11,360 --> 00:50:13,480 difficulty that impulse invariance does have-- 743 00:50:13,480 --> 00:50:14,950 namely, aliasing. 744 00:50:14,950 --> 00:50:19,880 That method is referred to the bilinear transformation, which 745 00:50:19,880 --> 00:50:24,090 we will define and utilize next time. 746 00:50:24,090 --> 00:50:25,340 Thank you.