1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high-quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,280 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,280 --> 00:00:18,450 at ocw.mit.edu. 8 00:00:22,790 --> 00:00:27,960 DENNIS FREEMAN: So today's topic is to talk about Z-transforms. 9 00:00:27,960 --> 00:00:29,910 But before I start, I want to mention 10 00:00:29,910 --> 00:00:33,990 that we've already covered a great deal of material. 11 00:00:33,990 --> 00:00:39,030 Here, I've made a map just of things we've done in DT, 12 00:00:39,030 --> 00:00:42,390 and there's a fair number of entries. 13 00:00:42,390 --> 00:00:45,510 More importantly, there's a fair number of connections 14 00:00:45,510 --> 00:00:47,610 between those entries. 15 00:00:47,610 --> 00:00:50,010 And if you're thinking about systems, 16 00:00:50,010 --> 00:00:51,900 you should be able to think about, 17 00:00:51,900 --> 00:00:55,950 what do each one of those arrows stand for? 18 00:00:55,950 --> 00:00:58,290 The reason that's important is that we 19 00:00:58,290 --> 00:01:01,480 like to have multiple representations for the system 20 00:01:01,480 --> 00:01:03,480 so that we're always able to choose the one that 21 00:01:03,480 --> 00:01:06,800 lets us work the most simply. 22 00:01:06,800 --> 00:01:10,410 But sometimes that involves moving between the squares. 23 00:01:10,410 --> 00:01:12,240 So the way the problem was posed to you 24 00:01:12,240 --> 00:01:14,220 may not be the simplest way to solve it, 25 00:01:14,220 --> 00:01:17,070 and that involves, then, going across one of those arrows. 26 00:01:17,070 --> 00:01:20,160 So for example, we know that there's 27 00:01:20,160 --> 00:01:22,800 a simple relationship between block diagrams and system 28 00:01:22,800 --> 00:01:24,510 functionals. 29 00:01:24,510 --> 00:01:26,560 All you need to do is think about delays 30 00:01:26,560 --> 00:01:29,910 as being the right shift operator. 31 00:01:29,910 --> 00:01:33,150 That's a way of thinking about the system functional is just 32 00:01:33,150 --> 00:01:40,600 a formula picture of the block diagram. 33 00:01:40,600 --> 00:01:43,270 Similarly, we can think about moving between the block 34 00:01:43,270 --> 00:01:44,980 diagram and the difference equation 35 00:01:44,980 --> 00:01:47,875 by thinking about delay being a shift of the index. 36 00:01:50,450 --> 00:01:53,120 So you should be able to think of ways of thinking about all 37 00:01:53,120 --> 00:01:55,630 of the transformations between the boxes. 38 00:01:55,630 --> 00:01:58,900 Over here, we thought about system functionals. 39 00:01:58,900 --> 00:02:03,190 How do you think about a system as a sequence of operations 40 00:02:03,190 --> 00:02:05,920 that you do to a signal? 41 00:02:05,920 --> 00:02:07,720 We also thought about system functions, 42 00:02:07,720 --> 00:02:10,300 and there's a very simple relationship between those. 43 00:02:10,300 --> 00:02:13,860 You think about the operator expression that involves 44 00:02:13,860 --> 00:02:18,070 Rs and replace each R with 1/z. 45 00:02:20,830 --> 00:02:23,140 OK, that was all a big set up, because now I 46 00:02:23,140 --> 00:02:25,300 want to ask a question. 47 00:02:25,300 --> 00:02:28,990 Using your vast knowledge of how all these things interrelate, 48 00:02:28,990 --> 00:02:32,290 what's the relationship between the system functional, 49 00:02:32,290 --> 00:02:37,180 a function of R, and the unit sample response? 50 00:02:41,897 --> 00:02:44,480 As usual, I'd like you to talk to your neighbor to figure this 51 00:02:44,480 --> 00:02:47,960 out, and as usual, you won't do that unless I tell you 52 00:02:47,960 --> 00:02:49,300 to do something trivial first. 53 00:02:49,300 --> 00:02:52,025 So look toward your neighbor, say, "Hi." 54 00:02:54,980 --> 00:02:56,300 Wonderful, good. 55 00:02:56,300 --> 00:02:57,900 Now, figure out this problem. 56 00:04:44,160 --> 00:04:45,432 OK, how many of you-- 57 00:04:45,432 --> 00:04:46,390 I want a show of hands. 58 00:04:46,390 --> 00:04:49,384 How many of you know a way of getting between those boxes? 59 00:04:49,384 --> 00:04:50,050 Raise your hand. 60 00:04:52,969 --> 00:04:54,760 How many of you are sitting beside somebody 61 00:04:54,760 --> 00:04:56,635 who knows the way to get between those boxes? 62 00:04:59,270 --> 00:04:59,770 No. 63 00:04:59,770 --> 00:05:01,450 OK, that didn't work either. 64 00:05:01,450 --> 00:05:04,190 OK, can somebody tell me, if I told you a system functional, 65 00:05:04,190 --> 00:05:07,810 say I told you that one, how would I figure out 66 00:05:07,810 --> 00:05:10,230 the unit sample response from the system function-- 67 00:05:10,230 --> 00:05:11,880 functional? 68 00:05:11,880 --> 00:05:12,611 Yes. 69 00:05:12,611 --> 00:05:14,770 AUDIENCE: Don't you take the inverse Laplace of it? 70 00:05:14,770 --> 00:05:16,630 DENNIS FREEMAN: Inverse Laplace. 71 00:05:16,630 --> 00:05:19,600 OK, that's kind of right, but not quite. 72 00:05:22,366 --> 00:05:26,110 So usually, the answer to the question 73 00:05:26,110 --> 00:05:28,570 is either the slide before this one or the slide 74 00:05:28,570 --> 00:05:30,290 after this one. 75 00:05:30,290 --> 00:05:31,955 So that doesn't quite fit this. 76 00:05:31,955 --> 00:05:34,771 The slide before this one said something about Z-transforms. 77 00:05:39,946 --> 00:05:41,320 Can somebody think of a way, if I 78 00:05:41,320 --> 00:05:43,690 told you the system functional is this thing, 79 00:05:43,690 --> 00:05:49,190 how would you derive the unit sample response? 80 00:05:49,190 --> 00:05:49,928 Yes. 81 00:05:49,928 --> 00:05:51,920 AUDIENCE: You could convert it into a difference equation 82 00:05:51,920 --> 00:05:53,420 and then just work it out manually. 83 00:05:53,420 --> 00:05:54,920 DENNIS FREEMAN: Convert it into a difference equation. 84 00:05:54,920 --> 00:05:55,590 That would work. 85 00:05:55,590 --> 00:06:01,012 So what you could do is go from this box to that box. 86 00:06:01,012 --> 00:06:02,470 Then, if you had this box how would 87 00:06:02,470 --> 00:06:03,844 you get the unit sample response? 88 00:06:08,610 --> 00:06:10,056 Yes. 89 00:06:10,056 --> 00:06:13,350 AUDIENCE: [INAUDIBLE] delta. 90 00:06:13,350 --> 00:06:15,350 DENNIS FREEMAN: Put a delta into the difference. 91 00:06:15,350 --> 00:06:17,690 Think about the difference equation being a system. 92 00:06:17,690 --> 00:06:19,970 Put a delta in and see what comes out. 93 00:06:19,970 --> 00:06:21,316 Exactly. 94 00:06:21,316 --> 00:06:22,940 Can somebody think of a more direct way 95 00:06:22,940 --> 00:06:25,250 that doesn't skirt through the difference equation? 96 00:06:25,250 --> 00:06:25,990 Yes. 97 00:06:25,990 --> 00:06:29,070 AUDIENCE: Multiply the system functional by a delta function. 98 00:06:29,070 --> 00:06:31,650 DENNIS FREEMAN: Multiply the functional by a delta function. 99 00:06:31,650 --> 00:06:33,240 I'm not quite sure what that means, 100 00:06:33,240 --> 00:06:35,250 but I'm willing to try anything. 101 00:06:35,250 --> 00:06:38,250 1 over 1 minus R minus R squared. 102 00:06:38,250 --> 00:06:40,050 Multiply that times a delta function. 103 00:06:44,050 --> 00:06:44,800 What do I do next? 104 00:06:50,040 --> 00:06:51,900 OK, as I said, the answer to the question 105 00:06:51,900 --> 00:06:53,580 is usually something that we just did 106 00:06:53,580 --> 00:06:55,584 or something we're just about to do. 107 00:06:55,584 --> 00:06:57,500 Where did we spend the last three weeks doing? 108 00:06:57,500 --> 00:06:58,630 No, it's not three weeks. 109 00:06:58,630 --> 00:07:02,990 It's only a week, but, you know, I exaggerate. 110 00:07:02,990 --> 00:07:05,010 What have we been doing? 111 00:07:05,010 --> 00:07:06,274 Yes. 112 00:07:06,274 --> 00:07:08,990 AUDIENCE: We could use long division [INAUDIBLE] 113 00:07:08,990 --> 00:07:11,127 DENNIS FREEMAN: One way would be use long division. 114 00:07:11,127 --> 00:07:11,960 How would that work? 115 00:07:11,960 --> 00:07:12,840 If we did 1/1-- 116 00:07:12,840 --> 00:07:18,650 maybe I'd better use a bigger blackboard space-- 117 00:07:18,650 --> 00:07:24,309 if I tried to do 1 over 1 minus R minus R squared, if I tried 118 00:07:24,309 --> 00:07:25,850 to think about that by long division, 119 00:07:25,850 --> 00:07:30,110 I would do 1 minus R minus R squared into 1. 120 00:07:30,110 --> 00:07:33,070 How would that work? 121 00:07:33,070 --> 00:07:33,800 Well, I'd go 1-- 122 00:07:33,800 --> 00:07:36,290 I'd go 1 goes into 1 once. 123 00:07:36,290 --> 00:07:38,690 I get 1 minus R minus R squared. 124 00:07:38,690 --> 00:07:41,660 This minus this gives me R plus R squared. 125 00:07:41,660 --> 00:07:48,690 This goes into this plus R. That gives me whatever. 126 00:07:48,690 --> 00:07:52,790 Having done it in the solitude of my own breakfast 127 00:07:52,790 --> 00:07:57,130 this morning, here's an answer that I got there. 128 00:07:57,130 --> 00:08:00,050 So if I perform long division on the functional, 129 00:08:00,050 --> 00:08:03,410 I get this kind of an answer. 130 00:08:03,410 --> 00:08:05,000 The question was, how do I relate 131 00:08:05,000 --> 00:08:07,400 the functional to the unit sample response? 132 00:08:07,400 --> 00:08:10,869 How do I-- how do I relate this to the unit sample response? 133 00:08:10,869 --> 00:08:12,410 AUDIENCE: The coefficient [INAUDIBLE] 134 00:08:12,410 --> 00:08:12,710 DENNIS FREEMAN: Excuse me? 135 00:08:12,710 --> 00:08:14,720 AUDIENCE: The coefficients [INAUDIBLE] 136 00:08:14,720 --> 00:08:17,130 DENNIS FREEMAN: The coefficients of this expression 137 00:08:17,130 --> 00:08:19,745 are the unit sample response. 138 00:08:22,330 --> 00:08:23,440 OK? 139 00:08:23,440 --> 00:08:25,210 So there's a very straightforward way. 140 00:08:25,210 --> 00:08:27,610 If I tell you the system functional, 141 00:08:27,610 --> 00:08:29,360 if I tell you this representation, 142 00:08:29,360 --> 00:08:32,679 there's a very simple way that we've thought about 143 00:08:32,679 --> 00:08:36,500 by which you can automatically calculate what 144 00:08:36,500 --> 00:08:39,160 is the unit sample response? 145 00:08:39,160 --> 00:08:40,090 OK? 146 00:08:40,090 --> 00:08:45,610 In fact, it's so simple that we can write a formula for it. 147 00:08:45,610 --> 00:08:49,710 Think about the unit sample response being h of n. 148 00:08:49,710 --> 00:08:52,340 Associate the h 0 term with a constant, 149 00:08:52,340 --> 00:08:56,550 the h 1 term with an R, with the h 2 term with an R squared. 150 00:08:56,550 --> 00:09:01,270 Add them all together, that's the answer. 151 00:09:01,270 --> 00:09:06,260 So the way you get between these representations is very simple. 152 00:09:06,260 --> 00:09:08,950 Here's the unit sample response, here's the system functional, 153 00:09:08,950 --> 00:09:11,920 and here's an equation that relates the two 154 00:09:11,920 --> 00:09:12,910 representations. 155 00:09:12,910 --> 00:09:15,530 That all completely clear? 156 00:09:15,530 --> 00:09:16,030 Simple. 157 00:09:16,030 --> 00:09:18,380 Yes, hi. 158 00:09:18,380 --> 00:09:20,642 Everything clear? 159 00:09:20,642 --> 00:09:21,475 Questions, comments? 160 00:09:24,980 --> 00:09:27,800 OK, a follow-up question. 161 00:09:27,800 --> 00:09:30,990 So here's a relationship between these two representations. 162 00:09:30,990 --> 00:09:33,765 How about a relationship between these two representations? 163 00:09:38,710 --> 00:09:39,210 Yes. 164 00:09:39,210 --> 00:09:40,376 AUDIENCE: Same relationship. 165 00:09:40,376 --> 00:09:42,390 Just use one of the place markers. 166 00:09:42,390 --> 00:09:44,020 DENNIS FREEMAN: Precisely. 167 00:09:44,020 --> 00:09:47,000 So it's really-- so we got this one in the previous step. 168 00:09:47,000 --> 00:09:50,264 R is just 1/z, so we get this. 169 00:09:50,264 --> 00:09:52,180 So we get a relationship that looks like that. 170 00:09:54,810 --> 00:09:58,600 OK, that relationship is the basis of today's lecture. 171 00:09:58,600 --> 00:10:01,280 We call that relationship-- 172 00:10:01,280 --> 00:10:04,680 so that relationship represents a mathematical relationship 173 00:10:04,680 --> 00:10:12,300 between a function of z and a function of n, 174 00:10:12,300 --> 00:10:16,460 and we call that relationship the Z-transform. 175 00:10:16,460 --> 00:10:17,710 So that's the topic for today. 176 00:10:17,710 --> 00:10:22,140 We're going to think about systems, not as an operator, 177 00:10:22,140 --> 00:10:26,160 but as a mathematical function, h of z. 178 00:10:26,160 --> 00:10:29,820 So the first thing to realize is that this transform then, 179 00:10:29,820 --> 00:10:32,070 this thing that we're going to be worried about today 180 00:10:32,070 --> 00:10:37,810 is a map between a discrete function 181 00:10:37,810 --> 00:10:42,180 and a continuous function of z. 182 00:10:42,180 --> 00:10:43,680 And even though it's been motivated, 183 00:10:43,680 --> 00:10:45,210 I motivated it by thinking about how 184 00:10:45,210 --> 00:10:48,780 it would relate to a system, that relationship 185 00:10:48,780 --> 00:10:50,790 between a function of z and a function of n 186 00:10:50,790 --> 00:10:55,830 is something that you could do on any discrete time signal. 187 00:10:55,830 --> 00:10:58,380 So in fact, if you have any signal x of n, 188 00:10:58,380 --> 00:11:00,270 we can think about the Z-transform of it 189 00:11:00,270 --> 00:11:03,180 by simply thinking about what that equation is telling us. 190 00:11:03,180 --> 00:11:09,040 How does it map from a function of n to a function of z? 191 00:11:09,040 --> 00:11:10,990 I've written it in a little-- 192 00:11:10,990 --> 00:11:12,890 in a way that you might not have anticipated. 193 00:11:12,890 --> 00:11:15,487 You might have thought I would have started at 0. 194 00:11:15,487 --> 00:11:17,320 That's just because we're going to do what's 195 00:11:17,320 --> 00:11:19,690 called the bilateral transform. 196 00:11:19,690 --> 00:11:22,210 There are many kinds of Z-transforms. 197 00:11:22,210 --> 00:11:23,920 We're going to do this particular one. 198 00:11:23,920 --> 00:11:25,390 Other classes that you might take 199 00:11:25,390 --> 00:11:28,360 may use something called a unilateral. 200 00:11:28,360 --> 00:11:29,830 If you're not taking such a class, 201 00:11:29,830 --> 00:11:31,663 there's no good reason for you to know that. 202 00:11:31,663 --> 00:11:34,990 If you are taking such a class, we're doing bilateral. 203 00:11:34,990 --> 00:11:36,700 That just means two-sided, and we'll 204 00:11:36,700 --> 00:11:41,580 see by the end of the hour how that makes some things simpler. 205 00:11:41,580 --> 00:11:44,040 The big picture is identical whether you 206 00:11:44,040 --> 00:11:48,750 do unilateral or bilateral, but there are differences. 207 00:11:48,750 --> 00:11:49,710 OK. 208 00:11:49,710 --> 00:11:50,790 Simple Z-transforms. 209 00:11:50,790 --> 00:11:53,130 What's the Z-transform of the simplest 210 00:11:53,130 --> 00:11:54,630 signal that we can imagine? 211 00:11:54,630 --> 00:11:56,580 Simplest signal that we can imagine 212 00:11:56,580 --> 00:11:58,170 is the unit sample signal. 213 00:11:58,170 --> 00:12:00,390 It's 0 everywhere except n equals 0. 214 00:12:00,390 --> 00:12:03,720 At n equals 0, it's the simplest possible non-zero answer, 215 00:12:03,720 --> 00:12:04,285 which is 1. 216 00:12:06,852 --> 00:12:07,810 What's the Z-transform? 217 00:12:07,810 --> 00:12:09,080 Well, that's trivial. 218 00:12:09,080 --> 00:12:11,970 You stick it in the formula. 219 00:12:11,970 --> 00:12:15,520 The only non-zero answer is at n equals 0. 220 00:12:15,520 --> 00:12:17,870 Stick in n equals 0, and we get that the answer 221 00:12:17,870 --> 00:12:20,960 is the Z-transform is 1. 222 00:12:20,960 --> 00:12:22,080 What could be easier? 223 00:12:22,080 --> 00:12:22,760 Good. 224 00:12:22,760 --> 00:12:28,860 Simple signal in time, simple signal in the Z-transform. 225 00:12:28,860 --> 00:12:33,720 What's the Z-transform for a delayed unit sample signal? 226 00:12:33,720 --> 00:12:37,050 What happens if the only place that it's not 0 227 00:12:37,050 --> 00:12:39,910 is shifted to n equals 1? 228 00:12:39,910 --> 00:12:41,510 Not a big deal. 229 00:12:41,510 --> 00:12:43,950 Again, there's a single non-zero answer, 230 00:12:43,950 --> 00:12:47,420 it's just that it's now at n equals 1. 231 00:12:47,420 --> 00:12:49,870 So this answer is z to the minus 1. 232 00:12:49,870 --> 00:12:53,750 We took a signal that was a function of n, 233 00:12:53,750 --> 00:12:55,800 and we represent it by a Z-transform, 234 00:12:55,800 --> 00:12:58,040 which is a function of z. 235 00:12:58,040 --> 00:13:00,590 Delta of n is 1. 236 00:13:00,590 --> 00:13:04,800 Delta of n minus 1 is z to the minus 1. 237 00:13:04,800 --> 00:13:08,460 OK, with that vast knowledge of Z-transforms, 238 00:13:08,460 --> 00:13:11,850 figure out the Z-transform for a slightly more complicated 239 00:13:11,850 --> 00:13:14,040 sequence of the type-- of the type 240 00:13:14,040 --> 00:13:16,920 that we saw when we were looking at systems. 241 00:13:16,920 --> 00:13:19,020 What if the signal that we're interested in 242 00:13:19,020 --> 00:13:22,080 is 7/8 to the n, u of n, where I'm using the u of n 243 00:13:22,080 --> 00:13:26,500 just to cut off the negative parts of 7/8 to the n. 244 00:13:26,500 --> 00:13:28,350 So find the Z-transform, and find 245 00:13:28,350 --> 00:13:30,630 if it looks like one of those four answers 246 00:13:30,630 --> 00:13:32,080 or none, which is number five. 247 00:15:13,020 --> 00:15:17,850 So what answer best represents the Z-transform of the signal 248 00:15:17,850 --> 00:15:19,020 x? 249 00:15:19,020 --> 00:15:21,135 Raise your hand, number one through five. 250 00:15:25,020 --> 00:15:28,510 It's a participatory sport. 251 00:15:28,510 --> 00:15:32,260 Good 97% correct, I think. 252 00:15:32,260 --> 00:15:35,230 So OK. 253 00:15:35,230 --> 00:15:36,119 Everybody says two. 254 00:15:36,119 --> 00:15:36,910 How do you get two? 255 00:15:36,910 --> 00:15:38,000 What do you do? 256 00:15:38,000 --> 00:15:39,190 Plug in the formula, right? 257 00:15:39,190 --> 00:15:42,490 It's a very simple-minded thing. 258 00:15:42,490 --> 00:15:44,590 If you were to think about this signal 259 00:15:44,590 --> 00:15:47,710 and think about the definition of the Z-transform, 260 00:15:47,710 --> 00:15:51,910 just substitute this particular signal, 7/8 to the n, u of n, 261 00:15:51,910 --> 00:15:57,440 in where there would have been the x of n 262 00:15:57,440 --> 00:15:59,990 and try to close the sum, right? 263 00:15:59,990 --> 00:16:02,480 It's a geometric sequence. 264 00:16:02,480 --> 00:16:05,550 We've had lots of experience with geometric sequences. 265 00:16:05,550 --> 00:16:07,250 It was in the homework, and so we all 266 00:16:07,250 --> 00:16:10,000 know that the answer to summing a geometric-- 267 00:16:10,000 --> 00:16:13,400 a one-sided geometric sequence is something of the form 268 00:16:13,400 --> 00:16:15,340 1 over 1 minus a. 269 00:16:15,340 --> 00:16:16,640 OK, easy, right? 270 00:16:19,360 --> 00:16:22,660 So the idea, then, is that we understand 271 00:16:22,660 --> 00:16:25,570 a general rule by which we can map a function of time 272 00:16:25,570 --> 00:16:28,420 into a function of z. 273 00:16:28,420 --> 00:16:33,030 So the function of time has to make sense everywhere. 274 00:16:33,030 --> 00:16:36,180 That's-- we want that to be true. 275 00:16:36,180 --> 00:16:39,300 We want to know what was the system's unit sample 276 00:16:39,300 --> 00:16:42,400 response, for example, or what was the input to the system, 277 00:16:42,400 --> 00:16:42,900 or whatever. 278 00:16:42,900 --> 00:16:46,050 We want to know that at all possible values of n. 279 00:16:46,050 --> 00:16:47,310 How about the other case? 280 00:16:47,310 --> 00:16:49,140 Does it make sense? 281 00:16:49,140 --> 00:16:53,190 Is x of z makes sense for all possible values of z? 282 00:16:55,710 --> 00:16:59,940 And since I ask the question, the answer is no. 283 00:16:59,940 --> 00:17:01,120 Yes, of course. 284 00:17:01,120 --> 00:17:02,700 I wouldn't have asked the question 285 00:17:02,700 --> 00:17:05,369 if the answer were yes, right? 286 00:17:05,369 --> 00:17:06,839 So by the theory of lecturers, you 287 00:17:06,839 --> 00:17:09,660 can tell the answer has to have been no. 288 00:17:09,660 --> 00:17:12,945 So the question is, what values of z don't make sense? 289 00:17:17,300 --> 00:17:19,079 Shout. 290 00:17:19,079 --> 00:17:21,800 AUDIENCE: The summation of the [INAUDIBLE] 291 00:17:21,800 --> 00:17:24,109 DENNIS FREEMAN: The summation has to converge, 292 00:17:24,109 --> 00:17:29,540 so it's only going to be defined if this sum, which happens 293 00:17:29,540 --> 00:17:34,230 to be an infinite sum and therefore may not converge, 294 00:17:34,230 --> 00:17:36,620 it's going to have to be the case that that infinite sum 295 00:17:36,620 --> 00:17:37,370 does converge. 296 00:17:37,370 --> 00:17:39,740 Otherwise, it won't make sense to talk about it. 297 00:17:39,740 --> 00:17:41,930 So then the question is, for what values of z 298 00:17:41,930 --> 00:17:43,910 will that converge? 299 00:17:43,910 --> 00:17:45,530 And again, we know from our experience 300 00:17:45,530 --> 00:17:53,900 with geometric sequences that the base of the geometric, a, 301 00:17:53,900 --> 00:17:56,720 when we did homework one, the base of the geometric 302 00:17:56,720 --> 00:17:58,230 has to have what? 303 00:17:58,230 --> 00:17:59,972 What's the property of the base of the-- 304 00:17:59,972 --> 00:18:01,430 the geometric that'll make it work? 305 00:18:04,650 --> 00:18:08,690 So if I want to sum some series of the form n equals 0 306 00:18:08,690 --> 00:18:12,800 to infinity a to the n, what's the values-- 307 00:18:12,800 --> 00:18:16,425 what's the limitation on a that makes it work? 308 00:18:16,425 --> 00:18:21,274 AUDIENCE: [INAUDIBLE] 309 00:18:21,274 --> 00:18:22,940 DENNIS FREEMAN: OK, you're not saying it 310 00:18:22,940 --> 00:18:26,432 loud enough or clearly enough, or I'm too dense or something. 311 00:18:26,432 --> 00:18:27,747 AUDIENCE: Abs less than 1? 312 00:18:27,747 --> 00:18:29,330 DENNIS FREEMAN: So we need something-- 313 00:18:29,330 --> 00:18:30,770 we need abs less than 1, right. 314 00:18:33,440 --> 00:18:34,950 So we do the same thing here. 315 00:18:34,950 --> 00:18:43,180 We're going to have to have that the geometric base, 7/8 1 316 00:18:43,180 --> 00:18:47,660 over z has to be less-- has to have a magnitude that 317 00:18:47,660 --> 00:18:49,950 is less than 1. 318 00:18:49,950 --> 00:18:53,150 And if we torture our minds with inequalities and magnitude 319 00:18:53,150 --> 00:18:55,760 signs, that says that the magnitude of z 320 00:18:55,760 --> 00:18:58,700 has to be bigger than 7/8. 321 00:18:58,700 --> 00:18:59,950 OK? 322 00:18:59,950 --> 00:19:04,110 The important idea is that when we characterize 323 00:19:04,110 --> 00:19:08,380 the Z-transform, we should be expecting 324 00:19:08,380 --> 00:19:10,840 it to work for all values of n, but not necessarily 325 00:19:10,840 --> 00:19:13,030 all values of z. 326 00:19:13,030 --> 00:19:14,690 So we have to be cognizant of that. 327 00:19:14,690 --> 00:19:18,640 We have to know which values of z are you talking about. 328 00:19:18,640 --> 00:19:20,720 We call that idea the region of convergence. 329 00:19:20,720 --> 00:19:24,430 We say the Z-transform converged for all z inside the region 330 00:19:24,430 --> 00:19:25,790 of convergence. 331 00:19:25,790 --> 00:19:28,480 So when you specify a Z-transform, 332 00:19:28,480 --> 00:19:32,650 generally, you have to tell me not just some functional form, 333 00:19:32,650 --> 00:19:34,270 but you also have to tell me what 334 00:19:34,270 --> 00:19:38,780 was the region for which that functional form converged. 335 00:19:38,780 --> 00:19:41,956 So in general, you have to tell me the region of convergence. 336 00:19:41,956 --> 00:19:43,330 In this particular case, you have 337 00:19:43,330 --> 00:19:45,280 to tell me that you should restrict 338 00:19:45,280 --> 00:19:48,730 your attention to values of z whose magnitude is 339 00:19:48,730 --> 00:19:51,250 bigger than 7/8. 340 00:19:51,250 --> 00:19:53,380 OK? 341 00:19:53,380 --> 00:19:56,470 OK, that's completely the whole definition of Z-transforms. 342 00:19:56,470 --> 00:20:00,090 We're done from a point of view of mathematics, right? 343 00:20:00,090 --> 00:20:06,030 So all we need to know is that the Z-transform of a signal 344 00:20:06,030 --> 00:20:14,280 is some sum of the signal z to the minus n, and we're done. 345 00:20:14,280 --> 00:20:16,590 We need to know about the region of convergence, 346 00:20:16,590 --> 00:20:18,630 but mathematically, we've completely specified 347 00:20:18,630 --> 00:20:20,160 the problem at this point. 348 00:20:20,160 --> 00:20:22,410 To make it useful, however, what we need to do 349 00:20:22,410 --> 00:20:25,740 is investigate properties of that transformation. 350 00:20:25,740 --> 00:20:28,350 If the transformation were not easy to manipulate, 351 00:20:28,350 --> 00:20:31,090 we wouldn't bother with it. 352 00:20:31,090 --> 00:20:35,810 So we want to know what's easy to do with the Z-transform 353 00:20:35,810 --> 00:20:38,210 and what's not easy to do with the Z-transform. 354 00:20:38,210 --> 00:20:40,940 When something's easy to do with the Z-transform, we'll use it. 355 00:20:40,940 --> 00:20:42,920 When things are easier to do some other way, 356 00:20:42,920 --> 00:20:44,420 we use the other way. 357 00:20:44,420 --> 00:20:45,980 That's the game plan. 358 00:20:45,980 --> 00:20:48,650 So there's a number of properties of the Z-transform 359 00:20:48,650 --> 00:20:51,210 that make it easy or hard to do things. 360 00:20:51,210 --> 00:20:53,420 So we'll talk about the easy ones. 361 00:20:53,420 --> 00:20:58,610 The most fundamental one is linearity. 362 00:20:58,610 --> 00:21:03,590 Basically, this entire subject is about linear things. 363 00:21:03,590 --> 00:21:06,590 If it's not linear, largely, we don't talk about it. 364 00:21:06,590 --> 00:21:08,840 The reason is we have such powerful tools for thinking 365 00:21:08,840 --> 00:21:10,980 about things that are linear. 366 00:21:10,980 --> 00:21:13,090 This is one of them. 367 00:21:13,090 --> 00:21:16,100 If-- so we say that the Z-transform is 368 00:21:16,100 --> 00:21:17,990 a linear operator, and all that means 369 00:21:17,990 --> 00:21:23,120 is if you apply the Z-transform to the sum of two things, 370 00:21:23,120 --> 00:21:24,932 you get the sum of the things out. 371 00:21:24,932 --> 00:21:26,640 It's a little more complicated than that. 372 00:21:26,640 --> 00:21:28,264 If I were being a little more careful-- 373 00:21:28,264 --> 00:21:32,289 in fact, I didn't quite say a complete definition here. 374 00:21:32,289 --> 00:21:34,580 This is certainly true, but there are additional things 375 00:21:34,580 --> 00:21:35,870 that are also true. 376 00:21:35,870 --> 00:21:37,550 We say that the Z-transform is linear 377 00:21:37,550 --> 00:21:42,770 because if we knew the z-transform for X 1, 378 00:21:42,770 --> 00:21:46,760 that includes a functional form and a region of convergence, 379 00:21:46,760 --> 00:21:49,380 and if we knew the Z-transform for X 2, 380 00:21:49,380 --> 00:21:52,580 again, a functional form and a region of convergence, 381 00:21:52,580 --> 00:21:54,710 then by the linearity of the operator, 382 00:21:54,710 --> 00:21:58,520 we can figure out just from the two Z-transforms, what 383 00:21:58,520 --> 00:22:02,300 is the Z-transform of the sum. 384 00:22:02,300 --> 00:22:04,860 And that's trivial to see. 385 00:22:04,860 --> 00:22:07,340 It's easy to see why it ought to be that way. 386 00:22:07,340 --> 00:22:09,680 Just look at-- let's think about y, 387 00:22:09,680 --> 00:22:12,890 which is the signal, which is the sum. 388 00:22:12,890 --> 00:22:14,930 By the definition of Z-transform, 389 00:22:14,930 --> 00:22:18,010 the Z-transform of the sum is that formula. 390 00:22:21,500 --> 00:22:25,550 Y, by definition, is that thing. 391 00:22:25,550 --> 00:22:31,460 Z commutes-- distributes over addition, 392 00:22:31,460 --> 00:22:35,080 and the sum separates. 393 00:22:35,080 --> 00:22:38,050 So we can always reduce this to this, 394 00:22:38,050 --> 00:22:44,475 at least when the case that z is in both regions of convergence. 395 00:22:44,475 --> 00:22:45,850 Though later in the course, we'll 396 00:22:45,850 --> 00:22:47,830 see that that's a little overly restrictive. 397 00:22:47,830 --> 00:22:52,791 Sometimes it works for more zs than in the intersection. 398 00:22:52,791 --> 00:22:54,790 But it's guaranteed to work in the intersection, 399 00:22:54,790 --> 00:23:00,880 because I know that I can do this sum if z is in ROC 1. 400 00:23:00,880 --> 00:23:04,090 I know I can do this sum if z is in ROC 2. 401 00:23:04,090 --> 00:23:06,640 So I know I can do both of them if it's in the intersection. 402 00:23:06,640 --> 00:23:08,290 We'll see a little later that sometimes you 403 00:23:08,290 --> 00:23:10,340 can do it even when it's not in the intersection. 404 00:23:10,340 --> 00:23:12,214 But for the time being, we know you can do it 405 00:23:12,214 --> 00:23:14,980 if it's in the intersection. 406 00:23:14,980 --> 00:23:21,610 So because of linearity, we know that the Z-transform of a sum 407 00:23:21,610 --> 00:23:23,680 is the sum of the Z-transforms. 408 00:23:23,680 --> 00:23:26,190 I didn't realize when I made this slide, 409 00:23:26,190 --> 00:23:28,240 linearity implies more than that. 410 00:23:28,240 --> 00:23:29,640 I could have done a weighted sum. 411 00:23:33,110 --> 00:23:38,570 I could have said if X 1 goes to X1 and X 2 goes to X 2, 412 00:23:38,570 --> 00:23:42,650 then alpha 1, X 1 plus alpha 2, X 2 413 00:23:42,650 --> 00:23:46,190 goes to alpha 1, X 1 plus alpha 2, X 2. 414 00:23:46,190 --> 00:23:47,690 I should have put that in the slide, 415 00:23:47,690 --> 00:23:50,660 and it just didn't occur to me. 416 00:23:50,660 --> 00:23:53,480 So the idea of linearity is slightly more powerful 417 00:23:53,480 --> 00:23:56,300 than the example that I gave here. 418 00:23:56,300 --> 00:23:58,760 But again, linearity is the most fundamental property 419 00:23:58,760 --> 00:23:59,774 of Z-transforms. 420 00:23:59,774 --> 00:24:01,190 If the Z-transform weren't linear, 421 00:24:01,190 --> 00:24:02,910 we wouldn't bother with it. 422 00:24:02,910 --> 00:24:06,080 But it is, so we do. 423 00:24:06,080 --> 00:24:08,870 Another important property is the delay property. 424 00:24:12,930 --> 00:24:15,030 Think about what we do with discrete signals. 425 00:24:15,030 --> 00:24:18,660 We put them through discrete systems. 426 00:24:18,660 --> 00:24:21,000 Discrete systems of the type we've looked at so far 427 00:24:21,000 --> 00:24:24,570 have adders, delays, gains. 428 00:24:24,570 --> 00:24:25,730 Delays. 429 00:24:25,730 --> 00:24:28,050 If the Z-transform couldn't handle delays, 430 00:24:28,050 --> 00:24:30,690 again, we wouldn't do it, right? 431 00:24:30,690 --> 00:24:32,370 Delays are so fundamental to the way 432 00:24:32,370 --> 00:24:34,590 we think about discrete systems that it 433 00:24:34,590 --> 00:24:36,660 must be the case that Z-transforms deal 434 00:24:36,660 --> 00:24:39,310 with delays well, and they do. 435 00:24:39,310 --> 00:24:40,900 We've already seen two examples. 436 00:24:40,900 --> 00:24:45,340 The first example I did was the unit sample signal 437 00:24:45,340 --> 00:24:49,620 has a transform of 1, and the delayed unit sample's signal 438 00:24:49,620 --> 00:24:52,044 has transform of z to the minus 1. 439 00:24:52,044 --> 00:24:53,460 So there's a simple relationship-- 440 00:24:53,460 --> 00:24:57,060 if I knew the first, how I could find the second. 441 00:24:57,060 --> 00:25:02,830 More generally, if I had a signal X and I knew its 442 00:25:02,830 --> 00:25:09,710 transform X, then I could readily compute the transform 443 00:25:09,710 --> 00:25:13,500 of the shifted version just from the transform of the unshifted 444 00:25:13,500 --> 00:25:15,900 version, and that's-- you can see how that would work 445 00:25:15,900 --> 00:25:16,950 mathematically. 446 00:25:16,950 --> 00:25:20,402 Let's call the shifted version Y, 447 00:25:20,402 --> 00:25:22,110 then the transform of the shifted version 448 00:25:22,110 --> 00:25:25,390 is given by this expression. 449 00:25:25,390 --> 00:25:27,670 By the definition of shift, y of n 450 00:25:27,670 --> 00:25:30,862 is the same as x of n minus 1. 451 00:25:30,862 --> 00:25:35,350 And now all we need to do is massage the math 452 00:25:35,350 --> 00:25:39,550 so it ends up looking like the definition of a Z-transform. 453 00:25:39,550 --> 00:25:46,060 So the way to do that would be to substitute m for n minus 1. 454 00:25:46,060 --> 00:25:48,730 We get something that looks more like a Z-transform. 455 00:25:48,730 --> 00:25:50,950 And if we bring this minus 1 out front, 456 00:25:50,950 --> 00:25:53,620 it looks exactly like a Z-transform pre-multiplied by z 457 00:25:53,620 --> 00:25:56,070 to the minus 1. 458 00:25:56,070 --> 00:26:00,190 So the delay theorem just says if I already 459 00:26:00,190 --> 00:26:02,020 know the Z-transform of a signal, 460 00:26:02,020 --> 00:26:04,600 then to find the Z-transform of the delayed version 461 00:26:04,600 --> 00:26:07,930 of that signal, simply multiply the original Z-transform by z 462 00:26:07,930 --> 00:26:10,640 to the minus 1. 463 00:26:10,640 --> 00:26:11,140 OK? 464 00:26:11,140 --> 00:26:15,020 So so far, I've talked about two properties 465 00:26:15,020 --> 00:26:18,740 of Z-transforms, linearity and the delay property. 466 00:26:18,740 --> 00:26:24,530 And in combination, they let me do a lot of stuff 467 00:26:24,530 --> 00:26:27,470 with discrete systems. 468 00:26:27,470 --> 00:26:29,295 So think, for example, of a system, 469 00:26:29,295 --> 00:26:30,920 a discrete system that can be described 470 00:26:30,920 --> 00:26:33,528 by a linear difference equation with constant coefficients. 471 00:26:36,740 --> 00:26:39,290 If you can describe a system that way, 472 00:26:39,290 --> 00:26:42,560 then you can write the relationship between the input 473 00:26:42,560 --> 00:26:46,511 signal X and the output signal Y that looks like a difference 474 00:26:46,511 --> 00:26:47,010 equation. 475 00:26:50,810 --> 00:26:53,900 Then, if you take into account that the Z-transform 476 00:26:53,900 --> 00:27:00,460 is both linear and has a simple representation for delays, 477 00:27:00,460 --> 00:27:03,910 I can take the Z-transform of that difference equation 478 00:27:03,910 --> 00:27:08,030 and get a new expression. 479 00:27:08,030 --> 00:27:10,420 So the difference equation represents an equality 480 00:27:10,420 --> 00:27:15,730 between two sums of time domain signals. 481 00:27:15,730 --> 00:27:18,460 Taking the Z-transform of that equality 482 00:27:18,460 --> 00:27:21,670 tells me some equivalent relationship 483 00:27:21,670 --> 00:27:24,850 of the Z-transforms. 484 00:27:24,850 --> 00:27:29,890 So if I-- so I think about the left-hand side by linearity, 485 00:27:29,890 --> 00:27:33,520 I can find the Z-transform of this sum 486 00:27:33,520 --> 00:27:35,470 by finding the sum of the Z-transforms. 487 00:27:38,620 --> 00:27:41,500 And so this one's easy, right? 488 00:27:41,500 --> 00:27:46,690 The Z-transform of y of n is, by assumption, y of z. 489 00:27:46,690 --> 00:27:49,600 By linearity, the part I didn't show you, 490 00:27:49,600 --> 00:27:51,310 if I pre-multiply by b nought, it's 491 00:27:51,310 --> 00:27:52,450 pre-multiplied by b nought. 492 00:27:56,040 --> 00:27:58,830 Because it's linear, I can just add the next term to it. 493 00:27:58,830 --> 00:28:00,990 The next term looks a lot like the previous term 494 00:28:00,990 --> 00:28:03,150 except that it's shifted, but shift is easy. 495 00:28:03,150 --> 00:28:06,420 You just put z to the minus 1 in front. 496 00:28:06,420 --> 00:28:11,300 So this term just becomes b one, z to the minus 1. 497 00:28:11,300 --> 00:28:14,040 This just becomes b 2, z to the minus 2, 498 00:28:14,040 --> 00:28:17,190 and the whole thing factors. 499 00:28:17,190 --> 00:28:19,467 Same sort of thing happens on the X side, 500 00:28:19,467 --> 00:28:21,300 and you end up with a very simple statement. 501 00:28:21,300 --> 00:28:25,750 The Z-transform of a system that can be represented 502 00:28:25,750 --> 00:28:29,110 by a difference equation with constant coefficients is 503 00:28:29,110 --> 00:28:33,730 the ratio of two polynomials in z to the minus 1-- 504 00:28:33,730 --> 00:28:36,054 this is a polynomial in z to the minus 1. 505 00:28:36,054 --> 00:28:40,210 Or if I multiply top and bottom by z to the k, 506 00:28:40,210 --> 00:28:43,742 I can make it look like a ratio of polynomials in z. 507 00:28:43,742 --> 00:28:44,242 Yeah. 508 00:28:44,242 --> 00:28:49,274 AUDIENCE: Why is it y to the x on the b 0 [INAUDIBLE] 509 00:28:49,274 --> 00:28:50,690 DENNIS FREEMAN: Why is it y the x? 510 00:28:50,690 --> 00:28:53,390 Because of my poor typography. 511 00:28:53,390 --> 00:28:54,230 Thank you very much. 512 00:28:54,230 --> 00:28:57,500 That's completely wrong. 513 00:28:57,500 --> 00:29:00,970 OK, so I'll try to remember that when I get to my office 514 00:29:00,970 --> 00:29:02,720 and change it before I post it on the web. 515 00:29:02,720 --> 00:29:03,650 Thank you. 516 00:29:03,650 --> 00:29:07,370 So this is y of z, this is y of z, this is y of z. 517 00:29:07,370 --> 00:29:12,020 That is-- and you can see I used [? e-max, ?] so I copied 518 00:29:12,020 --> 00:29:17,660 the formula and I got the single error to turn into two. 519 00:29:17,660 --> 00:29:20,780 It's always very good when you can do that, right? 520 00:29:20,780 --> 00:29:22,160 Something like that. 521 00:29:22,160 --> 00:29:25,280 OK, so the point then is that simply 522 00:29:25,280 --> 00:29:28,540 by knowing that the Z-transform is linear and has a delay 523 00:29:28,540 --> 00:29:31,070 property, it results in a very simple statement 524 00:29:31,070 --> 00:29:34,030 about the Z-transform for a system that can be represented 525 00:29:34,030 --> 00:29:35,280 by such a difference equation. 526 00:29:38,120 --> 00:29:39,080 OK. 527 00:29:39,080 --> 00:29:43,040 Now we use a little bit of knowledge about polynomials. 528 00:29:43,040 --> 00:29:45,785 We use the idea that the fundamental theorem 529 00:29:45,785 --> 00:29:46,640 in algebra-- 530 00:29:46,640 --> 00:29:48,890 anybody have any idea what that is? 531 00:29:48,890 --> 00:29:49,889 Nah. 532 00:29:49,889 --> 00:29:51,430 Fundamental theorem of algebra, what? 533 00:29:51,430 --> 00:29:52,700 AUDIENCE: That was a long time ago. 534 00:29:52,700 --> 00:29:53,690 DENNIS FREEMAN: Long time ago. 535 00:29:53,690 --> 00:29:56,010 If it was long for you, think about when it was for me. 536 00:29:56,010 --> 00:29:57,290 [LAUGHTER] 537 00:29:57,290 --> 00:29:59,450 No, you can't think that way. 538 00:29:59,450 --> 00:30:00,279 Yes? 539 00:30:00,279 --> 00:30:06,270 AUDIENCE: Are we [INAUDIBLE] 540 00:30:06,270 --> 00:30:07,460 DENNIS FREEMAN: Wonderful. 541 00:30:07,460 --> 00:30:08,310 Everybody hear that? 542 00:30:08,310 --> 00:30:09,393 Because I can't repeat it. 543 00:30:09,393 --> 00:30:11,750 But everybody hear that? 544 00:30:11,750 --> 00:30:15,800 An nth order polynomial has n roots. 545 00:30:15,800 --> 00:30:17,690 Roughly speaking. 546 00:30:17,690 --> 00:30:19,040 I'm not a mathematician, right? 547 00:30:19,040 --> 00:30:20,120 I'm an engineer. 548 00:30:20,120 --> 00:30:20,930 Right? 549 00:30:20,930 --> 00:30:24,140 An nth order polynomial has n roots. 550 00:30:24,140 --> 00:30:26,780 OK, and then the factor theorem. 551 00:30:26,780 --> 00:30:31,170 Surprisingly enough, that says you can factor things. 552 00:30:31,170 --> 00:30:35,600 So the idea then is that if I can represent the Z-transform 553 00:30:35,600 --> 00:30:40,530 as a ratio of polynomials and z, there's a factored form. 554 00:30:40,530 --> 00:30:42,810 And that's the basis of a decomposition 555 00:30:42,810 --> 00:30:45,150 that we will make extensive use of. 556 00:30:45,150 --> 00:30:47,040 You've already seen, extensively, 557 00:30:47,040 --> 00:30:48,300 the roots of the denominator. 558 00:30:48,300 --> 00:30:49,560 They're the poles. 559 00:30:49,560 --> 00:30:52,110 We will similarly define, because of this manipulation, 560 00:30:52,110 --> 00:30:53,420 the roots of the numerator. 561 00:30:53,420 --> 00:30:55,661 They're the zeros. 562 00:30:55,661 --> 00:30:56,160 OK? 563 00:30:59,630 --> 00:31:02,960 So-- and it's pretty easy to think through, then, 564 00:31:02,960 --> 00:31:05,120 how there is a relationship-- 565 00:31:05,120 --> 00:31:06,974 it's all about relationships, right? 566 00:31:06,974 --> 00:31:08,390 When I introduce something, I want 567 00:31:08,390 --> 00:31:10,100 to think about how the thing I just said 568 00:31:10,100 --> 00:31:13,130 relates to everything else I've ever said. 569 00:31:13,130 --> 00:31:17,570 There's a simple relationship between poles and regions 570 00:31:17,570 --> 00:31:19,700 of convergence. 571 00:31:19,700 --> 00:31:22,010 Turns out the regions of convergence 572 00:31:22,010 --> 00:31:23,570 are always going to be circles. 573 00:31:23,570 --> 00:31:25,670 That has-- circles in the z plane. 574 00:31:25,670 --> 00:31:27,830 That has to do with things like convergence 575 00:31:27,830 --> 00:31:29,060 of geometric sequences. 576 00:31:32,490 --> 00:31:37,380 If I build my system out of adders and delays and gains, 577 00:31:37,380 --> 00:31:39,570 then I have that complex set of reasoning 578 00:31:39,570 --> 00:31:42,590 that gives rise to the idea that I have polynomials. 579 00:31:42,590 --> 00:31:45,300 That's going to be-- that's going to give rise to, 580 00:31:45,300 --> 00:31:51,180 if you think about poles, partial fractions, 581 00:31:51,180 --> 00:31:52,920 each of those is going to have some kind 582 00:31:52,920 --> 00:31:54,240 of a characteristic response. 583 00:31:54,240 --> 00:31:55,823 It's going to be a geometric sequence. 584 00:31:55,823 --> 00:31:58,260 Each of those is going to have a convergence property that 585 00:31:58,260 --> 00:32:00,370 has something to do with a circle in the z plane. 586 00:32:03,170 --> 00:32:07,580 Each of those circles is going to be bounded by a pole. 587 00:32:07,580 --> 00:32:10,670 So the upshot of all that stuff is 588 00:32:10,670 --> 00:32:12,320 there's a relationship between poles 589 00:32:12,320 --> 00:32:14,180 and regions of convergence. 590 00:32:14,180 --> 00:32:15,560 Regions of convergence are always 591 00:32:15,560 --> 00:32:17,510 going to be circles in the z plane, 592 00:32:17,510 --> 00:32:20,240 and they're always going to be bounded by a pole. 593 00:32:20,240 --> 00:32:22,430 And we've seen an example of that already. 594 00:32:22,430 --> 00:32:24,980 If we had a geometric sequence that 595 00:32:24,980 --> 00:32:29,800 was defined only to the right of n equals 0-- 596 00:32:29,800 --> 00:32:34,190 if it is non-zero, only at n equals 0 or bigger-- 597 00:32:34,190 --> 00:32:39,530 then we get convergence inside some region defined 598 00:32:39,530 --> 00:32:45,590 by when the base has an absolute value that's less than 1. 599 00:32:45,590 --> 00:32:50,120 That's a circle in the z plane. 600 00:32:50,120 --> 00:32:56,000 And the pole, which is alpha, turns into the edge 601 00:32:56,000 --> 00:32:59,100 of that circle of convergence. 602 00:32:59,100 --> 00:33:01,790 So the regions of convergence for these kinds of systems 603 00:33:01,790 --> 00:33:04,970 will always be circles in the z plane, always bounded 604 00:33:04,970 --> 00:33:05,500 by a pole. 605 00:33:09,750 --> 00:33:13,650 OK, enough of my talking. 606 00:33:13,650 --> 00:33:17,770 What DC-- what DT signal has the following Z-transform? 607 00:33:23,120 --> 00:33:27,170 I want to know a DT signal that has transform of the form 608 00:33:27,170 --> 00:33:31,310 z over z minus 7/8 with a region of convergence 609 00:33:31,310 --> 00:33:37,410 inside absolute value of z equals 7/8. 610 00:35:43,450 --> 00:35:48,032 So what's the DT signal that has that Z-transform? 611 00:35:48,032 --> 00:35:49,490 AUDIENCE: It would be Y to the n is 612 00:35:49,490 --> 00:35:53,055 equal to x to the n plus 7/8y to the n minus 1? 613 00:35:53,055 --> 00:35:55,180 DENNIS FREEMAN: Sounded like a difference equation. 614 00:35:55,180 --> 00:35:55,580 Say it again. 615 00:35:55,580 --> 00:35:55,975 AUDIENCE: Oh yeah. 616 00:35:55,975 --> 00:35:57,370 Isn't that what you're looking for? 617 00:35:57,370 --> 00:35:57,590 DENNIS FREEMAN: No. 618 00:35:57,590 --> 00:35:58,800 I wanted the signal. 619 00:35:58,800 --> 00:35:59,740 AUDIENCE: Oh, OK. 620 00:35:59,740 --> 00:36:01,150 DENNIS FREEMAN: So I want to think about-- 621 00:36:01,150 --> 00:36:02,274 this is a little confusing. 622 00:36:02,274 --> 00:36:04,220 I apologize if I didn't say this clearly. 623 00:36:04,220 --> 00:36:05,920 We motivated the idea of Z-transform 624 00:36:05,920 --> 00:36:11,740 by looking at systems, but the result, the Z-transform's 625 00:36:11,740 --> 00:36:12,700 just a relationship. 626 00:36:12,700 --> 00:36:15,050 It's a map between a function of n and a function of z. 627 00:36:15,050 --> 00:36:17,750 So we can do that for every signal. 628 00:36:17,750 --> 00:36:19,570 So if I tell you the function of z, 629 00:36:19,570 --> 00:36:21,550 you can figure out the function of n. 630 00:36:21,550 --> 00:36:24,250 The question here is intended-- what I intended was, 631 00:36:24,250 --> 00:36:27,910 what's the function of n that corresponds 632 00:36:27,910 --> 00:36:30,220 to that function of z? 633 00:36:36,482 --> 00:36:36,982 Yes? 634 00:36:36,982 --> 00:36:40,150 AUDIENCE: [INAUDIBLE] 635 00:36:40,150 --> 00:36:41,990 DENNIS FREEMAN: 7/8 to the n, u of n. 636 00:36:41,990 --> 00:36:45,650 That sounds like something we did before. 637 00:36:45,650 --> 00:36:49,166 7/8 to the n, u of n. 638 00:36:49,166 --> 00:36:51,290 That sounds like something we did before, actually. 639 00:36:54,590 --> 00:36:55,422 Yes, no? 640 00:36:55,422 --> 00:36:57,380 Something we did before is a good thing, right? 641 00:36:57,380 --> 00:36:59,005 That's one of the general rules, right? 642 00:36:59,005 --> 00:37:01,250 When I ask a question, look at what we did before. 643 00:37:01,250 --> 00:37:04,420 That's a good rule. 644 00:37:04,420 --> 00:37:07,090 Is that the same Z-transform we did before? 645 00:37:07,090 --> 00:37:07,590 Yes? 646 00:37:07,590 --> 00:37:10,360 AUDIENCE: [INAUDIBLE] 647 00:37:10,360 --> 00:37:12,310 DENNIS FREEMAN: Yes, yes. 648 00:37:12,310 --> 00:37:18,966 The region of convergence that we did before was outside 7/8, 649 00:37:18,966 --> 00:37:21,340 and the region of convergence I asked for in this problem 650 00:37:21,340 --> 00:37:28,150 is inside 7/8 So what's the effect of switching 651 00:37:28,150 --> 00:37:29,720 the region of convergence? 652 00:37:33,380 --> 00:37:36,490 What happens if I switch the region of convergence? 653 00:37:40,490 --> 00:37:42,490 AUDIENCE: Does z change? 654 00:37:42,490 --> 00:37:44,990 Does the value of z change? 655 00:37:44,990 --> 00:37:48,930 DENNIS FREEMAN: Z. It's hard to talk about the value of z, 656 00:37:48,930 --> 00:37:49,630 right? 657 00:37:49,630 --> 00:37:51,276 Z. 658 00:37:51,276 --> 00:37:52,687 AUDIENCE: [INAUDIBLE] 659 00:37:52,687 --> 00:37:53,770 DENNIS FREEMAN: Excuse me. 660 00:37:53,770 --> 00:37:55,390 So z is defined-- 661 00:37:55,390 --> 00:37:57,940 the Z-transform is defined this way. 662 00:37:57,940 --> 00:38:05,270 So I want to say that h of z is always z over z minus 7/8. 663 00:38:07,910 --> 00:38:11,410 What happens if I say the region switched? 664 00:38:11,410 --> 00:38:13,160 Well, it says something about convergence. 665 00:38:13,160 --> 00:38:14,618 What's convergence have to do with? 666 00:38:14,618 --> 00:38:20,840 Convergence has to do with, well, I'm thinking about h 667 00:38:20,840 --> 00:38:29,030 of z is some sum over n of h of n, z to the minus n. 668 00:38:29,030 --> 00:38:34,340 So convergent has to do with which ones of those ns 669 00:38:34,340 --> 00:38:39,250 can be in the sum, because some of these terms, z 670 00:38:39,250 --> 00:38:43,060 to the minus n, when I switch the region, 671 00:38:43,060 --> 00:38:46,900 I consider a different family of z. 672 00:38:46,900 --> 00:38:53,780 In the first one, that sum had to converge outside the circle. 673 00:38:53,780 --> 00:38:56,590 And in the question of interest now, it has to converge inside. 674 00:38:56,590 --> 00:38:59,410 So it's a different set of zs for which 675 00:38:59,410 --> 00:39:03,290 the sum has to converge. 676 00:39:03,290 --> 00:39:04,930 There's a way you can think about that. 677 00:39:04,930 --> 00:39:05,910 Think about that sum-- 678 00:39:05,910 --> 00:39:09,389 think about exploding that sum. 679 00:39:09,389 --> 00:39:11,430 In general, we're going to go from minus infinity 680 00:39:11,430 --> 00:39:13,020 to infinity, so explode that. 681 00:39:13,020 --> 00:39:16,290 So we get a whole bunch of stuff. 682 00:39:16,290 --> 00:39:21,780 Then we get up to h of minus 2, z squared. 683 00:39:21,780 --> 00:39:25,890 Then we have h of minus 1, z. 684 00:39:25,890 --> 00:39:30,790 Then we have h of 0, z to the 0, which is 1. 685 00:39:30,790 --> 00:39:36,060 Then we have h of 1, z to the minus 1. 686 00:39:36,060 --> 00:39:40,840 Then we have h of 2, z to the minus 2, et cetera. 687 00:39:40,840 --> 00:39:41,340 Right? 688 00:39:41,340 --> 00:39:44,160 That's what the Z-transform always looks like. 689 00:39:44,160 --> 00:39:49,160 Which of those terms are the most convergent 690 00:39:49,160 --> 00:39:51,760 when I have a z with a large magnitude? 691 00:39:57,030 --> 00:40:00,020 So I'm thinking about the n equals minus 2, 692 00:40:00,020 --> 00:40:02,410 n equals minus 1, n equals 0. 693 00:40:02,410 --> 00:40:05,500 I'm thinking about all the terms and Ys. 694 00:40:05,500 --> 00:40:09,730 Which of those terms is the most convergent 695 00:40:09,730 --> 00:40:14,800 if z has a large magnitude? 696 00:40:14,800 --> 00:40:17,275 AUDIENCE: I think it would be z of negative n. 697 00:40:17,275 --> 00:40:19,227 Like, any negative. 698 00:40:19,227 --> 00:40:21,560 DENNIS FREEMAN: So they get increasingly convergent as I 699 00:40:21,560 --> 00:40:24,360 go to the right. 700 00:40:24,360 --> 00:40:27,600 If I have z with a big magnitude, 701 00:40:27,600 --> 00:40:30,420 each term is getting increasingly convergent 702 00:40:30,420 --> 00:40:33,080 as I go to the right. 703 00:40:33,080 --> 00:40:39,760 That means that these numbers h 0, h 1, h 2, h 3, 704 00:40:39,760 --> 00:40:42,250 they can keep being some finite number. 705 00:40:42,250 --> 00:40:44,230 They don't need to be 0. 706 00:40:44,230 --> 00:40:47,650 And it will increasingly come closer to 0 707 00:40:47,650 --> 00:40:50,180 as I keep going to the right. 708 00:40:50,180 --> 00:40:51,760 The implication of that is something 709 00:40:51,760 --> 00:40:53,630 that's very important. 710 00:40:53,630 --> 00:40:59,485 So the implication of that is that a right-sided signal-- 711 00:41:03,730 --> 00:41:10,180 signal, not single-- has a-- 712 00:41:10,180 --> 00:41:11,710 maps to an outside region. 713 00:41:17,130 --> 00:41:22,720 If I want the sum to get increasingly convergent 714 00:41:22,720 --> 00:41:26,560 for large values of z, outside regions-- 715 00:41:26,560 --> 00:41:28,540 outside regions have to take all the values, 716 00:41:28,540 --> 00:41:31,180 no matter how big they get. 717 00:41:31,180 --> 00:41:32,860 OK, I want things to be on the right. 718 00:41:32,860 --> 00:41:35,710 I don't want things to be on the left. 719 00:41:35,710 --> 00:41:40,650 Things on the left become decreasingly convergent. 720 00:41:40,650 --> 00:41:49,270 And a corollary of that, left-sided signals 721 00:41:49,270 --> 00:41:54,370 map to inside regions. 722 00:41:54,370 --> 00:41:58,000 So in fact, it's not a fluke that this right-handed signal 723 00:41:58,000 --> 00:42:00,370 mapped to this outside region. 724 00:42:00,370 --> 00:42:02,740 That's where the convergence for things to the right 725 00:42:02,740 --> 00:42:05,291 are the best. 726 00:42:05,291 --> 00:42:05,790 OK. 727 00:42:05,790 --> 00:42:07,800 Now I've just told you everything you need to know. 728 00:42:07,800 --> 00:42:09,258 What's it going to-- what do I need 729 00:42:09,258 --> 00:42:13,380 to have happen if I want it to converge to the inside region? 730 00:42:16,540 --> 00:42:17,860 Flip the signal. 731 00:42:17,860 --> 00:42:20,860 More or less, I want it to go to the left. 732 00:42:20,860 --> 00:42:23,880 I don't want it to go to the right. 733 00:42:23,880 --> 00:42:28,400 So the way I can think about that 734 00:42:28,400 --> 00:42:31,850 is by thinking about the functional form 735 00:42:31,850 --> 00:42:38,550 is the same for the inside and the outside region. 736 00:42:38,550 --> 00:42:41,947 That means they have the same difference equation, 737 00:42:41,947 --> 00:42:43,780 because you can find the difference equation 738 00:42:43,780 --> 00:42:46,210 from the functional form. 739 00:42:46,210 --> 00:42:47,970 But the way I want to think about it 740 00:42:47,970 --> 00:42:50,800 is propagating the signal that came in to the right 741 00:42:50,800 --> 00:42:52,450 or to the left. 742 00:42:52,450 --> 00:42:55,390 So rather then iterating forward in time, 743 00:42:55,390 --> 00:42:58,900 which is what we did before, one way I can think about it is, 744 00:42:58,900 --> 00:43:01,690 let's iterate backwards in time. 745 00:43:01,690 --> 00:43:07,270 Rather than solving for y of n plus 1 in terms of y of n, 746 00:43:07,270 --> 00:43:11,050 solve instead for y of n in terms of y of n plus 1. 747 00:43:11,050 --> 00:43:13,180 Run it backwards. 748 00:43:13,180 --> 00:43:15,914 Difference equation doesn't care. 749 00:43:15,914 --> 00:43:17,580 If the difference equation doesn't care, 750 00:43:17,580 --> 00:43:21,180 the functional form will be the same. 751 00:43:21,180 --> 00:43:24,580 So run the difference equation backwards. 752 00:43:24,580 --> 00:43:30,290 So if you think about doing that, rest starts to say, 753 00:43:30,290 --> 00:43:32,670 the system starts-- 754 00:43:32,670 --> 00:43:37,620 starts means future times, OK, because we're flipped. 755 00:43:37,620 --> 00:43:39,000 So the signal starts at 0. 756 00:43:39,000 --> 00:43:39,960 That means it starts-- 757 00:43:39,960 --> 00:43:45,390 that means at large values of n, the signal y is 0. 758 00:43:45,390 --> 00:43:46,890 So I fill in this table with a bunch 759 00:43:46,890 --> 00:43:51,420 of zeros for big values of n. 760 00:43:51,420 --> 00:43:53,640 So n's decreasing this way. 761 00:43:53,640 --> 00:43:55,140 I'm working from toward-- 762 00:43:55,140 --> 00:43:58,340 0 toward the left. 763 00:43:58,340 --> 00:44:00,290 So I assume that I start at rest. 764 00:44:00,290 --> 00:44:03,950 That means the output is 0 for times on the right. 765 00:44:06,880 --> 00:44:10,300 I want to find the unit sample response, so x is 1 only at n 766 00:44:10,300 --> 00:44:13,070 equals 0. 767 00:44:13,070 --> 00:44:16,780 And now I just compute each entry 768 00:44:16,780 --> 00:44:19,610 by sticking it into the difference equation. 769 00:44:19,610 --> 00:44:20,970 And so I've done that. 770 00:44:20,970 --> 00:44:25,060 So you stick these values into the difference equation, 771 00:44:25,060 --> 00:44:29,210 and that lets you compute y of minus 1. 772 00:44:29,210 --> 00:44:31,300 So substitute y of minus 1. 773 00:44:31,300 --> 00:44:37,720 Y of minus 1, it depends on y of 0 and x of 0. 774 00:44:37,720 --> 00:44:40,840 Similarly for all the entries, and then 775 00:44:40,840 --> 00:44:42,910 I can make a plot of that, and I get a function 776 00:44:42,910 --> 00:44:43,800 that looks like this. 777 00:44:47,250 --> 00:44:48,130 It's a geometric. 778 00:44:48,130 --> 00:44:50,590 I'm not surprised by that. 779 00:44:50,590 --> 00:44:53,230 The geometric is kind of facing the wrong way. 780 00:44:53,230 --> 00:44:54,750 It was 7/8 to the end. 781 00:44:54,750 --> 00:44:57,130 It looked convergent. 782 00:44:57,130 --> 00:45:00,210 This one looks divergent that way, 783 00:45:00,210 --> 00:45:04,240 but I'm multiplying these h numbers, which 784 00:45:04,240 --> 00:45:08,110 are blowing up that way, by numbers 785 00:45:08,110 --> 00:45:09,739 that look like z squared. 786 00:45:09,739 --> 00:45:12,155 So if I make z squared small enough, it'll still converge. 787 00:45:15,320 --> 00:45:17,140 That's the idea. 788 00:45:17,140 --> 00:45:17,770 OK? 789 00:45:17,770 --> 00:45:20,290 By flipping the region of convergence, 790 00:45:20,290 --> 00:45:25,000 I've changed the magnitudes of these numbers, 791 00:45:25,000 --> 00:45:27,730 and I've changed which side converges best. 792 00:45:27,730 --> 00:45:30,600 I've made something that-- 793 00:45:30,600 --> 00:45:31,990 it's an inside region. 794 00:45:31,990 --> 00:45:36,280 It was converging for z values inside some region, 795 00:45:36,280 --> 00:45:38,440 so it's become left-sided. 796 00:45:38,440 --> 00:45:44,590 So the idea then is that I have two different kinds of time 797 00:45:44,590 --> 00:45:48,520 signals that are both associated with the same functional form 798 00:45:48,520 --> 00:45:49,270 of z. 799 00:45:49,270 --> 00:45:51,580 They differ by the region of convergence. 800 00:45:51,580 --> 00:45:54,220 That's why when you tell me a Z-transform, 801 00:45:54,220 --> 00:45:57,250 you have to tell me the region of convergence. 802 00:45:57,250 --> 00:45:57,750 OK? 803 00:46:00,770 --> 00:46:02,140 OK. 804 00:46:02,140 --> 00:46:04,480 So I've given some exercises. 805 00:46:04,480 --> 00:46:06,530 I'm running out of time. 806 00:46:06,530 --> 00:46:11,494 The best thing to do is to think about the exercises offline. 807 00:46:11,494 --> 00:46:12,910 I could lead you through the math. 808 00:46:12,910 --> 00:46:16,670 That's never very inspirational. 809 00:46:16,670 --> 00:46:20,880 So what I wanted to show you is that there's many ways 810 00:46:20,880 --> 00:46:26,800 that you can go about expressing a complicated form 811 00:46:26,800 --> 00:46:31,990 by partial fractions to get a simpler form that can then 812 00:46:31,990 --> 00:46:36,160 be inverted this way. 813 00:46:36,160 --> 00:46:38,560 So that's the point of this exercise. 814 00:46:38,560 --> 00:46:42,070 So I went through in the notes, and the notes are online. 815 00:46:42,070 --> 00:46:44,580 So in the notes, I went through three different ways 816 00:46:44,580 --> 00:46:48,000 you can think of the answer, and all of those answers 817 00:46:48,000 --> 00:46:50,760 give you the same functional form, right? 818 00:46:50,760 --> 00:46:53,370 So it was an exercise in thinking through how 819 00:46:53,370 --> 00:46:55,680 you do partial fractions. 820 00:46:55,680 --> 00:46:58,320 But there's one more thing that I want to talk about, 821 00:46:58,320 --> 00:47:01,860 and that is that a lot of these problems-- one of the biggest 822 00:47:01,860 --> 00:47:05,700 uses of Z-transforms is to solve difference equations. 823 00:47:08,290 --> 00:47:09,880 Z-transforms are great for that. 824 00:47:09,880 --> 00:47:11,500 I've already talked about that. 825 00:47:11,500 --> 00:47:14,170 With-- by using a Z-transform, you can take a difference 826 00:47:14,170 --> 00:47:17,680 equation, think about the difference equation, 827 00:47:17,680 --> 00:47:20,650 think about the input, take the Laplace transform of everything 828 00:47:20,650 --> 00:47:22,260 you get-- 829 00:47:22,260 --> 00:47:25,120 Laplace transform-- a Z-transform, sorry. 830 00:47:25,120 --> 00:47:25,810 Slipped. 831 00:47:25,810 --> 00:47:28,370 Next time it will be Laplace transform. 832 00:47:28,370 --> 00:47:29,800 Take a Z-transform. 833 00:47:29,800 --> 00:47:32,500 You end up with a Z-transform, and then the trick 834 00:47:32,500 --> 00:47:37,270 is to recognize the inverse Z-transform. 835 00:47:37,270 --> 00:47:44,680 There are ways of thinking about that as a mathematician. 836 00:47:44,680 --> 00:47:50,430 Those ways are not easy, so by and large, we 837 00:47:50,430 --> 00:47:52,410 will never do this. 838 00:47:52,410 --> 00:47:56,730 If you would like to do that, I highly recommend course 18. 839 00:47:56,730 --> 00:47:57,660 OK? 840 00:47:57,660 --> 00:47:59,940 It is certainly something that people in course 18 841 00:47:59,940 --> 00:48:03,210 do all the time, but there are always simpler ways 842 00:48:03,210 --> 00:48:06,270 that we will do it, and those simpler ways 843 00:48:06,270 --> 00:48:10,440 derive from thinking about properties of the z-transform. 844 00:48:10,440 --> 00:48:13,300 And we'll think more about those as we go forward in the course. 845 00:48:13,300 --> 00:48:16,350 So the point of today was to emphasize 846 00:48:16,350 --> 00:48:19,140 that there are lots of different ways of thinking 847 00:48:19,140 --> 00:48:21,540 about DT systems. 848 00:48:21,540 --> 00:48:24,270 You should be able to think of all the relations between them. 849 00:48:24,270 --> 00:48:25,800 And in particular, today we talked 850 00:48:25,800 --> 00:48:28,560 about this thing, a mathematical relationship 851 00:48:28,560 --> 00:48:31,320 for how you can go from a unit sample response 852 00:48:31,320 --> 00:48:32,250 to a system function. 853 00:48:35,850 --> 00:48:38,000 See you next week.