1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,690 continue to offer high-quality educational resources for free. 5 00:00:10,690 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,318 at ocw.mit.edu. 8 00:00:26,403 --> 00:00:28,570 PROFESSOR: Thank you for coming out here in the rain 9 00:00:28,570 --> 00:00:33,800 and the day before a quiz, but this is stuff we need to know. 10 00:00:33,800 --> 00:00:38,410 So I'm going to be talking about a powerful class of models 11 00:00:38,410 --> 00:00:40,780 for communication channels. 12 00:00:40,780 --> 00:00:44,630 We've already seen the kind of setup that we're talking about. 13 00:00:44,630 --> 00:00:46,300 And so what we're looking to do is 14 00:00:46,300 --> 00:00:51,310 model a channel between this point xn, 15 00:00:51,310 --> 00:00:53,860 I think my pointer is-- 16 00:00:53,860 --> 00:00:54,760 OK, there we go. 17 00:00:54,760 --> 00:00:58,540 Between xn and yn out there, so what 18 00:00:58,540 --> 00:01:01,600 we refer to as the baseband channel. 19 00:01:01,600 --> 00:01:09,640 So we've got xn coming in, various things being done 20 00:01:09,640 --> 00:01:12,310 to it, and then yn coming out. 21 00:01:17,590 --> 00:01:21,430 And we refer to this as the baseband channel. 22 00:01:26,050 --> 00:01:28,570 So what's happening in here is things like-- 23 00:01:28,570 --> 00:01:33,520 let's see-- D to A conversion, Digital to Analog. 24 00:01:33,520 --> 00:01:35,740 And then there is the modulation. 25 00:01:40,640 --> 00:01:42,265 And then there is the physical channel. 26 00:01:47,500 --> 00:01:49,570 I may not have left enough space in this box. 27 00:01:49,570 --> 00:01:59,590 But here is the demodulation and whatever filtering, 28 00:01:59,590 --> 00:02:03,910 demodulation and filtering that happens in there. 29 00:02:03,910 --> 00:02:05,990 So there is distortion-- oh, I'm sorry. 30 00:02:05,990 --> 00:02:09,130 I forgot the A to D, didn't I? 31 00:02:09,130 --> 00:02:11,940 So let's stick that back in here. 32 00:02:14,700 --> 00:02:17,860 We're doing all our demodulation and filtering in discrete time, 33 00:02:17,860 --> 00:02:20,220 so we have an A to D converter here, and then 34 00:02:20,220 --> 00:02:27,180 D mod and filtering. 35 00:02:27,180 --> 00:02:29,260 And there is various places here that you 36 00:02:29,260 --> 00:02:30,630 can get distortion and noise. 37 00:02:30,630 --> 00:02:34,860 So for instance, the physical channel is a source of noise. 38 00:02:34,860 --> 00:02:36,810 But the discrete time operations as well, 39 00:02:36,810 --> 00:02:39,332 the computational pieces can also introduce noise. 40 00:02:39,332 --> 00:02:41,790 You could have numerical noise, because you're sounding off 41 00:02:41,790 --> 00:02:43,660 numbers, and so on. 42 00:02:43,660 --> 00:02:46,290 So there are various places that noise can originate. 43 00:02:46,290 --> 00:02:50,100 And there are various places that distortion of the signal 44 00:02:50,100 --> 00:02:50,980 can originate. 45 00:02:50,980 --> 00:02:54,690 So in the filtering process, for instance, 46 00:02:54,690 --> 00:02:59,490 or the channel process, you can get 47 00:02:59,490 --> 00:03:02,790 phenomena that will take what started out as a straight edge 48 00:03:02,790 --> 00:03:10,530 here and cause it to now get a little bit spread out and not 49 00:03:10,530 --> 00:03:11,560 so clean at the edge. 50 00:03:14,600 --> 00:03:17,005 OK, so that's what we refer to as a distortion. 51 00:03:17,005 --> 00:03:18,630 So there is all sorts of things in here 52 00:03:18,630 --> 00:03:20,010 that can account for that. 53 00:03:20,010 --> 00:03:21,630 Now, when we say baseband channel, 54 00:03:21,630 --> 00:03:24,600 we're actually trying to distinguish it from the channel 55 00:03:24,600 --> 00:03:28,050 that you see after the modulation. 56 00:03:28,050 --> 00:03:30,480 So once you've modulated, you typically 57 00:03:30,480 --> 00:03:32,880 move things to some other frequency range. 58 00:03:32,880 --> 00:03:35,700 And so the actual transmission across the physical channel 59 00:03:35,700 --> 00:03:38,260 happens in some other frequency range. 60 00:03:38,260 --> 00:03:40,860 And so the word "baseband" here is 61 00:03:40,860 --> 00:03:44,220 used to distinguish the channel that we're talking 62 00:03:44,220 --> 00:03:47,133 about from that channel. 63 00:03:47,133 --> 00:03:49,050 So this is what we're going to be focusing on. 64 00:03:49,050 --> 00:03:52,920 And then will later come back to talking about the modulation 65 00:03:52,920 --> 00:03:55,140 and demodulation pieces. 66 00:03:55,140 --> 00:04:00,090 So last time, I introduced a way to represent 67 00:04:00,090 --> 00:04:05,850 such models just as systems with an input and an output. 68 00:04:05,850 --> 00:04:08,730 One thing I made a point of saying 69 00:04:08,730 --> 00:04:13,917 was that when we look at a figure like this-- 70 00:04:13,917 --> 00:04:14,625 here is a system. 71 00:04:17,459 --> 00:04:20,670 We've got some input sequence that's actually going in 72 00:04:20,670 --> 00:04:23,680 and maps to some output sequence. 73 00:04:23,680 --> 00:04:25,140 So I use this notation with a dot 74 00:04:25,140 --> 00:04:28,680 there to indicate the entire time function. 75 00:04:28,680 --> 00:04:31,410 So I've got some entire time function here 76 00:04:31,410 --> 00:04:35,045 that goes through the system and gets mapped to some entire time 77 00:04:35,045 --> 00:04:35,670 function there. 78 00:04:35,670 --> 00:04:37,128 And I'm not telling you the details 79 00:04:37,128 --> 00:04:39,270 of how that mapping happens yet, but this 80 00:04:39,270 --> 00:04:40,680 is my abstract picture. 81 00:04:40,680 --> 00:04:44,826 Now, in many places you'll see people writing-- 82 00:04:44,826 --> 00:04:46,620 and again, I said this last time. 83 00:04:46,620 --> 00:04:49,050 But I want to remind you, you'll see 84 00:04:49,050 --> 00:04:52,980 them labeling xn going into the system and yn coming out. 85 00:04:52,980 --> 00:04:54,390 And when you see that, you've got 86 00:04:54,390 --> 00:04:55,890 to think that what you're looking at 87 00:04:55,890 --> 00:04:57,720 is just a snapshot at time n. 88 00:04:57,720 --> 00:05:07,500 So this picture is what you get in a snapshot at time n, 89 00:05:07,500 --> 00:05:09,360 whereas this picture is the picture that 90 00:05:09,360 --> 00:05:16,320 refers to actually mapping the input signal, the entire input 91 00:05:16,320 --> 00:05:21,780 signal to the output signal. 92 00:05:26,500 --> 00:05:31,230 OK, so these are two different ways of representing things. 93 00:05:33,840 --> 00:05:36,640 In this system, I'm not taking the value of time n 94 00:05:36,640 --> 00:05:38,310 and producing a value at time n. 95 00:05:38,310 --> 00:05:41,220 I typically will need to look at lots of values of the input 96 00:05:41,220 --> 00:05:43,920 to figure out any particular value of the output. 97 00:05:48,000 --> 00:05:53,130 All right, I did mention briefly the notion of causality. 98 00:05:53,130 --> 00:05:54,910 And we'll come back to that later. 99 00:05:54,910 --> 00:05:57,090 But the rough notion is that-- 100 00:05:57,090 --> 00:06:00,960 or a good enough notion is that the system is called causal 101 00:06:00,960 --> 00:06:03,480 if the response at any time depends only 102 00:06:03,480 --> 00:06:07,590 on present and past inputs and not on future inputs. 103 00:06:07,590 --> 00:06:08,550 That's easy enough. 104 00:06:12,600 --> 00:06:13,800 And then there were-- 105 00:06:13,800 --> 00:06:15,600 we were going to specialize, actually, 106 00:06:15,600 --> 00:06:18,410 to the case of linear and time-invariant systems. 107 00:06:18,410 --> 00:06:22,770 And so I want to first introduce the notion of time invariance. 108 00:06:22,770 --> 00:06:25,740 Time invariance says basically that, if you shift the input 109 00:06:25,740 --> 00:06:28,938 by a certain amount, then the output gets just shifted 110 00:06:28,938 --> 00:06:29,730 by the same amount. 111 00:06:29,730 --> 00:06:32,797 But the same input-output pair works as before. 112 00:06:32,797 --> 00:06:34,380 So what you're really trying to get at 113 00:06:34,380 --> 00:06:39,060 is a time-invariant system is one where the laws by which you 114 00:06:39,060 --> 00:06:42,360 compose the values of the input to get the output 115 00:06:42,360 --> 00:06:45,740 don't change with time. 116 00:06:45,740 --> 00:06:47,500 So let's see. 117 00:06:47,500 --> 00:06:50,600 Let me give you an example here. 118 00:06:50,600 --> 00:06:55,630 Suppose I had a system whose input and output 119 00:06:55,630 --> 00:06:57,000 were related in this fashion. 120 00:07:02,750 --> 00:07:05,600 Would that, do you think, be a time-invariant system 121 00:07:05,600 --> 00:07:06,790 or a time-varying system? 122 00:07:14,020 --> 00:07:16,810 I seem to have functions of time in here. 123 00:07:16,810 --> 00:07:18,520 Does that make it a time-varying system? 124 00:07:18,520 --> 00:07:21,070 Or is it perhaps time invariant? 125 00:07:21,070 --> 00:07:21,885 Yeah? 126 00:07:21,885 --> 00:07:25,280 AUDIENCE: Time invariant, because of the law [INAUDIBLE].. 127 00:07:28,115 --> 00:07:29,490 PROFESSOR: OK, so time invariant, 128 00:07:29,490 --> 00:07:31,730 because the law by which you're composing things 129 00:07:31,730 --> 00:07:34,470 to get the output doesn't depend on time. 130 00:07:34,470 --> 00:07:39,515 So the point is that these coefficients are constant. 131 00:07:45,950 --> 00:07:47,630 So because these are constant, what you 132 00:07:47,630 --> 00:07:50,330 have is actually a time-invariant system. 133 00:07:50,330 --> 00:07:52,700 So to get the output at any time, 134 00:07:52,700 --> 00:07:55,490 you're taking 1/3 of the output of the previous time 135 00:07:55,490 --> 00:07:58,010 plus twice the input of the present time. 136 00:07:58,010 --> 00:08:01,350 And that prescription holds along the entire time axis. 137 00:08:01,350 --> 00:08:05,010 So the actual value of n doesn't matter. 138 00:08:05,010 --> 00:08:06,980 But if I had here some function of n, 139 00:08:06,980 --> 00:08:13,730 if, instead of 1/3, I had something like 1/3 to the n, 140 00:08:13,730 --> 00:08:16,190 now I've got a time-varying system, 141 00:08:16,190 --> 00:08:19,100 because the law by which I combine things actually 142 00:08:19,100 --> 00:08:21,320 depends on my position along the time axis. 143 00:08:26,650 --> 00:08:28,780 So this would be time invariant. 144 00:08:28,780 --> 00:08:30,640 This would be not. 145 00:08:35,669 --> 00:08:38,010 So that's what this is trying to get at. 146 00:08:38,010 --> 00:08:38,580 Easy enough. 147 00:08:41,539 --> 00:08:45,520 The other notion was that of linearity. 148 00:08:45,520 --> 00:08:47,092 By the way, if you read the chapter, 149 00:08:47,092 --> 00:08:48,550 you'll see some other examples that 150 00:08:48,550 --> 00:08:51,520 will help you hone your intuition for what's 151 00:08:51,520 --> 00:08:52,840 time invariant and what's not. 152 00:08:55,960 --> 00:08:57,910 For linearity, the basic idea was 153 00:08:57,910 --> 00:09:01,990 that you can superpose inputs and find 154 00:09:01,990 --> 00:09:04,940 the corresponding responses by superposition. 155 00:09:04,940 --> 00:09:07,600 So if you've got the results of two experiments, 156 00:09:07,600 --> 00:09:10,930 the input in one experiment and the output, the input 157 00:09:10,930 --> 00:09:13,192 in a second experiment and the output, 158 00:09:13,192 --> 00:09:15,400 and then you take a new experiment in which the input 159 00:09:15,400 --> 00:09:18,250 is a linear combination of the previous two ones, 160 00:09:18,250 --> 00:09:22,060 the response will be the same linear combination 161 00:09:22,060 --> 00:09:24,550 of the previous two responses. 162 00:09:24,550 --> 00:09:26,350 So that's the basic idea here. 163 00:09:26,350 --> 00:09:30,800 So linearity means that superposition works. 164 00:09:30,800 --> 00:09:33,460 And so this is another feature that we'll use. 165 00:09:33,460 --> 00:09:39,670 And for this example on top, do you think it's linear or not? 166 00:09:45,590 --> 00:09:46,750 So what you really-- 167 00:09:46,750 --> 00:09:48,670 the way to think about it is, suppose 168 00:09:48,670 --> 00:09:51,970 I had an experiment A in which my output was 169 00:09:51,970 --> 00:09:57,610 y, in which I fed in xA, some time signal, and I got 170 00:09:57,610 --> 00:10:01,520 a response, some time signal. 171 00:10:01,520 --> 00:10:03,940 So what that means is that this is true. 172 00:10:08,860 --> 00:10:11,200 This is what it means to say that this 173 00:10:11,200 --> 00:10:13,690 is an input-output pair in experiment A. 174 00:10:13,690 --> 00:10:16,330 And now in experiment B, similarly, I 175 00:10:16,330 --> 00:10:25,465 have yB n satisfying this equation. 176 00:10:28,660 --> 00:10:30,490 So the subscript here just means experiment 177 00:10:30,490 --> 00:10:43,222 A, experiment B. So this is an experiment A and experiment B. 178 00:10:43,222 --> 00:10:44,680 So now the question you want to ask 179 00:10:44,680 --> 00:10:51,790 yourself is, is it true that, if I defined a new input xn to be, 180 00:10:51,790 --> 00:10:52,990 let's say-- 181 00:10:52,990 --> 00:10:56,350 what notation did I use there? 182 00:10:56,350 --> 00:10:58,180 Well, I didn't want an A and a B, did I? 183 00:11:00,840 --> 00:11:03,120 OK, if you ignore the notation on my slides, 184 00:11:03,120 --> 00:11:14,460 let's say that this is an alpha x A plus beta x B. 185 00:11:14,460 --> 00:11:16,170 OK, so here is a new experiment in which 186 00:11:16,170 --> 00:11:19,260 I'm going to use an input that's a linear combination 187 00:11:19,260 --> 00:11:23,880 of the previous two inputs with some arbitrary weights alpha 188 00:11:23,880 --> 00:11:25,110 and beta. 189 00:11:25,110 --> 00:11:30,660 And the question then is, is the corresponding combination 190 00:11:30,660 --> 00:11:33,330 of the outputs in the previous experiment, 191 00:11:33,330 --> 00:11:44,580 so alpha yA n plus beta yB n, does this x and y end pair 192 00:11:44,580 --> 00:11:47,300 satisfy the same equation? 193 00:11:47,300 --> 00:11:52,000 OK, so what we want to check now is, is it true-- 194 00:11:52,000 --> 00:11:55,080 well, is it true that the xn here, 195 00:11:55,080 --> 00:11:58,530 the yn here will satisfy the equation on top? 196 00:11:58,530 --> 00:12:01,540 And you can see very quickly that it will. 197 00:12:01,540 --> 00:12:04,260 And the reason is that yn here is 198 00:12:04,260 --> 00:12:07,860 expressed as a linear function of the yn minus 1 199 00:12:07,860 --> 00:12:08,940 and xn minus 1. 200 00:12:08,940 --> 00:12:10,800 So when you substitute these in, you'll 201 00:12:10,800 --> 00:12:14,610 find that xn defined this way and yn defined this way will 202 00:12:14,610 --> 00:12:17,760 actually satisfy that equation. 203 00:12:17,760 --> 00:12:20,990 So this is what superposition requires you to test. 204 00:12:20,990 --> 00:12:23,920 So if it's true for every possible pair of experiments 205 00:12:23,920 --> 00:12:29,110 here and every pair of weights alpha and beta 206 00:12:29,110 --> 00:12:33,407 that the superposition satisfies the equations governing 207 00:12:33,407 --> 00:12:35,490 the system, then what you have is a linear system. 208 00:12:41,120 --> 00:12:45,770 What about if I had to change this to 1/3 to the power n? 209 00:12:45,770 --> 00:12:49,085 So I had a time-varying expression of this type. 210 00:12:49,085 --> 00:12:51,380 So I have a time-varying system, do you 211 00:12:51,380 --> 00:12:54,470 think this system would still be linear? 212 00:12:57,225 --> 00:12:58,600 So if you work through it, you'll 213 00:12:58,600 --> 00:13:01,520 see, for the same reason, that superposition still works. 214 00:13:01,520 --> 00:13:04,933 So if I had 1/3 to the n there instead of 1/3, 215 00:13:04,933 --> 00:13:06,850 I get a time-varying system, but I could still 216 00:13:06,850 --> 00:13:07,810 superimpose solutions. 217 00:13:07,810 --> 00:13:09,520 It would be a linear time-varying system. 218 00:13:12,930 --> 00:13:16,230 Now, we don't want to spend too much time 219 00:13:16,230 --> 00:13:19,620 teasing all these apart, because what we'll be focused on 220 00:13:19,620 --> 00:13:21,480 is linear and time-invariant systems. 221 00:13:21,480 --> 00:13:25,530 And you'll actually come quickly to recognize them. 222 00:13:25,530 --> 00:13:31,980 OK, I defined last time also a pair of special signals 223 00:13:31,980 --> 00:13:34,800 which you've seen before, the unit sample signal 224 00:13:34,800 --> 00:13:38,100 which has the value 1 just at one point and the unit step 225 00:13:38,100 --> 00:13:39,790 signal. 226 00:13:39,790 --> 00:13:42,670 So let me just sketch them out for you here. 227 00:13:42,670 --> 00:13:52,180 So the unit sample, this is a signal delta n 228 00:13:52,180 --> 00:13:53,590 which is an entire signal. 229 00:13:53,590 --> 00:13:56,410 It's not just the number 1 at time 0. 230 00:13:56,410 --> 00:13:58,730 It's the entire signal. 231 00:13:58,730 --> 00:14:01,000 That's the unit sample function. 232 00:14:01,000 --> 00:14:03,320 There is another notation that's also sometimes used, 233 00:14:03,320 --> 00:14:06,945 which is delta sub 0 and dot. 234 00:14:06,945 --> 00:14:11,540 So this notation is a little bit more evocative of a function, 235 00:14:11,540 --> 00:14:13,240 whereas here, you are often tempted 236 00:14:13,240 --> 00:14:15,400 to think of it as a number. 237 00:14:15,400 --> 00:14:18,260 This says, what I'm looking at is a function. 238 00:14:18,260 --> 00:14:20,860 It's a unit sample function. 239 00:14:20,860 --> 00:14:23,380 And the 1 is at the value 0. 240 00:14:26,050 --> 00:14:29,830 So if you had delta of n minus 3, 241 00:14:29,830 --> 00:14:34,060 that would be this function shifted from 0 to 1, 2, 3. 242 00:14:34,060 --> 00:14:38,110 So the 1, value 1 would sit at time 3. 243 00:14:38,110 --> 00:14:42,940 Another notation for that would have been this. 244 00:14:42,940 --> 00:14:44,830 Sometimes that notation is useful also 245 00:14:44,830 --> 00:14:49,630 in making sense of expressions that you're looking at. 246 00:14:49,630 --> 00:14:54,550 OK, so this was the unit sample function. 247 00:14:54,550 --> 00:15:05,360 And then the unit step function steps up from 0 to 1 at time 0. 248 00:15:12,450 --> 00:15:13,700 That's the unit step function. 249 00:15:19,730 --> 00:15:23,930 And we also talked about the response to these two inputs. 250 00:15:23,930 --> 00:15:25,620 So you see them up there. 251 00:15:25,620 --> 00:15:33,290 And now my question is, if a unit sample signal at the input 252 00:15:33,290 --> 00:15:43,390 produces the unit sample response hn at the output 253 00:15:43,390 --> 00:15:48,110 and un produces the step response, 254 00:15:48,110 --> 00:15:52,570 and if what you have is an LTI system in here-- 255 00:15:52,570 --> 00:15:55,240 so it's the same LTI system that we're talking about-- 256 00:15:55,240 --> 00:15:57,410 can you actually relate the two? 257 00:15:57,410 --> 00:16:01,780 So the question is, can you relate the unit sample 258 00:16:01,780 --> 00:16:04,150 response and the step response? 259 00:16:04,150 --> 00:16:06,430 Do I need to give you both if I have an LTI system? 260 00:16:06,430 --> 00:16:07,888 Or does it suffice to give you one? 261 00:16:13,710 --> 00:16:15,750 So here is one way to think of that. 262 00:16:15,750 --> 00:16:17,530 This, by the way, is the same LTI system. 263 00:16:17,530 --> 00:16:23,190 Maybe I should indicate that more explicitly by, 264 00:16:23,190 --> 00:16:27,570 let's say, it's a specific system, system zero. 265 00:16:27,570 --> 00:16:29,790 And with the same system, I'm trying 266 00:16:29,790 --> 00:16:34,603 to deduce the results of another experiment. 267 00:16:40,030 --> 00:16:41,640 So if we're thinking superposition, 268 00:16:41,640 --> 00:16:45,810 can you tell me how to write the unit sample 269 00:16:45,810 --> 00:16:52,800 function as a linear combination of unit step functions, 270 00:16:52,800 --> 00:16:54,390 maybe delayed unit step functions, 271 00:16:54,390 --> 00:16:55,860 scaled unit step functions? 272 00:16:55,860 --> 00:16:56,836 Yeah? 273 00:16:56,836 --> 00:16:59,276 AUDIENCE: [INAUDIBLE] 274 00:17:02,700 --> 00:17:06,644 PROFESSOR: OK, would it be-- 275 00:17:06,644 --> 00:17:12,960 you said un minus un plus 1? 276 00:17:12,960 --> 00:17:15,210 Or is it un minus 1? 277 00:17:15,210 --> 00:17:15,990 n minus 1? 278 00:17:18,579 --> 00:17:22,119 So what we're saying is, take this unit step 279 00:17:22,119 --> 00:17:26,950 and then subtract from it a unit step delayed by 1. 280 00:17:29,500 --> 00:17:33,310 OK, so here is u of n minus 1. 281 00:17:33,310 --> 00:17:35,560 If we took the unit step and subtracted from it 282 00:17:35,560 --> 00:17:38,830 a delayed unit step, delayed by 1, 283 00:17:38,830 --> 00:17:42,760 the result will be just that value 1 at time 0 will survive. 284 00:17:42,760 --> 00:17:44,530 Everything else will cancel out. 285 00:17:44,530 --> 00:17:45,950 Is that what you had in mind? 286 00:17:45,950 --> 00:17:47,020 OK. 287 00:17:47,020 --> 00:17:51,310 So if delta of n can be written as that linear combination 288 00:17:51,310 --> 00:17:54,910 of unit steps, can you tell me how 289 00:17:54,910 --> 00:18:03,050 to write hn in terms of unit step responses? 290 00:18:03,050 --> 00:18:04,760 We're talking about an LTI system. 291 00:18:07,550 --> 00:18:10,830 I took that out, but we're still talking about an LTI system 292 00:18:10,830 --> 00:18:11,330 here. 293 00:18:15,260 --> 00:18:16,760 Somebody who hasn't spoken maybe? 294 00:18:19,640 --> 00:18:20,300 Yeah? 295 00:18:20,300 --> 00:18:23,390 AUDIENCE: It should just be x of n minus s of n minus 1. 296 00:18:23,390 --> 00:18:24,620 PROFESSOR: Yeah, OK. 297 00:18:24,620 --> 00:18:27,560 So superposition says that, if you've 298 00:18:27,560 --> 00:18:30,860 got an input that's a linear combination of inputs 299 00:18:30,860 --> 00:18:33,230 for which you know the results of the experiment, 300 00:18:33,230 --> 00:18:36,560 then the corresponding output is the same linear combination 301 00:18:36,560 --> 00:18:38,310 of the outputs for that experiment. 302 00:18:38,310 --> 00:18:43,100 So this is going to be sn minus s n minus 1. 303 00:18:48,390 --> 00:18:50,280 So you can actually deduce the unit sample 304 00:18:50,280 --> 00:18:54,280 response, given the unit setup response for an LTI system. 305 00:18:54,280 --> 00:18:55,050 So let's see. 306 00:18:55,050 --> 00:18:56,310 We've used linearity. 307 00:18:56,310 --> 00:18:59,010 Have we use time invariance? 308 00:18:59,010 --> 00:19:00,780 We used linearity because we said, 309 00:19:00,780 --> 00:19:03,570 here is an experiment in which the input 310 00:19:03,570 --> 00:19:06,150 is a linear combination of inputs 311 00:19:06,150 --> 00:19:07,950 that we know the responses to. 312 00:19:07,950 --> 00:19:10,175 Where have we invoked time invariance? 313 00:19:17,930 --> 00:19:18,430 Anyone? 314 00:19:21,520 --> 00:19:24,820 The superposition idea was part of the definition of linearity. 315 00:19:24,820 --> 00:19:28,030 Because a system is linear, if the input is 316 00:19:28,030 --> 00:19:31,060 a superposition of two inputs for which you know 317 00:19:31,060 --> 00:19:34,780 the response, then the output is the corresponding superposition 318 00:19:34,780 --> 00:19:36,550 of the responses. 319 00:19:36,550 --> 00:19:40,210 That seems like I've only used superposition there. 320 00:19:40,210 --> 00:19:42,100 Have I actually use time invariance as well? 321 00:19:42,100 --> 00:19:44,460 Yeah? 322 00:19:44,460 --> 00:19:44,960 Sorry? 323 00:19:44,960 --> 00:19:47,850 AUDIENCE: [INAUDIBLE] 324 00:19:47,850 --> 00:19:50,220 PROFESSOR: I've used it in concluding 325 00:19:50,220 --> 00:19:52,770 that, if I put in u of n minus 1, 326 00:19:52,770 --> 00:19:55,380 the response is s of n minus 1. 327 00:19:55,380 --> 00:19:58,680 So I've used time invariance as well as linearity 328 00:19:58,680 --> 00:20:01,230 here to come up with this statement. 329 00:20:01,230 --> 00:20:03,990 OK, good. 330 00:20:03,990 --> 00:20:07,860 So this is what I have on the slide. 331 00:20:07,860 --> 00:20:09,810 And you've figured it all out already. 332 00:20:09,810 --> 00:20:13,290 We've arrived at this equation. 333 00:20:13,290 --> 00:20:16,620 Now, if I want to turn it around and write sn 334 00:20:16,620 --> 00:20:18,540 in terms of the unit sample response, 335 00:20:18,540 --> 00:20:20,940 I can do that as well, except this 336 00:20:20,940 --> 00:20:23,640 is analogous to integrating a differential equation. 337 00:20:23,640 --> 00:20:25,890 What we have is a difference equation here. 338 00:20:25,890 --> 00:20:28,050 And when you come to integrate, well, 339 00:20:28,050 --> 00:20:29,940 in discrete time, what you do is summation 340 00:20:29,940 --> 00:20:31,500 instead of integration. 341 00:20:31,500 --> 00:20:34,480 You need to assume an initial condition of some kind. 342 00:20:34,480 --> 00:20:35,880 And so it turns out if you assume 343 00:20:35,880 --> 00:20:39,210 that, way back in the past, the value of the step response 344 00:20:39,210 --> 00:20:45,600 was 0, then you can actually go from this description 345 00:20:45,600 --> 00:20:48,780 to a description the other way, relating the step response 346 00:20:48,780 --> 00:20:50,490 to the unit sample response. 347 00:20:50,490 --> 00:20:55,590 OK, so if I have a causal system, for instance, so 348 00:20:55,590 --> 00:20:58,350 the causal system, it's got no response 349 00:20:58,350 --> 00:20:59,880 until the input hits it. 350 00:20:59,880 --> 00:21:02,940 So when I put a unit step in, I'm 351 00:21:02,940 --> 00:21:04,870 not going to get a response until time 0. 352 00:21:04,870 --> 00:21:08,610 And so I know at minus infinity, the step response was 0. 353 00:21:08,610 --> 00:21:11,550 And I can move forward from there. 354 00:21:11,550 --> 00:21:15,180 OK, so you can actually relate the step response 355 00:21:15,180 --> 00:21:17,520 to the unit sample response the other way 356 00:21:17,520 --> 00:21:20,040 as well here, where the summation is 357 00:21:20,040 --> 00:21:24,390 from minus infinity to the n that you're interested in. 358 00:21:24,390 --> 00:21:27,885 We'll be dealing right through with causal systems. 359 00:21:30,390 --> 00:21:32,980 If there are any deviations from that, we'll point them out. 360 00:21:32,980 --> 00:21:36,090 But basically, we'll be dealing with causal systems. 361 00:21:36,090 --> 00:21:40,530 OK, so let's-- this is an identity we'll be wanting 362 00:21:40,530 --> 00:21:42,030 to play with a bit. 363 00:21:45,770 --> 00:21:47,110 So let me put it up here. 364 00:21:47,110 --> 00:21:53,380 So the step response, let's say, for a causal system 365 00:21:53,380 --> 00:21:57,700 is going to be summation from k equals minus infinity 366 00:21:57,700 --> 00:22:01,510 to n h of k. 367 00:22:01,510 --> 00:22:05,380 So I take all the values of the unit sample response 368 00:22:05,380 --> 00:22:07,812 up to the present time and sum them together 369 00:22:07,812 --> 00:22:10,270 to get the value of the setup response at the present time. 370 00:22:13,430 --> 00:22:16,930 OK, so let's look at an example here. 371 00:22:16,930 --> 00:22:23,090 Here is the unit sample response of a particular LTI system. 372 00:22:23,090 --> 00:22:25,940 Is this a causal system? 373 00:22:31,050 --> 00:22:33,230 So this is the response to a unit sample. 374 00:22:33,230 --> 00:22:37,610 So the input was 0 everywhere except for a value of 1 here. 375 00:22:37,610 --> 00:22:40,100 And you see that the response actually happens 376 00:22:40,100 --> 00:22:41,700 subsequent to that input. 377 00:22:41,700 --> 00:22:44,090 So if the response starts at time 0 378 00:22:44,090 --> 00:22:47,780 or later for an input that started at time 0 or later, 379 00:22:47,780 --> 00:22:51,110 then what you are looking at is a causal system here, 380 00:22:51,110 --> 00:22:54,200 certainly in the case of a unit sample response. 381 00:22:54,200 --> 00:22:58,220 OK, so what's the step response going to look like, then? 382 00:23:04,510 --> 00:23:06,960 Anyone want to say in words? 383 00:23:06,960 --> 00:23:07,460 Yeah? 384 00:23:07,460 --> 00:23:12,810 AUDIENCE: [INAUDIBLE] 385 00:23:12,810 --> 00:23:14,820 PROFESSOR: OK, so the step response, 386 00:23:14,820 --> 00:23:16,800 if we're evaluating it at times over here, 387 00:23:16,800 --> 00:23:19,980 we're summing all the values of hk from minus infinity 388 00:23:19,980 --> 00:23:21,410 up to the present time. 389 00:23:21,410 --> 00:23:24,240 So the step response is 0 here, is 0 here, is 0 here. 390 00:23:24,240 --> 00:23:25,920 And then a time 3, the step response 391 00:23:25,920 --> 00:23:31,530 jumps to 1, and from then on stays at 1. 392 00:23:31,530 --> 00:23:36,030 So the step response is just that delayed step. 393 00:23:36,030 --> 00:23:39,750 And it kind of makes sense, because the kind of system 394 00:23:39,750 --> 00:23:43,530 we're talking about must be a delayed by 3 system 395 00:23:43,530 --> 00:23:51,940 here, because we put in a unit sample input, a unit sample 396 00:23:51,940 --> 00:23:52,440 function. 397 00:23:52,440 --> 00:23:54,600 And what came out, if you look at it, 398 00:23:54,600 --> 00:23:57,540 was actually delta of n minus 3. 399 00:23:57,540 --> 00:24:01,170 It was the unit sample function delayed by 3 steps. 400 00:24:01,170 --> 00:24:03,240 Is the height-- the height is unchanged, right? 401 00:24:03,240 --> 00:24:04,330 The height of still 1. 402 00:24:04,330 --> 00:24:07,110 So this must be a delay by 3 system we're looking at. 403 00:24:07,110 --> 00:24:10,430 And sure, if we put in a unit step, we're getting-- 404 00:24:10,430 --> 00:24:12,300 or sorry, yeah, if we put in the unit step, 405 00:24:12,300 --> 00:24:13,800 we're going to get a response that's 406 00:24:13,800 --> 00:24:15,600 just the step delayed by 3. 407 00:24:19,180 --> 00:24:20,764 I'm going-- sorry, yeah? 408 00:24:20,764 --> 00:24:23,134 AUDIENCE: Yeah, maybe I'm just a little confused here. 409 00:24:23,134 --> 00:24:24,880 But why is it from negative infinity 410 00:24:24,880 --> 00:24:27,970 to n and not like from n to positive infinity. 411 00:24:27,970 --> 00:24:37,030 PROFESSOR: This is because of my assumption assuming s of minus 412 00:24:37,030 --> 00:24:39,070 infinity was 0. 413 00:24:39,070 --> 00:24:41,650 So I need to have a boundary condition 414 00:24:41,650 --> 00:24:44,410 from which I start inverting. 415 00:24:44,410 --> 00:24:48,470 So just to go back-- let me just go back a second here. 416 00:24:48,470 --> 00:24:50,410 Oh, where am I going? 417 00:24:54,380 --> 00:24:58,070 OK, so we derived this first expression. 418 00:24:58,070 --> 00:25:03,020 If we want to turn it around, well, sn is hn plus sn minus 1. 419 00:25:03,020 --> 00:25:05,210 And then I can solve for sn minus 1. 420 00:25:05,210 --> 00:25:06,890 I can keep stepping backwards. 421 00:25:06,890 --> 00:25:08,850 But at some point, I need an actual value 422 00:25:08,850 --> 00:25:11,630 so that I can close off that expression. 423 00:25:11,630 --> 00:25:14,003 And if you're talking about a causal system, 424 00:25:14,003 --> 00:25:15,920 then what you're guaranteed is that the step-- 425 00:25:15,920 --> 00:25:19,670 if you're talking about a causal linear system, 426 00:25:19,670 --> 00:25:22,550 because the all-zero input produces the all-zero output, 427 00:25:22,550 --> 00:25:24,080 and it's causal, you can actually 428 00:25:24,080 --> 00:25:27,890 deduce that the step response at time minus infinity must be 0. 429 00:25:27,890 --> 00:25:29,100 The input hasn't yet arrived. 430 00:25:29,100 --> 00:25:30,433 Therefore, the output must be 0. 431 00:25:30,433 --> 00:25:37,380 AUDIENCE: So h of 5, does that mean [INAUDIBLE]?? 432 00:25:37,380 --> 00:25:40,330 PROFESSOR: H of 5 is just a number. 433 00:25:40,330 --> 00:25:41,880 It's not a function, right? 434 00:25:41,880 --> 00:25:45,370 If I write something like h of 5, it's just a number. 435 00:25:45,370 --> 00:25:50,040 So it means the value of the unit sample response at time 5. 436 00:25:50,040 --> 00:25:52,290 OK, this takes a little getting used to, 437 00:25:52,290 --> 00:25:55,540 but let's do another example here. 438 00:25:55,540 --> 00:25:57,600 So here is another unit sample response. 439 00:25:57,600 --> 00:25:59,590 This is more complicated, though. 440 00:25:59,590 --> 00:26:00,870 I put in a unit sample. 441 00:26:00,870 --> 00:26:05,670 And what comes out is a response that-- 442 00:26:05,670 --> 00:26:07,140 well, it still starts at time 0. 443 00:26:07,140 --> 00:26:09,840 So I'm talking about a causal system. 444 00:26:09,840 --> 00:26:11,970 Everything to the left is 0. 445 00:26:11,970 --> 00:26:15,940 And the stakes a value 0.2 for some number of steps 446 00:26:15,940 --> 00:26:18,210 and then settles to 0. 447 00:26:18,210 --> 00:26:22,043 So the question then is, what is the step response? 448 00:26:22,043 --> 00:26:23,460 So if you imagine that what you're 449 00:26:23,460 --> 00:26:25,830 doing to find the step response at any time 450 00:26:25,830 --> 00:26:29,530 is summing this from minus infinity up to that time, 451 00:26:29,530 --> 00:26:31,830 you will see that the step response 452 00:26:31,830 --> 00:26:35,970 is linearized like that. 453 00:26:35,970 --> 00:26:37,740 And then it settles out. 454 00:26:37,740 --> 00:26:40,688 OK, so you can get one or the other. 455 00:26:40,688 --> 00:26:42,480 And there are other examples on the slides. 456 00:26:42,480 --> 00:26:43,830 I won't go through all of them. 457 00:26:43,830 --> 00:26:45,497 I'm going a little slow here, because we 458 00:26:45,497 --> 00:26:47,520 miss a recitation tomorrow. 459 00:26:47,520 --> 00:26:49,225 Recitations tomorrow are office hours, 460 00:26:49,225 --> 00:26:52,120 so I wanted to actually give you a few examples. 461 00:26:52,120 --> 00:26:56,700 Here is a case where the unit sample response increases 462 00:26:56,700 --> 00:26:59,100 linearly and then stops. 463 00:26:59,100 --> 00:27:01,260 And so the unit step response actually 464 00:27:01,260 --> 00:27:04,980 starts to accelerate quadratically and then stops. 465 00:27:04,980 --> 00:27:07,763 This is the discrete time version of integration 466 00:27:07,763 --> 00:27:08,680 that we're looking at. 467 00:27:08,680 --> 00:27:09,441 Yeah? 468 00:27:09,441 --> 00:27:12,267 AUDIENCE: [INAUDIBLE] 469 00:27:14,275 --> 00:27:15,650 PROFESSOR: Oh, sorry, this thing? 470 00:27:15,650 --> 00:27:16,275 AUDIENCE: Yeah. 471 00:27:16,275 --> 00:27:20,030 PROFESSOR: Ah, OK, ignore that notation. 472 00:27:20,030 --> 00:27:21,410 First of all, it's bad notation. 473 00:27:21,410 --> 00:27:23,510 But these are figures that I got from somewhere. 474 00:27:23,510 --> 00:27:27,920 If I was doing it from scratch, I wouldn't have put it in. 475 00:27:27,920 --> 00:27:29,630 But I'll explain it. 476 00:27:29,630 --> 00:27:31,830 That's the notation for convolution. 477 00:27:31,830 --> 00:27:34,970 I don't actually like that notation. 478 00:27:34,970 --> 00:27:37,010 OK, examples of this type-- 479 00:27:37,010 --> 00:27:41,210 now, here is one important thing for you to get a feel for. 480 00:27:41,210 --> 00:27:45,110 Notice in all these examples, the unit sample response 481 00:27:45,110 --> 00:27:47,930 settles down to zero after some time. 482 00:27:47,930 --> 00:27:50,930 So you hit the system with a unit sample function 483 00:27:50,930 --> 00:27:51,650 at the input. 484 00:27:51,650 --> 00:27:55,180 So you hit it with a value 1 at time 0 and nothing else. 485 00:27:55,180 --> 00:27:56,060 And it responds. 486 00:27:56,060 --> 00:27:58,208 And it response for a while and settles. 487 00:27:58,208 --> 00:27:59,750 Now, that's not true for all systems, 488 00:27:59,750 --> 00:28:02,020 that they settle in finite time. 489 00:28:02,020 --> 00:28:05,000 A typical system might ring indefinitely, 490 00:28:05,000 --> 00:28:07,880 might respond indefinitely to a kick. 491 00:28:07,880 --> 00:28:09,920 Here, are all these examples are ones 492 00:28:09,920 --> 00:28:14,930 where the system has a transient and then settles down. 493 00:28:14,930 --> 00:28:17,135 And so what you expect to see in the step response 494 00:28:17,135 --> 00:28:18,980 is there is a transient. 495 00:28:18,980 --> 00:28:20,480 And then it settles down. 496 00:28:20,480 --> 00:28:22,310 The difference is, in the step response, 497 00:28:22,310 --> 00:28:23,550 it settles to another value. 498 00:28:23,550 --> 00:28:25,010 It doesn't come back down to 0. 499 00:28:25,010 --> 00:28:27,260 So when this comes back down to zero, 500 00:28:27,260 --> 00:28:31,430 what this is settled up to is sort of the integral of this. 501 00:28:31,430 --> 00:28:33,303 It's the area under this, but we're 502 00:28:33,303 --> 00:28:35,720 talking about discrete time functions, not continuous time 503 00:28:35,720 --> 00:28:36,680 functions. 504 00:28:36,680 --> 00:28:40,130 So the value that it's settled to here is the area under this. 505 00:28:40,130 --> 00:28:43,250 So the duration of a unit sample response 506 00:28:43,250 --> 00:28:46,580 gives you some feel for how long a transient lasts. 507 00:28:46,580 --> 00:28:49,790 So if you've got a channel and you hit it with an input, 508 00:28:49,790 --> 00:28:52,370 you know that the transient will last about 509 00:28:52,370 --> 00:28:55,250 as long as the unit sample response lasts. 510 00:28:55,250 --> 00:28:58,130 So the transient and the step response shows that clearly. 511 00:29:01,830 --> 00:29:06,840 You can get more elaborate sorts of unit sample responses. 512 00:29:06,840 --> 00:29:09,030 Here is one that changes sign. 513 00:29:09,030 --> 00:29:11,790 And correspondingly, what you find with the step response 514 00:29:11,790 --> 00:29:14,910 is that it's not a monotonic increase 515 00:29:14,910 --> 00:29:16,320 to the final steady state. 516 00:29:16,320 --> 00:29:17,760 There is actually some oscillation 517 00:29:17,760 --> 00:29:20,210 before it settles down. 518 00:29:20,210 --> 00:29:21,260 But it's the same idea. 519 00:29:21,260 --> 00:29:23,580 You're computing the area under this, 520 00:29:23,580 --> 00:29:25,380 if you like, but the area goes positive, 521 00:29:25,380 --> 00:29:27,960 and then slightly negative, and so on-- 522 00:29:27,960 --> 00:29:30,500 well, positive and then less positive. 523 00:29:30,500 --> 00:29:32,250 And so that's what you're seeing up there. 524 00:29:34,860 --> 00:29:38,860 Now, why do we talk about step responses so much? 525 00:29:38,860 --> 00:29:40,800 Well, it turns out that for a lot of what 526 00:29:40,800 --> 00:29:44,040 we do with signaling on communication channels, 527 00:29:44,040 --> 00:29:48,450 we're signaling with signals of this type, on-off-type signals, 528 00:29:48,450 --> 00:29:51,750 or plus-minus signals, the sort of square-wave-type signals 529 00:29:51,750 --> 00:29:54,330 or rectangular-wave signals. 530 00:29:54,330 --> 00:29:57,150 And these can be thought of as combinations 531 00:29:57,150 --> 00:29:58,920 of unit step functions. 532 00:29:58,920 --> 00:30:01,400 You may have seen this in recitation last time as well. 533 00:30:01,400 --> 00:30:03,330 So you can take an input of this type 534 00:30:03,330 --> 00:30:06,930 and write it as a linear combination of unit step 535 00:30:06,930 --> 00:30:08,340 functions. 536 00:30:08,340 --> 00:30:11,190 A unit step function that has its step at 0, 537 00:30:11,190 --> 00:30:15,150 minus 1 that has its step at 4, plus 1 that has its step at 12, 538 00:30:15,150 --> 00:30:18,700 minus 1 that has its step at 24. 539 00:30:18,700 --> 00:30:22,500 So if you combine those, if you add up all of these, 540 00:30:22,500 --> 00:30:24,550 you're going to get that input. 541 00:30:24,550 --> 00:30:27,870 So then it's back to this game again. 542 00:30:27,870 --> 00:30:29,760 If the input is a linear combination 543 00:30:29,760 --> 00:30:34,380 of unit steps scaled and delayed, 544 00:30:34,380 --> 00:30:37,780 then the response is going to be the same combination of unit 545 00:30:37,780 --> 00:30:40,650 steps. 546 00:30:40,650 --> 00:30:42,790 So that's what the response will look like. 547 00:30:42,790 --> 00:30:47,340 So here is the step un gives rise to sn. 548 00:30:47,340 --> 00:30:51,142 Therefore, minus un minus 4 will give rise to minus s 549 00:30:51,142 --> 00:30:53,740 of n minus 4, and so on. 550 00:30:53,740 --> 00:30:55,770 So knowing the step response, you 551 00:30:55,770 --> 00:30:58,745 can actually say what the response of the channel 552 00:30:58,745 --> 00:30:59,370 is going to be. 553 00:31:04,040 --> 00:31:05,810 All right, we've seen this visually too. 554 00:31:05,810 --> 00:31:08,210 I did an example last time where what 555 00:31:08,210 --> 00:31:12,050 went in was that square wave and what came out 556 00:31:12,050 --> 00:31:14,330 after we had done the demodulation of the filtering 557 00:31:14,330 --> 00:31:17,600 was a response that sort of had the features of what went in, 558 00:31:17,600 --> 00:31:20,400 but it was a little distorted. 559 00:31:20,400 --> 00:31:22,790 So you can see that what we're looking at here, 560 00:31:22,790 --> 00:31:27,020 for instance, is the character of the step response, 561 00:31:27,020 --> 00:31:29,420 because what went in at this point-- 562 00:31:29,420 --> 00:31:32,900 right now the input on the output are at rest. 563 00:31:32,900 --> 00:31:35,540 You've forgotten about what happened before. 564 00:31:35,540 --> 00:31:37,365 And now the input jumps up. 565 00:31:37,365 --> 00:31:39,740 Well, the output doesn't jump up all the way immediately. 566 00:31:39,740 --> 00:31:42,482 It's got a little transient before it settles. 567 00:31:42,482 --> 00:31:43,940 So what you're looking at is really 568 00:31:43,940 --> 00:31:47,300 the step response of the channel, where the channel 569 00:31:47,300 --> 00:31:48,890 includes all these pieces. 570 00:31:48,890 --> 00:31:52,610 It's everything including the filtering. 571 00:31:52,610 --> 00:31:54,710 In this particular example, if you 572 00:31:54,710 --> 00:31:56,360 go back and look at those slides, 573 00:31:56,360 --> 00:32:00,860 this was all entirely due to the local averaging 574 00:32:00,860 --> 00:32:03,050 that we were doing in the filtering here. 575 00:32:03,050 --> 00:32:06,120 But it does give you some kind of a distortion. 576 00:32:06,120 --> 00:32:10,670 OK, so the step response is important to figuring out 577 00:32:10,670 --> 00:32:13,880 the shape of the output of a channel. 578 00:32:13,880 --> 00:32:16,880 Here is another example that has a more rounded kind of step 579 00:32:16,880 --> 00:32:19,460 response, but it's still the setup response 580 00:32:19,460 --> 00:32:20,190 we're looking at. 581 00:32:20,190 --> 00:32:22,520 So here is the input step. 582 00:32:22,520 --> 00:32:24,950 And here is the response to the step. 583 00:32:24,950 --> 00:32:26,660 Again, there is this notation. 584 00:32:26,660 --> 00:32:28,970 And I've said, ignore this for now. 585 00:32:28,970 --> 00:32:30,110 We'll explain it shortly. 586 00:32:32,630 --> 00:32:35,120 OK, and once you've got the step response at the output, 587 00:32:35,120 --> 00:32:36,920 you're ready to start thinking about how 588 00:32:36,920 --> 00:32:40,520 you'll detect whether it was a 0 or a 1 that went in. 589 00:32:40,520 --> 00:32:44,090 So you might set a threshold, pick times at which 590 00:32:44,090 --> 00:32:47,060 you're going to sample. 591 00:32:47,060 --> 00:32:51,350 And then you come up with your call 592 00:32:51,350 --> 00:32:57,280 of what the input is, so 1, 0, 0, and so on. 593 00:32:57,280 --> 00:32:59,500 So this seems all benign enough. 594 00:32:59,500 --> 00:33:01,060 But now what if you decide you want 595 00:33:01,060 --> 00:33:05,000 to get that information across the channel faster? 596 00:33:05,000 --> 00:33:07,100 So you want to signal faster? 597 00:33:07,100 --> 00:33:08,800 So what you're going to want to do 598 00:33:08,800 --> 00:33:16,440 is put that same information, the transition from 1 599 00:33:16,440 --> 00:33:19,680 to 0 to 1, 1, 1, 0, 1, and so on, 600 00:33:19,680 --> 00:33:23,520 you want to squeeze that into a shorter length of time. 601 00:33:23,520 --> 00:33:26,700 So suppose this is what you send over that same channel. 602 00:33:26,700 --> 00:33:29,280 Well, now you, again, are going to superpose the step 603 00:33:29,280 --> 00:33:30,533 responses. 604 00:33:30,533 --> 00:33:31,950 But what's happening now is you've 605 00:33:31,950 --> 00:33:33,810 gotten so ambitious with how fast you 606 00:33:33,810 --> 00:33:37,140 want to get the bits across that you're not giving the step 607 00:33:37,140 --> 00:33:38,460 response time to settle. 608 00:33:38,460 --> 00:33:41,130 So over here, yes, there is time. 609 00:33:41,130 --> 00:33:42,990 The step response went up and settled 610 00:33:42,990 --> 00:33:45,990 because you had three 1's in a row over there. 611 00:33:45,990 --> 00:33:48,990 But now you're going down to 0 for one time instant 612 00:33:48,990 --> 00:33:51,060 and then jumping right back again. 613 00:33:51,060 --> 00:33:54,352 Well, here is the flipped over step response. 614 00:33:54,352 --> 00:33:56,560 And it doesn't have time to make it all the way down. 615 00:33:56,560 --> 00:33:57,990 It's jumped up again. 616 00:33:57,990 --> 00:34:00,960 OK, so if you get very ambitious with your signaling 617 00:34:00,960 --> 00:34:04,050 to try and get more of the bits across, 618 00:34:04,050 --> 00:34:06,450 you're going to start seeing the limitations imposed 619 00:34:06,450 --> 00:34:07,590 by the channel. 620 00:34:07,590 --> 00:34:09,449 The channel can only respond so fast. 621 00:34:09,449 --> 00:34:12,840 And you can drive it faster. 622 00:34:12,840 --> 00:34:15,370 So it's important to have a feel for that as well. 623 00:34:18,250 --> 00:34:20,730 So when the channel starts to respond like this, 624 00:34:20,730 --> 00:34:22,929 you become much more susceptible to noise. 625 00:34:22,929 --> 00:34:25,860 So for instance, if there was a noise spike at this point, 626 00:34:25,860 --> 00:34:27,850 you could well end up with a received sample 627 00:34:27,850 --> 00:34:29,460 that was above the threshold. 628 00:34:29,460 --> 00:34:33,090 And then you'd wrongly decode the 0 as a 1. 629 00:34:33,090 --> 00:34:35,340 So taking account of the channel characteristics 630 00:34:35,340 --> 00:34:38,370 is important when you're setting a signaling rate. 631 00:34:38,370 --> 00:34:40,500 You might want to get information across quickly, 632 00:34:40,500 --> 00:34:42,719 but you have to take account of the fact 633 00:34:42,719 --> 00:34:47,070 that the channel needs some time. 634 00:34:47,070 --> 00:34:48,389 OK, so much for steps. 635 00:34:48,389 --> 00:34:50,940 We'll come back to that later. 636 00:34:50,940 --> 00:34:55,290 We can do the same kind of thing with unit samples. 637 00:34:55,290 --> 00:34:59,230 So here is-- and in fact, the rest of the lecture, 638 00:34:59,230 --> 00:35:03,360 we're going to be talking about making up 639 00:35:03,360 --> 00:35:06,420 a signal as a weighted combination of unit sample 640 00:35:06,420 --> 00:35:07,020 functions. 641 00:35:07,020 --> 00:35:09,600 So take an arbitrary signal like this. 642 00:35:09,600 --> 00:35:11,940 Think of it as-- 643 00:35:11,940 --> 00:35:16,770 let's see, this starts with the value of something, 0.75, 644 00:35:16,770 --> 00:35:20,295 I guess, at time minus 2, and then a value of minus 0.5 645 00:35:20,295 --> 00:35:21,980 at minus 1, and so on. 646 00:35:21,980 --> 00:35:24,510 So here is your input signal xn. 647 00:35:24,510 --> 00:35:26,250 And I want to think of it as made up 648 00:35:26,250 --> 00:35:28,660 of a bunch of unit sample functions. 649 00:35:28,660 --> 00:35:30,300 So what are the unit sample functions? 650 00:35:30,300 --> 00:35:33,960 Well, here is one that's centered at minus 2 651 00:35:33,960 --> 00:35:36,390 but scaled by the value that the input 652 00:35:36,390 --> 00:35:39,150 signal has at time minus 2. 653 00:35:39,150 --> 00:35:42,450 Here is another one centered at minus 1, 654 00:35:42,450 --> 00:35:45,100 but scaled by the value that the input signal has 655 00:35:45,100 --> 00:35:47,530 at time minus 1, and so on. 656 00:35:47,530 --> 00:35:52,380 So what I'm basically doing is decomposing the input 657 00:35:52,380 --> 00:35:57,270 into a weighted combination of unit samples. 658 00:35:57,270 --> 00:35:58,540 And you can always do that. 659 00:35:58,540 --> 00:36:02,400 And it looks a little magical when you put it into notation 660 00:36:02,400 --> 00:36:05,973 like this, but that's basically all that it's saying. 661 00:36:05,973 --> 00:36:07,890 So to make sense of this, think, for instance, 662 00:36:07,890 --> 00:36:10,120 of putting in an actual number here. 663 00:36:10,120 --> 00:36:13,170 So if I wanted x at time 3, well, I'm 664 00:36:13,170 --> 00:36:15,990 going to set n equals 3 on the right-hand side 665 00:36:15,990 --> 00:36:18,150 and evaluate the sum. 666 00:36:18,150 --> 00:36:20,280 Well, the only value that survives 667 00:36:20,280 --> 00:36:23,250 is the value for which k equals 3. 668 00:36:23,250 --> 00:36:25,230 So I'll pull out x3. 669 00:36:25,230 --> 00:36:28,830 So this kind of seems tautologous. 670 00:36:28,830 --> 00:36:32,370 But it's a way to represent a general input 671 00:36:32,370 --> 00:36:36,780 as a weighted combination of delayed unit samples. 672 00:36:36,780 --> 00:36:41,310 OK, so if that was what went in, you're in the position now 673 00:36:41,310 --> 00:36:42,540 to tell me what comes out. 674 00:36:46,230 --> 00:36:48,255 So I'm talking about an LTI system. 675 00:36:54,620 --> 00:36:57,560 I'm talking about an LTI system. 676 00:36:57,560 --> 00:37:02,900 And the input xn is a weighted combination 677 00:37:02,900 --> 00:37:08,453 with these weights of a bunch of unit sample functions. 678 00:37:16,860 --> 00:37:19,560 Well, let's actually-- well, let me actually 679 00:37:19,560 --> 00:37:22,000 write this the other way as well. 680 00:37:22,000 --> 00:37:25,560 Another way to say this is, here is a time function going in. 681 00:37:25,560 --> 00:37:28,380 It's a weighted combination over all possible values 682 00:37:28,380 --> 00:37:33,040 of k of xk times-- 683 00:37:33,040 --> 00:37:34,830 and this is my other notation, remember, 684 00:37:34,830 --> 00:37:36,550 for unit sample functions. 685 00:37:36,550 --> 00:37:37,800 So I'm saying this is a unit-- 686 00:37:37,800 --> 00:37:39,420 sorry, this should be delta sub n. 687 00:37:46,240 --> 00:37:47,080 What should it be? 688 00:37:49,930 --> 00:37:55,810 Yeah, OK, so this is another way to write the same thing. 689 00:37:55,810 --> 00:37:57,290 We've chosen to write it this way. 690 00:37:57,290 --> 00:37:58,850 And actually, I find that simpler. 691 00:37:58,850 --> 00:38:01,060 But if you want to be reminded that what we're 692 00:38:01,060 --> 00:38:03,375 talking about here is an entire time function, 693 00:38:03,375 --> 00:38:05,250 then this is a notation that you might go to. 694 00:38:05,250 --> 00:38:06,290 Yeah? 695 00:38:06,290 --> 00:38:13,480 AUDIENCE: Why is the first sum [INAUDIBLE] 696 00:38:13,480 --> 00:38:14,980 PROFESSOR: OK, good question. 697 00:38:14,980 --> 00:38:19,750 Because I'm right now allowing my input to have values 698 00:38:19,750 --> 00:38:22,220 that extend from minus infinity to plus infinity. 699 00:38:22,220 --> 00:38:24,055 So I'm taking an arbitrary function. 700 00:38:24,055 --> 00:38:26,020 If we're talking about an experiment in which 701 00:38:26,020 --> 00:38:30,340 the input starts at time 0, then we can actually simplify these. 702 00:38:30,340 --> 00:38:31,840 I'll show you that. 703 00:38:31,840 --> 00:38:35,680 OK, let me actually erase this, because I don't 704 00:38:35,680 --> 00:38:37,400 want to confuse you with that. 705 00:38:37,400 --> 00:38:40,210 OK, so if this is what goes in-- 706 00:38:40,210 --> 00:38:48,310 it's a weighted combination of unit sample functions delayed-- 707 00:38:48,310 --> 00:38:51,850 what is it that must come out? 708 00:38:51,850 --> 00:38:53,480 OK, so what are we working with? 709 00:38:53,480 --> 00:38:56,740 We're working with the fact that, if delta of n 710 00:38:56,740 --> 00:39:00,610 goes into our system, what comes out is hn. 711 00:39:03,940 --> 00:39:05,860 So if it's a weighted combination 712 00:39:05,860 --> 00:39:08,530 of deltas that goes in, what's the response, given 713 00:39:08,530 --> 00:39:09,910 that this is an LTI system? 714 00:39:16,235 --> 00:39:17,860 Someone who hasn't spoken today, maybe? 715 00:39:22,690 --> 00:39:24,166 Do you want to try? yeah? 716 00:39:24,166 --> 00:39:25,990 AUDIENCE: [INAUDIBLE] 717 00:39:25,990 --> 00:39:27,860 PROFESSOR: The Same weighted combination 718 00:39:27,860 --> 00:39:31,100 of those responses-- so it's going to be summation 719 00:39:31,100 --> 00:39:35,080 over all k, the same weight. 720 00:39:35,080 --> 00:39:36,980 So it's going to be the xk's. 721 00:39:36,980 --> 00:39:39,470 But now here is the responses. 722 00:39:46,700 --> 00:39:48,360 So what have we been able to do? 723 00:39:48,360 --> 00:39:51,380 We've been able to write down what the output looks 724 00:39:51,380 --> 00:39:58,050 like for an arbitrary input in terms of the unit sample 725 00:39:58,050 --> 00:39:58,550 response. 726 00:39:58,550 --> 00:40:01,880 If you give me the unit sample response for an LTI system, 727 00:40:01,880 --> 00:40:04,370 I can write down the general response, the response 728 00:40:04,370 --> 00:40:05,330 to a general input. 729 00:40:09,860 --> 00:40:17,210 And this is what we refer to as a convolution or a convolution 730 00:40:17,210 --> 00:40:17,710 sum. 731 00:40:21,100 --> 00:40:22,602 That's a convolution. 732 00:40:34,670 --> 00:40:35,630 It may look mysterious. 733 00:40:35,630 --> 00:40:38,330 So let's actually do it. 734 00:40:38,330 --> 00:40:43,190 Let's do it step by step again. 735 00:40:43,190 --> 00:40:44,255 Here is our LTI system. 736 00:40:47,450 --> 00:40:51,422 If I put in a unit sample function-- 737 00:40:55,580 --> 00:40:58,430 OK, so this is the unit sample function going in-- 738 00:40:58,430 --> 00:40:59,345 I get some response. 739 00:41:08,690 --> 00:41:10,170 And let's say this is 0. 740 00:41:10,170 --> 00:41:12,380 This is 1, 2, and so on. 741 00:41:16,520 --> 00:41:20,660 The response, we refer to as the unit sample response hn. 742 00:41:20,660 --> 00:41:21,800 So what is this value? 743 00:41:21,800 --> 00:41:24,320 This is the value h0. 744 00:41:24,320 --> 00:41:27,050 This is the value h1. 745 00:41:27,050 --> 00:41:30,250 This is the value h2, and so on. 746 00:41:32,810 --> 00:41:37,670 OK, what if what goes in is actually 747 00:41:37,670 --> 00:41:44,610 the value x0 at this time and 0 everywhere else? 748 00:41:48,730 --> 00:41:50,230 What's the response in that case? 749 00:41:54,430 --> 00:41:57,300 So this is just a scaled version of the unit sample function. 750 00:41:57,300 --> 00:42:00,510 Instead of 1 going in, I'm having x0 go in. 751 00:42:00,510 --> 00:42:02,190 So the response is going to be-- 752 00:42:06,253 --> 00:42:06,920 what do we have? 753 00:42:10,608 --> 00:42:12,960 The same response? 754 00:42:12,960 --> 00:42:16,090 Twice the response? 755 00:42:16,090 --> 00:42:18,133 What comes out? 756 00:42:18,133 --> 00:42:19,888 AUDIENCE: x0 times that. 757 00:42:19,888 --> 00:42:20,930 PROFESSOR: I didn't hear. 758 00:42:20,930 --> 00:42:21,610 Where did that come from? 759 00:42:21,610 --> 00:42:21,930 Yeah? 760 00:42:21,930 --> 00:42:22,997 AUDIENCE: x0 times that. 761 00:42:22,997 --> 00:42:24,080 PROFESSOR: X0 times that-- 762 00:42:24,080 --> 00:42:27,770 OK, so what we'll get is x0 times h0 763 00:42:27,770 --> 00:42:30,740 coming out at the first time, and then 764 00:42:30,740 --> 00:42:35,870 x0 times h1 coming out of the second time, 765 00:42:35,870 --> 00:42:42,350 and then x0 h2 at the next time. 766 00:42:42,350 --> 00:42:55,720 And if I keep going, I get x0, let's say, hn at this time, 767 00:42:55,720 --> 00:42:56,340 and so on. 768 00:42:59,790 --> 00:43:03,930 What happens if now it's not that, 769 00:43:03,930 --> 00:43:08,790 but it's some value x1 going in at time 1 and 0 770 00:43:08,790 --> 00:43:09,480 everywhere else? 771 00:43:14,320 --> 00:43:16,750 So this is starting-- this is centered at a time 1, 772 00:43:16,750 --> 00:43:18,090 not at time 0. 773 00:43:18,090 --> 00:43:21,135 And it's scaled by x1. 774 00:43:21,135 --> 00:43:23,010 So what is it I'm going to see at the output? 775 00:43:25,707 --> 00:43:27,540 There was a hand somewhere there previously. 776 00:43:27,540 --> 00:43:28,370 Maybe you can answer now. 777 00:43:28,370 --> 00:43:29,033 Yeah? 778 00:43:29,033 --> 00:43:31,751 AUDIENCE: The same graph translated 1 over 779 00:43:31,751 --> 00:43:32,897 and scaled by x1. 780 00:43:32,897 --> 00:43:33,980 PROFESSOR: Right, exactly. 781 00:43:33,980 --> 00:43:37,040 So what's going to happen is, nothing will happen here. 782 00:43:37,040 --> 00:43:53,910 I'll get x1 h0, x1 h1, and so on, x1 h of n minus 1. 783 00:43:53,910 --> 00:43:54,960 And it keeps going. 784 00:43:58,380 --> 00:44:01,500 And you keep going here as well. 785 00:44:01,500 --> 00:44:04,626 You keep stringing in these. 786 00:44:04,626 --> 00:44:07,620 Each one of these will fire off a scale of the unit sample 787 00:44:07,620 --> 00:44:10,570 response, but delayed appropriately. 788 00:44:10,570 --> 00:44:13,770 And so at the next time, what you're going to get here 789 00:44:13,770 --> 00:44:18,780 is x2 h of n minus 2. 790 00:44:18,780 --> 00:44:19,740 And it keeps going. 791 00:44:23,210 --> 00:44:26,900 And what if you're interested in the value at time n? 792 00:44:26,900 --> 00:44:28,460 OK, so you look along here. 793 00:44:28,460 --> 00:44:32,300 And you've come to the value of time n. 794 00:44:32,300 --> 00:44:34,550 So it's going to be the sum of all of these, 795 00:44:34,550 --> 00:44:37,460 if your input is the sum of all of these, right? 796 00:44:37,460 --> 00:44:40,730 If your input is the sum of all of these x's, your response 797 00:44:40,730 --> 00:44:43,063 is going to be the sum of all of these. 798 00:44:43,063 --> 00:44:44,480 So what's the sum of all of these? 799 00:44:44,480 --> 00:44:49,497 Well, xk h of n minus k. 800 00:44:49,497 --> 00:44:50,330 That's all there is. 801 00:44:50,330 --> 00:44:52,205 There is nothing-- there is no magic to this. 802 00:44:52,205 --> 00:44:54,800 It's just invoking linearity-- 803 00:44:54,800 --> 00:44:56,690 that's the scaling part of it-- 804 00:44:56,690 --> 00:44:59,030 and time invariance, which is the delaying part of it. 805 00:45:02,830 --> 00:45:05,240 It's as simple as that. 806 00:45:05,240 --> 00:45:07,720 All right, so we'll be seeing this notation a lot. 807 00:45:07,720 --> 00:45:10,480 You probably recognize this kind of notation 808 00:45:10,480 --> 00:45:12,460 from the convolutional coder as well. 809 00:45:16,280 --> 00:45:18,160 And we don't want to keep writing these sums. 810 00:45:18,160 --> 00:45:19,720 So here is the notation that we use. 811 00:45:19,720 --> 00:45:24,250 We say that x is convolved with h. 812 00:45:24,250 --> 00:45:28,090 And we are interested in the value at time n. 813 00:45:28,090 --> 00:45:31,300 So this operation of-- 814 00:45:31,300 --> 00:45:35,330 this summation here is referred to as a convolution, as I said. 815 00:45:35,330 --> 00:45:38,830 I'm telling you what value of time I'm interested 816 00:45:38,830 --> 00:45:39,700 and the response at. 817 00:45:39,700 --> 00:45:40,810 That's the n. 818 00:45:40,810 --> 00:45:44,010 So that's what this notation is. 819 00:45:44,010 --> 00:45:46,083 The k here is just a dummy index. 820 00:45:46,083 --> 00:45:47,125 We're summing over the k. 821 00:45:47,125 --> 00:45:48,583 It doesn't matter what I called it. 822 00:45:48,583 --> 00:45:49,620 I can call it j. 823 00:45:49,620 --> 00:45:50,340 I can call it l. 824 00:45:50,340 --> 00:45:51,340 It doesn't matter. 825 00:45:51,340 --> 00:45:54,450 The important thing is this n here tells me at what time 826 00:45:54,450 --> 00:45:56,730 I'm looking for the response. 827 00:45:56,730 --> 00:45:59,370 And that's why that's the argument that I stick in here. 828 00:46:05,080 --> 00:46:06,910 All right, so all that's on the slides, 829 00:46:06,910 --> 00:46:09,760 but we've actually derived it ourselves here. 830 00:46:14,270 --> 00:46:21,590 Now, again, some gripes about notation-- 831 00:46:21,590 --> 00:46:24,260 you'll find, if you look in most engineering textbooks, 832 00:46:24,260 --> 00:46:32,030 that this would be written xn star hn. 833 00:46:32,030 --> 00:46:34,370 And I can't tell you how much I detest that notation. 834 00:46:34,370 --> 00:46:38,360 You'll never find it in a math book. 835 00:46:38,360 --> 00:46:40,610 The problem here is that this n is being 836 00:46:40,610 --> 00:46:42,260 asked to do too many things. 837 00:46:42,260 --> 00:46:45,080 The n is supposed to suggest-- 838 00:46:45,080 --> 00:46:46,580 the xn here is supposed to suggest 839 00:46:46,580 --> 00:46:49,190 we're interested in the whole time function. 840 00:46:49,190 --> 00:46:50,930 You would have been better off calling it 841 00:46:50,930 --> 00:46:54,320 x dot, but, OK, we're used to thinking of xn 842 00:46:54,320 --> 00:46:56,720 as also denoting an entire time function. 843 00:46:56,720 --> 00:46:59,210 This h is supposed-- the h of n is supposed to denote 844 00:46:59,210 --> 00:47:00,920 an entire time function. 845 00:47:00,920 --> 00:47:03,260 But the n is also supposed to tell you at what time 846 00:47:03,260 --> 00:47:04,980 you're interested in the response. 847 00:47:04,980 --> 00:47:07,940 So that index is just doing too much work. 848 00:47:07,940 --> 00:47:13,230 And it ends up being confused and confusing notation. 849 00:47:13,230 --> 00:47:17,228 So when you're in your downstream classes from here, 850 00:47:17,228 --> 00:47:18,770 if you find an instructor using that, 851 00:47:18,770 --> 00:47:20,767 make sure that you give him or her 852 00:47:20,767 --> 00:47:22,850 grief and say you really can't make sense of that, 853 00:47:22,850 --> 00:47:25,910 because this is much cleaner notation. 854 00:47:25,910 --> 00:47:30,260 This is what conveys what's actually going on. 855 00:47:30,260 --> 00:47:38,500 All right, I'm going to skip over a few things. 856 00:47:38,500 --> 00:47:42,460 I just want to suggest some properties here. 857 00:47:42,460 --> 00:47:46,300 And then we'll come back to more of this next time. 858 00:47:50,090 --> 00:47:52,130 OK, so it turns out that convolution 859 00:47:52,130 --> 00:47:54,260 has nice properties. 860 00:47:54,260 --> 00:47:56,910 For instance, the order doesn't matter. 861 00:47:56,910 --> 00:48:02,090 You can write x star h here, but it's the same as h star x. 862 00:48:02,090 --> 00:48:05,120 And that just comes from making a change of variables in here. 863 00:48:05,120 --> 00:48:11,690 If I call this m, then k is equal to n minus m. 864 00:48:11,690 --> 00:48:15,478 And I get something that looks different. 865 00:48:15,478 --> 00:48:16,770 But it's really the same thing. 866 00:48:16,770 --> 00:48:18,410 So this is the same as h star x. 867 00:48:22,000 --> 00:48:24,435 So convolution, you can interchange orders. 868 00:48:24,435 --> 00:48:25,810 There is some conditions on this, 869 00:48:25,810 --> 00:48:28,070 but we can talk about them later. 870 00:48:28,070 --> 00:48:29,140 You can associate them. 871 00:48:29,140 --> 00:48:31,000 You can group them arbitrarily. 872 00:48:31,000 --> 00:48:32,650 And you can distribute convolution 873 00:48:32,650 --> 00:48:34,460 over additional functions. 874 00:48:34,460 --> 00:48:36,735 So all of this actually makes it-- 875 00:48:36,735 --> 00:48:38,110 this is very powerful, because it 876 00:48:38,110 --> 00:48:41,612 allows you to deal with combinations of systems. 877 00:48:41,612 --> 00:48:43,070 And I'll just give you one example. 878 00:48:43,070 --> 00:48:45,100 And then we'll quit. 879 00:48:45,100 --> 00:48:49,400 So here is an example of the kind of thing you can do. 880 00:48:49,400 --> 00:48:53,350 Suppose you have an input going into one system, LTI, 881 00:48:53,350 --> 00:48:56,440 with a unit sample response h1, and then 882 00:48:56,440 --> 00:48:59,680 the output of that going into a second system, LTI with unit 883 00:48:59,680 --> 00:49:04,870 sample response h2, and then producing an overall output yn. 884 00:49:04,870 --> 00:49:07,060 Well, so how do you get y? 885 00:49:07,060 --> 00:49:09,370 It's h2 convolved with w. 886 00:49:09,370 --> 00:49:11,320 I've dropped the argument n because I 887 00:49:11,320 --> 00:49:14,230 want to do this just for general values. 888 00:49:14,230 --> 00:49:18,940 But w itself is h1 convolved with x. 889 00:49:18,940 --> 00:49:20,780 Now, I can group these any way I want. 890 00:49:20,780 --> 00:49:23,620 So I can, because convolution is associative, 891 00:49:23,620 --> 00:49:25,700 I can put those parentheses where I want. 892 00:49:25,700 --> 00:49:29,020 So this is equal to the expression at the end. 893 00:49:29,020 --> 00:49:30,640 But that's the same result I'd get 894 00:49:30,640 --> 00:49:35,530 by putting this input into a single LTI system whose 895 00:49:35,530 --> 00:49:38,140 unit sample response was the convolution of the two 896 00:49:38,140 --> 00:49:39,280 individual ones. 897 00:49:39,280 --> 00:49:43,450 So I can start to collapse two systems into one equivalent LTI 898 00:49:43,450 --> 00:49:44,230 system. 899 00:49:44,230 --> 00:49:47,080 And that kind of thing ends up being powerful. 900 00:49:47,080 --> 00:49:49,040 But I can also interchange orders. 901 00:49:49,040 --> 00:49:52,180 So from here, you can go to this, which then tells you 902 00:49:52,180 --> 00:49:56,260 that, for an LTI system, if you've got systems in cascade, 903 00:49:56,260 --> 00:49:59,080 if you've got two LTI systems in cascade, actually, 904 00:49:59,080 --> 00:50:02,140 the effect on the output is the same, whatever 905 00:50:02,140 --> 00:50:03,160 order the input-- 906 00:50:03,160 --> 00:50:05,620 the systems are connected in. 907 00:50:05,620 --> 00:50:08,080 You might ask yourself whether the same is true if this 908 00:50:08,080 --> 00:50:11,380 was linear but time varying. 909 00:50:11,380 --> 00:50:13,270 And you should hopefully find out 910 00:50:13,270 --> 00:50:15,820 that, in general, for linear but time-varying systems, 911 00:50:15,820 --> 00:50:16,570 you can't do this. 912 00:50:16,570 --> 00:50:19,000 So really, linearity and time invariance 913 00:50:19,000 --> 00:50:22,120 is what it takes to be able to attain this. 914 00:50:22,120 --> 00:50:23,720 OK, let's leave it at this for now. 915 00:50:23,720 --> 00:50:25,360 And we'll pick up again-- 916 00:50:25,360 --> 00:50:28,240 well, you pick up some in problem set four and also 917 00:50:28,240 --> 00:50:30,060 next week.