1 00:00:00,000 --> 00:00:11,420 2 00:00:11,420 --> 00:00:12,240 PROFESSOR: Hi. 3 00:00:12,240 --> 00:00:14,420 Today I'd like to talk about signals and systems again. 4 00:00:14,420 --> 00:00:16,960 At this point, you're probably familiar with the motivation 5 00:00:16,960 --> 00:00:19,410 for why we're talking about discrete linear time and 6 00:00:19,410 --> 00:00:22,330 variant systems, and also with a few of the representations 7 00:00:22,330 --> 00:00:24,800 that we're going to end up using in this course. 8 00:00:24,800 --> 00:00:28,190 But you're still not sure what it is that we're trying to 9 00:00:28,190 --> 00:00:28,770 accomplish. 10 00:00:28,770 --> 00:00:31,260 Or where's the part where we get to predict the future 11 00:00:31,260 --> 00:00:33,110 based on the fact that we are capable of 12 00:00:33,110 --> 00:00:34,990 manipulating these systems? 13 00:00:34,990 --> 00:00:36,940 Well, we actually have to be capable of 14 00:00:36,940 --> 00:00:37,940 manipulating these systems. 15 00:00:37,940 --> 00:00:40,020 And at this point, we can describe this system as we see 16 00:00:40,020 --> 00:00:43,025 it, but we can't also manipulate its representation 17 00:00:43,025 --> 00:00:45,380 in ways that make sense to us. 18 00:00:45,380 --> 00:00:47,420 So the thing that I'm going to do today is talk about 19 00:00:47,420 --> 00:00:50,650 different system equivalences and how to take a system and 20 00:00:50,650 --> 00:00:53,680 solve for an expression that represents a complex system 21 00:00:53,680 --> 00:00:57,200 and also how if you know that some things in your system are 22 00:00:57,200 --> 00:01:00,340 equivalent, how you can convert between them. 23 00:01:00,340 --> 00:01:02,560 At that point, we should be able to talk about poles, 24 00:01:02,560 --> 00:01:04,890 which is how we're going to actually predict the future. 25 00:01:04,890 --> 00:01:09,950 So different equivalences that I'd like to talk about. 26 00:01:09,950 --> 00:01:12,750 I'm first going to briefly review the facts that last 27 00:01:12,750 --> 00:01:17,190 time we discovered the notion of system function, right? 28 00:01:17,190 --> 00:01:21,090 We can take a representation of a system and abstract it 29 00:01:21,090 --> 00:01:24,750 away into some sort of function, where we take the 30 00:01:24,750 --> 00:01:28,410 input as it's given to us and then multiply it by this 31 00:01:28,410 --> 00:01:30,510 function and then get the output that 32 00:01:30,510 --> 00:01:33,110 we're interested in. 33 00:01:33,110 --> 00:01:34,470 How do we deal with something more complex? 34 00:01:34,470 --> 00:01:37,120 35 00:01:37,120 --> 00:01:38,990 I mean, y is all the way over here. 36 00:01:38,990 --> 00:01:40,680 And we've got multiple system functions. 37 00:01:40,680 --> 00:01:43,960 And I don't even know what happens here, but it doesn't 38 00:01:43,960 --> 00:01:45,090 have to be that scary. 39 00:01:45,090 --> 00:01:46,340 Let's break it down. 40 00:01:46,340 --> 00:01:48,280 41 00:01:48,280 --> 00:01:51,630 One of the easiest ways to approach something like this 42 00:01:51,630 --> 00:01:55,230 is to identify each position where you have a new signal, 43 00:01:55,230 --> 00:01:58,530 or if you were to sample here, you would have a new signal, 44 00:01:58,530 --> 00:01:59,980 and label those values appropriately. 45 00:01:59,980 --> 00:02:09,009 46 00:02:09,009 --> 00:02:12,210 You can then start with your final output and then back 47 00:02:12,210 --> 00:02:14,670 solve for the values that you're interested in as a 48 00:02:14,670 --> 00:02:18,090 consequence of that final output. 49 00:02:18,090 --> 00:02:23,606 In this particular example, y is going to be y2 plus y3. 50 00:02:23,606 --> 00:02:28,100 51 00:02:28,100 --> 00:02:40,820 y2 is going to be y1 times H2, where H2 52 00:02:40,820 --> 00:02:42,980 is some system function. 53 00:02:42,980 --> 00:02:46,520 And it probably is abstracting away some combination of 54 00:02:46,520 --> 00:02:48,320 gains, delays, and adders like this one here. 55 00:02:48,320 --> 00:02:53,460 56 00:02:53,460 --> 00:03:01,944 y1 is going to be x times H1. 57 00:03:01,944 --> 00:03:05,402 58 00:03:05,402 --> 00:03:09,120 And Y3 is going to be x times H3. 59 00:03:09,120 --> 00:03:20,160 60 00:03:20,160 --> 00:03:24,200 Now I've got all my expressions in terms of either 61 00:03:24,200 --> 00:03:30,310 y or something for which I have an equivalent 62 00:03:30,310 --> 00:03:31,930 expression for x. 63 00:03:31,930 --> 00:03:35,240 So I can do my substitutions, come up for an expression for 64 00:03:35,240 --> 00:03:38,270 y over x, in terms of H1, H2, and H3. 65 00:03:38,270 --> 00:04:18,620 66 00:04:18,620 --> 00:04:20,480 Here I've just made the substitutions of the equations 67 00:04:20,480 --> 00:04:48,760 above and factored out the x. 68 00:04:48,760 --> 00:04:50,820 If I wanted the system function, I would then just 69 00:04:50,820 --> 00:04:51,620 divide by x. 70 00:04:51,620 --> 00:04:56,720 And then I would have y over x is equal to this expression. 71 00:04:56,720 --> 00:05:01,850 The thing I wanted to indicate is that if I wanted to 72 00:05:01,850 --> 00:05:05,320 abstract this away into its own box-- maybe I wanted like 73 00:05:05,320 --> 00:05:07,840 a big H or an H0 or something like that-- 74 00:05:07,840 --> 00:05:12,570 and it represented what was happening in this top line, 75 00:05:12,570 --> 00:05:17,170 cascading two system functions is the functional equivalent 76 00:05:17,170 --> 00:05:18,600 of multiplying them together. 77 00:05:18,600 --> 00:05:21,420 So if I have an expression for H1 and I have an expression 78 00:05:21,420 --> 00:05:24,590 for H2, and I want the expression that is equal to 79 00:05:24,590 --> 00:05:27,430 cascading H1 and H2, I just multiply them together. 80 00:05:27,430 --> 00:05:30,280 81 00:05:30,280 --> 00:05:33,200 Likewise, if I want an expression for the linear 82 00:05:33,200 --> 00:05:37,080 combination of two system functions applied to an input 83 00:05:37,080 --> 00:05:46,580 individually, like the combination H1 and H2, and H3, 84 00:05:46,580 --> 00:05:50,990 it's a summation of those two values which 85 00:05:50,990 --> 00:05:52,670 is expressed here. 86 00:05:52,670 --> 00:05:55,670 This is the same as the relationship that we reviewed 87 00:05:55,670 --> 00:05:58,080 in a very basic sense when we were originally doing the 88 00:05:58,080 --> 00:05:59,910 accumulator. 89 00:05:59,910 --> 00:06:02,817 The only thing I'm attempting to indicate is that, that 90 00:06:02,817 --> 00:06:06,450 relationship scales to an arbitrary level of complexity. 91 00:06:06,450 --> 00:06:11,670 So if you need to, you could shift around these values, if 92 00:06:11,670 --> 00:06:14,960 you can find some sort of equivalence. 93 00:06:14,960 --> 00:06:17,300 Let's see what happens when H2 is equal to H3. 94 00:06:17,300 --> 00:06:22,690 95 00:06:22,690 --> 00:06:23,940 I'm going to take my operator equation. 96 00:06:23,940 --> 00:07:03,980 97 00:07:03,980 --> 00:07:08,110 What this means is that if I wanted to rewrite this block 98 00:07:08,110 --> 00:07:14,450 diagram, I could do so by doing-- 99 00:07:14,450 --> 00:07:41,980 100 00:07:41,980 --> 00:07:43,760 This is really similar to bubble pushing. 101 00:07:43,760 --> 00:07:47,300 If you've done 6.004 or 6.002 and have experience with logic 102 00:07:47,300 --> 00:07:49,990 gates, I just wanted to indicate that it's also a 103 00:07:49,990 --> 00:07:51,220 thing that you can do for block 104 00:07:51,220 --> 00:07:52,470 diagrams and system functions. 105 00:07:52,470 --> 00:08:00,940 106 00:08:00,940 --> 00:08:02,450 There's one more type of equivalence that I want to 107 00:08:02,450 --> 00:08:03,850 talk about. 108 00:08:03,850 --> 00:08:05,155 I call it feedback equivalence. 109 00:08:05,155 --> 00:08:09,300 110 00:08:09,300 --> 00:08:10,820 Here's our normal accumulator rate. 111 00:08:10,820 --> 00:08:20,700 112 00:08:20,700 --> 00:08:24,190 If I wanted to represent this feedback system as a feed 113 00:08:24,190 --> 00:08:26,440 forward system, what would I have to do? 114 00:08:26,440 --> 00:08:29,310 115 00:08:29,310 --> 00:08:33,080 Well, the first time that I sampled from x, it 116 00:08:33,080 --> 00:08:35,390 would just be y. 117 00:08:35,390 --> 00:08:38,620 So right now this diagram matches for 118 00:08:38,620 --> 00:08:39,919 the first time step. 119 00:08:39,919 --> 00:08:45,230 On the second time step, if I had an input from x from the 120 00:08:45,230 --> 00:08:50,060 previous time step, I would also want to account for it by 121 00:08:50,060 --> 00:08:55,800 putting in a delay and then summing it with the current 122 00:08:55,800 --> 00:09:01,110 value of x in order to get y. 123 00:09:01,110 --> 00:09:08,660 At the second time step, I would want access to the 124 00:09:08,660 --> 00:09:18,660 starting value, the value from the previous time step, and 125 00:09:18,660 --> 00:09:22,150 the value from the current time step. 126 00:09:22,150 --> 00:09:26,260 And one more time, to exhaust the example, at the third time 127 00:09:26,260 --> 00:09:45,680 step, my output would be a linear combination of the 128 00:09:45,680 --> 00:09:49,520 starting value, the value from the first time step, the value 129 00:09:49,520 --> 00:09:53,950 from the second time step, and the value from the current 130 00:09:53,950 --> 00:09:55,200 third time step. 131 00:09:55,200 --> 00:10:05,750 132 00:10:05,750 --> 00:10:09,030 We'd end up doing this ad nauseum to model 133 00:10:09,030 --> 00:10:11,930 our feedback system. 134 00:10:11,930 --> 00:10:17,230 So it's difficult to do on paper, but it turns out 135 00:10:17,230 --> 00:10:21,190 there's a great relationship between these two equivalences 136 00:10:21,190 --> 00:10:22,605 and things that we already know from-- 137 00:10:22,605 --> 00:10:25,400 138 00:10:25,400 --> 00:10:26,690 I want to say high school calculus or 139 00:10:26,690 --> 00:10:30,370 possibly 18.01, 18.02. 140 00:10:30,370 --> 00:10:31,620 Geometric sequences. 141 00:10:31,620 --> 00:10:35,500 142 00:10:35,500 --> 00:10:38,210 When we solved for the system function, we found an 143 00:10:38,210 --> 00:10:44,120 expression for our feedback system. 144 00:10:44,120 --> 00:10:52,570 145 00:10:52,570 --> 00:10:54,960 If I wanted to find an equivalent expression using 146 00:10:54,960 --> 00:10:59,400 this feed forward system, I would look at this infinite 147 00:10:59,400 --> 00:11:06,130 summation of x terms. 148 00:11:06,130 --> 00:11:18,170 149 00:11:18,170 --> 00:11:21,290 So if I wanted to know something about the long-term 150 00:11:21,290 --> 00:11:26,710 behavior of the system, in terms of this system function, 151 00:11:26,710 --> 00:11:30,810 I would solve for this expression and then using my 152 00:11:30,810 --> 00:11:39,475 knowledge of geometric sequences, in order to express 153 00:11:39,475 --> 00:11:41,790 the long-term behavior. 154 00:11:41,790 --> 00:11:44,690 In the general sense, in this course, we're going to be 155 00:11:44,690 --> 00:11:47,450 looking at the unit sample response of a system. 156 00:11:47,450 --> 00:11:50,360 157 00:11:50,360 --> 00:12:11,710 What that means is, if the only thing I ever do for input 158 00:12:11,710 --> 00:12:22,930 is a single value of 1 at time 0, then what does 159 00:12:22,930 --> 00:12:25,650 my output look like? 160 00:12:25,650 --> 00:12:27,510 The reason we're looking at the unit sample response is 161 00:12:27,510 --> 00:12:30,540 because it's (a) the simplest way to look at the long-term 162 00:12:30,540 --> 00:12:33,990 behavior of a discrete linear time invariant system. 163 00:12:33,990 --> 00:12:37,220 But the other reason (b) is -- once we have this, we can also 164 00:12:37,220 --> 00:12:39,970 use it to do things like to make predictions about the 165 00:12:39,970 --> 00:12:44,300 long-term step response and other more 166 00:12:44,300 --> 00:12:45,710 complicated input signals. 167 00:12:45,710 --> 00:12:49,650 168 00:12:49,650 --> 00:12:54,280 In the case of the accumulator, if I input 1 at 169 00:12:54,280 --> 00:13:08,140 time 0, my output is going to be 1 forever more. 170 00:13:08,140 --> 00:13:18,130 171 00:13:18,130 --> 00:13:28,050 That's reflected in the coefficient of 172 00:13:28,050 --> 00:13:29,300 my geometric sequence. 173 00:13:29,300 --> 00:13:31,800 174 00:13:31,800 --> 00:13:35,940 If I want to know what my long-term response is going to 175 00:13:35,940 --> 00:13:41,690 look like, I can look at the coefficient of R and make a 176 00:13:41,690 --> 00:13:45,630 decision about whether or not I'm going to diverge or 177 00:13:45,630 --> 00:13:46,970 converge or do neither. 178 00:13:46,970 --> 00:13:50,070 179 00:13:50,070 --> 00:14:47,460 So if I put a coefficient on R, whatever p0 converges to is 180 00:14:47,460 --> 00:14:51,510 what my system is going to converge to. 181 00:14:51,510 --> 00:14:55,510 So using my knowledge of p0, I can make long-term predictions 182 00:14:55,510 --> 00:14:58,920 about the behavior of the system. 183 00:14:58,920 --> 00:15:01,440 Next time I'm going to go over some general classifications 184 00:15:01,440 --> 00:15:04,180 of those behaviors for the system and how to more 185 00:15:04,180 --> 00:15:07,040 effectively use our knowledge of p0 and how to deal with 186 00:15:07,040 --> 00:15:08,290 things like second order systems. 187 00:15:08,290 --> 00:15:09,717