1 00:00:00,012 --> 00:00:06,556 Okay, in this video we're going to talk about the basic ideas behind systems. 2 00:00:06,556 --> 00:00:13,868 We'll first talk about some very simple systems, systems that the operations they 3 00:00:13,868 --> 00:00:19,473 perform in the input signal to boost an output are quite simple and 4 00:00:19,473 --> 00:00:23,596 straightforward. We're then going to talk about how you 5 00:00:23,596 --> 00:00:29,126 can book systems together, to make more complicated input, output relationships. 6 00:00:29,126 --> 00:00:32,558 The relationship between the input and the output . 7 00:00:32,558 --> 00:00:37,545 Then finally we're going to talk about linear systems, and time invariant 8 00:00:37,545 --> 00:00:40,682 systems. And the ones that are both linear. 9 00:00:40,682 --> 00:00:46,831 And time invariant are going to be extremely important to us, as we're going 10 00:00:46,831 --> 00:00:51,578 to see. Alright, so let's talk about some simple 11 00:00:51,578 --> 00:00:58,892 systems, ones that are pretty easy to understand. And the first one is quite 12 00:00:58,892 --> 00:01:03,175 simple. It's called a gain, and what the system 13 00:01:03,175 --> 00:01:07,881 does is, the output is equal to the input times a gain. 14 00:01:07,881 --> 00:01:12,488 A number G, a scalar. So I think that's pretty easy to 15 00:01:12,488 --> 00:01:17,218 understand it. It just takes the input, multiplies it 16 00:01:17,218 --> 00:01:24,193 times a number and that's its So, we can use the unit step as a example input to 17 00:01:24,193 --> 00:01:29,481 see how this works. So the input here, is I've drawn it as a 18 00:01:29,481 --> 00:01:33,708 unit step. And then the output is the same unit 19 00:01:33,708 --> 00:01:40,682 step, except it's now got an amplitude of G instead of 1, whatever that is. 20 00:01:40,682 --> 00:01:46,369 In this special case that I've drawn, I've indicated here of an amplifier has 21 00:01:46,369 --> 00:01:52,123 to do with the fact that if g is bigger than 1, the terminology is, that it is an 22 00:01:52,123 --> 00:01:54,788 amplifier. It makes things bigger. 23 00:01:54,788 --> 00:01:58,765 It amplifies. There are cases in which g is less than 1 24 00:01:58,765 --> 00:02:02,302 in which the case it serves as an attenuator. 25 00:02:02,302 --> 00:02:07,977 It makes things smaller. And finally, a somewhat confusing 26 00:02:07,977 --> 00:02:13,353 terminology, if G is negative, it's called an inverter. 27 00:02:13,353 --> 00:02:20,765 So it does not mean take the reciprocal. So if we use our unit step input for 28 00:02:20,765 --> 00:02:26,367 example here. So, we bring in new step, and we look at 29 00:02:26,367 --> 00:02:33,667 what the output is going to be. If it G was negative, the output, would 30 00:02:33,667 --> 00:02:39,467 be like that. So, it doesn't mean take the reciprocal 31 00:02:39,467 --> 00:02:45,792 it means change the sign. And I have no idea where the terminology 32 00:02:45,792 --> 00:02:49,542 inverter came from, but that's what it is. 33 00:02:49,542 --> 00:02:56,142 Okay, let's talk about some more systems, and this one we've seen already. 34 00:02:56,142 --> 00:03:02,042 The time delay, the output is equal to the input a little bit later, it's been 35 00:03:02,042 --> 00:03:08,959 delayed, and again using unit step as our example, you delay the output is also a 36 00:03:08,959 --> 00:03:14,318 unit step that is now delayed. It seems like I said, already talking 37 00:03:14,318 --> 00:03:19,004 about signals, pretty easy to understand what it does. 38 00:03:19,004 --> 00:03:25,735 Turns out is a more complicated to build. So, delay boxes turn out to not be easy. 39 00:03:25,735 --> 00:03:30,292 We do want to point out that they do occur naturally. 40 00:03:30,292 --> 00:03:35,988 Light, in fact, all electrical signals travel at a finite speed. 41 00:03:35,988 --> 00:03:40,602 They don't travel from one place or another instant. 42 00:03:40,602 --> 00:03:46,999 So, if something happens at the sun, a change in the brightness of the sun, it 43 00:03:46,999 --> 00:03:51,717 won't reach the earth for about eight and a half minutes. 44 00:03:51,717 --> 00:03:55,351 So that's a delay. That's a physical delay. 45 00:03:55,351 --> 00:04:00,332 Just run an electrical signal through a long piece of wire. 46 00:04:00,332 --> 00:04:06,108 Introduces the delay so in some cases it's pretty easy, it's very hard to have 47 00:04:06,108 --> 00:04:10,776 a controllable delay but that could be a bit more complicated. 48 00:04:10,776 --> 00:04:16,241 So like I said last time if you have tau greater than 0 that's called a time 49 00:04:16,241 --> 00:04:21,892 delay, if tau is negative it's a time advance that occurs earlier in time. 50 00:04:21,892 --> 00:04:31,137 turns out such systems are pretty hard to build, because, again using arguments 51 00:04:31,137 --> 00:04:39,021 step as an example, if time were negative, the output would come out 52 00:04:39,021 --> 00:04:46,345 before the change in the input occured. So if this is, for example, -1, this 53 00:04:46,345 --> 00:04:52,549 output at time 1, -1 is going to have to occur before it occurs at the input. 54 00:04:52,549 --> 00:04:59,230 So, somehow the system would have to know the future value, the future behavior of 55 00:04:59,230 --> 00:05:03,232 the input signal. Well, that's a little weird. 56 00:05:03,232 --> 00:05:09,117 And so, time advances are great mathematical things, but building them, 57 00:05:09,117 --> 00:05:15,288 is, more than just a little complicated. And another example that's along similar 58 00:05:15,288 --> 00:05:21,341 lines, something that's easy to write, but hard to build, and that's called time 59 00:05:21,341 --> 00:05:25,805 reversal. So here, the, using it again our unit 60 00:05:25,805 --> 00:05:31,794 step input, you put in the unit step, and it comes out, delayed. 61 00:05:31,794 --> 00:05:37,683 Not delayed, I'm sorry, my mistake it comes out time-reverse. 62 00:05:37,683 --> 00:05:44,585 So notice, that again, the value at, lets say time-1 is going to be the value of 63 00:05:44,585 --> 00:05:48,601 the input at time 1. The system would have to know what was 64 00:05:48,601 --> 00:05:53,163 going to happen 2 seconds ahead in that case to produce an output. 65 00:05:53,163 --> 00:05:57,569 Well it gets even worse. Let's clean this out, what happened at 66 00:05:57,569 --> 00:06:01,742 time 10 would have to come out at the output at time minus 10. 67 00:06:01,742 --> 00:06:06,907 Which means they would have to be able to predict way ahead in the future, in fact, 68 00:06:06,907 --> 00:06:11,997 have to predict the entire waveform into the infinite future, and then put it out 69 00:06:11,997 --> 00:06:16,667 at the infinite past, so time reverse systems are very weird, very hard to 70 00:06:16,667 --> 00:06:19,462 build. however, they have a very simple 71 00:06:19,462 --> 00:06:25,137 mathematical description. not all simple systems can be built. 72 00:06:25,137 --> 00:06:32,177 Well, let's talk about some systems that are maybe not so simple, but easy to 73 00:06:32,177 --> 00:06:36,577 understand, and that's the derivative system. 74 00:06:36,577 --> 00:06:44,228 So, what this system does is simply take the input and evaluate its derivative, 75 00:06:44,228 --> 00:06:49,010 and that produces the output. So, easy, understanding actually 76 00:06:49,010 --> 00:06:52,530 derivative systems are actually easy to build. 77 00:06:52,530 --> 00:06:57,802 We can also talk about, systems that integrate, and I'm going to point out 78 00:06:57,802 --> 00:07:04,012 something about this, again these are easy to build There's a convention in 79 00:07:04,012 --> 00:07:10,368 this course that's important and that is there are no things such as indefinite 80 00:07:10,368 --> 00:07:15,673 integrals in this course. Every integral is a definite integral. 81 00:07:15,673 --> 00:07:21,558 And most of our integrals will start at minus infinity and go up to time t. 82 00:07:21,558 --> 00:07:26,175 I tend to use Greek letters for variables of integration. 83 00:07:26,175 --> 00:07:31,616 It's just a convention I have to make it clear that it is a variable of 84 00:07:31,616 --> 00:07:35,162 integration. This will come up very frequently. 85 00:07:35,162 --> 00:07:40,497 So, the output y is equal to the integral over all previous time of the input x. 86 00:07:40,497 --> 00:07:46,866 This [INAUDIBLE] and like I said there is no trouble building this system. 87 00:07:46,866 --> 00:07:50,135 Okay. Those were our simple systems. 88 00:07:50,135 --> 00:07:55,187 Let's put them together. So, let's talk about some simple 89 00:07:55,187 --> 00:07:59,347 structures. Perhaps the simplest is called the 90 00:07:59,347 --> 00:08:06,222 cascade structure, and essentially, all it has is one system after another. 91 00:08:06,222 --> 00:08:12,467 We've already seen this in the fundamental model communication. 92 00:08:12,467 --> 00:08:17,831 It is a cascade of several systems, in fact five of them. 93 00:08:17,831 --> 00:08:25,198 So the input to the entire chain is x, goes through s1, produces w and then w in 94 00:08:25,198 --> 00:08:30,787 turn produces y. So we can think about this as one system, 95 00:08:30,787 --> 00:08:35,478 which is constructed as a cascade of two simpler systems. 96 00:08:35,478 --> 00:08:40,169 In fact, that's exactly what cascade structures are for. 97 00:08:40,169 --> 00:08:45,672 For example, suppose you wanted a system that delayed and inverted. 98 00:08:45,672 --> 00:08:50,365 And I could do that by building an inverter let's say here. 99 00:08:50,365 --> 00:08:56,655 Then passing that, the output of inverter into delay.And then the overall 100 00:08:56,655 --> 00:09:01,207 relationship between x and y would be an inverted delay. 101 00:09:01,207 --> 00:09:07,232 So, that's the way you get something a bit more complicated from simpler. 102 00:09:07,232 --> 00:09:10,553 Systems. Another way you can hook systems together 103 00:09:10,553 --> 00:09:15,054 to do interesting things is what's called a parallel connection. 104 00:09:15,054 --> 00:09:19,173 And I think it's pretty easy to see why they're in parallel. 105 00:09:19,173 --> 00:09:23,098 You have two systems here that are in parallel to each other. 106 00:09:23,098 --> 00:09:27,452 There are a couple of conventions here we need to talk about. 107 00:09:27,452 --> 00:09:32,492 I want you to notice here that I have x coming in, and it gets split. 108 00:09:32,492 --> 00:09:36,762 At least that's what the diagram would seem to indicate. 109 00:09:36,762 --> 00:09:42,907 In system theory, the convention is that when such as, a split occurs, it doesn't 110 00:09:42,907 --> 00:09:48,777 mean split the signal in two or anything. What it means is the same signal is 111 00:09:48,777 --> 00:09:53,522 applied to both systems after the split, so just a convention. 112 00:09:53,522 --> 00:09:57,062 That's why I labeled this very carefully x(t). 113 00:09:57,062 --> 00:10:03,257 When we get to talking about this in special cases, you're not going to see me 114 00:10:03,257 --> 00:10:07,102 write this. It will be understood that the input to 115 00:10:07,102 --> 00:10:11,878 s1 and s2 is x(t). That's what this diagram Diagram means. 116 00:10:11,878 --> 00:10:18,748 We also have a new system over here. We have a two input one output system. 117 00:10:18,748 --> 00:10:25,989 And I think it's pretty obvious what it is, it's an adder, so if we were to call 118 00:10:25,989 --> 00:10:32,680 this y 1, the operative of that system, and that y2 and y = y1 + y2. 119 00:10:32,680 --> 00:10:42,400 So, and that's what an adder does, I think it's pretty self-explanatory in 120 00:10:42,400 --> 00:10:47,748 terms of this what's called block diagram. 121 00:10:47,748 --> 00:10:54,379 So a parallel system. And there's two systems in parallel each 122 00:10:54,379 --> 00:11:01,898 having the same input x and you just add up their outputs to produce the total 123 00:11:01,898 --> 00:11:06,304 output. So for example suppose you wanted to 124 00:11:06,304 --> 00:11:12,030 model an echo chamber. So in that case the system s1, we let's 125 00:11:12,030 --> 00:11:15,774 say do nothing. It's output is equal to the input. 126 00:11:15,774 --> 00:11:19,727 The output of s2 or rather s2 would be a delay box. 127 00:11:19,727 --> 00:11:24,722 It would delay it's input. After you add them together, you get the 128 00:11:24,722 --> 00:11:30,221 same signal, plus it's self delayed. That's what happens when you have an 129 00:11:30,221 --> 00:11:33,777 echo. You get the same, you get a signal, plus 130 00:11:33,777 --> 00:11:38,582 its delayed version. So I could build, construct something out 131 00:11:38,582 --> 00:11:42,597 of very simple systems, it's a bit more complicated. 132 00:11:42,597 --> 00:11:48,742 Okay, finally, we have a the most complicated structure that we're going to 133 00:11:48,742 --> 00:11:54,188 encounter the feedback. And I'll explain how it's used in a 134 00:11:54,188 --> 00:12:00,117 second, but let's go through this. So we have the input and here's the 135 00:12:00,117 --> 00:12:04,272 output. The input is going to go through an adder 136 00:12:04,272 --> 00:12:10,262 and the other input to the adder comes from something derived from y. 137 00:12:10,262 --> 00:12:17,117 So, the way we think about this is that this output of this adder is an error 138 00:12:17,117 --> 00:12:21,332 signal. Because you can see this little minus 139 00:12:21,332 --> 00:12:26,937 sign sitting here. And that's the standard convention which 140 00:12:26,937 --> 00:12:31,849 means that the e is equal to x minus the output of S2. 141 00:12:31,849 --> 00:12:38,276 So that little minus sign, right there corresponds to that minus sign. 142 00:12:38,276 --> 00:12:43,417 So that's how you indicate a difference instead of a sum. 143 00:12:43,417 --> 00:12:50,187 So what this feedback configuration does, is it takes e passes it through S1 to 144 00:12:50,187 --> 00:12:54,672 produce the output. But that output gets sent back through S2 145 00:12:54,672 --> 00:12:59,262 to occur at the input. That gets subtracted from x, and produces 146 00:12:59,262 --> 00:13:04,192 the air signal, which goes around, and round, and round, and round. 147 00:13:04,192 --> 00:13:07,722 So this is the way a cruise control works on a car. 148 00:13:07,722 --> 00:13:13,433 You may choose the setting for the velocity which you want the car to go 149 00:13:13,433 --> 00:13:19,696 and, if that is different than it's current velocity, that produces an error 150 00:13:19,696 --> 00:13:23,308 signal. That causes the engine to rub. 151 00:13:23,308 --> 00:13:29,472 Let's clean that to describe the engine. Causes and output velocity y. 152 00:13:29,472 --> 00:13:33,877 Well that goes through the control system as two to produce the air signal. 153 00:13:33,877 --> 00:13:38,437 And finally when the air goes to zero, the cruise control kind of settles down 154 00:13:38,437 --> 00:13:43,127 and doesn't accelerate the car any more. So, that's a great example of feedback 155 00:13:43,127 --> 00:13:45,782 system. I want to point out that everything I 156 00:13:45,782 --> 00:13:48,342 just said in that example wasn't electrical. 157 00:13:48,342 --> 00:13:53,426 Turns out systems can describe a very general class of signal. 158 00:13:53,426 --> 00:13:58,485 The fundamental reason is, they're just mathematical constructs. 159 00:13:58,485 --> 00:14:05,158 And system theory and signal theory for that matter applies to electrical signals 160 00:14:05,158 --> 00:14:11,577 and beyond, so It's interesting that you can use these kind of models to describe 161 00:14:11,577 --> 00:14:16,861 much more general things. Alright, let's talk about special cases, 162 00:14:16,861 --> 00:14:22,717 special systems, and perhaps one of the most important, or linear systems. 163 00:14:22,717 --> 00:14:26,162 So let's look at this. So we have a system S. 164 00:14:26,162 --> 00:14:35,568 If it is linear, if you take an input that consists of a sum of inputs, sum of 165 00:14:35,568 --> 00:14:42,175 the signals. So x1 is a signal, x2 is a signal, and a2 166 00:14:42,175 --> 00:14:49,471 and a1 are scalars, constants. The output of that sum, is the sum of the 167 00:14:49,471 --> 00:14:55,307 outputs to the simple sequence. So, if you put in x1 by itself in S, you 168 00:14:55,307 --> 00:14:59,613 get that for an output, put x2 by itself you get that. 169 00:14:59,613 --> 00:15:05,262 You add those two together with the waiting cost and that's the same. 170 00:15:05,262 --> 00:15:10,196 As if the system saw this sum, presented to it as an input. 171 00:15:10,196 --> 00:15:15,274 So this is a very important property of linear systems. 172 00:15:15,274 --> 00:15:19,135 It's called the principle of superposition. 173 00:15:19,135 --> 00:15:24,820 We're going to use this a lot. We've already seen when we talked about 174 00:15:24,820 --> 00:15:32,117 signals that is very convenient, in many ways, to think about signals as a sum of 175 00:15:32,117 --> 00:15:37,767 simpler signals. So, for example, that triangular signal 176 00:15:37,767 --> 00:15:43,267 we saw in the signals video. We wrote as a sum of three signals. 177 00:15:43,267 --> 00:15:49,888 Well, to find out what the output to sum system is to that signal, all I have to 178 00:15:49,888 --> 00:15:55,573 do is figure out the output to a step, and to a ramp, and a delayed ramp, add 179 00:15:55,573 --> 00:16:01,355 them all up, and that's the going to be output to our complicated signal. 180 00:16:01,355 --> 00:16:06,127 So, this principle of superposition is very, very important. 181 00:16:06,127 --> 00:16:13,342 It helps decompose problems, once you can decompose a signal into simpler parts, 182 00:16:13,342 --> 00:16:20,437 you now can find the output because you probably can easily find the output to a 183 00:16:20,437 --> 00:16:25,962 simple signal. So, here's the some special cases of the 184 00:16:25,962 --> 00:16:31,177 linear system. Let's assume that the linear system here 185 00:16:31,177 --> 00:16:36,127 as is the linear. If you multiply the input by a number, it 186 00:16:36,127 --> 00:16:42,752 changes the input too, now the output is going to be the output you had before 187 00:16:42,752 --> 00:16:49,004 multiplied by the same gain factor and there was a way, almost sloppy way of 188 00:16:49,004 --> 00:16:52,785 saying this, you double the input, you double the output. 189 00:16:52,785 --> 00:16:56,012 If that happens, then the system could be linear. 190 00:16:56,012 --> 00:17:01,117 If that does not happen, if doubling the input does not double the output, then 191 00:17:01,117 --> 00:17:04,416 the system can't be linear. It violates the rules. 192 00:17:04,416 --> 00:17:11,305 That's a special case of what I wrote up here Because, here we have a 2 0, for the 193 00:17:11,305 --> 00:17:19,214 case that I'm showing down here. Okay, then the other case is, just, as I 194 00:17:19,214 --> 00:17:27,476 described in some detail, if you just, decompose a signal into a sum, it's 195 00:17:27,476 --> 00:17:34,177 output Of the system, the system's output to that sum is the sum of the output's to 196 00:17:34,177 --> 00:17:39,552 each, so that, goes just into detail of what we've said before. 197 00:17:39,552 --> 00:17:45,302 But, this result turns out has a very interesting consequence, so let's 198 00:17:45,302 --> 00:17:50,367 consider a special case. A2 is the negative of a1. 199 00:17:50,367 --> 00:17:55,949 And let's say that x1 and x2 are the same thing. 200 00:17:55,949 --> 00:18:03,577 We're just going to call that x. So, a1 and a2 are negatives of each 201 00:18:03,577 --> 00:18:10,267 other, and these two signals has the same thing. 202 00:18:10,267 --> 00:18:21,262 Well, what that gives us, is that the, input to our system is a1x- a1x. 203 00:18:21,262 --> 00:18:28,417 Well that's called 0. But now, let's look at what the linear 204 00:18:28,417 --> 00:18:32,577 system equation says in that special case. 205 00:18:32,577 --> 00:18:39,982 Now, since x1 and x2 are the same, we have S(x) and S(x), and they're weighted 206 00:18:39,982 --> 00:18:44,602 by the same a1, but they're subtracted to get 0. 207 00:18:44,602 --> 00:18:49,062 So, if you have. The system is linear. 208 00:18:49,062 --> 00:18:57,662 You stick in 0, identically 0, into a linear system, what you get out is 0. 209 00:18:57,662 --> 00:19:04,632 Any system which doesn't obey that rule, cannot be linear. 210 00:19:04,632 --> 00:19:10,296 It's sort of a very simple test, but it's not what's called, suffiecient, it's a 211 00:19:10,296 --> 00:19:15,939 neccessary property, but not suffiecient. The real test of wheter something is 212 00:19:15,939 --> 00:19:19,690 linear or not, goes back to the original definetion. 213 00:19:19,690 --> 00:19:25,052 And by the way, in detail, this Principles super position has to apply 214 00:19:25,052 --> 00:19:29,567 for all signals to XY and X2, and all constants A1 and A2. 215 00:19:29,567 --> 00:19:36,982 So, it's a pretty, demanding requirement, that, when you have a linear system, it's 216 00:19:36,982 --> 00:19:39,952 very, very nice. Another special. 217 00:19:39,952 --> 00:19:44,042 A class of systems or so-called timing variance systems. 218 00:19:44,042 --> 00:19:49,742 So what these kind of systems are are is that if you come into the lab, let's say, 219 00:19:49,742 --> 00:19:55,417 and you build a device and you put in an input and you measure the output end, you 220 00:19:55,417 --> 00:20:01,304 leave the system alone, you come back the next day, you put in the same signal And, 221 00:20:01,304 --> 00:20:07,149 what you should get out is the, exactly the same output, as you had the previous 222 00:20:07,149 --> 00:20:10,661 day. In that scenario, I just described, means 223 00:20:10,661 --> 00:20:16,415 that, I delay the input to the next day, and what happened? I got out that day, 224 00:20:16,415 --> 00:20:20,752 the same thing as what I got the previous day, but delayed. 225 00:20:20,752 --> 00:20:23,599 [INAUDIBLE] day, you have the information. 226 00:20:23,599 --> 00:20:27,719 So systems that do not change their behavior with time, are called 227 00:20:27,719 --> 00:20:31,720 Time-Invariant Systems. That does not mean, signals can't vary 228 00:20:31,720 --> 00:20:36,699 with time, that's certainly not the case. What it does mean, the system doesn't 229 00:20:36,699 --> 00:20:39,912 change its behavior, with time, stays the same. 230 00:20:39,912 --> 00:20:45,062 So, let's go through a little, set of examples, and see if we can, figure out 231 00:20:45,062 --> 00:20:49,387 how to classify signals according to whether they're linear, or 232 00:20:49,387 --> 00:20:52,587 time-invariant. So, here we have the game box. 233 00:20:52,587 --> 00:20:57,412 Multiply the signal the, and amplifier, let's say g is bigger than 1 and 234 00:20:57,412 --> 00:21:03,744 positive, it's an amplifier. Is that linear? Well, if I change x to x1 235 00:21:03,744 --> 00:21:09,318 + x2, do I get G * x1 + G * x2? I certainly do. 236 00:21:09,318 --> 00:21:16,375 It's linear. Is it time invariant? Well, I think it's 237 00:21:16,375 --> 00:21:22,907 pretty obvious that it is but I just change T to T minus tau. 238 00:21:22,907 --> 00:21:28,872 What I get is the previous expression for Y, if I stick in T minus tau there. 239 00:21:28,872 --> 00:21:33,607 So this is linear and time invariant. No problem with that. 240 00:21:33,607 --> 00:21:37,932 Okay, another example. How about the derivative box. 241 00:21:37,932 --> 00:21:44,402 Something that takes the derivative. Is that linear? If you take the 242 00:21:44,402 --> 00:21:52,563 derivative, of a sum, is that the sum of the derivatives? And I think you will 243 00:21:52,563 --> 00:21:58,012 agree, yeah sure, that's true. So that's, linear. 244 00:21:58,012 --> 00:22:04,862 How about time-invariant? Once the derivative of x / t minus tau, well, 245 00:22:04,862 --> 00:22:11,543 that's, in calculus you learn that's the derivative of x evaluated at t minus tau. 246 00:22:11,543 --> 00:22:17,973 Well that makes it time invariant. So, that's we got two special cases here 247 00:22:17,973 --> 00:22:24,144 of linear and time invariant systems. How about this system? How about this 248 00:22:24,144 --> 00:22:30,727 squaring system? So the output equals to the input squared. 249 00:22:30,727 --> 00:22:39,846 While that is certainly not linear. Alright, because the square of X 1 plus X 250 00:22:39,846 --> 00:22:45,202 2 is not equal to X 1 squared plus X 2 squared. 251 00:22:45,202 --> 00:22:51,327 How about time-invariant? If I, replace t, by t minus tau, is that the same 252 00:22:51,327 --> 00:22:57,177 formula I'd get if I replaced t minus tau in this, in this relationship, 253 00:22:57,177 --> 00:23:02,932 input/output relationship, and the answer is, it certainly is. 254 00:23:02,932 --> 00:23:08,532 This is an example of a non-linear, but Time-Invariant System. 255 00:23:08,532 --> 00:23:15,707 And finally, how about this system, which is called a modulator? So, what we have 256 00:23:15,707 --> 00:23:22,857 is our input is here, and what happens is that the system multiplies it by cosine 2 257 00:23:22,857 --> 00:23:30,907 pi f ct to produce an output. Well, is it linear? So if I have x1 + x2, 258 00:23:30,907 --> 00:23:37,632 will I get out, cosine x1 + cosine x2? And I certainly do. 259 00:23:37,632 --> 00:23:44,348 So, that's linear. How about time invariant? So, the thing 260 00:23:44,348 --> 00:23:50,025 to notice is that the delay applies only to x. 261 00:23:50,025 --> 00:23:59,381 So if I replace x here by t minus tau. You don't replace it over here because 262 00:23:59,381 --> 00:24:06,156 that's inside the system. That's not the input delay. 263 00:24:06,156 --> 00:24:11,739 Well that's not the same expression that you get, if you replace t minus tau over 264 00:24:11,739 --> 00:24:17,145 here in this formula, you have to replace the t in both places by t minus tau. 265 00:24:17,145 --> 00:24:23,231 So this system is not time-invariant, due to delay of the input you do not get the 266 00:24:23,231 --> 00:24:28,651 outputs we had before, delayed because this co sign is changing behavior with 267 00:24:28,651 --> 00:24:32,096 time. So we have the cases where both of these 268 00:24:32,096 --> 00:24:36,867 examples you going to have systems also that are not linear and not time 269 00:24:36,867 --> 00:24:40,911 invariant of course. And the ones that are really going to be 270 00:24:40,911 --> 00:24:45,553 special are the ones that are linear and And time invariant. 271 00:24:45,553 --> 00:24:48,754 We're going to have to deal with this one. 272 00:24:48,754 --> 00:24:54,938 But we're clearly can't use the linear time invariant system theory to talk 273 00:24:54,938 --> 00:24:58,536 about it, because it is not time invariant. 274 00:24:58,536 --> 00:25:03,493 We'll see that in a second. Well okay, we now know a lot about 275 00:25:03,493 --> 00:25:07,049 systems. What systems do, is they operate on input 276 00:25:07,049 --> 00:25:11,439 to produce an output for some reason. You might want to amplify it. 277 00:25:11,439 --> 00:25:15,950 You might want to delay the signal. Take it's derivative, whatever. 278 00:25:15,950 --> 00:25:19,706 That's what systems do. They can do all kinds of things to 279 00:25:19,706 --> 00:25:25,342 signals that are interesting and by building up structures in simple systems, 280 00:25:25,342 --> 00:25:31,412 you can make something that's a lot more complicated than any of the components. 281 00:25:31,412 --> 00:25:36,022 And that's basically the way electrical engineers do a lot of design. 282 00:25:36,022 --> 00:25:39,282 Breakdwon a complicated input and output relationship into a simpler set that I 283 00:25:39,282 --> 00:25:43,922 can realize by cascade, parallel, and feedback structures. 284 00:25:43,922 --> 00:25:49,359 And I keep emphasizing that linear time-invariant systems are very 285 00:25:49,359 --> 00:25:52,471 important. What we're going to call LTI. 286 00:25:52,471 --> 00:25:58,695 LTI systems really are important. They occur a lot, and as what we see when 287 00:25:58,695 --> 00:26:05,026 we start talking about circuits, we're going to talk about linear time invariant 288 00:26:05,026 --> 00:26:08,637 circuits. When we get to signal processing we'll 289 00:26:08,637 --> 00:26:12,197 talk about linear time-invariant signal processing. 290 00:26:12,197 --> 00:26:17,347 Those special systems are very important because you can do a lot of very 291 00:26:17,347 --> 00:26:23,107 interesting things and have a very good theory for them when we start talking 292 00:26:23,107 --> 00:26:25,430 about them. That comes very soon.