WEBVTT 00:00:00.040 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:03.880 Commons license. 00:00:03.880 --> 00:00:06.920 Your support will help MIT OpenCourseWare continue to 00:00:06.920 --> 00:00:10.570 offer high quality educational resources for free. 00:00:10.570 --> 00:00:13.470 To make a donation or view additional materials from 00:00:13.470 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.540 ocw.mit.edu. 00:00:59.745 --> 00:01:02.330 PROFESSOR: in the last lecture, we discussed 00:01:02.330 --> 00:01:06.370 sinusoidal and real and complex exponential signals 00:01:06.370 --> 00:01:10.110 both for continuous time and discrete time. 00:01:10.110 --> 00:01:14.520 And those signals will form very important building blocks 00:01:14.520 --> 00:01:19.100 when we return to a discussion of Fourier 00:01:19.100 --> 00:01:22.540 analysis in a later lecture. 00:01:22.540 --> 00:01:25.830 In today's lecture, I'd like to introduce some additional 00:01:25.830 --> 00:01:29.480 basic signals, specifically the unit step and 00:01:29.480 --> 00:01:31.830 unit impulse signal. 00:01:31.830 --> 00:01:35.760 Let's begin with discrete time and the discrete-time unit 00:01:35.760 --> 00:01:38.820 step and unit impulse. 00:01:38.820 --> 00:01:42.740 The discrete-time unit step is a sequence as I've indicated 00:01:42.740 --> 00:01:50.570 here, specifically a sequence which is 0 for negative values 00:01:50.570 --> 00:01:56.520 of its argument, and equal to 1 for positive values of its 00:01:56.520 --> 00:01:58.970 argument and 0. 00:01:58.970 --> 00:02:06.230 So mathematically, the unit step sequence is 1 for n 00:02:06.230 --> 00:02:10.460 greater than or equal to 0 and 0 for n less than 0. 00:02:13.660 --> 00:02:17.730 The unit impulse sequence, likewise, is defined in a 00:02:17.730 --> 00:02:19.430 straightforward way. 00:02:19.430 --> 00:02:25.630 The unit impulse sequence is a sequence which is 0 for all 00:02:25.630 --> 00:02:31.210 values of its argument except for n = 0. 00:02:31.210 --> 00:02:35.520 So the unit step and unit impulse sequence are defined 00:02:35.520 --> 00:02:38.210 in a straightforward way mathematically. 00:02:38.210 --> 00:02:42.430 And in fact, they are also related to each other in a 00:02:42.430 --> 00:02:45.170 straightforward way mathematically. 00:02:45.170 --> 00:02:50.050 Specifically, the unit impulse can be related to the unit 00:02:50.050 --> 00:02:55.590 step through the relationship that I've indicated here-- 00:02:55.590 --> 00:03:01.000 delta of n, the unit impulse, equal to a unit step minus the 00:03:01.000 --> 00:03:02.740 unit step delayed. 00:03:02.740 --> 00:03:06.930 So mathematically, the relationship is what is 00:03:06.930 --> 00:03:11.075 referred to as a first difference. 00:03:14.180 --> 00:03:19.330 And to see the validity of this expression, we can simply 00:03:19.330 --> 00:03:24.380 look at the unit step and its delayed version. 00:03:24.380 --> 00:03:29.400 So here, we show the unit step, u[n]. 00:03:29.400 --> 00:03:34.060 Here, we show the unit step delayed by 1. 00:03:34.060 --> 00:03:37.890 So it's 0 for n less than or equal to 0. 00:03:37.890 --> 00:03:42.700 And clearly, if we subtract the delayed step from the 00:03:42.700 --> 00:03:49.350 original unit step, everything subtracts out except at n = 0, 00:03:49.350 --> 00:03:52.070 at which point the difference is equal to 1. 00:03:52.070 --> 00:03:55.540 And so the difference between u[n] 00:03:55.540 --> 00:03:57.430 and u[n-1] 00:03:57.430 --> 00:04:02.180 is simply the unit impulse, sometimes incidentally also 00:04:02.180 --> 00:04:03.880 referred to as the unit sample. 00:04:06.490 --> 00:04:12.500 Now, in a similar way, we can express the unit step in terms 00:04:12.500 --> 00:04:13.550 of the unit impulse. 00:04:13.550 --> 00:04:16.579 And there are several ways of doing this. 00:04:16.579 --> 00:04:21.110 One way is through a relationship referred to as a 00:04:21.110 --> 00:04:23.080 running sum. 00:04:23.080 --> 00:04:25.740 What I mean by that is the following expression. 00:04:28.260 --> 00:04:37.960 If we think of forming the sum from minus infinity up to some 00:04:37.960 --> 00:04:45.620 value n of a unit impulse or unit sample, then this running 00:04:45.620 --> 00:04:50.480 sum, in fact, is equal to the unit step. 00:04:50.480 --> 00:04:55.000 And we can see that in a fairly straightforward way, by 00:04:55.000 --> 00:05:01.260 simply observing that in this expression, when n is less 00:05:01.260 --> 00:05:05.750 than 0, there's nothing accumulated in the sum. 00:05:05.750 --> 00:05:07.530 And we can see that graphically 00:05:07.530 --> 00:05:09.880 as I've shown here. 00:05:09.880 --> 00:05:12.530 So for n less than 0, so we accumulate no 00:05:12.530 --> 00:05:14.340 terms in the sum. 00:05:14.340 --> 00:05:20.800 Whereas for n greater than 0, we accumulate 1 non-zero value 00:05:20.800 --> 00:05:26.900 in the sum, namely the value of the unit sample at n = 0. 00:05:26.900 --> 00:05:30.520 So we have, then, one expression for the 00:05:30.520 --> 00:05:34.700 relationship between the unit step and the unit sample. 00:05:34.700 --> 00:05:41.810 We can also develop another relationship by observing in 00:05:41.810 --> 00:05:46.460 essence that if we look at the unit step sequence, as I've 00:05:46.460 --> 00:05:51.850 returned to here, we can, in effect, think of the unit step 00:05:51.850 --> 00:05:56.890 sequence as a succession of unit impulses, 00:05:56.890 --> 00:06:00.320 one following another. 00:06:00.320 --> 00:06:14.270 So if we consider forming a sum of delayed impulses, as I 00:06:14.270 --> 00:06:18.430 indicate mathematically here, and as I indicate graphically 00:06:18.430 --> 00:06:24.680 down below, we have an impulse here at n = 0 and an impulse 00:06:24.680 --> 00:06:30.160 here at n = 1, an impulse here at n = 2, et cetera. 00:06:30.160 --> 00:06:34.930 And when we continue to add these up, then what they add 00:06:34.930 --> 00:06:39.740 up to is the unit step sequence. 00:06:39.740 --> 00:06:43.840 And so mathematically, then, that would correspond to an 00:06:43.840 --> 00:06:48.640 impulse at n = 0 plus an impulse at n = 1 plus an 00:06:48.640 --> 00:06:50.960 impulse at n = 2, et cetera. 00:06:53.770 --> 00:06:57.980 Now, in continuous time, we have a very more or less 00:06:57.980 --> 00:07:00.270 similar situation. 00:07:00.270 --> 00:07:05.330 We will find it equally useful to talk about a unit step 00:07:05.330 --> 00:07:08.540 continuous-time signal and a unit impulse 00:07:08.540 --> 00:07:10.550 continuous-time signal. 00:07:10.550 --> 00:07:16.670 Let's begin with the continuous-time unit step. 00:07:16.670 --> 00:07:22.090 The continuous-time unit step function is graphically 00:07:22.090 --> 00:07:24.870 indicated as I've shown here. 00:07:24.870 --> 00:07:30.850 It's a time function which is 0 for t less than 0. 00:07:30.850 --> 00:07:34.360 And it's 1 for t greater than 0. 00:07:34.360 --> 00:07:38.970 And so mathematically, what it corresponds to is u[t] 00:07:38.970 --> 00:07:44.720 t defined as a time function which is 0 for t less than 0, 00:07:44.720 --> 00:07:48.250 1 for t greater than 0. 00:07:48.250 --> 00:07:53.710 Now, an obvious question is, what happens at t = 0? 00:07:53.710 --> 00:07:56.890 And the difficulty here-- which is not a difficulty that 00:07:56.890 --> 00:07:59.360 arises in the discrete-time case-- 00:07:59.360 --> 00:08:03.550 is that at t = 0, the units step function is in fact 00:08:03.550 --> 00:08:06.770 discontinuous, which generates a variety of 00:08:06.770 --> 00:08:08.460 mathematical problems. 00:08:08.460 --> 00:08:14.350 And one can define the unit step at t = 0 in a variety of 00:08:14.350 --> 00:08:19.020 ways, but the essential point is that the unit step function 00:08:19.020 --> 00:08:21.220 is discontinuous. 00:08:21.220 --> 00:08:25.190 So in effect, what we need to do is think of the unit step 00:08:25.190 --> 00:08:29.980 function as the limit of a continuous function. 00:08:29.980 --> 00:08:34.549 And so we can define a function, which I specify here 00:08:34.549 --> 00:08:38.480 as u_delta(t) (u sub delta of t). 00:08:38.480 --> 00:08:44.420 And u_delta(t) is a time function which is 0 for t less 00:08:44.420 --> 00:08:49.950 than 0, linearly increases to time delta which would 00:08:49.950 --> 00:08:52.720 correspond to this break point, and 00:08:52.720 --> 00:08:55.450 then 1 following that. 00:08:55.450 --> 00:09:01.620 And so we can think then of the discontinuous unit step as 00:09:01.620 --> 00:09:08.350 the limiting form of u_delta(t) as delta goes to 0. 00:09:08.350 --> 00:09:13.880 Now, we also want to define a unit impulse function. 00:09:13.880 --> 00:09:16.880 And it had a fairly straightforward definition in 00:09:16.880 --> 00:09:17.840 discrete time. 00:09:17.840 --> 00:09:22.130 In continuous time, things get slightly more difficult. 00:09:22.130 --> 00:09:25.370 And to motivate the definition, let me return to 00:09:25.370 --> 00:09:28.980 the discrete-time definition-- 00:09:28.980 --> 00:09:33.130 or rather, the discrete-time relationship between the unit 00:09:33.130 --> 00:09:37.010 step and the unit impulse function. 00:09:37.010 --> 00:09:44.660 In discrete time, we saw that the unit impulse function is 00:09:44.660 --> 00:09:50.790 the first difference of the unit step function. 00:09:50.790 --> 00:09:56.400 Well, similarly in continuous time, we can talk about an 00:09:56.400 --> 00:10:02.560 impulse function, which is the first derivative of a unit 00:10:02.560 --> 00:10:05.030 step function. 00:10:05.030 --> 00:10:09.320 So the unit impulse, as we want to define it, is the 00:10:09.320 --> 00:10:11.850 derivative of the unit step. 00:10:11.850 --> 00:10:15.640 Of course, we just finished discussing the fact that the 00:10:15.640 --> 00:10:21.710 unit step function is, in fact, discontinuous at t = 0. 00:10:21.710 --> 00:10:26.960 But we can think of its derivative as related to this 00:10:26.960 --> 00:10:29.840 approximation to the unit step. 00:10:29.840 --> 00:10:34.310 And specifically, we will think of the continuous-time 00:10:34.310 --> 00:10:42.540 impulse as the derivative of u_delta(t) as delta goes to 0. 00:10:42.540 --> 00:10:47.590 So to define the unit impulse, we think of the derivative of 00:10:47.590 --> 00:10:51.950 this approximation to the unit step and then observe what 00:10:51.950 --> 00:10:55.440 happens as delta goes to 0. 00:10:55.440 --> 00:11:04.270 We have, then, the definition of the unit impulse function 00:11:04.270 --> 00:11:09.100 more or less formally defined as the first derivative of the 00:11:09.100 --> 00:11:17.320 unit step, or thought of as the limiting form of the 00:11:17.320 --> 00:11:22.360 derivative of the approximation to the unit step 00:11:22.360 --> 00:11:27.380 in the limit, as delta, the duration of the 00:11:27.380 --> 00:11:29.210 discontinuity, goes to 0. 00:11:32.500 --> 00:11:35.780 Well, let's look at that. 00:11:35.780 --> 00:11:42.820 If we think about the derivative of u_delta(t), the 00:11:42.820 --> 00:11:47.230 derivative, of course, is 0 for t less than 0. 00:11:47.230 --> 00:11:51.270 It's equal to a constant during this linear slope, and 00:11:51.270 --> 00:11:55.220 then, it's 0 for t greater than delta. 00:11:55.220 --> 00:12:01.070 So the derivative of u_delta(t) will then be as 00:12:01.070 --> 00:12:04.000 I've indicated here. 00:12:04.000 --> 00:12:09.110 And it's simply a rectangle with a height, which is 1 / 00:12:09.110 --> 00:12:13.260 delta, and a width, which is equal to delta. 00:12:13.260 --> 00:12:17.460 And observe that no matter what the value of delta is, 00:12:17.460 --> 00:12:22.260 the area is always equal to 1. 00:12:22.260 --> 00:12:27.340 Now, as we let delta go to 0, what happens is that the width 00:12:27.340 --> 00:12:30.270 of the rectangle gets smaller, the height of the rectangle 00:12:30.270 --> 00:12:33.900 gets bigger, the area still remains 1. 00:12:33.900 --> 00:12:38.210 As delta goes to 0, of course, the width goes to 0, and the 00:12:38.210 --> 00:12:40.470 height goes to infinity. 00:12:40.470 --> 00:12:48.550 And graphically, we choose to depict that as an arrow, where 00:12:48.550 --> 00:12:53.240 the arrow indicates the fact that we have an impulse 00:12:53.240 --> 00:12:58.890 occurring at t = 0, and the height of the impulse is used 00:12:58.890 --> 00:13:03.380 to represent what the area of the impulse is. 00:13:03.380 --> 00:13:06.350 And in this case, since we took the derivative of a unit 00:13:06.350 --> 00:13:10.010 step, the height is equal to 1. 00:13:13.300 --> 00:13:19.360 So the impulse has 0 width, infinite height, area 1. 00:13:19.360 --> 00:13:24.060 It's mathematically not terribly comfortable because 00:13:24.060 --> 00:13:27.310 what we've done is taken the derivative of the unit step, 00:13:27.310 --> 00:13:30.970 which has a discontinuity at the origin, and there's some 00:13:30.970 --> 00:13:33.780 mathematical difficulties in doing that. 00:13:33.780 --> 00:13:37.700 We'll in fact return to another interpretation of the 00:13:37.700 --> 00:13:43.990 impulse later to emphasize some of the discomfort with 00:13:43.990 --> 00:13:45.460 the impulse. 00:13:45.460 --> 00:13:49.820 I remember something that Sam Mason used to say that his 00:13:49.820 --> 00:13:52.990 students said about the unit impulse. 00:13:52.990 --> 00:13:56.810 His definition was the unit impulse is something that's so 00:13:56.810 --> 00:14:00.470 small every place I can't see it except at one point where 00:14:00.470 --> 00:14:01.870 it's so big I can't see it. 00:14:01.870 --> 00:14:04.090 In other words, I can't see it at. 00:14:04.090 --> 00:14:10.440 Well, accept some informality with the unit impulse function 00:14:10.440 --> 00:14:15.410 in the continuous time case, and generally, we'll see that 00:14:15.410 --> 00:14:17.505 we won't get into particular difficulty. 00:14:20.040 --> 00:14:25.230 OK, now the impulse is the derivative of the step. 00:14:25.230 --> 00:14:29.510 We saw in the discrete-time case that the step could be 00:14:29.510 --> 00:14:32.952 recovered from the impulse through a running sum. 00:14:32.952 --> 00:14:36.250 In continuous time, if the impulse is the derivative of a 00:14:36.250 --> 00:14:39.900 step, we would more or less reasonably expect that the 00:14:39.900 --> 00:14:43.420 step would be like an integral of the impulse. 00:14:43.420 --> 00:14:45.370 And indeed, that's true. 00:14:45.370 --> 00:14:52.350 In fact, mathematically the relationship is that the unit 00:14:52.350 --> 00:14:59.340 step function in continuous time is the running integral 00:14:59.340 --> 00:15:02.230 of the unit impulse function. 00:15:02.230 --> 00:15:07.330 And so it's the integral from minus infinity up to time t, 00:15:07.330 --> 00:15:13.900 where t is the argument at which we're examining u(t). 00:15:13.900 --> 00:15:20.270 And so, just as with the discrete-time case, if we are 00:15:20.270 --> 00:15:24.580 looking at t less than 0, there's no area accumulated in 00:15:24.580 --> 00:15:25.780 the integral. 00:15:25.780 --> 00:15:29.620 If we're looking at t greater than 0, then we accumulate the 00:15:29.620 --> 00:15:31.255 area under the impulse. 00:15:35.760 --> 00:15:43.160 OK, now we'll shortly be returning to a further 00:15:43.160 --> 00:15:48.390 discussion on use of impulses and step functions. 00:15:48.390 --> 00:15:52.580 And in particular, what we'll see is that they provide a 00:15:52.580 --> 00:15:58.300 very convenient and powerful mechanism for describing a 00:15:58.300 --> 00:16:01.900 particular class of systems referred to as linear 00:16:01.900 --> 00:16:04.650 time-invariant systems. 00:16:04.650 --> 00:16:07.710 To lead up to that discussion, which will be the principal 00:16:07.710 --> 00:16:12.470 focus of the next lecture, let's for the remainder of 00:16:12.470 --> 00:16:15.980 this lecture talk about systems in general and then 00:16:15.980 --> 00:16:17.860 some properties of systems. 00:16:17.860 --> 00:16:23.380 And as the lecture proceeds, the specific properties that I 00:16:23.380 --> 00:16:26.085 want to get to are properties of linearity and time 00:16:26.085 --> 00:16:28.010 invariance. 00:16:28.010 --> 00:16:31.820 So let's first talk about systems in general. 00:16:31.820 --> 00:16:36.850 And a system in general, in its most general definition, 00:16:36.850 --> 00:16:41.770 is simply a transformation from an input signal to an 00:16:41.770 --> 00:16:43.840 output signal. 00:16:43.840 --> 00:16:48.360 So in a continuous-time case, we would have a 00:16:48.360 --> 00:16:53.740 continuous-time system, the input, x[t], 00:16:53.740 --> 00:16:56.140 and the output, y[t]. 00:16:56.140 --> 00:17:00.410 And the box, in essence, is used to denote a 00:17:00.410 --> 00:17:02.600 transformation from x[t] 00:17:02.600 --> 00:17:04.740 to y[t]. 00:17:04.740 --> 00:17:09.079 And sometimes, we'll also use a shorthand notation along the 00:17:09.079 --> 00:17:14.240 lines of indicating that the input, x[t], is transformed to 00:17:14.240 --> 00:17:17.700 the output, y[t]. 00:17:17.700 --> 00:17:21.540 Now, we have exactly the same kind of definition in the 00:17:21.540 --> 00:17:23.349 discrete-time case. 00:17:23.349 --> 00:17:27.020 In the discrete-time case, of course, the inputs are 00:17:27.020 --> 00:17:31.000 sequences, and the outputs are sequences. 00:17:31.000 --> 00:17:33.450 And our shorthand notation is similar. 00:17:36.250 --> 00:17:39.280 Often when we talk about systems, we'll want to talk 00:17:39.280 --> 00:17:41.380 about interconnections of systems. 00:17:41.380 --> 00:17:45.080 And we'll see again in later lectures that interconnections 00:17:45.080 --> 00:17:48.080 become very important and powerful. 00:17:48.080 --> 00:17:53.660 And in the way of introducing terminology, let me introduce 00:17:53.660 --> 00:17:56.490 the terminology for a few basic and important 00:17:56.490 --> 00:17:59.840 interconnections of systems. 00:17:59.840 --> 00:18:07.230 The first is what's referred to as a cascade of systems, or 00:18:07.230 --> 00:18:11.110 sometimes as a series interconnection of systems. 00:18:11.110 --> 00:18:14.720 And putting two systems in cascade, as I've indicated 00:18:14.720 --> 00:18:19.860 here, means taking the output of one system-- 00:18:19.860 --> 00:18:21.780 let's say system 1-- 00:18:21.780 --> 00:18:25.890 and using that as the input to the second system, which I've 00:18:25.890 --> 00:18:27.880 denoted as system 2. 00:18:27.880 --> 00:18:34.420 So in this cascade with system 1 first and system 2 second, 00:18:34.420 --> 00:18:37.740 we have the output of system 1 going into the 00:18:37.740 --> 00:18:40.710 input of system 2. 00:18:40.710 --> 00:18:46.000 Now, we could, of course, take those two systems and cascade 00:18:46.000 --> 00:18:48.570 them in the reverse order. 00:18:48.570 --> 00:18:52.630 Namely, take the output of system 2 and put it into the 00:18:52.630 --> 00:18:54.650 input of system 1. 00:18:54.650 --> 00:18:56.995 And I've indicated that here. 00:18:56.995 --> 00:19:00.330 And we have system 2 first in the cascade 00:19:00.330 --> 00:19:04.440 followed by system 1. 00:19:04.440 --> 00:19:09.370 It's important to keep in mind that in general, for general 00:19:09.370 --> 00:19:13.480 systems, the order in which you cascade the systems is 00:19:13.480 --> 00:19:15.440 very important. 00:19:15.440 --> 00:19:19.640 And in fact, the overall system transformation will be 00:19:19.640 --> 00:19:22.710 different depending on which system came first and which 00:19:22.710 --> 00:19:24.840 system came second. 00:19:24.840 --> 00:19:29.130 For example, if system 1 was, let's say, a system for which 00:19:29.130 --> 00:19:35.390 the output doubles the input, and system 2, the output is 00:19:35.390 --> 00:19:39.550 the square root of the input, clearly doubling first and 00:19:39.550 --> 00:19:43.080 then taking the square root is different than taking the 00:19:43.080 --> 00:19:46.680 square root first and then doubling. 00:19:46.680 --> 00:19:52.570 Now, kind of amazingly, what we'll see again when we get to 00:19:52.570 --> 00:19:55.140 this issue of linearity and time invariance-- 00:19:55.140 --> 00:19:58.340 which are a class of systems that we'll focus in on-- 00:19:58.340 --> 00:20:02.640 somewhat amazingly, it turns out that for that specific 00:20:02.640 --> 00:20:07.480 class of systems, the overall system transformation is 00:20:07.480 --> 00:20:09.680 independent of the order in which 00:20:09.680 --> 00:20:12.280 the systems are cascaded. 00:20:12.280 --> 00:20:16.280 And we'll see that, of course, in more detail later on. 00:20:16.280 --> 00:20:19.110 All right, well, that's a cascade or a series 00:20:19.110 --> 00:20:20.970 interconnection. 00:20:20.970 --> 00:20:24.240 Let's look at another interconnection, which is a 00:20:24.240 --> 00:20:27.270 parallel interconnection of systems. 00:20:27.270 --> 00:20:33.110 And here, what's meant by the interconnection is that the 00:20:33.110 --> 00:20:39.220 input is fed simultaneously into system 1 00:20:39.220 --> 00:20:42.260 and into system 2. 00:20:42.260 --> 00:20:47.410 And then, the outputs of these two systems are added to give 00:20:47.410 --> 00:20:50.900 the overall system output. 00:20:50.900 --> 00:20:57.670 Now, in this particular case, in contrast to a cascade, no 00:20:57.670 --> 00:21:02.710 matter what the systems are, it turns out that a parallel 00:21:02.710 --> 00:21:06.520 combination, the order in which they're 00:21:06.520 --> 00:21:09.740 put in parallel is-- 00:21:09.740 --> 00:21:13.660 the overall transformation is independent of the order. 00:21:13.660 --> 00:21:16.230 And of course, that follows from the fact that we're 00:21:16.230 --> 00:21:18.460 simply adding up outputs. 00:21:18.460 --> 00:21:22.120 And the outputs can be added in any order because of the 00:21:22.120 --> 00:21:25.000 fact that addition doesn't care in which order you're 00:21:25.000 --> 00:21:27.840 adding things up. 00:21:27.840 --> 00:21:30.880 OK, so that's the parallel interconnection. 00:21:30.880 --> 00:21:35.460 And let's look at one more interconnection, which is 00:21:35.460 --> 00:21:39.320 what's referred to as a feedback interconnection. 00:21:39.320 --> 00:21:43.530 And this, again, is an interconnection that will 00:21:43.530 --> 00:21:47.320 become a very important topic much later in the course. 00:21:47.320 --> 00:21:51.940 But let me just indicate at this point what I mean by it. 00:21:51.940 --> 00:21:55.000 What a feedback interconnection means is that 00:21:55.000 --> 00:22:05.090 we have one system, system 1, with an input and an output. 00:22:05.090 --> 00:22:09.710 The output of system 1 is the output of the overall system. 00:22:09.710 --> 00:22:16.120 And that output is fed into system 2. 00:22:16.120 --> 00:22:22.260 The output of system 2 is then added to the input to the 00:22:22.260 --> 00:22:23.860 overall system. 00:22:23.860 --> 00:22:27.790 So in essence, what happens in a feedback interconnection is 00:22:27.790 --> 00:22:29.910 we have one system. 00:22:29.910 --> 00:22:35.410 The output of that system is fed back through system 2, 00:22:35.410 --> 00:22:37.290 added to the overall input. 00:22:37.290 --> 00:22:40.770 And then, that sum is what forms the input of system 1. 00:22:40.770 --> 00:22:45.870 And we'll see that there are lots of uses for feedback and, 00:22:45.870 --> 00:22:49.680 of course, also lots of ways that feedback gets in the way. 00:22:49.680 --> 00:22:53.700 In fact, probably some of you are already familiar with some 00:22:53.700 --> 00:22:55.580 of the issues in feedback-- 00:22:55.580 --> 00:22:57.910 for example, in audio systems or whatever. 00:22:57.910 --> 00:23:01.750 But this will be a topic that we'll devote a considerable 00:23:01.750 --> 00:23:05.580 amount of time to later in the course. 00:23:05.580 --> 00:23:10.170 And then, of course, there are lots of other interconnections 00:23:10.170 --> 00:23:12.640 of systems. 00:23:12.640 --> 00:23:15.900 And as the course progresses, we'll see lots of ways in 00:23:15.900 --> 00:23:19.270 which systems get interconnected both in series 00:23:19.270 --> 00:23:22.700 and in parallel and feedback interconnections, et cetera, 00:23:22.700 --> 00:23:28.040 to achieve a wide variety of things. 00:23:28.040 --> 00:23:33.720 Now, what I've said so far relates to systems in general. 00:23:33.720 --> 00:23:37.240 And you can't say much about systems when you try to treat 00:23:37.240 --> 00:23:39.470 them in their most general form. 00:23:39.470 --> 00:23:44.845 So it's useful and important to focus in on properties that 00:23:44.845 --> 00:23:47.160 a system may or may not have. 00:23:47.160 --> 00:23:51.730 So what I'd like to do now is turn our attention to system 00:23:51.730 --> 00:23:53.000 properties. 00:23:53.000 --> 00:23:56.120 And we'll define a number of them. 00:23:56.120 --> 00:23:59.220 Some of them, we'll want to impose on a system. 00:23:59.220 --> 00:24:02.460 Some of them, we may not want to impose on a system. 00:24:02.460 --> 00:24:08.810 But as things progress, we'll tend to find it useful to ask 00:24:08.810 --> 00:24:10.540 whether a system does or doesn't 00:24:10.540 --> 00:24:13.200 have a certain property. 00:24:13.200 --> 00:24:18.640 Well, let's begin with a property which I refer to here 00:24:18.640 --> 00:24:20.690 as memoryless. 00:24:20.690 --> 00:24:26.540 And what I mean by a system being memoryless is that the 00:24:26.540 --> 00:24:35.590 output at any given time, t_0, depends only on the input at 00:24:35.590 --> 00:24:36.550 the same time. 00:24:36.550 --> 00:24:41.560 And so what I'm suggesting is that a memoryless system is 00:24:41.560 --> 00:24:48.320 one for which the output at a specific time depends only on 00:24:48.320 --> 00:24:50.520 the input at that time. 00:24:50.520 --> 00:24:56.080 And that statement is true, or that definition applies, both 00:24:56.080 --> 00:25:03.130 for continuous time, as I've indicated here, and also for 00:25:03.130 --> 00:25:05.390 discrete time, as I've indicated here. 00:25:05.390 --> 00:25:10.670 And so we have a similar definition in discrete time, 00:25:10.670 --> 00:25:16.650 that the system is memoryless if the output at any given 00:25:16.650 --> 00:25:20.750 time depends only on the input at that time. 00:25:23.490 --> 00:25:27.000 Well, I have a number of examples here. 00:25:27.000 --> 00:25:31.760 Let's look at the first example in which the output is 00:25:31.760 --> 00:25:33.520 the square of the input. 00:25:33.520 --> 00:25:39.170 And that is a possible system either in continuous time or 00:25:39.170 --> 00:25:40.680 discrete time. 00:25:40.680 --> 00:25:45.410 And that system, as you would expect, is commonly referred 00:25:45.410 --> 00:25:48.670 to a squarer. 00:25:48.670 --> 00:25:53.480 And since the square of a signal at any time depends 00:25:53.480 --> 00:25:58.220 only on the value of the signal at that time, clearly a 00:25:58.220 --> 00:26:01.190 squarer is a memoryless system. 00:26:01.190 --> 00:26:06.870 So in fact, we can indicate here that this system is 00:26:06.870 --> 00:26:08.120 memoryless. 00:26:10.850 --> 00:26:15.980 Another important system is what is referred to as an 00:26:15.980 --> 00:26:17.230 integrator. 00:26:18.990 --> 00:26:26.460 The output is equal to the integral of the input. 00:26:26.460 --> 00:26:31.240 Here, as I've indicated it, it's not just integrating 00:26:31.240 --> 00:26:35.100 x(t), but it's integrating the square of x(t). 00:26:35.100 --> 00:26:40.610 And whether we square before we integrate or not, the 00:26:40.610 --> 00:26:46.060 essential point is that since we're integrating the input, 00:26:46.060 --> 00:26:50.800 the value of the output at any time is an accumulation of 00:26:50.800 --> 00:26:52.900 past history of the input. 00:26:52.900 --> 00:26:55.690 Well, an accumulation of past history in 00:26:55.690 --> 00:26:57.580 essence implies memory. 00:26:57.580 --> 00:27:04.580 To get the output at a given time requires the input over 00:27:04.580 --> 00:27:08.330 an interval, specifically for longer than that time. 00:27:08.330 --> 00:27:12.340 So this system, in fact, is not memoryless. 00:27:16.350 --> 00:27:21.970 And now, I have a third system defined here. 00:27:21.970 --> 00:27:26.530 The third system is a system, a discrete-time system, in 00:27:26.530 --> 00:27:34.210 which the output is equal to the input but not quite. 00:27:34.210 --> 00:27:36.190 The output at some time-- let's say for 00:27:36.190 --> 00:27:38.690 example at n = 0-- 00:27:38.690 --> 00:27:44.260 is equal to the input at one time sample, or instant, or 00:27:44.260 --> 00:27:47.260 value of the index before that. 00:27:47.260 --> 00:27:52.930 And so this, in fact, is a system for which the output is 00:27:52.930 --> 00:27:57.930 simply the input delayed or shifted by one sample. 00:27:57.930 --> 00:28:05.610 So this system is referred to as a unit delay. 00:28:05.610 --> 00:28:10.190 And now, the question is, is that system memoryless? 00:28:12.830 --> 00:28:16.530 Well, the output depends only on the input-- 00:28:16.530 --> 00:28:20.530 the output at any instant depends only on the input at 00:28:20.530 --> 00:28:22.130 one instant. 00:28:22.130 --> 00:28:27.960 But since it depends on an instant prior to the time at 00:28:27.960 --> 00:28:30.040 which we're looking, or different than the time at 00:28:30.040 --> 00:28:33.310 which we're looking, it violates the definition of 00:28:33.310 --> 00:28:35.950 memoryless that we introduced. 00:28:35.950 --> 00:28:40.350 And so, in fact, the unit delay is a system that has 00:28:40.350 --> 00:28:43.990 memory, and so let's indicate that here. 00:28:43.990 --> 00:28:50.120 So this is not a memoryless system because of the fact 00:28:50.120 --> 00:28:52.540 that there is 1 unit of delay. 00:28:52.540 --> 00:28:54.800 And in essence, delay requires memory. 00:28:58.260 --> 00:29:04.500 OK, so that's the issue of memory and memoryless systems. 00:29:04.500 --> 00:29:10.080 Let's now turn to another property which a system may or 00:29:10.080 --> 00:29:14.155 may not have, the property referred to as invertibility. 00:29:16.700 --> 00:29:21.330 Now, essentially what invertibility means is that 00:29:21.330 --> 00:29:25.120 given the output of the system, you can figure out 00:29:25.120 --> 00:29:27.140 uniquely what the input was. 00:29:27.140 --> 00:29:30.740 That's one definition for invertibility. 00:29:30.740 --> 00:29:34.640 Said another way, invertibility means that given 00:29:34.640 --> 00:29:37.100 the output, there's only one input that 00:29:37.100 --> 00:29:38.180 could have caused it. 00:29:38.180 --> 00:29:43.180 That's another common definition for invertibility. 00:29:43.180 --> 00:29:46.700 Another way of looking at it is, in fact, to look at it in 00:29:46.700 --> 00:29:50.275 the context of a cascade of systems. 00:29:53.040 --> 00:29:55.810 So let's consider a system. 00:29:55.810 --> 00:30:03.490 And here is a system which I refer to as system A. And this 00:30:03.490 --> 00:30:06.260 could be continuous-time or discrete-time. 00:30:06.260 --> 00:30:11.270 It has an input, x_1(t) or x_1[n], depending on whether 00:30:11.270 --> 00:30:13.200 it's continuous-time or discrete-time that we're 00:30:13.200 --> 00:30:18.760 talking about, and an associated output. 00:30:18.760 --> 00:30:24.970 And here, we have system B with its 00:30:24.970 --> 00:30:29.510 associated input and output. 00:30:29.510 --> 00:30:34.490 And now, let's put these two systems in cascade. 00:30:34.490 --> 00:30:40.040 So we'll take the output of system 1 and feed it into the 00:30:40.040 --> 00:30:43.280 input of system 2-- 00:30:43.280 --> 00:30:48.590 or system B. So the output of system A goes into the input 00:30:48.590 --> 00:30:56.625 of system B. And if system A is invertible and system B is 00:30:56.625 --> 00:31:04.030 its inverse, then the consequence is that the output 00:31:04.030 --> 00:31:10.730 of system B is equal to the input of system A. 00:31:10.730 --> 00:31:15.560 Now, I know there are a lot of inputs and outputs and 00:31:15.560 --> 00:31:17.080 inverses in there. 00:31:17.080 --> 00:31:22.050 But essentially, what we mean by what I've just said is that 00:31:22.050 --> 00:31:30.160 if we have system A and it's invertible, and if we cascade 00:31:30.160 --> 00:31:37.650 it with its inverse, system B, then the overall cascade of 00:31:37.650 --> 00:31:43.710 these two systems is simply what's referred to as the 00:31:43.710 --> 00:31:45.110 identity system. 00:31:48.180 --> 00:31:52.280 And the identity system is simply a system which if you 00:31:52.280 --> 00:31:56.280 put a signal into it, you get the same signal out of it. 00:31:56.280 --> 00:31:59.200 In other words, the overall system is no 00:31:59.200 --> 00:32:00.830 transformation at all. 00:32:00.830 --> 00:32:07.330 And clearly, of course, for a system to be able to be 00:32:07.330 --> 00:32:11.400 cascaded with another system to generate the identity 00:32:11.400 --> 00:32:15.920 system requires that the first system, system A, be 00:32:15.920 --> 00:32:18.250 invertible. 00:32:18.250 --> 00:32:23.730 So let's look at some examples. 00:32:23.730 --> 00:32:30.090 If we had system A as I've indicated here, where now the 00:32:30.090 --> 00:32:36.980 output is the running integral of the input-- 00:32:36.980 --> 00:32:39.420 and remember that we saw the running integral when we 00:32:39.420 --> 00:32:42.990 talked about the relationship between steps and impulses-- 00:32:42.990 --> 00:32:45.970 if this happened to be an impulse, then the 00:32:45.970 --> 00:32:48.640 output would be a step. 00:32:48.640 --> 00:32:54.790 This system is referred to, of course, as an integrator since 00:32:54.790 --> 00:33:01.180 the output is the running integral of the input. 00:33:01.180 --> 00:33:09.570 And the integrator, in fact, is an invertible system. 00:33:09.570 --> 00:33:14.450 And its inverse is a system for which the output is the 00:33:14.450 --> 00:33:16.350 derivative of the input. 00:33:16.350 --> 00:33:22.700 So the inverse of system A, if system A is an integrator, is 00:33:22.700 --> 00:33:27.970 a system for which the output is equal to the derivative of 00:33:27.970 --> 00:33:32.360 the input which, not surprisingly, is referred to 00:33:32.360 --> 00:33:35.515 as a differentiator. 00:33:40.870 --> 00:33:46.690 So an integrator is invertible. 00:33:46.690 --> 00:33:50.270 Its inverse is a differentiator. 00:33:50.270 --> 00:33:52.810 What you might want to think about is the question of 00:33:52.810 --> 00:33:56.550 whether a differentiator is invertible. 00:33:56.550 --> 00:34:00.130 Now, to answer that, what you would ask yourself is, if you 00:34:00.130 --> 00:34:02.390 always knew what the derivative of the signal is, 00:34:02.390 --> 00:34:06.410 would you necessarily know what the signal was? 00:34:06.410 --> 00:34:10.090 In other words, if you have a differentiator and you have 00:34:10.090 --> 00:34:12.060 the output of the differentiator, could you 00:34:12.060 --> 00:34:14.550 always figure out what the input was? 00:34:14.550 --> 00:34:17.429 If you could, the system would be invertible. 00:34:17.429 --> 00:34:21.639 If you couldn't, the system would not be invertible. 00:34:21.639 --> 00:34:24.060 So you might just want to think about that. 00:34:24.060 --> 00:34:26.630 I guess I won't tell you right now. 00:34:26.630 --> 00:34:32.260 But I'm sure that you'll think about that more, particularly 00:34:32.260 --> 00:34:36.760 with the guidance of the video manual. 00:34:36.760 --> 00:34:41.639 OK, well let's look at one last system. 00:34:41.639 --> 00:34:48.000 I've indicated here a system for which the output is 00:34:48.000 --> 00:34:50.620 related to the input through this curve. 00:34:50.620 --> 00:34:56.800 And what I mean by this curve, which wasn't done quite as 00:34:56.800 --> 00:35:04.210 well as I might have, is that the output is equal to the 00:35:04.210 --> 00:35:06.530 square of the input. 00:35:06.530 --> 00:35:13.070 So this system is our squarer as we talked about before. 00:35:13.070 --> 00:35:17.510 We saw previously, or discussed the fact previously, 00:35:17.510 --> 00:35:22.315 that the squarer is a memoryless system. 00:35:25.500 --> 00:35:30.440 And now the question is, is a squarer an invertible system? 00:35:30.440 --> 00:35:35.020 Well, the question then is, if you're given the square of a 00:35:35.020 --> 00:35:37.990 signal, can you figure out what the signal is? 00:35:37.990 --> 00:35:42.800 And as I'm sure you've already suspected, the answer to that 00:35:42.800 --> 00:35:47.030 is no, because obviously if the signal was negative or 00:35:47.030 --> 00:35:49.540 positive, you wouldn't be able to figure that out after 00:35:49.540 --> 00:35:50.820 you've squared. 00:35:50.820 --> 00:35:55.085 So in fact, the squarer is not invertible. 00:35:57.670 --> 00:36:00.650 All right, so we've introduced several properties. 00:36:00.650 --> 00:36:04.260 And by the way, as we've gone through it, also introduced 00:36:04.260 --> 00:36:07.190 some systems that will turn out to be useful 00:36:07.190 --> 00:36:09.100 and important systems. 00:36:09.100 --> 00:36:13.930 And now, let's continue with some other properties. 00:36:13.930 --> 00:36:18.720 A property that we'll find useful to make reference to, 00:36:18.720 --> 00:36:21.610 from time to time, and will, in fact, play a fairly 00:36:21.610 --> 00:36:26.130 important role in a variety of discussions during the course, 00:36:26.130 --> 00:36:30.830 is a property which is referred to as causality. 00:36:30.830 --> 00:36:36.410 Now, in essence what causality means is the following. 00:36:36.410 --> 00:36:41.930 A system is set to be causal if, as one way of saying it, 00:36:41.930 --> 00:36:45.340 it only responds when you kick it. 00:36:45.340 --> 00:36:49.450 Is another way of saying it, its response at any time only 00:36:49.450 --> 00:36:54.020 depends on values of the input prior to that time. 00:36:54.020 --> 00:36:58.110 So a causal system, both continuous time and discrete 00:36:58.110 --> 00:37:04.320 time, is defined sometimes as a system which has the 00:37:04.320 --> 00:37:10.470 property that the output at any time depends only on the 00:37:10.470 --> 00:37:15.520 input prior or equal to that time. 00:37:15.520 --> 00:37:19.220 Essentially what we're saying is that the system can't 00:37:19.220 --> 00:37:22.550 anticipate future inputs. 00:37:22.550 --> 00:37:27.450 And so that, in fact, is another possible definition 00:37:27.450 --> 00:37:32.930 for causality, that the system is causal if it can't 00:37:32.930 --> 00:37:36.490 anticipate future inputs. 00:37:36.490 --> 00:37:39.110 Finally, an alternative way of saying it 00:37:39.110 --> 00:37:42.630 mathematically is as follows. 00:37:42.630 --> 00:37:50.920 If I have two signals, x_1(t) and x_2(t), with their 00:37:50.920 --> 00:37:58.070 associated outputs, y_1(t) and y_2(t), then a system is said 00:37:58.070 --> 00:38:03.870 to be causal if and only if it has the property that if those 00:38:03.870 --> 00:38:08.520 two inputs are identical up until some time, then the 00:38:08.520 --> 00:38:12.300 outputs are identical up until the same time. 00:38:12.300 --> 00:38:18.060 So if we have two signals that are exactly the same up until 00:38:18.060 --> 00:38:22.050 some time and perhaps do something different later on, 00:38:22.050 --> 00:38:26.980 causality requires that the outputs not anticipate the 00:38:26.980 --> 00:38:31.150 fact that those inputs are at some future time going to do 00:38:31.150 --> 00:38:32.330 something different. 00:38:32.330 --> 00:38:36.140 And that, in fact, is the most useful mathematical definition 00:38:36.140 --> 00:38:37.620 of causality. 00:38:37.620 --> 00:38:43.830 And of course, I've written that here for continuous time. 00:38:43.830 --> 00:38:52.050 And the same definition of causality also applies for 00:38:52.050 --> 00:38:53.300 discrete time. 00:38:55.520 --> 00:39:00.560 OK, well let's look at an example. 00:39:00.560 --> 00:39:06.290 Let's take an example which is a system which is the 00:39:06.290 --> 00:39:09.380 following discrete-time system. 00:39:09.380 --> 00:39:17.980 The output at any given time is the sum of x[n], x[n] 00:39:17.980 --> 00:39:20.270 delayed, and x[n] 00:39:20.270 --> 00:39:22.800 anticipated. 00:39:22.800 --> 00:39:27.570 And this is, in fact, a system that's very useful and 00:39:27.570 --> 00:39:30.735 referred to as a moving average. 00:39:34.210 --> 00:39:41.720 And so if we think of a moving average, if we have here a 00:39:41.720 --> 00:39:44.510 sequence, x[n] 00:39:44.510 --> 00:39:45.880 and x[n] 00:39:45.880 --> 00:39:53.770 with other values going off in both directions, for any 00:39:53.770 --> 00:39:57.680 value, n_0, at which we're computing the output-- 00:39:57.680 --> 00:40:01.050 and this is y[n], the output-- 00:40:01.050 --> 00:40:10.040 we take x[n 0], x[n 0-1], and x[n 0+1]. 00:40:10.040 --> 00:40:14.590 And so to form that moving average, we would take these 00:40:14.590 --> 00:40:19.580 three values and combine them together, adding them and then 00:40:19.580 --> 00:40:23.540 dividing by 3 to get that. 00:40:23.540 --> 00:40:27.100 Well, is the system causal? 00:40:27.100 --> 00:40:31.360 One way to answer that is to determine whether the output 00:40:31.360 --> 00:40:34.890 at any given time depends on future values of the input. 00:40:34.890 --> 00:40:38.370 And clearly, if you look at this, what you see is that the 00:40:38.370 --> 00:40:43.940 output at time n_0 depends both on past values and on 00:40:43.940 --> 00:40:46.290 future values. 00:40:46.290 --> 00:40:50.890 As opposed to another moving average, which I've indicated 00:40:50.890 --> 00:40:55.100 here, where I simply shifted the values 00:40:55.100 --> 00:40:57.210 that I combined together. 00:40:57.210 --> 00:41:01.490 And in this case, because of the way in which I picked the 00:41:01.490 --> 00:41:09.270 values, the output depends on the value at n_0, the value at 00:41:09.270 --> 00:41:14.380 n_0 - 1, and the value at n_0 - 2. 00:41:14.380 --> 00:41:16.160 So y[n 0] 00:41:16.160 --> 00:41:23.430 would depend here on x[n 0], x[n 0-1], and x[n 0-2]. 00:41:23.430 --> 00:41:31.210 And so in this case, the system is not causal. 00:41:31.210 --> 00:41:36.660 And in this case, the system is causal. 00:41:39.250 --> 00:41:42.900 All right, now let's turn to another system property, the 00:41:42.900 --> 00:41:45.975 property referred to as stability. 00:41:48.650 --> 00:41:53.730 Now, there are lots of definitions of stability, and 00:41:53.730 --> 00:41:57.720 some of them get very mathematical and formal. 00:41:57.720 --> 00:42:01.180 But we've chosen, and what we'll use as our definition of 00:42:01.180 --> 00:42:04.100 stability, is what's called bounded-input 00:42:04.100 --> 00:42:07.000 bounded-output stability. 00:42:07.000 --> 00:42:11.610 And essentially, the definition is that a system is 00:42:11.610 --> 00:42:16.880 stable if and only if for every bounded input, the 00:42:16.880 --> 00:42:18.650 output is bounded. 00:42:18.650 --> 00:42:23.180 So the notion is if you have a system and the input never 00:42:23.180 --> 00:42:29.520 gets above some finite value, then stability requires that 00:42:29.520 --> 00:42:33.600 the output also stay within some bounded values. 00:42:33.600 --> 00:42:36.150 And I'm sure that stability and instability are things 00:42:36.150 --> 00:42:40.030 that you're kind of informally familiar with. 00:42:40.030 --> 00:42:45.840 Let me just emphasize the point with something that I 00:42:45.840 --> 00:42:50.410 borrowed actually from my son with some 00:42:50.410 --> 00:42:53.360 reluctance on his part. 00:42:53.360 --> 00:42:57.170 If we, for example, take a system like this, which is in 00:42:57.170 --> 00:43:01.140 essence a pendulum, this system as I'm holding it here 00:43:01.140 --> 00:43:05.040 is stable because if I put in a bounded input, which is a 00:43:05.040 --> 00:43:09.170 displacement, the output, which is the movement of it, 00:43:09.170 --> 00:43:11.680 remains bounded. 00:43:11.680 --> 00:43:15.510 Now on the other hand, if I put the system like this, 00:43:15.510 --> 00:43:18.200 which is, in fact, what's referred to as an inverted 00:43:18.200 --> 00:43:22.990 pendulum, although we could conceivably get this to 00:43:22.990 --> 00:43:27.550 balance, just a slight displacement because of the 00:43:27.550 --> 00:43:30.050 fact that the pendulum is inverted, a slight 00:43:30.050 --> 00:43:35.130 displacement and the output becomes unbounded. 00:43:35.130 --> 00:43:37.630 Now, an interesting thing with the inverted pendulum, by the 00:43:37.630 --> 00:43:42.560 way, which I'm sure all of you, if you were anything like 00:43:42.560 --> 00:43:46.150 me, were intrigued with as a kid, was the notion that you 00:43:46.150 --> 00:43:50.920 could take an inverted pendulum and in effect turn it 00:43:50.920 --> 00:43:55.360 back into a stable system by using what I'm doing right 00:43:55.360 --> 00:43:58.730 now, which is feedback. 00:43:58.730 --> 00:44:03.200 What I've done in that case is I've stabilized the system by 00:44:03.200 --> 00:44:06.390 using feedback, visual feedback, from 00:44:06.390 --> 00:44:08.830 my eye to my hand. 00:44:08.830 --> 00:44:11.070 And in fact, one of the very important things that we'll 00:44:11.070 --> 00:44:13.850 see about feedback when we talk about feedback systems 00:44:13.850 --> 00:44:16.670 much later in the course is that one of their very 00:44:16.670 --> 00:44:22.330 important applications is in stabilizing unstable systems. 00:44:22.330 --> 00:44:25.960 By the way, one of their problems is that if not used 00:44:25.960 --> 00:44:30.370 correctly, it can destabilize stable systems. 00:44:30.370 --> 00:44:32.720 OK, well let's continue on with 00:44:32.720 --> 00:44:36.160 our property of stability. 00:44:36.160 --> 00:44:40.950 I have here, again, the example of an integrator. 00:44:40.950 --> 00:44:46.380 And as I indicate here, if we have an integrator and we put 00:44:46.380 --> 00:44:50.900 a step function into it or a step signal into it, the 00:44:50.900 --> 00:44:55.330 output is what's referred to as a ramp signal. 00:44:55.330 --> 00:44:58.240 It linearly increases. 00:44:58.240 --> 00:45:01.860 Now, the question is, is a ramp unbounded? 00:45:01.860 --> 00:45:03.680 The input is bounded. 00:45:03.680 --> 00:45:05.580 The step input is bounded. 00:45:05.580 --> 00:45:10.590 The ramp is unbounded because if you try to establish any 00:45:10.590 --> 00:45:15.820 bound on it, you can always go out far enough in time so that 00:45:15.820 --> 00:45:18.430 the output will exceed that bound. 00:45:18.430 --> 00:45:24.320 So in fact, the integrator is not stable. 00:45:29.680 --> 00:45:34.960 OK, now finally, I'd like to turn to two properties that 00:45:34.960 --> 00:45:39.180 we'll make considerable use of as the course goes on, the 00:45:39.180 --> 00:45:42.630 properties of time invariance and linearity. 00:45:45.210 --> 00:45:52.630 Time invariance, in essence, says that the system doesn't 00:45:52.630 --> 00:45:56.360 really care what you call the origin. 00:45:56.360 --> 00:46:00.430 In other words, it says if you take the input and you shift 00:46:00.430 --> 00:46:04.370 it in time, all that you've done is taken the output and 00:46:04.370 --> 00:46:07.630 shifted it in time by the same amount. 00:46:07.630 --> 00:46:12.680 Somewhat more formally as I've indicated here, if in 00:46:12.680 --> 00:46:16.270 continuous time we have an input, x(t), which generate an 00:46:16.270 --> 00:46:22.990 output, y(t), then time invariance requires that if 00:46:22.990 --> 00:46:26.820 the input is shifted by any amount of time, the output is 00:46:26.820 --> 00:46:29.610 shifted by the same amount of time. 00:46:29.610 --> 00:46:33.560 And exactly the same applies in discrete time. 00:46:36.370 --> 00:46:41.340 For example, if we have a system which is a system I've 00:46:41.340 --> 00:46:45.170 shown here, which by the way is the system that we talked 00:46:45.170 --> 00:46:48.930 about previously to go from a step sequence-- 00:46:48.930 --> 00:46:51.940 I'm sorry, from an impulse sequence to a step sequence. 00:46:51.940 --> 00:46:53.880 We called it a running sum. 00:46:53.880 --> 00:46:57.890 And actually, what it's also often called is an 00:46:57.890 --> 00:46:59.450 accumulator. 00:46:59.450 --> 00:47:04.580 What it does is accumulate past values of the input. 00:47:04.580 --> 00:47:10.590 Well, is the accumulator time-invariant? 00:47:10.590 --> 00:47:13.080 The best way to establish that is to work through the 00:47:13.080 --> 00:47:17.880 equations and verify that it either does or doesn't satisfy 00:47:17.880 --> 00:47:20.920 the formal definition of time invariance. 00:47:20.920 --> 00:47:26.130 Informally, if you think about it, it makes intuitive sense 00:47:26.130 --> 00:47:30.640 that the accumulator is time-invariant because if 00:47:30.640 --> 00:47:35.620 you're accumulating values and if you delay the values that 00:47:35.620 --> 00:47:40.010 you're putting into the accumulator, then the 00:47:40.010 --> 00:47:42.880 associated values that come out will be delayed by the 00:47:42.880 --> 00:47:43.550 same amount. 00:47:43.550 --> 00:47:47.930 The accumulator doesn't care really if you shift the input. 00:47:47.930 --> 00:47:51.730 It'll just simply shift the associated output. 00:47:51.730 --> 00:47:55.630 But more generally, if you're trying to test time 00:47:55.630 --> 00:47:58.940 invariance, it's important to return to the definition. 00:47:58.940 --> 00:48:01.690 And that's what you're required to do in the examples 00:48:01.690 --> 00:48:06.140 in the video manual. 00:48:06.140 --> 00:48:10.700 OK, well, I indicate another example. 00:48:10.700 --> 00:48:14.130 We had the example of an accumulator. 00:48:14.130 --> 00:48:20.430 Here's another example which, in fact, as we'll see later is 00:48:20.430 --> 00:48:23.500 a system which is a modulator. 00:48:23.500 --> 00:48:26.820 The output is the input, modulated. 00:48:26.820 --> 00:48:32.840 And although you might think at first that this system is 00:48:32.840 --> 00:48:40.330 time invariant, in fact it is not, because the input shifted 00:48:40.330 --> 00:48:46.240 generates an output which is the input shifted times the 00:48:46.240 --> 00:48:49.180 same modulation function. 00:48:49.180 --> 00:48:55.170 Whereas if we were to take the output of the system, we have 00:48:55.170 --> 00:49:03.220 x(t) is the input, then what that would correspond to is 00:49:03.220 --> 00:49:10.220 sin(t-t_0) * x(t-t_0). 00:49:10.220 --> 00:49:13.770 And since these two are not equal, this 00:49:13.770 --> 00:49:18.190 system is not time invariant. 00:49:18.190 --> 00:49:22.000 And this is an example that often causes a slight amount 00:49:22.000 --> 00:49:27.030 of difficulty because it seems like when you look 00:49:27.030 --> 00:49:29.180 at it ought to be. 00:49:29.180 --> 00:49:33.480 And so I strongly encourage you, in the context of working 00:49:33.480 --> 00:49:37.860 problems in the manual, that you think very carefully about 00:49:37.860 --> 00:49:42.030 this and at least believe that what I told you 00:49:42.030 --> 00:49:44.690 is the right answer. 00:49:44.690 --> 00:49:48.500 OK, now the final property that I want to introduce today 00:49:48.500 --> 00:49:52.580 is the property of linearity. 00:49:52.580 --> 00:49:58.290 And linearity is defined in a manner similar for continuous 00:49:58.290 --> 00:50:00.790 time and discrete time. 00:50:00.790 --> 00:50:07.090 And what it says is that if we have some inputs with 00:50:07.090 --> 00:50:13.300 associated outputs, let's say x_1(t) and x_2(t), then a 00:50:13.300 --> 00:50:19.930 system is linear if it has the property that the output to a 00:50:19.930 --> 00:50:24.000 linear combination of those inputs is the same linear 00:50:24.000 --> 00:50:27.250 combination of the associated outputs. 00:50:27.250 --> 00:50:32.650 And so that's what I've indicated here, that if we now 00:50:32.650 --> 00:50:36.290 put into the system a linear combination of those inputs, 00:50:36.290 --> 00:50:40.610 then for linearity, we require that the output is the same 00:50:40.610 --> 00:50:42.320 linear combination. 00:50:42.320 --> 00:50:47.110 And exactly the same applies in discrete time. 00:50:47.110 --> 00:50:50.960 And you can show from this definition that if a system is 00:50:50.960 --> 00:50:54.930 linear with two inputs, then it's linear in terms of an 00:50:54.930 --> 00:50:58.330 arbitrary number of inputs. 00:50:58.330 --> 00:51:01.280 I have a number of examples. 00:51:01.280 --> 00:51:07.410 And these are examples that, again, I ask you to think 00:51:07.410 --> 00:51:11.250 about as you look at the video manual. 00:51:11.250 --> 00:51:12.940 Just to suggest the answer-- 00:51:12.940 --> 00:51:16.420 well, not to suggest but to tell you the answer, the 00:51:16.420 --> 00:51:19.400 integrator as we have here is linear. 00:51:23.550 --> 00:51:26.890 This system in which the output is double the input 00:51:26.890 --> 00:51:30.570 plus a constant, you would kind of think it's linear 00:51:30.570 --> 00:51:33.030 because it's a straight line. 00:51:33.030 --> 00:51:35.780 But one has to be careful. 00:51:35.780 --> 00:51:40.630 And in fact, as it turns out, this is not linear. 00:51:40.630 --> 00:51:43.950 There is a qualifier attached to it because it has a 00:51:43.950 --> 00:51:47.470 property referred to as incrementally linear, which is 00:51:47.470 --> 00:51:51.590 discussed somewhat more in the text. 00:51:51.590 --> 00:51:57.340 And finally, we have a system which is the squarer that I've 00:51:57.340 --> 00:51:59.980 indicated again here. 00:51:59.980 --> 00:52:03.830 And squaring it is definitely not a linear operation. 00:52:07.080 --> 00:52:12.200 OK, so what we've done, then, is to introduce a number of 00:52:12.200 --> 00:52:15.570 properties of systems. 00:52:15.570 --> 00:52:19.370 And we've also, by the way, as I've stressed previously, as 00:52:19.370 --> 00:52:23.090 we've gone along introduced also a number of important and 00:52:23.090 --> 00:52:25.730 useful systems, like the accumulator, the integrator, 00:52:25.730 --> 00:52:29.550 the differentiator, et cetera. 00:52:29.550 --> 00:52:33.110 What we'll see is that the properties of linearity and 00:52:33.110 --> 00:52:38.640 time invariance in particular become central and important 00:52:38.640 --> 00:52:40.260 properties. 00:52:40.260 --> 00:52:45.930 And in the next lecture, what we'll show is that with 00:52:45.930 --> 00:52:51.890 systems that are linear and time-invariant, the use of the 00:52:51.890 --> 00:52:55.310 impulse function, both in continuous time and discrete 00:52:55.310 --> 00:53:00.150 time, provides an extraordinarily important and 00:53:00.150 --> 00:53:04.300 useful mechanism for characterizing those systems. 00:53:04.300 --> 00:53:05.550 Thank you.