WEBVTT 00:00:42.500 --> 00:00:45.620 PROFESSOR: Last time, we talked about the 00:00:45.620 --> 00:00:49.600 representation of linear time-invariant systems through 00:00:49.600 --> 00:00:52.400 the convolution sum in the discrete-time case and the 00:00:52.400 --> 00:00:56.120 convolution integral in the continuous-time case. 00:00:56.120 --> 00:00:58.520 Now, although the derivation was relatively 00:00:58.520 --> 00:01:01.970 straightforward, in fact, the result is really kind of 00:01:01.970 --> 00:01:06.250 amazing because what it tells us is that for linear 00:01:06.250 --> 00:01:09.580 time-invariant systems, if we know the response of the 00:01:09.580 --> 00:01:14.540 system to a single impulse at t = 0, or in fact, at any 00:01:14.540 --> 00:01:18.710 other time, then we can determine from that its 00:01:18.710 --> 00:01:22.590 response to an arbitrary input through the use of 00:01:22.590 --> 00:01:24.900 convolution. 00:01:24.900 --> 00:01:28.650 Furthermore, what we'll see as the course develops is that, 00:01:28.650 --> 00:01:31.660 in fact, the class of linear time-invariant systems is a 00:01:31.660 --> 00:01:33.720 very rich class. 00:01:33.720 --> 00:01:36.770 There are lots of systems that have that property. 00:01:36.770 --> 00:01:39.650 And in addition, there are lots of very interesting 00:01:39.650 --> 00:01:42.280 things that you can do with linear time-invariant systems. 00:01:44.850 --> 00:01:49.510 In today's lecture, what I'd like to begin with is focusing 00:01:49.510 --> 00:01:53.730 on convolution as an algebraic operation. 00:01:53.730 --> 00:01:57.370 And we'll see that it has a number of algebraic properties 00:01:57.370 --> 00:02:02.150 that in turn have important implications for linear 00:02:02.150 --> 00:02:04.120 time-invariant systems. 00:02:04.120 --> 00:02:08.009 Then we'll turn to a discussion of what the 00:02:08.009 --> 00:02:11.680 characterization of linear time-invariant systems through 00:02:11.680 --> 00:02:16.900 convolution implies, in terms of the relationship, of 00:02:16.900 --> 00:02:19.110 various other system properties 00:02:19.110 --> 00:02:21.240 to the impulse response. 00:02:21.240 --> 00:02:26.220 Let me begin by reminding you of the basic result that we 00:02:26.220 --> 00:02:31.180 developed last time, which is the convolution sum in 00:02:31.180 --> 00:02:37.740 discrete time, as I indicate here, and the convolution 00:02:37.740 --> 00:02:40.820 integral in continuous time. 00:02:40.820 --> 00:02:44.850 And what the convolution sum, or the convolution integral, 00:02:44.850 --> 00:02:51.850 tells us is how to relate the output to the input and to the 00:02:51.850 --> 00:02:53.940 system impulse response. 00:02:53.940 --> 00:02:58.750 And the expression looks basically the same in 00:02:58.750 --> 00:03:01.320 continuous time and discrete time. 00:03:01.320 --> 00:03:04.650 And I remind you also that we talked about a graphical 00:03:04.650 --> 00:03:09.940 interpretation, where essentially, to graphically 00:03:09.940 --> 00:03:15.760 interpret convolution required, or was developed, in 00:03:15.760 --> 00:03:21.160 terms of taking the system impulse response, flipping it, 00:03:21.160 --> 00:03:25.460 sliding it past the input, and positioned appropriately, 00:03:25.460 --> 00:03:29.500 depending on the value of the independent variable for which 00:03:29.500 --> 00:03:32.340 we're computing the convolution, and then 00:03:32.340 --> 00:03:36.230 multiplying and summing in the discrete-time case, or 00:03:36.230 --> 00:03:38.390 integrating in the continuous-time case. 00:03:41.160 --> 00:03:47.070 Now convolution, as an algebraic operation, has a 00:03:47.070 --> 00:03:51.000 number of important properties. 00:03:51.000 --> 00:03:55.370 One of the properties of convolution is that it is what 00:03:55.370 --> 00:03:58.300 is referred to as commutative. 00:03:58.300 --> 00:04:05.330 Commutative means that we can think either of convolving x 00:04:05.330 --> 00:04:11.760 with h, or h with x, and the order in which that's done 00:04:11.760 --> 00:04:14.590 doesn't affect the output result. 00:04:14.590 --> 00:04:19.649 So summarized here is what the commutative operation is in 00:04:19.649 --> 00:04:23.990 discrete time, or in continuous time. 00:04:23.990 --> 00:04:28.050 And it says, as I just indicated, that x[n] 00:04:28.050 --> 00:04:30.240 convolved with h[n] 00:04:30.240 --> 00:04:32.500 is equal to h[n] 00:04:32.500 --> 00:04:34.750 convolved with x[n]. 00:04:34.750 --> 00:04:38.930 Or the same, of course, in continuous time. 00:04:38.930 --> 00:04:44.200 And in fact, in the lecture last time, we worked an 00:04:44.200 --> 00:04:48.000 example where we had, in discrete time, an impulse 00:04:48.000 --> 00:04:51.980 response, which was an exponential, and an input, 00:04:51.980 --> 00:04:54.760 which is a unit step. 00:04:54.760 --> 00:04:59.520 And in the text, what you'll find is the same example 00:04:59.520 --> 00:05:04.110 worked, except in that case, the input is the exponential. 00:05:04.110 --> 00:05:07.370 And the system impulse response is the step. 00:05:07.370 --> 00:05:11.380 And that corresponds to the example in the text, which is 00:05:11.380 --> 00:05:15.720 example 3.1. 00:05:15.720 --> 00:05:20.130 And what happens in that example is that, in fact, what 00:05:20.130 --> 00:05:24.850 you'll see is that the same result occurs in example 3.1 00:05:24.850 --> 00:05:27.400 as we generated in the lecture. 00:05:27.400 --> 00:05:31.090 And there's a similar comparison in continuous time. 00:05:31.090 --> 00:05:34.680 This example was worked in lecture. 00:05:34.680 --> 00:05:38.740 And this example is worked in the text. 00:05:38.740 --> 00:05:43.370 In other words, the text works the example where the system 00:05:43.370 --> 00:05:47.960 impulse response is a unit step. 00:05:47.960 --> 00:05:52.130 And the input is an exponential. 00:05:52.130 --> 00:05:55.220 All right, now the commutative property, as I said, tells us 00:05:55.220 --> 00:05:59.700 that the order in which we do convolution doesn't affect the 00:05:59.700 --> 00:06:01.710 result of the convolution. 00:06:01.710 --> 00:06:03.560 The same is true for continuous time 00:06:03.560 --> 00:06:04.940 and discrete time. 00:06:04.940 --> 00:06:07.830 And in fact, for the other algebraic properties that I'll 00:06:07.830 --> 00:06:12.310 talk about, the results are exactly the same for 00:06:12.310 --> 00:06:15.040 continuous time and discrete time. 00:06:15.040 --> 00:06:22.210 So in fact, what we can do is drop the independent variable 00:06:22.210 --> 00:06:26.720 as an argument so that we suppress any kind of 00:06:26.720 --> 00:06:29.820 difference between continuous and discrete time. 00:06:29.820 --> 00:06:34.220 And suppressing the independent variable, we then 00:06:34.220 --> 00:06:39.170 state the commutative property as I've rewritten it here. 00:06:39.170 --> 00:06:42.970 Just x convolved with h equals h convolved with x. 00:06:42.970 --> 00:06:47.360 The same in continuous time and discrete time. 00:06:47.360 --> 00:06:52.950 Now, the derivation of the commutative property is, more 00:06:52.950 --> 00:06:54.900 or less, some algebra which you can follow 00:06:54.900 --> 00:06:55.960 through in the book. 00:06:55.960 --> 00:07:01.810 It involves some changes of variables and some 00:07:01.810 --> 00:07:04.310 things of that sort. 00:07:04.310 --> 00:07:08.440 What I'd like to focus on with that and other properties is 00:07:08.440 --> 00:07:12.520 not the derivation, which you can see in the text, but 00:07:12.520 --> 00:07:14.760 rather the interpretation. 00:07:14.760 --> 00:07:18.050 So we have the commutative property, and now there are 00:07:18.050 --> 00:07:21.090 two other important algebraic properties. 00:07:21.090 --> 00:07:25.760 One being what is referred to as the associative property, 00:07:25.760 --> 00:07:32.590 which tells us that if we have x convolved with the result of 00:07:32.590 --> 00:07:37.570 convolving h1 with h2, that's exactly the same as x 00:07:37.570 --> 00:07:42.890 convolved with h1, and that result convolved with h2. 00:07:42.890 --> 00:07:49.590 And what this permits is for us to write, for example, x 00:07:49.590 --> 00:07:56.470 convolved with h1 convolved with h2 without any ambiguity 00:07:56.470 --> 00:08:00.750 because it doesn't matter from the associative property how 00:08:00.750 --> 00:08:04.470 we group the terms together. 00:08:04.470 --> 00:08:07.950 The third important property is what is referred to as the 00:08:07.950 --> 00:08:11.670 distributive property, namely the fact that convolution 00:08:11.670 --> 00:08:13.520 distributes over addition. 00:08:13.520 --> 00:08:17.560 And what I mean by that is what I've indicated here on 00:08:17.560 --> 00:08:24.650 the slide: that if I think of x convolved with the sum of h1 00:08:24.650 --> 00:08:31.380 and h2, that's identical to first convolving x with h1, 00:08:31.380 --> 00:08:37.140 also convolving x with h2, and then adding the two together. 00:08:37.140 --> 00:08:39.960 And that result will be the same as this result. 00:08:42.770 --> 00:08:47.550 So convolution is commutative, associative, and it 00:08:47.550 --> 00:08:49.050 distributes over addition. 00:08:49.050 --> 00:08:52.080 Three very important algebraic properties. 00:08:52.080 --> 00:08:56.710 And by the way, there are other algebraic operations 00:08:56.710 --> 00:08:58.330 that have that same property. 00:08:58.330 --> 00:09:02.900 For example, multiplication of numbers is likewise 00:09:02.900 --> 00:09:07.000 commutative, associative, and distributive. 00:09:07.000 --> 00:09:11.890 Now let's look at what these three properties imply 00:09:11.890 --> 00:09:15.170 specifically for linear time-invariant systems. 00:09:15.170 --> 00:09:19.770 And as we'll see, the implications are both very 00:09:19.770 --> 00:09:22.260 interesting and very important. 00:09:22.260 --> 00:09:26.910 Let's begin with the commutative property. 00:09:26.910 --> 00:09:32.860 And consider, in particular, a system with an impulse 00:09:32.860 --> 00:09:34.630 response h. 00:09:34.630 --> 00:09:39.450 And I represent that by simply writing the h inside the box. 00:09:39.450 --> 00:09:46.100 An input x and an output, then, which is x * h. 00:09:46.100 --> 00:09:50.430 Now, since this operation is commutative, I can write 00:09:50.430 --> 00:09:55.020 instead of x * h, I can write h * x. 00:09:55.020 --> 00:10:00.210 And that would correspond to a system with impulse response 00:10:00.210 --> 00:10:07.880 x, and input h, and output then h * x. 00:10:07.880 --> 00:10:13.620 So the commutative property tells us that for a linear 00:10:13.620 --> 00:10:20.040 time-invariant system, the system output is independent 00:10:20.040 --> 00:10:24.290 of which function we call the input and which function we 00:10:24.290 --> 00:10:25.830 call the impulse response. 00:10:25.830 --> 00:10:27.360 Kind of amazing actually. 00:10:27.360 --> 00:10:31.090 We can interchange the role of input and impulse response. 00:10:31.090 --> 00:10:36.480 And from an output point of view, the output or the system 00:10:36.480 --> 00:10:38.660 doesn't care. 00:10:38.660 --> 00:10:44.100 Now furthermore, if we combine the commutative property with 00:10:44.100 --> 00:10:46.270 the associative property, we get another 00:10:46.270 --> 00:10:48.130 very interesting result. 00:10:48.130 --> 00:10:52.020 Namely that if we have two linear time-invariant systems 00:10:52.020 --> 00:10:57.330 in cascade, the overall system is independent of the order in 00:10:57.330 --> 00:10:58.890 which they're cascaded. 00:10:58.890 --> 00:11:02.740 And in fact, in either case, the cascade can be collapsed 00:11:02.740 --> 00:11:05.660 into a single system. 00:11:05.660 --> 00:11:13.120 To see this, let's first consider the cascade of two 00:11:13.120 --> 00:11:16.140 systems, one with impulse response h1, the other with 00:11:16.140 --> 00:11:18.330 impulse response h2. 00:11:18.330 --> 00:11:25.770 And the output of the first system is then x * h1. 00:11:25.770 --> 00:11:29.470 And then that is the input to the second system. 00:11:29.470 --> 00:11:36.920 And so the output of that is that result convolved with h2. 00:11:36.920 --> 00:11:40.500 So this is the result of cascading the two. 00:11:40.500 --> 00:11:45.190 And now we can use the associative property to 00:11:45.190 --> 00:11:55.000 rewrite this as x * (h1 * h2), where we group 00:11:55.000 --> 00:11:58.160 these two terms together. 00:11:58.160 --> 00:12:03.200 And so using the associative property, we now can collapse 00:12:03.200 --> 00:12:07.860 that into a single system with an input, which is x, and 00:12:07.860 --> 00:12:11.360 impulse response, which is h1 * h2. 00:12:11.360 --> 00:12:15.040 And the output is then x convolved with the result of 00:12:15.040 --> 00:12:17.880 those two convolved. 00:12:17.880 --> 00:12:20.880 Next, we can apply the commutative property. 00:12:20.880 --> 00:12:24.590 And the commutative property says we could write this 00:12:24.590 --> 00:12:28.970 impulse response that way, or we could write it this way. 00:12:28.970 --> 00:12:32.200 And since convolution is commutative, the resulting 00:12:32.200 --> 00:12:34.100 output will be exactly the same. 00:12:34.100 --> 00:12:39.200 And so these resulting outputs will be exactly the same. 00:12:39.200 --> 00:12:44.560 And now, once again we can use the associative property to 00:12:44.560 --> 00:12:48.670 group these two terms together. 00:12:48.670 --> 00:12:54.920 And x * h2 corresponds to putting x first through the 00:12:54.920 --> 00:12:59.270 system h2 and then that output through the system h1. 00:12:59.270 --> 00:13:03.320 And so finally applying the associative property again, as 00:13:03.320 --> 00:13:08.330 I just outlined, we can expand that system back into two 00:13:08.330 --> 00:13:16.160 systems in cascade with h2 first and h1 second, 00:13:16.160 --> 00:13:21.220 OK, well that involves a certain amount of algebraic 00:13:21.220 --> 00:13:22.490 manipulation. 00:13:22.490 --> 00:13:24.620 And that is not the algebraic 00:13:24.620 --> 00:13:26.140 manipulation that is important. 00:13:26.140 --> 00:13:28.250 It's the result that it's important. 00:13:28.250 --> 00:13:31.330 And what the result says, to reiterate, is if I have two 00:13:31.330 --> 00:13:35.750 linear time-invariant systems in cascade, I can cascade them 00:13:35.750 --> 00:13:39.740 in any order, and the result is the same. 00:13:39.740 --> 00:13:43.630 Now you might think, well gee, maybe that actually applies to 00:13:43.630 --> 00:13:46.710 systems in general, whether you put them 00:13:46.710 --> 00:13:48.020 this way or that way. 00:13:48.020 --> 00:13:52.140 But in fact, as we talked about last time, and I 00:13:52.140 --> 00:13:58.680 illustrated with an example, in general, if the systems are 00:13:58.680 --> 00:14:02.000 not linear and time-invariant, then the order in which 00:14:02.000 --> 00:14:05.490 they're cascaded is important to the interpretation of the 00:14:05.490 --> 00:14:06.690 overall system. 00:14:06.690 --> 00:14:11.070 For example, if one system took the square root and the 00:14:11.070 --> 00:14:16.080 other system doubled the input, taking the square root 00:14:16.080 --> 00:14:20.320 and then doubling gives us a different answer than doubling 00:14:20.320 --> 00:14:22.150 first and then taking the square root. 00:14:22.150 --> 00:14:25.570 And of course, one can construct much more elaborate 00:14:25.570 --> 00:14:27.330 examples than that. 00:14:27.330 --> 00:14:30.280 So it's a property very particular to linear 00:14:30.280 --> 00:14:31.570 time-invariant systems. 00:14:31.570 --> 00:14:37.590 And also one that we will exploit many, many times as we 00:14:37.590 --> 00:14:40.900 go through this material. 00:14:40.900 --> 00:14:45.440 The final property related to an interconnection of systems 00:14:45.440 --> 00:14:49.470 that I want to just indicate develops out of the 00:14:49.470 --> 00:14:51.270 distributive property. 00:14:51.270 --> 00:14:55.320 And what it applies to is an interpretation of the 00:14:55.320 --> 00:14:59.020 interconnection of systems in parallel. 00:14:59.020 --> 00:15:03.520 Recall that the parallel combination of systems 00:15:03.520 --> 00:15:09.030 corresponds, as I indicate here, to a system in which we 00:15:09.030 --> 00:15:13.900 simultaneously feed the input into h1 and h2, these 00:15:13.900 --> 00:15:16.680 representing the impulse responses. 00:15:16.680 --> 00:15:19.410 And then, the outputs are summed to 00:15:19.410 --> 00:15:22.740 form the overall output. 00:15:22.740 --> 00:15:28.050 And using the fact that convolution distributes over 00:15:28.050 --> 00:15:37.750 addition, we can rewrite this as x * (h1 + h2). 00:15:37.750 --> 00:15:41.980 And when we do that then, we can recognize this as the 00:15:41.980 --> 00:15:47.280 output of a system with input x and impulse response, which 00:15:47.280 --> 00:15:50.300 is the sum of these two impulse responses. 00:15:50.300 --> 00:15:55.240 So for linear time-invariant systems in parallel, we can, 00:15:55.240 --> 00:16:00.600 if we choose, replace that interconnection by a single 00:16:00.600 --> 00:16:04.920 system whose impulse response is simply the sum of those 00:16:04.920 --> 00:16:06.170 impulse responses. 00:16:09.854 --> 00:16:16.460 OK, now we have this very powerful representation for 00:16:16.460 --> 00:16:20.530 linear time-invariant systems in terms of convolution. 00:16:20.530 --> 00:16:25.010 And we've seen so far in this lecture how convolution and 00:16:25.010 --> 00:16:28.860 the representation through the impulse response leads to some 00:16:28.860 --> 00:16:31.570 important implications for system interconnections. 00:16:34.090 --> 00:16:39.100 What I'd like to turn to now are other system properties 00:16:39.100 --> 00:16:43.400 and see how, for linear time-invariant systems in 00:16:43.400 --> 00:16:49.490 particular, other system properties can be associated 00:16:49.490 --> 00:16:53.500 with particular properties or characteristics of the system 00:16:53.500 --> 00:16:55.160 impulse response. 00:16:55.160 --> 00:16:58.130 And what we'll talk about are a variety of properties. 00:16:58.130 --> 00:17:01.200 We'll talk about the issue of memory, we'll talk about the 00:17:01.200 --> 00:17:04.780 issue of invertibility, and we'll talk about the issue of 00:17:04.780 --> 00:17:07.004 causality and also stability. 00:17:10.210 --> 00:17:13.230 Well, let's begin with the issue of memory. 00:17:13.230 --> 00:17:18.660 And the question now is what are the implications for the 00:17:18.660 --> 00:17:22.490 system impulse response for a linear time-invariant system? 00:17:22.490 --> 00:17:25.440 Remember that we're always imposing that on the system. 00:17:25.440 --> 00:17:28.319 What are the implications on the impulse response if the 00:17:28.319 --> 00:17:32.350 system does or does not have memory? 00:17:32.350 --> 00:17:36.440 Well, we can answer that by looking at 00:17:36.440 --> 00:17:39.860 the convolution property. 00:17:39.860 --> 00:17:43.660 And we have here, as a reminder, the convolution 00:17:43.660 --> 00:17:51.230 integral, which tells us how x(tau) and h(t - tau) are 00:17:51.230 --> 00:17:56.020 combined to give us y(t). 00:17:56.020 --> 00:18:00.940 And what I've illustrated above is a 00:18:00.940 --> 00:18:02.730 general kind of example. 00:18:02.730 --> 00:18:04.890 Here is x(tau). 00:18:04.890 --> 00:18:08.670 Here is h(t - tau). 00:18:08.670 --> 00:18:13.450 And to compute the output at any time t, we would take 00:18:13.450 --> 00:18:18.160 these two, multiply them together, and integrate from 00:18:18.160 --> 00:18:19.730 -infinity to +infinity. 00:18:22.740 --> 00:18:27.050 So the question then is what can we say about h(t), the 00:18:27.050 --> 00:18:33.020 impulse response in order to guarantee, let's say, that the 00:18:33.020 --> 00:18:37.090 output depends only on the input at time t. 00:18:37.090 --> 00:18:43.900 Well, it's pretty much obvious from looking at the graphs. 00:18:43.900 --> 00:18:51.740 If we only want the output to depend on x(tau) at tau = t, 00:18:51.740 --> 00:18:59.360 then h(t - tau) better be non-zero only at tau = t. 00:18:59.360 --> 00:19:04.990 And so the implication then is that for the system to be 00:19:04.990 --> 00:19:12.100 memoryless, what we require is that h(t - tau) be non-zero 00:19:12.100 --> 00:19:16.500 only at tau = t. 00:19:16.500 --> 00:19:19.960 So we want the impulse response to be non-zero at 00:19:19.960 --> 00:19:22.230 only one point. 00:19:22.230 --> 00:19:24.670 We want it to contribute something after we multiply 00:19:24.670 --> 00:19:26.490 and go through an integral. 00:19:26.490 --> 00:19:29.410 And in effect, what that says is the only thing that it can 00:19:29.410 --> 00:19:30.680 be and meet all those 00:19:30.680 --> 00:19:33.400 conditions is a scaled impulse. 00:19:33.400 --> 00:19:39.310 So if the system is to be memoryless, then that requires 00:19:39.310 --> 00:19:43.020 that the impulse response be a scaled impulse. 00:19:43.020 --> 00:19:49.190 Any other impulse response then, in essence, requires 00:19:49.190 --> 00:19:52.010 that the system have memory, or implies that the system 00:19:52.010 --> 00:19:54.210 have memory. 00:19:54.210 --> 00:19:59.950 So for the continuous-time case then, memoryless would 00:19:59.950 --> 00:20:03.760 correspond only to the impulse response being proportional to 00:20:03.760 --> 00:20:05.060 an impulse. 00:20:05.060 --> 00:20:09.720 And in the discrete-time case, a similar statement, in which 00:20:09.720 --> 00:20:14.590 case, the output is just proportional to the input, 00:20:14.590 --> 00:20:17.730 again either in the continuous-time or in the 00:20:17.730 --> 00:20:18.980 discrete-time case. 00:20:22.430 --> 00:20:22.670 All right. 00:20:22.670 --> 00:20:27.650 Now we can turn our attention to the issue of system 00:20:27.650 --> 00:20:29.410 invertibility. 00:20:29.410 --> 00:20:35.460 And recall that what is meant by invertibility of a system, 00:20:35.460 --> 00:20:37.690 or the inverse of a system. 00:20:37.690 --> 00:20:41.080 The inverse of a system is a system, which when we cascade 00:20:41.080 --> 00:20:45.360 it with the one that we're inquiring about, the overall 00:20:45.360 --> 00:20:48.110 cascade is the identity system. 00:20:48.110 --> 00:20:53.370 In other words, the output is equal to the input. 00:20:53.370 --> 00:20:56.980 So let's consider a system with impulse 00:20:56.980 --> 00:21:00.440 response h, input is x. 00:21:00.440 --> 00:21:04.480 And let's say that the impulse response of the inverse system 00:21:04.480 --> 00:21:07.930 is h_i, and the output is y. 00:21:07.930 --> 00:21:16.850 Then, the output of this system is x * (h * h_i). 00:21:16.850 --> 00:21:21.830 And we want this to come out equal to x. 00:21:21.830 --> 00:21:28.740 And what that requires than is that this convolution just 00:21:28.740 --> 00:21:34.590 simply be equal to an impulse, either in the discrete-time 00:21:34.590 --> 00:21:38.270 case or in the continuous-time case. 00:21:38.270 --> 00:21:42.380 And under those conditions then, h_i is equal to the 00:21:42.380 --> 00:21:44.630 inverse of h. 00:21:44.630 --> 00:21:49.760 Notationally, by the way, it's often convenient to write 00:21:49.760 --> 00:21:53.620 instead of h_i as the impulse response of the inverse, 00:21:53.620 --> 00:21:57.330 you'll find it convenient often and more typical to 00:21:57.330 --> 00:22:01.970 write as the inverse, instead of h_i, h^(-1). 00:22:05.500 --> 00:22:10.670 And we mean by that the inverse impulse response. 00:22:10.670 --> 00:22:14.120 And one has to be careful not to misinterpret this as the 00:22:14.120 --> 00:22:16.610 reciprocal of h(t) or h(n). 00:22:16.610 --> 00:22:20.260 What's meant in this notation is the inverse system. 00:22:24.250 --> 00:22:33.930 Now, if h_i is the inverse of h, is h the inverse of h_i? 00:22:33.930 --> 00:22:38.710 Well, it seems like that ought to be plausible or perhaps 00:22:38.710 --> 00:22:40.060 make sense. 00:22:40.060 --> 00:22:43.470 The question, if you believe that the answer is yes, is 00:22:43.470 --> 00:22:47.300 how, in fact, do you verify that? 00:22:47.300 --> 00:22:49.860 And I'll leave it to you to think about it. 00:22:49.860 --> 00:22:51.020 The answer is yes, 00:22:51.020 --> 00:22:54.320 that if h_i is the inverse of h, then h is 00:22:54.320 --> 00:22:56.140 the inverse of h_i. 00:22:56.140 --> 00:23:00.800 And the key to showing that is to exploit the fact that when 00:23:00.800 --> 00:23:03.800 we take these systems and cascade them, we can cascade 00:23:03.800 --> 00:23:05.050 them in either order. 00:23:07.920 --> 00:23:12.200 All right now let's turn to another system property, the 00:23:12.200 --> 00:23:14.760 property of stability. 00:23:14.760 --> 00:23:21.160 And again, we can tie that property directly to issues 00:23:21.160 --> 00:23:27.010 related, in particular, to the system impulse response. 00:23:27.010 --> 00:23:30.845 Now, stability is defined as we've chosen to define it and 00:23:30.845 --> 00:23:34.530 as I've defined it previously, as bounded-input 00:23:34.530 --> 00:23:35.940 bounded-output stability. 00:23:35.940 --> 00:23:38.140 In other words, for every bounded input 00:23:38.140 --> 00:23:40.750 is a bounded output. 00:23:40.750 --> 00:23:41.850 What you can show-- 00:23:41.850 --> 00:23:45.690 and I won't go through the algebra here; it's gone 00:23:45.690 --> 00:23:47.700 through in the book-- 00:23:47.700 --> 00:23:53.160 is that a necessary and sufficient condition for a 00:23:53.160 --> 00:23:57.550 linear time-invariant system to be stable in the 00:23:57.550 --> 00:24:03.800 bounded-input bounded-output sense is that the impulse 00:24:03.800 --> 00:24:08.360 response be what is referred to as absolutely summable. 00:24:08.360 --> 00:24:12.410 In other words, if you take the absolute values and sum 00:24:12.410 --> 00:24:16.300 them over infinite limits, that's finite. 00:24:16.300 --> 00:24:21.300 Or in the continuous-time case, that the impulse 00:24:21.300 --> 00:24:23.530 response is absolutely integrable. 00:24:23.530 --> 00:24:27.450 In other words, if you take the absolute values of h(t) 00:24:27.450 --> 00:24:29.690 and integrate, that's finite. 00:24:29.690 --> 00:24:34.370 And under those conditions, the system is stable. 00:24:34.370 --> 00:24:39.070 If those conditions are violated, then for sure, as 00:24:39.070 --> 00:24:42.950 you'll see in the text, the system is unstable. 00:24:42.950 --> 00:24:46.010 So stability can also be tied to the 00:24:46.010 --> 00:24:47.390 system impulse response. 00:24:50.470 --> 00:24:54.130 Now, the next property that I want to talk about is the 00:24:54.130 --> 00:24:56.670 property of causality. 00:24:56.670 --> 00:25:01.890 And before I do, what I'd like to do is introduce a 00:25:01.890 --> 00:25:04.690 peripheral result that we'll then use-- 00:25:04.690 --> 00:25:06.870 when we talked about causality-- 00:25:06.870 --> 00:25:10.690 namely what's referred to as the zero input response of a 00:25:10.690 --> 00:25:11.940 linear system. 00:25:14.820 --> 00:25:18.570 The basic result, which is a very interesting and useful 00:25:18.570 --> 00:25:21.180 one, is that for a linear system-- 00:25:21.180 --> 00:25:24.350 and in fact, it's whether it's time-invariant or not, that 00:25:24.350 --> 00:25:26.600 this applies-- 00:25:26.600 --> 00:25:31.640 if you put nothing into it, you get nothing out of it. 00:25:31.640 --> 00:25:40.460 So if we have an input x(t) which is 0 for all t, and if 00:25:40.460 --> 00:25:49.040 the output of that system is y(t), if the input is 0 for 00:25:49.040 --> 00:25:54.440 all time, then the output likewise is 0 for all time. 00:25:54.440 --> 00:26:00.380 That's true for continuous time, and it's also true for 00:26:00.380 --> 00:26:01.630 discrete time. 00:26:04.750 --> 00:26:10.380 And in fact, to show that result is pretty much 00:26:10.380 --> 00:26:11.280 straightforward. 00:26:11.280 --> 00:26:15.570 We could do it either by using convolution, which would, of 00:26:15.570 --> 00:26:18.780 course, be associated with linearity and time invariance. 00:26:18.780 --> 00:26:22.960 But in fact, we can show that property relatively easily by 00:26:22.960 --> 00:26:27.350 simply using the fact that, for a linear system, what we 00:26:27.350 --> 00:26:35.180 know is that if we have an input x(t) with an output 00:26:35.180 --> 00:26:42.390 y(t), then if we scale the input, then the output scales 00:26:42.390 --> 00:26:43.940 accordingly. 00:26:43.940 --> 00:26:51.290 Well, we can simply choose, as the scale factor, a = 0. 00:26:51.290 --> 00:26:54.950 And if we do that, it says put nothing in, 00:26:54.950 --> 00:26:57.370 you get nothing out. 00:26:57.370 --> 00:27:02.380 And what we'll see is that has some important implications in 00:27:02.380 --> 00:27:05.420 terms of causality. 00:27:05.420 --> 00:27:09.400 It's important, though, while we're on it, to stress that 00:27:09.400 --> 00:27:11.750 not every system, obviously, has that property. 00:27:11.750 --> 00:27:14.380 That if you put nothing in, you get nothing out. 00:27:14.380 --> 00:27:19.980 A simple example is, let's say, a battery, let's say not 00:27:19.980 --> 00:27:21.330 connected to anything. 00:27:21.330 --> 00:27:25.250 The output is six volts no matter what the input is. 00:27:25.250 --> 00:27:29.970 And it of course then doesn't have this zero response to a 00:27:29.970 --> 00:27:31.150 zero input. 00:27:31.150 --> 00:27:35.170 It's very particular to linear systems. 00:27:35.170 --> 00:27:40.640 All right, well now let's see what this means for causality. 00:27:40.640 --> 00:27:45.500 To remind you, causality says, in effect, that the system 00:27:45.500 --> 00:27:47.060 can't anticipate the input. 00:27:47.060 --> 00:27:50.260 That's what, basically, causality means. 00:27:50.260 --> 00:27:54.470 When we talked about it previously, we defined it in a 00:27:54.470 --> 00:27:59.100 variety of ways, one of which was the statement that if two 00:27:59.100 --> 00:28:05.130 inputs are identical up until some time, then the outputs 00:28:05.130 --> 00:28:08.140 must be identical up until the same time. 00:28:08.140 --> 00:28:11.880 The reason, kind of intuitively, is that if the 00:28:11.880 --> 00:28:13.190 system is causal-- 00:28:13.190 --> 00:28:15.290 so it can't anticipate the future-- 00:28:15.290 --> 00:28:18.970 it can't anticipate whether these two identical inputs are 00:28:18.970 --> 00:28:22.530 sometime later going to change from each other or not. 00:28:25.670 --> 00:28:30.410 So causality, in general, is simply this statement, either 00:28:30.410 --> 00:28:34.030 continuous-time or discrete-time. 00:28:34.030 --> 00:28:37.200 And now, so let's look at what that means 00:28:37.200 --> 00:28:39.400 for a linear system. 00:28:39.400 --> 00:28:44.020 For a linear system, what that corresponds to or could be 00:28:44.020 --> 00:28:51.500 translated to is a statement that says that if x(t) is 0, 00:28:51.500 --> 00:28:58.050 for t less than t_0, then y(t) must be 0 for t 00:28:58.050 --> 00:29:00.580 less than t_0 also. 00:29:00.580 --> 00:29:07.790 And so what that, in effect, says, is that the system-- 00:29:07.790 --> 00:29:14.010 for a linear system to be causal, it must have the 00:29:14.010 --> 00:29:19.200 property sometimes referred to as initial rest, meaning it 00:29:19.200 --> 00:29:23.850 doesn't respond until there's some input that happens. 00:29:23.850 --> 00:29:26.250 That it's initially at rest until the 00:29:26.250 --> 00:29:29.680 input becomes non-zero. 00:29:29.680 --> 00:29:31.050 Now, why is this true? 00:29:31.050 --> 00:29:37.690 Why is this a consequence of causality for linear systems? 00:29:37.690 --> 00:29:40.980 Well, the reason is we know that if we put nothing in, we 00:29:40.980 --> 00:29:43.630 get nothing out. 00:29:43.630 --> 00:29:48.320 If we have an input that's 0 for t less than t_0, and the 00:29:48.320 --> 00:29:51.830 system can't anticipate whether that input is going to 00:29:51.830 --> 00:29:58.950 change from 0 or not, then the system must generate an output 00:29:58.950 --> 00:30:03.940 that's 0 up until that time, following the principle that 00:30:03.940 --> 00:30:07.360 if two inputs are identical up until some time, the outputs 00:30:07.360 --> 00:30:10.660 must be identical up until the same time. 00:30:10.660 --> 00:30:18.750 So this basic result for linear systems is essentially 00:30:18.750 --> 00:30:24.590 a consequence of the statement that for a linear system, zero 00:30:24.590 --> 00:30:29.010 in gives us zero out. 00:30:33.490 --> 00:30:40.390 Now, that tells us how to interpret 00:30:40.390 --> 00:30:43.190 causality for linear systems. 00:30:43.190 --> 00:30:46.360 Now, let's proceed to linear time-invariant systems. 00:30:46.360 --> 00:30:50.900 And in fact, we can carry the point one step further. 00:30:50.900 --> 00:30:55.380 In particular, a necessary and sufficient condition for 00:30:55.380 --> 00:30:59.840 causality in the case of linear time-invariant systems 00:30:59.840 --> 00:31:08.570 is that the impulse response be 0, for t less than 0 in the 00:31:08.570 --> 00:31:13.030 continuous-time case, or for n less than 0 in the 00:31:13.030 --> 00:31:14.630 discrete-time case. 00:31:14.630 --> 00:31:18.660 So for linear time-invariant systems, causality is 00:31:18.660 --> 00:31:24.310 equivalent to the impulse response being 0 up until t or 00:31:24.310 --> 00:31:27.090 n equal to 0. 00:31:27.090 --> 00:31:35.330 Now, to show this essentially follows by first considering 00:31:35.330 --> 00:31:42.710 why causality would imply that this is true. 00:31:42.710 --> 00:31:47.790 And that follows because of the straightforward fact that 00:31:47.790 --> 00:31:50.910 the impulse itself is 0 for t less than 0. 00:31:53.610 --> 00:31:58.480 And what we saw before is that for any linear system, 00:31:58.480 --> 00:32:02.280 causality requires that if the input is 0 up until some time, 00:32:02.280 --> 00:32:06.170 the output must be 0 up until the same time. 00:32:06.170 --> 00:32:11.740 And so that's showing the result in one direction. 00:32:11.740 --> 00:32:15.775 To show the result in the other direction, namely to 00:32:15.775 --> 00:32:20.820 show that if, in fact, the impulse response satisfies 00:32:20.820 --> 00:32:23.860 that condition, then the system is causal. 00:32:23.860 --> 00:32:28.170 While I won't work through it in detail, it essentially 00:32:28.170 --> 00:32:36.830 boils down to recognizing that in the convolution sum, or in 00:32:36.830 --> 00:32:41.880 the convolution integral, if, in fact, that condition is 00:32:41.880 --> 00:32:46.950 satisfied on the impulse response, then the upper limit 00:32:46.950 --> 00:32:48.880 on the sum, in the discrete-time 00:32:48.880 --> 00:32:50.900 case, changes to n. 00:32:50.900 --> 00:32:54.370 And in the continuous-time case, changes to t. 00:32:54.370 --> 00:33:00.940 And that, in effect, says that values of the input only from 00:33:00.940 --> 00:33:05.840 -infinity up to time n are used in computing y[n]. 00:33:05.840 --> 00:33:09.770 And a similar kind of result for the 00:33:09.770 --> 00:33:11.140 continuous-time case y(t). 00:33:15.380 --> 00:33:20.330 OK, so we've seen how the impulse response, or rather 00:33:20.330 --> 00:33:23.170 how certain system properties in the linear time-invariant 00:33:23.170 --> 00:33:28.920 case can, be converted into various requirements on the 00:33:28.920 --> 00:33:31.670 impulse response of a linear time-invariant system, the 00:33:31.670 --> 00:33:35.440 impulse response being a complete characterization. 00:33:35.440 --> 00:33:39.390 Let's look at some particular examples just to kind of 00:33:39.390 --> 00:33:42.140 cement the ideas further. 00:33:42.140 --> 00:33:46.190 And let's begin with a system that you've seen previously, 00:33:46.190 --> 00:33:48.165 which is an accumulator. 00:33:48.165 --> 00:33:55.860 An accumulator, as you recall, has an output y[n], which is 00:33:55.860 --> 00:34:03.080 the accumulated value of the input from -infinity up to n. 00:34:03.080 --> 00:34:07.660 Now, you've seen in the impulse in a previous lecture, 00:34:07.660 --> 00:34:11.239 or rather in the video course manual for a previous lecture, 00:34:11.239 --> 00:34:15.429 that an accumulator is a linear time-invariant system. 00:34:15.429 --> 00:34:19.760 And in fact, its impulse response is the accumulated 00:34:19.760 --> 00:34:21.370 values of an impulse. 00:34:21.370 --> 00:34:24.855 Namely, the impulse response is equal to a step. 00:34:27.650 --> 00:34:34.010 So what we want to answer is, knowing what that impulse 00:34:34.010 --> 00:34:36.929 response is, what some properties are of the 00:34:36.929 --> 00:34:38.280 accumulator. 00:34:38.280 --> 00:34:41.420 And let's first ask about memory. 00:34:41.420 --> 00:34:46.679 Well, we recognize that the impulse response is not simply 00:34:46.679 --> 00:34:47.300 an impulse. 00:34:47.300 --> 00:34:48.830 In fact, it's a step. 00:34:48.830 --> 00:34:51.560 And so this implies what? 00:34:51.560 --> 00:34:55.530 Well, it implies that the system has memory. 00:35:00.050 --> 00:35:06.780 Second, the impulse response is 0 for n less than 0. 00:35:06.780 --> 00:35:11.180 That implies that the system is causal. 00:35:15.180 --> 00:35:22.530 And finally, if we look at the sum of the absolute values of 00:35:22.530 --> 00:35:24.930 the impulse response from -infinity to 00:35:24.930 --> 00:35:27.380 +infinity, this is a step. 00:35:27.380 --> 00:35:30.790 If we accumulate those values over infinite limits, then 00:35:30.790 --> 00:35:35.210 that in fact comes out to be infinite. 00:35:35.210 --> 00:35:40.140 And so what that implies, then, is that the accumulator 00:35:40.140 --> 00:35:43.490 is not stable in the bounded-input 00:35:43.490 --> 00:35:44.780 bounded-output sense. 00:35:48.240 --> 00:35:50.410 Now I want to turn to some other systems. 00:35:50.410 --> 00:35:52.630 But while we're on the accumulator, I just want to 00:35:52.630 --> 00:35:55.900 draw your attention to the fact, which will kind of come 00:35:55.900 --> 00:36:01.350 up in a variety of ways again later, that we can rewrite the 00:36:01.350 --> 00:36:04.840 equation for an accumulator, the difference equation, by 00:36:04.840 --> 00:36:10.460 recognizing that we could, in fact, write the output as the 00:36:10.460 --> 00:36:14.540 accumulated values up to time n - 1 and then 00:36:14.540 --> 00:36:17.120 add on the last value. 00:36:17.120 --> 00:36:23.510 And in fact, if we do that, this corresponds to y[n-1]. 00:36:23.510 --> 00:36:28.330 And so we could rewrite this difference equation as y[n] 00:36:28.330 --> 00:36:30.090 = y[n-1] 00:36:30.090 --> 00:36:31.340 + x[n]. 00:36:31.340 --> 00:36:34.630 So the output is the previously-computed output 00:36:34.630 --> 00:36:36.800 plus the input. 00:36:36.800 --> 00:36:42.170 Expressed that way, what that corresponds to is what is 00:36:42.170 --> 00:36:44.400 called a recursive difference equation. 00:36:44.400 --> 00:36:47.980 And different equations will be a topic of considerable 00:36:47.980 --> 00:36:51.290 emphasis in the next lecture. 00:36:51.290 --> 00:36:53.520 Now, does an accumulator have an inverse? 00:36:53.520 --> 00:36:57.100 Well, the answer is, in fact, yes. 00:36:57.100 --> 00:37:02.190 And let's look at what the inverse of the accumulator is. 00:37:02.190 --> 00:37:06.290 The impulse response of the accumulator is a step. 00:37:06.290 --> 00:37:11.100 To inquire about its inverse, we inquire about whether 00:37:11.100 --> 00:37:15.780 there's a system, which when we cascade the accumulator 00:37:15.780 --> 00:37:20.510 with that system, which I'm calling its inverse, we get an 00:37:20.510 --> 00:37:22.410 impulse out. 00:37:22.410 --> 00:37:23.800 Well, let's see. 00:37:23.800 --> 00:37:28.060 The impulse response of the accumulator is a step. 00:37:28.060 --> 00:37:29.820 We want to put the step into something 00:37:29.820 --> 00:37:32.470 and get out an impulse. 00:37:32.470 --> 00:37:37.500 And in fact, what you recall from the lecture in which we 00:37:37.500 --> 00:37:42.120 introduced steps and impulses, the impulse is, in fact, the 00:37:42.120 --> 00:37:44.630 first difference of the units step. 00:37:44.630 --> 00:37:50.210 So we have a difference equation that describes for us 00:37:50.210 --> 00:37:54.660 how the impulse is related to the step. 00:37:54.660 --> 00:37:59.770 And so if this system does this, the output will be that, 00:37:59.770 --> 00:38:01.420 an impulse. 00:38:01.420 --> 00:38:04.465 And so if we think of x_2[n] 00:38:07.580 --> 00:38:09.470 as the input and y_2[n] 00:38:09.470 --> 00:38:14.410 as the output, then the difference equation for the 00:38:14.410 --> 00:38:18.330 inverse system is what I've indicated here. 00:38:18.330 --> 00:38:22.200 And if we want to look at the impulse response of that, we 00:38:22.200 --> 00:38:25.120 can then inquire as to what the response is with an 00:38:25.120 --> 00:38:26.830 impulse in. 00:38:26.830 --> 00:38:31.290 And what develops in a straightforward way then is 00:38:31.290 --> 00:38:37.580 delta[n], which is our impulse input, minus delta[n-1] 00:38:37.580 --> 00:38:40.530 is equal to the impulse response 00:38:40.530 --> 00:38:42.730 of the inverse system. 00:38:42.730 --> 00:38:45.260 So I'll write that as h^(-1)[n] 00:38:45.260 --> 00:38:47.410 (h-inverse of n). 00:38:47.410 --> 00:38:52.420 Now, we have then that the accumulator has an inverse. 00:38:52.420 --> 00:38:54.420 And this is the inverse. 00:38:54.420 --> 00:38:57.910 And you can examine issues of memory, stability, 00:38:57.910 --> 00:38:59.430 causality, et cetera. 00:38:59.430 --> 00:39:05.170 What you'll find is that the system has memory, the inverse 00:39:05.170 --> 00:39:05.810 accumulator. 00:39:05.810 --> 00:39:09.360 It's stable, and it's causal. 00:39:09.360 --> 00:39:12.360 And it's interesting to note, by the way, that although the 00:39:12.360 --> 00:39:17.620 accumulator was an unstable system, the inverse of the 00:39:17.620 --> 00:39:19.930 accumulator is a stable system. 00:39:19.930 --> 00:39:24.680 In general, if the system is stable, its inverse does not 00:39:24.680 --> 00:39:26.940 have to be stable or vice versa. 00:39:26.940 --> 00:39:28.365 And the same thing with causality. 00:39:32.210 --> 00:39:38.360 OK now, there are a number of other examples, which, of 00:39:38.360 --> 00:39:40.200 course, we could discuss. 00:39:40.200 --> 00:39:46.830 And let me just quickly point to one example, which is a 00:39:46.830 --> 00:39:50.640 difference equation, as I've indicated here. 00:39:50.640 --> 00:39:55.980 And as we'll talk about in more detail in our next 00:39:55.980 --> 00:39:59.480 lecture, where we'll get involved in a fairly detailed 00:39:59.480 --> 00:40:01.360 discussion of linear constant-coefficient 00:40:01.360 --> 00:40:04.690 difference and differential equations, this falls into 00:40:04.690 --> 00:40:06.290 that category. 00:40:06.290 --> 00:40:10.960 And under the imposition of what's referred to as initial 00:40:10.960 --> 00:40:17.970 rest, which corresponds to the response being 0 up until the 00:40:17.970 --> 00:40:22.450 time that the input becomes non-zero, the impulse response 00:40:22.450 --> 00:40:25.410 is a^n times u[n]. 00:40:25.410 --> 00:40:29.490 And something that you'll be asked to think about in the 00:40:29.490 --> 00:40:33.490 video course manual is whether that system has memory, 00:40:33.490 --> 00:40:36.550 whether it's causal, and whether it's stable. 00:40:36.550 --> 00:40:41.700 And likewise, for a linear constant coefficient 00:40:41.700 --> 00:40:46.070 differential equation, the specific one that I've 00:40:46.070 --> 00:40:50.020 indicated here, under the assumption of initial rest, 00:40:50.020 --> 00:40:54.710 the impulse response is e^(-2t) times u(t). 00:40:54.710 --> 00:40:58.810 And in the video course manual again, you'll be asked to 00:40:58.810 --> 00:41:02.930 examine whether the system has memory, whether it's causal, 00:41:02.930 --> 00:41:06.410 and whether it's stable. 00:41:06.410 --> 00:41:10.420 OK well, as I've indicated, in the next lecture we'll return 00:41:10.420 --> 00:41:13.270 to a much more detailed discussion of linear 00:41:13.270 --> 00:41:15.180 constant-coefficient differential 00:41:15.180 --> 00:41:17.190 and difference equations. 00:41:17.190 --> 00:41:22.480 Now, what I'd like to finally do in this lecture is use the 00:41:22.480 --> 00:41:26.410 notion of convolution in a much different way to help us 00:41:26.410 --> 00:41:30.030 with a problem that I alluded to earlier. 00:41:30.030 --> 00:41:33.800 In particular, the issue of how to deal with some of the 00:41:33.800 --> 00:41:37.200 mathematical difficulties associated with 00:41:37.200 --> 00:41:39.620 impulses and steps. 00:41:39.620 --> 00:41:43.540 Now, let me begin by illustrating kind of what the 00:41:43.540 --> 00:41:49.845 problem is and an example of the kind of paradox that you 00:41:49.845 --> 00:41:53.480 sort of run into when dealing with impulse functions and 00:41:53.480 --> 00:41:54.730 step functions. 00:41:57.000 --> 00:42:00.980 Let's consider, first of all, a system, which is the 00:42:00.980 --> 00:42:02.520 identity system. 00:42:02.520 --> 00:42:07.090 And so the output is, of course, equal to the input. 00:42:07.090 --> 00:42:12.180 And again, we can talk about that either in continuous time 00:42:12.180 --> 00:42:14.800 or in discrete time. 00:42:14.800 --> 00:42:18.990 Well, we know that the function that you convolve 00:42:18.990 --> 00:42:23.610 with a signal to retain the signal is an impulse. 00:42:23.610 --> 00:42:26.650 And so that means that the impulse response of an 00:42:26.650 --> 00:42:28.740 identity system is an impulse. 00:42:28.740 --> 00:42:32.220 Makes logical sense. 00:42:32.220 --> 00:42:36.300 Furthermore, if I take two identity systems and cascade 00:42:36.300 --> 00:42:40.240 them, I put in an input, get the same thing out of the 00:42:40.240 --> 00:42:40.900 first system. 00:42:40.900 --> 00:42:42.190 That goes into the second system. 00:42:42.190 --> 00:42:44.125 Get the same thing out of the second. 00:42:44.125 --> 00:42:51.540 In other words, if I have two identity systems in cascade, 00:42:51.540 --> 00:42:55.840 the cascade, likewise, is an identity system. 00:42:55.840 --> 00:42:59.020 In other words, this overall system is 00:42:59.020 --> 00:43:03.280 also an identity system. 00:43:03.280 --> 00:43:07.610 And the implication there is that the impulse response of 00:43:07.610 --> 00:43:09.110 this is an impulse. 00:43:09.110 --> 00:43:11.750 The impulse response of this is an impulse. 00:43:11.750 --> 00:43:16.560 And the convolution of those two is also an impulse. 00:43:16.560 --> 00:43:21.860 So for continuous time, we require, then, that an impulse 00:43:21.860 --> 00:43:24.550 convolved with itself is an impulse. 00:43:24.550 --> 00:43:28.850 And the same thing for discrete time. 00:43:28.850 --> 00:43:33.280 Now, in discrete time, we don't have any particular 00:43:33.280 --> 00:43:34.220 problem with that. 00:43:34.220 --> 00:43:37.130 If you think about convolving these together, it's a 00:43:37.130 --> 00:43:42.850 straightforward mathematical operation since the impulse in 00:43:42.850 --> 00:43:47.300 discrete time is very nicely defined. 00:43:47.300 --> 00:43:51.070 However, in continuous time, we have to be somewhat careful 00:43:51.070 --> 00:43:54.710 about the definition of the impulse because it was the 00:43:54.710 --> 00:43:55.870 derivative of a step. 00:43:55.870 --> 00:43:57.930 A step has a discontinuity. 00:43:57.930 --> 00:44:01.190 You can't really differentiate at a discontinuity. 00:44:01.190 --> 00:44:06.020 And the way that we dealt with that was to expand out the 00:44:06.020 --> 00:44:09.570 discontinuity so that it had some finite time region in 00:44:09.570 --> 00:44:11.090 which it happened. 00:44:11.090 --> 00:44:14.080 When we did that, we ended up with a definition for the 00:44:14.080 --> 00:44:18.200 impulse, which was the limiting form of this 00:44:18.200 --> 00:44:23.510 function, which is a rectangle of width Delta, and height 1 / 00:44:23.510 --> 00:44:27.120 Delta, and an area equal to 1. 00:44:27.120 --> 00:44:33.680 Now, if we think of convolving this signal with itself, the 00:44:33.680 --> 00:44:37.480 impulse being the limiting form of this, then the 00:44:37.480 --> 00:44:42.690 convolution of this with itself is a triangle of width 00:44:42.690 --> 00:44:47.290 2 Delta, height 1 / Delta, and area 1. 00:44:47.290 --> 00:44:51.560 In other words, this triangular function is this 00:44:51.560 --> 00:44:59.030 approximation delta_Delta(t) convolved with delta_Delta(t). 00:44:59.030 --> 00:45:05.000 And since the limit of this would correspond to the 00:45:05.000 --> 00:45:11.480 impulse response of the identity system convolved with 00:45:11.480 --> 00:45:17.470 itself, the implication is that not only should the top 00:45:17.470 --> 00:45:23.600 function, this one, correspond in its limiting form to an 00:45:23.600 --> 00:45:28.130 impulse, but also this should correspond in its limiting 00:45:28.130 --> 00:45:29.990 form to an impulse. 00:45:29.990 --> 00:45:32.750 So one could wonder well, what is an impulse? 00:45:32.750 --> 00:45:34.320 Is it this one in the limit? 00:45:34.320 --> 00:45:35.650 Or is it this one in the limit? 00:45:42.880 --> 00:45:45.420 Now, beyond that-- so kind of what this suggests is that in 00:45:45.420 --> 00:45:48.360 the limiting form, you kind of run into a contradiction 00:45:48.360 --> 00:45:50.920 unless you don't try to distinguish between this 00:45:50.920 --> 00:45:53.170 rectangle and the triangle. 00:45:53.170 --> 00:45:57.050 Things get even worse when you think about what happens when 00:45:57.050 --> 00:45:59.160 you put an impulse into a differentiator. 00:45:59.160 --> 00:46:03.630 And a differentiator is a very commonly occurring system. 00:46:03.630 --> 00:46:08.350 In particular, suppose we had a system for which the output 00:46:08.350 --> 00:46:11.010 was the derivative of the input. 00:46:11.010 --> 00:46:17.500 So if we put in x(t), we got out dx(t) / dt. 00:46:17.500 --> 00:46:20.360 If I put in an impulse, or if I talked about the impulse 00:46:20.360 --> 00:46:23.130 response, what is that? 00:46:23.130 --> 00:46:27.610 And of course, the problem is that if you think that the 00:46:27.610 --> 00:46:31.880 impulse itself is very badly behaved, then what about its 00:46:31.880 --> 00:46:36.110 derivative, which is not only infinitely big, but there's a 00:46:36.110 --> 00:46:38.350 positive-going one, and a negative-going one, and the 00:46:38.350 --> 00:46:40.440 difference between there has some area. 00:46:40.440 --> 00:46:42.910 And you end up in a lot of difficulty. 00:46:45.930 --> 00:46:51.250 Well, the way around this, formally, is through a set of 00:46:51.250 --> 00:46:56.140 mathematics referred to as generalized functions. 00:46:56.140 --> 00:46:58.110 We won't be quite that formal. 00:46:58.110 --> 00:47:01.340 But I'd like to, at least, suggest what the essence of 00:47:01.340 --> 00:47:02.530 that formality is. 00:47:02.530 --> 00:47:07.360 And it really helps us in interpreting the impulses in 00:47:07.360 --> 00:47:09.620 steps and functions of that type. 00:47:09.620 --> 00:47:13.660 And what it is is an operational definition of 00:47:13.660 --> 00:47:16.560 steps, impulses, and their derivatives in 00:47:16.560 --> 00:47:18.360 the following sense. 00:47:18.360 --> 00:47:21.600 Usually when we talk about a function, we talk about what 00:47:21.600 --> 00:47:24.910 the value of the function is at any instant of time. 00:47:24.910 --> 00:47:26.810 And of course, the trouble with an impulse is it's 00:47:26.810 --> 00:47:30.930 infinitely big, in zero width, and has some area, et cetera. 00:47:30.930 --> 00:47:35.310 What we can turn to is what is referred to as an operational 00:47:35.310 --> 00:47:41.550 definition where the operational definition is 00:47:41.550 --> 00:47:45.790 related not to what the impulse is, but to what the 00:47:45.790 --> 00:47:50.760 impulse does under the operation of convolution. 00:47:50.760 --> 00:47:52.540 So what is an impulse? 00:47:52.540 --> 00:47:56.110 An impulse is something, which under 00:47:56.110 --> 00:47:59.460 convolution, retains the function. 00:47:59.460 --> 00:48:04.200 And that then can serve as a definition of the impulse. 00:48:04.200 --> 00:48:06.410 Well, let's see where that gets us. 00:48:06.410 --> 00:48:11.060 Suppose that we now want to talk about the derivative of 00:48:11.060 --> 00:48:13.370 the impulse. 00:48:13.370 --> 00:48:18.970 Well, what we ask about is what it is operationally. 00:48:18.970 --> 00:48:25.510 And so if we have a system, which is a differentiator, and 00:48:25.510 --> 00:48:28.600 we inquire about its impulse response, which let's say we 00:48:28.600 --> 00:48:32.630 define notationally as u_1(t). 00:48:32.630 --> 00:48:37.650 What's important about this function u_1(t) is not what it 00:48:37.650 --> 00:48:42.620 is at each value of time but what it does under 00:48:42.620 --> 00:48:43.840 convolution. 00:48:43.840 --> 00:48:45.690 What does it do under convolution? 00:48:45.690 --> 00:48:49.610 Well, the output of the differentiator is the 00:48:49.610 --> 00:48:53.190 convolution of the input with the impulse response. 00:48:53.190 --> 00:48:58.100 And so what u_1(t) does under convolution is to 00:48:58.100 --> 00:48:59.780 differentiate. 00:48:59.780 --> 00:49:03.770 And that is the operational definition. 00:49:03.770 --> 00:49:07.450 And now, of course, we can think of extending that. 00:49:07.450 --> 00:49:11.540 Not only would we want to think about differentiating an 00:49:11.540 --> 00:49:15.960 impulse, but we would also want to think about 00:49:15.960 --> 00:49:18.740 differentiating the derivative of an impulse. 00:49:18.740 --> 00:49:23.000 We'll define that as a function u_2(t). 00:49:23.000 --> 00:49:24.690 u_2(t)-- 00:49:24.690 --> 00:49:28.050 because we have this impulse response convolved with this 00:49:28.050 --> 00:49:31.700 one is u_1(t) * u_1(t). 00:49:31.700 --> 00:49:36.800 And what is u_2(t) operationally? 00:49:36.800 --> 00:49:42.800 It is the operation such that when you convolve that with 00:49:42.800 --> 00:49:48.540 x(t), what you get is the second derivative. 00:49:48.540 --> 00:49:52.110 OK now, we can carry this further and, in fact, talk 00:49:52.110 --> 00:49:59.410 about the result of convolving u_1(t) with itself more times. 00:49:59.410 --> 00:50:03.870 In fact, if we think of the convulution of u_1(t) with 00:50:03.870 --> 00:50:07.460 itself k times, then logically we would 00:50:07.460 --> 00:50:11.330 define that as u_k(t). 00:50:11.330 --> 00:50:14.800 Again, we would interpret that operationally. 00:50:14.800 --> 00:50:20.440 And the operational definition is through convolution, where 00:50:20.440 --> 00:50:26.600 this corresponds to u_k(t) being the impulse response of 00:50:26.600 --> 00:50:29.630 k differentiators in cascade. 00:50:29.630 --> 00:50:32.090 So what is the operational definition? 00:50:32.090 --> 00:50:39.800 Well, it's simply that x(t) * u_k(t) is the k 00:50:39.800 --> 00:50:42.440 derivative of x(t). 00:50:45.080 --> 00:50:49.480 And this now gives us a set of what are referred to as 00:50:49.480 --> 00:50:50.720 singularity functions. 00:50:50.720 --> 00:50:55.360 Very badly behaved mathematically in a sense, but 00:50:55.360 --> 00:50:58.700 as we've seen, reasonably well defined under an operational 00:50:58.700 --> 00:51:00.780 definition. 00:51:00.780 --> 00:51:06.010 With k = 0, incidentally, that's the same as what we 00:51:06.010 --> 00:51:08.560 have referred to previously as the impulse. 00:51:08.560 --> 00:51:13.260 So with k 0, that's just delta(t). 00:51:13.260 --> 00:51:17.150 Now to be complete, we can also go the other way and talk 00:51:17.150 --> 00:51:20.940 about the impulse response of a string of integrators 00:51:20.940 --> 00:51:23.460 instead of a string of differentiators. 00:51:23.460 --> 00:51:25.280 Of course, the impulse response of a single 00:51:25.280 --> 00:51:27.580 integrator is a unit step. 00:51:27.580 --> 00:51:30.120 Two integrators together is the integral of a 00:51:30.120 --> 00:51:32.680 unit step, et cetera. 00:51:32.680 --> 00:51:38.110 And that, likewise, corresponds to a set of what 00:51:38.110 --> 00:51:40.290 are called singularity functions. 00:51:40.290 --> 00:51:46.410 In particular, if I take a string of m integrators in 00:51:46.410 --> 00:51:52.610 cascade, then the impulse response of that is denoted as 00:51:52.610 --> 00:51:56.000 u sub minus m of t. 00:51:56.000 --> 00:52:00.380 And for example, with a single integrator, u sub minus 1 of t 00:52:00.380 --> 00:52:06.950 corresponds to our unit step as we talked about previously. 00:52:06.950 --> 00:52:14.430 u sub minus 2 of t corresponds to a unit ramp, et cetera. 00:52:14.430 --> 00:52:18.060 And there is, in fact, a reason for choosing negative 00:52:18.060 --> 00:52:22.850 values of the argument when going in one direction near 00:52:22.850 --> 00:52:25.370 integration as compared with positive values of the 00:52:25.370 --> 00:52:28.130 argument when going in the other direction, namely 00:52:28.130 --> 00:52:30.120 differentiation. 00:52:30.120 --> 00:52:38.110 In particular, we know that with u sub minus m of t, the 00:52:38.110 --> 00:52:44.520 operational definition is the mth running integral. 00:52:44.520 --> 00:52:48.370 And likewise, u_k(t)-- 00:52:48.370 --> 00:52:51.040 so with a positive sub script-- 00:52:51.040 --> 00:52:56.240 has an operational definition, which is the derivative. 00:52:56.240 --> 00:53:01.290 So it's the kth derivative of x(t). 00:53:01.290 --> 00:53:08.070 And partly as a consequence of that, if we take u_k(t) and 00:53:08.070 --> 00:53:13.400 convolve it with u_l(t), the result is the singularity 00:53:13.400 --> 00:53:17.170 function with the subscript, which is the sum of k and l. 00:53:17.170 --> 00:53:21.470 And that holds whether this is positive values of the 00:53:21.470 --> 00:53:24.610 subscript or negative values of the subscript. 00:53:24.610 --> 00:53:28.550 So just to summarize this last discussion, we've used an 00:53:28.550 --> 00:53:35.520 operational definition to talk about derivatives of impulses 00:53:35.520 --> 00:53:37.940 and integrals of impulses. 00:53:37.940 --> 00:53:40.705 This led to a set of singularity functions-- what 00:53:40.705 --> 00:53:42.270 I've called singularity functions-- 00:53:42.270 --> 00:53:45.400 of which the impulse and the step are two examples. 00:53:45.400 --> 00:53:49.700 But using an operational definition through convolution 00:53:49.700 --> 00:53:55.500 allows us to define, at least in an operational sense, these 00:53:55.500 --> 00:53:57.880 functions that otherwise are very badly behaved. 00:54:01.110 --> 00:54:06.330 OK now, in this lecture and previous lectures, for the 00:54:06.330 --> 00:54:10.890 most part, our discussion has been about linear 00:54:10.890 --> 00:54:14.910 time-invariant systems in fairly general terms. 00:54:14.910 --> 00:54:18.480 And we've seen a variety of properties, representation 00:54:18.480 --> 00:54:22.390 through convolution, and properties as they can be 00:54:22.390 --> 00:54:25.750 associated with the impulse response. 00:54:25.750 --> 00:54:28.970 In the next lecture, we'll turn our attention to a very 00:54:28.970 --> 00:54:34.150 important subclass of those systems, namely systems that 00:54:34.150 --> 00:54:37.410 are describable by linear constant-coefficient 00:54:37.410 --> 00:54:40.640 difference equations in the discrete-time case, and linear 00:54:40.640 --> 00:54:43.540 constant-coefficient differential equations in the 00:54:43.540 --> 00:54:45.410 continuous-time case. 00:54:45.410 --> 00:54:50.800 Those classes, while not forming all of the class of 00:54:50.800 --> 00:54:53.080 linear time-invariant systems, are a very 00:54:53.080 --> 00:54:54.930 important sub class. 00:54:54.930 --> 00:54:57.470 And we'll focus in on those specifically next time. 00:54:57.470 --> 00:54:58.720 Thank you.