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 --> 00:00:21.518 ocw.mit.edu. 00:00:21.518 --> 00:00:56.170 [MUSIC PLAYING] 00:00:56.170 --> 00:00:58.350 PROFESSOR: Over the last series of lectures, in 00:00:58.350 --> 00:01:00.940 discussing filtering, modulation, and sampling, 00:01:00.940 --> 00:01:03.720 we've seen how powerful and useful the 00:01:03.720 --> 00:01:06.490 Fourier transform is. 00:01:06.490 --> 00:01:09.210 Beginning with this lecture, and over the next several 00:01:09.210 --> 00:01:14.400 lectures, I'd like to develop and exploit a generalization 00:01:14.400 --> 00:01:18.600 of the Fourier transform, which will not only lead to 00:01:18.600 --> 00:01:22.240 some important new insights about signals and systems, but 00:01:22.240 --> 00:01:27.110 also will remove some of the restrictions that we've had 00:01:27.110 --> 00:01:30.300 with the Fourier transform. 00:01:30.300 --> 00:01:34.510 The generalization that we'll be talking about in the 00:01:34.510 --> 00:01:38.330 continuous time case is referred to as the Laplace 00:01:38.330 --> 00:01:42.060 transform, and in the discrete time case, is referred to as 00:01:42.060 --> 00:01:43.680 the z transform. 00:01:43.680 --> 00:01:46.810 What I'd like to do in today's lecture is begin on the 00:01:46.810 --> 00:01:49.110 continuous time case, namely a discussion 00:01:49.110 --> 00:01:51.010 of the Laplace transform. 00:01:51.010 --> 00:01:55.130 Continue that into the next lecture, and following that 00:01:55.130 --> 00:01:58.690 develop the z transform for discrete time. 00:01:58.690 --> 00:02:02.150 And also, as we go along, exploit 00:02:02.150 --> 00:02:05.080 the two notions together. 00:02:05.080 --> 00:02:09.630 Now, to introduce the notion of the Laplace transform, let 00:02:09.630 --> 00:02:12.250 me remind you again of what led us 00:02:12.250 --> 00:02:15.380 into the Fourier transform. 00:02:15.380 --> 00:02:19.910 We developed the Fourier transform by considering the 00:02:19.910 --> 00:02:26.210 idea of representing signals as linear combinations of 00:02:26.210 --> 00:02:27.660 basic signals. 00:02:27.660 --> 00:02:33.060 And in the Fourier transform, in the continuous time case, 00:02:33.060 --> 00:02:36.190 the basic signals that we picked in the representation 00:02:36.190 --> 00:02:38.800 were complex exponentials. 00:02:38.800 --> 00:02:43.650 And in what we had referred to as the synthesis equation, the 00:02:43.650 --> 00:02:48.020 synthesis equation corresponded to, in effect, a 00:02:48.020 --> 00:02:52.230 decomposition as a linear combination, a decomposition 00:02:52.230 --> 00:02:55.350 of x of t as a linear combination of complex 00:02:55.350 --> 00:02:56.980 exponentials. 00:02:56.980 --> 00:03:00.440 And of course, associated with this was the corresponding 00:03:00.440 --> 00:03:05.010 analysis equation that, in effect, gave us the amplitudes 00:03:05.010 --> 00:03:08.850 associated with the complex exponentials. 00:03:08.850 --> 00:03:11.550 Now, why did we pick complex exponentials? 00:03:11.550 --> 00:03:16.760 Well, recall that the reason was that complex exponentials 00:03:16.760 --> 00:03:20.250 are eigenfunctions of linear time-invariant systems, and 00:03:20.250 --> 00:03:21.900 that was very convenient. 00:03:21.900 --> 00:03:27.900 Specifically, if we have a linear time-invariant system 00:03:27.900 --> 00:03:34.080 with an impulse response h of t, what we had shown is that 00:03:34.080 --> 00:03:37.750 that class of systems has the property that if we put in a 00:03:37.750 --> 00:03:43.070 complex exponential, we get out a complex exponential at 00:03:43.070 --> 00:03:47.100 the same frequency and with a change in amplitude. 00:03:47.100 --> 00:03:51.790 And this change in amplitude, in fact, corresponded as we 00:03:51.790 --> 00:03:55.180 showed as the discussion went along, to the Fourier 00:03:55.180 --> 00:04:00.380 transform of the system impulse response. 00:04:00.380 --> 00:04:06.630 So the notion of decomposing signals into complex 00:04:06.630 --> 00:04:09.710 exponentials was very intimately connected, and the 00:04:09.710 --> 00:04:13.470 Fourier transform was very intimately connected, with the 00:04:13.470 --> 00:04:16.810 eigenfunction property of complex exponentials for 00:04:16.810 --> 00:04:20.070 linear time-invariant systems. 00:04:20.070 --> 00:04:24.370 Well, complex exponentials of that type are not the only 00:04:24.370 --> 00:04:28.470 eigenfunctions for linear time-invariant systems. 00:04:28.470 --> 00:04:33.210 In fact, what you've seen previously is that if we took 00:04:33.210 --> 00:04:39.290 a more general exponential, e to the st, where s is a more 00:04:39.290 --> 00:04:40.760 general complex number. 00:04:40.760 --> 00:04:45.670 Not just j omega, but in fact sigma plus j omega. 00:04:45.670 --> 00:04:50.110 For any value of s, the complex exponential is an 00:04:50.110 --> 00:04:51.370 eigenfunction. 00:04:51.370 --> 00:04:55.350 And we can justify that simply by substitution into the 00:04:55.350 --> 00:04:57.040 convolution integral. 00:04:57.040 --> 00:05:01.060 In other words, the response to this complex exponential is 00:05:01.060 --> 00:05:05.570 the convolution of the impulse response with the excitation. 00:05:05.570 --> 00:05:12.070 And notice that we can break this term into a product, e to 00:05:12.070 --> 00:05:16.030 the st e to the minus s tau. 00:05:16.030 --> 00:05:21.300 And the e to the st term can come outside the integration. 00:05:21.300 --> 00:05:25.160 And consequently, just carrying through that algebra, 00:05:25.160 --> 00:05:32.760 would reduce this integral to an integral with an e to the 00:05:32.760 --> 00:05:35.040 st factor outside. 00:05:35.040 --> 00:05:38.130 So just simply carrying through the algebra, what we 00:05:38.130 --> 00:05:42.360 would conclude is that a complex exponential with any 00:05:42.360 --> 00:05:47.110 complex number s would generate, as an output, a 00:05:47.110 --> 00:05:51.960 complex exponential of the same form multiplied by 00:05:51.960 --> 00:05:53.460 whatever this integral is. 00:05:53.460 --> 00:05:56.520 And this integral, of course, will depend on what 00:05:56.520 --> 00:05:58.110 the value of s is. 00:05:58.110 --> 00:06:00.030 But that's all that it will depend on. 00:06:00.030 --> 00:06:06.340 Or said another way, what this all can be denoted as is some 00:06:06.340 --> 00:06:11.560 function h of s that depends on the value of s. 00:06:11.560 --> 00:06:16.370 So finally then, e to the st as an excitation to a linear 00:06:16.370 --> 00:06:20.430 time-invariant system generates a response, which is 00:06:20.430 --> 00:06:26.840 a complex constant depending on s, multiplying the same 00:06:26.840 --> 00:06:30.340 function that excited the system. 00:06:30.340 --> 00:06:35.840 So what we have then is the eigenfunction property, more 00:06:35.840 --> 00:06:41.300 generally, in terms of a more general complex exponential 00:06:41.300 --> 00:06:46.660 where the complex factor is given by this integral. 00:06:46.660 --> 00:06:51.830 Well, in fact, what that integral corresponds to is 00:06:51.830 --> 00:06:57.120 what we will define as the Laplace transform of the 00:06:57.120 --> 00:06:58.860 impulse response. 00:06:58.860 --> 00:07:04.710 And in fact, we can apply this transformation to a more 00:07:04.710 --> 00:07:08.240 general time function that may or may not be the impulse 00:07:08.240 --> 00:07:12.370 response of a linear time-invariant system. 00:07:12.370 --> 00:07:17.130 And so, in general, it is this transformation on a time 00:07:17.130 --> 00:07:22.020 function which is the Laplace transform of that time 00:07:22.020 --> 00:07:25.050 function, and it's a function of s. 00:07:25.050 --> 00:07:31.120 So the definition of the Laplace transform is that the 00:07:31.120 --> 00:07:37.120 Laplace transform of a time function x of t is the result 00:07:37.120 --> 00:07:39.990 of this transformation on x of t. 00:07:39.990 --> 00:07:45.280 It's denoted as x of s, and as a shorthand notation as we had 00:07:45.280 --> 00:07:49.040 with the Fourier transform, then we have in the time 00:07:49.040 --> 00:07:52.640 domain, the time function x of t, and in the Laplace 00:07:52.640 --> 00:07:57.170 transform domain, the function x of s. 00:07:57.170 --> 00:08:01.560 And these then represent a transform pair. 00:08:01.560 --> 00:08:07.190 Now, let me remind you that the development of that 00:08:07.190 --> 00:08:11.170 mapping is exactly the process the we went through initially 00:08:11.170 --> 00:08:14.670 in developing a mapping that ended up giving 00:08:14.670 --> 00:08:17.260 us the Fourier transform. 00:08:17.260 --> 00:08:20.710 Essentially, what we've done is just broadened our horizon 00:08:20.710 --> 00:08:24.140 somewhat, or our notation somewhat. 00:08:24.140 --> 00:08:27.750 And rather than pushing just a complex exponential through 00:08:27.750 --> 00:08:31.830 the system, we've pushed a more general time function e 00:08:31.830 --> 00:08:35.760 to the st, where s is a complex number with both a 00:08:35.760 --> 00:08:39.080 real part and an imaginary part. 00:08:39.080 --> 00:08:41.440 Well, the discussion that we've gone through so far, of 00:08:41.440 --> 00:08:45.440 course, is very closely related to what we went 00:08:45.440 --> 00:08:48.020 through for the Fourier transform. 00:08:48.020 --> 00:08:49.860 The mapping that we've ended up with is 00:08:49.860 --> 00:08:51.850 called the Laplace transform. 00:08:51.850 --> 00:08:54.490 And as you can well imagine and perhaps, may have 00:08:54.490 --> 00:08:59.200 recognized already, there's a very close connection between 00:08:59.200 --> 00:09:03.500 the Laplace transform and the Fourier transform. 00:09:03.500 --> 00:09:06.440 Well, to see one of the connections, what we can 00:09:06.440 --> 00:09:11.240 observe is that if we look at the Fourier transform 00:09:11.240 --> 00:09:16.020 expression and if we look at the Laplace transform 00:09:16.020 --> 00:09:21.720 expression, where s is now a general complex number sigma 00:09:21.720 --> 00:09:25.720 plus j omega, these two expressions, in fact, are 00:09:25.720 --> 00:09:31.730 identical if, in fact, sigma is equal to 0. 00:09:31.730 --> 00:09:36.530 If sigma is equal to 0 so that s is just j omega, then all 00:09:36.530 --> 00:09:40.550 that this transformation is, is the same as that. 00:09:40.550 --> 00:09:44.490 Substitute in s equals j omega and this is what we get. 00:09:47.530 --> 00:09:51.190 What this then tells us is that if we have the Laplace 00:09:51.190 --> 00:09:57.390 transform, and if we look at the Laplace transform at s 00:09:57.390 --> 00:10:05.350 equals j omega, then that, in fact, corresponds to the 00:10:05.350 --> 00:10:09.420 Fourier transform of x of t. 00:10:09.420 --> 00:10:13.910 Now, there is a slight notational issue that this 00:10:13.910 --> 00:10:15.650 raises, and it's very 00:10:15.650 --> 00:10:17.530 straightforward to clean it up. 00:10:17.530 --> 00:10:19.520 But it's something that it's-- 00:10:19.520 --> 00:10:23.980 you have to just kind of focus on for a second to understand 00:10:23.980 --> 00:10:25.470 what the issue is. 00:10:25.470 --> 00:10:30.510 Notice that on the left-hand side of this equation, x of s 00:10:30.510 --> 00:10:33.550 representing the Laplace transform. 00:10:33.550 --> 00:10:36.690 When we look at that with sigma equal to 0 or s equal to 00:10:36.690 --> 00:10:41.100 j omega, our natural inclination is to write that 00:10:41.100 --> 00:10:44.140 as x of j omega, of course. 00:10:44.140 --> 00:10:46.470 On the other hand, the right-hand side of the 00:10:46.470 --> 00:10:49.970 equation, namely the Fourier transform of x of t, we've 00:10:49.970 --> 00:10:55.500 typically written as x of omega. 00:10:55.500 --> 00:10:58.210 Focusing on the fact that it's a function of 00:10:58.210 --> 00:11:00.580 this variable omega. 00:11:00.580 --> 00:11:03.250 Well, there's a slight awkwardness here because here 00:11:03.250 --> 00:11:05.740 we're talking about an argument j omega, here we're 00:11:05.740 --> 00:11:07.920 talking about an argument omega. 00:11:07.920 --> 00:11:11.370 And a very straightforward way of dealing with that is to 00:11:11.370 --> 00:11:16.760 simply change our notation for the Fourier transform, 00:11:16.760 --> 00:11:19.850 recognizing that the Fourier transform, of course, is a 00:11:19.850 --> 00:11:22.740 function of omega, but it's also, in fact, a 00:11:22.740 --> 00:11:24.470 function of j omega. 00:11:24.470 --> 00:11:28.060 And if we write it that way, then the two 00:11:28.060 --> 00:11:29.440 notations come together. 00:11:29.440 --> 00:11:33.520 In other words, the Laplace transform at s equals j omega 00:11:33.520 --> 00:11:36.260 just simply reduces both mathematically and 00:11:36.260 --> 00:11:39.100 notationally to the Fourier transform. 00:11:39.100 --> 00:11:43.400 So the notation that we'll now be adopting for the Fourier 00:11:43.400 --> 00:11:48.110 transform is the notation whereby we express the Fourier 00:11:48.110 --> 00:11:53.190 transform no longer simply as x of omega, but choosing as 00:11:53.190 --> 00:11:55.760 the argument j omega. 00:11:55.760 --> 00:11:59.620 Simple notational change. 00:11:59.620 --> 00:12:02.360 Now, here we see one relationship between the 00:12:02.360 --> 00:12:06.560 Fourier transform and the Laplace transform. 00:12:06.560 --> 00:12:10.180 Namely that the Laplace transform for s equals j omega 00:12:10.180 --> 00:12:12.770 reduces to the Fourier transform. 00:12:12.770 --> 00:12:17.340 We also have another important relationship. 00:12:17.340 --> 00:12:22.880 In particular, the fact that the Laplace transform can be 00:12:22.880 --> 00:12:28.160 interpreted as the Fourier transform of a modified 00:12:28.160 --> 00:12:29.710 version of x of t. 00:12:29.710 --> 00:12:32.820 Let me show you what I mean. 00:12:32.820 --> 00:12:35.000 Here, of course, we have the relationship 00:12:35.000 --> 00:12:36.190 that we just developed. 00:12:36.190 --> 00:12:38.225 Namely that s equals j omega. 00:12:38.225 --> 00:12:42.540 The Laplace transform reduces to the Fourier transform. 00:12:42.540 --> 00:12:45.810 But now let's look at the more general Laplace transform 00:12:45.810 --> 00:12:47.440 expression. 00:12:47.440 --> 00:12:51.800 And if we substitute in s equals sigma plus j omega, 00:12:51.800 --> 00:12:57.500 which is the general form for this complex variable s, and 00:12:57.500 --> 00:13:04.820 we carry through some of the algebra, breaking this into 00:13:04.820 --> 00:13:06.840 the product of two exponentials, z to the minus 00:13:06.840 --> 00:13:10.360 sigma t times z to the minus j omega t. 00:13:10.360 --> 00:13:15.530 We now have this expression where, of course, in both of 00:13:15.530 --> 00:13:20.140 these there is a dt. 00:13:20.140 --> 00:13:26.380 And now when we look at this, what we observe is that this, 00:13:26.380 --> 00:13:30.930 in fact, is the Fourier transform of something. 00:13:30.930 --> 00:13:32.160 What's the something? 00:13:32.160 --> 00:13:35.830 It's not x of t anymore, it's the Fourier transform of x of 00:13:35.830 --> 00:13:40.730 t multiplied by e to the minus sigma t. 00:13:40.730 --> 00:13:44.790 So if we think of these two terms together, this integral 00:13:44.790 --> 00:13:46.930 is just the Fourier transform. 00:13:46.930 --> 00:13:51.070 It's the Fourier transform of x of t multiplied by an 00:13:51.070 --> 00:13:52.170 exponential. 00:13:52.170 --> 00:13:57.600 If sigma is greater than 0, it's an exponential that 00:13:57.600 --> 00:13:59.720 decays with time. 00:13:59.720 --> 00:14:02.540 If sigma is less than 0, it's an exponential 00:14:02.540 --> 00:14:05.380 that grows with time. 00:14:05.380 --> 00:14:09.720 So we have then this additional relationship, which 00:14:09.720 --> 00:14:13.010 tells us that the Laplace transform is the Fourier 00:14:13.010 --> 00:14:19.630 transform of an exponentially weighted time function. 00:14:19.630 --> 00:14:24.290 Now, this exponential weighting has some important 00:14:24.290 --> 00:14:26.000 significance. 00:14:26.000 --> 00:14:29.800 In particular, recall that there were issues of 00:14:29.800 --> 00:14:31.840 convergence with the Fourier transform. 00:14:31.840 --> 00:14:35.590 In particular, the Fourier transform 00:14:35.590 --> 00:14:36.940 may or may not converge. 00:14:36.940 --> 00:14:40.700 And for convergence, in fact, what's required is that the 00:14:40.700 --> 00:14:43.250 time function that we're transforming be absolutely 00:14:43.250 --> 00:14:45.140 integrable. 00:14:45.140 --> 00:14:49.920 Now, we can have a time function that isn't absolutely 00:14:49.920 --> 00:14:52.930 integrable because, let's say, it grows 00:14:52.930 --> 00:14:56.760 exponentially as time increases. 00:14:56.760 --> 00:15:00.940 But when we multiply it by this exponential factor that's 00:15:00.940 --> 00:15:05.050 embodied in the Laplace transform, in fact that brings 00:15:05.050 --> 00:15:07.970 the function back down for positive time. 00:15:07.970 --> 00:15:13.430 And we'll impose absolute integrability on the product 00:15:13.430 --> 00:15:17.190 of x of t times e to the minus sigma t. 00:15:17.190 --> 00:15:22.300 And so the conclusion, an important point is that the 00:15:22.300 --> 00:15:25.610 Laplace transform, the Fourier transform of this product may 00:15:25.610 --> 00:15:27.490 converge, even though the Fourier 00:15:27.490 --> 00:15:29.530 transform of x of t doesn't. 00:15:29.530 --> 00:15:33.460 In other words, the Laplace transform may converge even 00:15:33.460 --> 00:15:35.970 when the Fourier transform doesn't converge. 00:15:35.970 --> 00:15:38.030 And we'll see that and we'll see examples of it as the 00:15:38.030 --> 00:15:40.830 discussion goes along. 00:15:40.830 --> 00:15:43.500 Now let me also draw your attention to the fact, 00:15:43.500 --> 00:15:48.720 although we won't be working through this in detail. 00:15:48.720 --> 00:15:53.520 To the fact that this equation, in effect, provides 00:15:53.520 --> 00:16:00.990 the basis for us to figure out how to express x of t in terms 00:16:00.990 --> 00:16:03.100 of the Laplace transform. 00:16:03.100 --> 00:16:06.350 In effect, we can apply the inverse Fourier transform to 00:16:06.350 --> 00:16:10.870 this, thereby to this, account for the exponential factor by 00:16:10.870 --> 00:16:12.980 bringing it over to the other side. 00:16:12.980 --> 00:16:17.260 And if you go through this, and in fact, you'll have an 00:16:17.260 --> 00:16:19.860 opportunity to go through this both in the video course 00:16:19.860 --> 00:16:26.180 manual and also it's carried through in the text, what you 00:16:26.180 --> 00:16:30.600 end up with is a synthesis equation, an expression for x 00:16:30.600 --> 00:16:34.070 of t in terms of x of s which corresponds 00:16:34.070 --> 00:16:35.730 to a synthesis equation. 00:16:35.730 --> 00:16:40.070 And which now builds x of t out of a linear combination of 00:16:40.070 --> 00:16:44.570 not necessarily functions of the form e to the j omega t, 00:16:44.570 --> 00:16:50.930 but in terms of functions or basic signals which are more 00:16:50.930 --> 00:16:56.160 general exponentials e to the st. 00:16:56.160 --> 00:17:01.720 OK, well, let's just look at some examples of the Laplace 00:17:01.720 --> 00:17:03.320 transform of some time functions. 00:17:03.320 --> 00:17:07.980 And these examples that I'll go through are all examples 00:17:07.980 --> 00:17:10.410 that are worked out in the text. 00:17:10.410 --> 00:17:12.790 And so I don't want to focus on the algebra. 00:17:12.790 --> 00:17:15.990 What I'd like to focus on are some of the issues and the 00:17:15.990 --> 00:17:18.450 interpretation. 00:17:18.450 --> 00:17:24.010 Let's first of all, look at the example in the text, which 00:17:24.010 --> 00:17:26.329 is Example 9.1. 00:17:26.329 --> 00:17:31.720 If we take the Fourier transform of this exponential, 00:17:31.720 --> 00:17:35.745 then, as you well know, the result we have is 1 over j 00:17:35.745 --> 00:17:37.500 omega plus a. 00:17:37.500 --> 00:17:41.010 And that can't converge for any a. 00:17:41.010 --> 00:17:44.680 In particular, it's only for a greater than 0. 00:17:44.680 --> 00:17:48.090 What that really means is that for convergence of the Fourier 00:17:48.090 --> 00:17:52.840 transform, this has to be a decaying exponential. 00:17:52.840 --> 00:17:56.620 It can't be an increasing exponential. 00:17:56.620 --> 00:18:01.620 If instead we apply the Laplace transform to this, 00:18:01.620 --> 00:18:06.030 applying the Laplace transform is the same as taking the 00:18:06.030 --> 00:18:11.430 Fourier transform of x of t times an exponential, and the 00:18:11.430 --> 00:18:14.160 exponent that we would multiply by is e to 00:18:14.160 --> 00:18:15.710 the minus sigma t. 00:18:15.710 --> 00:18:21.100 So in effect, taking the Laplace transform of this is 00:18:21.100 --> 00:18:26.032 like taking the Fourier transform of e to the minus at 00:18:26.032 --> 00:18:29.860 e to the minus sigma t. 00:18:29.860 --> 00:18:34.490 And if we carry that through, just working through the 00:18:34.490 --> 00:18:38.750 integral, we end up with a Laplace transform, which is 1 00:18:38.750 --> 00:18:42.200 over s plus a. 00:18:42.200 --> 00:18:47.710 But just as in the Fourier transform, the Fourier 00:18:47.710 --> 00:18:50.980 transform won't converge for any a. 00:18:50.980 --> 00:18:56.110 Now what happens is that the Laplace transform will only 00:18:56.110 --> 00:19:00.530 converge when the Fourier transform of this converges. 00:19:00.530 --> 00:19:04.430 Said another way, it's when the combination of a plus 00:19:04.430 --> 00:19:07.490 sigma is greater than 0. 00:19:07.490 --> 00:19:11.860 So we would require that, if I write it over here, a plus 00:19:11.860 --> 00:19:14.000 sigma is greater than 0. 00:19:14.000 --> 00:19:19.510 Or that sigma is greater than minus a. 00:19:19.510 --> 00:19:24.460 So in fact, in the Laplace transform of this, we have an 00:19:24.460 --> 00:19:26.880 expression 1 over s plus a. 00:19:26.880 --> 00:19:32.260 But we also require, in interpreting that, that the 00:19:32.260 --> 00:19:37.430 real part of s be greater than minus a. 00:19:37.430 --> 00:19:42.120 So that, essentially, the Fourier transform of x of t 00:19:42.120 --> 00:19:45.300 times e to the minus sigma t converges. 00:19:45.300 --> 00:19:49.960 So it's important to recognize that the algebraic expression 00:19:49.960 --> 00:19:55.040 that we get is only valid for certain values of the 00:19:55.040 --> 00:19:56.580 real part of s. 00:19:56.580 --> 00:20:01.180 And so, for this example, we can summarize it as this 00:20:01.180 --> 00:20:05.110 exponential has a Laplace transform, which is 1 over s 00:20:05.110 --> 00:20:09.560 plus a, where s is restricted to the range the real part of 00:20:09.560 --> 00:20:11.430 s greater than minus a. 00:20:14.020 --> 00:20:18.900 Now, we haven't had this issue before of restrictions on what 00:20:18.900 --> 00:20:20.210 the value of s is. 00:20:20.210 --> 00:20:22.600 With the Fourier transform, either it converged or it 00:20:22.600 --> 00:20:23.880 didn't converge. 00:20:23.880 --> 00:20:27.760 With the Laplace transform, there are certain values of s. 00:20:27.760 --> 00:20:30.620 We now have more flexibility, and so there's certain values 00:20:30.620 --> 00:20:33.690 of the real part of s for which it converges and certain 00:20:33.690 --> 00:20:36.360 values for which it doesn't. 00:20:36.360 --> 00:20:39.110 The values of s for which the Laplace 00:20:39.110 --> 00:20:42.880 transform converges is-- 00:20:42.880 --> 00:20:47.570 the values are referred to as the region of convergence of 00:20:47.570 --> 00:20:50.580 the Laplace transform. 00:20:50.580 --> 00:20:56.130 And it's important to recognize that in specifying 00:20:56.130 --> 00:20:59.370 the Laplace transform, what's required is not only the 00:20:59.370 --> 00:21:05.370 algebraic expression, but also the domain or set of values of 00:21:05.370 --> 00:21:12.210 s for which that algebraic expression is valid. 00:21:12.210 --> 00:21:16.670 Just to underscore that point, let me draw your attention to 00:21:16.670 --> 00:21:21.890 another example in the text, which is Example 9.2. 00:21:21.890 --> 00:21:27.445 In Example 9.2, we have an exponential for negative time, 00:21:27.445 --> 00:21:30.040 0 for positive time. 00:21:30.040 --> 00:21:33.380 And if you carry through the algebra there, you end up with 00:21:33.380 --> 00:21:35.950 a Laplace transform expression, which is again 1 00:21:35.950 --> 00:21:38.660 over s plus a. 00:21:38.660 --> 00:21:42.280 Exactly the same algebraic expression as we had for the 00:21:42.280 --> 00:21:44.490 previous example. 00:21:44.490 --> 00:21:49.660 The important distinction is that now the real part of s is 00:21:49.660 --> 00:21:52.170 restricted to be less than minus a. 00:21:52.170 --> 00:21:56.750 And so, in fact, if you compare this example with the 00:21:56.750 --> 00:22:00.180 one above it, and let's just look back at the answer that 00:22:00.180 --> 00:22:03.350 we had there. 00:22:03.350 --> 00:22:07.240 If you compare those two examples, here the algebraic 00:22:07.240 --> 00:22:11.820 expression is 1 over s plus a with a certain region of 00:22:11.820 --> 00:22:13.130 convergence. 00:22:13.130 --> 00:22:18.260 Here the algebraic expression is 1 over s plus a. 00:22:18.260 --> 00:22:21.260 And the only difference between those two is the 00:22:21.260 --> 00:22:23.180 domain or region of convergence. 00:22:23.180 --> 00:22:27.840 So there is another complication, or twist, now. 00:22:27.840 --> 00:22:32.040 Not only do we need to generate the algebraic 00:22:32.040 --> 00:22:37.200 expression, but we also have to be careful to specify the 00:22:37.200 --> 00:22:41.500 region of convergence over which that algebraic 00:22:41.500 --> 00:22:44.280 expression is valid. 00:22:44.280 --> 00:22:49.450 Now, later on in this lecture, and actually also as the 00:22:49.450 --> 00:22:52.830 discussion of the Laplace transform goes on, we'll begin 00:22:52.830 --> 00:22:56.350 to see and understand more about how the region of 00:22:56.350 --> 00:22:59.360 convergence relates to various 00:22:59.360 --> 00:23:03.420 properties of the time function. 00:23:03.420 --> 00:23:07.070 Well, let's finally look at one additional example from 00:23:07.070 --> 00:23:10.550 the text, And this is Example 9.3. 00:23:10.550 --> 00:23:15.420 And what it consists of is the time function, which is the 00:23:15.420 --> 00:23:18.620 sum of two exponentials. 00:23:18.620 --> 00:23:22.190 And although we haven't formally talked about 00:23:22.190 --> 00:23:24.920 properties of the Laplace transform yet, one of the 00:23:24.920 --> 00:23:26.210 properties that we'll see-- and it's 00:23:26.210 --> 00:23:28.050 relatively easy to develop-- 00:23:28.050 --> 00:23:32.260 is the fact that the Laplace transform of a sum is the sum 00:23:32.260 --> 00:23:33.430 of the Laplace transform. 00:23:33.430 --> 00:23:38.330 So, in fact, we can get the Laplace transform of the sum 00:23:38.330 --> 00:23:44.910 of these two terms as the sum of the Laplace transforms. 00:23:44.910 --> 00:23:49.390 So for this one, we know from the example that we looked at 00:23:49.390 --> 00:23:53.810 previously, Example 9.1, that this is of the form 1 over s 00:23:53.810 --> 00:23:57.940 plus 1 with a region of convergence, which is the real 00:23:57.940 --> 00:24:01.180 part of s greater than minus 1. 00:24:01.180 --> 00:24:04.030 For this one, we have a Laplace transform which is 1 00:24:04.030 --> 00:24:08.810 over s plus 2 with a region of convergence which is the real 00:24:08.810 --> 00:24:12.270 part of s greater than minus 2. 00:24:12.270 --> 00:24:16.010 So for the two of them together, we have to take the 00:24:16.010 --> 00:24:18.140 overlap of those two regions. 00:24:18.140 --> 00:24:21.540 In other words, we have to take the region that 00:24:21.540 --> 00:24:24.810 encompasses both the real part of s greater than minus 1 and 00:24:24.810 --> 00:24:27.230 the real part of s greater than minus 2. 00:24:27.230 --> 00:24:31.870 And if we put those together, then we have a combined region 00:24:31.870 --> 00:24:34.140 of convergence, which is the real part of s 00:24:34.140 --> 00:24:36.610 greater than minus 1. 00:24:36.610 --> 00:24:39.140 So this is the expression. 00:24:39.140 --> 00:24:44.270 And for this particular example, what we have is a 00:24:44.270 --> 00:24:46.450 ratio of polynomials. 00:24:46.450 --> 00:24:49.830 The ratio of polynomials, there's a numerator polynomial 00:24:49.830 --> 00:24:52.080 and a denominator polynomial. 00:24:52.080 --> 00:24:59.020 And it's convenient to summarize these by plotting 00:24:59.020 --> 00:25:02.180 the roots of the numerator polynomial and the roots of 00:25:02.180 --> 00:25:06.110 the denominator polynomial in the complex plane. 00:25:06.110 --> 00:25:10.650 And the complex plane which they're plotted is referred to 00:25:10.650 --> 00:25:13.350 the s-plane. 00:25:13.350 --> 00:25:18.590 So we can, for example, take the denominator polynomial and 00:25:18.590 --> 00:25:24.230 summarize it by specifying the fact, or by representing the 00:25:24.230 --> 00:25:28.710 fact that it has roots at s equals minus 1 and at s 00:25:28.710 --> 00:25:30.190 equals minus 2. 00:25:30.190 --> 00:25:33.950 And I've done that in this picture by putting an x where 00:25:33.950 --> 00:25:37.850 the roots of the denominator polynomial are. 00:25:37.850 --> 00:25:42.720 The numerator polynomial has a root at s equals minus 3/2, 00:25:42.720 --> 00:25:45.070 and I've represented that by a circle. 00:25:45.070 --> 00:25:48.510 So these are the roots of the denominator polynomial and 00:25:48.510 --> 00:25:50.070 this is the root of the numerator 00:25:50.070 --> 00:25:52.220 polynomial for this example. 00:25:52.220 --> 00:25:56.540 And also, for this example, we can represent the region of 00:25:56.540 --> 00:26:00.300 convergence, which is the real part of s 00:26:00.300 --> 00:26:01.600 greater than minus 1. 00:26:01.600 --> 00:26:06.720 And so that's, in fact, the region over here. 00:26:06.720 --> 00:26:09.850 There is also, if I draw these, just the roots of the 00:26:09.850 --> 00:26:13.180 numerator and denominator of polynomials, I would need an 00:26:13.180 --> 00:26:16.820 additional piece of information to specify the 00:26:16.820 --> 00:26:18.460 algebraic expression completely. 00:26:18.460 --> 00:26:21.250 Namely, a multiplying constant out in front 00:26:21.250 --> 00:26:24.190 of the whole thing. 00:26:24.190 --> 00:26:31.220 Well, this particular example, has the Laplace transform as a 00:26:31.220 --> 00:26:32.600 rational function. 00:26:32.600 --> 00:26:35.950 Namely, it's one polynomial in the numerator and another 00:26:35.950 --> 00:26:38.040 polynomial in the denominator. 00:26:38.040 --> 00:26:41.610 And in fact, as we'll see, Laplace transforms, which are 00:26:41.610 --> 00:26:45.170 ratios of polynomials, form a very important class. 00:26:45.170 --> 00:26:48.430 They, in fact, represent systems that are describable 00:26:48.430 --> 00:26:51.810 by linear constant coefficient differential equations. 00:26:51.810 --> 00:26:53.510 You shouldn't necessarily-- 00:26:53.510 --> 00:26:55.420 in fact, for sure you shouldn't see 00:26:55.420 --> 00:26:56.850 why that's true now. 00:26:56.850 --> 00:26:59.050 We'll see that later. 00:26:59.050 --> 00:27:06.060 But that means that Laplace transforms that are rational 00:27:06.060 --> 00:27:10.280 functions, namely, the ratio of a numerator polynomial 00:27:10.280 --> 00:27:14.180 divided by the denominator polynomial, become very 00:27:14.180 --> 00:27:18.640 important in the discussion that follows. 00:27:18.640 --> 00:27:22.760 And in fact, we have some terminology for this. 00:27:22.760 --> 00:27:27.030 The roots of the numerator polynomial are referred to as 00:27:27.030 --> 00:27:30.750 the zeroes of the Laplace transform. 00:27:30.750 --> 00:27:34.165 Because, of course, those are the values of s at which x of 00:27:34.165 --> 00:27:36.790 s becomes 0. 00:27:36.790 --> 00:27:40.130 And the roots of the denominator polynomial are 00:27:40.130 --> 00:27:44.900 referred to as the poles of the Laplace transform. 00:27:44.900 --> 00:27:49.840 And those are the values of s at which the Laplace 00:27:49.840 --> 00:27:52.090 transform blows up. 00:27:52.090 --> 00:27:53.600 Namely, becomes infinite. 00:27:53.600 --> 00:27:57.430 If you think of setting s equal to a value where this 00:27:57.430 --> 00:28:00.780 denominator polynomial goes to 0, of course, x 00:28:00.780 --> 00:28:02.880 of s becomes infinite. 00:28:02.880 --> 00:28:06.860 And what we would expect and, of course, we'll see 00:28:06.860 --> 00:28:09.360 that this is true. 00:28:09.360 --> 00:28:12.670 What we would expect is that wherever that happens, there 00:28:12.670 --> 00:28:14.750 must be some problem with convergence 00:28:14.750 --> 00:28:15.930 of the Laplace transform. 00:28:15.930 --> 00:28:18.160 And indeed, the Laplace transform doesn't 00:28:18.160 --> 00:28:20.960 converge at the poles. 00:28:20.960 --> 00:28:24.390 Namely, at the roots of the denominator polynomial. 00:28:24.390 --> 00:28:28.120 So, in fact, let's focus in on that a little further. 00:28:28.120 --> 00:28:33.980 Let's examine and talk about the region of convergence of 00:28:33.980 --> 00:28:38.200 the Laplace transform, and how it's associated both with 00:28:38.200 --> 00:28:41.970 properties of the time function, and also with the 00:28:41.970 --> 00:28:46.720 location of the poles of the Laplace transform. 00:28:46.720 --> 00:28:50.320 And as we'll see, there are some very specific and 00:28:50.320 --> 00:28:54.500 important relationships and conclusions that we can draw 00:28:54.500 --> 00:28:58.790 about how the region of convergence is constrained and 00:28:58.790 --> 00:29:05.790 associated with the locations of the poles in the s-plane. 00:29:05.790 --> 00:29:11.310 Well, to begin with, we can, of course, make the statement 00:29:11.310 --> 00:29:14.350 as I've just made that the region of 00:29:14.350 --> 00:29:17.260 convergence contains no poles. 00:29:17.260 --> 00:29:22.600 In particular, if I think of this general rational 00:29:22.600 --> 00:29:29.310 function, the poles of x of s are the values of s at which 00:29:29.310 --> 00:29:30.870 the denominator is 0. 00:29:30.870 --> 00:29:34.560 Or equivalently, x of s blows up. 00:29:34.560 --> 00:29:38.160 And of course then, that implies that the expression 00:29:38.160 --> 00:29:41.460 has no longer converged. 00:29:41.460 --> 00:29:43.370 Well, that's one statement that we can make. 00:29:43.370 --> 00:29:46.160 Now, there are some others. 00:29:46.160 --> 00:29:51.170 And one, for example, is the statement that if I have a 00:29:51.170 --> 00:29:55.820 point in the s-plane that corresponds to convergence, 00:29:55.820 --> 00:30:00.680 then in fact any line in the s-plane with that same real 00:30:00.680 --> 00:30:05.100 part will also be a set of values for which the Laplace 00:30:05.100 --> 00:30:07.110 transform converges. 00:30:07.110 --> 00:30:09.450 And what's the reason for that? 00:30:09.450 --> 00:30:15.910 The reason for that is that s is sigma plus j omega and 00:30:15.910 --> 00:30:20.840 convergence of the Laplace transform is associated with 00:30:20.840 --> 00:30:25.580 convergence of the Fourier transform of e to the minus 00:30:25.580 --> 00:30:28.390 sigma t times x of t. 00:30:28.390 --> 00:30:31.950 And so the convergence only depends on sigma. 00:30:31.950 --> 00:30:37.010 If it only depends on sigma, then if it converges for one 00:30:37.010 --> 00:30:39.200 value of sigma-- 00:30:39.200 --> 00:30:43.980 I'm sorry, for a value of sigma for some value of omega, 00:30:43.980 --> 00:30:47.840 then it will converge for that same sigma for 00:30:47.840 --> 00:30:50.130 any value of omega. 00:30:50.130 --> 00:30:55.650 The conclusion then is that the region of convergence, if 00:30:55.650 --> 00:30:58.110 I have a point, then I also have a line. 00:30:58.110 --> 00:31:01.080 And so what that suggests is that as we look at the region 00:31:01.080 --> 00:31:05.930 of convergence, it in fact corresponds to strips in the 00:31:05.930 --> 00:31:09.360 complex plane. 00:31:09.360 --> 00:31:12.930 Now, finally we can tie together the region of 00:31:12.930 --> 00:31:17.980 convergence to the convergence of the Fourier transform. 00:31:17.980 --> 00:31:23.220 In particular, since we know that the Laplace transform 00:31:23.220 --> 00:31:28.480 reduces to the Fourier transform when the complex 00:31:28.480 --> 00:31:34.770 variable s is equal to j omega, the implication is that 00:31:34.770 --> 00:31:40.000 if we have the Laplace transform and if the Laplace 00:31:40.000 --> 00:31:43.300 transform reduces to the Fourier transform when 00:31:43.300 --> 00:31:44.440 sigma equals 0. 00:31:44.440 --> 00:31:48.140 In other words, when s is equal to j omega, then the 00:31:48.140 --> 00:31:52.600 Fourier transform of x of t converging is equivalent to 00:31:52.600 --> 00:31:57.530 the statement that the Laplace transform converges for sigma 00:31:57.530 --> 00:31:58.930 equal to 0. 00:31:58.930 --> 00:32:00.310 In other words, that the region of 00:32:00.310 --> 00:32:04.340 convergence includes what? 00:32:04.340 --> 00:32:06.690 The j omega axis in the s-plane. 00:32:10.880 --> 00:32:14.230 So we have then some statements that kind of tie 00:32:14.230 --> 00:32:17.680 together the location of the poles and the region of 00:32:17.680 --> 00:32:18.270 convergence. 00:32:18.270 --> 00:32:21.340 Let me make one other statement, which is a much 00:32:21.340 --> 00:32:22.800 harder statement to justify. 00:32:22.800 --> 00:32:25.240 And I won't try to, I'll just simply state it. 00:32:25.240 --> 00:32:27.810 And that is that the region of convergence of the Laplace 00:32:27.810 --> 00:32:31.320 transform is a connected region. 00:32:31.320 --> 00:32:37.330 In other words, if the entire region consists of a single 00:32:37.330 --> 00:32:42.480 strip in the s-plane, it can't consist of a strip over here, 00:32:42.480 --> 00:32:44.650 for example, and a strip over there. 00:32:44.650 --> 00:32:50.420 Well, let me emphasize some of those points a little further. 00:32:50.420 --> 00:32:59.060 Let's suppose that I have a Laplace transform, and the 00:32:59.060 --> 00:33:03.280 Laplace transform that I'm talking about is a rational 00:33:03.280 --> 00:33:09.610 function, which is 1 over s plus 1 times s plus 2. 00:33:09.610 --> 00:33:14.460 Then the pole-zero pattern, as it's referred to, in the 00:33:14.460 --> 00:33:18.160 s-plane, the location of the roots of the numerator and 00:33:18.160 --> 00:33:19.490 denominator polynomials. 00:33:19.490 --> 00:33:22.230 Of course, there is no numerator polynomial. 00:33:22.230 --> 00:33:25.850 The denominator polynomial roots, which I've represented 00:33:25.850 --> 00:33:28.440 by these x's, are shown here. 00:33:28.440 --> 00:33:33.430 And so this is the pole-zero pattern. 00:33:33.430 --> 00:33:37.690 And from what I've said, the region of convergence can't 00:33:37.690 --> 00:33:42.590 include any poles and it must correspond to 00:33:42.590 --> 00:33:44.860 strips in the s-plane. 00:33:44.860 --> 00:33:49.160 And furthermore, it must be just one connected region 00:33:49.160 --> 00:33:51.640 rather than multiple regions. 00:33:51.640 --> 00:33:57.090 And so with this algebraic expression then, the possible 00:33:57.090 --> 00:34:00.250 choices for the region of convergence consistent with 00:34:00.250 --> 00:34:02.550 those properties are the following. 00:34:02.550 --> 00:34:05.830 One of them would be a region of convergence to the 00:34:05.830 --> 00:34:09.489 right of this pole. 00:34:09.489 --> 00:34:16.190 A second would be a region of convergence which lies between 00:34:16.190 --> 00:34:21.570 the two poles as I show here. 00:34:21.570 --> 00:34:27.350 And a third is a region of convergence which is to the 00:34:27.350 --> 00:34:31.199 left of this pole. 00:34:31.199 --> 00:34:36.010 And because of the fact that I said without proof that the 00:34:36.010 --> 00:34:39.870 region of convergence must be a single strip, it can't be 00:34:39.870 --> 00:34:41.159 multiple strips. 00:34:41.159 --> 00:34:44.460 In fact, we could not consider, as a possible region 00:34:44.460 --> 00:34:48.060 of convergence, what I show here. 00:34:48.060 --> 00:34:52.810 So, in fact, this is not a valid region of convergence. 00:34:52.810 --> 00:34:56.750 There are only three possibilities associated with 00:34:56.750 --> 00:34:58.900 this pole-zero pattern. 00:34:58.900 --> 00:35:02.270 Namely, to the right of this pole, between the two poles, 00:35:02.270 --> 00:35:06.500 and to the left of this pole. 00:35:06.500 --> 00:35:11.250 Now, to carry the discussion further, we can, in fact, 00:35:11.250 --> 00:35:16.490 associate the region of convergence of the Laplace 00:35:16.490 --> 00:35:21.180 transform with some very specific characteristics of 00:35:21.180 --> 00:35:22.540 the time function. 00:35:22.540 --> 00:35:28.790 And what this will do is to help us understand how for 00:35:28.790 --> 00:35:32.320 various choices of the region of convergence, the 00:35:32.320 --> 00:35:35.740 interpretation that we can impose on the 00:35:35.740 --> 00:35:37.310 related time function. 00:35:37.310 --> 00:35:40.360 Let me show you what I mean. 00:35:40.360 --> 00:35:46.860 Suppose that we start with a time function as I indicate 00:35:46.860 --> 00:35:53.820 here, which is a finite duration time function. 00:35:53.820 --> 00:35:58.230 In other words, it's 0 except in some time interval. 00:35:58.230 --> 00:36:03.200 Now, recall that the Fourier transform converges if the 00:36:03.200 --> 00:36:06.580 time function has the property that it's absolutely 00:36:06.580 --> 00:36:07.450 integrable. 00:36:07.450 --> 00:36:10.870 And as long as everything's stays finite in terms of 00:36:10.870 --> 00:36:14.130 amplitudes in a finite duration signal, there's no 00:36:14.130 --> 00:36:17.830 difficulty that we're going to run into here. 00:36:17.830 --> 00:36:21.040 Now, here the Fourier transform will converge. 00:36:21.040 --> 00:36:25.920 And now the question is, what can we say about the region of 00:36:25.920 --> 00:36:29.770 convergence of the Laplace transform? 00:36:29.770 --> 00:36:34.700 Well, the Laplace transform is the Fourier transform of the 00:36:34.700 --> 00:36:37.810 time function multiplied by an exponential. 00:36:37.810 --> 00:36:41.820 And so we can ask about whether we can destroy the 00:36:41.820 --> 00:36:45.560 absolute integrability of this by multiplying by an 00:36:45.560 --> 00:36:49.510 exponential that grows to fast or decays 00:36:49.510 --> 00:36:50.490 too fast, or whatever. 00:36:50.490 --> 00:36:53.130 And let's take a look at that. 00:36:53.130 --> 00:36:57.310 Suppose that this time function is absolutely 00:36:57.310 --> 00:36:59.230 integrable. 00:36:59.230 --> 00:37:05.550 And let's multiply it by a decaying exponential. 00:37:05.550 --> 00:37:09.540 So this is now x of t times z to the minus sigma t if I 00:37:09.540 --> 00:37:11.960 think of multiplying these two together. 00:37:11.960 --> 00:37:16.260 And what you can see is that for positive time, sort of 00:37:16.260 --> 00:37:20.140 thinking informally, I'm helping the integrability of 00:37:20.140 --> 00:37:23.680 the product because I'm pushing this part down. 00:37:23.680 --> 00:37:25.960 For negative time, unfortunately, I'm making 00:37:25.960 --> 00:37:28.030 things grow. 00:37:28.030 --> 00:37:31.710 But I don't let them grow indefinitely because there's 00:37:31.710 --> 00:37:36.180 some time before which this is equal to 0. 00:37:36.180 --> 00:37:44.660 Likewise, if I had a growing exponential, then for a 00:37:44.660 --> 00:37:48.960 growing exponential for negative time, or for this 00:37:48.960 --> 00:37:52.720 part, I'm making things smaller. 00:37:52.720 --> 00:37:55.580 For positive time, eventually this exponential is growing 00:37:55.580 --> 00:37:56.930 without bound. 00:37:56.930 --> 00:38:01.370 But the time function stops at some point. 00:38:01.370 --> 00:38:08.580 So the idea then kind of is that for a finite duration 00:38:08.580 --> 00:38:12.760 time function, no matter what kind of exponential I multiply 00:38:12.760 --> 00:38:16.720 by, whether it's going this way or going this way, because 00:38:16.720 --> 00:38:19.870 of the fact that essentially the limits on the integral are 00:38:19.870 --> 00:38:24.940 finite, I'm guaranteed that I'll always maintain absolute 00:38:24.940 --> 00:38:26.370 integrability. 00:38:26.370 --> 00:38:30.890 And so, in fact then, for a finite duration time function, 00:38:30.890 --> 00:38:33.800 the region of convergence is the entire s-plane. 00:38:37.100 --> 00:38:41.890 Now, we can also make statements about other kinds 00:38:41.890 --> 00:38:43.470 of time functions. 00:38:43.470 --> 00:38:52.280 And let's look at a time function which I define as a 00:38:52.280 --> 00:38:55.470 right-sided time function. 00:38:55.470 --> 00:39:01.720 And a right-sided time function is one which is 0 up 00:39:01.720 --> 00:39:06.200 until some time, and then it goes on after that, presumably 00:39:06.200 --> 00:39:08.050 off to infinity. 00:39:08.050 --> 00:39:12.850 Now, let me remind you that the whole issue here with the 00:39:12.850 --> 00:39:18.540 region of convergence has to do with exponentials that we 00:39:18.540 --> 00:39:23.850 can multiply a time function by and have the product end up 00:39:23.850 --> 00:39:26.430 being absolutely integrable. 00:39:26.430 --> 00:39:32.020 Well, suppose that when I multiply this time function by 00:39:32.020 --> 00:39:35.400 an exponential which, let's say decays. 00:39:35.400 --> 00:39:39.860 But an exponential e to the minus sigma 0 t, what you can 00:39:39.860 --> 00:39:43.730 see sort of intuitively is that if this product is 00:39:43.730 --> 00:39:49.030 absolutely integrable, if I were to increase sigma 0, then 00:39:49.030 --> 00:39:52.090 I'm making things even better for positive time because I'm 00:39:52.090 --> 00:39:53.640 pushing them down. 00:39:53.640 --> 00:39:57.730 And whereas they might be worse for negative time, that 00:39:57.730 --> 00:40:01.170 doesn't matter because before some time the 00:40:01.170 --> 00:40:02.540 product is equal to 0. 00:40:02.540 --> 00:40:08.200 So if this product is absolutely integrable, then if 00:40:08.200 --> 00:40:12.840 I chose an exponential e to the minus sigma 1t where sigma 00:40:12.840 --> 00:40:17.370 1 is greater than sigma 0, then that product will also be 00:40:17.370 --> 00:40:19.470 absolutely integrable. 00:40:19.470 --> 00:40:23.270 And we can draw an important conclusion about that, about 00:40:23.270 --> 00:40:25.420 the region of convergence from that. 00:40:25.420 --> 00:40:30.700 In particular, we can make the statement that if the time 00:40:30.700 --> 00:40:38.310 function is right-sided and if convergence occurs for some 00:40:38.310 --> 00:40:44.250 value sigma 0, then in fact, we will have convergence of 00:40:44.250 --> 00:40:49.700 the Laplace transform for all values of the real part of s 00:40:49.700 --> 00:40:51.770 greater than sigma 0. 00:40:51.770 --> 00:40:56.610 The reason, of course, being that if sigma 0 increases, 00:40:56.610 --> 00:41:03.210 then the exponential decays even faster for positive time. 00:41:03.210 --> 00:41:06.070 Now what that says then thinking another way, in terms 00:41:06.070 --> 00:41:09.700 of the region of convergence as we might draw it in the 00:41:09.700 --> 00:41:13.080 s-plane, is that if we have a point that's in the region of 00:41:13.080 --> 00:41:17.450 convergence corresponding to some value sigma 0, then all 00:41:17.450 --> 00:41:21.580 values of s to the right of that in the s-plane will also 00:41:21.580 --> 00:41:24.330 be in the region of convergence. 00:41:24.330 --> 00:41:26.800 We can also combine that with the statement that for 00:41:26.800 --> 00:41:31.490 rational functions we know that there can't be any poles 00:41:31.490 --> 00:41:33.190 in the region of convergence. 00:41:33.190 --> 00:41:36.190 If you put those two statements together, then we 00:41:36.190 --> 00:41:41.830 end up with a statement that if x of t is right-sided and 00:41:41.830 --> 00:41:46.060 if its Laplace transform is rational, then the region of 00:41:46.060 --> 00:41:51.280 convergence is to the right of the rightmost pole. 00:41:51.280 --> 00:41:55.130 So we have here a very important insight, which tells 00:41:55.130 --> 00:41:59.980 us that we can infer some property about the time 00:41:59.980 --> 00:42:02.640 function from the region of convergence. 00:42:02.640 --> 00:42:05.880 Or conversely, if we know something about the time 00:42:05.880 --> 00:42:09.440 function, namely being right-sided, then we can infer 00:42:09.440 --> 00:42:12.570 something about the region of convergence. 00:42:12.570 --> 00:42:17.090 Well, in addition to right-sided signals, we can 00:42:17.090 --> 00:42:19.510 also have left-sided signals. 00:42:19.510 --> 00:42:22.370 And a left-sided signal is essentially a right-sided 00:42:22.370 --> 00:42:23.910 signal turned around. 00:42:23.910 --> 00:42:27.480 In other words, a left-sided signal is one that is 00:42:27.480 --> 00:42:30.460 0 after some time. 00:42:30.460 --> 00:42:32.850 Well, we can carry out exactly the same 00:42:32.850 --> 00:42:34.580 kind of argument there. 00:42:34.580 --> 00:42:37.800 Namely, if the signal goes off to infinity in the negative 00:42:37.800 --> 00:42:42.690 time direction and stops some place for positive time, if I 00:42:42.690 --> 00:42:46.110 have an exponential that I can multiply it by and have that 00:42:46.110 --> 00:42:48.620 product be absolutely integrable. 00:42:48.620 --> 00:42:51.590 And if I choose an exponential that decays even faster for 00:42:51.590 --> 00:42:54.560 negative time so that I'm pushing the stuff way out 00:42:54.560 --> 00:42:58.830 there down even further, then I enhance the 00:42:58.830 --> 00:43:00.850 integrability even more. 00:43:00.850 --> 00:43:03.830 And you might have to think through that a little bit, but 00:43:03.830 --> 00:43:07.170 it's exactly the flip side of the argument 00:43:07.170 --> 00:43:08.890 for right-sided signals. 00:43:08.890 --> 00:43:14.470 And the conclusion then is that if we have a left-sided 00:43:14.470 --> 00:43:20.010 signal and we have a point, a value of the real part of s 00:43:20.010 --> 00:43:23.400 which is in the region of convergence, then in fact, all 00:43:23.400 --> 00:43:29.140 values to the left of that point in the s-plane will also 00:43:29.140 --> 00:43:32.500 be in the region of convergence. 00:43:32.500 --> 00:43:35.330 Now, similar to the statement that we made for right-sided 00:43:35.330 --> 00:43:39.230 signals, if x of t is left-sided and, in fact, we're 00:43:39.230 --> 00:43:41.720 talking about a rational Laplace transform, which we 00:43:41.720 --> 00:43:43.830 most typically will. 00:43:43.830 --> 00:43:47.730 Then, in fact, we can make the statement that the region of 00:43:47.730 --> 00:43:53.290 convergence is to the left of the leftmost pole because we 00:43:53.290 --> 00:43:56.730 know if we find a point that's in the region of convergence, 00:43:56.730 --> 00:43:58.930 everything to the left of that has to be in the region of 00:43:58.930 --> 00:43:59.870 convergence. 00:43:59.870 --> 00:44:02.480 We can't have any poles in the region of convergence. 00:44:02.480 --> 00:44:04.960 You put those two statements together and it says it's to 00:44:04.960 --> 00:44:07.710 the left of the leftmost pole. 00:44:07.710 --> 00:44:12.340 Now the final situation is the situation where we have a 00:44:12.340 --> 00:44:14.680 signal which is neither right-sided nor left-sided. 00:44:14.680 --> 00:44:18.010 It goes off to infinity for positive time and it goes off 00:44:18.010 --> 00:44:20.100 to infinity for negative time. 00:44:20.100 --> 00:44:23.460 And there the thing to kind of recognize is that if you 00:44:23.460 --> 00:44:28.370 multiply by an exponential, and it's decaying very fast 00:44:28.370 --> 00:44:30.750 for positive time, it's going to be growing very fast for 00:44:30.750 --> 00:44:32.060 negative time. 00:44:32.060 --> 00:44:35.060 Conversely, if it's decaying very fast for negative time, 00:44:35.060 --> 00:44:37.530 it's growing very fast for positive time. 00:44:37.530 --> 00:44:39.840 And there's this notion of trying to balance 00:44:39.840 --> 00:44:41.710 the value of sigma. 00:44:41.710 --> 00:44:44.620 And in effect, what that says is that the region of 00:44:44.620 --> 00:44:47.810 convergence can't extent too far to the left or 00:44:47.810 --> 00:44:49.450 too far to the right. 00:44:49.450 --> 00:44:55.790 Said another way for a two-sided signal, if we have a 00:44:55.790 --> 00:44:59.560 point which is in the region of convergence, then that 00:44:59.560 --> 00:45:06.530 point defines a strip in the s-plane that takes that point 00:45:06.530 --> 00:45:09.960 and extends it to the left until you bump into a pole, 00:45:09.960 --> 00:45:12.990 and extends it to the right until you bump it into a pole. 00:45:15.845 --> 00:45:20.460 So you begin to then see that we can tie together some 00:45:20.460 --> 00:45:24.040 properties of the region of convergence and the 00:45:24.040 --> 00:45:25.940 right-sidedness, or left-sidedness, or 00:45:25.940 --> 00:45:29.030 two-sidedness of the time function. 00:45:29.030 --> 00:45:32.750 And you'll have a chance to examine that in more detail in 00:45:32.750 --> 00:45:35.010 the video course manual. 00:45:35.010 --> 00:45:41.440 Let's conclude this lecture by talking about how we might get 00:45:41.440 --> 00:45:46.040 the time function given the appliance transform. 00:45:46.040 --> 00:45:49.830 Well, if we have a Laplace transform, we can, in 00:45:49.830 --> 00:45:53.010 principle, get the time function back again by 00:45:53.010 --> 00:45:56.230 recognizing this relationship between the Laplace transform 00:45:56.230 --> 00:45:59.360 and the Fourier transform, and using the formal Fourier 00:45:59.360 --> 00:46:00.650 transform expression. 00:46:00.650 --> 00:46:04.510 Or equivalently, the formal inverse Laplace transform 00:46:04.510 --> 00:46:07.190 expression, which is in the text. 00:46:07.190 --> 00:46:11.040 But more typically what we would do is what we've done 00:46:11.040 --> 00:46:15.910 also with the Fourier transform, which is to use 00:46:15.910 --> 00:46:21.580 simple Laplace transform pairs together with the notion of 00:46:21.580 --> 00:46:23.490 the partial fraction expansion. 00:46:23.490 --> 00:46:28.020 And let's just go through that with an example. 00:46:28.020 --> 00:46:34.350 Let's suppose that I have a Laplace transform as I 00:46:34.350 --> 00:46:38.380 indicated here in its pole-zero plot and a region of 00:46:38.380 --> 00:46:42.060 convergence which is to the right of this pole. 00:46:42.060 --> 00:46:45.140 And what we can identify from the region of convergence, in 00:46:45.140 --> 00:46:47.980 fact, is that we're talking about a 00:46:47.980 --> 00:46:50.660 right-sided time function. 00:46:50.660 --> 00:46:54.300 So the region of convergence is the real part of s greater 00:46:54.300 --> 00:46:55.900 than minus 1. 00:46:55.900 --> 00:47:01.050 And now looking down at the algebraic expression, we have 00:47:01.050 --> 00:47:04.610 the algebraic expression for this, as I indicated here, 00:47:04.610 --> 00:47:09.560 equivalently expanded in a partial fraction expansion, as 00:47:09.560 --> 00:47:10.250 I show below. 00:47:10.250 --> 00:47:13.590 So if you just simply combine these together, that's the 00:47:13.590 --> 00:47:15.030 same as this. 00:47:15.030 --> 00:47:19.360 And the region of convergence is the real part of s greater 00:47:19.360 --> 00:47:21.410 than minus 1. 00:47:21.410 --> 00:47:24.830 Now, the region of convergence of-- 00:47:24.830 --> 00:47:28.360 this is the sum of two terms, so the time function is the 00:47:28.360 --> 00:47:30.320 sum of two time functions. 00:47:30.320 --> 00:47:37.760 And the region of convergence of the combination must be the 00:47:37.760 --> 00:47:39.850 intersection of the region of convergence 00:47:39.850 --> 00:47:41.960 associated with each one. 00:47:41.960 --> 00:47:46.990 Recognizing that this is to the right of the poles, that 00:47:46.990 --> 00:47:51.290 tells us immediately that each of these two then would 00:47:51.290 --> 00:47:55.110 correspond to the Laplace transform of a right-sided 00:47:55.110 --> 00:47:57.180 time function. 00:47:57.180 --> 00:48:00.910 Well, let's look at it term by term. 00:48:00.910 --> 00:48:06.620 The first term is the factor 1 over s plus 1 with a region of 00:48:06.620 --> 00:48:10.620 convergence to the right of this pole. 00:48:10.620 --> 00:48:15.070 And this algebraically corresponds 00:48:15.070 --> 00:48:16.510 to what I've indicated. 00:48:16.510 --> 00:48:21.610 And this, in fact, is similar to, or a special case of the 00:48:21.610 --> 00:48:24.940 example that we pointed to at the beginning of the lecture. 00:48:24.940 --> 00:48:27.010 Namely, Example 9.1. 00:48:27.010 --> 00:48:29.370 And so we can just simply use that result. 00:48:29.370 --> 00:48:32.120 If you think back to that example or refer to your 00:48:32.120 --> 00:48:37.570 notes, we know that time function of the form e to the 00:48:37.570 --> 00:48:41.360 minus a t gives us the Laplace transform, which is 1 over s 00:48:41.360 --> 00:48:47.530 plus a with the real part of s greater than minus a. 00:48:47.530 --> 00:48:50.620 And so this is the Laplace transform of the first. 00:48:50.620 --> 00:48:53.080 Or, I'm sorry, this is the inverse Laplace transform of 00:48:53.080 --> 00:48:55.310 the first term. 00:48:55.310 --> 00:49:03.610 If we now consider the pole at s equals minus 2, and here is 00:49:03.610 --> 00:49:06.790 the region of convergence that we originally began with. 00:49:06.790 --> 00:49:12.370 In fact, we can having removed the pole at minus 1, extend 00:49:12.370 --> 00:49:16.370 this region of convergence to this pole. 00:49:16.370 --> 00:49:21.820 And we now have an algebraic expression, which is minus 1 00:49:21.820 --> 00:49:26.370 over s plus 2, the real part of s greater than minus 1. 00:49:26.370 --> 00:49:29.190 Although, in fact, we can extend the region of 00:49:29.190 --> 00:49:31.870 convergence up to the pole. 00:49:31.870 --> 00:49:37.250 And the inverse transform of this is now, again, referring 00:49:37.250 --> 00:49:40.840 to the same example, minus e to the minus 2t 00:49:40.840 --> 00:49:43.210 times the unit step. 00:49:43.210 --> 00:49:49.530 And if we simply put the two terms together then, adding 00:49:49.530 --> 00:49:54.100 the one that we have here to what we had before, we have a 00:49:54.100 --> 00:50:00.860 total inverse Laplace transform, which is that. 00:50:00.860 --> 00:50:05.890 So essentially, what's happened is that each of the 00:50:05.890 --> 00:50:09.540 poles has contributed an exponential factor. 00:50:09.540 --> 00:50:13.030 And because of the region of convergence being to the right 00:50:13.030 --> 00:50:18.000 of all those poles, that is consistent with the notion 00:50:18.000 --> 00:50:20.080 that both of those terms correspond to 00:50:20.080 --> 00:50:22.720 right-sided time functions. 00:50:22.720 --> 00:50:29.260 Well, let's just focus for a second or two on the same 00:50:29.260 --> 00:50:31.260 pole-zero pattern. 00:50:31.260 --> 00:50:36.620 But instead of a region of convergence which is to the 00:50:36.620 --> 00:50:40.030 right of the poles as we had before, we'll now take a 00:50:40.030 --> 00:50:43.310 region of convergence which is between the two poles. 00:50:43.310 --> 00:50:47.700 And I'll let you work through this more leisurely in the 00:50:47.700 --> 00:50:49.510 video course manual. 00:50:49.510 --> 00:50:53.200 But when we carry out the partial fraction expansion, as 00:50:53.200 --> 00:51:00.060 I've done below, we would now associate with this pole a 00:51:00.060 --> 00:51:02.490 region of convergence to the right. 00:51:02.490 --> 00:51:07.200 With this pole, a region of convergence to the left. 00:51:07.200 --> 00:51:13.610 And so what we would have is the sum of a right-sided time 00:51:13.610 --> 00:51:15.520 function due to this pole. 00:51:15.520 --> 00:51:20.160 And in fact it's of the form e to the minus t for t positive. 00:51:20.160 --> 00:51:26.330 And a left-sided time function due to this pole. 00:51:26.330 --> 00:51:28.590 And in fact, that's of the form e to the 00:51:28.590 --> 00:51:30.970 minus 2t for t negative. 00:51:30.970 --> 00:51:35.280 And so, in fact, the answer that we will get when we 00:51:35.280 --> 00:51:39.120 decompose this, use the partial fraction expansion, 00:51:39.120 --> 00:51:42.870 being very careful about associating the region of 00:51:42.870 --> 00:51:46.230 convergence of this pole to the right and of this pole to 00:51:46.230 --> 00:51:50.530 the left, we'll have then, when we're all done, a time 00:51:50.530 --> 00:51:55.570 function which will be of the form e to the minus t times 00:51:55.570 --> 00:51:58.660 the unit step for t positive. 00:51:58.660 --> 00:52:03.380 And then we'll have a term of the form e to the-- 00:52:03.380 --> 00:52:06.750 I'm sorry, this would be e to the minus 2t since this is at 00:52:06.750 --> 00:52:09.150 minus 2 and this is at minus 1. 00:52:09.150 --> 00:52:11.960 This would be a plus sign and this would be minus e to the 00:52:11.960 --> 00:52:15.300 minus t for t negative. 00:52:15.300 --> 00:52:20.120 And you'll look at that a little more carefully when you 00:52:20.120 --> 00:52:21.970 sit down with the video course manual. 00:52:24.950 --> 00:52:28.920 OK, well, what we've gone through, rather quickly, is an 00:52:28.920 --> 00:52:32.510 introduction to the Laplace transform. 00:52:32.510 --> 00:52:36.970 And a couple of points to underscore again, is the fact 00:52:36.970 --> 00:52:41.190 that the Laplace transform is very closely associated with 00:52:41.190 --> 00:52:42.690 the Fourier transform. 00:52:42.690 --> 00:52:47.280 And in fact, the Laplace transform for s equals j omega 00:52:47.280 --> 00:52:49.570 reduces to the Fourier transform. 00:52:49.570 --> 00:52:52.790 But more generally, the Laplace transform is the 00:52:52.790 --> 00:52:59.760 Fourier transform of x of t with an exponential weighting. 00:52:59.760 --> 00:53:02.870 And there are some exponentials for which that 00:53:02.870 --> 00:53:05.010 product converges. 00:53:05.010 --> 00:53:08.430 There are other exponentials for which that product has a 00:53:08.430 --> 00:53:10.780 Fourier transform that doesn't converge. 00:53:10.780 --> 00:53:14.370 That then imposes on the discussion of the Laplace 00:53:14.370 --> 00:53:18.860 transform what we refer to as the region of convergence. 00:53:18.860 --> 00:53:22.670 And it's very important to understand that in specifying 00:53:22.670 --> 00:53:28.670 a Laplace transform, it's important to identify not only 00:53:28.670 --> 00:53:34.870 the algebraic expression, but also the values of s for which 00:53:34.870 --> 00:53:35.620 it's valid. 00:53:35.620 --> 00:53:37.740 Namely, the region of convergence 00:53:37.740 --> 00:53:39.490 of the Laplace transform. 00:53:39.490 --> 00:53:44.350 Finally what we did was to tie together some properties of a 00:53:44.350 --> 00:53:48.200 time function with things that we can say about the region of 00:53:48.200 --> 00:53:52.060 convergence of its Laplace transform. 00:53:52.060 --> 00:53:54.910 Now, just as with the Fourier transform, the Laplace 00:53:54.910 --> 00:53:58.670 transform has some very important properties. 00:53:58.670 --> 00:54:04.140 And out of these properties, both are some mechanisms for 00:54:04.140 --> 00:54:09.760 using the Laplace transform for such systems as those 00:54:09.760 --> 00:54:12.280 described by linear constant coefficient 00:54:12.280 --> 00:54:14.080 differential equations. 00:54:14.080 --> 00:54:17.900 But more importantly, the properties will help us. 00:54:17.900 --> 00:54:23.550 As we understand them further, will help us in using and 00:54:23.550 --> 00:54:27.890 exploiting the Laplace transform to study and 00:54:27.890 --> 00:54:31.700 understand linear time-invariant systems. 00:54:31.700 --> 00:54:34.970 And that's what we'll go on to next time. 00:54:34.970 --> 00:54:38.530 In particular, talking about properties, and then 00:54:38.530 --> 00:54:42.610 associating with linear time-invariant systems much of 00:54:42.610 --> 00:54:46.610 the discussion that we've had today relating 00:54:46.610 --> 00:54:48.040 to the Laplace transform. 00:54:48.040 --> 00:54:49.290 Thank you.