WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.720 continue to offer high quality educational resources for free. 00:00:10.720 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.280 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.280 --> 00:00:18.450 at ocw.mit.edu. 00:00:20.320 --> 00:00:22.320 DENNIS FREEMAN: So for the last couple of times, 00:00:22.320 --> 00:00:25.740 we've been looking at Fourier series 00:00:25.740 --> 00:00:29.190 as a way of looking at signals as a way of being composed out 00:00:29.190 --> 00:00:31.140 of sinusoids, much the way we had previously 00:00:31.140 --> 00:00:36.990 looked at frequency responses as a way of thinking about systems 00:00:36.990 --> 00:00:38.970 characterized by sinusoids. 00:00:38.970 --> 00:00:42.600 And for the past two sessions, we've looked at Fourier series 00:00:42.600 --> 00:00:44.220 not because they were terribly useful, 00:00:44.220 --> 00:00:47.170 but because they were terribly simple. 00:00:47.170 --> 00:00:49.770 Today, I want to do the much more difficult task, 00:00:49.770 --> 00:00:51.810 but much more interesting task of thinking 00:00:51.810 --> 00:00:54.180 about the general case for thinking 00:00:54.180 --> 00:00:58.560 about a sinusoidal decomposition of an arbitrary signal, one 00:00:58.560 --> 00:01:00.870 that is not necessarily periodic. 00:01:00.870 --> 00:01:03.180 So I should say upfront, what I'm going 00:01:03.180 --> 00:01:05.790 to talk about is motivational. 00:01:05.790 --> 00:01:06.990 It's not a proof. 00:01:06.990 --> 00:01:09.450 Proving Fourier series convergence 00:01:09.450 --> 00:01:11.247 is actually very complicated. 00:01:11.247 --> 00:01:13.080 It's something that mathematicians worked on 00:01:13.080 --> 00:01:14.950 for about 100 years. 00:01:14.950 --> 00:01:17.730 So I am not going to try to prove things 00:01:17.730 --> 00:01:19.200 in any rigorous fashion, but I am 00:01:19.200 --> 00:01:22.290 going to try to motivate things so 00:01:22.290 --> 00:01:26.400 that you should at least expect that such a thing should exist. 00:01:26.400 --> 00:01:29.580 So the idea, the motivation is going to be 00:01:29.580 --> 00:01:33.810 how can I think about an aperiodic signal 00:01:33.810 --> 00:01:36.030 within a periodic framework because I already have 00:01:36.030 --> 00:01:38.400 worked out all the details. 00:01:38.400 --> 00:01:42.090 The details for Fourier series are relatively simple, well, 00:01:42.090 --> 00:01:44.700 at least compared to a Fourier transform, which is harder. 00:01:44.700 --> 00:01:48.370 Fourier series themselves are not that easy. 00:01:48.370 --> 00:01:50.820 But if I believe the Fourier series idea, 00:01:50.820 --> 00:01:52.710 is there a way to leverage that to think 00:01:52.710 --> 00:01:54.300 about aperiodic signals? 00:01:54.300 --> 00:01:57.840 And the idea is going to be let's take an aperiodic signal. 00:01:57.840 --> 00:02:00.130 I've tried to choose something terribly simple. 00:02:00.130 --> 00:02:04.230 It's the simplest thing that I could think of it isn't zero. 00:02:04.230 --> 00:02:08.600 So it's one for a while and zero most of the time. 00:02:08.600 --> 00:02:10.110 But I could make that signal, which 00:02:10.110 --> 00:02:13.950 is clearly not periodic, by thinking about periodically 00:02:13.950 --> 00:02:15.330 extending it. 00:02:15.330 --> 00:02:16.170 Copy it. 00:02:16.170 --> 00:02:19.800 Add it to itself many times, each time 00:02:19.800 --> 00:02:21.720 shifted by a capital T. 00:02:21.720 --> 00:02:24.770 This signal is obviously periodic. 00:02:24.770 --> 00:02:26.640 This transformation is obviously going 00:02:26.640 --> 00:02:29.130 to take any signal regardless and turn it 00:02:29.130 --> 00:02:32.550 into something that is periodic in cap T. 00:02:32.550 --> 00:02:36.270 So if I did that, then I could kind of trivially say, 00:02:36.270 --> 00:02:37.950 well, the aperiodic thing is just 00:02:37.950 --> 00:02:41.790 the limit when capital T goes to infinity of the periodic thing. 00:02:41.790 --> 00:02:43.560 OK, that's pretty trivial. 00:02:43.560 --> 00:02:46.080 OK, that's obviously true. 00:02:46.080 --> 00:02:51.000 The trick is, what if I took a Fourier series in the middle? 00:02:51.000 --> 00:02:53.760 What if I periodically extended this thing 00:02:53.760 --> 00:02:55.770 to get something that is periodic, 00:02:55.770 --> 00:02:58.567 take a Fourier series of this thing, 00:02:58.567 --> 00:03:00.150 and then take the limit of the series? 00:03:02.670 --> 00:03:06.270 So that's the thing I'm going to do over the next three slides. 00:03:06.270 --> 00:03:10.770 So think about a general, aperiodic signal, periodically 00:03:10.770 --> 00:03:15.605 extended so it's now periodic in cap T, take a Fourier series-- 00:03:19.132 --> 00:03:21.090 just to motivate the kind of math that happens, 00:03:21.090 --> 00:03:25.020 I've written out the math for this particularly simple signal 00:03:25.020 --> 00:03:28.416 that is one for a while and zero most of the time. 00:03:28.416 --> 00:03:31.190 The Fourier series coefficient, a sub k 00:03:31.190 --> 00:03:34.227 is obviously 1 over t, the period, 00:03:34.227 --> 00:03:35.310 integral over the period-- 00:03:35.310 --> 00:03:39.590 I took the symmetric period because it's the easiest one-- 00:03:39.590 --> 00:03:42.965 signal of interest, basis function integrated over time. 00:03:45.730 --> 00:03:48.140 And that's pretty trivially-- 00:03:48.140 --> 00:03:49.310 those integrals are easy. 00:03:49.310 --> 00:03:50.750 That was chosen that way. 00:03:50.750 --> 00:03:53.050 And so I get an answer that looks like that. 00:03:53.050 --> 00:03:57.170 The thing I want you to see about the answer 00:03:57.170 --> 00:04:01.760 is that I can think about it as a function of omega or k. 00:04:01.760 --> 00:04:04.610 And that's what I've plotted here. 00:04:04.610 --> 00:04:08.630 In particular, if I multiply a sub k by capital T, 00:04:08.630 --> 00:04:11.720 so as to kill this 1 over t thing, 00:04:11.720 --> 00:04:19.750 and if I plot t times a sub k, I get a relationship 2 sine omega 00:04:19.750 --> 00:04:27.790 S over omega, omega being k 2pi over t. 00:04:27.790 --> 00:04:32.830 But for the Fourier series, that only exists for k and integer. 00:04:32.830 --> 00:04:36.800 So that's what's represented by the blue bars. 00:04:36.800 --> 00:04:39.670 But what I want you to see is just from the math, 00:04:39.670 --> 00:04:45.250 the envelope doesn't depend on t. 00:04:45.250 --> 00:04:46.880 OK, that's the trick. 00:04:46.880 --> 00:04:49.750 So the idea is I'm plotting the Fourier coefficients a sub 00:04:49.750 --> 00:04:52.570 k as a function of k. 00:04:52.570 --> 00:04:58.990 So k equals 0, 1, 2, 3, 4, 5, et cetera. 00:04:58.990 --> 00:05:02.050 But I notice that the envelope can be written strictly 00:05:02.050 --> 00:05:05.320 as a function of omega where there is a simple relationship 00:05:05.320 --> 00:05:06.880 between omega and k. 00:05:06.880 --> 00:05:09.760 But omega is defined across the entire axis, 00:05:09.760 --> 00:05:12.640 and it's represented by this light black curve. 00:05:12.640 --> 00:05:16.990 That's more apparent if I think about increasing capital 00:05:16.990 --> 00:05:21.300 T. What if I were to keep the base waveform the same, 00:05:21.300 --> 00:05:23.000 but change capital T? 00:05:23.000 --> 00:05:27.850 Say I double capital T. The thing that happens 00:05:27.850 --> 00:05:31.930 is the envelope stays the same, but the spacing 00:05:31.930 --> 00:05:35.740 of the k's becomes condensed. 00:05:35.740 --> 00:05:40.140 There's more k's in a given number of hertzes, 00:05:40.140 --> 00:05:42.100 in frequency, than there was before. 00:05:42.100 --> 00:05:44.890 And if I double it again, it doubles again. 00:05:44.890 --> 00:05:46.480 The envelope didn't change. 00:05:46.480 --> 00:05:48.220 The k's did. 00:05:50.706 --> 00:05:52.580 The interesting thing about that construction 00:05:52.580 --> 00:05:56.330 is that it has separated out the part that depends on capital T 00:05:56.330 --> 00:05:59.180 from the part that doesn't depend on capital T. Capital 00:05:59.180 --> 00:06:02.390 T was this arbitrary thing that I 00:06:02.390 --> 00:06:04.640 used to take an aperiodic signal and turn it 00:06:04.640 --> 00:06:06.260 into a periodic signal. 00:06:06.260 --> 00:06:09.900 And it has an effect on the answer 00:06:09.900 --> 00:06:14.420 that can be separated from the other part of the answer. 00:06:14.420 --> 00:06:19.070 Some part of the answer depends on the base waveform. 00:06:19.070 --> 00:06:23.510 Some other part of the answer depends on capital T. Well, 00:06:23.510 --> 00:06:26.000 that's nice because now if I think about taking the limit 00:06:26.000 --> 00:06:29.060 as capital T goes to infinity, I have a prayer of interpreting 00:06:29.060 --> 00:06:32.540 things because part of my answer is changing with t and part 00:06:32.540 --> 00:06:33.470 of it isn't. 00:06:33.470 --> 00:06:36.590 So all I need to do now is focus on the part that is changing 00:06:36.590 --> 00:06:38.420 with t and separate it from the part that's 00:06:38.420 --> 00:06:41.160 not changing with t. 00:06:41.160 --> 00:06:43.100 So now, I can think about taking-- 00:06:43.100 --> 00:06:47.390 so I just plug in this expression 00:06:47.390 --> 00:06:51.950 here for this integral. 00:06:51.950 --> 00:06:53.960 And what I get then is something that 00:06:53.960 --> 00:07:00.180 looks a lot like a Fourier series or even 00:07:00.180 --> 00:07:01.820 a Laplace transform. 00:07:01.820 --> 00:07:05.570 I get an integral-- ignore the limit part for a moment. 00:07:05.570 --> 00:07:08.030 I get an integral of something times 00:07:08.030 --> 00:07:11.000 some sort of a weighting function. 00:07:11.000 --> 00:07:13.760 And I get something over here where the integral 00:07:13.760 --> 00:07:17.980 was over time, but the function over here 00:07:17.980 --> 00:07:19.690 doesn't have time in it. 00:07:19.690 --> 00:07:22.000 It only has omega in it. 00:07:22.000 --> 00:07:23.500 That's the sense in which it sort of 00:07:23.500 --> 00:07:30.100 looks like the analysis formula for either Fourier 00:07:30.100 --> 00:07:31.984 series or Laplace transforms. 00:07:31.984 --> 00:07:33.400 It looks like the analysis formula 00:07:33.400 --> 00:07:40.020 because I'm calculating a T ak, the components of the series, 00:07:40.020 --> 00:07:43.530 or this new thing, E of omega, that doesn't depend-- 00:07:43.530 --> 00:07:45.180 neither of those depended on t. 00:07:47.970 --> 00:07:50.340 OK, so the idea is that when I do 00:07:50.340 --> 00:07:52.200 this kind of a limiting operation 00:07:52.200 --> 00:07:54.750 on the periodic extension, I get something 00:07:54.750 --> 00:08:01.590 that ends up looking like a transform relationship. 00:08:01.590 --> 00:08:03.990 And if I think about going the other way, 00:08:03.990 --> 00:08:06.450 doing a synthesis operation, I can 00:08:06.450 --> 00:08:08.010 think about how I would construct 00:08:08.010 --> 00:08:11.077 x of t out of the Fourier coefficients 00:08:11.077 --> 00:08:12.660 But now, there's a simple relationship 00:08:12.660 --> 00:08:15.630 between the Fourier series coefficients, 00:08:15.630 --> 00:08:24.230 a sub k, and this thing Ew, which I've represented here. 00:08:24.230 --> 00:08:26.320 And I don't like the t. 00:08:26.320 --> 00:08:29.620 So I'll do a substitution from here. 00:08:29.620 --> 00:08:32.740 t can be written as omega 0 over 2pi. 00:08:32.740 --> 00:08:34.690 1 over t can be written as omega 0 ever 2pi. 00:08:37.872 --> 00:08:39.330 And now, I've got everything I need 00:08:39.330 --> 00:08:44.760 to think about how that sum approaches an integral 00:08:44.760 --> 00:08:47.070 in a Riemann sum kind of sense. 00:08:47.070 --> 00:08:49.770 Think about as I add more and more-- 00:08:49.770 --> 00:08:52.920 as I make capital T get bigger and bigger, 00:08:52.920 --> 00:08:54.630 omega 0 gets smaller and smaller. 00:08:57.840 --> 00:09:01.020 As I make the capital T bigger and bigger, 00:09:01.020 --> 00:09:03.720 the spacing gets smaller and smaller. 00:09:03.720 --> 00:09:08.640 Increasingly, I can think about this function E of omega 00:09:08.640 --> 00:09:12.630 as being smooth and increasingly constant 00:09:12.630 --> 00:09:16.360 over the small interval between the bars. 00:09:16.360 --> 00:09:20.820 So I can think about the sum as a Riemann sum passed 00:09:20.820 --> 00:09:23.970 to the integral as the limit. 00:09:23.970 --> 00:09:28.590 So when I do that, omega 0 is the spacing 00:09:28.590 --> 00:09:29.705 between the two adjacents. 00:09:29.705 --> 00:09:31.080 It's the region over which I want 00:09:31.080 --> 00:09:35.730 to think about that integrand being constant. 00:09:35.730 --> 00:09:41.730 And so that in the limit, this omega 0 passes to d omega. 00:09:41.730 --> 00:09:44.760 And I'm left with something that looks like a synthesis 00:09:44.760 --> 00:09:47.190 equation. 00:09:47.190 --> 00:09:51.410 So if I just write those equations here and think 00:09:51.410 --> 00:09:54.800 about this Ew thing being some kind of a transform, which I'll 00:09:54.800 --> 00:10:00.000 mysteriously write as x of j omega, 00:10:00.000 --> 00:10:04.430 then the result for the aperiodic case 00:10:04.430 --> 00:10:07.250 has a structure that looks very much like the Fourier series, 00:10:07.250 --> 00:10:10.140 or for that matter like the Laplace transform. 00:10:10.140 --> 00:10:16.460 What it says is that I can synthesize an arbitrary x of t 00:10:16.460 --> 00:10:19.970 by adding together a whole bunch of components 00:10:19.970 --> 00:10:25.250 that already depend on omega weighted by some weighting 00:10:25.250 --> 00:10:25.750 function. 00:10:25.750 --> 00:10:27.910 s this looks like a synthesis equation 00:10:27.910 --> 00:10:31.160 very much like the synthesis equation for Fourier series 00:10:31.160 --> 00:10:33.700 or for Laplace transforms. 00:10:33.700 --> 00:10:36.900 And I get an analysis equation that similarly 00:10:36.900 --> 00:10:38.410 has the same form again. 00:10:38.410 --> 00:10:42.430 I take the x of t and figure out the component 00:10:42.430 --> 00:10:46.450 that should be at omega by multiplying 00:10:46.450 --> 00:10:50.980 by a complex exponential and integrating. 00:10:50.980 --> 00:10:54.270 OK, I have to emphasize this is not a proof. 00:10:54.270 --> 00:10:57.040 All I wanted to do was kind of motivate 00:10:57.040 --> 00:11:00.100 the way you can think about an aperiodic signal 00:11:00.100 --> 00:11:02.620 as being periodic in some time interval 00:11:02.620 --> 00:11:04.240 and passed to the limit. 00:11:04.240 --> 00:11:05.890 And if you do that, you can sort of 00:11:05.890 --> 00:11:09.900 see where the equations are coming from. 00:11:09.900 --> 00:11:11.540 OK? 00:11:11.540 --> 00:11:13.821 So the idea then-- 00:11:13.821 --> 00:11:14.320 whoops. 00:11:18.000 --> 00:11:20.570 So the idea then is that we will use 00:11:20.570 --> 00:11:24.410 these relationships to define an analysis and synthesis 00:11:24.410 --> 00:11:26.810 of aperiodic signals. 00:11:26.810 --> 00:11:29.960 And we'll refer to that as a Fourier transform. 00:11:29.960 --> 00:11:32.540 The Fourier transform will let us 00:11:32.540 --> 00:11:36.690 have insights that are completely 00:11:36.690 --> 00:11:39.030 analogous to the Fourier series, except they now 00:11:39.030 --> 00:11:40.450 apply for aperiodic signals. 00:11:40.450 --> 00:11:42.120 So in particular, we'll be able to think 00:11:42.120 --> 00:11:44.160 about a signal being composed of a bunch 00:11:44.160 --> 00:11:45.240 of sinusoidal components. 00:11:45.240 --> 00:11:49.240 And we'll be able to think about systems as filters. 00:11:52.020 --> 00:11:54.390 OK, so I've already alluded to the fact 00:11:54.390 --> 00:11:58.350 that the Fourier transform relations 00:11:58.350 --> 00:12:03.840 look very similar in form to the Laplace transform relations. 00:12:03.840 --> 00:12:08.010 And so I've illustrated the analysis equations here just 00:12:08.010 --> 00:12:12.130 to emphasize the similarity. 00:12:12.130 --> 00:12:14.750 The Laplace transform, you'll remember, had the-- 00:12:14.750 --> 00:12:17.610 we integrated some signal that was 00:12:17.610 --> 00:12:20.670 a function of time times a complex exponential integrated 00:12:20.670 --> 00:12:26.010 over time to get a Laplace transform that was 00:12:26.010 --> 00:12:31.860 a function of s, not time. 00:12:31.860 --> 00:12:35.670 It was a way of having an alternative representation 00:12:35.670 --> 00:12:36.930 for the signal. 00:12:36.930 --> 00:12:38.214 There was no new information. 00:12:38.214 --> 00:12:39.630 The same information was contained 00:12:39.630 --> 00:12:41.910 in s of x as was contained in x of t. 00:12:41.910 --> 00:12:44.880 Except now, where it was organized by time, 00:12:44.880 --> 00:12:47.959 now, it's organized by s. 00:12:47.959 --> 00:12:50.250 We get the same sort of thing with a Fourier transform. 00:12:50.250 --> 00:12:54.470 And in fact, this gives away the mysterious reason 00:12:54.470 --> 00:12:58.882 for calling it x of j omega in the previous slide. 00:12:58.882 --> 00:13:01.340 You can see that a different way to think about the Fourier 00:13:01.340 --> 00:13:04.490 transform is that it's simply-- 00:13:04.490 --> 00:13:07.100 a trivial way to think about it, it's the value-- 00:13:07.100 --> 00:13:12.010 the Fourier transform is the value of the Laplace transform 00:13:12.010 --> 00:13:14.690 evaluated s equals j omega. 00:13:14.690 --> 00:13:20.270 All you do is you take this expression for the Laplace 00:13:20.270 --> 00:13:24.230 transform, and every place there was an s, make s equal j omega. 00:13:24.230 --> 00:13:27.300 And you get this equation. 00:13:27.300 --> 00:13:29.670 So that's the reason we like the notation. 00:13:29.670 --> 00:13:33.350 The Fourier transform is x of j omega. 00:13:33.350 --> 00:13:35.120 There are confusions that arise by that. 00:13:35.120 --> 00:13:36.839 And I'll talk about those in a moment. 00:13:36.839 --> 00:13:38.630 But for the time being, the important thing 00:13:38.630 --> 00:13:41.750 is that the Fourier transform can 00:13:41.750 --> 00:13:46.820 be viewed as a special case looking at the j omega 00:13:46.820 --> 00:13:51.040 axis of the Laplace transform. 00:13:51.040 --> 00:13:53.140 OK? 00:13:53.140 --> 00:13:57.250 So that view points out two things. 00:13:57.250 --> 00:13:59.900 There's a lot of similarities, and there are some differences. 00:13:59.900 --> 00:14:02.620 First, the similarities-- because you 00:14:02.620 --> 00:14:06.490 can regard the Fourier transform as kind of a special case-- 00:14:06.490 --> 00:14:07.524 that's not really true. 00:14:07.524 --> 00:14:09.940 And I will say something about that by the end of the hour 00:14:09.940 --> 00:14:10.820 as well. 00:14:10.820 --> 00:14:13.420 But because it's kind of a special case of the Laplace 00:14:13.420 --> 00:14:18.004 transform, the Fourier transform inherits 00:14:18.004 --> 00:14:19.920 a lot of the important properties of a Laplace 00:14:19.920 --> 00:14:21.270 transform. 00:14:21.270 --> 00:14:24.950 In particular, the two things that we looked at most 00:14:24.950 --> 00:14:27.750 has been linearity. 00:14:27.750 --> 00:14:30.000 Because the Laplace transform is linear, 00:14:30.000 --> 00:14:34.120 we can do all manner of things with it. 00:14:34.120 --> 00:14:38.240 The same as we use the properties of linear systems 00:14:38.240 --> 00:14:41.480 to simplify our view of how to think about a system, 00:14:41.480 --> 00:14:44.480 we could, for example, because systems are linear, 00:14:44.480 --> 00:14:47.450 we can look at the response of a system 00:14:47.450 --> 00:14:50.690 to a sum of inputs as the sum of the responses 00:14:50.690 --> 00:14:51.710 to the individuals. 00:14:51.710 --> 00:14:54.830 That's a very important property that we used of systems 00:14:54.830 --> 00:14:56.300 as a result of linearity. 00:14:56.300 --> 00:14:59.870 We did the same thing with Laplace transforms. 00:14:59.870 --> 00:15:01.785 The Laplace transform of a sum is the sum 00:15:01.785 --> 00:15:04.340 of a Laplace transforms. 00:15:04.340 --> 00:15:07.676 And in conjunction with the differentiation roll 00:15:07.676 --> 00:15:09.050 by which we knew that the Laplace 00:15:09.050 --> 00:15:12.610 transform of a derivative is s times the Laplace transform 00:15:12.610 --> 00:15:17.090 the function, the combination of linearity 00:15:17.090 --> 00:15:19.040 and the differentiation role allowed 00:15:19.040 --> 00:15:23.360 us to apply Laplace transforms to turn differential equations 00:15:23.360 --> 00:15:26.870 into algebraic equations. 00:15:26.870 --> 00:15:32.580 Precisely the same thing will work with Fourier transforms. 00:15:32.580 --> 00:15:34.620 For reasons that should be clear, 00:15:34.620 --> 00:15:37.330 if the Laplace transform has the property of linearity, 00:15:37.330 --> 00:15:40.530 so does the Fourier. 00:15:40.530 --> 00:15:43.560 And if the Laplace transform is simply 00:15:43.560 --> 00:15:44.964 related to the Fourier transform, 00:15:44.964 --> 00:15:46.380 then there's a simple relationship 00:15:46.380 --> 00:15:48.690 between the Fourier transform of a derivative 00:15:48.690 --> 00:15:52.480 and the Fourier transform of the underlying function. 00:15:52.480 --> 00:15:55.600 So in the Laplace transform, you multiply by s. 00:15:55.600 --> 00:15:57.600 Not very surprisingly, in the Fourier transform, 00:15:57.600 --> 00:16:00.300 you multiply by j omega. 00:16:00.300 --> 00:16:02.850 So there's enormous similarity. 00:16:02.850 --> 00:16:05.670 And in fact, most of what you know about Laplace, 00:16:05.670 --> 00:16:09.420 you can immediately carry over into Fourier. 00:16:09.420 --> 00:16:12.496 There are some differences. 00:16:12.496 --> 00:16:13.870 And if there weren't differences, 00:16:13.870 --> 00:16:16.090 we probably wouldn't bother with talking about both of them. 00:16:16.090 --> 00:16:16.589 Right? 00:16:16.589 --> 00:16:19.030 There are some things that will be easy to think about 00:16:19.030 --> 00:16:20.110 with Fourier transforms. 00:16:20.110 --> 00:16:21.627 And that's the reason we do it. 00:16:21.627 --> 00:16:23.710 There are some things that are easy to think about 00:16:23.710 --> 00:16:24.760 with Laplace transforms. 00:16:24.760 --> 00:16:28.670 Otherwise, we would have just skipped straight to Fourier. 00:16:28.670 --> 00:16:34.012 So there are some things that Fourier and Laplace share. 00:16:34.012 --> 00:16:35.720 There are some things that are different. 00:16:35.720 --> 00:16:39.744 One of the biggest differences is the domain. 00:16:39.744 --> 00:16:41.410 When we think about a Laplace transform, 00:16:41.410 --> 00:16:44.860 we think about x of s. 00:16:44.860 --> 00:16:49.300 The domain or the Laplace transform is the domain of s. 00:16:49.300 --> 00:16:52.510 The domain of s, s is a complex number. 00:16:52.510 --> 00:16:55.060 For that reason, when we thought out Laplace transforms, 00:16:55.060 --> 00:16:59.560 we always talked about what does the Laplace transform 00:16:59.560 --> 00:17:02.590 look like in the s plane. 00:17:02.590 --> 00:17:04.770 And we thought about the real part of the s 00:17:04.770 --> 00:17:08.390 and the imaginary part of s. 00:17:08.390 --> 00:17:11.950 When we think about Fourier transforms, 00:17:11.950 --> 00:17:16.720 we're thinking about a transform with real domain. 00:17:16.720 --> 00:17:21.369 Rather than thinking about x of s as a complex number, 00:17:21.369 --> 00:17:23.770 we're going to think of x of j omega, omega 00:17:23.770 --> 00:17:30.550 a real number that's a little confusing, right? 00:17:30.550 --> 00:17:35.620 Just sort of to confuse you, we rewrite the one 00:17:35.620 --> 00:17:40.500 that is a complex number as s-- 00:17:40.500 --> 00:17:43.474 no indication whatever that it's complex. 00:17:43.474 --> 00:17:44.890 And the one that is a real number, 00:17:44.890 --> 00:17:49.070 we put a j in front of it to remind you that it's real. 00:17:49.070 --> 00:17:52.930 I apologize, I don't know why we do this. 00:17:52.930 --> 00:17:56.280 So just remember that s, which looks kind of real isn't. 00:17:56.280 --> 00:17:58.830 It's complex. 00:17:58.830 --> 00:18:01.016 And j omega, which looks kind of complex, 00:18:01.016 --> 00:18:02.640 well, it's the omega part that matters. 00:18:02.640 --> 00:18:03.620 It's real. 00:18:03.620 --> 00:18:06.630 OK, so the important thing is the Laplace transform, 00:18:06.630 --> 00:18:09.680 the domain of a Laplace transform 00:18:09.680 --> 00:18:13.190 is complex number s, real and imaginary parts, 00:18:13.190 --> 00:18:16.330 characterized by a plane. 00:18:16.330 --> 00:18:19.880 The domain of the Fourier transform is real. 00:18:19.880 --> 00:18:21.880 That's enormously important. 00:18:21.880 --> 00:18:25.110 And we'll come back to that over and over again. 00:18:25.110 --> 00:18:27.430 But just to drive home the point, 00:18:27.430 --> 00:18:29.110 one of the things we thought about 00:18:29.110 --> 00:18:33.280 with the Laplace transform was this idea of eigenfunctions 00:18:33.280 --> 00:18:35.590 and eigenvalues. 00:18:35.590 --> 00:18:37.360 It was an idea of linearity. 00:18:37.360 --> 00:18:41.820 It was the idea that we can think about a system 00:18:41.820 --> 00:18:45.190 by how you put in a function, like E to the st, 00:18:45.190 --> 00:18:46.580 and calculate the output. 00:18:46.580 --> 00:18:50.350 Well, if the output, if the system is linear time invariant 00:18:50.350 --> 00:18:54.970 and can be characterized by a Laplace transform h of s, 00:18:54.970 --> 00:18:58.970 what's the output of that system when the input is e to the st? 00:19:03.950 --> 00:19:04.630 Everybody shout. 00:19:04.630 --> 00:19:07.765 It will make me feel much better. 00:19:07.765 --> 00:19:09.140 If you all shout at once, I won't 00:19:09.140 --> 00:19:10.520 be able to understand a word you said, 00:19:10.520 --> 00:19:12.228 and I'll assume you said the right thing. 00:19:16.474 --> 00:19:18.140 OK I didn't understand a thing you said. 00:19:18.140 --> 00:19:25.950 So I assume you all said h of s e to the st. Right, 00:19:25.950 --> 00:19:31.240 e to the st is an eigenfunction of a linear time invariant 00:19:31.240 --> 00:19:33.010 system. 00:19:33.010 --> 00:19:35.140 Eigenfunction means the function in 00:19:35.140 --> 00:19:37.420 is the same form as the function out 00:19:37.420 --> 00:19:39.740 except it could be multiplied by a constant. 00:19:39.740 --> 00:19:41.500 The constant is the eigenvalue. 00:19:41.500 --> 00:19:45.640 The eigenvalue is h of s. 00:19:45.640 --> 00:19:49.180 If we wanted to know, for example, if we wanted 00:19:49.180 --> 00:19:52.690 to characterize a very simple system, 00:19:52.690 --> 00:19:54.910 we might have a system of the form 1 over 1 plus s. 00:19:54.910 --> 00:19:57.444 We might have a signal of the form 1 over 1 plus s. 00:19:57.444 --> 00:19:58.860 So let's say we have a system now. 00:19:58.860 --> 00:20:01.930 Let's say that x represents some kind of a system. 00:20:01.930 --> 00:20:04.270 Then we would have said that that's a pole. 00:20:04.270 --> 00:20:07.490 Where's the pole? 00:20:07.490 --> 00:20:10.410 Minus 1-- we would have said we have a system 00:20:10.410 --> 00:20:13.518 with a single pole at minus 1. 00:20:13.518 --> 00:20:16.440 I would never have drawn this complicated picture 00:20:16.440 --> 00:20:18.780 at the bottom because it would be frightening. 00:20:18.780 --> 00:20:20.820 I would always draw something friendly 00:20:20.820 --> 00:20:22.830 like the picture over here. 00:20:22.830 --> 00:20:28.110 Right, the entire system can be understood by a single x. 00:20:28.110 --> 00:20:31.650 OK well, if you were computing eigenfunctions and eigenvalues, 00:20:31.650 --> 00:20:34.185 you would like to know what's the magnitude and phase. 00:20:36.920 --> 00:20:40.970 Sorry, the x of s is a complex valued 00:20:40.970 --> 00:20:44.100 function of complex domain. 00:20:44.100 --> 00:20:48.360 s is a complex number, and the answer is a complex number. 00:20:48.360 --> 00:20:51.260 So we'd like to know the real and imaginary parts 00:20:51.260 --> 00:20:56.037 of h of s or the magnitude and phase equivalently. 00:20:56.037 --> 00:20:58.370 If we want to know the magnitude and phase, for example, 00:20:58.370 --> 00:21:02.180 of h of s, in principle, we need to know what magnitude could it 00:21:02.180 --> 00:21:04.280 be for all the different s's. 00:21:04.280 --> 00:21:06.740 So what's plotted here is a picture 00:21:06.740 --> 00:21:11.987 of the magnitude of this function 00:21:11.987 --> 00:21:13.570 as a function of all the different s's 00:21:13.570 --> 00:21:17.860 that can be an eigenfunction. 00:21:17.860 --> 00:21:20.470 Right, so for all of the-- so any 00:21:20.470 --> 00:21:24.420 s is an eigenfunction of the system. 00:21:24.420 --> 00:21:27.000 And that plot plots the magnitude 00:21:27.000 --> 00:21:30.690 of the associated eigenvalue. 00:21:30.690 --> 00:21:35.460 The point is that I have to tell you a complex plane 00:21:35.460 --> 00:21:38.301 number of values. 00:21:38.301 --> 00:21:38.800 Right? 00:21:38.800 --> 00:21:42.900 There a value for s equals 1, s equals minus 1, s equals 2, 00:21:42.900 --> 00:21:49.127 s equals minus 2, s equals j, s equals 2j, s equals 17 plus 5j. 00:21:49.127 --> 00:21:51.210 All the different values, all the different points 00:21:51.210 --> 00:21:54.120 in the s plane have a different associated eigenvalue. 00:21:54.120 --> 00:21:55.920 And to completely characterize this system, 00:21:55.920 --> 00:21:58.050 I have to tell you all of those. 00:21:58.050 --> 00:22:01.130 By contrast, if I think about the Fourier transform, 00:22:01.130 --> 00:22:02.910 the Fourier transform maps a function 00:22:02.910 --> 00:22:08.870 of time to a function of omega. 00:22:08.870 --> 00:22:11.280 The complete characterization of the Fourier transform 00:22:11.280 --> 00:22:13.220 is showed here. 00:22:13.220 --> 00:22:15.230 All I need to worry about is what 00:22:15.230 --> 00:22:17.630 are all the possible values of omega. 00:22:17.630 --> 00:22:19.520 I'm thinking now instead of thinking s, 00:22:19.520 --> 00:22:24.050 I'm thinking how would I compose x of t 00:22:24.050 --> 00:22:26.949 by summing together a bunch of sine waves. 00:22:26.949 --> 00:22:28.490 The reason I want to think about that 00:22:28.490 --> 00:22:31.280 is because I want to think about systems in terms 00:22:31.280 --> 00:22:33.420 of frequency responses. 00:22:33.420 --> 00:22:36.200 So I want to know which frequencies are amplified, 00:22:36.200 --> 00:22:38.780 which ones are attenuated, which ones are phase delayed, 00:22:38.780 --> 00:22:41.690 which ones are phase advanced. 00:22:41.690 --> 00:22:44.060 And in order to do that kind of construction, 00:22:44.060 --> 00:22:47.030 all I need to know is what's the magnitude 00:22:47.030 --> 00:22:52.220 and angle of the system function for all possible values 00:22:52.220 --> 00:22:52.720 of omega. 00:22:55.540 --> 00:22:57.150 So that's an enormous difference. 00:22:57.150 --> 00:23:00.340 Instead of having, in the previous case, 00:23:00.340 --> 00:23:05.640 I had a function of time turning into a function of two space. 00:23:05.640 --> 00:23:09.030 Function of one space turned into a function of two space. 00:23:09.030 --> 00:23:10.890 Here I have a function of one space turning 00:23:10.890 --> 00:23:13.990 into a function of one space. 00:23:13.990 --> 00:23:17.670 So that is conceptually a whole lot simpler. 00:23:17.670 --> 00:23:19.890 Even more importantly, it is going 00:23:19.890 --> 00:23:21.430 to give rise to something that we'll 00:23:21.430 --> 00:23:24.330 spend most of the time for the rest of the term on-- 00:23:24.330 --> 00:23:29.190 the notion of signal processing where we can alternatively 00:23:29.190 --> 00:23:35.010 represent a signal x not by its time samples, 00:23:35.010 --> 00:23:38.930 but instead by its frequency samples. 00:23:38.930 --> 00:23:42.130 It would be very difficult to use that technique. 00:23:42.130 --> 00:23:46.560 Although it would work perfectly, 00:23:46.560 --> 00:23:48.330 there would be an explosion of information 00:23:48.330 --> 00:23:53.070 if we tried to use a signal processing technique with this 00:23:53.070 --> 00:23:55.140 where we represent this one-dimensional signal 00:23:55.140 --> 00:24:00.510 by a two-dimensional transform because we would be exploding 00:24:00.510 --> 00:24:01.950 the amount of information. 00:24:01.950 --> 00:24:04.140 We would be increasing substantially 00:24:04.140 --> 00:24:08.310 the amount of information required to specify the signal. 00:24:08.310 --> 00:24:13.120 When we do the Fourier, there is no such explosion. 00:24:13.120 --> 00:24:15.280 It was a one-dimensional function of time. 00:24:15.280 --> 00:24:17.810 It is a one-dimensional function of omega. 00:24:21.520 --> 00:24:26.067 OK, OK, I've been talking too much. 00:24:26.067 --> 00:24:27.650 I would like you to make sure that you 00:24:27.650 --> 00:24:30.270 understand the mechanics of what I've just said. 00:24:30.270 --> 00:24:33.590 So here's a signal, x1 of t. 00:24:33.590 --> 00:24:36.050 Which of these, if any, represents 00:24:36.050 --> 00:24:39.544 the Fourier transform? 00:24:39.544 --> 00:24:40.460 You're all very quiet. 00:24:40.460 --> 00:24:41.660 Look at your neighbor. 00:24:41.660 --> 00:24:42.590 Don't be quiet. 00:24:42.590 --> 00:24:43.430 And then start. 00:24:43.430 --> 00:24:45.195 And then you can go back to being quiet. 00:24:45.195 --> 00:24:47.020 [SIDE CONVERSATIONS] 00:26:05.020 --> 00:26:05.640 So it's quiet. 00:26:05.640 --> 00:26:07.220 So I assume that means convergence. 00:26:07.220 --> 00:26:12.747 So which function represents the Fourier transform of x1 of t? 00:26:12.747 --> 00:26:13.830 Everybody raise your hand. 00:26:13.830 --> 00:26:15.960 Indicate by a number of fingers. 00:26:15.960 --> 00:26:18.900 And it's overwhelmingly correct, which is wonderful. 00:26:18.900 --> 00:26:20.230 That's the point. 00:26:20.230 --> 00:26:22.170 The point is Fourier transforms are easy. 00:26:22.170 --> 00:26:23.820 And you've all got it. 00:26:23.820 --> 00:26:29.100 So it's trivial to run this kind of an integral. 00:26:29.100 --> 00:26:31.590 It's not very different from doing a Laplace transform. 00:26:31.590 --> 00:26:33.760 Here I've indicated the Laplace transform. 00:26:33.760 --> 00:26:34.260 Right? 00:26:34.260 --> 00:26:37.050 We do e to the minus t. x of t is 1 or 0. 00:26:37.050 --> 00:26:40.860 We change the limits to indicate the 1 or zeroness. 00:26:40.860 --> 00:26:42.990 Very trivial here, we get a slightly different 00:26:42.990 --> 00:26:45.330 looking answer because instead of e to the st, 00:26:45.330 --> 00:26:48.090 we have e to the j omega t. 00:26:48.090 --> 00:26:51.700 But otherwise, it's pretty much the same. 00:26:51.700 --> 00:26:55.810 The big difference, though, is again the domain. 00:26:55.810 --> 00:26:57.890 So if you think about the answer-- 00:26:57.890 --> 00:27:00.234 so the answer is four like all of you said-- 00:27:00.234 --> 00:27:02.400 if you think about the answer from the point of view 00:27:02.400 --> 00:27:07.830 of eigenfunctions and eigenvalues, 00:27:07.830 --> 00:27:10.390 you have to think about a two space. 00:27:10.390 --> 00:27:13.380 The two space for even that simple function, 00:27:13.380 --> 00:27:18.990 sort of the least complicated function I could think of, 00:27:18.990 --> 00:27:21.460 is illustrated here. 00:27:21.460 --> 00:27:23.070 And what you're supposed to see there 00:27:23.070 --> 00:27:27.970 is if I were to integrate x of t e to the minus st 00:27:27.970 --> 00:27:36.670 dt to get x of s, if I think about s as sigma plus j omega-- 00:27:36.670 --> 00:27:39.480 it has a real part and an imaginary part-- 00:27:39.480 --> 00:27:43.000 the real part, as I make the real part big, e to the st 00:27:43.000 --> 00:27:45.780 becomes something that explodes. 00:27:45.780 --> 00:27:49.020 And you can see that manifest here over in this region. 00:27:49.020 --> 00:27:51.030 So this is the real axis this way. 00:27:51.030 --> 00:27:53.550 This is the imaginary axis that way. 00:27:53.550 --> 00:27:56.220 You can see that as you go to bigger 00:27:56.220 --> 00:27:59.310 numbers in the positive real direction, 00:27:59.310 --> 00:28:02.480 the magnitude explodes. 00:28:02.480 --> 00:28:05.000 If you go in the negative direction 00:28:05.000 --> 00:28:09.242 because there was a sum here, the magnitude explodes again. 00:28:09.242 --> 00:28:10.700 You get this horrible function that 00:28:10.700 --> 00:28:12.990 spends a lot of its time near infinity. 00:28:12.990 --> 00:28:15.020 Right? 00:28:15.020 --> 00:28:18.170 So that's a complicated picture by comparison to the picture 00:28:18.170 --> 00:28:20.314 that you get if you look at Fourier transform. 00:28:20.314 --> 00:28:21.980 So if you look at the Fourier transform, 00:28:21.980 --> 00:28:24.530 you get something that's relatively simpler. 00:28:24.530 --> 00:28:27.980 We're only looking along the imaginary axis now. 00:28:27.980 --> 00:28:30.770 Furthermore, there's an easy way to interpret this. 00:28:30.770 --> 00:28:33.830 This is explicitly telling us if you put a certain frequency 00:28:33.830 --> 00:28:36.800 into the system, say this represented a system function, 00:28:36.800 --> 00:28:38.420 if this represented a system function, 00:28:38.420 --> 00:28:41.570 it's telling you that there's a simple way of thinking 00:28:41.570 --> 00:28:44.150 about how it amplifies or attenuates frequencies. 00:28:44.150 --> 00:28:44.780 Right? 00:28:44.780 --> 00:28:47.240 It likes frequencies near the middle. 00:28:47.240 --> 00:28:49.020 There's a lot of frequencies-- 00:28:49.020 --> 00:28:52.240 so if this represented a system function, 00:28:52.240 --> 00:28:56.834 it would pass with a gain of two frequencies near 0. 00:28:56.834 --> 00:28:59.000 And the magnitude would be smaller for these others. 00:28:59.000 --> 00:29:00.583 And there is a phase relationship too. 00:29:00.583 --> 00:29:04.770 So there's insights that you can get from this Fourier 00:29:04.770 --> 00:29:07.920 representation that are less easy to get from the Laplace. 00:29:07.920 --> 00:29:10.860 I mean the Laplace was a complete specification 00:29:10.860 --> 00:29:14.357 of a signal or a system, either. 00:29:14.357 --> 00:29:15.690 So all the information is there. 00:29:15.690 --> 00:29:18.326 It's just that it's more apparent-- 00:29:18.326 --> 00:29:20.700 some of the information is more apparent-- in the Fourier 00:29:20.700 --> 00:29:23.910 representation. 00:29:23.910 --> 00:29:29.960 OK, second question, what if I stretched the time axis? 00:29:29.960 --> 00:29:35.060 x1 was 1 between minus 1 and 1. x2 is 1 between minus 2 and 2. 00:29:35.060 --> 00:29:37.460 So all I'm doing, stretching the axis. 00:29:37.460 --> 00:29:40.760 What happens to the Fourier transform? 00:29:40.760 --> 00:29:41.720 Look at your neighbor. 00:29:41.720 --> 00:29:43.500 Choose a number. 00:29:43.500 --> 00:29:45.480 [SIDE CONVERSATIONS] 00:30:34.520 --> 00:30:39.194 OK, which answer tells me what happens when I stretch time? 00:30:39.194 --> 00:30:40.860 So everybody raise your hand and tell me 00:30:40.860 --> 00:30:46.140 some number between 0 and 5, 1 and 5 actually. 00:30:46.140 --> 00:30:50.190 OK, 20% correct. 00:30:50.190 --> 00:30:52.890 S 00:30:52.890 --> 00:30:55.180 So what's going to happen? 00:30:55.180 --> 00:31:01.440 Well, it's pretty easy to simply do out the integral again. 00:31:01.440 --> 00:31:03.450 Right, so that's the sort of most primitive way 00:31:03.450 --> 00:31:05.100 you can think about it. 00:31:05.100 --> 00:31:08.810 If you simply run the integral, I've 00:31:08.810 --> 00:31:10.610 written it in a kind of funny way. 00:31:10.610 --> 00:31:11.750 Right? 00:31:11.750 --> 00:31:15.430 So a lot of you said one for the answer. 00:31:18.804 --> 00:31:19.970 This kind of looks like one. 00:31:19.970 --> 00:31:22.314 Why is that not one? 00:31:22.314 --> 00:31:23.230 That's actually three. 00:31:26.840 --> 00:31:30.200 Why do I like to write it as-- 00:31:30.200 --> 00:31:32.180 instead of writing 2 since 2 omega over omega 00:31:32.180 --> 00:31:34.460 I like to write 4 signed 2 omega over 2 omega. 00:31:34.460 --> 00:31:37.070 Why do I like that? 00:31:37.070 --> 00:31:38.495 Because I'm completely random. 00:31:41.169 --> 00:31:42.710 AUDIENCE: Omega is the same that way? 00:31:42.710 --> 00:31:43.205 DENNIS FREEMAN: Excuse me. 00:31:43.205 --> 00:31:45.185 AUDIENCE: Omega can be the same-- like you 00:31:45.185 --> 00:31:47.189 can have omega absorb the 2. 00:31:47.189 --> 00:31:48.730 DENNIS FREEMAN: That's kind of right. 00:31:48.730 --> 00:31:51.050 So can you unscramble the sentence slightly? 00:31:53.850 --> 00:31:54.900 What is more-- yes. 00:31:54.900 --> 00:31:57.440 AUDIENCE: Aren't they the same form that we use? 00:31:57.440 --> 00:31:59.700 DENNIS FREEMAN: It's the same form in what sense? 00:31:59.700 --> 00:32:01.140 I mean what's the same about it? 00:32:01.140 --> 00:32:02.330 Yes. 00:32:02.330 --> 00:32:03.152 Yes. 00:32:03.152 --> 00:32:06.104 AUDIENCE: Like I thought I was going to say omega 00:32:06.104 --> 00:32:08.056 is near 0 when number two is 4. 00:32:08.056 --> 00:32:09.430 DENNIS FREEMAN: Correct, correct. 00:32:09.430 --> 00:32:13.830 If you think about what happens for omega near 0, 00:32:13.830 --> 00:32:16.530 I've got the sine of 2 omega, which is-- 00:32:19.450 --> 00:32:22.540 what's the sine of 2 omega when you make it 0? 00:32:22.540 --> 00:32:23.140 0. 00:32:23.140 --> 00:32:24.580 So I have 0 over 0. 00:32:24.580 --> 00:32:25.984 Bad. 00:32:25.984 --> 00:32:26.650 So what do I do? 00:32:29.980 --> 00:32:31.050 L'Hospital. 00:32:31.050 --> 00:32:33.120 So if I do L'Hospital's rule, then I 00:32:33.120 --> 00:32:35.490 can make this thing look like one. 00:32:35.490 --> 00:32:39.510 And if I write it in the form sine 2 omega over 2 omega, 00:32:39.510 --> 00:32:43.350 that has a value near 0 that approaches 1. 00:32:43.350 --> 00:32:45.516 So the amplitude is 4. 00:32:45.516 --> 00:32:46.890 So that's a way of separating out 00:32:46.890 --> 00:32:50.400 the part that's unity amplitude from the part that 00:32:50.400 --> 00:32:52.750 is the constant that multiplies the amplitude. 00:32:52.750 --> 00:32:55.230 So the amplitude is 4. 00:32:55.230 --> 00:32:59.790 And frequency, which had been pi, moves to pi over 2. 00:32:59.790 --> 00:33:02.660 So the point is that-- 00:33:02.660 --> 00:33:05.210 so the answer is number three. 00:33:05.210 --> 00:33:11.750 The peak increases, and the frequency spacing decreases. 00:33:11.750 --> 00:33:17.650 But more generally, the point is that if I stretched time, 00:33:17.650 --> 00:33:19.782 I compress frequency. 00:33:19.782 --> 00:33:21.740 But I compress frequency in a very special way. 00:33:21.740 --> 00:33:28.360 I compress frequency in an area-preserving way. 00:33:28.360 --> 00:33:32.080 That's why the peak popped up. 00:33:32.080 --> 00:33:34.980 So what I'd like to do is think about a general scaling rule. 00:33:34.980 --> 00:33:40.050 If I wanted to think about scaling x1 into x2, 00:33:40.050 --> 00:33:45.340 such that x2 is a scaled version of time compared to x1, 00:33:45.340 --> 00:33:49.460 so if I wanted x2 of t to be x1 of at, 00:33:49.460 --> 00:33:53.160 and if I wanted to stretch x1 to turn it into x2, 00:33:53.160 --> 00:33:54.570 should I make a1-- 00:33:54.570 --> 00:33:56.810 should I make a bigger or less than 1? 00:34:01.634 --> 00:34:03.800 I'm trying to generalize the result that I just did. 00:34:03.800 --> 00:34:04.880 Right? 00:34:04.880 --> 00:34:06.800 So I stretched x1 into x2. 00:34:06.800 --> 00:34:12.587 And what I saw is that frequency shrunk, and amplitude went up. 00:34:12.587 --> 00:34:14.420 So now, I'm thinking about what would happen 00:34:14.420 --> 00:34:16.230 if I did that in general. 00:34:16.230 --> 00:34:19.850 If I took x1, and I stretched it by setting 00:34:19.850 --> 00:34:23.750 x2 equal to x1 of at, would I want 00:34:23.750 --> 00:34:28.219 a to be bigger or less than 1 if I want to stretch x1 00:34:28.219 --> 00:34:30.324 to turn it into x2? 00:34:30.324 --> 00:34:31.710 AUDIENCE: Less than 1. 00:34:31.710 --> 00:34:34.530 DENNIS FREEMAN: Less than 1 because then the logic 00:34:34.530 --> 00:34:40.050 is that if I wanted, for example, x2 of 2 to be x1 of 1, 00:34:40.050 --> 00:34:42.199 stretch x2-- 00:34:42.199 --> 00:34:48.690 stretch x1, sorry, so that it's value x2 at the position 2 00:34:48.690 --> 00:34:53.230 is the same as the original function x1 at 1. 00:34:53.230 --> 00:34:55.000 If I stretched it, then I clearly 00:34:55.000 --> 00:34:57.070 have to have a equal to a half in that case. 00:34:57.070 --> 00:35:00.890 And in general, stretching would correspond to a less than 1. 00:35:00.890 --> 00:35:03.010 And now, I can think about where that fits 00:35:03.010 --> 00:35:07.300 in the transform relationship. 00:35:07.300 --> 00:35:11.990 Think about finding the Fourier transform of x2, 00:35:11.990 --> 00:35:18.350 and substituting x1 of at for x2, 00:35:18.350 --> 00:35:21.350 and then making this relationship look 00:35:21.350 --> 00:35:24.450 more like a Fourier transform. 00:35:24.450 --> 00:35:26.330 So I don't want the at to be here. 00:35:26.330 --> 00:35:28.730 I want function of t. 00:35:28.730 --> 00:35:31.535 So I can rewrite at as tau. 00:35:34.450 --> 00:35:37.930 Now, this looks like a Fourier transform except that 00:35:37.930 --> 00:35:40.720 I've changed all my t's to tau's. 00:35:40.720 --> 00:35:43.660 And the point is that that transformation of tau 00:35:43.660 --> 00:35:47.200 equals at shows up in two places. 00:35:47.200 --> 00:35:50.230 There's an explicit time here. 00:35:50.230 --> 00:35:52.075 And there's a time dependence in the dt. 00:35:54.830 --> 00:35:57.710 So the dt one is the one that gives me the shrinking 00:35:57.710 --> 00:36:01.280 and swelling of the axis. 00:36:01.280 --> 00:36:06.320 And the 1 over a from here is the one that gives me 00:36:06.320 --> 00:36:09.410 the changing amplitude. 00:36:09.410 --> 00:36:12.000 That's how you get the area-preserving property. 00:36:12.000 --> 00:36:14.990 However much it got compressed-- however 00:36:14.990 --> 00:36:18.110 much it stretched in time so that it became compressed 00:36:18.110 --> 00:36:21.110 in frequency, whatever the factor is that compressed it 00:36:21.110 --> 00:36:23.694 in frequency, it also makes, by the same factor, 00:36:23.694 --> 00:36:24.485 it makes it taller. 00:36:26.829 --> 00:36:29.370 We'd like to build up intuition for how the Fourier transform 00:36:29.370 --> 00:36:29.580 works. 00:36:29.580 --> 00:36:31.829 That's the reason for doing these kinds of properties. 00:36:34.820 --> 00:36:38.680 So now, there's another way of thinking about that same thing 00:36:38.680 --> 00:36:41.475 by thinking about what we call the moment theorems. 00:36:45.730 --> 00:36:47.800 Here what we think about is what would 00:36:47.800 --> 00:36:50.270 happen if we evaluated the Fourier transform at omega 00:36:50.270 --> 00:36:52.610 equals 0. 00:36:52.610 --> 00:36:56.400 Well, omega equals 0 is associated with a particularly 00:36:56.400 --> 00:37:00.780 simple complex exponential. 00:37:00.780 --> 00:37:03.840 If the frequency is 0, e to the j0 t is 1. 00:37:06.560 --> 00:37:09.260 So what you see is that the value 00:37:09.260 --> 00:37:10.960 of the Fourier transform at omega 00:37:10.960 --> 00:37:13.335 equals 0 is the area under the curve. 00:37:17.200 --> 00:37:20.640 So the idea then is that if I took 00:37:20.640 --> 00:37:25.270 an x of t, which was x1, which was 1 between minus 1 and 1, 00:37:25.270 --> 00:37:26.550 there's an area of 2. 00:37:26.550 --> 00:37:28.300 And that's a way of directly saying, well, 00:37:28.300 --> 00:37:30.240 the Fourier transform at 0 better be 2. 00:37:32.572 --> 00:37:34.030 And the intuitive thing that you're 00:37:34.030 --> 00:37:35.590 supposed to take away from that is 00:37:35.590 --> 00:37:38.620 when you look at a Fourier transform, the value at 0 00:37:38.620 --> 00:37:40.450 is the dc. 00:37:40.450 --> 00:37:46.500 How much constant is there in that signal? 00:37:46.500 --> 00:37:48.660 So there's a very explicit representation 00:37:48.660 --> 00:37:50.520 for the frequency content. 00:37:50.520 --> 00:37:52.770 I mean that's what the Fourier transform is all about. 00:37:52.770 --> 00:37:55.200 And in particular, the zero frequency is dc. 00:37:55.200 --> 00:37:56.240 It's the average value. 00:37:58.770 --> 00:38:01.870 That kind of a relationship works both ways. 00:38:01.870 --> 00:38:06.150 If you were to use the synthesis formula 00:38:06.150 --> 00:38:11.565 and think about how do you synthesize x of 0, 00:38:11.565 --> 00:38:12.940 well, it's the same sort of thing 00:38:12.940 --> 00:38:17.050 except now the t is 0 instead of the omega being 0. 00:38:17.050 --> 00:38:19.660 And what we get is 1 over 2pi times 00:38:19.660 --> 00:38:25.170 the area under the transform. 00:38:25.170 --> 00:38:28.310 So what that says is whatever is going on 00:38:28.310 --> 00:38:33.800 over in this wiggly thing, the net area, the average value, 00:38:33.800 --> 00:38:38.690 divided by 2pi has to equal the value at x1 of 0. 00:38:38.690 --> 00:38:41.540 x1 of 0 is clearly 1. 00:38:41.540 --> 00:38:46.370 So that means the area under this thing must be 2pi. 00:38:46.370 --> 00:38:48.026 That wasn't particularly clear. 00:38:48.026 --> 00:38:49.400 I mean I don't know automatically 00:38:49.400 --> 00:38:51.140 the area under that curve. 00:38:51.140 --> 00:38:53.840 But that's such a frequently recurring thing 00:38:53.840 --> 00:38:59.240 that it's useful to notice that the area under this funny curve 00:38:59.240 --> 00:39:02.285 happens to be precisely the area of this inscribed triangle. 00:39:04.870 --> 00:39:06.910 So the height of the triangle is 2. 00:39:06.910 --> 00:39:10.300 Half the base is pi. 00:39:10.300 --> 00:39:11.440 So the area is 2pi. 00:39:14.080 --> 00:39:20.400 I'm sure some Greek knew this. 00:39:20.400 --> 00:39:21.390 But I don't. 00:39:21.390 --> 00:39:22.800 So if somebody can think of a way 00:39:22.800 --> 00:39:25.760 to derive that answer without using Fourier transforms-- 00:39:25.760 --> 00:39:27.630 I can do it with Fourier transforms. 00:39:27.630 --> 00:39:31.350 And I can look it up in books where the authors also 00:39:31.350 --> 00:39:32.940 use Fourier transforms. 00:39:32.940 --> 00:39:36.030 But I'm sure some ancient Greek can do this. 00:39:36.030 --> 00:39:38.160 So the question is if anybody can figure out 00:39:38.160 --> 00:39:41.320 how the ancient Greeks would have come to that conclusion, 00:39:41.320 --> 00:39:43.430 I would be very interested to know. 00:39:43.430 --> 00:39:47.494 Does everybody get-- so areas of inscribed whatevers, 00:39:47.494 --> 00:39:48.660 right, that's what they did. 00:39:48.660 --> 00:39:49.170 Right? 00:39:49.170 --> 00:39:51.780 So I would like to know how to get 00:39:51.780 --> 00:39:55.620 the fact that the area under this wiggly function 2 sine 00:39:55.620 --> 00:40:00.450 omega over omega is 2pi without knowing Fourier transforms. 00:40:00.450 --> 00:40:01.790 So that's an open challenge. 00:40:01.790 --> 00:40:04.620 So try to figure out how to prove that without using 00:40:04.620 --> 00:40:08.310 Fourier transforms. 00:40:08.310 --> 00:40:11.910 So if we use the moment idea, and we think about this scaling 00:40:11.910 --> 00:40:16.560 thing, we come up with a very interesting result 00:40:16.560 --> 00:40:19.110 that if you were to stretch x1, which 00:40:19.110 --> 00:40:21.330 had been 1 just between minus 1 and 1, 00:40:21.330 --> 00:40:24.960 to turn it into x2, which was 1 between minus 2 and 2, 00:40:24.960 --> 00:40:30.222 and just keep stretching, what would happen? 00:40:30.222 --> 00:40:31.680 Well, it gets skinnier and skinnier 00:40:31.680 --> 00:40:34.150 and skinnier and skinnier. 00:40:34.150 --> 00:40:37.312 But in a very special way, the area is the same. 00:40:37.312 --> 00:40:39.520 Even though it got skinnier and skinnier and skinnier 00:40:39.520 --> 00:40:43.030 and skinnier, the area is the same. 00:40:43.030 --> 00:40:47.720 If you keep doing that, it turns into an impulse. 00:40:47.720 --> 00:40:50.460 Well, that's pretty interesting. 00:40:50.460 --> 00:40:52.970 That's an alternative way of deriving an impulse. 00:40:52.970 --> 00:40:55.935 An impulse, we think about an impulse 00:40:55.935 --> 00:40:58.120 as a generalized function. 00:40:58.120 --> 00:41:00.250 Any function that has the property 00:41:00.250 --> 00:41:06.650 that in some kind of a limit, the area shrinks towards 0, 00:41:06.650 --> 00:41:10.030 but the area doesn't change, that 00:41:10.030 --> 00:41:11.339 turns into a delta function. 00:41:11.339 --> 00:41:13.630 That's a different way of thinking about the definition 00:41:13.630 --> 00:41:15.140 of a delta function. 00:41:15.140 --> 00:41:17.560 And so we just found something very interesting. 00:41:17.560 --> 00:41:23.880 The Fourier transform of the constant 1 00:41:23.880 --> 00:41:28.680 seems to be a delta function as 0 of area 2pi. 00:41:28.680 --> 00:41:30.030 Well, that's pretty interesting. 00:41:30.030 --> 00:41:31.446 What's the Laplace transform of 1? 00:41:38.450 --> 00:41:39.950 Too shocking. 00:41:39.950 --> 00:41:41.900 What's the Laplace transform of 1? 00:41:44.738 --> 00:41:46.160 AUDIENCE: It's a delta function. 00:41:46.160 --> 00:41:48.844 DENNIS FREEMAN: Delta function. 00:41:48.844 --> 00:41:50.260 What's the Laplace transform of 1? 00:41:50.260 --> 00:41:54.980 And So Laplace transform, right, so x of s, 00:41:54.980 --> 00:42:00.240 integral 1e to the minus st dt. 00:42:00.240 --> 00:42:02.070 What's the Laplace transform of 1? 00:42:12.300 --> 00:42:13.139 AUDIENCE: 1 over s. 00:42:13.139 --> 00:42:14.180 DENNIS FREEMAN: 1 over s. 00:42:14.180 --> 00:42:15.240 How about 1 over s? 00:42:18.990 --> 00:42:21.415 Yes? 00:42:21.415 --> 00:42:21.915 No? 00:42:25.880 --> 00:42:28.000 1 over s-- yes. 00:42:28.000 --> 00:42:28.800 1 over s-- no. 00:42:31.380 --> 00:42:32.710 Me. 00:42:32.710 --> 00:42:34.570 1 over s, no. 00:42:34.570 --> 00:42:37.345 Why not? 00:42:37.345 --> 00:42:41.170 AUDIENCE: You would just use the su of 1 u of t. 00:42:41.170 --> 00:42:42.500 DENNIS FREEMAN: It's 1, yes. 00:42:42.500 --> 00:42:48.501 So the Laplace transform of u of t is 1 over s. 00:42:48.501 --> 00:42:49.001 Right? 00:42:52.005 --> 00:42:54.380 You remember there was a region of convergence associated 00:42:54.380 --> 00:42:57.080 with Laplace transforms. 00:42:57.080 --> 00:42:59.180 The region of convergence, we thought about 00:42:59.180 --> 00:43:06.399 like if you had a time function like u of t, 00:43:06.399 --> 00:43:07.940 then it would converge as long as you 00:43:07.940 --> 00:43:15.020 multiplied by some factor that generally attenuated. 00:43:15.020 --> 00:43:18.050 So that bounded what kinds of s's worked. 00:43:18.050 --> 00:43:22.340 We needed the real part of s bigger than 0. 00:43:22.340 --> 00:43:25.790 Because if the real part of s was the other way, 00:43:25.790 --> 00:43:27.450 the interval would diverge-- 00:43:27.450 --> 00:43:29.180 bad. 00:43:29.180 --> 00:43:33.900 Right so we could find the Laplace transform of u of t-- 00:43:38.630 --> 00:43:41.600 real part of s bigger than 0. 00:43:41.600 --> 00:43:45.260 Or we could find the Laplace transform of a backward step. 00:43:48.130 --> 00:43:52.000 The region of convergence would flip. 00:43:52.000 --> 00:43:54.240 And we got a sign change. 00:43:56.770 --> 00:43:59.430 But the Laplace transform of 1 doesn't exist. 00:44:02.500 --> 00:44:06.940 There is no region of convergence for the function 1. 00:44:06.940 --> 00:44:10.030 That's a big difference between Fourier and Laplace as well. 00:44:10.030 --> 00:44:16.300 Even though Fourier, is in some sense, a subset of Laplace, 00:44:16.300 --> 00:44:18.730 there are some signals that have Fourier transforms 00:44:18.730 --> 00:44:21.980 and not Laplace transforms, and so in that sense, 00:44:21.980 --> 00:44:23.951 Laplace is a subset of Fourier. 00:44:23.951 --> 00:44:25.450 So in fact, you better think of them 00:44:25.450 --> 00:44:28.720 as Venn diagrams that overlap. 00:44:28.720 --> 00:44:32.470 So there are some signals that have both, 00:44:32.470 --> 00:44:36.880 but there are some signals that have one and not the other. 00:44:36.880 --> 00:44:45.370 OK, so the final and maybe most important property 00:44:45.370 --> 00:44:48.010 of Laplace transforms is that they have 00:44:48.010 --> 00:44:51.620 a simple, inverse relationship. 00:44:51.620 --> 00:44:53.370 You may remember that I talked about there 00:44:53.370 --> 00:44:55.120 being an inverse relationship for Laplace. 00:44:59.100 --> 00:45:01.930 So you can think about x of t being 00:45:01.930 --> 00:45:06.860 1 over j 2pi, the integral over some sort of a contour of x 00:45:06.860 --> 00:45:11.400 of s e to the st ds. 00:45:11.400 --> 00:45:13.240 And I told you don't ever try to do that 00:45:13.240 --> 00:45:17.710 without going over to math and talking to those folks first. 00:45:17.710 --> 00:45:20.170 That's complicated. 00:45:20.170 --> 00:45:22.210 The interesting thing about this relationship 00:45:22.210 --> 00:45:25.690 is that it's really simple. 00:45:25.690 --> 00:45:29.560 So there's a very simple relationship between a Fourier 00:45:29.560 --> 00:45:33.860 transform and its inverse. 00:45:33.860 --> 00:45:38.380 OK, so I think I'll defer talking about that 00:45:38.380 --> 00:45:44.150 until next time, the reason being that I want 00:45:44.150 --> 00:45:45.690 to end a little earlier today. 00:45:45.690 --> 00:45:48.890 So I'll finish talking about the rest of the slides 00:45:48.890 --> 00:45:51.100 on the next lecture.