WEBVTT 00:00:00.120 --> 00:00:02.520 The following content is provided under a Creative 00:00:02.520 --> 00:00:04.059 Commons license. 00:00:04.059 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.750 continue to offer high quality educational resources for free. 00:00:10.750 --> 00:00:13.350 To make a donation or view additional materials 00:00:13.350 --> 00:00:17.310 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.310 --> 00:00:18.480 at ocw.mit.edu. 00:00:21.780 --> 00:00:24.505 DENNIS FREEMAN: Hello. 00:00:24.505 --> 00:00:29.890 So there's almost certainly no need to show this, 00:00:29.890 --> 00:00:33.160 but the good news is this is the last time I'll ever 00:00:33.160 --> 00:00:35.420 show a slide like this. 00:00:35.420 --> 00:00:36.626 It'll never happen again. 00:00:36.626 --> 00:00:37.750 At least not to this crowd. 00:00:37.750 --> 00:00:41.110 So you can all rest assured this is the last time you'll 00:00:41.110 --> 00:00:43.780 see something like this. 00:00:43.780 --> 00:00:46.675 Any questions on this, everybody's happy? 00:00:49.180 --> 00:00:52.210 Those are two separate questions, I guess. 00:00:52.210 --> 00:00:53.590 Any questions on this? 00:00:56.120 --> 00:00:57.940 OK, good. 00:00:57.940 --> 00:01:01.390 So today I want to finish up talking 00:01:01.390 --> 00:01:04.450 about Fourier transforms by comparing a lot of stuff we've 00:01:04.450 --> 00:01:05.129 already done. 00:01:05.129 --> 00:01:07.840 So there's nothing new today. 00:01:07.840 --> 00:01:09.370 This is intended, if anything, to be 00:01:09.370 --> 00:01:12.010 a bit of a review in light of things 00:01:12.010 --> 00:01:14.740 that might happen tomorrow. 00:01:14.740 --> 00:01:18.250 And also just to make things look simpler. 00:01:18.250 --> 00:01:20.860 Now, the best way to make things look simpler 00:01:20.860 --> 00:01:24.220 is to start with something that looks incredibly complicated. 00:01:24.220 --> 00:01:27.020 OK, so this is my thing that looks very complicated. 00:01:27.020 --> 00:01:31.581 This is simply a summary of the transforms 00:01:31.581 --> 00:01:32.830 that we've talked about today. 00:01:32.830 --> 00:01:35.420 We've talked about four of them. 00:01:35.420 --> 00:01:38.980 And the point of the slide is first, 00:01:38.980 --> 00:01:41.020 it looks a little complicated. 00:01:41.020 --> 00:01:43.930 But much more importantly, by the end of the hour, 00:01:43.930 --> 00:01:45.500 it's supposed to look very simple. 00:01:45.500 --> 00:01:46.000 Yes? 00:01:46.000 --> 00:01:48.445 AUDIENCE: Could we increase the volume? 00:01:48.445 --> 00:01:50.570 DENNIS FREEMAN: Could we increase the volume a bit? 00:01:50.570 --> 00:01:53.860 Well, I know that I can shout more. 00:01:53.860 --> 00:01:57.580 And if they notice me, they'll turn the volume up, too. 00:01:57.580 --> 00:01:58.870 So I will try to shout more. 00:01:58.870 --> 00:02:04.190 So if I don't, please tell me to do so. 00:02:04.190 --> 00:02:09.639 So the idea, then, is to try to look for some structure. 00:02:09.639 --> 00:02:11.530 Not only within one of the transforms, 00:02:11.530 --> 00:02:16.660 but also across all of the transforms. 00:02:16.660 --> 00:02:20.470 And to start this off, let's just think about it 00:02:20.470 --> 00:02:22.120 at the most basic level. 00:02:22.120 --> 00:02:26.480 What were the kinds of signals that we thought about? 00:02:26.480 --> 00:02:27.980 The kinds of signals that we thought 00:02:27.980 --> 00:02:30.850 about when we did DT Fourier series, DT Fourier 00:02:30.850 --> 00:02:34.940 transforms, CT Fourier series, and CT Fourier transforms? 00:02:34.940 --> 00:02:40.570 The kinds of signals look different, right? 00:02:40.570 --> 00:02:44.290 So in the top ones, we had discrete time. 00:02:44.290 --> 00:02:47.470 In the bottom ones, we have continuous time. 00:02:47.470 --> 00:02:51.070 In the left we have things that are periodic. 00:02:51.070 --> 00:02:54.047 In the right we had things that were not periodic. 00:02:54.047 --> 00:02:55.380 Four different kinds of signals. 00:02:55.380 --> 00:02:57.463 Not surprising there would be four different kinds 00:02:57.463 --> 00:02:59.590 of transforms. 00:02:59.590 --> 00:03:04.210 An interesting thing happens if you look at the Fourier domain. 00:03:04.210 --> 00:03:05.770 I haven't switched around the order. 00:03:05.770 --> 00:03:09.720 It was DT Fourier, DT Fourier transform, CT series, 00:03:09.720 --> 00:03:10.974 CT transform. 00:03:10.974 --> 00:03:11.640 It's still that. 00:03:14.530 --> 00:03:16.750 I've drawn a sort of iconic view of how 00:03:16.750 --> 00:03:22.510 we would draw a picture of the Fourier transform of interest. 00:03:22.510 --> 00:03:24.150 Where is discrete? 00:03:28.080 --> 00:03:29.760 So discrete in the previous picture-- 00:03:29.760 --> 00:03:32.220 which was a picture of time-- 00:03:32.220 --> 00:03:33.762 NNTT. 00:03:33.762 --> 00:03:35.220 Where was discrete in this picture? 00:03:39.070 --> 00:03:40.300 Up. 00:03:40.300 --> 00:03:44.290 Where is discrete in this picture? 00:03:44.290 --> 00:03:45.910 Left, that's weird. 00:03:48.970 --> 00:03:50.695 Where is periodic in this picture? 00:03:53.770 --> 00:03:56.130 Left. 00:03:56.130 --> 00:03:57.800 Where is periodic in this picture? 00:04:02.110 --> 00:04:05.531 Is there a periodic in this picture? 00:04:05.531 --> 00:04:06.030 Top. 00:04:06.030 --> 00:04:08.760 Why do you say top? 00:04:08.760 --> 00:04:10.950 Ah, unit circle! 00:04:10.950 --> 00:04:13.050 Unit circle is periodic. 00:04:13.050 --> 00:04:17.490 So there's discreteness and periodicity in both pictures, 00:04:17.490 --> 00:04:20.890 but they're not at the same place. 00:04:20.890 --> 00:04:23.050 Well, that's kind of interesting, right? 00:04:23.050 --> 00:04:25.049 Somehow, discrete and periodic map 00:04:25.049 --> 00:04:26.590 to things like discrete and periodic, 00:04:26.590 --> 00:04:29.110 but they don't map directly. 00:04:29.110 --> 00:04:30.380 That's kind of interesting. 00:04:30.380 --> 00:04:31.960 So part of the goal today is to try 00:04:31.960 --> 00:04:33.168 to get to the bottom of that. 00:04:33.168 --> 00:04:35.380 To understand why that's true. 00:04:35.380 --> 00:04:39.670 You already know a lot about how different transformations are 00:04:39.670 --> 00:04:41.210 related. 00:04:41.210 --> 00:04:43.000 So let's think of a simple example 00:04:43.000 --> 00:04:45.690 of the relation between Fourier series and Fourier transforms. 00:04:45.690 --> 00:04:48.148 Imagine that we have one of those signals that was periodic 00:04:48.148 --> 00:04:51.400 in time, continuous in time. 00:04:51.400 --> 00:04:52.870 Then there'd be a Fourier series-- 00:04:52.870 --> 00:04:55.360 represented this way. 00:04:55.360 --> 00:04:57.220 So I'd end up with a Fourier series that has 00:04:57.220 --> 00:04:59.620 a bunch of coefficients, a, k. 00:04:59.620 --> 00:05:01.870 Can somebody tell me, what would the Fourier transform 00:05:01.870 --> 00:05:02.370 look like? 00:05:05.322 --> 00:05:06.780 Fourier transform-- are you allowed 00:05:06.780 --> 00:05:08.863 to take the Fourier transform of a periodic thing? 00:05:14.054 --> 00:05:14.554 Yeah? 00:05:14.554 --> 00:05:17.637 AUDIENCE: Could we take the same thing with impulses? 00:05:17.637 --> 00:05:19.720 DENNIS FREEMAN: Take the same thing with impulses. 00:05:19.720 --> 00:05:21.650 OK, I like lots of those words. 00:05:21.650 --> 00:05:25.182 Could you put them together in slightly a different order? 00:05:25.182 --> 00:05:26.515 Where should I put the impulses? 00:05:29.776 --> 00:05:30.758 Yeah? 00:05:30.758 --> 00:05:33.166 AUDIENCE: At the k omega not. 00:05:33.166 --> 00:05:35.247 DENNIS FREEMAN: At the k omega not. 00:05:35.247 --> 00:05:36.080 How do you get that? 00:05:36.080 --> 00:05:36.800 That's exactly right. 00:05:36.800 --> 00:05:37.591 How'd you get that? 00:05:37.591 --> 00:05:41.324 AUDIENCE: Because the [INAUDIBLE].. 00:05:53.670 --> 00:05:55.170 DENNIS FREEMAN: So there's some kind 00:05:55.170 --> 00:06:00.999 of a relationship between the k's, in the ak's, and frequency 00:06:00.999 --> 00:06:02.790 that ought to be represented as an impulse, 00:06:02.790 --> 00:06:05.250 is what you're saying? 00:06:05.250 --> 00:06:07.410 One way you can see that is to take the Fourier 00:06:07.410 --> 00:06:08.710 transform of this expression. 00:06:08.710 --> 00:06:10.876 What would you get if you took the Fourier transform 00:06:10.876 --> 00:06:12.420 of x of t, x of j omega? 00:06:12.420 --> 00:06:13.500 That's pretty easy. 00:06:13.500 --> 00:06:16.083 What would you get if you took the Fourier transform of a sum? 00:06:18.675 --> 00:06:20.175 What do you always get when you take 00:06:20.175 --> 00:06:22.387 the Fourier transfer of a sum? 00:06:22.387 --> 00:06:24.095 The sum of the Fourier transforms, right? 00:06:24.095 --> 00:06:25.850 It's within your operator. 00:06:25.850 --> 00:06:28.880 That's one of the most powerful reasons we like it. 00:06:28.880 --> 00:06:32.780 So sum of the functions, take the transform of each one. 00:06:32.780 --> 00:06:34.160 Add them back up. 00:06:34.160 --> 00:06:36.980 How do you take the Fourier transform of e to the j omega 00:06:36.980 --> 00:06:38.510 not t? 00:06:38.510 --> 00:06:41.170 What's the Fourier transform? e to the j omega 00:06:41.170 --> 00:06:44.570 not t transforms to what? 00:06:44.570 --> 00:06:50.840 Fourier transform-- impulse. 00:06:50.840 --> 00:06:53.000 So how do I know that? 00:06:53.000 --> 00:06:57.700 So impulse-- what else, anything? 00:06:57.700 --> 00:06:58.230 Am I done? 00:07:00.589 --> 00:07:01.880 Where should I put the impulse? 00:07:07.140 --> 00:07:08.260 What should I put-- 00:07:08.260 --> 00:07:11.800 how should I indicate where I put it? 00:07:11.800 --> 00:07:15.640 So I need to say omega minus omega not, or something 00:07:15.640 --> 00:07:17.510 like that, right? 00:07:17.510 --> 00:07:20.940 So how do I know that? 00:07:20.940 --> 00:07:23.970 And am I done? 00:07:23.970 --> 00:07:24.606 Yeah? 00:07:24.606 --> 00:07:25.500 AUDIENCE: Multiply by 2 pi. 00:07:25.500 --> 00:07:26.910 DENNIS FREEMAN: Multiply by 2 pi. 00:07:26.910 --> 00:07:28.285 And a good way to do that is just 00:07:28.285 --> 00:07:31.684 to memorize an absurd number of equations, right? 00:07:31.684 --> 00:07:33.600 Let's think about how you would remember that. 00:07:33.600 --> 00:07:36.700 So if you wanted to take the Fourier transform of anything, 00:07:36.700 --> 00:07:37.200 right? 00:07:37.200 --> 00:07:43.111 You would say, h of j omega is some integral of anything e 00:07:43.111 --> 00:07:47.550 to the minus j omega t dt, right? 00:07:47.550 --> 00:07:50.620 What happens if I plug x of t equals e to the j omega 00:07:50.620 --> 00:07:52.433 not t into this expression? 00:07:55.540 --> 00:07:58.554 Can you close the integral? 00:07:58.554 --> 00:08:00.470 I mean, that's the way you would do it, right? 00:08:00.470 --> 00:08:02.803 I want to take the Fourier transform of e to the j omega 00:08:02.803 --> 00:08:04.340 not t. 00:08:04.340 --> 00:08:06.800 You stick it in the formula, and it should come out, right? 00:08:10.862 --> 00:08:12.820 So how many of you can integrate this function? 00:08:18.000 --> 00:08:19.080 No one, good. 00:08:19.080 --> 00:08:22.137 That's good, because I can't either. 00:08:22.137 --> 00:08:23.970 What's hard about integrating that function? 00:08:29.675 --> 00:08:31.300 That was a much easier question, right? 00:08:31.300 --> 00:08:33.880 Can you integrate was a hard question. 00:08:33.880 --> 00:08:34.870 Well, maybe not. 00:08:34.870 --> 00:08:36.760 It's binary. 00:08:36.760 --> 00:08:38.409 What's difficult about integrating 00:08:38.409 --> 00:08:40.330 that function of time? 00:08:40.330 --> 00:08:44.169 What if I try to integrate e to the j omega not 00:08:44.169 --> 00:08:47.339 t e to the minus j omega t dt? 00:08:47.339 --> 00:08:49.380 What's difficult about integrating that function? 00:08:55.200 --> 00:08:58.460 What do we do when we see these sorts of things-- 00:08:58.460 --> 00:08:59.870 e to the j omega not t? 00:08:59.870 --> 00:09:03.050 How do you think about e to the-- 00:09:03.050 --> 00:09:05.680 complex exponential. 00:09:05.680 --> 00:09:08.696 You can name out things like Euler's equation, right? 00:09:08.696 --> 00:09:09.820 So e to the j omega not t-- 00:09:09.820 --> 00:09:11.903 you might want to say, well, that's the same thing 00:09:11.903 --> 00:09:15.190 as cosine omega not t plus j sine omega not t. 00:09:20.190 --> 00:09:22.550 If I took just the first term-- cosine omega not t, 00:09:22.550 --> 00:09:24.081 does it have a Laplace transform? 00:09:26.790 --> 00:09:28.790 Doesn't have a Laplace transform, hmm. 00:09:28.790 --> 00:09:32.770 How about a Fourier transform? 00:09:32.770 --> 00:09:35.020 Why does it have a Fourier transform and not a Laplace 00:09:35.020 --> 00:09:35.856 transform? 00:09:41.320 --> 00:09:44.600 So Laplace transforms-- there's only a Laplace transform 00:09:44.600 --> 00:09:47.600 if there's some kind of a multiplicative kernel-- 00:09:47.600 --> 00:09:50.760 e to the minus st-- 00:09:50.760 --> 00:09:54.920 that has some values that make the integral converge. 00:09:54.920 --> 00:09:57.960 There's no value of s for which e to the j omega 00:09:57.960 --> 00:10:00.380 not t will converge. 00:10:00.380 --> 00:10:03.417 The amplitude of e to the j omega not t-- 00:10:03.417 --> 00:10:05.750 the real part has an amplitude that fluctuates like this 00:10:05.750 --> 00:10:07.700 and never decays. 00:10:07.700 --> 00:10:09.980 Goes on forever and ever in both directions. 00:10:09.980 --> 00:10:12.230 There's no exponential function that you 00:10:12.230 --> 00:10:17.510 can multiply times that function to make it converge. 00:10:17.510 --> 00:10:20.635 There is no Laplace transform. 00:10:20.635 --> 00:10:22.120 Is there a Fourier transform? 00:10:22.120 --> 00:10:23.882 Yeah. 00:10:23.882 --> 00:10:24.590 What's different? 00:10:24.590 --> 00:10:26.780 What makes us be able to do a Fourier transform 00:10:26.780 --> 00:10:30.700 and not a Laplace transform? 00:10:30.700 --> 00:10:33.660 Delta functions. 00:10:33.660 --> 00:10:35.675 We never write a Laplace transform 00:10:35.675 --> 00:10:37.830 with a delta function, ever. 00:10:37.830 --> 00:10:47.450 If you write on your exam h of s is some function of delta of s, 00:10:47.450 --> 00:10:49.390 that's definitely wrong, right? 00:10:49.390 --> 00:10:50.490 We just never do that. 00:10:53.130 --> 00:10:55.800 We're much more liberal about using delta functions 00:10:55.800 --> 00:10:58.320 in Fourier transforms. 00:10:58.320 --> 00:11:02.130 That's one of the reasons they're powerful. 00:11:02.130 --> 00:11:05.280 And one of the ways you can think about that 00:11:05.280 --> 00:11:07.930 is that delta functions are easy to integrate. 00:11:07.930 --> 00:11:09.129 That's not quite enough. 00:11:09.129 --> 00:11:10.920 The other thing that's important to realize 00:11:10.920 --> 00:11:14.580 is that we've got this integral here-- 00:11:14.580 --> 00:11:18.150 it's either going to be zero or infinity. 00:11:18.150 --> 00:11:22.230 There's going to be certain values of omega 00:11:22.230 --> 00:11:23.730 that when we multiply this together, 00:11:23.730 --> 00:11:26.063 we're going to get something that goes on and on forever 00:11:26.063 --> 00:11:28.156 and ever and never decays. 00:11:28.156 --> 00:11:29.530 That's going to give us infinity. 00:11:29.530 --> 00:11:32.260 And there's going to be other values 00:11:32.260 --> 00:11:35.970 that the long-term integral's going to go to zero. 00:11:35.970 --> 00:11:40.540 And that information is summarized by the impulses, 00:11:40.540 --> 00:11:41.340 right? 00:11:41.340 --> 00:11:44.310 So if we choose omega not-- 00:11:44.310 --> 00:11:48.220 if we choose omega to be omega not-- 00:11:48.220 --> 00:11:50.470 then when we do the integral, which comes out 00:11:50.470 --> 00:11:52.645 a function of omega, we're going to get something 00:11:52.645 --> 00:11:54.074 that goes to infinity. 00:11:56.780 --> 00:11:58.129 OK? 00:11:58.129 --> 00:12:00.670 The easy way to think about that is to do the inverse Fourier 00:12:00.670 --> 00:12:02.950 transform, right? 00:12:02.950 --> 00:12:08.660 The inverse Fourier transform is x of t 00:12:08.660 --> 00:12:14.440 is 1 over 2 pi in the integral x of j omega e to the j omega t 00:12:14.440 --> 00:12:17.500 d omega. 00:12:17.500 --> 00:12:22.470 Now, if we put the delta function in this, it's easy. 00:12:22.470 --> 00:12:24.900 Because the delta function over here-- 00:12:24.900 --> 00:12:27.222 if we say that-- 00:12:27.222 --> 00:12:28.997 it's x over here and it's h over there. 00:12:28.997 --> 00:12:29.580 But who cares? 00:12:29.580 --> 00:12:30.580 That should have been x. 00:12:38.790 --> 00:12:41.160 If we put over there that the x of j 00:12:41.160 --> 00:12:44.700 omega is 2 pi delta omega minus omega not, 00:12:44.700 --> 00:12:46.980 then we can see that the function of the delta 00:12:46.980 --> 00:12:49.095 is just to sift out a particular value of e 00:12:49.095 --> 00:12:53.490 to the j omega t, the one at omega not. 00:12:53.490 --> 00:12:55.860 And the 2 pi's obvious, too. 00:12:55.860 --> 00:12:58.039 The 2 pi came from the 1 over 2 pi, right? 00:12:58.039 --> 00:13:00.330 We had to have a 2 pi out front so that the 1 over 2 pi 00:13:00.330 --> 00:13:01.620 killed it. 00:13:01.620 --> 00:13:03.630 OK, anyway-- so the idea, then, is 00:13:03.630 --> 00:13:06.390 that simply by taking the Fourier transform of this, 00:13:06.390 --> 00:13:09.300 we can get an expression for the Fourier transform. 00:13:09.300 --> 00:13:12.270 And that tells us something about the great utility 00:13:12.270 --> 00:13:14.730 of using impulses, right? 00:13:14.730 --> 00:13:16.980 Because we use impulses in the Fourier transform, 00:13:16.980 --> 00:13:19.080 we can represent a lot more signals 00:13:19.080 --> 00:13:21.810 than we could have represented in the Laplace transform. 00:13:21.810 --> 00:13:24.535 That's one of the utilities of a Fourier. 00:13:24.535 --> 00:13:26.910 And there's a big relationship between the Fourier series 00:13:26.910 --> 00:13:29.980 and the Fourier transform. 00:13:29.980 --> 00:13:33.040 So in particular, if we have a signal that's periodic in time, 00:13:33.040 --> 00:13:36.084 it'll have both a series and a transform. 00:13:36.084 --> 00:13:38.250 The transform is a bunch of delta functions weighted 00:13:38.250 --> 00:13:43.110 by the Fourier series coefficients and 2 pi. 00:13:43.110 --> 00:13:45.600 And what that says is, had we been 00:13:45.600 --> 00:13:48.600 willing to put in impulses to begin with, 00:13:48.600 --> 00:13:51.030 we would never have had to define the series. 00:13:51.030 --> 00:13:53.940 We did the series just because it was easy. 00:13:53.940 --> 00:13:57.500 Convergence issues were more clear. 00:13:57.500 --> 00:14:00.690 But if we had been willing to think about delta functions 00:14:00.690 --> 00:14:03.060 in the transforms-- which would have been an extremely 00:14:03.060 --> 00:14:06.780 foreign concept coming straight from Laplace, where 00:14:06.780 --> 00:14:09.090 you would never see that. 00:14:09.090 --> 00:14:11.190 Had we been willing to accept delta functions 00:14:11.190 --> 00:14:12.780 in the transforms from the outset, 00:14:12.780 --> 00:14:14.969 we never would have needed series. 00:14:14.969 --> 00:14:17.010 So the relationship between series and transforms 00:14:17.010 --> 00:14:21.510 has to do with the fact that the series represents 00:14:21.510 --> 00:14:22.440 periodic signals. 00:14:22.440 --> 00:14:25.500 Periodic signals go on forever. 00:14:25.500 --> 00:14:29.160 There's never a Laplace transform, 00:14:29.160 --> 00:14:30.210 and that means that-- 00:14:30.210 --> 00:14:31.830 there's never a Laplace transform. 00:14:31.830 --> 00:14:34.455 That means the Fourier transform has a bunch of delta functions 00:14:34.455 --> 00:14:38.600 in it, OK? 00:14:38.600 --> 00:14:40.340 OK, so what I want to do, then, is 00:14:40.340 --> 00:14:41.840 think through other relationships 00:14:41.840 --> 00:14:43.145 that are on this table. 00:14:43.145 --> 00:14:45.020 And in particular, I want to think about them 00:14:45.020 --> 00:14:48.780 by thinking about the underlying structure of the signals. 00:14:48.780 --> 00:15:05.890 So think about having DTFS, DTFT, CTFS, and CTFT. 00:15:05.890 --> 00:15:08.470 So what I want to do is think about the relationships 00:15:08.470 --> 00:15:12.640 among those four transforms by thinking about how 00:15:12.640 --> 00:15:15.300 they differ in time, right? 00:15:15.300 --> 00:15:18.790 So the top guys are discrete in time. 00:15:18.790 --> 00:15:20.590 The bottom guys are continuous in time. 00:15:20.590 --> 00:15:24.670 The left guys are periodic in time. 00:15:24.670 --> 00:15:26.830 The right ones are aperiodic in time. 00:15:26.830 --> 00:15:28.480 So what I want to do to think about how 00:15:28.480 --> 00:15:32.650 they relate to each other, is think about a base case. 00:15:32.650 --> 00:15:36.927 Think about a signal that's down here-- aperiodic CT. 00:15:36.927 --> 00:15:39.260 And think about what you would have to do to the signal, 00:15:39.260 --> 00:15:43.540 and therefore what would you have to do to the transform. 00:15:43.540 --> 00:15:45.370 If you wanted to move this way-- 00:15:45.370 --> 00:15:48.470 I'll do that one first. 00:15:48.470 --> 00:15:50.090 If you move up-- 00:15:50.090 --> 00:15:52.430 I'll do that one second. 00:15:52.430 --> 00:15:53.810 Or if you move that way-- 00:15:53.810 --> 00:15:55.070 I'll do that one third. 00:15:55.070 --> 00:15:58.150 So really, the rest of the hour is just doing three examples. 00:15:58.150 --> 00:16:00.320 Well, four examples. 00:16:00.320 --> 00:16:03.290 What's the transform of a base case? 00:16:03.290 --> 00:16:06.730 Then what would happen as I move through this table? 00:16:06.730 --> 00:16:09.530 OK, everybody's with the game plan? 00:16:09.530 --> 00:16:10.917 So here's my base case. 00:16:10.917 --> 00:16:13.250 I want to think about the Fourier transform of a signal, 00:16:13.250 --> 00:16:16.760 x of t, that's a triangle. 00:16:16.760 --> 00:16:18.657 We could calculate the Fourier transform just 00:16:18.657 --> 00:16:19.490 from the definition. 00:16:19.490 --> 00:16:21.730 That's trivial, right? 00:16:21.730 --> 00:16:23.360 Good way, bad way. 00:16:23.360 --> 00:16:24.135 Good way. 00:16:27.860 --> 00:16:29.780 Smile. 00:16:29.780 --> 00:16:30.320 Better. 00:16:30.320 --> 00:16:33.140 OK, bad way-- 00:16:33.140 --> 00:16:35.352 I can think of a better way to do it. 00:16:35.352 --> 00:16:36.560 What's a better way to do it? 00:16:40.560 --> 00:16:41.060 Yeah? 00:16:41.060 --> 00:16:42.060 AUDIENCE: [INAUDIBLE]. 00:16:59.837 --> 00:17:01.420 DENNIS FREEMAN: So one way is to think 00:17:01.420 --> 00:17:06.614 about turning straight lines into constants via derivatives 00:17:06.614 --> 00:17:08.030 and integrals and stuff like that. 00:17:08.030 --> 00:17:08.988 That's a very good way. 00:17:08.988 --> 00:17:10.994 Can somebody think of a different way? 00:17:10.994 --> 00:17:12.160 That'll work, it'll be fine. 00:17:12.160 --> 00:17:12.400 Yes? 00:17:12.400 --> 00:17:14.585 AUDIENCE: Is that the convolution of two square 00:17:14.585 --> 00:17:15.460 pulses? 00:17:15.460 --> 00:17:17.501 DENNIS FREEMAN: Convolution of two square pulses, 00:17:17.501 --> 00:17:18.880 that's wonderful. 00:17:18.880 --> 00:17:23.530 So a different way we can do it is to convolve a square pulse 00:17:23.530 --> 00:17:24.510 with a square pulse. 00:17:24.510 --> 00:17:26.560 That's particularly good for this example, 00:17:26.560 --> 00:17:30.886 because a square pulse is one of our canonical forms. 00:17:30.886 --> 00:17:32.510 So if we think about what's the Fourier 00:17:32.510 --> 00:17:33.676 transform of a square pulse? 00:17:33.676 --> 00:17:36.200 We all know that it's something that has the form sine 00:17:36.200 --> 00:17:38.030 omega over omega. 00:17:38.030 --> 00:17:41.660 So all we really need to do is fill in the constants. 00:17:41.660 --> 00:17:43.040 Filling in the constants is easy, 00:17:43.040 --> 00:17:45.410 because we know the moment theorems. 00:17:45.410 --> 00:17:47.060 The moment theorem says that whatever 00:17:47.060 --> 00:17:49.670 the area is under this curve, it has 00:17:49.670 --> 00:17:54.810 to be the height of that curve at 0. 00:17:54.810 --> 00:17:56.400 Whatever is the area under this curve, 00:17:56.400 --> 00:17:57.900 it has to be the height of that one. 00:17:57.900 --> 00:17:59.524 Except that you have to divide by 2 pi, 00:17:59.524 --> 00:18:02.260 because the annoying 2 pi factor. 00:18:02.260 --> 00:18:04.950 The annoying 2 pi factor is something 00:18:04.950 --> 00:18:06.690 that's in the inverse transform and not 00:18:06.690 --> 00:18:07.800 in the Fourier transform. 00:18:07.800 --> 00:18:10.830 That makes it asymmetric. 00:18:10.830 --> 00:18:14.910 So in order to choose the constants-- the area under this 00:18:14.910 --> 00:18:19.610 is 1, so that height must be 1. 00:18:19.610 --> 00:18:21.740 The area under this, by magic, turns 00:18:21.740 --> 00:18:24.980 into the area of that triangle, which is 2 pi. 00:18:24.980 --> 00:18:26.750 Divide by 2 pi is 1. 00:18:26.750 --> 00:18:29.480 So that means that this frequency must be 2 pi 00:18:29.480 --> 00:18:32.900 in order to make that constant be 1. 00:18:32.900 --> 00:18:34.160 So we're done. 00:18:34.160 --> 00:18:38.250 So we know, then, that the transform of this signal, 00:18:38.250 --> 00:18:42.400 which is the convolution of this signal with itself, 00:18:42.400 --> 00:18:46.570 is just the product of this function with itself. 00:18:46.570 --> 00:18:50.830 Convolution in time maps to multiplication in frequency. 00:18:50.830 --> 00:18:52.550 OK? 00:18:52.550 --> 00:18:53.550 So that's the base case. 00:18:53.550 --> 00:18:55.260 Now what I want to think about is 00:18:55.260 --> 00:18:57.000 how I would march through the table. 00:18:57.000 --> 00:18:58.580 That's my case down here. 00:18:58.580 --> 00:19:00.330 So what I want to do first is think about, 00:19:00.330 --> 00:19:03.180 how would I convert that this way? 00:19:03.180 --> 00:19:05.430 And what's the implication for that conversion 00:19:05.430 --> 00:19:11.260 in terms of time, and in terms of frequency? 00:19:11.260 --> 00:19:14.520 So one way to think about the transformation from a transform 00:19:14.520 --> 00:19:15.690 to a series-- 00:19:15.690 --> 00:19:17.891 you can only series with things that are periodic. 00:19:17.891 --> 00:19:19.890 The transformation that we talked about in class 00:19:19.890 --> 00:19:22.170 was periodic extension. 00:19:22.170 --> 00:19:24.300 So you think about taking the base case, which 00:19:24.300 --> 00:19:27.250 was a single triangle wave, and turning it 00:19:27.250 --> 00:19:29.670 into a sequence of triangle waves. 00:19:29.670 --> 00:19:31.800 There's lots of ways to think about that. 00:19:31.800 --> 00:19:34.870 But the way I want to motivate is to think about it-- 00:19:34.870 --> 00:19:37.260 well, no, backing up. 00:19:37.260 --> 00:19:39.480 So if we think about the triangle 00:19:39.480 --> 00:19:42.000 wave and transformation and make it into a periodic triangle 00:19:42.000 --> 00:19:43.770 wave, we could just substitute this 00:19:43.770 --> 00:19:45.660 into the definition of transform. 00:19:45.660 --> 00:19:46.760 That's ugly, why? 00:19:52.510 --> 00:19:55.270 For the same reason that one was ugly. 00:19:55.270 --> 00:19:56.310 What's ugly about this? 00:19:59.630 --> 00:20:02.916 The signal goes on and on and on and on and on. 00:20:02.916 --> 00:20:04.290 There will be certain frequencies 00:20:04.290 --> 00:20:06.330 where it'll blow up. 00:20:06.330 --> 00:20:08.160 Those will be infinity. 00:20:08.160 --> 00:20:11.190 There will be other frequencies where it is 0. 00:20:11.190 --> 00:20:13.470 There could be frequencies where it's minus infinity, 00:20:13.470 --> 00:20:17.047 and those are the only three options. 00:20:17.047 --> 00:20:18.630 Since it goes on and on and on and on, 00:20:18.630 --> 00:20:20.046 if it's anything different from 0, 00:20:20.046 --> 00:20:24.057 it has to be either infinity or minus infinity. 00:20:24.057 --> 00:20:26.390 It's going to be a train of impulses for the same reason 00:20:26.390 --> 00:20:28.670 that thing was some kind of a train of impulses. 00:20:28.670 --> 00:20:30.795 So we want to have a good way of thinking about it. 00:20:30.795 --> 00:20:32.900 I told you one way to think about this-- 00:20:32.900 --> 00:20:34.650 by taking the inverse transform. 00:20:34.650 --> 00:20:40.440 This is a little harder to guess the transforms, 00:20:40.440 --> 00:20:44.040 so you can verify it with the inverse. 00:20:44.040 --> 00:20:46.950 But you can form this by thinking 00:20:46.950 --> 00:20:55.350 about convolving the base waveform with an impulse train. 00:20:55.350 --> 00:20:57.225 So what if I had an impulse train? 00:21:00.420 --> 00:21:02.130 So I'm-- oh, it's on the next slide. 00:21:02.130 --> 00:21:05.910 What if I had an impulse train like so? 00:21:05.910 --> 00:21:07.860 If I think about having an impulse train, 00:21:07.860 --> 00:21:14.400 if I had the base waveform and an appropriate impulse train, 00:21:14.400 --> 00:21:20.550 how could I manufacture the signal of interest, which 00:21:20.550 --> 00:21:22.440 is this periodic extension? 00:21:22.440 --> 00:21:23.940 So if I have these two signals, what 00:21:23.940 --> 00:21:27.520 would I do to get that signal? 00:21:27.520 --> 00:21:29.270 Convolve. 00:21:29.270 --> 00:21:33.000 So if I knew the transform of this, 00:21:33.000 --> 00:21:36.902 which I do because this is the base case which we just did. 00:21:36.902 --> 00:21:38.610 If I knew the transform of the base case, 00:21:38.610 --> 00:21:41.550 and if I knew the transform of the impulse train-- 00:21:41.550 --> 00:21:44.360 impulse train is something that goes to function of time. 00:21:44.360 --> 00:21:46.390 0-- let's say that they're separated by t. 00:21:46.390 --> 00:21:48.450 Let's say each one has a height of 1. 00:21:48.450 --> 00:21:51.620 What's the Fourier transform of an impulse train? 00:22:00.232 --> 00:22:02.190 I can see you're all on the edge of your seats. 00:22:06.000 --> 00:22:09.640 How do you find the Fourier transform of an impulse train? 00:22:12.414 --> 00:22:13.330 OK, ask your neighbor. 00:24:07.010 --> 00:24:10.310 So how do you find the Fourier transform of an impulse train? 00:24:14.230 --> 00:24:17.430 Look it up in the tables! 00:24:17.430 --> 00:24:18.045 Boo, hiss. 00:24:21.280 --> 00:24:24.420 How do you find the Fourier transform of an impulse train? 00:24:28.630 --> 00:24:30.970 Is there anything easy about an impulse train? 00:24:30.970 --> 00:24:33.700 What a horrendous thing, right? 00:24:33.700 --> 00:24:36.010 It has an infinite number of impulses, yuck. 00:24:36.010 --> 00:24:41.790 Each impulse is horribly misbehaved. 00:24:41.790 --> 00:24:43.652 How would I find the Fourier transform 00:24:43.652 --> 00:24:44.610 of a train of impulses? 00:24:44.610 --> 00:24:44.950 Yes? 00:24:44.950 --> 00:24:46.200 AUDIENCE: So if you actually just plug 00:24:46.200 --> 00:24:47.950 that in to the formula, then you get 00:24:47.950 --> 00:24:51.100 different exponents and like, [INAUDIBLE],, 00:24:51.100 --> 00:24:53.780 and then, since there's negative infinity to infinity, 00:24:53.780 --> 00:24:56.750 they add up to the [INAUDIBLE]. 00:24:56.750 --> 00:24:58.980 DENNIS FREEMAN: So you're saying that somehow I 00:24:58.980 --> 00:25:00.510 should stick it into-- 00:25:00.510 --> 00:25:05.520 so I'm thinking about finding the Fourier transform by going 00:25:05.520 --> 00:25:08.860 back to the direct formula. 00:25:08.860 --> 00:25:13.910 So if I put an infinite train of impulses here, what would I do? 00:25:13.910 --> 00:25:19.740 So I'd have a sum of delta of t minus, 00:25:19.740 --> 00:25:27.020 say, n times t sum over n e to the minus j omega t dt. 00:25:34.780 --> 00:25:35.590 That's a good step. 00:25:38.630 --> 00:25:40.850 So now, what I might do is interchange 00:25:40.850 --> 00:25:43.780 the order, because that's what everybody does, right? 00:25:43.780 --> 00:25:45.990 And it's a little dicey doing that, right? 00:25:45.990 --> 00:25:48.680 You're always safe to do that if the integrand is absolutely 00:25:48.680 --> 00:25:49.400 summable. 00:25:49.400 --> 00:25:53.430 That's not precisely absolutely summable. 00:25:53.430 --> 00:25:56.310 Everybody knows what absolutely summable means? 00:25:59.260 --> 00:26:01.200 So it means if you took the absolute value 00:26:01.200 --> 00:26:02.730 and you summed it over all of time, 00:26:02.730 --> 00:26:04.521 you would get something less than infinity. 00:26:04.521 --> 00:26:06.720 That's not quite the truth here, right? 00:26:06.720 --> 00:26:09.810 So some of the sums in Fourier series and Fourier transforms 00:26:09.810 --> 00:26:11.210 can be hard, OK? 00:26:11.210 --> 00:26:13.830 But throwing caution to the wind, 00:26:13.830 --> 00:26:16.390 we would probably get something like that. 00:26:21.940 --> 00:26:25.270 And then we could say, well, that just samples it. 00:26:25.270 --> 00:26:27.815 We could sample this thing at t equals nt. 00:26:27.815 --> 00:26:33.210 So that would be e to the minus j omega nt. 00:26:35.860 --> 00:26:38.650 Then I end up summing a large number 00:26:38.650 --> 00:26:41.320 of complex exponentials-- 00:26:41.320 --> 00:26:43.948 large being infinite. 00:26:43.948 --> 00:26:46.235 It's kind of ugly. 00:26:46.235 --> 00:26:46.735 Yes? 00:26:46.735 --> 00:26:49.705 AUDIENCE: Could you now [INAUDIBLE].. 00:26:56.150 --> 00:26:58.310 So I could split this into two sums-- 00:26:58.310 --> 00:27:02.170 n less than 0, and a similar sum-- n bigger than 0. 00:27:02.170 --> 00:27:06.414 And then throw in the 0 term, just for good measure. 00:27:06.414 --> 00:27:09.495 AUDIENCE: And then we can see you have a bunch of sines-- 00:27:09.495 --> 00:27:11.870 actually, in this case, you have a bunch of cosine terms. 00:27:11.870 --> 00:27:15.838 So you know that-- by having the cosine terms, [INAUDIBLE].. 00:27:18.350 --> 00:27:20.600 DENNIS FREEMAN: So you just said something that seemed 00:27:20.600 --> 00:27:21.725 to take a left turn, there. 00:27:21.725 --> 00:27:23.990 What was the thing about series? 00:27:23.990 --> 00:27:25.752 AUDIENCE: So if you take your Fourier 00:27:25.752 --> 00:27:30.570 series of a sum of cosines, it's just the coefficient 00:27:30.570 --> 00:27:32.030 of the same sign? 00:27:32.030 --> 00:27:35.300 DENNIS FREEMAN: So, that sounds good. 00:27:35.300 --> 00:27:37.610 Could we use that idea of taking a Fourier series 00:27:37.610 --> 00:27:38.990 a little earlier in the proof? 00:27:42.200 --> 00:27:45.810 How would I take the Fourier transform of that? 00:27:45.810 --> 00:27:49.900 How would I take the Fourier series of that? 00:27:49.900 --> 00:27:51.358 Yeah? 00:27:51.358 --> 00:27:54.760 AUDIENCE: All 1's. 00:27:54.760 --> 00:27:56.260 All 1's. 00:27:56.260 --> 00:27:59.020 DENNIS FREEMAN: All the 1's, yes, yes. 00:27:59.020 --> 00:28:00.010 This is periodic. 00:28:00.010 --> 00:28:02.170 It has a series. 00:28:02.170 --> 00:28:03.840 Trick? 00:28:03.840 --> 00:28:08.590 No, no, no-- clever observation, not a trick. 00:28:08.590 --> 00:28:10.450 So it's a periodic waveform. 00:28:10.450 --> 00:28:12.730 It has a series. 00:28:12.730 --> 00:28:14.470 OK, why is that good? 00:28:14.470 --> 00:28:17.680 Periodic waveform series, I only need to look at one period. 00:28:17.680 --> 00:28:21.850 I reduce an infinite number of impulses to one. 00:28:21.850 --> 00:28:24.890 That's a good move. 00:28:24.890 --> 00:28:26.880 So all I need to do is-- 00:28:26.880 --> 00:28:28.430 to find the series, all I need to do 00:28:28.430 --> 00:28:32.600 is think about the Fourier series sum. 00:28:32.600 --> 00:28:35.420 Which is almost the same thing, except that now I've 00:28:35.420 --> 00:28:38.450 got something that has k's in it, right? 00:28:38.450 --> 00:28:43.024 Because a series only has harmonic frequencies. 00:28:43.024 --> 00:28:44.690 So what I can do is, instead of thinking 00:28:44.690 --> 00:28:47.630 about a Fourier series-- 00:28:47.630 --> 00:28:50.620 so now I can say, instead of finding the Fourier transform, 00:28:50.620 --> 00:28:52.610 let's find the Fourier series. 00:28:52.610 --> 00:28:56.050 That's find a sub k, which would be 1 over t 00:28:56.050 --> 00:29:01.630 to the integral on t of x of t e to the minus j 2 pi 00:29:01.630 --> 00:29:05.210 kt by t on t, OK? 00:29:05.210 --> 00:29:08.990 So now I only need to find the series. 00:29:08.990 --> 00:29:10.850 And by integrating over t-- 00:29:10.850 --> 00:29:12.469 OK, well I'll do the easy one, right? 00:29:12.469 --> 00:29:13.760 Never do something that's hard. 00:29:13.760 --> 00:29:15.380 I can choose any interval of t. 00:29:15.380 --> 00:29:18.140 Let me take that one. 00:29:18.140 --> 00:29:20.240 So the only thing that persists, then-- 00:29:20.240 --> 00:29:21.770 so if I do minus-- 00:29:21.770 --> 00:29:27.190 whatever, t by 2 to t by 2, then this becomes delta of t. 00:29:29.860 --> 00:29:35.815 And delta of t sifts out the 0 value of this guy. 00:29:35.815 --> 00:29:38.760 But e to 0 is 1. 00:29:38.760 --> 00:29:43.800 So this is 1 over t for all k. 00:29:43.800 --> 00:29:45.090 We knew that, right? 00:29:45.090 --> 00:29:49.950 The Fourier series for a unit impulse is 1 everywhere. 00:29:49.950 --> 00:29:52.320 All the harmonics have the same size-- 00:29:52.320 --> 00:29:56.580 same size, same magnitude, same phase everywhere. 00:29:56.580 --> 00:29:59.010 Then, since we know the series, we 00:29:59.010 --> 00:30:02.490 can trivially convert it into a Fourier transform, right? 00:30:02.490 --> 00:30:06.540 Using the deep insight from a few slides ago, right? 00:30:06.540 --> 00:30:10.680 So how do you take a series and turn it into a transform? 00:30:10.680 --> 00:30:12.090 Substitute for every k. 00:30:14.725 --> 00:30:18.000 AUDIENCE: [INAUDIBLE]. 00:30:18.000 --> 00:30:20.970 DENNIS FREEMAN: Put an impulse in place of every k 00:30:20.970 --> 00:30:23.070 at the frequency-- omega-- 00:30:23.070 --> 00:30:26.560 that corresponds to that k, OK? 00:30:26.560 --> 00:30:30.242 The k's-- 0, 1, 2, 3, are the 0, 1, 2, 3 harmonics. 00:30:30.242 --> 00:30:32.200 The harmonics are harmonics of the fundamental. 00:30:32.200 --> 00:30:36.460 The fundamental is 2 pi over t. 00:30:36.460 --> 00:30:39.890 So you put one impulse at every multiple of 2 pi 00:30:39.890 --> 00:30:45.730 over capital T. And you multiply them all by 2 pi. 00:30:45.730 --> 00:30:47.226 So here's your answer. 00:30:47.226 --> 00:30:48.850 You start with this thing you recognize 00:30:48.850 --> 00:30:50.500 as a periodic waveform. 00:30:50.500 --> 00:30:53.400 You turn it into a series. 00:30:53.400 --> 00:30:56.310 Then you convert every one of the harmonics 00:30:56.310 --> 00:30:59.340 into a continuous frequency representation 00:30:59.340 --> 00:31:05.250 so that the spacing is the base period 2 pi over cap T. 00:31:05.250 --> 00:31:07.020 The height of these guys was 1 over T, 00:31:07.020 --> 00:31:09.030 because of the annoying 1 over T factor in front 00:31:09.030 --> 00:31:12.590 of the Fourier series formula. 00:31:12.590 --> 00:31:15.380 And you get an extra 2 pi, because the annoying 1 over 2 00:31:15.380 --> 00:31:21.390 pi in the inverse Fourier formula, OK? 00:31:21.390 --> 00:31:23.625 So what we see, then, is that the Fourier transform 00:31:23.625 --> 00:31:27.190 of an impulse train is an impulse train. 00:31:27.190 --> 00:31:29.820 And so we can use that, then, to find the Fourier transform 00:31:29.820 --> 00:31:31.260 of this mass, remember? 00:31:31.260 --> 00:31:32.760 Oh, I've got-- oh, here it is. 00:31:32.760 --> 00:31:34.320 I was taking my base waveform, which 00:31:34.320 --> 00:31:37.290 was a triangle, trying to figure out 00:31:37.290 --> 00:31:39.780 what would happen if I did an infinite extension of that-- 00:31:39.780 --> 00:31:41.970 an infinite periodic extension. 00:31:41.970 --> 00:31:45.390 I think about the extension as being convolving the base 00:31:45.390 --> 00:31:47.520 waveform. 00:31:47.520 --> 00:31:48.460 Cover that up. 00:31:48.460 --> 00:31:52.684 If I think about convolving the base wave, form-- 00:31:52.684 --> 00:31:54.450 oh, no, it's gone. 00:31:54.450 --> 00:31:56.700 If I think about convolving the base waveform 00:31:56.700 --> 00:32:00.750 with periodic train of impulses, that 00:32:00.750 --> 00:32:02.700 means that-- so the time waveform is 00:32:02.700 --> 00:32:04.390 the convolution of two things. 00:32:04.390 --> 00:32:06.900 That means that the Fourier transform is 00:32:06.900 --> 00:32:10.420 the multiplying of two things. 00:32:10.420 --> 00:32:14.620 So I write down the transform of the base waveform-- 00:32:14.620 --> 00:32:16.690 thing that we did a long time ago. 00:32:16.690 --> 00:32:19.170 I write down the Fourier transform of the impulse train. 00:32:25.804 --> 00:32:27.220 The period of the impulse train is 00:32:27.220 --> 00:32:29.511 the same as the period of the periodic extension, which 00:32:29.511 --> 00:32:30.560 is capital T. 00:32:30.560 --> 00:32:32.350 It was 4. 00:32:32.350 --> 00:32:35.050 So that means the impulses are separated 00:32:35.050 --> 00:32:39.330 by 2 pi over 4 pi over 2. 00:32:39.330 --> 00:32:42.480 And the height of the impulse is always 00:32:42.480 --> 00:32:43.600 equal to their separation. 00:32:43.600 --> 00:32:47.200 So they're also pi over 2. 00:32:47.200 --> 00:32:49.600 So then all I do is multiply this times this, 00:32:49.600 --> 00:32:51.400 and that gives me the transform. 00:32:51.400 --> 00:32:54.700 Convolving time, multiplying frequency. 00:32:54.700 --> 00:32:56.980 The important thing that I found is that the effect 00:32:56.980 --> 00:32:58.540 of periodic extension-- 00:33:01.100 --> 00:33:03.230 I took a base waveform and turned it 00:33:03.230 --> 00:33:04.820 into a string of waveforms, right? 00:33:04.820 --> 00:33:08.930 So over here, I took a base waveform 00:33:08.930 --> 00:33:16.940 and turned it into a repeated thing, like so, in time. 00:33:16.940 --> 00:33:19.610 The effect of the periodic extension, 00:33:19.610 --> 00:33:25.720 which we understand trivially in time, is to sample frequency. 00:33:25.720 --> 00:33:30.640 What I get in the frequency representation is samples. 00:33:30.640 --> 00:33:33.720 The samples being controlled by how long 00:33:33.720 --> 00:33:37.160 is the periodic extension. 00:33:37.160 --> 00:33:39.050 So what I get is samples. 00:33:39.050 --> 00:33:41.270 Instead of getting all of these, I just get part. 00:33:41.270 --> 00:33:45.340 I just get a uniformly-spaced sample set separated by pi 00:33:45.340 --> 00:33:48.320 over 2. 00:33:48.320 --> 00:33:52.260 So then the effect of the periodic extension 00:33:52.260 --> 00:33:58.940 was to sample the Fourier transform of the base waveform. 00:33:58.940 --> 00:34:03.380 And that's the reason we can think about the Fourier series 00:34:03.380 --> 00:34:05.070 as being just a sequence of numbers. 00:34:05.070 --> 00:34:07.930 In fact, those numbers are the same. 00:34:07.930 --> 00:34:10.310 Or, they differ by 2 pi, right? 00:34:10.310 --> 00:34:13.370 So I have to multiply these guys by 2 pi 00:34:13.370 --> 00:34:16.150 to get the Fourier transform coefficients, 00:34:16.150 --> 00:34:19.310 because the Fourier transform of cosine of e to the j omega 00:34:19.310 --> 00:34:25.290 not t, I should say, is 2 pi delta, OK? 00:34:25.290 --> 00:34:27.989 So the reason the Fourier series coefficients 00:34:27.989 --> 00:34:33.270 are related the way they are, the reason they sample 00:34:33.270 --> 00:34:35.280 the Fourier transform, is because we 00:34:35.280 --> 00:34:37.940 can regard the periodic extension, which 00:34:37.940 --> 00:34:39.420 was convolved by an impulse train 00:34:39.420 --> 00:34:41.840 as being equivalently multiplying by an impulse 00:34:41.840 --> 00:34:44.433 train and frequency. 00:34:44.433 --> 00:34:45.906 OK? 00:34:45.906 --> 00:34:47.239 That's what I wanted you to see. 00:34:47.239 --> 00:34:50.020 I wanted you to see that if you think 00:34:50.020 --> 00:34:54.639 about periodic extension-- 00:34:54.639 --> 00:34:56.437 which is moving left in this diagram. 00:34:56.437 --> 00:34:58.770 If you think about moving left, what you're really doing 00:34:58.770 --> 00:35:01.750 is sampling in frequency. 00:35:01.750 --> 00:35:05.490 So that's the reason it came out having a sampled representation 00:35:05.490 --> 00:35:06.030 on the left. 00:35:08.750 --> 00:35:11.610 The periodic extension convolving with an impulse 00:35:11.610 --> 00:35:13.610 train was the same as multiplying by the impulse 00:35:13.610 --> 00:35:17.690 train and frequency, which has the effect of sampling 00:35:17.690 --> 00:35:19.990 the frequency response. 00:35:19.990 --> 00:35:24.440 OK, so now the effect of moving this way 00:35:24.440 --> 00:35:26.090 was the same sampling in frequency. 00:35:26.090 --> 00:35:27.506 Well, what happens if you actually 00:35:27.506 --> 00:35:30.211 sample in time, which is what you do when you go up? 00:35:33.182 --> 00:35:35.140 It's got to be an [INAUDIBLE] like this, right? 00:35:37.770 --> 00:35:40.885 So, the most useful thing about the Fourier transform 00:35:40.885 --> 00:35:42.760 is that the Fourier transform and the inverse 00:35:42.760 --> 00:35:45.490 transform look almost the same. 00:35:45.490 --> 00:35:48.230 That's the most useful property of the transform. 00:35:48.230 --> 00:35:51.610 Now, the Fourier transform has a simple inverse relationship, 00:35:51.610 --> 00:35:56.520 and is almost precisely the same except for the annoying 2 pi 00:35:56.520 --> 00:35:59.470 and except for a change of sign-- 00:35:59.470 --> 00:36:03.790 e to the j omega t, compared to e to the minus j omega t. 00:36:03.790 --> 00:36:05.420 Except for those trivial differences, 00:36:05.420 --> 00:36:06.760 they're the same thing. 00:36:06.760 --> 00:36:10.270 So we might expect that if we understood the effect of moving 00:36:10.270 --> 00:36:16.210 that way, which is convolve in time, multiply in frequency, 00:36:16.210 --> 00:36:19.330 you might expect an analogous thing to happen this way. 00:36:19.330 --> 00:36:21.730 And of course, it does. 00:36:21.730 --> 00:36:24.760 So here, let's think about the base waveform-- 00:36:24.760 --> 00:36:26.575 same base waveform. 00:36:26.575 --> 00:36:28.450 But now what we're doing is sampling in time. 00:36:28.450 --> 00:36:30.100 And I'll just arbitrarily, for the sake 00:36:30.100 --> 00:36:34.480 of illustration, sample it with capital T equals a half. 00:36:34.480 --> 00:36:38.380 Question is-- what would that sampling in time 00:36:38.380 --> 00:36:40.180 do to the Fourier transform? 00:36:40.180 --> 00:36:41.200 OK, sampling is time. 00:36:41.200 --> 00:36:43.480 That's something we understand completely in the time domain. 00:36:43.480 --> 00:36:45.146 What would happen in the Fourier domain? 00:36:47.610 --> 00:36:49.730 OK, well that's a little hard to compare, 00:36:49.730 --> 00:36:52.810 because the domain of these two signals-- 00:36:52.810 --> 00:36:56.920 this is continuous domain, T. This is discrete domain. 00:36:56.920 --> 00:36:58.917 And that's kind of annoying. 00:36:58.917 --> 00:37:01.000 So it's kind of hard, because they're not on equal 00:37:01.000 --> 00:37:02.890 footing to compare them. 00:37:02.890 --> 00:37:05.170 So what I can do is map the same information 00:37:05.170 --> 00:37:09.670 that was here into a representation over here that 00:37:09.670 --> 00:37:12.630 is continuous in time. 00:37:12.630 --> 00:37:15.970 The information over here was 0 everywhere, 00:37:15.970 --> 00:37:18.790 except one half-and-half. 00:37:18.790 --> 00:37:22.240 Here it's 0 everywhere except one half-and-half. 00:37:22.240 --> 00:37:25.810 Anyone know why I used impulses instead of just 00:37:25.810 --> 00:37:28.150 making the function different? 00:37:28.150 --> 00:37:30.850 Why didn't I, instead of using the complicated-- wouldn't it 00:37:30.850 --> 00:37:34.760 be much easier if I didn't use the impulses? 00:37:34.760 --> 00:37:42.490 And if instead, I just said, OK, my xp of t-- 00:37:42.490 --> 00:37:46.300 why didn't I say xp of t is almost everywhere 00:37:46.300 --> 00:37:53.565 0, except at that point, at that point, and at that point? 00:37:53.565 --> 00:37:54.440 Why didn't I do that? 00:38:03.136 --> 00:38:03.636 Yes? 00:38:03.636 --> 00:38:05.088 AUDIENCE: [INAUDIBLE]. 00:38:08.970 --> 00:38:11.480 DENNIS FREEMAN: Right, let's assume that it is continuous. 00:38:11.480 --> 00:38:19.240 And that there's exactly three points that differ 00:38:19.240 --> 00:38:21.690 from the straight line at 0. 00:38:21.690 --> 00:38:23.150 Yes? 00:38:23.150 --> 00:38:25.320 The integral's 0. 00:38:25.320 --> 00:38:28.610 That never works. 00:38:28.610 --> 00:38:30.590 In fact, the most trivial representation 00:38:30.590 --> 00:38:35.830 that only has energy at three places is three impulses. 00:38:35.830 --> 00:38:39.390 That's the only way to get non-trivial energy. 00:38:39.390 --> 00:38:41.230 OK, so I can't do that. 00:38:41.230 --> 00:38:43.190 I'm forced to do something like this. 00:38:43.190 --> 00:38:44.312 So now the idea is-- 00:38:44.312 --> 00:38:46.020 I switched to this kind of representation 00:38:46.020 --> 00:38:50.260 that has the same information, but has the same domain-- 00:38:50.260 --> 00:38:52.887 CT, continuous time. 00:38:52.887 --> 00:38:54.970 So that now I can think about-- what's the Fourier 00:38:54.970 --> 00:38:57.140 transform of this guy? 00:38:57.140 --> 00:38:59.350 Well, it's the same idea. 00:38:59.350 --> 00:39:05.620 The way I can think about computing 00:39:05.620 --> 00:39:10.147 this thing from that thing is to multiply in time by an impulse 00:39:10.147 --> 00:39:10.794 train. 00:39:13.540 --> 00:39:16.300 What's the effect of multiplying time by an impulse train? 00:39:16.300 --> 00:39:19.570 What's the effect in frequency? 00:39:19.570 --> 00:39:21.670 Multiply in time by an impulse train, 00:39:21.670 --> 00:39:23.600 what are you doing to frequency? 00:39:23.600 --> 00:39:26.130 Convolving frequency. 00:39:26.130 --> 00:39:28.320 So we take the original base waveform, 00:39:28.320 --> 00:39:29.380 but now we convolve it. 00:39:31.890 --> 00:39:34.970 And lo and behold, we convolve this signal-- 00:39:34.970 --> 00:39:37.470 which could be anything-- with this signal-- which is always 00:39:37.470 --> 00:39:38.310 periodic. 00:39:38.310 --> 00:39:42.630 If you convolve in anything with a periodic, what do you get? 00:39:42.630 --> 00:39:46.240 Periodic, inevitable. 00:39:46.240 --> 00:39:49.380 So the fact that I sampled in time 00:39:49.380 --> 00:39:51.630 leads to periodic in frequency. 00:39:51.630 --> 00:39:53.980 That's the point. 00:39:53.980 --> 00:39:57.460 Sampling in time leads to periodic in frequency. 00:39:57.460 --> 00:39:59.750 That's always going to be true. 00:39:59.750 --> 00:40:01.870 So we end up with a relationship here. 00:40:01.870 --> 00:40:04.600 The sampled waveform is going to be periodic. 00:40:04.600 --> 00:40:07.300 In here it's periodic, and 4 pi. 00:40:07.300 --> 00:40:09.670 That's a little different from what we've seen before. 00:40:09.670 --> 00:40:12.130 If we had looked at the x of n, we 00:40:12.130 --> 00:40:14.830 would have seen something that was periodic in 2 pi, 00:40:14.830 --> 00:40:19.250 but that's because the frequencies are different. 00:40:19.250 --> 00:40:20.320 One 00:40:20.320 --> 00:40:22.580 The first one I showed is a function 00:40:22.580 --> 00:40:27.420 of little omega, which has the units radians per second. 00:40:27.420 --> 00:40:31.130 This one is a function of cap Omega, which has the units 00:40:31.130 --> 00:40:32.750 radians, they're different. 00:40:32.750 --> 00:40:34.610 One's frequency in discrete time. 00:40:34.610 --> 00:40:39.350 One's frequency in continuous time. 00:40:39.350 --> 00:40:41.660 And there's a relationship, and that relationship 00:40:41.660 --> 00:40:44.510 is always this. 00:40:44.510 --> 00:40:48.420 That's the reason I write the two frequencies differently. 00:40:48.420 --> 00:40:52.100 One way to relate the two frequencies is via sampling. 00:40:52.100 --> 00:40:54.470 That's a way of taking a continuous time signal 00:40:54.470 --> 00:40:56.300 and turning it into a discrete time signal. 00:40:56.300 --> 00:40:59.480 When you do that, if you sample by the sampling 00:40:59.480 --> 00:41:02.120 interval capital T, that dictates 00:41:02.120 --> 00:41:04.350 that this relationship's going to be like that. 00:41:04.350 --> 00:41:06.400 And it's very general. 00:41:06.400 --> 00:41:09.350 All you need to do is look at the definitions. 00:41:09.350 --> 00:41:16.220 If you look at the definition of the DT Fourier transform-- 00:41:16.220 --> 00:41:22.465 that's this-- the CT Fourier transform-- that's this-- 00:41:25.190 --> 00:41:28.370 and just remember that we form this thing 00:41:28.370 --> 00:41:30.710 by samples of that thing. 00:41:34.260 --> 00:41:36.750 So now, if you think about flipping the order-- 00:41:36.750 --> 00:41:40.272 and again, not worrying about whether or not 00:41:40.272 --> 00:41:40.980 it will converge. 00:41:40.980 --> 00:41:43.380 You just assume it will for the time being. 00:41:43.380 --> 00:41:47.340 Just flip the order of the summation and the integration. 00:41:47.340 --> 00:41:55.340 This then sifts out the value of little t equals n cap T, 00:41:55.340 --> 00:41:58.550 and we get a formula for the continuous time Fourier 00:41:58.550 --> 00:42:02.000 transform that looks just like the formula 00:42:02.000 --> 00:42:04.100 for the discrete time Fourier transform. 00:42:06.720 --> 00:42:10.120 The only difference in the two is that one 00:42:10.120 --> 00:42:11.800 has discrete domain, n. 00:42:11.800 --> 00:42:14.200 The other has continuous domain, time. 00:42:14.200 --> 00:42:17.020 And by sampling, we convert continuous domain, time, 00:42:17.020 --> 00:42:19.840 into discrete domain, n. 00:42:19.840 --> 00:42:22.930 So we're always going to end up with this relationship that 00:42:22.930 --> 00:42:26.770 the discrete frequency-- that's generated by sampling-- 00:42:26.770 --> 00:42:28.180 the discrete frequency is related 00:42:28.180 --> 00:42:31.110 to the continuous frequency multiplied by capital 00:42:31.110 --> 00:42:33.970 T. Capital T has the units of seconds. 00:42:33.970 --> 00:42:39.080 So that takes radians per second and turns it into radians. 00:42:39.080 --> 00:42:43.840 OK, almost done, one more. 00:42:43.840 --> 00:42:47.860 So I've gone from here to here. 00:42:47.860 --> 00:42:49.960 What happens if I do a periodic extension? 00:42:49.960 --> 00:42:53.740 The answer was that you convolve in time, 00:42:53.740 --> 00:42:56.380 which is the same as multiply in frequency 00:42:56.380 --> 00:42:59.230 by an impulse train, which is the same as sampling 00:42:59.230 --> 00:43:01.740 in frequency. 00:43:01.740 --> 00:43:04.200 Convolving time, sampling frequency. 00:43:04.200 --> 00:43:06.450 That's what happened when I did this one. 00:43:06.450 --> 00:43:09.270 I just now did this one. 00:43:09.270 --> 00:43:12.910 What happens if you sample in time? 00:43:12.910 --> 00:43:16.976 Sample in time is multiplied by an impulse train. 00:43:16.976 --> 00:43:19.225 Multiplied by an impulse train is the same as convolve 00:43:19.225 --> 00:43:21.880 in frequency. 00:43:21.880 --> 00:43:25.640 Convolve in frequency makes it periodic. 00:43:25.640 --> 00:43:28.290 Sample in time, periodic in frequency. 00:43:28.290 --> 00:43:32.620 The last example is just to do this direction. 00:43:32.620 --> 00:43:36.210 What happens if I take the DTFT and try 00:43:36.210 --> 00:43:39.252 to turn that into a DTFS? 00:43:39.252 --> 00:43:40.710 Not surprisingly, what I have to do 00:43:40.710 --> 00:43:43.110 is the same thing I did in continuous time. 00:43:43.110 --> 00:43:45.270 If I want to change this DT signal-- which 00:43:45.270 --> 00:43:48.210 is aperiodic-- into a periodic DT signal, 00:43:48.210 --> 00:43:51.360 I do periodic extension. 00:43:51.360 --> 00:43:54.090 I can think about the periodically-extended waveform, 00:43:54.090 --> 00:43:58.850 here extended by multiples of 8, as being 00:43:58.850 --> 00:44:03.180 the original waveform convolved with that unit sample train. 00:44:07.120 --> 00:44:10.420 And so if I convolve by the unit sample train, what 00:44:10.420 --> 00:44:11.410 do I do in frequency? 00:44:15.320 --> 00:44:19.520 Convolve by unit sample train in time, 00:44:19.520 --> 00:44:22.590 gives [INAUDIBLE] and frequency. 00:44:22.590 --> 00:44:26.220 Multiply-- so I'm expecting that the answer will be sampled, 00:44:26.220 --> 00:44:28.290 and that's what happens. 00:44:28.290 --> 00:44:34.880 So I take the Fourier transform of this case. 00:44:34.880 --> 00:44:36.800 Which, like all Fourier transforms, 00:44:36.800 --> 00:44:41.860 is periodic in 2 pi. 00:44:41.860 --> 00:44:44.800 And now I think about convolving it 00:44:44.800 --> 00:44:49.850 with this unit sample train, which 00:44:49.850 --> 00:44:52.300 has the same relationship as the impulse train. 00:44:52.300 --> 00:44:54.950 And the impulse train-- the train 00:44:54.950 --> 00:44:57.710 of impulses separated by capital T, turned 00:44:57.710 --> 00:45:01.040 into a train of impulses separated by 2 pi over t. 00:45:01.040 --> 00:45:06.080 Here separated by capital N, turns into impulse trains 00:45:06.080 --> 00:45:08.090 separated by 2 pi over n-- 00:45:08.090 --> 00:45:10.230 same thing. 00:45:10.230 --> 00:45:13.820 So if this is 8 over here, I do 2 pi 00:45:13.820 --> 00:45:16.850 over 8, which is pi over 4. 00:45:16.850 --> 00:45:20.780 So convolving in discrete time is the same 00:45:20.780 --> 00:45:24.335 as multiplying by this impulse train, which gives me, 00:45:24.335 --> 00:45:29.410 then, a string of samples from that waveform. 00:45:29.410 --> 00:45:31.410 So the effect of going this way, which 00:45:31.410 --> 00:45:34.860 is periodic extension, which is the same as convolving in time, 00:45:34.860 --> 00:45:38.410 is sampling in frequency. 00:45:38.410 --> 00:45:40.820 OK, so the idea, then, is that that's 00:45:40.820 --> 00:45:43.860 a way of rationalizing how the Fourier series would 00:45:43.860 --> 00:45:44.360 have worked. 00:45:46.970 --> 00:45:49.400 So the Fourier series was a discrete set of numbers 00:45:49.400 --> 00:45:51.475 exactly for that reason. 00:45:51.475 --> 00:45:52.850 I started out with something that 00:45:52.850 --> 00:45:56.030 was continuous in frequency, but I sampled it 00:45:56.030 --> 00:46:02.440 because I was multiplying by an impulse train. 00:46:02.440 --> 00:46:07.257 So what I hope this did, OK, was dragged you 00:46:07.257 --> 00:46:09.340 through a couple of examples that will make things 00:46:09.340 --> 00:46:11.542 a little bit easier tomorrow. 00:46:11.542 --> 00:46:13.000 But I also hope that you understand 00:46:13.000 --> 00:46:15.490 how we went from a diagram that looks kind 00:46:15.490 --> 00:46:18.310 of complicated an arbitrary-- 00:46:18.310 --> 00:46:21.490 here we had DT on the top and CT on the bottom, 00:46:21.490 --> 00:46:25.640 periodic on the left and not periodic on the right. 00:46:25.640 --> 00:46:28.930 The transforms had all that things turn 90 degrees. 00:46:28.930 --> 00:46:33.760 We ended up with discrete on the left instead of on the top, 00:46:33.760 --> 00:46:36.790 and we ended up with periodic in the top 00:46:36.790 --> 00:46:39.260 rather than on the left. 00:46:39.260 --> 00:46:46.180 And that's entirely because of the relationships 00:46:46.180 --> 00:46:48.460 between the various Fourier transforms. 00:46:48.460 --> 00:46:49.689 So the idea is-- 00:46:49.689 --> 00:46:51.730 and you can understand all of those relationships 00:46:51.730 --> 00:46:54.880 by simply thinking about impulse trains. 00:46:54.880 --> 00:46:57.810 We take an aperiodic signal, turn it into a periodic 00:46:57.810 --> 00:47:00.190 by convolution in time, which is the same as multiplying 00:47:00.190 --> 00:47:03.570 frequency, which is the same as sampling. 00:47:03.570 --> 00:47:06.570 We sample in time by multiplying by an impulse train, which 00:47:06.570 --> 00:47:08.340 is the same as convolving in frequency, 00:47:08.340 --> 00:47:10.530 which makes it periodic. 00:47:10.530 --> 00:47:15.360 So the hope is that this looks like more than just-- 00:47:15.360 --> 00:47:18.450 1, 2, 3, 4, 5, 6, 7, 8-- eight independent equations, 00:47:18.450 --> 00:47:20.000 they're not. 00:47:20.000 --> 00:47:23.730 It's really two equations-- the transform and the inverse 00:47:23.730 --> 00:47:27.710 transform-- and they're almost the same equation, 00:47:27.710 --> 00:47:30.250 except for taking into account the special properties 00:47:30.250 --> 00:47:31.950 of the different domains. 00:47:31.950 --> 00:47:35.716 OK, have a good time, and see you tomorrow.