WEBVTT 00:00:00.030 --> 00:00:02.285 The following content is provided under a Creative 00:00:02.285 --> 00:00:03.610 Commons license. 00:00:03.610 --> 00:00:05.460 Your support will help MIT OpenCourseWare 00:00:05.460 --> 00:00:09.940 continue to offer high-quality educational resources for free. 00:00:09.940 --> 00:00:12.530 To make a donation, or to view additional materials 00:00:12.530 --> 00:00:15.840 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:15.840 --> 00:00:20.150 at ocw.mit.edu. 00:00:20.150 --> 00:00:21.840 PROFESSOR STRANG: OK, hi. 00:00:21.840 --> 00:00:25.810 So I've got homework nine for you. 00:00:25.810 --> 00:00:28.020 Ready to return at the end. 00:00:28.020 --> 00:00:31.440 Also, the department asked me to do evaluations, 00:00:31.440 --> 00:00:33.490 but that's the end of the lecture. 00:00:33.490 --> 00:00:39.250 Then, so everybody knows there's a quiz tomorrow night. 00:00:39.250 --> 00:00:45.030 And shall I just remember the four questions on the quiz? 00:00:45.030 --> 00:00:49.580 I mean, not the details but the general idea of the questions. 00:00:49.580 --> 00:00:51.110 Details OK too. 00:00:51.110 --> 00:00:53.070 Yes. 00:00:53.070 --> 00:00:58.270 Yeah, so there'll be one question on a Fourier series. 00:00:58.270 --> 00:01:03.870 And you should know the energy equality 00:01:03.870 --> 00:01:09.460 for all of these possibilities, connecting the function squared 00:01:09.460 --> 00:01:11.740 with the coefficient squared. 00:01:11.740 --> 00:01:17.630 A second one on the discrete Fourier transform. 00:01:17.630 --> 00:01:19.200 Cyclic stuff. 00:01:19.200 --> 00:01:25.060 The third one on the Fourier integral. 00:01:25.060 --> 00:01:35.490 And have a look at the applications to solving an ODE. 00:01:35.490 --> 00:01:38.920 I did one in class. 00:01:38.920 --> 00:01:42.310 The one in class was the one in the book 00:01:42.310 --> 00:01:47.800 -u''+a^2*u=f(x), so this will be. 00:01:47.800 --> 00:01:50.610 So have a look at that application. 00:01:50.610 --> 00:01:53.770 This is, of course, on minus infinity to infinity. 00:01:53.770 --> 00:02:00.310 and then a fourth question on convolution. 00:02:00.310 --> 00:02:01.000 OK. 00:02:01.000 --> 00:02:03.610 And this afternoon, of course, I'll 00:02:03.610 --> 00:02:07.950 be here to answer any questions from the homework, 00:02:07.950 --> 00:02:13.270 from any source, for these topics. 00:02:13.270 --> 00:02:15.060 Are there any questions just now, though? 00:02:15.060 --> 00:02:19.950 I'm OK to take questions. 00:02:19.950 --> 00:02:24.660 I thought I'd discussed today a topic that 00:02:24.660 --> 00:02:27.870 involves both Fourier series and Fourier integrals. 00:02:27.870 --> 00:02:32.650 It's a kind of cool connection and it's 00:02:32.650 --> 00:02:36.300 linked to the name of Claude Shannon who 00:02:36.300 --> 00:02:41.440 created information theory, who was a Bell Labs guy and then 00:02:41.440 --> 00:02:43.790 an MIT professor. 00:02:43.790 --> 00:02:49.560 So I should put his name in. 00:02:49.560 --> 00:02:50.060 Shannon. 00:02:50.060 --> 00:02:54.400 OK, so this is, yeah. 00:02:54.400 --> 00:02:57.160 You'll see. 00:02:57.160 --> 00:02:59.680 So it's not on the quiz but it gives me 00:02:59.680 --> 00:03:03.790 a chance to say something important, and at the same time 00:03:03.790 --> 00:03:07.140 review Fourier series and Fourier integrals. 00:03:07.140 --> 00:03:11.130 So let me start with the problem. 00:03:11.130 --> 00:03:17.250 The problem comes for an A to D converter. 00:03:17.250 --> 00:03:19.620 So what does that mean? 00:03:19.620 --> 00:03:22.550 That means, this A is analog. 00:03:22.550 --> 00:03:28.080 That means we have a function, A for analog. 00:03:28.080 --> 00:03:30.590 And D for digital. 00:03:30.590 --> 00:03:37.270 So we have a function, like-- So f(x), say, 00:03:37.270 --> 00:03:39.650 all the way minus infinity to infinity, 00:03:39.650 --> 00:03:43.300 so we'll be doing, that's where the Fourier 00:03:43.300 --> 00:03:45.190 integral's going to come up. 00:03:45.190 --> 00:03:47.380 So that's analog. 00:03:47.380 --> 00:03:51.210 All x, it's some curve. 00:03:51.210 --> 00:03:55.940 And people build, and you can buy, 00:03:55.940 --> 00:04:01.530 and they're sold in large quantities, something that 00:04:01.530 --> 00:04:03.900 just samples that function. 00:04:03.900 --> 00:04:05.750 Say, at the integers. 00:04:05.750 --> 00:04:11.300 So now I'll sample that function and let 00:04:11.300 --> 00:04:14.880 me take the period of the sample to be one, so that I'm 00:04:14.880 --> 00:04:20.010 going to take the values f(n). 00:04:20.010 --> 00:04:22.350 So now I've got something digital 00:04:22.350 --> 00:04:26.510 that I can work with, that I can compute with. 00:04:26.510 --> 00:04:36.370 And, so the sampling theorem-- Well, I mean, the question is-- 00:04:36.370 --> 00:04:40.450 Yeah, the sampling theorem is about this question, 00:04:40.450 --> 00:04:45.410 and it seems a crazy question, when do these numbers -- 00:04:45.410 --> 00:04:47.420 That's just a sequence of numbers. 00:04:47.420 --> 00:04:52.690 This x was all the way from minus infinity to infinity. 00:04:52.690 --> 00:04:58.520 And similarly, n is numbers all the way from minus infinity 00:04:58.520 --> 00:05:01.520 to infinity, they're just samples. 00:05:01.520 --> 00:05:06.800 When does that tell me the function? 00:05:06.800 --> 00:05:10.260 When can I learn from those samples, 00:05:10.260 --> 00:05:13.950 when do I have total information about the function? 00:05:13.950 --> 00:05:16.170 Now, you'll say impossible. 00:05:16.170 --> 00:05:16.990 Right? 00:05:16.990 --> 00:05:24.410 So suppose I draw a function f(x), OK? 00:05:24.410 --> 00:05:31.560 And I'm going to sample it at these points. 00:05:31.560 --> 00:05:37.150 All the way, so these are the numbers, these are my f(n). 00:05:37.150 --> 00:05:42.780 Sampling at equal intervals, because if we 00:05:42.780 --> 00:05:45.510 want to use Fourier ideas, equal spacing 00:05:45.510 --> 00:05:47.220 is the right thing to have. 00:05:47.220 --> 00:05:50.340 So when could I recover the function in between? 00:05:50.340 --> 00:05:52.580 Well, you'd say, never. 00:05:52.580 --> 00:05:55.640 Because how do I know what that function 00:05:55.640 --> 00:05:57.530 could be doing in between. 00:05:57.530 --> 00:06:04.640 So let me take the case when all the samples are zero. 00:06:04.640 --> 00:06:06.540 And let's think about that case. 00:06:06.540 --> 00:06:12.700 Suppose, what could the function be if all the samples, 00:06:12.700 --> 00:06:18.500 if these are all zeroes, forever. 00:06:18.500 --> 00:06:21.900 OK, well there's one leading candidate 00:06:21.900 --> 00:06:26.830 for the function, the zero function. 00:06:26.830 --> 00:06:31.370 Now, you'll see the whole point of the sampling theorem 00:06:31.370 --> 00:06:36.140 if you think about other, what other functions? 00:06:36.140 --> 00:06:37.460 Familiar functions, yeah. 00:06:37.460 --> 00:06:42.030 I mean, we could, obviously, any, all sorts of things. 00:06:42.030 --> 00:06:45.670 But since we're doing Fourier, we 00:06:45.670 --> 00:06:51.110 like to pick on the sines, cosines, the special functions, 00:06:51.110 --> 00:06:53.440 and think about those in particular. 00:06:53.440 --> 00:06:57.100 So, somebody said sines. 00:06:57.100 --> 00:07:02.850 Now, what function, so a sine function certainly, 00:07:02.850 --> 00:07:07.300 the sine function hits zero infinitely often. 00:07:07.300 --> 00:07:14.160 What frequency, so sine of what would give me the same answer? 00:07:14.160 --> 00:07:16.440 The same samples. 00:07:16.440 --> 00:07:19.930 If I put this sine function that you're going to tell me, 00:07:19.930 --> 00:07:24.450 so you're going to tell me it's sine of something 00:07:24.450 --> 00:07:27.540 will have these zero values at all 00:07:27.540 --> 00:07:32.390 the integers, at zero, one, two, minus one, minus two, and so 00:07:32.390 --> 00:07:32.940 on. 00:07:32.940 --> 00:07:36.780 So what would do the job? 00:07:36.780 --> 00:07:37.730 Sine of? 00:07:37.730 --> 00:07:45.390 Of what will hit zero. 00:07:45.390 --> 00:07:48.790 So I'm looking for a sine function, 00:07:48.790 --> 00:07:51.620 I guess I'm looking first for the function that 00:07:51.620 --> 00:07:54.130 just does that. 00:07:54.130 --> 00:07:56.880 And what is it? sin(pi*x). 00:07:56.880 --> 00:08:01.160 And now tell me some more. 00:08:01.160 --> 00:08:03.150 Tell me another function. 00:08:03.150 --> 00:08:07.360 Which will also, it won't be that graph. 00:08:07.360 --> 00:08:09.620 sin(2pi*x). 00:08:09.620 --> 00:08:14.800 And all the rest, OK? 00:08:14.800 --> 00:08:18.500 Let me just use a word that's kind of a handy word. 00:08:18.500 --> 00:08:22.300 Of course, let's put zero on the list here. 00:08:22.300 --> 00:08:24.210 OK. 00:08:24.210 --> 00:08:29.000 So this is where k, the frequency, usually appears. 00:08:29.000 --> 00:08:33.790 This is where k-- 00:08:33.790 --> 00:08:37.390 The word I want to introduce is alias. 00:08:37.390 --> 00:08:44.670 This frequency, pi, is an alias for this at frequency zero. 00:08:44.670 --> 00:08:49.110 Here's the, it's a different function 00:08:49.110 --> 00:08:51.370 but yet the samples are the same. 00:08:51.370 --> 00:08:55.200 So if you're only looking at the samples 00:08:55.200 --> 00:08:59.360 you're getting the same answer but somehow 00:08:59.360 --> 00:09:01.070 the function has a different name. 00:09:01.070 --> 00:09:05.360 So that frequency and this frequency, and all those others 00:09:05.360 --> 00:09:06.880 would be alias. 00:09:06.880 --> 00:09:11.010 Can I just write that word down, because you see it often. 00:09:11.010 --> 00:09:13.760 Alias. 00:09:13.760 --> 00:09:16.850 That means two frequencies, like pi and 2pi, 00:09:16.850 --> 00:09:20.580 and zero or whatever, that give you the same samples. 00:09:20.580 --> 00:09:27.980 OK, so now comes Shannon's question. 00:09:27.980 --> 00:09:32.810 So we have to make some assumption on the function. 00:09:32.810 --> 00:09:38.730 To knock out those possibilities. 00:09:38.730 --> 00:09:44.160 We want to know a limited class of functions. 00:09:44.160 --> 00:09:47.370 Which don't include these guys. 00:09:47.370 --> 00:09:50.290 So that within this limited class of functions, 00:09:50.290 --> 00:09:54.900 this is the only candidate and we have this possibility 00:09:54.900 --> 00:09:57.110 of doing the impossible. 00:09:57.110 --> 00:10:02.340 Of determining that if I know zeroes here, 00:10:02.340 --> 00:10:05.670 the function has to be zero everywhere. 00:10:05.670 --> 00:10:08.620 OK, now the question is what class of functions? 00:10:08.620 --> 00:10:13.140 We want to eliminate these guys, and sort of, your instinct 00:10:13.140 --> 00:10:19.260 is, you want to eliminate functions that, you know, 00:10:19.260 --> 00:10:21.770 if it's not zero then it's got to get up and back 00:10:21.770 --> 00:10:24.400 down in every thing. 00:10:24.400 --> 00:10:26.980 It could do different things in different intervals. 00:10:26.980 --> 00:10:33.160 But somehow it's got to have some of these frequencies. 00:10:33.160 --> 00:10:36.370 Pi or higher. 00:10:36.370 --> 00:10:37.810 Would have to be in there. 00:10:37.810 --> 00:10:40.450 So this is the instinct. 00:10:40.450 --> 00:10:48.870 That if I limit the frequency band, so I'm going to say f(x) 00:10:48.870 --> 00:10:53.560 is band-limited, can I introduce that word? 00:10:53.560 --> 00:10:55.740 I'll maybe take a moment just ask you 00:10:55.740 --> 00:10:59.480 if you've seen that word before. 00:10:59.480 --> 00:11:01.850 How many have seen this word, band-limited? 00:11:01.850 --> 00:11:04.410 Quite a few but not half. 00:11:04.410 --> 00:11:05.070 OK. 00:11:05.070 --> 00:11:10.600 Band-limited means the band is a band of frequencies. 00:11:10.600 --> 00:11:14.290 So the function's band-limited when 00:11:14.290 --> 00:11:19.270 its transform, this tells me how much of each frequency 00:11:19.270 --> 00:11:20.190 there is. 00:11:20.190 --> 00:11:23.420 If this is zero, in some band. 00:11:23.420 --> 00:11:32.480 In some band, let's say, all frequencies below something. 00:11:32.480 --> 00:11:36.110 And let's not even put equal in there. 00:11:36.110 --> 00:11:37.580 OK. 00:11:37.580 --> 00:11:40.190 But that's not critical. 00:11:40.190 --> 00:11:44.120 Band-limited, I have to tell you the size of the band. 00:11:44.120 --> 00:11:48.590 And the size of the band, the limiting frequency 00:11:48.590 --> 00:11:51.130 is this famous Nyquist frequency, 00:11:51.130 --> 00:11:53.270 so Nyquist is a guy's name. 00:11:53.270 --> 00:11:57.420 And the Nyquist frequency in our problem here is pi. 00:11:57.420 --> 00:12:00.090 This is the Nyquist frequency. 00:12:00.090 --> 00:12:08.780 If we let that frequency, that's the borderline frequency. 00:12:08.780 --> 00:12:12.510 And there would be a similar Nyquist sampling rate. 00:12:12.510 --> 00:12:16.680 So Nyquist is the guy who studied 00:12:16.680 --> 00:12:20.220 the sort of borderline case. 00:12:20.220 --> 00:12:26.940 So the point is that if our, say, band-limited by pi, 00:12:26.940 --> 00:12:29.710 I have to tell you, so band-limited means there's 00:12:29.710 --> 00:12:31.550 some limit on the band. 00:12:31.550 --> 00:12:37.030 And our interest is when that limit is the Nyquist frequency. 00:12:37.030 --> 00:12:43.310 The one we don't want to allow, so we-- This is the point. 00:12:43.310 --> 00:12:45.320 So this will be the idea. 00:12:45.320 --> 00:12:52.570 That if we take this class of function, 00:12:52.570 --> 00:12:56.600 that band-limited-- Those are called band-limited functions, 00:12:56.600 --> 00:12:58.770 and they're band-limited specifically 00:12:58.770 --> 00:13:01.940 by the Nyquist limit. 00:13:01.940 --> 00:13:05.050 If we take those, then the idea is 00:13:05.050 --> 00:13:13.990 that then we can reconstruct from the samples. 00:13:13.990 --> 00:13:17.980 Because the only function that has zero samples in that 00:13:17.980 --> 00:13:20.560 class is the zero function. 00:13:20.560 --> 00:13:22.490 You see that class has knocked out, 00:13:22.490 --> 00:13:26.130 is not allowing these guys. 00:13:26.130 --> 00:13:29.270 Of course, haven't proved anything yet. 00:13:29.270 --> 00:13:31.920 And I haven't shown how to reconstruct. 00:13:31.920 --> 00:13:35.790 Well, of course, we quickly reconstructed the zero function 00:13:35.790 --> 00:13:41.200 out of those zeroes, but now let me take another obviously 00:13:41.200 --> 00:13:43.570 important possible sample. 00:13:43.570 --> 00:13:48.560 Suppose I get zero samples except at that point, where 00:13:48.560 --> 00:13:51.910 it's one. 00:13:51.910 --> 00:14:00.270 OK, now the question is what function, f(x)-- 00:14:00.270 --> 00:14:04.490 Can I fill in, in between zero, zero, zero, zero, one, zero, 00:14:04.490 --> 00:14:11.010 zero, zero, can I fill in exactly one function that comes 00:14:11.010 --> 00:14:14.040 from this class? 00:14:14.040 --> 00:14:19.030 So that I have now the answer for this highly important 00:14:19.030 --> 00:14:19.930 sample? 00:14:19.930 --> 00:14:23.310 The sample that's all zeroes except for the delta sample, 00:14:23.310 --> 00:14:24.420 you could say. 00:14:24.420 --> 00:14:27.130 OK, so I'm looking for the function now 00:14:27.130 --> 00:14:30.760 which is one at that point. 00:14:30.760 --> 00:14:32.780 And zero at the others. 00:14:32.780 --> 00:14:36.150 So here's a key function. 00:14:36.150 --> 00:14:39.370 And I'll show you what it is. 00:14:39.370 --> 00:14:43.980 So the function, this function that I'm going to mention, 00:14:43.980 --> 00:14:49.860 will get down here, it'll oscillate, it'll go forever. 00:14:49.860 --> 00:14:52.450 It's not like a spline. 00:14:52.450 --> 00:14:58.030 Splines made it to zero and stayed there, right? 00:14:58.030 --> 00:15:00.140 The cubic spline, for example. 00:15:00.140 --> 00:15:08.480 OK, but I guess, yeah, that somehow that function, 00:15:08.480 --> 00:15:11.160 we're not in that league. 00:15:11.160 --> 00:15:13.650 We're in this band-limited league. 00:15:13.650 --> 00:15:20.080 In a way you could say that, I mean, 00:15:20.080 --> 00:15:24.140 what's the key connection between dropoff of the Fourier 00:15:24.140 --> 00:15:25.280 transform? 00:15:25.280 --> 00:15:28.790 So if the Fourier transform drops off fast, 00:15:28.790 --> 00:15:31.640 what does that tell me about the function? 00:15:31.640 --> 00:15:32.850 It's smooth, thanks. 00:15:32.850 --> 00:15:34.260 That's exactly the right word. 00:15:34.260 --> 00:15:36.280 If the Fourier transform drops off fast, 00:15:36.280 --> 00:15:38.080 the function is smooth. 00:15:38.080 --> 00:15:42.060 OK, this is really an extreme case. 00:15:42.060 --> 00:15:46.470 That it's dropped off totally. 00:15:46.470 --> 00:15:50.770 You know, it's not just decay rate, it's just zonk, out. 00:15:50.770 --> 00:15:53.870 Beyond this band of frequencies. 00:15:53.870 --> 00:15:57.700 And so that gives, you could say, sort 00:15:57.700 --> 00:15:59.620 of a hyper-smooth function. 00:15:59.620 --> 00:16:01.890 I mean, so smooth that you know everything 00:16:01.890 --> 00:16:03.130 by knowing the sample. 00:16:03.130 --> 00:16:07.710 OK, now I'm ready to write down the key function, 00:16:07.710 --> 00:16:11.420 a famous function that has those samples. 00:16:11.420 --> 00:16:17.960 And that function is sin(pi*x)/(pi*x). 00:16:21.820 --> 00:16:24.450 I don't know if you ever thought about this function, 00:16:24.450 --> 00:16:25.640 and it has a name. 00:16:25.640 --> 00:16:27.700 Do you know its name? 00:16:27.700 --> 00:16:28.220 Sinc. 00:16:28.220 --> 00:16:29.980 It's the sinc function. 00:16:29.980 --> 00:16:32.060 Which is a little-- you know, the name's 00:16:32.060 --> 00:16:34.390 a little unfortunate. 00:16:34.390 --> 00:16:37.680 Mainly because you know, you're using those same letters 00:16:37.680 --> 00:16:41.550 S I N but it's a c that turns it, 00:16:41.550 --> 00:16:46.120 that gives it-- So this is called the sinc function. 00:16:46.120 --> 00:16:47.660 sinc(x). 00:16:47.660 --> 00:16:50.530 But the main thing is its formula. 00:16:50.530 --> 00:16:56.410 OK, well everybody sees that at x=1, the sin(pi) is zero, 00:16:56.410 --> 00:17:00.600 at x=2, the sin(pi) is zero, all these ones we've seen already. 00:17:00.600 --> 00:17:03.230 And now what happens at equals zero? 00:17:03.230 --> 00:17:07.460 Do you recognize that this function, as x goes to zero, 00:17:07.460 --> 00:17:10.230 is a perfectly good function? 00:17:10.230 --> 00:17:14.750 I mean, it becomes 0/0 at x=0. 00:17:14.750 --> 00:17:17.860 But the limit, we know to be one. 00:17:17.860 --> 00:17:18.360 Right? 00:17:18.360 --> 00:17:25.550 This sin(theta)/theta is one as theta approaches zero, right? 00:17:25.550 --> 00:17:28.630 So that does have that correct sample. 00:17:28.630 --> 00:17:32.130 And now what's the, I claim that that 00:17:32.130 --> 00:17:33.690 is a band-limited function. 00:17:33.690 --> 00:17:40.650 And you'll see that that function pushes the limit. 00:17:40.650 --> 00:17:46.620 It's right-- Nyquist barely lets it in. 00:17:46.620 --> 00:17:48.190 Now here's a calculation. 00:17:48.190 --> 00:17:50.470 So this is our practice. 00:17:50.470 --> 00:17:59.360 What is f hat of k for that function? 00:17:59.360 --> 00:18:01.450 Now, let me think how to do this one. 00:18:01.450 --> 00:18:07.720 So just to understand this better, 00:18:07.720 --> 00:18:10.390 I want to see that that is a band-limited function 00:18:10.390 --> 00:18:14.130 and what is its Fourier transform. 00:18:14.130 --> 00:18:18.980 OK, now once again here we have a function where if I want to 00:18:18.980 --> 00:18:25.220 do-- How best to do this? 00:18:25.220 --> 00:18:33.010 You could say well, just do it. 00:18:33.010 --> 00:18:35.830 As I would say on the quiz, just go ahead and do it. 00:18:35.830 --> 00:18:41.470 But you'll see I'm going to have a problem, I think. 00:18:41.470 --> 00:18:44.040 But this gives us a chance to remember the formula. 00:18:44.040 --> 00:18:47.450 So what's the formula for the Fourier integral transform 00:18:47.450 --> 00:18:48.680 of this particular? 00:18:48.680 --> 00:18:51.640 So my function is the sinc function, 00:18:51.640 --> 00:18:55.400 sin(pi*x) over sin(pi*x). 00:18:55.400 --> 00:19:03.000 So how do I get its, I do a what here? e^(-ikx), 00:19:03.000 --> 00:19:05.650 and am I doing dx or dk? 00:19:05.650 --> 00:19:07.290 dx, right? 00:19:07.290 --> 00:19:10.790 And I'm going from minus infinity to infinity. 00:19:10.790 --> 00:19:14.640 And am I, do I have a 2pi? 00:19:14.640 --> 00:19:16.160 Yes or no? 00:19:16.160 --> 00:19:17.850 Who knows, anyway? 00:19:17.850 --> 00:19:21.150 Right. 00:19:21.150 --> 00:19:22.960 In the book I didn't? 00:19:22.960 --> 00:19:25.750 OK. 00:19:25.750 --> 00:19:33.100 Now, well, I don't know the answer. 00:19:33.100 --> 00:19:37.640 But so let's, it's much better to start with the answer, 00:19:37.640 --> 00:19:42.350 right, and check that-- So let me 00:19:42.350 --> 00:19:44.470 say what I think the answer is. 00:19:44.470 --> 00:19:51.340 I think the answer is, it's a function that's 00:19:51.340 --> 00:19:59.060 exactly as I say, it pushes the limit from, this is k. 00:19:59.060 --> 00:20:03.770 It's the square wave, it's zero, the height is one. 00:20:03.770 --> 00:20:08.010 It's the function that's zero all the way here, 00:20:08.010 --> 00:20:09.660 all the way there. 00:20:09.660 --> 00:20:13.740 I think that that's the Fourier transform of that function. 00:20:13.740 --> 00:20:20.370 And just before we check it, see how is Nyquist got really 00:20:20.370 --> 00:20:22.550 pushed up to the wall, right? 00:20:22.550 --> 00:20:26.590 Because the frequency is non-zero right, 00:20:26.590 --> 00:20:30.170 all the way through pi. 00:20:30.170 --> 00:20:36.020 But pi is just one point there. 00:20:36.020 --> 00:20:41.460 And anyway, I think we won't get into philosophical discussion 00:20:41.460 --> 00:20:45.460 about whether, you know, is that limited to pi? 00:20:45.460 --> 00:20:49.440 I don't know whether to put, you saw me chicken out here. 00:20:49.440 --> 00:20:52.620 I didn't know whether to put less or equal or not, 00:20:52.620 --> 00:20:55.640 and I still don't. 00:20:55.640 --> 00:21:00.770 But this is making the point that that's the key frequency. 00:21:00.770 --> 00:21:03.260 So this particular function, what I'm saying 00:21:03.260 --> 00:21:07.080 is this particular function has all the frequencies 00:21:07.080 --> 00:21:12.480 in equal amounts over a band, and nothing outside that band. 00:21:12.480 --> 00:21:14.260 And that's the Nyquist band. 00:21:14.260 --> 00:21:18.250 OK, now why is this the correct answer here? 00:21:18.250 --> 00:21:25.080 I guess the smart way would be, this is a good function, 00:21:25.080 --> 00:21:26.140 easy function. 00:21:26.140 --> 00:21:32.270 So let's take the transform in the other direction. 00:21:32.270 --> 00:21:34.710 Start from here and get to here. 00:21:34.710 --> 00:21:35.210 Right? 00:21:35.210 --> 00:21:36.500 That would be convincing. 00:21:36.500 --> 00:21:41.470 Because we do know that that pair of formulas 00:21:41.470 --> 00:21:46.220 for f connected to f hat connected to f, 00:21:46.220 --> 00:21:47.620 they go together. 00:21:47.620 --> 00:21:54.750 So if I can show that I go from here, that that Fourier 00:21:54.750 --> 00:21:56.710 integral takes me from here to there, 00:21:56.710 --> 00:21:59.080 then this guy will take me back. 00:21:59.080 --> 00:22:00.550 So let me just do that. 00:22:00.550 --> 00:22:05.120 Because that's a very very important one that you 00:22:05.120 --> 00:22:07.470 should be prepared for. 00:22:07.470 --> 00:22:07.970 Right. 00:22:07.970 --> 00:22:10.580 So now what do I want to do? 00:22:10.580 --> 00:22:23.780 Here's my function of k, and I'm hoping that I recall-- Now, 00:22:23.780 --> 00:22:27.240 what do I do when I want to do the transform 00:22:27.240 --> 00:22:29.010 in the opposite direction? 00:22:29.010 --> 00:22:37.120 It'll be an e^(+ikx), right? d what? dk, now. 00:22:37.120 --> 00:22:39.700 From k equal minus infinity to to infinity. 00:22:39.700 --> 00:22:43.800 And now I think I do put in the 2pi, is that right? 00:22:43.800 --> 00:22:50.740 And the question is, does that bring back the sinc function? 00:22:50.740 --> 00:22:55.640 If it does, then this was OK in the other direction. 00:22:55.640 --> 00:22:57.790 If the transform's correct in one direction then 00:22:57.790 --> 00:22:59.430 the inverse transform will be correct. 00:22:59.430 --> 00:23:03.370 So I just plan to do this integral. f 00:23:03.370 --> 00:23:07.380 hat of k, of course, is an easy integral now. f hat of k 00:23:07.380 --> 00:23:13.320 is one over, between minus pi and pi, 00:23:13.320 --> 00:23:17.040 so I only have to do over that range where f hat of k 00:23:17.040 --> 00:23:20.480 is just a one. 00:23:20.480 --> 00:23:23.660 And now that's an integral we can certainly do. 00:23:23.660 --> 00:23:30.490 So I have 1/(2pi), integrating e^(ikx) will give me e^(ikx) 00:23:30.490 --> 00:23:33.100 over ix. 00:23:33.100 --> 00:23:35.680 Now, remember I'm integrating dk. 00:23:35.680 --> 00:23:36.510 Oh, look. 00:23:36.510 --> 00:23:40.690 See, we're showing this x now in the denominator. 00:23:40.690 --> 00:23:44.340 That we're hoping for. 00:23:44.340 --> 00:23:50.360 And now I have to do that between k is minus pi and pi. 00:23:50.360 --> 00:23:56.550 So this is like, so I get 1/(2pi), 00:23:56.550 --> 00:24:06.390 e^(i*pi*k) minus e^(-i*pi*x), right? 00:24:06.390 --> 00:24:07.790 Over the ix. 00:24:10.350 --> 00:24:13.560 OK so far? 00:24:13.560 --> 00:24:17.710 I was doing a k integral and I get an x answer. 00:24:17.710 --> 00:24:21.350 And I want to be sure that this x answer is the x answer I 00:24:21.350 --> 00:24:23.890 want, it's the sinc function. 00:24:23.890 --> 00:24:24.890 OK, it is. 00:24:24.890 --> 00:24:25.900 Right? 00:24:25.900 --> 00:24:31.290 I recognize the sine, e^(i*theta)-e^(-i*theta), 00:24:31.290 --> 00:24:36.530 divided by two, I guess. 00:24:36.530 --> 00:24:38.260 Is the sine, right? 00:24:38.260 --> 00:24:40.700 So I have 1/(2pi). 00:24:40.700 --> 00:24:46.740 And here is ix-- Well, no, the i is part of that sine. 00:24:46.740 --> 00:24:54.930 So I'm just using the fact that e^(i*theta)-e^(-i*theta) is, 00:24:54.930 --> 00:24:59.650 this is cosine plus i sine, subtract cosine. 00:24:59.650 --> 00:25:05.630 But subtract minus i sine, so that will be 2i*sin(theta), 00:25:05.630 --> 00:25:07.080 right? 00:25:07.080 --> 00:25:11.530 We all know, and now theta is pi*x here. 00:25:11.530 --> 00:25:17.780 So I have two-- let me keep the i there, and 2i*sin(pi*x). 00:25:17.780 --> 00:25:20.780 You see it works. 00:25:20.780 --> 00:25:25.170 Just using this, replacing this by the sine, 00:25:25.170 --> 00:25:27.440 the two cancels the two. 00:25:27.440 --> 00:25:32.060 The i cancels the i, and I have sin(pi*x) over pi*x, 00:25:32.060 --> 00:25:34.010 that's the sinc function. 00:25:34.010 --> 00:25:42.380 OK, so that's the function we've now checked. 00:25:42.380 --> 00:25:44.860 We've checked two things about this function. 00:25:44.860 --> 00:25:48.190 It has the right samples, zero, zero, zero, zero, 00:25:48.190 --> 00:25:51.790 one at x=0 and then back to zero. 00:25:51.790 --> 00:25:55.650 It is band-limited, so it's the guy, 00:25:55.650 --> 00:26:00.570 if this was my sample, if this was my f(n), zero, zero, zero, 00:26:00.570 --> 00:26:05.820 one, zero, zero, zero, then I've got it. 00:26:05.820 --> 00:26:10.950 It's the right function. 00:26:10.950 --> 00:26:15.840 OK, we can create Shannon's sampling formula. 00:26:15.840 --> 00:26:20.830 Shannon's sampling formula gives me the f(x) for any f(n). 00:26:23.380 --> 00:26:25.370 Maybe you can spot that, now. 00:26:25.370 --> 00:26:28.200 So this is going to use the shift invariance. 00:26:28.200 --> 00:26:34.120 Oh yeah, let's-- Tell me what the, let's take one step here. 00:26:34.120 --> 00:26:39.300 Suppose my f(n)'s were zero, zero, zero, zero, and a one 00:26:39.300 --> 00:26:40.780 there. 00:26:40.780 --> 00:26:46.720 So suppose-- This is, for the exam too, this idea of shifting 00:26:46.720 --> 00:26:49.890 is simple and basic. 00:26:49.890 --> 00:26:52.270 And it's a great thing to be able to do. 00:26:52.270 --> 00:26:56.050 So this wouldn't be the right answer. 00:26:56.050 --> 00:26:59.230 That function produced the one at zero. 00:26:59.230 --> 00:27:04.500 Tell me what function, copying this idea, 00:27:04.500 --> 00:27:11.570 will produce all zeroes except for a one at this point. 00:27:11.570 --> 00:27:17.400 So I'll put this question over here. 00:27:17.400 --> 00:27:24.220 Suppose I'm all zeroes at that point but at this point I'm up 00:27:24.220 --> 00:27:26.400 and then I go back to zeroes. 00:27:26.400 --> 00:27:33.440 What function is giving me that? 00:27:33.440 --> 00:27:36.820 You see it does decay because of the x in the denominator, 00:27:36.820 --> 00:27:40.580 it kind of goes to zero but not very fast. 00:27:40.580 --> 00:27:45.370 OK, what's that function? 00:27:45.370 --> 00:27:48.810 I replace x by x-1, right. 00:27:48.810 --> 00:27:53.940 So the function here is sin(pi(x-1)), 00:27:53.940 --> 00:27:56.360 divided by pi(x-1). 00:27:56.360 --> 00:27:59.230 I just change the x to x-1. 00:27:59.230 --> 00:28:05.940 Now again, at x=0, this is sin(pi), sin(-pi), 00:28:05.940 --> 00:28:10.500 it's safely zero, but at x=1 that's now the point where 00:28:10.500 --> 00:28:13.660 I'm getting 0/0. 00:28:13.660 --> 00:28:18.080 And the numbers are right to give me the exact answer, one. 00:28:18.080 --> 00:28:24.110 OK, so now we see what to do if the sampling turned out 00:28:24.110 --> 00:28:25.790 to give us this answer. 00:28:25.790 --> 00:28:29.440 And now can you tell me the whole formula? 00:28:29.440 --> 00:28:31.660 Can you tell me the whole formula, 00:28:31.660 --> 00:28:35.210 so now I'm ready for Shannon's sampling theorem. 00:28:35.210 --> 00:28:41.200 Is that f(x), if f(x) is band-limited, 00:28:41.200 --> 00:28:45.040 then I can tell you what it is at all x. 00:28:45.040 --> 00:28:48.730 So I'm going back to the beginning of this lecture. 00:28:48.730 --> 00:28:52.880 It's a miracle that this is possible. 00:28:52.880 --> 00:28:57.690 That we can write down a formula for f(x) at all x, 00:28:57.690 --> 00:29:01.530 only using f(n)'s. 00:29:01.530 --> 00:29:05.550 OK, now it's going to be a sum. 00:29:05.550 --> 00:29:09.250 From n equal minus infinity to infinity, 00:29:09.250 --> 00:29:12.970 because I'm going to use all the f(n)'s to produce the f(x). 00:29:12.970 --> 00:29:17.910 And now what do I put in there? 00:29:17.910 --> 00:29:23.500 So I want my formula to be correct. 00:29:23.500 --> 00:29:28.810 I want my formula to be correct in this case, 00:29:28.810 --> 00:29:34.130 so that if all the f's were zero except for the middle one, 00:29:34.130 --> 00:29:37.200 then I want to put sin(pi*x)/(pi*x) in there. 00:29:37.200 --> 00:29:39.120 The sinc function. 00:29:39.120 --> 00:29:41.500 And I also want to get this one right. 00:29:41.500 --> 00:29:45.650 If all the f's are zero, so there'll only be one term. 00:29:45.650 --> 00:29:48.920 If this is the term, I want that to show up. 00:29:48.920 --> 00:29:53.710 OK, what do I-- Yeah, you can tell me. 00:29:53.710 --> 00:29:57.740 Suppose this is hitting at n. 00:29:57.740 --> 00:30:00.640 We'll just fix this and then you'll see it. 00:30:00.640 --> 00:30:06.460 Suppose that all the others, n, n-1, n+1, all those, it's zero, 00:30:06.460 --> 00:30:09.630 but at x=n, it's one. 00:30:09.630 --> 00:30:11.730 Now what should I have chosen? 00:30:11.730 --> 00:30:14.420 What's the correct-- I'm just going 00:30:14.420 --> 00:30:19.190 to make it easy for all of us to, yes. 00:30:19.190 --> 00:30:25.460 What's the good sinc function which peaks at a point n? 00:30:25.460 --> 00:30:27.790 Again, I'm just shifting it over. 00:30:27.790 --> 00:30:30.810 So what do I do? 00:30:30.810 --> 00:30:34.920 Put in, what do I write here? n. 00:30:34.920 --> 00:30:36.500 I shift the whole thing by n. 00:30:36.500 --> 00:30:39.660 So that's the right answer when this hits at n. 00:30:39.660 --> 00:30:42.150 So now maybe you see that this is 00:30:42.150 --> 00:30:51.970 going to be the right answer for all of them. 00:30:51.970 --> 00:30:56.130 You see that, we're using linearity and shift invariance. 00:30:56.130 --> 00:31:00.240 The shift invariance is telling us this answer 00:31:00.240 --> 00:31:03.620 for wherever the one hits. 00:31:03.620 --> 00:31:05.350 That's what we need. 00:31:05.350 --> 00:31:08.740 And then by linearity, I put together 00:31:08.740 --> 00:31:12.200 whatever the f is at that point, that 00:31:12.200 --> 00:31:14.320 would just amplify the sinc. 00:31:14.320 --> 00:31:17.800 And then I have to put them in for all the other values. 00:31:17.800 --> 00:31:20.410 That's the Shannon formula. 00:31:20.410 --> 00:31:25.940 That's the Shannon formula, and this function is band-limited, 00:31:25.940 --> 00:31:27.160 let's see. 00:31:27.160 --> 00:31:29.630 What's the-- Oh, yeah. 00:31:29.630 --> 00:31:34.120 What's the, do you see that this one, 00:31:34.120 --> 00:31:36.180 that this guy is band-limited? 00:31:36.180 --> 00:31:37.970 We checked, right? 00:31:37.970 --> 00:31:42.640 We checked that this one, sin(pi*x)/(pi*x), 00:31:42.640 --> 00:31:47.190 that was band-limited because we actually found the band. 00:31:47.190 --> 00:31:50.510 Now, that just gives us another chance to think. 00:31:50.510 --> 00:31:51.560 Allowed. 00:31:51.560 --> 00:31:55.280 What's the Fourier transform of this guy? 00:31:55.280 --> 00:31:58.560 My claim is that it's also in this band. 00:31:58.560 --> 00:32:04.200 Non-zero only in the band, and zero outside the Nyquist band. 00:32:04.200 --> 00:32:08.440 What is the transform of that? 00:32:08.440 --> 00:32:11.360 What happens if you shift a function, what 00:32:11.360 --> 00:32:14.280 happens to its transform? 00:32:14.280 --> 00:32:23.670 So that's one of the key rules that makes Fourier so special. 00:32:23.670 --> 00:32:28.300 If I took this sine, let me write this guy again. 00:32:28.300 --> 00:32:31.710 This was the un-shifted one. 00:32:31.710 --> 00:32:35.180 That connected to the, what am I going 00:32:35.180 --> 00:32:40.440 to call that, the box function. 00:32:40.440 --> 00:32:42.950 The box function, the square wave. 00:32:42.950 --> 00:32:45.380 Well, box is good. 00:32:45.380 --> 00:32:49.040 Now, what if I shift the function? 00:32:49.040 --> 00:32:50.670 If I shift a function, what happens 00:32:50.670 --> 00:32:53.120 to its Fourier transform? 00:32:53.120 --> 00:32:55.300 Anybody remember? 00:32:55.300 --> 00:33:02.090 You multiply it by, so if I shift the function, 00:33:02.090 --> 00:33:04.410 I just multiply this box function, 00:33:04.410 --> 00:33:07.790 this is a box function in the k, times something, 00:33:07.790 --> 00:33:19.140 e to the i shift, and the shift was one, right? 00:33:19.140 --> 00:33:24.210 Is it just e to the ik, d being the shift distance. 00:33:24.210 --> 00:33:26.340 Oh, the shift distance was n. 00:33:26.340 --> 00:33:27.970 Right. 00:33:27.970 --> 00:33:30.430 And possibly minus, who knows. 00:33:30.430 --> 00:33:33.500 OK, but what's the point here? 00:33:33.500 --> 00:33:38.100 The point is that it's still zero outside the box. 00:33:38.100 --> 00:33:43.700 Inside the box, instead of being one, it's this complex guy. 00:33:43.700 --> 00:33:45.660 But no change. 00:33:45.660 --> 00:33:47.810 It's still zero outside the box. 00:33:47.810 --> 00:33:49.370 It's still band-limited. 00:33:49.370 --> 00:33:54.010 So this is the transform of this guy. 00:33:54.010 --> 00:33:56.820 And then the transform of this combination 00:33:56.820 --> 00:34:00.960 would be still in the box, multiplied 00:34:00.960 --> 00:34:05.130 by some messy expression. 00:34:05.130 --> 00:34:07.000 So what was I doing there? 00:34:07.000 --> 00:34:10.000 I was just checking that, sure enough, this guy 00:34:10.000 --> 00:34:11.420 is band-limited. 00:34:11.420 --> 00:34:20.770 And it's band-limited, it gives us the right f(n)'s, of course. 00:34:20.770 --> 00:34:29.180 Everybody sees that at x=n, let's just have a look now. 00:34:29.180 --> 00:34:30.900 We've got this great formula. 00:34:30.900 --> 00:34:33.330 Plug in x=n. 00:34:33.330 --> 00:34:37.660 What happens when you plug in at one of the samples, 00:34:37.660 --> 00:34:43.710 you look to see what this A to D converter produced at time n, 00:34:43.710 --> 00:34:46.210 and let's just see. 00:34:46.210 --> 00:34:49.200 So at x=n, the left side is f(n). 00:34:49.200 --> 00:34:52.130 Why is the right side f of that n? 00:34:52.130 --> 00:34:53.440 That particular n? 00:34:53.440 --> 00:34:59.070 Maybe I should give a specific letter to that n. 00:34:59.070 --> 00:35:04.840 So at that particular sample, this left side is f(N), 00:35:04.840 --> 00:35:06.940 and I hope that the right side gives me 00:35:06.940 --> 00:35:10.670 f at that capital N. That particular one. 00:35:10.670 --> 00:35:12.400 Why does it? 00:35:12.400 --> 00:35:14.510 You're all seeing that. 00:35:14.510 --> 00:35:19.170 At x equal capital N, these guys are all zero, 00:35:19.170 --> 00:35:21.920 except for one of them. 00:35:21.920 --> 00:35:27.700 Except for the one when little n and capital N are the same. 00:35:27.700 --> 00:35:29.880 Then that becomes the one. 00:35:29.880 --> 00:35:34.440 And I'm getting f at capital N. So it will give me that, 00:35:34.440 --> 00:35:39.030 for the n=1, the one term, yeah. 00:35:39.030 --> 00:35:41.900 I don't know if it was necessary to say that. 00:35:41.900 --> 00:35:48.390 You've got the idea of the sampling formula. 00:35:48.390 --> 00:35:51.770 I could say more about the sampling, 00:35:51.770 --> 00:36:03.570 just to realize that the technology, communication 00:36:03.570 --> 00:36:07.340 theory is always trying to, like, 00:36:07.340 --> 00:36:10.480 to have a greater bandwidth. 00:36:10.480 --> 00:36:12.510 You always want a greater bandwidth. 00:36:12.510 --> 00:36:17.220 But if the bandwidth, which is this, is increased, 00:36:17.220 --> 00:36:19.240 well by the way, what does happen? 00:36:19.240 --> 00:36:30.190 Suppose it's band-limited by pi, oh, by pi/T. 00:36:30.190 --> 00:36:35.610 Let's just, I normalize things to choose samples 00:36:35.610 --> 00:36:37.400 every integer. 00:36:37.400 --> 00:36:39.380 Zero, one, two, three. 00:36:39.380 --> 00:36:43.290 And that turned out that the Nyquist frequency was pi. 00:36:43.290 --> 00:36:50.260 Now, what sampling rate would correspond 00:36:50.260 --> 00:36:54.980 to this band, which could be-- Well, let 00:36:54.980 --> 00:36:56.860 me just say what it is. 00:36:56.860 --> 00:36:59.730 That would be the Nyquist frequency 00:36:59.730 --> 00:37:10.710 for sampling every T. Instead of a sampling interval of one, 00:37:10.710 --> 00:37:17.920 if I sample every T, 2T, 3T, -T, my sampling rate is T, 00:37:17.920 --> 00:37:21.870 so if T is small, I'm sampling much more. 00:37:21.870 --> 00:37:24.910 Suppose T is 1/4. 00:37:24.910 --> 00:37:29.430 If T is 1/4, then I'm doing four samplings. 00:37:29.430 --> 00:37:33.040 I'm taking four samples, I'm paying more for this A 00:37:33.040 --> 00:37:35.720 to D converter, because it's taking four samples where 00:37:35.720 --> 00:37:39.780 previously it took one. 00:37:39.780 --> 00:37:41.420 How do I get paid back? 00:37:41.420 --> 00:37:43.240 What's the reward? 00:37:43.240 --> 00:37:51.850 The reward is if T is 1/4, then the Nyquist limit is 4pi. 00:37:51.850 --> 00:37:55.390 I can get a broader band of signals 00:37:55.390 --> 00:37:58.390 by sampling them more often. 00:37:58.390 --> 00:38:00.490 Let me just say that again, because that's 00:38:00.490 --> 00:38:03.070 the fundamental idea behind it. 00:38:03.070 --> 00:38:08.350 If I sample more often, say, so fast sampling would be small-- 00:38:08.350 --> 00:38:20.850 Fast samples would be small t and then a higher Nyquist. 00:38:20.850 --> 00:38:23.140 A higher band limit. 00:38:23.140 --> 00:38:24.390 More functions allowed. 00:38:24.390 --> 00:38:29.270 If I sample more often I'm able to catch on to more functions. 00:38:29.270 --> 00:38:33.190 If I sample-- And that's what, I mean, 00:38:33.190 --> 00:38:37.460 now, communications want wide bands. 00:38:37.460 --> 00:38:41.280 And this is where they get limited. 00:38:41.280 --> 00:38:46.060 I mean, this is, you could say, the fundamental, 00:38:46.060 --> 00:38:48.830 I don't know whether to say physical limit, sort 00:38:48.830 --> 00:38:53.610 of maybe Fourier limit on sampling theory. 00:38:53.610 --> 00:38:56.810 Is exactly this Nyquist frequency. 00:38:56.810 --> 00:39:00.160 OK, questions or discussion about that. 00:39:00.160 --> 00:39:04.110 OK. 00:39:04.110 --> 00:39:07.350 So that's an example that allowed 00:39:07.350 --> 00:39:13.110 us to do a lot of things. 00:39:13.110 --> 00:39:16.380 I did want to ask for your help doing these evaluations. 00:39:16.380 --> 00:39:19.840 Let me say what I'm going to do this afternoon. 00:39:19.840 --> 00:39:24.260 I'm going to answer all the questions I can, 00:39:24.260 --> 00:39:27.180 and I planned, when there is a pause, 00:39:27.180 --> 00:39:34.990 and nobody else asks, I plan to compute the Fourier integral 00:39:34.990 --> 00:39:38.580 and Fourier series, say, Fourier series, 00:39:38.580 --> 00:39:48.510 for a function that has, it's going to be like the one today 00:39:48.510 --> 00:39:53.820 except this is going to have a height of 1/h, 00:39:53.820 --> 00:39:56.490 and a width of h. 00:39:56.490 --> 00:40:03.630 So that's, in case you're not able to be here this afternoon, 00:40:03.630 --> 00:40:06.520 I thought I'd just say in advance what calculations 00:40:06.520 --> 00:40:07.740 I thought I would do. 00:40:07.740 --> 00:40:11.570 So there's a particular function f(x), 00:40:11.570 --> 00:40:13.880 it happens to be an even function. 00:40:13.880 --> 00:40:16.520 We'll compute its Fourier coefficients, in 00:40:16.520 --> 00:40:20.870 and we'll let h go to zero. 00:40:20.870 --> 00:40:21.930 To see what happens. 00:40:21.930 --> 00:40:23.540 It's just a good example that you 00:40:23.540 --> 00:40:27.180 may have seen on older exams. 00:40:27.180 --> 00:40:32.740 OK, well can I just say a personal word 00:40:32.740 --> 00:40:36.210 before I pass out-- So evaluations, 00:40:36.210 --> 00:40:39.640 if you're willing to help, and just leave them on the table, 00:40:39.640 --> 00:40:41.930 would be much appreciated. 00:40:41.930 --> 00:40:45.630 I'll stretch out the homeworks. 00:40:45.630 --> 00:40:48.570 I just want to say I've enjoyed teaching you guys. 00:40:48.570 --> 00:40:49.960 Very much. 00:40:49.960 --> 00:40:52.730 Thank you all, and-- Thanks.