WEBVTT 00:00:00.040 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:03.880 Commons license. 00:00:03.880 --> 00:00:06.920 Your support will help MIT OpenCourseWare continue to 00:00:06.920 --> 00:00:10.570 offer high-quality educational resources for free. 00:00:10.570 --> 00:00:13.470 To make a donation or view additional materials from 00:00:13.470 --> 00:00:17.400 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:17.400 --> 00:00:18.650 ocw.mit.edu. 00:00:20.524 --> 00:00:55.940 [MUSIC PLAYING] 00:00:55.940 --> 00:00:58.190 PROFESSOR: We concluded the last lecture with the 00:00:58.190 --> 00:00:59.960 statement of the sampling theorem. 00:00:59.960 --> 00:01:05.090 And just as a quick reminder, the sampling theorem said that 00:01:05.090 --> 00:01:10.170 if we have a continuous-time signal and we have equally 00:01:10.170 --> 00:01:15.460 spaced samples of that signal, sampled at a sampling period, 00:01:15.460 --> 00:01:20.860 which I indicate is capital T and if x of t is 00:01:20.860 --> 00:01:25.810 band-limited-- in other words, the Fourier transform is zero 00:01:25.810 --> 00:01:29.750 outside some band where omega sub m is the highest 00:01:29.750 --> 00:01:35.480 frequency-- then under the condition that the sampling 00:01:35.480 --> 00:01:39.010 frequency, which is 2 pi divided by the period, is 00:01:39.010 --> 00:01:42.770 greater than twice the highest frequency. 00:01:42.770 --> 00:01:47.520 The original signal is uniquely recoverable from the 00:01:47.520 --> 00:01:50.350 set of samples. 00:01:50.350 --> 00:01:54.990 And the sampling theorem essentially was derived by 00:01:54.990 --> 00:02:00.100 observing or using the notion that sampling could be done by 00:02:00.100 --> 00:02:03.830 multiplication or modulation with an impulse train. 00:02:03.830 --> 00:02:07.900 And the sampling theorem developed by examining the 00:02:07.900 --> 00:02:12.060 consequence of the modulation property in the context of the 00:02:12.060 --> 00:02:13.940 Fourier transform. 00:02:13.940 --> 00:02:19.370 In particular, if we have our signal x of t and if 00:02:19.370 --> 00:02:25.200 multiplied by an impulse train to give us a sampled signal-- 00:02:25.200 --> 00:02:30.570 another impulse train whose values or areas are samples of 00:02:30.570 --> 00:02:36.220 the original time function, as I indicate here-- then in 00:02:36.220 --> 00:02:40.120 fact, if we examine this equation or equivalently, 00:02:40.120 --> 00:02:45.770 bringing x of t inside this sum, if we examine either of 00:02:45.770 --> 00:02:50.200 these equations in the frequency domain, the Fourier 00:02:50.200 --> 00:02:57.210 transform of x of p of t is the convolution of the Fourier 00:02:57.210 --> 00:03:00.410 transform of the original signal and the Fourier 00:03:00.410 --> 00:03:04.600 transform of the impulse train. 00:03:04.600 --> 00:03:07.110 Now the impulse train is a periodic signal. 00:03:07.110 --> 00:03:08.850 It's Fourier transform. 00:03:08.850 --> 00:03:11.950 Therefore, as we talked about with Fourier transforms is 00:03:11.950 --> 00:03:13.880 itself an impulse train. 00:03:13.880 --> 00:03:17.860 And when we do this convolution, then using the 00:03:17.860 --> 00:03:21.020 fact that the Fourier transform, the impulse train 00:03:21.020 --> 00:03:23.480 is an impulse train. 00:03:23.480 --> 00:03:28.100 The result of this convolution, then tells us 00:03:28.100 --> 00:03:33.540 that the Fourier transform of the sample signal or the 00:03:33.540 --> 00:03:37.610 impulse train, which represents the samples, is a 00:03:37.610 --> 00:03:43.220 sum of frequency-shifted replications of the Fourier 00:03:43.220 --> 00:03:47.030 transform of the original signal. 00:03:47.030 --> 00:03:50.040 So mathematically, that's the relationship. 00:03:50.040 --> 00:03:53.760 It essentially says that after sampling or modulation with an 00:03:53.760 --> 00:03:57.210 impulse train, the resulting spectrum is the original 00:03:57.210 --> 00:04:02.510 spectrum added to itself, shifted by integer multiples 00:04:02.510 --> 00:04:04.550 of the sampling frequency. 00:04:04.550 --> 00:04:06.740 Well, let's see that as we did last 00:04:06.740 --> 00:04:09.280 time in terms of pictures. 00:04:09.280 --> 00:04:14.240 And again, to remind you of the basic picture involved, if 00:04:14.240 --> 00:04:17.980 we have an original signal with a spectrum as I indicated 00:04:17.980 --> 00:04:21.600 here-- where it's band-limited with the highest frequency 00:04:21.600 --> 00:04:28.030 omega sub m-- and if the time function is sampled so that in 00:04:28.030 --> 00:04:32.630 the frequency domain we convolve this spectrum with 00:04:32.630 --> 00:04:38.990 the spectrum shown below, which is the spectrum of the 00:04:38.990 --> 00:04:45.260 impulse train, the convolution of these two is then the 00:04:45.260 --> 00:04:51.600 Fourier transform or spectrum of the sample time function. 00:04:51.600 --> 00:04:55.750 And so that's what we end up with here. 00:04:55.750 --> 00:04:59.480 And then as you recall, to recover the original time 00:04:59.480 --> 00:05:03.790 function from this-- as long as these individual triangles 00:05:03.790 --> 00:05:09.890 don't overlap--to recover it just simply involves passing 00:05:09.890 --> 00:05:13.820 the impulse train through a low-pass filter, in effect 00:05:13.820 --> 00:05:18.800 extracting just one of these replications of 00:05:18.800 --> 00:05:21.430 the original spectrum. 00:05:21.430 --> 00:05:29.880 So the overall system then for doing the sampling and then 00:05:29.880 --> 00:05:32.940 the reconstruction of the original signal from the 00:05:32.940 --> 00:05:37.600 samples, consists of multiplying the original time 00:05:37.600 --> 00:05:41.070 function by an impulse train. 00:05:41.070 --> 00:05:45.990 And that gives us then the sampled signal. 00:05:45.990 --> 00:05:51.380 The Fourier transform I show here of the original signal 00:05:51.380 --> 00:05:55.520 and after modulation with the impulse train, the resulting 00:05:55.520 --> 00:06:01.510 spectrum that we have is that replicated around integer 00:06:01.510 --> 00:06:04.460 multiples of the sampling frequency. 00:06:04.460 --> 00:06:09.440 And then finally, to recover the original signal or to 00:06:09.440 --> 00:06:14.470 generate a reconstructed signal, we then multiply this 00:06:14.470 --> 00:06:19.100 in the frequency domain by the frequency response of an ideal 00:06:19.100 --> 00:06:20.700 low-pass filter. 00:06:20.700 --> 00:06:25.820 And what that accomplishes for us then is recovering the 00:06:25.820 --> 00:06:28.290 original signal. 00:06:28.290 --> 00:06:31.560 Now in this picture, an important point that I raised 00:06:31.560 --> 00:06:35.370 last time, relates to the fact that in doing the 00:06:35.370 --> 00:06:39.270 reconstruction--well we've assumed-- is that in 00:06:39.270 --> 00:06:45.840 replicating these individual versions of the original 00:06:45.840 --> 00:06:49.710 signal, those replications don't overlap and so by 00:06:49.710 --> 00:06:52.340 passing this through a low-pass filter in fact, we 00:06:52.340 --> 00:06:54.970 can recover the original signal. 00:06:54.970 --> 00:07:00.560 Well, what that requires is that this frequency, omega sub 00:07:00.560 --> 00:07:05.420 m, be less than this frequency. 00:07:05.420 --> 00:07:12.310 And this frequency is omega sub s minus omega sub m. 00:07:12.310 --> 00:07:16.850 And so what we require is that the frequency omega sub m be 00:07:16.850 --> 00:07:20.530 less than omega sub s minus omega sub m. 00:07:20.530 --> 00:07:24.600 Or equivalently, what we require is that the sampling 00:07:24.600 --> 00:07:29.450 frequency be greater than twice the highest frequency in 00:07:29.450 --> 00:07:32.190 the original signal. 00:07:32.190 --> 00:07:38.360 Now, if in fact that condition is violated, then we end up 00:07:38.360 --> 00:07:40.950 with a very important effect. 00:07:40.950 --> 00:07:44.840 And that effect is referred to as aliasing. 00:07:44.840 --> 00:07:52.150 In particular, if we look back at our original example--we 00:07:52.150 --> 00:07:57.010 are here-- we were able to recover our original spectrum 00:07:57.010 --> 00:07:59.710 by low-pass filtering. 00:07:59.710 --> 00:08:05.860 If in fact the sampling frequency is not high enough 00:08:05.860 --> 00:08:10.680 to avoid aliasing, then what happens in that case is that 00:08:10.680 --> 00:08:17.480 the individual replications of the Fourier transform of the 00:08:17.480 --> 00:08:23.210 original signal overlap and what we end up with is some 00:08:23.210 --> 00:08:24.500 distortion. 00:08:24.500 --> 00:08:27.920 As you can see, if we try to pass this through a low-pass 00:08:27.920 --> 00:08:32.650 filter to recover the original signal, in fact we won't 00:08:32.650 --> 00:08:36.070 recover the original signal since these individual 00:08:36.070 --> 00:08:38.059 replications have overlapped. 00:08:38.059 --> 00:08:42.370 And this is the case where omega sub s minus omega sub m 00:08:42.370 --> 00:08:44.810 is less than omega sub s. 00:08:44.810 --> 00:08:48.040 In other words, the sampling frequency is not greater in 00:08:48.040 --> 00:08:52.110 this case than twice the highest frequency. 00:08:52.110 --> 00:08:56.370 So what happens here then is that in effect, higher 00:08:56.370 --> 00:08:59.880 frequencies get folded down into lower frequencies. 00:08:59.880 --> 00:09:02.390 What would come out of the low-pass filter is the 00:09:02.390 --> 00:09:04.960 reflection of some higher frequencies into lower 00:09:04.960 --> 00:09:06.380 frequencies. 00:09:06.380 --> 00:09:10.030 As I suggested a minute ago, that effect is 00:09:10.030 --> 00:09:12.290 referred to as aliasing. 00:09:12.290 --> 00:09:15.970 And in order to both understand that term better 00:09:15.970 --> 00:09:20.540 and to understand in fact the effect better, it's useful to 00:09:20.540 --> 00:09:25.310 examine this a little more closely for the specific 00:09:25.310 --> 00:09:27.760 example of a sinusoidal signal. 00:09:27.760 --> 00:09:29.500 So let's concentrate on that. 00:09:29.500 --> 00:09:36.490 And what we want to look at is the effect of aliasing when 00:09:36.490 --> 00:09:39.920 our input signal is a sinusoidal signal. 00:09:39.920 --> 00:09:44.750 Now to do that, what I want to show shortly is a 00:09:44.750 --> 00:09:47.690 computer-generated movie that we've made. 00:09:47.690 --> 00:09:51.650 And let's first walk through a few frames of it to give you-- 00:09:51.650 --> 00:09:57.490 first of all, to set up our notation and to suggest what 00:09:57.490 --> 00:10:00.530 it is that we're trying to demonstrate. 00:10:00.530 --> 00:10:06.250 Well, what we have is an input signal-- is 00:10:06.250 --> 00:10:07.490 a sinusoidal signal. 00:10:07.490 --> 00:10:12.720 And the spectrum or Fourier transform of that is an 00:10:12.720 --> 00:10:15.840 impulse in the frequency domain at the 00:10:15.840 --> 00:10:18.190 frequency of the sinusoid. 00:10:18.190 --> 00:10:22.380 We then have samples of that and when we sample that-- and 00:10:22.380 --> 00:10:25.250 for this particular example, it's sampled at 10 kilohertz-- 00:10:25.250 --> 00:10:31.980 this spectrum is then replicated at multiples of the 00:10:31.980 --> 00:10:33.640 sampling frequency. 00:10:33.640 --> 00:10:36.590 And I haven't shown negative frequencies here, but the 00:10:36.590 --> 00:10:42.040 contribution due to the negative frequency is at 10 00:10:42.040 --> 00:10:46.620 kilohertz minus the input sinusoid. 00:10:46.620 --> 00:10:51.320 We then carry out a reconstruction with an ideal 00:10:51.320 --> 00:10:52.490 low-pass filter. 00:10:52.490 --> 00:10:55.980 And the ideal low-pass filter is set at half the sampling 00:10:55.980 --> 00:10:58.450 frequency or 5 kilohertz. 00:10:58.450 --> 00:11:06.410 So what we have then is the input signal x of t and the 00:11:06.410 --> 00:11:09.840 impulse train x of p of t. 00:11:09.840 --> 00:11:15.100 And then the reconstructed signal is the output from the 00:11:15.100 --> 00:11:21.120 low-pass filter which I denote as x of r of t. 00:11:21.120 --> 00:11:26.320 Now as the input frequency x of t increases, this impulse 00:11:26.320 --> 00:11:30.110 moves up in frequency, but this impulse 00:11:30.110 --> 00:11:32.120 moves down in frequency. 00:11:32.120 --> 00:11:35.440 And so let's just look at a few frames as the input 00:11:35.440 --> 00:11:37.190 frequency increases. 00:11:37.190 --> 00:11:43.290 So we have here a case where the input frequency has moved 00:11:43.290 --> 00:11:46.760 up close to 5 kilohertz. 00:11:46.760 --> 00:11:52.260 As we continue further, these two impulses will cross and 00:11:52.260 --> 00:11:56.120 what we'll end up with, as I indicated, is aliasing. 00:11:56.120 --> 00:12:00.650 So here now is a case where we have aliasing. 00:12:00.650 --> 00:12:04.710 The replication of the negative frequency has crossed 00:12:04.710 --> 00:12:08.710 into the passband of the filter and the reconstructed 00:12:08.710 --> 00:12:13.910 sinusoid will now be the frequency associated with this 00:12:13.910 --> 00:12:18.980 impulse rather than the frequency associated with the 00:12:18.980 --> 00:12:20.520 original sinusoid. 00:12:20.520 --> 00:12:27.240 And to dramatize that even further, here is the example 00:12:27.240 --> 00:12:31.600 where now the input frequency has moved up close to 10 00:12:31.600 --> 00:12:36.840 kilohertz, but what comes out of the low-pass filter is a 00:12:36.840 --> 00:12:38.760 much lower frequency. 00:12:38.760 --> 00:12:42.200 And in fact, you can see that here is the reconstructed 00:12:42.200 --> 00:12:46.600 sinusoid, whereas here we have the input sinusoid. 00:12:49.180 --> 00:12:53.010 Well, now what I'm going to want to do is demonstrate this 00:12:53.010 --> 00:12:56.910 as I indicated with a computer-generated movie. 00:12:56.910 --> 00:13:05.380 And what we'll see is the effect of reconstructing from 00:13:05.380 --> 00:13:10.590 the samples using a low-pass filter for an input which 00:13:10.590 --> 00:13:12.910 changes in frequency and with a 00:13:12.910 --> 00:13:15.730 sampling rate of 10 kilohertz. 00:13:15.730 --> 00:13:19.050 And what we'll see in the first part of this movie is 00:13:19.050 --> 00:13:23.530 the input x of t and the reconstructed signal x of r of 00:13:23.530 --> 00:13:27.290 t without explicitly showing the samples. 00:13:27.290 --> 00:13:30.270 And then, at a later point, we'll also show this and 00:13:30.270 --> 00:13:33.380 indicate that in fact the samples of those two are 00:13:33.380 --> 00:13:37.580 equal, even though they themselves are not. 00:13:37.580 --> 00:13:42.260 So at the top, we'll have the input sinusoid without showing 00:13:42.260 --> 00:13:43.840 the samples. 00:13:43.840 --> 00:13:48.510 And its Fourier transform is an impulse in the frequency 00:13:48.510 --> 00:13:50.500 domain as we've indicated. 00:13:50.500 --> 00:13:55.670 And if we sample it, that impulse then gets replicated. 00:13:55.670 --> 00:13:59.700 And so its samples, in particular, will have a 00:13:59.700 --> 00:14:03.500 Fourier transform not only with an impulse at the input 00:14:03.500 --> 00:14:06.595 sinusoidal frequency, but also at 10 00:14:06.595 --> 00:14:09.260 kilohertz minus that frequency. 00:14:09.260 --> 00:14:13.330 Now for the reconstruction, we passed the samples through an 00:14:13.330 --> 00:14:14.690 ideal low-pass filter. 00:14:14.690 --> 00:14:18.620 I picked the cutoff frequency of the low-pass filter at half 00:14:18.620 --> 00:14:21.530 the sampling frequency, namely 5 kilohertz. 00:14:21.530 --> 00:14:25.810 And here, what we see is that the output reconstructed 00:14:25.810 --> 00:14:29.540 signal in fact matches in frequency the input signal. 00:14:29.540 --> 00:14:34.380 Now as we change the input frequency, the reconstructed 00:14:34.380 --> 00:14:41.010 sinusoid is identical until we get to an input frequency, 00:14:41.010 --> 00:14:44.480 which exceeds half the sampling frequency. 00:14:44.480 --> 00:14:48.760 At that point we have aliasing and while the input frequency 00:14:48.760 --> 00:14:52.990 is increasing, the output frequency in fact is 00:14:52.990 --> 00:14:56.920 decreasing because that's what's inside the passband of 00:14:56.920 --> 00:14:58.960 the filter. 00:14:58.960 --> 00:15:01.310 Now let's sweep it back. 00:15:01.310 --> 00:15:04.630 And as the input frequency decreases, the output 00:15:04.630 --> 00:15:08.440 frequency increases until there's no aliasing and now 00:15:08.440 --> 00:15:11.410 the output reconstructed signal is equal to the input. 00:15:14.320 --> 00:15:18.670 So we've sampled a signal and then reconstructed the signal 00:15:18.670 --> 00:15:20.240 from the samples. 00:15:20.240 --> 00:15:24.740 And keep in mind, that given a set of samples, there are lots 00:15:24.740 --> 00:15:26.980 of continuous curves that we can thread 00:15:26.980 --> 00:15:28.640 through the set of samples. 00:15:28.640 --> 00:15:32.650 The one that we picked, of course, is the one consistent 00:15:32.650 --> 00:15:35.900 with the assumption about the signal bandwidth. 00:15:35.900 --> 00:15:40.130 In particular, we've reconstructed the signal whose 00:15:40.130 --> 00:15:44.430 spectrum falls within the passband of the filter. 00:15:44.430 --> 00:15:49.570 Now what I'd like to show is the same reconstruction and 00:15:49.570 --> 00:15:53.290 input as I showed before, but now let's look at the samples 00:15:53.290 --> 00:15:57.920 and what we'll see is that when there's aliasing, even 00:15:57.920 --> 00:16:02.420 though the output-- the reconstructed signal-- is not 00:16:02.420 --> 00:16:04.690 identical to the input. 00:16:04.690 --> 00:16:08.420 In fact it's consistent with the input samples that is 00:16:08.420 --> 00:16:11.620 sampling the reconstructed signal. 00:16:11.620 --> 00:16:14.110 It gives a set of samples identical to the samples of 00:16:14.110 --> 00:16:17.920 the input and it's just that the interpolation in between 00:16:17.920 --> 00:16:21.565 those samples is an interpolation consistent with 00:16:21.565 --> 00:16:24.690 the assumed bandwidth of the input based on 00:16:24.690 --> 00:16:25.940 the sampling theorem. 00:16:25.940 --> 00:16:31.080 So let's now look at that with the samples also shown along 00:16:31.080 --> 00:16:33.760 with the sinusoid. 00:16:33.760 --> 00:16:37.320 So at the top, we have the input sinusoid together with 00:16:37.320 --> 00:16:38.480 its samples. 00:16:38.480 --> 00:16:40.890 The bottom trace is the Fourier transform of the 00:16:40.890 --> 00:16:42.860 sampled waveform. 00:16:42.860 --> 00:16:47.260 The middle trace is the reconstructed sinusoid 00:16:47.260 --> 00:16:48.970 together with its samples. 00:16:48.970 --> 00:16:52.200 And notice, of course, that the samples of the input or 00:16:52.200 --> 00:16:55.830 reconstructed signal are identical. 00:16:55.830 --> 00:17:01.820 And also the input sinusoidal frequency and the output 00:17:01.820 --> 00:17:04.900 sinusoidal frequency are identical. 00:17:04.900 --> 00:17:09.510 And we now increase the frequency at the input. 00:17:09.510 --> 00:17:13.150 The reconstructed sinusoid tracks the input in frequency 00:17:13.150 --> 00:17:16.970 and, of course, the samples of the two are identical. 00:17:16.970 --> 00:17:20.829 The interpolation in between the samples is identical 00:17:20.829 --> 00:17:25.200 because of the fact that the input frequency is still less 00:17:25.200 --> 00:17:26.490 than half the sampling frequency. 00:17:31.500 --> 00:17:35.460 And so, as long as the input is frequency is less than half 00:17:35.460 --> 00:17:38.570 the sampling frequency, not only will the samples be 00:17:38.570 --> 00:17:42.550 identical, but also the reconstructed continuous 00:17:42.550 --> 00:17:45.725 waveform will match the input waveform. 00:17:56.820 --> 00:17:59.880 Now when we get to half the sampling frequency, we're just 00:17:59.880 --> 00:18:01.440 on the verge of aliasing. 00:18:01.440 --> 00:18:05.680 This isn't aliasing quite yet, but any increase in the input 00:18:05.680 --> 00:18:09.870 frequency will now generate aliasing. 00:18:09.870 --> 00:18:13.460 We now have aliasing, the output frequency is lower than 00:18:13.460 --> 00:18:15.840 the input frequency, but notice that 00:18:15.840 --> 00:18:19.640 the samples are identical. 00:18:19.640 --> 00:18:23.410 Now the low-pass filter is interpolating in between those 00:18:23.410 --> 00:18:27.600 samples with a sinusoid that falls within the passband of 00:18:27.600 --> 00:18:31.260 the low-pass filter, which no longer matches the frequency 00:18:31.260 --> 00:18:34.140 of the input sinusoid. 00:18:34.140 --> 00:18:37.420 But the important point is that even when we have 00:18:37.420 --> 00:18:42.290 aliasing, the samples of the reconstructed waveform are 00:18:42.290 --> 00:18:47.290 identical to the samples of the original waveform. 00:18:47.290 --> 00:18:51.340 And notice that as the input frequency increases, in fact 00:18:51.340 --> 00:18:55.280 the interpolated output, the reconstructed output has 00:18:55.280 --> 00:18:58.270 decreased in frequency. 00:18:58.270 --> 00:19:01.895 Now as the input frequency begins to get closer to 10 00:19:01.895 --> 00:19:07.020 kilohertz-- in fact your eye tends to also interpolate 00:19:07.020 --> 00:19:12.770 between the samples with a frequency that is lower than 00:19:12.770 --> 00:19:13.910 the input frequency. 00:19:13.910 --> 00:19:16.800 And that's particularly evident here. 00:19:16.800 --> 00:19:21.050 Notice that the input samples in fact look like they would 00:19:21.050 --> 00:19:25.310 be associated with a much lower frequency sinusoid, than 00:19:25.310 --> 00:19:28.630 in fact was the sinusoid that generated them. 00:19:28.630 --> 00:19:32.010 The lower-frequency sinusoid in fact corresponds to the 00:19:32.010 --> 00:19:34.070 reconstructed one. 00:19:34.070 --> 00:19:38.650 Now as we sweep back down, the aliasing eventually disappears 00:19:38.650 --> 00:19:40.450 and the output sinusoid tracks the 00:19:40.450 --> 00:19:41.760 input sinusoid in frequency. 00:19:44.490 --> 00:19:47.820 So we've seen the effect of aliasing for sinusoidal 00:19:47.820 --> 00:19:50.170 signals in terms of waveforms. 00:19:50.170 --> 00:19:53.270 Now let's hear how it sounds. 00:19:53.270 --> 00:19:59.760 Now what we have for this demonstration is an oscillator 00:19:59.760 --> 00:20:02.330 and a sampler. 00:20:02.330 --> 00:20:06.460 And the output of the sampler goes into a low-pass filter. 00:20:06.460 --> 00:20:12.240 So the input from the oscillator goes into the 00:20:12.240 --> 00:20:16.930 sampler and the output of the sampler goes into 00:20:16.930 --> 00:20:18.840 the low-pass filter. 00:20:18.840 --> 00:20:23.600 The sampler frequency is 10 kilohertz. 00:20:23.600 --> 00:20:28.880 And so the low-pass filter has a cutoff frequency as I 00:20:28.880 --> 00:20:32.740 indicate here, of 5 kilohertz. 00:20:32.740 --> 00:20:39.740 And what we'll listen to is the reconstructed output as 00:20:39.740 --> 00:20:44.280 the oscillator input frequency varies. 00:20:44.280 --> 00:20:48.850 And recall that what should happen is that when the 00:20:48.850 --> 00:20:52.340 oscillator input frequency gets past half the sampling 00:20:52.340 --> 00:20:56.600 frequency, we should hear aliasing. 00:20:56.600 --> 00:20:58.430 So we'll start the oscillator at 2 kilohertz. 00:20:58.430 --> 00:20:59.460 [OSCILLATOR SOUND IN BACKGROUND] 00:20:59.460 --> 00:21:02.420 PROFESSOR: And keep in mind that what you see on the dial 00:21:02.420 --> 00:21:04.960 is the input frequency, what you hear 00:21:04.960 --> 00:21:06.850 is the output frequency. 00:21:06.850 --> 00:21:09.880 As long as the input frequency is less than half the sampling 00:21:09.880 --> 00:21:13.220 frequency-- in other words, 5 kilohertz -- the reconstructed 00:21:13.220 --> 00:21:17.150 signal sounds identical to the input. 00:21:17.150 --> 00:21:20.660 Now at 5 kilohertz, we're right on the verge of 00:21:20.660 --> 00:21:24.505 aliasing, and when we increase the input frequency past 5 00:21:24.505 --> 00:21:27.330 kilohertz, the reconstructed frequency 00:21:27.330 --> 00:21:29.370 in fact will decrease. 00:21:29.370 --> 00:21:33.740 So as we move, for example, from 5 kilohertz up to let's 00:21:33.740 --> 00:21:36.440 say, 6 kilohertz. 00:21:36.440 --> 00:21:41.450 6 kilohertz in fact gets aliased down to, what? 00:21:41.450 --> 00:21:45.580 It gets aliased down to 4 kilohertz. 00:21:45.580 --> 00:21:52.190 So 6 kilohertz at the input is 4 kilohertz at the output. 00:21:52.190 --> 00:21:55.630 Now, if we move up even further, 7 kilohertz at the 00:21:55.630 --> 00:22:02.380 input gets aliased down to 3 kilohertz at the output. 00:22:02.380 --> 00:22:07.540 So that, then is an audio demonstration of aliasing. 00:22:07.540 --> 00:22:12.680 So to summarize, if we sample a signal and then reconstruct 00:22:12.680 --> 00:22:16.020 from the samples using a low-pass filter, as long as 00:22:16.020 --> 00:22:18.770 the sampling frequency is greater than twice the highest 00:22:18.770 --> 00:22:22.300 frequency in the signal we reconstruct exactly. 00:22:22.300 --> 00:22:25.900 If on the other hand, the sampling frequency is too low, 00:22:25.900 --> 00:22:27.160 less than twice the highest 00:22:27.160 --> 00:22:30.310 frequency, then we get aliasing. 00:22:30.310 --> 00:22:34.270 In other words, higher frequencies get folded or 00:22:34.270 --> 00:22:37.220 reflected down into lower frequencies as they come 00:22:37.220 --> 00:22:40.430 through the low-pass filter. 00:22:40.430 --> 00:22:44.560 Now, one of the common applications of the whole 00:22:44.560 --> 00:22:49.880 concept of sampling is the use of sampling to convert a 00:22:49.880 --> 00:22:55.080 continuous-time signal into a discrete-time signal to carry 00:22:55.080 --> 00:22:59.460 out what's often referred to as discrete-time processing of 00:22:59.460 --> 00:23:01.690 continuous-time signals. 00:23:01.690 --> 00:23:05.810 And this in fact is something that we'll be talking about in 00:23:05.810 --> 00:23:07.520 a fair amount of detail, beginning 00:23:07.520 --> 00:23:09.290 with the next lecture. 00:23:09.290 --> 00:23:14.270 But let me indicate that for that kind of processing, 00:23:14.270 --> 00:23:17.710 essentially what happens, is that we begin with the 00:23:17.710 --> 00:23:21.970 continuous-time signal and convert it to a discrete-time 00:23:21.970 --> 00:23:25.930 signal, carry out the discrete-time processing, and 00:23:25.930 --> 00:23:29.200 then convert back to continuous-time. 00:23:29.200 --> 00:23:33.770 And the conversion from a continuous-time signal to a 00:23:33.770 --> 00:23:39.240 discrete-time signal in fact, is done by exploiting 00:23:39.240 --> 00:23:44.160 sampling, specifically by sampling the continuous-time 00:23:44.160 --> 00:23:49.210 signal with an impulse train and then converting the 00:23:49.210 --> 00:23:54.600 impulse train into a sequence in a matter that I'll talk 00:23:54.600 --> 00:23:57.790 about in more detail next time. 00:23:57.790 --> 00:24:01.800 Now in doing that-- of course, as you can imagine-- it's 00:24:01.800 --> 00:24:05.480 important since we want an accurate representation of the 00:24:05.480 --> 00:24:08.850 original continuous-time signal, to choose the sampling 00:24:08.850 --> 00:24:12.800 frequency, to very carefully avoid aliasing. 00:24:12.800 --> 00:24:16.560 And so in fact, in that context and in many other 00:24:16.560 --> 00:24:19.890 contexts, aliasing is something that we're very 00:24:19.890 --> 00:24:21.750 eager to avoid. 00:24:21.750 --> 00:24:25.880 However, it's also important to understand that aliasing 00:24:25.880 --> 00:24:27.410 isn't all bad. 00:24:27.410 --> 00:24:30.560 And there are some very specific contexts in which 00:24:30.560 --> 00:24:36.230 aliasing is very useful and very heavily exploited. 00:24:36.230 --> 00:24:42.280 One example of a very useful context of aliasing is when 00:24:42.280 --> 00:24:45.820 you want to look at things that happen at frequencies 00:24:45.820 --> 00:24:48.550 that you can't look at, for one reason or another. 00:24:48.550 --> 00:24:51.800 And sampling and aliasing is used to map those into lower 00:24:51.800 --> 00:24:52.950 frequencies. 00:24:52.950 --> 00:24:56.930 One very common example of that is the use of the 00:24:56.930 --> 00:25:03.340 stroboscope which was invented by Dr. Harold Edgerton at MIT. 00:25:03.340 --> 00:25:07.810 And sometime earlier, in fact we had the opportunity to 00:25:07.810 --> 00:25:12.180 visit Dr. Edgerton's laboratory at MIT and see some 00:25:12.180 --> 00:25:13.590 examples of this. 00:25:13.590 --> 00:25:17.535 So I'd like to-- as a conclusion to this lecture-- 00:25:17.535 --> 00:25:22.195 take you on a visit to the strobe lab at MIT. 00:25:24.900 --> 00:25:28.200 In the lecture-- in discussing aliasing-- we've stressed the 00:25:28.200 --> 00:25:31.380 fact that in most situations, it's something that 00:25:31.380 --> 00:25:33.950 we'd like to avoid. 00:25:33.950 --> 00:25:38.340 However, right now we're at MIT, in Strobe Alley as it's 00:25:38.340 --> 00:25:42.860 called, on the way to visit the laboratory of my MIT 00:25:42.860 --> 00:25:45.850 colleague, Professor Harold Edgerton, where in fact 00:25:45.850 --> 00:25:49.430 aliasing is an everyday occurrence. 00:25:49.430 --> 00:25:54.970 Basically, the idea is the following-- that if in fact 00:25:54.970 --> 00:25:58.600 you want to make measurements at frequencies that, for one 00:25:58.600 --> 00:26:03.480 reason or another, you can't measure, then sampling and, 00:26:03.480 --> 00:26:07.140 consequently, aliasing can be used to bring those 00:26:07.140 --> 00:26:09.770 frequencies down into a frequency range 00:26:09.770 --> 00:26:12.200 that you can measure. 00:26:12.200 --> 00:26:18.230 Well, Professor Edgerton alias Doc Edgerton invented the 00:26:18.230 --> 00:26:20.700 stroboscope for exactly that reason. 00:26:20.700 --> 00:26:23.460 And, kind of, the idea is the following. 00:26:23.460 --> 00:26:26.720 The eye, essentially, is a low-pass filter and so there 00:26:26.720 --> 00:26:31.590 are things that happen at frequencies above which your 00:26:31.590 --> 00:26:33.650 eye can track. 00:26:33.650 --> 00:26:39.800 And by sampling with light pulses, sampling in time, what 00:26:39.800 --> 00:26:44.430 in effect you're able to do is sample in such a way that 00:26:44.430 --> 00:26:48.810 higher frequencies get aliased down to lower frequencies so 00:26:48.810 --> 00:26:51.660 that, in fact, your eye can track them. 00:26:51.660 --> 00:26:57.390 So let's take a look inside the lab and in fact see an 00:26:57.390 --> 00:27:00.600 illustration of this strobe and some of its effects. 00:27:05.310 --> 00:27:10.260 Let me introduce you to my MIT colleague, Doc Edgerton. 00:27:10.260 --> 00:27:13.760 Also by the way, this is a great place for kids of all 00:27:13.760 --> 00:27:17.710 ages and so my daughter, Justine, insisted on coming 00:27:17.710 --> 00:27:19.580 along to also help out. 00:27:19.580 --> 00:27:22.600 Doc, maybe we could begin with you just saying a little bit 00:27:22.600 --> 00:27:26.160 about what the strobe is and what some of the history is? 00:27:26.160 --> 00:27:28.950 DR. HAROLD EDGERTON: Sure, it's a very simple application 00:27:28.950 --> 00:27:30.390 of intermittent light. 00:27:30.390 --> 00:27:36.960 And this is a xenon lamp that flashes in a controlled rate 00:27:36.960 --> 00:27:39.840 depending on this knob which Justine's going to turn. 00:27:39.840 --> 00:27:43.740 And we're going to look at a motor that's driving an 00:27:43.740 --> 00:27:46.050 unbalanced weight to set up some [INAUDIBLE] 00:27:46.050 --> 00:27:48.420 oscillations in the spring. 00:27:48.420 --> 00:27:50.456 I'll turn on the motor. 00:27:50.456 --> 00:27:51.932 I'll turn on the strobe. 00:27:51.932 --> 00:27:53.182 [STROBOSCOPE SOUND IN BACKGROUND] 00:28:10.136 --> 00:28:12.838 DR. HAROLD EDGERTON: Just get the right range. 00:28:12.838 --> 00:28:15.295 All right, Justine, turn that now, until it stops. 00:28:17.905 --> 00:28:20.710 See that, Justine, the frequency is that the light, 00:28:20.710 --> 00:28:22.890 which corresponds to the frequency of the motor. 00:28:22.890 --> 00:28:25.820 And it's a little less or a little more, when you lean to 00:28:25.820 --> 00:28:27.380 go forward to backwards. 00:28:27.380 --> 00:28:29.480 PROFESSOR: Doc, maybe we could turn this 00:28:29.480 --> 00:28:30.560 strobe off for minute. 00:28:30.560 --> 00:28:38.230 And let me point out, by the way, the fact that when we're 00:28:38.230 --> 00:28:40.770 looking at this without the strobe on, what we're seeing 00:28:40.770 --> 00:28:43.560 essentially are frequencies that our eye can't track. 00:28:43.560 --> 00:28:48.170 So we can't see the motor turning and we can't really 00:28:48.170 --> 00:28:49.930 see other than with a blur. 00:28:49.930 --> 00:28:52.190 We can't see the movement of the spring. 00:28:52.190 --> 00:28:54.770 And so I guess, your point is that when we put the strobe 00:28:54.770 --> 00:28:59.070 on, we're essentially sampling this. 00:28:59.070 --> 00:29:03.950 And now we brought this down to a frequency that our eye is 00:29:03.950 --> 00:29:06.080 able to track. 00:29:06.080 --> 00:29:13.210 In fact, I guess if we turn the incandescent light off, 00:29:13.210 --> 00:29:18.600 what we'll be able to really bring out are the alias 00:29:18.600 --> 00:29:19.220 frequencies. 00:29:19.220 --> 00:29:23.010 So now, what we're looking at in fact are the alias 00:29:23.010 --> 00:29:24.430 frequencies. 00:29:24.430 --> 00:29:26.430 The spring, of course, is moving a lot faster than we 00:29:26.430 --> 00:29:27.640 see it, isn't that right? 00:29:27.640 --> 00:29:29.640 DR. HAROLD EDGERTON: Yes, it's going 00:29:29.640 --> 00:29:32.781 approximately 30 times a second. 00:29:32.781 --> 00:29:36.150 The motor is going far from 30 times a second. 00:29:36.150 --> 00:29:39.150 I will speed this up while I hit the next mode, where I get 00:29:39.150 --> 00:29:41.105 a figure 8 out of this thing. 00:29:41.105 --> 00:29:42.310 You want to see that now? 00:29:42.310 --> 00:29:44.542 PROFESSOR: Yeah, great. 00:29:44.542 --> 00:29:47.500 DR. HAROLD EDGERTON: [INAUDIBLE] 00:29:47.500 --> 00:30:01.330 [MACHINE NOISE GETS LOUDER] 00:30:01.330 --> 00:30:03.660 PROFESSOR: So what we'll be seeing now is essentially a 00:30:03.660 --> 00:30:04.995 second harmonic, is it? 00:30:04.995 --> 00:30:07.590 DR. HAROLD EDGERTON: Yes, that's the second harmonic. 00:30:07.590 --> 00:30:09.960 PROFESSOR: Justine, you think you could make that spring 00:30:09.960 --> 00:30:13.890 dance around a little bit by changing the strobe frequency? 00:30:13.890 --> 00:30:15.247 DR. HAROLD EDGERTON: Yeah, you need to go around that way. 00:30:15.247 --> 00:30:17.190 You go around this way. 00:30:17.190 --> 00:30:18.340 PROFESSOR: Hey, that's really neat. 00:30:18.340 --> 00:30:20.820 Let's turn the lights back on if we can. 00:30:20.820 --> 00:30:22.400 DR. HAROLD EDGERTON: Tomorrow [INAUDIBLE] 00:30:22.400 --> 00:30:24.790 it's periodic, it has to be periodic. 00:30:24.790 --> 00:30:27.090 PROFESSOR: And what's interesting now, if we look at 00:30:27.090 --> 00:30:31.280 this in a-- let's see, can you flip this strobe off again? 00:30:31.280 --> 00:30:33.580 DR. HAROLD EDGERTON: Sure. 00:30:33.580 --> 00:30:35.370 DR. HAROLD EDGERTON: Notice, Justine, when we look at the 00:30:35.370 --> 00:30:37.790 spring now, all that we can see is a blur. 00:30:37.790 --> 00:30:40.890 And you really can't see-- because your eye can't track 00:30:40.890 --> 00:30:42.580 it, you can't see things happening 00:30:42.580 --> 00:30:45.650 spatially in frequency. 00:30:45.650 --> 00:30:48.250 You said, by the way, that this was originally 00:30:48.250 --> 00:30:50.190 demonstrated at the World's Fair. 00:30:50.190 --> 00:30:52.540 DR. HAROLD EDGERTON: This particular instrument was made 00:30:52.540 --> 00:30:55.370 the World's Fair in Chicago-- not the last one, but the one 00:30:55.370 --> 00:30:55.900 before that. 00:30:55.900 --> 00:30:56.225 PROFESSOR: Wow. 00:30:56.225 --> 00:30:59.268 DR. HAROLD EDGERTON: It was a--you see it all scratched up 00:30:59.268 --> 00:31:01.900 because it's a-- the [INAUDIBLE] 00:31:01.900 --> 00:31:04.880 use this thing is to break the springs. 00:31:04.880 --> 00:31:09.080 Because of the uses, you try to find the parts that fail. 00:31:09.080 --> 00:31:09.510 PROFESSOR: I see. 00:31:09.510 --> 00:31:11.210 You put them under stress and fatigue and-- 00:31:11.210 --> 00:31:12.946 DR. HAROLD EDGERTON: If I run this for half an hour and so, 00:31:12.946 --> 00:31:15.010 the spring will break. 00:31:15.010 --> 00:31:19.000 And they work on automobiles, they run them 00:31:19.000 --> 00:31:20.175 until something vibrates. 00:31:20.175 --> 00:31:22.172 Then they find out what the part is and what 00:31:22.172 --> 00:31:24.300 frequency it is. 00:31:24.300 --> 00:31:26.670 PROFESSOR: Well let's--by the way, I bet you run this for a 00:31:26.670 --> 00:31:28.486 lot more than half an hour in this state. 00:31:28.486 --> 00:31:29.320 DR. HAROLD EDGERTON: Oh, yeah, we've broken many, many 00:31:29.320 --> 00:31:33.100 springs in this thing-- and it's continuous. 00:31:33.100 --> 00:31:36.680 We experiment, try new things on it. 00:31:36.680 --> 00:31:39.200 PROFESSOR: Maybe we could look at a couple of other things. 00:31:39.200 --> 00:31:40.570 How about the fan? 00:31:40.570 --> 00:31:41.070 Maybe-- 00:31:41.070 --> 00:31:42.905 DR. HAROLD EDGERTON: Sure, I'll plug this fan and this is 00:31:42.905 --> 00:31:44.886 a classic experiment for the strobe. 00:31:44.886 --> 00:31:45.690 [MACHINE NOISE STOPS] 00:31:45.690 --> 00:31:46.680 DR. HAROLD EDGERTON: That's a good idea. 00:31:46.680 --> 00:31:47.946 Get that thing off. 00:31:47.946 --> 00:31:50.562 Makes too much noise. 00:31:50.562 --> 00:31:52.825 PROFESSOR: Guess, we move that over here. 00:31:52.825 --> 00:31:55.750 DR. HAROLD EDGERTON: This is just an ordinary electric fan, 00:31:55.750 --> 00:31:58.720 but it has a mark on one blade, so that you can 00:31:58.720 --> 00:32:00.810 identify it. 00:32:00.810 --> 00:32:02.060 We'll plug it in, get it up to speed. 00:32:09.410 --> 00:32:11.650 PROFESSOR: This looks like a fan that was also demonstrated 00:32:11.650 --> 00:32:14.775 in the World's Fair, a few years ago. 00:32:14.775 --> 00:32:16.025 DR. HAROLD EDGERTON: Yeah, could've been. 00:32:18.600 --> 00:32:20.700 There was a movie Quicker'n a Wink had this 00:32:20.700 --> 00:32:22.170 thing in there and-- 00:32:22.170 --> 00:32:24.790 PROFESSOR: With this very fan? 00:32:24.790 --> 00:32:26.090 DR. HAROLD EDGERTON: Well, one like it. 00:32:26.090 --> 00:32:28.220 It was loaned to MGM. 00:32:28.220 --> 00:32:35.900 And Pete Smith, he said he wanted me to throw out a 00:32:35.900 --> 00:32:37.030 custard pie into it. 00:32:37.030 --> 00:32:40.680 I said, no, I'm a serious scientist. 00:32:40.680 --> 00:32:44.730 So he says, let's compromise on the egg. 00:32:44.730 --> 00:32:47.820 So we dropped an egg into it and you would see a high-speed 00:32:47.820 --> 00:32:49.780 movie of the egg dropping. 00:32:49.780 --> 00:32:52.090 No, not with the strobe, but with this [INAUDIBLE] 00:32:52.090 --> 00:32:54.065 PROFESSOR: That was with the high-speed photography. 00:32:54.065 --> 00:32:56.860 DR. HAROLD EDGERTON: High-speed movies, yeah. 00:32:56.860 --> 00:32:59.080 PROFESSOR: So again, I guess, without the strobe, when we 00:32:59.080 --> 00:33:04.470 look at it, what we're looking at are frequencies that are 00:33:04.470 --> 00:33:06.020 much higher than the eye can follow. 00:33:06.020 --> 00:33:09.750 And now, with the strobe on, you can see both the alias 00:33:09.750 --> 00:33:12.470 frequency and you can also see the original frequency because 00:33:12.470 --> 00:33:15.970 we had the incandescent light on. 00:33:15.970 --> 00:33:20.210 Let's turn down the background light again. 00:33:20.210 --> 00:33:22.580 And then, really all that we're able to see are the 00:33:22.580 --> 00:33:25.770 aliasing frequencies. 00:33:25.770 --> 00:33:30.650 And I guess when we see more than one mark, that means that 00:33:30.650 --> 00:33:32.910 we're actually running it at-- 00:33:32.910 --> 00:33:34.330 DR. HAROLD EDGERTON: Four times the speed of the fan. 00:33:34.330 --> 00:33:35.870 PROFESSOR: --four times the speed the fan, yeah. 00:33:35.870 --> 00:33:38.140 DR. HAROLD EDGERTON: You see a little variation in the-- 00:33:38.140 --> 00:33:38.515 PROFESSOR: Oh, yeah. 00:33:38.515 --> 00:33:38.650 Right. 00:33:38.650 --> 00:33:39.990 DR. HAROLD EDGERTON: It's because the blades aren't 00:33:39.990 --> 00:33:41.050 exactly the same. 00:33:41.050 --> 00:33:42.450 PROFESSOR: Actually, this gives me a chance to 00:33:42.450 --> 00:33:46.780 illustrate another important point related to the lecture. 00:33:46.780 --> 00:33:50.210 Let's see, can we bring it down to a frequency so that we 00:33:50.210 --> 00:33:51.620 only get one mark? 00:33:51.620 --> 00:33:52.870 DR. HAROLD EDGERTON: Sure. 00:33:56.171 --> 00:33:58.540 You may miss this because it's too lowered to just 00:33:58.540 --> 00:34:00.550 one blade there now. 00:34:00.550 --> 00:34:02.580 PROFESSOR: So the way we have it now, we've essentially 00:34:02.580 --> 00:34:05.800 aliased the fan's speed down so that it's just a little 00:34:05.800 --> 00:34:07.950 higher than DC. 00:34:07.950 --> 00:34:11.380 And now, I'm right at DC. 00:34:11.380 --> 00:34:14.810 And now, if I go down just a little further, in fact it 00:34:14.810 --> 00:34:17.770 looks like the fan is turning backwards. 00:34:17.770 --> 00:34:22.159 And if you think of this in the context of aliasing, it's 00:34:22.159 --> 00:34:25.670 like the two impulses in the frequency domain 00:34:25.670 --> 00:34:26.929 have crossed over. 00:34:26.929 --> 00:34:30.540 And what you get in effect, if you analyze it mathematically, 00:34:30.540 --> 00:34:32.770 is you get a phase reversal. 00:34:32.770 --> 00:34:37.870 And it wasn't until I first understood about aliasing, by 00:34:37.870 --> 00:34:40.790 the way, Doc, that I understood why when I went to 00:34:40.790 --> 00:34:44.420 Western movies, every once in a while you'd see the wagon 00:34:44.420 --> 00:34:46.570 wheels turning backwards. 00:34:46.570 --> 00:34:49.600 Then there's the wagon wheels of the Western movie going 00:34:49.600 --> 00:34:51.600 backwards, I guess. 00:34:51.600 --> 00:34:53.879 And, Justine, why don't you see if you can-- 00:34:53.879 --> 00:34:54.980 DR. HAROLD EDGERTON: Too much flicker there. 00:34:54.980 --> 00:34:57.460 Why don't you bring it up so you get two marks. 00:34:57.460 --> 00:34:59.740 PROFESSOR: See if you can bring the frequency up so that 00:34:59.740 --> 00:35:00.990 you get two marks. 00:35:03.240 --> 00:35:04.060 DR. HAROLD EDGERTON: You turn that, Justine. 00:35:04.060 --> 00:35:08.100 Grab right ahold of that and give it a big twist. 00:35:08.100 --> 00:35:09.350 You went past it. 00:35:11.940 --> 00:35:14.320 They're not regular there now. 00:35:14.320 --> 00:35:14.910 Here we are. 00:35:14.910 --> 00:35:16.150 Now hold it right there. 00:35:16.150 --> 00:35:19.290 Put your finger on there, hold the dial. 00:35:19.290 --> 00:35:21.200 It's flashing twice per revolution now, Al. 00:35:21.200 --> 00:35:23.620 PROFESSOR: I guess another thing that this demonstrates 00:35:23.620 --> 00:35:27.500 is something that I've heard a long time ago, which is that 00:35:27.500 --> 00:35:30.850 you should never use a power saw with a fluorescent light 00:35:30.850 --> 00:35:32.940 because the fluorescent light gives you a little bit of a 00:35:32.940 --> 00:35:36.230 strobe effect and you could actually convince yourself 00:35:36.230 --> 00:35:39.670 that that's standing still and make the mistake of trying to 00:35:39.670 --> 00:35:40.870 put your finger between the blades. 00:35:40.870 --> 00:35:42.850 DR. HAROLD EDGERTON: You want to stick your finger in there? 00:35:42.850 --> 00:35:44.330 PROFESSOR: No, I don't think I want to try it. 00:35:44.330 --> 00:35:45.290 How about you, Justine? 00:35:45.290 --> 00:35:45.800 What do you think? 00:35:45.800 --> 00:35:47.910 Is that standing still or is that moving? 00:35:47.910 --> 00:35:49.800 DR. HAROLD EDGERTON: She knows it's going. 00:35:49.800 --> 00:35:53.140 We won't let her get close to that fan. 00:35:53.140 --> 00:35:56.580 PROFESSOR: Actually if we turn the lights back on again, what 00:35:56.580 --> 00:35:59.470 that will let us see once again is that we can see both 00:35:59.470 --> 00:36:03.240 the alias frequencies when we do that and we can also see 00:36:03.240 --> 00:36:08.070 the higher frequencies because of the incandescent lighting. 00:36:08.070 --> 00:36:12.010 Maybe what we can do now is take a look at 00:36:12.010 --> 00:36:13.260 some other fun things. 00:36:13.260 --> 00:36:16.160 And one I guess I'm curious about is the disk that you 00:36:16.160 --> 00:36:17.870 have over there. 00:36:17.870 --> 00:36:22.120 Doc, maybe you can tell us what we have here? 00:36:22.120 --> 00:36:22.610 DR. HAROLD EDGERTON: Sure, Al. 00:36:22.610 --> 00:36:26.530 This is a disk to show how you can get motion pictures out of 00:36:26.530 --> 00:36:29.240 a series of still pictures. 00:36:29.240 --> 00:36:31.970 This circle is repeated 12 times. 00:36:31.970 --> 00:36:35.820 The white dot goes from the outer part of it on this side 00:36:35.820 --> 00:36:37.600 to the inner part on the other. 00:36:37.600 --> 00:36:40.740 If I flash one time per revolution on this, you'll see 00:36:40.740 --> 00:36:42.315 it exactly as it is. 00:36:42.315 --> 00:36:46.060 But if I skip one picture each time, then you get the 00:36:46.060 --> 00:36:48.930 relative motion of this ball. 00:36:48.930 --> 00:36:52.030 Well, the object is to show the ball rotating either this 00:36:52.030 --> 00:36:54.490 way or that way depending on whether the strobe was going 00:36:54.490 --> 00:36:56.506 faster or slower than the other. 00:36:59.110 --> 00:37:01.970 This way motion pictures were developed hundred years ago, 00:37:01.970 --> 00:37:03.140 long before photography. 00:37:03.140 --> 00:37:05.580 They drew pictures of people in different 00:37:05.580 --> 00:37:07.180 poses, animated pictures. 00:37:07.180 --> 00:37:07.930 Like to see it run? 00:37:07.930 --> 00:37:09.230 PROFESSOR: Yeah, great. 00:37:09.230 --> 00:37:13.090 It's actually, the title, kind of, is "Aliasing Can Be Fun." 00:37:13.090 --> 00:37:14.340 DR. HAROLD EDGERTON: That's right. 00:37:17.000 --> 00:37:20.015 Let me get it up to speed. 00:37:20.015 --> 00:37:22.480 On the way up, you get a lot of different, sort of, 00:37:22.480 --> 00:37:24.160 patterns as it goes through. 00:37:24.160 --> 00:37:27.513 When it eventually reaches its speed, which is about 1,100 00:37:27.513 --> 00:37:31.400 per minute, you'll see it stop. 00:37:31.400 --> 00:37:33.550 PROFESSOR: And the background blur, basically at the high 00:37:33.550 --> 00:37:37.530 frequencies that the eye can't follow and then, kind of, 00:37:37.530 --> 00:37:41.520 superimposed on that again, we can see the frequencies that 00:37:41.520 --> 00:37:42.530 are aliased down. 00:37:42.530 --> 00:37:44.306 And that's what the eye can follow. 00:37:44.306 --> 00:37:46.320 DR. HAROLD EDGERTON: Right now, we have one flash per 00:37:46.320 --> 00:37:49.145 revolution, so you can see the part of the disk that's 00:37:49.145 --> 00:37:51.790 illuminated with the strobe exactly as if it 00:37:51.790 --> 00:37:53.015 was standing still. 00:37:53.015 --> 00:37:57.260 Now if I increase the frequency, so they skip one 00:37:57.260 --> 00:37:59.780 circle, then you get the illusion 00:37:59.780 --> 00:38:01.300 that, that dot is moving. 00:38:01.300 --> 00:38:04.960 PROFESSOR: In fact, let's really enhance the revolution, 00:38:04.960 --> 00:38:09.100 let's turn the incandescent lights off again. 00:38:09.100 --> 00:38:11.690 And now, now what we see really are the alias 00:38:11.690 --> 00:38:13.780 frequencies. 00:38:13.780 --> 00:38:15.510 What do you think of this Justine? 00:38:15.510 --> 00:38:17.310 JUSTINE: Neat. 00:38:17.310 --> 00:38:19.190 DR. HAROLD EDGERTON: It looks like magic. 00:38:19.190 --> 00:38:22.517 I still have great joy in watching this thing, though 00:38:22.517 --> 00:38:23.770 it's so simple. 00:38:23.770 --> 00:38:25.420 PROFESSOR: Now, while we're watching this, something also 00:38:25.420 --> 00:38:31.780 I might point out for the lecture--for the course-- is 00:38:31.780 --> 00:38:34.620 that actually there really are two sampling frequencies that 00:38:34.620 --> 00:38:35.100 we're seeing. 00:38:35.100 --> 00:38:37.190 One is the strobe, which is the 00:38:37.190 --> 00:38:38.910 strobe that you're running. 00:38:38.910 --> 00:38:42.290 The other is the inherent frame rate for the TV, that's 00:38:42.290 --> 00:38:44.400 running at 30 frames a second. 00:38:44.400 --> 00:38:47.070 And that's one of the reasons, by the way, that people 00:38:47.070 --> 00:38:51.750 watching this on the video course are in fact seeing a 00:38:51.750 --> 00:38:56.690 flicker or a beating or modulation between the two 00:38:56.690 --> 00:38:59.060 unsynchronized frame rates. 00:38:59.060 --> 00:39:00.450 DR. HAROLD EDGERTON: I'll run the frequencies of the strobe 00:39:00.450 --> 00:39:04.195 up, so we get two of them in there. 00:39:04.195 --> 00:39:05.660 You keep watching, we had all these 00:39:05.660 --> 00:39:06.910 other interesting patterns. 00:39:09.235 --> 00:39:09.700 There's two now. 00:39:09.700 --> 00:39:12.360 And I'll make the two bounce on each other. 00:39:19.260 --> 00:39:21.640 You get all these patterns for free. 00:39:21.640 --> 00:39:24.225 You design a disk to show one thing and then when you run 00:39:24.225 --> 00:39:26.480 it, you find all the other patterns. 00:39:26.480 --> 00:39:29.000 PROFESSOR: I think it would be a terrific homework problem 00:39:29.000 --> 00:39:32.310 for the video course, to have them all sit down and analyze 00:39:32.310 --> 00:39:34.130 all the frequencies that they're seeing and what 00:39:34.130 --> 00:39:35.140 they're being aliased to. 00:39:35.140 --> 00:39:35.760 What do you think of that? 00:39:35.760 --> 00:39:37.670 DR. HAROLD EDGERTON: That's a good idea. 00:39:37.670 --> 00:39:40.780 As a teacher, I love to give quizzes. 00:39:40.780 --> 00:39:42.410 Find out whether the students are listening. 00:39:42.410 --> 00:39:44.000 PROFESSOR: I think that'll chase a few people away from 00:39:44.000 --> 00:39:45.190 the course, that's what I like-- 00:39:45.190 --> 00:39:47.900 DR. HAROLD EDGERTON: No, it attracts them because you get 00:39:47.900 --> 00:39:51.540 involved in these optical things, there's no limit on 00:39:51.540 --> 00:39:54.380 what you can do. 00:39:54.380 --> 00:39:56.960 PROFESSOR: Let's bring the incandescent lights back up 00:39:56.960 --> 00:39:59.880 again, just to remind everybody that in back of all 00:39:59.880 --> 00:40:03.920 these are some frequencies that are a lot higher than the 00:40:03.920 --> 00:40:05.640 ones that we begin to get the 00:40:05.640 --> 00:40:07.100 impression that we're watching. 00:40:07.100 --> 00:40:09.270 DR. HAROLD EDGERTON: It's just a motor running at constant 00:40:09.270 --> 00:40:11.440 speed with a pattern on it. 00:40:14.150 --> 00:40:17.110 PROFESSOR: Doc, I have to say that there aren't many people 00:40:17.110 --> 00:40:19.700 I know that have as much fun in their work as you do. 00:40:19.700 --> 00:40:21.805 DR. HAROLD EDGERTON: Well, I'm a lucky man. 00:40:21.805 --> 00:40:25.260 PROFESSOR: Well, what I'd like to do now, maybe, is take a 00:40:25.260 --> 00:40:26.950 look at one last experiment, if you could. 00:40:26.950 --> 00:40:27.870 DR. HAROLD EDGERTON: Sure. 00:40:27.870 --> 00:40:32.290 PROFESSOR: And what I'd like to do is go take a look at, I 00:40:32.290 --> 00:40:35.310 guess, what sometimes is called the--well, not the 00:40:35.310 --> 00:40:37.320 water drop experiment-- what's the name of the-- 00:40:37.320 --> 00:40:38.080 DR. HAROLD EDGERTON: You mean the Double Piddler Hydraulic 00:40:38.080 --> 00:40:39.710 Happening Machine? 00:40:39.710 --> 00:40:40.750 PROFESSOR: That's the one I was thinking of. 00:40:40.750 --> 00:40:42.970 Let's take a look over there. 00:40:42.970 --> 00:40:43.680 DR. HAROLD EDGERTON: Come on, Justine, let's go 00:40:43.680 --> 00:40:44.930 and turn on the water. 00:40:48.620 --> 00:40:51.990 PROFESSOR: So, Doc, this is the--what did you call it 00:40:51.990 --> 00:40:54.370 DPHHM for Double Piddler Hydraulic Happening Machine? 00:40:57.590 --> 00:41:00.000 Got it. 00:41:00.000 --> 00:41:01.110 DR. HAROLD EDGERTON: It looks like a continuous 00:41:01.110 --> 00:41:02.620 stream, but it's not. 00:41:02.620 --> 00:41:04.365 It's a pump over there. 00:41:04.365 --> 00:41:07.480 It's pumping 60 pulses a second. 00:41:07.480 --> 00:41:09.680 The water is coming out in spurts. 00:41:09.680 --> 00:41:13.790 PROFESSOR: So actually, again it's the 60 pulses a second 00:41:13.790 --> 00:41:15.000 your eye can't follow. 00:41:15.000 --> 00:41:16.910 DR. HAROLD EDGERTON: Your eye's no good at 60 a second. 00:41:16.910 --> 00:41:18.450 PROFESSOR: Basically looks like a blur. 00:41:18.450 --> 00:41:20.850 DR. HAROLD EDGERTON: It is a blur, a nice juicy blur. 00:41:20.850 --> 00:41:22.100 Now we put the strobe on. 00:41:27.970 --> 00:41:31.270 PROFESSOR: So again, I guess we have this essentially 00:41:31.270 --> 00:41:32.390 aliased down. 00:41:32.390 --> 00:41:34.060 And again with the incandescent light, you can 00:41:34.060 --> 00:41:39.210 see both the high frequency and the alias frequency. 00:41:39.210 --> 00:41:43.930 And let's see, I guess that's what the frequency close to DC 00:41:43.930 --> 00:41:47.480 and we can adjust it so that it's stopped. 00:41:47.480 --> 00:41:48.640 DR. HAROLD EDGERTON: All right, make the water go up. 00:41:48.640 --> 00:41:51.410 PROFESSOR: And then we can actually make it go up. 00:41:51.410 --> 00:41:53.950 DR. HAROLD EDGERTON: Of course, nobody believes that. 00:41:53.950 --> 00:41:57.580 PROFESSOR: Yeah, in fact, let me just again, to stress this 00:41:57.580 --> 00:41:59.110 point to the class. 00:41:59.110 --> 00:42:02.930 The idea here of the phase reversal-- of course, you can 00:42:02.930 --> 00:42:05.990 see it in the time domain-- you just think about when the 00:42:05.990 --> 00:42:08.040 flashes of light come. 00:42:08.040 --> 00:42:12.770 But if you think of these impulses that we have in the 00:42:12.770 --> 00:42:16.580 frequency domain and we're aliasing as we change the 00:42:16.580 --> 00:42:19.540 sampling frequency, what happens is that these impulses 00:42:19.540 --> 00:42:23.390 cross over and what that means is that we get a phase 00:42:23.390 --> 00:42:28.270 reversal depending on which phases are associated with 00:42:28.270 --> 00:42:31.130 which side of DC so that's kind of the idea 00:42:31.130 --> 00:42:32.380 of the phase reversal. 00:42:34.640 --> 00:42:36.150 Let's turn the-- 00:42:36.150 --> 00:42:37.710 DR. HAROLD EDGERTON: Well, we tried to have Justine put her 00:42:37.710 --> 00:42:39.090 finger in between those two drops. 00:42:39.090 --> 00:42:40.090 PROFESSOR: Yeah, let's turn the 00:42:40.090 --> 00:42:41.990 incandescent light off first. 00:42:41.990 --> 00:42:42.260 And-- 00:42:42.260 --> 00:42:43.610 DR. HAROLD EDGERTON: Take one finger out now. 00:42:43.610 --> 00:42:45.460 Put it right in between those two drops. 00:42:45.460 --> 00:42:46.330 PROFESSOR: Justine,you think you can do that? 00:42:46.330 --> 00:42:47.420 DR. HAROLD EDGERTON: Better get on the other side. 00:42:47.420 --> 00:42:49.525 Use your other hand, so they can see with it. 00:42:49.525 --> 00:42:50.330 You can-- 00:42:50.330 --> 00:42:52.620 PROFESSOR: Think you can get your finger in there? 00:42:52.620 --> 00:42:55.530 PROFESSOR: Whoop, there's water there all the time. 00:42:55.530 --> 00:42:56.790 PROFESSOR: Well I don't know, Doc. 00:42:56.790 --> 00:43:00.520 It seems to me if we-- can't we just adjust this so that 00:43:00.520 --> 00:43:02.030 the dots just go through each other? 00:43:02.030 --> 00:43:03.230 DR. HAROLD EDGERTON: Sure. 00:43:03.230 --> 00:43:05.038 PROFESSOR: Now if the dots can do it, Justine, how come you 00:43:05.038 --> 00:43:06.490 can't get your finger in there? 00:43:06.490 --> 00:43:06.720 JUSTINE: I don't know. 00:43:06.720 --> 00:43:08.060 PROFESSOR: Why don't you try that once more? 00:43:10.910 --> 00:43:12.020 I guess not. 00:43:12.020 --> 00:43:13.030 DR. HAROLD EDGERTON: No, that's one thing you 00:43:13.030 --> 00:43:14.450 can't do with it. 00:43:14.450 --> 00:43:18.080 PROFESSOR: Well let's bring the lights back up and again, 00:43:18.080 --> 00:43:23.480 just to stress the point, here we are at DC, here we are at a 00:43:23.480 --> 00:43:28.580 frequency that's just a little above DC, and we can go back 00:43:28.580 --> 00:43:31.560 down to DC and we can actually get a phase reversal. 00:43:31.560 --> 00:43:34.850 And I guess, if we do this long enough, we can empty out 00:43:34.850 --> 00:43:36.580 the whole ocean and put it back in 00:43:36.580 --> 00:43:37.490 wherever it comes from. 00:43:37.490 --> 00:43:37.950 Isn't that right? 00:43:37.950 --> 00:43:39.010 DR. HAROLD EDGERTON: And we caution the students when they 00:43:39.010 --> 00:43:40.370 run this, not to run it too long-- 00:43:40.370 --> 00:43:40.480 PROFESSOR: That's right. 00:43:40.480 --> 00:43:41.820 DR. HAROLD EDGERTON: We've got the bucket here. 00:43:41.820 --> 00:43:42.730 PROFESSOR: You have to be careful-- 00:43:42.730 --> 00:43:43.200 DR. HAROLD EDGERTON: --and it's been a while since 00:43:43.200 --> 00:43:45.130 somebody believes me. 00:43:45.130 --> 00:43:47.130 PROFESSOR: Well, I don't know about them, but I guess I 00:43:47.130 --> 00:43:48.862 believe you, Doc. 00:43:48.862 --> 00:43:50.580 DR. HAROLD EDGERTON: I'll put a little more pressure on so 00:43:50.580 --> 00:43:53.876 we get little more interesting patterns. 00:43:53.876 --> 00:43:57.200 Little patterns or surface tension that's pulling in 00:43:57.200 --> 00:43:58.450 things together. 00:44:01.250 --> 00:44:04.412 We have these machines, they're all over the place. 00:44:04.412 --> 00:44:05.990 They're a lot of fun. 00:44:05.990 --> 00:44:07.790 PROFESSOR: Well, Doc, this is really terrific. 00:44:07.790 --> 00:44:14.410 I think that this whole idea of using aliasing and strobes 00:44:14.410 --> 00:44:17.430 and the kinds of things that you do with 00:44:17.430 --> 00:44:19.050 them are just fantastic. 00:44:19.050 --> 00:44:22.630 And we really appreciate the chance to come in here and see 00:44:22.630 --> 00:44:23.432 the demonstration. 00:44:23.432 --> 00:44:24.750 DR. HAROLD EDGERTON: Well, that's the whole game. 00:44:24.750 --> 00:44:25.680 We've [? been happy to use ?] 00:44:25.680 --> 00:44:29.660 them for years and probably will for many years to come. 00:44:29.660 --> 00:44:32.970 PROFESSOR: So as I emphasized at the beginning, in lots of 00:44:32.970 --> 00:44:36.490 situations aliasing can, in fact, be very useful. 00:44:36.490 --> 00:44:40.020 Also what this demonstrates is that particularly when you 00:44:40.020 --> 00:44:43.470 have a colleague, like Doc Edgerton, aliasing and for 00:44:43.470 --> 00:44:45.260 that matter science, in general, can be an 00:44:45.260 --> 00:44:46.130 awful lot of fun. 00:44:46.130 --> 00:44:47.670 DR. HAROLD EDGERTON: Thanks for coming in. 00:44:47.670 --> 00:44:48.920 PROFESSOR: Thanks a lot, Doc. 00:44:48.920 --> 00:44:49.370 DR. HAROLD EDGERTON: See you again. 00:44:49.370 --> 00:44:50.580 PROFESSOR: And thank you, Justine. 00:44:50.580 --> 00:44:51.830 JUSTINE: You're welcome. 00:44:54.300 --> 00:44:58.610 PROFESSOR: Well, I have to say that visit was an awful lot of 00:44:58.610 --> 00:45:03.450 fun for me and for Justine and in fact, for the whole camera 00:45:03.450 --> 00:45:04.850 crew that was there. 00:45:04.850 --> 00:45:08.330 And hopefully, all of you at some point will also have a 00:45:08.330 --> 00:45:12.130 chance to visit at Strobe Alley. 00:45:12.130 --> 00:45:16.480 Well, hopefully what we've gone through today gives you a 00:45:16.480 --> 00:45:20.680 good feeling for the concepts of sampling and aliasing and 00:45:20.680 --> 00:45:23.540 both, why it might be useful and why we might 00:45:23.540 --> 00:45:25.910 want to avoid it. 00:45:25.910 --> 00:45:29.260 In the next lecture, we'll continue on the 00:45:29.260 --> 00:45:31.200 discussion of sampling. 00:45:31.200 --> 00:45:34.840 And in particular, what I'll be talking about is the 00:45:34.840 --> 00:45:39.020 interpretation of the reconstruction process not in 00:45:39.020 --> 00:45:42.930 the frequency domain, but in a time domain and interpretation 00:45:42.930 --> 00:45:46.950 specifically associated with the concept of interpolating 00:45:46.950 --> 00:45:48.450 between the samples. 00:45:48.450 --> 00:45:51.970 We'll then proceed from there to a discussion as I've 00:45:51.970 --> 00:45:57.070 alluded to in several lectures of what I've referred to as 00:45:57.070 --> 00:46:01.150 discrete-time processing of continuous-time signals, very 00:46:01.150 --> 00:46:04.560 heavily exploiting the concept and issues 00:46:04.560 --> 00:46:06.110 associated with sampling. 00:46:06.110 --> 00:46:07.360 Thank you.