1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high-quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation or view additional materials from 6 00:00:13,470 --> 00:00:17,400 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,400 --> 00:00:18,650 ocw.mit.edu. 8 00:00:20,524 --> 00:00:55,940 [MUSIC PLAYING] 9 00:00:55,940 --> 00:00:58,190 PROFESSOR: We concluded the last lecture with the 10 00:00:58,190 --> 00:00:59,960 statement of the sampling theorem. 11 00:00:59,960 --> 00:01:05,090 And just as a quick reminder, the sampling theorem said that 12 00:01:05,090 --> 00:01:10,170 if we have a continuous-time signal and we have equally 13 00:01:10,170 --> 00:01:15,460 spaced samples of that signal, sampled at a sampling period, 14 00:01:15,460 --> 00:01:20,860 which I indicate is capital T and if x of t is 15 00:01:20,860 --> 00:01:25,810 band-limited-- in other words, the Fourier transform is zero 16 00:01:25,810 --> 00:01:29,750 outside some band where omega sub m is the highest 17 00:01:29,750 --> 00:01:35,480 frequency-- then under the condition that the sampling 18 00:01:35,480 --> 00:01:39,010 frequency, which is 2 pi divided by the period, is 19 00:01:39,010 --> 00:01:42,770 greater than twice the highest frequency. 20 00:01:42,770 --> 00:01:47,520 The original signal is uniquely recoverable from the 21 00:01:47,520 --> 00:01:50,350 set of samples. 22 00:01:50,350 --> 00:01:54,990 And the sampling theorem essentially was derived by 23 00:01:54,990 --> 00:02:00,100 observing or using the notion that sampling could be done by 24 00:02:00,100 --> 00:02:03,830 multiplication or modulation with an impulse train. 25 00:02:03,830 --> 00:02:07,900 And the sampling theorem developed by examining the 26 00:02:07,900 --> 00:02:12,060 consequence of the modulation property in the context of the 27 00:02:12,060 --> 00:02:13,940 Fourier transform. 28 00:02:13,940 --> 00:02:19,370 In particular, if we have our signal x of t and if 29 00:02:19,370 --> 00:02:25,200 multiplied by an impulse train to give us a sampled signal-- 30 00:02:25,200 --> 00:02:30,570 another impulse train whose values or areas are samples of 31 00:02:30,570 --> 00:02:36,220 the original time function, as I indicate here-- then in 32 00:02:36,220 --> 00:02:40,120 fact, if we examine this equation or equivalently, 33 00:02:40,120 --> 00:02:45,770 bringing x of t inside this sum, if we examine either of 34 00:02:45,770 --> 00:02:50,200 these equations in the frequency domain, the Fourier 35 00:02:50,200 --> 00:02:57,210 transform of x of p of t is the convolution of the Fourier 36 00:02:57,210 --> 00:03:00,410 transform of the original signal and the Fourier 37 00:03:00,410 --> 00:03:04,600 transform of the impulse train. 38 00:03:04,600 --> 00:03:07,110 Now the impulse train is a periodic signal. 39 00:03:07,110 --> 00:03:08,850 It's Fourier transform. 40 00:03:08,850 --> 00:03:11,950 Therefore, as we talked about with Fourier transforms is 41 00:03:11,950 --> 00:03:13,880 itself an impulse train. 42 00:03:13,880 --> 00:03:17,860 And when we do this convolution, then using the 43 00:03:17,860 --> 00:03:21,020 fact that the Fourier transform, the impulse train 44 00:03:21,020 --> 00:03:23,480 is an impulse train. 45 00:03:23,480 --> 00:03:28,100 The result of this convolution, then tells us 46 00:03:28,100 --> 00:03:33,540 that the Fourier transform of the sample signal or the 47 00:03:33,540 --> 00:03:37,610 impulse train, which represents the samples, is a 48 00:03:37,610 --> 00:03:43,220 sum of frequency-shifted replications of the Fourier 49 00:03:43,220 --> 00:03:47,030 transform of the original signal. 50 00:03:47,030 --> 00:03:50,040 So mathematically, that's the relationship. 51 00:03:50,040 --> 00:03:53,760 It essentially says that after sampling or modulation with an 52 00:03:53,760 --> 00:03:57,210 impulse train, the resulting spectrum is the original 53 00:03:57,210 --> 00:04:02,510 spectrum added to itself, shifted by integer multiples 54 00:04:02,510 --> 00:04:04,550 of the sampling frequency. 55 00:04:04,550 --> 00:04:06,740 Well, let's see that as we did last 56 00:04:06,740 --> 00:04:09,280 time in terms of pictures. 57 00:04:09,280 --> 00:04:14,240 And again, to remind you of the basic picture involved, if 58 00:04:14,240 --> 00:04:17,980 we have an original signal with a spectrum as I indicated 59 00:04:17,980 --> 00:04:21,600 here-- where it's band-limited with the highest frequency 60 00:04:21,600 --> 00:04:28,030 omega sub m-- and if the time function is sampled so that in 61 00:04:28,030 --> 00:04:32,630 the frequency domain we convolve this spectrum with 62 00:04:32,630 --> 00:04:38,990 the spectrum shown below, which is the spectrum of the 63 00:04:38,990 --> 00:04:45,260 impulse train, the convolution of these two is then the 64 00:04:45,260 --> 00:04:51,600 Fourier transform or spectrum of the sample time function. 65 00:04:51,600 --> 00:04:55,750 And so that's what we end up with here. 66 00:04:55,750 --> 00:04:59,480 And then as you recall, to recover the original time 67 00:04:59,480 --> 00:05:03,790 function from this-- as long as these individual triangles 68 00:05:03,790 --> 00:05:09,890 don't overlap--to recover it just simply involves passing 69 00:05:09,890 --> 00:05:13,820 the impulse train through a low-pass filter, in effect 70 00:05:13,820 --> 00:05:18,800 extracting just one of these replications of 71 00:05:18,800 --> 00:05:21,430 the original spectrum. 72 00:05:21,430 --> 00:05:29,880 So the overall system then for doing the sampling and then 73 00:05:29,880 --> 00:05:32,940 the reconstruction of the original signal from the 74 00:05:32,940 --> 00:05:37,600 samples, consists of multiplying the original time 75 00:05:37,600 --> 00:05:41,070 function by an impulse train. 76 00:05:41,070 --> 00:05:45,990 And that gives us then the sampled signal. 77 00:05:45,990 --> 00:05:51,380 The Fourier transform I show here of the original signal 78 00:05:51,380 --> 00:05:55,520 and after modulation with the impulse train, the resulting 79 00:05:55,520 --> 00:06:01,510 spectrum that we have is that replicated around integer 80 00:06:01,510 --> 00:06:04,460 multiples of the sampling frequency. 81 00:06:04,460 --> 00:06:09,440 And then finally, to recover the original signal or to 82 00:06:09,440 --> 00:06:14,470 generate a reconstructed signal, we then multiply this 83 00:06:14,470 --> 00:06:19,100 in the frequency domain by the frequency response of an ideal 84 00:06:19,100 --> 00:06:20,700 low-pass filter. 85 00:06:20,700 --> 00:06:25,820 And what that accomplishes for us then is recovering the 86 00:06:25,820 --> 00:06:28,290 original signal. 87 00:06:28,290 --> 00:06:31,560 Now in this picture, an important point that I raised 88 00:06:31,560 --> 00:06:35,370 last time, relates to the fact that in doing the 89 00:06:35,370 --> 00:06:39,270 reconstruction--well we've assumed-- is that in 90 00:06:39,270 --> 00:06:45,840 replicating these individual versions of the original 91 00:06:45,840 --> 00:06:49,710 signal, those replications don't overlap and so by 92 00:06:49,710 --> 00:06:52,340 passing this through a low-pass filter in fact, we 93 00:06:52,340 --> 00:06:54,970 can recover the original signal. 94 00:06:54,970 --> 00:07:00,560 Well, what that requires is that this frequency, omega sub 95 00:07:00,560 --> 00:07:05,420 m, be less than this frequency. 96 00:07:05,420 --> 00:07:12,310 And this frequency is omega sub s minus omega sub m. 97 00:07:12,310 --> 00:07:16,850 And so what we require is that the frequency omega sub m be 98 00:07:16,850 --> 00:07:20,530 less than omega sub s minus omega sub m. 99 00:07:20,530 --> 00:07:24,600 Or equivalently, what we require is that the sampling 100 00:07:24,600 --> 00:07:29,450 frequency be greater than twice the highest frequency in 101 00:07:29,450 --> 00:07:32,190 the original signal. 102 00:07:32,190 --> 00:07:38,360 Now, if in fact that condition is violated, then we end up 103 00:07:38,360 --> 00:07:40,950 with a very important effect. 104 00:07:40,950 --> 00:07:44,840 And that effect is referred to as aliasing. 105 00:07:44,840 --> 00:07:52,150 In particular, if we look back at our original example--we 106 00:07:52,150 --> 00:07:57,010 are here-- we were able to recover our original spectrum 107 00:07:57,010 --> 00:07:59,710 by low-pass filtering. 108 00:07:59,710 --> 00:08:05,860 If in fact the sampling frequency is not high enough 109 00:08:05,860 --> 00:08:10,680 to avoid aliasing, then what happens in that case is that 110 00:08:10,680 --> 00:08:17,480 the individual replications of the Fourier transform of the 111 00:08:17,480 --> 00:08:23,210 original signal overlap and what we end up with is some 112 00:08:23,210 --> 00:08:24,500 distortion. 113 00:08:24,500 --> 00:08:27,920 As you can see, if we try to pass this through a low-pass 114 00:08:27,920 --> 00:08:32,650 filter to recover the original signal, in fact we won't 115 00:08:32,650 --> 00:08:36,070 recover the original signal since these individual 116 00:08:36,070 --> 00:08:38,059 replications have overlapped. 117 00:08:38,059 --> 00:08:42,370 And this is the case where omega sub s minus omega sub m 118 00:08:42,370 --> 00:08:44,810 is less than omega sub s. 119 00:08:44,810 --> 00:08:48,040 In other words, the sampling frequency is not greater in 120 00:08:48,040 --> 00:08:52,110 this case than twice the highest frequency. 121 00:08:52,110 --> 00:08:56,370 So what happens here then is that in effect, higher 122 00:08:56,370 --> 00:08:59,880 frequencies get folded down into lower frequencies. 123 00:08:59,880 --> 00:09:02,390 What would come out of the low-pass filter is the 124 00:09:02,390 --> 00:09:04,960 reflection of some higher frequencies into lower 125 00:09:04,960 --> 00:09:06,380 frequencies. 126 00:09:06,380 --> 00:09:10,030 As I suggested a minute ago, that effect is 127 00:09:10,030 --> 00:09:12,290 referred to as aliasing. 128 00:09:12,290 --> 00:09:15,970 And in order to both understand that term better 129 00:09:15,970 --> 00:09:20,540 and to understand in fact the effect better, it's useful to 130 00:09:20,540 --> 00:09:25,310 examine this a little more closely for the specific 131 00:09:25,310 --> 00:09:27,760 example of a sinusoidal signal. 132 00:09:27,760 --> 00:09:29,500 So let's concentrate on that. 133 00:09:29,500 --> 00:09:36,490 And what we want to look at is the effect of aliasing when 134 00:09:36,490 --> 00:09:39,920 our input signal is a sinusoidal signal. 135 00:09:39,920 --> 00:09:44,750 Now to do that, what I want to show shortly is a 136 00:09:44,750 --> 00:09:47,690 computer-generated movie that we've made. 137 00:09:47,690 --> 00:09:51,650 And let's first walk through a few frames of it to give you-- 138 00:09:51,650 --> 00:09:57,490 first of all, to set up our notation and to suggest what 139 00:09:57,490 --> 00:10:00,530 it is that we're trying to demonstrate. 140 00:10:00,530 --> 00:10:06,250 Well, what we have is an input signal-- is 141 00:10:06,250 --> 00:10:07,490 a sinusoidal signal. 142 00:10:07,490 --> 00:10:12,720 And the spectrum or Fourier transform of that is an 143 00:10:12,720 --> 00:10:15,840 impulse in the frequency domain at the 144 00:10:15,840 --> 00:10:18,190 frequency of the sinusoid. 145 00:10:18,190 --> 00:10:22,380 We then have samples of that and when we sample that-- and 146 00:10:22,380 --> 00:10:25,250 for this particular example, it's sampled at 10 kilohertz-- 147 00:10:25,250 --> 00:10:31,980 this spectrum is then replicated at multiples of the 148 00:10:31,980 --> 00:10:33,640 sampling frequency. 149 00:10:33,640 --> 00:10:36,590 And I haven't shown negative frequencies here, but the 150 00:10:36,590 --> 00:10:42,040 contribution due to the negative frequency is at 10 151 00:10:42,040 --> 00:10:46,620 kilohertz minus the input sinusoid. 152 00:10:46,620 --> 00:10:51,320 We then carry out a reconstruction with an ideal 153 00:10:51,320 --> 00:10:52,490 low-pass filter. 154 00:10:52,490 --> 00:10:55,980 And the ideal low-pass filter is set at half the sampling 155 00:10:55,980 --> 00:10:58,450 frequency or 5 kilohertz. 156 00:10:58,450 --> 00:11:06,410 So what we have then is the input signal x of t and the 157 00:11:06,410 --> 00:11:09,840 impulse train x of p of t. 158 00:11:09,840 --> 00:11:15,100 And then the reconstructed signal is the output from the 159 00:11:15,100 --> 00:11:21,120 low-pass filter which I denote as x of r of t. 160 00:11:21,120 --> 00:11:26,320 Now as the input frequency x of t increases, this impulse 161 00:11:26,320 --> 00:11:30,110 moves up in frequency, but this impulse 162 00:11:30,110 --> 00:11:32,120 moves down in frequency. 163 00:11:32,120 --> 00:11:35,440 And so let's just look at a few frames as the input 164 00:11:35,440 --> 00:11:37,190 frequency increases. 165 00:11:37,190 --> 00:11:43,290 So we have here a case where the input frequency has moved 166 00:11:43,290 --> 00:11:46,760 up close to 5 kilohertz. 167 00:11:46,760 --> 00:11:52,260 As we continue further, these two impulses will cross and 168 00:11:52,260 --> 00:11:56,120 what we'll end up with, as I indicated, is aliasing. 169 00:11:56,120 --> 00:12:00,650 So here now is a case where we have aliasing. 170 00:12:00,650 --> 00:12:04,710 The replication of the negative frequency has crossed 171 00:12:04,710 --> 00:12:08,710 into the passband of the filter and the reconstructed 172 00:12:08,710 --> 00:12:13,910 sinusoid will now be the frequency associated with this 173 00:12:13,910 --> 00:12:18,980 impulse rather than the frequency associated with the 174 00:12:18,980 --> 00:12:20,520 original sinusoid. 175 00:12:20,520 --> 00:12:27,240 And to dramatize that even further, here is the example 176 00:12:27,240 --> 00:12:31,600 where now the input frequency has moved up close to 10 177 00:12:31,600 --> 00:12:36,840 kilohertz, but what comes out of the low-pass filter is a 178 00:12:36,840 --> 00:12:38,760 much lower frequency. 179 00:12:38,760 --> 00:12:42,200 And in fact, you can see that here is the reconstructed 180 00:12:42,200 --> 00:12:46,600 sinusoid, whereas here we have the input sinusoid. 181 00:12:49,180 --> 00:12:53,010 Well, now what I'm going to want to do is demonstrate this 182 00:12:53,010 --> 00:12:56,910 as I indicated with a computer-generated movie. 183 00:12:56,910 --> 00:13:05,380 And what we'll see is the effect of reconstructing from 184 00:13:05,380 --> 00:13:10,590 the samples using a low-pass filter for an input which 185 00:13:10,590 --> 00:13:12,910 changes in frequency and with a 186 00:13:12,910 --> 00:13:15,730 sampling rate of 10 kilohertz. 187 00:13:15,730 --> 00:13:19,050 And what we'll see in the first part of this movie is 188 00:13:19,050 --> 00:13:23,530 the input x of t and the reconstructed signal x of r of 189 00:13:23,530 --> 00:13:27,290 t without explicitly showing the samples. 190 00:13:27,290 --> 00:13:30,270 And then, at a later point, we'll also show this and 191 00:13:30,270 --> 00:13:33,380 indicate that in fact the samples of those two are 192 00:13:33,380 --> 00:13:37,580 equal, even though they themselves are not. 193 00:13:37,580 --> 00:13:42,260 So at the top, we'll have the input sinusoid without showing 194 00:13:42,260 --> 00:13:43,840 the samples. 195 00:13:43,840 --> 00:13:48,510 And its Fourier transform is an impulse in the frequency 196 00:13:48,510 --> 00:13:50,500 domain as we've indicated. 197 00:13:50,500 --> 00:13:55,670 And if we sample it, that impulse then gets replicated. 198 00:13:55,670 --> 00:13:59,700 And so its samples, in particular, will have a 199 00:13:59,700 --> 00:14:03,500 Fourier transform not only with an impulse at the input 200 00:14:03,500 --> 00:14:06,595 sinusoidal frequency, but also at 10 201 00:14:06,595 --> 00:14:09,260 kilohertz minus that frequency. 202 00:14:09,260 --> 00:14:13,330 Now for the reconstruction, we passed the samples through an 203 00:14:13,330 --> 00:14:14,690 ideal low-pass filter. 204 00:14:14,690 --> 00:14:18,620 I picked the cutoff frequency of the low-pass filter at half 205 00:14:18,620 --> 00:14:21,530 the sampling frequency, namely 5 kilohertz. 206 00:14:21,530 --> 00:14:25,810 And here, what we see is that the output reconstructed 207 00:14:25,810 --> 00:14:29,540 signal in fact matches in frequency the input signal. 208 00:14:29,540 --> 00:14:34,380 Now as we change the input frequency, the reconstructed 209 00:14:34,380 --> 00:14:41,010 sinusoid is identical until we get to an input frequency, 210 00:14:41,010 --> 00:14:44,480 which exceeds half the sampling frequency. 211 00:14:44,480 --> 00:14:48,760 At that point we have aliasing and while the input frequency 212 00:14:48,760 --> 00:14:52,990 is increasing, the output frequency in fact is 213 00:14:52,990 --> 00:14:56,920 decreasing because that's what's inside the passband of 214 00:14:56,920 --> 00:14:58,960 the filter. 215 00:14:58,960 --> 00:15:01,310 Now let's sweep it back. 216 00:15:01,310 --> 00:15:04,630 And as the input frequency decreases, the output 217 00:15:04,630 --> 00:15:08,440 frequency increases until there's no aliasing and now 218 00:15:08,440 --> 00:15:11,410 the output reconstructed signal is equal to the input. 219 00:15:14,320 --> 00:15:18,670 So we've sampled a signal and then reconstructed the signal 220 00:15:18,670 --> 00:15:20,240 from the samples. 221 00:15:20,240 --> 00:15:24,740 And keep in mind, that given a set of samples, there are lots 222 00:15:24,740 --> 00:15:26,980 of continuous curves that we can thread 223 00:15:26,980 --> 00:15:28,640 through the set of samples. 224 00:15:28,640 --> 00:15:32,650 The one that we picked, of course, is the one consistent 225 00:15:32,650 --> 00:15:35,900 with the assumption about the signal bandwidth. 226 00:15:35,900 --> 00:15:40,130 In particular, we've reconstructed the signal whose 227 00:15:40,130 --> 00:15:44,430 spectrum falls within the passband of the filter. 228 00:15:44,430 --> 00:15:49,570 Now what I'd like to show is the same reconstruction and 229 00:15:49,570 --> 00:15:53,290 input as I showed before, but now let's look at the samples 230 00:15:53,290 --> 00:15:57,920 and what we'll see is that when there's aliasing, even 231 00:15:57,920 --> 00:16:02,420 though the output-- the reconstructed signal-- is not 232 00:16:02,420 --> 00:16:04,690 identical to the input. 233 00:16:04,690 --> 00:16:08,420 In fact it's consistent with the input samples that is 234 00:16:08,420 --> 00:16:11,620 sampling the reconstructed signal. 235 00:16:11,620 --> 00:16:14,110 It gives a set of samples identical to the samples of 236 00:16:14,110 --> 00:16:17,920 the input and it's just that the interpolation in between 237 00:16:17,920 --> 00:16:21,565 those samples is an interpolation consistent with 238 00:16:21,565 --> 00:16:24,690 the assumed bandwidth of the input based on 239 00:16:24,690 --> 00:16:25,940 the sampling theorem. 240 00:16:25,940 --> 00:16:31,080 So let's now look at that with the samples also shown along 241 00:16:31,080 --> 00:16:33,760 with the sinusoid. 242 00:16:33,760 --> 00:16:37,320 So at the top, we have the input sinusoid together with 243 00:16:37,320 --> 00:16:38,480 its samples. 244 00:16:38,480 --> 00:16:40,890 The bottom trace is the Fourier transform of the 245 00:16:40,890 --> 00:16:42,860 sampled waveform. 246 00:16:42,860 --> 00:16:47,260 The middle trace is the reconstructed sinusoid 247 00:16:47,260 --> 00:16:48,970 together with its samples. 248 00:16:48,970 --> 00:16:52,200 And notice, of course, that the samples of the input or 249 00:16:52,200 --> 00:16:55,830 reconstructed signal are identical. 250 00:16:55,830 --> 00:17:01,820 And also the input sinusoidal frequency and the output 251 00:17:01,820 --> 00:17:04,900 sinusoidal frequency are identical. 252 00:17:04,900 --> 00:17:09,510 And we now increase the frequency at the input. 253 00:17:09,510 --> 00:17:13,150 The reconstructed sinusoid tracks the input in frequency 254 00:17:13,150 --> 00:17:16,970 and, of course, the samples of the two are identical. 255 00:17:16,970 --> 00:17:20,829 The interpolation in between the samples is identical 256 00:17:20,829 --> 00:17:25,200 because of the fact that the input frequency is still less 257 00:17:25,200 --> 00:17:26,490 than half the sampling frequency. 258 00:17:31,500 --> 00:17:35,460 And so, as long as the input is frequency is less than half 259 00:17:35,460 --> 00:17:38,570 the sampling frequency, not only will the samples be 260 00:17:38,570 --> 00:17:42,550 identical, but also the reconstructed continuous 261 00:17:42,550 --> 00:17:45,725 waveform will match the input waveform. 262 00:17:56,820 --> 00:17:59,880 Now when we get to half the sampling frequency, we're just 263 00:17:59,880 --> 00:18:01,440 on the verge of aliasing. 264 00:18:01,440 --> 00:18:05,680 This isn't aliasing quite yet, but any increase in the input 265 00:18:05,680 --> 00:18:09,870 frequency will now generate aliasing. 266 00:18:09,870 --> 00:18:13,460 We now have aliasing, the output frequency is lower than 267 00:18:13,460 --> 00:18:15,840 the input frequency, but notice that 268 00:18:15,840 --> 00:18:19,640 the samples are identical. 269 00:18:19,640 --> 00:18:23,410 Now the low-pass filter is interpolating in between those 270 00:18:23,410 --> 00:18:27,600 samples with a sinusoid that falls within the passband of 271 00:18:27,600 --> 00:18:31,260 the low-pass filter, which no longer matches the frequency 272 00:18:31,260 --> 00:18:34,140 of the input sinusoid. 273 00:18:34,140 --> 00:18:37,420 But the important point is that even when we have 274 00:18:37,420 --> 00:18:42,290 aliasing, the samples of the reconstructed waveform are 275 00:18:42,290 --> 00:18:47,290 identical to the samples of the original waveform. 276 00:18:47,290 --> 00:18:51,340 And notice that as the input frequency increases, in fact 277 00:18:51,340 --> 00:18:55,280 the interpolated output, the reconstructed output has 278 00:18:55,280 --> 00:18:58,270 decreased in frequency. 279 00:18:58,270 --> 00:19:01,895 Now as the input frequency begins to get closer to 10 280 00:19:01,895 --> 00:19:07,020 kilohertz-- in fact your eye tends to also interpolate 281 00:19:07,020 --> 00:19:12,770 between the samples with a frequency that is lower than 282 00:19:12,770 --> 00:19:13,910 the input frequency. 283 00:19:13,910 --> 00:19:16,800 And that's particularly evident here. 284 00:19:16,800 --> 00:19:21,050 Notice that the input samples in fact look like they would 285 00:19:21,050 --> 00:19:25,310 be associated with a much lower frequency sinusoid, than 286 00:19:25,310 --> 00:19:28,630 in fact was the sinusoid that generated them. 287 00:19:28,630 --> 00:19:32,010 The lower-frequency sinusoid in fact corresponds to the 288 00:19:32,010 --> 00:19:34,070 reconstructed one. 289 00:19:34,070 --> 00:19:38,650 Now as we sweep back down, the aliasing eventually disappears 290 00:19:38,650 --> 00:19:40,450 and the output sinusoid tracks the 291 00:19:40,450 --> 00:19:41,760 input sinusoid in frequency. 292 00:19:44,490 --> 00:19:47,820 So we've seen the effect of aliasing for sinusoidal 293 00:19:47,820 --> 00:19:50,170 signals in terms of waveforms. 294 00:19:50,170 --> 00:19:53,270 Now let's hear how it sounds. 295 00:19:53,270 --> 00:19:59,760 Now what we have for this demonstration is an oscillator 296 00:19:59,760 --> 00:20:02,330 and a sampler. 297 00:20:02,330 --> 00:20:06,460 And the output of the sampler goes into a low-pass filter. 298 00:20:06,460 --> 00:20:12,240 So the input from the oscillator goes into the 299 00:20:12,240 --> 00:20:16,930 sampler and the output of the sampler goes into 300 00:20:16,930 --> 00:20:18,840 the low-pass filter. 301 00:20:18,840 --> 00:20:23,600 The sampler frequency is 10 kilohertz. 302 00:20:23,600 --> 00:20:28,880 And so the low-pass filter has a cutoff frequency as I 303 00:20:28,880 --> 00:20:32,740 indicate here, of 5 kilohertz. 304 00:20:32,740 --> 00:20:39,740 And what we'll listen to is the reconstructed output as 305 00:20:39,740 --> 00:20:44,280 the oscillator input frequency varies. 306 00:20:44,280 --> 00:20:48,850 And recall that what should happen is that when the 307 00:20:48,850 --> 00:20:52,340 oscillator input frequency gets past half the sampling 308 00:20:52,340 --> 00:20:56,600 frequency, we should hear aliasing. 309 00:20:56,600 --> 00:20:58,430 So we'll start the oscillator at 2 kilohertz. 310 00:20:58,430 --> 00:20:59,460 [OSCILLATOR SOUND IN BACKGROUND] 311 00:20:59,460 --> 00:21:02,420 PROFESSOR: And keep in mind that what you see on the dial 312 00:21:02,420 --> 00:21:04,960 is the input frequency, what you hear 313 00:21:04,960 --> 00:21:06,850 is the output frequency. 314 00:21:06,850 --> 00:21:09,880 As long as the input frequency is less than half the sampling 315 00:21:09,880 --> 00:21:13,220 frequency-- in other words, 5 kilohertz -- the reconstructed 316 00:21:13,220 --> 00:21:17,150 signal sounds identical to the input. 317 00:21:17,150 --> 00:21:20,660 Now at 5 kilohertz, we're right on the verge of 318 00:21:20,660 --> 00:21:24,505 aliasing, and when we increase the input frequency past 5 319 00:21:24,505 --> 00:21:27,330 kilohertz, the reconstructed frequency 320 00:21:27,330 --> 00:21:29,370 in fact will decrease. 321 00:21:29,370 --> 00:21:33,740 So as we move, for example, from 5 kilohertz up to let's 322 00:21:33,740 --> 00:21:36,440 say, 6 kilohertz. 323 00:21:36,440 --> 00:21:41,450 6 kilohertz in fact gets aliased down to, what? 324 00:21:41,450 --> 00:21:45,580 It gets aliased down to 4 kilohertz. 325 00:21:45,580 --> 00:21:52,190 So 6 kilohertz at the input is 4 kilohertz at the output. 326 00:21:52,190 --> 00:21:55,630 Now, if we move up even further, 7 kilohertz at the 327 00:21:55,630 --> 00:22:02,380 input gets aliased down to 3 kilohertz at the output. 328 00:22:02,380 --> 00:22:07,540 So that, then is an audio demonstration of aliasing. 329 00:22:07,540 --> 00:22:12,680 So to summarize, if we sample a signal and then reconstruct 330 00:22:12,680 --> 00:22:16,020 from the samples using a low-pass filter, as long as 331 00:22:16,020 --> 00:22:18,770 the sampling frequency is greater than twice the highest 332 00:22:18,770 --> 00:22:22,300 frequency in the signal we reconstruct exactly. 333 00:22:22,300 --> 00:22:25,900 If on the other hand, the sampling frequency is too low, 334 00:22:25,900 --> 00:22:27,160 less than twice the highest 335 00:22:27,160 --> 00:22:30,310 frequency, then we get aliasing. 336 00:22:30,310 --> 00:22:34,270 In other words, higher frequencies get folded or 337 00:22:34,270 --> 00:22:37,220 reflected down into lower frequencies as they come 338 00:22:37,220 --> 00:22:40,430 through the low-pass filter. 339 00:22:40,430 --> 00:22:44,560 Now, one of the common applications of the whole 340 00:22:44,560 --> 00:22:49,880 concept of sampling is the use of sampling to convert a 341 00:22:49,880 --> 00:22:55,080 continuous-time signal into a discrete-time signal to carry 342 00:22:55,080 --> 00:22:59,460 out what's often referred to as discrete-time processing of 343 00:22:59,460 --> 00:23:01,690 continuous-time signals. 344 00:23:01,690 --> 00:23:05,810 And this in fact is something that we'll be talking about in 345 00:23:05,810 --> 00:23:07,520 a fair amount of detail, beginning 346 00:23:07,520 --> 00:23:09,290 with the next lecture. 347 00:23:09,290 --> 00:23:14,270 But let me indicate that for that kind of processing, 348 00:23:14,270 --> 00:23:17,710 essentially what happens, is that we begin with the 349 00:23:17,710 --> 00:23:21,970 continuous-time signal and convert it to a discrete-time 350 00:23:21,970 --> 00:23:25,930 signal, carry out the discrete-time processing, and 351 00:23:25,930 --> 00:23:29,200 then convert back to continuous-time. 352 00:23:29,200 --> 00:23:33,770 And the conversion from a continuous-time signal to a 353 00:23:33,770 --> 00:23:39,240 discrete-time signal in fact, is done by exploiting 354 00:23:39,240 --> 00:23:44,160 sampling, specifically by sampling the continuous-time 355 00:23:44,160 --> 00:23:49,210 signal with an impulse train and then converting the 356 00:23:49,210 --> 00:23:54,600 impulse train into a sequence in a matter that I'll talk 357 00:23:54,600 --> 00:23:57,790 about in more detail next time. 358 00:23:57,790 --> 00:24:01,800 Now in doing that-- of course, as you can imagine-- it's 359 00:24:01,800 --> 00:24:05,480 important since we want an accurate representation of the 360 00:24:05,480 --> 00:24:08,850 original continuous-time signal, to choose the sampling 361 00:24:08,850 --> 00:24:12,800 frequency, to very carefully avoid aliasing. 362 00:24:12,800 --> 00:24:16,560 And so in fact, in that context and in many other 363 00:24:16,560 --> 00:24:19,890 contexts, aliasing is something that we're very 364 00:24:19,890 --> 00:24:21,750 eager to avoid. 365 00:24:21,750 --> 00:24:25,880 However, it's also important to understand that aliasing 366 00:24:25,880 --> 00:24:27,410 isn't all bad. 367 00:24:27,410 --> 00:24:30,560 And there are some very specific contexts in which 368 00:24:30,560 --> 00:24:36,230 aliasing is very useful and very heavily exploited. 369 00:24:36,230 --> 00:24:42,280 One example of a very useful context of aliasing is when 370 00:24:42,280 --> 00:24:45,820 you want to look at things that happen at frequencies 371 00:24:45,820 --> 00:24:48,550 that you can't look at, for one reason or another. 372 00:24:48,550 --> 00:24:51,800 And sampling and aliasing is used to map those into lower 373 00:24:51,800 --> 00:24:52,950 frequencies. 374 00:24:52,950 --> 00:24:56,930 One very common example of that is the use of the 375 00:24:56,930 --> 00:25:03,340 stroboscope which was invented by Dr. Harold Edgerton at MIT. 376 00:25:03,340 --> 00:25:07,810 And sometime earlier, in fact we had the opportunity to 377 00:25:07,810 --> 00:25:12,180 visit Dr. Edgerton's laboratory at MIT and see some 378 00:25:12,180 --> 00:25:13,590 examples of this. 379 00:25:13,590 --> 00:25:17,535 So I'd like to-- as a conclusion to this lecture-- 380 00:25:17,535 --> 00:25:22,195 take you on a visit to the strobe lab at MIT. 381 00:25:24,900 --> 00:25:28,200 In the lecture-- in discussing aliasing-- we've stressed the 382 00:25:28,200 --> 00:25:31,380 fact that in most situations, it's something that 383 00:25:31,380 --> 00:25:33,950 we'd like to avoid. 384 00:25:33,950 --> 00:25:38,340 However, right now we're at MIT, in Strobe Alley as it's 385 00:25:38,340 --> 00:25:42,860 called, on the way to visit the laboratory of my MIT 386 00:25:42,860 --> 00:25:45,850 colleague, Professor Harold Edgerton, where in fact 387 00:25:45,850 --> 00:25:49,430 aliasing is an everyday occurrence. 388 00:25:49,430 --> 00:25:54,970 Basically, the idea is the following-- that if in fact 389 00:25:54,970 --> 00:25:58,600 you want to make measurements at frequencies that, for one 390 00:25:58,600 --> 00:26:03,480 reason or another, you can't measure, then sampling and, 391 00:26:03,480 --> 00:26:07,140 consequently, aliasing can be used to bring those 392 00:26:07,140 --> 00:26:09,770 frequencies down into a frequency range 393 00:26:09,770 --> 00:26:12,200 that you can measure. 394 00:26:12,200 --> 00:26:18,230 Well, Professor Edgerton alias Doc Edgerton invented the 395 00:26:18,230 --> 00:26:20,700 stroboscope for exactly that reason. 396 00:26:20,700 --> 00:26:23,460 And, kind of, the idea is the following. 397 00:26:23,460 --> 00:26:26,720 The eye, essentially, is a low-pass filter and so there 398 00:26:26,720 --> 00:26:31,590 are things that happen at frequencies above which your 399 00:26:31,590 --> 00:26:33,650 eye can track. 400 00:26:33,650 --> 00:26:39,800 And by sampling with light pulses, sampling in time, what 401 00:26:39,800 --> 00:26:44,430 in effect you're able to do is sample in such a way that 402 00:26:44,430 --> 00:26:48,810 higher frequencies get aliased down to lower frequencies so 403 00:26:48,810 --> 00:26:51,660 that, in fact, your eye can track them. 404 00:26:51,660 --> 00:26:57,390 So let's take a look inside the lab and in fact see an 405 00:26:57,390 --> 00:27:00,600 illustration of this strobe and some of its effects. 406 00:27:05,310 --> 00:27:10,260 Let me introduce you to my MIT colleague, Doc Edgerton. 407 00:27:10,260 --> 00:27:13,760 Also by the way, this is a great place for kids of all 408 00:27:13,760 --> 00:27:17,710 ages and so my daughter, Justine, insisted on coming 409 00:27:17,710 --> 00:27:19,580 along to also help out. 410 00:27:19,580 --> 00:27:22,600 Doc, maybe we could begin with you just saying a little bit 411 00:27:22,600 --> 00:27:26,160 about what the strobe is and what some of the history is? 412 00:27:26,160 --> 00:27:28,950 DR. HAROLD EDGERTON: Sure, it's a very simple application 413 00:27:28,950 --> 00:27:30,390 of intermittent light. 414 00:27:30,390 --> 00:27:36,960 And this is a xenon lamp that flashes in a controlled rate 415 00:27:36,960 --> 00:27:39,840 depending on this knob which Justine's going to turn. 416 00:27:39,840 --> 00:27:43,740 And we're going to look at a motor that's driving an 417 00:27:43,740 --> 00:27:46,050 unbalanced weight to set up some [INAUDIBLE] 418 00:27:46,050 --> 00:27:48,420 oscillations in the spring. 419 00:27:48,420 --> 00:27:50,456 I'll turn on the motor. 420 00:27:50,456 --> 00:27:51,932 I'll turn on the strobe. 421 00:27:51,932 --> 00:27:53,182 [STROBOSCOPE SOUND IN BACKGROUND] 422 00:28:10,136 --> 00:28:12,838 DR. HAROLD EDGERTON: Just get the right range. 423 00:28:12,838 --> 00:28:15,295 All right, Justine, turn that now, until it stops. 424 00:28:17,905 --> 00:28:20,710 See that, Justine, the frequency is that the light, 425 00:28:20,710 --> 00:28:22,890 which corresponds to the frequency of the motor. 426 00:28:22,890 --> 00:28:25,820 And it's a little less or a little more, when you lean to 427 00:28:25,820 --> 00:28:27,380 go forward to backwards. 428 00:28:27,380 --> 00:28:29,480 PROFESSOR: Doc, maybe we could turn this 429 00:28:29,480 --> 00:28:30,560 strobe off for minute. 430 00:28:30,560 --> 00:28:38,230 And let me point out, by the way, the fact that when we're 431 00:28:38,230 --> 00:28:40,770 looking at this without the strobe on, what we're seeing 432 00:28:40,770 --> 00:28:43,560 essentially are frequencies that our eye can't track. 433 00:28:43,560 --> 00:28:48,170 So we can't see the motor turning and we can't really 434 00:28:48,170 --> 00:28:49,930 see other than with a blur. 435 00:28:49,930 --> 00:28:52,190 We can't see the movement of the spring. 436 00:28:52,190 --> 00:28:54,770 And so I guess, your point is that when we put the strobe 437 00:28:54,770 --> 00:28:59,070 on, we're essentially sampling this. 438 00:28:59,070 --> 00:29:03,950 And now we brought this down to a frequency that our eye is 439 00:29:03,950 --> 00:29:06,080 able to track. 440 00:29:06,080 --> 00:29:13,210 In fact, I guess if we turn the incandescent light off, 441 00:29:13,210 --> 00:29:18,600 what we'll be able to really bring out are the alias 442 00:29:18,600 --> 00:29:19,220 frequencies. 443 00:29:19,220 --> 00:29:23,010 So now, what we're looking at in fact are the alias 444 00:29:23,010 --> 00:29:24,430 frequencies. 445 00:29:24,430 --> 00:29:26,430 The spring, of course, is moving a lot faster than we 446 00:29:26,430 --> 00:29:27,640 see it, isn't that right? 447 00:29:27,640 --> 00:29:29,640 DR. HAROLD EDGERTON: Yes, it's going 448 00:29:29,640 --> 00:29:32,781 approximately 30 times a second. 449 00:29:32,781 --> 00:29:36,150 The motor is going far from 30 times a second. 450 00:29:36,150 --> 00:29:39,150 I will speed this up while I hit the next mode, where I get 451 00:29:39,150 --> 00:29:41,105 a figure 8 out of this thing. 452 00:29:41,105 --> 00:29:42,310 You want to see that now? 453 00:29:42,310 --> 00:29:44,542 PROFESSOR: Yeah, great. 454 00:29:44,542 --> 00:29:47,500 DR. HAROLD EDGERTON: [INAUDIBLE] 455 00:29:47,500 --> 00:30:01,330 [MACHINE NOISE GETS LOUDER] 456 00:30:01,330 --> 00:30:03,660 PROFESSOR: So what we'll be seeing now is essentially a 457 00:30:03,660 --> 00:30:04,995 second harmonic, is it? 458 00:30:04,995 --> 00:30:07,590 DR. HAROLD EDGERTON: Yes, that's the second harmonic. 459 00:30:07,590 --> 00:30:09,960 PROFESSOR: Justine, you think you could make that spring 460 00:30:09,960 --> 00:30:13,890 dance around a little bit by changing the strobe frequency? 461 00:30:13,890 --> 00:30:15,247 DR. HAROLD EDGERTON: Yeah, you need to go around that way. 462 00:30:15,247 --> 00:30:17,190 You go around this way. 463 00:30:17,190 --> 00:30:18,340 PROFESSOR: Hey, that's really neat. 464 00:30:18,340 --> 00:30:20,820 Let's turn the lights back on if we can. 465 00:30:20,820 --> 00:30:22,400 DR. HAROLD EDGERTON: Tomorrow [INAUDIBLE] 466 00:30:22,400 --> 00:30:24,790 it's periodic, it has to be periodic. 467 00:30:24,790 --> 00:30:27,090 PROFESSOR: And what's interesting now, if we look at 468 00:30:27,090 --> 00:30:31,280 this in a-- let's see, can you flip this strobe off again? 469 00:30:31,280 --> 00:30:33,580 DR. HAROLD EDGERTON: Sure. 470 00:30:33,580 --> 00:30:35,370 DR. HAROLD EDGERTON: Notice, Justine, when we look at the 471 00:30:35,370 --> 00:30:37,790 spring now, all that we can see is a blur. 472 00:30:37,790 --> 00:30:40,890 And you really can't see-- because your eye can't track 473 00:30:40,890 --> 00:30:42,580 it, you can't see things happening 474 00:30:42,580 --> 00:30:45,650 spatially in frequency. 475 00:30:45,650 --> 00:30:48,250 You said, by the way, that this was originally 476 00:30:48,250 --> 00:30:50,190 demonstrated at the World's Fair. 477 00:30:50,190 --> 00:30:52,540 DR. HAROLD EDGERTON: This particular instrument was made 478 00:30:52,540 --> 00:30:55,370 the World's Fair in Chicago-- not the last one, but the one 479 00:30:55,370 --> 00:30:55,900 before that. 480 00:30:55,900 --> 00:30:56,225 PROFESSOR: Wow. 481 00:30:56,225 --> 00:30:59,268 DR. HAROLD EDGERTON: It was a--you see it all scratched up 482 00:30:59,268 --> 00:31:01,900 because it's a-- the [INAUDIBLE] 483 00:31:01,900 --> 00:31:04,880 use this thing is to break the springs. 484 00:31:04,880 --> 00:31:09,080 Because of the uses, you try to find the parts that fail. 485 00:31:09,080 --> 00:31:09,510 PROFESSOR: I see. 486 00:31:09,510 --> 00:31:11,210 You put them under stress and fatigue and-- 487 00:31:11,210 --> 00:31:12,946 DR. HAROLD EDGERTON: If I run this for half an hour and so, 488 00:31:12,946 --> 00:31:15,010 the spring will break. 489 00:31:15,010 --> 00:31:19,000 And they work on automobiles, they run them 490 00:31:19,000 --> 00:31:20,175 until something vibrates. 491 00:31:20,175 --> 00:31:22,172 Then they find out what the part is and what 492 00:31:22,172 --> 00:31:24,300 frequency it is. 493 00:31:24,300 --> 00:31:26,670 PROFESSOR: Well let's--by the way, I bet you run this for a 494 00:31:26,670 --> 00:31:28,486 lot more than half an hour in this state. 495 00:31:28,486 --> 00:31:29,320 DR. HAROLD EDGERTON: Oh, yeah, we've broken many, many 496 00:31:29,320 --> 00:31:33,100 springs in this thing-- and it's continuous. 497 00:31:33,100 --> 00:31:36,680 We experiment, try new things on it. 498 00:31:36,680 --> 00:31:39,200 PROFESSOR: Maybe we could look at a couple of other things. 499 00:31:39,200 --> 00:31:40,570 How about the fan? 500 00:31:40,570 --> 00:31:41,070 Maybe-- 501 00:31:41,070 --> 00:31:42,905 DR. HAROLD EDGERTON: Sure, I'll plug this fan and this is 502 00:31:42,905 --> 00:31:44,886 a classic experiment for the strobe. 503 00:31:44,886 --> 00:31:45,690 [MACHINE NOISE STOPS] 504 00:31:45,690 --> 00:31:46,680 DR. HAROLD EDGERTON: That's a good idea. 505 00:31:46,680 --> 00:31:47,946 Get that thing off. 506 00:31:47,946 --> 00:31:50,562 Makes too much noise. 507 00:31:50,562 --> 00:31:52,825 PROFESSOR: Guess, we move that over here. 508 00:31:52,825 --> 00:31:55,750 DR. HAROLD EDGERTON: This is just an ordinary electric fan, 509 00:31:55,750 --> 00:31:58,720 but it has a mark on one blade, so that you can 510 00:31:58,720 --> 00:32:00,810 identify it. 511 00:32:00,810 --> 00:32:02,060 We'll plug it in, get it up to speed. 512 00:32:09,410 --> 00:32:11,650 PROFESSOR: This looks like a fan that was also demonstrated 513 00:32:11,650 --> 00:32:14,775 in the World's Fair, a few years ago. 514 00:32:14,775 --> 00:32:16,025 DR. HAROLD EDGERTON: Yeah, could've been. 515 00:32:18,600 --> 00:32:20,700 There was a movie Quicker'n a Wink had this 516 00:32:20,700 --> 00:32:22,170 thing in there and-- 517 00:32:22,170 --> 00:32:24,790 PROFESSOR: With this very fan? 518 00:32:24,790 --> 00:32:26,090 DR. HAROLD EDGERTON: Well, one like it. 519 00:32:26,090 --> 00:32:28,220 It was loaned to MGM. 520 00:32:28,220 --> 00:32:35,900 And Pete Smith, he said he wanted me to throw out a 521 00:32:35,900 --> 00:32:37,030 custard pie into it. 522 00:32:37,030 --> 00:32:40,680 I said, no, I'm a serious scientist. 523 00:32:40,680 --> 00:32:44,730 So he says, let's compromise on the egg. 524 00:32:44,730 --> 00:32:47,820 So we dropped an egg into it and you would see a high-speed 525 00:32:47,820 --> 00:32:49,780 movie of the egg dropping. 526 00:32:49,780 --> 00:32:52,090 No, not with the strobe, but with this [INAUDIBLE] 527 00:32:52,090 --> 00:32:54,065 PROFESSOR: That was with the high-speed photography. 528 00:32:54,065 --> 00:32:56,860 DR. HAROLD EDGERTON: High-speed movies, yeah. 529 00:32:56,860 --> 00:32:59,080 PROFESSOR: So again, I guess, without the strobe, when we 530 00:32:59,080 --> 00:33:04,470 look at it, what we're looking at are frequencies that are 531 00:33:04,470 --> 00:33:06,020 much higher than the eye can follow. 532 00:33:06,020 --> 00:33:09,750 And now, with the strobe on, you can see both the alias 533 00:33:09,750 --> 00:33:12,470 frequency and you can also see the original frequency because 534 00:33:12,470 --> 00:33:15,970 we had the incandescent light on. 535 00:33:15,970 --> 00:33:20,210 Let's turn down the background light again. 536 00:33:20,210 --> 00:33:22,580 And then, really all that we're able to see are the 537 00:33:22,580 --> 00:33:25,770 aliasing frequencies. 538 00:33:25,770 --> 00:33:30,650 And I guess when we see more than one mark, that means that 539 00:33:30,650 --> 00:33:32,910 we're actually running it at-- 540 00:33:32,910 --> 00:33:34,330 DR. HAROLD EDGERTON: Four times the speed of the fan. 541 00:33:34,330 --> 00:33:35,870 PROFESSOR: --four times the speed the fan, yeah. 542 00:33:35,870 --> 00:33:38,140 DR. HAROLD EDGERTON: You see a little variation in the-- 543 00:33:38,140 --> 00:33:38,515 PROFESSOR: Oh, yeah. 544 00:33:38,515 --> 00:33:38,650 Right. 545 00:33:38,650 --> 00:33:39,990 DR. HAROLD EDGERTON: It's because the blades aren't 546 00:33:39,990 --> 00:33:41,050 exactly the same. 547 00:33:41,050 --> 00:33:42,450 PROFESSOR: Actually, this gives me a chance to 548 00:33:42,450 --> 00:33:46,780 illustrate another important point related to the lecture. 549 00:33:46,780 --> 00:33:50,210 Let's see, can we bring it down to a frequency so that we 550 00:33:50,210 --> 00:33:51,620 only get one mark? 551 00:33:51,620 --> 00:33:52,870 DR. HAROLD EDGERTON: Sure. 552 00:33:56,171 --> 00:33:58,540 You may miss this because it's too lowered to just 553 00:33:58,540 --> 00:34:00,550 one blade there now. 554 00:34:00,550 --> 00:34:02,580 PROFESSOR: So the way we have it now, we've essentially 555 00:34:02,580 --> 00:34:05,800 aliased the fan's speed down so that it's just a little 556 00:34:05,800 --> 00:34:07,950 higher than DC. 557 00:34:07,950 --> 00:34:11,380 And now, I'm right at DC. 558 00:34:11,380 --> 00:34:14,810 And now, if I go down just a little further, in fact it 559 00:34:14,810 --> 00:34:17,770 looks like the fan is turning backwards. 560 00:34:17,770 --> 00:34:22,159 And if you think of this in the context of aliasing, it's 561 00:34:22,159 --> 00:34:25,670 like the two impulses in the frequency domain 562 00:34:25,670 --> 00:34:26,929 have crossed over. 563 00:34:26,929 --> 00:34:30,540 And what you get in effect, if you analyze it mathematically, 564 00:34:30,540 --> 00:34:32,770 is you get a phase reversal. 565 00:34:32,770 --> 00:34:37,870 And it wasn't until I first understood about aliasing, by 566 00:34:37,870 --> 00:34:40,790 the way, Doc, that I understood why when I went to 567 00:34:40,790 --> 00:34:44,420 Western movies, every once in a while you'd see the wagon 568 00:34:44,420 --> 00:34:46,570 wheels turning backwards. 569 00:34:46,570 --> 00:34:49,600 Then there's the wagon wheels of the Western movie going 570 00:34:49,600 --> 00:34:51,600 backwards, I guess. 571 00:34:51,600 --> 00:34:53,879 And, Justine, why don't you see if you can-- 572 00:34:53,879 --> 00:34:54,980 DR. HAROLD EDGERTON: Too much flicker there. 573 00:34:54,980 --> 00:34:57,460 Why don't you bring it up so you get two marks. 574 00:34:57,460 --> 00:34:59,740 PROFESSOR: See if you can bring the frequency up so that 575 00:34:59,740 --> 00:35:00,990 you get two marks. 576 00:35:03,240 --> 00:35:04,060 DR. HAROLD EDGERTON: You turn that, Justine. 577 00:35:04,060 --> 00:35:08,100 Grab right ahold of that and give it a big twist. 578 00:35:08,100 --> 00:35:09,350 You went past it. 579 00:35:11,940 --> 00:35:14,320 They're not regular there now. 580 00:35:14,320 --> 00:35:14,910 Here we are. 581 00:35:14,910 --> 00:35:16,150 Now hold it right there. 582 00:35:16,150 --> 00:35:19,290 Put your finger on there, hold the dial. 583 00:35:19,290 --> 00:35:21,200 It's flashing twice per revolution now, Al. 584 00:35:21,200 --> 00:35:23,620 PROFESSOR: I guess another thing that this demonstrates 585 00:35:23,620 --> 00:35:27,500 is something that I've heard a long time ago, which is that 586 00:35:27,500 --> 00:35:30,850 you should never use a power saw with a fluorescent light 587 00:35:30,850 --> 00:35:32,940 because the fluorescent light gives you a little bit of a 588 00:35:32,940 --> 00:35:36,230 strobe effect and you could actually convince yourself 589 00:35:36,230 --> 00:35:39,670 that that's standing still and make the mistake of trying to 590 00:35:39,670 --> 00:35:40,870 put your finger between the blades. 591 00:35:40,870 --> 00:35:42,850 DR. HAROLD EDGERTON: You want to stick your finger in there? 592 00:35:42,850 --> 00:35:44,330 PROFESSOR: No, I don't think I want to try it. 593 00:35:44,330 --> 00:35:45,290 How about you, Justine? 594 00:35:45,290 --> 00:35:45,800 What do you think? 595 00:35:45,800 --> 00:35:47,910 Is that standing still or is that moving? 596 00:35:47,910 --> 00:35:49,800 DR. HAROLD EDGERTON: She knows it's going. 597 00:35:49,800 --> 00:35:53,140 We won't let her get close to that fan. 598 00:35:53,140 --> 00:35:56,580 PROFESSOR: Actually if we turn the lights back on again, what 599 00:35:56,580 --> 00:35:59,470 that will let us see once again is that we can see both 600 00:35:59,470 --> 00:36:03,240 the alias frequencies when we do that and we can also see 601 00:36:03,240 --> 00:36:08,070 the higher frequencies because of the incandescent lighting. 602 00:36:08,070 --> 00:36:12,010 Maybe what we can do now is take a look at 603 00:36:12,010 --> 00:36:13,260 some other fun things. 604 00:36:13,260 --> 00:36:16,160 And one I guess I'm curious about is the disk that you 605 00:36:16,160 --> 00:36:17,870 have over there. 606 00:36:17,870 --> 00:36:22,120 Doc, maybe you can tell us what we have here? 607 00:36:22,120 --> 00:36:22,610 DR. HAROLD EDGERTON: Sure, Al. 608 00:36:22,610 --> 00:36:26,530 This is a disk to show how you can get motion pictures out of 609 00:36:26,530 --> 00:36:29,240 a series of still pictures. 610 00:36:29,240 --> 00:36:31,970 This circle is repeated 12 times. 611 00:36:31,970 --> 00:36:35,820 The white dot goes from the outer part of it on this side 612 00:36:35,820 --> 00:36:37,600 to the inner part on the other. 613 00:36:37,600 --> 00:36:40,740 If I flash one time per revolution on this, you'll see 614 00:36:40,740 --> 00:36:42,315 it exactly as it is. 615 00:36:42,315 --> 00:36:46,060 But if I skip one picture each time, then you get the 616 00:36:46,060 --> 00:36:48,930 relative motion of this ball. 617 00:36:48,930 --> 00:36:52,030 Well, the object is to show the ball rotating either this 618 00:36:52,030 --> 00:36:54,490 way or that way depending on whether the strobe was going 619 00:36:54,490 --> 00:36:56,506 faster or slower than the other. 620 00:36:59,110 --> 00:37:01,970 This way motion pictures were developed hundred years ago, 621 00:37:01,970 --> 00:37:03,140 long before photography. 622 00:37:03,140 --> 00:37:05,580 They drew pictures of people in different 623 00:37:05,580 --> 00:37:07,180 poses, animated pictures. 624 00:37:07,180 --> 00:37:07,930 Like to see it run? 625 00:37:07,930 --> 00:37:09,230 PROFESSOR: Yeah, great. 626 00:37:09,230 --> 00:37:13,090 It's actually, the title, kind of, is "Aliasing Can Be Fun." 627 00:37:13,090 --> 00:37:14,340 DR. HAROLD EDGERTON: That's right. 628 00:37:17,000 --> 00:37:20,015 Let me get it up to speed. 629 00:37:20,015 --> 00:37:22,480 On the way up, you get a lot of different, sort of, 630 00:37:22,480 --> 00:37:24,160 patterns as it goes through. 631 00:37:24,160 --> 00:37:27,513 When it eventually reaches its speed, which is about 1,100 632 00:37:27,513 --> 00:37:31,400 per minute, you'll see it stop. 633 00:37:31,400 --> 00:37:33,550 PROFESSOR: And the background blur, basically at the high 634 00:37:33,550 --> 00:37:37,530 frequencies that the eye can't follow and then, kind of, 635 00:37:37,530 --> 00:37:41,520 superimposed on that again, we can see the frequencies that 636 00:37:41,520 --> 00:37:42,530 are aliased down. 637 00:37:42,530 --> 00:37:44,306 And that's what the eye can follow. 638 00:37:44,306 --> 00:37:46,320 DR. HAROLD EDGERTON: Right now, we have one flash per 639 00:37:46,320 --> 00:37:49,145 revolution, so you can see the part of the disk that's 640 00:37:49,145 --> 00:37:51,790 illuminated with the strobe exactly as if it 641 00:37:51,790 --> 00:37:53,015 was standing still. 642 00:37:53,015 --> 00:37:57,260 Now if I increase the frequency, so they skip one 643 00:37:57,260 --> 00:37:59,780 circle, then you get the illusion 644 00:37:59,780 --> 00:38:01,300 that, that dot is moving. 645 00:38:01,300 --> 00:38:04,960 PROFESSOR: In fact, let's really enhance the revolution, 646 00:38:04,960 --> 00:38:09,100 let's turn the incandescent lights off again. 647 00:38:09,100 --> 00:38:11,690 And now, now what we see really are the alias 648 00:38:11,690 --> 00:38:13,780 frequencies. 649 00:38:13,780 --> 00:38:15,510 What do you think of this Justine? 650 00:38:15,510 --> 00:38:17,310 JUSTINE: Neat. 651 00:38:17,310 --> 00:38:19,190 DR. HAROLD EDGERTON: It looks like magic. 652 00:38:19,190 --> 00:38:22,517 I still have great joy in watching this thing, though 653 00:38:22,517 --> 00:38:23,770 it's so simple. 654 00:38:23,770 --> 00:38:25,420 PROFESSOR: Now, while we're watching this, something also 655 00:38:25,420 --> 00:38:31,780 I might point out for the lecture--for the course-- is 656 00:38:31,780 --> 00:38:34,620 that actually there really are two sampling frequencies that 657 00:38:34,620 --> 00:38:35,100 we're seeing. 658 00:38:35,100 --> 00:38:37,190 One is the strobe, which is the 659 00:38:37,190 --> 00:38:38,910 strobe that you're running. 660 00:38:38,910 --> 00:38:42,290 The other is the inherent frame rate for the TV, that's 661 00:38:42,290 --> 00:38:44,400 running at 30 frames a second. 662 00:38:44,400 --> 00:38:47,070 And that's one of the reasons, by the way, that people 663 00:38:47,070 --> 00:38:51,750 watching this on the video course are in fact seeing a 664 00:38:51,750 --> 00:38:56,690 flicker or a beating or modulation between the two 665 00:38:56,690 --> 00:38:59,060 unsynchronized frame rates. 666 00:38:59,060 --> 00:39:00,450 DR. HAROLD EDGERTON: I'll run the frequencies of the strobe 667 00:39:00,450 --> 00:39:04,195 up, so we get two of them in there. 668 00:39:04,195 --> 00:39:05,660 You keep watching, we had all these 669 00:39:05,660 --> 00:39:06,910 other interesting patterns. 670 00:39:09,235 --> 00:39:09,700 There's two now. 671 00:39:09,700 --> 00:39:12,360 And I'll make the two bounce on each other. 672 00:39:19,260 --> 00:39:21,640 You get all these patterns for free. 673 00:39:21,640 --> 00:39:24,225 You design a disk to show one thing and then when you run 674 00:39:24,225 --> 00:39:26,480 it, you find all the other patterns. 675 00:39:26,480 --> 00:39:29,000 PROFESSOR: I think it would be a terrific homework problem 676 00:39:29,000 --> 00:39:32,310 for the video course, to have them all sit down and analyze 677 00:39:32,310 --> 00:39:34,130 all the frequencies that they're seeing and what 678 00:39:34,130 --> 00:39:35,140 they're being aliased to. 679 00:39:35,140 --> 00:39:35,760 What do you think of that? 680 00:39:35,760 --> 00:39:37,670 DR. HAROLD EDGERTON: That's a good idea. 681 00:39:37,670 --> 00:39:40,780 As a teacher, I love to give quizzes. 682 00:39:40,780 --> 00:39:42,410 Find out whether the students are listening. 683 00:39:42,410 --> 00:39:44,000 PROFESSOR: I think that'll chase a few people away from 684 00:39:44,000 --> 00:39:45,190 the course, that's what I like-- 685 00:39:45,190 --> 00:39:47,900 DR. HAROLD EDGERTON: No, it attracts them because you get 686 00:39:47,900 --> 00:39:51,540 involved in these optical things, there's no limit on 687 00:39:51,540 --> 00:39:54,380 what you can do. 688 00:39:54,380 --> 00:39:56,960 PROFESSOR: Let's bring the incandescent lights back up 689 00:39:56,960 --> 00:39:59,880 again, just to remind everybody that in back of all 690 00:39:59,880 --> 00:40:03,920 these are some frequencies that are a lot higher than the 691 00:40:03,920 --> 00:40:05,640 ones that we begin to get the 692 00:40:05,640 --> 00:40:07,100 impression that we're watching. 693 00:40:07,100 --> 00:40:09,270 DR. HAROLD EDGERTON: It's just a motor running at constant 694 00:40:09,270 --> 00:40:11,440 speed with a pattern on it. 695 00:40:14,150 --> 00:40:17,110 PROFESSOR: Doc, I have to say that there aren't many people 696 00:40:17,110 --> 00:40:19,700 I know that have as much fun in their work as you do. 697 00:40:19,700 --> 00:40:21,805 DR. HAROLD EDGERTON: Well, I'm a lucky man. 698 00:40:21,805 --> 00:40:25,260 PROFESSOR: Well, what I'd like to do now, maybe, is take a 699 00:40:25,260 --> 00:40:26,950 look at one last experiment, if you could. 700 00:40:26,950 --> 00:40:27,870 DR. HAROLD EDGERTON: Sure. 701 00:40:27,870 --> 00:40:32,290 PROFESSOR: And what I'd like to do is go take a look at, I 702 00:40:32,290 --> 00:40:35,310 guess, what sometimes is called the--well, not the 703 00:40:35,310 --> 00:40:37,320 water drop experiment-- what's the name of the-- 704 00:40:37,320 --> 00:40:38,080 DR. HAROLD EDGERTON: You mean the Double Piddler Hydraulic 705 00:40:38,080 --> 00:40:39,710 Happening Machine? 706 00:40:39,710 --> 00:40:40,750 PROFESSOR: That's the one I was thinking of. 707 00:40:40,750 --> 00:40:42,970 Let's take a look over there. 708 00:40:42,970 --> 00:40:43,680 DR. HAROLD EDGERTON: Come on, Justine, let's go 709 00:40:43,680 --> 00:40:44,930 and turn on the water. 710 00:40:48,620 --> 00:40:51,990 PROFESSOR: So, Doc, this is the--what did you call it 711 00:40:51,990 --> 00:40:54,370 DPHHM for Double Piddler Hydraulic Happening Machine? 712 00:40:57,590 --> 00:41:00,000 Got it. 713 00:41:00,000 --> 00:41:01,110 DR. HAROLD EDGERTON: It looks like a continuous 714 00:41:01,110 --> 00:41:02,620 stream, but it's not. 715 00:41:02,620 --> 00:41:04,365 It's a pump over there. 716 00:41:04,365 --> 00:41:07,480 It's pumping 60 pulses a second. 717 00:41:07,480 --> 00:41:09,680 The water is coming out in spurts. 718 00:41:09,680 --> 00:41:13,790 PROFESSOR: So actually, again it's the 60 pulses a second 719 00:41:13,790 --> 00:41:15,000 your eye can't follow. 720 00:41:15,000 --> 00:41:16,910 DR. HAROLD EDGERTON: Your eye's no good at 60 a second. 721 00:41:16,910 --> 00:41:18,450 PROFESSOR: Basically looks like a blur. 722 00:41:18,450 --> 00:41:20,850 DR. HAROLD EDGERTON: It is a blur, a nice juicy blur. 723 00:41:20,850 --> 00:41:22,100 Now we put the strobe on. 724 00:41:27,970 --> 00:41:31,270 PROFESSOR: So again, I guess we have this essentially 725 00:41:31,270 --> 00:41:32,390 aliased down. 726 00:41:32,390 --> 00:41:34,060 And again with the incandescent light, you can 727 00:41:34,060 --> 00:41:39,210 see both the high frequency and the alias frequency. 728 00:41:39,210 --> 00:41:43,930 And let's see, I guess that's what the frequency close to DC 729 00:41:43,930 --> 00:41:47,480 and we can adjust it so that it's stopped. 730 00:41:47,480 --> 00:41:48,640 DR. HAROLD EDGERTON: All right, make the water go up. 731 00:41:48,640 --> 00:41:51,410 PROFESSOR: And then we can actually make it go up. 732 00:41:51,410 --> 00:41:53,950 DR. HAROLD EDGERTON: Of course, nobody believes that. 733 00:41:53,950 --> 00:41:57,580 PROFESSOR: Yeah, in fact, let me just again, to stress this 734 00:41:57,580 --> 00:41:59,110 point to the class. 735 00:41:59,110 --> 00:42:02,930 The idea here of the phase reversal-- of course, you can 736 00:42:02,930 --> 00:42:05,990 see it in the time domain-- you just think about when the 737 00:42:05,990 --> 00:42:08,040 flashes of light come. 738 00:42:08,040 --> 00:42:12,770 But if you think of these impulses that we have in the 739 00:42:12,770 --> 00:42:16,580 frequency domain and we're aliasing as we change the 740 00:42:16,580 --> 00:42:19,540 sampling frequency, what happens is that these impulses 741 00:42:19,540 --> 00:42:23,390 cross over and what that means is that we get a phase 742 00:42:23,390 --> 00:42:28,270 reversal depending on which phases are associated with 743 00:42:28,270 --> 00:42:31,130 which side of DC so that's kind of the idea 744 00:42:31,130 --> 00:42:32,380 of the phase reversal. 745 00:42:34,640 --> 00:42:36,150 Let's turn the-- 746 00:42:36,150 --> 00:42:37,710 DR. HAROLD EDGERTON: Well, we tried to have Justine put her 747 00:42:37,710 --> 00:42:39,090 finger in between those two drops. 748 00:42:39,090 --> 00:42:40,090 PROFESSOR: Yeah, let's turn the 749 00:42:40,090 --> 00:42:41,990 incandescent light off first. 750 00:42:41,990 --> 00:42:42,260 And-- 751 00:42:42,260 --> 00:42:43,610 DR. HAROLD EDGERTON: Take one finger out now. 752 00:42:43,610 --> 00:42:45,460 Put it right in between those two drops. 753 00:42:45,460 --> 00:42:46,330 PROFESSOR: Justine,you think you can do that? 754 00:42:46,330 --> 00:42:47,420 DR. HAROLD EDGERTON: Better get on the other side. 755 00:42:47,420 --> 00:42:49,525 Use your other hand, so they can see with it. 756 00:42:49,525 --> 00:42:50,330 You can-- 757 00:42:50,330 --> 00:42:52,620 PROFESSOR: Think you can get your finger in there? 758 00:42:52,620 --> 00:42:55,530 PROFESSOR: Whoop, there's water there all the time. 759 00:42:55,530 --> 00:42:56,790 PROFESSOR: Well I don't know, Doc. 760 00:42:56,790 --> 00:43:00,520 It seems to me if we-- can't we just adjust this so that 761 00:43:00,520 --> 00:43:02,030 the dots just go through each other? 762 00:43:02,030 --> 00:43:03,230 DR. HAROLD EDGERTON: Sure. 763 00:43:03,230 --> 00:43:05,038 PROFESSOR: Now if the dots can do it, Justine, how come you 764 00:43:05,038 --> 00:43:06,490 can't get your finger in there? 765 00:43:06,490 --> 00:43:06,720 JUSTINE: I don't know. 766 00:43:06,720 --> 00:43:08,060 PROFESSOR: Why don't you try that once more? 767 00:43:10,910 --> 00:43:12,020 I guess not. 768 00:43:12,020 --> 00:43:13,030 DR. HAROLD EDGERTON: No, that's one thing you 769 00:43:13,030 --> 00:43:14,450 can't do with it. 770 00:43:14,450 --> 00:43:18,080 PROFESSOR: Well let's bring the lights back up and again, 771 00:43:18,080 --> 00:43:23,480 just to stress the point, here we are at DC, here we are at a 772 00:43:23,480 --> 00:43:28,580 frequency that's just a little above DC, and we can go back 773 00:43:28,580 --> 00:43:31,560 down to DC and we can actually get a phase reversal. 774 00:43:31,560 --> 00:43:34,850 And I guess, if we do this long enough, we can empty out 775 00:43:34,850 --> 00:43:36,580 the whole ocean and put it back in 776 00:43:36,580 --> 00:43:37,490 wherever it comes from. 777 00:43:37,490 --> 00:43:37,950 Isn't that right? 778 00:43:37,950 --> 00:43:39,010 DR. HAROLD EDGERTON: And we caution the students when they 779 00:43:39,010 --> 00:43:40,370 run this, not to run it too long-- 780 00:43:40,370 --> 00:43:40,480 PROFESSOR: That's right. 781 00:43:40,480 --> 00:43:41,820 DR. HAROLD EDGERTON: We've got the bucket here. 782 00:43:41,820 --> 00:43:42,730 PROFESSOR: You have to be careful-- 783 00:43:42,730 --> 00:43:43,200 DR. HAROLD EDGERTON: --and it's been a while since 784 00:43:43,200 --> 00:43:45,130 somebody believes me. 785 00:43:45,130 --> 00:43:47,130 PROFESSOR: Well, I don't know about them, but I guess I 786 00:43:47,130 --> 00:43:48,862 believe you, Doc. 787 00:43:48,862 --> 00:43:50,580 DR. HAROLD EDGERTON: I'll put a little more pressure on so 788 00:43:50,580 --> 00:43:53,876 we get little more interesting patterns. 789 00:43:53,876 --> 00:43:57,200 Little patterns or surface tension that's pulling in 790 00:43:57,200 --> 00:43:58,450 things together. 791 00:44:01,250 --> 00:44:04,412 We have these machines, they're all over the place. 792 00:44:04,412 --> 00:44:05,990 They're a lot of fun. 793 00:44:05,990 --> 00:44:07,790 PROFESSOR: Well, Doc, this is really terrific. 794 00:44:07,790 --> 00:44:14,410 I think that this whole idea of using aliasing and strobes 795 00:44:14,410 --> 00:44:17,430 and the kinds of things that you do with 796 00:44:17,430 --> 00:44:19,050 them are just fantastic. 797 00:44:19,050 --> 00:44:22,630 And we really appreciate the chance to come in here and see 798 00:44:22,630 --> 00:44:23,432 the demonstration. 799 00:44:23,432 --> 00:44:24,750 DR. HAROLD EDGERTON: Well, that's the whole game. 800 00:44:24,750 --> 00:44:25,680 We've [? been happy to use ?] 801 00:44:25,680 --> 00:44:29,660 them for years and probably will for many years to come. 802 00:44:29,660 --> 00:44:32,970 PROFESSOR: So as I emphasized at the beginning, in lots of 803 00:44:32,970 --> 00:44:36,490 situations aliasing can, in fact, be very useful. 804 00:44:36,490 --> 00:44:40,020 Also what this demonstrates is that particularly when you 805 00:44:40,020 --> 00:44:43,470 have a colleague, like Doc Edgerton, aliasing and for 806 00:44:43,470 --> 00:44:45,260 that matter science, in general, can be an 807 00:44:45,260 --> 00:44:46,130 awful lot of fun. 808 00:44:46,130 --> 00:44:47,670 DR. HAROLD EDGERTON: Thanks for coming in. 809 00:44:47,670 --> 00:44:48,920 PROFESSOR: Thanks a lot, Doc. 810 00:44:48,920 --> 00:44:49,370 DR. HAROLD EDGERTON: See you again. 811 00:44:49,370 --> 00:44:50,580 PROFESSOR: And thank you, Justine. 812 00:44:50,580 --> 00:44:51,830 JUSTINE: You're welcome. 813 00:44:54,300 --> 00:44:58,610 PROFESSOR: Well, I have to say that visit was an awful lot of 814 00:44:58,610 --> 00:45:03,450 fun for me and for Justine and in fact, for the whole camera 815 00:45:03,450 --> 00:45:04,850 crew that was there. 816 00:45:04,850 --> 00:45:08,330 And hopefully, all of you at some point will also have a 817 00:45:08,330 --> 00:45:12,130 chance to visit at Strobe Alley. 818 00:45:12,130 --> 00:45:16,480 Well, hopefully what we've gone through today gives you a 819 00:45:16,480 --> 00:45:20,680 good feeling for the concepts of sampling and aliasing and 820 00:45:20,680 --> 00:45:23,540 both, why it might be useful and why we might 821 00:45:23,540 --> 00:45:25,910 want to avoid it. 822 00:45:25,910 --> 00:45:29,260 In the next lecture, we'll continue on the 823 00:45:29,260 --> 00:45:31,200 discussion of sampling. 824 00:45:31,200 --> 00:45:34,840 And in particular, what I'll be talking about is the 825 00:45:34,840 --> 00:45:39,020 interpretation of the reconstruction process not in 826 00:45:39,020 --> 00:45:42,930 the frequency domain, but in a time domain and interpretation 827 00:45:42,930 --> 00:45:46,950 specifically associated with the concept of interpolating 828 00:45:46,950 --> 00:45:48,450 between the samples. 829 00:45:48,450 --> 00:45:51,970 We'll then proceed from there to a discussion as I've 830 00:45:51,970 --> 00:45:57,070 alluded to in several lectures of what I've referred to as 831 00:45:57,070 --> 00:46:01,150 discrete-time processing of continuous-time signals, very 832 00:46:01,150 --> 00:46:04,560 heavily exploiting the concept and issues 833 00:46:04,560 --> 00:46:06,110 associated with sampling. 834 00:46:06,110 --> 00:46:07,360 Thank you.