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 00:00:55,619 --> 00:00:58,000 PROFESSOR: In discussing the continuous-time and 9 00:00:58,000 --> 00:01:01,540 discrete-time Fourier transforms, we developed a 10 00:01:01,540 --> 00:01:04,050 number of important properties. 11 00:01:04,050 --> 00:01:07,230 Two particularly significant ones, as I mentioned at the 12 00:01:07,230 --> 00:01:10,490 time, are the modulation property and 13 00:01:10,490 --> 00:01:12,860 the convolution property. 14 00:01:12,860 --> 00:01:15,760 Starting with the next lecture, the one after this 15 00:01:15,760 --> 00:01:19,190 one, we'll be developing and exploiting some of the 16 00:01:19,190 --> 00:01:22,260 consequences of the modulation property. 17 00:01:22,260 --> 00:01:27,620 In today's lecture though, I'd like to review and expand on 18 00:01:27,620 --> 00:01:31,100 the notion of filtering, which, as I had mentioned, 19 00:01:31,100 --> 00:01:36,700 flows more or less directly from the convolution property. 20 00:01:36,700 --> 00:01:39,820 To begin, let me just quickly review what the convolution 21 00:01:39,820 --> 00:01:41,260 property is. 22 00:01:41,260 --> 00:01:45,780 Both for continuous-time and for discrete-time, the 23 00:01:45,780 --> 00:01:51,400 convolution property tells us that the Fourier transform of 24 00:01:51,400 --> 00:01:56,400 the convolution of two time functions is the product of 25 00:01:56,400 --> 00:01:59,190 the Fourier transforms. 26 00:01:59,190 --> 00:02:04,450 Now, what this means in terms of linear time-invariant 27 00:02:04,450 --> 00:02:08,889 filters, since we know that in the time domain the output of 28 00:02:08,889 --> 00:02:12,890 a linear time-invariant filter is the convolution of the 29 00:02:12,890 --> 00:02:17,080 input and the impulse response, it says essentially 30 00:02:17,080 --> 00:02:20,790 then in the frequency domain that the Fourier transform of 31 00:02:20,790 --> 00:02:24,740 the output is the product the Fourier transform of the 32 00:02:24,740 --> 00:02:28,500 impulse response, namely the frequency response, and the 33 00:02:28,500 --> 00:02:31,850 Fourier transform of the input. 34 00:02:31,850 --> 00:02:36,370 So the output is described through that product. 35 00:02:36,370 --> 00:02:40,760 Now, recall also that in developing the Fourier 36 00:02:40,760 --> 00:02:45,250 transform, I interpreted the Fourier transform as the 37 00:02:45,250 --> 00:02:49,320 complex amplitude of a decomposition of the signal in 38 00:02:49,320 --> 00:02:52,120 terms of a set of complex exponentials. 39 00:02:52,120 --> 00:02:55,230 And the frequency response or the convolution property, in 40 00:02:55,230 --> 00:03:01,560 effect, tells us how to modify the amplitudes of each of 41 00:03:01,560 --> 00:03:05,320 those complex exponentials as they go through the system. 42 00:03:05,320 --> 00:03:09,190 Now, this led to the notion of filtering, where the basic 43 00:03:09,190 --> 00:03:14,780 concept was that since we can modify the amplitudes of each 44 00:03:14,780 --> 00:03:19,530 of the complex exponential components separately, we can, 45 00:03:19,530 --> 00:03:23,370 for example, retain some of them and 46 00:03:23,370 --> 00:03:25,040 totally eliminate others. 47 00:03:25,040 --> 00:03:27,810 And this is the basic notion of filtering. 48 00:03:27,810 --> 00:03:33,000 So we have, as you recall, first of all the notion in 49 00:03:33,000 --> 00:03:38,190 continuous-time of an ideal filter, for example, I 50 00:03:38,190 --> 00:03:43,730 illustrate here an ideal lowpass filter where we pass 51 00:03:43,730 --> 00:03:49,280 exactly frequency components in one band and reject totally 52 00:03:49,280 --> 00:03:51,940 frequency components in another band. 53 00:03:51,940 --> 00:03:54,620 The band being passed, of course, referred to as the 54 00:03:54,620 --> 00:04:00,030 passband, and the band rejected as the stopband. 55 00:04:00,030 --> 00:04:02,220 I illustrated here a lowpass filter. 56 00:04:02,220 --> 00:04:07,310 We can, of course, reject the low frequencies and retain the 57 00:04:07,310 --> 00:04:08,930 high frequencies. 58 00:04:08,930 --> 00:04:14,150 And that then corresponds to an ideal highpass filter. 59 00:04:14,150 --> 00:04:17,810 Or we can just retain frequencies within a band. 60 00:04:17,810 --> 00:04:22,640 And so I show below what is referred to commonly as a 61 00:04:22,640 --> 00:04:25,240 bandpass filter. 62 00:04:25,240 --> 00:04:29,560 Now, this is what the ideal filters looked like for 63 00:04:29,560 --> 00:04:31,070 continuous-time. 64 00:04:31,070 --> 00:04:35,200 For discrete-time, we have exactly the same situation. 65 00:04:35,200 --> 00:04:39,870 Namely, we have an ideal discrete-time lowpass filter, 66 00:04:39,870 --> 00:04:43,820 which passes exactly frequencies which are the low 67 00:04:43,820 --> 00:04:45,120 frequencies. 68 00:04:45,120 --> 00:04:48,080 Low frequencies, of course, being around 0, and because of 69 00:04:48,080 --> 00:04:52,260 the periodicity, also around 2pi. 70 00:04:52,260 --> 00:04:56,630 We show also an ideal highpass filter. 71 00:04:56,630 --> 00:05:00,180 And a highpass filter, as I indicated last time, passes 72 00:05:00,180 --> 00:05:02,740 frequencies around pi. 73 00:05:02,740 --> 00:05:10,280 And finally, below that, I show an ideal bandpass filter 74 00:05:10,280 --> 00:05:16,210 passing frequencies someplace in the range between 0 and pi. 75 00:05:16,210 --> 00:05:20,330 And recall also that the basic difference between 76 00:05:20,330 --> 00:05:22,610 continuous-time a discrete-time for these 77 00:05:22,610 --> 00:05:26,240 filters is that the discrete-time versions are, of 78 00:05:26,240 --> 00:05:29,960 course, periodic in frequency. 79 00:05:29,960 --> 00:05:34,600 Now, let's look at these ideal filters, and in particular the 80 00:05:34,600 --> 00:05:39,320 ideal lowpass filter in the time domain. 81 00:05:39,320 --> 00:05:45,290 We have the frequency response of the ideal lowpass filter. 82 00:05:45,290 --> 00:05:49,580 And shown below it is the impulse response. 83 00:05:49,580 --> 00:05:54,270 So here is the frequency response and below it the 84 00:05:54,270 --> 00:05:59,020 impulse response of the ideal lowpass filter. 85 00:05:59,020 --> 00:06:02,270 And this, of course, is a sine x over x 86 00:06:02,270 --> 00:06:04,270 form of impulse response. 87 00:06:04,270 --> 00:06:08,820 And recognize also or recall that since this frequency 88 00:06:08,820 --> 00:06:14,890 response is real-valued, the impulse response, in other 89 00:06:14,890 --> 00:06:19,150 words, the inverse transform is an even function of time. 90 00:06:19,150 --> 00:06:24,840 And notice also, since I want to refer back to this, that 91 00:06:24,840 --> 00:06:29,270 the impulse response of an ideal lowpass filter, in fact, 92 00:06:29,270 --> 00:06:30,530 is non-causal. 93 00:06:30,530 --> 00:06:33,420 That follows, from among other things, from the fact that 94 00:06:33,420 --> 00:06:34,940 it's an even function. 95 00:06:34,940 --> 00:06:39,160 But keep in mind, in fact, that a sine x over x function 96 00:06:39,160 --> 00:06:41,680 goes off to infinity in both directions. 97 00:06:41,680 --> 00:06:45,720 So the impulse response of the ideal lowpass filter is 98 00:06:45,720 --> 00:06:49,920 symmetric and continues to have tails off to plus and 99 00:06:49,920 --> 00:06:52,680 minus infinity. 100 00:06:52,680 --> 00:06:56,850 Now, the situation is basically the same in the 101 00:06:56,850 --> 00:06:59,610 discrete-time case. 102 00:06:59,610 --> 00:07:03,340 Let's look at the frequency response and associated 103 00:07:03,340 --> 00:07:08,200 impulse response for an ideal discrete-time lowpass filter. 104 00:07:08,200 --> 00:07:13,940 So once again, here is the frequency response of the 105 00:07:13,940 --> 00:07:16,230 ideal lowpass filter. 106 00:07:16,230 --> 00:07:20,000 And below what I show the impulse response. 107 00:07:20,000 --> 00:07:24,690 Again, it's a sine x over x type of impulse response. 108 00:07:24,690 --> 00:07:30,290 And again, we recognize that since in the frequency domain, 109 00:07:30,290 --> 00:07:34,760 this frequency response is real-valued. 110 00:07:34,760 --> 00:07:39,060 That means, as a consequence of the properties of the 111 00:07:39,060 --> 00:07:43,240 Fourier transform and inverse Fourier transform, that the 112 00:07:43,240 --> 00:07:47,800 impulse response is an even function in the time domain. 113 00:07:47,800 --> 00:07:52,180 And also, incidentally, the sine x over x function goes 114 00:07:52,180 --> 00:07:54,270 off to infinity, again, in both directions. 115 00:07:56,790 --> 00:08:01,650 Now, we've talked about ideal filters in this discussion. 116 00:08:01,650 --> 00:08:05,150 And ideal filters all are, in fact, ideal 117 00:08:05,150 --> 00:08:06,960 in a certain sense. 118 00:08:06,960 --> 00:08:10,550 What they do ideally is they pass a certain band of 119 00:08:10,550 --> 00:08:16,060 frequencies exactly and they reject a band 120 00:08:16,060 --> 00:08:19,010 of frequencies exactly. 121 00:08:19,010 --> 00:08:21,900 On the other hand, there are many filtering problems in 122 00:08:21,900 --> 00:08:26,330 which, generally, we don't have a sharp distinction 123 00:08:26,330 --> 00:08:28,670 between the frequencies we want to pass and the 124 00:08:28,670 --> 00:08:31,260 frequencies we want to reject. 125 00:08:31,260 --> 00:08:34,990 One example of this that's elaborated on in the text is 126 00:08:34,990 --> 00:08:37,610 the design of an automotive suspension system, which, in 127 00:08:37,610 --> 00:08:42,490 fact, is the design of a lowpass filter. 128 00:08:42,490 --> 00:08:46,550 And basically what you want to do in a case like that is 129 00:08:46,550 --> 00:08:51,920 filter out or attenuate very rapid road variations and keep 130 00:08:51,920 --> 00:08:56,020 the lower variations in, of course, elevation of the 131 00:08:56,020 --> 00:08:58,350 highway or road. 132 00:08:58,350 --> 00:09:02,250 And what you can see intuitively is that there 133 00:09:02,250 --> 00:09:05,910 isn't really a very sharp distinction or sharp cut-off 134 00:09:05,910 --> 00:09:08,670 between what you would logically call the low 135 00:09:08,670 --> 00:09:13,220 frequencies and what you would call the high frequencies. 136 00:09:13,220 --> 00:09:18,000 Now, also somewhat related to this is the fact that as we've 137 00:09:18,000 --> 00:09:22,350 seen in the time domain, these ideal filters have a very 138 00:09:22,350 --> 00:09:24,300 particular kind of character. 139 00:09:24,300 --> 00:09:29,900 For example, let's look back at the ideal lowpass filter. 140 00:09:29,900 --> 00:09:34,190 And we saw the impulse response. 141 00:09:34,190 --> 00:09:37,090 The impulse response is what we had shown here. 142 00:09:37,090 --> 00:09:41,860 Let's now look at the step response of the discrete-time 143 00:09:41,860 --> 00:09:44,110 ideal lowpass filter. 144 00:09:44,110 --> 00:09:47,590 And notice the fact that it has a tail that oscillates. 145 00:09:47,590 --> 00:09:50,300 And when the step hits, in fact, it has 146 00:09:50,300 --> 00:09:52,720 an oscillatory behavior. 147 00:09:52,720 --> 00:09:57,670 Now, exactly the same situation occurs in 148 00:09:57,670 --> 00:09:59,100 continuous-time. 149 00:09:59,100 --> 00:10:03,650 Let's look at the step response of the 150 00:10:03,650 --> 00:10:07,310 continuous-time ideal lowpass filter. 151 00:10:07,310 --> 00:10:12,540 And what we see is that when a step hits then, in fact, we 152 00:10:12,540 --> 00:10:14,480 get an oscillation. 153 00:10:14,480 --> 00:10:18,740 And very often, that oscillation is something 154 00:10:18,740 --> 00:10:19,670 that's undesirable. 155 00:10:19,670 --> 00:10:22,460 For example, if you were designing an automotive 156 00:10:22,460 --> 00:10:27,100 suspension system and you hit a curve, which is a step 157 00:10:27,100 --> 00:10:31,740 input, in fact, you probably would not like to have the 158 00:10:31,740 --> 00:10:38,260 automobile oscillating, dying down in oscillation. 159 00:10:38,260 --> 00:10:41,790 Now there's another very important point, which again, 160 00:10:41,790 --> 00:10:43,620 we can see either in continuous-time or 161 00:10:43,620 --> 00:10:49,200 discrete-time, which is that even if we want it to have an 162 00:10:49,200 --> 00:10:54,880 ideal filter, the ideal filter has another problem if we want 163 00:10:54,880 --> 00:10:57,960 to attempt to implement it in real time. 164 00:10:57,960 --> 00:10:59,180 What's the problem? 165 00:10:59,180 --> 00:11:04,420 The problem is that since the impulse response is even and, 166 00:11:04,420 --> 00:11:08,410 in fact, has tails that go off to plus and minus infinity, 167 00:11:08,410 --> 00:11:10,190 it's non-causal. 168 00:11:10,190 --> 00:11:15,320 So if, in fact, we want to build a filter and the filter 169 00:11:15,320 --> 00:11:20,290 is restricted to operate in real time, then, in fact, we 170 00:11:20,290 --> 00:11:23,920 can't build an ideal filter. 171 00:11:23,920 --> 00:11:28,080 So what that says is that, in practice, although ideal 172 00:11:28,080 --> 00:11:32,260 filters are nice to think about and perhaps relate to 173 00:11:32,260 --> 00:11:37,780 practical problems, more typically what we consider are 174 00:11:37,780 --> 00:11:43,780 nonideal filters and in the discrete-time case, a nonideal 175 00:11:43,780 --> 00:11:48,850 filter then we would have a characteristic somewhat like 176 00:11:48,850 --> 00:11:50,430 I've indicated here. 177 00:11:50,430 --> 00:11:55,300 Where instead of a very rapid transition from passband to 178 00:11:55,300 --> 00:12:00,330 stopband, there would be a more gradual transition with a 179 00:12:00,330 --> 00:12:06,960 passband cutoff frequency and a stopband cutoff frequency. 180 00:12:06,960 --> 00:12:11,170 And perhaps also instead of having an exactly flat 181 00:12:11,170 --> 00:12:15,210 characteristic in the stopband in the passband, we would 182 00:12:15,210 --> 00:12:18,180 allow a certain amount of ripple. 183 00:12:18,180 --> 00:12:23,350 We also have exactly the same situation in continuous-time, 184 00:12:23,350 --> 00:12:27,560 where here we'll just simply change our frequency axis to a 185 00:12:27,560 --> 00:12:30,480 continuous frequency axis instead of the discrete 186 00:12:30,480 --> 00:12:31,920 frequency axis. 187 00:12:31,920 --> 00:12:35,460 Again, we would think in terms of an allowable passband 188 00:12:35,460 --> 00:12:41,910 ripple, a transition from passband to stopband with a 189 00:12:41,910 --> 00:12:46,940 passband cutoff frequency and a stopband cutoff frequency. 190 00:12:46,940 --> 00:12:52,120 So the notion here is that, again, ideal filters are ideal 191 00:12:52,120 --> 00:12:55,100 in some respects, not ideal in other respects. 192 00:12:55,100 --> 00:12:57,850 And for many practical problems, we 193 00:12:57,850 --> 00:12:59,360 may not want them. 194 00:12:59,360 --> 00:13:02,400 And even if we did want them, we may not be able to get 195 00:13:02,400 --> 00:13:06,280 them, perhaps because of this issue of causality. 196 00:13:06,280 --> 00:13:11,050 Even if causality is not an issue, what happens in filter 197 00:13:11,050 --> 00:13:16,660 design and implementation, in fact, is that the sharper you 198 00:13:16,660 --> 00:13:20,720 attempt to make the cutoff, the more expensive, in some 199 00:13:20,720 --> 00:13:24,460 sense, the filter becomes, either in terms of components, 200 00:13:24,460 --> 00:13:27,960 in continuous-time, or in terms of computation in 201 00:13:27,960 --> 00:13:29,130 discrete-time. 202 00:13:29,130 --> 00:13:34,970 And so there are these whole variety of issues that really 203 00:13:34,970 --> 00:13:37,990 make it important to understand the 204 00:13:37,990 --> 00:13:41,790 notion nonideal filters. 205 00:13:41,790 --> 00:13:47,520 Now, just to illustrate as an example, let me remind you of 206 00:13:47,520 --> 00:13:53,640 one example of what, in fact, is a nonideal lowpass filter. 207 00:13:53,640 --> 00:13:57,820 And we have looked previously at the associated 208 00:13:57,820 --> 00:13:59,660 differential equation. 209 00:13:59,660 --> 00:14:04,270 Let me now, in fact, relate it to a circuit, and in 210 00:14:04,270 --> 00:14:08,730 particular an RC circuit, where the output could either 211 00:14:08,730 --> 00:14:12,120 be across the capacitor or the output can 212 00:14:12,120 --> 00:14:13,840 be across the resistor. 213 00:14:13,840 --> 00:14:16,340 So in effect, we have two systems here. 214 00:14:16,340 --> 00:14:20,540 We have a system, which is the system function from the 215 00:14:20,540 --> 00:14:24,490 voltage source input to the capacitor output, the system 216 00:14:24,490 --> 00:14:29,450 from the voltage source input to the resistor output. 217 00:14:29,450 --> 00:14:32,630 And, in fact, just applying Kirchhoff's Voltage Law to 218 00:14:32,630 --> 00:14:35,840 this, we can relate those in a very straightforward way. 219 00:14:35,840 --> 00:14:41,540 It's very straightforward to verify that the system from 220 00:14:41,540 --> 00:14:46,430 input to resistor output is simply the identity system 221 00:14:46,430 --> 00:14:51,480 with the capacitor output subtracted from it. 222 00:14:51,480 --> 00:14:54,850 Now, we can write the differential equation for 223 00:14:54,850 --> 00:14:59,360 either of these systems and, as we talked about last time 224 00:14:59,360 --> 00:15:03,860 in the last several lectures, solve that equation using and 225 00:15:03,860 --> 00:15:06,730 exploiting the properties of the Fourier transform. 226 00:15:06,730 --> 00:15:11,790 And in fact, if we look at the differential equation relating 227 00:15:11,790 --> 00:15:16,830 the capacitor output to the voltage source input, we 228 00:15:16,830 --> 00:15:20,000 recognize that this is an example that, in effect, we've 229 00:15:20,000 --> 00:15:21,740 solved previously. 230 00:15:21,740 --> 00:15:26,290 And so just working our way down, applying the Fourier 231 00:15:26,290 --> 00:15:30,110 transform to the differential equation and generating the 232 00:15:30,110 --> 00:15:34,500 system function by taking the ratio of the capacitor voltage 233 00:15:34,500 --> 00:15:37,360 or its Fourier transform to the Fourier transform of the 234 00:15:37,360 --> 00:15:42,030 source, we then have the system function associated 235 00:15:42,030 --> 00:15:44,650 with the system for which the output is 236 00:15:44,650 --> 00:15:46,550 the capacitor voltage. 237 00:15:46,550 --> 00:15:50,710 Or if we solve instead for the system function associated 238 00:15:50,710 --> 00:15:54,450 with the resistor output, we can simply 239 00:15:54,450 --> 00:15:57,390 subtract H1 from unity. 240 00:15:57,390 --> 00:16:00,510 And the system function that we get in that case is the 241 00:16:00,510 --> 00:16:03,040 system function that I show here. 242 00:16:03,040 --> 00:16:07,510 So we have, now, two system functions, one for the 243 00:16:07,510 --> 00:16:11,480 capacitor output, the other for the resistor output. 244 00:16:11,480 --> 00:16:15,540 And one, the first, corresponding to the capacitor 245 00:16:15,540 --> 00:16:20,520 output, in fact, if we plot it on a linear amplitude scale, 246 00:16:20,520 --> 00:16:21,430 looks like this. 247 00:16:21,430 --> 00:16:24,760 And as you can see, and as we saw last time, is an 248 00:16:24,760 --> 00:16:27,630 approximation to a lowpass filter. 249 00:16:27,630 --> 00:16:33,000 It is, in fact, and nonideal lowpass filter, whereas the 250 00:16:33,000 --> 00:16:37,920 resistor output is an approximation to a highpass 251 00:16:37,920 --> 00:16:42,200 filter, or in effect, a nonideal highpass filter. 252 00:16:42,200 --> 00:16:46,230 So in one case, just comparing the two, we have a lowpass 253 00:16:46,230 --> 00:16:50,210 filter as the capacitor output associated with the capacitor 254 00:16:50,210 --> 00:16:53,700 output, and a highpass filter associated with 255 00:16:53,700 --> 00:16:56,100 the resistor output. 256 00:16:56,100 --> 00:17:00,110 Let's just quickly look at that example now, looking on a 257 00:17:00,110 --> 00:17:05,069 Bode plot, instead of on the linear scale 258 00:17:05,069 --> 00:17:06,640 that we showed before. 259 00:17:06,640 --> 00:17:11,119 And recall incidentally, and be aware incidentally, of the 260 00:17:11,119 --> 00:17:16,380 fact that we can, of course, cascade several filters of 261 00:17:16,380 --> 00:17:18,940 this type and improve the characteristics. 262 00:17:18,940 --> 00:17:28,040 So I have shown at the top a Bode plot of the system 263 00:17:28,040 --> 00:17:30,940 function associated with the capacitor output. 264 00:17:30,940 --> 00:17:35,810 It's flat out to a frequency corresponding to 1 over the 265 00:17:35,810 --> 00:17:38,590 time constant, RC. 266 00:17:38,590 --> 00:17:43,820 And then it falls off at 10 dB per decade, a decade being a 267 00:17:43,820 --> 00:17:45,640 factor of 10. 268 00:17:45,640 --> 00:17:49,390 Or if instead we look at the system function associated 269 00:17:49,390 --> 00:17:54,300 with the resistor output, that corresponds to a 10 dB per 270 00:17:54,300 --> 00:17:59,020 decade increase in frequency up to approximately the 271 00:17:59,020 --> 00:18:02,510 reciprocal of the time constant, and then approaching 272 00:18:02,510 --> 00:18:05,550 a flat characteristic after that. 273 00:18:05,550 --> 00:18:11,400 And if we consider either one of these, looking back again 274 00:18:11,400 --> 00:18:15,980 at the lowpass filter, if we were to cascade several 275 00:18:15,980 --> 00:18:20,660 filters with this frequency response, then because we have 276 00:18:20,660 --> 00:18:24,330 things plotted on a Bode plot, the Bode plot for the cascade 277 00:18:24,330 --> 00:18:26,010 would simply be summing these. 278 00:18:26,010 --> 00:18:30,330 And so if we cascaded, for example, two stages instead of 279 00:18:30,330 --> 00:18:34,210 a roll-off at 10 dB per decade, it would roll off at 280 00:18:34,210 --> 00:18:37,780 20 dB per decade. 281 00:18:37,780 --> 00:18:42,100 Now, filters in this type, RC filters, perhaps several of 282 00:18:42,100 --> 00:18:47,360 them in cascade, are in fact very prevalent. 283 00:18:47,360 --> 00:18:53,465 And in fact, in an environment like this, where we're, in 284 00:18:53,465 --> 00:19:00,150 fact, doing recording, we see there are filters of that type 285 00:19:00,150 --> 00:19:03,790 that show up very commonly both in the audio and the 286 00:19:03,790 --> 00:19:08,140 video portion of the signal processing that's associated 287 00:19:08,140 --> 00:19:10,470 with making this set of tapes. 288 00:19:10,470 --> 00:19:14,350 In fact, let's take a look in the control room. 289 00:19:14,350 --> 00:19:19,070 And what I'll be able to show you in the control room is the 290 00:19:19,070 --> 00:19:23,200 audio portion of the processing that's done and the 291 00:19:23,200 --> 00:19:26,890 kinds of filters, very much of the type we just talked about, 292 00:19:26,890 --> 00:19:30,560 that are associated with the signal processing that's done 293 00:19:30,560 --> 00:19:33,140 in preparing the audio for the tapes. 294 00:19:33,140 --> 00:19:36,120 So let's just take a walk into the control room 295 00:19:36,120 --> 00:19:37,370 and see what we see. 296 00:19:40,110 --> 00:19:42,280 This is the control room that's 297 00:19:42,280 --> 00:19:44,260 used for camera switching. 298 00:19:44,260 --> 00:19:48,210 It's used for computer editing and also audio control. 299 00:19:48,210 --> 00:19:51,150 You can see the monitors, and these are used 300 00:19:51,150 --> 00:19:53,090 for the camera switching. 301 00:19:53,090 --> 00:19:56,870 And this is the computer editing console that's used 302 00:19:56,870 --> 00:19:59,890 for online and offline computer editing. 303 00:19:59,890 --> 00:20:02,520 What I really want to demonstrate though, in the 304 00:20:02,520 --> 00:20:06,760 context of the lecture is the audio control panel, which 305 00:20:06,760 --> 00:20:10,680 contains, among other things, a variety of filters for high 306 00:20:10,680 --> 00:20:13,280 frequency, low frequencies, et cetera, basically 307 00:20:13,280 --> 00:20:15,280 equalization filters. 308 00:20:15,280 --> 00:20:20,380 And what we have in the way of filtering is, first of all, 309 00:20:20,380 --> 00:20:23,390 what's referred to as a graphic equalizer, which 310 00:20:23,390 --> 00:20:26,500 consists of a set of bandpass filters, which I'll describe a 311 00:20:26,500 --> 00:20:28,340 little more carefully in a minute. 312 00:20:28,340 --> 00:20:32,820 And then also, an audio control panel, which is down 313 00:20:32,820 --> 00:20:36,350 here and which contains separate equalizer circuits 314 00:20:36,350 --> 00:20:39,860 for each of a whole set of channels and also lots of 315 00:20:39,860 --> 00:20:41,170 controls on them. 316 00:20:41,170 --> 00:20:46,690 Well, let me begin in the demonstration by demonstrating 317 00:20:46,690 --> 00:20:51,380 a little bit of what the graphic equalizer does. 318 00:20:51,380 --> 00:20:55,780 Well, what we have is a set of bandpass filters. 319 00:20:55,780 --> 00:20:58,880 And what's indicated up here are the center frequencies of 320 00:20:58,880 --> 00:21:02,430 the filters, and then a slider switch for each one that lets 321 00:21:02,430 --> 00:21:04,570 us attenuate or amplify. 322 00:21:04,570 --> 00:21:06,550 And this is a dB scale. 323 00:21:06,550 --> 00:21:12,460 So essentially, if you look across this bank of filters 324 00:21:12,460 --> 00:21:15,800 with the total output of the equalizer just being the sum 325 00:21:15,800 --> 00:21:19,550 of the outputs from each of these filters, interestingly 326 00:21:19,550 --> 00:21:22,320 the position of the slider switches as you move across 327 00:21:22,320 --> 00:21:25,680 here, in effect, shows you what the frequency response of 328 00:21:25,680 --> 00:21:27,090 the equalizer is. 329 00:21:27,090 --> 00:21:31,460 So you can change the overall shaping of the filter by 330 00:21:31,460 --> 00:21:33,870 moving the switches up and down. 331 00:21:33,870 --> 00:21:35,780 Right now the equalizer is out. 332 00:21:35,780 --> 00:21:38,480 Let's put the equalizer into the circuit. 333 00:21:38,480 --> 00:21:42,000 And now I put in this filtering characteristic. 334 00:21:42,000 --> 00:21:45,910 And what I'd like to demonstrate is filtering with 335 00:21:45,910 --> 00:21:49,240 this, when we do things that are a little more dramatic 336 00:21:49,240 --> 00:21:51,560 than what would normally be done in a typical audio 337 00:21:51,560 --> 00:21:53,250 recording setting. 338 00:21:53,250 --> 00:21:58,130 And to do this, let's add to my voice some music to make it 339 00:21:58,130 --> 00:22:00,120 more interesting. 340 00:22:00,120 --> 00:22:02,340 Not that my voice isn't interesting as it is. 341 00:22:02,340 --> 00:22:04,980 But in any case, let's bring some music up. 342 00:22:04,980 --> 00:22:05,660 [MUSIC PLAYING] 343 00:22:05,660 --> 00:22:12,900 And now what I'll do is set the low frequencies flat. 344 00:22:12,900 --> 00:22:18,430 And let me take out the high frequencies above 800 cycles. 345 00:22:18,430 --> 00:22:21,160 And so now what we have, effectively, 346 00:22:21,160 --> 00:22:23,940 is a lowpass filter. 347 00:22:23,940 --> 00:22:28,470 And now with the lowpass filter, let me now bring the 348 00:22:28,470 --> 00:22:31,180 highs back up. 349 00:22:31,180 --> 00:22:35,310 And so I'm bringing up those bandpass filters. 350 00:22:35,310 --> 00:22:39,490 And now let me cut out the lows. 351 00:22:39,490 --> 00:22:43,230 And you'll hear the lows disappearing and, in effect, 352 00:22:43,230 --> 00:22:47,200 keeping the highs in effectively crispens the 353 00:22:47,200 --> 00:22:50,050 sound, either my voice or the music. 354 00:22:50,050 --> 00:22:55,250 And finally, let me go back to 0 dB equalization on each of 355 00:22:55,250 --> 00:22:56,770 the filters. 356 00:22:56,770 --> 00:23:01,660 And what I'll also do now is take the equalizer out of the 357 00:23:01,660 --> 00:23:02,910 circuit totally. 358 00:23:05,860 --> 00:23:10,490 Now, let's take a look at the audio master control panel. 359 00:23:10,490 --> 00:23:15,000 And this panel has, of course, for each channel and, for 360 00:23:15,000 --> 00:23:16,460 example, the channel that we're working 361 00:23:16,460 --> 00:23:18,580 on, of a volume control. 362 00:23:18,580 --> 00:23:23,990 I can turn the volume down, and I can turn the volume up. 363 00:23:23,990 --> 00:23:29,100 And it also has, for this particular equalizer circuit, 364 00:23:29,100 --> 00:23:35,930 it has a set of three bandpass filters and knobs which let us 365 00:23:35,930 --> 00:23:41,480 either put in up to 12 dB gain or 12 dB attenuation in each 366 00:23:41,480 --> 00:23:44,990 of the bands, and also a selector switch that lets us 367 00:23:44,990 --> 00:23:46,780 select the center the band. 368 00:23:46,780 --> 00:23:49,630 So let me just again demonstrate a 369 00:23:49,630 --> 00:23:50,810 little bit with this. 370 00:23:50,810 --> 00:23:54,100 And let's get a close up of this panel. 371 00:23:54,100 --> 00:23:57,450 So what we have, as I indicated, is 372 00:23:57,450 --> 00:23:59,310 three bandpass filters. 373 00:23:59,310 --> 00:24:03,460 And these knobs that I'm pointing to here are controls 374 00:24:03,460 --> 00:24:07,200 that allow us for each of the filters to put in up to 12 dB 375 00:24:07,200 --> 00:24:10,310 gain or 12 dB attenuation. 376 00:24:10,310 --> 00:24:14,510 There are also with each of the filters a selector switch 377 00:24:14,510 --> 00:24:18,000 that lets us adjust the center frequency of the filter. 378 00:24:18,000 --> 00:24:21,660 Basically it's a two-position switch. 379 00:24:21,660 --> 00:24:27,200 There also, as you can see, is a button that let's us either 380 00:24:27,200 --> 00:24:28,830 put the equalization in or out. 381 00:24:28,830 --> 00:24:31,290 Currently the equalization is out. 382 00:24:31,290 --> 00:24:32,820 Let's put the equalization in. 383 00:24:32,820 --> 00:24:35,770 We won't hear any effect from that, because the gain 384 00:24:35,770 --> 00:24:38,420 controls are all set at 0 dB. 385 00:24:38,420 --> 00:24:41,820 And I'll want to illustrate shortly the effect of these. 386 00:24:41,820 --> 00:24:45,630 But before I do, let me draw your attention to one other 387 00:24:45,630 --> 00:24:49,660 filter, which is this white switch. 388 00:24:49,660 --> 00:24:55,870 And this switch is a highpass filter that essentially cuts 389 00:24:55,870 --> 00:24:58,740 out frequencies below about 100 cycles. 390 00:24:58,740 --> 00:25:02,510 So what it means is that if I put this switch in, everything 391 00:25:02,510 --> 00:25:05,080 is more or less flat above 100 cycles. 392 00:25:05,080 --> 00:25:09,960 And what that's used for, basically, is to eliminate 393 00:25:09,960 --> 00:25:13,990 perhaps 60 cycle noise, if that's present, or some low 394 00:25:13,990 --> 00:25:16,100 frequency hum or whatever. 395 00:25:16,100 --> 00:25:18,370 Well, we won't really demonstrate 396 00:25:18,370 --> 00:25:19,780 anything with that. 397 00:25:19,780 --> 00:25:24,130 Let's [? go ?] now with the equalization in, demonstrate 398 00:25:24,130 --> 00:25:28,250 the effect of boosting or attenuating the low and high 399 00:25:28,250 --> 00:25:29,250 frequencies. 400 00:25:29,250 --> 00:25:33,030 And again, I think to demonstrate this, it 401 00:25:33,030 --> 00:25:36,020 illustrates the point the best if we have a 402 00:25:36,020 --> 00:25:37,110 little background music. 403 00:25:37,110 --> 00:25:39,750 So maestro, if you can bring that up. 404 00:25:39,750 --> 00:25:41,050 [MUSIC PLAYING] 405 00:25:41,050 --> 00:25:45,330 And so now what I'm going to do is first boost the low 406 00:25:45,330 --> 00:25:46,760 frequencies. 407 00:25:46,760 --> 00:25:50,040 And that's what this potentiometer knob will do. 408 00:25:50,040 --> 00:25:54,980 So now, increasing the low frequency gain and, in fact, 409 00:25:54,980 --> 00:25:58,720 all the way up to 12 dB when I have the knob over as far as 410 00:25:58,720 --> 00:26:00,200 I've gone here. 411 00:26:00,200 --> 00:26:02,610 And so that has a very bassy sound. 412 00:26:02,610 --> 00:26:06,370 And in fact, we can make it even bassier by taking the 413 00:26:06,370 --> 00:26:11,230 high frequencies and attenuating those by 12 dB. 414 00:26:15,930 --> 00:26:21,230 OK well, let's put some of the high frequencies back in. 415 00:26:21,230 --> 00:26:26,300 And now let's turn the low-frequency gain first 416 00:26:26,300 --> 00:26:30,080 back down to 0. 417 00:26:30,080 --> 00:26:32,560 And now we're back to flat equalization. 418 00:26:32,560 --> 00:26:37,180 And now I can turn the low frequency gain down so that I 419 00:26:37,180 --> 00:26:41,090 attenuate the low frequencies by much as 12 dB. 420 00:26:41,090 --> 00:26:42,540 And that's where we are now. 421 00:26:42,540 --> 00:26:47,170 And so this has, of course, a much crisper sound. 422 00:26:47,170 --> 00:26:51,984 And to enhance the highs even more, I can, in addition to 423 00:26:51,984 --> 00:26:55,890 cutting out the lows, boost the highs by putting in, 424 00:26:55,890 --> 00:26:57,140 again, as much as 12 dB. 425 00:27:00,410 --> 00:27:07,050 OK well, let's turn down the music now and go back to no 426 00:27:07,050 --> 00:27:10,210 equalization by setting these knobs to 0 dB. 427 00:27:10,210 --> 00:27:13,200 And in fact, we can take the equalizer out. 428 00:27:13,200 --> 00:27:17,580 Well, that's a quick look at some real-world filters. 429 00:27:17,580 --> 00:27:21,160 Now let's stop having so much fun, and let's 430 00:27:21,160 --> 00:27:22,410 go back to the lecture. 431 00:27:29,150 --> 00:27:33,010 OK well, that's a little behind-the-scenes look. 432 00:27:33,010 --> 00:27:37,070 What I'd like to do now is turn our attention to 433 00:27:37,070 --> 00:27:39,130 discrete-time filters. 434 00:27:39,130 --> 00:27:45,720 And as I've meant in previous lectures, there are basically 435 00:27:45,720 --> 00:27:50,810 two classes of discrete-time filters or discrete-time 436 00:27:50,810 --> 00:27:53,280 difference equations. 437 00:27:53,280 --> 00:27:58,390 One class is referred to a non-recursive or moving 438 00:27:58,390 --> 00:28:00,930 average filter. 439 00:28:00,930 --> 00:28:04,740 And the basic idea with a moving average filter is 440 00:28:04,740 --> 00:28:07,840 something that perhaps you're somewhat familiar with 441 00:28:07,840 --> 00:28:09,490 intuitively. 442 00:28:09,490 --> 00:28:14,520 Think of the notion of taking a data sequence, and let's 443 00:28:14,520 --> 00:28:17,810 suppose that what we wanted to do was apply some smoothing to 444 00:28:17,810 --> 00:28:19,290 the data sequence. 445 00:28:19,290 --> 00:28:23,220 We could, for example, think of taking adjacent points, 446 00:28:23,220 --> 00:28:27,600 averaging them together, and then moving that average along 447 00:28:27,600 --> 00:28:28,960 the data sequence. 448 00:28:28,960 --> 00:28:32,000 And what you can kind of see intuitively is that that would 449 00:28:32,000 --> 00:28:34,020 apply some smoothing. 450 00:28:34,020 --> 00:28:36,720 So in fact, the difference equation, let's say, for 451 00:28:36,720 --> 00:28:39,730 three-point moving average would be the difference 452 00:28:39,730 --> 00:28:44,240 equation that I indicate here, just simply taking a data 453 00:28:44,240 --> 00:28:49,340 point and the two data points adjacent to it and forming an 454 00:28:49,340 --> 00:28:51,540 average of those three. 455 00:28:51,540 --> 00:28:55,260 So if we thought of the processing involved, if we're 456 00:28:55,260 --> 00:29:00,530 forming an output sequence value, we would take three 457 00:29:00,530 --> 00:29:02,240 adjacent points and average them. 458 00:29:02,240 --> 00:29:06,440 That would give us the output add the associated time. 459 00:29:06,440 --> 00:29:09,630 And then to compute the next output point, we would just 460 00:29:09,630 --> 00:29:14,590 simply slide this by one point, average these together, 461 00:29:14,590 --> 00:29:17,070 and that would give us the next output point. 462 00:29:17,070 --> 00:29:20,900 And we would continue along, just simply sliding and 463 00:29:20,900 --> 00:29:25,110 averaging to form the output data sequence. 464 00:29:25,110 --> 00:29:29,490 Now, that's an example of what's commonly referred to a 465 00:29:29,490 --> 00:29:31,370 three-point moving average. 466 00:29:31,370 --> 00:29:34,770 In fact, we can generalize that notion 467 00:29:34,770 --> 00:29:36,110 in a number of ways. 468 00:29:36,110 --> 00:29:39,920 One way of generalizing the notion of a moving average 469 00:29:39,920 --> 00:29:43,030 from the three-point moving average, which I summarize 470 00:29:43,030 --> 00:29:47,110 again here, is to think of extending that to a larger 471 00:29:47,110 --> 00:29:52,150 number of points, and in fact applying weights to that as I 472 00:29:52,150 --> 00:29:56,090 indicated here, so that, in addition to just summing up 473 00:29:56,090 --> 00:29:59,590 the points and dividing by the number of points summed, we 474 00:29:59,590 --> 00:30:03,870 can, in fact, apply individual weights to the points so that 475 00:30:03,870 --> 00:30:08,590 it's what is often referred to as a weighting moving average. 476 00:30:08,590 --> 00:30:15,130 And I show below one possible curve that might result, where 477 00:30:15,130 --> 00:30:19,450 these would be essentially the weights associated with this 478 00:30:19,450 --> 00:30:21,410 weighted moving average. 479 00:30:21,410 --> 00:30:25,670 And in fact, it's easy to verify that this indeed 480 00:30:25,670 --> 00:30:30,890 corresponds to the impulse response of the filter. 481 00:30:30,890 --> 00:30:34,910 Well, just to cement this notion, let me show you an 482 00:30:34,910 --> 00:30:36,580 example or two. 483 00:30:36,580 --> 00:30:41,410 Here is an example of a five-point moving average. 484 00:30:41,410 --> 00:30:45,610 A five-point moving average would have an impulse response 485 00:30:45,610 --> 00:30:50,510 that just consists of a rectangle of length five. 486 00:30:50,510 --> 00:30:54,510 And if this is convolved with a data sequence, that would 487 00:30:54,510 --> 00:30:57,880 correspond to taking five adjacent points and, in 488 00:30:57,880 --> 00:30:59,870 effect, averaging them. 489 00:30:59,870 --> 00:31:02,890 We've looked previously at the Fourier transform of this 490 00:31:02,890 --> 00:31:04,240 rectangular sequence. 491 00:31:04,240 --> 00:31:08,490 And the Fourier transform of that, in fact, is of the form 492 00:31:08,490 --> 00:31:12,300 of a sine n x over sine x curve. 493 00:31:12,300 --> 00:31:17,720 And as you can see, that is some approximation to a 494 00:31:17,720 --> 00:31:18,860 lowpass filter. 495 00:31:18,860 --> 00:31:23,320 And so this, again, is the impulse response and frequency 496 00:31:23,320 --> 00:31:28,430 response of a nonideal lowpass filter. 497 00:31:28,430 --> 00:31:33,770 Now, there are a variety of algorithms that, in fact, tell 498 00:31:33,770 --> 00:31:37,840 you how to choose the weights associated with a weighted 499 00:31:37,840 --> 00:31:41,080 moving average to, in some sense, design better 500 00:31:41,080 --> 00:31:45,060 approximations and without going into the details of any 501 00:31:45,060 --> 00:31:46,510 of those algorithms. 502 00:31:46,510 --> 00:31:51,920 Let me just show the result of choosing the weights for the 503 00:31:51,920 --> 00:31:58,610 design of a 251-point moving average filter, where the 504 00:31:58,610 --> 00:32:02,430 weights are chosen using an optimum algorithm to generate 505 00:32:02,430 --> 00:32:06,070 as sharp a cutoff as can possibly be generated. 506 00:32:06,070 --> 00:32:11,750 And so what I show here is the frequency response of the 507 00:32:11,750 --> 00:32:14,990 resulting filter on a logarithmic amplitude scale 508 00:32:14,990 --> 00:32:17,560 and a linear frequency scale. 509 00:32:17,560 --> 00:32:20,860 Notice that on this scale, the passband is very flat. 510 00:32:20,860 --> 00:32:23,500 Although here is an expanded view of it. 511 00:32:23,500 --> 00:32:27,240 And in fact, it has what's referred to as an equal-ripple 512 00:32:27,240 --> 00:32:29,350 characteristic. 513 00:32:29,350 --> 00:32:31,970 And then here is the transition band. 514 00:32:31,970 --> 00:32:35,750 And here we have to stopband, which in fact is down somewhat 515 00:32:35,750 --> 00:32:40,150 more than 80 dB and, again, has what's referred to as an 516 00:32:40,150 --> 00:32:41,400 equal-ripple characteristic. 517 00:32:43,770 --> 00:32:48,470 Now, the notion of a moving average for filtering is 518 00:32:48,470 --> 00:32:53,010 something that is very commonly used. 519 00:32:53,010 --> 00:32:56,780 I had shown last time actually the result of some filtering 520 00:32:56,780 --> 00:32:59,250 on a particular data sequence, the Dow 521 00:32:59,250 --> 00:33:01,260 Jones Industrial Average. 522 00:33:01,260 --> 00:33:08,180 And very often, in looking at various kinds of stock market 523 00:33:08,180 --> 00:33:13,810 publications, what you will see is the Dow Jones average 524 00:33:13,810 --> 00:33:16,630 shown in its raw form as a data sequence. 525 00:33:16,630 --> 00:33:21,780 And then very typically, you'll see also the result of 526 00:33:21,780 --> 00:33:25,520 a moving average, where the moving average might be on the 527 00:33:25,520 --> 00:33:28,930 order of day, or it might be on the order of months. 528 00:33:28,930 --> 00:33:32,950 The whole notion being to take some of the random high 529 00:33:32,950 --> 00:33:36,900 frequency fluctuations out of the average and show the low 530 00:33:36,900 --> 00:33:43,060 frequency, or trends, over some period of time. 531 00:33:43,060 --> 00:33:46,940 So let's, in fact, go back to the Dow Jones average. 532 00:33:46,940 --> 00:33:52,450 And let me now show you what the result of filtering with a 533 00:33:52,450 --> 00:33:56,330 moving average filter would look like on the same Dow 534 00:33:56,330 --> 00:33:58,040 Jones industrial average sequence that 535 00:33:58,040 --> 00:34:00,580 I showed last time. 536 00:34:00,580 --> 00:34:05,340 So once again, we have the Dow Jones average from 1927 to 537 00:34:05,340 --> 00:34:07,380 roughly 1932. 538 00:34:07,380 --> 00:34:10,460 At the top, we see the impulse response 539 00:34:10,460 --> 00:34:12,080 for the moving average. 540 00:34:12,080 --> 00:34:15,790 Again, I remind you on an expanded time scale, and 541 00:34:15,790 --> 00:34:18,030 what's shown here is the moving average 542 00:34:18,030 --> 00:34:19,630 with just one point. 543 00:34:19,630 --> 00:34:24,340 So the output on the bottom trace is just simply identical 544 00:34:24,340 --> 00:34:25,790 to the input. 545 00:34:25,790 --> 00:34:28,000 Now, let's increase the length of the moving 546 00:34:28,000 --> 00:34:29,389 average to two points. 547 00:34:29,389 --> 00:34:32,889 And we see that there is a small amount of smoothing, 548 00:34:32,889 --> 00:34:34,690 three points and just a little more 549 00:34:34,690 --> 00:34:37,489 smoothing, that gets inserted. 550 00:34:37,489 --> 00:34:43,590 Now a four-point moving average, and next the 551 00:34:43,590 --> 00:34:47,710 five-point moving average, and a six-point 552 00:34:47,710 --> 00:34:49,320 moving average next. 553 00:34:49,320 --> 00:34:51,690 And we see that the smoothing increases. 554 00:34:51,690 --> 00:34:54,639 Now, let's increase the length of the moving average filter 555 00:34:54,639 --> 00:35:00,310 much more rapidly and watch how the output is more and 556 00:35:00,310 --> 00:35:03,340 more smooth in relation to the input. 557 00:35:03,340 --> 00:35:07,620 Again, I emphasize that the time scale for the impulse 558 00:35:07,620 --> 00:35:11,890 response is significantly expanded in relationship to 559 00:35:11,890 --> 00:35:16,450 the time scale for both the input and the output. 560 00:35:16,450 --> 00:35:20,000 And once again, through the magic of filtering, we've been 561 00:35:20,000 --> 00:35:23,240 able to eliminate the 1929 Stock Market Crash. 562 00:35:26,530 --> 00:35:30,490 All right, so we've seen moving average filters, or 563 00:35:30,490 --> 00:35:35,520 what are sometimes referred to as non-recursive filters. 564 00:35:35,520 --> 00:35:39,580 And they are, as I stressed, a very important class of 565 00:35:39,580 --> 00:35:41,880 discrete-time filters. 566 00:35:41,880 --> 00:35:45,550 Another very important class of discrete-time filters are 567 00:35:45,550 --> 00:35:49,530 what are referred to as recursive filters. 568 00:35:49,530 --> 00:35:53,320 Recursive filters are filters for which the difference 569 00:35:53,320 --> 00:35:58,850 equation has feedback from the output back into the input. 570 00:35:58,850 --> 00:36:03,320 In other words, the output depends not only on the input, 571 00:36:03,320 --> 00:36:06,910 but also on previous values of the output. 572 00:36:06,910 --> 00:36:10,300 So for example, as I've stressed previously, a 573 00:36:10,300 --> 00:36:14,930 recursive difference equation has the general form that I 574 00:36:14,930 --> 00:36:18,710 indicate here, a linear combination of weighted 575 00:36:18,710 --> 00:36:21,950 outputs on the left-hand side and linear combination of 576 00:36:21,950 --> 00:36:24,720 weighted inputs on the right-hand side. 577 00:36:24,720 --> 00:36:30,370 And as we've talked about, we can solve this equation for 578 00:36:30,370 --> 00:36:34,790 the current output y of n in terms of current and past 579 00:36:34,790 --> 00:36:38,610 inputs and past outputs. 580 00:36:38,610 --> 00:36:42,550 For example, just to interpret this, focus on the 581 00:36:42,550 --> 00:36:47,050 interpretation of this as a filter, let's look at a first 582 00:36:47,050 --> 00:36:50,280 order difference equation, which we've talked about and 583 00:36:50,280 --> 00:36:52,770 generated the solution to previously. 584 00:36:52,770 --> 00:36:56,810 So the first order difference equation would be as I 585 00:36:56,810 --> 00:36:59,220 indicated here. 586 00:36:59,220 --> 00:37:04,150 And imposing causality on this, so that we assume that 587 00:37:04,150 --> 00:37:08,090 we are running this as a recursive forward in time, we 588 00:37:08,090 --> 00:37:13,020 can solve this for y of n in terms of x of n and y of n 589 00:37:13,020 --> 00:37:16,520 minus 1 weighted by the factor a. 590 00:37:16,520 --> 00:37:20,970 And I simply indicate the block diagram for this. 591 00:37:20,970 --> 00:37:24,120 But what we want to examine now for this first order 592 00:37:24,120 --> 00:37:28,400 recursion is the frequency response and see its 593 00:37:28,400 --> 00:37:30,840 interpretation as a filter. 594 00:37:30,840 --> 00:37:34,330 Well in fact, again, the mathematics for this we've 595 00:37:34,330 --> 00:37:37,540 gone through in the last lecture. 596 00:37:37,540 --> 00:37:41,210 And so interpreting the first order difference equation as a 597 00:37:41,210 --> 00:37:48,130 system, what we're attempting to generate is the frequency 598 00:37:48,130 --> 00:37:50,730 response, which is the Fourier transform 599 00:37:50,730 --> 00:37:52,760 of the impulse response. 600 00:37:52,760 --> 00:37:56,080 And from the difference equation, we can, of course, 601 00:37:56,080 --> 00:38:00,790 solve for either one of those by using the properties, 602 00:38:00,790 --> 00:38:03,770 exploiting the properties, of Fourier transform. 603 00:38:03,770 --> 00:38:07,760 Applying the Fourier transform to the difference equation, we 604 00:38:07,760 --> 00:38:12,060 will end up with the Fourier transform of the output equal 605 00:38:12,060 --> 00:38:15,530 to the Fourier transform of the input times this factor, 606 00:38:15,530 --> 00:38:19,160 which we know from the convolution property, in fact, 607 00:38:19,160 --> 00:38:24,980 is the frequency response of the system. 608 00:38:24,980 --> 00:38:27,440 So this is the frequency response. 609 00:38:27,440 --> 00:38:31,290 And of course, the inverse Fourier transform of that, 610 00:38:31,290 --> 00:38:37,910 which I indicate below, is the system impulse response. 611 00:38:37,910 --> 00:38:41,430 So we have the frequency response obtained by applying 612 00:38:41,430 --> 00:38:44,150 the Fourier transform to the difference equation, the 613 00:38:44,150 --> 00:38:46,190 impulse response. 614 00:38:46,190 --> 00:38:54,200 And, as we did last time, we can look at that in terms of a 615 00:38:54,200 --> 00:38:56,990 frequency response characteristic. 616 00:38:56,990 --> 00:39:01,470 And recall that, depending on whether the factor a is 617 00:39:01,470 --> 00:39:05,560 positive or negative, we either get a lowpass filter or 618 00:39:05,560 --> 00:39:07,840 a highpass filter. 619 00:39:07,840 --> 00:39:13,170 And if, in fact, we look at the frequency response for the 620 00:39:13,170 --> 00:39:16,640 factor a being positive, then we see that this is an 621 00:39:16,640 --> 00:39:22,080 approximation to a lowpass filter, whereas below it I 622 00:39:22,080 --> 00:39:26,390 show the frequency response for a negative. 623 00:39:26,390 --> 00:39:32,300 And there this corresponds to a highpass filter, because 624 00:39:32,300 --> 00:39:36,820 we're attenuating low frequencies and retaining the 625 00:39:36,820 --> 00:39:39,350 high frequencies. 626 00:39:39,350 --> 00:39:45,110 And recall also that we illustrated this 627 00:39:45,110 --> 00:39:49,170 characteristic as a lowpass or highpass filter for the first 628 00:39:49,170 --> 00:39:55,660 order recursion by looking at how it worked as a filter in 629 00:39:55,660 --> 00:39:59,030 both cases when the input was the Dow Jones average. 630 00:39:59,030 --> 00:40:02,630 And indeed, we saw that it generated both lowpass and 631 00:40:02,630 --> 00:40:06,380 highpass filtering in the appropriate cases. 632 00:40:06,380 --> 00:40:09,880 So for discrete-time, we have the two classes, moving 633 00:40:09,880 --> 00:40:14,210 average and recursive filters. 634 00:40:14,210 --> 00:40:17,270 And there are a variety of issues discussed in the text 635 00:40:17,270 --> 00:40:19,960 about why, in certain contexts, one might want to 636 00:40:19,960 --> 00:40:21,390 use one of the other. 637 00:40:21,390 --> 00:40:24,870 Basically, what happens is that for the moving average 638 00:40:24,870 --> 00:40:28,660 filter, for a given set a filter specifications, there 639 00:40:28,660 --> 00:40:31,440 are many more multiplications required than 640 00:40:31,440 --> 00:40:33,190 for a recursive filter. 641 00:40:33,190 --> 00:40:36,250 But there are, in certain contexts, some very important 642 00:40:36,250 --> 00:40:41,450 compensating benefits for the moving average filter. 643 00:40:41,450 --> 00:40:47,320 Now, this concludes, pretty much, what I want to say in 644 00:40:47,320 --> 00:40:51,200 detail about filtering, the concept of filtering, in the 645 00:40:51,200 --> 00:40:52,880 set of lectures. 646 00:40:52,880 --> 00:40:57,830 This is only a very quick glimpse into a very important 647 00:40:57,830 --> 00:41:01,440 and very rich topic, and one, of course, that can be studied 648 00:41:01,440 --> 00:41:05,250 on its own in an considerable amount of detail. 649 00:41:05,250 --> 00:41:09,760 As the lectures go on, what we'll find is that the basic 650 00:41:09,760 --> 00:41:14,570 concept of filtering, both ideal and nonideal filtering, 651 00:41:14,570 --> 00:41:19,180 will be a very important part of what we do. 652 00:41:19,180 --> 00:41:24,370 And in particular, beginning with the next lecture, we'll 653 00:41:24,370 --> 00:41:29,930 turn to a discussion of modulation, exploiting the 654 00:41:29,930 --> 00:41:33,490 property of modulation as it relates to 655 00:41:33,490 --> 00:41:35,340 some practical problems. 656 00:41:35,340 --> 00:41:39,280 And what we'll find when we do that is that a very important 657 00:41:39,280 --> 00:41:42,660 part of that discussion and, in fact, a very important part 658 00:41:42,660 --> 00:41:48,120 of the use of modulation also just naturally incorporates 659 00:41:48,120 --> 00:41:51,270 the concept and properties of filtering. 660 00:41:51,270 --> 00:41:52,520 Thank you.