WEBVTT 00:00:00.040 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:03.880 Commons license. 00:00:03.880 --> 00:00:06.920 Your support will help MIT OpenCourseWare continue to 00:00:06.920 --> 00:00:10.570 offer high quality educational resources for free. 00:00:10.570 --> 00:00:13.470 To make a donation or view additional materials from 00:00:13.470 --> 00:00:17.400 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:17.400 --> 00:00:18.650 ocw.mit.edu. 00:00:55.619 --> 00:00:58.000 PROFESSOR: In discussing the continuous-time and 00:00:58.000 --> 00:01:01.540 discrete-time Fourier transforms, we developed a 00:01:01.540 --> 00:01:04.050 number of important properties. 00:01:04.050 --> 00:01:07.230 Two particularly significant ones, as I mentioned at the 00:01:07.230 --> 00:01:10.490 time, are the modulation property and 00:01:10.490 --> 00:01:12.860 the convolution property. 00:01:12.860 --> 00:01:15.760 Starting with the next lecture, the one after this 00:01:15.760 --> 00:01:19.190 one, we'll be developing and exploiting some of the 00:01:19.190 --> 00:01:22.260 consequences of the modulation property. 00:01:22.260 --> 00:01:27.620 In today's lecture though, I'd like to review and expand on 00:01:27.620 --> 00:01:31.100 the notion of filtering, which, as I had mentioned, 00:01:31.100 --> 00:01:36.700 flows more or less directly from the convolution property. 00:01:36.700 --> 00:01:39.820 To begin, let me just quickly review what the convolution 00:01:39.820 --> 00:01:41.260 property is. 00:01:41.260 --> 00:01:45.780 Both for continuous-time and for discrete-time, the 00:01:45.780 --> 00:01:51.400 convolution property tells us that the Fourier transform of 00:01:51.400 --> 00:01:56.400 the convolution of two time functions is the product of 00:01:56.400 --> 00:01:59.190 the Fourier transforms. 00:01:59.190 --> 00:02:04.450 Now, what this means in terms of linear time-invariant 00:02:04.450 --> 00:02:08.889 filters, since we know that in the time domain the output of 00:02:08.889 --> 00:02:12.890 a linear time-invariant filter is the convolution of the 00:02:12.890 --> 00:02:17.080 input and the impulse response, it says essentially 00:02:17.080 --> 00:02:20.790 then in the frequency domain that the Fourier transform of 00:02:20.790 --> 00:02:24.740 the output is the product the Fourier transform of the 00:02:24.740 --> 00:02:28.500 impulse response, namely the frequency response, and the 00:02:28.500 --> 00:02:31.850 Fourier transform of the input. 00:02:31.850 --> 00:02:36.370 So the output is described through that product. 00:02:36.370 --> 00:02:40.760 Now, recall also that in developing the Fourier 00:02:40.760 --> 00:02:45.250 transform, I interpreted the Fourier transform as the 00:02:45.250 --> 00:02:49.320 complex amplitude of a decomposition of the signal in 00:02:49.320 --> 00:02:52.120 terms of a set of complex exponentials. 00:02:52.120 --> 00:02:55.230 And the frequency response or the convolution property, in 00:02:55.230 --> 00:03:01.560 effect, tells us how to modify the amplitudes of each of 00:03:01.560 --> 00:03:05.320 those complex exponentials as they go through the system. 00:03:05.320 --> 00:03:09.190 Now, this led to the notion of filtering, where the basic 00:03:09.190 --> 00:03:14.780 concept was that since we can modify the amplitudes of each 00:03:14.780 --> 00:03:19.530 of the complex exponential components separately, we can, 00:03:19.530 --> 00:03:23.370 for example, retain some of them and 00:03:23.370 --> 00:03:25.040 totally eliminate others. 00:03:25.040 --> 00:03:27.810 And this is the basic notion of filtering. 00:03:27.810 --> 00:03:33.000 So we have, as you recall, first of all the notion in 00:03:33.000 --> 00:03:38.190 continuous-time of an ideal filter, for example, I 00:03:38.190 --> 00:03:43.730 illustrate here an ideal lowpass filter where we pass 00:03:43.730 --> 00:03:49.280 exactly frequency components in one band and reject totally 00:03:49.280 --> 00:03:51.940 frequency components in another band. 00:03:51.940 --> 00:03:54.620 The band being passed, of course, referred to as the 00:03:54.620 --> 00:04:00.030 passband, and the band rejected as the stopband. 00:04:00.030 --> 00:04:02.220 I illustrated here a lowpass filter. 00:04:02.220 --> 00:04:07.310 We can, of course, reject the low frequencies and retain the 00:04:07.310 --> 00:04:08.930 high frequencies. 00:04:08.930 --> 00:04:14.150 And that then corresponds to an ideal highpass filter. 00:04:14.150 --> 00:04:17.810 Or we can just retain frequencies within a band. 00:04:17.810 --> 00:04:22.640 And so I show below what is referred to commonly as a 00:04:22.640 --> 00:04:25.240 bandpass filter. 00:04:25.240 --> 00:04:29.560 Now, this is what the ideal filters looked like for 00:04:29.560 --> 00:04:31.070 continuous-time. 00:04:31.070 --> 00:04:35.200 For discrete-time, we have exactly the same situation. 00:04:35.200 --> 00:04:39.870 Namely, we have an ideal discrete-time lowpass filter, 00:04:39.870 --> 00:04:43.820 which passes exactly frequencies which are the low 00:04:43.820 --> 00:04:45.120 frequencies. 00:04:45.120 --> 00:04:48.080 Low frequencies, of course, being around 0, and because of 00:04:48.080 --> 00:04:52.260 the periodicity, also around 2pi. 00:04:52.260 --> 00:04:56.630 We show also an ideal highpass filter. 00:04:56.630 --> 00:05:00.180 And a highpass filter, as I indicated last time, passes 00:05:00.180 --> 00:05:02.740 frequencies around pi. 00:05:02.740 --> 00:05:10.280 And finally, below that, I show an ideal bandpass filter 00:05:10.280 --> 00:05:16.210 passing frequencies someplace in the range between 0 and pi. 00:05:16.210 --> 00:05:20.330 And recall also that the basic difference between 00:05:20.330 --> 00:05:22.610 continuous-time a discrete-time for these 00:05:22.610 --> 00:05:26.240 filters is that the discrete-time versions are, of 00:05:26.240 --> 00:05:29.960 course, periodic in frequency. 00:05:29.960 --> 00:05:34.600 Now, let's look at these ideal filters, and in particular the 00:05:34.600 --> 00:05:39.320 ideal lowpass filter in the time domain. 00:05:39.320 --> 00:05:45.290 We have the frequency response of the ideal lowpass filter. 00:05:45.290 --> 00:05:49.580 And shown below it is the impulse response. 00:05:49.580 --> 00:05:54.270 So here is the frequency response and below it the 00:05:54.270 --> 00:05:59.020 impulse response of the ideal lowpass filter. 00:05:59.020 --> 00:06:02.270 And this, of course, is a sine x over x 00:06:02.270 --> 00:06:04.270 form of impulse response. 00:06:04.270 --> 00:06:08.820 And recognize also or recall that since this frequency 00:06:08.820 --> 00:06:14.890 response is real-valued, the impulse response, in other 00:06:14.890 --> 00:06:19.150 words, the inverse transform is an even function of time. 00:06:19.150 --> 00:06:24.840 And notice also, since I want to refer back to this, that 00:06:24.840 --> 00:06:29.270 the impulse response of an ideal lowpass filter, in fact, 00:06:29.270 --> 00:06:30.530 is non-causal. 00:06:30.530 --> 00:06:33.420 That follows, from among other things, from the fact that 00:06:33.420 --> 00:06:34.940 it's an even function. 00:06:34.940 --> 00:06:39.160 But keep in mind, in fact, that a sine x over x function 00:06:39.160 --> 00:06:41.680 goes off to infinity in both directions. 00:06:41.680 --> 00:06:45.720 So the impulse response of the ideal lowpass filter is 00:06:45.720 --> 00:06:49.920 symmetric and continues to have tails off to plus and 00:06:49.920 --> 00:06:52.680 minus infinity. 00:06:52.680 --> 00:06:56.850 Now, the situation is basically the same in the 00:06:56.850 --> 00:06:59.610 discrete-time case. 00:06:59.610 --> 00:07:03.340 Let's look at the frequency response and associated 00:07:03.340 --> 00:07:08.200 impulse response for an ideal discrete-time lowpass filter. 00:07:08.200 --> 00:07:13.940 So once again, here is the frequency response of the 00:07:13.940 --> 00:07:16.230 ideal lowpass filter. 00:07:16.230 --> 00:07:20.000 And below what I show the impulse response. 00:07:20.000 --> 00:07:24.690 Again, it's a sine x over x type of impulse response. 00:07:24.690 --> 00:07:30.290 And again, we recognize that since in the frequency domain, 00:07:30.290 --> 00:07:34.760 this frequency response is real-valued. 00:07:34.760 --> 00:07:39.060 That means, as a consequence of the properties of the 00:07:39.060 --> 00:07:43.240 Fourier transform and inverse Fourier transform, that the 00:07:43.240 --> 00:07:47.800 impulse response is an even function in the time domain. 00:07:47.800 --> 00:07:52.180 And also, incidentally, the sine x over x function goes 00:07:52.180 --> 00:07:54.270 off to infinity, again, in both directions. 00:07:56.790 --> 00:08:01.650 Now, we've talked about ideal filters in this discussion. 00:08:01.650 --> 00:08:05.150 And ideal filters all are, in fact, ideal 00:08:05.150 --> 00:08:06.960 in a certain sense. 00:08:06.960 --> 00:08:10.550 What they do ideally is they pass a certain band of 00:08:10.550 --> 00:08:16.060 frequencies exactly and they reject a band 00:08:16.060 --> 00:08:19.010 of frequencies exactly. 00:08:19.010 --> 00:08:21.900 On the other hand, there are many filtering problems in 00:08:21.900 --> 00:08:26.330 which, generally, we don't have a sharp distinction 00:08:26.330 --> 00:08:28.670 between the frequencies we want to pass and the 00:08:28.670 --> 00:08:31.260 frequencies we want to reject. 00:08:31.260 --> 00:08:34.990 One example of this that's elaborated on in the text is 00:08:34.990 --> 00:08:37.610 the design of an automotive suspension system, which, in 00:08:37.610 --> 00:08:42.490 fact, is the design of a lowpass filter. 00:08:42.490 --> 00:08:46.550 And basically what you want to do in a case like that is 00:08:46.550 --> 00:08:51.920 filter out or attenuate very rapid road variations and keep 00:08:51.920 --> 00:08:56.020 the lower variations in, of course, elevation of the 00:08:56.020 --> 00:08:58.350 highway or road. 00:08:58.350 --> 00:09:02.250 And what you can see intuitively is that there 00:09:02.250 --> 00:09:05.910 isn't really a very sharp distinction or sharp cut-off 00:09:05.910 --> 00:09:08.670 between what you would logically call the low 00:09:08.670 --> 00:09:13.220 frequencies and what you would call the high frequencies. 00:09:13.220 --> 00:09:18.000 Now, also somewhat related to this is the fact that as we've 00:09:18.000 --> 00:09:22.350 seen in the time domain, these ideal filters have a very 00:09:22.350 --> 00:09:24.300 particular kind of character. 00:09:24.300 --> 00:09:29.900 For example, let's look back at the ideal lowpass filter. 00:09:29.900 --> 00:09:34.190 And we saw the impulse response. 00:09:34.190 --> 00:09:37.090 The impulse response is what we had shown here. 00:09:37.090 --> 00:09:41.860 Let's now look at the step response of the discrete-time 00:09:41.860 --> 00:09:44.110 ideal lowpass filter. 00:09:44.110 --> 00:09:47.590 And notice the fact that it has a tail that oscillates. 00:09:47.590 --> 00:09:50.300 And when the step hits, in fact, it has 00:09:50.300 --> 00:09:52.720 an oscillatory behavior. 00:09:52.720 --> 00:09:57.670 Now, exactly the same situation occurs in 00:09:57.670 --> 00:09:59.100 continuous-time. 00:09:59.100 --> 00:10:03.650 Let's look at the step response of the 00:10:03.650 --> 00:10:07.310 continuous-time ideal lowpass filter. 00:10:07.310 --> 00:10:12.540 And what we see is that when a step hits then, in fact, we 00:10:12.540 --> 00:10:14.480 get an oscillation. 00:10:14.480 --> 00:10:18.740 And very often, that oscillation is something 00:10:18.740 --> 00:10:19.670 that's undesirable. 00:10:19.670 --> 00:10:22.460 For example, if you were designing an automotive 00:10:22.460 --> 00:10:27.100 suspension system and you hit a curve, which is a step 00:10:27.100 --> 00:10:31.740 input, in fact, you probably would not like to have the 00:10:31.740 --> 00:10:38.260 automobile oscillating, dying down in oscillation. 00:10:38.260 --> 00:10:41.790 Now there's another very important point, which again, 00:10:41.790 --> 00:10:43.620 we can see either in continuous-time or 00:10:43.620 --> 00:10:49.200 discrete-time, which is that even if we want it to have an 00:10:49.200 --> 00:10:54.880 ideal filter, the ideal filter has another problem if we want 00:10:54.880 --> 00:10:57.960 to attempt to implement it in real time. 00:10:57.960 --> 00:10:59.180 What's the problem? 00:10:59.180 --> 00:11:04.420 The problem is that since the impulse response is even and, 00:11:04.420 --> 00:11:08.410 in fact, has tails that go off to plus and minus infinity, 00:11:08.410 --> 00:11:10.190 it's non-causal. 00:11:10.190 --> 00:11:15.320 So if, in fact, we want to build a filter and the filter 00:11:15.320 --> 00:11:20.290 is restricted to operate in real time, then, in fact, we 00:11:20.290 --> 00:11:23.920 can't build an ideal filter. 00:11:23.920 --> 00:11:28.080 So what that says is that, in practice, although ideal 00:11:28.080 --> 00:11:32.260 filters are nice to think about and perhaps relate to 00:11:32.260 --> 00:11:37.780 practical problems, more typically what we consider are 00:11:37.780 --> 00:11:43.780 nonideal filters and in the discrete-time case, a nonideal 00:11:43.780 --> 00:11:48.850 filter then we would have a characteristic somewhat like 00:11:48.850 --> 00:11:50.430 I've indicated here. 00:11:50.430 --> 00:11:55.300 Where instead of a very rapid transition from passband to 00:11:55.300 --> 00:12:00.330 stopband, there would be a more gradual transition with a 00:12:00.330 --> 00:12:06.960 passband cutoff frequency and a stopband cutoff frequency. 00:12:06.960 --> 00:12:11.170 And perhaps also instead of having an exactly flat 00:12:11.170 --> 00:12:15.210 characteristic in the stopband in the passband, we would 00:12:15.210 --> 00:12:18.180 allow a certain amount of ripple. 00:12:18.180 --> 00:12:23.350 We also have exactly the same situation in continuous-time, 00:12:23.350 --> 00:12:27.560 where here we'll just simply change our frequency axis to a 00:12:27.560 --> 00:12:30.480 continuous frequency axis instead of the discrete 00:12:30.480 --> 00:12:31.920 frequency axis. 00:12:31.920 --> 00:12:35.460 Again, we would think in terms of an allowable passband 00:12:35.460 --> 00:12:41.910 ripple, a transition from passband to stopband with a 00:12:41.910 --> 00:12:46.940 passband cutoff frequency and a stopband cutoff frequency. 00:12:46.940 --> 00:12:52.120 So the notion here is that, again, ideal filters are ideal 00:12:52.120 --> 00:12:55.100 in some respects, not ideal in other respects. 00:12:55.100 --> 00:12:57.850 And for many practical problems, we 00:12:57.850 --> 00:12:59.360 may not want them. 00:12:59.360 --> 00:13:02.400 And even if we did want them, we may not be able to get 00:13:02.400 --> 00:13:06.280 them, perhaps because of this issue of causality. 00:13:06.280 --> 00:13:11.050 Even if causality is not an issue, what happens in filter 00:13:11.050 --> 00:13:16.660 design and implementation, in fact, is that the sharper you 00:13:16.660 --> 00:13:20.720 attempt to make the cutoff, the more expensive, in some 00:13:20.720 --> 00:13:24.460 sense, the filter becomes, either in terms of components, 00:13:24.460 --> 00:13:27.960 in continuous-time, or in terms of computation in 00:13:27.960 --> 00:13:29.130 discrete-time. 00:13:29.130 --> 00:13:34.970 And so there are these whole variety of issues that really 00:13:34.970 --> 00:13:37.990 make it important to understand the 00:13:37.990 --> 00:13:41.790 notion nonideal filters. 00:13:41.790 --> 00:13:47.520 Now, just to illustrate as an example, let me remind you of 00:13:47.520 --> 00:13:53.640 one example of what, in fact, is a nonideal lowpass filter. 00:13:53.640 --> 00:13:57.820 And we have looked previously at the associated 00:13:57.820 --> 00:13:59.660 differential equation. 00:13:59.660 --> 00:14:04.270 Let me now, in fact, relate it to a circuit, and in 00:14:04.270 --> 00:14:08.730 particular an RC circuit, where the output could either 00:14:08.730 --> 00:14:12.120 be across the capacitor or the output can 00:14:12.120 --> 00:14:13.840 be across the resistor. 00:14:13.840 --> 00:14:16.340 So in effect, we have two systems here. 00:14:16.340 --> 00:14:20.540 We have a system, which is the system function from the 00:14:20.540 --> 00:14:24.490 voltage source input to the capacitor output, the system 00:14:24.490 --> 00:14:29.450 from the voltage source input to the resistor output. 00:14:29.450 --> 00:14:32.630 And, in fact, just applying Kirchhoff's Voltage Law to 00:14:32.630 --> 00:14:35.840 this, we can relate those in a very straightforward way. 00:14:35.840 --> 00:14:41.540 It's very straightforward to verify that the system from 00:14:41.540 --> 00:14:46.430 input to resistor output is simply the identity system 00:14:46.430 --> 00:14:51.480 with the capacitor output subtracted from it. 00:14:51.480 --> 00:14:54.850 Now, we can write the differential equation for 00:14:54.850 --> 00:14:59.360 either of these systems and, as we talked about last time 00:14:59.360 --> 00:15:03.860 in the last several lectures, solve that equation using and 00:15:03.860 --> 00:15:06.730 exploiting the properties of the Fourier transform. 00:15:06.730 --> 00:15:11.790 And in fact, if we look at the differential equation relating 00:15:11.790 --> 00:15:16.830 the capacitor output to the voltage source input, we 00:15:16.830 --> 00:15:20.000 recognize that this is an example that, in effect, we've 00:15:20.000 --> 00:15:21.740 solved previously. 00:15:21.740 --> 00:15:26.290 And so just working our way down, applying the Fourier 00:15:26.290 --> 00:15:30.110 transform to the differential equation and generating the 00:15:30.110 --> 00:15:34.500 system function by taking the ratio of the capacitor voltage 00:15:34.500 --> 00:15:37.360 or its Fourier transform to the Fourier transform of the 00:15:37.360 --> 00:15:42.030 source, we then have the system function associated 00:15:42.030 --> 00:15:44.650 with the system for which the output is 00:15:44.650 --> 00:15:46.550 the capacitor voltage. 00:15:46.550 --> 00:15:50.710 Or if we solve instead for the system function associated 00:15:50.710 --> 00:15:54.450 with the resistor output, we can simply 00:15:54.450 --> 00:15:57.390 subtract H1 from unity. 00:15:57.390 --> 00:16:00.510 And the system function that we get in that case is the 00:16:00.510 --> 00:16:03.040 system function that I show here. 00:16:03.040 --> 00:16:07.510 So we have, now, two system functions, one for the 00:16:07.510 --> 00:16:11.480 capacitor output, the other for the resistor output. 00:16:11.480 --> 00:16:15.540 And one, the first, corresponding to the capacitor 00:16:15.540 --> 00:16:20.520 output, in fact, if we plot it on a linear amplitude scale, 00:16:20.520 --> 00:16:21.430 looks like this. 00:16:21.430 --> 00:16:24.760 And as you can see, and as we saw last time, is an 00:16:24.760 --> 00:16:27.630 approximation to a lowpass filter. 00:16:27.630 --> 00:16:33.000 It is, in fact, and nonideal lowpass filter, whereas the 00:16:33.000 --> 00:16:37.920 resistor output is an approximation to a highpass 00:16:37.920 --> 00:16:42.200 filter, or in effect, a nonideal highpass filter. 00:16:42.200 --> 00:16:46.230 So in one case, just comparing the two, we have a lowpass 00:16:46.230 --> 00:16:50.210 filter as the capacitor output associated with the capacitor 00:16:50.210 --> 00:16:53.700 output, and a highpass filter associated with 00:16:53.700 --> 00:16:56.100 the resistor output. 00:16:56.100 --> 00:17:00.110 Let's just quickly look at that example now, looking on a 00:17:00.110 --> 00:17:05.069 Bode plot, instead of on the linear scale 00:17:05.069 --> 00:17:06.640 that we showed before. 00:17:06.640 --> 00:17:11.119 And recall incidentally, and be aware incidentally, of the 00:17:11.119 --> 00:17:16.380 fact that we can, of course, cascade several filters of 00:17:16.380 --> 00:17:18.940 this type and improve the characteristics. 00:17:18.940 --> 00:17:28.040 So I have shown at the top a Bode plot of the system 00:17:28.040 --> 00:17:30.940 function associated with the capacitor output. 00:17:30.940 --> 00:17:35.810 It's flat out to a frequency corresponding to 1 over the 00:17:35.810 --> 00:17:38.590 time constant, RC. 00:17:38.590 --> 00:17:43.820 And then it falls off at 10 dB per decade, a decade being a 00:17:43.820 --> 00:17:45.640 factor of 10. 00:17:45.640 --> 00:17:49.390 Or if instead we look at the system function associated 00:17:49.390 --> 00:17:54.300 with the resistor output, that corresponds to a 10 dB per 00:17:54.300 --> 00:17:59.020 decade increase in frequency up to approximately the 00:17:59.020 --> 00:18:02.510 reciprocal of the time constant, and then approaching 00:18:02.510 --> 00:18:05.550 a flat characteristic after that. 00:18:05.550 --> 00:18:11.400 And if we consider either one of these, looking back again 00:18:11.400 --> 00:18:15.980 at the lowpass filter, if we were to cascade several 00:18:15.980 --> 00:18:20.660 filters with this frequency response, then because we have 00:18:20.660 --> 00:18:24.330 things plotted on a Bode plot, the Bode plot for the cascade 00:18:24.330 --> 00:18:26.010 would simply be summing these. 00:18:26.010 --> 00:18:30.330 And so if we cascaded, for example, two stages instead of 00:18:30.330 --> 00:18:34.210 a roll-off at 10 dB per decade, it would roll off at 00:18:34.210 --> 00:18:37.780 20 dB per decade. 00:18:37.780 --> 00:18:42.100 Now, filters in this type, RC filters, perhaps several of 00:18:42.100 --> 00:18:47.360 them in cascade, are in fact very prevalent. 00:18:47.360 --> 00:18:53.465 And in fact, in an environment like this, where we're, in 00:18:53.465 --> 00:19:00.150 fact, doing recording, we see there are filters of that type 00:19:00.150 --> 00:19:03.790 that show up very commonly both in the audio and the 00:19:03.790 --> 00:19:08.140 video portion of the signal processing that's associated 00:19:08.140 --> 00:19:10.470 with making this set of tapes. 00:19:10.470 --> 00:19:14.350 In fact, let's take a look in the control room. 00:19:14.350 --> 00:19:19.070 And what I'll be able to show you in the control room is the 00:19:19.070 --> 00:19:23.200 audio portion of the processing that's done and the 00:19:23.200 --> 00:19:26.890 kinds of filters, very much of the type we just talked about, 00:19:26.890 --> 00:19:30.560 that are associated with the signal processing that's done 00:19:30.560 --> 00:19:33.140 in preparing the audio for the tapes. 00:19:33.140 --> 00:19:36.120 So let's just take a walk into the control room 00:19:36.120 --> 00:19:37.370 and see what we see. 00:19:40.110 --> 00:19:42.280 This is the control room that's 00:19:42.280 --> 00:19:44.260 used for camera switching. 00:19:44.260 --> 00:19:48.210 It's used for computer editing and also audio control. 00:19:48.210 --> 00:19:51.150 You can see the monitors, and these are used 00:19:51.150 --> 00:19:53.090 for the camera switching. 00:19:53.090 --> 00:19:56.870 And this is the computer editing console that's used 00:19:56.870 --> 00:19:59.890 for online and offline computer editing. 00:19:59.890 --> 00:20:02.520 What I really want to demonstrate though, in the 00:20:02.520 --> 00:20:06.760 context of the lecture is the audio control panel, which 00:20:06.760 --> 00:20:10.680 contains, among other things, a variety of filters for high 00:20:10.680 --> 00:20:13.280 frequency, low frequencies, et cetera, basically 00:20:13.280 --> 00:20:15.280 equalization filters. 00:20:15.280 --> 00:20:20.380 And what we have in the way of filtering is, first of all, 00:20:20.380 --> 00:20:23.390 what's referred to as a graphic equalizer, which 00:20:23.390 --> 00:20:26.500 consists of a set of bandpass filters, which I'll describe a 00:20:26.500 --> 00:20:28.340 little more carefully in a minute. 00:20:28.340 --> 00:20:32.820 And then also, an audio control panel, which is down 00:20:32.820 --> 00:20:36.350 here and which contains separate equalizer circuits 00:20:36.350 --> 00:20:39.860 for each of a whole set of channels and also lots of 00:20:39.860 --> 00:20:41.170 controls on them. 00:20:41.170 --> 00:20:46.690 Well, let me begin in the demonstration by demonstrating 00:20:46.690 --> 00:20:51.380 a little bit of what the graphic equalizer does. 00:20:51.380 --> 00:20:55.780 Well, what we have is a set of bandpass filters. 00:20:55.780 --> 00:20:58.880 And what's indicated up here are the center frequencies of 00:20:58.880 --> 00:21:02.430 the filters, and then a slider switch for each one that lets 00:21:02.430 --> 00:21:04.570 us attenuate or amplify. 00:21:04.570 --> 00:21:06.550 And this is a dB scale. 00:21:06.550 --> 00:21:12.460 So essentially, if you look across this bank of filters 00:21:12.460 --> 00:21:15.800 with the total output of the equalizer just being the sum 00:21:15.800 --> 00:21:19.550 of the outputs from each of these filters, interestingly 00:21:19.550 --> 00:21:22.320 the position of the slider switches as you move across 00:21:22.320 --> 00:21:25.680 here, in effect, shows you what the frequency response of 00:21:25.680 --> 00:21:27.090 the equalizer is. 00:21:27.090 --> 00:21:31.460 So you can change the overall shaping of the filter by 00:21:31.460 --> 00:21:33.870 moving the switches up and down. 00:21:33.870 --> 00:21:35.780 Right now the equalizer is out. 00:21:35.780 --> 00:21:38.480 Let's put the equalizer into the circuit. 00:21:38.480 --> 00:21:42.000 And now I put in this filtering characteristic. 00:21:42.000 --> 00:21:45.910 And what I'd like to demonstrate is filtering with 00:21:45.910 --> 00:21:49.240 this, when we do things that are a little more dramatic 00:21:49.240 --> 00:21:51.560 than what would normally be done in a typical audio 00:21:51.560 --> 00:21:53.250 recording setting. 00:21:53.250 --> 00:21:58.130 And to do this, let's add to my voice some music to make it 00:21:58.130 --> 00:22:00.120 more interesting. 00:22:00.120 --> 00:22:02.340 Not that my voice isn't interesting as it is. 00:22:02.340 --> 00:22:04.980 But in any case, let's bring some music up. 00:22:04.980 --> 00:22:05.660 [MUSIC PLAYING] 00:22:05.660 --> 00:22:12.900 And now what I'll do is set the low frequencies flat. 00:22:12.900 --> 00:22:18.430 And let me take out the high frequencies above 800 cycles. 00:22:18.430 --> 00:22:21.160 And so now what we have, effectively, 00:22:21.160 --> 00:22:23.940 is a lowpass filter. 00:22:23.940 --> 00:22:28.470 And now with the lowpass filter, let me now bring the 00:22:28.470 --> 00:22:31.180 highs back up. 00:22:31.180 --> 00:22:35.310 And so I'm bringing up those bandpass filters. 00:22:35.310 --> 00:22:39.490 And now let me cut out the lows. 00:22:39.490 --> 00:22:43.230 And you'll hear the lows disappearing and, in effect, 00:22:43.230 --> 00:22:47.200 keeping the highs in effectively crispens the 00:22:47.200 --> 00:22:50.050 sound, either my voice or the music. 00:22:50.050 --> 00:22:55.250 And finally, let me go back to 0 dB equalization on each of 00:22:55.250 --> 00:22:56.770 the filters. 00:22:56.770 --> 00:23:01.660 And what I'll also do now is take the equalizer out of the 00:23:01.660 --> 00:23:02.910 circuit totally. 00:23:05.860 --> 00:23:10.490 Now, let's take a look at the audio master control panel. 00:23:10.490 --> 00:23:15.000 And this panel has, of course, for each channel and, for 00:23:15.000 --> 00:23:16.460 example, the channel that we're working 00:23:16.460 --> 00:23:18.580 on, of a volume control. 00:23:18.580 --> 00:23:23.990 I can turn the volume down, and I can turn the volume up. 00:23:23.990 --> 00:23:29.100 And it also has, for this particular equalizer circuit, 00:23:29.100 --> 00:23:35.930 it has a set of three bandpass filters and knobs which let us 00:23:35.930 --> 00:23:41.480 either put in up to 12 dB gain or 12 dB attenuation in each 00:23:41.480 --> 00:23:44.990 of the bands, and also a selector switch that lets us 00:23:44.990 --> 00:23:46.780 select the center the band. 00:23:46.780 --> 00:23:49.630 So let me just again demonstrate a 00:23:49.630 --> 00:23:50.810 little bit with this. 00:23:50.810 --> 00:23:54.100 And let's get a close up of this panel. 00:23:54.100 --> 00:23:57.450 So what we have, as I indicated, is 00:23:57.450 --> 00:23:59.310 three bandpass filters. 00:23:59.310 --> 00:24:03.460 And these knobs that I'm pointing to here are controls 00:24:03.460 --> 00:24:07.200 that allow us for each of the filters to put in up to 12 dB 00:24:07.200 --> 00:24:10.310 gain or 12 dB attenuation. 00:24:10.310 --> 00:24:14.510 There are also with each of the filters a selector switch 00:24:14.510 --> 00:24:18.000 that lets us adjust the center frequency of the filter. 00:24:18.000 --> 00:24:21.660 Basically it's a two-position switch. 00:24:21.660 --> 00:24:27.200 There also, as you can see, is a button that let's us either 00:24:27.200 --> 00:24:28.830 put the equalization in or out. 00:24:28.830 --> 00:24:31.290 Currently the equalization is out. 00:24:31.290 --> 00:24:32.820 Let's put the equalization in. 00:24:32.820 --> 00:24:35.770 We won't hear any effect from that, because the gain 00:24:35.770 --> 00:24:38.420 controls are all set at 0 dB. 00:24:38.420 --> 00:24:41.820 And I'll want to illustrate shortly the effect of these. 00:24:41.820 --> 00:24:45.630 But before I do, let me draw your attention to one other 00:24:45.630 --> 00:24:49.660 filter, which is this white switch. 00:24:49.660 --> 00:24:55.870 And this switch is a highpass filter that essentially cuts 00:24:55.870 --> 00:24:58.740 out frequencies below about 100 cycles. 00:24:58.740 --> 00:25:02.510 So what it means is that if I put this switch in, everything 00:25:02.510 --> 00:25:05.080 is more or less flat above 100 cycles. 00:25:05.080 --> 00:25:09.960 And what that's used for, basically, is to eliminate 00:25:09.960 --> 00:25:13.990 perhaps 60 cycle noise, if that's present, or some low 00:25:13.990 --> 00:25:16.100 frequency hum or whatever. 00:25:16.100 --> 00:25:18.370 Well, we won't really demonstrate 00:25:18.370 --> 00:25:19.780 anything with that. 00:25:19.780 --> 00:25:24.130 Let's [? go ?] now with the equalization in, demonstrate 00:25:24.130 --> 00:25:28.250 the effect of boosting or attenuating the low and high 00:25:28.250 --> 00:25:29.250 frequencies. 00:25:29.250 --> 00:25:33.030 And again, I think to demonstrate this, it 00:25:33.030 --> 00:25:36.020 illustrates the point the best if we have a 00:25:36.020 --> 00:25:37.110 little background music. 00:25:37.110 --> 00:25:39.750 So maestro, if you can bring that up. 00:25:39.750 --> 00:25:41.050 [MUSIC PLAYING] 00:25:41.050 --> 00:25:45.330 And so now what I'm going to do is first boost the low 00:25:45.330 --> 00:25:46.760 frequencies. 00:25:46.760 --> 00:25:50.040 And that's what this potentiometer knob will do. 00:25:50.040 --> 00:25:54.980 So now, increasing the low frequency gain and, in fact, 00:25:54.980 --> 00:25:58.720 all the way up to 12 dB when I have the knob over as far as 00:25:58.720 --> 00:26:00.200 I've gone here. 00:26:00.200 --> 00:26:02.610 And so that has a very bassy sound. 00:26:02.610 --> 00:26:06.370 And in fact, we can make it even bassier by taking the 00:26:06.370 --> 00:26:11.230 high frequencies and attenuating those by 12 dB. 00:26:15.930 --> 00:26:21.230 OK well, let's put some of the high frequencies back in. 00:26:21.230 --> 00:26:26.300 And now let's turn the low-frequency gain first 00:26:26.300 --> 00:26:30.080 back down to 0. 00:26:30.080 --> 00:26:32.560 And now we're back to flat equalization. 00:26:32.560 --> 00:26:37.180 And now I can turn the low frequency gain down so that I 00:26:37.180 --> 00:26:41.090 attenuate the low frequencies by much as 12 dB. 00:26:41.090 --> 00:26:42.540 And that's where we are now. 00:26:42.540 --> 00:26:47.170 And so this has, of course, a much crisper sound. 00:26:47.170 --> 00:26:51.984 And to enhance the highs even more, I can, in addition to 00:26:51.984 --> 00:26:55.890 cutting out the lows, boost the highs by putting in, 00:26:55.890 --> 00:26:57.140 again, as much as 12 dB. 00:27:00.410 --> 00:27:07.050 OK well, let's turn down the music now and go back to no 00:27:07.050 --> 00:27:10.210 equalization by setting these knobs to 0 dB. 00:27:10.210 --> 00:27:13.200 And in fact, we can take the equalizer out. 00:27:13.200 --> 00:27:17.580 Well, that's a quick look at some real-world filters. 00:27:17.580 --> 00:27:21.160 Now let's stop having so much fun, and let's 00:27:21.160 --> 00:27:22.410 go back to the lecture. 00:27:29.150 --> 00:27:33.010 OK well, that's a little behind-the-scenes look. 00:27:33.010 --> 00:27:37.070 What I'd like to do now is turn our attention to 00:27:37.070 --> 00:27:39.130 discrete-time filters. 00:27:39.130 --> 00:27:45.720 And as I've meant in previous lectures, there are basically 00:27:45.720 --> 00:27:50.810 two classes of discrete-time filters or discrete-time 00:27:50.810 --> 00:27:53.280 difference equations. 00:27:53.280 --> 00:27:58.390 One class is referred to a non-recursive or moving 00:27:58.390 --> 00:28:00.930 average filter. 00:28:00.930 --> 00:28:04.740 And the basic idea with a moving average filter is 00:28:04.740 --> 00:28:07.840 something that perhaps you're somewhat familiar with 00:28:07.840 --> 00:28:09.490 intuitively. 00:28:09.490 --> 00:28:14.520 Think of the notion of taking a data sequence, and let's 00:28:14.520 --> 00:28:17.810 suppose that what we wanted to do was apply some smoothing to 00:28:17.810 --> 00:28:19.290 the data sequence. 00:28:19.290 --> 00:28:23.220 We could, for example, think of taking adjacent points, 00:28:23.220 --> 00:28:27.600 averaging them together, and then moving that average along 00:28:27.600 --> 00:28:28.960 the data sequence. 00:28:28.960 --> 00:28:32.000 And what you can kind of see intuitively is that that would 00:28:32.000 --> 00:28:34.020 apply some smoothing. 00:28:34.020 --> 00:28:36.720 So in fact, the difference equation, let's say, for 00:28:36.720 --> 00:28:39.730 three-point moving average would be the difference 00:28:39.730 --> 00:28:44.240 equation that I indicate here, just simply taking a data 00:28:44.240 --> 00:28:49.340 point and the two data points adjacent to it and forming an 00:28:49.340 --> 00:28:51.540 average of those three. 00:28:51.540 --> 00:28:55.260 So if we thought of the processing involved, if we're 00:28:55.260 --> 00:29:00.530 forming an output sequence value, we would take three 00:29:00.530 --> 00:29:02.240 adjacent points and average them. 00:29:02.240 --> 00:29:06.440 That would give us the output add the associated time. 00:29:06.440 --> 00:29:09.630 And then to compute the next output point, we would just 00:29:09.630 --> 00:29:14.590 simply slide this by one point, average these together, 00:29:14.590 --> 00:29:17.070 and that would give us the next output point. 00:29:17.070 --> 00:29:20.900 And we would continue along, just simply sliding and 00:29:20.900 --> 00:29:25.110 averaging to form the output data sequence. 00:29:25.110 --> 00:29:29.490 Now, that's an example of what's commonly referred to a 00:29:29.490 --> 00:29:31.370 three-point moving average. 00:29:31.370 --> 00:29:34.770 In fact, we can generalize that notion 00:29:34.770 --> 00:29:36.110 in a number of ways. 00:29:36.110 --> 00:29:39.920 One way of generalizing the notion of a moving average 00:29:39.920 --> 00:29:43.030 from the three-point moving average, which I summarize 00:29:43.030 --> 00:29:47.110 again here, is to think of extending that to a larger 00:29:47.110 --> 00:29:52.150 number of points, and in fact applying weights to that as I 00:29:52.150 --> 00:29:56.090 indicated here, so that, in addition to just summing up 00:29:56.090 --> 00:29:59.590 the points and dividing by the number of points summed, we 00:29:59.590 --> 00:30:03.870 can, in fact, apply individual weights to the points so that 00:30:03.870 --> 00:30:08.590 it's what is often referred to as a weighting moving average. 00:30:08.590 --> 00:30:15.130 And I show below one possible curve that might result, where 00:30:15.130 --> 00:30:19.450 these would be essentially the weights associated with this 00:30:19.450 --> 00:30:21.410 weighted moving average. 00:30:21.410 --> 00:30:25.670 And in fact, it's easy to verify that this indeed 00:30:25.670 --> 00:30:30.890 corresponds to the impulse response of the filter. 00:30:30.890 --> 00:30:34.910 Well, just to cement this notion, let me show you an 00:30:34.910 --> 00:30:36.580 example or two. 00:30:36.580 --> 00:30:41.410 Here is an example of a five-point moving average. 00:30:41.410 --> 00:30:45.610 A five-point moving average would have an impulse response 00:30:45.610 --> 00:30:50.510 that just consists of a rectangle of length five. 00:30:50.510 --> 00:30:54.510 And if this is convolved with a data sequence, that would 00:30:54.510 --> 00:30:57.880 correspond to taking five adjacent points and, in 00:30:57.880 --> 00:30:59.870 effect, averaging them. 00:30:59.870 --> 00:31:02.890 We've looked previously at the Fourier transform of this 00:31:02.890 --> 00:31:04.240 rectangular sequence. 00:31:04.240 --> 00:31:08.490 And the Fourier transform of that, in fact, is of the form 00:31:08.490 --> 00:31:12.300 of a sine n x over sine x curve. 00:31:12.300 --> 00:31:17.720 And as you can see, that is some approximation to a 00:31:17.720 --> 00:31:18.860 lowpass filter. 00:31:18.860 --> 00:31:23.320 And so this, again, is the impulse response and frequency 00:31:23.320 --> 00:31:28.430 response of a nonideal lowpass filter. 00:31:28.430 --> 00:31:33.770 Now, there are a variety of algorithms that, in fact, tell 00:31:33.770 --> 00:31:37.840 you how to choose the weights associated with a weighted 00:31:37.840 --> 00:31:41.080 moving average to, in some sense, design better 00:31:41.080 --> 00:31:45.060 approximations and without going into the details of any 00:31:45.060 --> 00:31:46.510 of those algorithms. 00:31:46.510 --> 00:31:51.920 Let me just show the result of choosing the weights for the 00:31:51.920 --> 00:31:58.610 design of a 251-point moving average filter, where the 00:31:58.610 --> 00:32:02.430 weights are chosen using an optimum algorithm to generate 00:32:02.430 --> 00:32:06.070 as sharp a cutoff as can possibly be generated. 00:32:06.070 --> 00:32:11.750 And so what I show here is the frequency response of the 00:32:11.750 --> 00:32:14.990 resulting filter on a logarithmic amplitude scale 00:32:14.990 --> 00:32:17.560 and a linear frequency scale. 00:32:17.560 --> 00:32:20.860 Notice that on this scale, the passband is very flat. 00:32:20.860 --> 00:32:23.500 Although here is an expanded view of it. 00:32:23.500 --> 00:32:27.240 And in fact, it has what's referred to as an equal-ripple 00:32:27.240 --> 00:32:29.350 characteristic. 00:32:29.350 --> 00:32:31.970 And then here is the transition band. 00:32:31.970 --> 00:32:35.750 And here we have to stopband, which in fact is down somewhat 00:32:35.750 --> 00:32:40.150 more than 80 dB and, again, has what's referred to as an 00:32:40.150 --> 00:32:41.400 equal-ripple characteristic. 00:32:43.770 --> 00:32:48.470 Now, the notion of a moving average for filtering is 00:32:48.470 --> 00:32:53.010 something that is very commonly used. 00:32:53.010 --> 00:32:56.780 I had shown last time actually the result of some filtering 00:32:56.780 --> 00:32:59.250 on a particular data sequence, the Dow 00:32:59.250 --> 00:33:01.260 Jones Industrial Average. 00:33:01.260 --> 00:33:08.180 And very often, in looking at various kinds of stock market 00:33:08.180 --> 00:33:13.810 publications, what you will see is the Dow Jones average 00:33:13.810 --> 00:33:16.630 shown in its raw form as a data sequence. 00:33:16.630 --> 00:33:21.780 And then very typically, you'll see also the result of 00:33:21.780 --> 00:33:25.520 a moving average, where the moving average might be on the 00:33:25.520 --> 00:33:28.930 order of day, or it might be on the order of months. 00:33:28.930 --> 00:33:32.950 The whole notion being to take some of the random high 00:33:32.950 --> 00:33:36.900 frequency fluctuations out of the average and show the low 00:33:36.900 --> 00:33:43.060 frequency, or trends, over some period of time. 00:33:43.060 --> 00:33:46.940 So let's, in fact, go back to the Dow Jones average. 00:33:46.940 --> 00:33:52.450 And let me now show you what the result of filtering with a 00:33:52.450 --> 00:33:56.330 moving average filter would look like on the same Dow 00:33:56.330 --> 00:33:58.040 Jones industrial average sequence that 00:33:58.040 --> 00:34:00.580 I showed last time. 00:34:00.580 --> 00:34:05.340 So once again, we have the Dow Jones average from 1927 to 00:34:05.340 --> 00:34:07.380 roughly 1932. 00:34:07.380 --> 00:34:10.460 At the top, we see the impulse response 00:34:10.460 --> 00:34:12.080 for the moving average. 00:34:12.080 --> 00:34:15.790 Again, I remind you on an expanded time scale, and 00:34:15.790 --> 00:34:18.030 what's shown here is the moving average 00:34:18.030 --> 00:34:19.630 with just one point. 00:34:19.630 --> 00:34:24.340 So the output on the bottom trace is just simply identical 00:34:24.340 --> 00:34:25.790 to the input. 00:34:25.790 --> 00:34:28.000 Now, let's increase the length of the moving 00:34:28.000 --> 00:34:29.389 average to two points. 00:34:29.389 --> 00:34:32.889 And we see that there is a small amount of smoothing, 00:34:32.889 --> 00:34:34.690 three points and just a little more 00:34:34.690 --> 00:34:37.489 smoothing, that gets inserted. 00:34:37.489 --> 00:34:43.590 Now a four-point moving average, and next the 00:34:43.590 --> 00:34:47.710 five-point moving average, and a six-point 00:34:47.710 --> 00:34:49.320 moving average next. 00:34:49.320 --> 00:34:51.690 And we see that the smoothing increases. 00:34:51.690 --> 00:34:54.639 Now, let's increase the length of the moving average filter 00:34:54.639 --> 00:35:00.310 much more rapidly and watch how the output is more and 00:35:00.310 --> 00:35:03.340 more smooth in relation to the input. 00:35:03.340 --> 00:35:07.620 Again, I emphasize that the time scale for the impulse 00:35:07.620 --> 00:35:11.890 response is significantly expanded in relationship to 00:35:11.890 --> 00:35:16.450 the time scale for both the input and the output. 00:35:16.450 --> 00:35:20.000 And once again, through the magic of filtering, we've been 00:35:20.000 --> 00:35:23.240 able to eliminate the 1929 Stock Market Crash. 00:35:26.530 --> 00:35:30.490 All right, so we've seen moving average filters, or 00:35:30.490 --> 00:35:35.520 what are sometimes referred to as non-recursive filters. 00:35:35.520 --> 00:35:39.580 And they are, as I stressed, a very important class of 00:35:39.580 --> 00:35:41.880 discrete-time filters. 00:35:41.880 --> 00:35:45.550 Another very important class of discrete-time filters are 00:35:45.550 --> 00:35:49.530 what are referred to as recursive filters. 00:35:49.530 --> 00:35:53.320 Recursive filters are filters for which the difference 00:35:53.320 --> 00:35:58.850 equation has feedback from the output back into the input. 00:35:58.850 --> 00:36:03.320 In other words, the output depends not only on the input, 00:36:03.320 --> 00:36:06.910 but also on previous values of the output. 00:36:06.910 --> 00:36:10.300 So for example, as I've stressed previously, a 00:36:10.300 --> 00:36:14.930 recursive difference equation has the general form that I 00:36:14.930 --> 00:36:18.710 indicate here, a linear combination of weighted 00:36:18.710 --> 00:36:21.950 outputs on the left-hand side and linear combination of 00:36:21.950 --> 00:36:24.720 weighted inputs on the right-hand side. 00:36:24.720 --> 00:36:30.370 And as we've talked about, we can solve this equation for 00:36:30.370 --> 00:36:34.790 the current output y of n in terms of current and past 00:36:34.790 --> 00:36:38.610 inputs and past outputs. 00:36:38.610 --> 00:36:42.550 For example, just to interpret this, focus on the 00:36:42.550 --> 00:36:47.050 interpretation of this as a filter, let's look at a first 00:36:47.050 --> 00:36:50.280 order difference equation, which we've talked about and 00:36:50.280 --> 00:36:52.770 generated the solution to previously. 00:36:52.770 --> 00:36:56.810 So the first order difference equation would be as I 00:36:56.810 --> 00:36:59.220 indicated here. 00:36:59.220 --> 00:37:04.150 And imposing causality on this, so that we assume that 00:37:04.150 --> 00:37:08.090 we are running this as a recursive forward in time, we 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 00:37:13.020 --> 00:37:16.520 minus 1 weighted by the factor a. 00:37:16.520 --> 00:37:20.970 And I simply indicate the block diagram for this. 00:37:20.970 --> 00:37:24.120 But what we want to examine now for this first order 00:37:24.120 --> 00:37:28.400 recursion is the frequency response and see its 00:37:28.400 --> 00:37:30.840 interpretation as a filter. 00:37:30.840 --> 00:37:34.330 Well in fact, again, the mathematics for this we've 00:37:34.330 --> 00:37:37.540 gone through in the last lecture. 00:37:37.540 --> 00:37:41.210 And so interpreting the first order difference equation as a 00:37:41.210 --> 00:37:48.130 system, what we're attempting to generate is the frequency 00:37:48.130 --> 00:37:50.730 response, which is the Fourier transform 00:37:50.730 --> 00:37:52.760 of the impulse response. 00:37:52.760 --> 00:37:56.080 And from the difference equation, we can, of course, 00:37:56.080 --> 00:38:00.790 solve for either one of those by using the properties, 00:38:00.790 --> 00:38:03.770 exploiting the properties, of Fourier transform. 00:38:03.770 --> 00:38:07.760 Applying the Fourier transform to the difference equation, we 00:38:07.760 --> 00:38:12.060 will end up with the Fourier transform of the output equal 00:38:12.060 --> 00:38:15.530 to the Fourier transform of the input times this factor, 00:38:15.530 --> 00:38:19.160 which we know from the convolution property, in fact, 00:38:19.160 --> 00:38:24.980 is the frequency response of the system. 00:38:24.980 --> 00:38:27.440 So this is the frequency response. 00:38:27.440 --> 00:38:31.290 And of course, the inverse Fourier transform of that, 00:38:31.290 --> 00:38:37.910 which I indicate below, is the system impulse response. 00:38:37.910 --> 00:38:41.430 So we have the frequency response obtained by applying 00:38:41.430 --> 00:38:44.150 the Fourier transform to the difference equation, the 00:38:44.150 --> 00:38:46.190 impulse response. 00:38:46.190 --> 00:38:54.200 And, as we did last time, we can look at that in terms of a 00:38:54.200 --> 00:38:56.990 frequency response characteristic. 00:38:56.990 --> 00:39:01.470 And recall that, depending on whether the factor a is 00:39:01.470 --> 00:39:05.560 positive or negative, we either get a lowpass filter or 00:39:05.560 --> 00:39:07.840 a highpass filter. 00:39:07.840 --> 00:39:13.170 And if, in fact, we look at the frequency response for the 00:39:13.170 --> 00:39:16.640 factor a being positive, then we see that this is an 00:39:16.640 --> 00:39:22.080 approximation to a lowpass filter, whereas below it I 00:39:22.080 --> 00:39:26.390 show the frequency response for a negative. 00:39:26.390 --> 00:39:32.300 And there this corresponds to a highpass filter, because 00:39:32.300 --> 00:39:36.820 we're attenuating low frequencies and retaining the 00:39:36.820 --> 00:39:39.350 high frequencies. 00:39:39.350 --> 00:39:45.110 And recall also that we illustrated this 00:39:45.110 --> 00:39:49.170 characteristic as a lowpass or highpass filter for the first 00:39:49.170 --> 00:39:55.660 order recursion by looking at how it worked as a filter in 00:39:55.660 --> 00:39:59.030 both cases when the input was the Dow Jones average. 00:39:59.030 --> 00:40:02.630 And indeed, we saw that it generated both lowpass and 00:40:02.630 --> 00:40:06.380 highpass filtering in the appropriate cases. 00:40:06.380 --> 00:40:09.880 So for discrete-time, we have the two classes, moving 00:40:09.880 --> 00:40:14.210 average and recursive filters. 00:40:14.210 --> 00:40:17.270 And there are a variety of issues discussed in the text 00:40:17.270 --> 00:40:19.960 about why, in certain contexts, one might want to 00:40:19.960 --> 00:40:21.390 use one of the other. 00:40:21.390 --> 00:40:24.870 Basically, what happens is that for the moving average 00:40:24.870 --> 00:40:28.660 filter, for a given set a filter specifications, there 00:40:28.660 --> 00:40:31.440 are many more multiplications required than 00:40:31.440 --> 00:40:33.190 for a recursive filter. 00:40:33.190 --> 00:40:36.250 But there are, in certain contexts, some very important 00:40:36.250 --> 00:40:41.450 compensating benefits for the moving average filter. 00:40:41.450 --> 00:40:47.320 Now, this concludes, pretty much, what I want to say in 00:40:47.320 --> 00:40:51.200 detail about filtering, the concept of filtering, in the 00:40:51.200 --> 00:40:52.880 set of lectures. 00:40:52.880 --> 00:40:57.830 This is only a very quick glimpse into a very important 00:40:57.830 --> 00:41:01.440 and very rich topic, and one, of course, that can be studied 00:41:01.440 --> 00:41:05.250 on its own in an considerable amount of detail. 00:41:05.250 --> 00:41:09.760 As the lectures go on, what we'll find is that the basic 00:41:09.760 --> 00:41:14.570 concept of filtering, both ideal and nonideal filtering, 00:41:14.570 --> 00:41:19.180 will be a very important part of what we do. 00:41:19.180 --> 00:41:24.370 And in particular, beginning with the next lecture, we'll 00:41:24.370 --> 00:41:29.930 turn to a discussion of modulation, exploiting the 00:41:29.930 --> 00:41:33.490 property of modulation as it relates to 00:41:33.490 --> 00:41:35.340 some practical problems. 00:41:35.340 --> 00:41:39.280 And what we'll find when we do that is that a very important 00:41:39.280 --> 00:41:42.660 part of that discussion and, in fact, a very important part 00:41:42.660 --> 00:41:48.120 of the use of modulation also just naturally incorporates 00:41:48.120 --> 00:41:51.270 the concept and properties of filtering. 00:41:51.270 --> 00:41:52.520 Thank you.