WEBVTT 00:00:06.770 --> 00:00:42.070 [MUSIC PLAYING] 00:00:42.070 --> 00:00:46.020 PROFESSOR: Last time, we began the discussion of 00:00:46.020 --> 00:00:50.510 discreet-time processing of continuous-time signals. 00:00:50.510 --> 00:00:56.170 And, as a reminder, let me review the basic notion. 00:00:56.170 --> 00:01:03.880 The idea was that we convert from a continuous-time signal 00:01:03.880 --> 00:01:09.900 to a sequence through an operation which I represent as 00:01:09.900 --> 00:01:12.890 a continuous to discrete time converter. 00:01:12.890 --> 00:01:22.740 And then that sequence is used as the input to an appropriate 00:01:22.740 --> 00:01:25.000 discreet-time system. 00:01:25.000 --> 00:01:29.250 And after appropriate discreet-time processing, that 00:01:29.250 --> 00:01:35.820 sequence is converted back to a continuous-time signal 00:01:35.820 --> 00:01:40.960 through an operation which I label as a discrete to 00:01:40.960 --> 00:01:44.180 continuous time converter. 00:01:44.180 --> 00:01:49.630 Now, in the lecture last time, we carried out some analysis 00:01:49.630 --> 00:01:54.910 which related for us the spectra in the first step of 00:01:54.910 --> 00:01:56.250 this operation. 00:01:56.250 --> 00:02:00.730 Namely in the transformation from a continuous-time time 00:02:00.730 --> 00:02:03.520 signal to a sequence. 00:02:03.520 --> 00:02:07.290 And let me, by the way, draw your attention to the fact 00:02:07.290 --> 00:02:12.970 that, in the real world, this operation is essentially 00:02:12.970 --> 00:02:17.540 implemented by what you would typically label as an analog 00:02:17.540 --> 00:02:21.430 to digital converter, if in fact the discreet-time 00:02:21.430 --> 00:02:23.990 processing is being done digitally. 00:02:23.990 --> 00:02:26.290 Now, it's important to emphasize that it's not 00:02:26.290 --> 00:02:30.290 exactly what an analog to digital converter does, but, 00:02:30.290 --> 00:02:34.060 in some sense at least, you should think of this mapping 00:02:34.060 --> 00:02:37.180 from continuous-time to discreet-time in very much the 00:02:37.180 --> 00:02:39.070 same way that one would think of an 00:02:39.070 --> 00:02:40.760 analog to digital converter. 00:02:40.760 --> 00:02:47.790 And the mapping back then corresponds, in some sense, to 00:02:47.790 --> 00:02:52.330 what would happen with a digital to analog converter. 00:02:52.330 --> 00:02:56.620 Well, let me review what is involved in the mapping from 00:02:56.620 --> 00:02:59.460 the continuous-time signal to the sequence. 00:02:59.460 --> 00:03:04.470 And, let me stress again that, this operation is basically-- 00:03:04.470 --> 00:03:07.400 in the continuous to discreet-time conversion-- 00:03:07.400 --> 00:03:10.170 a two-step process. 00:03:10.170 --> 00:03:13.940 In the first part of the process, the continuous-time 00:03:13.940 --> 00:03:20.740 signal is modulated with impulse train, where the 00:03:20.740 --> 00:03:25.510 period of the impulse train is capital T. And so we have a 00:03:25.510 --> 00:03:29.980 continuous-time time impulse train signal which captures 00:03:29.980 --> 00:03:35.470 the samples of the original continuous-time signal. 00:03:35.470 --> 00:03:42.280 That impulse train is then put through an operation which, 00:03:42.280 --> 00:03:49.100 essentially, re-labels the samples so that the sample 00:03:49.100 --> 00:03:55.400 values, the impulse areas, are re-labeled as sequence values. 00:03:55.400 --> 00:03:59.520 And the result of that conversion is then the 00:03:59.520 --> 00:04:02.020 sequence x of n. 00:04:02.020 --> 00:04:06.930 So the overall process, then, is a sampling process, 00:04:06.930 --> 00:04:10.610 followed by what is simply, in this box, 00:04:10.610 --> 00:04:12.510 a re-labeling process. 00:04:12.510 --> 00:04:18.110 And, although as I indicated just a minute ago, that is 00:04:18.110 --> 00:04:21.100 essentially what an analog to digital converter does. 00:04:21.100 --> 00:04:24.770 An analog to digital converter doesn't necessarily carry it 00:04:24.770 --> 00:04:30.280 out in those two steps, but particularly, in terms of 00:04:30.280 --> 00:04:33.610 carrying through an analysis, thinking of it as a two-step 00:04:33.610 --> 00:04:36.890 process is particularly convenient. 00:04:36.890 --> 00:04:43.200 Now we talked last time about what this mapping from 00:04:43.200 --> 00:04:46.370 continuous-time to discreet-time means, both in 00:04:46.370 --> 00:04:49.210 the time domain, and terms of the spectra. 00:04:49.210 --> 00:04:53.530 And in particular, in the time domain we begin with the 00:04:53.530 --> 00:04:58.550 continuous-time signal, which is then sampled with an 00:04:58.550 --> 00:05:04.010 impulse train and converted to a sequence by simply 00:05:04.010 --> 00:05:08.020 generating a sequence whose values are the 00:05:08.020 --> 00:05:10.490 areas of the impulses. 00:05:10.490 --> 00:05:14.420 And I stress the fact that what this corresponds to, 00:05:14.420 --> 00:05:17.620 essentially, is a normalization of the time 00:05:17.620 --> 00:05:23.950 axis, essentially, by dividing the time axis by capital T. 00:05:23.950 --> 00:05:29.430 In the frequency domain, then, we had the spectrum of the 00:05:29.430 --> 00:05:33.150 original signal, which, because of the sampling 00:05:33.150 --> 00:05:40.210 process, is replicated at integer multiples of the 00:05:40.210 --> 00:05:45.780 sampling frequency omega sub s, or 2 pi over capital T. And 00:05:45.780 --> 00:05:51.480 then, in converting the impulses to a sequence, we are 00:05:51.480 --> 00:05:57.150 essentially normalizing the frequency axis, so that the 00:05:57.150 --> 00:06:01.970 frequency 2 pi over capital T gets re-labeled as 2 pi. 00:06:01.970 --> 00:06:05.130 And the resulting discreet-time spectrum looks 00:06:05.130 --> 00:06:07.110 like I indicate here. 00:06:07.110 --> 00:06:12.310 Which really is nothing more than a frequency scaling 00:06:12.310 --> 00:06:15.140 corresponding to the associated time scaling. 00:06:15.140 --> 00:06:19.070 So the mapping from the impulse train spectrum to the 00:06:19.070 --> 00:06:23.450 discreet-time spectrum corresponds to a mapping 00:06:23.450 --> 00:06:27.620 specified by capital omega equal to small omega times 00:06:27.620 --> 00:06:29.800 capital T. 00:06:29.800 --> 00:06:35.470 And equivalently, it's the frequency 2 pi over capital T, 00:06:35.470 --> 00:06:39.670 which is, of course, the sampling frequency which gets 00:06:39.670 --> 00:06:44.730 normalized to the frequency 2 pi. 00:06:44.730 --> 00:06:49.070 And so, in the frequency domain, there is a frequency 00:06:49.070 --> 00:06:55.620 normalization associated with the fact that corresponding to 00:06:55.620 --> 00:07:01.290 this spectrum is a time sequence or discreet-time 00:07:01.290 --> 00:07:05.030 sequence, as I showed previously, and the 00:07:05.030 --> 00:07:10.900 discreet-time sequence is related to the original 00:07:10.900 --> 00:07:16.600 continuous-time signal through a time normalization. 00:07:16.600 --> 00:07:22.840 In other words, these sequence values are simply samples of 00:07:22.840 --> 00:07:26.510 the continuous-time signal with the time axis 00:07:26.510 --> 00:07:27.760 renormalized. 00:07:30.090 --> 00:07:31.910 Now, what we want to consider-- 00:07:31.910 --> 00:07:34.070 this is the conversion from continuous-time to 00:07:34.070 --> 00:07:35.260 discreet-time-- 00:07:35.260 --> 00:07:39.960 what we want to consider now is the overall system which 00:07:39.960 --> 00:07:42.960 implements not just the conversion, but filtering, and 00:07:42.960 --> 00:07:45.360 then coming back out of the conversion back to 00:07:45.360 --> 00:07:47.630 continuous-time. 00:07:47.630 --> 00:07:51.400 So let's look at the overall system. 00:07:51.400 --> 00:07:55.610 And the overall system, of course, as I've stressed 00:07:55.610 --> 00:08:01.080 several times in the past, consists of first, the 00:08:01.080 --> 00:08:09.980 sampling process, conversion to an impulse train, and the 00:08:09.980 --> 00:08:13.530 impulse train converted to a sequence. 00:08:13.530 --> 00:08:18.430 That sequence is then processed through our 00:08:18.430 --> 00:08:21.180 discreet-time filter. 00:08:21.180 --> 00:08:25.770 And after the discreet-time time processing, the result of 00:08:25.770 --> 00:08:30.700 that is converted back to an impulse train. 00:08:30.700 --> 00:08:35.710 So this resulting process sequence is then converted 00:08:35.710 --> 00:08:37.980 back to an impulse train. 00:08:37.980 --> 00:08:43.700 And then, finally, we carry out the desampling process by 00:08:43.700 --> 00:08:49.900 simply using a low-pass filter with a cutoff associated with 00:08:49.900 --> 00:08:53.170 the sampling frequency that we used. 00:08:53.170 --> 00:08:57.620 Now, typically in a system like that-- which implements 00:08:57.620 --> 00:09:01.690 discreet-time processing of continuous-time time signals-- 00:09:01.690 --> 00:09:07.030 we need to ensure in one way or another that the bandwidth 00:09:07.030 --> 00:09:10.100 of the input is sufficiently limited, so 00:09:10.100 --> 00:09:12.010 that we avoid aliasing. 00:09:12.010 --> 00:09:16.300 One way to do that is to force it one way or another, or 00:09:16.300 --> 00:09:19.190 simply know that our signal satisfies the bandwidth 00:09:19.190 --> 00:09:20.160 constraint. 00:09:20.160 --> 00:09:24.190 Although, a fairly typical thing to do in addition to 00:09:24.190 --> 00:09:30.770 this sampling process is to include what is referred to an 00:09:30.770 --> 00:09:33.440 anti-aliasing filter. 00:09:33.440 --> 00:09:37.580 In other words, this is a filter that would band-limit 00:09:37.580 --> 00:09:43.090 the input at at least half the sampling frequency, so that we 00:09:43.090 --> 00:09:47.850 are guaranteed, then, that there is no aliasing that's 00:09:47.850 --> 00:09:50.400 carried out in this process. 00:09:50.400 --> 00:09:55.000 And it's important to stress that, in this kind of 00:09:55.000 --> 00:09:55.610 processing-- 00:09:55.610 --> 00:09:59.190 discreet-time processing of continuous-time signals-- 00:09:59.190 --> 00:10:02.400 except in certain special situations, it's very 00:10:02.400 --> 00:10:06.750 important to avoid aliasing because we're going to want to 00:10:06.750 --> 00:10:09.660 do a reconstruction after we do the sampling and 00:10:09.660 --> 00:10:10.910 processing. 00:10:13.260 --> 00:10:17.070 OK, now, this is the sequence of steps in the time domain. 00:10:17.070 --> 00:10:20.610 Let's examine what happens as a consequence of this in the 00:10:20.610 --> 00:10:23.240 frequency domain. 00:10:23.240 --> 00:10:26.730 Well, let's choose some type of simple 00:10:26.730 --> 00:10:28.500 representative spectrum. 00:10:28.500 --> 00:10:32.320 And, of course, what's important about it is that the 00:10:32.320 --> 00:10:35.620 spectrum we choose is band-limited, or that there's 00:10:35.620 --> 00:10:37.470 an anti-aliasing filter. 00:10:37.470 --> 00:10:41.080 And it's not the shape, of course, that is critical. 00:10:41.080 --> 00:10:45.140 And as we work our way through the system, this is the 00:10:45.140 --> 00:10:47.460 continuous-time spectrum. 00:10:47.460 --> 00:10:53.530 After sampling, that spectrum is replicated at multiples of 00:10:53.530 --> 00:10:54.690 the sampling frequency-- 00:10:54.690 --> 00:10:56.840 integer multiples of the sampling frequency-- 00:10:56.840 --> 00:10:59.560 and so there would be another one over here, and another one 00:10:59.560 --> 00:11:01.380 over here, et cetera. 00:11:01.380 --> 00:11:05.370 And then, in converting to a discreet-time sequence, there 00:11:05.370 --> 00:11:10.500 is the associated frequency normalization, so that the 00:11:10.500 --> 00:11:14.360 sampling frequency gets normalized to a 00:11:14.360 --> 00:11:17.230 frequency of 2 pi. 00:11:17.230 --> 00:11:26.670 OK, now, at that point, where we are in the system is at 00:11:26.670 --> 00:11:30.750 this point, so that we've converted to a sequence. 00:11:30.750 --> 00:11:34.350 We now want to carry out some filtering, and then, after 00:11:34.350 --> 00:11:40.060 that filtering, convert back to a continuous-time signal. 00:11:40.060 --> 00:11:45.790 All right, so, here we are at the spectrum associated with 00:11:45.790 --> 00:11:49.040 the sequence. 00:11:49.040 --> 00:11:53.410 And now, the processing that we're carrying out is linear 00:11:53.410 --> 00:11:57.380 time and variant filtering in the discreet-time domain. 00:11:57.380 --> 00:12:01.560 And what that corresponds to, then, is multiplying this 00:12:01.560 --> 00:12:05.080 spectrum by the filter frequency response. 00:12:05.080 --> 00:12:08.250 And I've chosen a particular shape. 00:12:08.250 --> 00:12:10.430 And again, it's not the shape that's important to the 00:12:10.430 --> 00:12:14.310 discussion, but the fact, for example, that it has a 00:12:14.310 --> 00:12:18.070 particular cutoff frequency, which we will track as we work 00:12:18.070 --> 00:12:19.790 through this. 00:12:19.790 --> 00:12:24.550 And so now, the spectrum of y of n, the output of the 00:12:24.550 --> 00:12:30.120 digital filter, is the product of this spectrum, and the 00:12:30.120 --> 00:12:32.940 Fourier transform, or frequency response, of the 00:12:32.940 --> 00:12:34.190 digital filter. 00:12:36.320 --> 00:12:39.060 Now, in working our way through, we're going to take 00:12:39.060 --> 00:12:43.880 the output of the filter and undo the two-step process. 00:12:43.880 --> 00:12:46.700 So we now want to take that sequence, convert it to an 00:12:46.700 --> 00:12:50.600 impulse train, and then take that impulse train and 00:12:50.600 --> 00:12:53.460 desample through a low-pass filter. 00:12:53.460 --> 00:12:59.220 So, here we are now at the output of the digital filter. 00:12:59.220 --> 00:13:02.340 We then convert that to an impulse train. 00:13:02.340 --> 00:13:05.730 Well that's really undoing the original time normalization. 00:13:05.730 --> 00:13:10.520 And so, what that means, is that we are undoing the 00:13:10.520 --> 00:13:12.080 frequency normalization. 00:13:12.080 --> 00:13:17.320 In particular, we're dividing the frequency axis by capital 00:13:17.320 --> 00:13:24.610 T. Whereas, this point in y of omega was 2 pi, now it's 2 pi 00:13:24.610 --> 00:13:29.540 over capital T. What that means is that, equivalently, 00:13:29.540 --> 00:13:33.550 we're multiplying this spectrum by the frequency 00:13:33.550 --> 00:13:36.270 response of the digital filter, but now 00:13:36.270 --> 00:13:40.225 linearly-scaled in frequency, so that what was a cutoff 00:13:40.225 --> 00:13:44.830 frequency of omega sub c is now cutoff frequency of omega 00:13:44.830 --> 00:13:48.590 sub c, divided by capital T. 00:13:48.590 --> 00:13:53.760 So now, the next step in the process is the reconstructing 00:13:53.760 --> 00:13:55.290 low-pass filter. 00:13:55.290 --> 00:13:59.090 And what that extracts is simply the portion of this 00:13:59.090 --> 00:14:02.020 periodic spectrum around the origin. 00:14:02.020 --> 00:14:06.550 And so finally, then, the spectrum of the output of the 00:14:06.550 --> 00:14:11.920 overall system will be the spectrum of the input 00:14:11.920 --> 00:14:19.680 multiplied by a frequency response, which is the digital 00:14:19.680 --> 00:14:25.040 filter frequency response frequency scaled by dividing 00:14:25.040 --> 00:14:30.360 that digital filter frequency axis by capital T. 00:14:30.360 --> 00:14:35.580 OK, now, what we can ask is, we've got this processing-- 00:14:35.580 --> 00:14:37.390 we've converted to discreet-time, and we've gone 00:14:37.390 --> 00:14:41.590 back to continuous-time, and one can ask now what 00:14:41.590 --> 00:14:45.010 equivalent, overall continuous-time system does 00:14:45.010 --> 00:14:46.970 that correspond to? 00:14:46.970 --> 00:14:48.550 In other words, if we-- 00:14:48.550 --> 00:14:50.730 that, of course, is a continuous-time system, it's a 00:14:50.730 --> 00:14:53.790 continuous-time input and continuous-time time output-- 00:14:53.790 --> 00:14:58.980 and the overall system, then, would be one that would give 00:14:58.980 --> 00:15:02.540 us exactly the same output spectrum as we're getting. 00:15:02.540 --> 00:15:04.200 Well, what is that? 00:15:04.200 --> 00:15:08.480 What we have is an output spectrum, which is the product 00:15:08.480 --> 00:15:12.790 of the input spectrum and the digital filter frequency 00:15:12.790 --> 00:15:16.150 characteristic frequency-scaled. 00:15:16.150 --> 00:15:23.420 And so, in fact, the resulting continuous-time time filter is 00:15:23.420 --> 00:15:27.210 simply the digital filter with an 00:15:27.210 --> 00:15:29.120 appropriate frequency scaling. 00:15:29.120 --> 00:15:31.710 In other words, with the frequency axis divided by 00:15:31.710 --> 00:15:42.200 capital T. So said another way, if we show here the 00:15:42.200 --> 00:15:51.090 frequency response of the original digital filter, then 00:15:51.090 --> 00:15:56.150 the corresponding continuous-time filter would 00:15:56.150 --> 00:15:59.470 be this, frequency-scaled. 00:15:59.470 --> 00:16:02.930 And then, because of the associated low-pass filtering 00:16:02.930 --> 00:16:06.810 and the reconstruction, we would select out just one of 00:16:06.810 --> 00:16:10.070 these periods-- in particular, the portion around the origin. 00:16:10.070 --> 00:16:13.270 And the essential consequence of that is that the 00:16:13.270 --> 00:16:15.900 corresponding continuous-time filter, 00:16:15.900 --> 00:16:18.790 then, is given by this. 00:16:18.790 --> 00:16:24.150 And these two are related simply by a linear scaling of 00:16:24.150 --> 00:16:25.840 the frequency axis. 00:16:25.840 --> 00:16:28.830 And note that, where the digital filter has a cutoff 00:16:28.830 --> 00:16:33.100 frequency of omega sub c, the continuous-time filter has a 00:16:33.100 --> 00:16:38.560 cutoff frequency of omega sub c divided by capital T. 00:16:38.560 --> 00:16:41.750 So that's the linear frequency scaling. 00:16:41.750 --> 00:16:45.690 And, by the way, plant away for now-- and we'll return to 00:16:45.690 --> 00:16:48.170 this point later-- 00:16:48.170 --> 00:16:52.960 the observation that even if the digital filter frequency 00:16:52.960 --> 00:16:57.720 response is fixed, which we would assume it is, by 00:16:57.720 --> 00:17:00.740 changing the sampling frequency, in fact, what we're 00:17:00.740 --> 00:17:05.510 able to do is affect a linear scaling all of the equivalent 00:17:05.510 --> 00:17:09.170 continuous-time filter. 00:17:09.170 --> 00:17:12.960 OK, well, this is pretty much the process and the analysis, 00:17:12.960 --> 00:17:17.390 but to highlight a number of the issues and emphasize these 00:17:17.390 --> 00:17:24.530 points, what I'd like to do is illustrate some of this with a 00:17:24.530 --> 00:17:29.840 videotape demonstration that, in fact, was made originally 00:17:29.840 --> 00:17:35.390 as part of another course-- a course devoted entirely to 00:17:35.390 --> 00:17:38.120 digital signal processing, which essentially is 00:17:38.120 --> 00:17:41.200 discreet-time time processing, whether or not it's related to 00:17:41.200 --> 00:17:43.060 continuous-time signals. 00:17:43.060 --> 00:17:48.240 And what I'd like to now focus on are some of the details of 00:17:48.240 --> 00:17:50.980 that demonstration. 00:17:50.980 --> 00:17:57.450 In the demonstration, the specific impulse response that 00:17:57.450 --> 00:18:01.600 is used for the digital filter, or discreet-time 00:18:01.600 --> 00:18:06.160 filter, is the one that I show here. 00:18:06.160 --> 00:18:12.360 And the associated frequency response is the frequency 00:18:12.360 --> 00:18:16.460 response of a discreet-time, low-pass filter, 00:18:16.460 --> 00:18:18.800 as I indicate below. 00:18:18.800 --> 00:18:21.520 And the cutoff frequency of that filter-- 00:18:21.520 --> 00:18:24.890 as I indicate, the filter was designed as a discreet-time 00:18:24.890 --> 00:18:28.680 filter with a cutoff frequency of pi over 5. 00:18:28.680 --> 00:18:32.540 And let me just draw your attention to the fact that pi 00:18:32.540 --> 00:18:39.200 over 5 is also a 10th of 2 pi. 00:18:39.200 --> 00:18:44.140 And so in fact the digital or discreet-time filter cutoff 00:18:44.140 --> 00:18:47.250 frequency is a 10th of 2 pi. 00:18:47.250 --> 00:18:51.230 And as I'll stress again shortly, remember that, in the 00:18:51.230 --> 00:18:55.670 frequency normalization or unnormalization, 2 pi 00:18:55.670 --> 00:18:58.940 represents, in effect, the sampling frequency. 00:18:58.940 --> 00:19:03.080 And so the consequence of that is that the cutoff frequency 00:19:03.080 --> 00:19:06.750 really, is going to be associated with a 10th of the 00:19:06.750 --> 00:19:07.760 sampling frequency. 00:19:07.760 --> 00:19:09.720 But, for now, keep in mind that it's just 00:19:09.720 --> 00:19:13.040 simply a 10th of 2 pi. 00:19:13.040 --> 00:19:19.040 Now, the equivalent continuous-time system, in 00:19:19.040 --> 00:19:23.600 terms of the impulse response, is, of course, a band-limited 00:19:23.600 --> 00:19:31.350 interpolation of the impulse response associated with the 00:19:31.350 --> 00:19:33.380 discreet-time filter. 00:19:33.380 --> 00:19:37.080 And in the frequency domain, the frequency response is 00:19:37.080 --> 00:19:44.380 correspondingly a time-scaled, frequency scaled version of 00:19:44.380 --> 00:19:46.290 the frequency response. 00:19:46.290 --> 00:19:50.900 So, in fact, in the frequency domain and in the time domain 00:19:50.900 --> 00:19:57.320 related to the continuous-time signal, the associated impulse 00:19:57.320 --> 00:20:00.540 response is what I indicate here-- 00:20:00.540 --> 00:20:04.950 a band-limited interpolation of the discreet-time impulse 00:20:04.950 --> 00:20:09.340 response and time scale, in fact. 00:20:09.340 --> 00:20:11.000 And the frequency response-- 00:20:11.000 --> 00:20:12.680 following the discussion that we've 00:20:12.680 --> 00:20:14.410 previously gone through-- 00:20:14.410 --> 00:20:18.390 is a frequency-scaled version of the one associated with the 00:20:18.390 --> 00:20:20.630 digital filter. 00:20:20.630 --> 00:20:24.190 Well, the first thing that I'll want to look at is the 00:20:24.190 --> 00:20:25.980 impulse response. 00:20:25.980 --> 00:20:32.370 And when we do, let me just indicate that in the actual 00:20:32.370 --> 00:20:36.350 implementation things are slightly different than they 00:20:36.350 --> 00:20:39.720 are associated with the ideal analysis. 00:20:39.720 --> 00:20:46.920 In particular, in converting from a discreet-time sequence 00:20:46.920 --> 00:20:52.070 to the continuous-time signal, whereas this way of looking at 00:20:52.070 --> 00:20:56.530 it is convenient in the context of the analysis, in 00:20:56.530 --> 00:21:02.830 fact, the way it's done is using a more or less standard 00:21:02.830 --> 00:21:04.840 digital to analog converter. 00:21:04.840 --> 00:21:08.530 And what a digital to analog converter does, as I indicated 00:21:08.530 --> 00:21:13.900 in the previous lecture, is to convert the sequence not to an 00:21:13.900 --> 00:21:17.400 impulse train but, in fact, to go directly through a 00:21:17.400 --> 00:21:18.930 zero-order hold. 00:21:18.930 --> 00:21:22.140 And so, usually what comes out of a digital to analog 00:21:22.140 --> 00:21:27.590 converter is a staircase type of signal associated with a 00:21:27.590 --> 00:21:29.150 zero-order hold. 00:21:29.150 --> 00:21:33.260 And then, the result of that is low-pass filtered to do the 00:21:33.260 --> 00:21:34.590 reconstruction. 00:21:34.590 --> 00:21:39.410 So what we'll want to look at, then, is that reconstruction, 00:21:39.410 --> 00:21:42.170 first with just an impulse input. 00:21:42.170 --> 00:21:47.420 And so, what we'll see after the low-pass filter, for the 00:21:47.420 --> 00:21:52.900 impulse response, is a smooth curve like this. 00:21:52.900 --> 00:21:57.320 But also, as part of the demonstration, what I'll do, 00:21:57.320 --> 00:22:01.300 just to show the zero-order hold, is to take the low-pass 00:22:01.300 --> 00:22:05.610 filter out temporarily and then put it back in. 00:22:05.610 --> 00:22:11.010 So first, let's just look at the filter impulse response. 00:22:11.010 --> 00:22:14.790 What we see here is the impulse response of the 00:22:14.790 --> 00:22:17.040 overall system. 00:22:17.040 --> 00:22:20.600 And we observe, for one thing, that it's a symmetrical 00:22:20.600 --> 00:22:21.620 impulse response. 00:22:21.620 --> 00:22:25.170 In other words, corresponds to a linear phase filter. 00:22:25.170 --> 00:22:28.190 We could also look at the impulse response before the 00:22:28.190 --> 00:22:31.340 desampling low-pass filter-- let's take out the desampling 00:22:31.340 --> 00:22:34.050 low-pass filter slowly-- 00:22:34.050 --> 00:22:38.700 and what we observe is, basically, the output of the 00:22:38.700 --> 00:22:40.570 digital to analog converter. 00:22:40.570 --> 00:22:43.440 Which, of course, is a staircase, or boxcar, 00:22:43.440 --> 00:22:45.425 function, not an impulse train. 00:22:45.425 --> 00:22:48.640 In the real world, the output of a D to A converter, 00:22:48.640 --> 00:22:51.590 generally, is a boxcar type of function. 00:22:51.590 --> 00:22:55.990 We can put the desampling filter back in now and notice 00:22:55.990 --> 00:22:58.330 that the effect of the desampling filter is, 00:22:58.330 --> 00:23:04.210 basically, to smooth out the rough edges in the boxcar 00:23:04.210 --> 00:23:05.605 output from the D to A converter. 00:23:08.570 --> 00:23:12.210 OK, so, that's the impulse response of the system. 00:23:12.210 --> 00:23:15.910 Now, what I'd like to show is the frequency 00:23:15.910 --> 00:23:17.650 response of the system. 00:23:17.650 --> 00:23:20.390 And to measure the frequency response, of course, what we 00:23:20.390 --> 00:23:28.260 can do is put a sine wave into the system and look at the 00:23:28.260 --> 00:23:29.780 sinusoidal output. 00:23:29.780 --> 00:23:33.590 So, in particular now, what will happen is that, with the 00:23:33.590 --> 00:23:38.540 system, we will put in a continuous-time sinusoid, 00:23:38.540 --> 00:23:43.350 which is sampled, converted to a sequence. 00:23:43.350 --> 00:23:46.630 The sampled continuous-time sinusoid is a 00:23:46.630 --> 00:23:48.670 discreet-time sinusoid. 00:23:48.670 --> 00:23:52.650 That goes through the digital filter and gets attenuated, or 00:23:52.650 --> 00:23:55.110 amplified appropriately. 00:23:55.110 --> 00:23:59.400 And then the output of that is converted back-- and that's, 00:23:59.400 --> 00:24:01.450 again, a sinusoidal output-- 00:24:01.450 --> 00:24:08.000 that gets converted back to a continuous-time sinusoid. 00:24:08.000 --> 00:24:11.110 Theoretically, as I indicate here-- but again, as we just 00:24:11.110 --> 00:24:15.520 saw, really, represented by a zero-order hold, followed by a 00:24:15.520 --> 00:24:17.680 low-pass filter. 00:24:17.680 --> 00:24:22.120 So, that's the overall operation, with one 00:24:22.120 --> 00:24:25.310 modification from the diagram that we have here. 00:24:25.310 --> 00:24:30.430 In this particular diagram I've included and 00:24:30.430 --> 00:24:32.540 anti-aliasing filter. 00:24:32.540 --> 00:24:35.540 In fact, in the demonstration there is no 00:24:35.540 --> 00:24:37.510 anti-aliasing filter. 00:24:37.510 --> 00:24:43.750 And so, in fact, the input is a sinusoidal input which is 00:24:43.750 --> 00:24:48.990 not band-limited by virtue of an anti-aliasing filter. 00:24:48.990 --> 00:24:53.610 It's only, of course, band-limited appropriately if 00:24:53.610 --> 00:24:57.530 we choose the sinusoidal frequency that way. 00:24:57.530 --> 00:24:59.900 So, there is no anti-aliasing filter, 00:24:59.900 --> 00:25:02.070 and this is the system. 00:25:02.070 --> 00:25:09.020 And one consequence of that is that, in fact, if we sweep the 00:25:09.020 --> 00:25:13.900 input sinusoid only up to half the sampling frequency, we'll 00:25:13.900 --> 00:25:15.490 see no aliasing. 00:25:15.490 --> 00:25:17.360 But if we let it sweep past that, we're 00:25:17.360 --> 00:25:19.100 going to get aliasing. 00:25:19.100 --> 00:25:23.560 Now, in the demonstration, the sampling rate that's picked 00:25:23.560 --> 00:25:25.950 for this part of the demonstration is a 20 00:25:25.950 --> 00:25:28.790 kilohertz sampling rate. 00:25:28.790 --> 00:25:31.950 That means, based on the sampling theorem, that as long 00:25:31.950 --> 00:25:34.940 as the input frequency is below 10 00:25:34.940 --> 00:25:39.300 kilohertz we get no aliasing. 00:25:39.300 --> 00:25:43.900 When the input frequency goes beyond 10 kilohertz , that 00:25:43.900 --> 00:25:46.760 higher frequency is going to get aliased down 00:25:46.760 --> 00:25:48.670 into a lower frequency. 00:25:48.670 --> 00:25:51.590 A consequence of that, then, is that as we go through the 00:25:51.590 --> 00:25:55.910 processing, and we demonstrate the frequency response of the 00:25:55.910 --> 00:26:02.310 system, what we'll see in the output is no aliasing when the 00:26:02.310 --> 00:26:04.660 input is below 10 kilohertz. 00:26:04.660 --> 00:26:08.310 As the input sweeps past 10 kilohertz-- 00:26:08.310 --> 00:26:10.450 when we let it, which we will eventually in the 00:26:10.450 --> 00:26:11.880 demonstration-- 00:26:11.880 --> 00:26:16.460 then, in fact, that frequency, as it finally shows up here, 00:26:16.460 --> 00:26:21.870 will begin to be aliased down into a lower frequency. 00:26:21.870 --> 00:26:26.640 Another way of thinking about that is that, when we watch 00:26:26.640 --> 00:26:33.090 the frequency response of the system, as we look at the 00:26:33.090 --> 00:26:37.750 digital filter frequency response, what we're sweeping 00:26:37.750 --> 00:26:45.180 as we go from 0 up to 10 kilohertz in the input 00:26:45.180 --> 00:26:49.010 frequency is this portion of the frequency response. 00:26:49.010 --> 00:26:53.170 As we sweep from 10 kilohertz out to 20 kilohertz, what 00:26:53.170 --> 00:26:57.170 we'll see is this portion of the frequency response. 00:26:57.170 --> 00:27:00.880 In other words, we'll see it periodically replicated. 00:27:00.880 --> 00:27:06.520 Or, if we look at the corresponding continuous-time 00:27:06.520 --> 00:27:10.350 frequency response, what it means, really, is that 00:27:10.350 --> 00:27:14.510 sweeping from 0 to 10 kilohertz is 00:27:14.510 --> 00:27:16.190 moving up this way. 00:27:16.190 --> 00:27:20.360 And then sweeping from 10 kilohertz to 20 kilohertz on 00:27:20.360 --> 00:27:26.330 the input, really because of the aliasing, reflects itself 00:27:26.330 --> 00:27:30.460 in the digital filter by looking back toward lower 00:27:30.460 --> 00:27:31.590 frequencies. 00:27:31.590 --> 00:27:36.310 And so the continuous-time filter sweeps back down from 00:27:36.310 --> 00:27:40.520 10 kilohertz back to 0. 00:27:40.520 --> 00:27:44.020 OK, so, that's what we'll see, and we'll see it in several 00:27:44.020 --> 00:27:46.560 different ways as explained in the demonstration. 00:27:46.560 --> 00:27:49.130 So now let's look at the frequency 00:27:49.130 --> 00:27:51.930 response of the filter. 00:27:51.930 --> 00:27:54.440 Now what we'd like to illustrate is the frequency 00:27:54.440 --> 00:27:58.610 response of the equivalent continuous-time filter. 00:27:58.610 --> 00:28:01.010 And we can do that by sweeping the filter 00:28:01.010 --> 00:28:02.880 with sinusoidal input. 00:28:02.880 --> 00:28:06.890 So, what we'll see in this demonstration is, on the upper 00:28:06.890 --> 00:28:10.180 trace, the input sinusoid, on the lower trace, the output 00:28:10.180 --> 00:28:14.740 sinusoid, using a 20 kilohertz sampling rate, and a sweep 00:28:14.740 --> 00:28:16.420 from 0 to 10 kilohertz. 00:28:16.420 --> 00:28:21.080 In other words, a sweep from 0 to, effectively, pi, in terms 00:28:21.080 --> 00:28:23.320 of the digital filter. 00:28:23.320 --> 00:28:28.430 So what we'll observe as the input frequency increases, is 00:28:28.430 --> 00:28:31.450 that the output sinusoid will have, essentially, constant 00:28:31.450 --> 00:28:35.020 amplitude up to the cutoff frequency of the filter, and 00:28:35.020 --> 00:28:38.170 then approximately zero amplitude past. 00:28:38.170 --> 00:28:41.525 So let's now sweep the filter frequency response. 00:28:46.580 --> 00:28:50.410 And there is the filter cutoff frequency. 00:28:55.300 --> 00:28:59.310 Now, we can also observe the filter frequency response in 00:28:59.310 --> 00:29:00.840 several other ways. 00:29:00.840 --> 00:29:04.180 One way in which we can observe it is by looking, 00:29:04.180 --> 00:29:09.880 also, at the amplitude of the output sinusoid as a function 00:29:09.880 --> 00:29:12.980 of frequency, rather than as a function of time. 00:29:12.980 --> 00:29:16.530 And so we'll observe that on the left-hand scope. 00:29:16.530 --> 00:29:19.280 While on the right-hand scope, we'll have the same trace the 00:29:19.280 --> 00:29:21.760 we just saw, namely two traces-- 00:29:21.760 --> 00:29:24.990 the upper trace is the inputs sinusoid, the lower trace is 00:29:24.990 --> 00:29:26.800 the output sinusoid. 00:29:26.800 --> 00:29:30.730 And, in addition to observing the frequency response, let's 00:29:30.730 --> 00:29:35.430 also listen to the output sinusoid and observe the 00:29:35.430 --> 00:29:38.730 attenuation in the output as we go from the filter passband 00:29:38.730 --> 00:29:40.520 to the filter stopband. 00:29:40.520 --> 00:29:44.550 Again, a 20 kilohertz sampling rate and a sweep range from 0 00:29:44.550 --> 00:29:45.800 to 10 kilohertz. 00:29:53.290 --> 00:29:54.850 Now, of course, we're in the filter stopband. 00:29:58.360 --> 00:30:04.310 Now, if we increase the sweep range from 10 kilohertz the 20 00:30:04.310 --> 00:30:07.880 kilohertz, so that the sweep range is equal to the sampling 00:30:07.880 --> 00:30:11.310 frequency, in essence, that corresponds to sweeping out 00:30:11.310 --> 00:30:15.110 the digital filter from 0 to 2 pi. 00:30:15.110 --> 00:30:18.810 And, in that case, we'll begin to see some of the periodicity 00:30:18.810 --> 00:30:21.210 in the digital filter frequency response. 00:30:21.210 --> 00:30:25.890 So let's do that now with a 20 kilohertz sampling rate and a 00:30:25.890 --> 00:30:27.870 sweep range of 0 to 20 kilohertz. 00:30:32.790 --> 00:30:41.140 Now as we come near 2 pi, we get back the past-band. 00:30:41.140 --> 00:30:45.780 And, finally, back to a 0 to 10 kilohertz sweep, so that 00:30:45.780 --> 00:30:50.130 we're again sweeping only from 0 to pi with regard to the 00:30:50.130 --> 00:30:51.380 digital filter. 00:31:05.070 --> 00:31:10.560 Now, an important observation is that, with the digital or 00:31:10.560 --> 00:31:15.040 discreet-time filter cutoff frequency fixed as I've 00:31:15.040 --> 00:31:16.130 indicated here-- 00:31:16.130 --> 00:31:20.110 and I remind you that what the cutoff frequency is, 00:31:20.110 --> 00:31:22.370 is a 10th of 2 pi-- 00:31:22.370 --> 00:31:26.060 with that cutoff frequency fixed, because of the 00:31:26.060 --> 00:31:30.470 normalization that we get as we come back to a 00:31:30.470 --> 00:31:35.335 continuous-time filter, in fact, what we have is a cutoff 00:31:35.335 --> 00:31:41.140 frequency that is dependent on the sampling frequency or on 00:31:41.140 --> 00:31:42.620 the sampling period. 00:31:42.620 --> 00:31:46.580 And, more specifically, since the discreet-time, or digital, 00:31:46.580 --> 00:31:50.010 filter or has a cutoff frequency which is a 10th of 2 00:31:50.010 --> 00:31:56.630 pi, the normalization, as you recall, is that 2 pi, in 00:31:56.630 --> 00:32:00.780 discreet-time frequency, corresponds to omega sub s, 00:32:00.780 --> 00:32:02.750 the sampling frequency, in terms of 00:32:02.750 --> 00:32:05.180 continuous-time frequency. 00:32:05.180 --> 00:32:09.540 The consequence is that this cutoff frequency, in 00:32:09.540 --> 00:32:13.970 fact, is 1/10 of-- 00:32:13.970 --> 00:32:16.680 not 2 pi now because of the normalization-- 00:32:16.680 --> 00:32:20.180 it's 1/10 of the sampling frequency. 00:32:20.180 --> 00:32:25.980 So, consequently, as we change the sampling frequency, what 00:32:25.980 --> 00:32:29.010 will happen is that, even with the discreet-time filter 00:32:29.010 --> 00:32:33.870 cutoff fixed, the cutoff frequency of the equivalent 00:32:33.870 --> 00:32:37.810 continuous-time filter will change. 00:32:37.810 --> 00:32:40.570 Now, that's what I want to demonstrate. 00:32:40.570 --> 00:32:44.790 But let me again stress and ask you to keep in mind that 00:32:44.790 --> 00:32:47.010 this demonstration is done without an 00:32:47.010 --> 00:32:49.480 anti-aliasing filter in. 00:32:49.480 --> 00:32:53.590 And we are going to be changing the sampling 00:32:53.590 --> 00:33:00.650 frequency and, so keep in mind that, as we look at this, as 00:33:00.650 --> 00:33:04.940 the input frequency sweeps past half the sampling 00:33:04.940 --> 00:33:05.560 frequency-- 00:33:05.560 --> 00:33:08.680 whatever sampling frequency we happen to be looking at-- 00:33:08.680 --> 00:33:11.550 then, because of the fact that there's no anti-aliasing 00:33:11.550 --> 00:33:14.360 filter we'll get aliasing. 00:33:14.360 --> 00:33:17.230 In other words, the frequency and the digital filter, or 00:33:17.230 --> 00:33:21.040 discreet-time filter, as we sweep the input frequency up, 00:33:21.040 --> 00:33:25.280 will move up in frequency until we get past half the 00:33:25.280 --> 00:33:28.130 sampling frequency and then essentially will move back 00:33:28.130 --> 00:33:31.170 down in frequency. 00:33:31.170 --> 00:33:33.570 Consequently, what we'll get, then-- 00:33:33.570 --> 00:33:36.340 or what we'll see-- are periodic replications of the 00:33:36.340 --> 00:33:39.370 frequency response when we swept past half 00:33:39.370 --> 00:33:40.910 the sampling frequency. 00:33:40.910 --> 00:33:45.440 All right, so now, let's look at the same digital filter, 00:33:45.440 --> 00:33:50.040 but the frequency response, as we change 00:33:50.040 --> 00:33:52.980 the sampling frequency. 00:33:52.980 --> 00:33:56.070 Now, what we would like to demonstrate is the effect of 00:33:56.070 --> 00:33:58.450 changing the sampling frequency. 00:33:58.450 --> 00:34:03.930 And we know that the effective filter cutoff frequency is 00:34:03.930 --> 00:34:08.139 tied to the sampling frequency and, for this particular 00:34:08.139 --> 00:34:12.719 filter, corresponds to a 10th of the sampling frequency. 00:34:12.719 --> 00:34:15.690 Consequently, if we double the sampling frequency, we should 00:34:15.690 --> 00:34:20.690 double the effective filter passband width, or double the 00:34:20.690 --> 00:34:22.730 filter cutoff frequency. 00:34:22.730 --> 00:34:25.100 And, so, let's do that now. 00:34:25.100 --> 00:34:29.070 Again a 0 to 10 kilohertz sweep range, but a 40 00:34:29.070 --> 00:34:30.355 kilohertz sampling frequency. 00:34:34.690 --> 00:34:38.560 And we should observe that the filter cutoff frequency has 00:34:38.560 --> 00:34:40.574 now doubled out to four kilohertz. 00:34:43.469 --> 00:34:47.889 Now, let's begin to decrease the filter sampling frequency. 00:34:47.889 --> 00:34:50.110 So from 40, let's change the sampling 00:34:50.110 --> 00:34:52.730 frequency to 20 kilohertz. 00:34:52.730 --> 00:34:55.489 We should see the cutoff frequency cut in half. 00:35:05.970 --> 00:35:07.710 Now, we can go even further. 00:35:07.710 --> 00:35:08.810 We can cut the sampling 00:35:08.810 --> 00:35:10.650 frequency down to 10 kilohertz. 00:35:10.650 --> 00:35:13.800 And remember that the sweep range is 0 to 10 kilohertz. 00:35:13.800 --> 00:35:16.080 So now we'll be sweeping from 0 to 2 pi. 00:35:23.830 --> 00:35:29.240 So as we get close to 2 pi, we'll see the passband again. 00:35:29.240 --> 00:35:33.040 And, now, let's cut down the sampling frequency even 00:35:33.040 --> 00:35:34.965 further, to 5 kilohertz. 00:35:40.990 --> 00:35:42.240 Here we are at 2 pi. 00:35:45.020 --> 00:35:46.830 And then at 4pi. 00:35:49.390 --> 00:35:52.110 All right, so, that illustrates the effect of 00:35:52.110 --> 00:35:54.610 changing the sampling frequency. 00:35:54.610 --> 00:35:57.920 Now let's conclude this demonstration of the effect of 00:35:57.920 --> 00:36:01.180 the sampling frequency on the filter cutoff frequency by 00:36:01.180 --> 00:36:04.810 carrying out some filtering on some live audio. 00:36:04.810 --> 00:36:09.870 What we'll watch, in this case, is the output audio 00:36:09.870 --> 00:36:15.220 waveform as a function of time on the single tray scope, and 00:36:15.220 --> 00:36:18.220 also we'll listen to the output. 00:36:18.220 --> 00:36:21.870 We'll begin it with a 40 kilohertz sampling rate, then 00:36:21.870 --> 00:36:23.870 reduce that to 20 kilohertz, 10 00:36:23.870 --> 00:36:26.100 kilohertz, and then 5 kilohertz. 00:36:26.100 --> 00:36:29.040 And in each of those cases, the effective filter cutoff 00:36:29.040 --> 00:36:33.130 frequency, then, is cut in half from 4 kilohertz, to 2 00:36:33.130 --> 00:36:37.080 kilohertz, to 1 kilohertz, and then to 500 cycles. 00:36:37.080 --> 00:36:40.120 So let's begin with a 40 kilohertz sampling frequency, 00:36:40.120 --> 00:36:43.375 or an effective filter cutoff frequency of 4 kilohertz. 00:36:49.950 --> 00:36:54.090 Now, let's reduce that a 20 kilohertz sampling frequency, 00:36:54.090 --> 00:36:55.340 or a 2 kilohertz filter. 00:37:00.090 --> 00:37:02.655 Then a 10 kilohertz sampling frequency. 00:37:07.950 --> 00:37:11.130 And, finally, a 5 kilohertz sampling frequency 00:37:11.130 --> 00:37:15.615 corresponding to a 500 cycle equivalent analog filter. 00:37:20.280 --> 00:37:24.760 Alright, now, let's finally conclude by returning to a 00:37:24.760 --> 00:37:27.916 little higher quality ragtime by changing the sampling 00:37:27.916 --> 00:37:30.406 frequency back to 40 kilohertz. 00:37:37.880 --> 00:37:40.700 Alright, well, hopefully what you've seen in the 00:37:40.700 --> 00:37:46.775 demonstration and in this lecture gives you a sense and 00:37:46.775 --> 00:37:52.780 a feeling for the analysis and the use of discreet-time 00:37:52.780 --> 00:37:56.160 filters for processing continuous-time signals. 00:37:56.160 --> 00:38:00.480 And as you may be aware, and as I've tried to indicate 00:38:00.480 --> 00:38:04.020 previously in the past, this, in fact, is one very 00:38:04.020 --> 00:38:08.670 important-- but not the only-- but one very important context 00:38:08.670 --> 00:38:12.000 in which discreet-time filtering is used. 00:38:12.000 --> 00:38:16.480 And this, in fact, is an area that is developing rapidly 00:38:16.480 --> 00:38:20.800 because of the fact that microprocessors, digital 00:38:20.800 --> 00:38:25.100 technology, computers, et cetera afford considerable 00:38:25.100 --> 00:38:30.710 flexibility in carrying out digital processing of signals. 00:38:30.710 --> 00:38:37.540 And when digital processing is used, that naturally 00:38:37.540 --> 00:38:44.190 corresponds to implementing the processing and analyzing 00:38:44.190 --> 00:38:45.440 it in discreet-time. 00:38:48.100 --> 00:38:52.120 Now, in the next lecture we'll be 00:38:52.120 --> 00:38:54.980 continuing on another aspect-- 00:38:54.980 --> 00:38:56.460 developing another aspect-- 00:38:56.460 --> 00:38:58.400 of sampling. 00:38:58.400 --> 00:39:02.550 And, in particular, what we'll be talking about is sampling 00:39:02.550 --> 00:39:04.760 of discreet-time signals. 00:39:04.760 --> 00:39:08.440 As I'll indicate there, one of the contexts in which 00:39:08.440 --> 00:39:12.890 discreet-time sampling, in fact, plays an important role 00:39:12.890 --> 00:39:18.180 is in the context in which we are processing continuous-time 00:39:18.180 --> 00:39:21.170 signals using discreet-time processing. 00:39:21.170 --> 00:39:24.510 Where, in fact, one step that we might want to take-- in 00:39:24.510 --> 00:39:27.130 addition to the steps so we've talked about here-- 00:39:27.130 --> 00:39:31.470 is an additional sampling process following whatever 00:39:31.470 --> 00:39:33.770 kinds of filtering that we do. 00:39:33.770 --> 00:39:36.650 Well, that's a discussion and a topic that we'll be going 00:39:36.650 --> 00:39:37.930 into in the next lecture. 00:39:37.930 --> 00:39:39.180 Thank you.