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 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.730 ocw.mit.edu. 00:00:20.730 --> 00:00:55.220 [MUSIC PLAYING] 00:00:55.220 --> 00:00:58.180 PROFESSOR: In discussing the sampling theorem, we saw that 00:00:58.180 --> 00:01:01.650 for a band limited signal, which is sampled at a 00:01:01.650 --> 00:01:05.660 frequency that is at least twice the highest frequency, 00:01:05.660 --> 00:01:09.320 we can implement exact reconstruction of the original 00:01:09.320 --> 00:01:16.240 signal by low pass filtering an impulse train, whose areas 00:01:16.240 --> 00:01:19.640 are identical to the sample values. 00:01:19.640 --> 00:01:23.200 Well essentially, this low pass filtering operation 00:01:23.200 --> 00:01:26.980 provides for us an interpolation in between the 00:01:26.980 --> 00:01:28.440 sampled values. 00:01:28.440 --> 00:01:32.160 In other words, the output of a low pass filter, in fact, is 00:01:32.160 --> 00:01:36.820 a continuous curve, which fits between the sampled values 00:01:36.820 --> 00:01:39.980 some continuous function. 00:01:39.980 --> 00:01:44.170 Now, I'm sure that many of you are familiar with other kinds 00:01:44.170 --> 00:01:47.330 of interpolation that we could potentially provide in between 00:01:47.330 --> 00:01:48.810 sampled values. 00:01:48.810 --> 00:01:52.280 And in fact, in today's lecture what I would like to 00:01:52.280 --> 00:01:58.640 do is first of all developed the interpretation of the 00:01:58.640 --> 00:02:03.640 reconstruction as an interpolation process and then 00:02:03.640 --> 00:02:07.370 also see how this exact interpolation, using a low 00:02:07.370 --> 00:02:12.350 pass filter, relates to other kinds of interpolation, such 00:02:12.350 --> 00:02:14.520 as linear interpolation that you may 00:02:14.520 --> 00:02:16.330 already be familiar with. 00:02:16.330 --> 00:02:21.220 Well to begin, let's again review what the overall system 00:02:21.220 --> 00:02:25.290 is for exact sampling and reconstruction. 00:02:25.290 --> 00:02:28.670 And so let me remind you that the overall system for 00:02:28.670 --> 00:02:31.960 sampling and desampling, or reconstruction, is as I 00:02:31.960 --> 00:02:34.040 indicate here. 00:02:34.040 --> 00:02:37.150 The sampling process consists of multiplying 00:02:37.150 --> 00:02:38.960 by an impulse train. 00:02:38.960 --> 00:02:42.200 And then the reconstruction process corresponds to 00:02:42.200 --> 00:02:46.590 processing that impulse train with a low pass filter. 00:02:46.590 --> 00:02:52.420 So if the spectrum of the original signal is what I 00:02:52.420 --> 00:02:57.890 indicate in this diagram, then after sampling with an impulse 00:02:57.890 --> 00:03:02.020 train, that spectrum is replicated. 00:03:02.020 --> 00:03:07.960 And this replicated spectrum for reconstruction is then 00:03:07.960 --> 00:03:10.610 processed through a low pass filter. 00:03:10.610 --> 00:03:14.770 And so, in fact, if this frequency response is an ideal 00:03:14.770 --> 00:03:20.560 low pass filter, as I indicate on the diagram below, then 00:03:20.560 --> 00:03:25.540 multiplying the spectrum of the sample signal by this 00:03:25.540 --> 00:03:31.350 extracts for us just the portion of the spectrum 00:03:31.350 --> 00:03:33.210 centered around the origin. 00:03:33.210 --> 00:03:36.920 And what we're left with, then, is the spectrum, 00:03:36.920 --> 00:03:41.490 finally, of the reconstructed signal, which for the case of 00:03:41.490 --> 00:03:46.070 an ideal low pass filter is exactly equal to the spectrum 00:03:46.070 --> 00:03:48.160 of the original signal. 00:03:48.160 --> 00:03:55.780 Now, that is the frequency domain picture of the sampling 00:03:55.780 --> 00:03:57.220 and reconstruction. 00:03:57.220 --> 00:04:00.600 Let's also look at, basically, the same process. 00:04:00.600 --> 00:04:04.920 But let's examine it now in the time domain. 00:04:04.920 --> 00:04:11.270 Well in the time domain, what we have is our original signal 00:04:11.270 --> 00:04:14.110 multiplied by an impulse train. 00:04:14.110 --> 00:04:19.920 And this then is the sample signal, or the impulse train 00:04:19.920 --> 00:04:23.630 whose areas are equal to the sample values. 00:04:23.630 --> 00:04:27.620 And because of the fact that this is an impulse train, in 00:04:27.620 --> 00:04:34.160 fact, we can take this term inside the summation. 00:04:34.160 --> 00:04:37.130 And of course, what counts about x of t in this 00:04:37.130 --> 00:04:41.400 expression is just as values at the sampling instance, 00:04:41.400 --> 00:04:47.220 which are displaced in time by capital T. And so what we can 00:04:47.220 --> 00:04:52.350 equivalently write is the expression for the impulse 00:04:52.350 --> 00:04:55.880 train samples, or impulse train of samples, as I've 00:04:55.880 --> 00:04:57.010 indicated here. 00:04:57.010 --> 00:05:03.410 Simply an impulse train, whose areas are the sampled values. 00:05:03.410 --> 00:05:08.650 Now, in the reconstruction we process that impulse train 00:05:08.650 --> 00:05:10.940 with a low pass filter. 00:05:10.940 --> 00:05:13.880 That's the basic notion of the reconstruction. 00:05:13.880 --> 00:05:18.810 And so in the time domain, the reconstructed signal is 00:05:18.810 --> 00:05:23.850 related to the impulse train of samples through a 00:05:23.850 --> 00:05:28.280 convolution with the filter impulse response. 00:05:28.280 --> 00:05:32.670 And carrying out this convolution, since this is 00:05:32.670 --> 00:05:35.880 just a train of pulses, in effect, what happens in this 00:05:35.880 --> 00:05:40.640 convolution is that this impulse response gets 00:05:40.640 --> 00:05:45.820 reproduced at each of the locations of the impulses in x 00:05:45.820 --> 00:05:48.790 of p of t with the appropriate area. 00:05:48.790 --> 00:05:52.580 And finally, then, in the time domain, the reconstructed 00:05:52.580 --> 00:05:59.140 signal is simply a linear combination of shifted 00:05:59.140 --> 00:06:03.280 versions of the impulse response with amplitudes, 00:06:03.280 --> 00:06:05.080 which are the sample values. 00:06:05.080 --> 00:06:09.410 And so this expression, in fact then, is our basic 00:06:09.410 --> 00:06:14.840 reconstruction expression in the time domain. 00:06:14.840 --> 00:06:20.350 Well in terms of a diagram, we can think of the original 00:06:20.350 --> 00:06:23.150 waveform as I've shown here. 00:06:23.150 --> 00:06:30.100 And the red arrows denote the sampled wave form, or the 00:06:30.100 --> 00:06:35.790 train of impulses, whose amplitudes are the sampled 00:06:35.790 --> 00:06:40.310 values of the original continuous time signal. 00:06:40.310 --> 00:06:45.330 And then, I've shown here what might be a typical impulse 00:06:45.330 --> 00:06:49.310 response, particularly typical in the case where we're 00:06:49.310 --> 00:06:53.280 talking about reconstruction with an ideal low pass filter. 00:06:53.280 --> 00:06:58.250 Now, what happens in the reconstruction is that the 00:06:58.250 --> 00:07:04.140 convolution of these impulses with this impulse response 00:07:04.140 --> 00:07:09.240 means that in the reconstruction, we superimpose 00:07:09.240 --> 00:07:10.890 one of these impulse responses-- 00:07:10.890 --> 00:07:13.810 whatever the filter impulse response happens to be-- 00:07:13.810 --> 00:07:16.570 at each of these time instance. 00:07:16.570 --> 00:07:20.310 And in doing that, then those are added up. 00:07:20.310 --> 00:07:23.660 And that gives us the total reconstructed signal. 00:07:23.660 --> 00:07:28.730 Of course, for the case in which the filter is an ideal 00:07:28.730 --> 00:07:34.110 low pass filter, then what we know is that in that case, the 00:07:34.110 --> 00:07:38.090 impulse response is of the form of a sync function. 00:07:38.090 --> 00:07:41.880 But generally, we may want to consider other kinds of 00:07:41.880 --> 00:07:43.070 impulse responses. 00:07:43.070 --> 00:07:47.420 And so in fact, the interpolating impulse response 00:07:47.420 --> 00:07:50.160 may have and will have, as this discussion goes along, 00:07:50.160 --> 00:07:52.830 some different shapes. 00:07:52.830 --> 00:07:56.750 Now what I'd like to do is illustrate, or demonstrate, 00:07:56.750 --> 00:08:04.240 this process of effectively doing the interpolation by 00:08:04.240 --> 00:08:07.530 replacing each of the impulses by an appropriate 00:08:07.530 --> 00:08:10.250 interpolating impulse response and adding these up. 00:08:10.250 --> 00:08:12.690 And I'd like to do this with a computer 00:08:12.690 --> 00:08:14.690 movie that we generated. 00:08:14.690 --> 00:08:17.860 And what you'll see in the computer movie is, 00:08:17.860 --> 00:08:20.040 essentially, an original wave form, which is 00:08:20.040 --> 00:08:21.870 a continuous curve. 00:08:21.870 --> 00:08:29.000 And then below that in the movie is a train of samples. 00:08:29.000 --> 00:08:33.500 And then below that will be the reconstructed signal. 00:08:33.500 --> 00:08:38.110 And the reconstruction will be carried out by showing the 00:08:38.110 --> 00:08:41.530 location of the impulse response as it moves along in 00:08:41.530 --> 00:08:42.950 the wave form. 00:08:42.950 --> 00:08:46.650 And then the reconstructed curve is simply the summation 00:08:46.650 --> 00:08:49.350 of those as that impulse response moves along. 00:08:49.350 --> 00:08:53.930 So what you'll see then is an impulse response like this-- 00:08:53.930 --> 00:08:58.340 for the particular case of an ideal low pass filter for the 00:08:58.340 --> 00:08:59.590 reconstruction-- 00:09:02.140 --> 00:09:07.480 placed successively at the locations of these impulses. 00:09:07.480 --> 00:09:10.400 And that is the convolution process. 00:09:10.400 --> 00:09:14.490 And below that then will be the summation of these. 00:09:14.490 --> 00:09:17.220 And the summation of those will then be the 00:09:17.220 --> 00:09:18.740 reconstructed signal. 00:09:18.740 --> 00:09:23.250 So let's take a look at, first of all that reconstruction 00:09:23.250 --> 00:09:26.150 where the impulse response corresponds to the impulse 00:09:26.150 --> 00:09:29.800 response of an ideal low pass filter. 00:09:29.800 --> 00:09:34.460 Shown here, first, is the continuous time signal, which 00:09:34.460 --> 00:09:39.310 we want to sample and then reconstruct using band limited 00:09:39.310 --> 00:09:42.860 interpolation, or equivalently, ideal low pass 00:09:42.860 --> 00:09:45.120 filtering on the set of samples. 00:09:45.120 --> 00:09:48.230 So the first step then is to sample this 00:09:48.230 --> 00:09:49.940 continuous time signal. 00:09:49.940 --> 00:09:53.850 And we see here now the set of samples. 00:09:53.850 --> 00:09:56.720 And superimposed on the samples are the original 00:09:56.720 --> 00:10:01.180 continuous time signal to focus on the fact that those 00:10:01.180 --> 00:10:03.420 are samples of the top curve. 00:10:03.420 --> 00:10:08.250 Let's now remove the continuous time envelope of 00:10:08.250 --> 00:10:09.420 the samples. 00:10:09.420 --> 00:10:12.380 And it's this set of samples that we then want to use for 00:10:12.380 --> 00:10:14.490 the reconstruction. 00:10:14.490 --> 00:10:17.900 The reconstruction process, interpreted as interpolation, 00:10:17.900 --> 00:10:20.600 consists of replacing each sample with a 00:10:20.600 --> 00:10:22.970 sine x over x function. 00:10:22.970 --> 00:10:28.940 And so let's first consider the sample at t equals 0. 00:10:28.940 --> 00:10:32.800 And here is the interpolating sine x over x function 00:10:32.800 --> 00:10:35.530 associated with that sample. 00:10:35.530 --> 00:10:41.610 Now, the more general process then is to place a sine x over 00:10:41.610 --> 00:10:45.150 x function at the time location of each sample and 00:10:45.150 --> 00:10:46.400 superimpose those. 00:10:48.980 --> 00:10:54.330 Let's begin that process at the left-hand set of samples. 00:10:54.330 --> 00:10:58.800 And in the bottom curve, we'll build up the reconstruction as 00:10:58.800 --> 00:11:02.150 those sine x over x functions are added together. 00:11:02.150 --> 00:11:04.470 So we begin with the left-hand sample. 00:11:04.470 --> 00:11:07.470 And we see there the sine x over x function on the bottom 00:11:07.470 --> 00:11:11.900 curve is the first step in the reconstruction. 00:11:11.900 --> 00:11:14.710 We now have the sine x over x function associated with the 00:11:14.710 --> 00:11:16.070 second sample. 00:11:16.070 --> 00:11:18.620 Let's add that in. 00:11:18.620 --> 00:11:22.680 Now we move on to the third sample. 00:11:22.680 --> 00:11:27.980 And that sine x over x function is added in. 00:11:27.980 --> 00:11:33.860 Continuing on, the next sample generates a sine x over x 00:11:33.860 --> 00:11:36.670 function, which is superimposed on the result 00:11:36.670 --> 00:11:39.750 that we've accumulated so far. 00:11:39.750 --> 00:11:42.720 And now let's just speed up the process. 00:11:42.720 --> 00:11:45.490 We'll move on to the fifth sample. 00:11:45.490 --> 00:11:46.540 Add that in. 00:11:46.540 --> 00:11:48.710 The sixth sample, add that in. 00:11:48.710 --> 00:11:51.440 And continue on through the set of samples. 00:11:51.440 --> 00:11:54.960 And keep in mind the fact that, basically, what we're 00:11:54.960 --> 00:11:58.860 doing explicitly here is the convolution of the impulse 00:11:58.860 --> 00:12:02.240 train with a sine x over x function. 00:12:02.240 --> 00:12:05.730 And because the set of samples that we started with were 00:12:05.730 --> 00:12:09.610 samples of an exactly band limited function, what we are 00:12:09.610 --> 00:12:14.530 reconstructing exactly is the original continuous time 00:12:14.530 --> 00:12:16.980 signal that we have on the top trace. 00:12:24.900 --> 00:12:31.060 OK, so that then kind of gives you the picture of doing 00:12:31.060 --> 00:12:34.720 interpolation by replacing the impulses by 00:12:34.720 --> 00:12:35.870 a continuous curve. 00:12:35.870 --> 00:12:40.010 And that's the way we're fitting a continuous curve to 00:12:40.010 --> 00:12:42.260 the original impulse train. 00:12:42.260 --> 00:12:47.870 And let me stress that this reconstruction process-- 00:12:47.870 --> 00:12:51.320 by putting the impulses through a filter-- 00:12:54.690 --> 00:12:59.320 follows this relationship whether or not this impulse 00:12:59.320 --> 00:13:03.700 response, in fact, corresponds to an ideal low pass filter. 00:13:03.700 --> 00:13:07.210 What this expression always says is that reconstructing 00:13:07.210 --> 00:13:12.440 this way corresponds to replacing the impulses by a 00:13:12.440 --> 00:13:17.940 shifted impulse response with an amplitude that is an 00:13:17.940 --> 00:13:22.220 amplitude corresponding to the sample value. 00:13:22.220 --> 00:13:27.120 Now the kind of reconstruction that we've just talked about, 00:13:27.120 --> 00:13:32.320 and the ideal reconstruction, is often referred to as band 00:13:32.320 --> 00:13:36.420 limited interpolation because we're interpolating in between 00:13:36.420 --> 00:13:39.840 the samples by making the assumption that the signal is 00:13:39.840 --> 00:13:43.840 band limited and using the impulse response for an ideal 00:13:43.840 --> 00:13:48.780 low pass filter, which has a cut off frequency consistent 00:13:48.780 --> 00:13:51.500 with the assumed bandwidth for the signal. 00:13:51.500 --> 00:13:57.590 So if we look here, for example, at the impulse train, 00:13:57.590 --> 00:14:02.170 then in the demonstration that you just saw, we built up the 00:14:02.170 --> 00:14:05.420 reconstructed curve by replacing each of these 00:14:05.420 --> 00:14:07.980 impulses with the sync function. 00:14:07.980 --> 00:14:13.570 And the sum of those built up the reconstructed curve. 00:14:13.570 --> 00:14:18.230 Well, there are lots of other kinds of interpolation that 00:14:18.230 --> 00:14:23.180 are perhaps maybe not as exact but often easier to implement. 00:14:23.180 --> 00:14:25.080 And what I'd like to do is focus our 00:14:25.080 --> 00:14:28.300 attention on two of these. 00:14:28.300 --> 00:14:31.440 The first that I want to mention is what's referred to 00:14:31.440 --> 00:14:34.760 as the zero order hold, where in effect, we do the 00:14:34.760 --> 00:14:39.100 interpolation in between these sample values by simply 00:14:39.100 --> 00:14:43.610 holding the sample value until the next sampling instant. 00:14:43.610 --> 00:14:48.070 And the reconstruction that we end up, in that case, will 00:14:48.070 --> 00:14:49.260 look something like this. 00:14:49.260 --> 00:14:54.590 It's a staircase, or box car, kind of function where we've 00:14:54.590 --> 00:14:57.660 simply held the sample value until the next sampling 00:14:57.660 --> 00:15:01.420 instant and then replaced by that value, held it until the 00:15:01.420 --> 00:15:04.040 next sampling instant, et cetera. 00:15:04.040 --> 00:15:07.750 Now that's one kind of interpolation. 00:15:07.750 --> 00:15:11.030 Another kind of very common interpolation is what's 00:15:11.030 --> 00:15:13.290 referred to as linear interpolation, where we simply 00:15:13.290 --> 00:15:16.910 fit a straight line between the sampled values. 00:15:16.910 --> 00:15:21.240 And in that case, the type of reconstruction that we would 00:15:21.240 --> 00:15:25.060 get would look something like I indicate here, where we take 00:15:25.060 --> 00:15:30.540 a sample value, and the following sample value, and 00:15:30.540 --> 00:15:34.170 simply fit an interpolated curve between them, which is a 00:15:34.170 --> 00:15:36.430 straight line. 00:15:36.430 --> 00:15:42.470 Now interestingly, in fact, both the zero order hold and 00:15:42.470 --> 00:15:46.155 the linear interpolation, which is often referred to as 00:15:46.155 --> 00:15:50.600 a first order hold, can also be either implemented or 00:15:50.600 --> 00:15:53.820 interpreted, both implemented and interpreted, in the 00:15:53.820 --> 00:15:56.670 context of the equation that we just developed. 00:15:56.670 --> 00:16:00.450 In particular, the processing of the impulse train of 00:16:00.450 --> 00:16:05.520 samples by a linear time invariant filter. 00:16:05.520 --> 00:16:12.550 Specifically, if we consider a system where the impulse 00:16:12.550 --> 00:16:21.030 response is a rectangular function, then in fact, if we 00:16:21.030 --> 00:16:25.570 processed the train of samples through a filter with this 00:16:25.570 --> 00:16:29.610 impulse response, exactly the reconstruction that we would 00:16:29.610 --> 00:16:32.450 get is what I've shown here. 00:16:32.450 --> 00:16:38.810 Alternatively, if we chose an impulse response which was a 00:16:38.810 --> 00:16:44.980 triangular impulse response, then what in effect happens is 00:16:44.980 --> 00:16:49.690 that each of these impulses activates this triangle. 00:16:49.690 --> 00:16:53.210 And when we add up those triangles at successive 00:16:53.210 --> 00:16:57.630 locations, in fact, what we generate is this linear 00:16:57.630 --> 00:16:59.730 interpolation. 00:16:59.730 --> 00:17:06.540 So what this says, in fact, is that either a zero order hold, 00:17:06.540 --> 00:17:09.849 which holds the value, or linear interpolation can 00:17:09.849 --> 00:17:15.190 likewise be interpreted as a process of convulving the 00:17:15.190 --> 00:17:18.380 impulse train of samples with an appropriate 00:17:18.380 --> 00:17:21.380 filter impulse response. 00:17:21.380 --> 00:17:25.480 Well, what I'd like to do is demonstrate, as we did with 00:17:25.480 --> 00:17:29.860 the band limited interpolation or the sync interpolation as 00:17:29.860 --> 00:17:31.500 it's sometimes called-- interpolating with 00:17:31.500 --> 00:17:33.100 a sine x over x-- 00:17:33.100 --> 00:17:35.260 let me now show the process. 00:17:35.260 --> 00:17:41.820 First of all, where we have a zero order hold as 00:17:41.820 --> 00:17:44.280 corresponding to this impulse response. 00:17:44.280 --> 00:17:48.160 In which case, we'll see basically the same process as 00:17:48.160 --> 00:17:52.340 we saw in the computer generated movie previously. 00:17:52.340 --> 00:17:56.080 But now, rather than a sync function replacing each of 00:17:56.080 --> 00:17:59.770 these impulses, we'll have a rectangular function. 00:17:59.770 --> 00:18:04.200 That will generate then our approximation, which is a zero 00:18:04.200 --> 00:18:06.130 order hold. 00:18:06.130 --> 00:18:08.900 And following that, we'll do exactly the same thing with 00:18:08.900 --> 00:18:12.820 the same wave form, using a first order hold or a 00:18:12.820 --> 00:18:14.960 triangular impulse response. 00:18:14.960 --> 00:18:18.330 In which case, what we'll see again is that as the triangle 00:18:18.330 --> 00:18:23.150 moves along here, and we build up the running sum or the 00:18:23.150 --> 00:18:27.900 convolution, then we'll, in fact, fit the original curve 00:18:27.900 --> 00:18:29.550 with a linear curve. 00:18:29.550 --> 00:18:33.980 So now let's again look at that, remembering that at the 00:18:33.980 --> 00:18:36.910 top we'll see the original continuous curve, exactly the 00:18:36.910 --> 00:18:38.820 one that we had before. 00:18:38.820 --> 00:18:42.680 Below it, the set of samples together with the impulse 00:18:42.680 --> 00:18:45.070 response moving along. 00:18:45.070 --> 00:18:48.650 And then finally below that, the accumulation of those 00:18:48.650 --> 00:18:52.370 impulse responses, or equivalently the convolution, 00:18:52.370 --> 00:18:56.050 or equivalently the reconstruction. 00:18:56.050 --> 00:18:59.600 So we have the same continuous time signal that we use 00:18:59.600 --> 00:19:03.740 previously with band limited interpolation. 00:19:03.740 --> 00:19:08.130 And in this case now, we want to sample and then interpolate 00:19:08.130 --> 00:19:10.810 first with a zero order hold and then with 00:19:10.810 --> 00:19:11.790 a first order hold. 00:19:11.790 --> 00:19:15.250 So the first step then is to sample the 00:19:15.250 --> 00:19:17.170 continuous time signal. 00:19:17.170 --> 00:19:21.360 And we show here the set of samples, once again, 00:19:21.360 --> 00:19:25.360 superimposed on which we have the continuous time signal, 00:19:25.360 --> 00:19:28.490 which of course is exactly the same curve as 00:19:28.490 --> 00:19:30.650 we have in the top. 00:19:30.650 --> 00:19:34.020 Well, let's remove that envelope so that we focus 00:19:34.020 --> 00:19:37.790 attention on the samples that we're using to interpolate. 00:19:37.790 --> 00:19:41.690 And the interpolation process consists of replacing each 00:19:41.690 --> 00:19:47.450 sample by a rectangular signal, whose amplitude is 00:19:47.450 --> 00:19:49.570 equal to the sample size. 00:19:49.570 --> 00:19:53.510 So let's put one, first of all, at t equals 0 associated 00:19:53.510 --> 00:19:56.550 with that sample. 00:19:56.550 --> 00:20:00.660 And that then would be the interpolating rectangle 00:20:00.660 --> 00:20:04.370 associated with the sample at t equals 0. 00:20:04.370 --> 00:20:07.780 Now to build up the interpolation, what we'll have 00:20:07.780 --> 00:20:09.740 is one of those at each sample time, and 00:20:09.740 --> 00:20:11.890 those are added together. 00:20:11.890 --> 00:20:14.930 We'll start that process, as we did before, at the 00:20:14.930 --> 00:20:18.762 left-hand end of the set of samples and build the 00:20:18.762 --> 00:20:20.390 interpolating signal on the bottom. 00:20:20.390 --> 00:20:25.650 So with the left-hand sample, we have first the rectangle 00:20:25.650 --> 00:20:27.640 associated with that. 00:20:27.640 --> 00:20:30.510 That's shown now on the bottom curve. 00:20:30.510 --> 00:20:36.740 We now have an interpolating rectangle with a second sample 00:20:36.740 --> 00:20:40.200 that gets added into the bottom curve. 00:20:40.200 --> 00:20:44.210 Similarly, an interpolating rectangle with the zero order 00:20:44.210 --> 00:20:46.430 hold with the third sample. 00:20:46.430 --> 00:20:49.440 We add that into the bottom curve. 00:20:49.440 --> 00:20:52.570 And as we proceed, we're building a staircase 00:20:52.570 --> 00:20:53.920 approximation. 00:20:53.920 --> 00:20:58.630 On to the next sample, that gets added in as we see there. 00:20:58.630 --> 00:21:00.720 And now let's speed up the process. 00:21:00.720 --> 00:21:04.920 And we'll see the staircase approximation building up. 00:21:04.920 --> 00:21:09.390 And notice in this case, as in the previous case, that what 00:21:09.390 --> 00:21:13.780 we're basically watching dynamically is the convolution 00:21:13.780 --> 00:21:17.370 of the impulse train of samples with the impulse 00:21:17.370 --> 00:21:20.420 response of the interpolating filter, which in this 00:21:20.420 --> 00:21:24.800 particular case is just a rectangular pulse. 00:21:24.800 --> 00:21:27.140 And so this staircase approximation that we're 00:21:27.140 --> 00:21:32.040 generating is the zero order hold interpolation between the 00:21:32.040 --> 00:21:37.070 samples of the band limited signal, which is at the top. 00:21:41.890 --> 00:21:44.800 Now let's do the same thing with a first order hold. 00:21:44.800 --> 00:21:49.610 So in this case, we want to interpolate using a triangular 00:21:49.610 --> 00:21:53.760 impulse response rather then the sine x over x, or 00:21:53.760 --> 00:21:56.940 rectangular impulse responses that we showed previously. 00:21:56.940 --> 00:22:02.670 So first, let's say with the sample at t equals 0, we would 00:22:02.670 --> 00:22:08.520 replace that with a triangular interpolating function. 00:22:08.520 --> 00:22:12.050 And more generally, each impulse or sample is replaced 00:22:12.050 --> 00:22:14.950 with a triangular interpolating function of a 00:22:14.950 --> 00:22:17.550 height equal to the sample type. 00:22:17.550 --> 00:22:20.050 And these are superimposed to generate the linear 00:22:20.050 --> 00:22:21.680 interpolation. 00:22:21.680 --> 00:22:26.010 We'll begin this process with the leftmost sample. 00:22:26.010 --> 00:22:29.950 And we'll build the superposition below in the 00:22:29.950 --> 00:22:31.240 bottom curve. 00:22:31.240 --> 00:22:34.810 So here is the interpolating triangle for 00:22:34.810 --> 00:22:36.970 the leftmost sample. 00:22:36.970 --> 00:22:40.750 And now it's reproduced below. 00:22:40.750 --> 00:22:43.760 With the second sample, we have an interpolating 00:22:43.760 --> 00:22:47.490 triangle, which is added into the bottom curve. 00:22:47.490 --> 00:22:50.670 And now on to the third sample. 00:22:50.670 --> 00:22:55.800 And again, that interpolating triangle will be added on to 00:22:55.800 --> 00:22:59.550 the curve that we've developed so far. 00:22:59.550 --> 00:23:03.300 And now onto the next sample. 00:23:03.300 --> 00:23:05.630 We add that in. 00:23:05.630 --> 00:23:07.480 Then we'll speed up the process. 00:23:07.480 --> 00:23:13.700 And as we proceed through, we are building, basically, a 00:23:13.700 --> 00:23:18.670 linear interpolation in between the sample points, 00:23:18.670 --> 00:23:20.550 essentially corresponding to-- 00:23:20.550 --> 00:23:22.290 if one wants to think of it this way-- 00:23:22.290 --> 00:23:23.730 connecting the dots. 00:23:23.730 --> 00:23:26.550 And what you're watching, once again, is essentially the 00:23:26.550 --> 00:23:31.220 convolution process convulving the impulse train with the 00:23:31.220 --> 00:23:34.640 impulse response of the interpolating filter. 00:23:34.640 --> 00:23:39.010 And what we're generating, then, is a linear 00:23:39.010 --> 00:23:44.850 approximation to the band limited continuous time curve 00:23:44.850 --> 00:23:46.100 at the top. 00:23:48.230 --> 00:23:54.500 OK, so what we have then is several other kinds of 00:23:54.500 --> 00:23:59.940 interpolation, which fit within the same context as 00:23:59.940 --> 00:24:02.210 exact band limited interpolation. 00:24:02.210 --> 00:24:06.020 One being interpolation in the time domain with an impulse 00:24:06.020 --> 00:24:08.130 response, which is a rectangle. 00:24:08.130 --> 00:24:12.420 The second being interpolation in the time domain with an 00:24:12.420 --> 00:24:15.520 impulse response, which is a triangle. 00:24:15.520 --> 00:24:19.380 And in fact, it's interesting to also look at the 00:24:19.380 --> 00:24:22.710 relationship between that and band limited interpolation. 00:24:22.710 --> 00:24:27.190 Look at it, specifically, in the frequency domain. 00:24:27.190 --> 00:24:32.170 Well, in the frequency domain, what we know, of course, is 00:24:32.170 --> 00:24:36.810 that for exact interpolation, what we want as our 00:24:36.810 --> 00:24:40.780 interpolating filter is an ideal low pass filter. 00:24:40.780 --> 00:24:43.850 Now keep in mind, by the way, that an ideal low pass filter 00:24:43.850 --> 00:24:46.350 is an abstraction, as I've stressed several 00:24:46.350 --> 00:24:47.850 times in the past. 00:24:47.850 --> 00:24:52.700 An ideal low pass filter is a non-causal filter and, in 00:24:52.700 --> 00:24:55.730 fact, infinite extent, which is one of the reasons why in 00:24:55.730 --> 00:24:58.710 any case we would use some approximation to it. 00:24:58.710 --> 00:25:05.700 But here, what we have is the exact interpolating filter. 00:25:05.700 --> 00:25:11.030 And that corresponds to an ideal low pass filter. 00:25:11.030 --> 00:25:17.260 If, instead, we carried out the interpolating using the 00:25:17.260 --> 00:25:21.280 zero order hold, the zero order hold has a rectangular 00:25:21.280 --> 00:25:22.740 impulse response. 00:25:22.740 --> 00:25:25.600 And that means in the frequency domain, its 00:25:25.600 --> 00:25:30.410 frequency response is of the form of a sync function, or 00:25:30.410 --> 00:25:31.890 sine x over x. 00:25:31.890 --> 00:25:35.910 And so this, in fact, when we're doing the reconstruction 00:25:35.910 --> 00:25:40.230 with a zero order hold, is the associated frequency response. 00:25:40.230 --> 00:25:43.040 Now notice that it does some 00:25:43.040 --> 00:25:44.930 approximate low pass filtering. 00:25:44.930 --> 00:25:51.430 But of course, it permits significant energy outside the 00:25:51.430 --> 00:25:53.670 past band of the filter. 00:25:53.670 --> 00:25:56.680 Well, instead of the zero order hold, if we used the 00:25:56.680 --> 00:25:59.610 first order hold corresponding to the triangular impulse 00:25:59.610 --> 00:26:03.630 response, in that case then in the frequency domain, the 00:26:03.630 --> 00:26:07.560 associated frequency response would be the Fourier transform 00:26:07.560 --> 00:26:09.090 of the triangle. 00:26:09.090 --> 00:26:13.520 And the Fourier transform of a triangle is a sine squared x 00:26:13.520 --> 00:26:16.310 over x squared kind of function. 00:26:16.310 --> 00:26:20.770 And so in that case, what we would have for the frequency 00:26:20.770 --> 00:26:24.070 response, associated with the first order hold, is a 00:26:24.070 --> 00:26:27.750 frequency response as I show here. 00:26:27.750 --> 00:26:33.330 And the fact that there's somewhat more attenuation 00:26:33.330 --> 00:26:37.620 outside the past band of the ideal filter is what suggests, 00:26:37.620 --> 00:26:42.270 in fact, that the first order hold, or linear interpolation, 00:26:42.270 --> 00:26:46.280 gives us a somewhat smoother approximation to the original 00:26:46.280 --> 00:26:49.620 signal than the zero order hold does. 00:26:49.620 --> 00:26:53.680 And so, in fact, just to compare these two, we can see 00:26:53.680 --> 00:26:57.990 that here is the ideal filter. 00:26:57.990 --> 00:27:03.300 Here is the zero order hold, corresponding to generating a 00:27:03.300 --> 00:27:05.790 box car kind of reconstruction. 00:27:05.790 --> 00:27:09.710 And here is the first order hold, corresponding to a 00:27:09.710 --> 00:27:12.500 linear interpolation. 00:27:12.500 --> 00:27:17.460 Now in fact, in many sampling systems, in any sampling 00:27:17.460 --> 00:27:22.200 system really, we need to use some approximation to the low 00:27:22.200 --> 00:27:23.520 pass filter. 00:27:23.520 --> 00:27:27.340 And very often, in fact, what is done in many sampling 00:27:27.340 --> 00:27:32.130 systems, is to first use just the zero order hold, and then 00:27:32.130 --> 00:27:34.300 follow the zero order hold with some 00:27:34.300 --> 00:27:37.830 additional low pass filtering. 00:27:37.830 --> 00:27:43.240 Well, to illustrate some of these ideas and the notion of 00:27:43.240 --> 00:27:46.380 doing a reconstruction with a zero order hold or first order 00:27:46.380 --> 00:27:50.700 hold and then in fact adding to that some additional low 00:27:50.700 --> 00:27:56.220 pass filtering, what I'd like to do is demonstrate, or 00:27:56.220 --> 00:27:59.820 illustrate, sampling and interpolation in the context 00:27:59.820 --> 00:28:01.230 of some images. 00:28:01.230 --> 00:28:05.380 An image, of course, is a two-dimensional signal. 00:28:05.380 --> 00:28:07.490 The independent variables are spatial 00:28:07.490 --> 00:28:09.650 variables not time variables. 00:28:09.650 --> 00:28:13.050 And of course, we can sample in both of the spatial 00:28:13.050 --> 00:28:16.020 dimensions, both in x and y. 00:28:16.020 --> 00:28:22.000 And what I've chosen as a possibly appropriate choice 00:28:22.000 --> 00:28:26.140 for an image is, again, our friend and 00:28:26.140 --> 00:28:29.350 colleague J.B.J. Fourier. 00:28:29.350 --> 00:28:33.800 So let's begin with the original image, which we then 00:28:33.800 --> 00:28:36.350 want to sample and reconstruct. 00:28:36.350 --> 00:28:40.590 And the sampling is done by effectively multiplying by a 00:28:40.590 --> 00:28:43.510 pulse both horizontally and vertically. 00:28:43.510 --> 00:28:47.840 The sample picture is then the next one that I show. 00:28:47.840 --> 00:28:51.930 And as you can see, this corresponds, in effect, to 00:28:51.930 --> 00:28:54.280 extracting small brightness elements out of 00:28:54.280 --> 00:28:55.080 the original image. 00:28:55.080 --> 00:28:58.200 In fact, let's look in a little closer. 00:28:58.200 --> 00:29:01.500 And what you can see, essentially, is that what we 00:29:01.500 --> 00:29:05.220 have, of course, are not impulses spatially but small 00:29:05.220 --> 00:29:10.490 spatial pillars that implement the sampling for us. 00:29:10.490 --> 00:29:15.450 OK, now going back to the original sample picture, we 00:29:15.450 --> 00:29:18.920 know that a picture can be reconstructed by low pass 00:29:18.920 --> 00:29:20.200 filtering from the samples. 00:29:20.200 --> 00:29:23.850 And in fact, we can do that optically in this case by 00:29:23.850 --> 00:29:26.480 simply defocusing the camera. 00:29:26.480 --> 00:29:30.270 And when we do that, what happens is that we smear out 00:29:30.270 --> 00:29:34.430 the picture, or effectively convulve the impulses with the 00:29:34.430 --> 00:29:36.620 point spread function of the optical system. 00:29:36.620 --> 00:29:40.340 And this then is not too bad a reconstruction. 00:29:40.340 --> 00:29:44.420 So that's an approximate reconstruction. 00:29:44.420 --> 00:29:47.840 And focusing back now what we have again 00:29:47.840 --> 00:29:49.830 is the sample picture. 00:29:53.990 --> 00:29:58.590 Now these images are, in fact, taken off a computer display. 00:29:58.590 --> 00:30:02.620 And a common procedure in computer generated or 00:30:02.620 --> 00:30:08.090 displayed images is in fact the use of a zero order hold. 00:30:08.090 --> 00:30:11.640 And if the sampling rate is high enough, then that 00:30:11.640 --> 00:30:13.490 actually works reasonably well. 00:30:13.490 --> 00:30:16.650 So now let's look at the result of applying a zero 00:30:16.650 --> 00:30:21.350 order hold to the sample image that I just showed. 00:30:24.580 --> 00:30:27.760 The zero order hold corresponds to replacing the 00:30:27.760 --> 00:30:30.910 impulses by rectangles. 00:30:30.910 --> 00:30:35.180 And you can see that what that generates is a mosaic effect, 00:30:35.180 --> 00:30:37.490 as you would expect. 00:30:37.490 --> 00:30:41.860 And in fact, let's go in a little closer and emphasize 00:30:41.860 --> 00:30:42.840 the mosaic effect. 00:30:42.840 --> 00:30:45.390 You can see that, essentially, where there were impulses 00:30:45.390 --> 00:30:49.200 previously, there are now rectangles with those 00:30:49.200 --> 00:30:50.870 brightness values. 00:30:50.870 --> 00:30:55.300 A very common procedure with computer generated images is 00:30:55.300 --> 00:30:59.110 to first do a zero order hold, as we've done here, and then 00:30:59.110 --> 00:31:02.460 follow that with some additional low pass filtering. 00:31:02.460 --> 00:31:05.900 And fact, we can do that low pass filtering now again by 00:31:05.900 --> 00:31:07.930 defocusing the camera. 00:31:07.930 --> 00:31:12.890 And you can begin to see that with the zero order hold plus 00:31:12.890 --> 00:31:14.140 the low pass filtering, the 00:31:14.140 --> 00:31:17.230 reconstruction is not that bad. 00:31:17.230 --> 00:31:21.450 Well, let's go back to the full image with 00:31:21.450 --> 00:31:23.930 the zero order hold. 00:31:23.930 --> 00:31:29.200 And again, now the effect of low pass filtering will be 00:31:29.200 --> 00:31:30.100 somewhat better. 00:31:30.100 --> 00:31:33.310 And let's defocus again here. 00:31:33.310 --> 00:31:37.210 And you can begin to see that this is a reasonable 00:31:37.210 --> 00:31:39.370 reconstruction. 00:31:39.370 --> 00:31:44.050 With the mosaic, in fact, with this back in focus, you can 00:31:44.050 --> 00:31:47.440 apply your own low pass filtering to it either by 00:31:47.440 --> 00:31:51.040 squinting, or if you have the right or wrong kind of 00:31:51.040 --> 00:31:55.800 eyeglasses, either taking them off or putting them on. 00:31:55.800 --> 00:31:59.780 Now, in addition to the zero order hold, we can, of course, 00:31:59.780 --> 00:32:01.260 apply a first order hold. 00:32:01.260 --> 00:32:05.550 And that would correspond to replacing the impulses, 00:32:05.550 --> 00:32:09.310 instead of with rectangles as we have here, replacing them 00:32:09.310 --> 00:32:10.880 with triangles. 00:32:10.880 --> 00:32:14.930 And so now let's take a look at the result of a first order 00:32:14.930 --> 00:32:18.430 hold applied to the original samples. 00:32:18.430 --> 00:32:22.410 And you can see now that the reconstruction is somewhat 00:32:22.410 --> 00:32:24.960 smoother because of the fact that we're using an impulse 00:32:24.960 --> 00:32:28.750 response that's somewhat smoother or a corresponding 00:32:28.750 --> 00:32:31.310 frequency response that has a sharper cut off. 00:32:31.310 --> 00:32:34.580 I emphasize again that this is a somewhat low pass filtered 00:32:34.580 --> 00:32:38.220 version of the original because we have under sampled 00:32:38.220 --> 00:32:41.870 somewhat spatially to bring out the point that I want to 00:32:41.870 --> 00:32:43.120 illustrate. 00:32:46.840 --> 00:32:52.120 OK, to emphasize these effects even more, what I'd like to do 00:32:52.120 --> 00:32:55.750 is go through, basically, the same sequence again. 00:32:55.750 --> 00:32:58.990 But in this case, what we'll do is double the sample 00:32:58.990 --> 00:33:02.830 spacing both horizontal and vertically. 00:33:02.830 --> 00:33:05.550 This of course, means that we'll be even more highly 00:33:05.550 --> 00:33:09.970 under sampled than in the ones I previously showed. 00:33:09.970 --> 00:33:12.970 And so the result of the reconstructions with some low 00:33:12.970 --> 00:33:18.910 pass filtering will be a much more low pass filtered image. 00:33:18.910 --> 00:33:21.590 So we now have the sampled picture. 00:33:21.590 --> 00:33:24.900 But I've now under sampled considerably more. 00:33:24.900 --> 00:33:28.460 And you can see the effect of the sampling. 00:33:28.460 --> 00:33:33.510 And if we now apply a zero order hold to this picture, we 00:33:33.510 --> 00:33:34.910 will again get a mosaic. 00:33:34.910 --> 00:33:37.200 And let's look at that. 00:33:37.200 --> 00:33:40.300 And that mosaic, of course, looks even 00:33:40.300 --> 00:33:42.290 blockier than the original. 00:33:42.290 --> 00:33:46.110 And again, it emphasizes the fact that the zero order hold 00:33:46.110 --> 00:33:50.140 simply corresponds to filling in squares, or replacing the 00:33:50.140 --> 00:33:53.120 impulses, by squares, with the 00:33:53.120 --> 00:33:55.990 corresponding brightness values. 00:33:55.990 --> 00:34:00.540 Finally, if we, instead of a zero order hold, use a first 00:34:00.540 --> 00:34:04.910 order hold, corresponding to two dimensional triangles in 00:34:04.910 --> 00:34:06.750 place of these original blocks. 00:34:06.750 --> 00:34:09.920 What we get is the next image. 00:34:09.920 --> 00:34:13.840 And that, again, is a smoother reconstruction consistent with 00:34:13.840 --> 00:34:16.040 the fact that the triangles are smoother than the 00:34:16.040 --> 00:34:17.270 rectangles. 00:34:17.270 --> 00:34:20.530 Again, I emphasize that this looks so highly low pass 00:34:20.530 --> 00:34:24.280 filtered because of the fact that we've under sampled so 00:34:24.280 --> 00:34:27.720 severely to essentially emphasize the effect. 00:34:31.110 --> 00:34:35.469 As I mentioned, the images that we just looked at were 00:34:35.469 --> 00:34:38.250 taken from a computer, although of course the 00:34:38.250 --> 00:34:41.659 original images were continuous time images or more 00:34:41.659 --> 00:34:43.880 specifically, continuous space. 00:34:43.880 --> 00:34:45.280 That is the independent variable 00:34:45.280 --> 00:34:47.830 is a spatial variable. 00:34:47.830 --> 00:34:52.900 Now, computer processing of signals, pictures, speech, or 00:34:52.900 --> 00:34:57.040 whatever the signals are is very important and useful 00:34:57.040 --> 00:35:00.070 because it offers a lot of flexibility. 00:35:00.070 --> 00:35:02.220 And in fact, the kinds of things that I showed with 00:35:02.220 --> 00:35:07.120 these pictures would have been very hard to do without, in 00:35:07.120 --> 00:35:10.620 fact, doing computer processing. 00:35:10.620 --> 00:35:15.250 Well, in computer processing of any kind of signal, 00:35:15.250 --> 00:35:20.800 basically what's required is that we do the processing in 00:35:20.800 --> 00:35:24.280 the context of discrete time signals and discrete time 00:35:24.280 --> 00:35:27.860 processing because of the fact that a computer 00:35:27.860 --> 00:35:29.310 is run off a clock. 00:35:29.310 --> 00:35:35.630 And essentially, things happen in the computer as a sequence 00:35:35.630 --> 00:35:39.400 of numbers and as a sequence of events. 00:35:39.400 --> 00:35:43.290 Well, it turns out that the sampling theorem, in fact, as 00:35:43.290 --> 00:35:47.280 I've indicated previously, provides us with a very nice 00:35:47.280 --> 00:35:53.170 mechanism for converting our continuous time signals into 00:35:53.170 --> 00:35:54.830 discrete time signals. 00:35:54.830 --> 00:35:58.810 For example, for computer processing or, in fact, if 00:35:58.810 --> 00:36:02.230 it's not a computer for some other kind of discrete time or 00:36:02.230 --> 00:36:04.930 perhaps digital processing. 00:36:04.930 --> 00:36:11.700 Well, the basic idea, as I've indicated previously, is to 00:36:11.700 --> 00:36:15.520 carry out discrete time processing of continuous time 00:36:15.520 --> 00:36:22.010 signals by first converting the continuous time signal to 00:36:22.010 --> 00:36:27.890 a discrete time signal, carry out the appropriate discrete 00:36:27.890 --> 00:36:33.310 time processing of the discrete time signal, and then 00:36:33.310 --> 00:36:37.210 after we're done with that processing, converting from 00:36:37.210 --> 00:36:41.840 the discrete time sequence back to a continuous time 00:36:41.840 --> 00:36:45.400 signal, corresponding to the output that we have here. 00:36:48.010 --> 00:36:51.500 Well in the remainder of this lecture, what I'd like to 00:36:51.500 --> 00:36:55.830 analyze is the first step in that process, namely the 00:36:55.830 --> 00:36:59.710 conversion from a continuous time signal to a discrete time 00:36:59.710 --> 00:37:05.630 signal and understand how the two relate both in the time 00:37:05.630 --> 00:37:07.900 domain and in the frequency domain. 00:37:07.900 --> 00:37:11.410 And in the next lecture, we'll be analyzing and demonstrating 00:37:11.410 --> 00:37:15.750 the overall system, including some intermediate processing. 00:37:15.750 --> 00:37:20.560 So the first step in the process is the conversion from 00:37:20.560 --> 00:37:24.670 a continuous time signal to a discrete time signal. 00:37:24.670 --> 00:37:28.160 And that can be thought of as a process that involves two 00:37:28.160 --> 00:37:31.490 steps, although in practical terms it may not be 00:37:31.490 --> 00:37:34.680 implemented specifically as these two steps. 00:37:34.680 --> 00:37:39.500 The two steps are to first convert from the continuous 00:37:39.500 --> 00:37:45.010 time, or continuous time continuous signal, to an 00:37:45.010 --> 00:37:52.150 impulse train through a sampling process and then to 00:37:52.150 --> 00:37:56.950 convert that impulse train to a discrete time sequence. 00:37:56.950 --> 00:38:02.640 And the discrete time sequence x of n is simply then a 00:38:02.640 --> 00:38:08.320 sequence of values which are the samples of the continuous 00:38:08.320 --> 00:38:09.450 time signal. 00:38:09.450 --> 00:38:12.550 And as we'll see as we walk through this, basically the 00:38:12.550 --> 00:38:16.960 step of going from the impulse train to the sequence 00:38:16.960 --> 00:38:22.320 corresponds principally to a relabeling step where we pick 00:38:22.320 --> 00:38:29.250 off the impulse values and use those as the sequence values 00:38:29.250 --> 00:38:31.670 for the discrete time signal. 00:38:31.670 --> 00:38:35.880 So what I'd like to do as a first step in understanding 00:38:35.880 --> 00:38:40.980 this process is to analyze it in particular with our 00:38:40.980 --> 00:38:44.160 attention focused on trying to understand what the 00:38:44.160 --> 00:38:48.040 relationship is in the frequency domain between the 00:38:48.040 --> 00:38:52.370 discrete time Fourier transform of the sequence, 00:38:52.370 --> 00:38:56.380 discrete time signal, and the continuous time Fourier 00:38:56.380 --> 00:39:00.630 transform of the original unsampled, and then the 00:39:00.630 --> 00:39:02.170 sampled signal. 00:39:02.170 --> 00:39:04.430 So let's go through that. 00:39:04.430 --> 00:39:10.760 And in particular, what we have is a process where the 00:39:10.760 --> 00:39:15.020 continuous time signal is, of course, modulated or 00:39:15.020 --> 00:39:17.740 multiplied by an impulse train. 00:39:17.740 --> 00:39:20.060 And that gives us, then, another 00:39:20.060 --> 00:39:21.290 continuous time signal. 00:39:21.290 --> 00:39:23.470 We're still in the continuous time domain. 00:39:23.470 --> 00:39:26.200 It gives us another continuous time signal, which is an 00:39:26.200 --> 00:39:28.120 impulse train. 00:39:28.120 --> 00:39:31.250 And in fact, we've gone through this analysis 00:39:31.250 --> 00:39:32.540 previously. 00:39:32.540 --> 00:39:37.830 And what we have is this multiplication or taking this 00:39:37.830 --> 00:39:42.630 term inside the summation and recognizing that the impulse 00:39:42.630 --> 00:39:48.660 train is simply an impulse train with areas of the 00:39:48.660 --> 00:39:51.680 impulses, which are the samples of the 00:39:51.680 --> 00:39:53.510 continuous time function. 00:39:53.510 --> 00:39:56.190 We can then carry out the analysis in 00:39:56.190 --> 00:39:59.540 the frequency domain. 00:39:59.540 --> 00:40:03.080 Now in the time domain, we have a multiplication process. 00:40:03.080 --> 00:40:06.680 So in the frequency domain, we have a convolution of the 00:40:06.680 --> 00:40:12.010 Fourier transform of the continuous time signal, the 00:40:12.010 --> 00:40:14.820 original signal, and the Fourier transform of the 00:40:14.820 --> 00:40:18.590 impulse train, which is itself an impulse train. 00:40:18.590 --> 00:40:22.950 So in the frequency domain then, the Fourier transform of 00:40:22.950 --> 00:40:26.590 the sampled signal, which is an impulse train, is the 00:40:26.590 --> 00:40:30.780 convolution of the Fourier transform of the sampling 00:40:30.780 --> 00:40:34.310 function P of t and the Fourier transform of the 00:40:34.310 --> 00:40:36.440 sampled signal. 00:40:36.440 --> 00:40:41.710 Since the sampling signal is a periodic impulse train, its 00:40:41.710 --> 00:40:44.750 Fourier transform is an impulse train. 00:40:44.750 --> 00:40:49.700 And consequently, carrying out this convolution in effect 00:40:49.700 --> 00:40:53.300 says that this Fourier transform simply gets 00:40:53.300 --> 00:40:57.600 replicated at each of the locations of these impulses. 00:40:57.600 --> 00:41:02.390 And finally, what we end up with then is a Fourier 00:41:02.390 --> 00:41:08.730 transform after the sampling process, which is the original 00:41:08.730 --> 00:41:13.650 Fourier transform of the continuous signal but added to 00:41:13.650 --> 00:41:17.750 itself shifted by integer multiples of 00:41:17.750 --> 00:41:19.350 the sampling frequency. 00:41:19.350 --> 00:41:23.040 And so this is the basic equation then that tells us in 00:41:23.040 --> 00:41:28.280 the frequency domain what happens through the first part 00:41:28.280 --> 00:41:30.510 of this two step process. 00:41:30.510 --> 00:41:33.030 Now I emphasize that it's a two step process. 00:41:33.030 --> 00:41:38.500 The first process is sampling, where we're still essentially 00:41:38.500 --> 00:41:41.240 in the continuous time world. 00:41:41.240 --> 00:41:46.050 The next step is essentially a relabeling process, where we 00:41:46.050 --> 00:41:50.650 convert that impulse train simply to a sequence. 00:41:50.650 --> 00:41:53.770 So let's look at the next step. 00:41:53.770 --> 00:41:57.780 The next step is to take the impulse train and convert it 00:41:57.780 --> 00:42:01.850 through a process to a sequence. 00:42:01.850 --> 00:42:06.880 And the sequence values are simply then samples of the 00:42:06.880 --> 00:42:09.550 original continuous signal. 00:42:09.550 --> 00:42:13.840 And so now we can analyze this. 00:42:13.840 --> 00:42:18.860 And what we want to relate is the discrete time Fourier 00:42:18.860 --> 00:42:22.930 transform of this and the continuous time Fourier 00:42:22.930 --> 00:42:25.920 transform of this, or in fact, the continuous time Fourier 00:42:25.920 --> 00:42:30.660 transform of x of C of T. 00:42:30.660 --> 00:42:34.930 OK, we have the impulse train. 00:42:34.930 --> 00:42:40.570 And it's Fourier transform we can get by simply evaluating 00:42:40.570 --> 00:42:42.000 the Fourier transform. 00:42:42.000 --> 00:42:45.430 And since the Fourier transform of this-- 00:42:45.430 --> 00:42:48.170 since this corresponds to an impulse train-- 00:42:48.170 --> 00:42:51.140 the Fourier transform, by the time we change some sums and 00:42:51.140 --> 00:42:56.340 integrals, will then have this impulse replaced by the 00:42:56.340 --> 00:42:59.420 Fourier transform of the shifted impulse, which is this 00:42:59.420 --> 00:43:01.570 exponential factor. 00:43:01.570 --> 00:43:04.950 So this expression is the Fourier transform of the 00:43:04.950 --> 00:43:08.850 impulse train, the continuous time Fourier transform. 00:43:08.850 --> 00:43:13.750 And alternatively, we can look at the Fourier transform of 00:43:13.750 --> 00:43:15.080 the sequence. 00:43:15.080 --> 00:43:17.540 And this, of course, is a discrete 00:43:17.540 --> 00:43:21.110 time Fourier transform. 00:43:21.110 --> 00:43:25.310 So we have the continuous time Fourier transform of the 00:43:25.310 --> 00:43:29.570 impulse train, we have the discrete Fourier transform of 00:43:29.570 --> 00:43:30.360 the sequence. 00:43:30.360 --> 00:43:33.970 And now we want to look at how those two relate. 00:43:33.970 --> 00:43:37.750 Well, it pretty much falls out of just comparing these two 00:43:37.750 --> 00:43:39.020 summations. 00:43:39.020 --> 00:43:44.870 In particular, this term and this term are identical. 00:43:44.870 --> 00:43:52.100 That's just a relabeling of what the sequence values are. 00:43:52.100 --> 00:43:55.540 And notice that when we compare these exponential 00:43:55.540 --> 00:44:01.350 factors, they're identical as long as we associate capital 00:44:01.350 --> 00:44:05.650 omega with little omega times capital T. In other words, if 00:44:05.650 --> 00:44:09.370 we were to replace here capital omega by little omega 00:44:09.370 --> 00:44:15.060 times capital T, and replace x of n by x of c of nt, then 00:44:15.060 --> 00:44:20.220 this expression would be identical to this expression. 00:44:20.220 --> 00:44:26.010 So in fact, these two are equal with a relabeling, or 00:44:26.010 --> 00:44:29.970 with a transformation, between small omega and capital omega. 00:44:29.970 --> 00:44:34.350 And so in fact, the relationship that we have is 00:44:34.350 --> 00:44:39.000 that the discrete time Fourier transform of the sequence of 00:44:39.000 --> 00:44:43.990 samples is equal to the continuous time Fourier 00:44:43.990 --> 00:44:50.440 transform of the impulse train of samples where we associate 00:44:50.440 --> 00:44:54.460 the continuous time frequency variable and the discrete time 00:44:54.460 --> 00:44:58.200 frequency variable through a frequency scaling as I 00:44:58.200 --> 00:44:59.520 indicate here. 00:44:59.520 --> 00:45:04.130 Or said another way, the discrete time spectrum is the 00:45:04.130 --> 00:45:10.090 continuous time spectrum of the samples with small omega 00:45:10.090 --> 00:45:12.440 replaced by capital omega divided by 00:45:12.440 --> 00:45:15.480 capital T. All right. 00:45:15.480 --> 00:45:19.660 So we have then this two step process. 00:45:19.660 --> 00:45:22.960 The first step is taking the continuous time signal, 00:45:22.960 --> 00:45:26.130 sampling it with an impulse train. 00:45:26.130 --> 00:45:31.480 In the frequency domain, that corresponds to replicating the 00:45:31.480 --> 00:45:34.200 Fourier transform of the original 00:45:34.200 --> 00:45:36.540 continuous time signal. 00:45:36.540 --> 00:45:41.780 The second step is relabeling that, in effect turning it 00:45:41.780 --> 00:45:43.170 into a sequence. 00:45:43.170 --> 00:45:46.580 And what that does in the frequency domain is provide us 00:45:46.580 --> 00:45:50.140 with a rescaling of the frequency axis, or as we'll 00:45:50.140 --> 00:45:55.040 see a frequency normalization, which is associated with the 00:45:55.040 --> 00:45:56.130 corresponding time 00:45:56.130 --> 00:45:58.740 normalization in the time domain. 00:45:58.740 --> 00:46:01.810 Well, let's look at those statements a little more 00:46:01.810 --> 00:46:04.350 specifically. 00:46:04.350 --> 00:46:06.900 What I show here is the original 00:46:06.900 --> 00:46:09.750 continuous time signal. 00:46:09.750 --> 00:46:16.020 And then below it is the sampled signal. 00:46:16.020 --> 00:46:20.770 And these two are signals in the continuous time domain. 00:46:20.770 --> 00:46:23.310 Now, what is the conversion from this 00:46:23.310 --> 00:46:25.660 impulse train to a sequence? 00:46:25.660 --> 00:46:31.040 Well, it's simply taking these impulse areas, or these sample 00:46:31.040 --> 00:46:39.070 values, and relabeling them, in effect as I show below, as 00:46:39.070 --> 00:46:41.850 sequence values. 00:46:41.850 --> 00:46:48.610 And essentially, I'm now replacing the impulse by the 00:46:48.610 --> 00:46:50.700 designation of a sequence value. 00:46:50.700 --> 00:46:51.830 That's one step. 00:46:51.830 --> 00:46:56.930 But the other important step to focus on is that whereas in 00:46:56.930 --> 00:47:00.370 the impulse train, these impulses are spaced by integer 00:47:00.370 --> 00:47:05.000 multiples of the sampling period capital T. In the 00:47:05.000 --> 00:47:08.390 sequence, of course, because of the way that we label 00:47:08.390 --> 00:47:13.210 sequences, these are always spaced by simply integer 00:47:13.210 --> 00:47:14.750 multiples of one. 00:47:14.750 --> 00:47:18.160 So in effect, you could say that the step in going from 00:47:18.160 --> 00:47:23.180 here to here corresponds to normalizing out in the time 00:47:23.180 --> 00:47:27.590 domain the sampling period capital T. 00:47:27.590 --> 00:47:31.570 To stress that another way, if the sampling period were 00:47:31.570 --> 00:47:35.810 doubled so that in this picture, the spacing stretched 00:47:35.810 --> 00:47:38.280 out by a factor of two. 00:47:38.280 --> 00:47:45.320 Nevertheless, for the discrete time signal, the spacing would 00:47:45.320 --> 00:47:46.740 remain as one. 00:47:46.740 --> 00:47:52.470 And essentially, it's the envelope of those sequence 00:47:52.470 --> 00:47:56.460 values that would then get compressed in time. 00:47:56.460 --> 00:48:00.370 So you can think of the step in going from the impulse 00:48:00.370 --> 00:48:04.430 train to the samples as, essentially, a time 00:48:04.430 --> 00:48:05.740 normalization. 00:48:05.740 --> 00:48:08.470 Now let's look at this in the frequency domain. 00:48:08.470 --> 00:48:11.320 In the frequency domain, what we have is the Fourier 00:48:11.320 --> 00:48:16.110 transform of our original continuous signal. 00:48:16.110 --> 00:48:21.220 After sampling with an impulse train, this spectrum retains 00:48:21.220 --> 00:48:26.280 its shape but is replicated at integer multiples of the 00:48:26.280 --> 00:48:31.350 sampling frequency 2 pi over capital T, as I indicate here. 00:48:31.350 --> 00:48:38.000 Now, we know that a discrete time spectrum must be periodic 00:48:38.000 --> 00:48:40.580 in frequency with a period of 2 pi. 00:48:40.580 --> 00:48:42.560 Here, we have the periodicity. 00:48:42.560 --> 00:48:45.590 But it's not periodic with a period of 2 pi. 00:48:45.590 --> 00:48:47.750 It's periodic with a period, which is equal to 00:48:47.750 --> 00:48:50.680 the sampling frequency. 00:48:50.680 --> 00:48:55.930 However, in converting from the samples to the sequence 00:48:55.930 --> 00:48:58.610 values, we go through another step. 00:48:58.610 --> 00:48:59.490 What's the other step? 00:48:59.490 --> 00:49:02.940 The other step is a time normalization, where we take 00:49:02.940 --> 00:49:05.890 the impulses, which are spaced by the sampling period. 00:49:05.890 --> 00:49:08.810 And we rescale that, essentially in the time 00:49:08.810 --> 00:49:11.400 domain, to a spacing which is unity. 00:49:11.400 --> 00:49:19.990 So we're dividing out in the time domain by a factor, which 00:49:19.990 --> 00:49:23.050 is equal to the sampling period. 00:49:23.050 --> 00:49:28.740 Well, dividing out in the time domain by capital T would 00:49:28.740 --> 00:49:33.510 correspond to multiplying in the frequency domain the 00:49:33.510 --> 00:49:38.060 frequency axis by capital T. And indeed, what happens is 00:49:38.060 --> 00:49:42.270 that in going from the impulse train to the sequence values, 00:49:42.270 --> 00:49:49.250 we now rescale this axis so that, in fact, the axis gets 00:49:49.250 --> 00:49:53.550 stretched by capital T. And the frequency, which 00:49:53.550 --> 00:49:57.380 corresponded to 2 pi over capital T, now gets 00:49:57.380 --> 00:50:00.980 renormalized to 2 pi. 00:50:00.980 --> 00:50:04.740 So just looking at this again, and perhaps with the overall 00:50:04.740 --> 00:50:08.940 picture, in the time domain, we've gone from a continuous 00:50:08.940 --> 00:50:13.380 curve to samples, relabeled those, and in effect 00:50:13.380 --> 00:50:15.780 implemented a time normalization. 00:50:15.780 --> 00:50:20.180 Corresponding in the frequency domain, we have replicated the 00:50:20.180 --> 00:50:24.690 spectrum through the initial sampling process and then 00:50:24.690 --> 00:50:29.020 rescaled the frequency axis so that, in fact, now this 00:50:29.020 --> 00:50:33.480 periodicity corresponds to a periodicity here, which is 2 00:50:33.480 --> 00:50:37.870 pi, and here, which is the sampling frequency. 00:50:37.870 --> 00:50:41.470 So very often, in fact-- and we'll be 00:50:41.470 --> 00:50:44.000 doing this next time-- 00:50:44.000 --> 00:50:49.630 when you think of continuous time signals, which have been 00:50:49.630 --> 00:50:52.280 converted to discrete time signals, when you look at the 00:50:52.280 --> 00:50:56.520 discrete time frequency axis, the frequency 2 pi is 00:50:56.520 --> 00:51:03.290 associated with the sampling frequency as it was applied to 00:51:03.290 --> 00:51:06.530 the original continuous time signal. 00:51:06.530 --> 00:51:13.010 Now as I indicated, what we'll want to go on to from here is 00:51:13.010 --> 00:51:15.880 an understanding of what happens when we take a 00:51:15.880 --> 00:51:19.180 continuous time signal, convert it to a discrete time 00:51:19.180 --> 00:51:22.440 signal as I've just gone through, do some discrete time 00:51:22.440 --> 00:51:25.890 processing with a linear time invariant system, and then 00:51:25.890 --> 00:51:29.520 carry that back into the continuous time world. 00:51:29.520 --> 00:51:34.190 That is a procedure that we'll go through, and analyze, and 00:51:34.190 --> 00:51:38.660 in fact, illustrate in some detail next time. 00:51:38.660 --> 00:51:42.970 In preparation for that, what I would be eager to encourage 00:51:42.970 --> 00:51:46.650 you to do using the study guide and in reviewing this 00:51:46.650 --> 00:51:53.340 lecture, is to begin the next lecture with a careful and 00:51:53.340 --> 00:51:56.220 thorough understanding of the arguments that 00:51:56.220 --> 00:51:57.500 I've just gone through. 00:51:57.500 --> 00:52:02.210 In particular, understanding the process that's involved in 00:52:02.210 --> 00:52:06.320 going from a continuous time signal through sampling to a 00:52:06.320 --> 00:52:08.390 discrete time signal. 00:52:08.390 --> 00:52:11.860 And what that means in the frequency domain in terms of 00:52:11.860 --> 00:52:15.710 taking the original spectrum, replicating it because of the 00:52:15.710 --> 00:52:20.590 sampling process, and then rescaling that so that the 00:52:20.590 --> 00:52:24.710 periodicity gets rescaled so that it's periodic with a 00:52:24.710 --> 00:52:26.010 period of 2 pi. 00:52:26.010 --> 00:52:31.120 So we'll continue with that next time, focusing now on the 00:52:31.120 --> 00:52:33.610 subsequent steps in the processing. 00:52:33.610 --> 00:52:34.860 Thank you.