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 --> 00:00:13.460 To make a donation or view additional materials from 00:00:13.460 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.540 ocw.mit.edu. 00:00:55.450 --> 00:00:57.940 PROFESSOR: In the last lecture, we discussed discrete 00:00:57.940 --> 00:01:00.760 time processing of continuous time signals. 00:01:00.760 --> 00:01:04.069 And, as you know, the basis for that arises essentially 00:01:04.069 --> 00:01:06.150 out of a sampling theorem. 00:01:06.150 --> 00:01:10.240 Now in that context, and also in its own right, another 00:01:10.240 --> 00:01:14.850 important sampling issue is the sampling of discrete time 00:01:14.850 --> 00:01:18.640 signals, in other words, the sampling of a sequence. 00:01:18.640 --> 00:01:23.000 One common context in which this arises, for example, is, 00:01:23.000 --> 00:01:26.690 if we've converted from a continuous time signal to a 00:01:26.690 --> 00:01:32.150 sequence, and we then carry out some additional filtering, 00:01:32.150 --> 00:01:35.260 then there's the possibility that we can resample that 00:01:35.260 --> 00:01:38.460 sequence, and as we'll see as we go through the discussion, 00:01:38.460 --> 00:01:42.300 save something in the way of storage or whatever. 00:01:42.300 --> 00:01:47.800 So discrete time sampling, as I indicated, has important 00:01:47.800 --> 00:01:51.930 application in a context referred to here, namely 00:01:51.930 --> 00:01:55.830 resampling after discrete time filtering. 00:01:55.830 --> 00:01:59.370 And closely related to that, as we'll indicate in this 00:01:59.370 --> 00:02:04.270 lecture, is the concept of using discrete time sampling 00:02:04.270 --> 00:02:08.740 for what's referred to as sampling rate conversion. 00:02:08.740 --> 00:02:15.220 And also closely associated with both of those ideas is a 00:02:15.220 --> 00:02:18.120 set of ideas that I'll bring up in today's lecture, 00:02:18.120 --> 00:02:22.980 referred to as decimation and interpolation of discrete time 00:02:22.980 --> 00:02:25.420 signals or sequences. 00:02:25.420 --> 00:02:30.550 Now the basic process for discrete time sampling is the 00:02:30.550 --> 00:02:33.720 same as it is for continuous time sampling. 00:02:33.720 --> 00:02:38.880 Namely, we can analyze it and set it up on the basis of 00:02:38.880 --> 00:02:43.330 multiplying or modulating a discrete time signal by an 00:02:43.330 --> 00:02:47.430 impulse train, the impulse train essentially, or pulse 00:02:47.430 --> 00:02:51.140 train, pulling out sequence values at the times that we 00:02:51.140 --> 00:02:52.950 want to sample. 00:02:52.950 --> 00:02:58.400 So the basic block diagram for the sampling process is to 00:02:58.400 --> 00:03:01.430 modulate or multiply the sequence that we want to 00:03:01.430 --> 00:03:05.520 sample by an impulse train. 00:03:05.520 --> 00:03:11.380 And here, the impulse train has impulses spaced by integer 00:03:11.380 --> 00:03:14.160 multiples of capital N. This then becomes 00:03:14.160 --> 00:03:16.170 the sampling period. 00:03:16.170 --> 00:03:20.010 And the result of that modulation is then the sample 00:03:20.010 --> 00:03:23.230 sequence x of p of n. 00:03:23.230 --> 00:03:28.250 So if we just look at what a sequence and a sampled version 00:03:28.250 --> 00:03:33.750 of that sequence might look like, what we have here is an 00:03:33.750 --> 00:03:36.220 original sequence x of n. 00:03:36.220 --> 00:03:41.790 And then we have the sampling impulse train, or sampling 00:03:41.790 --> 00:03:46.120 sequence, and it's the modulation or product of these 00:03:46.120 --> 00:03:50.840 two that gives us the sample sequence x of p of n. 00:03:50.840 --> 00:03:55.090 And so, as you can see, multiplying this by this 00:03:55.090 --> 00:03:59.640 essentially pulls out of the original sequence sample 00:03:59.640 --> 00:04:02.740 values at the times that this pulse train is on. 00:04:02.740 --> 00:04:06.330 And of course here, I've drawn this for the case where 00:04:06.330 --> 00:04:12.570 capital N, the sampling period, is equal to 3. 00:04:12.570 --> 00:04:18.540 Now the analysis of discrete time sampling is very similar 00:04:18.540 --> 00:04:21.579 to the analysis of continuous time sampling. 00:04:21.579 --> 00:04:24.100 And let's just quickly look through the 00:04:24.100 --> 00:04:26.990 steps that are involved. 00:04:26.990 --> 00:04:28.380 We're modulating or 00:04:28.380 --> 00:04:30.430 multiplying in the time domain. 00:04:30.430 --> 00:04:33.870 And what that corresponds to in the frequency domain is a 00:04:33.870 --> 00:04:35.330 convolution. 00:04:35.330 --> 00:04:43.020 And so the spectrum of the sampled sequence is the 00:04:43.020 --> 00:04:47.710 periodic convolution of the spectrum of the sampling 00:04:47.710 --> 00:04:51.700 sequence and the spectrum of the 00:04:51.700 --> 00:04:54.210 sequence that we're sampling. 00:04:54.210 --> 00:04:59.680 And since the sampling sequence is an impulse train, 00:04:59.680 --> 00:05:03.500 as we know, the Fourier transform of an impulse train 00:05:03.500 --> 00:05:05.560 is itself an impulse train. 00:05:05.560 --> 00:05:09.500 And so this is the Fourier transform of 00:05:09.500 --> 00:05:11.990 the sampling sequence. 00:05:11.990 --> 00:05:17.340 And now, finally, the Fourier transform of the resulting 00:05:17.340 --> 00:05:23.080 sample sequence, being the convolution of this with the 00:05:23.080 --> 00:05:26.300 Fourier transform of the sequence that we're sampling, 00:05:26.300 --> 00:05:32.090 gives us then a spectrum which consists of a sum of 00:05:32.090 --> 00:05:35.370 replicated versions of the Fourier transform of the 00:05:35.370 --> 00:05:37.970 sequence that we're sampling. 00:05:37.970 --> 00:05:40.940 In other words, what we're doing, very much as we did in 00:05:40.940 --> 00:05:44.790 continuous time, is, through the sampling process when we 00:05:44.790 --> 00:05:49.460 look at it in the frequency domain, taking the spectrum of 00:05:49.460 --> 00:05:53.190 the sequence there were sampling and shifting it and 00:05:53.190 --> 00:05:54.540 then adding it in-- 00:05:54.540 --> 00:05:57.470 shifting it by integer multiples of 00:05:57.470 --> 00:05:58.900 the sampling frequency. 00:05:58.900 --> 00:06:02.620 In particular, looking back at this equation, what we 00:06:02.620 --> 00:06:09.120 recognize is that this term, k times 2 pi over capital N, is 00:06:09.120 --> 00:06:14.460 in fact an integer multiple of the sampling frequency. 00:06:14.460 --> 00:06:17.260 And the same thing is true here. 00:06:17.260 --> 00:06:23.080 This is k times omega sub s, where omega sub s, the 00:06:23.080 --> 00:06:30.580 sampling frequency, is 2 pi divided by capital N. 00:06:30.580 --> 00:06:34.720 All right, so now let's look at what this means pictorially 00:06:34.720 --> 00:06:37.520 or graphically in the frequency domain. 00:06:37.520 --> 00:06:42.110 And as you can imagine, since the analysis and algebra is 00:06:42.110 --> 00:06:45.740 similar to what happens in continuous time, we would 00:06:45.740 --> 00:06:49.410 expect the pictures to more or less be identical to what 00:06:49.410 --> 00:06:51.570 we've seen previously for continuous time. 00:06:51.570 --> 00:06:53.660 And indeed that's the case. 00:06:53.660 --> 00:06:58.890 So here we have the spectrum of the signal 00:06:58.890 --> 00:07:00.460 that's we're sampling. 00:07:00.460 --> 00:07:04.160 This is its Fourier transform, with an assumed highest 00:07:04.160 --> 00:07:08.700 frequency omega sub m, highest frequency over a 2pi range, or 00:07:08.700 --> 00:07:10.810 over a range of pi, rather. 00:07:10.810 --> 00:07:18.520 And now the spectrum of the sampling signal is what I show 00:07:18.520 --> 00:07:22.680 below, which is an impulse train with impulses occurring 00:07:22.680 --> 00:07:27.260 at integer multiples of the sampling frequency. 00:07:27.260 --> 00:07:31.930 And then finally, the convolution of these two is 00:07:31.930 --> 00:07:35.760 simply this one replicated at the 00:07:35.760 --> 00:07:37.910 locations of these impulses. 00:07:37.910 --> 00:07:41.610 And so that's finally what I show below. 00:07:41.610 --> 00:07:46.770 And here I made one particular choice for 00:07:46.770 --> 00:07:48.460 the sampling period. 00:07:48.460 --> 00:07:54.760 This in particular corresponds to a sampling period which is 00:07:54.760 --> 00:08:00.570 capital N equal to 3. 00:08:00.570 --> 00:08:04.520 And so the sampling frequency, omega sub s, is 00:08:04.520 --> 00:08:07.210 2pi divided by 3. 00:08:09.740 --> 00:08:16.060 Now when we look at this, what we recognize is that we have 00:08:16.060 --> 00:08:20.060 basically the same issue here as we had in continuous time, 00:08:20.060 --> 00:08:25.730 in the sense that when these individual replications of the 00:08:25.730 --> 00:08:30.600 Fourier transform, when the sampling frequency is chosen 00:08:30.600 --> 00:08:34.370 high enough so that they don't overlap, then we see the 00:08:34.370 --> 00:08:37.539 potential for being able to get one of them back. 00:08:37.539 --> 00:08:40.659 On the other hand, when they do overlap then what we'll 00:08:40.659 --> 00:08:43.309 have is aliasing, in particular, discrete time 00:08:43.309 --> 00:08:46.460 aliasing, much as we had continuous time aliasing in 00:08:46.460 --> 00:08:48.030 the continuous time case. 00:08:48.030 --> 00:08:53.330 Well notice in this picture that what we have is we've 00:08:53.330 --> 00:08:59.370 chosen this picture so that omega sub s minus omega sub m 00:08:59.370 --> 00:09:04.360 is greater than omega sub m, or equivalently, so that omega 00:09:04.360 --> 00:09:11.890 sub s is greater than 2 omega sub m. 00:09:11.890 --> 00:09:15.240 And so with omega sub s greater than 2 omega sum m, 00:09:15.240 --> 00:09:17.600 that corresponds to this picture. 00:09:17.600 --> 00:09:23.330 Whereas, if that condition is violated then, in fact, the 00:09:23.330 --> 00:09:27.160 picture that we would have is a picture that looks like. 00:09:27.160 --> 00:09:32.030 And in this picture, the individual replications of the 00:09:32.030 --> 00:09:36.700 Fourier transform of the original signal overlap. 00:09:36.700 --> 00:09:41.770 And we can no longer recover the Fourier transform of the 00:09:41.770 --> 00:09:43.000 original signal. 00:09:43.000 --> 00:09:47.390 And this, just as it was in continuous time, is referred 00:09:47.390 --> 00:09:50.860 to as aliasing. 00:09:50.860 --> 00:09:54.870 Now let's look more closely at the situation in which there 00:09:54.870 --> 00:09:56.120 is no aliasing. 00:09:58.680 --> 00:10:07.280 So in that case, what we have is a Fourier transform for the 00:10:07.280 --> 00:10:13.980 sampled signal, which is as I indicated here, and the 00:10:13.980 --> 00:10:17.380 Fourier transform for the original signal, as I indicate 00:10:17.380 --> 00:10:18.710 at the top. 00:10:18.710 --> 00:10:22.190 And the question now is how do we recover this 00:10:22.190 --> 00:10:23.590 one from this one. 00:10:23.590 --> 00:10:27.340 Well, the way that we do that, just as we did in a continuous 00:10:27.340 --> 00:10:31.010 time case, is by a low pass filtering. 00:10:31.010 --> 00:10:34.520 In particular, processing in the time domain or in the 00:10:34.520 --> 00:10:40.590 frequency domain, this with an ideal low pass filter has the 00:10:40.590 --> 00:10:46.080 effect of extracting that part of the spectrum that in fact 00:10:46.080 --> 00:10:53.400 we identify with the original signal that we began with. 00:10:53.400 --> 00:10:57.470 So what we see, again, is that the process is 00:10:57.470 --> 00:10:58.460 very much the same. 00:10:58.460 --> 00:11:01.170 As long as there's no aliasing, we can recover the 00:11:01.170 --> 00:11:06.150 original signal by ideal low pass filtering. 00:11:06.150 --> 00:11:13.290 So the overall system is, just to reiterate, a system which 00:11:13.290 --> 00:11:20.400 consists of modulating the original sequence with a pulse 00:11:20.400 --> 00:11:22.960 train or impulse train. 00:11:22.960 --> 00:11:27.160 And then that is going to be processed 00:11:27.160 --> 00:11:30.370 with a low pass filter. 00:11:30.370 --> 00:11:34.140 The spectrum of the original signal x of n 00:11:34.140 --> 00:11:36.770 is what I show here. 00:11:36.770 --> 00:11:40.020 The spectrum of the sampled signal, where I'm drawing the 00:11:40.020 --> 00:11:43.830 picture on the assumption that the sampling period is 3, is 00:11:43.830 --> 00:11:47.470 now what's indicated, where these are replicated, where 00:11:47.470 --> 00:11:50.320 the original spectrum is replicated. 00:11:50.320 --> 00:11:56.740 This is now processed through a filter which, for exact 00:11:56.740 --> 00:12:01.220 reconstruction, is an ideal low pass filter. 00:12:01.220 --> 00:12:04.820 And so we would multiply this spectrum by this one. 00:12:04.820 --> 00:12:08.980 And the result, after doing that, will generate a 00:12:08.980 --> 00:12:11.890 reconstructed spectrum which, in fact, is 00:12:11.890 --> 00:12:14.300 identical to the original. 00:12:14.300 --> 00:12:21.450 So the frequency domain picture is the same. 00:12:21.450 --> 00:12:24.740 And what we would expect then is that the time domain 00:12:24.740 --> 00:12:26.110 picture would be the same. 00:12:26.110 --> 00:12:29.380 Well, let's in fact look at the time domain. 00:12:29.380 --> 00:12:34.780 And in the time domain, what we have is an analysis more or 00:12:34.780 --> 00:12:38.340 less identical to what we had in continuous time. 00:12:38.340 --> 00:12:41.360 We of course have the same system. 00:12:41.360 --> 00:12:45.260 And in the time domain, we are multiplying 00:12:45.260 --> 00:12:47.550 by an impulse train. 00:12:47.550 --> 00:12:54.340 Consequently, the sample sequence is an impulse train 00:12:54.340 --> 00:12:58.820 whose values are samples of x of at integer multiples of 00:12:58.820 --> 00:13:05.310 capital N. For the reconstruction, this is now 00:13:05.310 --> 00:13:10.220 processed through an ideal low pass filter. 00:13:10.220 --> 00:13:11.610 And that implements a 00:13:11.610 --> 00:13:13.900 convolution in the time domain. 00:13:13.900 --> 00:13:17.700 And so the reconstructed signal is the convolution of 00:13:17.700 --> 00:13:22.720 the sample sequence and the filter impulse response. 00:13:22.720 --> 00:13:26.010 And expressed another way, namely writing out the 00:13:26.010 --> 00:13:29.250 convolution as a sum, we have this expression. 00:13:29.250 --> 00:13:34.010 And so it says then that the reconstruction is carried out 00:13:34.010 --> 00:13:41.430 by replacing the impulses here, these impulses, by 00:13:41.430 --> 00:13:44.460 versions of the filter impulse response. 00:13:47.300 --> 00:13:53.360 Well, if the filter is an ideal low pass filter, then 00:13:53.360 --> 00:13:58.020 that corresponds in the time domain to sine nx over sine x 00:13:58.020 --> 00:13:58.840 kind of function. 00:13:58.840 --> 00:14:03.570 And that is the interpolation in between the samples to do 00:14:03.570 --> 00:14:05.800 the reconstruction. 00:14:05.800 --> 00:14:10.870 Also, as is discussed somewhat in the text, we can consider 00:14:10.870 --> 00:14:14.680 other kinds of interpolation, for example discrete time zero 00:14:14.680 --> 00:14:18.640 order hold or discrete time first order hold, just as we 00:14:18.640 --> 00:14:20.420 had in continuous time. 00:14:20.420 --> 00:14:23.530 And the issues and analysis for the discrete times zero 00:14:23.530 --> 00:14:28.170 order hold and first order hold are very similar to what 00:14:28.170 --> 00:14:30.950 they were in continuous time-- the zero order hold just 00:14:30.950 --> 00:14:34.340 simply holding the value until the next sampling instant, and 00:14:34.340 --> 00:14:38.160 the first order hold carrying out linear interpolation in 00:14:38.160 --> 00:14:39.410 between the samples. 00:14:42.400 --> 00:14:49.610 Now in this sampling process, if we look again at the wave 00:14:49.610 --> 00:14:53.910 forms involved, or sequences involved, the process 00:14:53.910 --> 00:15:01.090 consisted of taking a sequence and extracting from it 00:15:01.090 --> 00:15:03.980 individual values. 00:15:03.980 --> 00:15:08.560 And in between those values, we have sequence 00:15:08.560 --> 00:15:10.930 values equal to 0. 00:15:10.930 --> 00:15:18.030 So what we're doing in this case is retaining the same 00:15:18.030 --> 00:15:21.950 number of sequence values and simply setting some number of 00:15:21.950 --> 00:15:24.980 them equal to 0. 00:15:24.980 --> 00:15:28.380 Well, let's say, for example, that we want 00:15:28.380 --> 00:15:29.930 to carry out sampling. 00:15:29.930 --> 00:15:32.200 And what we're talking about is a sequence. 00:15:32.200 --> 00:15:36.170 And let's say this sequence is stored in a computer memory. 00:15:36.170 --> 00:15:40.150 As you can imagine, the notion of sampling it and actually 00:15:40.150 --> 00:15:44.250 replacing some of the values by zero is somewhat 00:15:44.250 --> 00:15:44.790 inefficient. 00:15:44.790 --> 00:15:47.550 Namely, it doesn't make sense to think of storing in the 00:15:47.550 --> 00:15:50.900 memory a lot of zeros, when in fact those are zeros that we 00:15:50.900 --> 00:15:52.460 can always put back in. 00:15:52.460 --> 00:15:54.670 We know exactly what the values are. 00:15:54.670 --> 00:15:57.920 And if we know what the sampling rate was in discrete 00:15:57.920 --> 00:16:00.760 time, then we would know when and how to put 00:16:00.760 --> 00:16:03.070 the zeros back in. 00:16:03.070 --> 00:16:07.700 So actually, in discrete time sampling, what we've talked 00:16:07.700 --> 00:16:10.710 about so far is really only one part or 00:16:10.710 --> 00:16:12.610 one step in the process. 00:16:12.610 --> 00:16:16.190 Basically, the other step is to take those zeros and just 00:16:16.190 --> 00:16:18.820 throw them away because we could put them in any time we 00:16:18.820 --> 00:16:22.790 want to and really only retain, for example in our 00:16:22.790 --> 00:16:26.800 computer memory or list of sequence values or whatever, 00:16:26.800 --> 00:16:30.830 only retain the non-zero values. 00:16:30.830 --> 00:16:34.540 So that process and the resulting sequence that we end 00:16:34.540 --> 00:16:39.020 up with is associated with a concept called decimation. 00:16:39.020 --> 00:16:42.050 What I mean by decimation is very simple. 00:16:42.050 --> 00:16:47.200 What we're doing is, instead of working with this sequence, 00:16:47.200 --> 00:16:50.190 we're going to work with this sequence. 00:16:50.190 --> 00:16:55.440 Namely, we'll toss out the zeros in between here and 00:16:55.440 --> 00:17:00.270 collapse the sequence down only to the sequence values 00:17:00.270 --> 00:17:03.860 that are associated with the original x of n. 00:17:03.860 --> 00:17:07.839 Now, in talking about a decimated sequence, we could 00:17:07.839 --> 00:17:12.450 of course do that directly from this step down to here, 00:17:12.450 --> 00:17:15.890 although again in the analysis it will be somewhat more 00:17:15.890 --> 00:17:20.650 convenient to carry that out by thinking, at least 00:17:20.650 --> 00:17:23.839 analytically, in terms of a 2-step process-- 00:17:23.839 --> 00:17:27.079 one being a sampling process, then the other being a 00:17:27.079 --> 00:17:28.079 decimation. 00:17:28.079 --> 00:17:32.930 But basically, this is a decimated version of that. 00:17:32.930 --> 00:17:38.040 Now for the grammatical purists out there, the word 00:17:38.040 --> 00:17:40.620 decimation of course means taking every tenth one. 00:17:40.620 --> 00:17:43.910 The implication is not that we're always sampling with a 00:17:43.910 --> 00:17:45.220 period of 10. 00:17:45.220 --> 00:17:49.650 The idea of decimating is to pick out every nth sample and 00:17:49.650 --> 00:17:53.390 end up with a collapsed sequence. 00:17:53.390 --> 00:17:59.410 Let's now look at a little bit of the analysis and understand 00:17:59.410 --> 00:18:02.870 what the consequence is in the frequency domain. 00:18:02.870 --> 00:18:07.690 In particular what we want to develop is how the Fourier 00:18:07.690 --> 00:18:12.000 transform of the decimated sequence is related to the 00:18:12.000 --> 00:18:15.060 Fourier transform of the original sequence or the 00:18:15.060 --> 00:18:16.540 sample sequence. 00:18:16.540 --> 00:18:19.125 So let's look at this in the frequency domain. 00:18:21.730 --> 00:18:28.040 So what we have is a decimated sequence, which consists of 00:18:28.040 --> 00:18:32.250 pulling out every capital Nth value of x of n. 00:18:32.250 --> 00:18:36.080 And of course that's the same as we can either decimate x of 00:18:36.080 --> 00:18:40.710 n or we can decimate the sample signal. 00:18:40.710 --> 00:18:45.960 Now in going through this analysis, I'll kind of go 00:18:45.960 --> 00:18:49.520 through it quickly because again there's the issue of 00:18:49.520 --> 00:18:51.170 some slight mental gymnastics. 00:18:51.170 --> 00:18:55.400 And if you're anything like I am, it's usually best to kind 00:18:55.400 --> 00:18:58.390 of try to absorb that by yourself quietly, rather than 00:18:58.390 --> 00:19:00.410 having somebody throw it at you. 00:19:00.410 --> 00:19:03.640 Let me say, though, that the steps that I'm following here 00:19:03.640 --> 00:19:06.090 are slightly different than the steps that 00:19:06.090 --> 00:19:07.120 I use in the text. 00:19:07.120 --> 00:19:11.250 It's a slightly different way of going through the analysis. 00:19:11.250 --> 00:19:13.790 I guess you could say for one thing that if we've gone 00:19:13.790 --> 00:19:16.560 through it twice, and it comes out the same, well of course 00:19:16.560 --> 00:19:18.930 it has to be right. 00:19:18.930 --> 00:19:24.760 Well anyway, here we have then the relationship between the 00:19:24.760 --> 00:19:29.320 decimated sequence, the original sequence, and the 00:19:29.320 --> 00:19:30.990 sampled sequence. 00:19:30.990 --> 00:19:34.260 And we know of course that the Fourier transform of the 00:19:34.260 --> 00:19:38.560 sample sequence is just simply this summation. 00:19:38.560 --> 00:19:43.320 And now kind of the idea in the analysis is that we can 00:19:43.320 --> 00:19:51.020 collapse this summation by recognizing that this term is 00:19:51.020 --> 00:19:55.420 only non-zero at every nth value. 00:19:55.420 --> 00:19:58.200 And so if we do that, essentially making a 00:19:58.200 --> 00:20:02.260 substitution of variables with n equal to small m times 00:20:02.260 --> 00:20:06.260 capital N, we can turn this into a summation on m. 00:20:06.260 --> 00:20:09.160 And that's what I've done here. 00:20:09.160 --> 00:20:13.170 And we've just simply used the fact that we can collapse the 00:20:13.170 --> 00:20:18.340 sum because of the fact that all but every nth value is 00:20:18.340 --> 00:20:20.080 equal to zero. 00:20:20.080 --> 00:20:23.110 So this then is the Fourier transform all 00:20:23.110 --> 00:20:25.590 of the sampled signal. 00:20:25.590 --> 00:20:29.920 And now if we look at the Fourier transform of the 00:20:29.920 --> 00:20:34.800 decimated signal, that Fourier transform, of course, is this 00:20:34.800 --> 00:20:38.680 summation on the decimated sequence. 00:20:38.680 --> 00:20:41.480 Well, what we want to look at is the correspondence between 00:20:41.480 --> 00:20:43.790 this equation and the one above it. 00:20:43.790 --> 00:20:47.780 So we want to compare this equation to this one. 00:20:47.780 --> 00:20:51.460 And recognizing that this decimated sequence is just 00:20:51.460 --> 00:20:59.070 simply related to the sample sequence this way, these two 00:20:59.070 --> 00:21:02.960 become equal under a substitution of variables. 00:21:02.960 --> 00:21:09.090 In particular, notice that if we replace in this equation 00:21:09.090 --> 00:21:14.650 omega by omega times capital N, then these two equations 00:21:14.650 --> 00:21:16.740 become equal. 00:21:16.740 --> 00:21:20.180 So the consequence of that, then, what it all boils down 00:21:20.180 --> 00:21:27.625 to and says, is that the relationship between the 00:21:27.625 --> 00:21:30.360 Fourier transform of the decimated sequence and the 00:21:30.360 --> 00:21:34.680 Fourier transform of the sampled sequence is simply a 00:21:34.680 --> 00:21:38.540 frequency scaling corresponding to dividing the 00:21:38.540 --> 00:21:41.350 frequency axis by capital N. 00:21:41.350 --> 00:21:44.040 So that's essentially what happens. 00:21:44.040 --> 00:21:45.680 That's really all that's involved in 00:21:45.680 --> 00:21:47.420 the decimation process. 00:21:47.420 --> 00:21:50.180 And now, again, let's look at that pictorially 00:21:50.180 --> 00:21:52.550 and see what it means. 00:21:52.550 --> 00:21:55.780 So what we want to look at, now that we've looked in the 00:21:55.780 --> 00:22:00.540 time domain in this particular view graph, we now want to 00:22:00.540 --> 00:22:04.470 look in the frequency domain. 00:22:04.470 --> 00:22:11.620 And in the frequency domain, we have, again, the Fourier 00:22:11.620 --> 00:22:17.470 transform of the original sequence and we have the 00:22:17.470 --> 00:22:23.350 Fourier transform of the sampled sequence. 00:22:23.350 --> 00:22:28.490 And now the Fourier transform of the decimated sequence is 00:22:28.490 --> 00:22:35.030 simply this spectrum with a linear frequency scaling. 00:22:35.030 --> 00:22:38.450 And in particular, it simply corresponds to multiplying 00:22:38.450 --> 00:22:44.150 this frequency axis by capital N. And notice that this 00:22:44.150 --> 00:22:48.840 frequency now, 2 pi over capital N, that frequency ends 00:22:48.840 --> 00:22:55.800 up getting rescaled to a frequency of 2 pi. 00:22:55.800 --> 00:23:00.980 So in fact now, in the rescaling, it's that this 00:23:00.980 --> 00:23:07.200 point in the decimation gets rescaled to this point. 00:23:07.200 --> 00:23:10.480 And correspondingly, of course, this whole spectrum 00:23:10.480 --> 00:23:11.650 broadens out. 00:23:11.650 --> 00:23:14.770 Now we can also look at that in the context of 00:23:14.770 --> 00:23:16.010 the original spectrum. 00:23:16.010 --> 00:23:19.130 And you can see that the relationship between the 00:23:19.130 --> 00:23:22.020 original spectrum and the spectrum of the decimated 00:23:22.020 --> 00:23:27.730 signal corresponds to simply linearly scaling this. 00:23:27.730 --> 00:23:32.740 But it's important also to keep in mind that that 00:23:32.740 --> 00:23:37.330 analysis, that particular relationship, assumes that 00:23:37.330 --> 00:23:38.850 we've avoided aliasing. 00:23:38.850 --> 00:23:41.870 The relationship between the spectrum of the decimated 00:23:41.870 --> 00:23:46.050 signal and the spectrum of the sample signal is true whether 00:23:46.050 --> 00:23:47.700 or not we have aliasing. 00:23:47.700 --> 00:23:51.820 But being able to clearly associate it with just simply 00:23:51.820 --> 00:23:56.480 scaling of this spectrum of the original signal assumes 00:23:56.480 --> 00:24:00.380 that the spectrum of the original signal, the shape of 00:24:00.380 --> 00:24:03.950 it, is preserved when we generate the sample signal. 00:24:06.940 --> 00:24:11.870 Well, when might discrete time sampling, and for that matter, 00:24:11.870 --> 00:24:14.080 decimation, be used? 00:24:14.080 --> 00:24:18.450 Well, I indicated one context in which it might be useful at 00:24:18.450 --> 00:24:19.710 the beginning of this lecture. 00:24:19.710 --> 00:24:22.320 And let me now focus in on that a little more 00:24:22.320 --> 00:24:23.570 specifically. 00:24:25.730 --> 00:24:33.450 In particular, suppose that we have gone through a process in 00:24:33.450 --> 00:24:39.730 which the continuous time signal has been converted to a 00:24:39.730 --> 00:24:41.810 discrete time signal. 00:24:41.810 --> 00:24:44.200 And we then carry out some additional 00:24:44.200 --> 00:24:45.690 discrete time filtering. 00:24:45.690 --> 00:24:49.060 So we have a situation where we've gone through a 00:24:49.060 --> 00:24:52.710 continuous to discrete time conversion. 00:24:52.710 --> 00:24:56.470 And after that conversion, we carry out some 00:24:56.470 --> 00:24:58.035 discrete time filtering. 00:25:01.380 --> 00:25:03.940 And in particular, in going through this part of the 00:25:03.940 --> 00:25:09.990 process, we choose the sampling rate for going from 00:25:09.990 --> 00:25:13.710 the continuous time signal to the sequence so that we don't 00:25:13.710 --> 00:25:15.750 violate the sampling theorem. 00:25:15.750 --> 00:25:19.500 Well let's suppose, then, that this is the spectrum of the 00:25:19.500 --> 00:25:22.380 continuous time signal. 00:25:22.380 --> 00:25:27.170 Below it, we have the spectrum of the output of the 00:25:27.170 --> 00:25:31.200 continuous to discrete time conversion. 00:25:31.200 --> 00:25:36.090 And I've chosen the sampling frequency to be just high 00:25:36.090 --> 00:25:38.560 enough so that I avoid aliasing. 00:25:41.350 --> 00:25:47.220 Well that then is the lowest sampling frequency I can pick. 00:25:47.220 --> 00:25:51.860 But now, if we go through some additional low pass filtering, 00:25:51.860 --> 00:25:53.580 then let's see what happens. 00:25:53.580 --> 00:25:59.790 If I now low pass filter the sequence x of n, then in 00:25:59.790 --> 00:26:03.340 effect, I'm multiplying the sequence 00:26:03.340 --> 00:26:06.490 spectrum by this filter. 00:26:06.490 --> 00:26:10.470 And so the result of that, the product of the filter 00:26:10.470 --> 00:26:14.380 frequency response and the Fourier transform of x of n 00:26:14.380 --> 00:26:20.130 would have a shape somewhat like I indicate below. 00:26:20.130 --> 00:26:25.490 Now notice that in this spectrum, although in the 00:26:25.490 --> 00:26:31.490 input to the filter this entire band was filled up, in 00:26:31.490 --> 00:26:38.380 the output of the filter, there is a band that in fact 00:26:38.380 --> 00:26:40.190 has zero energy in it. 00:26:40.190 --> 00:26:45.310 So what I can consider doing is taking the output sequence 00:26:45.310 --> 00:26:49.590 from the filter and in fact resampling it, in other words 00:26:49.590 --> 00:26:53.520 sampling it, which would be more or less associated with a 00:26:53.520 --> 00:26:56.230 different sampling rate for the continuous 00:26:56.230 --> 00:26:58.560 time signals involved. 00:26:58.560 --> 00:27:02.830 So I could now go through a process which is commonly 00:27:02.830 --> 00:27:05.900 referred to as down sampling that is lowering 00:27:05.900 --> 00:27:07.270 the sampling rate. 00:27:07.270 --> 00:27:10.570 When we do that, of course, what's going to happen is that 00:27:10.570 --> 00:27:13.690 in fact this spectral energy will now fill 00:27:13.690 --> 00:27:16.110 out more of the band. 00:27:16.110 --> 00:27:21.950 And for example, if this was a third, then in fact if I down 00:27:21.950 --> 00:27:25.470 sampled by a factor of three, then I would fill up the 00:27:25.470 --> 00:27:27.400 entire band with this energy. 00:27:27.400 --> 00:27:29.870 But since I've done some additional low pass filtering, 00:27:29.870 --> 00:27:34.260 as I indicate here, there's no problem with aliasing. 00:27:34.260 --> 00:27:38.230 If I had, let's say, down sampled by a factor of three 00:27:38.230 --> 00:27:40.880 and I'm now taking that signal and converting it back to a 00:27:40.880 --> 00:27:45.430 continuous time signal, then of course the way I can do 00:27:45.430 --> 00:27:51.170 that is by simply running my output clock for the discrete 00:27:51.170 --> 00:27:52.760 to continuous time converter. 00:27:52.760 --> 00:27:56.600 I can run my output clock at a third the rate 00:27:56.600 --> 00:27:58.380 of the input clock. 00:27:58.380 --> 00:28:00.250 And that, in effect, takes care of the 00:28:00.250 --> 00:28:01.500 bookkeeping for me. 00:28:03.930 --> 00:28:11.000 So here we have now the notion of sampling a sequence, and 00:28:11.000 --> 00:28:14.250 very closely tied in with that, the notion of decimating 00:28:14.250 --> 00:28:19.000 a sequence, and related to both of those, the notion of 00:28:19.000 --> 00:28:22.630 down sampling, that is changing the sampling rates so 00:28:22.630 --> 00:28:25.880 that, if we were trying this in with continuous time 00:28:25.880 --> 00:28:30.170 signals, we've essentially changed our clock rate. 00:28:30.170 --> 00:28:35.030 And we might also want to, and it's important to, consider 00:28:35.030 --> 00:28:36.800 the opposite of that. 00:28:36.800 --> 00:28:41.140 So now a question is what's the opposite of decimation. 00:28:41.140 --> 00:28:45.080 Suppose that we had a sequence and we decimate it. 00:28:45.080 --> 00:28:49.470 Thinking about it as a 2-step process, that would correspond 00:28:49.470 --> 00:28:52.170 to first multiplying by an impulse train, where there are 00:28:52.170 --> 00:28:56.920 bunch of zeros in there, and then choosing, throwing away 00:28:56.920 --> 00:29:00.880 the zeros and keeping only the values that are non-zero, 00:29:00.880 --> 00:29:03.370 because the zeros we can always recreate. 00:29:03.370 --> 00:29:06.570 Well, in fact, the inverse process is very specifically a 00:29:06.570 --> 00:29:10.955 process of recreating the zeros and then doing the 00:29:10.955 --> 00:29:12.860 desampling. 00:29:12.860 --> 00:29:22.610 So in the opposite operation, what we would do is undo the 00:29:22.610 --> 00:29:24.890 decimation step. 00:29:24.890 --> 00:29:28.330 And that would consist of converting the decimated 00:29:28.330 --> 00:29:35.790 sequence back to an impulse train and then processing that 00:29:35.790 --> 00:29:42.470 impulse train by an ideal low pass filter to do the 00:29:42.470 --> 00:29:46.500 interpolation or reconstruction, filling in the 00:29:46.500 --> 00:29:51.610 values which, in this impulse train, are equal to zero. 00:29:51.610 --> 00:29:53.640 So we now have the two steps. 00:29:53.640 --> 00:29:56.490 We take the decimated sequence and we expand it 00:29:56.490 --> 00:29:58.950 out, putting in zeros. 00:29:58.950 --> 00:30:03.380 And then we desample that by processing it through a low 00:30:03.380 --> 00:30:05.150 pass filter. 00:30:05.150 --> 00:30:12.680 So just kind of looking at sequences again, what we have 00:30:12.680 --> 00:30:18.360 is an original sequence, the sequence x of n. 00:30:18.360 --> 00:30:23.510 And then the sample sequence is simply a sequence which 00:30:23.510 --> 00:30:26.100 alternates, in this particular case, those 00:30:26.100 --> 00:30:28.300 sequence values was zero. 00:30:28.300 --> 00:30:35.610 Here what we're assuming is that the sampling period is 2. 00:30:35.610 --> 00:30:40.160 And so every other value here is equal to zero. 00:30:40.160 --> 00:30:45.350 The decimated sequence then is this sequence, collapsed as I 00:30:45.350 --> 00:30:48.400 show in the sequence above. 00:30:48.400 --> 00:30:52.960 And so it's, in effect, time compressing the sample 00:30:52.960 --> 00:30:57.160 sequence or the original sequence so that we throw out 00:30:57.160 --> 00:30:59.890 the sequence values which were equal to zero 00:30:59.890 --> 00:31:02.500 in the sample sequence. 00:31:02.500 --> 00:31:06.270 Now in recovering the original sequence from the decimated 00:31:06.270 --> 00:31:09.400 sequence, we can think of a 2-step process. 00:31:09.400 --> 00:31:15.640 Namely, we spread this out alternating with zeros, and 00:31:15.640 --> 00:31:18.050 again, keeping in mind that this is drawn for the case 00:31:18.050 --> 00:31:19.950 where capital N is 2. 00:31:19.950 --> 00:31:23.630 And then finally, we interpolate between the 00:31:23.630 --> 00:31:28.700 non-zero values here by going through a low pass filter to 00:31:28.700 --> 00:31:32.240 reconstruct the original sequence. 00:31:32.240 --> 00:31:36.650 And that's what we show finally on the bottom curve. 00:31:36.650 --> 00:31:39.730 So that's what we would see in the time domain. 00:31:39.730 --> 00:31:43.950 Let's look at what we would see in the frequency domain. 00:31:43.950 --> 00:31:48.860 In the frequency domain, we have to begin with the 00:31:48.860 --> 00:31:52.020 sequence on the bottom, or the spectrum on the bottom, which 00:31:52.020 --> 00:31:56.400 would correspond to the original spectrum. 00:31:56.400 --> 00:31:59.710 Then, through the sampling process, that is periodically 00:31:59.710 --> 00:32:00.910 replicated. 00:32:00.910 --> 00:32:04.990 Again, this is drawn on the assumption that the sampling 00:32:04.990 --> 00:32:09.870 frequency is pi or the sampling period is equal to 2. 00:32:09.870 --> 00:32:12.480 And so this is now replicated. 00:32:12.480 --> 00:32:17.310 And then, in going from this to the spectrum of the 00:32:17.310 --> 00:32:23.990 decimated sequence, we would rescale the frequency axis so 00:32:23.990 --> 00:32:29.590 that the frequency pi now gets rescaled in the spectrum for 00:32:29.590 --> 00:32:34.090 the decimated sequence to a frequency which is 2 pi. 00:32:34.090 --> 00:32:36.900 And so this now is the spectrum of 00:32:36.900 --> 00:32:38.970 the decimated sequence. 00:32:38.970 --> 00:32:44.990 If we now want to reconvert to the original sequence we would 00:32:44.990 --> 00:32:49.210 first intersperse in the time domain with zeros, 00:32:49.210 --> 00:32:54.910 corresponding to compressing in the frequency domain. 00:32:54.910 --> 00:32:58.610 This would then be low pass filtered. 00:32:58.610 --> 00:33:02.680 And the low pass filtering would consist of throwing away 00:33:02.680 --> 00:33:06.600 this replication, accounting for a factor which is the 00:33:06.600 --> 00:33:10.630 factor capital N, and extracting the portion of the 00:33:10.630 --> 00:33:16.680 spectrum which is associated with the spectrum of the 00:33:16.680 --> 00:33:20.110 original signal which we began with. 00:33:20.110 --> 00:33:24.420 So once again, we have decimation and interpolation. 00:33:24.420 --> 00:33:30.150 And the decimation can be thought of as a time 00:33:30.150 --> 00:33:34.250 compression that corresponds to a frequency expansion then. 00:33:34.250 --> 00:33:37.890 And the interpolation process is then just the reverse. 00:33:41.080 --> 00:33:45.130 Now there are lots of situations in which decimation 00:33:45.130 --> 00:33:48.980 and interpolation and discrete time sampling are useful. 00:33:48.980 --> 00:33:52.390 And one context that I just want to quickly draw your 00:33:52.390 --> 00:33:56.480 attention to is the use of decimation and interpolation 00:33:56.480 --> 00:34:00.030 in what is commonly referred to as sampling rate 00:34:00.030 --> 00:34:01.480 conversion. 00:34:01.480 --> 00:34:05.710 What the basic issue and sampling rate conversion is is 00:34:05.710 --> 00:34:10.650 that, in some situations, and a very common one is digital 00:34:10.650 --> 00:34:15.330 audio, a continuous time signal is sampled. 00:34:15.330 --> 00:34:18.850 And those sampled values are stored or whatever. 00:34:18.850 --> 00:34:22.699 And kind of the notion is that, perhaps when that is 00:34:22.699 --> 00:34:26.530 played back, it's played back through a different system. 00:34:26.530 --> 00:34:31.350 And the different system has a different assumed sampling 00:34:31.350 --> 00:34:33.480 frequency or sampling period. 00:34:33.480 --> 00:34:36.630 So that's kind of the issue and the idea. 00:34:36.630 --> 00:34:41.090 We have, let's say, a continuous time signal which 00:34:41.090 --> 00:34:45.070 we've converted to a sequence through a sampling process 00:34:45.070 --> 00:34:48.830 using an assumed sampling period of T1. 00:34:48.830 --> 00:34:54.420 And these sequence values may then, for example, be put into 00:34:54.420 --> 00:34:55.710 digital storage. 00:34:55.710 --> 00:34:59.360 In the case of a digital audio system, it may, for example, 00:34:59.360 --> 00:35:01.720 go onto a digital record. 00:35:01.720 --> 00:35:06.660 And it might be the output of this that we want to recreate. 00:35:06.660 --> 00:35:11.350 Or we might in fact follow that with some additional 00:35:11.350 --> 00:35:14.120 processing, whatever that additional processing is. 00:35:14.120 --> 00:35:17.400 And I'll kind of put a question mark in there because 00:35:17.400 --> 00:35:21.030 we don't know exactly what that might be. 00:35:21.030 --> 00:35:25.900 And then, in any case, the result of that is going to be 00:35:25.900 --> 00:35:30.540 converted back to a continuous time signal. 00:35:30.540 --> 00:35:36.470 But it might be converted through a system that has a 00:35:36.470 --> 00:35:40.660 different assumed sampling period. 00:35:40.660 --> 00:35:44.950 And so a very common issue, and it comes up as I indicated 00:35:44.950 --> 00:35:50.190 particularly in digital audio, a very common issue is to be 00:35:50.190 --> 00:35:55.490 able to convert from one assumed sampling period, T1, 00:35:55.490 --> 00:35:58.660 our sampling frequency, to another 00:35:58.660 --> 00:36:00.780 assumed sampling period. 00:36:00.780 --> 00:36:02.120 Now how do we do that? 00:36:02.120 --> 00:36:07.460 Well in fact, we do that by using the ideas of decimation 00:36:07.460 --> 00:36:09.140 and interpolation. 00:36:09.140 --> 00:36:14.110 In particular, if we had, for example, a situation where we 00:36:14.110 --> 00:36:19.180 wanted to convert from a sampling period, T1, to a 00:36:19.180 --> 00:36:24.210 sampling period which was twice as long as that, then 00:36:24.210 --> 00:36:29.340 essentially, we're going to take the sequence and process 00:36:29.340 --> 00:36:32.710 it in a way that would, in effect, correspond to assuming 00:36:32.710 --> 00:36:36.190 that we had sampled at half the original frequency. 00:36:36.190 --> 00:36:37.440 Well how do we do that? 00:36:37.440 --> 00:36:40.210 The way we do it is we take the sequence we have and we 00:36:40.210 --> 00:36:43.460 just throw away every other value. 00:36:43.460 --> 00:36:47.370 So in that case, we would then, for this sampling rate 00:36:47.370 --> 00:36:51.560 conversion, down sample and decimate. 00:36:51.560 --> 00:36:54.770 Or actually, we might not go through this step formally. 00:36:54.770 --> 00:36:57.520 We might just simply decimate. 00:36:57.520 --> 00:37:02.590 Now we might have an alternative situation where in 00:37:02.590 --> 00:37:05.680 fact the new sampling period, or the sampling period of the 00:37:05.680 --> 00:37:09.230 output, is half the sampling period of the input, 00:37:09.230 --> 00:37:13.670 corresponding to an assumed sampling frequency, which is 00:37:13.670 --> 00:37:15.550 twice as high. 00:37:15.550 --> 00:37:20.010 And in that case, then., we would go through a process of 00:37:20.010 --> 00:37:21.000 interpolation. 00:37:21.000 --> 00:37:25.730 And in particular, we would up sample and interpolate by a 00:37:25.730 --> 00:37:27.780 factor of 2 to one. 00:37:27.780 --> 00:37:31.090 So in one case, we're simply throwing 00:37:31.090 --> 00:37:32.260 away every other value. 00:37:32.260 --> 00:37:34.080 In the other case, what we're going to do is take our 00:37:34.080 --> 00:37:37.630 sequence, put in zeros, put it through a low pass filter to 00:37:37.630 --> 00:37:39.470 interpolate. 00:37:39.470 --> 00:37:43.300 Now life would be simple if everything happened in simple 00:37:43.300 --> 00:37:45.070 integer amounts like that. 00:37:45.070 --> 00:37:49.690 A more common situation is that we may have an assumed 00:37:49.690 --> 00:37:55.250 output sampling period which is 3/2 of the 00:37:55.250 --> 00:37:57.250 input sampling period. 00:37:57.250 --> 00:38:00.530 And now the question is what are we going to do to convert 00:38:00.530 --> 00:38:04.520 from this sampling period to this sampling period. 00:38:04.520 --> 00:38:09.850 Well, in fact, the answer to that is to use a combination 00:38:09.850 --> 00:38:13.090 of down sampling and up sampling, or up sampling and 00:38:13.090 --> 00:38:17.470 down sampling, equivalently interpolation and decimation. 00:38:17.470 --> 00:38:21.900 And for this particular case, in fact, what we would do is 00:38:21.900 --> 00:38:28.630 to first take the data, up sample by a factor of 2, and 00:38:28.630 --> 00:38:33.370 then down sample the result of that by a factor of 3. 00:38:33.370 --> 00:38:37.290 And what that would give us is a sampling rate conversion, 00:38:37.290 --> 00:38:42.750 overall, of 3/2, or a sampling period conversion of 3/2. 00:38:42.750 --> 00:38:45.730 And more generally, what you could think of is how you 00:38:45.730 --> 00:38:51.240 might do this if, in general, the relationship between the 00:38:51.240 --> 00:38:54.090 input and output sampling periods was some rational 00:38:54.090 --> 00:38:56.190 number p/q. 00:38:56.190 --> 00:39:00.350 And so in fact, in many systems, in hardware systems 00:39:00.350 --> 00:39:04.370 related to digital audio, very often the sampling rate 00:39:04.370 --> 00:39:07.160 conversion, most typically the sampling rate conversion, is 00:39:07.160 --> 00:39:12.570 done through a process of up sampling or interpolating and 00:39:12.570 --> 00:39:15.280 then down sampling by some other amount. 00:39:18.470 --> 00:39:23.970 Now what we've seen, what we talked about in a set of 00:39:23.970 --> 00:39:29.830 lectures, is the concepts of sampling a signal. 00:39:29.830 --> 00:39:32.640 And what we've seen is that the signal can be represented 00:39:32.640 --> 00:39:35.140 by samples under certain conditions. 00:39:35.140 --> 00:39:38.240 And the sampling that we've been talking about is sampling 00:39:38.240 --> 00:39:39.020 in the time domain. 00:39:39.020 --> 00:39:41.920 And we've done that for continuous time and we've done 00:39:41.920 --> 00:39:45.150 it for discrete time. 00:39:45.150 --> 00:39:49.450 Now we know that there is some type of duality both 00:39:49.450 --> 00:39:52.600 continuous time and discrete time, some type of duality, 00:39:52.600 --> 00:39:55.240 between the time domain and frequency domain. 00:39:55.240 --> 00:40:01.220 And so, as you can imagine, we can also talk about sampling 00:40:01.220 --> 00:40:07.720 in the frequency domain and expect that, more or less, the 00:40:07.720 --> 00:40:11.640 kinds of properties and analysis will be similar to 00:40:11.640 --> 00:40:15.400 those related to sampling in the time domain. 00:40:15.400 --> 00:40:21.020 Well I want to talk just briefly about that and leave 00:40:21.020 --> 00:40:24.810 the more detailed discussion to the text and 00:40:24.810 --> 00:40:26.550 video course manual. 00:40:26.550 --> 00:40:29.980 But let me indicate, for example, one context in which 00:40:29.980 --> 00:40:32.370 frequency domain sampling is important. 00:40:32.370 --> 00:40:38.220 Suppose that you have a signal and what you'd like to measure 00:40:38.220 --> 00:40:41.770 is its Fourier transform, its spectrum. 00:40:41.770 --> 00:40:46.130 Well of course, if you want to measure it or calculate it, 00:40:46.130 --> 00:40:49.150 you can never do that exactly at every single frequency. 00:40:49.150 --> 00:40:50.630 There are too many frequencies, namely, an 00:40:50.630 --> 00:40:52.280 infinite number of them. 00:40:52.280 --> 00:40:55.670 And so, in fact, all that you can really calculate or 00:40:55.670 --> 00:41:00.920 measure is the Fourier transform at a set of sample 00:41:00.920 --> 00:41:02.170 frequencies. 00:41:02.170 --> 00:41:06.570 So essentially, if you are going to look at a spectrum, 00:41:06.570 --> 00:41:09.950 continuous time or discrete time, you can only really look 00:41:09.950 --> 00:41:11.060 at samples. 00:41:11.060 --> 00:41:15.210 And a reasonable question to ask, then, is when does a set 00:41:15.210 --> 00:41:19.500 of samples in fact tell you everything that there is to 00:41:19.500 --> 00:41:23.450 know about the Fourier transform. 00:41:23.450 --> 00:41:27.100 That, and the answer to that, is very closely related to the 00:41:27.100 --> 00:41:31.140 concept of frequency domain sampling. 00:41:31.140 --> 00:41:33.770 Well, frequency domain sampling, just to kind of 00:41:33.770 --> 00:41:37.900 introduce the topic, corresponds and can be 00:41:37.900 --> 00:41:44.300 analyzed in terms doing modulation in the frequency 00:41:44.300 --> 00:41:48.280 domain, very much like the modulation that we carried out 00:41:48.280 --> 00:41:51.160 in the time domain for time domain sampling. 00:41:51.160 --> 00:41:57.440 And so we would multiply the Fourier transform of the 00:41:57.440 --> 00:42:02.200 signal whose spectrum is to be sampled by an 00:42:02.200 --> 00:42:05.060 impulse train in frequency. 00:42:05.060 --> 00:42:10.520 And so shown below is what might be a representative 00:42:10.520 --> 00:42:13.660 spectrum for the input signal. 00:42:13.660 --> 00:42:19.550 And the spectrum, then for the signal associated with the 00:42:19.550 --> 00:42:23.760 frequency domain sampling, consists of multiplying the 00:42:23.760 --> 00:42:27.670 frequency domain by this impulse train. 00:42:27.670 --> 00:42:32.950 Or correspondingly, the Fourier transform of the 00:42:32.950 --> 00:42:39.430 resulting signal is an impulse train in frequency with an 00:42:39.430 --> 00:42:42.670 envelope which is the original spectrum 00:42:42.670 --> 00:42:45.420 that we were sampling. 00:42:45.420 --> 00:42:48.360 Well, this of course is what we would do in 00:42:48.360 --> 00:42:49.620 the frequency domain. 00:42:49.620 --> 00:42:52.560 It's modulation by an impulse train. 00:42:52.560 --> 00:42:55.690 What does this mean in the time domain? 00:42:55.690 --> 00:42:57.070 Well, let's see. 00:42:57.070 --> 00:42:59.690 Multiplication in the time domain is convolution in the 00:42:59.690 --> 00:43:01.220 frequency domain. 00:43:01.220 --> 00:43:04.210 Convolution in the frequency domain is multiplication-- 00:43:04.210 --> 00:43:04.880 I'm sorry. 00:43:04.880 --> 00:43:07.700 Multiplication in the frequency domain, then, is 00:43:07.700 --> 00:43:09.660 convolution in the time domain. 00:43:09.660 --> 00:43:12.490 And in fact, the process in the time domain is a 00:43:12.490 --> 00:43:14.120 convolution process. 00:43:14.120 --> 00:43:22.020 Namely, the time domain signal is replicated at integer 00:43:22.020 --> 00:43:26.840 amounts of a particular time associated with the spacing in 00:43:26.840 --> 00:43:29.050 frequency under which we're doing the 00:43:29.050 --> 00:43:31.340 frequency domain sampling. 00:43:31.340 --> 00:43:39.590 So in fact, if we look at this in the time domain, the 00:43:39.590 --> 00:43:46.170 resulting picture corresponds to an original signal whose 00:43:46.170 --> 00:43:50.150 spectrum or Fourier transform we've sampled. 00:43:50.150 --> 00:43:53.980 And a consequence of the sampling is that the 00:43:53.980 --> 00:43:58.220 associated time domain signal is just like the original 00:43:58.220 --> 00:44:02.490 signal, but periodically replicated, in time now, not 00:44:02.490 --> 00:44:07.020 frequency, but in time, at integer multiples of 2 pi 00:44:07.020 --> 00:44:12.720 divided by the spectral sampling interval omega 0. 00:44:12.720 --> 00:44:18.050 And so this then is the time function associated with the 00:44:18.050 --> 00:44:20.630 sample frequency function. 00:44:20.630 --> 00:44:24.640 Now, that's not surprising because what we've done is 00:44:24.640 --> 00:44:27.750 generated an impulse train and frequency 00:44:27.750 --> 00:44:29.660 with a certain envelope. 00:44:29.660 --> 00:44:32.460 We know that an impulse train in frequency is the Fourier 00:44:32.460 --> 00:44:36.340 transform of a periodic time function. 00:44:36.340 --> 00:44:39.180 And so in fact, we have a periodic time function. 00:44:39.180 --> 00:44:43.240 We also know that the envelope of those impulses-- 00:44:43.240 --> 00:44:46.500 we know this from way back when we talked about Fourier 00:44:46.500 --> 00:44:47.670 transforms-- 00:44:47.670 --> 00:44:49.560 the envelope, in fact, is the Fourier 00:44:49.560 --> 00:44:51.060 transform of one period. 00:44:51.060 --> 00:44:54.400 And so all of this, of course, fits together as it should in 00:44:54.400 --> 00:44:56.620 a consistent way. 00:44:56.620 --> 00:45:02.340 Now given that we have this periodic time function whose 00:45:02.340 --> 00:45:05.460 Fourier transform is the samples in the frequency 00:45:05.460 --> 00:45:10.250 domain, how do we get back the original time function? 00:45:10.250 --> 00:45:17.070 Well, with time domain sampling, what we did was to 00:45:17.070 --> 00:45:21.370 multiply in the frequency domain by a gate, or window, 00:45:21.370 --> 00:45:24.060 to extract that part of the spectrum. 00:45:24.060 --> 00:45:29.080 What we do here is exactly the same thing, namely multiply in 00:45:29.080 --> 00:45:34.650 the time domain by a time window which extracts just one 00:45:34.650 --> 00:45:38.280 period of this periodic signal, which would then give 00:45:38.280 --> 00:45:42.930 us back the original signal that we started with. 00:45:42.930 --> 00:45:48.550 Now also let's keep in mind, going back to this time 00:45:48.550 --> 00:45:52.400 function and the relationship between them, then again, 00:45:52.400 --> 00:45:56.090 there is the potential, if this time function is too long 00:45:56.090 --> 00:46:00.780 in relation to 2 pi divided by omega 0, there's the potential 00:46:00.780 --> 00:46:02.420 for these to overlap. 00:46:02.420 --> 00:46:06.880 And so what this means is that, in fact, what we can end 00:46:06.880 --> 00:46:11.320 up with, if the sample spacing and the frequency is not small 00:46:11.320 --> 00:46:15.100 enough, what we can end up with is an overlap in the 00:46:15.100 --> 00:46:17.530 replication in the time domain. 00:46:17.530 --> 00:46:21.220 And what that corresponds to and what it's called is, in 00:46:21.220 --> 00:46:23.200 fact, time aliasing. 00:46:23.200 --> 00:46:27.590 So we can have time aliasing with frequency domain sampling 00:46:27.590 --> 00:46:30.440 just as we can have frequency aliasing 00:46:30.440 --> 00:46:33.440 with time domain sampling. 00:46:33.440 --> 00:46:37.730 Finally, let me just indicate very quickly that, although 00:46:37.730 --> 00:46:41.780 we're not going through this in any detail, the same basic 00:46:41.780 --> 00:46:44.820 idea applies in discrete time. 00:46:44.820 --> 00:46:50.280 Namely, if we have a discrete time signal and if the 00:46:50.280 --> 00:46:55.870 discrete time signal is a finite length, if we sample 00:46:55.870 --> 00:47:01.260 its Fourier transform, the time function associated with 00:47:01.260 --> 00:47:05.980 those samples is a periodic replication. 00:47:05.980 --> 00:47:11.270 And we can now extract, from this periodic signal, the 00:47:11.270 --> 00:47:16.080 original signal by multiplying by an appropriate time window, 00:47:16.080 --> 00:47:19.660 the product of that giving us the reconstructed time 00:47:19.660 --> 00:47:21.630 function as I indicate below. 00:47:24.730 --> 00:47:29.100 So we've now seen a little bit of the notion of frequency 00:47:29.100 --> 00:47:31.690 domain sampling, as well as time domain sampling. 00:47:31.690 --> 00:47:34.440 And let me stress that, although I haven't gone into 00:47:34.440 --> 00:47:38.800 this in a lot of detail, it's important. 00:47:38.800 --> 00:47:40.510 It's used very often. 00:47:40.510 --> 00:47:43.020 It's naturally important to understand it. 00:47:43.020 --> 00:47:46.550 But, in fact, there is so much duality between the time 00:47:46.550 --> 00:47:49.460 domain and frequency domain, that a thorough understanding 00:47:49.460 --> 00:47:53.270 of time domain sampling just naturally leads to a thorough 00:47:53.270 --> 00:47:55.520 understanding of frequency domain sampling. 00:47:58.810 --> 00:48:01.680 Now we've talked a lot about sampling. 00:48:01.680 --> 00:48:06.840 And this now concludes our discussion of sampling. 00:48:06.840 --> 00:48:10.170 I've stressed many times in the lectures associated with 00:48:10.170 --> 00:48:16.020 this that sampling is a very important topic in the context 00:48:16.020 --> 00:48:19.440 of our whole discussion, in part because it forms such an 00:48:19.440 --> 00:48:22.520 important bridge between continuous time and discrete 00:48:22.520 --> 00:48:23.890 time ideas. 00:48:23.890 --> 00:48:27.150 And your picture now should kind of be a global one that 00:48:27.150 --> 00:48:31.060 sees how continuous time and discrete time fit together, 00:48:31.060 --> 00:48:33.395 not just analytically, but also practically. 00:48:36.510 --> 00:48:42.560 Beginning in the next lecture, what I will introduce is the 00:48:42.560 --> 00:48:46.740 Laplace transform and, beyond that, the Z transform. 00:48:46.740 --> 00:48:51.460 And what those will correspond to are generalizations of the 00:48:51.460 --> 00:48:52.930 Fourier transform. 00:48:52.930 --> 00:48:55.730 So we now want to turn our attention back to some 00:48:55.730 --> 00:48:59.270 analytical tools, in particular developing some 00:48:59.270 --> 00:49:03.830 generalizations of the Fourier transform in both continuous 00:49:03.830 --> 00:49:05.780 time and discrete time. 00:49:05.780 --> 00:49:11.140 And what we'll see is that those generalizations provide 00:49:11.140 --> 00:49:16.510 us with considerably enhanced flexibility in dealing with 00:49:16.510 --> 00:49:21.210 and analyzing both signals and linear time invariant systems. 00:49:21.210 --> 00:49:22.460 Thank you.