WEBVTT 00:00:00.000 --> 00:00:42.327 [MUSIC PLAYING] 00:00:42.327 --> 00:00:45.850 PROFESSOR: I'm Al Oppenheim, and I'd like to welcome you to 00:00:45.850 --> 00:00:50.260 this videotape course on signals and systems. 00:00:50.260 --> 00:00:54.810 Signals, at least as an informal definition, are 00:00:54.810 --> 00:00:58.820 functions of one or more independent variables that 00:00:58.820 --> 00:01:02.460 typically carry some type of information. 00:01:02.460 --> 00:01:05.950 Systems, in our setting, would typically be 00:01:05.950 --> 00:01:08.860 used to process signals. 00:01:08.860 --> 00:01:12.740 One very common example of a signal might be, let's say, a 00:01:12.740 --> 00:01:14.130 speech signal. 00:01:14.130 --> 00:01:17.670 And you might think of the air pressure as a function of 00:01:17.670 --> 00:01:20.920 time, or perhaps the electrical signal after it 00:01:20.920 --> 00:01:23.580 goes through the microphone transducer as a function of 00:01:23.580 --> 00:01:27.150 time, as representing the speech signal. 00:01:27.150 --> 00:01:31.120 And we might see a typical speech signal looking 00:01:31.120 --> 00:01:33.990 something like I've indicated here. 00:01:33.990 --> 00:01:38.310 It's a function of time, in this particular case. 00:01:38.310 --> 00:01:42.970 And the independent variable, being time, is, in fact, 00:01:42.970 --> 00:01:44.290 continuous. 00:01:44.290 --> 00:01:47.780 And so a signal like this, we will typically be referring to 00:01:47.780 --> 00:01:51.570 as a continuous time signal. 00:01:51.570 --> 00:01:55.180 Now, it also, for this particular example, is a 00:01:55.180 --> 00:01:58.470 function of one independent variable. 00:01:58.470 --> 00:02:06.660 And that will be referred to as a one-dimensional signal 00:02:06.660 --> 00:02:10.470 corresponding to the fact that there's only one independent 00:02:10.470 --> 00:02:13.890 variable instead of several independent variables. 00:02:13.890 --> 00:02:18.696 So the speech signal is an example of a continuous time, 00:02:18.696 --> 00:02:19.946 one-dimensional signal. 00:02:21.520 --> 00:02:25.990 Now, signals can, of course, be multi-dimensional. 00:02:25.990 --> 00:02:28.740 And they may not have, as their independent variables, 00:02:28.740 --> 00:02:30.130 time variables. 00:02:30.130 --> 00:02:34.000 One very common example are the examples 00:02:34.000 --> 00:02:36.080 represented by images. 00:02:36.080 --> 00:02:41.340 Images, as signals, we might think of as representing 00:02:41.340 --> 00:02:46.020 brightness, as it varies in a horizontal 00:02:46.020 --> 00:02:48.500 and vertical direction. 00:02:48.500 --> 00:02:51.720 And so the brightness as a function of these two spatial 00:02:51.720 --> 00:02:55.990 variables is then a two-dimensional signal. 00:03:00.270 --> 00:03:07.460 And the independent variables would typically be continuous, 00:03:07.460 --> 00:03:09.730 but of course they're not time variables. 00:03:09.730 --> 00:03:14.690 And incidentally, it's worth just commenting that very 00:03:14.690 --> 00:03:18.950 often, simply for convenience, we'll have a tendency to refer 00:03:18.950 --> 00:03:22.230 to the independent variables when we talk about signals as 00:03:22.230 --> 00:03:27.170 time variables, whether or not they really do represent time. 00:03:27.170 --> 00:03:31.690 Well, let me illustrate one example of an image. 00:03:31.690 --> 00:03:37.790 And this is a picture of J. B. J. Fourier, who, perhaps, more 00:03:37.790 --> 00:03:42.240 than anyone else, is responsible for the elegance 00:03:42.240 --> 00:03:45.260 and beauty of a lot of the concepts that we'll be talking 00:03:45.260 --> 00:03:47.780 about throughout this course. 00:03:47.780 --> 00:03:51.470 And when you look at this, in addition to seeing Fourier 00:03:51.470 --> 00:03:54.720 himself, you should recognize that what you're looking at is 00:03:54.720 --> 00:03:58.530 basically a signal which is brightness as a function of 00:03:58.530 --> 00:04:03.400 the horizontal and vertical spatial variables. 00:04:03.400 --> 00:04:08.960 As another example of an image as a signal, let's look at an 00:04:08.960 --> 00:04:10.610 aerial photograph. 00:04:10.610 --> 00:04:16.040 This is an aerial photograph taken over a set of roads, 00:04:16.040 --> 00:04:19.019 which you can more or less recognize in the picture. 00:04:19.019 --> 00:04:23.600 And one of the difficulties with this signal is that the 00:04:23.600 --> 00:04:27.430 road system is somewhat obscured by cloud cover. 00:04:27.430 --> 00:04:31.990 And what I'll want to show later as an example of what a 00:04:31.990 --> 00:04:35.340 system might do to such a signal, in terms of processing 00:04:35.340 --> 00:04:41.540 it, is an attempt to at least compensate somewhat for the 00:04:41.540 --> 00:04:44.900 cloud cover that's represented in the photograph. 00:04:44.900 --> 00:04:48.140 Although in terms of the detailed analysis that we go 00:04:48.140 --> 00:04:52.550 through during the course, our focus of attention is pretty 00:04:52.550 --> 00:04:54.810 much restricted to one-dimensional signals. 00:04:54.810 --> 00:04:58.940 In fact, we will be using two-dimensional signals, more 00:04:58.940 --> 00:05:03.390 specifically images, very often to illustrate a variety 00:05:03.390 --> 00:05:06.090 of concepts. 00:05:06.090 --> 00:05:10.400 Now, speech and images are examples of what we've 00:05:10.400 --> 00:05:15.250 referred to as continuous-time signals in that they are 00:05:15.250 --> 00:05:18.990 functions of continuous variables. 00:05:18.990 --> 00:05:22.380 An equally important class of signals that we will be 00:05:22.380 --> 00:05:27.170 concentrating on in the course are signals that are 00:05:27.170 --> 00:05:33.750 discrete-time signals, where by discrete-time, what we mean 00:05:33.750 --> 00:05:38.770 is that the signal is a function of an integer 00:05:38.770 --> 00:05:44.570 variable, and so specifically only takes on values at 00:05:44.570 --> 00:05:48.150 integer values of the argument. 00:05:48.150 --> 00:05:52.820 So here is a graphical illustration of a 00:05:52.820 --> 00:05:54.540 discrete-time signal. 00:05:54.540 --> 00:05:58.290 And discrete-time signals arise in a variety of ways. 00:05:58.290 --> 00:06:04.040 One very common example that is seen fairly often is 00:06:04.040 --> 00:06:08.440 discrete-time signals in the context of economic time 00:06:08.440 --> 00:06:13.250 series, for example, stock market analysis. 00:06:13.250 --> 00:06:17.810 So what I show here is one very commonly occurring 00:06:17.810 --> 00:06:21.220 example of a discrete-time signal. 00:06:21.220 --> 00:06:25.330 It represents the weekly stock market index. 00:06:25.330 --> 00:06:30.390 The independent variable in this case is the week number. 00:06:30.390 --> 00:06:34.780 And we see what the stock market is doing over this 00:06:34.780 --> 00:06:37.830 particular period as a function of the 00:06:37.830 --> 00:06:38.830 number of the week. 00:06:38.830 --> 00:06:41.550 And, of course, along the vertical axis 00:06:41.550 --> 00:06:44.800 is the weekly index. 00:06:44.800 --> 00:06:47.660 Incidentally, this particular period was 00:06:47.660 --> 00:06:49.925 not chosen at random. 00:06:49.925 --> 00:06:54.260 It In fact captures a very interesting aspect of stock 00:06:54.260 --> 00:07:00.640 market history, namely the stock market crash in 1929, 00:07:00.640 --> 00:07:05.180 which, in fact, is represented by the behavior of this 00:07:05.180 --> 00:07:08.720 discrete-time signal, or sequence, in 00:07:08.720 --> 00:07:09.810 this particular area. 00:07:09.810 --> 00:07:12.960 So this dramatic dip, in fact, is the stock 00:07:12.960 --> 00:07:16.610 market crash of 1929. 00:07:16.610 --> 00:07:21.340 Well, the Dow Jones weekly average is an example of a 00:07:21.340 --> 00:07:24.520 one-dimensional discrete-time signal. 00:07:24.520 --> 00:07:29.150 And just as with continuous time, we had not just 00:07:29.150 --> 00:07:31.810 one-dimensional but multi-dimensional signals, 00:07:31.810 --> 00:07:35.950 likewise we have multi-dimensional signals in 00:07:35.950 --> 00:07:41.870 the discrete-time case where, in that case, then, the 00:07:41.870 --> 00:07:43.900 discrete-time signal that we're talking about, or 00:07:43.900 --> 00:07:48.160 sequence, is a function of two integer variables. 00:07:48.160 --> 00:07:52.230 And as one example, this might, let's say, represent a 00:07:52.230 --> 00:07:58.330 spatial antenna array where this is array number in, let's 00:07:58.330 --> 00:08:01.730 say, the horizontal direction, and this is array number in 00:08:01.730 --> 00:08:03.550 the vertical direction. 00:08:03.550 --> 00:08:06.070 Both classes of signals, continuous-time and 00:08:06.070 --> 00:08:10.100 discrete-time, as I've indicated, are very important. 00:08:10.100 --> 00:08:14.300 And it should be emphasized that the importance of 00:08:14.300 --> 00:08:17.680 discrete-time signals and associated processing 00:08:17.680 --> 00:08:22.490 continues to grow in large part because of the current 00:08:22.490 --> 00:08:27.400 and emerging technologies that permit, basically, the 00:08:27.400 --> 00:08:32.900 processing of continuous-time signals by first converting 00:08:32.900 --> 00:08:36.309 them to discrete-time signals and processing them with 00:08:36.309 --> 00:08:37.990 discrete-time systems. 00:08:37.990 --> 00:08:42.429 And that, in fact, is a topic that we will discuss in a fair 00:08:42.429 --> 00:08:44.710 amount of detail later on in the course. 00:08:47.550 --> 00:08:51.620 Let's now our attention to systems. 00:08:51.620 --> 00:08:58.190 And as I indicated, a system basically processes signals. 00:08:58.190 --> 00:09:02.830 And they have, of course, inputs and outputs. 00:09:02.830 --> 00:09:06.670 And depending on whether we're talking about continuous time 00:09:06.670 --> 00:09:11.340 or discrete time, the system may be a continuous-time 00:09:11.340 --> 00:09:14.130 system or a discrete-time system. 00:09:14.130 --> 00:09:18.030 So in the continuous-time case, I indicate here an input 00:09:18.030 --> 00:09:23.050 x(t) and an output y(t) If we were talking about a 00:09:23.050 --> 00:09:27.790 discrete-time system, I would represent the input in terms 00:09:27.790 --> 00:09:32.170 of a discrete-time variable, and, of course, the output in 00:09:32.170 --> 00:09:36.050 terms of a discrete-time variable also. 00:09:36.050 --> 00:09:40.700 Now, in very general terms, systems are hard to deal with 00:09:40.700 --> 00:09:45.530 because they are defined very broadly and very generally. 00:09:45.530 --> 00:09:49.770 And in dealing with systems and analyzing them, what we 00:09:49.770 --> 00:09:55.060 will do is attempt to exploit some very specific, and as 00:09:55.060 --> 00:09:59.410 we'll see, very useful system properties. 00:09:59.410 --> 00:10:04.160 To indicate what I mean and how things might be split up, 00:10:04.160 --> 00:10:07.660 we could talk about systems, and will talk about systems, 00:10:07.660 --> 00:10:10.300 that are linear. 00:10:10.300 --> 00:10:14.090 And we could divide systems, basically, into systems that 00:10:14.090 --> 00:10:20.510 are either linear or nonlinear, and we will, and 00:10:20.510 --> 00:10:24.590 also divide systems into systems that are what we'll 00:10:24.590 --> 00:10:29.420 refer to as time-invariant or time-varying systems. 00:10:29.420 --> 00:10:33.320 And these aren't terms that we've defined yet, of course, 00:10:33.320 --> 00:10:37.160 but we will be defining in the course very shortly. 00:10:37.160 --> 00:10:40.640 And while, in some sense, this division represents all 00:10:40.640 --> 00:10:45.260 systems, and this does, too, the focus of the course is 00:10:45.260 --> 00:10:49.040 really going to be principally on linear, 00:10:49.040 --> 00:10:50.860 time-invariant systems. 00:10:50.860 --> 00:10:55.560 So it's basically these systems that we will be 00:10:55.560 --> 00:10:57.120 focusing on. 00:10:57.120 --> 00:11:00.692 And we'll be referring to those systems as linear, 00:11:00.692 --> 00:11:04.880 time-invariant systems. 00:11:04.880 --> 00:11:08.100 Well, as a brief glimpse at some of the kinds of things 00:11:08.100 --> 00:11:14.140 that systems can do, let me illustrate, first in a 00:11:14.140 --> 00:11:17.570 one-dimensional continuous-time context, and 00:11:17.570 --> 00:11:21.840 then later with a discrete-time example, one 00:11:21.840 --> 00:11:27.020 example of some processing of signals with 00:11:27.020 --> 00:11:28.540 an appropriate system. 00:11:28.540 --> 00:11:32.630 The particular example that I want to illustrate relates to 00:11:32.630 --> 00:11:35.740 the restoration of old recordings. 00:11:35.740 --> 00:11:38.960 And this is some work that was done by Professor Thomas 00:11:38.960 --> 00:11:43.190 Stockham, who is at the University of Utah, and work 00:11:43.190 --> 00:11:46.800 that he had done a number of years ago relating to the fact 00:11:46.800 --> 00:11:53.380 that in old recordings, for example in Caruso recordings, 00:11:53.380 --> 00:11:57.380 the recording was done through a mechanical horn, and the 00:11:57.380 --> 00:11:59.990 characteristics of the horn tended to 00:11:59.990 --> 00:12:01.860 vary from day to day. 00:12:01.860 --> 00:12:04.900 And because of the characteristics of the horn, 00:12:04.900 --> 00:12:08.190 the recording tended to have a muffled quality, something 00:12:08.190 --> 00:12:12.360 like this, sort of the sense that you would get if you were 00:12:12.360 --> 00:12:15.120 speaking through a megaphone. 00:12:15.120 --> 00:12:20.360 What Professor Stockham did was develop a system to 00:12:20.360 --> 00:12:25.260 process these old recordings in such a way that a lot of 00:12:25.260 --> 00:12:28.650 the characteristics and distortion due to that 00:12:28.650 --> 00:12:30.710 recording system was removed. 00:12:30.710 --> 00:12:34.870 So I'd like to illustrate that as one example of some signal 00:12:34.870 --> 00:12:38.790 processing with an appropriate continuous-time system. 00:12:38.790 --> 00:12:43.400 And what you'll hear is a two-track recording. 00:12:43.400 --> 00:12:47.830 On the first track is the original, unrestored Caruso 00:12:47.830 --> 00:12:52.130 recording, and on the second track is the result of the 00:12:52.130 --> 00:12:53.280 restoration. 00:12:53.280 --> 00:12:56.910 And so as I switch back and forth from channel one to 00:12:56.910 --> 00:13:00.890 channel two, we'll be switching from the original to 00:13:00.890 --> 00:13:02.280 the restored. 00:13:02.280 --> 00:13:06.330 We'll begin the tape by playing the original. 00:13:06.330 --> 00:13:10.170 And then, as it proceeds, we'll switch. 00:13:10.170 --> 00:13:12.350 So we'll begin on channel one. 00:13:19.090 --> 00:13:19.300 [MUSIC PLAYING, MUFFLED] 00:13:19.300 --> 00:13:21.640 That's the original recording. 00:13:21.640 --> 00:13:22.600 And switch now to the processed. 00:13:22.600 --> 00:13:31.970 [MUSIC PLAYING, CLEARER] 00:13:31.970 --> 00:13:37.850 Now let's switch back, back to the original. 00:13:46.770 --> 00:13:48.020 Back to the restoration. 00:13:54.670 --> 00:13:58.030 And once again, back to the original. 00:14:04.880 --> 00:14:09.500 And presumably and hopefully, what you heard was that in the 00:14:09.500 --> 00:14:13.830 restoration, in fact, a lot of the muffled characteristics of 00:14:13.830 --> 00:14:19.470 the original recording were compensated for or removed. 00:14:19.470 --> 00:14:22.480 Now one of the interesting things that happened, in fact, 00:14:22.480 --> 00:14:26.620 in the work that Professor Stockham did is that in the 00:14:26.620 --> 00:14:28.060 process of the restoration-- 00:14:28.060 --> 00:14:30.160 and perhaps you heard this-- 00:14:30.160 --> 00:14:33.020 in the process of the restoration, in fact, some of 00:14:33.020 --> 00:14:37.580 the background noise on the recording was emphasized. 00:14:37.580 --> 00:14:43.310 And so he processed the signal further in an attempt to 00:14:43.310 --> 00:14:45.350 remove that background noise. 00:14:45.350 --> 00:14:48.250 And with that particular processing, the processing was 00:14:48.250 --> 00:14:51.060 very highly nonlinear. 00:14:51.060 --> 00:14:54.980 A very interesting thing happened, which was that not 00:14:54.980 --> 00:14:59.060 only in that processing was the background noise removed, 00:14:59.060 --> 00:15:02.180 but somewhat surprisingly, also the 00:15:02.180 --> 00:15:03.630 orchestra was removed. 00:15:03.630 --> 00:15:08.240 And let me just play that now as an example of some very 00:15:08.240 --> 00:15:11.930 sophisticated processing with a nonlinear system. 00:15:11.930 --> 00:15:17.515 What you'll hear on channel one is the restoration as we 00:15:17.515 --> 00:15:19.320 had just played it. 00:15:19.320 --> 00:15:23.610 When I switch to channel two, it will be after the 00:15:23.610 --> 00:15:27.986 processing with an attempt to remove the orchestra and the 00:15:27.986 --> 00:15:30.060 background noise. 00:15:30.060 --> 00:15:31.310 Channel one now. 00:15:38.696 --> 00:15:41.691 And now the noise and orchestra removed. 00:15:44.390 --> 00:15:46.293 Back to channel one. 00:15:58.460 --> 00:16:02.085 And finally, once again, with the orchestra removed. 00:16:12.220 --> 00:16:15.950 So that's an example of processing of a 00:16:15.950 --> 00:16:19.280 continuous-time signal with a corresponding 00:16:19.280 --> 00:16:21.160 continuous-time system. 00:16:21.160 --> 00:16:25.800 Now I'd like to illustrate an example of some processing on 00:16:25.800 --> 00:16:27.600 a discrete-time signal. 00:16:27.600 --> 00:16:30.730 And I'd like to do that in the context of the example that I 00:16:30.730 --> 00:16:33.260 showed before of a discrete-time signal, which 00:16:33.260 --> 00:16:38.660 was the Dow Jones Industrial weekly stock market index. 00:16:38.660 --> 00:16:44.880 I had shown it before, as I've shown it here again, over a 00:16:44.880 --> 00:16:49.550 period of slightly more than a year, where this is the number 00:16:49.550 --> 00:16:53.630 of weeks and this is the weekly index. 00:16:53.630 --> 00:16:58.640 And to illustrate some of the processing, what I'd like to 00:16:58.640 --> 00:17:03.020 do is show the stock market index, the weekly index, over 00:17:03.020 --> 00:17:07.750 a much longer time period, in particular, the weekly index 00:17:07.750 --> 00:17:10.119 over a 10 year period. 00:17:10.119 --> 00:17:12.290 And that's what I show here. 00:17:12.290 --> 00:17:19.329 So what this covers is roughly 1927 to 1937. 00:17:19.329 --> 00:17:22.950 And in this case, although this is still a discrete-time 00:17:22.950 --> 00:17:26.319 signal, just simply for the purposes of display, what 00:17:26.319 --> 00:17:31.010 we've done is to essentially connect the dots and draw a 00:17:31.010 --> 00:17:34.550 continuous curve through the points so that this picture 00:17:34.550 --> 00:17:37.110 isn't filled up with a lot of vertical lines. 00:17:37.110 --> 00:17:42.050 So this is the discrete-time sequence that represents the 00:17:42.050 --> 00:17:46.970 weekly Dow Jones Index over a 10 year period. 00:17:46.970 --> 00:17:53.250 And here, by the way, again, is the crash of 1929. 00:17:53.250 --> 00:17:56.990 It's interesting to note, by the way, that actually the 00:17:56.990 --> 00:18:02.050 disaster in the stock market wasn't so much the 1929 crash 00:18:02.050 --> 00:18:04.780 but the long downward trend that followed that. 00:18:04.780 --> 00:18:10.430 And you can see that here by filtering through, by eye, the 00:18:10.430 --> 00:18:13.220 rapid variations in the index. 00:18:13.220 --> 00:18:17.160 And what you see is this smooth downward trend 00:18:17.160 --> 00:18:23.000 followed, eventually, by an upward trend. 00:18:23.000 --> 00:18:27.800 Now, this issue of looking at something like this, looking 00:18:27.800 --> 00:18:33.180 at a sequence, and following the smoother parts of it, 00:18:33.180 --> 00:18:37.950 namely the long term trends, is, in fact, something that is 00:18:37.950 --> 00:18:40.470 done quite typically with economic 00:18:40.470 --> 00:18:42.325 time series like this. 00:18:42.325 --> 00:18:47.560 And in particular, what's done is to smooth it, or average 00:18:47.560 --> 00:18:53.820 over some time period, to emphasize the slow variations 00:18:53.820 --> 00:18:56.750 and de-emphasize the rapid variations. 00:18:56.750 --> 00:19:01.070 And that, in fact, is processing that is done with a 00:19:01.070 --> 00:19:02.560 discrete-time system. 00:19:02.560 --> 00:19:06.660 So when you hear referred to, let's say, in stock market 00:19:06.660 --> 00:19:11.070 reports, a 51-day moving average, that, in fact, is 00:19:11.070 --> 00:19:15.630 processing the stock market index with a particular 00:19:15.630 --> 00:19:17.460 discrete-time system. 00:19:17.460 --> 00:19:21.960 The result of doing that on this particular example 00:19:21.960 --> 00:19:25.090 generates a smooth version of the curve, 00:19:25.090 --> 00:19:27.040 which I overlay here. 00:19:27.040 --> 00:19:34.330 And the overlay, then, is really attempting to track the 00:19:34.330 --> 00:19:37.600 smoother variations and de-emphasize the more rapid 00:19:37.600 --> 00:19:39.370 variations. 00:19:39.370 --> 00:19:42.930 Let me just slightly offset that so that the difference 00:19:42.930 --> 00:19:44.680 stands out a little more. 00:19:44.680 --> 00:19:51.230 And so here you see what is the original weekly index. 00:19:51.230 --> 00:19:56.280 And this is the result of processing that sequence with 00:19:56.280 --> 00:20:00.990 an appropriate system to apply smoothing. 00:20:00.990 --> 00:20:04.110 And in fact, what it is is a moving average. 00:20:04.110 --> 00:20:08.330 And so here again, you can see, in the smoother curve, 00:20:08.330 --> 00:20:12.800 this general downward trend up until this time period, 00:20:12.800 --> 00:20:17.620 followed by, eventually, a recovery. 00:20:17.620 --> 00:20:23.560 Well, we've seen an example with a continuous-time signal, 00:20:23.560 --> 00:20:26.070 the Caruso recording, an example of the discrete-time 00:20:26.070 --> 00:20:28.900 signal, this stock market index. 00:20:28.900 --> 00:20:33.810 And what I'd also like to show is a third example, which is 00:20:33.810 --> 00:20:38.320 the result of some processing on an image, in particular the 00:20:38.320 --> 00:20:40.720 image that we talked about before, which was the aerial 00:20:40.720 --> 00:20:45.000 photograph that had the problem of some cloud cover. 00:20:45.000 --> 00:20:50.570 So once again, what we see here is the original aerial 00:20:50.570 --> 00:20:53.770 photograph with the cloud cover. 00:20:53.770 --> 00:20:58.260 And some processing was applied to this using a system 00:20:58.260 --> 00:21:02.520 which, in fact, was both nonlinear and quote 00:21:02.520 --> 00:21:06.280 "time-varying," or, in the case of these independent 00:21:06.280 --> 00:21:10.520 variables, we would refer to it as spatially-varying. 00:21:10.520 --> 00:21:16.100 And the result of applying that processing is shown in 00:21:16.100 --> 00:21:18.590 the adjoining picture. 00:21:18.590 --> 00:21:24.350 And what we see there is hopefully a reasonable attempt 00:21:24.350 --> 00:21:26.890 to compensate for the cloud cover. 00:21:26.890 --> 00:21:31.080 And this, by the way, was some work that was done by 00:21:31.080 --> 00:21:37.150 Professor Lim at MIT, and has been very successful type of 00:21:37.150 --> 00:21:40.160 processing for aerial photographs. 00:21:40.160 --> 00:21:44.480 I should say, also, that this particular example is one 00:21:44.480 --> 00:21:49.670 where, although the original signal was a signal that is 00:21:49.670 --> 00:21:52.790 continuous-time, that is, the independent variables are 00:21:52.790 --> 00:21:57.080 continuous, as they are in a spatial, aerial photograph, in 00:21:57.080 --> 00:22:01.180 fact, for the processing, that picture was first converted to 00:22:01.180 --> 00:22:05.050 a sequence through a process called sampling, which we'll 00:22:05.050 --> 00:22:06.620 be talking about later. 00:22:06.620 --> 00:22:09.180 And then the processing, in fact, was done 00:22:09.180 --> 00:22:10.430 on a digital computer. 00:22:12.890 --> 00:22:19.340 Well, these then are some examples of the use of some 00:22:19.340 --> 00:22:25.490 systems to process some signals, both in continuous 00:22:25.490 --> 00:22:30.290 time and discrete time, for one-dimensional signals and 00:22:30.290 --> 00:22:34.190 for multi-dimensional signals. 00:22:34.190 --> 00:22:38.580 And as I've referred to systems, we've thought of them 00:22:38.580 --> 00:22:42.310 as one big block with an appropriate, or associated, 00:22:42.310 --> 00:22:43.990 input and output. 00:22:43.990 --> 00:22:47.530 And as we'll be getting into in the first part of the 00:22:47.530 --> 00:22:52.320 course, very often, systems are interconnected together 00:22:52.320 --> 00:22:54.100 for a variety of reasons. 00:22:54.100 --> 00:22:56.010 Some of the kinds of interconnections that we'll 00:22:56.010 --> 00:23:02.300 talk about are connecting systems in what are called 00:23:02.300 --> 00:23:06.080 series, or cascade interconnections, parallel 00:23:06.080 --> 00:23:09.930 interconnections, feedback interconnections. 00:23:09.930 --> 00:23:14.590 And feedback interconnections, in fact, are very interesting, 00:23:14.590 --> 00:23:18.160 very important, and very useful, and will be a major 00:23:18.160 --> 00:23:21.340 topic toward the end of the course. 00:23:21.340 --> 00:23:27.280 Feedback, as you may or may not know, comes into play in a 00:23:27.280 --> 00:23:31.250 variety of situations, for example, in amplifier design, 00:23:31.250 --> 00:23:36.770 as we'll talk about, feedback plays an important role. 00:23:36.770 --> 00:23:42.310 In a situation where you have a basically unstable system, 00:23:42.310 --> 00:23:45.320 feedback is often used to stabilize the system. 00:23:45.320 --> 00:23:49.520 And feedback interconnections of systems in that sense are 00:23:49.520 --> 00:23:53.020 very often used in high performance aircraft, which 00:23:53.020 --> 00:23:56.830 are inherently unstable, and are stabilized through this 00:23:56.830 --> 00:23:59.770 kind of interconnection. 00:23:59.770 --> 00:24:03.760 Just to give you a little sense of this without going 00:24:03.760 --> 00:24:08.970 into any of the details, what I'd like to show you is an 00:24:08.970 --> 00:24:15.490 excerpt from a lecture that we'll be seeing toward the end 00:24:15.490 --> 00:24:20.340 of the course relating to the analysis of feedback systems 00:24:20.340 --> 00:24:22.100 and the uses of feedback. 00:24:22.100 --> 00:24:25.880 And this is in the context of what's referred to as the 00:24:25.880 --> 00:24:29.670 inverted pendulum, which is a system that's basically 00:24:29.670 --> 00:24:32.790 unstable, and feedback interconnections are used to 00:24:32.790 --> 00:24:34.340 stabilize it. 00:24:34.340 --> 00:24:39.720 The idea, as you'll see in this brief clip, is that there 00:24:39.720 --> 00:24:44.220 is a cart that's moving on a track with a rod that's 00:24:44.220 --> 00:24:45.880 pivoted at the base. 00:24:45.880 --> 00:24:49.500 And so that system, in the absence of anything, is 00:24:49.500 --> 00:24:52.390 unstable in that the rod would tend to fall. 00:24:52.390 --> 00:24:57.510 And as we go into in detail in the lecture later, we use 00:24:57.510 --> 00:25:01.000 feedback to position the cart under the 00:25:01.000 --> 00:25:03.680 pendulum to balance it. 00:25:03.680 --> 00:25:07.730 And in fact, that balancing can be done even when we 00:25:07.730 --> 00:25:10.690 modify the system in a variety of ways, as 00:25:10.690 --> 00:25:11.970 you'll see in this clip. 00:25:11.970 --> 00:25:15.850 So let's take a look at that, remembering that this is just 00:25:15.850 --> 00:25:19.340 a brief excerpt from a longer discussion. 00:25:19.340 --> 00:25:19.700 [VIDEO PLAYBACK] 00:25:19.700 --> 00:25:25.460 I can change the overall system even further by, let's 00:25:25.460 --> 00:25:28.810 say, for example, pouring a liquid in. 00:25:28.810 --> 00:25:32.870 And now, let me also comment that I've changed the physics 00:25:32.870 --> 00:25:36.800 of it a little bit because the liquid can slosh around a 00:25:36.800 --> 00:25:37.160 little bit. 00:25:37.160 --> 00:25:40.170 It becomes a little more complicated a system, but as 00:25:40.170 --> 00:25:43.080 you can see, it still remains balanced. 00:25:43.080 --> 00:25:45.560 [END VIDEO PLAYBACK] 00:25:45.560 --> 00:25:48.090 As you'll see when we get to it, by the way, that 00:25:48.090 --> 00:25:51.290 demonstration was a lot of fun to do. 00:25:51.290 --> 00:25:56.090 Now, in talking about signals and systems as we go through 00:25:56.090 --> 00:26:01.360 the course, there are several domains, two in particular, 00:26:01.360 --> 00:26:05.850 that we will find convenient for the analysis and 00:26:05.850 --> 00:26:10.070 representation of signals and systems. 00:26:10.070 --> 00:26:17.100 One is the time domain, which is what we tend to think of, 00:26:17.100 --> 00:26:21.910 and which we have kind of been focusing on in the discussion 00:26:21.910 --> 00:26:23.950 so far in this lecture. 00:26:23.950 --> 00:26:28.890 But equally important is what's referred to as the 00:26:28.890 --> 00:26:34.220 frequency domain as a representation for signals, 00:26:34.220 --> 00:26:37.630 and as a means for analysis for systems. 00:26:37.630 --> 00:26:40.600 And in the context of the frequency domain 00:26:40.600 --> 00:26:44.750 representation, some of the kinds of ideas and topics that 00:26:44.750 --> 00:26:50.270 we'll be exploring are the Fourier Transform, and the 00:26:50.270 --> 00:26:55.930 Laplace Transform, and a discrete-time counterpart of 00:26:55.930 --> 00:26:59.140 the Laplace Transform, which is the z-Transform. 00:26:59.140 --> 00:27:02.080 The Fourier Transform discussion we'll get into 00:27:02.080 --> 00:27:03.990 fairly early in the course. 00:27:03.990 --> 00:27:08.110 And the Laplace Transform and z-Transform represent 00:27:08.110 --> 00:27:11.200 extensions of the Fourier transform, and we'll be 00:27:11.200 --> 00:27:14.520 getting into that later in the course. 00:27:14.520 --> 00:27:18.550 Just initially to think about the time domain and frequency 00:27:18.550 --> 00:27:21.610 domain, you might think, for example, of 00:27:21.610 --> 00:27:23.490 a note being played. 00:27:23.490 --> 00:27:28.800 And the time-domain representation would be how 00:27:28.800 --> 00:27:32.250 the sound pressure, as a function 00:27:32.250 --> 00:27:34.330 of time, would change. 00:27:34.330 --> 00:27:37.560 And the frequency-domain representation would 00:27:37.560 --> 00:27:40.960 correspond to a representation of the frequency 00:27:40.960 --> 00:27:43.440 content of the note. 00:27:43.440 --> 00:27:46.720 And, in fact, what I'd like to do is illustrate that and 00:27:46.720 --> 00:27:51.720 those two domains simultaneously by playing a 00:27:51.720 --> 00:27:53.370 glockenspiel note. 00:27:53.370 --> 00:27:58.220 What you'll hear is the note repeated several times over. 00:27:58.220 --> 00:28:02.490 And at the same time, you'll see two displays, one on the 00:28:02.490 --> 00:28:05.950 left representing the time domain display a 00:28:05.950 --> 00:28:08.820 representation of the signal, and the one on the right 00:28:08.820 --> 00:28:11.650 representing the frequency domain. 00:28:11.650 --> 00:28:13.350 So let's look at that. 00:28:13.350 --> 00:28:16.130 And you'll hear the note and simultaneously 00:28:16.130 --> 00:28:17.380 see these two displays. 00:28:21.080 --> 00:28:25.850 So there's the note on the left, the time waveform. 00:28:25.850 --> 00:28:30.285 And on the right, what we see is the frequency content, in 00:28:30.285 --> 00:28:34.900 particular, indicating the fact that there are several 00:28:34.900 --> 00:28:36.420 harmonic lines in the tone. 00:28:54.540 --> 00:29:00.410 Well, what I've gone through in this lecture represents a 00:29:00.410 --> 00:29:04.450 brief overview of signals and systems. 00:29:04.450 --> 00:29:08.340 And beginning with the next lecture, we will be much more 00:29:08.340 --> 00:29:13.950 specific and precise, first discussing some basic signals, 00:29:13.950 --> 00:29:16.880 and then talking about systems, and system 00:29:16.880 --> 00:29:20.860 properties, and how to exploit them. 00:29:20.860 --> 00:29:26.890 As one final comment that I'd like to make in this lecture, 00:29:26.890 --> 00:29:34.640 I'd like to emphasize at the outset that the taped lectures 00:29:34.640 --> 00:29:38.180 represent only one component of the course. 00:29:38.180 --> 00:29:42.690 And equally important will be both the textbook and the 00:29:42.690 --> 00:29:44.670 video course manual. 00:29:44.670 --> 00:29:49.140 In particular, it's important not only to be viewing the 00:29:49.140 --> 00:29:55.470 tapes, but simultaneously, or in conjunction with that, 00:29:55.470 --> 00:29:59.510 doing the appropriate reading in the textbook and also 00:29:59.510 --> 00:30:02.500 working through the problems carefully in the 00:30:02.500 --> 00:30:04.940 video course manual. 00:30:04.940 --> 00:30:08.830 In a course like this, you basically only get out of it 00:30:08.830 --> 00:30:10.630 as much as you put into it. 00:30:10.630 --> 00:30:14.900 The hope is that if you put the right amount of time and 00:30:14.900 --> 00:30:18.910 effort into it, you'll find the course to be educational 00:30:18.910 --> 00:30:19.720 and interesting. 00:30:19.720 --> 00:30:22.250 And I certainly hope that that will be the case. 00:30:22.250 --> 00:30:23.500 Thank you.