WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.720 continue to offer high quality educational resources for free. 00:00:10.720 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.280 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.280 --> 00:00:22.970 at ocw.mit.edu 00:00:22.970 --> 00:00:26.150 PROFESSOR: Hello and welcome. 00:00:26.150 --> 00:00:30.710 So today is mostly distinguished by what happens tomorrow. 00:00:30.710 --> 00:00:33.650 Not surprisingly, but of course, you all know that. 00:00:33.650 --> 00:00:37.100 So tomorrow we have our first quiz. 00:00:37.100 --> 00:00:40.407 Tomorrow evening 7:30 to 9:30, no recitation. 00:00:40.407 --> 00:00:41.990 We've been through this several times. 00:00:41.990 --> 00:00:45.320 I won't spend any time on it other than to ask if there are 00:00:45.320 --> 00:00:49.100 any questions, so if I don't hear any questions, 00:00:49.100 --> 00:00:51.260 the idea is going to be at the end of this lecture, 00:00:51.260 --> 00:00:54.470 the next time we'll see you is in office hours, 00:00:54.470 --> 00:01:00.260 and, or tomorrow, Wednesday, 7:30 on the third floor, 00:01:00.260 --> 00:01:01.070 building 26. 00:01:03.690 --> 00:01:06.380 Questions or comments about the exam? 00:01:06.380 --> 00:01:07.050 Yep. 00:01:07.050 --> 00:01:11.650 AUDIENCE: [INAUDIBLE] 00:01:11.650 --> 00:01:14.910 PROFESSOR: So that one page of notes, 8 and 1/2 00:01:14.910 --> 00:01:17.680 by 11, front and back. 00:01:17.680 --> 00:01:20.790 You can write as small as you like. 00:01:20.790 --> 00:01:22.780 In fact, later in today's lecture, 00:01:22.780 --> 00:01:24.631 I'll show you how a microscope works 00:01:24.631 --> 00:01:26.880 and you're welcome to use a microscope because they're 00:01:26.880 --> 00:01:29.370 completely non-electronic. 00:01:29.370 --> 00:01:33.780 Oh, as long as you use an optical microscope. 00:01:33.780 --> 00:01:35.150 Other questions about the exam? 00:01:38.290 --> 00:01:44.020 OK, then for today, so far, since the beginning 00:01:44.020 --> 00:01:46.090 of the term, we thought about a number 00:01:46.090 --> 00:01:47.710 of different kinds of representations 00:01:47.710 --> 00:01:49.540 for both DT systems-- 00:01:49.540 --> 00:01:53.110 discrete time-- and CT systems-- continuous time systems-- 00:01:53.110 --> 00:01:55.660 and we saw that we were interested 00:01:55.660 --> 00:01:57.670 in that large number of representations, 00:01:57.670 --> 00:02:01.720 because each of them had some particular aspect that 00:02:01.720 --> 00:02:04.630 made it particularly convenient sometimes. 00:02:04.630 --> 00:02:08.229 So, for example, in both CT and DT we looked at verbal, 00:02:08.229 --> 00:02:09.100 but not so much. 00:02:09.100 --> 00:02:11.500 That was mostly in the homework. 00:02:11.500 --> 00:02:14.380 We looked at difference in differential equations, 00:02:14.380 --> 00:02:19.660 mostly because they were so compact, so concise, so 00:02:19.660 --> 00:02:22.320 precise, they told you exactly what the system does. 00:02:22.320 --> 00:02:25.630 No fluff, this is it. 00:02:25.630 --> 00:02:27.040 So that was nice. 00:02:27.040 --> 00:02:31.226 Block diagrams, by contrast, are less concise, 00:02:31.226 --> 00:02:33.100 but they tell you the way a signal propagates 00:02:33.100 --> 00:02:36.560 through the system on its way from the input to the output, 00:02:36.560 --> 00:02:38.620 and that can be very helpful for understanding 00:02:38.620 --> 00:02:41.500 why certain behaviors occur, especially when we talk 00:02:41.500 --> 00:02:43.750 about things like feedback. 00:02:43.750 --> 00:02:45.340 We looked at operator representations. 00:02:45.340 --> 00:02:47.950 They were nice because we could transform 00:02:47.950 --> 00:02:50.980 the way we think about systems into the way 00:02:50.980 --> 00:02:55.522 we think about polynomials, so we reduced a college level 00:02:55.522 --> 00:02:56.980 thing to a high school level thing. 00:02:56.980 --> 00:03:00.820 That's always nice, and then we looked at transforms. 00:03:00.820 --> 00:03:03.670 The distinguishing feature of transform representation 00:03:03.670 --> 00:03:06.550 was that we took an entire function of time, 00:03:06.550 --> 00:03:08.890 and turned it into an algebraic expression. 00:03:08.890 --> 00:03:10.900 So we turned a differential system 00:03:10.900 --> 00:03:13.390 that described a system of differential equations 00:03:13.390 --> 00:03:16.480 that describes a system into a system of algebraic equations 00:03:16.480 --> 00:03:18.160 that describes a system. 00:03:18.160 --> 00:03:20.050 All of those were useful for different ways 00:03:20.050 --> 00:03:24.640 and what I wanted to talk about today is yet another way 00:03:24.640 --> 00:03:27.580 to represent a system and that is to represent 00:03:27.580 --> 00:03:30.180 a system by a single signal. 00:03:33.066 --> 00:03:34.690 So in some sense we're going backwards, 00:03:34.690 --> 00:03:36.850 because we're taking what we would normally 00:03:36.850 --> 00:03:41.429 think of as an entire system and reducing it, 00:03:41.429 --> 00:03:42.970 getting rid of the system altogether, 00:03:42.970 --> 00:03:44.830 so all we're going to have is signals. 00:03:44.830 --> 00:03:47.230 That turns out to be a particular powerful way 00:03:47.230 --> 00:03:50.110 to do some sorts of operations, and is actually 00:03:50.110 --> 00:03:52.240 the first instance, the first step, 00:03:52.240 --> 00:03:55.630 in that we will take in a major field 00:03:55.630 --> 00:03:57.820 thought of signal processing. 00:03:57.820 --> 00:04:00.670 When you reduce the entire behavior of a system 00:04:00.670 --> 00:04:03.970 to a signal, we then regard the whole processing task 00:04:03.970 --> 00:04:05.320 as a signal processing task. 00:04:05.320 --> 00:04:07.750 Signal processing task, not system. 00:04:10.930 --> 00:04:13.750 So far, we have focused on the responses 00:04:13.750 --> 00:04:16.000 to the most elementary kinds of signals-- 00:04:16.000 --> 00:04:22.120 unit sample signal, unit impulse signal-- 00:04:22.120 --> 00:04:24.700 but generally speaking, we're interested in much more 00:04:24.700 --> 00:04:25.720 complicated signals. 00:04:25.720 --> 00:04:27.760 As you've already seen, I've already 00:04:27.760 --> 00:04:30.477 asked you to calculate things like unit step responses. 00:04:30.477 --> 00:04:32.560 So generally, we're going to be interested in much 00:04:32.560 --> 00:04:34.780 more complicated signals-- 00:04:34.780 --> 00:04:38.800 the responses of systems to much more complicated signals. 00:04:38.800 --> 00:04:40.930 That's not hard. 00:04:40.930 --> 00:04:42.760 The reason we skipped over it was 00:04:42.760 --> 00:04:46.030 that you can always figure out the response 00:04:46.030 --> 00:04:49.210 to a more complicated signal at least 00:04:49.210 --> 00:04:51.520 by falling back on some of our more primitive ways 00:04:51.520 --> 00:04:52.829 of thinking about systems. 00:04:52.829 --> 00:04:55.120 So for example, if we think about a difference equation 00:04:55.120 --> 00:04:57.580 or a block diagram, we can think about how 00:04:57.580 --> 00:05:03.430 a more complicated signal excites a response by simply 00:05:03.430 --> 00:05:05.970 thinking about the system operating on a sample 00:05:05.970 --> 00:05:08.200 by sample basis. 00:05:08.200 --> 00:05:09.650 So you, of course, all know that, 00:05:09.650 --> 00:05:11.150 and to prove that you all know that, 00:05:11.150 --> 00:05:13.190 answer the following question. 00:05:13.190 --> 00:05:14.215 Here is a system. 00:05:16.790 --> 00:05:19.700 Here is a signal that is more complicated than just a unit 00:05:19.700 --> 00:05:22.220 sample or unit sample signal. 00:05:22.220 --> 00:05:26.780 Figure out what is the third response 00:05:26.780 --> 00:05:29.280 of this system to that signal? 00:05:37.400 --> 00:05:40.310 You should be absolutely quiet. 00:05:40.310 --> 00:05:43.180 That was sarcasm, just so you know. 00:06:24.620 --> 00:06:28.950 Lots of self-satisfied looks so I assume everybody's done. 00:06:28.950 --> 00:06:29.870 So what's the answer? 00:06:29.870 --> 00:06:32.000 Raise the number of fingers that corresponds 00:06:32.000 --> 00:06:35.136 to the answer y of 3. 00:06:35.136 --> 00:06:36.760 Raise your hand so that I can see them. 00:06:39.390 --> 00:06:40.940 Drop the ones with the wrong answers. 00:06:40.940 --> 00:06:43.730 I don't want to see those. 00:06:43.730 --> 00:06:46.430 OK, about 50%, so maybe take another 10 seconds. 00:06:50.580 --> 00:06:53.105 Notice that I'm asking for y of 3. 00:07:27.490 --> 00:07:30.970 So what's the answer y of 3, and raise your hand. 00:07:30.970 --> 00:07:34.670 Everybody has the right answer, raise your hand. 00:07:34.670 --> 00:07:35.170 Much better. 00:07:35.170 --> 00:07:38.590 OK, so now the overwhelming majority says the answer is 2. 00:07:38.590 --> 00:07:40.930 That's not very hard. 00:07:40.930 --> 00:07:43.906 If we think about the block diagram representation, 00:07:43.906 --> 00:07:45.280 and if we think about propagating 00:07:45.280 --> 00:07:47.740 the more complicated signal represented over here 00:07:47.740 --> 00:07:51.280 through that signal, through that system, 00:07:51.280 --> 00:07:52.840 we start with the system at rest. 00:07:52.840 --> 00:07:58.140 That means there's 0's coming out of all the delays, 00:07:58.140 --> 00:08:03.390 at time n equals minus 1, x is 0, 00:08:03.390 --> 00:08:07.580 combined with the initial 0's, we get the answer is 0. 00:08:10.430 --> 00:08:17.270 And then, at time n equals 0, the input becomes 1, 00:08:17.270 --> 00:08:24.290 so now the output goes to 1, at time 1, it goes to 2. 00:08:24.290 --> 00:08:26.350 At time 2, it goes to 3. 00:08:26.350 --> 00:08:28.770 Times 3, it goes to 2. 00:08:28.770 --> 00:08:30.930 This is obvious, right? 00:08:30.930 --> 00:08:32.580 Et cetera. 00:08:32.580 --> 00:08:36.419 So the answer is 2. 00:08:36.419 --> 00:08:41.280 y of 3 is 2, and the point is that it's very trivial 00:08:41.280 --> 00:08:44.144 to think about by thinking about the system in a sort of sample 00:08:44.144 --> 00:08:47.140 by sample way. 00:08:47.140 --> 00:08:49.080 Not surprisingly, the point of today 00:08:49.080 --> 00:08:53.940 is to not think of the system sample by sample, 00:08:53.940 --> 00:08:58.710 but to elevate the conversation from samples to signals. 00:08:58.710 --> 00:09:01.770 The first step in thinking about it as signals 00:09:01.770 --> 00:09:04.350 is to realize that you can think about the response 00:09:04.350 --> 00:09:08.850 of the system by decomposing the input into additive parts. 00:09:08.850 --> 00:09:13.740 I can think about x which is this 3 sample signal. 00:09:13.740 --> 00:09:17.730 I can decompose it into single samples, 00:09:17.730 --> 00:09:22.130 and then think about the response to each of those, 00:09:22.130 --> 00:09:25.440 and it may not surprise you that if the system were linear, 00:09:25.440 --> 00:09:27.020 then the response to the sum would 00:09:27.020 --> 00:09:29.516 be the sum of the responses. 00:09:29.516 --> 00:09:30.890 So you can sort of see that there 00:09:30.890 --> 00:09:32.390 would be a way of adding together 00:09:32.390 --> 00:09:35.280 these rectangular pulses to get a triangle pulse. 00:09:38.180 --> 00:09:42.930 So that works simply because the system is linear. 00:09:42.930 --> 00:09:47.540 The system has the property that the output for a sum 00:09:47.540 --> 00:09:52.720 is the sum of the outputs for the individual components, 00:09:52.720 --> 00:09:56.500 and we can write that this way, so a system is linear. 00:09:56.500 --> 00:10:00.130 We can define linear in a more rigorous mathematical sense 00:10:00.130 --> 00:10:02.050 by saying that a system is linear 00:10:02.050 --> 00:10:04.840 if the response to a weighted sum of inputs 00:10:04.840 --> 00:10:08.200 is the similarly weighted sum of outputs. 00:10:08.200 --> 00:10:11.650 So imagine that I have a system whose 00:10:11.650 --> 00:10:16.680 output when the input is x1 is y1 00:10:16.680 --> 00:10:19.970 and whose output when the input is x2 is y2. 00:10:19.970 --> 00:10:21.900 We'll say the system is linear if 00:10:21.900 --> 00:10:27.480 and only if the weighted sum of inputs alpha x1 plus beta x2 00:10:27.480 --> 00:10:31.620 gives the same wakings alpha y1 plus beta y2 00:10:31.620 --> 00:10:34.650 for all possible values of alpha and beta. 00:10:34.650 --> 00:10:36.960 If that's true, then we'll say the system is linear. 00:10:41.810 --> 00:10:44.930 So then if it's linear, then we're 00:10:44.930 --> 00:10:46.467 allowed to do this decomposition, 00:10:46.467 --> 00:10:48.050 because all we're doing is decomposing 00:10:48.050 --> 00:10:49.820 the input into a sum of inputs. 00:10:52.370 --> 00:10:56.292 We'll always be able to do that operation of the decomposition 00:10:56.292 --> 00:10:58.500 if the system is linear according to the definition I 00:10:58.500 --> 00:10:59.480 already showed. 00:10:59.480 --> 00:11:01.220 If the system also has the property 00:11:01.220 --> 00:11:04.010 that we will call time invariance, 00:11:04.010 --> 00:11:05.810 then the response to the parts will be 00:11:05.810 --> 00:11:08.330 particularly easy to calculate. 00:11:08.330 --> 00:11:12.170 Time invariance has a similar formal definition. 00:11:12.170 --> 00:11:16.420 We will call a system time invariant 00:11:16.420 --> 00:11:18.670 if given that the input-- 00:11:18.670 --> 00:11:23.200 the response to x of n is y. 00:11:23.200 --> 00:11:26.800 Given that, we'll say the system is time invariant 00:11:26.800 --> 00:11:28.480 if a shifted version of the input 00:11:28.480 --> 00:11:32.770 simply shifts the response. 00:11:32.770 --> 00:11:35.750 That seems like a kind of gobbledygook sort of thing. 00:11:35.750 --> 00:11:39.010 It's a very simple minded notion that you should all 00:11:39.010 --> 00:11:40.330 have from common experience. 00:11:40.330 --> 00:11:43.990 All it says is that if I do something today, 00:11:43.990 --> 00:11:50.430 and I get a response, if I do the same thing tomorrow, 00:11:50.430 --> 00:11:54.210 I should get the same response just delayed by a day. 00:11:54.210 --> 00:11:55.200 That's all it's saying. 00:11:55.200 --> 00:11:58.350 So basically the system behaves sort of the same 00:11:58.350 --> 00:12:02.380 now as it did previously, and as it will in the future. 00:12:02.380 --> 00:12:05.580 That's what time invariance means. 00:12:05.580 --> 00:12:08.560 So if the response is time invariant, 00:12:08.560 --> 00:12:13.920 then we can compute the response to a shifted unit sample. 00:12:13.920 --> 00:12:16.750 Notice that this is the unit sample response. 00:12:16.750 --> 00:12:19.650 This is a shifted unit sample so if the system 00:12:19.650 --> 00:12:22.080 is time invariant, then the response to a shifted unit 00:12:22.080 --> 00:12:25.290 sample is the shifted unit sample response which 00:12:25.290 --> 00:12:27.990 you can see from the picture. 00:12:27.990 --> 00:12:30.570 So the idea then is that superposition 00:12:30.570 --> 00:12:34.470 is a very easy way to think about the response of a system. 00:12:34.470 --> 00:12:37.200 If the system is linear and time invariant, 00:12:37.200 --> 00:12:40.290 linearity let's us break it up, and think about the response 00:12:40.290 --> 00:12:41.580 to each part. 00:12:41.580 --> 00:12:43.200 Shift invariance, or time invariance, 00:12:43.200 --> 00:12:45.930 allows us to shift the input, and know automatically 00:12:45.930 --> 00:12:48.180 what's the response going to look like. 00:12:50.810 --> 00:12:54.360 And there's a formal way we can think about the way you 00:12:54.360 --> 00:12:56.970 do that operation. 00:12:56.970 --> 00:13:01.470 How do you implement this superposition thing? 00:13:01.470 --> 00:13:05.650 You think about a system as having a unit sample response. 00:13:05.650 --> 00:13:07.200 This is the unit sample signal. 00:13:07.200 --> 00:13:12.430 We'll call the unit sample response h of n, 00:13:12.430 --> 00:13:16.090 then a shifted unit sample will give a shifted unit 00:13:16.090 --> 00:13:17.130 sample response. 00:13:17.130 --> 00:13:18.340 That's time invariance. 00:13:20.950 --> 00:13:28.590 Then a weighted shifted impulse gives the same weighted shifted 00:13:28.590 --> 00:13:29.400 impulse response. 00:13:32.170 --> 00:13:42.260 Then the sum of such things gives the sum of such things. 00:13:42.260 --> 00:13:46.100 That's just a formal derivation of a process 00:13:46.100 --> 00:13:48.990 that we will call convolution. 00:13:48.990 --> 00:13:54.290 So the response to an arbitrary DT signal 00:13:54.290 --> 00:13:58.460 that excites a linear time invariant system 00:13:58.460 --> 00:14:02.660 can be described by the convolution of the input 00:14:02.660 --> 00:14:06.680 with the unit sample response. 00:14:06.680 --> 00:14:08.510 We'll call that formula-- 00:14:08.510 --> 00:14:12.590 we'll call that operation convolution. 00:14:12.590 --> 00:14:16.847 Convolution is completely straightforward. 00:14:16.847 --> 00:14:19.430 For that reason, we try to make it a little bit more confusing 00:14:19.430 --> 00:14:22.610 by using terribly confusing notation. 00:14:22.610 --> 00:14:24.140 That too is sarcastic, I mean. 00:14:28.140 --> 00:14:30.440 So the only thing that's at all confusing 00:14:30.440 --> 00:14:34.310 about convolution-- convolution is completely trivial. 00:14:34.310 --> 00:14:38.270 Here's the way we would write it. x convolves with h. 00:14:38.270 --> 00:14:41.995 The signal x convolves with the signal h to give a new signal. 00:14:44.560 --> 00:14:50.030 Being a signal, I can ask what's the nth sample look like, 00:14:50.030 --> 00:14:54.920 and what that symbol means is this sum. 00:14:54.920 --> 00:14:59.630 The confusing thing is that most people in the field 00:14:59.630 --> 00:15:03.190 write it this way. 00:15:03.190 --> 00:15:07.430 The signal x of n convolves with the signal h of n. 00:15:07.430 --> 00:15:11.690 The reason that's confusing, and the thing you will never do, 00:15:11.690 --> 00:15:13.592 because you are here. 00:15:13.592 --> 00:15:15.050 The thing that you will never do is 00:15:15.050 --> 00:15:18.830 confuse the meaning of that statement 00:15:18.830 --> 00:15:21.613 with what looks like an operation on samples. 00:15:24.650 --> 00:15:29.720 Had I said multiply, you would have said, 00:15:29.720 --> 00:15:32.420 if this were a multiply operator instead of a convolution 00:15:32.420 --> 00:15:35.270 multiplier operator-- 00:15:35.270 --> 00:15:38.000 if that were multiply instead of convolve, 00:15:38.000 --> 00:15:41.360 you would have said, oh, that means 00:15:41.360 --> 00:15:47.200 the oneth sample multiplied by the oneth sample 00:15:47.200 --> 00:15:48.940 is the product of the oneth samples. 00:15:52.180 --> 00:15:54.220 This is not true. 00:15:54.220 --> 00:15:55.990 This is not generally true. 00:15:58.720 --> 00:16:04.270 The convolution operation means take the whole signal x. 00:16:04.270 --> 00:16:07.030 That's why we think about it as an operation on signals, 00:16:07.030 --> 00:16:10.390 not an operation on samples. 00:16:10.390 --> 00:16:13.990 Convolution means take the whole signal x, 00:16:13.990 --> 00:16:16.870 and convolve it with the whole signal h 00:16:16.870 --> 00:16:21.820 to get a brand new signal x convolved h, 00:16:21.820 --> 00:16:25.400 and then take the nth sample. 00:16:25.400 --> 00:16:28.720 So the only thing that's at all confusing about convolution 00:16:28.720 --> 00:16:32.140 is remembering the convolution is an operation that is applied 00:16:32.140 --> 00:16:33.580 to signals, not samples. 00:16:40.160 --> 00:16:41.987 So structure of convolution. 00:16:41.987 --> 00:16:43.820 So I just showed you a mathematical formula. 00:16:43.820 --> 00:16:45.530 What I'd like you to have is a little bit 00:16:45.530 --> 00:16:50.270 of an intuition for what happens when you convolve two signals. 00:16:50.270 --> 00:16:52.930 So let's think about the structure of this operation. 00:16:52.930 --> 00:16:54.740 What are we doing? 00:16:54.740 --> 00:16:57.467 Imagine that we're going back to that original problem. 00:16:57.467 --> 00:17:00.050 What happens when you take x of n and convolve it with h of n? 00:17:03.110 --> 00:17:05.900 All we need to do is this formula, right? 00:17:05.900 --> 00:17:08.000 That's all we need to do. 00:17:08.000 --> 00:17:09.560 What's that formula say? 00:17:09.560 --> 00:17:15.060 Well let's think about how you would compute the 0-th output. 00:17:15.060 --> 00:17:20.280 According to that formula all I did was substitute n equal 0 00:17:20.280 --> 00:17:23.430 every place there was an n, and what 00:17:23.430 --> 00:17:27.240 I see is I have to multiply x of k times h of minus k, 00:17:27.240 --> 00:17:29.430 but I've got x of n in h of n. 00:17:29.430 --> 00:17:31.980 So the first thing I do is I flip the axises and I make them 00:17:31.980 --> 00:17:32.610 k's. 00:17:32.610 --> 00:17:34.260 That's not hard. 00:17:34.260 --> 00:17:39.570 Then the x looks OK, but the h doesn't. 00:17:39.570 --> 00:17:45.800 I need h of minus k, so I have to flip it. 00:17:45.800 --> 00:17:49.165 So I'm flipping about the n equals 0 axis. 00:17:52.120 --> 00:17:55.870 So that positive n becomes minus n, 00:17:55.870 --> 00:17:57.880 positive k becomes minus k, because I want 00:17:57.880 --> 00:18:01.200 this to be minus k up here. 00:18:01.200 --> 00:18:05.780 Then generally, I have some shift thing here. 00:18:05.780 --> 00:18:08.801 This 0, because I was looking for the 0-th sample. 00:18:08.801 --> 00:18:10.300 In general, that might be different. 00:18:10.300 --> 00:18:14.770 That might be 7 if I wanted to find y of 7. 00:18:14.770 --> 00:18:21.530 Then I'd have a 7 over here, and that number represents a shift. 00:18:21.530 --> 00:18:23.430 In the case of 0, it's a 0 shift. 00:18:26.630 --> 00:18:29.860 Then I have to multiply these two, 00:18:29.860 --> 00:18:32.380 so I just place this thing over here so I can multiply. 00:18:32.380 --> 00:18:34.510 I multiply down. 00:18:34.510 --> 00:18:39.400 You can see that there's only one sample that in the two 00:18:39.400 --> 00:18:42.280 is both non-zero. 00:18:42.280 --> 00:18:46.750 Therefore, I get a single non-zero answer, 00:18:46.750 --> 00:18:48.280 and then according to the formula, 00:18:48.280 --> 00:19:00.420 I have to sum so the 0-th answer is flip, shift, multiply, sum, 00:19:00.420 --> 00:19:02.800 and you just repeat that for all the different answers. 00:19:02.800 --> 00:19:06.390 So at time equals 0, the answer at time 0 00:19:06.390 --> 00:19:10.950 is flip, shift by 0, multiply, sum. 00:19:10.950 --> 00:19:13.030 The answer is 1. 00:19:13.030 --> 00:19:19.410 If I want to find the one answer now the shift is shift by 1. 00:19:19.410 --> 00:19:21.960 So now instead of having flip which 00:19:21.960 --> 00:19:25.590 would have put the 3 samples here, I shift by 1 00:19:25.590 --> 00:19:27.820 so now they're over there. 00:19:27.820 --> 00:19:31.500 So now when I do the multiply, I pick up 2 non-zero answers 00:19:31.500 --> 00:19:34.300 and the answer is 2. 00:19:34.300 --> 00:19:37.080 If I wanted y equals 2, I do the same thing 00:19:37.080 --> 00:19:39.840 but now I shift by 2. 00:19:39.840 --> 00:19:43.280 Flip, shift, multiply, sum, that's all I do. 00:19:43.280 --> 00:19:46.410 It's completely trivial. 00:19:46.410 --> 00:19:49.320 If I continue, the shift becomes larger 00:19:49.320 --> 00:19:52.950 and now it's falling off the end. 00:19:52.950 --> 00:19:57.540 Continue, continue, and I get in general 00:19:57.540 --> 00:20:00.460 that's the prescription. 00:20:00.460 --> 00:20:02.430 So what I've tried to show is two ways 00:20:02.430 --> 00:20:05.370 of thinking about this convolution thing. 00:20:05.370 --> 00:20:07.650 The first was by superposition, where I just 00:20:07.650 --> 00:20:11.364 think about breaking the input into a bunch of samples, 00:20:11.364 --> 00:20:13.530 thinking about the response to each of those samples 00:20:13.530 --> 00:20:14.029 and adding. 00:20:18.528 --> 00:20:21.570 That's an input centric way of thinking about things, 00:20:21.570 --> 00:20:23.820 because I think of the input being broken up 00:20:23.820 --> 00:20:25.590 by a bunch of samples. 00:20:25.590 --> 00:20:28.380 This convolution formula is an output centric way 00:20:28.380 --> 00:20:30.120 of thinking about things I tell you. 00:20:30.120 --> 00:20:35.450 I'd like to know the output at time p, 00:20:35.450 --> 00:20:37.760 and to compute the output of time p, 00:20:37.760 --> 00:20:39.290 you say, well that's easy. 00:20:39.290 --> 00:20:44.240 Flip, shift by p, multiply, sum. 00:20:44.240 --> 00:20:48.149 So input centric, that's the superposition way 00:20:48.149 --> 00:20:49.190 of thinking about things. 00:20:49.190 --> 00:20:51.080 Output centric, that's the convolution way 00:20:51.080 --> 00:20:53.550 of thinking about things. 00:20:53.550 --> 00:20:56.180 So now that you know about convolution, 00:20:56.180 --> 00:21:00.920 find which plot below 1, 2, 3, 4, or none of the above 00:21:00.920 --> 00:21:03.530 shows the result of convolving the two functions shown above. 00:21:26.554 --> 00:21:27.659 You're so quiet. 00:21:27.659 --> 00:21:29.700 I assume you're practicing for the exam tomorrow. 00:21:36.222 --> 00:21:37.180 You're allowed to talk. 00:22:58.510 --> 00:22:59.530 So which one's right? 00:22:59.530 --> 00:23:03.377 1, 2, 3, 4, or 5? 00:23:03.377 --> 00:23:03.960 See if I know. 00:23:08.790 --> 00:23:10.430 It looks good. 00:23:10.430 --> 00:23:18.530 About I only see one wrong, two wrong, so 95% or so. 00:23:18.530 --> 00:23:21.470 So how do I think about this? 00:23:21.470 --> 00:23:25.220 What's the way that I should think about convolving those? 00:23:25.220 --> 00:23:27.890 Easiest, most straightforward way, go back to the formula. 00:23:30.920 --> 00:23:31.890 That will always work. 00:23:34.440 --> 00:23:37.170 Can somebody tell me a more intuitive, insightful way 00:23:37.170 --> 00:23:40.350 of thinking about what will be the result of convolving 00:23:40.350 --> 00:23:42.060 those top two functions? 00:23:42.060 --> 00:23:44.758 Tell me a property of the result of convolving. 00:23:51.310 --> 00:23:54.190 Yes, yes, flip, shift, multiply, sum. 00:23:57.900 --> 00:23:59.860 What's the answer at n equals 1? 00:24:03.640 --> 00:24:04.990 1. 00:24:04.990 --> 00:24:09.850 So I've got two things that look kind of like geometrics. 00:24:09.850 --> 00:24:11.890 Imagine for the moment-- 00:24:11.890 --> 00:24:14.230 that was intended to be a hint-- 00:24:14.230 --> 00:24:17.800 imagine for the moment that the sequence looks like 1, 00:24:17.800 --> 00:24:25.180 2/3, 4/9, 8/27, blah, blah, blah. 00:24:25.180 --> 00:24:28.030 Imagine that it's a geometric sequence 00:24:28.030 --> 00:24:29.260 with the base of about 2/3. 00:24:32.870 --> 00:24:36.590 How would I compute the answer when I convolve that sequence 00:24:36.590 --> 00:24:40.300 with itself at zero? 00:24:40.300 --> 00:24:42.970 Flip, shift, multiply, divide, so I started out 00:24:42.970 --> 00:24:46.300 with two things that were both starting at zero. 00:24:46.300 --> 00:24:48.580 You flip one of them. 00:24:48.580 --> 00:24:51.130 How much overlap is there? 00:24:51.130 --> 00:24:52.690 Just the 1. 00:24:52.690 --> 00:24:55.660 Just the n equals 0, what's the answer at n equals 0? 00:24:59.580 --> 00:25:01.750 1, right? 00:25:01.750 --> 00:25:05.650 So if I imagine that this is the sequence after I flip it. 00:25:05.650 --> 00:25:07.750 There's a 1 under the 1. 00:25:07.750 --> 00:25:09.820 The 0's here kill the terms down here. 00:25:09.820 --> 00:25:12.010 The 0's here kill the terms up there. 00:25:12.010 --> 00:25:14.530 The only thing that lives is 1 times 1 is 1, 00:25:14.530 --> 00:25:16.570 so the answer at y equals 0 is 1. 00:25:16.570 --> 00:25:17.970 What's the answer at y equals 1? 00:25:22.470 --> 00:25:24.220 Flip, shift, multiply, sum. 00:25:29.820 --> 00:25:35.180 So I flip, shift. 00:25:35.180 --> 00:25:44.010 So when I shift, the new answer looks like 1, 2/3, 4/9, 8/27, 00:25:44.010 --> 00:25:46.130 et cetera. 00:25:46.130 --> 00:25:48.970 Multiply and sum, what's the answer? 00:25:48.970 --> 00:25:49.950 AUDIENCE: [INAUDIBLE] 00:25:49.950 --> 00:25:51.970 PROFESSOR: 4/3. 00:25:51.970 --> 00:25:57.095 So the only non-zero answers are 1 times 2/3 plus 2/3 times 1-- 00:25:57.095 --> 00:25:57.595 4/3. 00:26:00.360 --> 00:26:08.320 If I want to compute y of 0 1 2, I shift it further. 00:26:08.320 --> 00:26:14.810 So I do 1, 2/3, 4/9, 8/27, blah, blah, blah. 00:26:17.710 --> 00:26:20.220 So flip, shift, I shift 1 more, multiply, 00:26:20.220 --> 00:26:24.900 sum, multiply 1 times 4/9, and I get 4/9. 00:26:24.900 --> 00:26:29.160 Multiply 2/3 times 2/3, I get 4/9. 00:26:29.160 --> 00:26:32.920 Multiply 4/9 times 1, I get 4/9. 00:26:32.920 --> 00:26:36.670 The answer to the sum of those is 4/3. 00:26:36.670 --> 00:26:39.750 So I get one 4/3, 4/3. 00:26:39.750 --> 00:26:41.640 So you can see it's tracing out. 00:26:41.640 --> 00:26:44.340 This wave form so far, this is the only one that 00:26:44.340 --> 00:26:48.810 has up and then flat, and if I continue that process 00:26:48.810 --> 00:26:50.010 it will start to fall off. 00:26:53.250 --> 00:26:57.980 If you're exclusively mathematically minded, 00:26:57.980 --> 00:27:01.100 you can also just do it with math. 00:27:01.100 --> 00:27:03.800 All you do is think about a mathematical description 00:27:03.800 --> 00:27:06.430 of the left signal. 00:27:06.430 --> 00:27:12.440 Say 2/3 of the nu of n and the right signal, and now all 00:27:12.440 --> 00:27:15.810 I need to do is think about that formula. 00:27:15.810 --> 00:27:19.670 So do a sum, taking this, the function of n 00:27:19.670 --> 00:27:24.010 and turning it into a function of k. 00:27:24.010 --> 00:27:27.460 The second one, I want to make a function of n minus k. 00:27:27.460 --> 00:27:29.740 I have to shift both. 00:27:29.740 --> 00:27:34.090 I have to change the exponent as well as the index 00:27:34.090 --> 00:27:39.350 into the unit sample signal, same thing over here. 00:27:39.350 --> 00:27:42.910 Now when I think about multiplying them, u of k 00:27:42.910 --> 00:27:45.760 kills all the terms for k less than 0, 00:27:45.760 --> 00:27:50.660 so I can start at 0 instead of minus infinity. 00:27:50.660 --> 00:27:57.170 This u kills everything for which n minus k is less than 0. 00:27:57.170 --> 00:27:59.870 That means k less than n-- 00:27:59.870 --> 00:28:01.430 less than or equal to n. 00:28:01.430 --> 00:28:04.180 So I end up with this. 00:28:04.180 --> 00:28:06.180 This product is particularly easy 00:28:06.180 --> 00:28:10.060 because it's the the k into the minus k, 00:28:10.060 --> 00:28:15.590 so the answer is to the n, and now I'm summing over k, 00:28:15.590 --> 00:28:20.380 but there are no k's, so that's summing over 1. 00:28:20.380 --> 00:28:23.000 And so my answer is just n plus 1 and 2/3 00:28:23.000 --> 00:28:26.170 to the nu of n which is the same thing by thinking about it 00:28:26.170 --> 00:28:29.650 intuitively. 00:28:29.650 --> 00:28:36.060 So the point is that the operation is friendly, 00:28:36.060 --> 00:28:40.200 and so the idea then the big picture was convolution 00:28:40.200 --> 00:28:45.400 is a different way to represent a system. 00:28:45.400 --> 00:28:49.180 Using convolution, we represent an entire system 00:28:49.180 --> 00:28:53.120 by a single signal. 00:28:53.120 --> 00:28:55.670 That signal, the unit sample response, 00:28:55.670 --> 00:28:58.550 is sufficient to characterize the output of the system 00:28:58.550 --> 00:28:59.930 for any possible input. 00:28:59.930 --> 00:29:03.950 We just saw how the operation is called convolution, 00:29:03.950 --> 00:29:06.230 so that enables us the big picture. 00:29:06.230 --> 00:29:11.810 We've represented an entire system by a single signal, 00:29:11.810 --> 00:29:14.905 in this case h of n. 00:29:14.905 --> 00:29:16.030 That's what convolution is. 00:29:16.030 --> 00:29:18.710 It's a new representation. 00:29:18.710 --> 00:29:25.940 You can do exactly the same thing for a CT system, 00:29:25.940 --> 00:29:29.760 and the reason we use delta to represent the unit sample 00:29:29.760 --> 00:29:31.950 signal and the unit impulse response. 00:29:31.950 --> 00:29:35.590 The unit impulse signal is clear, 00:29:35.590 --> 00:29:41.350 because the representation of an arbitrary signal, 00:29:41.350 --> 00:29:45.480 in terms of delta functions, looks much the same 00:29:45.480 --> 00:29:49.450 in CT and in DT. 00:29:49.450 --> 00:29:53.200 You can get there by thinking about the limiting argument 00:29:53.200 --> 00:29:56.020 for how to interpret the unit impulse. 00:29:56.020 --> 00:29:59.110 The unit impulse function was a function 00:29:59.110 --> 00:30:02.030 that is easiest to think about in a limit. 00:30:02.030 --> 00:30:05.500 Imagine that I have a signal that 00:30:05.500 --> 00:30:10.780 is a square pulse whose area, regardless of width, is 1. 00:30:10.780 --> 00:30:15.130 That's what a unit impulse function is. 00:30:15.130 --> 00:30:18.560 Imagine how you would construct an arbitrary signal x of t 00:30:18.560 --> 00:30:21.160 by having such a signal. 00:30:21.160 --> 00:30:25.920 You could take a signal and it's shifted version, and come up 00:30:25.920 --> 00:30:29.970 with a weighted sum of impulse functions 00:30:29.970 --> 00:30:33.900 or rectangular approximations to impulse functions 00:30:33.900 --> 00:30:36.900 to represent an arbitrary signal. 00:30:36.900 --> 00:30:40.950 If you did that, you would get an approximation to the signal 00:30:40.950 --> 00:30:44.130 x which could be written as a limit. 00:30:47.040 --> 00:30:51.080 So if you think about each of these being of width capital 00:30:51.080 --> 00:30:54.420 delta, then the height has to be 1 over delta. 00:30:54.420 --> 00:30:59.780 So the area remains one. 00:30:59.780 --> 00:31:02.690 Then if I want to build an arbitrary function x 00:31:02.690 --> 00:31:06.710 out of such signals, I need a sum of them, 00:31:06.710 --> 00:31:10.950 and each one of these p's has to be multiplied by delta. 00:31:10.950 --> 00:31:15.560 So that when I multiply by the value of x at one 00:31:15.560 --> 00:31:17.330 point k delta. 00:31:17.330 --> 00:31:22.860 I get the right height independent of what is delta. 00:31:22.860 --> 00:31:26.330 So for a given delta, I get a sum that looks like that, 00:31:26.330 --> 00:31:28.280 and then in keeping with the idea of thinking 00:31:28.280 --> 00:31:30.440 about a unit impulse function as a limit, 00:31:30.440 --> 00:31:32.720 I take the limit of that. 00:31:32.720 --> 00:31:34.580 The result is a function that looks 00:31:34.580 --> 00:31:40.160 very much like the decomposition of a signal in terms 00:31:40.160 --> 00:31:43.070 of the unit sample. 00:31:43.070 --> 00:31:48.200 In the previous case, we sum together a weighted version 00:31:48.200 --> 00:31:51.050 of a unit sample signal. 00:31:51.050 --> 00:31:53.630 Here the sum is replaced by an integral, 00:31:53.630 --> 00:31:55.425 and is weighted just like it was before. 00:31:58.690 --> 00:32:04.180 The point is that the mathematics for CT and DT 00:32:04.180 --> 00:32:06.220 look very similar. 00:32:06.220 --> 00:32:11.950 I decompose in the case of the CT, and arbitrary signal x of t 00:32:11.950 --> 00:32:19.670 into an entire row of weighted unit impulse functions. 00:32:19.670 --> 00:32:21.830 Once I have it in that form, the argument's 00:32:21.830 --> 00:32:25.820 precisely the same for CT as it was in DT. 00:32:25.820 --> 00:32:28.940 Imagine that I have a linear time invariant system. 00:32:28.940 --> 00:32:33.530 Linear means that I can compute the response to a sum 00:32:33.530 --> 00:32:36.860 as the sum of the responses. 00:32:36.860 --> 00:32:39.950 Time invariant means that shifting the input 00:32:39.950 --> 00:32:41.680 merely shifts the output. 00:32:41.680 --> 00:32:43.820 Doing the experiment tomorrow is the same 00:32:43.820 --> 00:32:49.010 as doing the experiment today, except it's now a day later. 00:32:49.010 --> 00:32:53.330 So if the response of a system is 00:32:53.330 --> 00:32:56.510 h of t when the input is delta of t, 00:32:56.510 --> 00:32:59.495 if the system is shift invariant, shifting this by tau 00:32:59.495 --> 00:33:01.095 is the same as shifting that by tau. 00:33:04.240 --> 00:33:10.970 A weighted sum of such things is a weighted sum of such things, 00:33:10.970 --> 00:33:14.780 and a sum of such things is the sum of such things, 00:33:14.780 --> 00:33:16.760 so I get an expression which we'll 00:33:16.760 --> 00:33:20.570 think of as convolution for CT that looks just the same. 00:33:23.290 --> 00:33:28.120 So in DT, we thought about if you convolve x with h, 00:33:28.120 --> 00:33:31.030 you take the first index, x of n, 00:33:31.030 --> 00:33:34.750 and turn it into x of k, a dummy variable. 00:33:34.750 --> 00:33:37.900 You take the second one, and do n minus k. 00:33:37.900 --> 00:33:42.540 Here we do the same thing. t goes to tau, 00:33:42.540 --> 00:33:46.158 and the second one goes to t minus tau. 00:33:46.158 --> 00:33:48.476 The sum up here turns into an integral. 00:33:48.476 --> 00:33:50.100 Otherwise, it's exactly the same thing. 00:33:52.750 --> 00:33:58.610 So to show your mastery of such things, what signal would 00:33:58.610 --> 00:34:02.930 result if you convolved e to the minus tu of t with e 00:34:02.930 --> 00:34:04.185 to the minus tu of t? 00:34:04.185 --> 00:34:05.840 1, 2, 3, 4, or none? 00:35:53.430 --> 00:35:55.260 Well, the place is quiet so I assume 00:35:55.260 --> 00:35:56.940 that means you stopped talking, so that 00:35:56.940 --> 00:36:01.400 means you've all agreed, yes? 00:36:01.400 --> 00:36:03.980 So which wave form best represents 00:36:03.980 --> 00:36:08.350 the convolution of the top two signals, 1, 2, 3, or 4? 00:36:14.150 --> 00:36:15.650 Almost 100% correct. 00:36:15.650 --> 00:36:16.830 Most people say 4. 00:36:16.830 --> 00:36:17.720 How do you get 4? 00:36:23.900 --> 00:36:25.040 Yeah? 00:36:25.040 --> 00:36:28.470 AUDIENCE: Same reason [INAUDIBLE] 00:36:28.470 --> 00:36:31.144 PROFESSOR: So what would I do? 00:36:31.144 --> 00:36:32.310 What would be my first step? 00:36:34.830 --> 00:36:37.530 So I imagine that I want to think 00:36:37.530 --> 00:36:44.490 of flip so this gets multiplied by the flip of the other one. 00:36:44.490 --> 00:36:48.900 So at time t equals 0, the answer is-- 00:36:48.900 --> 00:36:49.729 AUDIENCE: 0 00:36:49.729 --> 00:36:51.520 PROFESSOR: --0, because there's no overlap. 00:36:54.316 --> 00:36:55.720 AUDIENCE: [INAUDIBLE] 00:36:55.720 --> 00:36:57.730 PROFESSOR: OK so that it starts at 0, 00:36:57.730 --> 00:37:00.540 so that means that this is out. 00:37:00.540 --> 00:37:04.590 OK, OK, OK, fine. 00:37:04.590 --> 00:37:05.380 Now shift. 00:37:05.380 --> 00:37:06.330 What do I shift? 00:37:06.330 --> 00:37:09.484 Which one do I shift which way? 00:37:09.484 --> 00:37:13.510 AUDIENCE: Why is there a [INAUDIBLE] flipping is there 00:37:13.510 --> 00:37:15.760 a value that equals 0? 00:37:15.760 --> 00:37:17.940 PROFESSOR: That's a valid question. 00:37:17.940 --> 00:37:22.170 So the question is if I'm integrating 00:37:22.170 --> 00:37:26.220 over a function that goes from minus infinity to 0, and from 0 00:37:26.220 --> 00:37:27.220 to infinity. 00:37:27.220 --> 00:37:30.780 Let's say the answer right at 0 is 1. 00:37:30.780 --> 00:37:35.190 You could say that there is a single point 00:37:35.190 --> 00:37:38.250 whose value is non-zero. 00:37:38.250 --> 00:37:41.940 What would happen if I integrated a function that is 0 00:37:41.940 --> 00:37:44.460 everywhere except at a point? 00:37:44.460 --> 00:37:49.800 So it's 0 everywhere up to here, then a 0 everywhere after that, 00:37:49.800 --> 00:37:51.540 and at zero it's not zero. 00:37:51.540 --> 00:37:54.360 What's the integral of a function that differs 00:37:54.360 --> 00:37:57.090 from 0 at a single point? 00:37:57.090 --> 00:37:58.400 AUDIENCE: [INAUDIBLE] 00:37:58.400 --> 00:38:01.380 PROFESSOR: 0, it's a little bit of a trick question, 00:38:01.380 --> 00:38:04.290 because we will later have some functions for which that's not 00:38:04.290 --> 00:38:06.090 true. 00:38:06.090 --> 00:38:09.004 What kind of a function would that not be true for? 00:38:09.004 --> 00:38:10.812 AUDIENCE: [INAUDIBLE] 00:38:10.812 --> 00:38:12.620 PROFESSOR: Delta. 00:38:12.620 --> 00:38:19.870 If I were convolving delta with delta, 00:38:19.870 --> 00:38:25.560 then you can integrate over an infinitesimal area region, 00:38:25.560 --> 00:38:28.230 and get something that's not 0. 00:38:28.230 --> 00:38:30.810 So a little bit of a caveat, so as long 00:38:30.810 --> 00:38:33.570 as the function that I'm convolving 00:38:33.570 --> 00:38:36.540 doesn't have an impulse in it or worse. 00:38:36.540 --> 00:38:38.730 We will talk later in the course about things worse 00:38:38.730 --> 00:38:40.650 than impulses. 00:38:40.650 --> 00:38:46.110 If there's nothing as high as an impulse or worse, 00:38:46.110 --> 00:38:48.770 so that has a step in it. 00:38:48.770 --> 00:38:51.090 We would think of a step as a singularity that 00:38:51.090 --> 00:38:56.550 is better, less ill behaved than an impulse. 00:38:56.550 --> 00:39:02.220 As long as the function does not have an impulse or a worse, 00:39:02.220 --> 00:39:07.160 when you flip it, you'll get zero contribution at zero. 00:39:07.160 --> 00:39:09.780 That all makes sense? 00:39:09.780 --> 00:39:12.860 So this is zero at zero, but the reasoning is a little bit 00:39:12.860 --> 00:39:13.820 complicated. 00:39:13.820 --> 00:39:18.170 So now what do I get when t gets a little bit bigger than 0? 00:39:18.170 --> 00:39:21.680 What's the result of convolving when the time is 00:39:21.680 --> 00:39:25.870 slightly bigger than time 0? 00:39:25.870 --> 00:39:29.920 All flip, shift, multiply, integrate. 00:39:29.920 --> 00:39:32.610 So I have to shift one of those, so now 00:39:32.610 --> 00:39:36.570 instead of having this one, I might have shifted a little bit 00:39:36.570 --> 00:39:39.370 to the right. 00:39:39.370 --> 00:39:41.190 So it might look like that. 00:39:41.190 --> 00:39:44.636 So now what happens when I multiply? 00:39:44.636 --> 00:39:47.360 Well you don't get 0 anymore. 00:39:47.360 --> 00:39:52.280 So as for small t for t on the order of epsilon, 00:39:52.280 --> 00:39:57.510 how does the integral grow with t? 00:39:57.510 --> 00:40:01.421 So if I want to make a plot of the convolution, 00:40:01.421 --> 00:40:03.170 so if I want to think about e to the minus 00:40:03.170 --> 00:40:11.990 tu of t convolved with e to the minus tu of t versus t, 00:40:11.990 --> 00:40:15.800 I already know that that's like that for t small. 00:40:15.800 --> 00:40:16.940 How will the function grow? 00:40:20.040 --> 00:40:26.850 Linear so if this is very small then 00:40:26.850 --> 00:40:34.050 the deviations from the height which is 1 is very small. 00:40:34.050 --> 00:40:36.600 So if this distance is small, the deviation 00:40:36.600 --> 00:40:39.870 is that little triangle which goes like t-square. 00:40:39.870 --> 00:40:45.015 So for t small t-square is very small compared to t, 00:40:45.015 --> 00:40:48.960 I can ignore it, and so the function 00:40:48.960 --> 00:40:52.940 is going to start going up like t, 00:40:52.940 --> 00:40:56.720 and if you work out the details it will eventually roll off. 00:40:56.720 --> 00:41:00.830 Because as you shift it further and further, 00:41:00.830 --> 00:41:05.840 the one exponential is in the tail of the other exponential, 00:41:05.840 --> 00:41:08.610 so one of the exponentials kills the other one, 00:41:08.610 --> 00:41:10.110 and so the response goes to zero. 00:41:12.830 --> 00:41:15.066 If you're more mathematically inclined, yes? 00:41:15.066 --> 00:41:17.298 AUDIENCE: [INAUDIBLE] you were saying 00:41:17.298 --> 00:41:20.026 that if both functions are left sided [INAUDIBLE] 00:41:20.026 --> 00:41:25.494 right sided it always starts out t equals 0 is always 0. 00:41:25.494 --> 00:41:27.910 PROFESSOR: That's not quite right because right sided just 00:41:27.910 --> 00:41:31.520 means that the left is 0. 00:41:31.520 --> 00:41:34.940 So if I tell you that a signal is right sided, all I've said 00:41:34.940 --> 00:41:38.430 is that all of the non-zero values are on the right, 00:41:38.430 --> 00:41:42.330 but I haven't told you whether they're impulses or not. 00:41:42.330 --> 00:41:44.310 Right sided says something about the left. 00:41:44.310 --> 00:41:46.620 The left is zero. 00:41:46.620 --> 00:41:50.025 Kind of weird, so if I'm right handed, 00:41:50.025 --> 00:41:51.900 I might as well not have a left hand when I'm 00:41:51.900 --> 00:41:54.910 writing so the left is zero. 00:41:54.910 --> 00:41:58.200 So right sided signals have zeros. 00:41:58.200 --> 00:42:02.430 The signals on the left of right sided signals are zero, 00:42:02.430 --> 00:42:04.305 but I haven't told you what was on the right. 00:42:04.305 --> 00:42:06.300 The right could be an impulse or worse. 00:42:09.580 --> 00:42:11.770 So I just inferred some properties 00:42:11.770 --> 00:42:13.750 of what this convolution is going to look like. 00:42:13.750 --> 00:42:15.250 If you were mathematically inclined, 00:42:15.250 --> 00:42:17.450 you could do it by math. 00:42:17.450 --> 00:42:18.950 The math doesn't look very different 00:42:18.950 --> 00:42:23.000 from the math for the discrete version. 00:42:23.000 --> 00:42:27.950 You simply write this as a function of tau rather than t. 00:42:27.950 --> 00:42:31.100 This one as a function of t minus tau rather than t. 00:42:34.300 --> 00:42:37.240 Recognize that the u's cut off parts of the integral. 00:42:37.240 --> 00:42:42.040 This u is 1 only if t is bigger than 0. 00:42:42.040 --> 00:42:45.010 So that lops off the t less than 0 part. 00:42:45.010 --> 00:42:49.210 This lopped off the part bigger than capital, bigger than t. 00:42:49.210 --> 00:42:53.460 So that leaves the integrals 0 to t. 00:42:53.460 --> 00:42:55.680 Putting these two together results 00:42:55.680 --> 00:42:58.120 in the tau parts killing each other, 00:42:58.120 --> 00:43:01.500 so I'm left with only a t part but the integrals on tau. 00:43:01.500 --> 00:43:04.980 So just like the other one, the integral 00:43:04.980 --> 00:43:08.990 goes to the integral of 1 over finite limit 00:43:08.990 --> 00:43:11.910 0 to t, so the answer is t. 00:43:11.910 --> 00:43:15.180 So the overall answer is te to the minus tu of t 00:43:15.180 --> 00:43:16.170 which is plotted here. 00:43:18.840 --> 00:43:22.320 So the point of today then is that this 00:43:22.320 --> 00:43:27.370 is a different representation for the way systems work. 00:43:27.370 --> 00:43:29.900 It's often computationally interesting. 00:43:29.900 --> 00:43:30.820 This is the way. 00:43:30.820 --> 00:43:32.320 This is a perfectly plausible way 00:43:32.320 --> 00:43:35.740 of doing discrete time signal processing. 00:43:35.740 --> 00:43:40.780 Represent the system by a signal h of n, 00:43:40.780 --> 00:43:45.610 and then compute the response to any signal by convolving h of n 00:43:45.610 --> 00:43:47.410 with that signal. 00:43:47.410 --> 00:43:50.570 Perfectly reasonable way to compute things. 00:43:50.570 --> 00:43:52.930 Honestly, we'll find better ways of computing things 00:43:52.930 --> 00:43:55.820 by the end of the course. 00:43:55.820 --> 00:43:59.710 The real reason for studying convolution is conceptually, 00:43:59.710 --> 00:44:01.540 you can think about how a system ought 00:44:01.540 --> 00:44:04.349 to work by thinking about convolution, 00:44:04.349 --> 00:44:05.890 and I want to show an example of that 00:44:05.890 --> 00:44:09.430 by thinking about systems that I'm interested in. 00:44:09.430 --> 00:44:10.370 I do work on hearing. 00:44:10.370 --> 00:44:14.560 I do work with microscopes, and we can regard a microscope 00:44:14.560 --> 00:44:15.550 as an LTI system. 00:44:15.550 --> 00:44:18.700 A linear time invariant system, and convolution 00:44:18.700 --> 00:44:22.735 is a very good way of thinking about such optical systems. 00:44:25.300 --> 00:44:30.040 So the idea is that even the best microscope 00:44:30.040 --> 00:44:36.306 gives blurry images, and that's very fundamental physics. 00:44:36.306 --> 00:44:37.680 It has to do with the diffraction 00:44:37.680 --> 00:44:39.549 limit of optical systems. 00:44:39.549 --> 00:44:41.340 If you're interested in that sort of thing, 00:44:41.340 --> 00:44:44.970 take an optics course in the physics department 00:44:44.970 --> 00:44:48.730 or come to my lab and do a [INAUDIBLE].. 00:44:48.730 --> 00:44:51.760 So the idea is that even the best microscopes in the world 00:44:51.760 --> 00:44:52.260 are blurred. 00:44:52.260 --> 00:44:55.470 We have the best microscopes in the world in my lab, 00:44:55.470 --> 00:44:57.240 and they fundamentally blur things, 00:44:57.240 --> 00:44:59.760 and we have to worry about that. 00:44:59.760 --> 00:45:02.580 The blurring is inversely related 00:45:02.580 --> 00:45:05.070 to the numerical aperture which has to do 00:45:05.070 --> 00:45:06.870 with the size of the optic. 00:45:06.870 --> 00:45:09.890 Big optics are good. 00:45:09.890 --> 00:45:14.180 So if you imagine that a target emits 00:45:14.180 --> 00:45:16.900 a spherical wave of light. 00:45:16.900 --> 00:45:19.870 So every point on the target emits a spherical wave, 00:45:19.870 --> 00:45:22.915 then there's some optic that's collecting all of those waves, 00:45:22.915 --> 00:45:26.590 , and relaying them back to a different point. 00:45:26.590 --> 00:45:28.150 The resolution of the picture goes 00:45:28.150 --> 00:45:31.540 with how many of those rays the optic system picked up. 00:45:34.210 --> 00:45:37.630 So if you make the optics smaller, 00:45:37.630 --> 00:45:40.420 the picture becomes blurrier. 00:45:40.420 --> 00:45:42.400 If you make the optic even smaller, 00:45:42.400 --> 00:45:44.620 the picture becomes even smaller, 00:45:44.620 --> 00:45:49.800 and the way we think about that is by convolution. 00:45:49.800 --> 00:45:53.810 We think about the microscope as an LTI system, 00:45:53.810 --> 00:45:57.920 so we characterize it by its point spread function. 00:45:57.920 --> 00:45:59.484 We don't like to use any words that 00:45:59.484 --> 00:46:00.650 come from a different field. 00:46:00.650 --> 00:46:04.230 Just like every other field, we like to invent our own. 00:46:04.230 --> 00:46:09.050 So in optics, the thing that we will call a impulse response 00:46:09.050 --> 00:46:11.450 is called a point spread function. 00:46:11.450 --> 00:46:14.570 It just says if you had an ideal point of light, 00:46:14.570 --> 00:46:17.750 what would the image look like? 00:46:17.750 --> 00:46:19.370 It's exactly the same as convolving, 00:46:19.370 --> 00:46:22.670 so you can think of the blurry image as the convolution 00:46:22.670 --> 00:46:25.760 of the effect of the microscope, the point spread function, 00:46:25.760 --> 00:46:28.655 the impulse response with the ideal target. 00:46:31.310 --> 00:46:35.450 So then as you change the size of the optic, 00:46:35.450 --> 00:46:38.810 it changes the size of the impulse response, 00:46:38.810 --> 00:46:40.490 the point spread function. 00:46:40.490 --> 00:46:43.820 Crummy optics, fat point spread motions. 00:46:43.820 --> 00:46:48.410 Fat point spread functions, blurry pictures, 00:46:48.410 --> 00:46:51.290 and so here's a picture of how our system works. 00:46:51.290 --> 00:46:55.610 This is a representation to scale 00:46:55.610 --> 00:46:59.910 of a tiny microscopic bead a fraction of a micron. 00:46:59.910 --> 00:47:03.660 So it's about six times smaller than the image. 00:47:03.660 --> 00:47:06.960 So this is an image taken with our microscope system, 00:47:06.960 --> 00:47:08.990 and you can see that most of the energy 00:47:08.990 --> 00:47:12.620 fits inside a region about half a micron. 00:47:12.620 --> 00:47:15.470 World's best microscope, you can't do better 00:47:15.470 --> 00:47:17.780 than this by physics. 00:47:17.780 --> 00:47:21.110 This is using 500 nanometer light, and the size of that 00:47:21.110 --> 00:47:24.800 has to do with the length scale of the light. 00:47:24.800 --> 00:47:27.500 So you end up with this particular microscope 00:47:27.500 --> 00:47:33.530 not being able to make images with less blurring than that. 00:47:33.530 --> 00:47:35.637 That's the point spread function. 00:47:35.637 --> 00:47:37.970 Now of course, the point spread function of a microscope 00:47:37.970 --> 00:47:40.370 is three dimensional. 00:47:40.370 --> 00:47:44.240 In this class, we're only talking about 1-D time. 00:47:44.240 --> 00:47:48.620 In a microscope is 3D, x, y, and z. 00:47:48.620 --> 00:47:53.240 So the impulse response has extent in x, y, and z. 00:47:53.240 --> 00:47:55.870 So here is a picture taken by Anthony Patire, who 00:47:55.870 --> 00:48:01.040 was a student in my lab of that same tiny little dot. 00:48:01.040 --> 00:48:04.210 When the in-focus plane shows the dot, 00:48:04.210 --> 00:48:06.350 and as you go out of focus, the dot 00:48:06.350 --> 00:48:10.060 gets bigger, smearier, and blurrier. 00:48:10.060 --> 00:48:12.980 What you can do then is assemble those pictures that 00:48:12.980 --> 00:48:16.820 were taken one of the time into a 3-D volume, 00:48:16.820 --> 00:48:20.750 and that 3-D volume then represents the point spread 00:48:20.750 --> 00:48:22.490 function or the three dimensional impulse 00:48:22.490 --> 00:48:26.420 response of the microscope. 00:48:26.420 --> 00:48:29.630 So the idea then is that convolution is a very good way 00:48:29.630 --> 00:48:31.770 to think about optical systems because they 00:48:31.770 --> 00:48:34.820 are very easy to relate to the underlying physics. 00:48:34.820 --> 00:48:39.860 The blurring is a direct result of a fundamental property 00:48:39.860 --> 00:48:40.580 of light. 00:48:40.580 --> 00:48:43.490 The diffraction limit, and you can very accurately 00:48:43.490 --> 00:48:45.350 represent the effect of the blurring 00:48:45.350 --> 00:48:48.590 as convolving with a point spread function and impulse 00:48:48.590 --> 00:48:50.150 response. 00:48:50.150 --> 00:48:53.240 Same sort of thing applies to optics at any scale. 00:48:53.240 --> 00:48:56.270 So going from the microscopic to the rather macroscopic-- that 00:48:56.270 --> 00:48:59.330 is to say the universe and beyond. 00:48:59.330 --> 00:49:02.000 We can think about the Hubble Space Telescope. 00:49:02.000 --> 00:49:04.030 Same thing. 00:49:04.030 --> 00:49:08.030 Light, that's all that matters, and here the issue 00:49:08.030 --> 00:49:10.910 is-- the reason they wanted to make a Space Telescope 00:49:10.910 --> 00:49:13.520 is that there are two principal sources of blurring 00:49:13.520 --> 00:49:16.940 for a ground based telescope. 00:49:16.940 --> 00:49:19.340 One is the atmosphere blurring, because we're 00:49:19.340 --> 00:49:22.250 looking through an atmosphere and other is blurring because 00:49:22.250 --> 00:49:26.600 of the property of the optic elements in the telescope, 00:49:26.600 --> 00:49:29.630 and it turns out pretty easy to show 00:49:29.630 --> 00:49:33.410 that the combined effect of the atmosphere and the lenses 00:49:33.410 --> 00:49:36.650 is the convolution of the individual parts. 00:49:36.650 --> 00:49:39.350 You can think about atmospheric blurring 00:49:39.350 --> 00:49:41.510 as convolving with the atmosphere's point spread 00:49:41.510 --> 00:49:46.350 function, and you can think of the blurring 00:49:46.350 --> 00:49:50.690 due to the microscope as convolving with the blurring 00:49:50.690 --> 00:49:53.360 function due to optics. 00:49:53.360 --> 00:49:56.330 The combined is the convolution of those 00:49:56.330 --> 00:49:59.360 which means that if you've got some amount 00:49:59.360 --> 00:50:04.820 of atmospheric blurring and a telescope made out 00:50:04.820 --> 00:50:11.310 of 12 centimeter optics, then the combined responses showed 00:50:11.310 --> 00:50:16.360 here not very different from each individual. 00:50:16.360 --> 00:50:19.620 But if you made a big telescope-- 00:50:19.620 --> 00:50:21.870 a one meter type telescope-- you might be 00:50:21.870 --> 00:50:24.210 expecting this much blurring. 00:50:24.210 --> 00:50:27.240 But because of the atmosphere, you get much more blurring. 00:50:27.240 --> 00:50:30.600 So the deviation between the small telescope, 00:50:30.600 --> 00:50:33.780 and what you actually measure is not very different. 00:50:33.780 --> 00:50:35.970 The atmosphere makes an enormous difference 00:50:35.970 --> 00:50:40.500 when you start talking about a high resolution telescope. 00:50:40.500 --> 00:50:42.250 That's the reason for putting it in space. 00:50:42.250 --> 00:50:43.830 You get rid of the atmosphere. 00:50:47.340 --> 00:50:50.190 The Hubble Space Telescope was made principally 00:50:50.190 --> 00:50:53.160 out of two big mirrors, both were parabolic, 00:50:53.160 --> 00:50:56.490 both were highly optimized, both were enormous. 00:50:56.490 --> 00:51:00.390 This is the main lens. 00:51:00.390 --> 00:51:05.110 The lens is 2.4 meters, about eight feet in diameter, 00:51:05.110 --> 00:51:07.930 and it was astonishing the thing. 00:51:07.930 --> 00:51:12.470 So in order for a lens to work perfectly like I illustrated, 00:51:12.470 --> 00:51:14.400 it's important that every reflection 00:51:14.400 --> 00:51:18.610 remain in phase coherence. 00:51:18.610 --> 00:51:21.660 So the length that every ray travels 00:51:21.660 --> 00:51:25.200 has to be precisely matched to what it's supposed to be. 00:51:25.200 --> 00:51:29.910 In the case of the Hubble, they matched the surface. 00:51:29.910 --> 00:51:33.250 The surface was controlled to within 10 nanometers. 00:51:33.250 --> 00:51:35.280 That's absurd. 00:51:35.280 --> 00:51:40.540 So the blurring of my microscope was about half a micron. 00:51:40.540 --> 00:51:43.050 So 50 times worse. 00:51:43.050 --> 00:51:45.360 The best I could see with my microscope 00:51:45.360 --> 00:51:47.820 is 50 times worse than was required 00:51:47.820 --> 00:51:49.890 for making this mirror. 00:51:49.890 --> 00:51:53.220 It was absolutely astonishing feat to make the mirror 00:51:53.220 --> 00:51:56.530 and they made a mistake. 00:51:56.530 --> 00:51:58.170 So when they put this in space, they 00:51:58.170 --> 00:52:00.165 were expecting to see pictures like this. 00:52:00.165 --> 00:52:03.090 This is a picture of a distant star. 00:52:03.090 --> 00:52:05.820 They were expecting the distant star would look like this. 00:52:05.820 --> 00:52:07.320 This is what they actually measured, 00:52:07.320 --> 00:52:10.680 and the reason was that the feedback 00:52:10.680 --> 00:52:13.230 system that they used to grind the lens 00:52:13.230 --> 00:52:17.980 made a mistake by 2.2 microns. 00:52:17.980 --> 00:52:21.420 2.2 microns would have been just barely resolvable 00:52:21.420 --> 00:52:25.300 on my microscope, but barely. 00:52:25.300 --> 00:52:26.970 So the hair? 00:52:26.970 --> 00:52:30.360 That's about 100 microns in diameter. 00:52:30.360 --> 00:52:35.310 They were off by 2.2 microns, and because of that, 00:52:35.310 --> 00:52:38.560 it was like a complete disaster. 00:52:38.560 --> 00:52:43.420 So that small error was enough to make the images terrible. 00:52:43.420 --> 00:52:46.770 So the solution-- it wasn't very practical to ship up 00:52:46.770 --> 00:52:50.910 a new lens, so they shipped up eyeglasses. 00:52:50.910 --> 00:52:53.400 The eyeglasses was another transformation, 00:52:53.400 --> 00:52:55.530 just like your eyeglasses. 00:52:55.530 --> 00:52:59.190 Your eyeglasses work by changing the point spread function that 00:52:59.190 --> 00:53:02.100 is determined by your retina and your lens 00:53:02.100 --> 00:53:04.500 into a new point spread function. 00:53:04.500 --> 00:53:06.840 They shipped up eyeglasses, and the result 00:53:06.840 --> 00:53:09.300 of putting the eyeglasses into Hubble 00:53:09.300 --> 00:53:12.090 was to turn this into that, and to give 00:53:12.090 --> 00:53:15.820 some of the most dazzling pictures we've ever had. 00:53:15.820 --> 00:53:19.710 So the point is that convolution is a complete way 00:53:19.710 --> 00:53:21.220 of describing a system. 00:53:21.220 --> 00:53:24.690 It's a very intuitive way for certain kinds of systems, 00:53:24.690 --> 00:53:26.670 and it's especially useful for systems 00:53:26.670 --> 00:53:29.460 like light based systems where blurring 00:53:29.460 --> 00:53:33.930 is a natural way of thinking about the way the system works. 00:53:33.930 --> 00:53:34.890 Have a good time. 00:53:34.890 --> 00:53:37.820 See you tomorrow at 7:30.