WEBVTT 00:00:00.040 --> 00:00:02.470 The following content is provided under a Creative 00:00:02.470 --> 00:00:03.880 Commons license. 00:00:03.880 --> 00:00:06.920 Your support will help MIT OpenCourseWare continue to 00:00:06.920 --> 00:00:10.570 offer high quality educational resources for free. 00:00:10.570 --> 00:00:13.470 To make a donation or view additional materials from 00:00:13.470 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.750 ocw.mit.edu. 00:00:20.750 --> 00:00:56.781 [MUSIC PLAYING] 00:00:56.781 --> 00:00:59.630 PROFESSOR: In the last lecture, we discussed a number 00:00:59.630 --> 00:01:04.610 of general properties for systems, which, as you recall, 00:01:04.610 --> 00:01:08.710 applied both to continuous-time and to 00:01:08.710 --> 00:01:12.060 discrete-time systems. 00:01:12.060 --> 00:01:15.590 These properties were the properties associated with a 00:01:15.590 --> 00:01:17.810 system having memory. 00:01:17.810 --> 00:01:22.390 The issue of whether a system is or isn't invertible, we 00:01:22.390 --> 00:01:27.180 talked about causality and stability, and finally we 00:01:27.180 --> 00:01:31.170 talked about when linearity and time invariance. 00:01:31.170 --> 00:01:34.710 In today's lecture, what I'd like to do is focus 00:01:34.710 --> 00:01:39.210 specifically on linearity and time invariance, and show how 00:01:39.210 --> 00:01:43.420 for systems that have those properties, we can exploit 00:01:43.420 --> 00:01:47.840 them to generate a general representation. 00:01:47.840 --> 00:01:50.610 Let me begin by just reviewing the two 00:01:50.610 --> 00:01:52.700 properties again quickly. 00:01:52.700 --> 00:01:57.480 Time invariance, as you recall, is a property that 00:01:57.480 --> 00:01:59.430 applied both to continuous-time and 00:01:59.430 --> 00:02:05.280 discrete-time systems, and in essence stated that for any 00:02:05.280 --> 00:02:10.070 given input and output relationship if we simply 00:02:10.070 --> 00:02:13.470 shift the input, then the output 00:02:13.470 --> 00:02:15.800 shifts by the same amount. 00:02:15.800 --> 00:02:20.320 And of course, exactly the same kind of statement applied 00:02:20.320 --> 00:02:22.590 in discrete time. 00:02:22.590 --> 00:02:25.510 So time invariance was a property that said that the 00:02:25.510 --> 00:02:28.850 system didn't care about what the time origin 00:02:28.850 --> 00:02:31.540 of the signal is. 00:02:31.540 --> 00:02:35.810 Linearity was a property related to the fact that if we 00:02:35.810 --> 00:02:39.390 have a set of outputs associated with a 00:02:39.390 --> 00:02:42.040 given set of inputs-- 00:02:42.040 --> 00:02:46.910 as I've indicated here with the inputs as phi_k and the 00:02:46.910 --> 00:02:53.160 outputs psi_k, then the property of linearity states 00:02:53.160 --> 00:02:58.250 that if we have an input which is a linear combination of 00:02:58.250 --> 00:03:04.470 those inputs, then the output is a linear combination of the 00:03:04.470 --> 00:03:06.250 associated outputs. 00:03:06.250 --> 00:03:11.590 So that linear combination of inputs generates, for a linear 00:03:11.590 --> 00:03:16.600 system, an output which is a linear combination of the 00:03:16.600 --> 00:03:19.600 associated outputs. 00:03:19.600 --> 00:03:23.940 Now the question is, how can we exploit the properties of 00:03:23.940 --> 00:03:27.530 linearity and time invariance? 00:03:27.530 --> 00:03:31.420 There's a basic strategy which will flow more or less through 00:03:31.420 --> 00:03:33.720 most of this course. 00:03:33.720 --> 00:03:39.660 The strategy is to attempt to decompose a signal, either 00:03:39.660 --> 00:03:43.590 continuous-time or discrete-time, into a set of 00:03:43.590 --> 00:03:46.630 basic signals. 00:03:46.630 --> 00:03:50.410 I've indicated that here. 00:03:50.410 --> 00:03:56.000 And the question then is, what basic signal should we pick? 00:03:56.000 --> 00:03:59.670 Well, the answer, kind of, is we should pick a set of basic 00:03:59.670 --> 00:04:03.210 signals that provide a certain degree of analytical 00:04:03.210 --> 00:04:04.180 convenience. 00:04:04.180 --> 00:04:08.200 So we choose a set of inputs for the decomposition that 00:04:08.200 --> 00:04:13.810 provide outputs that we can easily generate. 00:04:13.810 --> 00:04:20.279 Now, as we'll see, when we do this, there are two classes of 00:04:20.279 --> 00:04:24.890 inputs that are particularly suited to that strategy. 00:04:24.890 --> 00:04:31.170 One class is the set of delayed impulses, namely 00:04:31.170 --> 00:04:35.300 decomposing a signal into a linear combination of these. 00:04:35.300 --> 00:04:39.390 And as we'll see, that leads to a representation for linear 00:04:39.390 --> 00:04:45.840 time-invariant systems, which is referred to as convolution. 00:04:45.840 --> 00:04:50.650 The second is a decomposition of inputs into complex 00:04:50.650 --> 00:04:52.380 exponentials-- 00:04:52.380 --> 00:04:55.450 a linear combination of complex exponentials-- 00:04:55.450 --> 00:05:00.220 and that leads to a representation of signals and 00:05:00.220 --> 00:05:04.045 systems through what we'll refer to as Fourier analysis. 00:05:06.560 --> 00:05:10.290 Now, Fourier analysis will be a topic for a 00:05:10.290 --> 00:05:12.030 set of later lectures. 00:05:12.030 --> 00:05:15.150 What I'd like to begin with is the representation in terms of 00:05:15.150 --> 00:05:18.520 impulses and the associated description of linear 00:05:18.520 --> 00:05:22.520 time-invariant systems using convolution. 00:05:22.520 --> 00:05:27.040 So let's begin with a discussion of discrete-time 00:05:27.040 --> 00:05:31.410 signals, and in particular the issue of how discrete-time 00:05:31.410 --> 00:05:34.770 signals can be decomposed as a linear 00:05:34.770 --> 00:05:39.740 combination of delayed impulses. 00:05:39.740 --> 00:05:43.560 Well, in fact, it's relatively straightforward. 00:05:43.560 --> 00:05:49.140 What I've shown here is a general sequence with values 00:05:49.140 --> 00:05:52.640 which I've indicated at the top. 00:05:52.640 --> 00:05:57.100 And more or less as we did when we talked about 00:05:57.100 --> 00:06:01.660 representing a unit step in terms of impulses, we can 00:06:01.660 --> 00:06:08.170 think of this general sequence as a sequence of impulses-- 00:06:08.170 --> 00:06:14.060 delayed, namely, occurring at the appropriate time instant, 00:06:14.060 --> 00:06:16.950 and with the appropriate amplitude. 00:06:16.950 --> 00:06:21.950 So we can think of this general sequence and an 00:06:21.950 --> 00:06:28.780 impulse occurring at n = 0 and with a height of x[0], plus an 00:06:28.780 --> 00:06:31.960 impulse of height x[1] 00:06:31.960 --> 00:06:35.830 occurring at time n = 1, and so that's x[1] 00:06:35.830 --> 00:06:40.850 delta[n-1], an impulse at -1 with an 00:06:40.850 --> 00:06:42.310 amplitude of x[-1], etc. 00:06:45.030 --> 00:06:50.500 So if we continued to generate a set of weighted, delayed 00:06:50.500 --> 00:06:55.600 unit samples like that, and if we added all these together, 00:06:55.600 --> 00:07:00.120 then that will generate the total sequence. 00:07:00.120 --> 00:07:04.610 Algebraically, then, what that corresponds to is representing 00:07:04.610 --> 00:07:09.500 the sequence as a sum of individual terms as I've 00:07:09.500 --> 00:07:16.490 indicated here or in terms of a general sum, the sum of x[k] 00:07:16.490 --> 00:07:18.820 delta[n-k]. 00:07:18.820 --> 00:07:19.950 So that's our strategy-- 00:07:19.950 --> 00:07:25.400 the strategy is to decompose an arbitrary sequence into a 00:07:25.400 --> 00:07:29.540 linear combination of weighted, delayed impulses. 00:07:29.540 --> 00:07:34.150 And here again is the representation, which we just 00:07:34.150 --> 00:07:37.160 finished generating. 00:07:37.160 --> 00:07:41.370 Now, why is this representation useful? 00:07:41.370 --> 00:07:45.380 It's useful because we now have a decomposition of the 00:07:45.380 --> 00:07:49.790 sequence as a linear combination of basic 00:07:49.790 --> 00:07:53.080 sequences, namely the delayed impulses. 00:07:53.080 --> 00:07:58.920 And if we are talking about a linear system, the response to 00:07:58.920 --> 00:08:01.390 that linear combination is a linear 00:08:01.390 --> 00:08:04.270 combination of the responses. 00:08:04.270 --> 00:08:10.830 So if we denote the response to a delayed impulse as 00:08:10.830 --> 00:08:19.020 h_k[n], then the response to this general input is what 00:08:19.020 --> 00:08:24.720 I've indicated here, where y[n], of course, is the output 00:08:24.720 --> 00:08:27.900 due to the general input x[n]. 00:08:27.900 --> 00:08:28.720 h_k[n] 00:08:28.720 --> 00:08:34.169 in is the output due to the delayed impulse, and these are 00:08:34.169 --> 00:08:37.299 simply the coefficients in the weighting. 00:08:40.169 --> 00:08:45.340 So for a linear system, we have this representation. 00:08:45.340 --> 00:08:50.830 And if now, in addition, the system is time-invariant, we 00:08:50.830 --> 00:08:54.810 can, in fact, relate the outputs due to these 00:08:54.810 --> 00:08:57.240 individual delayed impulses. 00:08:57.240 --> 00:09:01.220 Specifically, if the system is time-invariant, then the 00:09:01.220 --> 00:09:08.330 response to an impulse at time k is exactly the same as the 00:09:08.330 --> 00:09:13.470 response to an impulse at time 0, shifted over to time k. 00:09:13.470 --> 00:09:16.890 Said another way, h_k[n] 00:09:16.890 --> 00:09:24.450 is simply h_0[n-k], where h_0 is the response of the system 00:09:24.450 --> 00:09:27.140 to an impulse at n = 0. 00:09:27.140 --> 00:09:32.060 And it's generally useful to, rather than carrying around 00:09:32.060 --> 00:09:36.800 h_0[n], just simply define h_0[n] 00:09:36.800 --> 00:09:42.700 as h[n], which is the unit sample or unit impulse 00:09:42.700 --> 00:09:45.070 response of the system. 00:09:45.070 --> 00:09:49.440 And so the consequence, then, is for a linear time-invariant 00:09:49.440 --> 00:09:54.520 system, the output can be expressed as 00:09:54.520 --> 00:09:58.510 this sum where h[n-k] 00:09:58.510 --> 00:10:03.600 is the response to an impulse occurring at time n = k. 00:10:03.600 --> 00:10:10.780 And this is referred to as the convolution sum. 00:10:10.780 --> 00:10:12.450 Now, we can-- 00:10:12.450 --> 00:10:17.570 just to emphasize how we've gone about this, let me show 00:10:17.570 --> 00:10:18.820 it from another perspective. 00:10:22.040 --> 00:10:28.080 We of course have taken the sequence x[n], we have 00:10:28.080 --> 00:10:33.180 decomposed it as a linear combination of these weighted, 00:10:33.180 --> 00:10:36.010 delayed impulses. 00:10:36.010 --> 00:10:39.400 When these are added together, those correspond to the 00:10:39.400 --> 00:10:42.810 original sequence x[n]. 00:10:42.810 --> 00:10:48.200 If this impulse, for example, generates a 00:10:48.200 --> 00:10:51.990 response which is x[0] 00:10:51.990 --> 00:10:53.290 h[n], where h[n] 00:10:53.290 --> 00:10:58.750 is the response to a unit impulse at n = 0, and the 00:10:58.750 --> 00:11:03.150 second one generates a delayed weighted response, and the 00:11:03.150 --> 00:11:07.770 third one similarly, and we generate these individual 00:11:07.770 --> 00:11:12.050 responses, these are all added together, and it's that linear 00:11:12.050 --> 00:11:16.180 combination that forms the final output. 00:11:16.180 --> 00:11:18.450 So that's really kind of the way we're thinking about it. 00:11:18.450 --> 00:11:22.870 We have a general sequence, we're thinking of each 00:11:22.870 --> 00:11:26.160 individual sample individually, each one of 00:11:26.160 --> 00:11:29.890 those pops the system, and because of linearity, the 00:11:29.890 --> 00:11:34.930 response is the sum of those individual responses. 00:11:34.930 --> 00:11:38.720 That's what happens in discrete time, and pretty much 00:11:38.720 --> 00:11:43.050 the same strategy works in continuous time. 00:11:43.050 --> 00:11:51.380 In particular, we can begin in continuous time with the 00:11:51.380 --> 00:11:57.600 notion of decomposing a continuous-time signal into a 00:11:57.600 --> 00:12:02.060 succession of arbitrarily narrow rectangles. 00:12:02.060 --> 00:12:06.670 And as the width of the rectangles goes to 0, the 00:12:06.670 --> 00:12:09.080 approximation gets better. 00:12:09.080 --> 00:12:11.410 Essentially what's going to happen is that each of those 00:12:11.410 --> 00:12:15.510 individual rectangles, as they get narrower and narrower, 00:12:15.510 --> 00:12:18.770 correspond more and more to an impulse. 00:12:18.770 --> 00:12:21.970 Let me show you what I mean. 00:12:21.970 --> 00:12:29.540 Here we have a continuous-time signal, and I've approximated 00:12:29.540 --> 00:12:33.530 it by a staircase. 00:12:33.530 --> 00:12:39.970 So in essence I can think of this as individual rectangles 00:12:39.970 --> 00:12:42.710 of heights associated with the height of the continuous 00:12:42.710 --> 00:12:46.220 curve, and so I've indicated that down below. 00:12:46.220 --> 00:12:50.120 Here, for example, is the impulse corresponding to the 00:12:50.120 --> 00:12:56.110 rectangle between t = -2 Delta and t = -Delta. 00:12:56.110 --> 00:13:01.830 Here's the one from -Delta to 0, and as we continue on down, 00:13:01.830 --> 00:13:06.410 we get impulses, or rather rectangles, from successive 00:13:06.410 --> 00:13:08.810 parts of the wave form. 00:13:08.810 --> 00:13:14.270 Now let's look specifically at the rectangle, for example, 00:13:14.270 --> 00:13:19.410 starting at 0 and ending at Delta, and the amplitude of it 00:13:19.410 --> 00:13:20.660 is x(Delta). 00:13:24.730 --> 00:13:26.760 So what we have-- 00:13:26.760 --> 00:13:32.040 actually, this should be x(0), and so let me just 00:13:32.040 --> 00:13:33.480 correct that here. 00:13:33.480 --> 00:13:35.280 That's x(0). 00:13:35.280 --> 00:13:40.660 And so we have a rectangle height x(0), and recall the 00:13:40.660 --> 00:13:46.740 function that I defined last time as delta_Delta(t), which 00:13:46.740 --> 00:13:50.010 had height 1 / delta and width delta. 00:13:50.010 --> 00:13:55.060 So multiplying finally by this last little Delta, then, this 00:13:55.060 --> 00:13:58.580 is a representation for the rectangle 00:13:58.580 --> 00:14:00.110 that I've shown there. 00:14:00.110 --> 00:14:03.010 Now there's a little bit of algebra there to kind of track 00:14:03.010 --> 00:14:07.510 through, but what we're really doing is just simply 00:14:07.510 --> 00:14:10.490 representing this in terms of rectangles. 00:14:10.490 --> 00:14:15.390 What I want to do is describe each rectangle as in terms of 00:14:15.390 --> 00:14:19.360 that function delta_Delta(t), which in the limit, then, 00:14:19.360 --> 00:14:21.330 becomes an impulse. 00:14:21.330 --> 00:14:23.290 So let's track that through a little further. 00:14:25.800 --> 00:14:29.490 When we have that linear combination, then, we're 00:14:29.490 --> 00:14:34.300 saying that x(t) can be represented by a sum as I 00:14:34.300 --> 00:14:42.180 indicate here, which I can then write more generally in 00:14:42.180 --> 00:14:46.650 this form, just indicating that this is an infinite sum. 00:14:46.650 --> 00:14:51.110 We now want to take the limit as Delta goes to 0, and as 00:14:51.110 --> 00:14:56.110 Delta goes to 0, notice that this term 00:14:56.110 --> 00:14:59.370 becomes arbitrarily narrow. 00:14:59.370 --> 00:15:02.820 This goes to our impulse function, and this, of course, 00:15:02.820 --> 00:15:05.150 goes to x of tau. 00:15:05.150 --> 00:15:10.840 And in fact, in the limit, a sum of this form is exactly 00:15:10.840 --> 00:15:13.270 the way an integral is defined. 00:15:13.270 --> 00:15:17.040 So we have an expression for y(t) in terms 00:15:17.040 --> 00:15:18.800 of an impulse function. 00:15:18.800 --> 00:15:22.460 There, I have to admit, is a little bit of detail to kind 00:15:22.460 --> 00:15:26.700 of focus on at your leisure, but this is the general flow 00:15:26.700 --> 00:15:28.300 of the strategy. 00:15:28.300 --> 00:15:32.500 So what we have now is an integral that tells us that 00:15:32.500 --> 00:15:38.640 tells us how x(t) can be described as a sum or linear 00:15:38.640 --> 00:15:43.120 combination involving impulses. 00:15:43.120 --> 00:15:46.430 This bottom equation, by the way, is often referred to as 00:15:46.430 --> 00:15:48.170 the sifting integral. 00:15:48.170 --> 00:15:52.130 In essence, what it says is that if I take a time function 00:15:52.130 --> 00:15:56.720 x(t) and put it through that integral, the impulse as it 00:15:56.720 --> 00:16:02.070 zips by generates x(t) all over again. 00:16:02.070 --> 00:16:06.000 Now, at first glance, what it could look like is that we've 00:16:06.000 --> 00:16:10.610 taken a time function x(t) and proceeded to represent it in a 00:16:10.610 --> 00:16:14.780 very complicated way, in terms of itself, and one could ask, 00:16:14.780 --> 00:16:16.720 why bother doing that? 00:16:16.720 --> 00:16:22.940 And the reason, going back to what our strategy was, is that 00:16:22.940 --> 00:16:29.240 what we want to do is exploit the property of linearity. 00:16:29.240 --> 00:16:34.630 So by describing a time function as a linear 00:16:34.630 --> 00:16:39.370 combination of weighted, delayed impulses, as in effect 00:16:39.370 --> 00:16:44.350 we've done through this summation that corresponds to 00:16:44.350 --> 00:16:49.580 a decomposition in terms of impulses, we can now exploit 00:16:49.580 --> 00:16:54.350 linearity, specifically recognizing that the output of 00:16:54.350 --> 00:17:00.440 a linear system is the sum of the responses to these 00:17:00.440 --> 00:17:02.120 individual inputs. 00:17:02.120 --> 00:17:09.750 So with h_kDelta(t) corresponding to the response 00:17:09.750 --> 00:17:15.770 to delta_Delta(t-kDelta) and the rest of this stuff, the 00:17:15.770 --> 00:17:18.700 x(kDelta) and this little Delta are 00:17:18.700 --> 00:17:21.140 basically scale factors-- 00:17:21.140 --> 00:17:24.839 for a linear system, then, if the input is expressed in this 00:17:24.839 --> 00:17:29.610 form, the output is expressed in this form, and again taking 00:17:29.610 --> 00:17:33.460 the limit as Delta goes to 0, by definition, this 00:17:33.460 --> 00:17:35.710 corresponds to an integral. 00:17:35.710 --> 00:17:40.380 It's the integral that I indicate here with h_tau(t) 00:17:40.380 --> 00:17:47.500 corresponding to the impulse response due to an impulse 00:17:47.500 --> 00:17:50.360 occurring at time tau. 00:17:50.360 --> 00:17:54.580 Now, again, we can do the same thing. 00:17:54.580 --> 00:17:58.760 In particular, if the system is time-invariant, then the 00:17:58.760 --> 00:18:02.920 response to each of these delayed impulses is simply a 00:18:02.920 --> 00:18:07.870 delayed version of the impulse response, and so we can relate 00:18:07.870 --> 00:18:10.160 these individual terms. 00:18:10.160 --> 00:18:13.500 And in particular, then, the response to an impulse 00:18:13.500 --> 00:18:18.850 occurring at time t = tau is simply the response to an 00:18:18.850 --> 00:18:23.020 impulse occurring at time 0 shifted over to the time 00:18:23.020 --> 00:18:24.840 origin tau. 00:18:24.840 --> 00:18:30.020 Again, as we did before, we'll drop this subscript h_0, so 00:18:30.020 --> 00:18:37.090 h_0(t) we'll simply define as h(t). 00:18:37.090 --> 00:18:42.110 What we're left with when we do that is the description of 00:18:42.110 --> 00:18:46.690 a linear time-invariant system through this integral, which 00:18:46.690 --> 00:18:53.240 tells us how the output is related to the input and to 00:18:53.240 --> 00:18:54.990 the impulse response. 00:18:54.990 --> 00:18:58.910 Again, let's just quickly look at this from another 00:18:58.910 --> 00:19:01.240 perspective as we did in discrete time. 00:19:03.840 --> 00:19:07.740 Recall that what we've done is to take the continuous 00:19:07.740 --> 00:19:14.510 function, decompose it in terms of rectangles, and then 00:19:14.510 --> 00:19:19.970 each of these rectangles generates its individual 00:19:19.970 --> 00:19:23.600 response, and then these individual 00:19:23.600 --> 00:19:25.225 responses are added together. 00:19:27.930 --> 00:19:31.480 And as we go through that process, of course, there's a 00:19:31.480 --> 00:19:37.260 process whereby we let the approximation go to the 00:19:37.260 --> 00:19:40.920 representation of a smooth curve. 00:19:40.920 --> 00:19:45.890 Now, again, I stress if there is certainly a fair amount to 00:19:45.890 --> 00:19:50.050 kind of examine carefully there, but it's important to 00:19:50.050 --> 00:19:53.980 also reflect on what we've done, which is really pretty 00:19:53.980 --> 00:19:55.240 significant. 00:19:55.240 --> 00:19:58.790 What we've managed to accomplish is to exploit the 00:19:58.790 --> 00:20:03.720 properties of linearity and time invariance, so that the 00:20:03.720 --> 00:20:09.840 system could be represented in terms only of its response to 00:20:09.840 --> 00:20:12.380 an impulse at time 0. 00:20:12.380 --> 00:20:14.410 So for a linear time-invariant system-- 00:20:14.410 --> 00:20:16.410 quite amazingly, actually-- 00:20:16.410 --> 00:20:20.670 if you know its response to an impulse at t = 0 or n = 0, 00:20:20.670 --> 00:20:24.490 depending on discrete or continuous time, then in fact, 00:20:24.490 --> 00:20:27.380 through the convolution sum in discrete time or the 00:20:27.380 --> 00:20:30.260 convolution integral in continuous time, you can 00:20:30.260 --> 00:20:33.110 generate the response to an arbitrary input. 00:20:36.130 --> 00:20:41.560 Let me just introduce a small amount of notation. 00:20:41.560 --> 00:20:46.660 Again, reminding you of the convolution sum in the 00:20:46.660 --> 00:20:51.080 discrete-time case, which looks as I've indicated here-- 00:20:51.080 --> 00:20:52.510 the sum of x[k] 00:20:52.510 --> 00:20:54.980 h[n-k] 00:20:54.980 --> 00:20:59.990 will have the requirement of making such frequent reference 00:20:59.990 --> 00:21:04.190 to convolution that it's convenient to notationally 00:21:04.190 --> 00:21:07.380 represent it as I have here with an asterisk. 00:21:07.380 --> 00:21:09.350 So x[n] 00:21:09.350 --> 00:21:12.200 * h[n] 00:21:12.200 --> 00:21:15.060 means or denotes the convolution of x[n] 00:21:15.060 --> 00:21:17.060 with h[n]. 00:21:17.060 --> 00:21:21.590 And correspondingly in the continuous-time case, we have 00:21:21.590 --> 00:21:26.230 the convolution integral, which here is the sifting 00:21:26.230 --> 00:21:26.670 integral 00:21:26.670 --> 00:21:30.660 as we talked about, representing x(t) in terms of 00:21:30.660 --> 00:21:34.800 itself as a linear combination of delayed impulses. 00:21:34.800 --> 00:21:38.580 Here we have the convolution integral, and again we'll use 00:21:38.580 --> 00:21:43.040 the asterisk to denote convolution. 00:21:43.040 --> 00:21:46.950 Now, there's a lot about convolution that we'll want to 00:21:46.950 --> 00:21:47.750 talk about. 00:21:47.750 --> 00:21:50.240 There are properties of convolution which tell us 00:21:50.240 --> 00:21:53.540 about properties of linear time-invariant systems. 00:21:53.540 --> 00:21:57.920 Also, it's important to focus on the mechanics of 00:21:57.920 --> 00:21:59.370 implementing a convolution-- 00:21:59.370 --> 00:22:02.690 in other words, understanding and generating some fluency 00:22:02.690 --> 00:22:08.860 and insight into what these particular sum and integral 00:22:08.860 --> 00:22:11.080 expressions mean. 00:22:11.080 --> 00:22:15.410 So let's first look at discrete-time convolution and 00:22:15.410 --> 00:22:20.680 examine more specifically what in essence the sum tells us to 00:22:20.680 --> 00:22:22.935 do in terms of manipulating the sequences. 00:22:25.960 --> 00:22:35.040 So returning to the expression for the convolution sum, as I 00:22:35.040 --> 00:22:39.960 show here, the sum of x[k] 00:22:39.960 --> 00:22:42.430 h[n-k]-- 00:22:42.430 --> 00:22:47.080 let's focus in on an example where we choose x[n] 00:22:47.080 --> 00:22:49.950 as a unit step and h[n] 00:22:49.950 --> 00:22:53.900 as a real exponential times the a unit step. 00:22:53.900 --> 00:22:56.090 So the sequence x[n] 00:22:56.090 --> 00:23:00.950 is as I indicate here, and the sequence h[n] 00:23:00.950 --> 00:23:05.760 is an exponential for positive time and 0 for negative time. 00:23:05.760 --> 00:23:07.105 So we have x[n] 00:23:07.105 --> 00:23:12.840 and h[n], but now let's look back at the equation and let 00:23:12.840 --> 00:23:16.960 me stress that what we want is not x[n] 00:23:16.960 --> 00:23:18.340 and h[n]-- 00:23:18.340 --> 00:23:21.440 we want x[k], because we're going to sum 00:23:21.440 --> 00:23:24.660 over k, and not h[k] 00:23:24.660 --> 00:23:25.910 but h[n-k]. 00:23:27.970 --> 00:23:33.970 So we have then from x[n], it's straightforward to 00:23:33.970 --> 00:23:35.220 generate x[k]. 00:23:35.220 --> 00:23:39.580 It's simply changing the index of summation. 00:23:39.580 --> 00:23:42.310 And what's h[n-k]? 00:23:42.310 --> 00:23:44.740 Well what's h[-k]? 00:23:44.740 --> 00:23:46.100 h[-k] 00:23:46.100 --> 00:23:47.640 is h[k] 00:23:47.640 --> 00:23:49.120 flipped over. 00:23:49.120 --> 00:23:53.550 So if this is what h[k] 00:23:53.550 --> 00:24:00.140 looks like, then this is what h[n-k] 00:24:00.140 --> 00:24:01.690 looks like. 00:24:01.690 --> 00:24:05.920 In essence, what the operation of convolution or the 00:24:05.920 --> 00:24:11.390 mechanics of convolution tells us to do is to take the 00:24:11.390 --> 00:24:16.710 sequence h[n-k], which is h[-k] 00:24:16.710 --> 00:24:21.870 positioned with its origin at k = n, and multiply this 00:24:21.870 --> 00:24:27.700 sequence by this sequence and sum the product from k = 00:24:27.700 --> 00:24:29.780 -infinity to +infinity. 00:24:29.780 --> 00:24:38.080 So if we were to compute, for example, the output at n = 0-- 00:24:38.080 --> 00:24:43.000 as I positioned this sequence here, this is at n = 0-- 00:24:43.000 --> 00:24:46.820 I would take this and multiply it by this and sum from 00:24:46.820 --> 00:24:49.020 -infinity to +infinity. 00:24:49.020 --> 00:24:52.900 Or, for n = 1, I would position it here, for n = 2, I 00:24:52.900 --> 00:24:55.470 would position it here. 00:24:55.470 --> 00:24:58.780 Well, you can kind of see what the idea is. 00:24:58.780 --> 00:25:02.770 Let's look at this a little more dynamically and see, in 00:25:02.770 --> 00:25:07.180 fact, how one sequence slides past the other, and how the 00:25:07.180 --> 00:25:08.590 output y[n] 00:25:08.590 --> 00:25:11.910 builds up to the correct answer. 00:25:14.950 --> 00:25:18.100 So the input that we're considering is a step input, 00:25:18.100 --> 00:25:22.680 which I show here, and the impulse response that we will 00:25:22.680 --> 00:25:26.830 convolve this with is a decaying exponential. 00:25:26.830 --> 00:25:31.340 Now, to form the convolution, we want the product of x[k]-- 00:25:31.340 --> 00:25:35.990 not with h[k], but with h[n-k], corresponding to 00:25:35.990 --> 00:25:37.330 taking h[k] 00:25:37.330 --> 00:25:41.020 and reflecting it about the origin and then shifting it 00:25:41.020 --> 00:25:42.130 appropriately. 00:25:42.130 --> 00:25:44.320 So here we see h[n-k] 00:25:44.320 --> 00:25:52.050 for n = 0, namely h[-k], and now h[1-k], which we'll show 00:25:52.050 --> 00:25:57.670 next, is this shifted to the right by one point. 00:25:57.670 --> 00:26:00.720 Here we have h[1-k]-- 00:26:00.720 --> 00:26:07.420 shifting to the right by one more point is h[2-k], and 00:26:07.420 --> 00:26:10.370 shifting again to the right we'll have h[3-k]. 00:26:12.880 --> 00:26:20.480 Now let's shift back to the left until n is negative, and 00:26:20.480 --> 00:26:22.210 then we'll begin the convolution. 00:26:22.210 --> 00:26:32.890 So here's n = 0, n = -1, n = -2, and n = -3. 00:26:32.890 --> 00:26:36.530 Now, to form the convolution, we want the product of x[k] 00:26:36.530 --> 00:26:38.470 with h[n-k] 00:26:38.470 --> 00:26:43.090 summed from -infinity to +infinity. 00:26:43.090 --> 00:26:47.360 For n negative, that product is 0, and therefore the result 00:26:47.360 --> 00:26:49.440 of the convolution is 0. 00:26:49.440 --> 00:26:52.130 As we shift to the right, we'll build up the 00:26:52.130 --> 00:26:55.510 convolution, and the result of the convolution will be shown 00:26:55.510 --> 00:26:57.330 on the bottom trace. 00:26:57.330 --> 00:27:01.780 So we begin the process with n negative, and 00:27:01.780 --> 00:27:03.730 here we have n = -1. 00:27:03.730 --> 00:27:09.210 At n = 0, we get our first non-zero contribution. 00:27:09.210 --> 00:27:13.310 Now as we shift further to the right corresponding to 00:27:13.310 --> 00:27:18.820 increasing n, we will accumulate more and more terms 00:27:18.820 --> 00:27:22.960 in the sum, and the convolution will build up. 00:27:22.960 --> 00:27:25.670 In particular for this example, the result of the 00:27:25.670 --> 00:27:30.040 convolution increases monotonically, asymptotically 00:27:30.040 --> 00:27:34.790 approaching a constant, and that constant, in fact, is 00:27:34.790 --> 00:27:38.630 just simply the accumulation of the values under the 00:27:38.630 --> 00:27:39.880 exponential. 00:27:48.780 --> 00:27:51.800 Now let's carry out the convolution this time with an 00:27:51.800 --> 00:27:55.870 input which is a rectangular pulse instead of a step input. 00:27:55.870 --> 00:27:58.510 Again, the same impulse response, namely a decaying 00:27:58.510 --> 00:28:03.370 exponential, and so we want to begin with h[n-k] 00:28:03.370 --> 00:28:06.750 and again with n negative shown here. 00:28:06.750 --> 00:28:10.710 Again, with n negative, there are no non-zero terms in the 00:28:10.710 --> 00:28:14.510 product, and so the convolution for n negative 00:28:14.510 --> 00:28:17.390 will be 0 as it was in the previous case. 00:28:17.390 --> 00:28:21.230 Again, on the bottom trace we'll show the result of the 00:28:21.230 --> 00:28:25.470 convolution as the impulse response slides along. 00:28:25.470 --> 00:28:30.040 At n = 0, we get our first non-zero term. 00:28:30.040 --> 00:28:35.860 As n increases past 0, we will begin to generate an output, 00:28:35.860 --> 00:28:39.350 basically the same as the output that we generated with 00:28:39.350 --> 00:28:45.100 a step input, until the impulse response reaches a 00:28:45.100 --> 00:28:50.280 point where as we slide further, we slide outside the 00:28:50.280 --> 00:28:53.490 interval where the rectangle is non-zero. 00:28:53.490 --> 00:28:57.690 So when we slide one point further from what's shown 00:28:57.690 --> 00:29:03.370 here, the output will now decay, corresponding to the 00:29:03.370 --> 00:29:09.100 fact that the impulse response is sliding outside the 00:29:09.100 --> 00:29:12.790 interval in which the input is non-zero. 00:29:12.790 --> 00:29:15.480 So, on the bottom trace we now see the result of the 00:29:15.480 --> 00:29:16.730 convolution. 00:29:20.470 --> 00:29:21.010 OK. 00:29:21.010 --> 00:29:27.450 So what you've seen, then, is an example of discrete-time 00:29:27.450 --> 00:29:29.010 convolution. 00:29:29.010 --> 00:29:33.470 Let's now look at an example of continuous-time 00:29:33.470 --> 00:29:35.340 convolution. 00:29:35.340 --> 00:29:38.140 As you might expect, continuous-time convolution 00:29:38.140 --> 00:29:42.030 operates in exactly the same way. 00:29:42.030 --> 00:29:44.530 Continuous-time convolution-- 00:29:44.530 --> 00:29:51.750 we have the expression again y(t) is an integral with now 00:29:51.750 --> 00:29:55.270 x(tau) and h(t-tau). 00:29:55.270 --> 00:30:01.450 It has exactly the same kind of form as we had previously 00:30:01.450 --> 00:30:05.190 for discrete-time convolution-- 00:30:05.190 --> 00:30:08.190 and in fact, the mechanics of the continuous-time 00:30:08.190 --> 00:30:10.930 convolution are identical. 00:30:10.930 --> 00:30:16.440 So here is our example with x(t) equal to a unit step and 00:30:16.440 --> 00:30:22.280 h(t) now a real exponential times a unit step. 00:30:22.280 --> 00:30:28.330 I show here x(t), which is the unit step function. 00:30:28.330 --> 00:30:34.590 Here we have h(t), which is an exponential for positive time 00:30:34.590 --> 00:30:38.670 and 0 for negative time. 00:30:38.670 --> 00:30:43.960 Again, looking back at the expression for convolution, 00:30:43.960 --> 00:30:48.770 it's not x(t) that we want, it x(tau) that we want. 00:30:48.770 --> 00:30:54.480 And it's not h(t) or h(tau) that we want, it's h(t-tau). 00:30:57.310 --> 00:31:01.880 We plan to multiply these together and integrate over 00:31:01.880 --> 00:31:06.270 the variable tau, and that gives us the output at any 00:31:06.270 --> 00:31:07.700 given time. 00:31:07.700 --> 00:31:12.260 If we want it at another time, we change the value of t as an 00:31:12.260 --> 00:31:14.580 argument inside this integral. 00:31:14.580 --> 00:31:21.340 So here we have x(t), and here we have h(t), which isn't 00:31:21.340 --> 00:31:23.210 quite what we wanted. 00:31:23.210 --> 00:31:26.430 Here we have x(tau), and that's fine-- it's just x(t) 00:31:26.430 --> 00:31:29.410 with t relabeled as tau. 00:31:29.410 --> 00:31:33.800 Now, what is h(t-tau)? 00:31:33.800 --> 00:31:39.590 Well, here's h(tau), and if we simply turn that 00:31:39.590 --> 00:31:43.820 over, here is h(t-tau). 00:31:43.820 --> 00:31:52.470 And h(t-tau) is positioned, then, at tau equal to t. 00:31:52.470 --> 00:31:56.320 As we change the value of t that change the position of 00:31:56.320 --> 00:32:03.020 this signal, now we multiply these two together and 00:32:03.020 --> 00:32:08.970 integrate from -infinity to +infinity with h(t-tau) 00:32:08.970 --> 00:32:11.370 positioned at the appropriate value of t. 00:32:14.600 --> 00:32:18.700 Again, it's best really to see this example and get the 00:32:18.700 --> 00:32:23.080 notion of the signal being flipped and the two signals 00:32:23.080 --> 00:32:28.020 sliding past each other, multiplying and integrating by 00:32:28.020 --> 00:32:31.260 looking at it dynamically and observing how the 00:32:31.260 --> 00:32:33.970 answer builds up. 00:32:33.970 --> 00:32:38.300 Again, the input that we consider is a step input. 00:32:38.300 --> 00:32:41.800 And again, we use an impulse response which is a decaying 00:32:41.800 --> 00:32:44.250 exponential. 00:32:44.250 --> 00:32:48.840 To form the convolution, we want the product of x(tau)-- 00:32:48.840 --> 00:32:52.040 not with h(tau), but with h(t-tau). 00:32:52.040 --> 00:32:56.890 So we want h(t) time-reversed, and then shifted appropriately 00:32:56.890 --> 00:32:59.190 depending on the value of t. 00:32:59.190 --> 00:33:04.480 Let's first just look at h(t-tau) for t positive 00:33:04.480 --> 00:33:09.510 corresponding to shifting h(-tau) out to the right, and 00:33:09.510 --> 00:33:12.155 here we have t increasing. 00:33:15.810 --> 00:33:20.310 Here is t decreasing, and we'll want to begin the 00:33:20.310 --> 00:33:25.430 convolution with t negative, corresponding to shifting 00:33:25.430 --> 00:33:29.030 h(-tau) to the left. 00:33:29.030 --> 00:33:30.660 Now to form the convolution, we want the 00:33:30.660 --> 00:33:32.290 product of these two. 00:33:32.290 --> 00:33:38.820 For t negative, there are no non-zero contributions to the 00:33:38.820 --> 00:33:42.580 integral, and so the convolution will be 0 for t 00:33:42.580 --> 00:33:44.090 less than 0. 00:33:44.090 --> 00:33:48.090 On the bottom trace, we show the result of the convolution, 00:33:48.090 --> 00:33:55.030 here for t negative, and for t less than 0, we will continue 00:33:55.030 --> 00:33:58.490 to have 0 in the convolution. 00:33:58.490 --> 00:34:04.910 Now as t increases past 0, we begin to get some non-zero 00:34:04.910 --> 00:34:09.040 contribution in the product, indicated by the fact that the 00:34:09.040 --> 00:34:09.989 convolution-- 00:34:09.989 --> 00:34:11.270 the result of the convolution-- 00:34:11.270 --> 00:34:14.370 starts to build up. 00:34:14.370 --> 00:34:19.780 As t increases further, we will get more and more 00:34:19.780 --> 00:34:23.540 non-zero contribution in the integrand. 00:34:23.540 --> 00:34:28.270 So, the result of the convolution will be a 00:34:28.270 --> 00:34:31.469 monotonically increasing function for this particular 00:34:31.469 --> 00:34:36.100 example, which asymptotically approaches a constant. 00:34:36.100 --> 00:34:40.320 That constant will be proportional to the area under 00:34:40.320 --> 00:34:43.360 the impulse response, because of the fact that we're 00:34:43.360 --> 00:34:45.090 convolving with a step input. 00:34:59.270 --> 00:35:02.160 Now let's carry out the convolution with an input 00:35:02.160 --> 00:35:05.110 which is a rectangular pulse-- 00:35:05.110 --> 00:35:08.420 again, an impulse response which is an exponential. 00:35:08.420 --> 00:35:13.710 So to form the convolution, we want x(tau) with h(t-tau)-- 00:35:13.710 --> 00:35:18.010 h(t-tau) shown here for t negative. 00:35:18.010 --> 00:35:21.880 To form the convolution, we take the integral of the 00:35:21.880 --> 00:35:25.920 product of these two, which again will be 0 00:35:25.920 --> 00:35:28.360 for t less than 0. 00:35:28.360 --> 00:35:31.360 The bottom trace shows the result of the convolution 00:35:31.360 --> 00:35:36.630 here, shown as 0, and it will continue to be 0 until t 00:35:36.630 --> 00:35:40.990 becomes positive, at which point we build up some 00:35:40.990 --> 00:35:47.040 non-zero term in the integrand. 00:35:47.040 --> 00:35:53.540 Now as we slide further, until the impulse response shifts 00:35:53.540 --> 00:35:58.870 outside the interval in which the pulse is non-zero, the 00:35:58.870 --> 00:36:00.670 output will build up. 00:36:00.670 --> 00:36:05.530 But here we've now begun to leave that interval, and so 00:36:05.530 --> 00:36:09.700 the output will start to decay exponentially. 00:36:09.700 --> 00:36:13.680 As the impulse response slides further and further 00:36:13.680 --> 00:36:20.410 corresponding to increasing t, then the output will decay 00:36:20.410 --> 00:36:24.980 exponentially, representing the fact that there is less 00:36:24.980 --> 00:36:31.200 and less area in the product of x(tau) and h(t-tau). 00:36:34.580 --> 00:36:37.680 Asymptotically, this output will then approach 0. 00:36:40.830 --> 00:36:41.140 OK. 00:36:41.140 --> 00:36:45.360 So you've seen convolution, you've seen the derivation of 00:36:45.360 --> 00:36:49.120 convolution, and kind of the graphical representation of 00:36:49.120 --> 00:36:50.820 convolution. 00:36:50.820 --> 00:36:56.880 Finally, let's work again with these two examples, and let's 00:36:56.880 --> 00:37:00.870 go through those two examples analytically so that we 00:37:00.870 --> 00:37:05.980 finally see how, analytically, the result develops for those 00:37:05.980 --> 00:37:07.230 same examples. 00:37:10.450 --> 00:37:13.400 Well, we have first the discrete-time case, and let's 00:37:13.400 --> 00:37:16.960 take our discrete-time example. 00:37:16.960 --> 00:37:23.490 In general, the convolution sum is as I've indicated here. 00:37:23.490 --> 00:37:27.860 This is just our expression from before, which is the 00:37:27.860 --> 00:37:31.140 convolution sum. 00:37:31.140 --> 00:37:33.080 If we take our two examples-- 00:37:33.080 --> 00:37:39.760 the example of an input which is a unit step, and an impulse 00:37:39.760 --> 00:37:44.220 response, which is a real exponential multiplied by a 00:37:44.220 --> 00:37:46.110 unit step-- 00:37:46.110 --> 00:37:50.990 we have then replacing x[k] 00:37:50.990 --> 00:37:56.230 by what we know the input to be, and h[n-k] 00:37:56.230 --> 00:38:00.700 by what we know the impulse response to be, the output is 00:38:00.700 --> 00:38:02.290 the expression that we have here. 00:38:05.020 --> 00:38:07.240 Now, in this expression-- 00:38:07.240 --> 00:38:11.430 and you'll see this very generally and with some more 00:38:11.430 --> 00:38:17.210 complicated examples when you look at the text-- 00:38:17.210 --> 00:38:22.720 as you go to evaluate these expressions, generally what 00:38:22.720 --> 00:38:26.050 happens is that the signals have different analytical 00:38:26.050 --> 00:38:28.380 forms in different regions. 00:38:28.380 --> 00:38:31.100 That's, in fact, what we have here. 00:38:31.100 --> 00:38:36.670 In particular, let's look at the sum, and what we observe 00:38:36.670 --> 00:38:42.660 first of all is that the limits on this sum are going 00:38:42.660 --> 00:38:47.200 to be modified, depending on where this unit 00:38:47.200 --> 00:38:50.390 step is 0 and non-zero. 00:38:50.390 --> 00:38:54.820 In particular, if we first consider what will turn out to 00:38:54.820 --> 00:38:55.980 be the simple case-- 00:38:55.980 --> 00:38:59.030 namely, n less than 0-- 00:38:59.030 --> 00:39:08.790 for n less than 0, this unit step is 0 for k 00:39:08.790 --> 00:39:10.640 greater than n. 00:39:10.640 --> 00:39:17.460 With n less than 0, that means that this unit step never is 00:39:17.460 --> 00:39:21.480 non-zero for k positive. 00:39:21.480 --> 00:39:27.360 On the other hand, this unit step is never non-zero or 00:39:27.360 --> 00:39:31.350 always 0 for k negative. 00:39:31.350 --> 00:39:34.480 Let me just stress that by looking at the particular 00:39:34.480 --> 00:39:40.840 graphs, here is the unit step u[k]. 00:39:40.840 --> 00:39:48.420 Here is the unit step u[n-k], and for n less than 0, so that 00:39:48.420 --> 00:39:52.550 this point comes before this point, the product of these 00:39:52.550 --> 00:39:54.960 two is equal to 0. 00:39:54.960 --> 00:39:58.730 That means there is no overlap between these two terms, and 00:39:58.730 --> 00:40:03.380 so it says that y[n], the output, is 0 00:40:03.380 --> 00:40:05.940 for n less than 0. 00:40:05.940 --> 00:40:08.980 Well, that was an easy one. 00:40:08.980 --> 00:40:13.320 For n greater than 0, it's not quite as straightforward as 00:40:13.320 --> 00:40:15.520 coming out with the answer 0. 00:40:15.520 --> 00:40:19.270 So now let's look at what happens when the two unit 00:40:19.270 --> 00:40:25.430 steps overlap, and this would correspond to what I've 00:40:25.430 --> 00:40:30.470 labeled here as interval 2, namely for n greater than 0. 00:40:30.470 --> 00:40:36.840 If we just look back at the summation that we had, the 00:40:36.840 --> 00:40:44.430 summation now corresponds to this unit step and this unit 00:40:44.430 --> 00:40:48.260 step, having some overlap. 00:40:48.260 --> 00:40:53.960 So for interval 2, corresponding to n greater 00:40:53.960 --> 00:41:00.150 than 0, we have u[k], the unit step. 00:41:00.150 --> 00:41:05.980 We have u[n-k], which is a unit step going backward in 00:41:05.980 --> 00:41:13.870 time, but which extends for positive values of n. 00:41:13.870 --> 00:41:17.970 If we think about multiplying these two together, we will 00:41:17.970 --> 00:41:24.320 get in the product unity for what values of k? 00:41:24.320 --> 00:41:27.900 Well, for k starting at 0 corresponding to one of the 00:41:27.900 --> 00:41:31.320 unit steps and ending at n corresponding to 00:41:31.320 --> 00:41:33.270 the other unit step. 00:41:33.270 --> 00:41:39.242 So we have an overlap between these for k equal to 0, et 00:41:39.242 --> 00:41:41.630 cetera, up through the value n. 00:41:44.630 --> 00:41:49.990 Now, that means that in terms of the original sum, we can 00:41:49.990 --> 00:41:53.170 get rid of the unit steps involved by simply changing 00:41:53.170 --> 00:41:56.210 the limits on the sum. 00:41:56.210 --> 00:42:02.790 The limits now are from 0 to n, of the term alpha^(n-k). 00:42:05.680 --> 00:42:07.250 We had before a u[k] 00:42:07.250 --> 00:42:11.360 and u[n-k], and that disappeared because we dealt 00:42:11.360 --> 00:42:14.740 with that simply by modifying the limits. 00:42:14.740 --> 00:42:19.590 We now pull out the term alpha^n, because the summation 00:42:19.590 --> 00:42:24.350 is on k, not on n, so we can simply pull 00:42:24.350 --> 00:42:25.750 that term of the sum. 00:42:29.190 --> 00:42:33.810 We now have alpha^(-k), which we can rewrite as 00:42:33.810 --> 00:42:36.480 (alpha^(-1))^k. 00:42:36.480 --> 00:42:40.100 The upshot of all of this is that y[n] 00:42:40.100 --> 00:42:45.600 now we can reexpress as alpha^n, the sum from 0 to n 00:42:45.600 --> 00:42:46.850 of (alpha^(-1)^k. 00:42:49.460 --> 00:42:54.850 The question is, how do we evaluate that? 00:42:54.850 --> 00:42:58.960 It essentially corresponds to a finite number of terms in a 00:42:58.960 --> 00:43:01.060 geometric series. 00:43:01.060 --> 00:43:05.140 That, by the way, is a summation that will recur over 00:43:05.140 --> 00:43:08.900 and over and over and over again, and it's one that you 00:43:08.900 --> 00:43:11.870 should write down, write on your back pocket, write on the 00:43:11.870 --> 00:43:15.650 palm of your hand, or whatever it takes to remember it. 00:43:15.650 --> 00:43:20.700 What you'll see is it that will recur more or less 00:43:20.700 --> 00:43:24.970 throughout the course, and so it's one worth remembering. 00:43:24.970 --> 00:43:31.520 In particular, what the sum of a geometric series is, is what 00:43:31.520 --> 00:43:32.950 I've indicated here. 00:43:32.950 --> 00:43:35.930 We have the sum from 0 to r, of beta^k. 00:43:38.830 --> 00:43:42.430 It's 1 - beta^(r+1)-- 00:43:42.430 --> 00:43:48.460 this is one more than the upper limit on the summation-- 00:43:48.460 --> 00:43:51.640 and in the denominator is 1 - beta. 00:43:51.640 --> 00:43:55.100 So, this equation is important. 00:43:55.100 --> 00:43:57.720 There's no point in attempting to derive it. 00:43:57.720 --> 00:44:03.460 However you get to it, it's important to retain it. 00:44:03.460 --> 00:44:08.595 We can now use that summation in the expression 00:44:08.595 --> 00:44:11.170 that we just developed. 00:44:11.170 --> 00:44:16.930 So let's proceed to evaluate that sum in closed form. 00:44:16.930 --> 00:44:21.520 We now go back to the expression that we just 00:44:21.520 --> 00:44:22.910 worked out-- y[n] 00:44:22.910 --> 00:44:27.150 is alpha^n, the sum from 0 to n, (alpha^(-1))^k. 00:44:31.950 --> 00:44:35.280 This plays the role of beta in the term that I just-- 00:44:35.280 --> 00:44:39.120 in the expression then I just presented. 00:44:39.120 --> 00:44:45.170 So, using that result, we can rewrite this summation as I 00:44:45.170 --> 00:44:46.420 indicate here. 00:44:49.000 --> 00:44:52.550 The final result that we end up with after a certain amount 00:44:52.550 --> 00:44:55.180 of algebra is y[n] 00:44:55.180 --> 00:45:02.230 equal to (1 - alpha^(n+1)) / (1 - alpha). 00:45:02.230 --> 00:45:09.560 Let me just kind of indicate with a few dots here that 00:45:09.560 --> 00:45:13.970 there is a certain amount of algebra required in going from 00:45:13.970 --> 00:45:18.470 this step to this step, and I'd like to leave you with the 00:45:18.470 --> 00:45:20.740 fun and opportunity of doing that at your leisure. 00:45:23.860 --> 00:45:28.770 The expression we have now for y[n], is y[n] 00:45:28.770 --> 00:45:33.560 = (1 - alpha^(n+1)) / (1 - alpha). 00:45:33.560 --> 00:45:35.900 That's for n greater than 0. 00:45:35.900 --> 00:45:37.980 We had found out previously there it was 0 00:45:37.980 --> 00:45:40.070 for n less than 0. 00:45:40.070 --> 00:45:45.000 Finally, if we were to plot this, what we would get is the 00:45:45.000 --> 00:45:48.310 graph that I indicate here. 00:45:48.310 --> 00:45:53.350 The first non-zero value occurs at n = 0, and it has a 00:45:53.350 --> 00:45:59.750 height of 1, and then the next non-zero value at 1, and this 00:45:59.750 --> 00:46:05.490 has a height of 1 + alpha, and this is 1 + alpha + alpha^2. 00:46:08.760 --> 00:46:13.960 The sequence continues on like that and asymptotically 00:46:13.960 --> 00:46:17.980 approaches, as n goes to infinity, asymptotically 00:46:17.980 --> 00:46:22.630 approaches 1 / (1 - alpha), which is consistent with the 00:46:22.630 --> 00:46:27.780 algebraic expression that we have, that we developed, and 00:46:27.780 --> 00:46:32.370 obviously of course is also consistent with the movie. 00:46:32.370 --> 00:46:35.760 That's our discrete-time example, which we kind of went 00:46:35.760 --> 00:46:39.720 through graphically with the transparencies, and we went 00:46:39.720 --> 00:46:41.870 through graphically with the movie, and now we've gone 00:46:41.870 --> 00:46:43.500 through analytically. 00:46:43.500 --> 00:46:48.690 Now let's look analytically at the continuous-time example, 00:46:48.690 --> 00:46:52.090 which pretty much flows in the same way as 00:46:52.090 --> 00:46:55.010 we've just gone through. 00:46:55.010 --> 00:46:59.820 Again, we have the convolution integral, which is the 00:46:59.820 --> 00:47:02.275 integral indicated at the top. 00:47:05.070 --> 00:47:11.250 Our example, you recall, was with x(t) as a unit step, and 00:47:11.250 --> 00:47:15.850 h(t) as an exponential times a unit step. 00:47:15.850 --> 00:47:20.830 So when we substitute those in, this then corresponds to 00:47:20.830 --> 00:47:24.730 x(t) and this corresponds to h(t-tau). 00:47:27.600 --> 00:47:33.230 Again, we have the same issue more or less, which is that 00:47:33.230 --> 00:47:36.690 inside that integral, there are two steps, one of them 00:47:36.690 --> 00:47:39.480 going forward in time and one of them going backward in 00:47:39.480 --> 00:47:44.340 time, and we need to examine when they overlap and when 00:47:44.340 --> 00:47:44.810 they don't. 00:47:44.810 --> 00:47:48.310 When they don't overlap, the product, of course, is 0, and 00:47:48.310 --> 00:47:50.920 there's no point doing any integration because the 00:47:50.920 --> 00:47:52.720 integrand is 0. 00:47:52.720 --> 00:47:58.160 So if we track it through, we have again Interval 1, which 00:47:58.160 --> 00:48:00.300 is t less than 0. 00:48:00.300 --> 00:48:06.940 For t less than 0, this unit step, which only begins at tau 00:48:06.940 --> 00:48:13.990 = 0, and this unit step which is 0, by the time tau gets up 00:48:13.990 --> 00:48:16.170 to t and beyond. 00:48:16.170 --> 00:48:22.860 For t less than 0, there's no overlap between the unit step 00:48:22.860 --> 00:48:25.040 going forward in time and the unit step 00:48:25.040 --> 00:48:27.210 going backward in time. 00:48:27.210 --> 00:48:31.940 Consequently, the integrand is equal to 0, and consequently, 00:48:31.940 --> 00:48:33.450 the output is equal to 0. 00:48:40.190 --> 00:48:45.520 We can likewise look at the interval where the two unit 00:48:45.520 --> 00:48:48.070 steps do overlap. 00:48:48.070 --> 00:48:51.380 In that case what happens again is that the overlap, in 00:48:51.380 --> 00:48:55.940 essence of the unit step, tells us, gives us a range on 00:48:55.940 --> 00:48:59.320 the integration-- 00:48:59.320 --> 00:49:03.600 in particular, the two steps overlap when t is greater than 00:49:03.600 --> 00:49:10.840 0 from tau = 0 to tau = t. 00:49:10.840 --> 00:49:15.140 For Interval 2, for t greater than 0-- 00:49:15.140 --> 00:49:19.370 again, of course, we have this expression. 00:49:19.370 --> 00:49:25.200 This product of this unit step and this unit step is equal to 00:49:25.200 --> 00:49:34.310 1 in this range, and so that allows us, then, to change the 00:49:34.310 --> 00:49:36.800 limits on the integral-- 00:49:36.800 --> 00:49:40.290 instead of from -infinity to +infinity, we know that the 00:49:40.290 --> 00:49:45.590 integrand is non-zero only over this range. 00:49:45.590 --> 00:49:48.990 Looking at this integral, we can now pull out the term 00:49:48.990 --> 00:49:55.240 which corresponds to e^-at, just as we pulled out a term 00:49:55.240 --> 00:49:56.725 in the discrete-time case. 00:49:59.310 --> 00:50:05.190 We notice in the integral that we have e^(-a*-tau), so that 00:50:05.190 --> 00:50:09.740 gives us the integral from 0 to t of e^(a*tau). 00:50:09.740 --> 00:50:13.600 If we perform that integration, we end up with 00:50:13.600 --> 00:50:15.220 this expression. 00:50:15.220 --> 00:50:22.500 Finally, multiplying this by e^-at, we have for y(t), for t 00:50:22.500 --> 00:50:28.270 greater than 0, the algebraic expression that I've indicated 00:50:28.270 --> 00:50:29.520 on the bottom. 00:50:33.540 --> 00:50:36.500 So we had worked out t less than 0, and we 00:50:36.500 --> 00:50:38.170 come out with 0. 00:50:38.170 --> 00:50:43.630 We work out t greater than 0, and we come out with this 00:50:43.630 --> 00:50:44.880 algebraic expression. 00:50:47.250 --> 00:50:50.560 If we plot this algebraic expression as a function of 00:50:50.560 --> 00:50:54.690 time, we find that what it corresponds to is an 00:50:54.690 --> 00:51:00.210 exponential behavior starting at zero and exponentially 00:51:00.210 --> 00:51:04.800 heading asymptotically toward the value 1 / a. 00:51:09.980 --> 00:51:12.430 We've gone through these examples several ways, and one 00:51:12.430 --> 00:51:14.470 is analytically. 00:51:14.470 --> 00:51:19.780 In order to develop a feel and fluency for convolution, it's 00:51:19.780 --> 00:51:24.000 absolutely essential to work through a variety of examples, 00:51:24.000 --> 00:51:27.410 both understanding them graphically and understanding 00:51:27.410 --> 00:51:31.790 them as we did here analytically. 00:51:31.790 --> 00:51:34.860 You'll have an opportunity to do that through the problems 00:51:34.860 --> 00:51:38.950 that I've suggested in the video course manual. 00:51:38.950 --> 00:51:43.980 In the next lecture, what we'll turn to are some general 00:51:43.980 --> 00:51:48.210 properties of convolution, and show how this rather amazing 00:51:48.210 --> 00:51:52.310 representation of linear time-invariant systems in fact 00:51:52.310 --> 00:51:55.580 leads to a variety of properties of linear 00:51:55.580 --> 00:51:57.620 time-invariant systems. 00:51:57.620 --> 00:52:02.690 We'll find that convolution is fairly rich in its properties, 00:52:02.690 --> 00:52:09.180 and what this leads to are some very nice and desirable 00:52:09.180 --> 00:52:12.680 and exploitable properties of linear time-invariant systems. 00:52:12.680 --> 00:52:13.930 Thank you.