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 --> 00:00:20.540 ocw.mit.edu. 00:00:55.306 --> 00:00:57.270 PROFESSOR: Over the past several lectures, we've 00:00:57.270 --> 00:01:01.880 developed a representation for linear time-invariant systems. 00:01:01.880 --> 00:01:05.500 A particularly important set of systems, which are linear 00:01:05.500 --> 00:01:08.700 and time-invariant, are those that are represented by linear 00:01:08.700 --> 00:01:11.240 constant-coefficient differential equations in 00:01:11.240 --> 00:01:14.950 continuous time or linear constant-coefficient 00:01:14.950 --> 00:01:17.630 difference equations in discrete time. 00:01:17.630 --> 00:01:20.780 For example, electrical circuits that are built, let's 00:01:20.780 --> 00:01:23.300 say, out of resistors, inductors, and capacitors, 00:01:23.300 --> 00:01:26.780 perhaps with op-amps, correspond to systems 00:01:26.780 --> 00:01:29.150 described by differential equations. 00:01:29.150 --> 00:01:31.690 Mechanical systems with springs and dashpots, 00:01:31.690 --> 00:01:35.010 likewise, are described by differential equations. 00:01:35.010 --> 00:01:39.300 And in the discrete-time case, things such as moving average 00:01:39.300 --> 00:01:44.470 filters, digital filters, and most simple kinds of data 00:01:44.470 --> 00:01:47.840 smoothing are all linear constant-coefficient 00:01:47.840 --> 00:01:50.110 difference equations. 00:01:50.110 --> 00:01:55.180 Now, presumably in a previous course, you've had some 00:01:55.180 --> 00:02:00.260 exposure to differential equations for continuous time, 00:02:00.260 --> 00:02:04.000 and their solution using notions like particular 00:02:04.000 --> 00:02:06.220 solution, and homogeneous solution, initial 00:02:06.220 --> 00:02:09.380 conditions, et cetera. 00:02:09.380 --> 00:02:13.760 Later on in the course, when we've developed the concept of 00:02:13.760 --> 00:02:17.450 the Fourier transform after that, the Laplace transform, 00:02:17.450 --> 00:02:21.500 we'll see some very efficient and useful ways of generating 00:02:21.500 --> 00:02:23.160 solutions, both for differential 00:02:23.160 --> 00:02:25.010 and difference equations. 00:02:25.010 --> 00:02:29.620 At this point, however, I'd like to just introduce linear 00:02:29.620 --> 00:02:33.020 constant-coefficient differential equations and 00:02:33.020 --> 00:02:35.130 their discrete-time counterpart. 00:02:35.130 --> 00:02:38.890 And address, among other things, the issue of when they 00:02:38.890 --> 00:02:43.650 do and don't correspond to linear time-invariant systems. 00:02:43.650 --> 00:02:48.420 Well, let's first consider what I refer to as an 00:02:48.420 --> 00:02:51.010 nth-order linear constant-coefficient 00:02:51.010 --> 00:02:54.270 differential equation, as I've indicated here. 00:02:54.270 --> 00:02:57.350 And what it consists of is a linear combination of 00:02:57.350 --> 00:03:01.560 derivatives of the system output, y(t), equal to a 00:03:01.560 --> 00:03:04.660 linear combination of derivatives of the system 00:03:04.660 --> 00:03:07.300 input x(t). 00:03:07.300 --> 00:03:12.470 And it's referred to as a constant-coefficient equation, 00:03:12.470 --> 00:03:15.550 of course, because the coefficients are constant. 00:03:15.550 --> 00:03:18.670 In other words, not assumed to be time-varying. 00:03:18.670 --> 00:03:25.200 And it's referred to as linear because it corresponds to a 00:03:25.200 --> 00:03:30.820 linear combination of these derivatives, not because it 00:03:30.820 --> 00:03:32.620 corresponds to a linear system. 00:03:32.620 --> 00:03:36.910 And, in fact, as we'll see, or as I'll indicate, this 00:03:36.910 --> 00:03:39.340 equation may or may not, in fact, 00:03:39.340 --> 00:03:42.410 correspond to a linear system. 00:03:42.410 --> 00:03:47.500 In the discrete-time case, the corresponding equation is a 00:03:47.500 --> 00:03:50.710 linear constant-coefficient difference equation. 00:03:50.710 --> 00:03:55.650 And that corresponds to, again, a linear combination of 00:03:55.650 --> 00:04:00.370 delayed versions of the output equal to a linear combination 00:04:00.370 --> 00:04:02.630 of delayed versions of the input. 00:04:02.630 --> 00:04:04.590 This equation is referred to an 00:04:04.590 --> 00:04:07.290 nth-order difference equation. 00:04:07.290 --> 00:04:10.750 The n referring to the number of delays of the output 00:04:10.750 --> 00:04:15.690 involved, just as an Nth-order differential equation, the n 00:04:15.690 --> 00:04:18.230 or the order of the equation refers to the number of 00:04:18.230 --> 00:04:21.050 derivatives of the output. 00:04:21.050 --> 00:04:25.150 Now, let's first begin with linear constant-coefficient 00:04:25.150 --> 00:04:27.170 differential equations. 00:04:27.170 --> 00:04:34.290 And the basic point of the solution for the differential 00:04:34.290 --> 00:04:40.300 equations is the fact that if we've generated some solution, 00:04:40.300 --> 00:04:44.600 which I refer to here as y_p(t), some solution to the 00:04:44.600 --> 00:04:49.890 equation for a given input, then, in fact, we can add to 00:04:49.890 --> 00:04:55.750 that solution any other solution which satisfies 00:04:55.750 --> 00:04:59.910 what's referred to as the homogeneous equation. 00:04:59.910 --> 00:05:06.620 So in fact, this differential equation by itself is not a 00:05:06.620 --> 00:05:10.300 unique specification of the system. 00:05:10.300 --> 00:05:15.360 If I have any solution, then I can add to that solution any 00:05:15.360 --> 00:05:18.950 other solution which satisfies the homogeneous equation, and 00:05:18.950 --> 00:05:22.140 the sum of those two will likewise be a solution. 00:05:22.140 --> 00:05:24.630 And that's very straightforward to verify. 00:05:24.630 --> 00:05:28.770 By simply substituting into the differential equation, the 00:05:28.770 --> 00:05:33.910 sum of a particular and the homogeneous solution, and what 00:05:33.910 --> 00:05:38.180 you'll see is that the homogeneous contribution, in 00:05:38.180 --> 00:05:41.950 fact, goes to zero by definition of what we mean by 00:05:41.950 --> 00:05:44.670 the homogeneous equation. 00:05:44.670 --> 00:05:48.430 Now, the homogeneous solution for a linear 00:05:48.430 --> 00:05:53.290 constant-coefficient differential equation is of 00:05:53.290 --> 00:05:58.030 the form that I indicate at the bottom. 00:05:58.030 --> 00:06:07.230 And it typically consists of a sum of N complex exponentials. 00:06:07.230 --> 00:06:11.650 And the constants are undetermined by 00:06:11.650 --> 00:06:14.000 the equation itself. 00:06:14.000 --> 00:06:17.240 And this form for the homogeneous solution, in 00:06:17.240 --> 00:06:23.860 essence, drops out of examining the homogeneous 00:06:23.860 --> 00:06:28.890 equation, where if we assume that the form of the 00:06:28.890 --> 00:06:34.210 homogeneous solution is a complex exponential with some 00:06:34.210 --> 00:06:38.570 unspecified amplitude and unspecified exponent. 00:06:38.570 --> 00:06:43.580 If we substitute this into our homogeneous equation, we end 00:06:43.580 --> 00:06:46.510 up with the equation that I've indicated here. 00:06:46.510 --> 00:06:50.780 The factor A and the e^(st) can in fact be canceled out, 00:06:50.780 --> 00:06:56.110 and we find that that equation is satisfied 00:06:56.110 --> 00:07:02.160 for N values of s. 00:07:02.160 --> 00:07:06.540 And that's true no matter what choice is made for these 00:07:06.540 --> 00:07:07.890 coefficients. 00:07:07.890 --> 00:07:12.510 And the essential consequence of all of that is that the 00:07:12.510 --> 00:07:16.660 homogeneous solution is of the form that I indicated 00:07:16.660 --> 00:07:18.160 previously. 00:07:18.160 --> 00:07:26.010 Namely it consists of a sum of N complex exponentials, where 00:07:26.010 --> 00:07:29.720 the coefficients, the N coefficients, attach to each 00:07:29.720 --> 00:07:33.320 of those complex exponential is undetermined or 00:07:33.320 --> 00:07:34.980 unspecified. 00:07:34.980 --> 00:07:39.710 So what this says is that in order to obtain the solution 00:07:39.710 --> 00:07:41.360 for a linear constant-coefficient 00:07:41.360 --> 00:07:44.750 differential equation, we need some kind of auxiliary 00:07:44.750 --> 00:07:49.760 information that tells us is how to obtain these N 00:07:49.760 --> 00:07:51.690 undetermined constants. 00:07:51.690 --> 00:07:55.550 And there are variety of ways of specifying this auxiliary 00:07:55.550 --> 00:07:57.930 information or auxiliary conditions. 00:07:57.930 --> 00:08:02.250 For example, in addition to the differential equation, 00:08:02.250 --> 00:08:10.000 what I can tell you is the value of the output and N - 1 00:08:10.000 --> 00:08:15.300 of its derivatives, at some specified time, t_0. 00:08:15.300 --> 00:08:20.590 And so the differential equation together with the 00:08:20.590 --> 00:08:25.020 auxiliary information, the initial conditions, then lets 00:08:25.020 --> 00:08:28.760 you determine the total solution, which namely lets 00:08:28.760 --> 00:08:31.750 you determine these previously-unspecified 00:08:31.750 --> 00:08:35.130 coefficients in the homogeneous solution. 00:08:35.130 --> 00:08:39.120 Now, depending on how the auxiliary information is 00:08:39.120 --> 00:08:42.350 stated or what auxiliary information is available, the 00:08:42.350 --> 00:08:47.080 system may or may not correspond to a linear system, 00:08:47.080 --> 00:08:49.360 and may or may not correspond to a linear 00:08:49.360 --> 00:08:51.880 time-invariant system. 00:08:51.880 --> 00:08:55.180 One essential condition for it to correspond to a linear 00:08:55.180 --> 00:09:01.110 system is that the initial conditions must be 0. 00:09:01.110 --> 00:09:06.550 And one can see the reason for that, if we refer back to the 00:09:06.550 --> 00:09:10.090 previous lecture in which we saw that for a linear system, 00:09:10.090 --> 00:09:13.090 if we put 0 in, we get 0 out. 00:09:13.090 --> 00:09:17.560 So if x(t), the input, is 0, the output must be 0. 00:09:17.560 --> 00:09:20.710 And so that, in essence, tells us that, at least for the 00:09:20.710 --> 00:09:26.220 system to be linear, these initial conditions must be 0. 00:09:26.220 --> 00:09:33.380 Now, beyond that, if we want the system to be causal and 00:09:33.380 --> 00:09:38.890 linear and time-invariant, then what's required on the 00:09:38.890 --> 00:09:42.840 initial conditions is that they be consistent with what's 00:09:42.840 --> 00:09:46.520 referred to as initial rest. 00:09:46.520 --> 00:09:53.770 Initial rest says that the output must be 0 up until the 00:09:53.770 --> 00:09:56.910 time that the input becomes non-zero. 00:09:56.910 --> 00:09:59.590 And we can see, of course, that that's consistent with 00:09:59.590 --> 00:10:03.150 the notion of causality, as we talked about in 00:10:03.150 --> 00:10:04.730 the previous lecture. 00:10:04.730 --> 00:10:10.400 And it's relatively straightforward to see that if 00:10:10.400 --> 00:10:14.920 the system is causal and linear and time-invariant, 00:10:14.920 --> 00:10:18.810 that will require initial rest. 00:10:18.810 --> 00:10:23.600 It's somewhat more difficult to see that if we specify 00:10:23.600 --> 00:10:27.410 initial rest, then that, in fact, is sufficient to 00:10:27.410 --> 00:10:31.040 determine that the system is both causal and linear and 00:10:31.040 --> 00:10:32.100 time invariant. 00:10:32.100 --> 00:10:35.730 But the essential point then is that it requires initial 00:10:35.730 --> 00:10:40.500 rest for both linearity and causality. 00:10:40.500 --> 00:10:46.360 OK, well, let's look at an example, and let's take the 00:10:46.360 --> 00:10:52.580 example of a first-order differential equation, as I've 00:10:52.580 --> 00:10:54.470 indicated here. 00:10:54.470 --> 00:10:58.623 So we have a first-order differential equation, dy(t) / 00:10:58.623 --> 00:11:03.350 dt + ay(t) is the input x(t). 00:11:03.350 --> 00:11:07.360 And let's first look at what the homogeneous solution of 00:11:07.360 --> 00:11:08.500 this equation is. 00:11:08.500 --> 00:11:13.370 And so, we consider the homogeneous equation, namely 00:11:13.370 --> 00:11:17.600 the equation that specifies solutions, which would 00:11:17.600 --> 00:11:20.650 correspond to 0 input. 00:11:20.650 --> 00:11:27.180 We, in essence, guess, or impose a solution of the form. 00:11:27.180 --> 00:11:31.590 The homogeneous solution is an amplitude factor times a 00:11:31.590 --> 00:11:33.450 complex exponential. 00:11:33.450 --> 00:11:38.390 Substituting this into the homogeneous equation, we then 00:11:38.390 --> 00:11:40.860 get the equation that I've indicated here. 00:11:40.860 --> 00:11:45.780 What you can see is that in this equation, I can cancel 00:11:45.780 --> 00:11:50.530 out the amplitude factor and this complex exponential. 00:11:50.530 --> 00:11:56.590 So let's just cancel those out on both sides of the equation. 00:11:56.590 --> 00:12:02.340 And what we're left with is an equation that specifies what 00:12:02.340 --> 00:12:04.760 the complex exponent must be. 00:12:04.760 --> 00:12:08.110 In particular, for the homogeneous solution, s must 00:12:08.110 --> 00:12:12.060 be equal to -a, and so finally, our homogeneous 00:12:12.060 --> 00:12:16.380 solution is as I've indicated here. 00:12:19.050 --> 00:12:23.150 Now, let's look at the solution for a specific input. 00:12:23.150 --> 00:12:25.990 Let's consider, for example, an input which is 00:12:25.990 --> 00:12:28.000 a scaled unit step. 00:12:28.000 --> 00:12:31.530 And although I won't work out the solution in detail and 00:12:31.530 --> 00:12:36.510 perhaps using what you've worked on previously, you know 00:12:36.510 --> 00:12:39.805 how to carry out the solution for that. 00:12:39.805 --> 00:12:44.950 A solution with a step input is what I've indicated here: a 00:12:44.950 --> 00:12:47.060 scalar, 1 minus an exponential, 00:12:47.060 --> 00:12:48.540 times a unit step. 00:12:48.540 --> 00:12:51.430 And you can verify that simply by substituting into the 00:12:51.430 --> 00:12:53.170 differential equation. 00:12:53.170 --> 00:12:56.930 Now, we know that there's a family of solutions. 00:12:56.930 --> 00:13:02.620 In other words, any solution with a homogeneous solution 00:13:02.620 --> 00:13:04.990 added to it, is, again, a solution. 00:13:04.990 --> 00:13:08.670 And so if we consider the solution that I just 00:13:08.670 --> 00:13:13.160 indicated, we generate the entire family of solutions by 00:13:13.160 --> 00:13:17.440 adding a homogeneous solution to it, and so this then 00:13:17.440 --> 00:13:21.400 corresponds to the entire family of solutions, where the 00:13:21.400 --> 00:13:27.640 constant A is unspecified so far, and needs to be specified 00:13:27.640 --> 00:13:31.470 through some type of auxiliary conditions. 00:13:31.470 --> 00:13:36.640 Now, a class of auxiliary conditions is the condition of 00:13:36.640 --> 00:13:41.310 initial rest, which as I indicated before, is 00:13:41.310 --> 00:13:44.830 equivalent to the statement that the system is causal and 00:13:44.830 --> 00:13:46.900 linear and time-invariant. 00:13:46.900 --> 00:13:52.080 And in that case, for the initial rest condition, we 00:13:52.080 --> 00:13:56.660 would then require in this equation above that this 00:13:56.660 --> 00:13:59.710 constant be equal to 0. 00:13:59.710 --> 00:14:04.860 And so finally, the response to a scaled step-- 00:14:04.860 --> 00:14:07.730 if the system is to correspond to a causal linear 00:14:07.730 --> 00:14:12.760 time-invariant system, is then just this term, namely, a 00:14:12.760 --> 00:14:18.870 constant times 1 minus an exponential, times the step. 00:14:18.870 --> 00:14:23.060 Now, if the system is a linear time-invariant system, it can, 00:14:23.060 --> 00:14:26.260 as we know, be described through its impulse response. 00:14:26.260 --> 00:14:30.420 And as you've worked out previously in the video course 00:14:30.420 --> 00:14:34.730 manual, for a linear time-invariant system, the 00:14:34.730 --> 00:14:36.480 impulse response is the 00:14:36.480 --> 00:14:38.750 derivative of the step response. 00:14:38.750 --> 00:14:41.730 And just to quickly remind you of where that 00:14:41.730 --> 00:14:43.960 result comes from. 00:14:43.960 --> 00:14:48.020 In essence, we can consider two linear time-invariant 00:14:48.020 --> 00:14:51.770 systems in cascade, one a differentiator, the other the 00:14:51.770 --> 00:14:55.060 system that we're talking about described by the 00:14:55.060 --> 00:14:56.850 differential equation. 00:14:56.850 --> 00:15:01.700 And a step in here then generates an impulse into our 00:15:01.700 --> 00:15:04.600 system, and out comes the impulse response. 00:15:04.600 --> 00:15:07.100 Well, just using the fact that these are both linear 00:15:07.100 --> 00:15:10.840 time-invariant systems, and they can be cascaded in either 00:15:10.840 --> 00:15:15.970 order, then means that if we have the step response to our 00:15:15.970 --> 00:15:20.380 system, and that goes through the differentiator, what must 00:15:20.380 --> 00:15:24.570 come out, again, is the impulse response. 00:15:24.570 --> 00:15:27.380 So differentiating the step response, we 00:15:27.380 --> 00:15:29.310 get the impulse response. 00:15:29.310 --> 00:15:33.240 Here again, is the step response as we just worked it 00:15:33.240 --> 00:15:35.990 out, this time for a unit step. 00:15:35.990 --> 00:15:40.760 If we differentiate, we have then, since the step response 00:15:40.760 --> 00:15:44.120 is the product of two terms, the derivative of a product is 00:15:44.120 --> 00:15:45.940 the sum of the derivatives. 00:15:45.940 --> 00:15:49.640 And carrying that algebra through, and using the fact 00:15:49.640 --> 00:15:54.350 that the derivative of the step is an impulse, finally we 00:15:54.350 --> 00:15:59.180 come down to this statement for the impulse response. 00:15:59.180 --> 00:16:01.820 And then recognizing that this is a time 00:16:01.820 --> 00:16:03.970 function times an impulse. 00:16:03.970 --> 00:16:07.800 And we know that if a time function times an impulse 00:16:07.800 --> 00:16:14.160 takes on the value at the time that the impulse occurs, then 00:16:14.160 --> 00:16:18.390 this term is simply 0. 00:16:18.390 --> 00:16:22.140 And the impulse response then finally is an 00:16:22.140 --> 00:16:24.420 exponential of this form. 00:16:24.420 --> 00:16:28.390 And this is the decaying exponential for a-positive, 00:16:28.390 --> 00:16:31.640 it's a growing exponential for a-negative. 00:16:31.640 --> 00:16:38.480 And recall that, as we talked about previously, a linear 00:16:38.480 --> 00:16:42.400 time-invariant system is stable if its impulse response 00:16:42.400 --> 00:16:44.610 is absolutely integrable. 00:16:44.610 --> 00:16:49.760 For this particular case, this impulse response is absolutely 00:16:49.760 --> 00:16:56.250 integrable provided that the exponential factor a is 00:16:56.250 --> 00:16:59.290 greater than 0. 00:16:59.290 --> 00:17:04.700 Okay, so what we've seen then is the impulse response for a 00:17:04.700 --> 00:17:07.230 system described by a linear constant-coefficient 00:17:07.230 --> 00:17:10.829 differential equation, where in addition, we would impose 00:17:10.829 --> 00:17:15.069 causality, linearity, and time-invariance, essentially, 00:17:15.069 --> 00:17:19.619 through the initial conditions of initial rest. 00:17:19.619 --> 00:17:23.609 Now, pretty much the same kinds of things happen with 00:17:23.609 --> 00:17:27.079 difference equations as we've gone through with 00:17:27.079 --> 00:17:29.740 differential equations. 00:17:29.740 --> 00:17:33.820 In particular, again, let me remind you of the form of an 00:17:33.820 --> 00:17:35.640 Nth order linear constant 00:17:35.640 --> 00:17:37.760 coefficient difference equation. 00:17:37.760 --> 00:17:40.820 It's as I indicate here. 00:17:40.820 --> 00:17:44.390 And, again, a linear combination, this time of 00:17:44.390 --> 00:17:48.220 delayed versions of the output equal to a linear combination 00:17:48.220 --> 00:17:51.520 of delayed versions of the input. 00:17:51.520 --> 00:17:57.210 Once again, difference equation is not a complete 00:17:57.210 --> 00:18:00.310 specification of the system because we can add to the 00:18:00.310 --> 00:18:03.820 response any homogeneous solution. 00:18:03.820 --> 00:18:07.500 In other words, any solution that satisfies the homogeneous 00:18:07.500 --> 00:18:12.230 equation and the sum of those will also satisfy the original 00:18:12.230 --> 00:18:13.910 difference equation. 00:18:13.910 --> 00:18:21.500 So if we have a particular response that satisfies the 00:18:21.500 --> 00:18:27.070 difference equation, then adding to that any response 00:18:27.070 --> 00:18:32.680 that is a solution to the homogeneous equation will also 00:18:32.680 --> 00:18:36.640 be a solution to the total equation. 00:18:36.640 --> 00:18:41.470 The homogeneous solution, again, is of the form of a 00:18:41.470 --> 00:18:44.650 linear combination of exponentials. 00:18:44.650 --> 00:18:48.300 Here, we have the homogeneous equation. 00:18:48.300 --> 00:18:53.390 As with differential equations, we can guess or 00:18:53.390 --> 00:18:58.010 impose solutions of the form A times an exponential. 00:18:58.010 --> 00:19:01.780 When we substitute this into the homogeneous equation, we 00:19:01.780 --> 00:19:05.830 then end up with the equation that I've indicated here. 00:19:05.830 --> 00:19:15.580 We recognize again that A, the amplitude, and z^n, this 00:19:15.580 --> 00:19:19.310 exponential factor, cancel out. 00:19:19.310 --> 00:19:27.000 And so this equation is satisfied for any values of z 00:19:27.000 --> 00:19:30.310 that satisfy this equation. 00:19:30.310 --> 00:19:34.310 And there are N roots, z_1 through z_N. 00:19:34.310 --> 00:19:38.830 And so finally, the form for the homogeneous solution is a 00:19:38.830 --> 00:19:44.240 linear combination of capital N exponentials, where capital 00:19:44.240 --> 00:19:46.930 N is the order of the equation. 00:19:46.930 --> 00:19:49.400 With each of those exponentials, the amplitude 00:19:49.400 --> 00:19:53.220 factor is undetermined and needs to be determined in some 00:19:53.220 --> 00:19:58.630 way through the imposition of appropriate initial conditions 00:19:58.630 --> 00:19:59.880 or boundary conditions. 00:20:01.990 --> 00:20:07.370 So the general form then for the solution to the difference 00:20:07.370 --> 00:20:13.590 equation is a sum of exponentials plus any 00:20:13.590 --> 00:20:15.830 particular solution. 00:20:15.830 --> 00:20:20.730 It's through auxiliary conditions that we determine 00:20:20.730 --> 00:20:23.150 these coefficients. 00:20:23.150 --> 00:20:28.420 We have N undetermined coefficients and, so we 00:20:28.420 --> 00:20:31.850 require N auxiliary conditions. 00:20:31.850 --> 00:20:37.520 For example, some set of values of the output at N 00:20:37.520 --> 00:20:41.210 distinct instance of time. 00:20:41.210 --> 00:20:44.200 Now this was the same as with differential equations. 00:20:44.200 --> 00:20:46.640 In the case of differential equations, we talked about 00:20:46.640 --> 00:20:50.510 specifying the value of the output and its derivatives. 00:20:50.510 --> 00:20:53.270 And there, we indicated that for 00:20:53.270 --> 00:20:56.890 linearity, what we required-- 00:20:56.890 --> 00:21:00.740 for linearity, what we required is that the auxiliary 00:21:00.740 --> 00:21:02.580 conditions be 0. 00:21:02.580 --> 00:21:05.830 And the same thing applies here for the same reason. 00:21:05.830 --> 00:21:09.350 Namely, if the system is to be linear, then the response, if 00:21:09.350 --> 00:21:13.440 there's no input, must be equal to 0. 00:21:13.440 --> 00:21:19.200 In addition, what we may want to impose on the system is 00:21:19.200 --> 00:21:22.290 that it be causal, and in addition to linear 00:21:22.290 --> 00:21:28.320 time-invariant, and what that requires, again, is that the 00:21:28.320 --> 00:21:32.220 auxiliary conditions be consistent with initial rest, 00:21:32.220 --> 00:21:37.690 namely, that if the input is 0 prior to some time, then the 00:21:37.690 --> 00:21:42.680 output is 0 prior to the same time. 00:21:42.680 --> 00:21:47.240 So we've seen a very direct parallel so far between 00:21:47.240 --> 00:21:52.210 differential equations and difference equations. 00:21:52.210 --> 00:21:55.930 In fact, one difference between them that, in some 00:21:55.930 --> 00:22:00.140 sense, makes difference equations easier to deal with 00:22:00.140 --> 00:22:04.000 in some situations, is that in contrast to a differential 00:22:04.000 --> 00:22:07.980 equation, a difference equation, if we assume 00:22:07.980 --> 00:22:13.580 causality, in fact is an explicit input-output 00:22:13.580 --> 00:22:15.080 relationship for the system. 00:22:15.080 --> 00:22:18.670 Now, let me show you what I mean. 00:22:18.670 --> 00:22:22.130 Let's consider the nth-order difference equation, as I've 00:22:22.130 --> 00:22:23.490 indicated here. 00:22:23.490 --> 00:22:28.590 And let's assume that we're imposing causality so that the 00:22:28.590 --> 00:22:33.000 output can only depend on prior values of the input, and 00:22:33.000 --> 00:22:35.730 therefore, aren't prior values of the output. 00:22:35.730 --> 00:22:41.450 Well, we can simply rearrange this equation solving for y[n] 00:22:41.450 --> 00:22:44.680 the leading term, with k = 0. 00:22:44.680 --> 00:22:47.660 Taking all of the other terms over to the right side of the 00:22:47.660 --> 00:22:54.740 equation, and we then have a recursive equation, namely, an 00:22:54.740 --> 00:22:59.070 equation that expresses the output in terms of prior 00:22:59.070 --> 00:23:03.850 values of the input, which is this term, and prior values of 00:23:03.850 --> 00:23:05.640 the output. 00:23:05.640 --> 00:23:10.050 And so if, in fact, we have this equation running, then 00:23:10.050 --> 00:23:13.800 once it started, we know how to compute the output for the 00:23:13.800 --> 00:23:16.740 next time instant. 00:23:16.740 --> 00:23:19.170 Well, how do we get it started? 00:23:19.170 --> 00:23:22.240 The way we get it started, of course, is through the 00:23:22.240 --> 00:23:26.240 appropriate set of initial conditions or boundary 00:23:26.240 --> 00:23:27.300 conditions. 00:23:27.300 --> 00:23:30.760 And, if for example, we assume initial rest corresponding to 00:23:30.760 --> 00:23:35.360 a causal linear time-invariant system, then if the input is 0 00:23:35.360 --> 00:23:39.560 up until some time, the output must be 0 up until that time. 00:23:39.560 --> 00:23:43.390 And that, in essence, helps us get the equation started. 00:23:43.390 --> 00:23:48.150 Well, let's look at this specifically in the context of 00:23:48.150 --> 00:23:50.160 a first-order difference equation. 00:23:50.160 --> 00:23:55.510 So let's take a first-order difference equation, as I've 00:23:55.510 --> 00:23:57.560 indicated here. 00:23:57.560 --> 00:24:00.650 And so, we have an equation that tells us that y[n] 00:24:00.650 --> 00:24:03.920 - ay[n-1] 00:24:03.920 --> 00:24:06.750 = x[n]. 00:24:06.750 --> 00:24:10.970 Now, if we want this to correspond to a causal linear 00:24:10.970 --> 00:24:16.780 time-invariant system, we impose initial rest on it. 00:24:16.780 --> 00:24:21.550 We can rewrite the first-order difference equation by taking 00:24:21.550 --> 00:24:24.280 the term involving y[n-1] 00:24:24.280 --> 00:24:25.945 over two the right hand side of the equation. 00:24:28.450 --> 00:24:33.560 Now, this gives us a recursive equation that expresses the 00:24:33.560 --> 00:24:38.500 output in terms of the input and past values of the output. 00:24:38.500 --> 00:24:43.610 And since we've imposed causality, and if we're 00:24:43.610 --> 00:24:46.820 talking about a linear time-invariant system, we can 00:24:46.820 --> 00:24:50.350 now inquire as to what the impulse response is. 00:24:50.350 --> 00:24:53.120 And we know, of course, the impulse response tells us 00:24:53.120 --> 00:24:56.910 everything that we need to know about the system. 00:24:56.910 --> 00:25:02.990 So let's choose an input, which is an impulse. 00:25:02.990 --> 00:25:07.070 So the impulse response is delta[n] 00:25:07.070 --> 00:25:09.040 corresponding to the x[n] 00:25:09.040 --> 00:25:13.490 up here, plus a delta of-- 00:25:13.490 --> 00:25:15.880 this should be h[n-1]. 00:25:15.880 --> 00:25:17.970 And let me just correct that. 00:25:17.970 --> 00:25:20.640 This is h[n-1]. 00:25:20.640 --> 00:25:24.270 So we have the impulse response as delta[n] 00:25:24.270 --> 00:25:25.520 plus a times h[n-1]. 00:25:28.470 --> 00:25:33.235 Now, from initial rest, we know that since the input, 00:25:33.235 --> 00:25:37.620 namely an impulse, is 0 for n less than 0, the impulse 00:25:37.620 --> 00:25:42.410 response is likewise 0 for n less than 0. 00:25:42.410 --> 00:25:45.620 And now, let's work out what h[0] 00:25:45.620 --> 00:25:46.870 is. 00:25:46.870 --> 00:25:52.900 Well, h of 0, with n = 0, is delta[0] 00:25:52.900 --> 00:25:55.940 plus a times h[n-1]. 00:25:55.940 --> 00:25:56.950 h[n-1] 00:25:56.950 --> 00:25:58.010 is 0. 00:25:58.010 --> 00:25:59.600 And so, h[0] 00:25:59.600 --> 00:26:01.440 is equal to 1. 00:26:05.490 --> 00:26:09.030 Now that we have h[0], we can figure out h[1] 00:26:09.030 --> 00:26:12.340 by running this recursive equation. 00:26:12.340 --> 00:26:14.640 So h[1] 00:26:14.640 --> 00:26:21.360 is delta[1], which is 0, plus a times h[0], which we just 00:26:21.360 --> 00:26:22.820 figured out is 1. 00:26:22.820 --> 00:26:24.410 So, h[1] 00:26:24.410 --> 00:26:26.120 is equal to a. 00:26:26.120 --> 00:26:29.380 And if we carry this through, we'll have h[2] 00:26:29.380 --> 00:26:34.180 equal to a^2, and this will continue on. 00:26:34.180 --> 00:26:37.660 And in fact, what we can recognize by looking at this 00:26:37.660 --> 00:26:41.860 and how we would expect these terms to build, we would see 00:26:41.860 --> 00:26:46.120 that the impulse response, h[n], in fact, is of the form 00:26:46.120 --> 00:26:50.630 a^n times u[n]. 00:26:50.630 --> 00:26:56.080 And we also can recognize then that this corresponds to a 00:26:56.080 --> 00:27:02.890 stable system, if and only if the impulse response, which is 00:27:02.890 --> 00:27:05.640 what we just figured out, if and only if the impulse 00:27:05.640 --> 00:27:08.150 response is absolutely summable. 00:27:08.150 --> 00:27:12.820 And what that will require is that the magnitude of a be 00:27:12.820 --> 00:27:14.070 less than 1. 00:27:17.400 --> 00:27:21.110 Now, we imposed causality, linearity, and 00:27:21.110 --> 00:27:25.750 time-invariance, and generated a solution recursively. 00:27:25.750 --> 00:27:29.590 And now, of course, if we want to generate the more general 00:27:29.590 --> 00:27:33.880 set of solutions to this difference equation, we can do 00:27:33.880 --> 00:27:38.650 that by adding all of the homogeneous solutions, namely, 00:27:38.650 --> 00:27:42.860 all the solutions that satisfy the homogeneous equation. 00:27:42.860 --> 00:27:46.230 So here, we have the causal linear 00:27:46.230 --> 00:27:49.330 time-invariant impulse response. 00:27:49.330 --> 00:27:55.220 In fact, with an impulse input, all of the possible 00:27:55.220 --> 00:28:00.380 solutions are that impulse response plus 00:28:00.380 --> 00:28:02.500 the homogeneous solutions. 00:28:02.500 --> 00:28:06.620 The homogeneous solution is the solution that satisfies 00:28:06.620 --> 00:28:09.660 the homogeneous equation. 00:28:09.660 --> 00:28:14.960 That will, in general, be of the form an amplitude factor 00:28:14.960 --> 00:28:19.950 times an exponential factor, and if we substitute this into 00:28:19.950 --> 00:28:24.610 this equation, then we see that the homogeneous equation 00:28:24.610 --> 00:28:30.480 is satisfied for any values of a and any values of z that 00:28:30.480 --> 00:28:33.780 satisfy this equation. 00:28:33.780 --> 00:28:36.180 Again, as we did with differential equations, the 00:28:36.180 --> 00:28:39.960 factor A cancels out. 00:28:39.960 --> 00:28:43.870 And also, in fact, I can cancel out a z^n 00:28:43.870 --> 00:28:48.100 there and a z^n here. 00:28:48.100 --> 00:28:52.420 And so what we're left with is a statement that tells us then 00:28:52.420 --> 00:28:58.890 that the homogeneous solution is of the form a(z^n), for any 00:28:58.890 --> 00:29:05.180 value of A and any value of z that satisfies this equation. 00:29:05.180 --> 00:29:10.200 And that value of z, in particular, is z equal to a. 00:29:10.200 --> 00:29:14.620 So the homogeneous solution then is any exponential of 00:29:14.620 --> 00:29:17.080 this form with any amplitude factor. 00:29:17.080 --> 00:29:20.670 And so, the family of solutions, with an impulse 00:29:20.670 --> 00:29:25.800 input is the solution corresponding to the system 00:29:25.800 --> 00:29:29.810 being causal, linear, and time-invariant, plus the 00:29:29.810 --> 00:29:32.010 homogeneous term. 00:29:32.010 --> 00:29:36.950 If we impose causality, linearity, and time-invariance 00:29:36.950 --> 00:29:41.430 on the system, then of course, that additional exponential 00:29:41.430 --> 00:29:43.170 factor will be 0. 00:29:43.170 --> 00:29:45.400 In other words, A is equal to 0. 00:29:48.970 --> 00:29:54.450 Now, we've seen the differential equations and 00:29:54.450 --> 00:29:59.880 difference equations in terms of the fact that there are 00:29:59.880 --> 00:30:04.040 families of solutions, and in order to get causality, 00:30:04.040 --> 00:30:07.350 linearity, and time-invariance requires imposing a particular 00:30:07.350 --> 00:30:10.930 set of initial conditions, namely, imposing initial rest 00:30:10.930 --> 00:30:12.480 on the system. 00:30:12.480 --> 00:30:17.310 Let's now look at the difference equation and then 00:30:17.310 --> 00:30:20.540 later, the differential equation, interpreted in block 00:30:20.540 --> 00:30:23.190 diagram terms. 00:30:23.190 --> 00:30:27.790 Now, the difference equation, as I just simply repeated 00:30:27.790 --> 00:30:30.780 here, is y[n] 00:30:30.780 --> 00:30:31.690 = x[n] 00:30:31.690 --> 00:30:37.360 + ay[n-1], where I've taken the delayed term over to the 00:30:37.360 --> 00:30:39.440 right hand side of the equation. 00:30:39.440 --> 00:30:42.200 So, in effect, what I'm imposing on 00:30:42.200 --> 00:30:44.460 this system is causality. 00:30:44.460 --> 00:30:47.990 I'm assuming that if I know the past history of the input 00:30:47.990 --> 00:30:52.190 and the output, I can determine the next value of 00:30:52.190 --> 00:30:53.860 the output. 00:30:53.860 --> 00:30:58.600 Well, we can, in fact, draw a block diagram that represents 00:30:58.600 --> 00:31:00.130 that equation. 00:31:00.130 --> 00:31:03.140 The equations says, we take x[n] 00:31:03.140 --> 00:31:07.890 for any given value of n, say n0, whatever value we're 00:31:07.890 --> 00:31:09.980 computing the output for. 00:31:09.980 --> 00:31:15.850 We take the input at that time, and add to it the factor 00:31:15.850 --> 00:31:21.390 a times the output value that we calculated last time. 00:31:21.390 --> 00:31:27.880 So if we have x[n], which is our input, and if we have 00:31:27.880 --> 00:31:34.160 y[n], which is our output, we, in fact, can get y of n by 00:31:34.160 --> 00:31:41.290 taking the last value of y[n], indicated here by putting y[n] 00:31:41.290 --> 00:31:52.070 through a delay, multiplying that by the factor a, and then 00:31:52.070 --> 00:31:59.930 adding that result to the input. 00:31:59.930 --> 00:32:05.430 And the result of doing that is y[n]. 00:32:05.430 --> 00:32:09.750 So the way this block diagram might be interpreted, for 00:32:09.750 --> 00:32:16.720 example, as an algorithm is to say that we take x[n], add to 00:32:16.720 --> 00:32:20.680 it a times the previous value the output. 00:32:20.680 --> 00:32:24.160 That sum gives us the current value of the output, which we 00:32:24.160 --> 00:32:29.440 then put out of the system, and also put into a delay 00:32:29.440 --> 00:32:33.630 element, or basically, into a storage register, to use on 00:32:33.630 --> 00:32:35.980 the next iteration or recursion. 00:32:35.980 --> 00:32:37.580 Now, how do we get this started? 00:32:37.580 --> 00:32:40.680 Well, we know that the difference equation requires 00:32:40.680 --> 00:32:42.500 initial conditions. 00:32:42.500 --> 00:32:47.560 And, in fact, the initial conditions correspond to what 00:32:47.560 --> 00:32:52.210 we store in the delay register when this block diagram or 00:32:52.210 --> 00:32:55.300 equation initially starts up. 00:32:55.300 --> 00:33:01.380 OK Now, let's look at this for the case of difference 00:33:01.380 --> 00:33:03.275 equations more generally. 00:33:06.650 --> 00:33:13.360 So, what we've said is that we can calculate the output by 00:33:13.360 --> 00:33:18.460 having previous values of the input, previous values of the 00:33:18.460 --> 00:33:22.270 output, and forming the appropriate linear 00:33:22.270 --> 00:33:23.850 combination. 00:33:23.850 --> 00:33:27.550 So let's just build up the more general block diagram 00:33:27.550 --> 00:33:28.960 that would correspond to this. 00:33:31.870 --> 00:33:35.900 And what it says is that we want to have a mechanism for 00:33:35.900 --> 00:33:40.130 storing past values of the input, and a mechanism for 00:33:40.130 --> 00:33:41.930 storing past values of the output. 00:33:41.930 --> 00:33:47.400 And I've indicated that on this figure, so far, by a 00:33:47.400 --> 00:33:55.610 chain of delay elements, indicating that what the 00:33:55.610 --> 00:34:02.090 output of each delay is, is the input delayed by one time 00:34:02.090 --> 00:34:05.050 instant or interval. 00:34:05.050 --> 00:34:09.560 And so what we see down this chain of delays are delayed 00:34:09.560 --> 00:34:13.010 replications of the input. 00:34:13.010 --> 00:34:19.159 And what we see on the other chain is delayed replications 00:34:19.159 --> 00:34:20.409 of the output. 00:34:22.960 --> 00:34:26.139 Now, the difference equation says that we want to take 00:34:26.139 --> 00:34:31.150 these, and multiply them by the appropriate coefficients, 00:34:31.150 --> 00:34:33.909 the coefficients in the difference equation, and so, 00:34:33.909 --> 00:34:37.690 we can do that as I've indicated here. 00:34:37.690 --> 00:34:42.880 So now, we have these delay elements, each multiplied by 00:34:42.880 --> 00:34:46.090 the appropriate coefficients on the input, and by 00:34:46.090 --> 00:34:50.139 appropriate coefficients on the output. 00:34:50.139 --> 00:34:56.300 Those are then summed together, and so we now will 00:34:56.300 --> 00:35:00.330 sum these and will sum these. 00:35:00.330 --> 00:35:03.990 After we've summed these, we want to add those together. 00:35:03.990 --> 00:35:07.200 And there's a factor of 1 / a_0 that comes in. 00:35:07.200 --> 00:35:11.300 And so that then generates our output. 00:35:11.300 --> 00:35:16.690 And so this, in fact, then represents a block diagram, 00:35:16.690 --> 00:35:20.370 which is a general block diagram for implementing or 00:35:20.370 --> 00:35:22.670 representing a linear constant-coefficient 00:35:22.670 --> 00:35:25.060 difference equation. 00:35:25.060 --> 00:35:28.190 Now, if you think about what it means in terms of, let's 00:35:28.190 --> 00:35:31.530 say, a computer algorithm or a piece of hardware, in fact, 00:35:31.530 --> 00:35:34.670 this block diagram is a recipe or algorithm for doing the 00:35:34.670 --> 00:35:36.410 implementation. 00:35:36.410 --> 00:35:41.680 But it's important to recognize, even at this point, 00:35:41.680 --> 00:35:45.980 that it's only one of many possible algorithms or 00:35:45.980 --> 00:35:48.990 implementations for this difference equation. 00:35:48.990 --> 00:35:57.280 Just for example, I can consider that equation for 00:35:57.280 --> 00:35:59.320 that block diagram. 00:35:59.320 --> 00:36:03.290 And here, I've re-drawn it. 00:36:03.290 --> 00:36:05.630 So here, are once again. 00:36:05.630 --> 00:36:09.140 I have the same block diagram that we just saw. 00:36:09.140 --> 00:36:13.810 And I can recognize, for example, that this, in 00:36:13.810 --> 00:36:19.620 essence, corresponds to two linear time-invariant systems 00:36:19.620 --> 00:36:22.180 in cascade. 00:36:22.180 --> 00:36:26.540 Now, that assumes, of course, that my initial conditions are 00:36:26.540 --> 00:36:30.020 such that the system is, in fact, linear. 00:36:30.020 --> 00:36:33.480 And that, in turn, requires that we're assuming initial 00:36:33.480 --> 00:36:35.300 rests, namely, before the input does 00:36:35.300 --> 00:36:37.880 anything other than 0. 00:36:37.880 --> 00:36:41.030 There are just 0 values stored in the registers. 00:36:41.030 --> 00:36:43.710 But assuming that it corresponds to a linear 00:36:43.710 --> 00:36:48.640 time-invariant system, this is a cascade of two linear 00:36:48.640 --> 00:36:50.540 time-invariant invriant systems. 00:36:50.540 --> 00:36:52.990 We know that two linear time-invariant systems can be 00:36:52.990 --> 00:36:55.160 cascaded in either order. 00:36:55.160 --> 00:36:59.090 So, in particular, I can consider breaking this cascade 00:36:59.090 --> 00:37:03.820 here, and moving this block over to the other side. 00:37:03.820 --> 00:37:05.520 And so let's just do that. 00:37:08.610 --> 00:37:16.160 And when I do, I then have this combination of systems, 00:37:16.160 --> 00:37:19.370 and, of course, you can ask what advantage there is to 00:37:19.370 --> 00:37:24.570 doing that, and the advantage arises because of the fact 00:37:24.570 --> 00:37:31.990 that in this form, exactly what is stored in these delays 00:37:31.990 --> 00:37:35.120 is also stored in these delay registers. 00:37:35.120 --> 00:37:38.270 In other words, it's this intermediate variable-- 00:37:38.270 --> 00:37:39.500 whatever it is-- 00:37:39.500 --> 00:37:42.850 down this chain of the delays and down this chain of delays, 00:37:42.850 --> 00:37:47.150 and so, in fact, I can collapse those delays into a 00:37:47.150 --> 00:37:49.000 single chain of delays. 00:37:49.000 --> 00:37:53.790 And the network that I'm left with is the network that I 00:37:53.790 --> 00:37:57.930 indicate on this view graph, where what I've done is to 00:37:57.930 --> 00:38:01.980 simply collapse that double chain of delays into a single 00:38:01.980 --> 00:38:04.580 change of delays. 00:38:04.580 --> 00:38:10.180 Now, one can ask, well, what's the advantage to doing that? 00:38:10.180 --> 00:38:13.020 And one advantage, simply stated, is that when you think 00:38:13.020 --> 00:38:16.160 in terms of an implementation of a difference equation, a 00:38:16.160 --> 00:38:21.340 delay corresponds to a storage register, a memory location, 00:38:21.340 --> 00:38:24.980 and by simply using the fact that we can interchange the 00:38:24.980 --> 00:38:27.200 order in which linear time-invariant systems are 00:38:27.200 --> 00:38:30.030 cascaded, we can reduce the amount of memory 00:38:30.030 --> 00:38:31.280 by a factor of 2. 00:38:35.280 --> 00:38:40.570 Now, an essentially similar procedure can also be used for 00:38:40.570 --> 00:38:44.910 differential equations, in terms of implementation using 00:38:44.910 --> 00:38:47.590 block diagrams or the interpretation of 00:38:47.590 --> 00:38:50.350 implementations using block diagrams. 00:38:50.350 --> 00:38:52.380 And let me first do that-- 00:38:52.380 --> 00:38:56.840 rather than in general-- let me first do it in the context 00:38:56.840 --> 00:39:00.870 of a specific example. 00:39:00.870 --> 00:39:06.830 So let's consider a linear constant-coefficient 00:39:06.830 --> 00:39:10.790 differential equation, as I've indicated here, and I have 00:39:10.790 --> 00:39:13.930 terms on the left side and terms on the right side. 00:39:13.930 --> 00:39:19.870 And with the differential equation, let's consider 00:39:19.870 --> 00:39:22.800 taking all the terms over to the right side of the 00:39:22.800 --> 00:39:26.030 equation, except for the highest 00:39:26.030 --> 00:39:28.950 derivative in the output. 00:39:28.950 --> 00:39:33.250 Next, we integrate both sides of the equation so that when 00:39:33.250 --> 00:39:35.540 we're done, we end up with on the left side of the 00:39:35.540 --> 00:39:37.480 equation with y(t). 00:39:37.480 --> 00:39:40.480 On the right side of the equation with the appropriate 00:39:40.480 --> 00:39:42.700 number of integrations. 00:39:42.700 --> 00:39:46.170 And so the integral equation that we'll get for this 00:39:46.170 --> 00:39:52.410 example y(t), the output, is x(t) plus b, the scale factor 00:39:52.410 --> 00:39:56.370 times the integral of the input, and minus a, that scale 00:39:56.370 --> 00:39:59.480 factor, times the integral of the output. 00:39:59.480 --> 00:40:05.600 So to form the output in the block diagram terms, we form a 00:40:05.600 --> 00:40:09.140 linear combination of the input, a scaled integral of 00:40:09.140 --> 00:40:12.970 the input, and a scaled integral of the output, all of 00:40:12.970 --> 00:40:14.820 that added together. 00:40:14.820 --> 00:40:20.540 So we need, in addition to the input, we need the 00:40:20.540 --> 00:40:22.070 integral of the input. 00:40:22.070 --> 00:40:25.250 And so this box indicates an integrator. 00:40:25.250 --> 00:40:28.260 In addition to the output, we need the 00:40:28.260 --> 00:40:30.670 integral of the output. 00:40:30.670 --> 00:40:38.490 And now, to form y(t), we multiply the integrated input 00:40:38.490 --> 00:40:41.540 by the scale factor, b. 00:40:41.540 --> 00:40:49.860 And add that to x(t), and we take the integrated output, 00:40:49.860 --> 00:40:57.330 multiply it by -a, and add to that the result of the 00:40:57.330 --> 00:41:00.530 previous addition, and according to the integral 00:41:00.530 --> 00:41:05.180 equation, then that forms the output. 00:41:05.180 --> 00:41:09.580 So just as we did with the difference equation, we've 00:41:09.580 --> 00:41:12.330 converted the differential equation to an integral 00:41:12.330 --> 00:41:17.200 equation, and we have a block diagram form very similar to 00:41:17.200 --> 00:41:20.100 what we had in the case of the difference equation. 00:41:20.100 --> 00:41:24.160 Now, the initial conditions, of course, are tied up in, 00:41:24.160 --> 00:41:27.430 again, how these integrators are initialized. 00:41:27.430 --> 00:41:30.980 Assuming that we impose initial rest on the system, we 00:41:30.980 --> 00:41:33.710 can think of the overall system as a linear 00:41:33.710 --> 00:41:37.740 time-invariant system, and it's a cascade of one linear 00:41:37.740 --> 00:41:40.470 time-invariant system with a second. 00:41:40.470 --> 00:41:47.190 So we can, in fact, break this, and consider 00:41:47.190 --> 00:41:49.620 interchanging the order in which these 00:41:49.620 --> 00:41:51.520 two systems are cascaded. 00:41:51.520 --> 00:41:54.580 And so I've indicated that down below. 00:41:54.580 --> 00:41:58.840 Here, I've simply taken the top block diagram, 00:41:58.840 --> 00:42:01.920 interchanged the order in which the 00:42:01.920 --> 00:42:05.590 two systems are cascaded. 00:42:05.590 --> 00:42:09.490 And here, again, we can ask what the advantages to this, 00:42:09.490 --> 00:42:12.090 as opposed to the previous one. 00:42:12.090 --> 00:42:14.830 And what you can see, just as we saw with the difference 00:42:14.830 --> 00:42:18.790 equation, is that now, the integrators-- 00:42:18.790 --> 00:42:19.780 both integrators-- 00:42:19.780 --> 00:42:21.730 are integrating the same thing. 00:42:21.730 --> 00:42:27.170 In particular, the input to this integrator and the input 00:42:27.170 --> 00:42:30.210 to this integrator are identical. 00:42:30.210 --> 00:42:34.660 So in fact, rather than using this one, we can simply tap 00:42:34.660 --> 00:42:36.970 off from here. 00:42:36.970 --> 00:42:40.660 We can, in fact, remove this integrator, break this 00:42:40.660 --> 00:42:45.610 connection, and tap in at this point. 00:42:45.610 --> 00:42:49.910 And so what we've done then, by interchanging the order in 00:42:49.910 --> 00:42:53.420 which the systems are cascaded, is reduced the 00:42:53.420 --> 00:42:56.730 implementation to the implementation with a single 00:42:56.730 --> 00:42:58.060 integrator. 00:42:58.060 --> 00:43:01.830 Very much similar to what we talked about in the case of 00:43:01.830 --> 00:43:03.630 the difference equation. 00:43:03.630 --> 00:43:07.600 Now, let's just, again, with the integral equation or the 00:43:07.600 --> 00:43:13.270 differential equation, look at this somewhat more generally. 00:43:13.270 --> 00:43:17.360 Again, if we take the differential equation, the 00:43:17.360 --> 00:43:20.480 general differential equation, integrate it a sufficient 00:43:20.480 --> 00:43:23.740 number of times to convert it to an integral equation. 00:43:23.740 --> 00:43:28.320 We would then have this cascade of systems. 00:43:28.320 --> 00:43:33.000 And again, if we assume initial rest, so that these 00:43:33.000 --> 00:43:36.520 are both linear time-invariant systems, we can interchange 00:43:36.520 --> 00:43:38.610 the order in which they're cascaded. 00:43:38.610 --> 00:43:45.990 Namely, take the second system, and move it to precede 00:43:45.990 --> 00:43:48.220 the first system. 00:43:48.220 --> 00:43:51.890 And then what we recognize is that the input to this chain 00:43:51.890 --> 00:43:54.530 of integrators and this chain of integrators 00:43:54.530 --> 00:43:56.220 is exactly the same. 00:43:56.220 --> 00:44:00.320 And so, in fact we can collapse these together using 00:44:00.320 --> 00:44:02.310 only one chain of integrators. 00:44:02.310 --> 00:44:07.890 And the system that we're left with then is a system that 00:44:07.890 --> 00:44:09.730 looks as I've indicated here. 00:44:09.730 --> 00:44:14.890 So we have now just a single chains of integrators instead 00:44:14.890 --> 00:44:18.330 of the two sets of integrators. 00:44:18.330 --> 00:44:21.670 So we've seen that the situation is very similar here 00:44:21.670 --> 00:44:24.510 as it was in the case of the difference equation. 00:44:24.510 --> 00:44:26.900 Again, why do we want to cut the number of 00:44:26.900 --> 00:44:28.310 integrators in half? 00:44:28.310 --> 00:44:32.140 Well, one reason is because integrators, in effect, 00:44:32.140 --> 00:44:33.710 represent hardware. 00:44:33.710 --> 00:44:38.155 And if we have half as many integrators, then we're using 00:44:38.155 --> 00:44:39.410 half as much hardware. 00:44:41.940 --> 00:44:44.440 Well, let me just conclude by 00:44:44.440 --> 00:44:47.550 summarizing a number of points. 00:44:47.550 --> 00:44:50.340 I indicated at the beginning that linear 00:44:50.340 --> 00:44:52.690 constant-coefficient differential equations and 00:44:52.690 --> 00:44:57.060 difference equations will play an important role as linear 00:44:57.060 --> 00:45:01.490 time-invariant systems throughout this course and 00:45:01.490 --> 00:45:04.160 throughout this set of lectures. 00:45:04.160 --> 00:45:09.210 I also stressed the fact that differential or difference 00:45:09.210 --> 00:45:13.990 equations, by themselves, are not a complete specification 00:45:13.990 --> 00:45:18.190 of the system because of the fact that we can add to any 00:45:18.190 --> 00:45:21.970 solution a homogeneous solution. 00:45:21.970 --> 00:45:26.080 How do we specify the appropriate initial conditions 00:45:26.080 --> 00:45:28.270 to ensure-- 00:45:28.270 --> 00:45:30.260 how do we specify the appropriate initial conditions 00:45:30.260 --> 00:45:34.250 to ensure that the system is linear and time-invariant? 00:45:34.250 --> 00:45:39.110 Well, the auxiliary information, namely, the 00:45:39.110 --> 00:45:45.100 initial conditions associated with the system being causal, 00:45:45.100 --> 00:45:47.790 linear, and time-invariant are the 00:45:47.790 --> 00:45:50.630 conditions of initial rest. 00:45:50.630 --> 00:45:54.130 And, in fact, for most of the course, what we'll be 00:45:54.130 --> 00:45:57.200 interested in are systems that are in fact, causal, linear, 00:45:57.200 --> 00:45:58.420 and time-invariant. 00:45:58.420 --> 00:46:01.150 And so we will, in fact, be assuming initial rest 00:46:01.150 --> 00:46:03.640 conditions. 00:46:03.640 --> 00:46:09.670 Now, as I also indicated, there are a variety of 00:46:09.670 --> 00:46:12.580 efficient procedures for solving differential and 00:46:12.580 --> 00:46:15.560 difference equations that we haven't yet addressed. 00:46:15.560 --> 00:46:18.990 And beginning with the next set of lectures, we'll be 00:46:18.990 --> 00:46:22.510 talking about the Fourier Transform and much later in 00:46:22.510 --> 00:46:26.470 the course, what's referred to as the Laplace Transform for 00:46:26.470 --> 00:46:30.070 continuous time and the Z-transform for discrete time. 00:46:30.070 --> 00:46:34.030 And what we'll see is that with the Fourier Transform and 00:46:34.030 --> 00:46:37.300 later with the Laplace and Z-transform, we'll have a 00:46:37.300 --> 00:46:42.480 number of efficient and very useful ways of generating the 00:46:42.480 --> 00:46:48.000 solution for differential and difference equations under the 00:46:48.000 --> 00:46:50.705 assumption that the system is causal, linear, and 00:46:50.705 --> 00:46:52.510 time-invariant. 00:46:52.510 --> 00:46:56.020 Also, we'll see in addition to the block diagram 00:46:56.020 --> 00:46:58.510 implementations of these systems that we've talked 00:46:58.510 --> 00:47:04.170 about so far, we'll see a number of other useful 00:47:04.170 --> 00:47:07.880 implementations that exploit a variety of properties 00:47:07.880 --> 00:47:10.870 associated with Fourier and Laplace Transforms. 00:47:10.870 --> 00:47:12.120 Thank you.