1 00:00:00,040 --> 00:00:02,470 The following content is provided under a Creative 2 00:00:02,470 --> 00:00:03,880 Commons license. 3 00:00:03,880 --> 00:00:06,920 Your support will help MIT OpenCourseWare continue to 4 00:00:06,920 --> 00:00:10,570 offer high quality educational resources for free. 5 00:00:10,570 --> 00:00:13,470 To make a donation or view additional materials from 6 00:00:13,470 --> 00:00:19,290 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:19,290 --> 00:00:20,540 ocw.mit.edu. 8 00:00:55,306 --> 00:00:57,270 PROFESSOR: Over the past several lectures, we've 9 00:00:57,270 --> 00:01:01,880 developed a representation for linear time-invariant systems. 10 00:01:01,880 --> 00:01:05,500 A particularly important set of systems, which are linear 11 00:01:05,500 --> 00:01:08,700 and time-invariant, are those that are represented by linear 12 00:01:08,700 --> 00:01:11,240 constant-coefficient differential equations in 13 00:01:11,240 --> 00:01:14,950 continuous time or linear constant-coefficient 14 00:01:14,950 --> 00:01:17,630 difference equations in discrete time. 15 00:01:17,630 --> 00:01:20,780 For example, electrical circuits that are built, let's 16 00:01:20,780 --> 00:01:23,300 say, out of resistors, inductors, and capacitors, 17 00:01:23,300 --> 00:01:26,780 perhaps with op-amps, correspond to systems 18 00:01:26,780 --> 00:01:29,150 described by differential equations. 19 00:01:29,150 --> 00:01:31,690 Mechanical systems with springs and dashpots, 20 00:01:31,690 --> 00:01:35,010 likewise, are described by differential equations. 21 00:01:35,010 --> 00:01:39,300 And in the discrete-time case, things such as moving average 22 00:01:39,300 --> 00:01:44,470 filters, digital filters, and most simple kinds of data 23 00:01:44,470 --> 00:01:47,840 smoothing are all linear constant-coefficient 24 00:01:47,840 --> 00:01:50,110 difference equations. 25 00:01:50,110 --> 00:01:55,180 Now, presumably in a previous course, you've had some 26 00:01:55,180 --> 00:02:00,260 exposure to differential equations for continuous time, 27 00:02:00,260 --> 00:02:04,000 and their solution using notions like particular 28 00:02:04,000 --> 00:02:06,220 solution, and homogeneous solution, initial 29 00:02:06,220 --> 00:02:09,380 conditions, et cetera. 30 00:02:09,380 --> 00:02:13,760 Later on in the course, when we've developed the concept of 31 00:02:13,760 --> 00:02:17,450 the Fourier transform after that, the Laplace transform, 32 00:02:17,450 --> 00:02:21,500 we'll see some very efficient and useful ways of generating 33 00:02:21,500 --> 00:02:23,160 solutions, both for differential 34 00:02:23,160 --> 00:02:25,010 and difference equations. 35 00:02:25,010 --> 00:02:29,620 At this point, however, I'd like to just introduce linear 36 00:02:29,620 --> 00:02:33,020 constant-coefficient differential equations and 37 00:02:33,020 --> 00:02:35,130 their discrete-time counterpart. 38 00:02:35,130 --> 00:02:38,890 And address, among other things, the issue of when they 39 00:02:38,890 --> 00:02:43,650 do and don't correspond to linear time-invariant systems. 40 00:02:43,650 --> 00:02:48,420 Well, let's first consider what I refer to as an 41 00:02:48,420 --> 00:02:51,010 nth-order linear constant-coefficient 42 00:02:51,010 --> 00:02:54,270 differential equation, as I've indicated here. 43 00:02:54,270 --> 00:02:57,350 And what it consists of is a linear combination of 44 00:02:57,350 --> 00:03:01,560 derivatives of the system output, y(t), equal to a 45 00:03:01,560 --> 00:03:04,660 linear combination of derivatives of the system 46 00:03:04,660 --> 00:03:07,300 input x(t). 47 00:03:07,300 --> 00:03:12,470 And it's referred to as a constant-coefficient equation, 48 00:03:12,470 --> 00:03:15,550 of course, because the coefficients are constant. 49 00:03:15,550 --> 00:03:18,670 In other words, not assumed to be time-varying. 50 00:03:18,670 --> 00:03:25,200 And it's referred to as linear because it corresponds to a 51 00:03:25,200 --> 00:03:30,820 linear combination of these derivatives, not because it 52 00:03:30,820 --> 00:03:32,620 corresponds to a linear system. 53 00:03:32,620 --> 00:03:36,910 And, in fact, as we'll see, or as I'll indicate, this 54 00:03:36,910 --> 00:03:39,340 equation may or may not, in fact, 55 00:03:39,340 --> 00:03:42,410 correspond to a linear system. 56 00:03:42,410 --> 00:03:47,500 In the discrete-time case, the corresponding equation is a 57 00:03:47,500 --> 00:03:50,710 linear constant-coefficient difference equation. 58 00:03:50,710 --> 00:03:55,650 And that corresponds to, again, a linear combination of 59 00:03:55,650 --> 00:04:00,370 delayed versions of the output equal to a linear combination 60 00:04:00,370 --> 00:04:02,630 of delayed versions of the input. 61 00:04:02,630 --> 00:04:04,590 This equation is referred to an 62 00:04:04,590 --> 00:04:07,290 nth-order difference equation. 63 00:04:07,290 --> 00:04:10,750 The n referring to the number of delays of the output 64 00:04:10,750 --> 00:04:15,690 involved, just as an Nth-order differential equation, the n 65 00:04:15,690 --> 00:04:18,230 or the order of the equation refers to the number of 66 00:04:18,230 --> 00:04:21,050 derivatives of the output. 67 00:04:21,050 --> 00:04:25,150 Now, let's first begin with linear constant-coefficient 68 00:04:25,150 --> 00:04:27,170 differential equations. 69 00:04:27,170 --> 00:04:34,290 And the basic point of the solution for the differential 70 00:04:34,290 --> 00:04:40,300 equations is the fact that if we've generated some solution, 71 00:04:40,300 --> 00:04:44,600 which I refer to here as y_p(t), some solution to the 72 00:04:44,600 --> 00:04:49,890 equation for a given input, then, in fact, we can add to 73 00:04:49,890 --> 00:04:55,750 that solution any other solution which satisfies 74 00:04:55,750 --> 00:04:59,910 what's referred to as the homogeneous equation. 75 00:04:59,910 --> 00:05:06,620 So in fact, this differential equation by itself is not a 76 00:05:06,620 --> 00:05:10,300 unique specification of the system. 77 00:05:10,300 --> 00:05:15,360 If I have any solution, then I can add to that solution any 78 00:05:15,360 --> 00:05:18,950 other solution which satisfies the homogeneous equation, and 79 00:05:18,950 --> 00:05:22,140 the sum of those two will likewise be a solution. 80 00:05:22,140 --> 00:05:24,630 And that's very straightforward to verify. 81 00:05:24,630 --> 00:05:28,770 By simply substituting into the differential equation, the 82 00:05:28,770 --> 00:05:33,910 sum of a particular and the homogeneous solution, and what 83 00:05:33,910 --> 00:05:38,180 you'll see is that the homogeneous contribution, in 84 00:05:38,180 --> 00:05:41,950 fact, goes to zero by definition of what we mean by 85 00:05:41,950 --> 00:05:44,670 the homogeneous equation. 86 00:05:44,670 --> 00:05:48,430 Now, the homogeneous solution for a linear 87 00:05:48,430 --> 00:05:53,290 constant-coefficient differential equation is of 88 00:05:53,290 --> 00:05:58,030 the form that I indicate at the bottom. 89 00:05:58,030 --> 00:06:07,230 And it typically consists of a sum of N complex exponentials. 90 00:06:07,230 --> 00:06:11,650 And the constants are undetermined by 91 00:06:11,650 --> 00:06:14,000 the equation itself. 92 00:06:14,000 --> 00:06:17,240 And this form for the homogeneous solution, in 93 00:06:17,240 --> 00:06:23,860 essence, drops out of examining the homogeneous 94 00:06:23,860 --> 00:06:28,890 equation, where if we assume that the form of the 95 00:06:28,890 --> 00:06:34,210 homogeneous solution is a complex exponential with some 96 00:06:34,210 --> 00:06:38,570 unspecified amplitude and unspecified exponent. 97 00:06:38,570 --> 00:06:43,580 If we substitute this into our homogeneous equation, we end 98 00:06:43,580 --> 00:06:46,510 up with the equation that I've indicated here. 99 00:06:46,510 --> 00:06:50,780 The factor A and the e^(st) can in fact be canceled out, 100 00:06:50,780 --> 00:06:56,110 and we find that that equation is satisfied 101 00:06:56,110 --> 00:07:02,160 for N values of s. 102 00:07:02,160 --> 00:07:06,540 And that's true no matter what choice is made for these 103 00:07:06,540 --> 00:07:07,890 coefficients. 104 00:07:07,890 --> 00:07:12,510 And the essential consequence of all of that is that the 105 00:07:12,510 --> 00:07:16,660 homogeneous solution is of the form that I indicated 106 00:07:16,660 --> 00:07:18,160 previously. 107 00:07:18,160 --> 00:07:26,010 Namely it consists of a sum of N complex exponentials, where 108 00:07:26,010 --> 00:07:29,720 the coefficients, the N coefficients, attach to each 109 00:07:29,720 --> 00:07:33,320 of those complex exponential is undetermined or 110 00:07:33,320 --> 00:07:34,980 unspecified. 111 00:07:34,980 --> 00:07:39,710 So what this says is that in order to obtain the solution 112 00:07:39,710 --> 00:07:41,360 for a linear constant-coefficient 113 00:07:41,360 --> 00:07:44,750 differential equation, we need some kind of auxiliary 114 00:07:44,750 --> 00:07:49,760 information that tells us is how to obtain these N 115 00:07:49,760 --> 00:07:51,690 undetermined constants. 116 00:07:51,690 --> 00:07:55,550 And there are variety of ways of specifying this auxiliary 117 00:07:55,550 --> 00:07:57,930 information or auxiliary conditions. 118 00:07:57,930 --> 00:08:02,250 For example, in addition to the differential equation, 119 00:08:02,250 --> 00:08:10,000 what I can tell you is the value of the output and N - 1 120 00:08:10,000 --> 00:08:15,300 of its derivatives, at some specified time, t_0. 121 00:08:15,300 --> 00:08:20,590 And so the differential equation together with the 122 00:08:20,590 --> 00:08:25,020 auxiliary information, the initial conditions, then lets 123 00:08:25,020 --> 00:08:28,760 you determine the total solution, which namely lets 124 00:08:28,760 --> 00:08:31,750 you determine these previously-unspecified 125 00:08:31,750 --> 00:08:35,130 coefficients in the homogeneous solution. 126 00:08:35,130 --> 00:08:39,120 Now, depending on how the auxiliary information is 127 00:08:39,120 --> 00:08:42,350 stated or what auxiliary information is available, the 128 00:08:42,350 --> 00:08:47,080 system may or may not correspond to a linear system, 129 00:08:47,080 --> 00:08:49,360 and may or may not correspond to a linear 130 00:08:49,360 --> 00:08:51,880 time-invariant system. 131 00:08:51,880 --> 00:08:55,180 One essential condition for it to correspond to a linear 132 00:08:55,180 --> 00:09:01,110 system is that the initial conditions must be 0. 133 00:09:01,110 --> 00:09:06,550 And one can see the reason for that, if we refer back to the 134 00:09:06,550 --> 00:09:10,090 previous lecture in which we saw that for a linear system, 135 00:09:10,090 --> 00:09:13,090 if we put 0 in, we get 0 out. 136 00:09:13,090 --> 00:09:17,560 So if x(t), the input, is 0, the output must be 0. 137 00:09:17,560 --> 00:09:20,710 And so that, in essence, tells us that, at least for the 138 00:09:20,710 --> 00:09:26,220 system to be linear, these initial conditions must be 0. 139 00:09:26,220 --> 00:09:33,380 Now, beyond that, if we want the system to be causal and 140 00:09:33,380 --> 00:09:38,890 linear and time-invariant, then what's required on the 141 00:09:38,890 --> 00:09:42,840 initial conditions is that they be consistent with what's 142 00:09:42,840 --> 00:09:46,520 referred to as initial rest. 143 00:09:46,520 --> 00:09:53,770 Initial rest says that the output must be 0 up until the 144 00:09:53,770 --> 00:09:56,910 time that the input becomes non-zero. 145 00:09:56,910 --> 00:09:59,590 And we can see, of course, that that's consistent with 146 00:09:59,590 --> 00:10:03,150 the notion of causality, as we talked about in 147 00:10:03,150 --> 00:10:04,730 the previous lecture. 148 00:10:04,730 --> 00:10:10,400 And it's relatively straightforward to see that if 149 00:10:10,400 --> 00:10:14,920 the system is causal and linear and time-invariant, 150 00:10:14,920 --> 00:10:18,810 that will require initial rest. 151 00:10:18,810 --> 00:10:23,600 It's somewhat more difficult to see that if we specify 152 00:10:23,600 --> 00:10:27,410 initial rest, then that, in fact, is sufficient to 153 00:10:27,410 --> 00:10:31,040 determine that the system is both causal and linear and 154 00:10:31,040 --> 00:10:32,100 time invariant. 155 00:10:32,100 --> 00:10:35,730 But the essential point then is that it requires initial 156 00:10:35,730 --> 00:10:40,500 rest for both linearity and causality. 157 00:10:40,500 --> 00:10:46,360 OK, well, let's look at an example, and let's take the 158 00:10:46,360 --> 00:10:52,580 example of a first-order differential equation, as I've 159 00:10:52,580 --> 00:10:54,470 indicated here. 160 00:10:54,470 --> 00:10:58,623 So we have a first-order differential equation, dy(t) / 161 00:10:58,623 --> 00:11:03,350 dt + ay(t) is the input x(t). 162 00:11:03,350 --> 00:11:07,360 And let's first look at what the homogeneous solution of 163 00:11:07,360 --> 00:11:08,500 this equation is. 164 00:11:08,500 --> 00:11:13,370 And so, we consider the homogeneous equation, namely 165 00:11:13,370 --> 00:11:17,600 the equation that specifies solutions, which would 166 00:11:17,600 --> 00:11:20,650 correspond to 0 input. 167 00:11:20,650 --> 00:11:27,180 We, in essence, guess, or impose a solution of the form. 168 00:11:27,180 --> 00:11:31,590 The homogeneous solution is an amplitude factor times a 169 00:11:31,590 --> 00:11:33,450 complex exponential. 170 00:11:33,450 --> 00:11:38,390 Substituting this into the homogeneous equation, we then 171 00:11:38,390 --> 00:11:40,860 get the equation that I've indicated here. 172 00:11:40,860 --> 00:11:45,780 What you can see is that in this equation, I can cancel 173 00:11:45,780 --> 00:11:50,530 out the amplitude factor and this complex exponential. 174 00:11:50,530 --> 00:11:56,590 So let's just cancel those out on both sides of the equation. 175 00:11:56,590 --> 00:12:02,340 And what we're left with is an equation that specifies what 176 00:12:02,340 --> 00:12:04,760 the complex exponent must be. 177 00:12:04,760 --> 00:12:08,110 In particular, for the homogeneous solution, s must 178 00:12:08,110 --> 00:12:12,060 be equal to -a, and so finally, our homogeneous 179 00:12:12,060 --> 00:12:16,380 solution is as I've indicated here. 180 00:12:19,050 --> 00:12:23,150 Now, let's look at the solution for a specific input. 181 00:12:23,150 --> 00:12:25,990 Let's consider, for example, an input which is 182 00:12:25,990 --> 00:12:28,000 a scaled unit step. 183 00:12:28,000 --> 00:12:31,530 And although I won't work out the solution in detail and 184 00:12:31,530 --> 00:12:36,510 perhaps using what you've worked on previously, you know 185 00:12:36,510 --> 00:12:39,805 how to carry out the solution for that. 186 00:12:39,805 --> 00:12:44,950 A solution with a step input is what I've indicated here: a 187 00:12:44,950 --> 00:12:47,060 scalar, 1 minus an exponential, 188 00:12:47,060 --> 00:12:48,540 times a unit step. 189 00:12:48,540 --> 00:12:51,430 And you can verify that simply by substituting into the 190 00:12:51,430 --> 00:12:53,170 differential equation. 191 00:12:53,170 --> 00:12:56,930 Now, we know that there's a family of solutions. 192 00:12:56,930 --> 00:13:02,620 In other words, any solution with a homogeneous solution 193 00:13:02,620 --> 00:13:04,990 added to it, is, again, a solution. 194 00:13:04,990 --> 00:13:08,670 And so if we consider the solution that I just 195 00:13:08,670 --> 00:13:13,160 indicated, we generate the entire family of solutions by 196 00:13:13,160 --> 00:13:17,440 adding a homogeneous solution to it, and so this then 197 00:13:17,440 --> 00:13:21,400 corresponds to the entire family of solutions, where the 198 00:13:21,400 --> 00:13:27,640 constant A is unspecified so far, and needs to be specified 199 00:13:27,640 --> 00:13:31,470 through some type of auxiliary conditions. 200 00:13:31,470 --> 00:13:36,640 Now, a class of auxiliary conditions is the condition of 201 00:13:36,640 --> 00:13:41,310 initial rest, which as I indicated before, is 202 00:13:41,310 --> 00:13:44,830 equivalent to the statement that the system is causal and 203 00:13:44,830 --> 00:13:46,900 linear and time-invariant. 204 00:13:46,900 --> 00:13:52,080 And in that case, for the initial rest condition, we 205 00:13:52,080 --> 00:13:56,660 would then require in this equation above that this 206 00:13:56,660 --> 00:13:59,710 constant be equal to 0. 207 00:13:59,710 --> 00:14:04,860 And so finally, the response to a scaled step-- 208 00:14:04,860 --> 00:14:07,730 if the system is to correspond to a causal linear 209 00:14:07,730 --> 00:14:12,760 time-invariant system, is then just this term, namely, a 210 00:14:12,760 --> 00:14:18,870 constant times 1 minus an exponential, times the step. 211 00:14:18,870 --> 00:14:23,060 Now, if the system is a linear time-invariant system, it can, 212 00:14:23,060 --> 00:14:26,260 as we know, be described through its impulse response. 213 00:14:26,260 --> 00:14:30,420 And as you've worked out previously in the video course 214 00:14:30,420 --> 00:14:34,730 manual, for a linear time-invariant system, the 215 00:14:34,730 --> 00:14:36,480 impulse response is the 216 00:14:36,480 --> 00:14:38,750 derivative of the step response. 217 00:14:38,750 --> 00:14:41,730 And just to quickly remind you of where that 218 00:14:41,730 --> 00:14:43,960 result comes from. 219 00:14:43,960 --> 00:14:48,020 In essence, we can consider two linear time-invariant 220 00:14:48,020 --> 00:14:51,770 systems in cascade, one a differentiator, the other the 221 00:14:51,770 --> 00:14:55,060 system that we're talking about described by the 222 00:14:55,060 --> 00:14:56,850 differential equation. 223 00:14:56,850 --> 00:15:01,700 And a step in here then generates an impulse into our 224 00:15:01,700 --> 00:15:04,600 system, and out comes the impulse response. 225 00:15:04,600 --> 00:15:07,100 Well, just using the fact that these are both linear 226 00:15:07,100 --> 00:15:10,840 time-invariant systems, and they can be cascaded in either 227 00:15:10,840 --> 00:15:15,970 order, then means that if we have the step response to our 228 00:15:15,970 --> 00:15:20,380 system, and that goes through the differentiator, what must 229 00:15:20,380 --> 00:15:24,570 come out, again, is the impulse response. 230 00:15:24,570 --> 00:15:27,380 So differentiating the step response, we 231 00:15:27,380 --> 00:15:29,310 get the impulse response. 232 00:15:29,310 --> 00:15:33,240 Here again, is the step response as we just worked it 233 00:15:33,240 --> 00:15:35,990 out, this time for a unit step. 234 00:15:35,990 --> 00:15:40,760 If we differentiate, we have then, since the step response 235 00:15:40,760 --> 00:15:44,120 is the product of two terms, the derivative of a product is 236 00:15:44,120 --> 00:15:45,940 the sum of the derivatives. 237 00:15:45,940 --> 00:15:49,640 And carrying that algebra through, and using the fact 238 00:15:49,640 --> 00:15:54,350 that the derivative of the step is an impulse, finally we 239 00:15:54,350 --> 00:15:59,180 come down to this statement for the impulse response. 240 00:15:59,180 --> 00:16:01,820 And then recognizing that this is a time 241 00:16:01,820 --> 00:16:03,970 function times an impulse. 242 00:16:03,970 --> 00:16:07,800 And we know that if a time function times an impulse 243 00:16:07,800 --> 00:16:14,160 takes on the value at the time that the impulse occurs, then 244 00:16:14,160 --> 00:16:18,390 this term is simply 0. 245 00:16:18,390 --> 00:16:22,140 And the impulse response then finally is an 246 00:16:22,140 --> 00:16:24,420 exponential of this form. 247 00:16:24,420 --> 00:16:28,390 And this is the decaying exponential for a-positive, 248 00:16:28,390 --> 00:16:31,640 it's a growing exponential for a-negative. 249 00:16:31,640 --> 00:16:38,480 And recall that, as we talked about previously, a linear 250 00:16:38,480 --> 00:16:42,400 time-invariant system is stable if its impulse response 251 00:16:42,400 --> 00:16:44,610 is absolutely integrable. 252 00:16:44,610 --> 00:16:49,760 For this particular case, this impulse response is absolutely 253 00:16:49,760 --> 00:16:56,250 integrable provided that the exponential factor a is 254 00:16:56,250 --> 00:16:59,290 greater than 0. 255 00:16:59,290 --> 00:17:04,700 Okay, so what we've seen then is the impulse response for a 256 00:17:04,700 --> 00:17:07,230 system described by a linear constant-coefficient 257 00:17:07,230 --> 00:17:10,829 differential equation, where in addition, we would impose 258 00:17:10,829 --> 00:17:15,069 causality, linearity, and time-invariance, essentially, 259 00:17:15,069 --> 00:17:19,619 through the initial conditions of initial rest. 260 00:17:19,619 --> 00:17:23,609 Now, pretty much the same kinds of things happen with 261 00:17:23,609 --> 00:17:27,079 difference equations as we've gone through with 262 00:17:27,079 --> 00:17:29,740 differential equations. 263 00:17:29,740 --> 00:17:33,820 In particular, again, let me remind you of the form of an 264 00:17:33,820 --> 00:17:35,640 Nth order linear constant 265 00:17:35,640 --> 00:17:37,760 coefficient difference equation. 266 00:17:37,760 --> 00:17:40,820 It's as I indicate here. 267 00:17:40,820 --> 00:17:44,390 And, again, a linear combination, this time of 268 00:17:44,390 --> 00:17:48,220 delayed versions of the output equal to a linear combination 269 00:17:48,220 --> 00:17:51,520 of delayed versions of the input. 270 00:17:51,520 --> 00:17:57,210 Once again, difference equation is not a complete 271 00:17:57,210 --> 00:18:00,310 specification of the system because we can add to the 272 00:18:00,310 --> 00:18:03,820 response any homogeneous solution. 273 00:18:03,820 --> 00:18:07,500 In other words, any solution that satisfies the homogeneous 274 00:18:07,500 --> 00:18:12,230 equation and the sum of those will also satisfy the original 275 00:18:12,230 --> 00:18:13,910 difference equation. 276 00:18:13,910 --> 00:18:21,500 So if we have a particular response that satisfies the 277 00:18:21,500 --> 00:18:27,070 difference equation, then adding to that any response 278 00:18:27,070 --> 00:18:32,680 that is a solution to the homogeneous equation will also 279 00:18:32,680 --> 00:18:36,640 be a solution to the total equation. 280 00:18:36,640 --> 00:18:41,470 The homogeneous solution, again, is of the form of a 281 00:18:41,470 --> 00:18:44,650 linear combination of exponentials. 282 00:18:44,650 --> 00:18:48,300 Here, we have the homogeneous equation. 283 00:18:48,300 --> 00:18:53,390 As with differential equations, we can guess or 284 00:18:53,390 --> 00:18:58,010 impose solutions of the form A times an exponential. 285 00:18:58,010 --> 00:19:01,780 When we substitute this into the homogeneous equation, we 286 00:19:01,780 --> 00:19:05,830 then end up with the equation that I've indicated here. 287 00:19:05,830 --> 00:19:15,580 We recognize again that A, the amplitude, and z^n, this 288 00:19:15,580 --> 00:19:19,310 exponential factor, cancel out. 289 00:19:19,310 --> 00:19:27,000 And so this equation is satisfied for any values of z 290 00:19:27,000 --> 00:19:30,310 that satisfy this equation. 291 00:19:30,310 --> 00:19:34,310 And there are N roots, z_1 through z_N. 292 00:19:34,310 --> 00:19:38,830 And so finally, the form for the homogeneous solution is a 293 00:19:38,830 --> 00:19:44,240 linear combination of capital N exponentials, where capital 294 00:19:44,240 --> 00:19:46,930 N is the order of the equation. 295 00:19:46,930 --> 00:19:49,400 With each of those exponentials, the amplitude 296 00:19:49,400 --> 00:19:53,220 factor is undetermined and needs to be determined in some 297 00:19:53,220 --> 00:19:58,630 way through the imposition of appropriate initial conditions 298 00:19:58,630 --> 00:19:59,880 or boundary conditions. 299 00:20:01,990 --> 00:20:07,370 So the general form then for the solution to the difference 300 00:20:07,370 --> 00:20:13,590 equation is a sum of exponentials plus any 301 00:20:13,590 --> 00:20:15,830 particular solution. 302 00:20:15,830 --> 00:20:20,730 It's through auxiliary conditions that we determine 303 00:20:20,730 --> 00:20:23,150 these coefficients. 304 00:20:23,150 --> 00:20:28,420 We have N undetermined coefficients and, so we 305 00:20:28,420 --> 00:20:31,850 require N auxiliary conditions. 306 00:20:31,850 --> 00:20:37,520 For example, some set of values of the output at N 307 00:20:37,520 --> 00:20:41,210 distinct instance of time. 308 00:20:41,210 --> 00:20:44,200 Now this was the same as with differential equations. 309 00:20:44,200 --> 00:20:46,640 In the case of differential equations, we talked about 310 00:20:46,640 --> 00:20:50,510 specifying the value of the output and its derivatives. 311 00:20:50,510 --> 00:20:53,270 And there, we indicated that for 312 00:20:53,270 --> 00:20:56,890 linearity, what we required-- 313 00:20:56,890 --> 00:21:00,740 for linearity, what we required is that the auxiliary 314 00:21:00,740 --> 00:21:02,580 conditions be 0. 315 00:21:02,580 --> 00:21:05,830 And the same thing applies here for the same reason. 316 00:21:05,830 --> 00:21:09,350 Namely, if the system is to be linear, then the response, if 317 00:21:09,350 --> 00:21:13,440 there's no input, must be equal to 0. 318 00:21:13,440 --> 00:21:19,200 In addition, what we may want to impose on the system is 319 00:21:19,200 --> 00:21:22,290 that it be causal, and in addition to linear 320 00:21:22,290 --> 00:21:28,320 time-invariant, and what that requires, again, is that the 321 00:21:28,320 --> 00:21:32,220 auxiliary conditions be consistent with initial rest, 322 00:21:32,220 --> 00:21:37,690 namely, that if the input is 0 prior to some time, then the 323 00:21:37,690 --> 00:21:42,680 output is 0 prior to the same time. 324 00:21:42,680 --> 00:21:47,240 So we've seen a very direct parallel so far between 325 00:21:47,240 --> 00:21:52,210 differential equations and difference equations. 326 00:21:52,210 --> 00:21:55,930 In fact, one difference between them that, in some 327 00:21:55,930 --> 00:22:00,140 sense, makes difference equations easier to deal with 328 00:22:00,140 --> 00:22:04,000 in some situations, is that in contrast to a differential 329 00:22:04,000 --> 00:22:07,980 equation, a difference equation, if we assume 330 00:22:07,980 --> 00:22:13,580 causality, in fact is an explicit input-output 331 00:22:13,580 --> 00:22:15,080 relationship for the system. 332 00:22:15,080 --> 00:22:18,670 Now, let me show you what I mean. 333 00:22:18,670 --> 00:22:22,130 Let's consider the nth-order difference equation, as I've 334 00:22:22,130 --> 00:22:23,490 indicated here. 335 00:22:23,490 --> 00:22:28,590 And let's assume that we're imposing causality so that the 336 00:22:28,590 --> 00:22:33,000 output can only depend on prior values of the input, and 337 00:22:33,000 --> 00:22:35,730 therefore, aren't prior values of the output. 338 00:22:35,730 --> 00:22:41,450 Well, we can simply rearrange this equation solving for y[n] 339 00:22:41,450 --> 00:22:44,680 the leading term, with k = 0. 340 00:22:44,680 --> 00:22:47,660 Taking all of the other terms over to the right side of the 341 00:22:47,660 --> 00:22:54,740 equation, and we then have a recursive equation, namely, an 342 00:22:54,740 --> 00:22:59,070 equation that expresses the output in terms of prior 343 00:22:59,070 --> 00:23:03,850 values of the input, which is this term, and prior values of 344 00:23:03,850 --> 00:23:05,640 the output. 345 00:23:05,640 --> 00:23:10,050 And so if, in fact, we have this equation running, then 346 00:23:10,050 --> 00:23:13,800 once it started, we know how to compute the output for the 347 00:23:13,800 --> 00:23:16,740 next time instant. 348 00:23:16,740 --> 00:23:19,170 Well, how do we get it started? 349 00:23:19,170 --> 00:23:22,240 The way we get it started, of course, is through the 350 00:23:22,240 --> 00:23:26,240 appropriate set of initial conditions or boundary 351 00:23:26,240 --> 00:23:27,300 conditions. 352 00:23:27,300 --> 00:23:30,760 And, if for example, we assume initial rest corresponding to 353 00:23:30,760 --> 00:23:35,360 a causal linear time-invariant system, then if the input is 0 354 00:23:35,360 --> 00:23:39,560 up until some time, the output must be 0 up until that time. 355 00:23:39,560 --> 00:23:43,390 And that, in essence, helps us get the equation started. 356 00:23:43,390 --> 00:23:48,150 Well, let's look at this specifically in the context of 357 00:23:48,150 --> 00:23:50,160 a first-order difference equation. 358 00:23:50,160 --> 00:23:55,510 So let's take a first-order difference equation, as I've 359 00:23:55,510 --> 00:23:57,560 indicated here. 360 00:23:57,560 --> 00:24:00,650 And so, we have an equation that tells us that y[n] 361 00:24:00,650 --> 00:24:03,920 - ay[n-1] 362 00:24:03,920 --> 00:24:06,750 = x[n]. 363 00:24:06,750 --> 00:24:10,970 Now, if we want this to correspond to a causal linear 364 00:24:10,970 --> 00:24:16,780 time-invariant system, we impose initial rest on it. 365 00:24:16,780 --> 00:24:21,550 We can rewrite the first-order difference equation by taking 366 00:24:21,550 --> 00:24:24,280 the term involving y[n-1] 367 00:24:24,280 --> 00:24:25,945 over two the right hand side of the equation. 368 00:24:28,450 --> 00:24:33,560 Now, this gives us a recursive equation that expresses the 369 00:24:33,560 --> 00:24:38,500 output in terms of the input and past values of the output. 370 00:24:38,500 --> 00:24:43,610 And since we've imposed causality, and if we're 371 00:24:43,610 --> 00:24:46,820 talking about a linear time-invariant system, we can 372 00:24:46,820 --> 00:24:50,350 now inquire as to what the impulse response is. 373 00:24:50,350 --> 00:24:53,120 And we know, of course, the impulse response tells us 374 00:24:53,120 --> 00:24:56,910 everything that we need to know about the system. 375 00:24:56,910 --> 00:25:02,990 So let's choose an input, which is an impulse. 376 00:25:02,990 --> 00:25:07,070 So the impulse response is delta[n] 377 00:25:07,070 --> 00:25:09,040 corresponding to the x[n] 378 00:25:09,040 --> 00:25:13,490 up here, plus a delta of-- 379 00:25:13,490 --> 00:25:15,880 this should be h[n-1]. 380 00:25:15,880 --> 00:25:17,970 And let me just correct that. 381 00:25:17,970 --> 00:25:20,640 This is h[n-1]. 382 00:25:20,640 --> 00:25:24,270 So we have the impulse response as delta[n] 383 00:25:24,270 --> 00:25:25,520 plus a times h[n-1]. 384 00:25:28,470 --> 00:25:33,235 Now, from initial rest, we know that since the input, 385 00:25:33,235 --> 00:25:37,620 namely an impulse, is 0 for n less than 0, the impulse 386 00:25:37,620 --> 00:25:42,410 response is likewise 0 for n less than 0. 387 00:25:42,410 --> 00:25:45,620 And now, let's work out what h[0] 388 00:25:45,620 --> 00:25:46,870 is. 389 00:25:46,870 --> 00:25:52,900 Well, h of 0, with n = 0, is delta[0] 390 00:25:52,900 --> 00:25:55,940 plus a times h[n-1]. 391 00:25:55,940 --> 00:25:56,950 h[n-1] 392 00:25:56,950 --> 00:25:58,010 is 0. 393 00:25:58,010 --> 00:25:59,600 And so, h[0] 394 00:25:59,600 --> 00:26:01,440 is equal to 1. 395 00:26:05,490 --> 00:26:09,030 Now that we have h[0], we can figure out h[1] 396 00:26:09,030 --> 00:26:12,340 by running this recursive equation. 397 00:26:12,340 --> 00:26:14,640 So h[1] 398 00:26:14,640 --> 00:26:21,360 is delta[1], which is 0, plus a times h[0], which we just 399 00:26:21,360 --> 00:26:22,820 figured out is 1. 400 00:26:22,820 --> 00:26:24,410 So, h[1] 401 00:26:24,410 --> 00:26:26,120 is equal to a. 402 00:26:26,120 --> 00:26:29,380 And if we carry this through, we'll have h[2] 403 00:26:29,380 --> 00:26:34,180 equal to a^2, and this will continue on. 404 00:26:34,180 --> 00:26:37,660 And in fact, what we can recognize by looking at this 405 00:26:37,660 --> 00:26:41,860 and how we would expect these terms to build, we would see 406 00:26:41,860 --> 00:26:46,120 that the impulse response, h[n], in fact, is of the form 407 00:26:46,120 --> 00:26:50,630 a^n times u[n]. 408 00:26:50,630 --> 00:26:56,080 And we also can recognize then that this corresponds to a 409 00:26:56,080 --> 00:27:02,890 stable system, if and only if the impulse response, which is 410 00:27:02,890 --> 00:27:05,640 what we just figured out, if and only if the impulse 411 00:27:05,640 --> 00:27:08,150 response is absolutely summable. 412 00:27:08,150 --> 00:27:12,820 And what that will require is that the magnitude of a be 413 00:27:12,820 --> 00:27:14,070 less than 1. 414 00:27:17,400 --> 00:27:21,110 Now, we imposed causality, linearity, and 415 00:27:21,110 --> 00:27:25,750 time-invariance, and generated a solution recursively. 416 00:27:25,750 --> 00:27:29,590 And now, of course, if we want to generate the more general 417 00:27:29,590 --> 00:27:33,880 set of solutions to this difference equation, we can do 418 00:27:33,880 --> 00:27:38,650 that by adding all of the homogeneous solutions, namely, 419 00:27:38,650 --> 00:27:42,860 all the solutions that satisfy the homogeneous equation. 420 00:27:42,860 --> 00:27:46,230 So here, we have the causal linear 421 00:27:46,230 --> 00:27:49,330 time-invariant impulse response. 422 00:27:49,330 --> 00:27:55,220 In fact, with an impulse input, all of the possible 423 00:27:55,220 --> 00:28:00,380 solutions are that impulse response plus 424 00:28:00,380 --> 00:28:02,500 the homogeneous solutions. 425 00:28:02,500 --> 00:28:06,620 The homogeneous solution is the solution that satisfies 426 00:28:06,620 --> 00:28:09,660 the homogeneous equation. 427 00:28:09,660 --> 00:28:14,960 That will, in general, be of the form an amplitude factor 428 00:28:14,960 --> 00:28:19,950 times an exponential factor, and if we substitute this into 429 00:28:19,950 --> 00:28:24,610 this equation, then we see that the homogeneous equation 430 00:28:24,610 --> 00:28:30,480 is satisfied for any values of a and any values of z that 431 00:28:30,480 --> 00:28:33,780 satisfy this equation. 432 00:28:33,780 --> 00:28:36,180 Again, as we did with differential equations, the 433 00:28:36,180 --> 00:28:39,960 factor A cancels out. 434 00:28:39,960 --> 00:28:43,870 And also, in fact, I can cancel out a z^n 435 00:28:43,870 --> 00:28:48,100 there and a z^n here. 436 00:28:48,100 --> 00:28:52,420 And so what we're left with is a statement that tells us then 437 00:28:52,420 --> 00:28:58,890 that the homogeneous solution is of the form a(z^n), for any 438 00:28:58,890 --> 00:29:05,180 value of A and any value of z that satisfies this equation. 439 00:29:05,180 --> 00:29:10,200 And that value of z, in particular, is z equal to a. 440 00:29:10,200 --> 00:29:14,620 So the homogeneous solution then is any exponential of 441 00:29:14,620 --> 00:29:17,080 this form with any amplitude factor. 442 00:29:17,080 --> 00:29:20,670 And so, the family of solutions, with an impulse 443 00:29:20,670 --> 00:29:25,800 input is the solution corresponding to the system 444 00:29:25,800 --> 00:29:29,810 being causal, linear, and time-invariant, plus the 445 00:29:29,810 --> 00:29:32,010 homogeneous term. 446 00:29:32,010 --> 00:29:36,950 If we impose causality, linearity, and time-invariance 447 00:29:36,950 --> 00:29:41,430 on the system, then of course, that additional exponential 448 00:29:41,430 --> 00:29:43,170 factor will be 0. 449 00:29:43,170 --> 00:29:45,400 In other words, A is equal to 0. 450 00:29:48,970 --> 00:29:54,450 Now, we've seen the differential equations and 451 00:29:54,450 --> 00:29:59,880 difference equations in terms of the fact that there are 452 00:29:59,880 --> 00:30:04,040 families of solutions, and in order to get causality, 453 00:30:04,040 --> 00:30:07,350 linearity, and time-invariance requires imposing a particular 454 00:30:07,350 --> 00:30:10,930 set of initial conditions, namely, imposing initial rest 455 00:30:10,930 --> 00:30:12,480 on the system. 456 00:30:12,480 --> 00:30:17,310 Let's now look at the difference equation and then 457 00:30:17,310 --> 00:30:20,540 later, the differential equation, interpreted in block 458 00:30:20,540 --> 00:30:23,190 diagram terms. 459 00:30:23,190 --> 00:30:27,790 Now, the difference equation, as I just simply repeated 460 00:30:27,790 --> 00:30:30,780 here, is y[n] 461 00:30:30,780 --> 00:30:31,690 = x[n] 462 00:30:31,690 --> 00:30:37,360 + ay[n-1], where I've taken the delayed term over to the 463 00:30:37,360 --> 00:30:39,440 right hand side of the equation. 464 00:30:39,440 --> 00:30:42,200 So, in effect, what I'm imposing on 465 00:30:42,200 --> 00:30:44,460 this system is causality. 466 00:30:44,460 --> 00:30:47,990 I'm assuming that if I know the past history of the input 467 00:30:47,990 --> 00:30:52,190 and the output, I can determine the next value of 468 00:30:52,190 --> 00:30:53,860 the output. 469 00:30:53,860 --> 00:30:58,600 Well, we can, in fact, draw a block diagram that represents 470 00:30:58,600 --> 00:31:00,130 that equation. 471 00:31:00,130 --> 00:31:03,140 The equations says, we take x[n] 472 00:31:03,140 --> 00:31:07,890 for any given value of n, say n0, whatever value we're 473 00:31:07,890 --> 00:31:09,980 computing the output for. 474 00:31:09,980 --> 00:31:15,850 We take the input at that time, and add to it the factor 475 00:31:15,850 --> 00:31:21,390 a times the output value that we calculated last time. 476 00:31:21,390 --> 00:31:27,880 So if we have x[n], which is our input, and if we have 477 00:31:27,880 --> 00:31:34,160 y[n], which is our output, we, in fact, can get y of n by 478 00:31:34,160 --> 00:31:41,290 taking the last value of y[n], indicated here by putting y[n] 479 00:31:41,290 --> 00:31:52,070 through a delay, multiplying that by the factor a, and then 480 00:31:52,070 --> 00:31:59,930 adding that result to the input. 481 00:31:59,930 --> 00:32:05,430 And the result of doing that is y[n]. 482 00:32:05,430 --> 00:32:09,750 So the way this block diagram might be interpreted, for 483 00:32:09,750 --> 00:32:16,720 example, as an algorithm is to say that we take x[n], add to 484 00:32:16,720 --> 00:32:20,680 it a times the previous value the output. 485 00:32:20,680 --> 00:32:24,160 That sum gives us the current value of the output, which we 486 00:32:24,160 --> 00:32:29,440 then put out of the system, and also put into a delay 487 00:32:29,440 --> 00:32:33,630 element, or basically, into a storage register, to use on 488 00:32:33,630 --> 00:32:35,980 the next iteration or recursion. 489 00:32:35,980 --> 00:32:37,580 Now, how do we get this started? 490 00:32:37,580 --> 00:32:40,680 Well, we know that the difference equation requires 491 00:32:40,680 --> 00:32:42,500 initial conditions. 492 00:32:42,500 --> 00:32:47,560 And, in fact, the initial conditions correspond to what 493 00:32:47,560 --> 00:32:52,210 we store in the delay register when this block diagram or 494 00:32:52,210 --> 00:32:55,300 equation initially starts up. 495 00:32:55,300 --> 00:33:01,380 OK Now, let's look at this for the case of difference 496 00:33:01,380 --> 00:33:03,275 equations more generally. 497 00:33:06,650 --> 00:33:13,360 So, what we've said is that we can calculate the output by 498 00:33:13,360 --> 00:33:18,460 having previous values of the input, previous values of the 499 00:33:18,460 --> 00:33:22,270 output, and forming the appropriate linear 500 00:33:22,270 --> 00:33:23,850 combination. 501 00:33:23,850 --> 00:33:27,550 So let's just build up the more general block diagram 502 00:33:27,550 --> 00:33:28,960 that would correspond to this. 503 00:33:31,870 --> 00:33:35,900 And what it says is that we want to have a mechanism for 504 00:33:35,900 --> 00:33:40,130 storing past values of the input, and a mechanism for 505 00:33:40,130 --> 00:33:41,930 storing past values of the output. 506 00:33:41,930 --> 00:33:47,400 And I've indicated that on this figure, so far, by a 507 00:33:47,400 --> 00:33:55,610 chain of delay elements, indicating that what the 508 00:33:55,610 --> 00:34:02,090 output of each delay is, is the input delayed by one time 509 00:34:02,090 --> 00:34:05,050 instant or interval. 510 00:34:05,050 --> 00:34:09,560 And so what we see down this chain of delays are delayed 511 00:34:09,560 --> 00:34:13,010 replications of the input. 512 00:34:13,010 --> 00:34:19,159 And what we see on the other chain is delayed replications 513 00:34:19,159 --> 00:34:20,409 of the output. 514 00:34:22,960 --> 00:34:26,139 Now, the difference equation says that we want to take 515 00:34:26,139 --> 00:34:31,150 these, and multiply them by the appropriate coefficients, 516 00:34:31,150 --> 00:34:33,909 the coefficients in the difference equation, and so, 517 00:34:33,909 --> 00:34:37,690 we can do that as I've indicated here. 518 00:34:37,690 --> 00:34:42,880 So now, we have these delay elements, each multiplied by 519 00:34:42,880 --> 00:34:46,090 the appropriate coefficients on the input, and by 520 00:34:46,090 --> 00:34:50,139 appropriate coefficients on the output. 521 00:34:50,139 --> 00:34:56,300 Those are then summed together, and so we now will 522 00:34:56,300 --> 00:35:00,330 sum these and will sum these. 523 00:35:00,330 --> 00:35:03,990 After we've summed these, we want to add those together. 524 00:35:03,990 --> 00:35:07,200 And there's a factor of 1 / a_0 that comes in. 525 00:35:07,200 --> 00:35:11,300 And so that then generates our output. 526 00:35:11,300 --> 00:35:16,690 And so this, in fact, then represents a block diagram, 527 00:35:16,690 --> 00:35:20,370 which is a general block diagram for implementing or 528 00:35:20,370 --> 00:35:22,670 representing a linear constant-coefficient 529 00:35:22,670 --> 00:35:25,060 difference equation. 530 00:35:25,060 --> 00:35:28,190 Now, if you think about what it means in terms of, let's 531 00:35:28,190 --> 00:35:31,530 say, a computer algorithm or a piece of hardware, in fact, 532 00:35:31,530 --> 00:35:34,670 this block diagram is a recipe or algorithm for doing the 533 00:35:34,670 --> 00:35:36,410 implementation. 534 00:35:36,410 --> 00:35:41,680 But it's important to recognize, even at this point, 535 00:35:41,680 --> 00:35:45,980 that it's only one of many possible algorithms or 536 00:35:45,980 --> 00:35:48,990 implementations for this difference equation. 537 00:35:48,990 --> 00:35:57,280 Just for example, I can consider that equation for 538 00:35:57,280 --> 00:35:59,320 that block diagram. 539 00:35:59,320 --> 00:36:03,290 And here, I've re-drawn it. 540 00:36:03,290 --> 00:36:05,630 So here, are once again. 541 00:36:05,630 --> 00:36:09,140 I have the same block diagram that we just saw. 542 00:36:09,140 --> 00:36:13,810 And I can recognize, for example, that this, in 543 00:36:13,810 --> 00:36:19,620 essence, corresponds to two linear time-invariant systems 544 00:36:19,620 --> 00:36:22,180 in cascade. 545 00:36:22,180 --> 00:36:26,540 Now, that assumes, of course, that my initial conditions are 546 00:36:26,540 --> 00:36:30,020 such that the system is, in fact, linear. 547 00:36:30,020 --> 00:36:33,480 And that, in turn, requires that we're assuming initial 548 00:36:33,480 --> 00:36:35,300 rests, namely, before the input does 549 00:36:35,300 --> 00:36:37,880 anything other than 0. 550 00:36:37,880 --> 00:36:41,030 There are just 0 values stored in the registers. 551 00:36:41,030 --> 00:36:43,710 But assuming that it corresponds to a linear 552 00:36:43,710 --> 00:36:48,640 time-invariant system, this is a cascade of two linear 553 00:36:48,640 --> 00:36:50,540 time-invariant invriant systems. 554 00:36:50,540 --> 00:36:52,990 We know that two linear time-invariant systems can be 555 00:36:52,990 --> 00:36:55,160 cascaded in either order. 556 00:36:55,160 --> 00:36:59,090 So, in particular, I can consider breaking this cascade 557 00:36:59,090 --> 00:37:03,820 here, and moving this block over to the other side. 558 00:37:03,820 --> 00:37:05,520 And so let's just do that. 559 00:37:08,610 --> 00:37:16,160 And when I do, I then have this combination of systems, 560 00:37:16,160 --> 00:37:19,370 and, of course, you can ask what advantage there is to 561 00:37:19,370 --> 00:37:24,570 doing that, and the advantage arises because of the fact 562 00:37:24,570 --> 00:37:31,990 that in this form, exactly what is stored in these delays 563 00:37:31,990 --> 00:37:35,120 is also stored in these delay registers. 564 00:37:35,120 --> 00:37:38,270 In other words, it's this intermediate variable-- 565 00:37:38,270 --> 00:37:39,500 whatever it is-- 566 00:37:39,500 --> 00:37:42,850 down this chain of the delays and down this chain of delays, 567 00:37:42,850 --> 00:37:47,150 and so, in fact, I can collapse those delays into a 568 00:37:47,150 --> 00:37:49,000 single chain of delays. 569 00:37:49,000 --> 00:37:53,790 And the network that I'm left with is the network that I 570 00:37:53,790 --> 00:37:57,930 indicate on this view graph, where what I've done is to 571 00:37:57,930 --> 00:38:01,980 simply collapse that double chain of delays into a single 572 00:38:01,980 --> 00:38:04,580 change of delays. 573 00:38:04,580 --> 00:38:10,180 Now, one can ask, well, what's the advantage to doing that? 574 00:38:10,180 --> 00:38:13,020 And one advantage, simply stated, is that when you think 575 00:38:13,020 --> 00:38:16,160 in terms of an implementation of a difference equation, a 576 00:38:16,160 --> 00:38:21,340 delay corresponds to a storage register, a memory location, 577 00:38:21,340 --> 00:38:24,980 and by simply using the fact that we can interchange the 578 00:38:24,980 --> 00:38:27,200 order in which linear time-invariant systems are 579 00:38:27,200 --> 00:38:30,030 cascaded, we can reduce the amount of memory 580 00:38:30,030 --> 00:38:31,280 by a factor of 2. 581 00:38:35,280 --> 00:38:40,570 Now, an essentially similar procedure can also be used for 582 00:38:40,570 --> 00:38:44,910 differential equations, in terms of implementation using 583 00:38:44,910 --> 00:38:47,590 block diagrams or the interpretation of 584 00:38:47,590 --> 00:38:50,350 implementations using block diagrams. 585 00:38:50,350 --> 00:38:52,380 And let me first do that-- 586 00:38:52,380 --> 00:38:56,840 rather than in general-- let me first do it in the context 587 00:38:56,840 --> 00:39:00,870 of a specific example. 588 00:39:00,870 --> 00:39:06,830 So let's consider a linear constant-coefficient 589 00:39:06,830 --> 00:39:10,790 differential equation, as I've indicated here, and I have 590 00:39:10,790 --> 00:39:13,930 terms on the left side and terms on the right side. 591 00:39:13,930 --> 00:39:19,870 And with the differential equation, let's consider 592 00:39:19,870 --> 00:39:22,800 taking all the terms over to the right side of the 593 00:39:22,800 --> 00:39:26,030 equation, except for the highest 594 00:39:26,030 --> 00:39:28,950 derivative in the output. 595 00:39:28,950 --> 00:39:33,250 Next, we integrate both sides of the equation so that when 596 00:39:33,250 --> 00:39:35,540 we're done, we end up with on the left side of the 597 00:39:35,540 --> 00:39:37,480 equation with y(t). 598 00:39:37,480 --> 00:39:40,480 On the right side of the equation with the appropriate 599 00:39:40,480 --> 00:39:42,700 number of integrations. 600 00:39:42,700 --> 00:39:46,170 And so the integral equation that we'll get for this 601 00:39:46,170 --> 00:39:52,410 example y(t), the output, is x(t) plus b, the scale factor 602 00:39:52,410 --> 00:39:56,370 times the integral of the input, and minus a, that scale 603 00:39:56,370 --> 00:39:59,480 factor, times the integral of the output. 604 00:39:59,480 --> 00:40:05,600 So to form the output in the block diagram terms, we form a 605 00:40:05,600 --> 00:40:09,140 linear combination of the input, a scaled integral of 606 00:40:09,140 --> 00:40:12,970 the input, and a scaled integral of the output, all of 607 00:40:12,970 --> 00:40:14,820 that added together. 608 00:40:14,820 --> 00:40:20,540 So we need, in addition to the input, we need the 609 00:40:20,540 --> 00:40:22,070 integral of the input. 610 00:40:22,070 --> 00:40:25,250 And so this box indicates an integrator. 611 00:40:25,250 --> 00:40:28,260 In addition to the output, we need the 612 00:40:28,260 --> 00:40:30,670 integral of the output. 613 00:40:30,670 --> 00:40:38,490 And now, to form y(t), we multiply the integrated input 614 00:40:38,490 --> 00:40:41,540 by the scale factor, b. 615 00:40:41,540 --> 00:40:49,860 And add that to x(t), and we take the integrated output, 616 00:40:49,860 --> 00:40:57,330 multiply it by -a, and add to that the result of the 617 00:40:57,330 --> 00:41:00,530 previous addition, and according to the integral 618 00:41:00,530 --> 00:41:05,180 equation, then that forms the output. 619 00:41:05,180 --> 00:41:09,580 So just as we did with the difference equation, we've 620 00:41:09,580 --> 00:41:12,330 converted the differential equation to an integral 621 00:41:12,330 --> 00:41:17,200 equation, and we have a block diagram form very similar to 622 00:41:17,200 --> 00:41:20,100 what we had in the case of the difference equation. 623 00:41:20,100 --> 00:41:24,160 Now, the initial conditions, of course, are tied up in, 624 00:41:24,160 --> 00:41:27,430 again, how these integrators are initialized. 625 00:41:27,430 --> 00:41:30,980 Assuming that we impose initial rest on the system, we 626 00:41:30,980 --> 00:41:33,710 can think of the overall system as a linear 627 00:41:33,710 --> 00:41:37,740 time-invariant system, and it's a cascade of one linear 628 00:41:37,740 --> 00:41:40,470 time-invariant system with a second. 629 00:41:40,470 --> 00:41:47,190 So we can, in fact, break this, and consider 630 00:41:47,190 --> 00:41:49,620 interchanging the order in which these 631 00:41:49,620 --> 00:41:51,520 two systems are cascaded. 632 00:41:51,520 --> 00:41:54,580 And so I've indicated that down below. 633 00:41:54,580 --> 00:41:58,840 Here, I've simply taken the top block diagram, 634 00:41:58,840 --> 00:42:01,920 interchanged the order in which the 635 00:42:01,920 --> 00:42:05,590 two systems are cascaded. 636 00:42:05,590 --> 00:42:09,490 And here, again, we can ask what the advantages to this, 637 00:42:09,490 --> 00:42:12,090 as opposed to the previous one. 638 00:42:12,090 --> 00:42:14,830 And what you can see, just as we saw with the difference 639 00:42:14,830 --> 00:42:18,790 equation, is that now, the integrators-- 640 00:42:18,790 --> 00:42:19,780 both integrators-- 641 00:42:19,780 --> 00:42:21,730 are integrating the same thing. 642 00:42:21,730 --> 00:42:27,170 In particular, the input to this integrator and the input 643 00:42:27,170 --> 00:42:30,210 to this integrator are identical. 644 00:42:30,210 --> 00:42:34,660 So in fact, rather than using this one, we can simply tap 645 00:42:34,660 --> 00:42:36,970 off from here. 646 00:42:36,970 --> 00:42:40,660 We can, in fact, remove this integrator, break this 647 00:42:40,660 --> 00:42:45,610 connection, and tap in at this point. 648 00:42:45,610 --> 00:42:49,910 And so what we've done then, by interchanging the order in 649 00:42:49,910 --> 00:42:53,420 which the systems are cascaded, is reduced the 650 00:42:53,420 --> 00:42:56,730 implementation to the implementation with a single 651 00:42:56,730 --> 00:42:58,060 integrator. 652 00:42:58,060 --> 00:43:01,830 Very much similar to what we talked about in the case of 653 00:43:01,830 --> 00:43:03,630 the difference equation. 654 00:43:03,630 --> 00:43:07,600 Now, let's just, again, with the integral equation or the 655 00:43:07,600 --> 00:43:13,270 differential equation, look at this somewhat more generally. 656 00:43:13,270 --> 00:43:17,360 Again, if we take the differential equation, the 657 00:43:17,360 --> 00:43:20,480 general differential equation, integrate it a sufficient 658 00:43:20,480 --> 00:43:23,740 number of times to convert it to an integral equation. 659 00:43:23,740 --> 00:43:28,320 We would then have this cascade of systems. 660 00:43:28,320 --> 00:43:33,000 And again, if we assume initial rest, so that these 661 00:43:33,000 --> 00:43:36,520 are both linear time-invariant systems, we can interchange 662 00:43:36,520 --> 00:43:38,610 the order in which they're cascaded. 663 00:43:38,610 --> 00:43:45,990 Namely, take the second system, and move it to precede 664 00:43:45,990 --> 00:43:48,220 the first system. 665 00:43:48,220 --> 00:43:51,890 And then what we recognize is that the input to this chain 666 00:43:51,890 --> 00:43:54,530 of integrators and this chain of integrators 667 00:43:54,530 --> 00:43:56,220 is exactly the same. 668 00:43:56,220 --> 00:44:00,320 And so, in fact we can collapse these together using 669 00:44:00,320 --> 00:44:02,310 only one chain of integrators. 670 00:44:02,310 --> 00:44:07,890 And the system that we're left with then is a system that 671 00:44:07,890 --> 00:44:09,730 looks as I've indicated here. 672 00:44:09,730 --> 00:44:14,890 So we have now just a single chains of integrators instead 673 00:44:14,890 --> 00:44:18,330 of the two sets of integrators. 674 00:44:18,330 --> 00:44:21,670 So we've seen that the situation is very similar here 675 00:44:21,670 --> 00:44:24,510 as it was in the case of the difference equation. 676 00:44:24,510 --> 00:44:26,900 Again, why do we want to cut the number of 677 00:44:26,900 --> 00:44:28,310 integrators in half? 678 00:44:28,310 --> 00:44:32,140 Well, one reason is because integrators, in effect, 679 00:44:32,140 --> 00:44:33,710 represent hardware. 680 00:44:33,710 --> 00:44:38,155 And if we have half as many integrators, then we're using 681 00:44:38,155 --> 00:44:39,410 half as much hardware. 682 00:44:41,940 --> 00:44:44,440 Well, let me just conclude by 683 00:44:44,440 --> 00:44:47,550 summarizing a number of points. 684 00:44:47,550 --> 00:44:50,340 I indicated at the beginning that linear 685 00:44:50,340 --> 00:44:52,690 constant-coefficient differential equations and 686 00:44:52,690 --> 00:44:57,060 difference equations will play an important role as linear 687 00:44:57,060 --> 00:45:01,490 time-invariant systems throughout this course and 688 00:45:01,490 --> 00:45:04,160 throughout this set of lectures. 689 00:45:04,160 --> 00:45:09,210 I also stressed the fact that differential or difference 690 00:45:09,210 --> 00:45:13,990 equations, by themselves, are not a complete specification 691 00:45:13,990 --> 00:45:18,190 of the system because of the fact that we can add to any 692 00:45:18,190 --> 00:45:21,970 solution a homogeneous solution. 693 00:45:21,970 --> 00:45:26,080 How do we specify the appropriate initial conditions 694 00:45:26,080 --> 00:45:28,270 to ensure-- 695 00:45:28,270 --> 00:45:30,260 how do we specify the appropriate initial conditions 696 00:45:30,260 --> 00:45:34,250 to ensure that the system is linear and time-invariant? 697 00:45:34,250 --> 00:45:39,110 Well, the auxiliary information, namely, the 698 00:45:39,110 --> 00:45:45,100 initial conditions associated with the system being causal, 699 00:45:45,100 --> 00:45:47,790 linear, and time-invariant are the 700 00:45:47,790 --> 00:45:50,630 conditions of initial rest. 701 00:45:50,630 --> 00:45:54,130 And, in fact, for most of the course, what we'll be 702 00:45:54,130 --> 00:45:57,200 interested in are systems that are in fact, causal, linear, 703 00:45:57,200 --> 00:45:58,420 and time-invariant. 704 00:45:58,420 --> 00:46:01,150 And so we will, in fact, be assuming initial rest 705 00:46:01,150 --> 00:46:03,640 conditions. 706 00:46:03,640 --> 00:46:09,670 Now, as I also indicated, there are a variety of 707 00:46:09,670 --> 00:46:12,580 efficient procedures for solving differential and 708 00:46:12,580 --> 00:46:15,560 difference equations that we haven't yet addressed. 709 00:46:15,560 --> 00:46:18,990 And beginning with the next set of lectures, we'll be 710 00:46:18,990 --> 00:46:22,510 talking about the Fourier Transform and much later in 711 00:46:22,510 --> 00:46:26,470 the course, what's referred to as the Laplace Transform for 712 00:46:26,470 --> 00:46:30,070 continuous time and the Z-transform for discrete time. 713 00:46:30,070 --> 00:46:34,030 And what we'll see is that with the Fourier Transform and 714 00:46:34,030 --> 00:46:37,300 later with the Laplace and Z-transform, we'll have a 715 00:46:37,300 --> 00:46:42,480 number of efficient and very useful ways of generating the 716 00:46:42,480 --> 00:46:48,000 solution for differential and difference equations under the 717 00:46:48,000 --> 00:46:50,705 assumption that the system is causal, linear, and 718 00:46:50,705 --> 00:46:52,510 time-invariant. 719 00:46:52,510 --> 00:46:56,020 Also, we'll see in addition to the block diagram 720 00:46:56,020 --> 00:46:58,510 implementations of these systems that we've talked 721 00:46:58,510 --> 00:47:04,170 about so far, we'll see a number of other useful 722 00:47:04,170 --> 00:47:07,880 implementations that exploit a variety of properties 723 00:47:07,880 --> 00:47:10,870 associated with Fourier and Laplace Transforms. 724 00:47:10,870 --> 00:47:12,120 Thank you.