1 00:00:42,500 --> 00:00:45,620 PROFESSOR: Last time, we talked about the 2 00:00:45,620 --> 00:00:49,600 representation of linear time-invariant systems through 3 00:00:49,600 --> 00:00:52,400 the convolution sum in the discrete-time case and the 4 00:00:52,400 --> 00:00:56,120 convolution integral in the continuous-time case. 5 00:00:56,120 --> 00:00:58,520 Now, although the derivation was relatively 6 00:00:58,520 --> 00:01:01,970 straightforward, in fact, the result is really kind of 7 00:01:01,970 --> 00:01:06,250 amazing because what it tells us is that for linear 8 00:01:06,250 --> 00:01:09,580 time-invariant systems, if we know the response of the 9 00:01:09,580 --> 00:01:14,540 system to a single impulse at t = 0, or in fact, at any 10 00:01:14,540 --> 00:01:18,710 other time, then we can determine from that its 11 00:01:18,710 --> 00:01:22,590 response to an arbitrary input through the use of 12 00:01:22,590 --> 00:01:24,900 convolution. 13 00:01:24,900 --> 00:01:28,650 Furthermore, what we'll see as the course develops is that, 14 00:01:28,650 --> 00:01:31,660 in fact, the class of linear time-invariant systems is a 15 00:01:31,660 --> 00:01:33,720 very rich class. 16 00:01:33,720 --> 00:01:36,770 There are lots of systems that have that property. 17 00:01:36,770 --> 00:01:39,650 And in addition, there are lots of very interesting 18 00:01:39,650 --> 00:01:42,280 things that you can do with linear time-invariant systems. 19 00:01:44,850 --> 00:01:49,510 In today's lecture, what I'd like to begin with is focusing 20 00:01:49,510 --> 00:01:53,730 on convolution as an algebraic operation. 21 00:01:53,730 --> 00:01:57,370 And we'll see that it has a number of algebraic properties 22 00:01:57,370 --> 00:02:02,150 that in turn have important implications for linear 23 00:02:02,150 --> 00:02:04,120 time-invariant systems. 24 00:02:04,120 --> 00:02:08,009 Then we'll turn to a discussion of what the 25 00:02:08,009 --> 00:02:11,680 characterization of linear time-invariant systems through 26 00:02:11,680 --> 00:02:16,900 convolution implies, in terms of the relationship, of 27 00:02:16,900 --> 00:02:19,110 various other system properties 28 00:02:19,110 --> 00:02:21,240 to the impulse response. 29 00:02:21,240 --> 00:02:26,220 Let me begin by reminding you of the basic result that we 30 00:02:26,220 --> 00:02:31,180 developed last time, which is the convolution sum in 31 00:02:31,180 --> 00:02:37,740 discrete time, as I indicate here, and the convolution 32 00:02:37,740 --> 00:02:40,820 integral in continuous time. 33 00:02:40,820 --> 00:02:44,850 And what the convolution sum, or the convolution integral, 34 00:02:44,850 --> 00:02:51,850 tells us is how to relate the output to the input and to the 35 00:02:51,850 --> 00:02:53,940 system impulse response. 36 00:02:53,940 --> 00:02:58,750 And the expression looks basically the same in 37 00:02:58,750 --> 00:03:01,320 continuous time and discrete time. 38 00:03:01,320 --> 00:03:04,650 And I remind you also that we talked about a graphical 39 00:03:04,650 --> 00:03:09,940 interpretation, where essentially, to graphically 40 00:03:09,940 --> 00:03:15,760 interpret convolution required, or was developed, in 41 00:03:15,760 --> 00:03:21,160 terms of taking the system impulse response, flipping it, 42 00:03:21,160 --> 00:03:25,460 sliding it past the input, and positioned appropriately, 43 00:03:25,460 --> 00:03:29,500 depending on the value of the independent variable for which 44 00:03:29,500 --> 00:03:32,340 we're computing the convolution, and then 45 00:03:32,340 --> 00:03:36,230 multiplying and summing in the discrete-time case, or 46 00:03:36,230 --> 00:03:38,390 integrating in the continuous-time case. 47 00:03:41,160 --> 00:03:47,070 Now convolution, as an algebraic operation, has a 48 00:03:47,070 --> 00:03:51,000 number of important properties. 49 00:03:51,000 --> 00:03:55,370 One of the properties of convolution is that it is what 50 00:03:55,370 --> 00:03:58,300 is referred to as commutative. 51 00:03:58,300 --> 00:04:05,330 Commutative means that we can think either of convolving x 52 00:04:05,330 --> 00:04:11,760 with h, or h with x, and the order in which that's done 53 00:04:11,760 --> 00:04:14,590 doesn't affect the output result. 54 00:04:14,590 --> 00:04:19,649 So summarized here is what the commutative operation is in 55 00:04:19,649 --> 00:04:23,990 discrete time, or in continuous time. 56 00:04:23,990 --> 00:04:28,050 And it says, as I just indicated, that x[n] 57 00:04:28,050 --> 00:04:30,240 convolved with h[n] 58 00:04:30,240 --> 00:04:32,500 is equal to h[n] 59 00:04:32,500 --> 00:04:34,750 convolved with x[n]. 60 00:04:34,750 --> 00:04:38,930 Or the same, of course, in continuous time. 61 00:04:38,930 --> 00:04:44,200 And in fact, in the lecture last time, we worked an 62 00:04:44,200 --> 00:04:48,000 example where we had, in discrete time, an impulse 63 00:04:48,000 --> 00:04:51,980 response, which was an exponential, and an input, 64 00:04:51,980 --> 00:04:54,760 which is a unit step. 65 00:04:54,760 --> 00:04:59,520 And in the text, what you'll find is the same example 66 00:04:59,520 --> 00:05:04,110 worked, except in that case, the input is the exponential. 67 00:05:04,110 --> 00:05:07,370 And the system impulse response is the step. 68 00:05:07,370 --> 00:05:11,380 And that corresponds to the example in the text, which is 69 00:05:11,380 --> 00:05:15,720 example 3.1. 70 00:05:15,720 --> 00:05:20,130 And what happens in that example is that, in fact, what 71 00:05:20,130 --> 00:05:24,850 you'll see is that the same result occurs in example 3.1 72 00:05:24,850 --> 00:05:27,400 as we generated in the lecture. 73 00:05:27,400 --> 00:05:31,090 And there's a similar comparison in continuous time. 74 00:05:31,090 --> 00:05:34,680 This example was worked in lecture. 75 00:05:34,680 --> 00:05:38,740 And this example is worked in the text. 76 00:05:38,740 --> 00:05:43,370 In other words, the text works the example where the system 77 00:05:43,370 --> 00:05:47,960 impulse response is a unit step. 78 00:05:47,960 --> 00:05:52,130 And the input is an exponential. 79 00:05:52,130 --> 00:05:55,220 All right, now the commutative property, as I said, tells us 80 00:05:55,220 --> 00:05:59,700 that the order in which we do convolution doesn't affect the 81 00:05:59,700 --> 00:06:01,710 result of the convolution. 82 00:06:01,710 --> 00:06:03,560 The same is true for continuous time 83 00:06:03,560 --> 00:06:04,940 and discrete time. 84 00:06:04,940 --> 00:06:07,830 And in fact, for the other algebraic properties that I'll 85 00:06:07,830 --> 00:06:12,310 talk about, the results are exactly the same for 86 00:06:12,310 --> 00:06:15,040 continuous time and discrete time. 87 00:06:15,040 --> 00:06:22,210 So in fact, what we can do is drop the independent variable 88 00:06:22,210 --> 00:06:26,720 as an argument so that we suppress any kind of 89 00:06:26,720 --> 00:06:29,820 difference between continuous and discrete time. 90 00:06:29,820 --> 00:06:34,220 And suppressing the independent variable, we then 91 00:06:34,220 --> 00:06:39,170 state the commutative property as I've rewritten it here. 92 00:06:39,170 --> 00:06:42,970 Just x convolved with h equals h convolved with x. 93 00:06:42,970 --> 00:06:47,360 The same in continuous time and discrete time. 94 00:06:47,360 --> 00:06:52,950 Now, the derivation of the commutative property is, more 95 00:06:52,950 --> 00:06:54,900 or less, some algebra which you can follow 96 00:06:54,900 --> 00:06:55,960 through in the book. 97 00:06:55,960 --> 00:07:01,810 It involves some changes of variables and some 98 00:07:01,810 --> 00:07:04,310 things of that sort. 99 00:07:04,310 --> 00:07:08,440 What I'd like to focus on with that and other properties is 100 00:07:08,440 --> 00:07:12,520 not the derivation, which you can see in the text, but 101 00:07:12,520 --> 00:07:14,760 rather the interpretation. 102 00:07:14,760 --> 00:07:18,050 So we have the commutative property, and now there are 103 00:07:18,050 --> 00:07:21,090 two other important algebraic properties. 104 00:07:21,090 --> 00:07:25,760 One being what is referred to as the associative property, 105 00:07:25,760 --> 00:07:32,590 which tells us that if we have x convolved with the result of 106 00:07:32,590 --> 00:07:37,570 convolving h1 with h2, that's exactly the same as x 107 00:07:37,570 --> 00:07:42,890 convolved with h1, and that result convolved with h2. 108 00:07:42,890 --> 00:07:49,590 And what this permits is for us to write, for example, x 109 00:07:49,590 --> 00:07:56,470 convolved with h1 convolved with h2 without any ambiguity 110 00:07:56,470 --> 00:08:00,750 because it doesn't matter from the associative property how 111 00:08:00,750 --> 00:08:04,470 we group the terms together. 112 00:08:04,470 --> 00:08:07,950 The third important property is what is referred to as the 113 00:08:07,950 --> 00:08:11,670 distributive property, namely the fact that convolution 114 00:08:11,670 --> 00:08:13,520 distributes over addition. 115 00:08:13,520 --> 00:08:17,560 And what I mean by that is what I've indicated here on 116 00:08:17,560 --> 00:08:24,650 the slide: that if I think of x convolved with the sum of h1 117 00:08:24,650 --> 00:08:31,380 and h2, that's identical to first convolving x with h1, 118 00:08:31,380 --> 00:08:37,140 also convolving x with h2, and then adding the two together. 119 00:08:37,140 --> 00:08:39,960 And that result will be the same as this result. 120 00:08:42,770 --> 00:08:47,550 So convolution is commutative, associative, and it 121 00:08:47,550 --> 00:08:49,050 distributes over addition. 122 00:08:49,050 --> 00:08:52,080 Three very important algebraic properties. 123 00:08:52,080 --> 00:08:56,710 And by the way, there are other algebraic operations 124 00:08:56,710 --> 00:08:58,330 that have that same property. 125 00:08:58,330 --> 00:09:02,900 For example, multiplication of numbers is likewise 126 00:09:02,900 --> 00:09:07,000 commutative, associative, and distributive. 127 00:09:07,000 --> 00:09:11,890 Now let's look at what these three properties imply 128 00:09:11,890 --> 00:09:15,170 specifically for linear time-invariant systems. 129 00:09:15,170 --> 00:09:19,770 And as we'll see, the implications are both very 130 00:09:19,770 --> 00:09:22,260 interesting and very important. 131 00:09:22,260 --> 00:09:26,910 Let's begin with the commutative property. 132 00:09:26,910 --> 00:09:32,860 And consider, in particular, a system with an impulse 133 00:09:32,860 --> 00:09:34,630 response h. 134 00:09:34,630 --> 00:09:39,450 And I represent that by simply writing the h inside the box. 135 00:09:39,450 --> 00:09:46,100 An input x and an output, then, which is x * h. 136 00:09:46,100 --> 00:09:50,430 Now, since this operation is commutative, I can write 137 00:09:50,430 --> 00:09:55,020 instead of x * h, I can write h * x. 138 00:09:55,020 --> 00:10:00,210 And that would correspond to a system with impulse response 139 00:10:00,210 --> 00:10:07,880 x, and input h, and output then h * x. 140 00:10:07,880 --> 00:10:13,620 So the commutative property tells us that for a linear 141 00:10:13,620 --> 00:10:20,040 time-invariant system, the system output is independent 142 00:10:20,040 --> 00:10:24,290 of which function we call the input and which function we 143 00:10:24,290 --> 00:10:25,830 call the impulse response. 144 00:10:25,830 --> 00:10:27,360 Kind of amazing actually. 145 00:10:27,360 --> 00:10:31,090 We can interchange the role of input and impulse response. 146 00:10:31,090 --> 00:10:36,480 And from an output point of view, the output or the system 147 00:10:36,480 --> 00:10:38,660 doesn't care. 148 00:10:38,660 --> 00:10:44,100 Now furthermore, if we combine the commutative property with 149 00:10:44,100 --> 00:10:46,270 the associative property, we get another 150 00:10:46,270 --> 00:10:48,130 very interesting result. 151 00:10:48,130 --> 00:10:52,020 Namely that if we have two linear time-invariant systems 152 00:10:52,020 --> 00:10:57,330 in cascade, the overall system is independent of the order in 153 00:10:57,330 --> 00:10:58,890 which they're cascaded. 154 00:10:58,890 --> 00:11:02,740 And in fact, in either case, the cascade can be collapsed 155 00:11:02,740 --> 00:11:05,660 into a single system. 156 00:11:05,660 --> 00:11:13,120 To see this, let's first consider the cascade of two 157 00:11:13,120 --> 00:11:16,140 systems, one with impulse response h1, the other with 158 00:11:16,140 --> 00:11:18,330 impulse response h2. 159 00:11:18,330 --> 00:11:25,770 And the output of the first system is then x * h1. 160 00:11:25,770 --> 00:11:29,470 And then that is the input to the second system. 161 00:11:29,470 --> 00:11:36,920 And so the output of that is that result convolved with h2. 162 00:11:36,920 --> 00:11:40,500 So this is the result of cascading the two. 163 00:11:40,500 --> 00:11:45,190 And now we can use the associative property to 164 00:11:45,190 --> 00:11:55,000 rewrite this as x * (h1 * h2), where we group 165 00:11:55,000 --> 00:11:58,160 these two terms together. 166 00:11:58,160 --> 00:12:03,200 And so using the associative property, we now can collapse 167 00:12:03,200 --> 00:12:07,860 that into a single system with an input, which is x, and 168 00:12:07,860 --> 00:12:11,360 impulse response, which is h1 * h2. 169 00:12:11,360 --> 00:12:15,040 And the output is then x convolved with the result of 170 00:12:15,040 --> 00:12:17,880 those two convolved. 171 00:12:17,880 --> 00:12:20,880 Next, we can apply the commutative property. 172 00:12:20,880 --> 00:12:24,590 And the commutative property says we could write this 173 00:12:24,590 --> 00:12:28,970 impulse response that way, or we could write it this way. 174 00:12:28,970 --> 00:12:32,200 And since convolution is commutative, the resulting 175 00:12:32,200 --> 00:12:34,100 output will be exactly the same. 176 00:12:34,100 --> 00:12:39,200 And so these resulting outputs will be exactly the same. 177 00:12:39,200 --> 00:12:44,560 And now, once again we can use the associative property to 178 00:12:44,560 --> 00:12:48,670 group these two terms together. 179 00:12:48,670 --> 00:12:54,920 And x * h2 corresponds to putting x first through the 180 00:12:54,920 --> 00:12:59,270 system h2 and then that output through the system h1. 181 00:12:59,270 --> 00:13:03,320 And so finally applying the associative property again, as 182 00:13:03,320 --> 00:13:08,330 I just outlined, we can expand that system back into two 183 00:13:08,330 --> 00:13:16,160 systems in cascade with h2 first and h1 second, 184 00:13:16,160 --> 00:13:21,220 OK, well that involves a certain amount of algebraic 185 00:13:21,220 --> 00:13:22,490 manipulation. 186 00:13:22,490 --> 00:13:24,620 And that is not the algebraic 187 00:13:24,620 --> 00:13:26,140 manipulation that is important. 188 00:13:26,140 --> 00:13:28,250 It's the result that it's important. 189 00:13:28,250 --> 00:13:31,330 And what the result says, to reiterate, is if I have two 190 00:13:31,330 --> 00:13:35,750 linear time-invariant systems in cascade, I can cascade them 191 00:13:35,750 --> 00:13:39,740 in any order, and the result is the same. 192 00:13:39,740 --> 00:13:43,630 Now you might think, well gee, maybe that actually applies to 193 00:13:43,630 --> 00:13:46,710 systems in general, whether you put them 194 00:13:46,710 --> 00:13:48,020 this way or that way. 195 00:13:48,020 --> 00:13:52,140 But in fact, as we talked about last time, and I 196 00:13:52,140 --> 00:13:58,680 illustrated with an example, in general, if the systems are 197 00:13:58,680 --> 00:14:02,000 not linear and time-invariant, then the order in which 198 00:14:02,000 --> 00:14:05,490 they're cascaded is important to the interpretation of the 199 00:14:05,490 --> 00:14:06,690 overall system. 200 00:14:06,690 --> 00:14:11,070 For example, if one system took the square root and the 201 00:14:11,070 --> 00:14:16,080 other system doubled the input, taking the square root 202 00:14:16,080 --> 00:14:20,320 and then doubling gives us a different answer than doubling 203 00:14:20,320 --> 00:14:22,150 first and then taking the square root. 204 00:14:22,150 --> 00:14:25,570 And of course, one can construct much more elaborate 205 00:14:25,570 --> 00:14:27,330 examples than that. 206 00:14:27,330 --> 00:14:30,280 So it's a property very particular to linear 207 00:14:30,280 --> 00:14:31,570 time-invariant systems. 208 00:14:31,570 --> 00:14:37,590 And also one that we will exploit many, many times as we 209 00:14:37,590 --> 00:14:40,900 go through this material. 210 00:14:40,900 --> 00:14:45,440 The final property related to an interconnection of systems 211 00:14:45,440 --> 00:14:49,470 that I want to just indicate develops out of the 212 00:14:49,470 --> 00:14:51,270 distributive property. 213 00:14:51,270 --> 00:14:55,320 And what it applies to is an interpretation of the 214 00:14:55,320 --> 00:14:59,020 interconnection of systems in parallel. 215 00:14:59,020 --> 00:15:03,520 Recall that the parallel combination of systems 216 00:15:03,520 --> 00:15:09,030 corresponds, as I indicate here, to a system in which we 217 00:15:09,030 --> 00:15:13,900 simultaneously feed the input into h1 and h2, these 218 00:15:13,900 --> 00:15:16,680 representing the impulse responses. 219 00:15:16,680 --> 00:15:19,410 And then, the outputs are summed to 220 00:15:19,410 --> 00:15:22,740 form the overall output. 221 00:15:22,740 --> 00:15:28,050 And using the fact that convolution distributes over 222 00:15:28,050 --> 00:15:37,750 addition, we can rewrite this as x * (h1 + h2). 223 00:15:37,750 --> 00:15:41,980 And when we do that then, we can recognize this as the 224 00:15:41,980 --> 00:15:47,280 output of a system with input x and impulse response, which 225 00:15:47,280 --> 00:15:50,300 is the sum of these two impulse responses. 226 00:15:50,300 --> 00:15:55,240 So for linear time-invariant systems in parallel, we can, 227 00:15:55,240 --> 00:16:00,600 if we choose, replace that interconnection by a single 228 00:16:00,600 --> 00:16:04,920 system whose impulse response is simply the sum of those 229 00:16:04,920 --> 00:16:06,170 impulse responses. 230 00:16:09,854 --> 00:16:16,460 OK, now we have this very powerful representation for 231 00:16:16,460 --> 00:16:20,530 linear time-invariant systems in terms of convolution. 232 00:16:20,530 --> 00:16:25,010 And we've seen so far in this lecture how convolution and 233 00:16:25,010 --> 00:16:28,860 the representation through the impulse response leads to some 234 00:16:28,860 --> 00:16:31,570 important implications for system interconnections. 235 00:16:34,090 --> 00:16:39,100 What I'd like to turn to now are other system properties 236 00:16:39,100 --> 00:16:43,400 and see how, for linear time-invariant systems in 237 00:16:43,400 --> 00:16:49,490 particular, other system properties can be associated 238 00:16:49,490 --> 00:16:53,500 with particular properties or characteristics of the system 239 00:16:53,500 --> 00:16:55,160 impulse response. 240 00:16:55,160 --> 00:16:58,130 And what we'll talk about are a variety of properties. 241 00:16:58,130 --> 00:17:01,200 We'll talk about the issue of memory, we'll talk about the 242 00:17:01,200 --> 00:17:04,780 issue of invertibility, and we'll talk about the issue of 243 00:17:04,780 --> 00:17:07,004 causality and also stability. 244 00:17:10,210 --> 00:17:13,230 Well, let's begin with the issue of memory. 245 00:17:13,230 --> 00:17:18,660 And the question now is what are the implications for the 246 00:17:18,660 --> 00:17:22,490 system impulse response for a linear time-invariant system? 247 00:17:22,490 --> 00:17:25,440 Remember that we're always imposing that on the system. 248 00:17:25,440 --> 00:17:28,319 What are the implications on the impulse response if the 249 00:17:28,319 --> 00:17:32,350 system does or does not have memory? 250 00:17:32,350 --> 00:17:36,440 Well, we can answer that by looking at 251 00:17:36,440 --> 00:17:39,860 the convolution property. 252 00:17:39,860 --> 00:17:43,660 And we have here, as a reminder, the convolution 253 00:17:43,660 --> 00:17:51,230 integral, which tells us how x(tau) and h(t - tau) are 254 00:17:51,230 --> 00:17:56,020 combined to give us y(t). 255 00:17:56,020 --> 00:18:00,940 And what I've illustrated above is a 256 00:18:00,940 --> 00:18:02,730 general kind of example. 257 00:18:02,730 --> 00:18:04,890 Here is x(tau). 258 00:18:04,890 --> 00:18:08,670 Here is h(t - tau). 259 00:18:08,670 --> 00:18:13,450 And to compute the output at any time t, we would take 260 00:18:13,450 --> 00:18:18,160 these two, multiply them together, and integrate from 261 00:18:18,160 --> 00:18:19,730 -infinity to +infinity. 262 00:18:22,740 --> 00:18:27,050 So the question then is what can we say about h(t), the 263 00:18:27,050 --> 00:18:33,020 impulse response in order to guarantee, let's say, that the 264 00:18:33,020 --> 00:18:37,090 output depends only on the input at time t. 265 00:18:37,090 --> 00:18:43,900 Well, it's pretty much obvious from looking at the graphs. 266 00:18:43,900 --> 00:18:51,740 If we only want the output to depend on x(tau) at tau = t, 267 00:18:51,740 --> 00:18:59,360 then h(t - tau) better be non-zero only at tau = t. 268 00:18:59,360 --> 00:19:04,990 And so the implication then is that for the system to be 269 00:19:04,990 --> 00:19:12,100 memoryless, what we require is that h(t - tau) be non-zero 270 00:19:12,100 --> 00:19:16,500 only at tau = t. 271 00:19:16,500 --> 00:19:19,960 So we want the impulse response to be non-zero at 272 00:19:19,960 --> 00:19:22,230 only one point. 273 00:19:22,230 --> 00:19:24,670 We want it to contribute something after we multiply 274 00:19:24,670 --> 00:19:26,490 and go through an integral. 275 00:19:26,490 --> 00:19:29,410 And in effect, what that says is the only thing that it can 276 00:19:29,410 --> 00:19:30,680 be and meet all those 277 00:19:30,680 --> 00:19:33,400 conditions is a scaled impulse. 278 00:19:33,400 --> 00:19:39,310 So if the system is to be memoryless, then that requires 279 00:19:39,310 --> 00:19:43,020 that the impulse response be a scaled impulse. 280 00:19:43,020 --> 00:19:49,190 Any other impulse response then, in essence, requires 281 00:19:49,190 --> 00:19:52,010 that the system have memory, or implies that the system 282 00:19:52,010 --> 00:19:54,210 have memory. 283 00:19:54,210 --> 00:19:59,950 So for the continuous-time case then, memoryless would 284 00:19:59,950 --> 00:20:03,760 correspond only to the impulse response being proportional to 285 00:20:03,760 --> 00:20:05,060 an impulse. 286 00:20:05,060 --> 00:20:09,720 And in the discrete-time case, a similar statement, in which 287 00:20:09,720 --> 00:20:14,590 case, the output is just proportional to the input, 288 00:20:14,590 --> 00:20:17,730 again either in the continuous-time or in the 289 00:20:17,730 --> 00:20:18,980 discrete-time case. 290 00:20:22,430 --> 00:20:22,670 All right. 291 00:20:22,670 --> 00:20:27,650 Now we can turn our attention to the issue of system 292 00:20:27,650 --> 00:20:29,410 invertibility. 293 00:20:29,410 --> 00:20:35,460 And recall that what is meant by invertibility of a system, 294 00:20:35,460 --> 00:20:37,690 or the inverse of a system. 295 00:20:37,690 --> 00:20:41,080 The inverse of a system is a system, which when we cascade 296 00:20:41,080 --> 00:20:45,360 it with the one that we're inquiring about, the overall 297 00:20:45,360 --> 00:20:48,110 cascade is the identity system. 298 00:20:48,110 --> 00:20:53,370 In other words, the output is equal to the input. 299 00:20:53,370 --> 00:20:56,980 So let's consider a system with impulse 300 00:20:56,980 --> 00:21:00,440 response h, input is x. 301 00:21:00,440 --> 00:21:04,480 And let's say that the impulse response of the inverse system 302 00:21:04,480 --> 00:21:07,930 is h_i, and the output is y. 303 00:21:07,930 --> 00:21:16,850 Then, the output of this system is x * (h * h_i). 304 00:21:16,850 --> 00:21:21,830 And we want this to come out equal to x. 305 00:21:21,830 --> 00:21:28,740 And what that requires than is that this convolution just 306 00:21:28,740 --> 00:21:34,590 simply be equal to an impulse, either in the discrete-time 307 00:21:34,590 --> 00:21:38,270 case or in the continuous-time case. 308 00:21:38,270 --> 00:21:42,380 And under those conditions then, h_i is equal to the 309 00:21:42,380 --> 00:21:44,630 inverse of h. 310 00:21:44,630 --> 00:21:49,760 Notationally, by the way, it's often convenient to write 311 00:21:49,760 --> 00:21:53,620 instead of h_i as the impulse response of the inverse, 312 00:21:53,620 --> 00:21:57,330 you'll find it convenient often and more typical to 313 00:21:57,330 --> 00:22:01,970 write as the inverse, instead of h_i, h^(-1). 314 00:22:05,500 --> 00:22:10,670 And we mean by that the inverse impulse response. 315 00:22:10,670 --> 00:22:14,120 And one has to be careful not to misinterpret this as the 316 00:22:14,120 --> 00:22:16,610 reciprocal of h(t) or h(n). 317 00:22:16,610 --> 00:22:20,260 What's meant in this notation is the inverse system. 318 00:22:24,250 --> 00:22:33,930 Now, if h_i is the inverse of h, is h the inverse of h_i? 319 00:22:33,930 --> 00:22:38,710 Well, it seems like that ought to be plausible or perhaps 320 00:22:38,710 --> 00:22:40,060 make sense. 321 00:22:40,060 --> 00:22:43,470 The question, if you believe that the answer is yes, is 322 00:22:43,470 --> 00:22:47,300 how, in fact, do you verify that? 323 00:22:47,300 --> 00:22:49,860 And I'll leave it to you to think about it. 324 00:22:49,860 --> 00:22:51,020 The answer is yes, 325 00:22:51,020 --> 00:22:54,320 that if h_i is the inverse of h, then h is 326 00:22:54,320 --> 00:22:56,140 the inverse of h_i. 327 00:22:56,140 --> 00:23:00,800 And the key to showing that is to exploit the fact that when 328 00:23:00,800 --> 00:23:03,800 we take these systems and cascade them, we can cascade 329 00:23:03,800 --> 00:23:05,050 them in either order. 330 00:23:07,920 --> 00:23:12,200 All right now let's turn to another system property, the 331 00:23:12,200 --> 00:23:14,760 property of stability. 332 00:23:14,760 --> 00:23:21,160 And again, we can tie that property directly to issues 333 00:23:21,160 --> 00:23:27,010 related, in particular, to the system impulse response. 334 00:23:27,010 --> 00:23:30,845 Now, stability is defined as we've chosen to define it and 335 00:23:30,845 --> 00:23:34,530 as I've defined it previously, as bounded-input 336 00:23:34,530 --> 00:23:35,940 bounded-output stability. 337 00:23:35,940 --> 00:23:38,140 In other words, for every bounded input 338 00:23:38,140 --> 00:23:40,750 is a bounded output. 339 00:23:40,750 --> 00:23:41,850 What you can show-- 340 00:23:41,850 --> 00:23:45,690 and I won't go through the algebra here; it's gone 341 00:23:45,690 --> 00:23:47,700 through in the book-- 342 00:23:47,700 --> 00:23:53,160 is that a necessary and sufficient condition for a 343 00:23:53,160 --> 00:23:57,550 linear time-invariant system to be stable in the 344 00:23:57,550 --> 00:24:03,800 bounded-input bounded-output sense is that the impulse 345 00:24:03,800 --> 00:24:08,360 response be what is referred to as absolutely summable. 346 00:24:08,360 --> 00:24:12,410 In other words, if you take the absolute values and sum 347 00:24:12,410 --> 00:24:16,300 them over infinite limits, that's finite. 348 00:24:16,300 --> 00:24:21,300 Or in the continuous-time case, that the impulse 349 00:24:21,300 --> 00:24:23,530 response is absolutely integrable. 350 00:24:23,530 --> 00:24:27,450 In other words, if you take the absolute values of h(t) 351 00:24:27,450 --> 00:24:29,690 and integrate, that's finite. 352 00:24:29,690 --> 00:24:34,370 And under those conditions, the system is stable. 353 00:24:34,370 --> 00:24:39,070 If those conditions are violated, then for sure, as 354 00:24:39,070 --> 00:24:42,950 you'll see in the text, the system is unstable. 355 00:24:42,950 --> 00:24:46,010 So stability can also be tied to the 356 00:24:46,010 --> 00:24:47,390 system impulse response. 357 00:24:50,470 --> 00:24:54,130 Now, the next property that I want to talk about is the 358 00:24:54,130 --> 00:24:56,670 property of causality. 359 00:24:56,670 --> 00:25:01,890 And before I do, what I'd like to do is introduce a 360 00:25:01,890 --> 00:25:04,690 peripheral result that we'll then use-- 361 00:25:04,690 --> 00:25:06,870 when we talked about causality-- 362 00:25:06,870 --> 00:25:10,690 namely what's referred to as the zero input response of a 363 00:25:10,690 --> 00:25:11,940 linear system. 364 00:25:14,820 --> 00:25:18,570 The basic result, which is a very interesting and useful 365 00:25:18,570 --> 00:25:21,180 one, is that for a linear system-- 366 00:25:21,180 --> 00:25:24,350 and in fact, it's whether it's time-invariant or not, that 367 00:25:24,350 --> 00:25:26,600 this applies-- 368 00:25:26,600 --> 00:25:31,640 if you put nothing into it, you get nothing out of it. 369 00:25:31,640 --> 00:25:40,460 So if we have an input x(t) which is 0 for all t, and if 370 00:25:40,460 --> 00:25:49,040 the output of that system is y(t), if the input is 0 for 371 00:25:49,040 --> 00:25:54,440 all time, then the output likewise is 0 for all time. 372 00:25:54,440 --> 00:26:00,380 That's true for continuous time, and it's also true for 373 00:26:00,380 --> 00:26:01,630 discrete time. 374 00:26:04,750 --> 00:26:10,380 And in fact, to show that result is pretty much 375 00:26:10,380 --> 00:26:11,280 straightforward. 376 00:26:11,280 --> 00:26:15,570 We could do it either by using convolution, which would, of 377 00:26:15,570 --> 00:26:18,780 course, be associated with linearity and time invariance. 378 00:26:18,780 --> 00:26:22,960 But in fact, we can show that property relatively easily by 379 00:26:22,960 --> 00:26:27,350 simply using the fact that, for a linear system, what we 380 00:26:27,350 --> 00:26:35,180 know is that if we have an input x(t) with an output 381 00:26:35,180 --> 00:26:42,390 y(t), then if we scale the input, then the output scales 382 00:26:42,390 --> 00:26:43,940 accordingly. 383 00:26:43,940 --> 00:26:51,290 Well, we can simply choose, as the scale factor, a = 0. 384 00:26:51,290 --> 00:26:54,950 And if we do that, it says put nothing in, 385 00:26:54,950 --> 00:26:57,370 you get nothing out. 386 00:26:57,370 --> 00:27:02,380 And what we'll see is that has some important implications in 387 00:27:02,380 --> 00:27:05,420 terms of causality. 388 00:27:05,420 --> 00:27:09,400 It's important, though, while we're on it, to stress that 389 00:27:09,400 --> 00:27:11,750 not every system, obviously, has that property. 390 00:27:11,750 --> 00:27:14,380 That if you put nothing in, you get nothing out. 391 00:27:14,380 --> 00:27:19,980 A simple example is, let's say, a battery, let's say not 392 00:27:19,980 --> 00:27:21,330 connected to anything. 393 00:27:21,330 --> 00:27:25,250 The output is six volts no matter what the input is. 394 00:27:25,250 --> 00:27:29,970 And it of course then doesn't have this zero response to a 395 00:27:29,970 --> 00:27:31,150 zero input. 396 00:27:31,150 --> 00:27:35,170 It's very particular to linear systems. 397 00:27:35,170 --> 00:27:40,640 All right, well now let's see what this means for causality. 398 00:27:40,640 --> 00:27:45,500 To remind you, causality says, in effect, that the system 399 00:27:45,500 --> 00:27:47,060 can't anticipate the input. 400 00:27:47,060 --> 00:27:50,260 That's what, basically, causality means. 401 00:27:50,260 --> 00:27:54,470 When we talked about it previously, we defined it in a 402 00:27:54,470 --> 00:27:59,100 variety of ways, one of which was the statement that if two 403 00:27:59,100 --> 00:28:05,130 inputs are identical up until some time, then the outputs 404 00:28:05,130 --> 00:28:08,140 must be identical up until the same time. 405 00:28:08,140 --> 00:28:11,880 The reason, kind of intuitively, is that if the 406 00:28:11,880 --> 00:28:13,190 system is causal-- 407 00:28:13,190 --> 00:28:15,290 so it can't anticipate the future-- 408 00:28:15,290 --> 00:28:18,970 it can't anticipate whether these two identical inputs are 409 00:28:18,970 --> 00:28:22,530 sometime later going to change from each other or not. 410 00:28:25,670 --> 00:28:30,410 So causality, in general, is simply this statement, either 411 00:28:30,410 --> 00:28:34,030 continuous-time or discrete-time. 412 00:28:34,030 --> 00:28:37,200 And now, so let's look at what that means 413 00:28:37,200 --> 00:28:39,400 for a linear system. 414 00:28:39,400 --> 00:28:44,020 For a linear system, what that corresponds to or could be 415 00:28:44,020 --> 00:28:51,500 translated to is a statement that says that if x(t) is 0, 416 00:28:51,500 --> 00:28:58,050 for t less than t_0, then y(t) must be 0 for t 417 00:28:58,050 --> 00:29:00,580 less than t_0 also. 418 00:29:00,580 --> 00:29:07,790 And so what that, in effect, says, is that the system-- 419 00:29:07,790 --> 00:29:14,010 for a linear system to be causal, it must have the 420 00:29:14,010 --> 00:29:19,200 property sometimes referred to as initial rest, meaning it 421 00:29:19,200 --> 00:29:23,850 doesn't respond until there's some input that happens. 422 00:29:23,850 --> 00:29:26,250 That it's initially at rest until the 423 00:29:26,250 --> 00:29:29,680 input becomes non-zero. 424 00:29:29,680 --> 00:29:31,050 Now, why is this true? 425 00:29:31,050 --> 00:29:37,690 Why is this a consequence of causality for linear systems? 426 00:29:37,690 --> 00:29:40,980 Well, the reason is we know that if we put nothing in, we 427 00:29:40,980 --> 00:29:43,630 get nothing out. 428 00:29:43,630 --> 00:29:48,320 If we have an input that's 0 for t less than t_0, and the 429 00:29:48,320 --> 00:29:51,830 system can't anticipate whether that input is going to 430 00:29:51,830 --> 00:29:58,950 change from 0 or not, then the system must generate an output 431 00:29:58,950 --> 00:30:03,940 that's 0 up until that time, following the principle that 432 00:30:03,940 --> 00:30:07,360 if two inputs are identical up until some time, the outputs 433 00:30:07,360 --> 00:30:10,660 must be identical up until the same time. 434 00:30:10,660 --> 00:30:18,750 So this basic result for linear systems is essentially 435 00:30:18,750 --> 00:30:24,590 a consequence of the statement that for a linear system, zero 436 00:30:24,590 --> 00:30:29,010 in gives us zero out. 437 00:30:33,490 --> 00:30:40,390 Now, that tells us how to interpret 438 00:30:40,390 --> 00:30:43,190 causality for linear systems. 439 00:30:43,190 --> 00:30:46,360 Now, let's proceed to linear time-invariant systems. 440 00:30:46,360 --> 00:30:50,900 And in fact, we can carry the point one step further. 441 00:30:50,900 --> 00:30:55,380 In particular, a necessary and sufficient condition for 442 00:30:55,380 --> 00:30:59,840 causality in the case of linear time-invariant systems 443 00:30:59,840 --> 00:31:08,570 is that the impulse response be 0, for t less than 0 in the 444 00:31:08,570 --> 00:31:13,030 continuous-time case, or for n less than 0 in the 445 00:31:13,030 --> 00:31:14,630 discrete-time case. 446 00:31:14,630 --> 00:31:18,660 So for linear time-invariant systems, causality is 447 00:31:18,660 --> 00:31:24,310 equivalent to the impulse response being 0 up until t or 448 00:31:24,310 --> 00:31:27,090 n equal to 0. 449 00:31:27,090 --> 00:31:35,330 Now, to show this essentially follows by first considering 450 00:31:35,330 --> 00:31:42,710 why causality would imply that this is true. 451 00:31:42,710 --> 00:31:47,790 And that follows because of the straightforward fact that 452 00:31:47,790 --> 00:31:50,910 the impulse itself is 0 for t less than 0. 453 00:31:53,610 --> 00:31:58,480 And what we saw before is that for any linear system, 454 00:31:58,480 --> 00:32:02,280 causality requires that if the input is 0 up until some time, 455 00:32:02,280 --> 00:32:06,170 the output must be 0 up until the same time. 456 00:32:06,170 --> 00:32:11,740 And so that's showing the result in one direction. 457 00:32:11,740 --> 00:32:15,775 To show the result in the other direction, namely to 458 00:32:15,775 --> 00:32:20,820 show that if, in fact, the impulse response satisfies 459 00:32:20,820 --> 00:32:23,860 that condition, then the system is causal. 460 00:32:23,860 --> 00:32:28,170 While I won't work through it in detail, it essentially 461 00:32:28,170 --> 00:32:36,830 boils down to recognizing that in the convolution sum, or in 462 00:32:36,830 --> 00:32:41,880 the convolution integral, if, in fact, that condition is 463 00:32:41,880 --> 00:32:46,950 satisfied on the impulse response, then the upper limit 464 00:32:46,950 --> 00:32:48,880 on the sum, in the discrete-time 465 00:32:48,880 --> 00:32:50,900 case, changes to n. 466 00:32:50,900 --> 00:32:54,370 And in the continuous-time case, changes to t. 467 00:32:54,370 --> 00:33:00,940 And that, in effect, says that values of the input only from 468 00:33:00,940 --> 00:33:05,840 -infinity up to time n are used in computing y[n]. 469 00:33:05,840 --> 00:33:09,770 And a similar kind of result for the 470 00:33:09,770 --> 00:33:11,140 continuous-time case y(t). 471 00:33:15,380 --> 00:33:20,330 OK, so we've seen how the impulse response, or rather 472 00:33:20,330 --> 00:33:23,170 how certain system properties in the linear time-invariant 473 00:33:23,170 --> 00:33:28,920 case can, be converted into various requirements on the 474 00:33:28,920 --> 00:33:31,670 impulse response of a linear time-invariant system, the 475 00:33:31,670 --> 00:33:35,440 impulse response being a complete characterization. 476 00:33:35,440 --> 00:33:39,390 Let's look at some particular examples just to kind of 477 00:33:39,390 --> 00:33:42,140 cement the ideas further. 478 00:33:42,140 --> 00:33:46,190 And let's begin with a system that you've seen previously, 479 00:33:46,190 --> 00:33:48,165 which is an accumulator. 480 00:33:48,165 --> 00:33:55,860 An accumulator, as you recall, has an output y[n], which is 481 00:33:55,860 --> 00:34:03,080 the accumulated value of the input from -infinity up to n. 482 00:34:03,080 --> 00:34:07,660 Now, you've seen in the impulse in a previous lecture, 483 00:34:07,660 --> 00:34:11,239 or rather in the video course manual for a previous lecture, 484 00:34:11,239 --> 00:34:15,429 that an accumulator is a linear time-invariant system. 485 00:34:15,429 --> 00:34:19,760 And in fact, its impulse response is the accumulated 486 00:34:19,760 --> 00:34:21,370 values of an impulse. 487 00:34:21,370 --> 00:34:24,855 Namely, the impulse response is equal to a step. 488 00:34:27,650 --> 00:34:34,010 So what we want to answer is, knowing what that impulse 489 00:34:34,010 --> 00:34:36,929 response is, what some properties are of the 490 00:34:36,929 --> 00:34:38,280 accumulator. 491 00:34:38,280 --> 00:34:41,420 And let's first ask about memory. 492 00:34:41,420 --> 00:34:46,679 Well, we recognize that the impulse response is not simply 493 00:34:46,679 --> 00:34:47,300 an impulse. 494 00:34:47,300 --> 00:34:48,830 In fact, it's a step. 495 00:34:48,830 --> 00:34:51,560 And so this implies what? 496 00:34:51,560 --> 00:34:55,530 Well, it implies that the system has memory. 497 00:35:00,050 --> 00:35:06,780 Second, the impulse response is 0 for n less than 0. 498 00:35:06,780 --> 00:35:11,180 That implies that the system is causal. 499 00:35:15,180 --> 00:35:22,530 And finally, if we look at the sum of the absolute values of 500 00:35:22,530 --> 00:35:24,930 the impulse response from -infinity to 501 00:35:24,930 --> 00:35:27,380 +infinity, this is a step. 502 00:35:27,380 --> 00:35:30,790 If we accumulate those values over infinite limits, then 503 00:35:30,790 --> 00:35:35,210 that in fact comes out to be infinite. 504 00:35:35,210 --> 00:35:40,140 And so what that implies, then, is that the accumulator 505 00:35:40,140 --> 00:35:43,490 is not stable in the bounded-input 506 00:35:43,490 --> 00:35:44,780 bounded-output sense. 507 00:35:48,240 --> 00:35:50,410 Now I want to turn to some other systems. 508 00:35:50,410 --> 00:35:52,630 But while we're on the accumulator, I just want to 509 00:35:52,630 --> 00:35:55,900 draw your attention to the fact, which will kind of come 510 00:35:55,900 --> 00:36:01,350 up in a variety of ways again later, that we can rewrite the 511 00:36:01,350 --> 00:36:04,840 equation for an accumulator, the difference equation, by 512 00:36:04,840 --> 00:36:10,460 recognizing that we could, in fact, write the output as the 513 00:36:10,460 --> 00:36:14,540 accumulated values up to time n - 1 and then 514 00:36:14,540 --> 00:36:17,120 add on the last value. 515 00:36:17,120 --> 00:36:23,510 And in fact, if we do that, this corresponds to y[n-1]. 516 00:36:23,510 --> 00:36:28,330 And so we could rewrite this difference equation as y[n] 517 00:36:28,330 --> 00:36:30,090 = y[n-1] 518 00:36:30,090 --> 00:36:31,340 + x[n]. 519 00:36:31,340 --> 00:36:34,630 So the output is the previously-computed output 520 00:36:34,630 --> 00:36:36,800 plus the input. 521 00:36:36,800 --> 00:36:42,170 Expressed that way, what that corresponds to is what is 522 00:36:42,170 --> 00:36:44,400 called a recursive difference equation. 523 00:36:44,400 --> 00:36:47,980 And different equations will be a topic of considerable 524 00:36:47,980 --> 00:36:51,290 emphasis in the next lecture. 525 00:36:51,290 --> 00:36:53,520 Now, does an accumulator have an inverse? 526 00:36:53,520 --> 00:36:57,100 Well, the answer is, in fact, yes. 527 00:36:57,100 --> 00:37:02,190 And let's look at what the inverse of the accumulator is. 528 00:37:02,190 --> 00:37:06,290 The impulse response of the accumulator is a step. 529 00:37:06,290 --> 00:37:11,100 To inquire about its inverse, we inquire about whether 530 00:37:11,100 --> 00:37:15,780 there's a system, which when we cascade the accumulator 531 00:37:15,780 --> 00:37:20,510 with that system, which I'm calling its inverse, we get an 532 00:37:20,510 --> 00:37:22,410 impulse out. 533 00:37:22,410 --> 00:37:23,800 Well, let's see. 534 00:37:23,800 --> 00:37:28,060 The impulse response of the accumulator is a step. 535 00:37:28,060 --> 00:37:29,820 We want to put the step into something 536 00:37:29,820 --> 00:37:32,470 and get out an impulse. 537 00:37:32,470 --> 00:37:37,500 And in fact, what you recall from the lecture in which we 538 00:37:37,500 --> 00:37:42,120 introduced steps and impulses, the impulse is, in fact, the 539 00:37:42,120 --> 00:37:44,630 first difference of the units step. 540 00:37:44,630 --> 00:37:50,210 So we have a difference equation that describes for us 541 00:37:50,210 --> 00:37:54,660 how the impulse is related to the step. 542 00:37:54,660 --> 00:37:59,770 And so if this system does this, the output will be that, 543 00:37:59,770 --> 00:38:01,420 an impulse. 544 00:38:01,420 --> 00:38:04,465 And so if we think of x_2[n] 545 00:38:07,580 --> 00:38:09,470 as the input and y_2[n] 546 00:38:09,470 --> 00:38:14,410 as the output, then the difference equation for the 547 00:38:14,410 --> 00:38:18,330 inverse system is what I've indicated here. 548 00:38:18,330 --> 00:38:22,200 And if we want to look at the impulse response of that, we 549 00:38:22,200 --> 00:38:25,120 can then inquire as to what the response is with an 550 00:38:25,120 --> 00:38:26,830 impulse in. 551 00:38:26,830 --> 00:38:31,290 And what develops in a straightforward way then is 552 00:38:31,290 --> 00:38:37,580 delta[n], which is our impulse input, minus delta[n-1] 553 00:38:37,580 --> 00:38:40,530 is equal to the impulse response 554 00:38:40,530 --> 00:38:42,730 of the inverse system. 555 00:38:42,730 --> 00:38:45,260 So I'll write that as h^(-1)[n] 556 00:38:45,260 --> 00:38:47,410 (h-inverse of n). 557 00:38:47,410 --> 00:38:52,420 Now, we have then that the accumulator has an inverse. 558 00:38:52,420 --> 00:38:54,420 And this is the inverse. 559 00:38:54,420 --> 00:38:57,910 And you can examine issues of memory, stability, 560 00:38:57,910 --> 00:38:59,430 causality, et cetera. 561 00:38:59,430 --> 00:39:05,170 What you'll find is that the system has memory, the inverse 562 00:39:05,170 --> 00:39:05,810 accumulator. 563 00:39:05,810 --> 00:39:09,360 It's stable, and it's causal. 564 00:39:09,360 --> 00:39:12,360 And it's interesting to note, by the way, that although the 565 00:39:12,360 --> 00:39:17,620 accumulator was an unstable system, the inverse of the 566 00:39:17,620 --> 00:39:19,930 accumulator is a stable system. 567 00:39:19,930 --> 00:39:24,680 In general, if the system is stable, its inverse does not 568 00:39:24,680 --> 00:39:26,940 have to be stable or vice versa. 569 00:39:26,940 --> 00:39:28,365 And the same thing with causality. 570 00:39:32,210 --> 00:39:38,360 OK now, there are a number of other examples, which, of 571 00:39:38,360 --> 00:39:40,200 course, we could discuss. 572 00:39:40,200 --> 00:39:46,830 And let me just quickly point to one example, which is a 573 00:39:46,830 --> 00:39:50,640 difference equation, as I've indicated here. 574 00:39:50,640 --> 00:39:55,980 And as we'll talk about in more detail in our next 575 00:39:55,980 --> 00:39:59,480 lecture, where we'll get involved in a fairly detailed 576 00:39:59,480 --> 00:40:01,360 discussion of linear constant-coefficient 577 00:40:01,360 --> 00:40:04,690 difference and differential equations, this falls into 578 00:40:04,690 --> 00:40:06,290 that category. 579 00:40:06,290 --> 00:40:10,960 And under the imposition of what's referred to as initial 580 00:40:10,960 --> 00:40:17,970 rest, which corresponds to the response being 0 up until the 581 00:40:17,970 --> 00:40:22,450 time that the input becomes non-zero, the impulse response 582 00:40:22,450 --> 00:40:25,410 is a^n times u[n]. 583 00:40:25,410 --> 00:40:29,490 And something that you'll be asked to think about in the 584 00:40:29,490 --> 00:40:33,490 video course manual is whether that system has memory, 585 00:40:33,490 --> 00:40:36,550 whether it's causal, and whether it's stable. 586 00:40:36,550 --> 00:40:41,700 And likewise, for a linear constant coefficient 587 00:40:41,700 --> 00:40:46,070 differential equation, the specific one that I've 588 00:40:46,070 --> 00:40:50,020 indicated here, under the assumption of initial rest, 589 00:40:50,020 --> 00:40:54,710 the impulse response is e^(-2t) times u(t). 590 00:40:54,710 --> 00:40:58,810 And in the video course manual again, you'll be asked to 591 00:40:58,810 --> 00:41:02,930 examine whether the system has memory, whether it's causal, 592 00:41:02,930 --> 00:41:06,410 and whether it's stable. 593 00:41:06,410 --> 00:41:10,420 OK well, as I've indicated, in the next lecture we'll return 594 00:41:10,420 --> 00:41:13,270 to a much more detailed discussion of linear 595 00:41:13,270 --> 00:41:15,180 constant-coefficient differential 596 00:41:15,180 --> 00:41:17,190 and difference equations. 597 00:41:17,190 --> 00:41:22,480 Now, what I'd like to finally do in this lecture is use the 598 00:41:22,480 --> 00:41:26,410 notion of convolution in a much different way to help us 599 00:41:26,410 --> 00:41:30,030 with a problem that I alluded to earlier. 600 00:41:30,030 --> 00:41:33,800 In particular, the issue of how to deal with some of the 601 00:41:33,800 --> 00:41:37,200 mathematical difficulties associated with 602 00:41:37,200 --> 00:41:39,620 impulses and steps. 603 00:41:39,620 --> 00:41:43,540 Now, let me begin by illustrating kind of what the 604 00:41:43,540 --> 00:41:49,845 problem is and an example of the kind of paradox that you 605 00:41:49,845 --> 00:41:53,480 sort of run into when dealing with impulse functions and 606 00:41:53,480 --> 00:41:54,730 step functions. 607 00:41:57,000 --> 00:42:00,980 Let's consider, first of all, a system, which is the 608 00:42:00,980 --> 00:42:02,520 identity system. 609 00:42:02,520 --> 00:42:07,090 And so the output is, of course, equal to the input. 610 00:42:07,090 --> 00:42:12,180 And again, we can talk about that either in continuous time 611 00:42:12,180 --> 00:42:14,800 or in discrete time. 612 00:42:14,800 --> 00:42:18,990 Well, we know that the function that you convolve 613 00:42:18,990 --> 00:42:23,610 with a signal to retain the signal is an impulse. 614 00:42:23,610 --> 00:42:26,650 And so that means that the impulse response of an 615 00:42:26,650 --> 00:42:28,740 identity system is an impulse. 616 00:42:28,740 --> 00:42:32,220 Makes logical sense. 617 00:42:32,220 --> 00:42:36,300 Furthermore, if I take two identity systems and cascade 618 00:42:36,300 --> 00:42:40,240 them, I put in an input, get the same thing out of the 619 00:42:40,240 --> 00:42:40,900 first system. 620 00:42:40,900 --> 00:42:42,190 That goes into the second system. 621 00:42:42,190 --> 00:42:44,125 Get the same thing out of the second. 622 00:42:44,125 --> 00:42:51,540 In other words, if I have two identity systems in cascade, 623 00:42:51,540 --> 00:42:55,840 the cascade, likewise, is an identity system. 624 00:42:55,840 --> 00:42:59,020 In other words, this overall system is 625 00:42:59,020 --> 00:43:03,280 also an identity system. 626 00:43:03,280 --> 00:43:07,610 And the implication there is that the impulse response of 627 00:43:07,610 --> 00:43:09,110 this is an impulse. 628 00:43:09,110 --> 00:43:11,750 The impulse response of this is an impulse. 629 00:43:11,750 --> 00:43:16,560 And the convolution of those two is also an impulse. 630 00:43:16,560 --> 00:43:21,860 So for continuous time, we require, then, that an impulse 631 00:43:21,860 --> 00:43:24,550 convolved with itself is an impulse. 632 00:43:24,550 --> 00:43:28,850 And the same thing for discrete time. 633 00:43:28,850 --> 00:43:33,280 Now, in discrete time, we don't have any particular 634 00:43:33,280 --> 00:43:34,220 problem with that. 635 00:43:34,220 --> 00:43:37,130 If you think about convolving these together, it's a 636 00:43:37,130 --> 00:43:42,850 straightforward mathematical operation since the impulse in 637 00:43:42,850 --> 00:43:47,300 discrete time is very nicely defined. 638 00:43:47,300 --> 00:43:51,070 However, in continuous time, we have to be somewhat careful 639 00:43:51,070 --> 00:43:54,710 about the definition of the impulse because it was the 640 00:43:54,710 --> 00:43:55,870 derivative of a step. 641 00:43:55,870 --> 00:43:57,930 A step has a discontinuity. 642 00:43:57,930 --> 00:44:01,190 You can't really differentiate at a discontinuity. 643 00:44:01,190 --> 00:44:06,020 And the way that we dealt with that was to expand out the 644 00:44:06,020 --> 00:44:09,570 discontinuity so that it had some finite time region in 645 00:44:09,570 --> 00:44:11,090 which it happened. 646 00:44:11,090 --> 00:44:14,080 When we did that, we ended up with a definition for the 647 00:44:14,080 --> 00:44:18,200 impulse, which was the limiting form of this 648 00:44:18,200 --> 00:44:23,510 function, which is a rectangle of width Delta, and height 1 / 649 00:44:23,510 --> 00:44:27,120 Delta, and an area equal to 1. 650 00:44:27,120 --> 00:44:33,680 Now, if we think of convolving this signal with itself, the 651 00:44:33,680 --> 00:44:37,480 impulse being the limiting form of this, then the 652 00:44:37,480 --> 00:44:42,690 convolution of this with itself is a triangle of width 653 00:44:42,690 --> 00:44:47,290 2 Delta, height 1 / Delta, and area 1. 654 00:44:47,290 --> 00:44:51,560 In other words, this triangular function is this 655 00:44:51,560 --> 00:44:59,030 approximation delta_Delta(t) convolved with delta_Delta(t). 656 00:44:59,030 --> 00:45:05,000 And since the limit of this would correspond to the 657 00:45:05,000 --> 00:45:11,480 impulse response of the identity system convolved with 658 00:45:11,480 --> 00:45:17,470 itself, the implication is that not only should the top 659 00:45:17,470 --> 00:45:23,600 function, this one, correspond in its limiting form to an 660 00:45:23,600 --> 00:45:28,130 impulse, but also this should correspond in its limiting 661 00:45:28,130 --> 00:45:29,990 form to an impulse. 662 00:45:29,990 --> 00:45:32,750 So one could wonder well, what is an impulse? 663 00:45:32,750 --> 00:45:34,320 Is it this one in the limit? 664 00:45:34,320 --> 00:45:35,650 Or is it this one in the limit? 665 00:45:42,880 --> 00:45:45,420 Now, beyond that-- so kind of what this suggests is that in 666 00:45:45,420 --> 00:45:48,360 the limiting form, you kind of run into a contradiction 667 00:45:48,360 --> 00:45:50,920 unless you don't try to distinguish between this 668 00:45:50,920 --> 00:45:53,170 rectangle and the triangle. 669 00:45:53,170 --> 00:45:57,050 Things get even worse when you think about what happens when 670 00:45:57,050 --> 00:45:59,160 you put an impulse into a differentiator. 671 00:45:59,160 --> 00:46:03,630 And a differentiator is a very commonly occurring system. 672 00:46:03,630 --> 00:46:08,350 In particular, suppose we had a system for which the output 673 00:46:08,350 --> 00:46:11,010 was the derivative of the input. 674 00:46:11,010 --> 00:46:17,500 So if we put in x(t), we got out dx(t) / dt. 675 00:46:17,500 --> 00:46:20,360 If I put in an impulse, or if I talked about the impulse 676 00:46:20,360 --> 00:46:23,130 response, what is that? 677 00:46:23,130 --> 00:46:27,610 And of course, the problem is that if you think that the 678 00:46:27,610 --> 00:46:31,880 impulse itself is very badly behaved, then what about its 679 00:46:31,880 --> 00:46:36,110 derivative, which is not only infinitely big, but there's a 680 00:46:36,110 --> 00:46:38,350 positive-going one, and a negative-going one, and the 681 00:46:38,350 --> 00:46:40,440 difference between there has some area. 682 00:46:40,440 --> 00:46:42,910 And you end up in a lot of difficulty. 683 00:46:45,930 --> 00:46:51,250 Well, the way around this, formally, is through a set of 684 00:46:51,250 --> 00:46:56,140 mathematics referred to as generalized functions. 685 00:46:56,140 --> 00:46:58,110 We won't be quite that formal. 686 00:46:58,110 --> 00:47:01,340 But I'd like to, at least, suggest what the essence of 687 00:47:01,340 --> 00:47:02,530 that formality is. 688 00:47:02,530 --> 00:47:07,360 And it really helps us in interpreting the impulses in 689 00:47:07,360 --> 00:47:09,620 steps and functions of that type. 690 00:47:09,620 --> 00:47:13,660 And what it is is an operational definition of 691 00:47:13,660 --> 00:47:16,560 steps, impulses, and their derivatives in 692 00:47:16,560 --> 00:47:18,360 the following sense. 693 00:47:18,360 --> 00:47:21,600 Usually when we talk about a function, we talk about what 694 00:47:21,600 --> 00:47:24,910 the value of the function is at any instant of time. 695 00:47:24,910 --> 00:47:26,810 And of course, the trouble with an impulse is it's 696 00:47:26,810 --> 00:47:30,930 infinitely big, in zero width, and has some area, et cetera. 697 00:47:30,930 --> 00:47:35,310 What we can turn to is what is referred to as an operational 698 00:47:35,310 --> 00:47:41,550 definition where the operational definition is 699 00:47:41,550 --> 00:47:45,790 related not to what the impulse is, but to what the 700 00:47:45,790 --> 00:47:50,760 impulse does under the operation of convolution. 701 00:47:50,760 --> 00:47:52,540 So what is an impulse? 702 00:47:52,540 --> 00:47:56,110 An impulse is something, which under 703 00:47:56,110 --> 00:47:59,460 convolution, retains the function. 704 00:47:59,460 --> 00:48:04,200 And that then can serve as a definition of the impulse. 705 00:48:04,200 --> 00:48:06,410 Well, let's see where that gets us. 706 00:48:06,410 --> 00:48:11,060 Suppose that we now want to talk about the derivative of 707 00:48:11,060 --> 00:48:13,370 the impulse. 708 00:48:13,370 --> 00:48:18,970 Well, what we ask about is what it is operationally. 709 00:48:18,970 --> 00:48:25,510 And so if we have a system, which is a differentiator, and 710 00:48:25,510 --> 00:48:28,600 we inquire about its impulse response, which let's say we 711 00:48:28,600 --> 00:48:32,630 define notationally as u_1(t). 712 00:48:32,630 --> 00:48:37,650 What's important about this function u_1(t) is not what it 713 00:48:37,650 --> 00:48:42,620 is at each value of time but what it does under 714 00:48:42,620 --> 00:48:43,840 convolution. 715 00:48:43,840 --> 00:48:45,690 What does it do under convolution? 716 00:48:45,690 --> 00:48:49,610 Well, the output of the differentiator is the 717 00:48:49,610 --> 00:48:53,190 convolution of the input with the impulse response. 718 00:48:53,190 --> 00:48:58,100 And so what u_1(t) does under convolution is to 719 00:48:58,100 --> 00:48:59,780 differentiate. 720 00:48:59,780 --> 00:49:03,770 And that is the operational definition. 721 00:49:03,770 --> 00:49:07,450 And now, of course, we can think of extending that. 722 00:49:07,450 --> 00:49:11,540 Not only would we want to think about differentiating an 723 00:49:11,540 --> 00:49:15,960 impulse, but we would also want to think about 724 00:49:15,960 --> 00:49:18,740 differentiating the derivative of an impulse. 725 00:49:18,740 --> 00:49:23,000 We'll define that as a function u_2(t). 726 00:49:23,000 --> 00:49:24,690 u_2(t)-- 727 00:49:24,690 --> 00:49:28,050 because we have this impulse response convolved with this 728 00:49:28,050 --> 00:49:31,700 one is u_1(t) * u_1(t). 729 00:49:31,700 --> 00:49:36,800 And what is u_2(t) operationally? 730 00:49:36,800 --> 00:49:42,800 It is the operation such that when you convolve that with 731 00:49:42,800 --> 00:49:48,540 x(t), what you get is the second derivative. 732 00:49:48,540 --> 00:49:52,110 OK now, we can carry this further and, in fact, talk 733 00:49:52,110 --> 00:49:59,410 about the result of convolving u_1(t) with itself more times. 734 00:49:59,410 --> 00:50:03,870 In fact, if we think of the convulution of u_1(t) with 735 00:50:03,870 --> 00:50:07,460 itself k times, then logically we would 736 00:50:07,460 --> 00:50:11,330 define that as u_k(t). 737 00:50:11,330 --> 00:50:14,800 Again, we would interpret that operationally. 738 00:50:14,800 --> 00:50:20,440 And the operational definition is through convolution, where 739 00:50:20,440 --> 00:50:26,600 this corresponds to u_k(t) being the impulse response of 740 00:50:26,600 --> 00:50:29,630 k differentiators in cascade. 741 00:50:29,630 --> 00:50:32,090 So what is the operational definition? 742 00:50:32,090 --> 00:50:39,800 Well, it's simply that x(t) * u_k(t) is the k 743 00:50:39,800 --> 00:50:42,440 derivative of x(t). 744 00:50:45,080 --> 00:50:49,480 And this now gives us a set of what are referred to as 745 00:50:49,480 --> 00:50:50,720 singularity functions. 746 00:50:50,720 --> 00:50:55,360 Very badly behaved mathematically in a sense, but 747 00:50:55,360 --> 00:50:58,700 as we've seen, reasonably well defined under an operational 748 00:50:58,700 --> 00:51:00,780 definition. 749 00:51:00,780 --> 00:51:06,010 With k = 0, incidentally, that's the same as what we 750 00:51:06,010 --> 00:51:08,560 have referred to previously as the impulse. 751 00:51:08,560 --> 00:51:13,260 So with k 0, that's just delta(t). 752 00:51:13,260 --> 00:51:17,150 Now to be complete, we can also go the other way and talk 753 00:51:17,150 --> 00:51:20,940 about the impulse response of a string of integrators 754 00:51:20,940 --> 00:51:23,460 instead of a string of differentiators. 755 00:51:23,460 --> 00:51:25,280 Of course, the impulse response of a single 756 00:51:25,280 --> 00:51:27,580 integrator is a unit step. 757 00:51:27,580 --> 00:51:30,120 Two integrators together is the integral of a 758 00:51:30,120 --> 00:51:32,680 unit step, et cetera. 759 00:51:32,680 --> 00:51:38,110 And that, likewise, corresponds to a set of what 760 00:51:38,110 --> 00:51:40,290 are called singularity functions. 761 00:51:40,290 --> 00:51:46,410 In particular, if I take a string of m integrators in 762 00:51:46,410 --> 00:51:52,610 cascade, then the impulse response of that is denoted as 763 00:51:52,610 --> 00:51:56,000 u sub minus m of t. 764 00:51:56,000 --> 00:52:00,380 And for example, with a single integrator, u sub minus 1 of t 765 00:52:00,380 --> 00:52:06,950 corresponds to our unit step as we talked about previously. 766 00:52:06,950 --> 00:52:14,430 u sub minus 2 of t corresponds to a unit ramp, et cetera. 767 00:52:14,430 --> 00:52:18,060 And there is, in fact, a reason for choosing negative 768 00:52:18,060 --> 00:52:22,850 values of the argument when going in one direction near 769 00:52:22,850 --> 00:52:25,370 integration as compared with positive values of the 770 00:52:25,370 --> 00:52:28,130 argument when going in the other direction, namely 771 00:52:28,130 --> 00:52:30,120 differentiation. 772 00:52:30,120 --> 00:52:38,110 In particular, we know that with u sub minus m of t, the 773 00:52:38,110 --> 00:52:44,520 operational definition is the mth running integral. 774 00:52:44,520 --> 00:52:48,370 And likewise, u_k(t)-- 775 00:52:48,370 --> 00:52:51,040 so with a positive sub script-- 776 00:52:51,040 --> 00:52:56,240 has an operational definition, which is the derivative. 777 00:52:56,240 --> 00:53:01,290 So it's the kth derivative of x(t). 778 00:53:01,290 --> 00:53:08,070 And partly as a consequence of that, if we take u_k(t) and 779 00:53:08,070 --> 00:53:13,400 convolve it with u_l(t), the result is the singularity 780 00:53:13,400 --> 00:53:17,170 function with the subscript, which is the sum of k and l. 781 00:53:17,170 --> 00:53:21,470 And that holds whether this is positive values of the 782 00:53:21,470 --> 00:53:24,610 subscript or negative values of the subscript. 783 00:53:24,610 --> 00:53:28,550 So just to summarize this last discussion, we've used an 784 00:53:28,550 --> 00:53:35,520 operational definition to talk about derivatives of impulses 785 00:53:35,520 --> 00:53:37,940 and integrals of impulses. 786 00:53:37,940 --> 00:53:40,705 This led to a set of singularity functions-- what 787 00:53:40,705 --> 00:53:42,270 I've called singularity functions-- 788 00:53:42,270 --> 00:53:45,400 of which the impulse and the step are two examples. 789 00:53:45,400 --> 00:53:49,700 But using an operational definition through convolution 790 00:53:49,700 --> 00:53:55,500 allows us to define, at least in an operational sense, these 791 00:53:55,500 --> 00:53:57,880 functions that otherwise are very badly behaved. 792 00:54:01,110 --> 00:54:06,330 OK now, in this lecture and previous lectures, for the 793 00:54:06,330 --> 00:54:10,890 most part, our discussion has been about linear 794 00:54:10,890 --> 00:54:14,910 time-invariant systems in fairly general terms. 795 00:54:14,910 --> 00:54:18,480 And we've seen a variety of properties, representation 796 00:54:18,480 --> 00:54:22,390 through convolution, and properties as they can be 797 00:54:22,390 --> 00:54:25,750 associated with the impulse response. 798 00:54:25,750 --> 00:54:28,970 In the next lecture, we'll turn our attention to a very 799 00:54:28,970 --> 00:54:34,150 important subclass of those systems, namely systems that 800 00:54:34,150 --> 00:54:37,410 are describable by linear constant-coefficient 801 00:54:37,410 --> 00:54:40,640 difference equations in the discrete-time case, and linear 802 00:54:40,640 --> 00:54:43,540 constant-coefficient differential equations in the 803 00:54:43,540 --> 00:54:45,410 continuous-time case. 804 00:54:45,410 --> 00:54:50,800 Those classes, while not forming all of the class of 805 00:54:50,800 --> 00:54:53,080 linear time-invariant systems, are a very 806 00:54:53,080 --> 00:54:54,930 important sub class. 807 00:54:54,930 --> 00:54:57,470 And we'll focus in on those specifically next time. 808 00:54:57,470 --> 00:54:58,720 Thank you.