1 00:00:00,000 --> 00:00:00,690 2 00:00:00,690 --> 00:00:04,179 In this video, I'm going to introduce you to the concept 3 00:00:04,179 --> 00:00:09,949 of the convolution, one of the first times a mathematician's 4 00:00:09,949 --> 00:00:11,809 actually named something similar to what 5 00:00:11,810 --> 00:00:12,589 it's actually doing. 6 00:00:12,589 --> 00:00:14,589 You're actually convoluting the functions. 7 00:00:14,589 --> 00:00:16,969 And in this video, I'm not going to dive into the 8 00:00:16,969 --> 00:00:20,829 intuition of the convolution, because there's a lot of 9 00:00:20,829 --> 00:00:21,769 different ways you can look at it. 10 00:00:21,769 --> 00:00:23,429 It has a lot of different applications, and if you 11 00:00:23,429 --> 00:00:26,320 become an engineer really of any kind, you're going to see 12 00:00:26,320 --> 00:00:29,949 the convolution in kind of a discreet form and a continuous 13 00:00:29,949 --> 00:00:31,619 form, and a bunch of different ways. 14 00:00:31,620 --> 00:00:35,620 But in this video I just want to make you comfortable with 15 00:00:35,619 --> 00:00:39,099 the idea of a convolution, especially in the context of 16 00:00:39,100 --> 00:00:41,289 taking Laplace transforms. 17 00:00:41,289 --> 00:00:44,759 So the convolution theorem-- well, actually, before I even 18 00:00:44,759 --> 00:00:46,969 go to the convolution theorem, let me define what a 19 00:00:46,969 --> 00:00:48,469 convolution is. 20 00:00:48,469 --> 00:00:52,100 So let's say that I have some function f of t. 21 00:00:52,100 --> 00:00:58,983 So if I convolute f with g-- so this means that I'm going 22 00:00:58,982 --> 00:01:02,250 to take the convolution of f and g, and this is going to be 23 00:01:02,250 --> 00:01:04,219 a function of t. 24 00:01:04,219 --> 00:01:06,609 And so far, nothing I've written should make any sense 25 00:01:06,609 --> 00:01:08,790 to you, because I haven't defined what this means. 26 00:01:08,790 --> 00:01:11,100 This is like those SAT problems where they say, like, 27 00:01:11,099 --> 00:01:18,059 you know, a triangle b means a plus b over 3, while you're 28 00:01:18,060 --> 00:01:19,840 standing on one leg or something like that. 29 00:01:19,840 --> 00:01:23,000 So I need to define this in some similar way. 30 00:01:23,000 --> 00:01:26,454 So let me undo this silliness that I just wrote there. 31 00:01:26,454 --> 00:01:28,959 And the definition of a convolution, we're going to do 32 00:01:28,959 --> 00:01:32,199 it over a-- well, there's several definitions you'll 33 00:01:32,200 --> 00:01:34,140 see, but the definition we're going to use in this, context 34 00:01:34,140 --> 00:01:36,150 there's actually one other definition you'll see in the 35 00:01:36,150 --> 00:01:45,290 continuous case, is the integral from 0 to t of f of t 36 00:01:45,290 --> 00:01:56,195 minus tau, times g of t-- let me just write it-- sorry, it's 37 00:01:56,194 --> 00:02:00,399 times g of tau d tau. 38 00:02:00,400 --> 00:02:04,969 Now, this might seem like a very bizarro thing to do, and 39 00:02:04,969 --> 00:02:07,560 you're like, Sal, how do I even compute 40 00:02:07,560 --> 00:02:08,520 one of these things? 41 00:02:08,520 --> 00:02:11,300 And to kind of give you that comfort, let's actually 42 00:02:11,300 --> 00:02:12,550 compute a convolution. 43 00:02:12,550 --> 00:02:15,790 44 00:02:15,789 --> 00:02:19,090 Actually, it was hard to find some functions that are very 45 00:02:19,090 --> 00:02:21,344 easy to analytically compute, and you're going to find that 46 00:02:21,344 --> 00:02:23,789 we're going to go into a lot of trig identities to actually 47 00:02:23,789 --> 00:02:24,859 compute this. 48 00:02:24,860 --> 00:02:31,680 But if I say that f of t, if I define f of t to be equal to 49 00:02:31,680 --> 00:02:36,980 the sine of t, and I define cosine of t-- let me do it in 50 00:02:36,979 --> 00:02:43,810 orange-- or I define g of t to be equal to the cosine of t. 51 00:02:43,810 --> 00:02:46,930 Now let's convolute the two functions. 52 00:02:46,930 --> 00:02:54,310 So the convolution of f with g, and this is going to be a 53 00:02:54,310 --> 00:02:59,000 function of t, it equals this. 54 00:02:59,000 --> 00:03:01,490 I'm just going to show you how to apply this integral. 55 00:03:01,490 --> 00:03:05,040 So it equals the integral-- I'll do it in purple-- the 56 00:03:05,039 --> 00:03:09,120 integral from 0 to t of f of t minus tau. 57 00:03:09,120 --> 00:03:10,090 This is my f of t. 58 00:03:10,090 --> 00:03:17,810 So it's is going to be sine of t minus tau times g of tau. 59 00:03:17,810 --> 00:03:22,500 Well, this is my g of t, so g of tau is cosine of tau, 60 00:03:22,500 --> 00:03:27,699 cosine of tau d tau. 61 00:03:27,699 --> 00:03:30,659 So that's the integral, and now to evaluate it, we're 62 00:03:30,659 --> 00:03:33,000 going to have to break out some trigonometry. 63 00:03:33,000 --> 00:03:34,759 So let's do that. 64 00:03:34,759 --> 00:03:37,099 This almost is just a very good trigonometry and 65 00:03:37,099 --> 00:03:38,509 integration review. 66 00:03:38,509 --> 00:03:39,509 So let's evaluate this. 67 00:03:39,509 --> 00:03:41,469 But I wanted to evaluate this in this video because I want 68 00:03:41,469 --> 00:03:43,359 to show you that this isn't some abstract thing, that you 69 00:03:43,360 --> 00:03:45,730 can actually evaluate these functions. 70 00:03:45,729 --> 00:03:47,834 So the first thing I want to do-- I mean, I don't know what 71 00:03:47,835 --> 00:03:49,150 the antiderivative of this is. 72 00:03:49,150 --> 00:03:51,450 It's tempting, you see a sine and a cosine, maybe they're 73 00:03:51,449 --> 00:03:52,799 the derivatives of each other, but this is the 74 00:03:52,800 --> 00:03:55,830 sine of t minus tau. 75 00:03:55,830 --> 00:03:58,980 So let me rewrite that sine of t minus tau, and we'll just 76 00:03:58,979 --> 00:04:05,209 use the trig identity, that the sine of t minus tau is 77 00:04:05,210 --> 00:04:15,010 just equal to the sine of t times the cosine of tau minus 78 00:04:15,009 --> 00:04:18,990 the sine of tau times the cosine of t. 79 00:04:18,990 --> 00:04:21,709 And actually, I just made a video where I go through all 80 00:04:21,709 --> 00:04:26,000 of these trig identities really just to review them for 81 00:04:26,000 --> 00:04:28,560 myself and actually to make a video in better quality on 82 00:04:28,560 --> 00:04:29,420 them as well. 83 00:04:29,420 --> 00:04:31,560 So if we make this subsitution, this you'll find 84 00:04:31,560 --> 00:04:35,389 on the inside cover of any trigonometry or calculus book, 85 00:04:35,389 --> 00:04:39,175 you get the convolution of f and g is equal to-- I'll just 86 00:04:39,175 --> 00:04:42,730 write that f-star g; I'll just write it with that-- is equal 87 00:04:42,730 --> 00:04:48,660 to the integral from 0 to t of, instead of sine of t minus 88 00:04:48,660 --> 00:04:50,780 tau, I'm going to write this thing right there. 89 00:04:50,779 --> 00:04:56,969 So I'm going to write the sine of t times the cosine of tau 90 00:04:56,970 --> 00:05:04,420 minus the sine of tau times the cosine of t, and then all 91 00:05:04,420 --> 00:05:06,355 of that's times the cosine of tau. 92 00:05:06,355 --> 00:05:09,450 93 00:05:09,449 --> 00:05:12,969 I have to be careful with my taus and t's, and let's see, t 94 00:05:12,970 --> 00:05:14,770 and tau, tau and t. 95 00:05:14,769 --> 00:05:16,490 Everything's working so far. 96 00:05:16,490 --> 00:05:19,810 So let's see, so then that's dt. 97 00:05:19,810 --> 00:05:20,860 Oh, sorry, d tau. 98 00:05:20,860 --> 00:05:22,949 Let me be very careful here. 99 00:05:22,949 --> 00:05:26,459 Now let's distribute this cosine of tau out, 100 00:05:26,459 --> 00:05:27,879 and what do we get? 101 00:05:27,879 --> 00:05:32,639 We get this is equal to-- so f convoluted with g, I guess we 102 00:05:32,639 --> 00:05:38,709 call it f-star g, is equal to the integral from 0 to t of 103 00:05:38,709 --> 00:05:43,560 sine of t times cosine of tau times cosine of tau. 104 00:05:43,560 --> 00:05:44,620 I'm just distributing this cosine of tau. 105 00:05:44,620 --> 00:05:51,449 So it's cosine squared of tau, and then minus-- let's rewrite 106 00:05:51,449 --> 00:05:53,870 the cosine of t first, and I'm doing that because we're 107 00:05:53,870 --> 00:05:55,680 integrating with respect to tau. 108 00:05:55,680 --> 00:05:58,910 So I'm just going to write my cosine of t first. So cosine 109 00:05:58,910 --> 00:06:09,950 of t times sine of tau times the cosine of tau d tau. 110 00:06:09,949 --> 00:06:13,420 And now, since we're taking the integral of really two 111 00:06:13,420 --> 00:06:15,530 things subtracting from each other, let's just turn this 112 00:06:15,529 --> 00:06:17,859 into two separate integrals. 113 00:06:17,860 --> 00:06:24,500 So this is equal to the integral from 0 to t, of sine 114 00:06:24,500 --> 00:06:32,009 of t, times the cosine squared of tau d tau minus the 115 00:06:32,009 --> 00:06:40,709 integral from 0 to t of cosine of t times sine of tau cosine 116 00:06:40,709 --> 00:06:43,909 of tau d tau. 117 00:06:43,910 --> 00:06:45,390 Now, what can we do? 118 00:06:45,389 --> 00:06:48,550 Well, to simplify it more, remember, we're integrating 119 00:06:48,550 --> 00:06:51,329 with respect to-- let me be careful here. 120 00:06:51,329 --> 00:06:53,500 We're integrating with respect to tau. 121 00:06:53,500 --> 00:06:54,699 I wrote a t there. 122 00:06:54,699 --> 00:06:57,319 We're integrating with respect to tau. 123 00:06:57,319 --> 00:06:59,579 So all of these, this cosine of t right 124 00:06:59,579 --> 00:07:01,479 here, that's a constant. 125 00:07:01,480 --> 00:07:02,650 The sine of t is a constant. 126 00:07:02,649 --> 00:07:04,319 For all I know, t could be equal to 5. 127 00:07:04,319 --> 00:07:06,189 It doesn't matter that one of the boundaries of our 128 00:07:06,189 --> 00:07:07,519 integration is also a t. 129 00:07:07,519 --> 00:07:09,609 That t would be a 5, in which case these 130 00:07:09,610 --> 00:07:10,730 are all just constants. 131 00:07:10,730 --> 00:07:13,730 We're integrating only with respect to the tau, so if 132 00:07:13,730 --> 00:07:17,500 cosine of 5, that's a constant, we can take it out 133 00:07:17,500 --> 00:07:17,959 of the integral. 134 00:07:17,959 --> 00:07:27,259 So this is equal to sine of t times the integral from 0 to t 135 00:07:27,259 --> 00:07:38,360 of cosine squared of tau d tau and then minus cosine of t-- 136 00:07:38,360 --> 00:07:41,500 that's just a constant; I'm bringing it out-- times the 137 00:07:41,500 --> 00:07:52,870 integral from 0 to t of sine of tau cosine of tau d tau. 138 00:07:52,870 --> 00:07:55,269 Now, this antiderivative is pretty straightforward. 139 00:07:55,269 --> 00:07:57,229 You could do u substitution. 140 00:07:57,230 --> 00:07:59,470 Let me do it here, instead of doing it in our heads. 141 00:07:59,470 --> 00:08:00,960 This is a complicated problem, so we don't 142 00:08:00,959 --> 00:08:02,259 want to skip steps. 143 00:08:02,259 --> 00:08:10,259 If we said u is equal to sine of tau, then du d tau is equal 144 00:08:10,259 --> 00:08:14,319 to the cosine of tau, just the derivative of sine. 145 00:08:14,319 --> 00:08:18,180 Or we could write that du is equal to the 146 00:08:18,180 --> 00:08:23,170 cosine of tau d tau. 147 00:08:23,170 --> 00:08:28,439 148 00:08:28,439 --> 00:08:32,120 We'll undo the substitution before we evaluate the 149 00:08:32,120 --> 00:08:33,149 endpoints here. 150 00:08:33,149 --> 00:08:35,519 But this was a little bit more of a conundrum. 151 00:08:35,519 --> 00:08:37,370 I don't know how to take the antiderivative of cosine 152 00:08:37,370 --> 00:08:38,360 squared of tau. 153 00:08:38,360 --> 00:08:41,269 It's not obvious what that is. 154 00:08:41,269 --> 00:08:43,740 So to do this, we're going to break out some more 155 00:08:43,740 --> 00:08:45,600 trigonometric identities. 156 00:08:45,600 --> 00:08:48,250 And in a video I just recorded, it might not be the 157 00:08:48,250 --> 00:08:57,830 last video in the playlist, I showed that the cosine squared 158 00:08:57,830 --> 00:09:02,960 of tau-- I'm just using tau as an example-- is equal to 1/2 159 00:09:02,960 --> 00:09:12,660 times 1 plus the cosine of 2 tau. 160 00:09:12,659 --> 00:09:15,689 And once again, this is just a trig identity that you'll find 161 00:09:15,690 --> 00:09:19,250 really in the inside cover of probably your calculus book. 162 00:09:19,250 --> 00:09:22,870 So we can make this substitution here, make this 163 00:09:22,870 --> 00:09:26,350 substitution right there, and then let's see what our 164 00:09:26,350 --> 00:09:29,930 integrals become. 165 00:09:29,929 --> 00:09:33,279 So the first one over here, let me just write it here. 166 00:09:33,279 --> 00:09:42,519 We get sine of t times the integral from 0 to t of this 167 00:09:42,519 --> 00:09:45,059 thing here. 168 00:09:45,059 --> 00:09:50,054 Let me just take the 1/2 out, to keep things simple. 169 00:09:50,054 --> 00:09:51,799 So I'll put the 1/2 out here. 170 00:09:51,799 --> 00:09:52,979 That's this 1/2. 171 00:09:52,980 --> 00:10:00,080 So 1 plus cosine of 2 tau and all of that is d tau. 172 00:10:00,080 --> 00:10:02,610 173 00:10:02,610 --> 00:10:04,529 That's this integral right there. 174 00:10:04,529 --> 00:10:11,559 And then we have this integral right here, minus cosine of t 175 00:10:11,559 --> 00:10:15,929 times the integral from-- let me be very clear. 176 00:10:15,929 --> 00:10:25,599 This is tau is equal to 0 to tau is equal to t. 177 00:10:25,600 --> 00:10:27,960 And then this thing right here, I did some u 178 00:10:27,960 --> 00:10:28,509 subsitution. 179 00:10:28,509 --> 00:10:33,179 If u is equal to sine of t, then this becomes u. 180 00:10:33,179 --> 00:10:36,709 And we showed that du is equal to cosine-- sorry, u is equal 181 00:10:36,710 --> 00:10:38,060 to sine of tau. 182 00:10:38,059 --> 00:10:42,929 And then we showed that du is equal to cosine tau d tau, so 183 00:10:42,929 --> 00:10:46,009 this thing right here is equal to du. 184 00:10:46,009 --> 00:10:53,289 So it's u du, and let's see if we can do anything useful now. 185 00:10:53,289 --> 00:11:01,209 So this integral right here, the antiderivative of this is 186 00:11:01,210 --> 00:11:03,305 pretty straightforward, so what are we going to get? 187 00:11:03,304 --> 00:11:05,109 Let me write this outside part. 188 00:11:05,110 --> 00:11:10,060 So we have 1/2 times the sine of t. 189 00:11:10,059 --> 00:11:13,219 And now let me take the antiderivative of this. 190 00:11:13,220 --> 00:11:17,399 This is going to be tau plus the antiderivative of this. 191 00:11:17,399 --> 00:11:20,465 It's going to be 1/2 sine of 2 tau. 192 00:11:20,465 --> 00:11:24,870 193 00:11:24,870 --> 00:11:26,549 I mean, we could have done the u substitution. 194 00:11:26,549 --> 00:11:29,719 we could have said u is equal to 2 tau and all of that, but 195 00:11:29,720 --> 00:11:31,460 I think you could do that from recognition, and if you don't 196 00:11:31,460 --> 00:11:34,720 believe me, you just have to take the derivative of this. 197 00:11:34,720 --> 00:11:38,230 1/2 sine of 2 tau is the derivative of this. 198 00:11:38,230 --> 00:11:41,550 You multiply, you take the derivative of the inside, so 199 00:11:41,549 --> 00:11:44,839 that's 2, so the 2 and the 1/2 cancel out, and the derivative 200 00:11:44,840 --> 00:11:47,450 of the outside, so cosine of 2 tau. 201 00:11:47,450 --> 00:11:52,910 And you're going to evaluate that from 0 to t. 202 00:11:52,909 --> 00:11:58,549 And then we have minus cosine of t. 203 00:11:58,549 --> 00:12:01,549 When we take the antiderivative of this-- let 204 00:12:01,549 --> 00:12:02,689 me do this on the side. 205 00:12:02,690 --> 00:12:07,540 So the integral of u du, that's trivially easy. 206 00:12:07,539 --> 00:12:12,319 That's 1/2 u squared. 207 00:12:12,320 --> 00:12:15,940 Now, that's 1/2 u squared, but what was u to begin with? 208 00:12:15,940 --> 00:12:18,610 It was sine of tau. 209 00:12:18,610 --> 00:12:22,970 So the antiderivative of this thing right here is 1/2 u 210 00:12:22,970 --> 00:12:25,290 squared, but u is sine of tau. 211 00:12:25,289 --> 00:12:33,490 So it's going to be 1/2u, which is sine of tau squared. 212 00:12:33,490 --> 00:12:39,129 And we're going to evaluate that from 0 to t. 213 00:12:39,129 --> 00:12:41,120 And we didn't even have to do all this u substitution. 214 00:12:41,120 --> 00:12:43,600 The way I often do it in my head, I see the sine of tau, 215 00:12:43,600 --> 00:12:44,790 cosine of tau. 216 00:12:44,789 --> 00:12:47,370 if I have a function and I have its derivative, I can 217 00:12:47,370 --> 00:12:50,139 treat that function just like as if I had an x there, so 218 00:12:50,139 --> 00:12:53,689 it'd be sine squared of tau over 2, which is exactly what 219 00:12:53,690 --> 00:12:54,980 we have there. 220 00:12:54,980 --> 00:12:56,896 So it looks like we're in the home stretch. 221 00:12:56,895 --> 00:12:59,809 222 00:12:59,809 --> 00:13:03,789 We're taking the convolution of sine of t with cosine of t. 223 00:13:03,789 --> 00:13:09,879 And so we get 1/2 sine of t. 224 00:13:09,879 --> 00:13:12,519 Now, if I evaluate this thing at t, what do I get? 225 00:13:12,519 --> 00:13:18,230 I get t plus 1/2 sine of 2t, that's when I 226 00:13:18,230 --> 00:13:19,800 evaluated it at t. 227 00:13:19,799 --> 00:13:23,159 And then from that I need to subtract it evaluated at 0, so 228 00:13:23,159 --> 00:13:30,719 minus 0 minus 1/2 sine of 2 times 0, which is 229 00:13:30,720 --> 00:13:33,170 just sine of 0. 230 00:13:33,169 --> 00:13:35,929 So this part right here, this whole thing right there, what 231 00:13:35,929 --> 00:13:37,459 does that simplify to? 232 00:13:37,460 --> 00:13:40,910 Well this is 0, sine of 0 is 0, so this is all 0. 233 00:13:40,909 --> 00:13:47,939 So this first integral right there simplifies to 1/2 sine 234 00:13:47,940 --> 00:13:56,010 of t times t plus 1/2 sine of 2t. 235 00:13:56,009 --> 00:14:00,740 236 00:14:00,740 --> 00:14:04,279 All right, now what does this one simplify to over here? 237 00:14:04,279 --> 00:14:12,190 Well, this one over here, you have minus cosine of t. 238 00:14:12,191 --> 00:14:14,899 And we're going to evaluate this whole thing at t, so you 239 00:14:14,899 --> 00:14:23,199 get 1/2 sine squared of t minus 1/2 the sine of 0 240 00:14:23,200 --> 00:14:26,680 squared, which is just 0, so that's just minus 0. 241 00:14:26,679 --> 00:14:32,179 So far, everything that we have written simplifies to-- 242 00:14:32,179 --> 00:14:34,549 let me multiply it all out. 243 00:14:34,549 --> 00:14:41,809 So I have 1/2-- let me just pick a good color-- 1/2t sine 244 00:14:41,809 --> 00:14:50,579 of t-- I'm just multiplying those out-- plus 1/4 sine of t 245 00:14:50,580 --> 00:14:53,110 sine of 2t. 246 00:14:53,110 --> 00:15:02,210 And then over here I have minus 1/2 sine squared t times 247 00:15:02,210 --> 00:15:03,639 cosine of t. 248 00:15:03,639 --> 00:15:05,909 I just took the minus cosine t and multiplied it through here 249 00:15:05,909 --> 00:15:08,419 and I got that. 250 00:15:08,419 --> 00:15:11,199 Now, this is a valid answer, but I suspect that we can 251 00:15:11,200 --> 00:15:15,000 simplify this more, maybe using some more trigonometric 252 00:15:15,000 --> 00:15:16,159 identities. 253 00:15:16,159 --> 00:15:20,719 And this guy right there looks ripe to simplify. 254 00:15:20,720 --> 00:15:26,529 And we know that the sine of 2t-- another trig identity 255 00:15:26,529 --> 00:15:28,789 you'll find in the inside cover of any of your books-- 256 00:15:28,789 --> 00:15:34,889 is 2 times the sine of t times the cosine of t. 257 00:15:34,889 --> 00:15:36,929 So if you substitute that there, what does our whole 258 00:15:36,929 --> 00:15:38,479 expression equal? 259 00:15:38,480 --> 00:15:40,705 You get this first term. 260 00:15:40,705 --> 00:15:42,889 Let me scroll down a little bit. 261 00:15:42,889 --> 00:15:54,279 You get 1/2t times the sine of t plus 1/4 sine of t times 262 00:15:54,279 --> 00:16:01,639 this thing in here, so times 2 sine of t cosine of t. 263 00:16:01,639 --> 00:16:04,230 Just a trig identity, nothing more than that. 264 00:16:04,230 --> 00:16:09,870 And then finally I have this minus 1/2 sine squared t 265 00:16:09,870 --> 00:16:11,519 cosine of t. 266 00:16:11,519 --> 00:16:13,590 No one ever said this was going to be easy, but 267 00:16:13,590 --> 00:16:15,610 hopefully it's instructive on some level. 268 00:16:15,610 --> 00:16:17,940 At least it shows you that you didn't memorize your trig 269 00:16:17,940 --> 00:16:20,230 identities for nothing. 270 00:16:20,230 --> 00:16:22,190 So let me rewrite the whole thing, or let me just 271 00:16:22,190 --> 00:16:23,420 rewrite this part. 272 00:16:23,419 --> 00:16:26,099 So this is equal to 1/4. 273 00:16:26,100 --> 00:16:29,810 Now, I have-- well let me see, 1/4 times 2. 274 00:16:29,809 --> 00:16:32,039 1/4 times 2 is 1/2. 275 00:16:32,039 --> 00:16:35,449 And then sine squared of t, right? 276 00:16:35,450 --> 00:16:39,490 This sine times this sine is sine squared of t cosine of t. 277 00:16:39,490 --> 00:16:45,180 And then this one over here is minus 1/2 sine squared of t 278 00:16:45,179 --> 00:16:46,929 cosine of t. 279 00:16:46,929 --> 00:16:50,919 And luckily for us, or lucky for us, these cancel out. 280 00:16:50,919 --> 00:16:52,899 And, of course, we had this guy out in the front. 281 00:16:52,899 --> 00:16:58,639 We had this 1/2t sine t out in front. 282 00:16:58,639 --> 00:17:01,899 Now, this guy cancels with this guy, and all we're left 283 00:17:01,899 --> 00:17:05,450 with, through this whole hairy problem, and this is pretty 284 00:17:05,450 --> 00:17:11,180 satisfying, is 1/2t sine of t. 285 00:17:11,180 --> 00:17:16,450 So we just showed you that the convolution-- if I define-- 286 00:17:16,450 --> 00:17:17,869 let me write our result. 287 00:17:17,868 --> 00:17:23,259 I feel like writing this in stone because 288 00:17:23,259 --> 00:17:25,118 this was so much work. 289 00:17:25,118 --> 00:17:32,799 But if we write that f of t is equal to sine of t, and g of t 290 00:17:32,799 --> 00:17:35,869 is equal to cosine of t, I just showed you that the 291 00:17:35,869 --> 00:17:41,039 convolution of f with g, which is a function of t, which is 292 00:17:41,039 --> 00:17:48,950 defined as the integral from 0 to t of f of t minus tau times 293 00:17:48,950 --> 00:17:54,180 g of tau d tau, which was equal to-- and I'll switch 294 00:17:54,180 --> 00:17:59,400 colors here-- which was equal to the integral from 0 to t of 295 00:17:59,400 --> 00:18:08,470 sine of t minus tau times g of tau d tau, that all of this 296 00:18:08,470 --> 00:18:12,180 mess, all of this convolution, it all equals-- and this is 297 00:18:12,180 --> 00:18:21,230 pretty satisfying-- it all equals 1/2t sine of t. 298 00:18:21,230 --> 00:18:23,400 And the whole reason why I went through all of this mess 299 00:18:23,400 --> 00:18:26,680 and kind of bringing out the neurons that had the trig 300 00:18:26,680 --> 00:18:29,279 identities memorized or having to reproof them or whatever 301 00:18:29,279 --> 00:18:32,309 else is to just show you that this convolution, it is 302 00:18:32,309 --> 00:18:34,740 convoluted and it seems a little bit bizarre, but you 303 00:18:34,740 --> 00:18:39,099 really can take the convolutions of actual 304 00:18:39,099 --> 00:18:41,389 functions and get an actual answer. 305 00:18:41,390 --> 00:18:44,250 So the convolution of sine of t with cosine of t is 306 00:18:44,250 --> 00:18:46,660 1/2t sine of t. 307 00:18:46,660 --> 00:18:49,060 So, hopefully, you have a little of intuition of-- well, 308 00:18:49,059 --> 00:18:52,369 not intuition, but you at least have a little bit of 309 00:18:52,369 --> 00:18:56,619 hands-on understanding of how the convolution can be 310 00:18:56,619 --> 00:18:57,869 calculated. 311 00:18:57,869 --> 00:18:58,239