1 00:00:00,012 --> 00:00:05,570 So, in this video, we are going to talk about signals again but from a very 2 00:00:05,570 --> 00:00:10,108 different view point. From all the previous videos when we 3 00:00:10,108 --> 00:00:15,877 talked about circuits, we discovered that if we could express a signal as a 4 00:00:15,877 --> 00:00:22,017 superposition of complex exponentials different thinking as a sinusoid, Then 5 00:00:22,017 --> 00:00:27,767 solving the circuit was easy. the complex exponential was a natural 6 00:00:27,767 --> 00:00:32,912 input to study the, the, circuit, circuit's behavior with. 7 00:00:32,912 --> 00:00:37,687 however, sinusoids by themselves are pretty limiting. 8 00:00:37,687 --> 00:00:42,182 you need to talk about more general, signals. 9 00:00:42,182 --> 00:00:47,402 It still would like to express a signal as a super position, a complex 10 00:00:47,402 --> 00:00:50,992 exponentials can we do that? The answer is yes. 11 00:00:50,992 --> 00:00:57,232 This, so we're going to begin by talking about periodic signals, something called 12 00:00:57,232 --> 00:01:03,815 the Fourier series, and what we're going to discover Which is a very interesting 13 00:01:03,815 --> 00:01:10,174 point, is that you can define a signal, as a function in time, or as a function 14 00:01:10,174 --> 00:01:16,594 of frequency, and we can get between the 2 representations of exactly the same 15 00:01:16,594 --> 00:01:22,206 quantity, using the Fourier series. And so we're going to start talking about 16 00:01:22,206 --> 00:01:27,246 signals and what we call domains, we can talk about them as functions of time or 17 00:01:27,246 --> 00:01:32,119 we can talk about them as functions of frequency, whichever seems to be most 18 00:01:32,119 --> 00:01:35,163 convenient. But to get started we need to find the 19 00:01:35,163 --> 00:01:38,157 Fourier series and understand it a little bit. 20 00:01:38,157 --> 00:01:42,247 So, this is the Fourier series for a periodic signal. 21 00:01:42,247 --> 00:01:46,702 This is very important. It's periodic with period T. 22 00:01:46,702 --> 00:01:52,827 And, Fourier claimed that at the beginning of the 19th century, that you 23 00:01:52,827 --> 00:01:59,472 could express a periodic signal as a superposition of complex exponentials. 24 00:01:59,472 --> 00:02:07,905 The frequency of which was K over T, so this is the Kth harmonic of the 25 00:02:07,905 --> 00:02:17,718 fundamental frequency 1 over T, so the fundamental is the first harmonic, then 26 00:02:17,718 --> 00:02:22,342 there's the second. The third and the fourth. 27 00:02:22,342 --> 00:02:28,732 You notice, K here goes negative, so we actually talk about the -1 harmonic, and 28 00:02:28,732 --> 00:02:34,557 -2 harmonic, etcetera, and we'll see how to handle that in just a second. 29 00:02:34,557 --> 00:02:41,122 The, coefficients of this expansion, are called the Fourier coefficients. 30 00:02:41,122 --> 00:02:47,349 So what Fourier claimed is that, if you give me a, time signal. 31 00:02:47,349 --> 00:02:51,414 There are some coefficients here, C sub k. 32 00:02:51,414 --> 00:02:58,220 That allow me to, express. This signal is a weighted superposition 33 00:02:58,220 --> 00:03:03,504 of complex exponentials. Now the question is what are those Cks? 34 00:03:03,504 --> 00:03:08,837 How do you find them? Well we're going to use properties of the complex exponential 35 00:03:08,837 --> 00:03:13,651 here, of what are called harmonically related complex exponentials, is 36 00:03:13,651 --> 00:03:17,775 extremely important. And, it goes by the technical term, 37 00:03:17,775 --> 00:03:21,502 orthogonality. So, I want to consider this interval 38 00:03:21,502 --> 00:03:28,892 Between, a complex exponential having a harmonic number of k. 39 00:03:28,892 --> 00:03:37,125 And one that has a harmonic number of -l. Well, let's do this interval. 40 00:03:37,125 --> 00:03:42,272 So, we have 1 over t, and we have the interval. 41 00:03:42,272 --> 00:03:54,080 I'm going to merge the two exponentials, and the exponents, okay, and do this 42 00:03:54,080 --> 00:04:06,837 intervals very easy we just have to bring down the That the, quantity multiplying 43 00:04:06,837 --> 00:04:10,651 t. Put that in the denominator. 44 00:04:10,651 --> 00:04:17,762 And the interval gives us back the exponential, of course. 45 00:04:17,762 --> 00:04:22,307 So we get 1 over t. 1 over j 2 pi, k. 46 00:04:22,307 --> 00:04:30,835 -l T * times the same thing. So now, it's available, that would be 0 47 00:04:30,835 --> 00:04:37,191 to 2. Well, what do we get when we evaluated at 48 00:04:37,191 --> 00:04:45,942 the upper limit here? Well, the T is capital T so these cancel. 49 00:04:45,942 --> 00:04:52,852 And what you're left with is e ^ +j 2 pi * k - l. 50 00:04:52,852 --> 00:04:59,662 And, evaluating at 0 gives us, 1 of course. 51 00:04:59,662 --> 00:05:07,094 Well, what is e ^ j 2 pi * an integer? For any integer. 52 00:05:07,094 --> 00:05:11,576 Well, I hope you said that the answer is 1. 53 00:05:11,576 --> 00:05:18,053 So, this term turns out to always be 1 for any value of K and N. 54 00:05:18,053 --> 00:05:23,936 The difference between 2 integers is another integer. 55 00:05:23,936 --> 00:05:28,778 So. You get the J2 pi and integer is gotta be 56 00:05:28,778 --> 00:05:33,327 1, so it looks like we get 0 for an answer. 57 00:05:33,327 --> 00:05:41,048 1-1 here, that seems to be kind of odd until we realize over here in the 58 00:05:41,048 --> 00:05:48,311 denominator that we have a k - l. So, when k = l The numerator here, given 59 00:05:48,311 --> 00:05:54,728 by this term, is 1, well the denominator is also zero, so we have a problem. 60 00:05:54,728 --> 00:06:00,948 Numerator is zero, denominator is zero, what's, how we figure this out. 61 00:06:00,948 --> 00:06:05,472 Let's go back and look at the original interval. 62 00:06:05,472 --> 00:06:14,772 When k=l, well, rewriting it just like I did before, k=l, this is 0, which makes 63 00:06:14,772 --> 00:06:19,772 this 1. And so the integral gets very easy. 64 00:06:19,772 --> 00:06:26,162 We get the integral of 1, from 0 to t, that's capital T. 65 00:06:26,162 --> 00:06:31,450 Divide by capital T, you get 1. So that's why this is true. 66 00:06:31,450 --> 00:06:37,100 As long as k does not equal l, then you get, this interval of 0. 67 00:06:37,100 --> 00:06:43,270 But when k and l agree, you get 1. Okay, well that's a very important 68 00:06:43,270 --> 00:06:56,853 property, it's very useful So, suppose I, take my expression for s(t). 69 00:06:56,853 --> 00:07:15,725 Alright, and i'm going to multiply both sides now by that, and integrate. 70 00:07:15,725 --> 00:07:33,440 Well, we know over here what happens. Over here the only time you get anything 71 00:07:33,440 --> 00:07:41,677 is when k=l, so what happens, all the terms in this sum disappear except for 72 00:07:41,677 --> 00:07:50,628 the l, whatever l is, and that leaves us with the following result, that The c sub 73 00:07:50,628 --> 00:08:00,180 l term is = to the original signal * a complex exponential, having a frequency 74 00:08:00,180 --> 00:08:09,382 of -l over t, integrate, and / capital T. So the idea is, in order for this. 75 00:08:09,382 --> 00:08:15,395 To be true, the c sub ks, the c sub ls, have to be given by this. 76 00:08:15,395 --> 00:08:22,366 Or else it cannot be true. This is the way, that given the, periodic 77 00:08:22,366 --> 00:08:29,672 signal, as a function of time, I can find, it's kth Fourier coefficient. 78 00:08:29,672 --> 00:08:37,942 So there's an equivalence here. Which is an important thing to realize. 79 00:08:37,942 --> 00:08:42,598 So. Given the signal, I can find the c sub 80 00:08:42,598 --> 00:08:50,333 Ks, by using this expression. And, given the c sub Ks, I can find the 81 00:08:50,333 --> 00:08:54,092 signal. I'm using this expression. 82 00:08:54,092 --> 00:08:59,757 So I can go back and forth as the arrow suggests here. 83 00:08:59,757 --> 00:09:07,747 And between the time domain, expression of the signal is that function of time, 84 00:09:07,747 --> 00:09:13,548 and the frequency domain The expression of the signal as a function here after 85 00:09:13,548 --> 00:09:18,420 harmonic index, where as a function of K, you have to give me the C sub K's, and 86 00:09:18,420 --> 00:09:22,904 then the complex exponentials are understood because we know the period 87 00:09:22,904 --> 00:09:25,431 T's. This is going to be a very important 88 00:09:25,431 --> 00:09:28,767 concept as we go through. An important result for us. 89 00:09:28,767 --> 00:09:32,599 Right now, because of the. Your position property. 90 00:09:32,599 --> 00:09:38,729 Well, let's do, let's figure out the Fourier series from, for some simple, 91 00:09:38,729 --> 00:09:42,385 wave forms. So here we have our, sine wave. 92 00:09:42,385 --> 00:09:47,309 And I've given it a phase this time to make it a bit more general. 93 00:09:47,309 --> 00:09:53,897 So the period is 1 over the frequency. So, this is the fundamental frequency. 94 00:09:53,897 --> 00:09:58,812 It's just f 0. And, I could do the integral, but I want 95 00:09:58,812 --> 00:10:03,527 to illustrate something by using Euler's formula. 96 00:10:03,527 --> 00:10:10,562 So, as we all know, Euler's formula says, that a cosine can be written as e ^ 97 00:10:10,562 --> 00:10:17,077 Minus, of a complex exponential remaining e to the plus, for a complex exponential. 98 00:10:17,077 --> 00:10:22,797 And, the a/2 is the scaling factors you need to make, you need to get it to, 99 00:10:22,797 --> 00:10:26,291 work. Well, lets look at the Fourier series, 100 00:10:26,291 --> 00:10:29,932 and consider what I call the syemetric terms. 101 00:10:29,932 --> 00:10:35,342 So here's the Fourier series, and I'm just writing it out, and I'm going to 102 00:10:35,342 --> 00:10:40,672 pick up the -kth and the kth terms here, and not worry about the other ones. 103 00:10:40,672 --> 00:10:46,317 So, this is what I mean by symmetry, I'm picking up the terms where the in, the, 104 00:10:46,317 --> 00:10:49,712 harmonic numbers are negatives to each other. 105 00:10:49,712 --> 00:10:54,501 Well, notice that this looks a lot like that. 106 00:10:54,501 --> 00:11:00,832 And that looks a lot like that. So I think, if we pick k= to 1. 107 00:11:00,832 --> 00:11:06,694 I think we have found our Fourier series coefficients. 108 00:11:06,694 --> 00:11:10,862 All the rest of these are going to be zero. 109 00:11:10,862 --> 00:11:17,484 Because we found how to express the, sine wave by this sum. 110 00:11:17,484 --> 00:11:26,474 And that's exactly what the answer is. Is that the, Fourier coefficients are all 111 00:11:26,474 --> 00:11:32,642 zero, except when k=1. And we get these, quantities. 112 00:11:32,642 --> 00:11:39,756 For the, harmonic fundamental frequency rather and it's negative counterpart. 113 00:11:39,756 --> 00:11:45,877 This is another interesting fact, in that, to, we've seen this already but to 114 00:11:45,877 --> 00:11:52,874 express a signal, as would now you'd be needing negative frequency term, in it's 115 00:11:52,874 --> 00:11:56,887 pi's a good frequency. And we're going to see, that their 116 00:11:56,887 --> 00:12:02,287 Fourier coefficients are related to each other in a very simple way, which we'll 117 00:12:02,287 --> 00:12:06,162 get to in just a second. Now I want to go on to a square wave 118 00:12:06,162 --> 00:12:09,387 example now. where were going to have to do the 119 00:12:09,387 --> 00:12:14,187 interval to see what we get. So here's our square wave and, you note 120 00:12:14,187 --> 00:12:19,431 that it's a discontinuous function. in fact, I didn't even define what it 121 00:12:19,431 --> 00:12:24,158 was, at the point of discontinuity. We'll see that's an interesting point in 122 00:12:24,158 --> 00:12:27,228 just a second. And, it turns out in order to do the 123 00:12:27,228 --> 00:12:30,971 integral we're going to need to express it in, mathematically. 124 00:12:30,971 --> 00:12:33,805 So, this is a mathematical expression for it. 125 00:12:33,805 --> 00:12:36,642 Notice I'm leaving open what it is at t over 2. 126 00:12:36,642 --> 00:12:45,667 Okay, so to do the integral, what we have to do is split the range of integration 127 00:12:45,667 --> 00:12:54,717 up so that within each range we have an m loop formula for what the signal is. 128 00:12:54,717 --> 00:13:00,937 So from 0 to t/2. It was just A from T/2 to t, it's minus 129 00:13:00,937 --> 00:13:06,067 A. So now we can do the integrals, and these 130 00:13:06,067 --> 00:13:14,383 integrals are pretty easy to do. I'm going to do the one on the left, so 131 00:13:14,383 --> 00:13:22,289 we get 1/T. And, and again we get a, and then we're 132 00:13:22,289 --> 00:13:33,992 going to get 1 over that exponent is j 2 pi, k/t * e to the -j2pi kt over t. 133 00:13:33,992 --> 00:13:40,022 And I evaluate with that from zero to T/2. 134 00:13:40,022 --> 00:13:51,087 Okay at T/2, the let's see the t, t's cancel and that 2 cancels and so what 135 00:13:51,087 --> 00:13:56,573 we're left with is e^-j pi k. Minus 1. 136 00:13:56,573 --> 00:14:08,159 The, exponential of 0 is, is 1. What is e^j*pi*k? What is that equal to? 137 00:14:08,159 --> 00:14:17,627 It's equal to something very simple. In particular, I would point out, that 138 00:14:17,627 --> 00:14:21,497 that's e, to the minus j pi, all to the k. 139 00:14:21,497 --> 00:14:26,772 That's the same thing. Well, in some quarters, this is 140 00:14:26,772 --> 00:14:34,602 considered one of the most interesting and important formulas in mathematics. 141 00:14:34,602 --> 00:14:42,291 E to the - j pi. Even e with the + j pi is -1. 142 00:14:42,291 --> 00:14:51,112 So, what we get here is that this is -1 to the k. 143 00:14:51,112 --> 00:14:58,238 And then you bring on the -1. Turns out this other integral is the same 144 00:14:58,238 --> 00:15:06,172 thing except with a - sign, and because of the - sign out here the two terms 145 00:15:06,172 --> 00:15:11,922 merge and what we get is the following expression, so. 146 00:15:11,922 --> 00:15:20,384 The Fourier coefficient for a square wave is given by this. 147 00:15:20,384 --> 00:15:25,882 And when K is an odd number like 1, 3, etc. 148 00:15:25,882 --> 00:15:33,002 this is mine, this is minus 1. You get minus 1 minus 1. 149 00:15:33,002 --> 00:15:40,613 Gives you a minus 2 and some things cancel leaving you with this. 150 00:15:40,613 --> 00:15:49,470 Oh when k is even, well that's one, and 1-1=0 so you always get zero here. 151 00:15:49,470 --> 00:15:54,301 So, the result is that. The spectrum. 152 00:15:54,301 --> 00:16:01,416 The coeffi-, Fourier coefficients of a square wave only consist of odd harmonic 153 00:16:01,416 --> 00:16:05,302 terms. And furthermore, their harmonics are 154 00:16:05,302 --> 00:16:07,417 decaying. As one over k. 155 00:16:07,417 --> 00:16:11,452 As the harmonic number gets bigger and bigger. 156 00:16:11,452 --> 00:16:17,912 The coefficient amplitude gets bigger, gets smaller and smaller, I'm sorry, 157 00:16:17,912 --> 00:16:22,987 smaller and smaller. So, lets plot it, lets see what it looks 158 00:16:22,987 --> 00:16:27,592 like. So, here's our, 4a coefficient formula, 159 00:16:27,592 --> 00:16:32,037 and, here I'm plotting, the spectrum Of a square root. 160 00:16:32,037 --> 00:16:36,362 I'm plotting the magnitude in phase of the Fourier coefficient. 161 00:16:36,362 --> 00:16:41,762 This should remind you of what we did for transfer functions where we plotted the 162 00:16:41,762 --> 00:16:44,887 magnitude and phase of the transfer function. 163 00:16:44,887 --> 00:16:50,287 And so I really am thinking about this Fourier coefficient though, as a function 164 00:16:50,287 --> 00:16:56,358 of frequency Just like I thought about the transfer function as a function of 165 00:16:56,358 --> 00:17:01,074 frequency. And, re-express this axis in frequency in 166 00:17:01,074 --> 00:17:03,875 just a second. So, the magnitude. 167 00:17:03,875 --> 00:17:10,392 You can see how the, coefficients go down, like 1 over k, when they're odd. 168 00:17:10,392 --> 00:17:16,220 And when they're even, you get zero, which is what the expression says. 169 00:17:16,220 --> 00:17:22,650 So this is the odd harmonic structure, if you will, of the superposition of, of 170 00:17:22,650 --> 00:17:26,714 complex exponentials that express a square wave. 171 00:17:26,714 --> 00:17:33,220 The phase is interesting. The Again, we have, something in Cartisan 172 00:17:33,220 --> 00:17:41,098 form, so what's the phase of that? We need to know the phase of 1/J, and there 173 00:17:41,098 --> 00:17:49,242 are lots of ways of thinking about that. That's equal to 1/e^j on over to And so 174 00:17:49,242 --> 00:17:56,108 when that goes upstairs you get even minus j pi/2 so the phase is minus pi/2 175 00:17:56,108 --> 00:18:03,261 okay? And since that term is the same for all of them the phase is a constant at 176 00:18:03,261 --> 00:18:10,982 the odd harmonic terms and at the zero everywhere else because the phase is not 177 00:18:10,982 --> 00:18:16,801 You can define when your magnitude is 0, you might as well call it zero. 178 00:18:16,801 --> 00:18:22,384 So, that's the spectrum, and a spectrum is a function of frequency. 179 00:18:22,384 --> 00:18:28,743 All I've done here, is relabel the axis, to ex, express exactly, what is, the 180 00:18:28,743 --> 00:18:33,387 spectrum is a function of F, this is in Hertz. 181 00:18:33,387 --> 00:18:43,652 Okay? So, it consists of something, a, complex exponential at 1/t, 3/t, 5/t, 182 00:18:43,652 --> 00:18:49,042 etc. So with the period was 1 millisecond. 183 00:18:49,042 --> 00:18:55,137 That'd make the fundamental frequency 1 kilohertz, so a square wave that has a 184 00:18:55,137 --> 00:19:01,257 period of a millisecond, as Forier series coefficients, has a spectrum in it, 185 00:19:01,257 --> 00:19:05,115 kilohertz, 3 kilohertz, 5 kilohertz, etcetera. 186 00:19:05,115 --> 00:19:11,119 And the amplitude is going down, at 1 over the harmonic Now it expresses, the 187 00:19:11,119 --> 00:19:16,929 superposition of complex exponentials, are what we need, to produce a square 188 00:19:16,929 --> 00:19:18,818 wave. Very interesting. 189 00:19:18,818 --> 00:19:24,115 And now let's do another example, that's a little bit easier to do. 190 00:19:24,115 --> 00:19:30,501 but it's a, this is a very important example it turns just a little bit later 191 00:19:30,501 --> 00:19:34,491 in this course. So, it's the periodic pulse sequence. 192 00:19:34,491 --> 00:19:39,811 So here's our well-known pulse. It has a duration of delta, and I'm going 193 00:19:39,811 --> 00:19:45,805 to repeat it every capital T, to make a, what's called a periodic pulse train. 194 00:19:45,805 --> 00:19:51,093 So again, I need a formula for it over The period, and here is our formula for 195 00:19:51,093 --> 00:19:54,824 the pulse. which I think is pretty straightforward. 196 00:19:54,824 --> 00:19:58,017 And doing the interval is very straightforward. 197 00:19:58,017 --> 00:20:01,554 You just plug it in, and you get the following answer. 198 00:20:01,554 --> 00:20:05,912 Well, I'd like to plot this like I did for the square wave example. 199 00:20:05,912 --> 00:20:09,229 But it isn't obvious to me at all what this looks like. 200 00:20:09,229 --> 00:20:13,870 I could just type the expression in to the computer and have it plotted. 201 00:20:13,870 --> 00:20:17,044 But I'd rather we get some more understanding here. 202 00:20:17,044 --> 00:20:19,631 And I want to show you a cute little trick. 203 00:20:19,631 --> 00:20:22,601 This is real important thing to appreciate. 204 00:20:22,601 --> 00:20:26,282 And it has to do with this little part of it right here. 205 00:20:26,282 --> 00:20:30,946 So, a little math. So, I want you to note, that I have a 206 00:20:30,946 --> 00:20:35,289 term that looks like 1 minus e to the minus j theta. 207 00:20:35,289 --> 00:20:39,607 Suppose I factor out, what we call half the phase. 208 00:20:39,607 --> 00:20:43,337 So, I am dividing this phase over here by 2. 209 00:20:43,337 --> 00:20:46,119 And I pull out. That factor. 210 00:20:46,119 --> 00:20:52,081 hope you agree that this and this are the same thing. 211 00:20:52,081 --> 00:20:59,241 Okay? That cancels that, and now I get my phase back to where it was. 212 00:20:59,241 --> 00:21:07,702 Well, that's very exciting because if I divide this by 2j Multiplied by 2 j. 213 00:21:07,702 --> 00:21:16,775 What's that? Well that turns out to be Euhler's formula for a sine wave. 214 00:21:16,775 --> 00:21:25,282 And so, after all is said and done, this term here is equal to that. 215 00:21:25,282 --> 00:21:34,887 And so there's a sine, buried in there, times a,uh,a complex exponential, a phase 216 00:21:34,887 --> 00:21:36,067 term. So. 217 00:21:36,067 --> 00:21:45,462 That means, using that result, I can plug that in, simplify the expression. 218 00:21:45,462 --> 00:21:52,279 And here's what I get for a final answer. For the Fourier coefficient of a periodic 219 00:21:52,279 --> 00:21:56,840 pulse train. It has a sined, what we call a sine x /x 220 00:21:56,840 --> 00:22:00,380 behavior, known as the function of k here. 221 00:22:00,380 --> 00:22:05,062 We get sine over the argument times the same thing. 222 00:22:05,062 --> 00:22:13,902 So if I take this and multiply it by delta and divide it by T, and I'll do the 223 00:22:13,902 --> 00:22:23,109 same thing up here, delta over T, then I have something that looks like sine X 224 00:22:23,109 --> 00:22:27,277 over X. And this has got a name, this is called 225 00:22:27,277 --> 00:22:32,067 the sinc function. And I need to know what a sinc function 226 00:22:32,067 --> 00:22:36,477 looks like. And, here's the plot of a sinc function. 227 00:22:36,477 --> 00:22:43,562 very important, for us, it turns it's going to reoccur in this course a lot. 228 00:22:43,562 --> 00:22:50,306 So, sinc function, the reasons come through as 0 of course. 229 00:22:50,306 --> 00:22:56,512 Because of the sinc, this amplitude is dying like 1 over X. 230 00:22:56,512 --> 00:23:03,722 At the origin, that's the interesting part, notice you get 0 for 0. 231 00:23:03,722 --> 00:23:09,377 And you have to use L'Hopital's rule to figure it out, and you'll quickly 232 00:23:09,377 --> 00:23:15,760 discover that it's equal to 1, so this is a plot of our, sinc function, 1 at the 233 00:23:15,760 --> 00:23:22,307 origin and then, it looks like a sinusoid with the amplitude going down at 1 over 234 00:23:22,307 --> 00:23:28,616 x, so that means we can plot. Now, our spectrum, for a, periodic pulse 235 00:23:28,616 --> 00:23:33,000 sequence. Again I'm plotting, this in terms of 236 00:23:33,000 --> 00:23:38,862 harmonic number, and frequency is k/t. So the t is understood. 237 00:23:38,862 --> 00:23:45,859 And I'm, this plot applies, for this special case, where the pulse width is 238 00:23:45,859 --> 00:23:50,303 1/5 of the period. Just, a special case to plot it. 239 00:23:50,303 --> 00:23:57,260 And now, we get, we see that all the, the Fourier coefficients are non 0, except in 240 00:23:57,260 --> 00:24:03,566 the case where they, are harmonically related to 1 over, delta over T. 241 00:24:03,566 --> 00:24:07,931 And that's the way, because of this sync behavior. 242 00:24:07,931 --> 00:24:11,169 Right? It goes to 0 at some places. Okay. 243 00:24:11,169 --> 00:24:16,557 So there's our sync function. Except we're plotting the magnitude, so 244 00:24:16,557 --> 00:24:22,211 those negative parts in the previous slide go above the axis, and we have to 245 00:24:22,211 --> 00:24:26,653 change the phase. But, the point is, again, the spectrum of 246 00:24:26,653 --> 00:24:31,281 periodic pulses. Consist of a lot of, of harmonic terms 247 00:24:31,281 --> 00:24:38,073 whose amplitudes are generally decaying but now they, they ripple up and down 248 00:24:38,073 --> 00:24:43,662 depending on what the values are but generally they are going down. 249 00:24:43,662 --> 00:24:49,113 The phase is interesting. it turns out it looks like the phase 250 00:24:49,113 --> 00:24:52,817 should. A linear function, where the phase is 251 00:24:52,817 --> 00:24:56,407 just that argument. But, but is multiplying J. 252 00:24:56,407 --> 00:24:59,752 And it looks like it's linearly decreasing. 253 00:24:59,752 --> 00:25:04,802 It's got a negative slope. It looks like it should do that forever. 254 00:25:04,802 --> 00:25:08,372 And why do I have this funny plot where it pops. 255 00:25:08,372 --> 00:25:15,037 It looks like it's doing what it's supposed to, then it pops up and does it 256 00:25:15,037 --> 00:25:19,601 again. And that has to do with a complication of 257 00:25:19,601 --> 00:25:25,359 the sync function. In this area where it goes negative, that 258 00:25:25,359 --> 00:25:31,151 adds a phase of pi to the in the phase term, right, because you 259 00:25:31,151 --> 00:25:36,170 can, a magnitude cannot go negative, so that pops into the phase. 260 00:25:36,170 --> 00:25:41,214 Well, the next value on this straight line is going to be minus pi. 261 00:25:41,214 --> 00:25:46,976 I add a phase of pi because it's going negative, and it pops it back So that's 262 00:25:46,976 --> 00:25:52,162 what all these have to do with these are returning to their original values 263 00:25:52,162 --> 00:25:56,671 because of the phase. and insert the phase of the sinc coming 264 00:25:56,671 --> 00:25:59,442 into play, as a kind of of a subtle phase. 265 00:25:59,442 --> 00:26:02,550 It may be real value but it still has a phase. 266 00:26:02,550 --> 00:26:06,292 Negative numbers have a phase of minus of not of pi. 267 00:26:06,292 --> 00:26:10,832 And -pi depending on how you want to think of it. 268 00:26:10,832 --> 00:26:14,262 Okay. So, lets think about this now. 269 00:26:14,262 --> 00:26:21,496 We have a signal, that's in both the time domain and the frequency domain. 270 00:26:21,496 --> 00:26:28,735 I can, express it, as a super position of Of sine itself of rather complex 271 00:26:28,735 --> 00:26:34,127 exponentials. Can i really do that, are they really 272 00:26:34,127 --> 00:26:39,713 equal to each other. So, i need to explore them in more 273 00:26:39,713 --> 00:26:46,835 details and what I am going to do is, look at Get what's called a finite 274 00:26:46,835 --> 00:26:53,012 approximation of this term, and see what happens as I let K get B. 275 00:26:53,012 --> 00:27:00,404 So, if we start with our square wave and look at it for some value K, well, this 276 00:27:00,404 --> 00:27:04,782 finite sum shouldn't look like a square wave. 277 00:27:04,782 --> 00:27:11,242 But as I take more and more and more and more turns in, it had better become the 278 00:27:11,242 --> 00:27:15,937 square way or I cant write, this equality is just wrong. 279 00:27:15,937 --> 00:27:20,582 Well let's see what happens, and this get interesting. 280 00:27:20,582 --> 00:27:26,312 So if you only go out to the. K=1 term, which is the first harmonic, 281 00:27:26,312 --> 00:27:31,482 you see that you get the sine wave like you should, you should. 282 00:27:31,482 --> 00:27:37,717 And it kind of looks like a square wave, but it's not certainly not equal to it. 283 00:27:37,717 --> 00:27:41,512 As k gets big, I go down to the 49th term here. 284 00:27:41,512 --> 00:27:45,172 You see that it's coming closer and closer. 285 00:27:45,172 --> 00:27:48,907 To the square waves. Really very interesting. 286 00:27:48,907 --> 00:27:52,437 You should plot this for yourself sometime. 287 00:27:52,437 --> 00:27:58,892 It's really quite exciting how it works. However, Josiah Gibbs, an American 288 00:27:58,892 --> 00:28:02,732 mathematical physicist pointed out something. 289 00:28:02,732 --> 00:28:07,309 He noticed. See these little bumps up up here? And it 290 00:28:07,309 --> 00:28:14,317 may look like, they're diminishing. The ripples inside here definitely are 291 00:28:14,317 --> 00:28:19,344 getting smaller. But those little bumps, which he called 292 00:28:19,344 --> 00:28:23,942 ears, they're known as Gibbs' ears, don't go away. 293 00:28:23,942 --> 00:28:27,412 And it turns out they will always be there. 294 00:28:27,412 --> 00:28:32,847 And they get closer and closer to the, point of discontinuity. 295 00:28:32,847 --> 00:28:36,426 Turns out the discontinuity's our problem. 296 00:28:36,426 --> 00:28:42,555 Well, let's think about this. if we start with a discontinuous signal, 297 00:28:42,555 --> 00:28:47,685 like a square wave. Well complex exponentials are all 298 00:28:47,685 --> 00:28:52,888 continuous. How can a sum of continuous things add up 299 00:28:52,888 --> 00:29:00,531 to be a discontinuous thing? How can this be true? And so we've got to call in to 300 00:29:00,531 --> 00:29:05,257 question the fact that this may not Work right. 301 00:29:05,257 --> 00:29:13,253 Does the gray seies really work? Well, it turns out, when Fourier presented his 302 00:29:13,253 --> 00:29:21,349 work in the theory of heat, to the French Academy of Sciences, he, his work was 303 00:29:21,349 --> 00:29:28,163 examined by, 3 mathematicians Lapluass. Legendre and Lagrange, three 304 00:29:28,163 --> 00:29:33,146 mathematicians you may have heard of, pretty, and they awarded him the prize 305 00:29:33,146 --> 00:29:38,129 for the best piece of work, but they really called into question what happens 306 00:29:38,129 --> 00:29:42,295 when s is discontinuous. It seemed like it'd work, like I showed 307 00:29:42,295 --> 00:29:48,242 you, but there's these little points And these, the skid here that things don't 308 00:29:48,242 --> 00:29:52,697 quite work right. How do you justify that? Well, we have to 309 00:29:52,697 --> 00:29:57,177 get, go off and do a little, do a little bit of mathematics. 310 00:29:57,177 --> 00:30:02,727 And it's pretty easy, so, it's been shown, that if you have an s(t) that's 311 00:30:02,727 --> 00:30:08,464 continuous, like a sine wave Or something else that's a, periodic, and has to be 312 00:30:08,464 --> 00:30:11,982 periodic. If it's continuous, the Fourier series, 313 00:30:11,982 --> 00:30:14,943 the technical word is converges for each t. 314 00:30:14,943 --> 00:30:20,307 In other words, this side and this side are equal, for every value of t, as long 315 00:30:20,307 --> 00:30:24,682 as s is continuous. The issue comes when it's discontinuous. 316 00:30:24,682 --> 00:30:31,600 And, if s of t has discontinuities, like the square wave, or the periodic pulses 317 00:30:31,600 --> 00:30:38,885 we looked at, the Fourier series does not converge, at the points of discontinuity. 318 00:30:38,885 --> 00:30:44,932 It turns out, that for a square wave, I've plotted a square wave here. 319 00:30:44,932 --> 00:30:52,232 for second, and you can evaluate the complex exponentials, all at T over 2, 320 00:30:52,232 --> 00:30:59,007 and add them up, and it turns out, it's going to give you a value, even though 321 00:30:59,007 --> 00:31:06,162 for the square wave, I didn't assign a value, and the value it's going to assign 322 00:31:06,162 --> 00:31:11,750 It turns out, is the, is the, average of the values on either side of the 323 00:31:11,750 --> 00:31:16,914 discontinuity, so it's going to assign a value of zero to what happens at 324 00:31:16,914 --> 00:31:22,510 precisely T over 2, even though I didn't define it, of course, so it doesn't 325 00:31:22,510 --> 00:31:26,072 converge. It gives a value, but it may not be the 326 00:31:26,072 --> 00:31:31,957 value that you'd want. So, how do we think about this. 327 00:31:31,957 --> 00:31:41,022 Well, do want to point out something. So, if you look at the difference between 328 00:31:41,022 --> 00:31:49,874 the signal you have and this finite gray series approximation You look at the 329 00:31:49,874 --> 00:31:55,548 power in that error. And you take the limit as k goes to 330 00:31:55,548 --> 00:32:00,275 infinity. It turns out you can prove that you 331 00:32:00,275 --> 00:32:05,707 always get zero. So, this is the error in your, your 332 00:32:05,707 --> 00:32:10,094 approximation. Case, k term approximation. 333 00:32:10,094 --> 00:32:16,449 And it turns out that error, as the number of terms gets big, has no power, 334 00:32:16,449 --> 00:32:22,827 has no energy, there's no, there's no difference between the 2 in terms of 335 00:32:22,827 --> 00:32:27,162 power. So, mathemeticians said, it does really 336 00:32:27,162 --> 00:32:34,632 Work, except you have to change your definition of, equality to mean this, 337 00:32:34,632 --> 00:32:42,318 okay? Two things are equal if their, the difference between the two has no power, 338 00:32:42,318 --> 00:32:49,922 and this is called, convergence in mean-square, or mean-square equality. 339 00:32:49,922 --> 00:32:54,857 And it really is the power in the error that becomes the important thing. 340 00:32:54,857 --> 00:32:59,716 So, for all intents and purposes, the Fourier series really does work. 341 00:32:59,716 --> 00:33:04,120 It's just that it doesn't quite work at points of discontinuity. 342 00:33:04,120 --> 00:33:07,942 But we don't care about points of discontinuity anyway. 343 00:33:07,942 --> 00:33:12,942 What do I care? But it is equal. So the Fourier series really does work. 344 00:33:12,942 --> 00:33:17,258 A far as were concerned. Alright. 345 00:33:17,258 --> 00:33:25,055 So, let's look at the properties of the forier coefficients. 346 00:33:25,055 --> 00:33:31,082 So, again, here's my time domain, if you will. 347 00:33:31,082 --> 00:33:35,027 That, representation of the signal as a function of t. 348 00:33:35,027 --> 00:33:39,782 Here's my frequency domain, representation as the function of k, 349 00:33:39,782 --> 00:33:45,017 which governs the, the Fourier coefficients, and here's how you get 350 00:33:45,017 --> 00:33:48,902 between the 2. And lets look at the properties. 351 00:33:48,902 --> 00:33:55,892 So, first of all, if S of T is real valued, you can show that the harmonics 352 00:33:55,892 --> 00:34:02,487 that are neglies of each other are complex conjugates of each other. 353 00:34:02,487 --> 00:34:08,929 We have seen this before. This says that the real part Of c minus 354 00:34:08,929 --> 00:34:17,402 k, is equal to the real part, of c k. And that the imaginary part, of c of 355 00:34:17,402 --> 00:34:25,333 minus k, is equal to the minus, the negative dimension part of c k. 356 00:34:25,333 --> 00:34:31,776 And so this was conjugate symmetry. That we talked about before. 357 00:34:31,776 --> 00:34:35,316 The real part is even. The imaginary part is odd. 358 00:34:35,316 --> 00:34:39,803 And so you can show that the Fourier coefficients evade is. 359 00:34:39,803 --> 00:34:44,714 And I think, if you check all the previous examples we've worked on, 360 00:34:44,714 --> 00:34:49,787 they've had this property. Furthermore, if s(t), itself, is real and 361 00:34:49,787 --> 00:34:53,649 even. It's an even, signal about the origin. 362 00:34:53,649 --> 00:34:59,550 Then the, c, c sub -k-=c sub k. Which means the c sub k's are always 363 00:34:59,550 --> 00:35:04,074 real, and they're also even, that's what even means. 364 00:35:04,074 --> 00:35:11,006 So, you have really an even, for the signal, the 4a coefficients are also real 365 00:35:11,006 --> 00:35:16,388 and even. And if you have an odd, signal, that's, a 366 00:35:16,388 --> 00:35:23,806 real value, and odd, and that's what odd means, it turns out the coefficients are 367 00:35:23,806 --> 00:35:30,238 also odd, but they're pure imaginary. Okay? That's what it takes to be 368 00:35:30,238 --> 00:35:36,269 consistent with this, this result. Up here, okay? So very important 369 00:35:36,269 --> 00:35:42,468 symmetries that you can use to simplify your life, and essentially, what I'm 370 00:35:42,468 --> 00:35:48,526 doing is figuring out what the negative frequency axis is from the positive 371 00:35:48,526 --> 00:35:53,062 frequency axis. So I can fill in what happens At negative 372 00:35:53,062 --> 00:35:59,291 harmonic numbers, negative frequency from what's having, happening at the positive 373 00:35:59,291 --> 00:36:03,067 indices. And one little cute property here, that's 374 00:36:03,067 --> 00:36:08,665 very important, suppose I delay the signal, what is its Fourier coefficient 375 00:36:08,665 --> 00:36:13,892 going to be? So, the idea is that I have let's say a square wave Okay. 376 00:36:13,892 --> 00:36:21,661 And, it turns out it, to make it periodic, we've gotta do that. 377 00:36:21,661 --> 00:36:29,982 And now, when I delay it, let's say by a little bit, so it does this. 378 00:36:29,982 --> 00:36:36,931 You would think that the Fourier coefficients would be related somehow, 379 00:36:36,931 --> 00:36:43,058 and it turns out they are, through this linear phase right here. 380 00:36:43,058 --> 00:36:48,222 The, phase turns out to be depending on the delay. 381 00:36:48,222 --> 00:36:56,112 And then the frequency of the Fourier coefficient enters in but it looks like a 382 00:36:56,112 --> 00:37:00,082 linear delay. I mean linear phase. 383 00:37:00,082 --> 00:37:08,589 Delay in time signal corresponds to multiplying the spectral Version of the 384 00:37:08,589 --> 00:37:14,797 signal by a linear phase. It's linear because, this argument up 385 00:37:14,797 --> 00:37:22,426 here, which is 2 pi, I'm going to write it a different way, tao*k, is a linear 386 00:37:22,426 --> 00:37:27,842 function of k. Okay, well, through the Fourier series, 387 00:37:27,842 --> 00:37:32,467 we can study a signal in both time and frequency. 388 00:37:32,467 --> 00:37:37,692 They're equivalent to each other. We can go back and forth. 389 00:37:37,692 --> 00:37:43,892 And this is going to be a very convenient thing for us to use to explore how 390 00:37:43,892 --> 00:37:50,602 signals ought, what their structures are and how systems operate on them. 391 00:37:50,602 --> 00:37:56,417 So the periodic signals, the Fourier series is the way we obtain the signal's 392 00:37:56,417 --> 00:38:00,457 spectrum. The signals can exist in either time or 393 00:38:00,457 --> 00:38:04,422 frequency domains. This is really important thing. 394 00:38:04,422 --> 00:38:10,152 You may start, we have started with signals as being defined as functions of 395 00:38:10,152 --> 00:38:14,502 time and we used the interval here to figure out what. 396 00:38:14,502 --> 00:38:18,352 What it's spectrum is. I'm going to give you an example in the 397 00:38:18,352 --> 00:38:23,607 next video, where we start with the super position, and that's the natural thing to 398 00:38:23,607 --> 00:38:26,352 do. Now we're start with a frequency domain 399 00:38:26,352 --> 00:38:31,562 specification to signal, and see how it's built, what the result of the signal is. 400 00:38:31,562 --> 00:38:37,315 So, either one, it's perfectly valid. And this is very exciting, and will help 401 00:38:37,315 --> 00:38:42,849 us figure out how signals and systems work in modern information systems.