1 00:00:00,012 --> 00:00:04,764 So in this video, we're going to continue our discussion of the frequency domain. 2 00:00:04,764 --> 00:00:08,559 We're thinking about signal structure, in the frequency domain. 3 00:00:08,559 --> 00:00:12,379 In addition to the time domain. We're first going to prove something 4 00:00:12,379 --> 00:00:16,939 called Parseval's Theorem, which shows us how to plot, compute the tower and a 5 00:00:16,939 --> 00:00:22,817 signal, entirely in the frequency domain. In many cases, it's easier to compute it 6 00:00:22,817 --> 00:00:27,391 that way than directly in the time limit. We'll then talk about constructing 7 00:00:27,391 --> 00:00:30,074 signals explicitly in the frequency domain. 8 00:00:30,074 --> 00:00:34,971 that's a digital communication example. It's a lot of fun to see a little bit, 9 00:00:34,971 --> 00:00:38,835 get a little glimpse of what digital communication is all about. 10 00:00:38,835 --> 00:00:42,673 And finally, we're going to filter a periodic signal. 11 00:00:42,673 --> 00:00:48,510 it's a fairly simple extension of what we've already done, already, when we 12 00:00:48,510 --> 00:00:53,320 talked about analog filters. Alright, so the crucial thing we're 13 00:00:53,320 --> 00:00:57,210 going to prove is something called Parseval's Theorem. 14 00:00:57,210 --> 00:01:03,545 And this is a very important result in the consideration of signals and of 15 00:01:03,545 --> 00:01:05,694 power. So just to review, 16 00:01:05,694 --> 00:01:09,378 this is the Fourier series of a periodic signal, S. 17 00:01:09,378 --> 00:01:15,136 It's got a period of capital T. We know these are the 48 coefficients, 18 00:01:15,136 --> 00:01:21,553 and it's expressed as a superposition of complex exponential signals whose 19 00:01:21,553 --> 00:01:26,957 frequencies are harmonics of 1/t. To find the Fourier coefficient we 20 00:01:26,957 --> 00:01:32,926 calculate this integral, we have to plug in what the signal is and just do the 21 00:01:32,926 --> 00:01:39,260 integral and find it for [INAUDIBLE] Okay, so here's Parseval's Theorem stated 22 00:01:39,260 --> 00:01:43,179 in words. The power in time-domain equals power 23 00:01:43,179 --> 00:01:48,289 calculated in the frequency domain. So let's see what that means. 24 00:01:48,289 --> 00:01:52,790 First of all power in a signal is given by this expression. 25 00:01:52,790 --> 00:01:58,733 Now, we have talked about power in terms of the circuits faultaging current. 26 00:01:58,733 --> 00:02:04,234 The idea here is that the average power of a periodic signal, Is going to be 27 00:02:04,234 --> 00:02:09,185 related to the integral of the square root of the signal, as if it were being 28 00:02:09,185 --> 00:02:14,015 passed through a one ohm resistor. So that's the idea, so, we look back at 29 00:02:14,015 --> 00:02:17,650 the power formulas. No matter if this is a voltage, or a 30 00:02:17,650 --> 00:02:23,064 current, this would be the average, power, expression, assuming it was, going 31 00:02:23,064 --> 00:02:27,221 through 1 ohm resistor. I hip, want to really point out, that 32 00:02:27,221 --> 00:02:32,261 this is a very general expression that the power over here is not in watts. It's 33 00:02:32,261 --> 00:02:35,230 proportional to the power expressed in watts. 34 00:02:35,230 --> 00:02:39,986 Of course it depends, if it's a voltage or current and what the value of the 35 00:02:39,986 --> 00:02:45,930 resistor it goes through as to whether this expression is numerically correct. 36 00:02:45,930 --> 00:02:51,620 It usually isn't but for theory we just we'll figure that out later, but this is 37 00:02:51,620 --> 00:02:55,808 the key element. The power in the signal of course depends 38 00:02:55,808 --> 00:02:59,990 on the signal wave form, how big it is, all kinds of things. 39 00:02:59,990 --> 00:03:03,702 So, how are we going to prove Parseval's Theorem? And 40 00:03:03,702 --> 00:03:10,535 the idea is, that I'm going to take this expression, for the, for s, as a Fourier 41 00:03:10,535 --> 00:03:17,112 series, and plug it into there. Now we're going to do a little bit of 42 00:03:17,112 --> 00:03:21,816 mathematics, because we have the square of the sum. 43 00:03:21,816 --> 00:03:28,828 So let's talk about, just in general. Suppose I have a sum, a k's, and I square 44 00:03:28,828 --> 00:03:33,198 them. What's the expression for that? Well it's 45 00:03:33,198 --> 00:03:37,206 not the sum, of ak squared. That's not the answer. 46 00:03:37,206 --> 00:03:43,577 It's the, it's a double sum, which I'm going to write as a sum over k and l, 47 00:03:43,577 --> 00:03:45,942 with a sub k, a sub l. 48 00:03:45,942 --> 00:03:52,827 And to see that, look at the very example of x+y^2. 49 00:03:52,827 --> 00:04:00,612 Well, that equal to x times x plus x times y is equal to all possible pairwise 50 00:04:00,612 --> 00:04:06,617 products. Right? So that's why this is equal to 51 00:04:06,617 --> 00:04:12,814 that. And, so, when I plug the, Fourier series 52 00:04:12,814 --> 00:04:20,665 into this expression, I'm going to make use of this property here. 53 00:04:20,665 --> 00:04:25,689 Okay. So, what is 1 divided by T, the integral, 54 00:04:25,689 --> 00:04:38,003 from 0 to t, of, the double sum, k and l, of c sub k, c sub l, e to the plus, and 55 00:04:38,003 --> 00:04:47,380 I'm going to combine the exponential parts. And this ought to look very 56 00:04:47,380 --> 00:04:53,967 familiar from the previous video, because this has to do with the orthogonality 57 00:04:53,967 --> 00:05:00,331 relationship I showed previously. What I'm going to do is push the interval 58 00:05:00,331 --> 00:05:07,669 inside, so it becomes clear, and now we know what the value of this interval This 59 00:05:07,669 --> 00:05:13,470 integral is equal to 0, if K does not equal -L. 60 00:05:13,470 --> 00:05:23,071 And with that situation you got 1, occurred when this term right here was 0. 61 00:05:23,071 --> 00:05:29,482 So, the integral is 0 when K does not equal -L. 62 00:05:29,482 --> 00:05:35,359 You see where the 1 from k=l. Well that's great. 63 00:05:35,359 --> 00:05:46,044 So this becomes, equal to, and by the way when, the 1/t makes this a 1 up here so, 64 00:05:46,044 --> 00:05:52,027 we have the sum over k. Of c sub k, c sub minus k. 65 00:05:52,027 --> 00:06:01,467 Well, remember our properties? What is c of minus k? That's equal to c k 66 00:06:01,467 --> 00:06:06,632 conjugate. Conjugate symmetry here. 67 00:06:06,632 --> 00:06:12,622 So, c sub k times c sub k conjugate, that's the magnitude squared of c sub k. 68 00:06:12,622 --> 00:06:17,947 So that's Parseval's Theorem, is that the power of a signal can be 69 00:06:17,947 --> 00:06:24,362 expressed, e, can, can be calculated, expressed, either in the time domain, or 70 00:06:24,362 --> 00:06:29,792 in the frequency domain. The expressions are very similar in that 71 00:06:29,792 --> 00:06:35,232 you square something and add them up. Integration is like adding in, in many 72 00:06:35,232 --> 00:06:40,532 ways, so, this is quite useful. You may decide that this is an easy way 73 00:06:40,532 --> 00:06:46,342 to calculate our, and this, and I'm about to give you some examples of when that's 74 00:06:46,342 --> 00:06:50,214 true. When that comes up, abd you think about 75 00:06:50,214 --> 00:06:56,555 using the, for a series, as an approximation when you have a finite 76 00:06:56,555 --> 00:07:01,266 number of terms. So, for all kinds of reasons you may 77 00:07:01,266 --> 00:07:08,419 decide I only want capital K terms in my, for a series Going from -K to +K is what 78 00:07:08,419 --> 00:07:13,334 I mean by capital K terms, rather than the full series. 79 00:07:13,334 --> 00:07:20,287 Well how good is that approximation? Well, we can talk about the error, which 80 00:07:20,287 --> 00:07:25,229 is this, the original signal minus the approximation. 81 00:07:25,229 --> 00:07:32,463 Well, when you subtract out essentially the middle terms out of this sum, 82 00:07:32,463 --> 00:07:38,855 what you get left are the terms from minus infinity to minus K, minus 1, and 83 00:07:38,855 --> 00:07:45,222 from K plus 1 to infinity. Well, you think about this another way, 84 00:07:45,222 --> 00:07:52,971 this is just this, these terms over here are just the Fourier series for, for the 85 00:07:52,971 --> 00:07:58,594 error signal, here. So how much power, is in the error? the 86 00:07:58,594 --> 00:08:03,820 RMS value, square root power, is given by this expression. 87 00:08:03,820 --> 00:08:11,328 So, I did a few things here to point out some We know that for negative frequency, 88 00:08:11,328 --> 00:08:16,759 negative index. It's C sub minus K equals C K conjugate. 89 00:08:16,759 --> 00:08:23,930 That means the magnitude of C minus K squared is equal to the magnitude of C 90 00:08:23,930 --> 00:08:27,610 sub K squared. So when I sum up That is Parseval's 91 00:08:27,610 --> 00:08:31,345 Theorme says, from minus, minus infinity to infinity. 92 00:08:31,345 --> 00:08:36,893 The negative index terms, negative frequency terms are going to be the same 93 00:08:36,893 --> 00:08:42,026 as the positive frequency terms. So you can write the whole thing and this 94 00:08:42,026 --> 00:08:48,376 is a very good example of why you want to do that is only being over the positive 95 00:08:48,376 --> 00:08:54,583 index parts and you put in a 2. Because you can, that's, your concluding 96 00:08:54,583 --> 00:08:57,796 the negative frequency terms. Okay, 97 00:08:57,796 --> 00:09:04,003 so now we have an idea of how well the Fourier series with K terms approximates 98 00:09:04,003 --> 00:09:08,322 some signal. We can look at the RMS value of the error 99 00:09:08,322 --> 00:09:12,132 signal. Great example of using the frequency 100 00:09:12,132 --> 00:09:17,802 domain to calculate power. So we go back to our approximation of the 101 00:09:17,802 --> 00:09:24,522 square wave. Here we have a plot of the magnitude squared of K, and you can see 102 00:09:24,522 --> 00:09:28,842 it decreases like 1 over k squared, like we would expect. 103 00:09:28,842 --> 00:09:33,102 Now this the squared, and now if you look at the RMS error 104 00:09:33,102 --> 00:09:39,367 that's calculated from the expression given above, you see it's going down but 105 00:09:39,367 --> 00:09:45,056 it doesn't go down very quickly. that's just, so this tells you that if 106 00:09:45,056 --> 00:09:50,412 you want to approximate the Fourier, want to approximate a square way with a 107 00:09:50,412 --> 00:09:53,854 Fourier series, you're going to need lots and lots of 108 00:09:53,854 --> 00:09:57,514 terms. 49 is a pretty high number and that gets 109 00:09:57,514 --> 00:10:01,502 the error down to right about there. I calculated it. 110 00:10:01,502 --> 00:10:05,549 So if you really wanted something with a very, very small area. 111 00:10:05,549 --> 00:10:08,884 you're going to need lots of terms in the Fourier series. 112 00:10:08,884 --> 00:10:12,201 It's a very interesting way of looking at it, I think. 113 00:10:12,201 --> 00:10:17,200 So there's an alternate view of using the Fourier series to approximate a signal, 114 00:10:17,200 --> 00:10:21,773 and that is to measure its distortion. So the idea is I have some periodic 115 00:10:21,773 --> 00:10:25,747 waveform. In this example it's just the square 116 00:10:25,747 --> 00:10:29,223 wave. And the question is, how close to a 117 00:10:29,223 --> 00:10:35,806 sinusoid is it? And this is called harmonic distortion, because this is used 118 00:10:35,806 --> 00:10:40,342 in audio to access how linear amplifiers really are. 119 00:10:40,342 --> 00:10:50,767 And the measure that's used is called the Total Harmonic Distortion, a very long 120 00:10:50,767 --> 00:11:00,592 term. So THD stands for Total Harmonic Distortion, and it has the following 121 00:11:00,592 --> 00:11:05,736 definition. The numerator is the power in the higher 122 00:11:05,736 --> 00:11:08,680 harmonics. Notice it starts at 2, 123 00:11:08,680 --> 00:11:15,869 and of course, this 2 has to do with the fact that we are including the negative, 124 00:11:15,869 --> 00:11:21,511 frequency terms as well. And, the denominator is the power in the 125 00:11:21,511 --> 00:11:25,369 signal. Now the, a little detail here is that you 126 00:11:25,369 --> 00:11:31,149 subtract off c0. And the reason for that is that c0 is known as the average value 127 00:11:31,149 --> 00:11:36,740 of the signal, and if you look at the formula for computing c0, you'll see it's 128 00:11:36,740 --> 00:11:41,682 just the integral of the signal over a period divided by the period. 129 00:11:41,682 --> 00:11:47,489 Well, that's called the average value. In computing harmonic distortion, I'm 130 00:11:47,489 --> 00:11:50,845 interested in how it departs, from a sine wave. 131 00:11:50,845 --> 00:11:56,247 I'm not really interested in whether there's a, a, com, a average value that 132 00:11:56,247 --> 00:11:58,842 isn't zero. It's shifted up or down. 133 00:11:58,842 --> 00:12:03,587 That's really not the issue. The issue is, how much does it depart, 134 00:12:03,587 --> 00:12:07,876 from a sine wave. So that's why this little term in here is 135 00:12:07,876 --> 00:12:13,231 kind of a detail and serving for my square wave, for example of c zero is 136 00:12:13,231 --> 00:12:16,336 zero, which you will notice the numerator 137 00:12:16,336 --> 00:12:20,288 ignores c zero as well. And so using our theory series 138 00:12:20,288 --> 00:12:26,101 expressions, through Parseval's Theorem, we see that the power in the signal, 139 00:12:26,101 --> 00:12:32,578 forgetting about the other value, is that, and, the numerator is that. 140 00:12:32,578 --> 00:12:39,876 Now, to compute the harmonic distortion, I would have to, plug in what the 4a 141 00:12:39,876 --> 00:12:45,785 series is, and do the sum. And, gotta tell you, that's really not a 142 00:12:45,785 --> 00:12:53,472 terribly good way of doing it, because those sums, by enlarge, don't have, do 143 00:12:53,472 --> 00:13:00,462 not have closed forms expressions. So let me point out something, that this 144 00:13:00,462 --> 00:13:07,650 is equal to the power in the original signal, which is all the terms summed up 145 00:13:07,650 --> 00:13:13,731 from 1 to infinity. That's the de, denominator minus the power in c1. 146 00:13:13,731 --> 00:13:18,324 The first harmonic is what that's supposed to mean. 147 00:13:18,324 --> 00:13:24,985 So I can use that to compute my numerator, only thing we have left is 148 00:13:24,985 --> 00:13:30,415 compute the power and the signal. Well, that's easily done, that's, you 149 00:13:30,415 --> 00:13:34,118 just square the signal. Which in the case of the square wave, is 150 00:13:34,118 --> 00:13:37,974 going to give us a value of 1. Integrate that over a period, divide by 151 00:13:37,974 --> 00:13:42,378 the length of the period, we get 1. So, we can find the power in the original 152 00:13:42,378 --> 00:13:48,248 signal, usually quite easily. So, c 1, if you remember what, the, 153 00:13:48,248 --> 00:13:54,877 Fourier series is, for a square wave, is just 2, over j pi. 154 00:13:54,877 --> 00:14:03,618 Which makes that numerator equal to one, minus two, times 2 over pi squared. 155 00:14:03,618 --> 00:14:10,339 And, this 1 Is the power in the signal which we calculated just by looking at 156 00:14:10,339 --> 00:14:14,205 what the waveform was and acting accordingly. 157 00:14:14,205 --> 00:14:18,735 Well you plug in the numbers, you do the calculations. 158 00:14:18,735 --> 00:14:22,720 You wind up with about 20% harmonic distortion. 159 00:14:22,720 --> 00:14:29,162 What that means is that the square wave has about 20% of its power in higher 160 00:14:29,162 --> 00:14:35,782 harmonics and another way of saying it is the sine wave approximates the square 161 00:14:35,782 --> 00:14:40,097 wave in such a way that it captures 80% of the power. 162 00:14:40,097 --> 00:14:46,502 Well it turns out that that is a spectacularly large harmonic distortion. 163 00:14:46,502 --> 00:14:53,357 the idea is in audio that you want to assess how linear an amplifier is. 164 00:14:53,357 --> 00:15:00,032 If the amplifier were perfectly linear if you put in a sign wave into your 165 00:15:00,032 --> 00:15:06,202 amplifier, you put in a sign wave, you should get out a sign wave. 166 00:15:06,202 --> 00:15:12,872 Perfect sine wave, which would make, the harmonic distortion 0, because all of 167 00:15:12,872 --> 00:15:16,922 these are 0. Well, amplifiers have some moderate 168 00:15:16,922 --> 00:15:23,567 amount of distortion and so you want one number that quantifies how linear it is 169 00:15:23,567 --> 00:15:28,767 and they use a total harmonic distortion for that through a series a test 170 00:15:28,767 --> 00:15:32,092 equipment. harmonic distortion of even 1% is 171 00:15:32,092 --> 00:15:36,117 considered pretty large. In fact, all audio amplifiers are 172 00:15:36,117 --> 00:15:39,267 specified through their harmonic distortion. 173 00:15:39,267 --> 00:15:44,592 Let's explore the frequency domain in another case of digital communications. 174 00:15:44,592 --> 00:15:48,622 Let me tell you How digital communication works. 175 00:15:48,622 --> 00:15:53,172 We're going to explore this in much more detail later. 176 00:15:53,172 --> 00:15:57,922 So, here's the idea. I want to send a bit, either a 0 or a 1 177 00:15:57,922 --> 00:16:03,647 to, to the sin, to the receiver. And one way of doing it, is either to 178 00:16:03,647 --> 00:16:08,404 send nothing. Over an, of really/g, interval of, T, to 179 00:16:08,404 --> 00:16:12,915 represent a 0, or you send a sine wave to represent a 1. 180 00:16:12,915 --> 00:16:19,030 And, to send a sequence of bits, you alternate between a 0 and a sine wave, 181 00:16:19,030 --> 00:16:24,132 depending on what the bit is you want to send at any one time. 182 00:16:24,132 --> 00:16:28,537 Okay, this is a one bit at a time scheme, so 183 00:16:28,537 --> 00:16:33,687 I'm sending 1 bit within the, interval 0 to T. 184 00:16:33,687 --> 00:16:40,712 Now you can think of a fancier scheme, that sends two bits at a time. 185 00:16:40,712 --> 00:16:47,317 So, in, instead of just thinking, you know, same T second interval, I can, send 186 00:16:47,317 --> 00:16:50,267 2 bit 0 0. I'll just send nothing. 187 00:16:50,267 --> 00:16:56,742 If I see zero one, I'll send a zero plus the sine wave of a given frequency, 188 00:16:56,742 --> 00:17:02,767 but if want to send 1 0, I'll send a sine wave of a different frequency. 189 00:17:02,767 --> 00:17:07,321 And finally if I have both bits of 1, I'll send to sum. 190 00:17:07,321 --> 00:17:13,172 Essentially, what I'm doing is constructing the transmitted signal in 191 00:17:13,172 --> 00:17:20,074 the frequency domain by essentially using super position and indicating which are 192 00:17:20,074 --> 00:17:26,652 the two bits are by choosing which frequencies are turned on and turned off. 193 00:17:26,652 --> 00:17:32,882 So that is a total construction of a signal in the frequency domain. 194 00:17:32,882 --> 00:17:40,182 Here's an example, where I got, here's the frequency f 1, it's over here and the 195 00:17:40,182 --> 00:17:46,172 frequency f 2 is over there, and here I have the sequence 1 0. 196 00:17:46,172 --> 00:17:52,717 Here's the sequence one one, here's the bit sequence, zero one, and, here's the, 197 00:17:52,717 --> 00:17:58,041 waveforms that are produced. Entirely constructed by thinking about 198 00:17:58,041 --> 00:18:03,834 encoding information, encoding bits, as freq, in the frequency domain, by 199 00:18:03,834 --> 00:18:06,932 representing them by frequencies. So. 200 00:18:06,932 --> 00:18:11,856 Again, you want to think about signals both in the time domain, or in the 201 00:18:11,856 --> 00:18:15,747 frequency domain depending on what the application is. 202 00:18:15,747 --> 00:18:21,292 Very useful, to be flexible and be able to go back and forth, depending on, what 203 00:18:21,292 --> 00:18:25,562 the application is. Okay, now we need to filter, a signal. 204 00:18:25,562 --> 00:18:30,738 So we know how this works for what we've done with circuits. 205 00:18:30,738 --> 00:18:37,474 We know that if the signal is a complex exponential and the filter is linear in 206 00:18:37,474 --> 00:18:45,015 timing variant, that the output is given by the transfer function times a complex 207 00:18:45,015 --> 00:18:52,213 exponential of the same frequency as the input, So, given a transfer function, 208 00:18:52,213 --> 00:18:56,068 you'll know what the filter is going to do. 209 00:18:56,068 --> 00:19:04,392 And, of course if the frequency is just harmonic of some one over t, we just plug 210 00:19:04,392 --> 00:19:09,951 in into the transferred function what that frequency is, and of course we carry 211 00:19:09,951 --> 00:19:14,971 along the frequency of the input. And as I showed you when we talked about 212 00:19:14,971 --> 00:19:20,684 circuits, if you have a superposition of two complex exponentials, the output is a 213 00:19:20,684 --> 00:19:24,813 superposition. This is the definition of a linear, 214 00:19:24,813 --> 00:19:31,033 system, so guess what happens when you have a Fourier series? It's just a 215 00:19:31,033 --> 00:19:38,004 superposition of lots and lots of complex exponentials, so the output is given by 216 00:19:38,004 --> 00:19:44,221 the transfer, sum of the transfer functions evaluated at those harmonic 217 00:19:44,221 --> 00:19:49,465 frequencies, times, the c sub k's, for the original 218 00:19:49,465 --> 00:19:52,944 signal. So, in essence, this is the Fourier 219 00:19:52,944 --> 00:19:58,264 coefficient, for the output. So what do you do? To figure out what a 220 00:19:58,264 --> 00:20:04,499 filter does to a signal, you start with the signal, usually in the time domain. 221 00:20:04,499 --> 00:20:07,926 You find it's Fourier series representation. 222 00:20:07,926 --> 00:20:12,434 You find those C sub K's. You plug that into this expression along 223 00:20:12,434 --> 00:20:17,764 with the transfer function, evaluate it for those harmonic frequencies, and 224 00:20:17,764 --> 00:20:22,168 reconstruct what y is. So, start in the time domain, go into the 225 00:20:22,168 --> 00:20:26,628 frequency domain, figure out what's going on there and then 226 00:20:26,628 --> 00:20:31,850 reconstruct in the time domain. This may seem tedious, but it turns out 227 00:20:31,850 --> 00:20:37,760 to really be a easy way of doing things and very efficient, as we'll see a little 228 00:20:37,760 --> 00:20:39,061 bit later. Okay. 229 00:20:39,061 --> 00:20:42,482 So let's do an example. So, here's a filter. 230 00:20:42,482 --> 00:20:46,282 And, the input I'm going to use, is a periodic pulse train. 231 00:20:46,282 --> 00:20:51,632 And we know that spectrum is, I'm not going to, write the expression again, 232 00:20:51,632 --> 00:20:55,932 it's pretty long-winded. But of course it has a Fourier series. 233 00:20:55,932 --> 00:21:00,657 And the filter I'm going to put it through, is our old friend the low-pass 234 00:21:00,657 --> 00:21:05,515 filter. So, if you recall The, transfer function 235 00:21:05,515 --> 00:21:13,830 is given by that expression, and if you plot the magnitude as a function, if you 236 00:21:13,830 --> 00:21:21,928 can see it rolls off and has a cut off frequency which in this case, is equal to 237 00:21:21,928 --> 00:21:27,422 1 over 2 pi rc. We showed this in previous videos. 238 00:21:27,422 --> 00:21:36,127 I want to figure out what happens when I put this periodic pulse train as the 239 00:21:36,127 --> 00:21:42,802 source signal so that's going to be x(t). The output, vout. 240 00:21:42,802 --> 00:21:47,512 That's going to be y of t. So, how do I find it? I use the, 241 00:21:47,512 --> 00:21:51,668 superposition principle. And this is what we get. 242 00:21:51,668 --> 00:21:57,487 here's the transfer function. And here it is evaluated at k over t. 243 00:21:57,487 --> 00:22:00,992 The kth harmonic. And multiply it times c. 244 00:22:00,992 --> 00:22:06,922 And then reconstruct the signal. Well, this reconstruction, to say the 245 00:22:06,922 --> 00:22:10,281 least, We can't do it, analytically. 246 00:22:10,281 --> 00:22:16,496 And so this is where you just, call upon a computer that can evaluate these things 247 00:22:16,496 --> 00:22:19,477 and add them up. And here's what you get, 248 00:22:19,477 --> 00:22:22,932 really kind of interesting. So, first of all, 249 00:22:22,932 --> 00:22:29,719 I had to tell you, or I had to pick what t was, the period of our periodic pulse 250 00:22:29,719 --> 00:22:32,520 train. I picked 1 millisecond. 251 00:22:32,520 --> 00:22:39,107 What does that make the fundamental frequency? The first harmonic of that 252 00:22:39,107 --> 00:22:47,255 signal? What is that frequency? And of course the answer is, it's 1/t so it's 1 253 00:22:47,255 --> 00:22:51,325 kilohertz. Let's keep that in mind. 254 00:22:51,325 --> 00:23:00,651 And along with the previous example that I gave for the periodic pulse train the 255 00:23:00,651 --> 00:23:05,873 width of the pulse is about 20% of the, period. 256 00:23:05,873 --> 00:23:10,578 Okay which corresponds to the examples I'll show you. 257 00:23:10,578 --> 00:23:17,661 So, I'm now am looking at what happens to the output and the outputs are shown down 258 00:23:17,661 --> 00:23:21,277 here. This is the spectrum of the output. 259 00:23:21,277 --> 00:23:28,122 So these are the Fourier coefficients. For filter of various frequencies. 260 00:23:28,122 --> 00:23:34,252 so I'm changing r times c to produce these various cut off frequencies. 261 00:23:34,252 --> 00:23:41,461 Okay, don't forget that the fundamental is at 1 kilohertz. So, you recall the 262 00:23:41,461 --> 00:23:48,739 Fourier series for periodic pulse train has power at all permonics of the 263 00:23:48,739 --> 00:23:53,600 fundamental. So it has power of 1 kilohertz, 2 264 00:23:53,600 --> 00:23:58,902 kilohertz, 3 kilohertz, 4 kilohertz, etc. Okay. 265 00:23:58,902 --> 00:24:07,707 So, picking a call frequency of a 100 hertz, that's really lowpass filtering. 266 00:24:07,707 --> 00:24:16,582 The the, first harmonic is well above, 10 times the cutoff frequency of the filter. 267 00:24:16,582 --> 00:24:24,458 So it attinuates the the spectrum of the input a lot, as you can see by this 268 00:24:24,458 --> 00:24:27,703 figure. When you get produced, a somewhat lumpy 269 00:24:27,703 --> 00:24:32,818 looking output, it's not even clear that their were pulses in there originally, 270 00:24:32,818 --> 00:24:37,188 it's really low past filtering. So low past filtering tends to produce, 271 00:24:37,188 --> 00:24:40,771 tends to produce signals that don't bury very much in time. 272 00:24:40,771 --> 00:24:45,133 They can't wiggle very quickly because the higher frequencies have been 273 00:24:45,133 --> 00:24:47,228 surpressed. By the filter. 274 00:24:47,228 --> 00:24:52,052 They're low pass, remember they tend to attenuate high frequencies. 275 00:24:52,052 --> 00:24:57,288 When the cut off frequency is now one kilohertz, which is right at the first 276 00:24:57,288 --> 00:25:00,764 harmonic. The first harmonic won't be suppressed 277 00:25:00,764 --> 00:25:04,622 very much, be suppressed by 1 of the square root of 2. 278 00:25:04,622 --> 00:25:10,801 You can sort of see the pulses coming out, kind of, it is not clear the origin 279 00:25:10,801 --> 00:25:17,164 of the pulses which tend to get which i call the short fins, it goes up the comes 280 00:25:17,164 --> 00:25:23,619 down, goes up and comes down, this is periodic, now this is periodic of course. 281 00:25:23,619 --> 00:25:31,423 And then when the cutoff frequency of the filter is well above the first harmonic, 282 00:25:31,423 --> 00:25:37,788 well above the, eighth harmonic, ninth harmonic, it's way up there. 283 00:25:37,788 --> 00:25:44,662 What you will see, is that the output very closely resembles the original, 284 00:25:44,662 --> 00:25:51,083 pulse, and there you can see, if you look very carefully, of course you can see it, 285 00:25:51,083 --> 00:25:57,353 that it's rounded, the edges here, both, at the tops and at the bottoms. 286 00:25:57,353 --> 00:26:04,013 So, it doesn't filter very much, but, if it's, the cutoff frequencies really low 287 00:26:04,013 --> 00:26:09,961 compared to the first harmonic. You get a lot of change of the output. 288 00:26:09,961 --> 00:26:16,767 So, and this is a common use for lowpass filters, is to change the wave form to be 289 00:26:16,767 --> 00:26:23,122 something that you want given some various simple wave form that you could 290 00:26:23,122 --> 00:26:26,182 probably produce in the lab. Okay. 291 00:26:26,182 --> 00:26:33,147 So, signals can be defined either in time or frequency. 292 00:26:33,147 --> 00:26:42,382 What I concentrated on in this video, frequency domain, characterizations of 293 00:26:42,382 --> 00:26:48,042 signals. So the idea was I started with s sub ks. 294 00:26:48,042 --> 00:26:53,179 And I use that to construct signals. In going the other way you can take your 295 00:26:53,179 --> 00:26:58,444 signal of course and figure out what its frequency domain representation is. 296 00:26:58,444 --> 00:27:03,302 And that really means you can go either way, you can start in time, go to 297 00:27:03,302 --> 00:27:06,852 frequency, start in frequency and then go to time. 298 00:27:06,852 --> 00:27:13,744 all, all choices are available to you. So, you can define them in either domain 299 00:27:13,744 --> 00:27:19,640 depending on the application. You can also study, the structure of the 300 00:27:19,640 --> 00:27:24,770 signal in either domain. You may be very interested in what the 301 00:27:24,770 --> 00:27:26,040 pulse width is for example, but you're also maybe interested in how high up, 302 00:27:26,040 --> 00:27:34,568 what's the highest harmonic that's needed to have a 90% approximation of the 303 00:27:34,568 --> 00:27:38,144 original pulse train. Either way of thinking about it it may be 304 00:27:38,144 --> 00:27:44,380 important, depending on the application. And, finally, for linear, time-invariant 305 00:27:44,380 --> 00:27:50,208 filters, we can determine their outputs for periodic inputs because we can use 306 00:27:50,208 --> 00:27:56,252 the Fourier series results that we've got in superposition to get the answer. 307 00:27:56,252 --> 00:27:59,664 So we're really becoming more sophisticated. 308 00:27:59,664 --> 00:28:03,392 Next video, what we're going to do is drop periodic. 309 00:28:03,392 --> 00:28:08,498 We're going to talk about how to generalize these results so we can talk 310 00:28:08,498 --> 00:28:14,474 in general about general signals, whether they be periodic or not, and that means 311 00:28:14,474 --> 00:28:18,049 we'll really know a lot about signals and systems.