1 00:00:02,540 --> 00:00:04,940 Welcome back to linear circuits. This is Dr Ferri. 2 00:00:06,180 --> 00:00:08,390 Today's lesson is on the frequency spectrum. 3 00:00:09,780 --> 00:00:15,910 So we are most of the way through this module, and in the previous class, we 4 00:00:15,910 --> 00:00:18,020 introduced the transfer function as a way of 5 00:00:18,020 --> 00:00:21,490 computing the circuit response to sinusoids of different frequencies. 6 00:00:23,220 --> 00:00:27,920 Today's lesson is to introduce frequency spectrum as a way of showing the frequency 7 00:00:27,920 --> 00:00:29,400 content of signals. 8 00:00:30,760 --> 00:00:32,680 We're going to be introducing both linear and 9 00:00:32,680 --> 00:00:36,050 log scales for displaying the frequency content and 10 00:00:36,050 --> 00:00:39,060 there, they have different advantages and disadvantages showing 11 00:00:39,060 --> 00:00:40,940 something on a linear scale versus a log scale. 12 00:00:43,290 --> 00:00:46,210 So the purpose is to look at more complicated signals. 13 00:00:46,210 --> 00:00:48,770 Ones that have multiple, frequencies to them. 14 00:00:48,770 --> 00:00:50,960 And let's start out with something very simple. 15 00:00:50,960 --> 00:00:52,325 A summation of sines. 16 00:00:52,325 --> 00:00:56,330 In this particular case, we've got a sine wave 17 00:00:58,410 --> 00:01:01,560 that has a frequency of omega in radians per second. 18 00:01:03,810 --> 00:01:08,790 That is equal to 2 pi times 2, but we've seen from other lessons that 19 00:01:08,790 --> 00:01:14,460 omega is equal to 2 pi f, where f is in hertz and in that case 20 00:01:15,500 --> 00:01:20,420 f is the value of two in hertz. So this is a single sine wave. 21 00:01:21,420 --> 00:01:26,870 With an amplitude of one, and a frequency in hertz of two. 22 00:01:26,870 --> 00:01:29,440 I can plot that. It's going 23 00:01:29,440 --> 00:01:31,360 to be called the frequency spectrum 24 00:01:35,160 --> 00:01:39,515 where we've got an amplitude of one and a frequency of two. 25 00:01:39,515 --> 00:01:42,870 And we look at another frequency. 26 00:01:42,870 --> 00:01:45,910 This is a higher frequency, smaller amplitude. 27 00:01:45,910 --> 00:01:49,245 Frequency of six and amplitude of two. 28 00:01:49,245 --> 00:01:56,270 So I can look at the spectrum that way. Amplitude of 0.2, freqency of six. 29 00:01:56,270 --> 00:02:00,390 Now, the point here is to look at. Multiples are 30 00:02:00,390 --> 00:02:03,365 summations, super positions of frequencies. 31 00:02:03,365 --> 00:02:07,510 If I look at this one, I add those two together. 32 00:02:07,510 --> 00:02:09,870 I get something that is primarily low frequency 33 00:02:12,570 --> 00:02:15,280 because the dominant signal is low frequency. 34 00:02:15,280 --> 00:02:17,720 This has a much higher amplitude to it than the high frequency. 35 00:02:17,720 --> 00:02:20,270 So this looks like a low frequency signal with 36 00:02:20,270 --> 00:02:22,920 a little bit of high frequency added on to it. 37 00:02:22,920 --> 00:02:25,980 But this is a lot more clear in the frequency domain, looking at the spectrum. 38 00:02:25,980 --> 00:02:27,604 It's clear that it dominates a low frequency, the, by the low frequency. 39 00:02:27,604 --> 00:02:28,556 So the frequency spectrum, by showing the 40 00:02:28,556 --> 00:02:30,420 different amplitudes, it tells us what dominates. 41 00:02:30,420 --> 00:02:39,911 This has got a much larger low frequency componnet 42 00:02:39,911 --> 00:02:45,090 to it. Than it does high frequency. 43 00:02:46,280 --> 00:02:49,740 Let me look at a different sign wave summation. 44 00:02:49,740 --> 00:02:53,640 This case, the same frequencies, but I've switched around the amplitudes. 45 00:02:53,640 --> 00:02:59,780 So now the low frequency the low frequency component has a much smaller amplitude. 46 00:02:59,780 --> 00:03:05,700 So 0.2 at a frequency of two. We've got 0.2 amplitude frequency of two. 47 00:03:05,700 --> 00:03:10,720 Here we've got an amplitude of one at a frequency of six. 48 00:03:10,720 --> 00:03:12,220 We see that over here. 49 00:03:12,220 --> 00:03:16,180 Putting them together, I see something very different then what I saw before. 50 00:03:16,180 --> 00:03:20,220 In the previous case it was dominated by the low frequency. 51 00:03:20,220 --> 00:03:22,320 This is dominated by the high frequency. 52 00:03:22,320 --> 00:03:25,750 So we see that in the time domain, but it, it appears more. 53 00:03:26,980 --> 00:03:30,840 More directly in the frequency domain. We have an amplitude of one, 54 00:03:30,840 --> 00:03:32,630 at a frequency six. 55 00:03:32,630 --> 00:03:34,640 It's clear that we're dominated by the 56 00:03:34,640 --> 00:03:37,060 high frequency for the purpose of the frequency 57 00:03:37,060 --> 00:03:39,930 spectrum is to very clearly see frequency content 58 00:03:39,930 --> 00:03:43,350 and signals, and more particularly see what dominates. 59 00:03:45,570 --> 00:03:48,400 So, let's look at a particular type of signal, one that 60 00:03:48,400 --> 00:03:50,485 has a lot of sine waves in it, not just two. 61 00:03:50,485 --> 00:03:53,505 And as sine waves are such that they're multiples of each other. 62 00:03:53,505 --> 00:03:58,480 So if I look at this particular signal, I've got 63 00:03:58,480 --> 00:04:02,010 what I'm going to call a DC component, that's the constant component. 64 00:04:03,410 --> 00:04:07,040 And then I've got Frequency starting out when k is equal 65 00:04:07,040 --> 00:04:11,510 to 1, I've got a omega naught frequency, and I'll call that 66 00:04:11,510 --> 00:04:12,690 the fundamental frequency. 67 00:04:22,490 --> 00:04:26,100 And then I've got these values when k is equal to two and higher. 68 00:04:26,100 --> 00:04:27,830 And those what are called the harmonics. 69 00:04:32,550 --> 00:04:35,230 For various values of k, their multiples. 70 00:04:35,230 --> 00:04:37,868 If I plot this in this vector it becomes a lot more clear. 71 00:04:37,868 --> 00:04:42,520 When DC component, I look at the amplitude and I plot that. 72 00:04:42,520 --> 00:04:43,660 That would be a zero. 73 00:04:43,660 --> 00:04:49,150 At the fundamental frequency of omega naught. 74 00:04:49,150 --> 00:04:55,710 I plot a sub one, at the second, I'm going to label this a second harmonic, 75 00:05:00,220 --> 00:05:02,796 right there I would plot a sub two. 76 00:05:02,796 --> 00:05:06,820 A sub three would be the third harmonic and so on. 77 00:05:14,890 --> 00:05:19,230 So in this case, we can see the harmonics are clearly multiples of the fundamental. 78 00:05:20,610 --> 00:05:22,600 And then you would also want to be able to see 79 00:05:22,600 --> 00:05:26,130 what the relative amplitudes are of these to see what dominates. 80 00:05:30,590 --> 00:05:35,580 Oftentimes, we plot the frequency spectrum on a log scale, not a linear scale. 81 00:05:35,580 --> 00:05:41,000 So far, my plots have been on a linear scale, but on a log scale, we oftentimes 82 00:05:41,000 --> 00:05:48,960 plot signals both in the vertical axis as well as a horizontal axis on a log scale. 83 00:05:48,960 --> 00:05:52,070 On the horizontal axis, equal divisions here, 84 00:05:53,690 --> 00:05:55,850 so that division and that division being equal 85 00:05:56,930 --> 00:06:01,710 vary by a order of magnitude, order times ten in 86 00:06:01,710 --> 00:06:04,940 other words, so this frequency is ten times that one. 87 00:06:04,940 --> 00:06:07,440 This frequency is going to be ten times this one. 88 00:06:09,140 --> 00:06:13,080 And, but they're have the same distance here. 89 00:06:13,080 --> 00:06:17,598 So on the vertical scale we get the log scale by taking 90 00:06:17,598 --> 00:06:22,110 out the amplitude, taking the log of it, log base ten multiplying 91 00:06:22,110 --> 00:06:23,290 it by 20. 92 00:06:23,290 --> 00:06:26,660 And the resulting is displayed in what we'll call decibels, 93 00:06:28,910 --> 00:06:33,480 or the units being db for short. Why do we do this? 94 00:06:33,480 --> 00:06:38,490 It just seems so much easier to plot things on a linear scale than a log scale. 95 00:06:38,490 --> 00:06:39,930 Well, it turns out that some frequency 96 00:06:39,930 --> 00:06:42,210 components are better viewed in the log scale. 97 00:06:42,210 --> 00:06:44,180 They just show up better. 98 00:06:44,180 --> 00:06:49,570 We also have a larger dynamic range using a Log Scale while maintaining resolution 99 00:06:49,570 --> 00:06:55,260 at low amplitude range and the other reason is just for historical purposes. 100 00:06:55,260 --> 00:06:59,010 We often times use, we started using Log Scales a 101 00:06:59,010 --> 00:07:03,060 long time ago, when grass had to be drawn by hand. 102 00:07:03,060 --> 00:07:04,720 And hand calculations were done. 103 00:07:06,110 --> 00:07:09,465 So some people still use log scales because that's just what we're used to. 104 00:07:09,465 --> 00:07:14,610 What I want to do is a specific example to show 105 00:07:14,610 --> 00:07:19,609 why log scales are really helpful. This is a particular signal, x of t. 106 00:07:24,248 --> 00:07:27,160 And you could see that it has an average value here. 107 00:07:27,160 --> 00:07:30,340 And that average value will be our DC component. 108 00:07:30,340 --> 00:07:34,630 When we plot the spectrum, we'll see DC component with a value of one. 109 00:07:34,630 --> 00:07:36,960 In addition we've got some noise on here. 110 00:07:36,960 --> 00:07:40,290 The noise looks sort of periodic but not exactly. 111 00:07:40,290 --> 00:07:42,810 And it's hard to tell, in the time domain. 112 00:07:42,810 --> 00:07:49,090 What that frequency content is. So then we can do the spectrum. 113 00:07:49,090 --> 00:07:54,544 This is a frequency spectrum, and in particular, 114 00:07:54,544 --> 00:08:00,024 this is a linear scale, so when we see an amplitude 115 00:08:00,024 --> 00:08:05,230 of one right there at our dc value, it's a value 116 00:08:05,230 --> 00:08:09,888 of frequency of zero, and everything else 117 00:08:09,888 --> 00:08:14,546 is really kind of small, it's really hard 118 00:08:14,546 --> 00:08:23,430 to see where the content of this noise is. There is a little bitty blip right there. 119 00:08:23,430 --> 00:08:25,920 At a value of 20 radians per second. 120 00:08:25,920 --> 00:08:29,800 Hard to tell and we never really do see much else. 121 00:08:29,800 --> 00:08:31,740 You know we're surroundiveness here. 122 00:08:31,740 --> 00:08:33,430 Always see is a single frequency right there. 123 00:08:33,430 --> 00:08:37,540 So in other words the linear scale doesn't show us that much. 124 00:08:37,540 --> 00:08:40,120 But if instead all I did was 125 00:08:40,120 --> 00:08:41,228 take my magnitude. 126 00:08:41,228 --> 00:08:47,210 And take 20 times the log of it and plot it this way, then it's very clear at that 127 00:08:47,210 --> 00:08:51,670 same frequency of 20, it's very clear at that same 128 00:08:51,670 --> 00:08:54,160 frequency of 20 I get a big peak right here. 129 00:08:55,480 --> 00:08:56,830 And then I also see a lot of 130 00:08:56,830 --> 00:08:59,940 other frequency content there, and that gives us the 131 00:08:59,940 --> 00:09:02,530 frequency content of this noise, which it turned 132 00:09:02,530 --> 00:09:05,160 out to be random noise, so I can see 133 00:09:05,160 --> 00:09:07,220 the frequency content a little bit better. 134 00:09:07,220 --> 00:09:10,016 On the log scale than I can on the linear scale. 135 00:09:10,016 --> 00:09:17,170 So in summary, a frequency spectrum is a plot of the frequency content of signals. 136 00:09:18,330 --> 00:09:22,550 We look at harmonics, they include the fundamental frequency and multiples of it. 137 00:09:23,710 --> 00:09:30,320 We often times use a log scale because it shows us content that is hard to see 138 00:09:30,320 --> 00:09:33,240 in a linear scale and also because of historical reasons. 139 00:09:33,240 --> 00:09:38,880 When we use the log scale, the units are decibels or dB. 140 00:09:38,880 --> 00:09:42,750 In our next lesson, we will do a lab demo of a guitar string application 141 00:09:42,750 --> 00:09:46,490 where you take an actual signal and look at the frequency content of that signal. 142 00:09:47,570 --> 00:09:51,070 Please visit the forums to ask questions and I'll see you online. 143 00:09:51,070 --> 00:09:51,730 Thank You.