1 00:00:00,012 --> 00:00:03,391 Okay. We need to talk about a slightly esoteric 2 00:00:03,391 --> 00:00:09,709 invention that's used in Electrical Engineering all the time, and that's the 3 00:00:09,709 --> 00:00:14,483 use of the word decibels. So, I'm sure you all know what the prefix 4 00:00:14,483 --> 00:00:19,614 deci means, it means a tenth of something, but the question is, a tenth 5 00:00:19,614 --> 00:00:23,626 of what? And apparently, it is a tenth of a bel. 6 00:00:23,626 --> 00:00:30,492 And we'll talk about the specific definition in a second, but who was the 7 00:00:30,492 --> 00:00:34,689 bel named for? Think about that for a second. 8 00:00:34,689 --> 00:00:41,692 Any idea who it might be? Of course, it was named for Alexander Graham Bell. 9 00:00:41,692 --> 00:00:49,162 Well, the definition of a decibel scale, what it, going into the details, well 10 00:00:49,162 --> 00:00:57,122 what it means is that you express a value logarithmically relative to a reference. 11 00:00:57,122 --> 00:01:04,732 So, we're going to see how that works. Here's the explicit definition power with 12 00:01:04,732 --> 00:01:09,807 signal s in decibels. So, we've got our signal s and it has 13 00:01:09,807 --> 00:01:14,057 some power. There's a reference signal s0 and we 14 00:01:14,057 --> 00:01:19,207 compute its power. It could be a standard like it's a 1 watt 15 00:01:19,207 --> 00:01:24,807 signal or 1 milliwatt signal. Any kind of standard reference value that 16 00:01:24,807 --> 00:01:31,497 is convenient for your application. You take the ratio between the two, you 17 00:01:31,497 --> 00:01:37,543 evaluate the log base 10. That ratio would multiply by 10, the 10 18 00:01:37,543 --> 00:01:42,790 is the deci part. And that is the definition of the power 19 00:01:42,790 --> 00:01:49,685 of a signal expressed in decibels. Now this second definition down here for 20 00:01:49,685 --> 00:01:54,873 specifing the amplitude of a signal in decibels looks, is different. 21 00:01:54,873 --> 00:02:00,546 Notice that it's 20 sitting out here, but it turns out that 20 is there because 22 00:02:00,546 --> 00:02:05,802 it's consistent with our relationship between power and amplitude. 23 00:02:05,802 --> 00:02:10,446 Because power is proportional to amplitude squared. 24 00:02:10,446 --> 00:02:16,932 If you insert this formula up into the first definition, you get amplitude 25 00:02:16,932 --> 00:02:22,711 squared divided by the amplitude squared of the reference. 26 00:02:22,711 --> 00:02:29,827 Those are squares because the log come out and multiply and give you a 20 log 10 27 00:02:29,827 --> 00:02:35,202 amplitude of the ratio. So, these are just different ways of 28 00:02:35,202 --> 00:02:41,435 thinking about the same thing. You can express the power of a signal in 29 00:02:41,435 --> 00:02:46,331 decibels or you can express its amplitude in decibels. 30 00:02:46,331 --> 00:02:54,877 And the main reason why we use decibels is because adding decibel values 31 00:02:54,877 --> 00:03:03,052 corresponds to multiplying either power or amplitude values, 32 00:03:03,052 --> 00:03:10,259 right? Because the log a times b, that's equal to the log(a)+log(b) and this has 33 00:03:10,259 --> 00:03:15,614 to do with transfer functions. We know that when you put systems in 34 00:03:15,614 --> 00:03:19,338 cascade, their transfer functions multiply. 35 00:03:19,338 --> 00:03:25,330 And so, you can talk about what the overall transfer function is in decibels 36 00:03:25,330 --> 00:03:31,851 by just adding the transfer function at each, expressed in decibels. 37 00:03:31,851 --> 00:03:39,470 Now, let's go over some interesting values for decibels that come up all the 38 00:03:39,470 --> 00:03:43,921 time. This is something everybody remembers. 39 00:03:43,921 --> 00:03:49,931 So, clearly, if the power ratio is 1, that is equivelent to 0 dB. 40 00:03:49,931 --> 00:03:56,378 If your power ratio was signal to noise ratio, signal power divided by reference 41 00:03:56,378 --> 00:04:02,624 noise power, 0 dB means there's just as much power in the signal as there is in 42 00:04:02,624 --> 00:04:08,532 the noise, and that's not good. That's a very low signal to noise ratio 43 00:04:08,532 --> 00:04:14,787 If the power ratio is 10, well that turns out to correspond to 10 dB. 44 00:04:14,787 --> 00:04:21,360 If the power ration is a tenth, the signal power is 1/10 the power of the 45 00:04:21,360 --> 00:04:26,929 reference, that's -10 dB. And, of course, the log of 1/x is the 46 00:04:26,929 --> 00:04:31,722 negative log of x, so that explains the minus sign. 47 00:04:31,722 --> 00:04:36,555 And here it is, we multiply two things together. 48 00:04:36,555 --> 00:04:43,049 Well, their decibel value is added. So, that's 10+10, gives you 20. 49 00:04:43,049 --> 00:04:52,078 Now, what's interesting is that when the power ratio 2 corresponds almost exactly 50 00:04:52,078 --> 00:04:56,367 to 3 dB. That turns out to be very, very close 51 00:04:56,367 --> 00:05:03,407 approximation so that, it turns out to be a very good one to memorize, that 3 dB 52 00:05:03,407 --> 00:05:09,527 corresponds to a factor of 2. Now, how do you get this, that a power 53 00:05:09,527 --> 00:05:15,947 ration of 5 has to be 7 dB? Think about that for a second. 54 00:05:15,947 --> 00:05:23,635 How would you get that? The answer, of course, is that 2*5, is 55 00:05:23,635 --> 00:05:33,788 10, well, since they multiply here, they add here and the sum has to be 10 because 56 00:05:33,788 --> 00:05:40,058 we've already seen that 10 is 10 dB. So right away, you can derive other 57 00:05:40,058 --> 00:05:45,718 decibel values from other ones by using the properties of the log. 58 00:05:45,718 --> 00:05:51,888 And then finally, I want to mention a very interesting and important one. 59 00:05:51,888 --> 00:05:59,035 2^10, 2^10 power is just about a 1000. The log base 10 of a 1000, is 3, which 60 00:05:59,035 --> 00:06:08,069 makes this decibel value very close to 30 dB, turns out to be, or exactly 30.1 dB. 61 00:06:08,069 --> 00:06:15,913 But you can roughly say that if you have a power ratio of 2^10, that's about 30 62 00:06:15,913 --> 00:06:21,297 dB, that's going to be very very important to remember as we go along. 63 00:06:21,297 --> 00:06:26,907 So, that's all there is to decibels. It's very important thing we use in 64 00:06:26,907 --> 00:06:32,987 Electrical Engineering to summarize how systems work and how they're behaving. 65 00:06:32,987 --> 00:06:36,870 Once you get used to it, you'll use it all the time, too.