 You have a handout which is labeled definitions and concepts it contains some acronyms that you can look through you don't have to remember them all But if you want to look up You can see some of the common ones will use some summary of units and prefixes That is the symbols that we're going to use through the course about Hertz watts seconds and so on and prefixes like Tera giga mega micro nano picco So you know all of these and if you don't then you'll take the online lesson I think the first lesson just reminds you about units and prefixes Logarithms just a reminder of some properties of logarithms When we looked at Shannon and Nyquist capacity, we saw that the log function was used there. So remember how to calculate logarithms and And especially we'll see in decibels and later a lot of the measures that we used or the items we measure Expressed on a logarithmic scale So we take advantage of some of these properties like the log of x times y is the same as the log of x plus the log of y That is the log of 20 is the same as the log of 2 plus the log of 10 And we'll use that property especially with decibels and some other properties are nice to look up and remember What we want to do in the last 15 or 20 minutes is to just introduce what we mean by DB or decibels With our signals, we've talked about signal strengths the amplitude the height So we've plotted the signals say the sine waves the signal strength is going up and down. It's varying there are two Units to measure signal strengths volts or watts. So we think of as a voltage or a power level They they're proportional to each other volts and watts, okay, we can convert between volts and watts when necessary So pat a signal level will measure in volts or watts I think from now on mainly we'll use watts. So we talk about power level or some signal level with some power P and Commonly we'll talk about a ratio between two power levels one power level is ten times another power level and we saw that with Shannon capacity We talked about signal to noise ratio ratio between two power levels decibels allow us to look at ratios, but on a logarithmic scale a Ratio is just one number divided by another Say one power level divided by another The DB form of that ratio is we take that ratio Take the logarithm in base 10 and multiply by 10 So remember this equation. Let's go through some examples and see it in use Some of them may be on the the rest of that document, but we'll just calculate a few quick examples Let's consider first our audio system or a simple audio system. We have an amplifier and an amplifier What does it do? It increases the signal strength so we can think we have some input power to an amplifier P in and What comes out is another signal with a output power? So say an audio amplifier My signal goes in it increases the signal strength and the signal comes out if we measure the power of the signal going in We'll give it the power level P in and the power of the signal coming out P out Well, we can talk about the ratio of those two power Values how much bigger is the output than the input? Let's take an example where What the power in is Let's give it a value a simple value one watt one w one watt is the input power and we'll look at How good our amplifier is when we have different output powers and Talk about some characteristics of that amplifier Let's say in the first example the output power is And measured in watts Actually, we'll write it the output power is one watt I've got an amplifier. I send a signal in with power one watt. What comes out is one watt. Is it a good amplifier? Is it amplifying? No, it's not changing the strength of the signal an amplifier would expect to increase the strength of the signal Okay Right not a good amplifier. What's the ratio of? The output to the input signal how much bigger is the output than the input well We can give that a name we can talk about There's a gain I'll denote as G the gain of that amplifier Can be calculated as the output power Divided by the input power How much does this amplifier? Amplify our signal Well, if we calculate that it in this case, it's one watt divided by one watt and gives us a gain of one Note that there are no units here One watt divided by one watt the watts cancel each other out. There are no units left It's just a gain of one You can think of it as a multiplier one times The output is one times that of the input This gain We've expressed as an absolute multiplier here We can also express it in a logarithmic scale and that's what we do when we use decibels The gain in db That same gain we can write as in decibels and the general equation is 10 log base 10 of The gain of the absolute value or in detail the output divided by the input So in this case The gain is one times The output is one times the input It's not really a gain in this. They're the same, but if this value is one log of one is what? In base 10 what log of one zero Okay, ten to ten to the power of zero is one so the log of one is zero zero times ten gives us zero So we say that the gain is zero db zero decibels So here we're just using decibels as another way to measure the gain These are the same value one and zero db are identical, but they're just using different scales One is the absolute scale. It's just a multiplier. One is on a logarithmic scale Why would we do that? We'll see through some other examples it sometimes easier to work on the logarithmic scale To do some operations we'll see But let's try for some better amplifiers. What if my amplifier Still takes an input of one watt. What if it outputs two watts? Well, the gain two divided by one is two that's easy. What's the gain in db? Calculate it You'll need your calculator for this one unless you can remember it If the gain is a factor of two that is the output is two times stronger than the input What's the equivalent gain in decibels? Calculate on your calculator log of in base ten of two is zero point three something something something Okay, you try on your calculator. It's about zero point three Log of two is about point three oh something. So log of two zero point three times by ten and you get about three Three point oh something something. All right, but I approximate here about three db. So log of Two is point three times by ten we get three db What does this mean? Our amplifier has a gain of two The output is two times the input or The amplifier has a gain of three db Just another measure of that game three decibels Let's try a few other values Well, let's double the output power Effectively double the gain If my amplifier I buy a different one it takes one what in for what's out It has a gain of a factor of four calculate The equivalent gain in decibels log of four times by ten log of four Zero point six oh two oh five Six point oh two db about six db. What if we had a better amplifier double the game it produced eight watts output a Gain of eight in db. What do we get? nine db This is a nice thing to remember that if you remember a gain of two, which is Something we'll commonly talk about double or half A gain of two means we double something then it's equivalent to about three db one of the properties of logarithms Four is two times two The log of two times two is the same as the log of two plus the log of two log of two is three If we multiply by ten Plus another three db. We get our six db. So in fact when we double we just add three db 16 would be 12 db But let's try some other values ten watts out The gain of ten log Log of ten is one times by ten we get ten db Let's jump up a bit. We have one watt in and jump up to one hundred watts out A gain of one hundred how many db a gain of ten is ten db 100 is ten times ten Using a logarithmic scale we'd get ten db plus another ten db. We get twenty db You can check with your calculator of course. Well, no, you don't need your calculator here log of a hundred is two times by ten but That property of the logarithm can be useful. We'll see another one in a moment 20 db 200 watts a multiplier of 200 How many db 100 is 20 db. We double that a factor another factor of two so we add three db 23 db. There's a shortcut Just the properties of logarithms We know 100 is 20 db if we multiply 100 by two We get 200 but because we're doing the log of these values the log of 100 times Two and then all multiplied by ten is the same as 20 db plus 3 db. We get 23 db Now you don't need to use that shortcut You can use your calculator in my exams. You can always have a calculator But sometimes when you start to deal with db a lot, it's nice to remember some shortcuts You can quickly approximate a thousand watts 30 db log of a thousand is three one more one million watts How many db? How many zeros? six log of a million is six sixty db and Another reason why we often use db when we have very big values in decibels are manageable numbers 60 db easier to write easier to do operations on as opposed to a number with six zeros on it a Billion and so on is just 90 db and In communication systems often we have multiple stages an amplifier another amplifier and Adding in db is often easier than multiplying large numbers So remember how to calculate decibels Last thing last two minutes a couple more examples Remember our input was one watt. What if the output was? 0.1 watt What's the game? Well 0.1 divided by 1 0.1 Is it a game? Not really Input is one output is lower We've gone down, but still it's a value So log of 0.1 Is minus one times by ten we get minus ten db this amplifier has a gain of minus ten db or It has a loss the loss is the inverse of the game The input power divided by the output power the loss is a factor of ten The output is ten times smaller than the input Convert to db a factor of ten is Ten db the purple value is the gain the green value for those that At the back the green value is loss again of 0.1 It goes It's the output is 0.1 times the input is the same as saying the output is ten times smaller than the input a loss of ten a factor of ten and convert it to db you'll see that that are the the negative We've swapped the sign minus ten db is the same as ten db. So we can talk about gain and loss These are the same same value ones We can say our amplifier has a gain of minus ten db or we could say it has a loss of ten db and That's a good place to to leave you We'll use db in some of the the following lectures. So remember that equation and how to do the conversion, okay?