 So we're talking about how to get data from one point to another point, and the way that we do it is we send signals. So we're now trying to look at, well what do the signals look like? And in the previous lecture we gave some examples of the square wave type signals where we transmit say a high voltage for some period of time, like plus one volt for half a second to represent one of our bits of digital data, like a bit one, then we transmit a low voltage to represent the other bit. And if we want to transmit a sequence of bits, zeros and ones, we can just change the voltage depending upon what's the data we want to send. So we had some examples on that, and we finished with an example with an analog signal. Instead of having the square wave, we had a sine wave. And the idea is to transmit digital data, zeros and ones, we change the shape of the sine wave, or generally we change the shape of the signal. One shape represents bit one, a different shape would represent bit zero. So we'll continue with those examples today and work through them. So the principles of designing signals is what we're on about. But in fact most of our lecture today is from the examples handout. So we'll go direct to that. So the communication signal examples, we went through some in the first page, last lecture now with looking at, I think we got to this one, that changing sine wave. And it's shown here what we got to. We said, well, one interpretation of this signal, the blue signal, we see it has a shape of a sine wave, not a complete sine wave, but you see the structure there, is let's say I tell you that the duration of a signal element, SE I use here as a signal element, in this case is 0.005 seconds. What that means is that we transmit a signal for that period of time and that element represents some data. And in this example the data it represents was a bit one. So the green bit at the top, this signal element represents a bit one. And the second signal element for the same duration represented a bit zero. Why did it represent one and zero? Well, we said that there's some mapping. Let's assume that sends the signal where we go up first, the first half of a sine wave, to represent bit one, or in other words a high signal represents bit one, and a low signal to represent bit zero. So if we define or the mapping to say high signal bit one, low bit zero, signal element duration 0.005 seconds, if we know that then given this signal we can work out what the data is. Or the other way, if I gave you some data you should be able to draw that signal. So we arrived at this sequence of 16 bits of data. You see the patterns quite obvious, or in this case 1, 0, 0, 1, 1 and so on. So just in one example. But important here we need to know that mapping. High means one, low means zero. It could be different. And we'll see some examples of different mappings in a moment. And we need to know the signal element duration. We'll look at some other examples and talk about data rate as well. The second example. Let's quickly do that. Given the same mapping, high is one, low is zero, and a signal element duration of 0.025 seconds, draw the signal. So I think you're the transmitter. You've got a sequence of four bits, 0, 0, 1, 0. You want to send them. The transmitter takes those bits and generates a signal. What is the signal for those four bits? Try and draw that. It should take a couple of minutes to draw a signal representing just 0, 0, 1, 0. While you're drawing it, I'll see if I can draw it as well. So we need some time scale. We have a signal element duration of what? 0.025 seconds. And we have four bits. So the first bit, we're going to have a signal for 0.025 seconds. Then the second bit, third bit, and the fourth bit. So a total duration four times this is 0.1 seconds. Let me try and mark on here the time. I'll spread it out a bit. Let's say we split this is 0.025 and this will be 0.0, 0.1 seconds. So that's our scale. We're going to have just four bits in this case. So draw the signal to represent the data 0, 1, 0, 0, 0, 1, 0. You have four bits of data to transmit, generate a signal. Well, we said for bit 0, low, so we could draw it. Mine will not be perfect. The first bit, 0, transmit half a period of a sine wave where we go down first. That's either a negative amplitude or a phase, which is non-zero. The second bit is also a 0. The third bit is a 1 and the last bit is a 0. So there's an example signal if we use the same mapping as the previous one. And the receiver, so the transmitter takes the data, generates a signal. The receiver receives the blue signal and then maps it back to data by saying, okay, for 0.025 seconds it's low, must be a bit 0 received. It's low again, bit 0, it's high, bit 1, it's low, bit 0. So the receiver maps it back to data. What's the data rate in this case? Well, how many bits per second? Well, there are four bits transmitted and the time it took to transmit four bits was 0.1 seconds. Four bits per 0.1 seconds, in one second 40 bits. So the data rate with this case is 40 bits per second. What about another one, a new thing? A signalling rate. The signalling rate will define as the number of signal elements per second. Remember a signal element duration was 0.025 seconds. So how many signal elements per second in this case? How many signal elements per second? Anyone? I don't think it's 20. What's a signal element? Well, it's the portion of the signal that we use to represent a sequence of data. So in this case, this is the duration of one signal element. This is the second signal element, the third element, the fourth element. Where in this case, every element represented one bit. So the signalling rate is the same as the data rate. The data rate was 40 bits per second. The signalling rate we could say 40 signal elements per second. I'll just shorten signal elements per second. It's not a common unit, we'll see. One thing you may have heard of is board rate, B-A-U-D, in the old modems. And that corresponds to the signalling rate here. But that's not of importance to us at this stage. We'll see this changes later. In this case, it's the same as the data rate. Look at your sine wave. I know it's not a full sine wave, but look at a signal element. If I kept going for this sine wave, what would the period of the sine wave be? The period will denote as uppercase T. What's the period of the sine wave? Just maybe focus on this part. Well, it's nice because if we keep going, this is one cycle of the sine wave. The period is the duration of that cycle. So the duration of this sine wave is 0.05 seconds. Half a cycle is 0.025, so the full cycle is 0.05 seconds. The period. If we know the period of a sine wave, we can find the frequency. It's the inverse. The frequency is how many cycles per second? The inverse of this, divided by 0.05, is 20 hertz. Hz. So the signal we're using in this example, if we look from the sine wave perspective, has a frequency of 20 hertz. If we kept going in one second, there would be 20 cycles of that signal. The data rate we've achieved is 40 bits per second. And the signaling rate is also 40 signal elements per second. In this case, we gave you the four bits of data, 0010, and I said generator signal. And we generate a signal where for every signal element, we drew half a sine wave. We could do it differently. The mapping from the bits to the signal may change. So let's do the same question again, but with a different signal. I'll draw it again and then we'll explain. Let me see what I need to draw. It'll be perfect, but you'll see the shape. There's my sine wave. Can you see the pattern there? Our same four bits, 0010, but the signal I generated in this case for every signal element is a full period of the sine wave. So for the first 0.025 seconds, bit zero, I generated a signal where it went down first. So it's the flipped sine wave. It goes down and then up, and it completes one cycle within that signal element duration. Next bit zero, the same shape. Bit one, a different shape. And that's the general procedure. That is a different shape signal for the different combinations of bits. Bit one, we go up and then down. Bit zero, down, then up. So this is our normal sine wave. This is the sine wave with an amplitude of minus one, or maybe a phase of pi over two. We can generate that by changing the parameters of the sine wave. This is what we say different mapping from the data to the signal. The previous example used a mapping of bit zero was half a sine wave going down. In this case, it's the full period of the sine wave. So we just changed the characteristics. The data rate, how many bits per second? Look at the previous one. Has it changed? No. Same duration, same number of bits. So it's still 40 bits per second. Signalling rate is the same. It's still 40 signal elements per second. The period in the previous example was 0.05. Here it's 0.025. That's changed. And the frequency is the inverse of the period, 40 hertz. So the thing that we've changed in this example compared to the previous one is the frequency of the sine wave. And we could change it to other parameters, or we could increase the frequency. But the key point is that for bit zero, you have one shape. And for bit one, a different shape. So when the receiver receives a signal, it can distinguish between bit zero or one being received. Any questions? If there's no questions, then do example C. C then D. In example C, I give you the signal element duration. What about the mapping? Well, here I'll let you guess. We need to actually define the mapping. What shape signal corresponds to bit zero and what shape corresponds to bit one? But I'll let you maybe guess, and then I'll choose what I guess in a moment. It may be different. So look at the signal element duration and maybe split it for every 0.01 seconds. Then determine the data. What do we say? Signal element duration 0.01 seconds. So on the time scale, the first signal element is here and finishes here. Then the second signal element goes to here. It's hard to visualize or draw. That's the first signal element. And then the second signal element is over this duration. And then do the same for the other, the rest of the signal. Look at of the what? Eight signal elements in this case. How many different shapes are there? How many different shapes of the signal are there in the eight different signal elements? You see in this first signal element, it's a side wave. The normal side wave, it goes up and then down within the signal element duration, the side wave has two cycles, two periods. The second signal element also a side wave, but it goes down first and completes two cycles in that duration. And then if you look at the rest, you'll see they are one of those first two. Either the side wave going up first or going down first. So there are two different shaped signals and we'll use them to represent our two bits, 0 and 1. Which signal you map to which bit needs to be defined up front. But let's keep it similar to the previous examples. If we go up first, bit 1, if we go down first, bit 0. Given that, find the data. The normal side wave corresponds to bit 1. The flipped side wave going down first corresponds to bit 0. So there are two different shapes. The normal side wave that goes up first and then the flipped one. So we're saying that this one will correspond to bit 1 and this bit 0. This mapping we need to know to know the correct sequence of bits. So it needs to be defined in advance. The transmitter and the receiver, which are trying to communicate, must both know that same mapping. First bit, first signal element goes up. So we have a bit 1. The second signal element goes down first, so bit 0. Then up, 1, 1, 0, 0, 1, 1. So here all we've done is same as before, but we have the signal has a higher frequency. Let's calculate those characteristics. Data rate, signaling rate and frequency. Here we have 8 bits transmitted in 0.08 seconds. So we can determine the data rate. 8 bits, 0.08 seconds. 1 bit in 0.01 seconds or 100 bits per second. Signaling rate. The signaling rate will be the same. The other way to think of the signaling rate is the inverse of the signal element duration. Signal element duration was 0.01 seconds. The inverse of 0.01 is 100. 100 signal elements per second. It's the same number as the data rate in this case. Stay here. The period of our sine wave. We start, say, at time zero. It goes up, comes back down and finishes here. This was 0.01, so this is half of 0.01. 0.005. The period. Frequency is the inverse of the period. 200 hertz. 100 bits per second. The signal frequency is 200 hertz in this case. Any questions before we do example D? Have a look at D and try that. And D is a trick question. We change something else. Let people get a chance to write that down. Before we look at D, let's come back to the mapping. How do we map signal to bits? This mapping, I drew it as going up is bit one and then down is bit zero. Then we define that maybe mathematically. Different ways to do it. Both of them are sine waves. This is our normal sine wave. The amplitude goes up to something like 1.3. Remember, our sine wave has three characteristics. An amplitude, the peak that it reaches. The frequency, how often it repeats within a second. And the third characteristic is the phase. Something about the offset. In both the bit zero and bit one, the frequency of the signal is the same. The frequency doesn't change. In fact, we calculated the frequency to be 200 hertz. How do we generate the sine wave which is upside down? What characteristic of the sine wave can we modify to get that shape? To give you a hint, the other characteristic we have here is the phase. When we have the normal sine wave that goes up at time zero, the phase is zero, there's no phase offset. When we have the bit zero, what's the sine wave? How do we generate an upside down sine wave? In fact, there are two ways. One is you can set the amplitude to be negative. If the peak amplitude is minus 1.3 and you try and plot that, then the shape of the sine wave with minus 1.3 times the sine function, it will go down first. So a negative amplitude for a sine function makes it go down and then up. So we could do that, set the amplitude to minus 1.3. Another way to think of it is that the zero sine wave is the same as the one sine wave but just shifted along in time. And that shift is performed by setting the phase to a particular value. So if you change the phase, then we can shift the sine wave along such that at this origin point it goes down first instead of up. So there's two different options. Either set the amplitude to be negative or set the phase to be some other value. It doesn't matter at this stage which one. I would choose amplitude the same, the phase to be different, and from memory it should be pi over 2. If you want to know why, go and plot a sine wave where the phase is pi over 2. Just a reminder, we mentioned this, we haven't said it today but last week, a general function for a sinusoid signal. Amplitude sine 2 pi f t, f is the frequency plus the phase. So we're saying there are three parameters there. The amplitude, a, the frequency, f, and the phase, that offset that we have. If you change, in our case, if we keep the amplitude 1.3, the frequency 200 hertz, but in one case set the phase to zero, and in another case set the phase to pi over 2, you'll generate those two different shapes. One's going up first, one's going down first. So that's how we could do it. We'll come back to that, we'll see those three characteristics come up in all communication signals. But let's do one last example of our signal mapping digital data to analog signal. D, signal element duration 0.01 seconds, same as C. So look at that picture, split it into signal elements of 0.01 seconds, and look at how many shapes of the signal there are. So try D. So split the signal D into signal elements, and then look at the different shapes of each element. Draw a line for every 0.01 seconds on the picture, on D. In example C, we saw there are two different shapes, and we said they mapped a bit 1 and 0. Try and think about it from example D's perspective, how many different shapes there are. Same signal element duration, the first signal element, then the second signal element, and so on. So there will be eight signal elements, the same as the previous example, but you notice something different. There are how many different shapes across those eight signal elements. There are four different shapes. Let's see if we can define them. I'll try and draw them first, and then we'll put some numbers to them. So this first signal element is the normal sine wave going up first, but it has a low amplitude compared to the other shapes. So we have one which is low, and then the second signal element also goes up first, but has a much higher amplitude. Then if you look at the third signal element, it's the same amplitude, it's the second one, but it's the flipped inverse. It goes down first from this point. I'm not going to fit them in. And the fourth one, if you come along here, again it goes down, but it's a lower amplitude. I'll draw that at the top. Two small ones, and two higher ones, high amplitude, and one of them of each are flipped. Let's put some numbers to them. What's the amplitude of the tall ones? What's the peak? It's about 1.3, in fact, here. They both go up to 1.3. What's the amplitude of the smaller ones? Up to something like 0.4. We don't have to be exact, but you see that there's a significant difference. What's the frequency of all of those signal elements? Try and calculate. If you look closely, they all have the same frequency. Within the same duration, they have the same number of cycles. Within half the signal element duration, there's one cycle, same here, and with all of them, they have the same period, and therefore the same frequency. The period is, again, what? 0.005 seconds. You look on your scale. The period of one cycle is 0.005 seconds. Therefore, we have a frequency, one divided by that, 200 hertz. It's the same as the previous example. So they all have a frequency of 200 hertz. They don't differ there. And the phase, that third characteristic. Well, the previous example, we saw that to represent a normal sine wave that goes up first, the phase is zero. To represent the sine wave, but it goes down first, you set the phase to pi divided by 2 in terms of radians. So that's what we have in these shapes here. Two of them are going up first, like a normal sine wave. Their phase is zero. The other two go down first, so the phase would be pi divided by 2. Going down, pi divided by 2. Going up, zero. Going up, zero. And pi over 2. Radians are the units there. You can convert to degrees if you want. So here we have four different signal elements. In all the previous examples, we just had two different signal elements. Each signal element represented one bit. What's the mapping? Let's choose a mapping of each of the signal elements to bits, so that we can map the signal to data. The first, or choose any one of those signal elements, what sequence of bits are you going to map it to? Maybe the first thing to answer is how many bits is represented by one signal element? In the previous examples, we just had one bit per signal element. But in this case, with four different signal elements, we could have two bits per signal element. Why? Because if we want to be able to generate any sequence of bits, then if we look at pairs of bits, there are four combinations, either 0, 0, 0, 1, 1, 0, or 1, 1. So we can generate any sequence of bits if we have four different signal elements. So each of these could be mapped to two bits in a row. And the mapping we may choose, it may be different as long as it's defined. So let's choose a mapping. Sorry. Let's say in no particular order, this is 0, 0, 0, 1, 0, 1, 1. Generate a random sequence of bits, 1,000 bits long, then split them into pairs of bits. And each pair of bits corresponds to one of the four signal elements. So therefore you can generate a signal as that data. If we only had three signal elements, we could not do this. We could not have three different signal elements and have each represent two bits, because we wouldn't be able to get any combination. So in this case we need at least four signal elements to represent two bits per signal element. In the original examples, we had one signal element, one bit. Here we have one for two bits, as long as we have four combinations. We can expand. If we have eight different shapes, eight different signal elements, each signal element could represent three bits and so on. So it's not as simple as the first cases. Right. So here the mapping from the shape or the combination of amplitude, frequency and phase to the pair of bits I just chose randomly in this case. The important point as long as both the transmitter and the receiver know that mapping. They need to know what it is. It could have been a different arrangement. It would still work. Now there may be better arrangements, but in this case I just chose randomly that mapping. As long as it's known at both points. What's our data now? Go through our signal and look at the shapes that you have. The first one, what do we have? Low amplitude, phase of zero, low amplitude, phase of zero, bits zero one. The second signal element, high amplitude, phase of zero, high amplitude, phase of zero, low amplitude, high amplitude, but the phase of pi over two. It's the inverse. High amplitude, phase of pi over two, one one. This portion, phase of zero, high amplitude, one zero. Low amplitude, phase of pi over two, is zero zero. Zero one. And the last one is one zero. If you had a different mapping or a different arrangement here, you get a different sequence of bits. But as long as the transmitter who takes the data and generates the signal and the receiver who receives the signal and gets the data back, as long as they use the same mapping, they're okay. What's the data rate? Find the data rate. Before we look at the data rate, just to note, these values of 0.4, 1.3 are not significant. Just in this example, the actual values, what's important is that they are different values, different amplitudes, different phases, so that the receiver can distinguish if they're different, they correspond to a different pair of bits. The values are just for this specific example. What's the rate? How many bits per second? Well, an easy way. Count the bits. 16. Divided by the time. 0.08. 200 bits per second. Signaling rate. Signal element duration is 0.01 seconds. Therefore, the number of signal elements per second is the inverse of that. Divided by 0.01 is 100 signal elements per second. And we've calculated the frequency before, the 200 hertz. But here we have different data rate to signaling rate. In previous examples, they were the same. We see in one signal element duration, by using four different signal elements, we're enabled to transmit two bits per duration. So we've effectively doubled our data rate by having four different levels. Under the same conditions, if we had eight different levels, we have three bits per signal element. So we'd get 300 bits per second. If we went up to 16 bits per... 16 levels, I have four bits per signal element duration, or 400 bits per second. Increasing the number of levels allows us to increase the data rate when all other conditions are fixed. And towards the end of this topic, not today, we'll come back to that and put an equation together that relates these things together. The data rate and the number of levels in our signal. This is just a quick illustration. How'd we go? Any questions before we move on? So we're just trying to see the characteristics now that we can represent our data as analog signals. In this case, very simple analog signals of sine waves. And we've seen some features of how to map the data to a signal. Concepts of data rate, signaling rate. And now we've seen by using more than two levels, we can transmit more bits within the same period of time. We can also do that with a digital signal. With our digital signal, we had high and low. It could also be four different levels. The same concept applies. In a later discussion, we'll see that increasing the number of levels from two to four has a disadvantage as well. We get some problems that arise. But let's look at... Let's come back to our general equation in the lecture notes and look at some other examples. Every sine wave, we can... has three key characteristics. The peak amplitude, the multiplier at the front, the frequency, f, how many times it repeats per second, and the phase, that offset from the point of origin. Our signals in our examples had a single sine wave. In real communication signals, they're more complex. They're not just a sine wave. They may be a very different shape. But it turns out, any communication signal, whether it's a sine wave, a square wave, or just a... some very strange shape, any signal, we can decompose down to a combination of sine waves. The summation of multiple sine waves. So that's why... And that's what we're going to go through in a moment and why it's important to remember those three characteristics. Amplitude, phase, and frequency of that sine wave. Because every signal we look at, we can simplify as just sine waves added together. So let's go through that. We've looked at that signal element, signal... in our examples and try to make the point, hopefully we'll finish today, that any signal, any periodic signal, can be decomposed in the summation of sine waves of sinusoid. So let's see some examples to illustrate that point. And on the next page in your examples will start very simple. Here's a sine wave, a signal, write an equation for it. And on signal 1A, write the equation for this sine wave. Signal 1A as a function of time equals what? Look at those three characteristics. Amplitude, frequency, and phase. What are their values? And then write the equation for this one. Remembering the general form of the equation we'll go back to it is this. Any sine wave, an amplitude, a frequency, and a phase. What are the values in this example? This is signal 1A as a function of time. From the the plot, identify the peak amplitude. You don't have to be exact. But you see the peak amplitude goes up to here, so about this point. It's about 1.3. Why did I set it to 1.3 or about 1.3? Would it be nicer if it was 1? It looks nice if it just goes up to 1. We'll see later there's some magic number involved here why it is 1.3. But for now it's not so important. It's about 1.3. If you want to be precise the answer of the value it's 4 divided by pi. There's the magic number. The significance of that number comes out maybe towards the end today. But about 1.3. So that's the peak amplitude. So when we write the equation that's the uppercase A at the front. Sine what? 2 times pi times the frequency what's the frequency? Well one period of the sine wave takes half a second. Inverse of a half is 2. So the frequency is 2 hertz. The general equation is 2 pi f t where f is the frequency. In this case the amplitude is 2. 2 pi times 2 t plus the phase. The phase is how much it shifted from the point of origin. In this case it's the normal shape at time zero the amplitude is zero. So there's zero phase in this case plus zero. Amplitude frequency and phase. We can determine the equation from the plot in this simple case. And you can simplify the equation as what? 4 divided by pi sine 4 pi t and I'll use that later the simplified one. Zero phase we can omit that for brevity. Let's give another sine wave and we'll draw it just below I'll write the equation and let you draw it. The second example handouts is another signal. We have written the equation for this sine wave later we'll come back to this plot we'll not use it yet. Move to the second example draw the plot given this equation 4 on pi times 1 third sine 12 pi t Given the equation draw the sine wave. Signal 1B 4 on pi times a third sine 12 pi t ran out of space. So from the equation identify the peak amplitude frequency and phase. Peak amplitude is the multiplier at the front and to draw it it maybe helps to compare it to the first plot. The peak amplitude is 1 third of the first plot. The first one was 4 over pi this is 4 over pi times 1 third so it's 1 third of the height. The phase is zero still. There is plus zero here so it will start at the origin here at time zero. What's the frequency? 12 pi t, what's f? General form is 2 pi f t we have 12 pi t so f is 6 2 pi f t 2 times pi times 6 t is the general form so the frequency of this signal is 6 hertz. Given that draw it I'm going to try and draw it try and do it on the same scale as the one above we can see the frequency should be 6 hertz the phase is zero and the peak amplitude is 4 on pi times a third compared to our first plot phase is the same so it's going to start at the origin point the amplitude of our second plot will be 1 third of the height of the first one and the frequency of the first plot was 2 hertz and the second one is 6 hertz so it's 3 times the frequency it's going to repeat 3 cycles in the time it takes the first one to repeat 1 cycle given that we can try and draw 1 third of 4 over pi my scale may not be exact but maybe it's about here this is a little bit slow there but you'll see the pattern and it keeps going we get there you must draw it neater than mine okay just try and identify the shape and importantly the height is 1 third of the previous one and the frequency on your handout where do you draw it, good question so on the printed handout in front of you this was the first signal signal 1a don't draw it under here save this space for another picture later not that one so on the next page sorry this signal signal 1b draw the signal in the time domain here the one below it, again we're not using yet we'll come back to that so just try and draw it here doesn't have to be exact the key point from the equation identify the frequency and the amplitude that's all and I tried to draw mine so it's three times as frequent as the top one within one period it repeats three times at the bottom and the height is about a third of the top one, that's what I tried to draw quickly draw it, doesn't have to be exact quick in five seconds draw it once you've drawn it the next signal signal 2 so we had signal 1a signal 1b signal 2 let's be more complicated instead of having one sine wave add the two together S1a plus 1b is signal 2 try and draw that and I think you may have it on your on your handouts because it's too hard to draw but it's hard to see but try and visualize what happens if you add these two together let me zoom out what if you add the two together what do you get well it's going to the one with the highest magnitude is going to influence the shape the most when you add them together you add the points the signal strength at the same point in time and you see what you get well it's hard to draw but if you look on your next page I think you have the plot it's this red one add these two together you get this you see the general shape follows the top one that's the one that influences the most but you see when it's the peak in the first signal we have some trough here in the second signal so that gives us this sort of dip down at the top here and you can find them of course it's easier if you have a computer to draw this for you but here's another, a new signal which is generated by adding two sine waves together let's write the equation for that so you have it on the next page again skipping this plot signal two the equation is quite simple just the previous two added together so we can write that in full and that's what we get when we plot it signal two as a function of time the first sine wave sine, 4 on pi sine 4 pi t we had 2 pi times 2t plus the next signal 12 pi t so the first point is that when we create our signals we just have a single sine wave in fact normally we do not here we add two sine waves together to get a different communication signal the two that I've added together I've chosen especially for this example it doesn't have to be those two but we'll see in the next hopefully 10 minutes why I chose these two particular signals there's something special about them one thing you may notice is this red signal has the same frequency as the first signal the first signal had a frequency of 2 hertz a period of 0.05 seconds if you look closely at the red one every 0.05 seconds it repeats so there's some relationship between the frequency of the first two signals and the resulting one so we say signal two is made up from adding two sine components together and in fact any signal that we deal with can be written as adding up different sine waves and that will lead to some important concepts in communication things like frequency spectrum and bandwidth but to get to that let's just expand this example let's add a third sine wave and I will not try and draw the third one I'll just give you the the shape here's a third one this signal is the same as the red one but we add a third component let's write it down the signal equation the first two components are the same as before 4 on pi sine 4 pi t plus one third 4 on pi sine 12 pi t and we add one more sine wave 4 on pi times one fifth sine pi t no 10 20 pi t there's some pattern here so this green plot is a plot of that equation three sine waves added together and you get this shape and you see again it resembles the previous ones but so we see this general shape of going up and then down but the peaks and troughs here differ a little bit we'll soon get to the significance of that but before let's just highlight for each of those components sine waves identify their amplitude and frequency the phase of these three is the same zero zero zero zero so focus on the amplitude the first component amplitude of 4 on pi that's a 4 sorry try and draw it again frequency of the first component remember was 2 hertz the base omit the symbols the units here the second component one 4 on pi times a third the frequency of the second component what was it 6 6 hertz and to make it a little bit more obvious 6 it was 3 times the first component 3 times the frequency of the first component and you'll see the pattern the third component 2 pi f t so f must be 10 which is in fact 2 times 5 the way that I've added the components follows some pattern such that the first sine wave had a frequency of 2 hertz the next component I added was 1 third the magnitude 5 3 times the frequency the next component I added was 5th the amplitude 5 times the frequency of the first there's a signal number 4 on your handouts where we add a fourth component which is 1 seventh of the amplitude and 7 times the frequency of the first one not all signals are like this but there's a reason for doing it in this pattern for this example this is signal 4 with 4 components we will not write the equation this is signal 30 with 30 components I added plus 1 ninth 9 times the frequency 1 11th, 1 13th and I went up to 30 components so I think 1 divided by 63 or something kept adding those components and we get this plot what does it look like what would we say it's almost a square wave except for these offshoots here we've almost got a square wave if we keep going instead of 30 components but maybe an infinite number of components you get a perfect square wave or from the other perspective given any signal such as a square wave we can decompose it back to a summation of sine waves where each component has its own frequency amplitude and phase and to finish today because we'll continue on that and include we'll come back to that and finish some of these plots tomorrow with the last thing today this one for example 3 components frequency of 2, 6 and 10 3 frequency components 2, 6 and 10 hertz minimum 2 hertz maximum 10 hertz we say the width of those 2 between the minimum and maximum is 8 hertz this signal ranges from 2 hertz up to 10 hertz a width of 8 hertz that's what we call the bandwidth of a signal so I think you may have heard of the word bandwidth in terms of communications so what we're going to arrive at that the bandwidth of a signal relates to the frequency of the components in that signal and then maybe tomorrow and next week we'll see that the data rate that we can achieve with a particular signal is somehow related to the bandwidth of the signal so that's why we're going through this set of equations but let's stop there tomorrow we'll come back recap on these and fill in those other plots those remaining plots