 We're trying to look at more complex communication signals than we've seen examples for. So we saw some examples with just single sine waves and now we're starting to work in and see we saw some examples where we add two sine waves together and it gives us a slightly different shape. And in general any communication signal, however complex it is, a key point is that we can decompose that signal into the summation of sine waves. So that's a key point in especially analyzing and designing signals. Think of anything, any shape, any periodic signal can be broken down as just one sine wave plus another plus another, some combination. And the example we went through, we started with a simple sine wave, we added another one and we get this other shape which we'll see in a moment. Then we added a third component, so I refer to those individual sine waves we add together as components. We added a third component and a fourth component and we saw the shape changing and we eventually got to, where was it? We started with a single sine wave, we added some, a second one, we haven't drawn that here but we added a second sine wave and we ended up with this red one. So we generate a different signal. And then we added a third component and we got this one and then a fourth component, so four sine waves added together, the signal four, we got this one and we said that if we keep going and there was some pattern there in adding components, this one had 30 components, 30 sine waves added together. And the point is that we're getting close to a digital signal, a square wave. And in theory, if you keep adding components following the pattern that we were following, it would become a perfect square wave. That is, these offshoots would not be here, it would be perfectly straight. Why do we do that? We'll now go backwards. The point is that given some signal, say a square wave, you can break it down to just the summation of sine waves. And each sine wave has a peak amplitude and a frequency and that's what we're interested in. Component sine waves that make up this signal, what are their peak amplitudes and frequencies? Any signal, doesn't matter what the shape is, we can just decompose into summation of sine waves and then for each of those sine waves, look at their peak amplitude and frequency and that's of importance to us and that's what we're getting to now. So a signal made up of component sinusoids or sine waves, easier to say for me, a signal made up of summing sine waves, those series of sine waves, we can look at different characteristics of them. We'll often look at the component which has the lowest frequency, maybe component one is 2 hertz, the second one is 6 hertz and then 10 hertz. So we talk about the one with the lowest frequency, 2 hertz and often those other components may be a multiple, an integer multiple of the first one. So we'll see in our examples that the component with the lowest frequency will refer to as the fundamental frequency and the components which are integer multiples of that fundamental frequency will refer to as harmonic frequencies. And because a signal now is made up of summing multiple sine waves each with their own frequencies we refer to that set of frequencies in the signal as the spectrum of the signal. So when we come back to our examples we'll see if a signal has components with 2, 6 and 10 hertz, the spectrum of that signal is 2, 6 and 10 hertz. It contains three components with those particular frequencies. And the width of that spectrum ranging from 2 hertz up to 10 hertz will refer to the bandwidth of the signal, so 8 hertz in that case. So these are important characteristics which are going to be useful when we look at real communication systems. And we've plotted our signals in the time domain where it's a signal amplitude versus time, that's the plots that we've seen all the time, but we can view the signal in the frequency domain from a different perspective. Because now we know the signals are made up of adding sine waves each sine wave has a particular frequency and a particular peak amplitude we can plot those components of peak amplitude versus frequency and that's a signal plot in the frequency domain, not versus time but versus frequency. And it turns out analysing designing signals in the frequency domain is usually much easier than in the time domain. It tells us some quick information about the signal of interest. So let's draw those plots. We started last lecture but let's continue and complete the plots of our signals in the frequency domain and then we'll look at the last set of characteristics. So this one, let's start with this one. I think we draw it but we'll draw it again. This signal, S2, you have it, so on your handouts it's this page. We have the signal equation. It was 4 over pi sine 4 pi t plus 1 third 4 over pi sine 12 pi t, two components and the characteristics of those two components, the amplitude of the first component is 4 over pi and the frequency was 2 hertz. Remember this general equation, 2 pi f t. We have 4 pi t, so f, the frequency must be 2 hertz. The second component has one third the amplitude of the first, 4 over pi times a third and a frequency of 6 hertz, 12 pi t implies f, the frequency is 6. So this is the plot of the signal in the time domain. Now let's plot it in the frequency domain and we did it before but let's do it again and you can draw it underneath here on this grid. Let's see if I can draw it. There's our axis and this is what we say the peak amplitude on this axis and we write it as uppercase s as a function of frequency and here this axis is not time but frequency in hertz in our case. So our signal has two components so we look at the peak amplitude and the frequency of those two components. Let's give some scale to this so let's say we have 2, 4, 6, 8, 10 and we may keep going. That's the frequencies and the peak amplitude let's say this point is 4 on pi and a third of that which is about right here is one third of 4 on pi. There's our axis let's draw our signal so our signal in the frequency domain we just draw impulses for each component where the impulse is at the frequency of that component and the height of the impulse is the peak amplitude of that component. So we have two components back to our equation. The first impulse is at frequency 2 hertz amplitude 4 on pi the second impulse is at one third 4 on pi frequency of 6 hertz at 2 hertz and at 6 hertz and just draw the dot there to indicate that's an impulse. There's our plot finished. It's just a different view of the same signal. Here we have a plot versus frequency and the one above is a plot versus time and the third representation is the equation. So we can write that signal as an equation, we can plot it in the time domain or we can plot it in the frequency domain. It's all the same signal. Why do that? Why plot in the frequency domain or why look at it in the frequency domain? It turns out it's easier to analyze and to design and to extract the key features of the signal and we'll see that shortly. Well we can see it in this example. Let's say in the exam I give you this plot. You don't know the equation but I give you the plot. I ask you to write the equation. Given the time domain plot write the equation for the signal well you need to work out how many components it has and for each component the peak amplitude and the frequency. It's not obvious from here. It's not obvious if you don't know the equation in advance what are the peak amplitudes of the individual components and what are the frequencies of those components. So that would be a hard exam question. But an easier one would be if I gave you this plot and I ask you write the equation for this signal. Well you'd know that this equation has two sine waves added together because there are two impulses. The first one would have a frequency of two hertz, the second one six hertz and you know the peak amplitudes of those two components. So you know A, you know F and you don't know anything about the phase so maybe you would assume the phase is zero. This doesn't capture the phase in the plot, it only captures the amplitude of frequency. So this one is easier to work back to the equation than the time domain plot and that's generally what we deal with when we look at communication signals, the frequency domain. But the other characteristics we can see from this plot, what's the spectrum? Actually let's start, what's the fundamental frequency? What's the fundamental frequency of this signal? Well we said it's the lowest frequency or another way we can think of it. When we have multiple components we have one frequency and the others are integer multiples of that one, then that one frequency is the fundamental frequency. So we have two hertz and six hertz or two hertz and three times two hertz. So we can say the fundamental frequency is two hertz and the harmonics, there's one harmonic in this case, harmonic frequencies in this case six hertz. The spectrum of this signal is the range of frequencies inside it, so we'll just list them, two and six hertz. So think of the spectrum as a list of frequencies. And the bandwidth is the width of that spectrum, the maximum minus the minimum. Those characteristics, especially the bandwidth, that's a key one of interest, is easily obtained by looking at the frequency domain plot. Bandwidth, just the difference between the minimum component and the maximum, four hertz. So you can quickly see the bandwidth from the frequency domain plot. Using those characteristics from the time domain plot, not so easy. Although one thing that we can see in the time domain plot, the fundamental frequency is two hertz. So the resulting signal, we have two components, the resulting signal has a frequency of two hertz. Let's look at the time domain. A frequency of two hertz means a period of one divided by two seconds, half a second. Is that true in our time domain plot? The period, the signal repeats every half a second. So that fundamental frequency tells us the frequency and period of our resulting signal. Let's continue and look at the frequency domain plots of our other signals. So complete them. I think you can try for signal three and four at least. I think maybe you did it last lecture, but just complete the frequency domain plots for the last two examples and also list those four characteristics. The frequency is the spectrum and the bandwidth. That is, you've done it for signal two. Make sure you complete it for signal three. Draw it here. List those characteristics. Signal four. As you can remember the pattern, the equation we went through. For the signal and the frequency domain. You could even try for signal 30. Maybe you don't want to draw it, but you could maybe calculate the characteristics. Signal three I think we did last week, last lecture. Do we have it? Right. We've done that. Signal three. Three components, frequencies of two, six and ten hertz and the peak amplitudes one times four over pi, one third times four over pi, one fifth times four over pi. The fundamental frequency, they're all multiples of two hertz. Two hertz, three times two, five times two. So when we plot in the frequency domain, three impulses at two, six and ten and the corresponding amplitudes. Fundamental frequency, I will not list the harmonics, we just list the spectrum which contains them and the bandwidth. They're all multiples of two hertz so the fundamental frequency is two hertz. The spectrum is just the range of frequencies, think of it as a list. And the bandwidth will be eight hertz in this case, the difference between two and ten. So we talk about, we say that this signal occupies a bandwidth of eight hertz. It uses a range of frequencies which is a width of eight hertz in this case. The previous signal, the red one, occupied a bandwidth of four hertz. And that will be a key characteristic of communication signals, the bandwidth. Any questions? Find the bandwidth of the signal four. The fundamental frequency in this case when all the components are integer multiples of one frequency, then that's the fundamental frequency. So in our case, they're all multiples of two. One times two, three times two, five times two. Two is our fundamental frequency. They don't have to be integer multiples of one of them. So in the examples we use, they are, but they don't have to be. Yep, sorry. The harmonics are the other ones. So the fundamental is two, the harmonics are the six and ten in this case. They're the multiples of two. So we can say that two is the fundamental frequency. If you look at the plot in the time domain, the frequency is two hertz. The harmonics are those other frequency components, six and ten hertz. But I'm not so interested in the harmonics in here because that information is captured in the spectrum. Remember that this time domain plot was created from adding three sine waves. The first sine wave, the very first one that we plotted, had a frequency of two hertz. And the resulting signal also has a frequency of two hertz. So it matches the first component. The other two components, which have a signal of six and ten hertz, cause these small deviations at the top and the bottom. Any other questions? The bandwidth is eight because we define the bandwidth as the width of the spectrum. The spectrum is the range of the components in the signal. So it ranges from two up to ten. Ten minus two is eight. So that's the definition in this case of the bandwidth of our signal. We'll soon see why it's very important. It's a key characteristic of our signal. Find the bandwidth of signal four. I'll let you plot it, I will not plot it. Signal four, and in particular find the bandwidth. And maybe also the bandwidth of signal 30. Signal four, I didn't write the full equation, but it had the first three components plus this fourth component. I will not plot it, but the characteristics, fundamental frequency, we had two, six, ten, and fourteen hertz. They're all multiples of two. The spectrum, two, six, ten, and now the fourth component, fourteen hertz. And the bandwidth, it goes from two to fourteen hertz. So a bandwidth of twelve hertz. You can plot those four impulses as well. And the bandwidth of signal 30, anyone calculate? One hundred and sixteen. The harmonics, so in this case the fundamental frequency is two hertz. The spectrum also contains six, ten, and fourteen. So the harmonics are six, ten, and fourteen. The others. First harmonic, the second harmonic, and the third harmonic. So six, ten, and fourteen are the harmonics. The key things we're generally interested in are the fundamental frequency and the bandwidth. The bandwidth will have an impact upon how good the signal is, the quality of the signal, and how much data we can send. And the fundamental frequency or the frequency of the signal will have an impact upon how that signal can propagate through different media. So you should be able to plot our signals in the frequency domain. For simple signals you can plot in the time domain, but maybe that's not so easy without a computer sometimes now. But you should be able to take a signal equation and plot in the frequency domain or vice-versa. Given the frequency domain plot, write the equation. And answer questions about characteristics like, what is the bandwidth of this signal? Simply from the frequency domain plot. Where was it? This is just an animation I'll go to from Wikipedia that shows, tries to illustrate the changing from time domain to frequency domain. Maybe I'll bring it up tomorrow. Makeup we'll discuss later, let's see if we can zoom in. So we have a signal in the time domain, the red one, we can think of it as made up of multiple components. This one has six components. So if we look at those individual components, look at their peak amplitudes, we can plot them in the frequency domain. So here's our frequency domain plot, the six impulses. So we'll let that loop through and it'll come back. This is just trying to illustrate how we convert from time domain, a signal with six components. We look and extract those six components, six sine waves. And then we can plot them, if we look at those individual six sine waves, we look at the frequency of each component and the peak amplitude and they give us six impulses. And that's our frequency domain plot. So that just tries to illustrate that step. Let's finish on our examples. This signal, S30 had 30 components. So the first component had a frequency of two hertz, so same as before, two, four, six, ten and it kept going. If you follow the pattern it will go two, four, six, ten, twelve, fourteen and so on. And there's 30 components, so you find the 30th and find the bandwidth. This signal would have a much higher bandwidth than the previous ones. So now we want to talk about why do we care about the bandwidth. These are just example signals. We can have any shape signal, break it into sine waves, find the bandwidth of that signal. And they're usually much more complex than this. What will we look at? In some signals, or in fact in practice, often there's not just two components, three components, there are many components or an infinite number of components for a digital signal. So when we have many impulses, rather than trying to draw many impulses, we approximate with a line to capture those impulses. I'll try and illustrate that and we'll come back and see what this shape represents. So let's say we have a signal, we will not draw it in the time domain, but in the frequency domain we have a signal, what will we draw? And it has many impulses, the shape of the signal that there are impulses at many different points. And I'll just draw a lot. And note that the signal doesn't have to be just going down like that. In fact often it's mirrored on both sides. So we may have many impulses and similar on the other side. Often the signals we deal with are not just on one direction but they are effectively mirrored on both sides. So this is just approximate, there's no significance to the values I'm using here, just the shape that I'm interested in. So that may be the frequency domain plot of another signal. And if there are many more components, so many more impulses, often we don't plot the individual impulses, we look at the pattern that it forms, the peaks of those impulses. So you can see the approximate shape on this one, it's sort of going very small, it goes up and it goes around and then down. So often instead of plotting just for real signals, instead of plotting just the impulses we'll plot that curve that captures the peaks and that's what we'll see in practice. So again it's just the plot of the signal and the frequency domain but not showing the actual impulses. And usually in practice it will become very, very small here and in some case it will go to infinity but we cannot plot to infinity. So we'll usually have some cut-off and say once the impulses are very, very small let's ignore them. They're very close to zero. And in different technologies they usually define where that cut-off is. Which components are so small that have very little impact upon the signal? The amplitude is almost zero. That's what I've tried to capture in this one, here's just another signal. The blue line instead of drawing the impulses usually we define some cut-offs and say any components less than this low cut-off or higher than the high cut-off let's ignore them. Let's assume that they're all zero just for simplicity and let's focus on the peaks, the high points and use those cut-offs to define the bandwidth. So the bandwidth is from the lowest point to the highest point. But if the lowest point goes to zero or one of the points goes to infinity we cannot work out the bandwidth, it's infinite. So in practice some of the impulses are so low that we can ignore them. So a common cut-off in practice is to say when the impulses are less than 70 percent of the peak the highest point let's ignore them. So these points let's ignore them anything beyond here, anything lower than the low cut-off and let's say the practical bandwidth of this blue signal is from this point to this point whatever the frequencies are. And there's a center frequency, the middle point and we're usually interested in that which is closely corresponds to our fundamental frequency. When we saw the fundamental frequency we really plotted just this half of the curve. Now it's mirrored on both sides. We don't want to go into the details of why we get this and why it's 70 percent but the main point is in real communication signals we can look at it in the frequency domain and usually we can identify approximately the bandwidth. The main range of frequencies used and that's a key point of our signal. And the other key point usually is the center frequency. So the center frequency and the bandwidth of a signal are very important in communication systems. So let's just list those characteristics. So in practical signals say when I'm sending a Wi-Fi signal from my laptop to the access point. If we look at that signal in the frequency domain there'll be many impulses and it'll be very a shape which is not easy to draw but we could observe that. And we'd see that there's approximate bandwidth and some center frequency. So the bandwidth so we define some cutoff frequencies the lowest and the highest frequency of interest to us. The bandwidth is the width between them and the center frequency is the middle point or the mean of those two cutoffs. And another term that we'll introduce is a channel or a communications channel is refers to a medium that will carry signals with a particular frequency and bandwidth. So when I transmit a signal using my Wi-Fi transmitter on my laptop to the access point I transmit a signal with a particular bandwidth and a particular center frequency and we talk about on transmitting on a channel. If anyone's configured their Wi-Fi access point you can usually select an option, channel 1, channel 6, channel 11. So those channels correspond to a range of or a center frequency and a particular bandwidth. And we'll see some examples using Wi-Fi maybe next week or the week after and look at some real signals. Today we focus on the theory. So what we've arrived at going from simple signals of a sine wave, any signal can be created by adding sine waves together and that resulting signal we've arrived at has a spectrum, a range of components and the things of interest of those frequency components the width of that spectrum, the bandwidth of the signal and the center frequency or the fundamental frequency of the signal of interest. Does anyone know the center frequency and bandwidth of Wi-Fi? 2.4 gigahertz. I think some of you if you've used Wi-Fi you may see references to 2.4 gigahertz. That's something the approximate center frequency of the signal that we use. In fact it varies. See if we can find an example. When you connect to a Wi-Fi access point my wireless LAN device on my laptop is using center frequency 2.462 gigahertz. So it's not exactly 2.4, it's 2.462 gigahertz. So that's the center frequency when I transmit. Because I'm connected to a particular, I'm using a particular channel. I don't know the channel number but in Wi-Fi there are usually 11 channels available. Channel 1 through to 11. When you set up the access point you can specify which channel to use and I'm using channel which has a frequency, center frequency of 2.462 gigahertz. So that's the center point. But if we looked at the signal, it's not shown here but if we observed it, it's actually a range of frequencies. It's not just one frequency. It's a range of frequencies which are transmitted. What is the width of that range? It's usually defined by the people who create Wi-Fi and in this case I know it's 20 megahertz. The bandwidth of the typical Wi-Fi signal is 20 million hertz, 20 megahertz. It's not shown here but we can look up that details to find that. So with Wi-Fi, here's the center frequency and the bandwidth is 20 megahertz. If we wanted to plot that, it would look like this. The center frequency would be the 2.462 gigahertz and the difference between this low cutoff and the high cutoff would be 20 megahertz if we looked at that signal. Before we look at the correspondence between bandwidth and data rate, any questions? Any questions about frequency domain plots? So the next quiz and especially the exam, there'll be questions of interpreting the plots and signal equations. Move on. Why do we care about a signal frequency or a signal bandwidth? Well, there's some different issues involved. When we transmit a signal, the range of frequencies that we can use is usually limited. We cannot use any frequency. And the electromagnetic spectrum is the range of frequencies available to send signals. I think in the next topic we'll show some plots of the spectrum but all the possible frequencies we can use, they are limited so it becomes a scarce resource. When something is limited and two people want to use it, the more that you use, maybe the more the cost involved. So it turns out that the more frequencies you use to transmit a signal, the higher the cost of doing that, whether it's a financial cost or some limitation in terms of laws. So generally the more frequencies our signal uses, the higher the cost. Therefore when we want to have a communication signal and we want low cost, transmit a signal with a low bandwidth, a low range of frequencies. Higher bandwidth, higher cost. There's some other factors though. The center frequency and the range of frequencies we transmit, the way that the signals propagate differ depending upon the frequencies. And the easiest example that I think you know is you've got your old TV remote control, what, infrared, maybe an infrared remote control. Does it work through walls? If you put the remote, take the remote into this room and the TV is in the other room and try and change the channels, generally it will not work. When you use Wi-Fi, if the access point is here and I go out the door or into the other room, can I connect to this access point? Generally yes. Wi-Fi, the signals will propagate through the walls. They'll be reduced in strength to some extent, but they will propagate through. Infrared generally will not go through walls because they are using different frequencies and the physical nature of the frequencies and how they hit different obstacles determines whether they'll go through the wall or not and other obstructions as well. So depending upon the frequency, the signals propagate differently. So if I want to build a communication system which just covers a few meters, maybe infrared is okay. But if I want to cover a whole building, infrared and the range of frequencies there is not appropriate. So the frequencies we choose impact on how our signal will propagate. The other factor, the more frequencies we use or the larger the bandwidth will generally impact upon the amount of data that we can transfer. The general rule is that the larger the bandwidth, the higher the data rate. So larger the bandwidth, higher the cost. We need to use more frequencies. Larger the bandwidth, higher the data rate. So there's a trade-off. We want low cost but high data rate generally. But also the frequencies we use depend upon how it can be work in the physical environment. So that's why we care about signal frequency and bandwidth. There's another issue. When we transmit a signal, the range of frequencies which are transmitted through a particular medium is usually limited. I cannot transmit any range of frequencies. There's usually a physical limit. If I send a signal through an electrical cable, the range of frequencies that can transfer is usually limited. Or a law or regulation that says you're not allowed to transmit a range of frequencies larger than a bandwidth of 20 megahertz. So there are limits on how much we can transmit. So in practice, when people design communication systems, they usually have a limit on the bandwidth that they can use. So we say, here, you can use a bandwidth of B hertz. You can't use any more. Then the challenge is, design a signal that gives us the highest data rate. We care about a high data rate. But is the best quality. And the quality will arise in minimizing errors, which we haven't talked about much yet. Let's try and illustrate that point. The idea is that the thing between the transmitter and receiver is called the medium. The medium between my laptop and the access point is the air. But the laws or the regulations of using Wi-Fi say that I cannot transmit a signal larger than 20 megahertz. The government creates a law or a regulation saying that's the limit when you use Wi-Fi. So there's a limit on the bandwidth of the medium. I think the thing that the signal goes through. If we have a limit of say B hertz, if we transmit a signal of some shape, if the signal has a bandwidth larger than B, not all components of the signal will be received. For example, if bandwidth is 20 megahertz, but I try to transmit a signal which is a bandwidth of 30 megahertz, not all 30 megahertz will be received. Only 20 megahertz will be received. The medium limits what can be received. The result is the received signal will look different from the transmitted signal. And that can be a problem. Generally we'd like the received signal and the received data to be the same as the transmitted signal, the transmitted data. But if we transmit a signal with a large bandwidth through a medium with a limited bandwidth, then what's received will not be the same as what's transmitted. And that can lead to errors. Now I'll just try and illustrate, and you've got it on your handout. We'll not draw them, but you've got it on, where is it? The next page here. Some explanation. This example, let's say that this is our transmitted signal, the square wave. Maybe it represents a sequence of 10 bits, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, similar to that. So here's our example transmitted signal. This is what I want to send to the other side, a square wave. We know that we can break that into the summation of sine waves. That's what we've just studied. So we can determine for each sine wave the frequency and amplitude and plot this signal in the frequency domain. Now what I've done using some simple software is taken this signal in the time domain and plotted it in the frequency domain. I'm not a good programmer, so it doesn't come out very nice, but you can sort of see the impulses here. It approximates. Instead of drawing a perfect impulse, you see the triangle type shape. You see that in the time domain, there are components, here's 0 to 20 kilohertz, there are components around 2 and 4 kilohertz, very high amplitude, and it gets lower as we go up to 100 kilohertz, so it's very low here. Here's our transmitted signal in the time domain and the frequency domain. I want to send that to someone. What's the bandwidth approximately of our transmitted signal? Goes from about 0 up to in this plot about 100 kilohertz. So it's at least 100 kilohertz. In fact it keeps going. If you kept plotting, it would keep going, but it'd be very small down here. It's in fact infinite bandwidth. Let's say it's 100 kilohertz. The bandwidth of our transmitted signal is 100 kilohertz. We transmit a signal. If we transmit it through a medium which has a limit of B kilohertz, what will be the received signal? The next page plots several cases. So what I do is I consider this transmitted signal through a medium of what if the bandwidth was 4 kilohertz or 8 or 12 or 16 up to 20 and that's what the next page plots. The top one, this is the time domain. The dashed line is our transmitted signal, the original signal. Had a bandwidth of about 100 kilohertz, but if we can only transmit 4 kilohertz, then really all that we get to transmit is the first components. The bandwidth which we can transmit is just the first 4 kilohertz. So this is the received signal in the frequency domain and the blue one is the received signal in the time domain. Is the received signal the same as the transmitted signal? Is the received signal the same as the transmitted signal? Receive is blue, transmitted is the dashed black line. No, it's not the same. Do you think it represents the correct data though? When you see some similarities, yes, you can see that it's sort of low at this point. It goes high here, then low here and so on. So it follows the general shape, but of course it's not identical and it's maybe not perfect. Maybe at this point, does it represent a 1 or a 0? Is it high or low? It may be hard to tell, it's sort of in the middle, whereas our transmitted signal is definitely low at this point, but our received signal is maybe up here at 0.4. So there may be some problems at the receiver in interpreting what this blue signal means. Does it mean 0 or 1? That's if our medium bandwidth was 4 kilohertz. The next plot is if the medium bandwidth is 8 kilohertz, a larger bandwidth. It means more of our transmitted signal is received. More components. The components of the first 8 kilohertz and in the time domain it looks like this blue one. Of the two time domain signals, 4 or 8 kilohertz, which one do you think is the more accurate representation of the transmitted signal? 4 or 8? Which one, which blue one looks closer to the original black one? It looks closer. By allowing us to transmit a larger range of frequencies, higher bandwidth, we can get a more accurate received signal. It's of course not identical, but the shape is getting closer. And if we increase the bandwidth that we're allowed to send, it doesn't change much in this case, but I think in some cases it gets a bit closer. The 16 is maybe even closer, especially at the peaks at the top. It seems to be flatter instead of going up and 20 kilohertz doesn't change much at all. But you can see for the later signals, the received signal is almost the same as transmitted. Similar shape. The point with a medium, with a limited bandwidth, the larger bandwidth we can use, the more accurate the received signal will be compared to the transmitted signal. That is, the received signal will be closer to the transmitted signal if we have a larger bandwidth. Or the other point, with a small bandwidth, there's a larger chance of getting errors. There's a larger chance in this case of the receiver thinking, ah, I've got a signal. It's high here. It means bit one. But what the transmitted signal was, in fact low, bit zero was transmitted. So there could be an error at the receiver. Larger the bandwidth, given all other things the same, less chance of errors at the receiver. So that example tries to illustrate that point. And maybe we can summarize some of those trade-offs that we're interested in with the signals. With bandwidth of our signal. Or the range of frequency we can use is a limited resource. We cannot use any frequency. Other people want to use them. So generally, the greater the bandwidth we use, the higher the cost involved. If you want to set up your own telephone network, maybe as a competitor to Tru and AIS, then you need to pay a license to use the range of frequencies. The larger the bandwidth of those frequencies, the more frequencies you use, the larger you pay, the more you pay for the license. Millions of baht in terms of the licenses to use the frequencies. More bandwidth, more cost. That's a problem. Generally, the more bandwidth we use, given everything else is the same, the more bits per second we can send. The greater the data rate. And if you've maybe dealt in the details with Wi-Fi, you can buy access points, the normal bandwidth is 20 megahertz. But you can buy devices which will double the bandwidth to 40 megahertz. And it doubles the data rate. Instead of being able to send it, say, 54 megabits per second, if you double the bandwidth, really use two channels at the same time, you get double the data rate, 108 megabits per second. So higher bandwidth, higher cost, but higher bandwidth, higher the data rate. So we have a trade-off there. And the last thing, if you can see down the bottom, the lower the bandwidth, the more chance of errors. We tried to illustrate that is, if we use a small bandwidth, the more chance that the receiver receives the wrong signal and makes an error in interpreting the data. High bandwidth gives us high data rate and high accuracy, but it costs more. So we need to consider those trade-offs. Everyone following? All right, so maybe if you missed the last 20 minutes or didn't capture all the points, these are the main points to pick up on. Trade-offs between bandwidth and cost and data rate and accuracy. That's why we care about bandwidth. We don't care about the frequency. Like I said, infrared will not go through walls, Wi-Fi will. Different frequencies propagate in different manners. Everything up until now we've assumed when we transmit a signal, our communication system is perfect. Nothing goes wrong, but we know in life everything can go wrong. So we need to look at what can go wrong in a transmission system. The impairments. When I transmit a signal, it propagates through some medium and then is received. What can go wrong between the transmitter and receiver? What can impair that signal? In a perfect communication system, we would assume that if we transmit a signal, the received signal is exactly the same. That would be a perfect communication system. But unfortunately, we don't have a perfect communication system. We have impairments. The real communication systems, the received signal is different from what's transmitted. Why is it different? Well, the things that call us the differences we call the impairments. Now, the three main impairments, attenuation, distortion and noise. In this topic, we're only going to talk about attenuation and noise. Distortion needs a little bit more time to explain, so we'll try and avoid that. And attenuation and noise are the two main things that we'll see arise in most systems that we deal with. So let's explain attenuation and noise. So in general, we now have a transmitter that's going to transmit a signal. So the transmitted signal comes out. The signal, we'll say, is attenuated. What does attenuation mean? Gets weaker. Something is attenuated, it gets weaker. So we'll explain that. There may be also distortion on the signal. So the signal is transmitted. It's attenuated and distorted. And then there may be other sources of energy nearby, which create noise. And we can think from the transmitted signal, which is then attenuated and distorted, we get some output signal. There's some noise. If we add the noise and the output signal together, that's what's received. So mathematically, if we look at the sine waves or the signal equations, we can say we start with a transmitted signal. It maybe gets weaker. We add in noise, and that's what we receive. And now the receiver must interpret that signal. Given that received signal, what's the data? What's attenuation? Quite simple. Signal gets weaker over distance. So I will not do it, but if I turned off my microphone, when I speak, there's a signal that comes out of my mouth of a particular amplitude. And as it propagates through the air, the people who are close out in the front will receive that signal slightly weaker from what it was when it came out of my mouth, because it gets weaker across distance. And the people at the back of the room, it will be even weaker. It'll be quieter at the back. And that's attenuation. As any signal propagates, it gets weaker and weaker. The signal strength reduces. So if this was our transmitted signal, it has an amplitude, it goes up to one here, approximately. And then we transmit this signal across some distance, and then we measure what's received. Maybe it looks like this. Same shape, but the amplitude is shrunk. Maybe down to 0.2 or whatever. So attenuation just means our signal strength gets weaker across distance. There are some details about how much it gets weaker, and it depends upon the frequency components. And different frequencies attenuate in different ways. Higher the frequency, attenuates more than lower frequency. And that leads to distortion, which we're not going to explain. So distortion is something about that the different frequencies are impaired in different ways. But we'll keep it simple and just look at attenuation. The amplitude gets smaller across distance. The further you go, the weaker it gets. Up to a point where the receiver will not be able to make any sense of the signal. So if I do turn off the microphone, you can hear me if it's off. So I'm transmitting a signal in all cases, and the signal was always propagating. It was going to your ears, but in the last case when I was talking so quietly, the signal propagated and got weaker and weaker, and when it got to your ears, it was at a level that your ears could not discern it, could not make out that signal. So there's a point such that a receiver, if the signal is too weak, it will not be able to recognize that that's a data signal. So we need to generally transmit a signal of a particular strength such that when it attenuates, the received signal is large enough such that the receiver can understand that. How much does a signal attenuate across distance? It depends upon different factors. If I talk in this room, then the signal propagates quite well through air. But when the signal hits the walls, it attenuates by a much larger factor. It gets very weak. Same as our Wi-Fi and our other signals. If you have your Wi-Fi inside this classroom, the signal will be quite good at the back. But in the neighbouring room, we may receive it, but it will be quite weak because when the signal tries to pass through the wall, it's attenuated by a large amount. There are different mathematical models and experimental models by how much the signal is attenuated, which we'll see in the next topic on transmission media. Approximately, the amount of attenuation is proportional to the distance squared. Doubling the distance quadruples the attenuation. Increasing the distance, so if you plot, you think the further the distance goes, it gets very weak quite quickly. But we'll see that in transmission media. Attenuation is the first impairment. Distortion will skip over. Interesting, but no time for this course. Noise is the other major thing we want to talk about. I transmit a signal to the receiver. The receiver receives my signal, but the receiver may also receive signals from other sources. Any other source that creates a signal that the receiver receives is called noise. Any unwanted input is noise. What's the noise in this room? When I'm talking to you, I'm sending a signal to you. What noise do you hear? The air conditioner. Other people talking, moving paper and so on. So that's all noise from your perspective. My signal is the data signal. Everything else that you hear is noise. In fact, if you turned off the air, no one spoke, there would still be noise from the environment. So there's what we call any, there's noise from phones and different sources outside. And in electrical systems, there's noise from the electrical components. So there's always sources of noise. There are different types of noise. Thermal, intermodulation, crosstalk and impulse. I think we don't need to, you don't need to remember the difference between all of these. We'll go through the four, but it's not so important to know the names of them. Just be aware that there are different sources. Thermal noise in a communication system is always present. Electrons move with a different temperature and that creates some very small amount of noise and you can calculate how much. So there's always thermal noise, but it's usually very small. Very, very low that it has little impact in many cases. Similar, if you turn off the air and the lights and everyone stopped talking, there would still be a small amount of noise basically from the environment. Intermodulation noise is when we transmit a signal which contains different frequency components. Some frequency components cause noise on others, interfere with others. So from a single signal transmitted, then some of the components can interfere with other components. Usually that's caused in practice if there's a problem in the transmitter. The transmitter has an error or is an electrical problem, it may cause such noise. Or we try to transmit too high. I will not do it, but if we turn the amplifier up very high, then the different components can cause some interference or noise on each other. Cross talk is when we have different sources transmitting. Quite simple, I'm talking and then someone else is talking at the same time, then the receiver is receiving my voice and the other voice at the same time and they interfere. So the other voice can be considered noise. When two transmitters transmit in the same vicinity, then the receiver can hear both transmissions and one of them is considered noise compared to the other. In communication systems, maybe you have a cable and there's a signal transmitted along the cable and there's another cable nearby lying next to it, which also has a signal transmitted. They may cause interference or noise on each other, cross talk is the name there. And the last thing is usually some impulses from some disturbances like lightning hits some communication system or some electrical disturbance and there's a high peak of noise and that can cause problems as well. We say noise is additive. So the noise from the air conditioner, from the lights, from people talking, from outside. What you receive, the total noise you receive is the summation of all those components that produce noise. We add them together to get the total noise. One way we can illustrate that, considering just attenuation and noise, we transmit a signal. This is the original signal, the black one. Attenuation means the signal gets weaker. So after attenuation, this blue one comes out here. Same shape, but just different height. Then there's maybe some noise sources, maybe from the environment. Maybe for some period of time there was some small electrical disturbance where the noise went high. Often the noise varies randomly. So this is a plot of the signal noise. If we add the red one to the blue one, we get the purple one. That's what's received. So we say it's additive here in that the attenuated signal plus the noise equals the receive signal. So the receiver receives this. Let's say the data transmitted was 1, 0, 1, 0. What's received? Well, the receiver needs to look at this and see, is the signal high? Okay, it looks high, one's received. Here it looks low most of the time, maybe zero's received. Although at this point it's high. So there may be a problem here. What's received may not tell us what the correct data is. So the receiver takes this signal and tries to interpret what was the data. The more noise, the more attenuation, the more chance the receiver will make errors. It will think it's got data which wasn't transmitted. And I think that's sufficient to cover the impairments that we need to look at. So primarily attenuation and noise. There are other components but that's in this introduction and that's the main impairments we'll look at. Next week we'll summarize on what was learned about signals and we'll finish on looking at some equations that relate bandwidth with data rate. Some capacity equations. Let's do that next week.