 We've introduced some concepts. What do we mean by data communications, links, networks? We've talked about signals and then we've looked at performance metrics in the previous topic. Now we're going to go into the details of the basics of communications and look really at signals. And look at signals, communication signals in a mathematical perspective in this topic. And then in the next topics, a few more details about how do we take, let's say our data, think of bits. If we have digital data, how do we take a sequence of bits and convert them into some signal that can be sent across some link as an electrical signal or generally electromagnetic signal. Many of the examples I'll use will talk about transmitting digital data. Zeroes and ones is the data we want to communicate whether it's a file, like a text file or a video. It can be converted to a binary and then that's the data. Then we convert it to a signal to be sent across a link. But the same signals are used for analog data as well. Like when I'm talking, I'm communicating analog data to you. It's not digital. So we'll see examples of data as we go through. Data transmission. Let's first introduce some simple terminology. We're focusing on a link, a single link. There's a transmitter and a receiver. And we want to get data from the transmitter to the receiver across some link or generally across some medium, the thing between the transmitter and receiver. And what's transferred, so we want to get data from transmitter to receiver, the way that we do it is we send some signal, some communication signal, which is in the form of electromagnetic waves, an electromagnetic signal. And the examples I'll use think of a sine wave as a very simple representation of electromagnetic wave. But electromagnetic actually contains two components. The electrical component, electro, as well as the magnetic component. A magnetic field and electric field. But we'll just think of it as sine wave and we'll use that as the concept. So get data by sending a signal from transmitter to receiver via some medium. What types of medium? Well, we've got classified the medium as either guided or unguided. Guided is when that signal is, as the name suggests, guided somehow by some materials in some direction. And really that means wired. So the signal may be electricity flowing through a copper conductor. And the signal effectively is maintained within that wire or with the insulation around that wire. The signal does not disperse much. So we'll look in the next topic about these media in detail like copper wires called twisted pair, the LAN cables, coaxial cable which you may use for cable internet or cable TV, an optical fiber, and so examples of guided media, wires or cables. The signal, think of the signal is guided within the conductor or within the wire. Unguided is wireless. So we don't have a material, we transmit our signal through the air. No wires. Through the air, through the water in some cases or in a theoretical case through a vacuum. So guided is wired, unguided is wireless. So we're focusing on getting from transmitter to receiver across a link that may be guided or unguided. Usually in most of our cases or examples we'll refer to one device sends to one other device, a point-to-point configuration. Two devices share the medium. We have a cable from one computer to another. The medium is that wire where our signal propagates across. It's shared by the two devices which are connected. So the wire connects into the LAN cable, LAN port of my laptop and into the LAN port of this PC. There's two devices sharing that medium. That's the most typical case we think of. But we can have multi-point configurations where there are more than two devices sharing a medium. An example? Someone? What's an example of a medium where we have multi-point communications? Air. Okay, wireless. Typically, not always, but we typically think of wireless communications as multi-point. Some cases we'll think of it as point-to-point. That is when I'm talking to you, I'm generating a signal coming out of my mouth and there are multiple receivers of that signal. So it's not one-to-one, it's not point-to-point, it's one-to-many or multiple points sharing this medium. So different configurations of our links. Point-to-point generally is the easiest. There's not much we need to do to get that to work. With multi-point, we need to deal with the fact that there may be some interference. It's also the case in point-to-point, but more so in multi-point communications that it becomes a little bit more complex to make sure it will work well. And when we communicate, we also look at the direction. And the terminology we use is simplex, half-duplex and full-duplex. Simplex is a communications medium where we send in one direction only. For example, broadcast TV. There's a TV station somewhere, they have a transmitter, some TV tower maybe, or if it's satellite TV, they go via the satellite and they transmit a signal and your TV with the antenna receives the signal. So it's going from the TV station to your TV. Your TV does not send anything back to the TV station. So it's simplex communications. Full-duplex at the other end is a, think of a link where we can send from A to B and at the same time be sending from B to A. So both directions at the same time. Half-duplex, both directions are possible but only one at a time. A can send to B but if A is sending to B, B cannot send back to A. They must wait until A is no longer sending to B. That's half-duplex. Handheld radios, sometimes called CB radios or private radios that are an example where you press a button to talk. You press the button, talk. The other person hears and listens. Then you release the button and then they press the button and talk back to you. So we can communicate in both directions but only one person at a time. That's half-duplex. Any questions about this introductory terminology? What is this lecture? Which medium, which configuration, which direction? The lecture, which medium do we use? Guided or unguided? When I'm talking to you, when we're lecturing first, medium unguided, we're using wireless communications. Well, there's some form of wired communications if we use the microphone but assume there's no microphone, if I just talk then wireless all the way from my mouth to your ears. Configuration, this lecture, point-to-point or multi-point? Multi-point. One transmitter, multiple receivers, more than two. Direction. Hands up for simplex. Hands up for full-duplex. Hands up for half-duplex. I prefer half-duplex. If I talk, you don't talk. But if you have a question then do so but don't interrupt other people. So we prefer half-duplex. Sometimes it turns out people get excited and talk a lot and it becomes full-duplex. But let's not make it simplex. Make sure you ask some questions during the lecture but try not to interrupt too many other people. For some reason I've never fixed the title of this slide. It's wrong. It's not appropriate. Maybe a better title for this slide, and you can change it, is electromagnetic signals. It's a copy and paste that I didn't fix at some stage. Electromagnetic signals. Those three things, a frequency spectrum and bandwidth, come up later. So the signals that we use to transmit the data from transmitter to receiver, the transmitter generates an electromagnetic signal. So my LAN card generates a signal and outputs it in the case of a LAN cable. The LAN cable has copper wires inside. So it outputs an electrical signal under those copper wires and that signal propagates to the receiver. The signal represents the data and I think we've mentioned before a simple case would be if you think of a signal, a high signal could represent bit one and a low signal, say in terms of voltage, could represent a bit zero. We'll see some other schemes. In this topic we're going to look at the mathematics of those signals and see that a complex communication signal can actually be broken down into simple signals that we can analyse and design. If I ask you to draw in an exam or a quiz a signal, what would you draw? A simple signal. Someone does this, a digital, a square wave, and another one, a sine wave, a sinusoid. So that's what we'll deal with and we'll see that even complex communication signals like sending a radio signal from a laptop to this access point, we can think in the perspective of just a set of sinusoids, sine waves. So that's what we'll go through today. The two domains we'll come to later. We'll focus on the time domain. I'll introduce the frequency domain after we cover the time domain. Very simple stuff today. Some waveforms or example signals we can distinguish between analog and digital. Analog. It varies in a continuous manner over time. Digital, discrete manner. So digital signal maintains some level for some period of time and then instantaneously changes to another level. And so on. So just the difference between analog and digital in terms of a signal. And if you look at these plots, this axis is time. So things are changing over time and this axis is shown as volts, sometimes also power watts. The signal amplitude, the height of the signal, measured in volts typically. If you think of an electrical signal coming out of my laptop, you can measure the signal amplitude. Any problems, analog versus digital in terms of signals? Continuously varying versus the discrete variations. Going back to simple signals. Some concepts. This is from high school physics, so this is nothing new to you. We can talk about periodic signals. These two are examples of periodic signals. They repeat. And these two are examples of aperiodic signals. There's no repetition. They continuously change. In much of our analysis, we'll look at periodic signals. But in practice, aperiodic signals are common. Nothing interesting there, period. Let's look at a simple sine wave. Again, a refresher from when? High school. Sinusoids. An equation for a sine wave. The signal strength has a function of time. I have my pen. I have a flat battery, I think. A mouse. The signal strength has a function of time. S is the signal strength of the amplitude in those previous plots. T is time. Generally, it can be written as the amplitude, or the peak amplitude, A, times by a sine, the sine function, of 2 pi f t plus where 2 times pi f will be the frequency of the signal. T is time. And phase c is related to the relative position of that signal to some starting point. To zero. Three components, or three key parameters in this equation. Sine is just a function. 2 and pi are just constants. T is the input parameter. We have peak amplitude A, which is a multiplier on the sine function. We have frequency f, which will be the rate at which the signal repeats. And we have the phase phi here. Let's just remind you of how those impact on the shape of a signal. These are four plots of that equation with different values of the f and the phase. And it's probably easier to read, maybe a little bit easier, but it's hard to read on the screen. The top left plot, if you read here that amplitude is 1, the frequency is 1, and the phase is 0. So if you plug those values into this equation, A is 1, f is 1, and phase is 0, it becomes what? Sine 2 pi t. Sine 2 pi t, where t is time. And in our plot, time varies in this case from 0 to 1.5 seconds. So sine of 2 pi t, if t is 0, sine of 0 is 0. If t is 0.25, then sine of 2 pi t is an output of 1. And we get our sine plot. What's the period of this signal? What's the period? 1 what? No? 1 is correct, 1 something. What is the value of the period? 1 second. The period, remember, is the duration in time of one cycle. So one wave. It goes up, comes back down, and back to the origin here. So the period is from 0 to 1. Period is 1 second. How many times does a cycle repeat per second? How many cycles per second? In 1 second, how many cycles? 1. Therefore the frequency is 1. The frequency is the measure of the number of cycles per second. The repetitions per second. Frequency is 1. The period is the inverse of the frequency. If a frequency is 1, the period is 1 divided by 1. Also 1 in this case. The amplitude, or actually I always call it amplitude, is the peak amplitude is the maximum height. It goes up to plus 1 and it comes down to minus 1 here. The phase we'll see as we get to the other ones. It's the offset from the origin. When time is 0, our signal is 0. If we change the phase, that will change. Move to the right. Same equation except peak amplitude A is 0.5. Same frequency, same period, same phase, but you see the amplitude is shrunk to half. That's all. Bottom left. Peak amplitude is 1. Frequency is 2. Within one second there are two repetitions. Two repetitions per second. Frequency of 2. Two cycles per second. How do we measure frequency? How do we measure frequency? What are the units? Hurts. A frequency of 2 hertz Z is the symbol of the abbreviation. Frequency of 2 hertz 2 repetitions per second. Period of what's the period of this bottom left signal? A half a second. Period is 1 divided by frequency. If the frequency is 2, the period is a half. This is nothing new to all of you, I'm sure. The time for one repetition is half a second. Any questions on the first three? They just illustrate the impact of the peak amplitude and the frequency. And the last one illustrates the impact of the phase. Ampitude 1, frequency 1, phase, what we call a phase offset of pi over 4. Phase is a measure of angle measured in radians. Pi divided by 4 radians. You see the impact on our waveform compared to the top left and the bottom right. It shifts it along. So think of this one. It's been shifted back some portion relative to this one. So we don't start at 0, we start at 0.7 something with this phase offset. The offset's in time. The original sine wave. Any questions so far on sine waves? Remember that equation. Let's have a look at some. Why do we introduce this? Because our signals will represent as sine waves, sinusoids. But more complex than just a single sinusoid, what we'll eventually come up with with communication signals is if you add multiple together, you get another wave. And if you add multiple together, you can start to generate communication signals to represent the data that we want to send. So that's what we'll end up with with this topic. We combine sinusoid together. Let's go through a few examples. And I'll try and plot some. And you can answer some questions about them. I need to remember how to do this. Objective. I've just got some mathematics software that will produce some plots of sine waves. Just so you can see the impact of the different parameters. And then we'll start to combine them. Just so everyone's clear. But to run this software I need to set things up. So just bear with me as I set up the picture so it's nice to display. We're going to create some plots. I'll do it a bit quickly. I'll copy and paste some commands that I did before and then it will make sense. Just set up the axes. The labels. And we're going to plot a sine wave from time 0 to 1 second. Very simple. And set the axes. Access values. And let's get ready. Start simple and then you don't have to worry about those commands. I was just setting up this picture. Now I'm going to create a plot. On the x-axis we'll have time. And then on the time here, on the y-axis we'll have the signal. The signal strength. Our sinusoid. The notation may be a little bit confusing to get started with but you'll make sense of it. 1 times sine 2 times pi times 1 times t plus 0. I'll zoom in then we'll zoom back out so people can see. I'm going to plot time versus our signal. Where the signal the peak amplitude a this multiplies 1 I'll set it to 1. Sine of 2 times pi the frequency I'll set to 1 and the phase to 0. The simplest signal we can start with. And I'll just make the color of that plot to be blue. Exciting? Same as what we saw before. Let's just vary a little bit. Let's change the frequency from 1 to 2. 2 times pi times 2t the general formula 2pi ft And you see the frequency changes and we can change maybe a green one we can change the peak amplitude to say 1.5 and also change the frequency to 3 what are we going to see? The frequency is 3 the peak amplitude is 1.5 try and think of what you're going to see. Within 1 second 0 to 1 the frequency is 3 so we see 3 repetitions of that sine wave and the peak amplitude is 1.5 it goes from 0 to plus 1.5 down to minus 1.5 Let's try and look at the phase a little bit because that's maybe the hardest one to visualize or to recognize what the impact will be. Let's start again with a clean plot. Let's try again our first one we had a phase of 0 plus 0 plus nothing here. Let's change the phase keep everything else the same a red one the phase is measured it's an angle it's measured in radians so not in degrees you can convert it to degrees if you like measured in radians pi over 4 how many degrees how many degrees pi over 4 5 a circle is 2 pi 360 degrees is 2 pi so pi is 180 pi on 2 is 90 pi on 4 will be 45 degrees so a phase offset you can see the shift you can think it's shifted back in time pi over 2 a phase of 90 degrees it looks it back even further another 45 degrees 3 pi on 4 and pi what will we get? a phase of pi what will our side wave look like? anyone plot it with their finger? a phase of pi what shape will it be? should be upside down that is goes down first then up all back to this point where we go down first and then up a phase of 2 pi 2 pi is the same as 0 all the way around 360 degrees is the same as 0 in terms of the phase offset in a side so we can vary let's summarize with a sinusoid we can vary 3 parameters the peak amplitude sometimes just referred to as the amplitude the frequency and the phase and in fact what we're going to study in signals is that there are many different approaches of varying those parameters to create different real signals who's listened to radio before? what channel? not internet radio FM, what's FM mean? frequency modulation the other form of radio AM, amplitude modulation so with radio there are two common approaches where to send the audio radio is audio, music and people talking to send that as a signal from the radio tower to your car or your radio receiver they with FM they apply some approach which involves changing the frequency of the some signal sent, frequency modulation and with AM they change the amplitude of the signal so we get amplitude modulation we'll see them in the signal encoding techniques and some other ones which again just changing these 3 parameters amplitude frequency and phase someone I asked before how do you draw a signal and someone drew a square wave can you create a square wave using sine waves? a square wave a digital wave form let's try because that's what we think of in terms of what we call digital signaling we can send plus some voltage for some time and maybe zero voltage or minus some voltage and hold that level to create this digital wave form let's clear this and try some other things now for these demonstrations let's set the phase to zero so I'll remove the plus zero just to ignore that and we'll set the in the initial one I'll set the peak amplitude instead of typing one times there's no need to type one times it's just sine just to save some space sine 2 pi and for an example let's set the frequency to 2 what color is k? black here's our base signal it has a frequency of 2, 2 hertz 2 repetitions within 1 second let's try a different one let's change the peak amplitude to be 1 third instead of 1, 1 third and let's change the frequency instead of 2 hertz to be 6 hertz so it's 2 pi 2 times 3t or 2 pi 6t go back to our lecture notes the general formula peak amplitude sine 2 pi ft 2 pi t times the frequency so in this specific example 2 pi 6t frequency is 6 you'll see why I write it as 2 times 3 in a moment that'll make more sense then just a difference sine wave instead of the height the peak amplitude is 1 third 3 times the frequency frequency of 6 so you see 3 times the number of repetitions in 1 second just 2 sine waves now we'll add them together so what do we get if we add the 2 together anyone want to predict or draw the shape add the 2 together that's a little bit harder to visualize but if you look at the shape at each point in time add the 2 values together so at this peak point it's the value there 1 plus whatever this is minus 0.3 so we 0.7 so add them together and see what we get so take the first one and I'll add the second one and it's the blue one is the shape that we get when we add those 2 sine at size together you can see it's not this smooth sine wave anymore the peak has some 2 humps at the top because of this 1 third small one added together produces that shape it starts at 0 it crosses 0 here goes down and then back up at 0 and so on what's the frequency of the blue signal what is the frequency of the blue signal in 1 second you can see it goes up it comes back to the origin and then comes back to the origin here 2 repetitions in 1 second it's a frequency of 2 hertz of that blue signal the first sine wave had a frequency of 2 hertz the second one had a frequency of 6 hertz when we added them together the resulting sine wave had a frequency of 2 hertz we'll see that we'll take advantage of that as we go through why do we do that let's see why we do it let's add some more to that one I'll draw it again but without the 2 components I'll draw that one again but just the resulting wave so that was the resulting wave just to be a bit clearer the frequency of 2 hertz 2 repetitions in 1 second peak amplitude is it 1? not quite okay so the peak have to be careful there I want to change the peak amplitude same plot exactly the same equation except I'm going to multiply everything on times by 4 over pi it's going to be 4 over pi sine 2 pi 2t plus 4 over pi times 1 third sine 2 pi 6t everything's multiplied by 4 over pi what will that do this will be a red one what will the red plot look like can you visualize what the red one when I press enter will look like multiplying by something out the front what does it do just changes the peak amplitude increases the height as we saw here we change the peak amplitude from 1 to 0.5 the height was cut in half here we're just changing the peak amplitude by multiplying by 4 over pi just move things up okay why did I do that it'll make sense soon so there's still a question why would I multiply by 4 over pi it'll make sense as we go through the next few steps so let's focus on the red one we've got 2 sin added together sine 2 pi 2t 1 third sine 2 pi 6t add them together multiply by 4 over pi those 2 sin I'll call them 2 components of the resulting signal the resulting signal is made up of 2 different components let's add a third component and see what shape we get and then we'll come back and explain why we're doing this so I'll clear this figure to make it a bit nicer we'll plot an original there's our original sine wave but for now I'm multiplying everything by 4 over pi so it's gone up above 1 there 4 over pi is about 1.3 and then our second one 2 components we added this red one we added an extra part of 1.3 of the original one and 2 pi 6t and that's what we got there 4 over pi sine 2 pi 2t plus 1.3 sine 2 pi 6t the way that I wrote it you'll start to see some pattern arriving 1.3 sine 2 pi 2 3t the 3 and the 3 here are related the way that I've chose them is that they're the same I want to add another component a third sine wave I'm going to add one which is one fifth of the original size a smaller one and the frequency will be 2 pi 2 times 5t and instead of typing I'll just copy and paste so I don't make mistakes it wraps around but the 3 sine waves sine 2 pi 2t 1.3 sine 2 pi 2 times 3t 1.5 sine 2 pi 2 times 5t and in green we get this what shape is the green one starting to resemble the original one the blue one was the original sine wave smooth there's a couple of humps at the top 2 humps there not as smooth as the sine wave the green one actually has 3 humps at the top the way that I've chosen adding these sine waves together is producing a shape and what I want to do is add more sine waves until I get a shape of a digital wave form square so that's why I've chosen the values in this way because I know that if I keep adding these sine waves in such a pattern the resulting signal will be a perfect square wave form trying to demonstrate the point that what we can do is create a digital wave form just by combining sinusoid and in fact any communication signal doesn't matter the shape or the holding signal we can think of it as made up of adding together sine waves and that makes the analysis of communication systems much easier let's add another component same as before 1 5th sine 2 pi 5 t 1 7th sine 2 pi 2 times 7 t you see the pattern 1 3rd 2 times 3 1 7th getting smaller and increasing the frequency of those individual components it's a bit harder to see there I can keep adding components but to illustrate the point I've created one with something like 30 odd components took me a long time to type this one goes up to I think 59 okay the same pattern 1 over 27 2 times 27 1 over 29 2 times 29 we'll see the pattern in a moment and we're close to a square wave there are some variations here but the black one we're getting close to a perfect square wave how do we get a perfect square wave how do we get it even nicer than this what do I do how do I get it so that those it doesn't go up and down just on those edges of the square wave sigma is just the sum of here I'm typing the actual sum plus plus plus and then do what add more what I call components add more these sine waves 1 over 61 sine 2 pi 2 times 61 t add another one how many should I add if I add an infinite number of components you'll get a perfect square wave but it would take me too long to type an infinite number of components so in theory adding an infinite number of components in this pattern produces a perfect square wave any communication signal can be made up of combining sinusoids together let's let's look at some of the properties of these resulting signals in all of them what's the frequency it's the same it's still 2 hertz didn't matter how many components we added there the resulting signal is still 2 hertz it's still 2 repetitions per second the period is still half a second this resulting signal why is it 2 hertz maybe you can see in the pattern if you can almost see here I type 2 pi 2 times 57 2 pi 2 times 59 2 times 53 and so on the original sine wave was 2 pi 2t the frequency with 2 each component is a multiple of 2 an integer multiple of 2 if we have such a resulting signal where each component is an integer multiple of the frequency of the first component then the resulting signal is the same as the first one and that resulting well that that first frequency these 2 hertz in this case is called the fundamental frequency and the other frequencies like 2 times 55 110 hertz are called harmonics, harmonic frequencies those details are not so important although the fundamental frequency will be important when we do some analysis what can we do next come back to our slides let's introduce some concepts make note of some things we'll do this again and then write down an equation to represent some of these start with this one this is the red one from the same as before let's write the equation for this and then look at some of the characteristics of this one try and write the s of t equals you can almost copy from the software there try and write the equation in a nicer way s of t equals 4 pi all of this 4 pi times by sine 2 pi 2t plus 1 third sine 2 pi 6t I'll just write that and then we'll analyze or talk about some characteristics of that equation I'll call it s2 of t because we'll have some others later we had 4 over pi was that multiplier and all inside we had the first component of sine what do we have 2 pi 2t or simply 4 pi t plus 1 third sine it was 2 pi 2 2 times 3t or 12t 12 pi t and close the square brackets that's the signal equation for the one that we just plotted with the two humps combination of two components but instead of the full 2 times pi times 2t just 4 times pi t and you see the first component has a peak amplitude if you expand the multiplication 4 over pi sine 4 pi t remember the general equation a sine 2 pi ft plus the phase ignores the phase the phase is 0 in these examples so a sine 2 pi ft what is a 4 over pi if we look at the two components the first one we'd say here a is 4 over pi and what is the frequency of this component someone said 4 someone said 1 this component this one we split it into two sine waves the general equation is a sine 2 pi ft f is the frequency so what value of f do we need to have 2 2 pi ft 2 times pi times 2 so the frequency of the first component is 2 hertz the amplitude I should write units of say volts but I'll be lazy and not write the units we don't care at this stage sometimes we represent as volts other times watts some power level the second component and the phase was 0 it's right at just to be complete the second component the peak amplitude is 1 over 3 4 times pi times 1 over 3 let's keep it as that frequency of 12 pi t is 6 hertz because the general formula 2 pi ft don't forget that 2 phase is also 0 so if we add 2 sinusoid with these parameters together we get this plot okay that's what we did there note let's look at the frequencies the first component has a frequency of 2 hertz the second component has a frequency of 6 hertz which is an integer multiple of the other this is 3 times 2 if we have such then we can say the resulting signal the fundamental frequency of our resulting signal s2 of t equals 2 hertz if the two components frequencies such that the others are integer multiples of 1 then that 1 is the fundamental frequency of that s2 of t or the frequency of that signal the frequency of s2 is 2 hertz and we see that here we can say now that s2 contains two components a component with frequency 2 hertz and a component with frequency 6 hertz and they have different amplitudes we chose the amplitudes to get that nice shape such that if we keep adding them in that way we would eventually get a square wave but it's not always like that the range of frequencies inside s2 or the set of frequencies sorry don't copy the set of frequencies the set of frequencies in s2 of t are 2 and 6 hertz that is s2 of t is made up of two sinusoids one has the frequency of 2 hertz one has the frequency of 6 hertz so we can say the set of frequencies in this signal are 2 and 6 hertz we call that the spectrum of the signal the set of frequencies inside a signal is called the spectrum so we can say the spectrum of s2 is 2 and 6 hertz so we're introducing some characteristics that we'll use to talk about signals the spectrum is one thing the fundamental frequency will refer to this sometimes the let's say the width of the spectrum is that the spectrum contains 2 and 6 hertz the width is the difference between the minimum frequency component and the maximum in this case there are just 2 2 and 6 what's the width what's the difference between 6 and 2 4 so we say the width of the spectrum the spectrum ranges from 2 all the way up to 6 the width how wide is it what's the name of that we refer to the width of the spectrum as bandwidth the width of the band of frequencies so we say the bandwidth of s2 is 4 hertz and this will be an important parameter when we talk about real signals the bandwidth of a signal so with any communication signal doesn't matter how complex the shape is in theory it can be broken down into the addition waves of sinusoids and each of those sinusoids have a frequency so we can determine the spectrum of that signal and the difference between the minimum frequency component and the maximum frequency component is the bandwidth of that signal so we can talk about for some signal it has some bandwidth and as we go through this topic we'll see that the bandwidth can impact upon some practical things like data rate how many bits can we send per second the bandwidth will have a role in that but we're not there yet any questions before we do a next example simple concept so far but just looking at the very fundamentals of signals let me know if there's some errors in the calculations or there's some questions about the writing let's go back to our plot that was with two components I'll do the green one with three components sine 2 pi 2T one third sine 2 pi 2 times 3T one fifth sine 2 pi 2 times 5T all multiplied by 4 over pi at the front the green one which has those three pumps at the top and we kept adding components until we got close to a square wave write the equation for the green one and determine its spectrum fundamental frequency and bandwidth the question how do we get the fundamental frequency S2 of T the red one had two components frequency of 2 hertz and 6 hertz this component is an integer multiple the frequency is an integer multiple of this component this frequency 6 hertz is 3 times 2 hertz in that case when all the components they're all integer multiples of another that one is the fundamental frequency this is 1 times 2 this is 3 times 2 the fundamental frequency therefore is 2 and the signals in these simple examples will have a fundamental frequency that we can easily determine write the equation for the green one write the equation determine the spectrum and bandwidth of the green signal it's the same as the previous one plus another component write it down just so you understand the structure of this signal and maybe if it's hard to read it's hard to read in the for this software I'll help you by writing it it's the same as the first one but there's a third component 1 over 5 see if we can fit it in let's call it s3 of t I'm going to have to squeeze some things in here because we'll run out of space 4 over pi was the multiplier at the front what do we have 4 pi t plus 1 third sine 12 pi t that was the same as before plus 1 fifth what's the the third component 1 fifth sine what if you can't remember from the software you can work it out by the pattern the first component was 2 pi 2t 2 pi 2 times 3t 2 pi 2 times 5t 20 hertz not 20 hertz 20 pi t 3 components the peak amplitudes of component 1, 2 and 3 just the multipliers out the front find the frequencies of each component the frequencies of component 1, 2 and 3 2, 6 and 10 from before 2 and 6 10, 20 pi t the general formula is 2 pi ft so f must be 10 10 hertz 6 hertz then find those 3 parameters of that resulting signal how do we get 20 pi if we look at our plot let's see if I can zoom in a little bit maybe hard to see but the way that I created the plot sine 2 pi the last 5 minutes let's keep it down and just explain so everyone's clear just wait 1 minute just so others can hear the way that I created it I followed some pattern ignore the multiplier for a moment sine 2 pi 2t our general formula is sine 2 pi ft so f is 2 in this case the next one was 1 third of the height and 3 times the frequency sine 2 pi 2 times 3t or sine 2 pi 6t that's the pattern I was following 1 third of the height 3 times the frequency of the first component and the next one and I have to zoom back out where to go 1 fifth sine 2 pi 2 times 5t see the pattern 1 third 3 times 2 1 fifth 2 and if I added another one it would be 1 seventh 7 times 2 the frequency of the second component is 6 2 times 3 of the third component the frequency is 10 2 times 5 not all signals have that pattern I just chose it so that if we kept going like that we would get a perfect square wave fundamental frequency component 1 2 hertz 2 frequency of 6 hertz component 3 10 hertz they're all integer multiples of 2 hertz 2 times 1 2 times 3 2 times 5 so they're all integer multiples of this other one f1 so the fundamental frequency is still 2 hertz in this case the spectrum is the set of frequencies inside this signal and I'll just write it as 2 6 and 10 hertz giving us a bandwidth of what so I've heard 2 numbers 8 and 4 I think 10 minus 2 the width so from the minimum to the maximum the bandwidth is 8 hertz and if you draw the next one with another component we will not 1 over 7 what would the fourth component be 1 over 7 2 pi 2 times 7 T which is what 28 I will not write it down but if we had 107 2 times pi 2 times 7 2 times 7 is 14 times 2 28 pi T the frequency of the fourth component would be 14 hertz 12 hertz so we'll stop there we'll continue next week and then we'll start to see that the bandwidth of a signal can impact upon the data rate how fast we can send bits and it may also impact upon the cost of using that signal so we'll come back to see why the bandwidth and spectrum are important next week questions