 So, the first thing to look at is when we transmit, we want to transmit some data between two devices, look at the very basics in terms of the physics of how do we transmit some information between devices, well we use some form of signals, some physical signals and that's what this topic is going to mainly look at, the types of signals, the issues with the designing communication signals that can be sent between two devices. First some terminology, think of in the simplest case we have two devices connected via some link. We transmit data from one device to another, the source device is called the transmitter, it transmits the data, the receiver device receives the data and in between the transmitter and receiver is the medium. What is actually sent between transmitter and receiver? Some signals which are in the form of electromagnetic waves, so in basic physics we have some wave form carrying energy and they are our signals and these signals must carry some information, for example bits, zeros and ones, so between transmitter and receiver is a medium, well we can classify this medium into two types, guided and unguided. A guided medium is when the signals is guided by some material, the example is wired mediums, wires, cables, so we have a copper wire inside a LAN cable, again I don't have the LAN cable but a copper wire inside a LAN cable with some plastic coating around that copper wire, if we transmit electricity across that copper wire, an electrical signal, then that signal is maintained along the copper wire and does not disperse, or does not disperse much, you can think that signal as some wave form is guided along the copper wire. Similar if we have optical fiber, our signal is light, with optical fiber we have glass or plastic fibers and we send light into that fiber and it bounces over both sides and traverses through the optical fiber, the light, the signal is contained within the optical fiber, it doesn't disperse or not much at least, so that's a guided medium in that the signal is guided along a particular material, so wires and cables, twisted pair, copper cables, coaxial cable, optical fiber, we'll talk about some of these technologies in the next topic on transmission media. Unguided is essentially wireless communications where we transmit some signal and transmit it across the air, maybe through water or vacuum in theory and those signals may disperse in multiple directions, they're not guided by a particular material, by some cable or some wire, so when my laptop is using Wi-Fi, wireless LAN, it sends a signal to the access point up on the wall and you can think the energy there's an antenna built into the laptop, the energy is transmitted out of the laptop, some signal and it disperses in fact in multiple directions, if we could see that signal we would see some energy is traveling as a wave form to the access point but also the signal is being transmitted in that direction up, down and essentially all around, so the signal is not guided inside a particular medium, it's unguided in this case, so it's actually wired versus wireless, guided or unguided communication mediums, so this is the link between transmitter and receiver, we're just classifying and introducing some terminology, this link that we set up between transmitter and receiver we can configure it in two different ways, it can be a point-to-point link where we have one medium, one transmitter and one receiver, so just two devices sharing that medium or a multi-point link where we have more than two devices, multiple devices sharing the medium, so commonly we will see point-to-point links, I connect a cable from my laptop to this PC, two devices sharing that link, sharing that medium but we also see cases of multi-point communications where one device transmits and multiple devices may receive and therefore we have multiple devices sharing that medium and depending upon what we want to achieve in our communications we may choose one or the other, they have advantages and disadvantages, in either case when we have a communications medium, a link, the direction of communications may be either simplex, half duplex or full duplex, simplex communications is that the medium transmits signals in one direction only, the communication system just transmits in one direction from A to B, an example television with TV there's a TV station that transmits some signal and your TV at home has an antenna maybe that receives the signal, your TV does not send anything back to the TV station, so that's a simplex communication system, the data is all traveling in one direction, full duplex at the other end point is that we use the medium to transmit in both directions and at the same time that is we can transmit from A to B and at the same time using that medium we can be transmitting data from B to A, so both directions at the same time and in the middle half duplex both directions but only one at a time, so we have a link, A can transmit to B, B can transmit to A but they cannot transmit at the same time it's either one or the other, one direction or the other, so the direction the different technologies and systems maybe either simplex, half duplex or full duplex, this lecture when I'm talking to you are we using a guided or unguided medium, imagine the microphone is off, are we using a guided or unguided medium, unguided the medium is air in this case if the microphone is off I'm talking and the audio signal is in fact going in all directions it's not going across a cable it's going in all directions and those within range will receive and hear it, point to point or multi-point this lecture hands up for point to point let's see who's following hands up for multi-point hands up if you don't know hands down if you don't know that confused you okay it's multi-point that is you think of when I talk there are multiple receivers okay it's not one device just one other I'm the transmitter we have 20 or so receivers in this communication medium so it's point to multi-point or simply multi-point medium simplex half duplex or full duplex this lecture half duplex sometimes it's full duplex sometimes it's simplex it should be half duplex this lecture it means if I'm talking then no one else should be talking but if you have a question then of course you can be sending information back to me full duplex would be I'm lecturing and you're talking to your friends okay we try and avoid full duplex simplex is just me lecturing and lecturing and you falling asleep so let's try and keep it half duplex for this lecture so you can ask questions so what we want to focus on really in this topic is the signals that are sent from transmitter to receiver what are these signals how are they designed how do they carry information or some electromagnetic signals sent across our medium and we use these signals to represent data if you think of the data if we want to send a file as a sequence of bits my computer wants to send bits to some destination then I need to generate some signal that goes out of the computer that represents those bits we'll look at different ways to do that what we're going to see is that the communication signals that people use to transmit across systems we usually design them to be made up of multiple simple component signals that is we take some simple signals and combine them together to get one more complex signal and we're going to look at the mathematics of that and see how that impacts on our signal designer and performance we'll see that we can analyse signals from two different perspectives two different domains the time domain and frequency domain time domain you'll be familiar with frequency domain we'll explain as we go through starting with time domain and let's look at a very basic communication signal actually we still got a few more some terms to introduce and I think you know this thing difference between analog and digital you've seen this in in computing courses and in general physics courses okay we can differentiate between analog and digital waveforms analog the signal is continuous over time it continuously varying a digital signal we think is that it's some constant level then instantaneously changes to some other level like this square way the difference between analog and digital a simple concept we can have periodic or a periodic signals periodic as a signal that repeats both of these examples are periodic signals a periodic is one that doesn't have repetition over some period of over some time frame there's not an example here with a periodic signal we can measure its period and we'll see that through some further examples so I want to send information from one computer device to another using some form of signals whether it's light electricity some radio signals over wireless the person who designs the the transmitter and receiver designs the signals that are sent and how they carry information and the very basic way in which they design signals is based upon a sine wave so think of the simplest signal and usually come up with a sine wave something that is varying like this so you know the shape of a sign so a very simple signal s as a function of time can be expressed as a sine wave and here we can vary the shape of this sine wave using different parameters there's a multiplier at the front which is the amplitude if you can imagine a sine wave if you multiply by some value at the front then you increase the height so the amplitude of that signal a sine wave continuously varying you know it has some frequency in some period we can change the frequency from a low frequency to a higher frequency by varying some parameters so the sine function sine as a function of time t so depending on time we'll get a different value as an output and the general form is two times pi times the frequency where frequency is measured in Hertz so if we want a high frequency signal we increase the value of f so think of f as a parameter in this signal if we want a high frequency signal to increase f low frequency decrease f and you'll see if you plot this and we'll plot some as we go increasing f will get a higher frequency of our sine wave and a third parameter is the phase phi here it's a bit harder to visualize for most people but that is a shift of the that sine wave relative to some origin point so if we think of some signal that is generated by a transmitter here's one example where we take over time we take say time equal to zero the sine if we set these three parameters the amplitude or the peak amplitude the frequency and the phase just constants then we generate a sine wave that varies in some manner over time where we can think of the peak amplitude is the signal strength so measured in volts generate some electricity measured in volts coming out of the transmitter the frequency is the number of times the signal repeats per second the Hertz measured in Hertz and the phase some relative relatively shift of this signal is measured usually in radians some other parameters that are related to these three parameters of our very basic sine wave is the period and you know the period is simply the inverse of the frequency a frequency of two Hertz gives a period of half a second and another parameter is the wavelength the distance occupied by one cycle of our signal and calculated as the speed of light C divided by the frequency gives us the wavelength of that signal so in fact starting with a very basic communication signal a sine wave we can have three parameters that will change the shape of that sine wave the amplitude or the peak amplitude the frequency and the phase and related to them are the period and the wavelength of that signal this shows four examples of a sine wave with varying parameters the first one so we see on this plot it on the time is ranging from zero seconds up to 1.5 seconds and as the caption here we see the amplitude is one the frequency is one f is one and the phase is zero so if we write that signal the top left one the s1 of t some signal as a function of time the equation for this amplitude is one times sine two times pi times the frequency in this case the frequency is one write that times one times t plus a phase in this case the phase is zero or quite simply sine two pi t so with this function as t increases we get this output so when t is zero the s of t is zero when t is point seven five the s of t is minus one and so on so we get the the sinusoid and it keeps repeating by changing the amplitude the frequency and the phase we just change the shape of that sinusoid and the top right one we see the same equation except we've changed the amplitude to 0.5 the peak amplitude is now half frequency is the same phase is the same and you see the shape is just shrunk it's condensed so the peak goes up to point five and to minus point five here this the bottom left plot shows the same as the first except the frequency is now two so it would be one times sine two pi times two t plus zero which is simply sine four pi t for the bottom left signal and we see the difference now is that within one second we get two repetitions of the the cycle that is we have a frequency of two hertz in one second two repetitions here we have a frequency of one hertz sorry from zero to one second the period of our top left signal is one second that is one cycle the duration is one second it repeats every one second the period of our bottom left signal is half a second because it repeats every half a second change the amplitude change the frequency last one to change the phase amplitude is the same as the first one amplitude of one frequency of one the phase here is pi divided by four the fact phase is an angle but measured in radians and what does it do you can think it shifts that sinusoid along so take this one and shift it back and instead of starting at zero and going up it starts at this point and goes up so it's as if this one has been shifted by how much well by the phase of pi divided by four and as you change the phase you'll see a different shift of that signal so for now all we're doing is covering some high school mathematics or physics of the sinusoid okay a sine function and we've got three parameters we can change to change the shape of that function the peak amplitude the frequency and the phase and this just visualizes those changes the effect of those changes so a very simple signal that we want to transmit to carry information we can generate a signal which is simply a sinusoid a sine a sine wave and think of that comes out and the sine sine wave representing that the the the energy being transmitted is sent across the cable and received by the destination we can easily change the shape of the sine wave by changing these three parameters in fact we can change more than one at a time we can change the amplitude and the frequency and we can do more complex things we'll see is that we can take multiple sine sine waves and add them together combine them to get more complex output signals and that's the basics of how communication signals are designed we start with sine waves allowing us to vary these three parameters and combine them together in different ways to generate the signal that we want to transmit across the system across the communications link so in the very basics we can think of all of our communication signals are made up of different sine waves any questions that everyone remembers the sine function ask you to plot in the exam or the next quiz well I wouldn't ask you to plot but I would ask you to be able to recognize given this plot tell me the frequency so given this bottom left plot tell me the frequency well you look and you see it's a sine sinusoid and it's repeating in one second two times so every one second there are two repetitions so it has a frequency of two hertz or given this plot write an equation for it at least in this simple case well you see the peak amplitude is one the frequency f is two the phase because it starts at zero there's no offset or shift is zero so this one would be the signal as a function of time is sine 4 pi f at 4 pi t so you need to remember this general function here the signal is the peak amplitude times sine 2 pi f t plus the phase and then if you can determine those three parameters the value of the peak amplitude frequency and phase then you can determine the equation for that signal 2 pi times the frequency if the frequency is 2 we at 4 pi times t t is the parameter of the function the time let's let's draw some signals and then do some different combinations of them let me just check what pictures we have okay so with our basic sine wave we can vary those three parameters to change the shape but we can do more for example we can add two different sine waves together this is an example signal s of t sine 200 pi t plus one-third sine 600 pi t so we're taking one sine wave plus another sine wave and we'll get some different shaped signal as an output we'll plot that in a moment or a similar one so we can combine these sine waves together to get different shaped outputs and the communication signals that we deal with we can think as being composed of many different component sinusoid signals that made up of many different sine waves and usually at different frequencies in this top equation we can say there are two sinusoids what's the frequency of the first one 100 what 100 Hertz okay from the first component let's call this the first component sine 200 pi t map it back to our general equation that's our general equation well let's map it back and see okay we have in fact we have a multiply here 4 divided by pi times sine 200 pi t what's the peak amplitude of the first component in fact this multiplier is multiplied by both components so the peak amplitude of the first component is 4 over pi the frequency of the first component well it's sine 200 pi t the general form is sine 2 pi f t so what is f well 2 pi f t we have 200 pi t means f must be 100 Hertz because 2 times pi times 100 gives us out 200 pi times by the time plus a phase and in this case there is zero phase the phase is zero in this case there's no additional component here but we have a second sinusoid at it we're adding two together so we can give the same parameters for the second one peak amplitude of the second component write it down peak amplitude frequency and phase of the second component try and determine them there's this extra multiplier at the front we'll see the purpose of it soon but if we multiply 4 divided by pi times by one-third sine 600 pi t then that peak amplitude is simply 4 divided by pi times one-third that's the multiplier of the sine function sine 600 pi t what is the frequency of the second component 300 because our general form is 2 pi f t we have 600 pi t f must be 300 Hertz and the phase again is zero because there's no additional component radians the phase is measured so in fact this signal s of t is made up of two sinusoid's added together and both of them have different values of these three parameters the amplitude the frequency in the phase and in general our communication signals that we transmit we can think as being made up of as a combination of different sinusoid's be more complex than this but this is the principle that we can add them together to get a different shaped output I'll show you a plot of this one in a moment where each of the sinusoid signals have a frequency so in this case we have two components with two different frequencies 100 Hertz and 300 Hertz and also different peak amplitudes let's plot this before we go on to the next part actually I have it will plot another one later this is the plot of those the first two are the plot of the individual components and the third plot is the addition of the two now the scale on here doesn't exactly match the the equation here it's a general scale where uppercase t is the period but the shape matches what we have here we have a frequency of 100 Hertz we have to adjust t here and then a frequency which is three times as much 300 Hertz well you see the frequency of the second plot is three times as much as the top one and it's also one-third of the peak amplitude you see if this is a peak amplitude of one this will be one-third of that so the shapes at least match our our components here when you add them together and plot them you get this shape okay there's these humps here two humps at each point here importantly in this case that the frequency of the resulting signal is the same as the top signal we see the repetitions here here and if we keep going which is the same as the top signal and that was the way that the the equation was set up to produce that if we take some signal and we make it up by combining different sinusoids together when all frequency components are entered a multiple of one frequency then that one frequency is called the fundamental frequency and the others the other components are harmonic frequencies and our example equation first component was 100 Hertz second component was three times 100 Hertz which matches this condition here we have two components where this is one times 100 Hertz the second component is three times 100 Hertz so we say the fundamental frequency right f subscript f is 100 Hertz and this is a harmonic frequency 300 Hertz when we add these sinusoids together which match this condition then the resulting signal has a period equal to that of the fundamental fundamental frequency component the resulting signal has a period and frequency the same as this 100 Hertz signal and that's illustrated on this resulting signal here we have the first component the second component which has three times the frequency as the first component and the resulting signal when we add them together has the same frequency as the first component so if we structure our signal by adding sinusoids together we can get a resulting signal of a particular shape and having properties in this case the properties are that the same frequency as the the fundamental frequency in general by combining sine waves with different amplitudes frequencies and phases we can design or construct any communication signal we want so any real signal that we transmit we can think of it as made up of individual sinusoids sine waves and people use this to design signals so the people who create your wireless transmitter or design the standard for the wireless LAN or your ADSL modem and so on they need to create some hardware that generates signals well how do they design that hardware well they design it based upon these fundamental concepts of we can think of a signal as just a combination of many sine waves and we can use that concept to design the signal and look at its performance let's try and plot some different examples I'm going to plot them on the computer I have some software just some mathematics software that will produce some plots okay so it's not so important how I do it the important will be output plot that we see so we're looking at some time frame from zero up to one and I'm going to create a sine function and plot it and we'll see how we can combine them let me remember on a plot on the x access the time and on the y access the signal s of t and the first one remember our general our general form of a sinusoid amplitude time sine 2 pi f t plus phase let's set the peak amplitude to be one one time sine 2 times pi times the frequency and we can choose the frequency so I've chosen a peak amplitude of one a equals 1 sine 2 times pi times a frequency and let's just for this example choose a frequency of 2 2 Hertz times the time plus a phase let's make the phase to start zero zero phase and let's plot that as a blue line this is just the software I need to specify I'm specifying the color see if it works okay here's our our signal peak amplitude of one so it goes from plus one to minus one two times pi times 2 t so frequency should be 2 Hertz within one second there are two repetitions so we see that between zero and one we get two repetitions and there's no phase offset in this case let's try a couple of variations I want to do first I'm going to change something set the axes to be currently my axes range from minus one to plus one I'm going to change that which means it's the same signal except I've just changed the plot to go up to mine it from minus two to plus two we can change the peak amplitude so a new signal a red one and if we change the amplitude to say 1.5 simply increases the height if we have a green one with the original one amplitude and change the frequency to four so four Hertz we see it's the same height as the blue one except it repeats four times in this period of one second the green signal in this case so changing the amplitude and the frequency usually obvious to to visualize let's look at the impact of the the phase the start again so our first signal again and now let's introduce some phase shift and see how that impacts so before I had plus zero let's add pi over four so this is measured in radians so we can map it back back to degrees if we want but the input is radians and let's change the color we see the difference between the red one now has a phase of pi over four we see visually it's shifted by introducing this phase we have this shift of the from some point of origin and we can keep changing the phase pi over two another one three pi over four yellow it's hard to see and simply pi can you see the last one yep so by changing the phase we see this shift of this signal in the in the time domain from some point of origin so that's the change of the three parameters now let's consider if we combine two together take our original signal and then add a new one which is a third of the amplitude and instead of a frequency of two hertz change it to six hertz so I've shrunk the peak amplitude by a third and increase the frequency by three for the red one and now if we add the two together what we get take the first one two times pi times two t plus the second component so now we have it we're going to plot a signal with two components and we see the green one same frequency as the blue the original component but we have this different shaped signal we'll add some more components shortly and see how that impacts and what that results in first alright we're sending signals now so we can use sine functions and combine them to generate a signal of a particular shape but let's say we want to send bits from computer a to b what what can I do to send zeros and ones how can I shape my signal any ideas okay so but considering these basics these basic sinusoids what can we use how can we use them to represent zeros and ones so assign an amplitude to bit zero so if I want to send a zero I send a signal at one amplitude and if I want to send a bit one I send a signal at a different amplitude so that I'll send these signals and when the receiver receives a signal it will measure if the amplitude is low for example that means that someone just transmitted a bit zero if the amplitude is high it means they transmitted a bit one so that's the basic a basic way that we can use the signals to represent data let's say I want to send the sequence sequence of data with four bits the data I want to send zero one one one there's four bits of course if we focus on these four bits we want to get them from a to b so what we do is we generate a signal that represent this data and let's use the scheme that says if we want to send a bit zero we send a sinusoid with a low amp amplitude if we want to send a bit one we send high amplitude in fact we can simplify that and try and draw a case that captured that information let's try and what we do is for some period of time we send a signal to represent the first bit and then for another period of time the same period we send a signal to represent the second bit and so on let's see if we can use just a sine wave to do that so for this case we need to send four different bits so I'm going to do that over a period of one second our time frame so we'll break that into four different chunks of time I've set this up before so it's a bit simpler I'll explain it as we go so for the first if we imagine this one second is broken into four chunks each a quarter of a second long so for the time set one t1 we're going to plot our sine wave and in this case the frequency is to the peak amplitude is still one let's change the phase to be pi you'll see why in a moment and let's set the axis it'll be correct and for the second time time slot I'll use the same phase and the third time slot I'll use a phase of 0 and the fourth time slot a phase of pi so what I've done is over time I've changed the phase of this sinusoid this sine function so I've used the same peak amplitude over four time slots one one one one the same frequency two hertz two two two but I've simply changed the phase in this example and this is the resulting signal that say transmitted across our link where we're using the scheme when it's low it represents bit zero and when it's high bit one so zero one one one and by changing the phase I get the shape of the signal that represents the data I want to send whenever I want to send a bit zero I set the phase so that we go down and bit one the phase so that it shows a positive peak negative and positive a very simplistic way to transmit bits as some signal it may not be like that in real life but it captures the basics how fast did I send bits well no think of bits now think of how do we measure the speed of bits last week in fact on Monday data rate was one measure of how many bits we send per second how many bits per second do we send in this very simple example four bits were sent bit one two three four in a time period of one second four bits per second what was the frequency of our signal in all cases remember if we look at the sign the peak amplitude the frequency in this case is two hertz in all cases it's two hertz so I set that so using this simple scheme a negative value to represent bit zero a positive value to represent bit one we transmit this signal the receiver should receive a signal with similar shape and what it does for the first quarter of a second it measures the amplitude okay it's close to minus one therefore it must mean a bit zero was transmitted then it measures the amplitude plus one plus one plus one it must mean a three bit ones were transmitted in this simple case using a sinusoid with one component with a frequency of two hertz we had a data rate of four bits per second let's try using a different signal and see what happens when we change the signals that we use how that impacts upon our data rate and our other performance metrics let's draw it again instead of having one sinusoid component I'm now going to generate a signal using two components and it has some structure which you'll soon notice I hope it works now we have a signal which has two components sine 2 pi 2 time one plus pi peak amplitude is one frequency of this component is 2 phase offset is pi and a second component with one third amplitude frequency of six and a phase offset of pi we're missing a bracket there let's see how it goes and then we'll do that for the other time periods same concept but now we're just using a different signal where again negative amplitude represents a bit zero positive amplitude represents bit one so zero one one one the only difference is now I've used a different signal to generate the resulting signal here I use two sine components and you see the way that I've structured it is that the second component is one third of the amplitude of the first and three times the frequency here was two hertz here is six hertz I can keep adding components so this has two components if I add a third component using a similar structure we can get a similar shape signal but slightly different or add one more actually we'll see in general what I've done come back to that what we're doing is similar to this this is a signal with two sine components you can see in the caption sine 2 pi FT generally one third sine 2 pi 3 FT we see the shape with these two humps at the top in this case it's positive negative positive negative this is if we add a third sine component and the pattern we see is now plus one fifth sine 2 pi FT five times the frequency is the first one fifth of the amplitude this is with four components the last one is one seventh of the amplitude and seven times the frequency this is with an infinite number of components using the same pattern and we see we get a perfect square waveform now we can use this same approach to send our zeros and ones instead of just sending a sinusoid we can send in theory perfect square waveform where we'd say for example plus one represents bit one minus one represents a bit bit zero and we'd get a plot I'm not going to do it on the computer but we'd get a plot like this for our zero one one over our period of one second represents bit zero it's a negative value bit one one and one again we'd use the same signal except we can change the phase to determine the output shape the one I draw on the board represents the same data as this one different shape signals different signal transmitted but are carrying the same four bits zero one one one and at the same data rate four bits in one second and in this one over there we got also four bits in one second so we can choose and design signals differently to carry our data we'll look and compare which one is better is this square one better than the one on the screen to send data or we'll see that generally it is in terms of quality because the receiver in the presence of errors if there are errors in this one it's more likely that receiver will be able to still determine the correct bits if their errors in this one there's more of a chance that they'll get the wrong bit and we've got a detailed example of that later let's step back and see what else is different amongst the different signals so we skipped over a few slides here what we've been doing is taking sinusoids and changing the shape and now combining combining them together adding them together in this case to get a different resulting shape we can look at the some of the parameters or performance metrics of these signals and the way that we can make this easier this is the signal in the time domain let's look at it in the frequency domain and their mathematics Fourier transforms that can convert a signal in the time domain to a frequency domain we're not going to cover that we'll just look at the end results here is a plot of the same so looking at the bottom signal here made up of two components note the difference between the two components one has a the second one is a third of the peak amplitude it's a third of the size but three times the frequency and we add them together and we get this resulting signal in the frequency domain what we do is we consider the individual components and we consider their peak amplitudes and their frequencies this is a plot of the same signal the bottom one on the previous slide but in the frequency domain how we interpret this is say that at one F we have one component with a particular peak amplitude and a frequency three times that we have a second component with a third of the peak amplitude so this is the same signal but looking from the frequency perspective not as a function of time and the reason we do this is that because it makes analysis and design much easier for the for the engineers that must make the devices let's try and do that with a real example and then we'll define some terminology to finish let's take a signal and well I'll create a signal and then we'll draw it right out of space here's a signal with three components 15 times sine 2 pi 4t so 2 times pi times 4 times the time plus 5 sine 2 pi 12t plus 3 sine 2 pi 20t so I've created this signal let's analyze the individual components from our general sinusoid equation so we have three components what's the amplitude of the first component 15 easy just to multiply frequency of the first component keep helping me for okay remember the general structure 2 pi ft what's the frequency in this case I've written it easily I could have written 8 pi t that frequency is 4 4 hertz phase in all cases there's no plus anything so the phase is 0 I won't I won't write the phase second component amplitude is 5 frequency is 12 third component amplitude is 3 frequency is 20 hertz plot this signal in the time domain well I won't ask you to do that because you need a computer to get accurate so if I ask you to plot this signal in the time domain I could do it on the computer but it's not exactly the same but it would look something like this this would be the shape now for you to plot that in the time domain is again you need a computer to get the exact shape there but let's analyze this from the frequency domain and and we can first plot it in the frequency domain in the frequency domain the plot shows the the peak amplitude as a function of the frequency for each of the components here we have three components we know their peak amplitudes we know their frequencies so at a particular frequency we plot an impulse a spike I won't try and plot it in the time domain but in the frequency domain here's frequency in hertz and here is the peak amplitude of the signal as a function of f this is an uppercase s notation so at what have we got at 4 hertz we have a component may not be the greatest scale at 12 hertz and at 20 hertz we have three components we know their frequencies we know their peak amplitudes 15 5 and 3 so we'd plot an impulse 15 5 and 3 that is the plot of this signal the equation but given in the frequency domain the reason for using the frequency domain is to make the analysis and and the design of the signals easier it's much easier to do operations in the frequency domain than time domain and many signals are analyzed from this perspective now to finish let's introduce some concepts that we can gain from this we can say this signal s of t has three components the frequencies of those components of 4 12 and 20 hertz for any signal the spectrum of that signal is the range of frequencies it contains so for our signal we can say the spectrum is 4 12 and 20 hertz this signal contains three components with these three frequencies that's the spectrum of that one signal another term the absolute bandwidth of this signal is the width of the spectrum our spectrum ranges from 4 hertz up to 20 hertz therefore the absolute bandwidth is 16 hertz from 4 to 20 that's the width so the bandwidth in this case 20 minus 4 16 hertz some signals may have a DC component we'll not cover that yet so there's two new terms that we've introduced spectrum and absolute bandwidth spectrum of a signal the set of frequencies the bandwidth is the the maximum frequency component minus the minimum in fact on this plot it's very easy to see the bandwidth and the spectrum it shows us the spectrum and the bandwidth the absolute bandwidth is just the maximum minus the minimum 16 hertz in this case we'll see and we'll show some examples that indicate that the bandwidth of a signal and of a communication system impacts upon the data rate the number of bits per second we can send and in generally in general the larger the bandwidth with other things fixed the higher the data rate we can achieve so if we have another signal which with a larger bandwidth we can increase our data rate but there are some costs of increasing the bandwidth let's see if we can summarize with one more or finish with one more example same example let's say I now have a another signal with just two components I remove the last one what is its absolute bandwidth so a second signal two components what is the absolute bandwidth of this second signal it's 8 hertz because now we have two components amplitude 15 frequency of the first component 4 hertz second component amplitude of 5 frequency at 12 hertz so the spectrum is 4 hertz and 12 hertz the bandwidth is just 8 hertz in this case so we'd say this signal occupies a bandwidth of 8 hertz if we transmit this signal through our system it uses 8 hertz bandwidth the first signal used a bandwidth of 16 hertz in general the larger the bandwidth the higher the data rate the faster we can send bits so that the signal with a higher bandwidth is better from that perspective but another trade-off is that the higher the bandwidth we consume the higher the cost involved of transmitting that signal so we'd like a signal with a low bandwidth it's cheaper but we'd like a signal with a high bandwidth because we can send more bits per second so there's a trade off there in that when someone designs a signal what signal do I transmit across my Wi-Fi my wireless LAN then they need to consider that trade-off of do we design one with a large bandwidth or do we limit it so the trade-off between data rate and cost next week we'll go through that trade-off and look at some other factors as well with it with a detailed example what we want to summarize for today so we haven't gone through all of these factors let's go back all of our communication signals can be made up of we can think of it as sine waves where a fundamental sine wave is the peak amplitude times sine 2 pi times the frequency times time plus a phase varying those three parameters change the shape of that sine wave combining sine waves together we can get signals of different shapes as we see in this case adding these two we get this different shape signal and in fact if we added in in the pattern that we saw if we add an infinite number of components using this pattern we eventually get a square wave perfect square wave we can use this signal to represent bits for example transmitter high level to represent a bit one a low level for a bit zero so from that depending upon the hour our frequency we can get some bit rate number of bits per second one bit another bit and so on and determine a data rate these signals are in the time domain most analysis of communication signals performed is performed in the frequency domain and just going back in the frequency domain domain as a plot we look at the individual components determine their peak amplitude and their frequency and plotted impulse for a spike at that particular frequency and with that peak amplitude so this is a plot of the signal in the frequency domain we can also write an equation for that I'll do some analysis mathematically and the two terms so far we've defined spectrum and absolute bandwidth spectrum is simply the range of frequencies in that signal absolute bandwidth is the width of the spectrum what we'll do next week is show an example that connects all them together in particular bandwidth data rate and other factors like cost and errors