 Welcome to this lecture on digital communication using GNU radio. My name is Kumar Appaya and I belong to the Department of Electrical Engineering at IIT Bombay. In this lecture, we are going to have a brief introduction to wireless communication and OFDM also known as Orthogonal Frequency Division Multiplexing. To give you some context of why we are going to this topic, in the previous lecture you have seen that if you have a channel that has an impulse response that affects your performance. One issue that you will face is that you have to perform equalization at the receiver. Now what if you could perform some bit of equalization right at the transmitter even before sending it, sending your signal and even without knowing what the channel is. So in that context OFDM is a nice way by which you can parallelize the channels and you end up with a very easy equalization process at the receiver. We are going to do that but I will first motivate it with a simple channel model for the wireless system. So we are going to just look at a simple approach to looking at wireless channels. Let me just give you a caveat, it's just not for wireless but the same kind of modeling approach can work for several channel models including optical channels or underwater fiber cables, underwater communication, free space, all those things. But wireless is just one motivation, it works for several other systems also. The key idea is we will be looking both at the time and frequency picture of the channels. We will then discuss the frequency response and the frequency selectivity which basically means how the channel behaves differently for different frequency ranges. And finally, we will present orthogonal frequency division multiplexing, there should be frequency over here. Orthogonal frequency division multiplexing as a pre equalization technique, in other words how it simplifies the equalization process at the receiver is something that we will see closely today. Now if you look, if you remember from your electromagnetic spectrum, this electromagnetic spectrum is taken from Wikipedia, you can just check that the visible light spectrum is between 400 and 700 nanometers, between 400 and 700 nanometers that is the visible spectrum. Now when you look at the frequency picture or the wavelength picture that tells you the kind of signals that you can come to expect, below 400 nanometer you have really high frequency and low wavelength signals, these are typically ultraviolet and X-rays. But most of your communication systems operate in the infrared region, that is if you look at something like 10 micron, 100 micron and so on, that is where if you look at say something like a 1.5 micron and so on, that is where optical communication happens. If you keep going and generally people switch to the frequency notation, if you look at several tens of megahertz or kilohertz, that is where a lot of wireless communication happens. Let me give you sideways picture of the same. Now if you look over here in the frequency picture, when you have these kinds of frequencies, so when you have something like gigahertz, this is gigahertz and tens of kilohertz, you have radar, radio, TV and so on, FM radio and when you go for some of those frequencies that is where propagation in air is very effective, that is you can essentially send signals at those frequencies or wavelengths and that is where the signal propagates for long distances without a significant loss. Now if you look at why you know you choose those frequencies or why you choose one frequency over the other, there are multiple choices of course. One is policy, for example the government may tell you to use only this particular channel or the government may say you have to pay for this particular frequency usage and you may choose based on that. You have to decide based on propagation loss, how much energy loss happens if I signal at that particular wavelength or that particular frequency. You have to look at the reflectivity of the waveforms, that is you will have to see how much of reflection you can encounter and sometimes reflections are good, sometimes reflections are bad. You have to look at attenuation in the atmosphere meaning propagation loss is one way to look at it, but in particular depending on atmospheric conditions you may face different losses at different wavelengths. For example, certain wavelengths are affected significantly by the presence of some parts of air's oxygen, some parts are affected because of water vapor, some parts because of foliage that is the presence of plants and so on. So some wavelengths are more preferable over others depending on the environment in which you want to deploy your communication system. Finally, you may also be interested in the directivity or line of sight that is you may have to look at whether the receiver and transmitter have to face each other directly. An example where you don't need line of sight is your traditional wireless communication system where you have something like your GSM or a 4G based handset. This handset works inside the building even if you don't have a direct visible path from your phone or from your device to the base station. Some for example, modern millimeter wave systems are designed in a way where you have to have the line of sight. They have to see each other in order to communicate effectively. That is something which you have to keep in mind. So let's look at the so frequency domain picture. So I'm going to look at the frequency domain picture and from the previous exercise we have been looking at the time domain picture where you had inter-symbol interference and so on. So you had inter-symbol interference then you know the channel became channel was no longer an impulse like channel did not have a flat frequency response it had some different frequency response. Let us just look at the frequency response picture directly this time because it will serve our purpose well to understand the new pre equalization technique that I'm talking about. So you have to select the appropriate frequency range based on what frequency you have the license to or permission to radiate on where there are losses which are acceptable where you have a transmitter receiver radio that is acceptable and so on. Finally you have decided on a frequency range. Let's say that you choose between you know around FC and let's say FC plus or minus W by 2 you have chosen. This is the band at which you are going to signal. Now obviously the moment you start signaling let's look at this particular thing as a baseband signal a flat spectrum means there is no data. See you chose a sync pulse or a root trace cosine pulse or something and you put data on it. The moment you put data right it's not going to be flat but the channel if you want it to be flat ok you are happy because the data is going to go through without any issue. But the problem is that channel ends up being not flat why? Because there are some reflections there are some losses at some frequencies more than others there are some frequencies which get reflected and come back and so you typically never have a flat channel. The non-flat channel is essentially the same as having inter-symbol interfere or some other or some other effect you know. So typically we hope it's a linear transformation but that's what we saw in the previous set of lectures also where we were discussing equalization that whenever the channel performs some transformation on your waveform then you have to undo it by performing some filtering called equalization. In the same way you have to look at this particular non-flat channel response as having as causing essentially a problem which needs to be equalized. Now in the baseband picture I'm calling it HB the baseband picture has let's say that you did you know you had your signal at FC and after the passing through the channel you brought it back to baseband the flat channel is not flat. One approach was to use your linear equalizer such as MMSE or your zero forcing or one of those adaptive equalization techniques to bring back the channel performance. But the receive typically receiver has to somehow learn and correct for the channel. In this case because the channel behaves differently at different frequencies we call it a frequency selective channel. This is a very commonly used kind of nomenclature frequency selective is the same as a channel that performs some filtering some non-trivial filtering of course scaling just if your channel just scales the input then that's not frequency selective. If it behaves differently at different frequencies we call it a frequency selective channel. So frequency selective channels you know you know behavior has to be learned carefully. Now let's look at a traditional wireless channel a typical wireless channel let us say around the several hundreds of megahertz to a few gigahertz. You can think of it as say 800 megahertz to about you can say two and half or three gigahertz you can say this is the behavior. So if you look at hundreds of megahertz to around five gigahertz in fact signals get reflected multiple times before they reach the receiver. The same model essentially carries over to other frequencies as well but in other frequencies you may have more or less of some impacts that's something which we are just going to say. So let us suppose that this is something like your cell phone which you have inside your building and it is not necessarily facing the base station. So this is your base station this is your cell phone. So typically the signal doesn't come directly it gets reflected it can get reflected from some building it can get reflected from some wall it can get reflected from some tree it can get reflected from the car which is going on the road or something like that. Notice that some of the objects that I talked about are static some of the objects that I talked about may be moving. By the way you must remember one thing if you think about something like a building right a building which is short may be okay but if you have a building which is very very tall then buildings also have some sway they also move a little maybe by a few maybe millimeters or centimeters you can say a few millimeters centimeters of movement is not much but a few millimeters centimeters of movement is actually significant in if you're dealing with signals which have tens of centimeters of wavelength which is essentially what you are you know these signals are see if you look at 1 gigahertz right 1 gigahertz is about 30 centimeters of wavelength if I'm not mistaken you can check. So that means that the environment around you essentially changes. Now typically you don't get one reflection you get another reflection from another source or the same from the same source at different points you get different reflections and you have another reflection as well. So what you end up having is this you have transmissions that reach you maybe or maybe there is a line of sight maybe there is not but you get these reflected copies also. Now typically the problem is that these reflected copies take a long time for you to get them in other words if you look at the first copy that comes at sometime second comes second one comes with slightly different gain slightly different delay the third one comes with a slightly different gain slightly different delay and each of these essentially start adding up and you have an effective impulse response that the channel produces at your system. So our aim is to handle this impulse response somehow learn it and correct for it and that should be the goal of your system that is what we've been seeing also. So you have generally in this case you have written two parts you may have 10 parts 12 parts 20 parts depending on the system that you're dealing with you more you may have a single direct path and you can ignore all those there are various kinds of scenarios that you can deal with. Now this frequency selective perspective okay now these this non-uniform kind of gain pattern across frequency is induced by those parts if you had only a single path then you can say it's a flat channel because you don't have any other gains but the moment you have one path and another delayed and part with another gain other delayed part with another gain then you have to handle this frequency selective channel. How do we do it? Of course perform some zero forcing or MMSE or adaptive equalization but what else can we do? What if we had a picture where we could divide the channel into you know this is the bandwidth essentially let's say this is minus W by 2 to W by 2 in the base band what if we could divide it into several smaller narrow band you know narrow band channels that is what if we could break it up into several smaller parts you break it into several smaller parts then there is a significant advantage what is the advantage if you split the channel to several smaller parts we can make an approximation the approximation that we can make is we can essentially assume that the if you if you take these narrow paths to be narrow enough you can assume that the gain is constant within that range of course you may argue that you've drawn it in a way where it is not but if you take it in a very very narrow manner see the channel is induced by a natural process and it doesn't have any abrupt changes in the frequency domain it moves very gradually if you can break it into smaller parts within each part then you can assume that the gain is roughly constant right and therefore within each part the gain is constant then you don't really need a an equalization which takes into account inter simple interference and so on why see if you the problem was if you had a wide bandwidth with a wide bandwidth you had to have this equalization to correct for the gain but if you make a very very narrow band channel then by the time in the frequency domain the channel variation is visible it's very very little it's very flat so an approximation that the channel is a flat works very well or in the time domain picture for that narrow band channel impulse response is close to just something like a delta there is no real multiple parts and so on you divided your frequency selective channel into multiple frequency flat channels combination of multiple frequency flat channels okay this looks like a good trick how do you do it the intuitive way is for us to use much much narrower pulses right for example sorry much much wider pulses rather so for example if you use the sink or a sink or a raised cosine or a root race cosine rather you would have chosen it based on this minus W by 2 to W by 2 that is what you would have chosen now however that that you know that means that you will have to then use another sink or another you know root race cosine with the same which operates at a slightly neighboring frequency in another name for this is called frequency division multiplexing that is essentially let me just correct this you're essentially saying rather than use my other news my this channel I'm just going to use this particular channel this narrow band channel then I'm going to use this narrow band channel and this narrow band channel each with a different small f's fc let's say so to speak right so you're going to use this you're going to use this you're going to use this and you're just going to shift them in the frequency domain by ever so slight amounts and this combination gives you what you want now that's one way so if you recall the way we signal was x t was summation b k p of t minus k t our p was the effective effective pulse that takes into account both the transmit pulse as well as the channel now this p of t determines the bandwidth usage remember the bandwidth usage is determined only by gtx because gtx convolved with p cannot use more frequency than gtx convolved with the channel rather cannot use more frequency than gtx because it's an lti system and whenever you perform convolution you don't you can never expand the frequency footprint so one approach is to make p of t wider which is what I said right if you make p of t wider and wider in the frequency domain it becomes narrower and narrower but that is a little cumbersome because you have to create multiple such parallel you know parallel streams of data which are then modulated so that they are placed exactly there you sort of read multiple radio frequency translations and so on right because you need to essentially take one part put in the next put the next at the same gap put next in the same gap and so on we don't like that approach is there a simpler approach so one approach is very simple why don't we just repeat b0 multiple times b1 multiple times okay now this sounds a little strange I'm saying use the same p of t that uses your bandwidth between minus w by 2 and w by 2 and you start repeating b0 b1 multiple times then that results in a slightly lower frequency usage pattern how so let's just do this exercise once so now let's say that hypothetically let's say that I have a sequence consisting of b0 b1 b2 etc and I now translate this to sequence that uses b0 b0 b0 b0 and then b1 4 times and so on okay hypothetically I mean you may argue that I'm now just making the data rate one fourth I agree with you but we will handle that eventually let's just live with this so I'm essentially repeating bk 4 times let's say what happens to the spectrum usage let's say that I use let's say that my p of t is essentially sync of t upon t which is equal to sync of w t okay because my w is one upon t let's say hypothetically now if I use if I look at bk sync of t plus let's say b0 but I'll write bk plus bk sync of remember I'm sending bk again with a delay of t plus bk so the same bk is being sent 4 times this means I can also assume that I have an equivalent p equivalent let's say p prime of t is equal to sync of t upon t plus sync of t minus t upon t plus sync of t minus 2 t upon t plus sync of t minus 3 t upon t okay I have this is my equivalent p of t and if you now look at the effect of this what is the what is the spectrum of this p of t so sync is between minus w and w so what about this so this p prime of t is equal to sync of w t convolution okay delta of t because there's no change plus delta of t minus capital t because I delayed by t because delta of t minus 2 t delta of 3 minus t minus 3 t so what is my p prime of f this is going to give me this rectangle minus w by 2 to w by 2 multiplied by what am I multiplying it by 1 plus e power minus j 2 pi f t this is my continuous time Fourier transform plus e power minus j 2 pi 2 f t plus e power minus j 2 pi 3 f t if I now simplify this a little bit okay I'll just do this part this is going to be I'm going to take e power minus j 3 pi f t common so I'm going to get e power j 3 pi f t plus e power j pi f t plus I'm going to write this term next e power minus j pi f t plus e power minus j 3 pi f t this is equal to e power minus j 3 pi f t I'll put 2 outside so I'll make this a cost make this a cost that will become 2 times cost of 3 pi f t plus cost of pi f t if I apply the cost a plus cost b formula okay cost a plus cost b I'll get 2 times so I took the 2 outside made it for cost of a plus b by 2 so 3 pi 4 pi by 2 2 pi f t cost of pi pi f t now if you look at cost of pi f t right cost of pi f t where does it go to 0 the first 0 occurs at when this is pi by 2 so I'm just going to write pi f t so it's like pi upon 2 by pi t is equal to 1 by 2 t so now okay let me just so this means that let's let's do this more carefully apologies so cost pi f t is 0 that is when pi f t is equal to pi by 2 okay so that means my f is equal to pi upon sorry I just take this pi away 1 upon 2 t okay so now let's look at the spectrum okay if you now look at the spectrum original spectrum was between minus w by 2 to w by 2 okay now you saw that the original spectrum was close to 1 between minus t by 2 and t by 2 now it so happens that this spectrum is going to go to 0 over okay sorry the this is fine I must look at this one also if you look at the 2 pi f t 2 pi f t is equal to pi by 2 sorry so that means f is equal to 1 by 4 t so now the problem is that you're going to get a spectrum that goes to 0 over here in other words you essentially said that the spectrum lies only in this region because it goes to 0 over here it's almost like you have used one fourth of the spectrum now the key idea with this is if you now use another spectral component that occupies the neighboring part another one that occupies the neighboring part can you fill in four frequencies over here now it turns out that you can and this is what leads to the concept of orthogonal frequency division multiplexing so the key concept is if you if you now repeat your symbol four times then you essentially make the spectral usage one fourth this is intuitive from your DSP also if you repeat your signals you reduce the spectral footprint of course it's not like this one fourth it's actually like this but still good enough and we will see that the interference related issues are not there so what we will see is that from the next lecture onwards we are going to formalize this notion and build this into what we call orthogonal frequency division multiplexing and this will translate into several benefits in terms of your equalization process at the receiver thank you.