 So, this is mixture number 3, so there was a mistake I made in the last class which was pointed out to me later, so the norm for a vector is defined as what, positive square root of, I think I forgot the square root when I wrote down the norm last time, so go back and correct this, come sit here if you cannot hear, people behind you do not seem to be complaining so, I am assuming it is audible, so positive square root is something I probably missed out and I think I missed it out, so make sure you have that, okay the norm square is the inner product, okay so let's go, let's keep going and the last thing was defining was baseband and passband signals, okay so that's where we are, okay, so I want to show some picture from a Fourier transform point of view, so if you have a baseband signal which is real like I said, the only signal we'll be dealing with in this course are real baseband and passband signals, okay, so if you have a baseband real signal how will it Fourier transform look, okay, so Fourier transform is going to have a real part and an imaginary part, the real part will be, what will the real part be, will it have some properties, will it be even symmetric, right, so you will have symmetry about zero, will the real part be real, what about the imaginary part, will it be real, yeah it will also be real, okay, so it's just the name, all right, so here's the, here's the, here's how a real part might look for an X of f which is baseband and real, okay, so baseband means it's going to be around the origin, right, so the non-zero values for the spectrum are going to be around the origin and say let's say I'm going to just roughly draw it like this and it's going to be within say minus w2 plus double, okay, so some such bandwidth and the imaginary part is also going to be similar, maybe I'll draw it like this, okay, but it'll have odd symmetry around the origin and it'll again die out between minus w2 and double, okay, so that's how the spectrum will look for a real baseband signal, so X of t is real and baseband, okay, so the moment a signal is real, it's enough if you specify the spectrum, you have to specify both the real and imaginary part, but it's enough if you specify it for the positive frequencies, okay, you don't have to specify it for the negative frequencies, which anyway don't exist, we just write them down, okay, so if you go to the real passband signal, if you think of drawing a possible spectrum for a real passband signal, it's going to look the real part of the spectrum, it's going to be once again even symmetric, but I'll draw it a little differently, you'll see why this kind of a picture is important, FC is going to be somewhere in the band where it's non-zero presumably, okay, and then it's going to die out before and after, what FC minus w, FC plus w, okay, so the part on this side is going to be a mirror image in exact same way, okay, so this is how the real part would look, so if you were to draw the imaginary part, so what is this, this is X of t being real and passband, okay, well I'm going to draw I guess minus FC here, then FC, okay, so this is the imaginary part, okay, so that's the way of visualizing how a real and imaginary part of the spectrum will look for real signals, okay, so the only thing that's important here is that X of t was real, okay, the baseband and passband, the way it worked out is the non-zero parts are closer to the origin and baseband, and non-zero parts are around a particular center frequency, FC in passband, okay, so normally you might be, might have used to FC being at the absolute center of the passband, all that is not required, the way I defined it, FC has to be somewhere in that band where it is non-zero typically for you to be, for you to deal with this comfortably, okay, so that's how we define these things, okay, so that's real baseband and real passband, okay, so like I said, I might have seen this before maybe, but I want to repeat it once again because it's very important for this course, a real passband signal can be represented using a complex baseband signal, okay, so what will be the difference between the spectrum of a complex baseband signal and a real baseband signal, what can be different, okay, the spectrum of a complex baseband signal will also have a real part and an imaginary part, okay, right, it's also a complex function, need not be symmetric, that's the only thing, okay, so if you take the spectrum of the, of a complex signal, the real part of the spectrum will still be real, imaginary part will still be real, there's no problem with the real nature, but it won't be symmetric, even symmetry and odd symmetry will not necessarily be there, okay, so we'll see that those things mean a few things, so how to go from here to there is the important idea here, but you can see why it's motivated, why you get the complex thing, you know, I mean, see if you notice here, if you think of a baseband signal zero being the center frequency, you have symmetry about the center frequency, for the passband case, if you look at only one side about the center frequency, there's no symmetry, okay, so when you kind of move the center frequency to baseband, you will not have symmetry, which means you'll, you'll probably have a complex signal, okay, that's one thing and the other thing is, if you look at the bandwidth for the real baseband signal, the actual real bandwidth in positive frequencies is only W, when you go to the real passband signal, what's the bandwidth in real positive frequencies, 2W, okay, so when you get it back to baseband, it's natural that you should get more than one real baseband signal, if you just get only one, then you can never have asymmetry around or you can't occupy this 2W bandwidth, it's the equivalence, okay, so those are motivations for why they should work out from an intuitive way to a complex baseband signal and not a real baseband signal, okay, so we'll see how to do it, it's a sequence of steps, you might have seen it, but I want you to pay attention because it's a little bit intricate, you might have seen it from the time domain, but I'm going to motivate it from the frequency domain, do the derivation from the frequency domain first and then maybe if time permits, maybe I won't see the time domain at all, the time domain is a little bit confusing, so I'm going to drop it and only do the frequency domain derivation, okay, so let's start with the real baseband signal and see how to break it down and represent it using a real baseband signal, okay, so that's what we're going to see now, okay, so the first definition will make, okay, so keep this picture in mind, right, when I say a real passband signal, this picture all the way at the bottom, okay, so that's the spectrum you should keep in mind, okay, all right, so the first definition, okay, once again, X of t is real and passband, okay, that's a spectrum like I showed in the previous picture, okay, so the first definition, okay, so I'll say center frequency is FC, once again, center frequency is used in the definition of passband signals as some frequency in the nonzero band, okay, so the first observation you can make is X of f is going to be equal to what, X star of minus f, there's going to be conjugate symmetry, right, so that I know because X of t is real, okay, so the next definition I'm going to make is to say only positive frequencies matter, so I'll define a new spectrum with only the positive frequency, okay, so that I'm going to call as X plus of f, I'm going to say X plus of f is equal to X of f when f is greater than zero and it'll be zero for f less than or equal to zero, so I'm going to take the spectrum X of f and retain only the positive side, okay, only the positive side, so now if I look at X plus of t which is the Fourier transform of, inverse Fourier transform of X plus of f, what kind of a signal will it be, it'll be complex, right, because you've lost the symmetry, right, you've kicked out the negative part, you've lost the symmetry, okay, so if you want to imagine a filter that achieves this, what will be the frequency response of the filter that will achieve this, sorry, yeah, I can think of the unit step if you want but it need not really be flat in the entire frequency range, you only need to be flat in the positive range from fc plus w to fc minus w, so if you have any filter like that, you will get this, okay, so this is the thing that's interesting, okay, so you get X plus of f, okay, so maybe I'll draw a general description of X plus of f, how it looks, okay, so if you look at the real part of X plus of f, it's going to look like this, I'll specifically write a minus fc to show that that part is zero, okay, and the imaginary part if you want to draw, okay, so it's going to be something here and then the minus fc part once again will be zeroed out, okay, so this is my X plus of f, okay, all right, so there's another way of writing it down if you want, if you like the unit step a lot, you might want to write it as what, X of f times u of f, okay, so that's another way of writing it, so many different ways of doing it, okay, so the next thing I'm going to do is I'm going to shift this to baseband, okay, so how do you shift in frequency to baseband? In frequency, okay, so all I have to do is just do, instead of f, do f plus fc, okay, so if you change my argument to f plus fc, the new plot is going to look closer to, it's going to be a baseband, okay, so that's the next thing I'll do, so I'll try to look at the real part of X plus of f plus fc, okay, which will look like this around zero, so remember this will be still between minus w and plus w, and the imaginary part will look like this, okay, so I can do this, I can imagine doing this, if you like in the time domain also there is an equivalent way of doing this, right, you'll multiply by a suitable exponential, you can always come back to baseband, okay, that's possible, but one thing I want you to want to point out before we go further is, if you look at X of f and X plus of f, one thing that has changed is I've chopped out the negative frequencies, what it means in reality is, you might say negative, nothing you should lose, but you'll lose something, right, what will you lose? You'll lose energy, okay, so the energy in X plus of f will be what of the energy in X of f, half of the energy, okay, so to compensate for that, typically people try to multiply this X plus by root 2, okay, so you multiply by root 2 so that the energy is equal, okay, so that you don't lose the energy in the equivalence, okay, you don't want to lose it, so you want to keep that going, so I'll introduce a root 2 term as we go along, that's one thing I'll do, okay, so maybe I'll write that down here, so if you look at the two norm of say X plus of f, okay, it's going to be 1 by root 2 times the two norm of X of f, okay, so that's one point you might want to note, okay, we'll adjust for this as we go along, as we go along, okay, so I'm going to take this X plus of f plus fc, okay, and then look at its inverse Fourier transform, okay, so that will happen to be the signal that is of interest to me, okay, so that's my definition finally, finally I'm ready to define what's called convex envelope of, not convex, complex envelope of, it's kind of a strange name, I'll justify it later, X of t is defined as that signal X tilde of t, X tilde of t, which is a Fourier transform with root 2 times X plus of X plus f plus fc, okay, so you take this X plus f plus fc, you multiply by root 2, but that root 2 is just a convenience factor to retain the energy as the same, if you're convinced about energy being the same, you can drop it, it's not a big deal, it doesn't change much, look at its inverse Fourier transform, that will give you your the complex envelope, okay, so if you go back and substitute it to the formula, you'll see this becomes root 2 X of f plus fc times U f plus fc, okay, so you shift X to the left and then multiply by the unit step, it's a suitable unit step, it's going to get rid of the negative, okay, so that's another way of visualizing it, visualizing what's happening, okay, if you want a relationship purely in terms of X, you want to get rid of this X plus, this is what happens, this is what happens, okay, seems like a simple enough description, but we'll have to study this in more detail, so what's happening, so X tilde of t for, first of all, will be a complex signal, right, so you can go back and see the symmetry has been completely lost in general, so you have a complex signal, okay, so I want to see more closely in what way does it exactly represent the X of t in base band, so it's also not too difficult to see why, okay, so as long as you can, so when can you say it's a complete representation, and you can go from X of t to X tilde of t and then go back again, you should be able to go back from X tilde of t to X of t, once I can do both, then I have something like a one to one mapping, right, so I can go from here to there and that back again, so I can say I have a valid representation from a real pass band signal, okay, so we'll see how to go back, it's a little bit involved, I'll try to do it without too much detail, but it's easy to see why that will happen again, okay, so I'll give you the formula first, so going back, I'll give you the formula first and then I'll give you a justification for why that is true, X of t you can show equals root 2 times real part of X tilde of t e power j 2 pi f c t, okay, so if I define X tilde of t like this, I can show if I take X tilde of t, which is a complex base band signal, multiply it with e power j 2 pi f c t which is the exponential with frequency f c and take its real part, well the root 2 adjustment is once again to adjust for the power of the energy, once you do that you get back your X of t exactly intact without any problem, okay, so in a way it should be possible because all the information that was in the spectrum is still retained in X tilde of t, so you should be able to go back by suitable playing around with the spectrum, so that's done by this nice looking neat operation, okay, so I'll justify this, the first thing is real part, okay, when you take real part what happens to the signal, okay, so that's something that you should know, if you have X of t being a Fourier transform pair with X of f, okay, so what happens to real part of X of t, how do you write real part of X of t, X of t plus X conjugate of t divided by 2, okay, so if you write it in this way you'll see this goes to X of f plus X conjugate of minus f by 2, okay, so that's the first thing, so you've taken care of the real part, when you do real part of a signal what happens is the spectrum gets added to its conjugate mirror image divided by 2, it has to happen, why, because when you go from complex to real your spectrum has to suddenly become conjugate symmetric, so you have to go from X of f to a conjugate symmetric form in all cases, so it has to happen, something like this has to happen, okay, so you get this, so that's the first thing and the next thing is e part j 2 pi of ct, what does it do in the spectrum? Yeah, it shifts to the right by fc, okay, and then you take the real part, you do X star of minus f, it'll be easy to justify this, okay, so I'll write one more equation, if you want a real justification I'll write one more step, that's basically this result, you can show, you need to also use this result, you have to use that X of f will be, I believe X plus of f plus X plus conjugate of minus f, okay, so X plus of f was only the positive frequency part, okay, you do a conjugate reverse of that, that will give you the negative frequency, right, so X of f is equal to this, so you use these two relationships, it's a very trivial step, one more step to show that this is exactly true, and you have to adjust for the root 2 carefully, there's a by 2 coming, there's also a root 2 in the definition of X plus, it'll work out properly, okay, so I'm not doing the complete derivation, so you use these two guys, these two facts, you can quickly show that this is true, okay, so what we have finally established is a real pass band signal X of t is equivalent to its equivalent or is represented by its convex envelope X tilde of t, how do you define it? You take the positive parts alone, shift it to base band and then take its inverse Fourier transform, that's your convex envelope in the definition and it's a proper representation because you can also go back from the convex envelope to a real pass band signal, okay, what about fc, what role did fc play in all of this? Yeah, it's just an arbitrary shift, so you can imagine one complex base band signal can represent an arbitrary number of real pass band signals, just change fc, you get a different pass band signal, okay, so that's another interesting relationship, if you want to map the set of all base pass band signals to the set of all base band signals, it'll be a many to one type relationship, okay, so fc plays a role which determines how it's shifted, okay, so that's what I want to establish, there's an equivalence between X of t and X tilde of t which hopefully you're convinced about, okay, so you can go back and forth between the two, okay, a particularly important thing here is to think about it from a practical point of view, why would something like this be really useful, okay, so one of the complications is X of t can be on a circuit, it can be carried by a single wire, right, well single wire to ground or just two wires, right, you only need one quantity, one voltage or something, what do you need for X tilde of t, you need two, okay, but is there an advantage, did you gain anything, why have you gone from one wire to two wires now in a circuit, what is the advantage, yeah, so the bandwidth is lesser now, so it's all base band, so you probably have better filters, better everything in base band and you can maybe do some smart things much more easily in lower frequency, so even though you've increased the number of signals you have to keep track of, when you go from pass band to base band, advantage is you've gained a so immense that it's worthwhile, okay, so this is pretty much done all the time, nobody really processes pass band signals, you always shift it to complex base band, keep track of two different guys and process them, okay, so this is something which is done all the time, alright, so even from a theory point of view it's useful, that's one notion from a practical circuit building point of view, but from a theory point of view it's again useful because go back to my formulation for the channel, what is the model for my channel, Y of t is X of t plus N of t, now I can restrict myself completely to only base band X of t, I don't have to look at any pass band X of t at all, right, I can restrict myself totally and completely to a base band X of t and do all my design and thinking around a base band X of t, maybe it's simple in some fashion and it will equally apply well enough to any pass band system because as long as I keep my X of t complex I have to allow it to be complex, right, so as long as I allow it to be complex then I can go to any pass band situation also just by multiplying by suitable exponentials shifting back and forth, it makes a lot of sense, both from theory and practice, so this is used all the time, okay, the next thing is, is there any question floating around, everybody's happy, no confusion as to how this happened, okay, so if, okay, so I want to comment about this name convex envelope, why is it called convex envelope and maybe show you a plot to justify why it's actually an envelope of X of t, but before that one interesting derivation that you can try is to write a time domain expression for X of t in terms of, no for X tilde of t in terms of X of t, okay, so I did not write a time domain expression, so I wrote it in frequency domain and I said do a Fourier transform and get it, okay, any answers what will play a role when you do a time domain, the Hilbert transform will play a role, okay, so you'll see the real part of X of t will have a very simple relationship with real part of X tilde of t will have a direct relationship with X of t, but the imaginary part you'll have to do a Hilbert transform, okay, so that's how the time domain representation works but in frequency domain it's easier to understand, okay, so let me just quickly motivate the convex envelope part of it, okay, so let's write down, let's look at this expression closely, which expression this, this expression going from, it's a root 2 real part of X tilde of t e power j 2 pi f c t, okay, so I'm going to now introduce more notation for talking about X tilde of t, X tilde of t I'll write as X i of t plus j X q of t, okay, write out its real part and imaginary part explicitly, this is typically called the in phase part and this is called the quadrature part, okay, this is standard terminology, the i part and the q part, you'll see people always referring it to the i and the q, okay, in phase and quadrature components of the convex envelope X tilde of t, so if I plug it back in and multiply it out with e power j 2 pi f c t and take the real part multiplied by root 2, you'll see X of t will work out to root 2 X i of t cosine 2 pi f c t minus root 2 X q t sine 2 pi f c t, okay, this is how X of t will work out to if you actually substitute it back into the formula and work it out, okay, so it's not too difficult to see this, okay, so remember the X i of t and X q of t are what type of signals, real and then baseband as well, okay, so X tilde of t is also baseband, so real and baseband signals, okay, so what else, okay, so and then X tilde of t, X of t is also a real signal, okay, so everything is real, so you see it's good to see a nice real expression, okay, so you have a lot of real part and tilde's floating around but it's finally good to see a nice real expression, this also gives you the motivation for why convex envelope and why real envelope and all that, okay, so why this convex envelope name came about, okay, so if you try to plot a signal like this, okay, where X i of t is baseband and then you have a cos 2 pi f c t and imagine fc being very, very large, how will such a signal look, it will be a cosine 2 pi f c t but modulated, okay, assume X i of t is positive just for simplicity, it will be modulated by this baseband X i of t, okay, so in that sense X i of t will be the envelope of this entire signal X i of t cos 2 pi f c t, okay, so it will, you can imagine something like this, right, so I have the, right, I am drawing it very, very poorly, I am sorry, so it should be equal like this if you assume it's all positive, right, it will look like this and your X i of, X i of t will be this guy, okay, assuming it's positive, it's negative, it will cross over and all that, this is in confusion, so I will just ditch that, so this is envelope is, right, this envelope is X i of t, okay and the frequency at which that oscillation comes about is X 2 is f c, right, cos 2 pi f c t, a similar story will be true for the other part also, X q of t sine 2 pi f c t but what will be the difference between the cos 2 pi f c t and sine 2 pi f c t, there will be a phase lag or lead or whatever you want to call it, of 90 degrees, okay, so while this starts at maximum here, where will that same point be as time 0 for sine, it will be at 0, so there will be a small phase lag between those two, there will significant phase difference between the two and then the envelope will be X q of t, okay, so you take the envelope corresponding to the cosine part, take the envelope corresponding to the sine part, put them, put the two together, you get the convex envelope of X of t, okay, okay, so that's the, that's the final story here, okay, so this equation itself is very interesting, as in any passband signal, real passband signal can be written in this form, okay, sine multiplying a baseband minus cosine multiplying another baseband signal, okay, those two are envelopes, you take those two envelopes, put them together with J floating around, you get a convex envelope, okay, so that the name envelope is justified because it is, it is actually an envelope of some amplitude modulated sine wave, okay, so that's the reason why this name comes about and I want to draw more pictures to illustrate what's happening here, okay, so the picture I want to draw is the following, okay, so this up conversion and down conversion picture and I want to have, add more comments also, okay, so that's the picture of how to go from the in-phase and quadrature parts to the passband signal, okay, so if you have X i of t and X q of t, okay, remember together what is this, X tilde of t, right, X tilde of t is actually two wires, one carrying X i of t, another carrying X q of t, so how do you go to the real passband signal, you multiply one by root two cosine omega, well I am writing omega, okay, so I think maybe you shouldn't write omega, sorry, 2 pi of ct, okay, multiply the other by root 2 sine 2 pi of ct, okay, so typically what people do is you will have one crystal and then you will phase shift it by 90 degrees, so that one becomes 90 degrees out of phase with the other, so you can think of one as cos and the other as sine, okay, so it comes out and then you add, well you add, actually you should add this and subtract this, right, you get X of t, okay, so this is what's called an upconversion picture, okay, so in practice you will see people referring to this as the i channel and the q channel, okay, so both the i and the q channels actually occupy the same frequency band, okay, the exact same frequency band around fc, but they are separable, okay, so how do you separate, here's the downconversion picture, okay, so you have an X of t coming in, so you first split it, multiply one by, so all these root twos are quite irrelevant, okay, so hopefully I am just writing them down for completeness but in practice you don't worry about things like root two, okay, you don't measure the amplitude to see if it is, okay, then what do you do? You do a low pass filter here and the low pass filter here, what will you get? You will get X i of t and X q of t, so even though these two signals, the i and q channel occupy the same frequency band, you can receive them separately, okay, so you can go through this downconversion process and separate them very easily, okay, so I don't want to go through this derivation, it's very easy to show that the signal will be a baseband component plus something at 4 pi fct or something, so you put a simple filter around the baseband frequencies, you will get back your X i of t and X q of t, okay, what is one significant assumption in this downconversion? Is there a huge assumption from a practical point of view, assuming this up and down conversion happen at physically different locations, what am I assuming? FCs have to be exact, okay, if the FC is not exact, what happens? There is going to be something in X i of t, there is going to be an e power j, there is going to be an extra, yeah, there is going to be an extra thing which is floating around, okay, and then all kinds of complications can occur because of that, okay, so assumption is this process is coherent, okay, so we will assume always that it's coherent, it's possible to transfer this FC exactly to the receiver using some nice single processing ideas, it's called carrier synchronization, okay, so you can achieve that in practice, so it's possible to assume that you have the same FC at some other locations, okay, so you get the X i of t and X q of t together, this form the complex envelope, X tilde of t, okay, so now go back to my, what should I say, to my channel picture, okay, so I have to convert a sequence of bits into a real signal X of t and send it across on a channel, okay, and then receive it and process it and so on, okay, if my channel is baseband, okay, then my signal X of t has to be real baseband, if my channel is passband what can I do, you can do two real baseband signals, do you see that, okay, so it's as almost as if if just by having a passband channel, you're capable of now dealing with two real baseband signals, so you can imagine two streams of bits being modulated into two real baseband signals which together make the convex envelope which then gets up converted into one passband signal, okay, so it somehow seems like you can do more in passband, why is that, there's more bandwidth, right, so there's more real bandwidth, okay, in baseband if I say w is the bandwidth, then I have only zero to w, in passband if I say two w then I have two w, entire two w I can use, okay, I don't have to maintain the symmetry around fc, okay, I have to only maintain a symmetry around zero, in baseband I have to maintain a symmetry around zero, so what's on the negative band has to be exactly equal to what's on the positive band, okay, in passband that's not true, so it's a very natural extension, so since you're using two times the bandwidth, you can expect two times the data to ride on the same bandwidth and it's separable, okay, so that's the way to think about it, so in practice people use this, it's called quadrature amplitude modulation, okay, so it's used in a thing called quadrature amplitude modulation, it's very very popular, today pretty much every decent digital communication system uses this quadrature amplitude modulation, okay, where you use in a passband channel you're able to send two streams of data together riding on the same frequency band around a center frequency, okay, so that's the idea, okay, QAM is called, all right, so I think that's pretty much all I wanted to say, let me just quickly see if I wanted to say anything more, I think that's pretty much it, okay, all right, so any questions on this baseband, passband and suddenly why are we getting the two times in passband alone, what's so great about that, it's probably clear, right, it's okay, all right, so we'll move along, so remember we're still in preliminaries, I want to wrap up the preliminaries quickly and then move further, the next thing we need is to move from continuous time to discrete time, I'll do the Nyquist sampling theorem formally because I'll use it several times in so many different guises, so I think you should know this a little bit formally as well, so I'll write down Nyquist sampling theorem, this is what enables this digital processing and sampling and all that, okay, okay, so you might have learnt it in various ways, so what's the standard way in which you talk about Nyquist sampling, if you have a signal band limited to a bandwidth, w, it's enough if you take samples one by two w away, you can accurately and completely reconstruct your original signal just by those samples, okay, so there are several ways of understanding it, one way of understanding it is it's possible to write a continuous time signal as long as it's limited in bandwidth as a sequence of numbers, okay, which means you're going, you're losing dimension, right, so you potentially have an infinite dimensional entity initially x of t, from there you're going to x0, x1, x2 so on, okay, you're losing dimension, okay, so it seems to be something that you're losing but still you're able to go back, okay, all that it means is when you have good signals which are limited to bandwidth, it's possible to think, it's possible to have a countable basis, okay, so even though it's infinite dimensional, you can have a basis for it, for it which is countable, okay, so it's like for your series and all these things that you think about, so it's enough to give a series of numbers from that you can go back to this signal, as long as it's limited in bandwidth that's very important, okay, so that's what I'm going to quickly do now, so you'll see how it works out, okay, so it's a very simple formula to write down, okay, so suppose you have a signal x of t, which I will say belongs to L2, okay, so I will need a finite energy signal, okay, so its spectrum additionally is going to be contained in a bandwidth minus W by 2 to W by 2, okay, so yeah, and then the spectrum, what will, what do we know about the spectrum, okay, it's also going to be in L2, right, so it'll be another finite energy signal in X of f, okay, so it'll be something like this, okay, so that's our look and you know this will also be in L2 because you had the original thing in L2, okay, so first thing I'll do is to look at the spectrum which is between minus W by 2 and W by 2 and expand it in a Fourier series, okay, so you know that's always possible, if you have any time limited function, any finite support function as in time limited function, it just doesn't vanish over a small interval, you can do a periodic extension of that if you want, then write down a Fourier series type thing for me, so I'm going to imagine a periodic extension of this X of f with period W, okay, and then writing a Fourier series expansion for it, okay, so that's the way, that's the way probably you've been taught, so that's what, that's what you're going to write, okay, so I'm going to expand this minus infinity to infinity Xn, I'll write e power minus j 2 pi f n by W, maybe you wrote it as plus but it's okay, I can write minus or plus, so it's going from minus infinity to infinity makes no difference, this is the Fourier series expansion which is valid in what, from mod f less than or equal to W by 2 technically, okay, right, the left hand side vanishes outside of minus W by 2 and W by 2, what about the right hand side, it's periodic, it repeats in every W interval, it's going to repeat, okay, so it's periodic forever, so I'll say I'll restrict my f to W by 2 just to make this thing exact, what is Xn now, what's Xn, okay, so maybe you're used to doing this with t and suddenly when I do Fourier series with f maybe you're scared, okay, so don't worry about t, t is just a variable, you can do Fourier series for any f of x, as long as it's contained in a finite support 0 to t you can do it, any f of x doesn't matter, right, e power minus j you can do it, don't worry about it too much, so what's the Fourier series formula now, 1 by W integral minus W by 2 to W by 2 X of f e power I'll do plus j 2 pi f n by W, I'm doing this plus and minus just because I inverted the thing, I mean you can do it in any way you want, okay, df, this is my sum i Fourier series coefficients, okay, so what is this now, you stare at it very closely, this integral is nothing but the inverse Fourier transform of X of f evaluated at n by W, so this is in very simply 1 by W X n by W, okay, so these Fourier series coefficients for X of f are what? Samples of X of t, you knew this, okay, there's nothing great about it, you must have seen this somewhere as long as, so why do you need this finite support, because only then Fourier transform will exist, right, you have to have a finite extent for your X of f, only then you can do this e power j 2 pi f by W, if you don't have finite you can't do that, okay, so that's why you need it and Fourier series coefficients are samples of X of t, which is again something you might expect, okay, it's not something very difficult to worry about, okay, so I'm going to write this guy back again in a slightly different form, I'll write X of f equals summation, I want to drop this mod f less than or equal to W by 2, I don't like it too much and then I also want to get rid of this Xn in terms of the original thing, so I'll write it as 1 by W X n by W, why don't you just come and sit here, this part is not very comfortable, 1 by W X of n by W, what will you have next, okay, so you're going to have e power minus j 2 pi f n by W, I want to drop this mod f less than or equal to W by 2, so instead of that I can say rect minus W by 2 W by 2, what f, okay, hopefully everybody's happy with that, it's just another way of saying mod f is less than or equal to W by 2, okay, so my original W by 2 went out of the screen, okay, I'm sorry about that, maybe you can still see it, okay, so that's my picture, so now what am I going to do, I have a picture completely in f, okay, I can do a inverse Fourier transform and go to a picture completely in t, okay, and that'll give you my Nyquist sampling theorem, okay, so what's the frequency Fourier transform for X of f, it's X of t, okay, then I have the summation, the reason why all these summations and integrations can be exchanged is because I said X of t is an L2, once you have L2 everything will work, there's no problem, you don't have to worry about arbitrary terms showing up here and there, it'll work out very nicely, infinity to infinity, X of n by W, if you do some work and convert that rect into a Fourier transform, there's also an e power minus j, okay, so you can see how it will look, okay, rect is going to go to a sink, okay, and then this e power minus j is going to make that sink shift, okay, so once it shifts and it's 1 by W you'll see it'll cancel with some or something else, so all that will go away, so you'll get ultimately sink, I've written here, okay, the way I've written here is W times, yeah, I think W t minus n by W, yeah, that's fine, that is fine, okay, so you can show this is true for all t, okay, so this is your famous Nyquist sampling theorem, which says X of t, if it's contained within a finite bandwidth minus W by 2 to W by 2, is completely reproducible using its samples spaced 1 by W seconds away, okay, you take samples forever, all you have to do is a simple sink interpolation to get back your X of t at every possible time instant t, okay, so this is an important result for us, okay, so if you don't use this result, it's difficult to justify so many things you do in digital communications, okay, so this is a very basic, okay, so I think this should be very familiar to you, you might have seen this in several forms, this is the form in which I'm going to introduce it, okay, so another thing that you might want to check, this is nothing but an orthogonal basis expansion, okay, so you might not have seen this before if maybe maybe you should know that this is an orthogonal basis expansion, okay, so these guys just like e power j 2 pi f 2 pi f n f by W, okay, so those things form a basis for finite support functions, these guys form a basis for finite bandwidth functions, okay, they're also orthogonal, you can check that this inner product will vanish, sink W t minus n by W and sink W t minus m by W will be what, you can show it will be 0 if m is not equal to n, okay, so it's a very simple thing to show, okay, you can prove it using Fourier transform, okay, so you go to the Fourier transform domain, you get the e power j's, they will always have this property, okay, so this is a, this is nothing but an orthogonal basis expansion but the nice thing about this orthogonal basis is the multiplying coefficients happen to be samples of X of t, if you use any other orthogonal basis, you won't necessarily get samples of X of t, you'll get a complicated integral formula, okay, which may not be very useful, but for this orthogonal basis, the multiplying terms happen to be samples of X of t, which is very, very useful in practice, okay, so like you might have already known, this sampling theorem justifies digital signal processing completely, okay, so you can take an X of t, okay, just to be sure, typically there is a low pass filter at W by 2, okay, and then what do you do? You sample it at rate 1 by W, you get your sequence X k, okay, so I'll use X of k here just to be consistent with the way I'm going to write down, okay, you'll get samples X of t, X of k, now what can you do with these samples, okay, with modern computers and all these things available, you can just take these samples and store them wherever, okay, once you store all the samples, you can process it using all the complicated algorithms that you know, okay, do anything you want with those numbers, add them, multiply them, subtract them, divide them, take tan inverse, take whatever, hyperbolic sign, whatever you want to do, you can do to those numbers very easily, if you want to do all those functions in X of t directly without sampling, it's difficult, okay, there are only so many things you can do in analog working with circuits, okay, once you sample them, get them as samples, you can do whatever you want to those numbers inside a computer, okay, so typically those that's called a processor, it's a digital signal processor, okay, so there are several of them available today, and then once you're happy with all the processing you've done, you're going to get y of k, okay, then what do you do? You can kind of interpolate this y of k very coarsely first, just maybe sample and hold for a while, and then you do the proper sync interpolation, which is, which you can show as nothing but an LPF, okay, you do a simple low pass filter, you get back a continuous time waveform, which is a processed form of X of t, okay, so this is a very very powerful idea which motivates this entire area of digital signal processing, and this is possible as long as the bandwidth that's coming in is low enough, you can do that, okay, so in pretty much all the processing that we will do at the receiver, okay, so the receiver you imagine y of t is a continuous time signal that's coming in, the first box will always be a low pass filter at a frequency and followed by a sample, okay, this is something that will be common to pretty much any digital communication system today, so you convert it into a discrete time signal and then do all your processing in discrete time, so you can implement complicated algorithms, okay, you can imagine having complicated algorithms, all that is possible because I know I can do it digitally without losing anything, I don't lose anything in the process, that's another thing to keep in mind, okay, all right, so I think we are getting, okay, so I think this is probably a good place to stop because the next preliminaries we need are maybe you can pass along this, so I think I was hoping to finish the preliminaries today but maybe it's not possible, maybe it's a good place to stop as well, so next week I'll begin by looking at discrete time Fourier transforms and Z transforms, just establishing the notation, making sure we quickly go through the main results there, and then we'll do something called spectral factorization which you maybe have not seen in digital signal processing, and then after that we'll push, okay.