 A warm welcome to the 14th session in the third module on signals and systems, we stated a very important principle. I dare say one of the most fundamental principles of sampling and reconstruction in the previous session. The principle said that a band limited signal, band limited to FM can be reconstructed perfectly from its samples taken at a rate more than 2 times FM and we saw why that is the case. Essentially the imposters can be removed and the original can be retained if we have ensured that this Nyquist principle namely FS greater than twice FM is obeyed. And in fact the proof of this theorem was constructive in the sense that it gave you a procedure by which you could reconstruct the signal from its samples. Let us recapitulate that procedure as it is a very important idea that we should appreciate very thoroughly. What we said was that if you had an original spectrum like this, let me draw it again for you. If you had this original spectrum of the signal band limited to FM, let us say looking something like this and you sampled it at a rate FS greater than twice FM, copies of the original spectrum would be created at every multiple of FS. And reconstruction essentially means retaining the original and cutting off the copies. So we need a system which retains this and cuts off all the rest. Now how do you do that? Suppose you pass this whole sample signal, this is the spectrum of the sample signal. Suppose you pass this sample signal through a linear shift invariant system which has a frequency response and writing the frequency response as a function of the cycles per second frequency note. And what is this frequency response that we desire, let us draw that explicitly. The desired frequency response needs to be one, the desired frequency response needs to be one between just a little before FM, I mean minus FM actually, just a little before minus FM on the negative side and just a little after FM on the positive side of F. So here I am writing in terms of the cycles per second or Hertz frequency and I am saying the desired HF needs to be one from let us say minus FM minus delta to FM plus delta. Essentially you must ensure that FM plus delta is less than FS minus FM, strictly less than. So that is not very difficult to do. If you go back to the previous drawing of the spectrum, you can see that that should be possible. Look at it here. So this is FS minus FM here and you have this margin as I pointed out in the previous discussion. As long as you have ensured that FM plus delta remains within the margin, you are doing well. So you are essentially saying here that FM plus delta keeps the cutoff or remains within the margin. Now, let us look at this drawing again of the desired HF. You have FM just a little before here and minus FM here. You know you could have always, I mean one could have argued that you could have kept this desired HF one up to FM and minus FM, you know between minus FM and plus FM and then 0 afterwards. But what is the problem there? The problem is that you might have a tonal component at FM, meaning there might be an impulse in the frequency domain at FM. When will that happen? That would happen when you have a pure sinusoid. You recall that there are impulses in the frequency domain when you have pure sinusoids. So there is a tonal component at FM and if you make the HF cutoff at FM, you could have trouble with that tonal component. So you need to keep FM plus delta as the cutoff. This is an important and a subtle point. In fact, we can understand it even better. If we take just that tonal component and focus upon it. So let me explain this point, it is a little subtle and we need to understand it well. See, consider a tonal component when you say a tonal component, you mean a sinusoid with frequency FM. Once we sample this at exactly twice FM, the sampling rate FS is exactly twice FM. What is going to happen? Let us see it pictorially. So you have the sinusoid. Now this condition FS is twice FM essentially means that you have two samples per period. Now that is very easy to see. Suppose for example, your 0 is located here, so you have taken one sample here. The next sample would come at the corresponding point on the negative cycle. So here and the next sample would come here and so on and so forth. You could keep doing this. So these would be the samples. Now suppose you are unlucky and you take the sample where I am showing you in green. You start sampling it here. What would happen? Your sample would come here and then here and then at every other 0 crossing. So in the unlucky event that you start sampling the sinusoid at a 0 crossing, all your samples are going to fall at 0 crossings. This is where you have trouble when you sample at exactly twice FM. That is why we said FS must be greater than twice FM. This was a subtle point. I am emphasizing that now because a tonal component at FM, that is what it is, a tonal component of sinusoid with precisely the frequency FM, if not sampled at a rate more than 2 FM can have this trouble of all the samples falling unfortunately on 0 locations. Then if all the samples fall on 0 locations, there is no way to distinguish between whether you had a zero signal in the first place or whether you had a sinusoid at all. Now there is another reason also and that reason is what we are seeing in this business of a margin here. You know you of course require that margin because you might want to retain this tonal component. There might be an impulse right at the boundary at FM and you do not want to cut off or you do not want the cut off to fall on that impulse. You want the cut off to be beyond that impulse. The impulse also comes in. That is one way to understand why you need, that is another way to understand why you need FS to be greater than 2 FM. But there is another more subtle reason which we will now proceed to derive. Let us go back to that frequency response that we drew, the desired HF as we called it. Let us go back and see what it looked like. This is what it looked like. Let us find out the corresponding impulse response. So, the corresponding impulse response can be found by taking the inverse Fourier transform. Let me work out the inverse Fourier transform. Let us do that little bit of work. It is very important. All of us are familiar with the expression for the inverse Fourier transform. We multiply the spectrum by e raised to the power j 2 pi f t and integrate with respect to f over all regions where the spectrum is nonzero. In this case, the spectrum happens to be 1 between minus FM minus delta and FM plus delta. Let us put FM plus delta equal to f c. It is a very easy integral to evaluate. Let us evaluate this. This is simply e raised to the power j 2 pi f t divided by j 2 pi t from minus f c to plus f c. It is a very easy integral to evaluate. Let us simplify this. So, it is e raised to the power j 2 pi f c t minus e raised to the power minus j 2 pi f c t by j 2 pi t, which is easy to simplify. It is essentially 2 j times sin 2 pi f c t divided by 2 j 2 pi t. And of course, we could multiply and divide by f c and cut off the 2 j's from the numerator and the denominator. So, where are we now? That gives us essentially 2 f c times sin 2 pi f c t divided by 2 pi f c t. And that is essentially 2 f c times what is called sinc. You know how to write sinc, so sinc 2 f c t. You will recall that sinc x is essentially sin pi x by pi x. So, we derive this impulse response here. Let us sketch it. Where does the sinc function go to 0? It goes to 0 at all the integers. So, sinc x is equal to 1 for x equal to 0 and 0 for all integer x, x not equal to 0. So, sinc of 1 sinc of minus 1 and so on. Now what are we really saying here? You see we are saying we have this function which is 1 at t equal to 0. So, if I locate this function at the point of sampling, it picks the sample as it is and the integers, where are the integers located? The integer located where 2 f c t is equal to an integer. What does that mean in terms of sample? We should analyze this in greater detail in the next discussion. It is going to take us a while to understand what this impulse response is doing. And that will hold the key, in fact, that will see a lot of things about the system that we are talking about too. It will tell us how we are reconstructing the signal from its samples and it will also tell us something rather disturbing about the system, about the realizability of the system. So, let us wait for the next discussion to look at the impulse response in greater depth. Thank you.