 A warm welcome to the thirteenth session in the third module on signals and systems. We are now well equipped to state a very important principle in the context of signals and systems. Perhaps a landmark which has led to the movement from analog to digital. In fact, I consider what we are going to state today as one of the central ideas in the context of sampling and reconstruction. And it is a well celebrated theorem that we are going to state well known by many in the field of electrical engineering communication and in fact, many other engineering disciplines as well. We are going to state what is called the Shannon, Wittaker, Nyquist sampling theorem. By the way, all of these are names, Shannon, Wittaker and Nyquist. And if I am not mistaken, Wittaker actually needs to have a double T, but I am leaving that open because I have seen different spellings. But anyway, the point is there are several different researchers who have contributed to the idea of what constitutes an ability to sample and reconstruct. Shannon, Wittaker, Nyquist are among those people who have been associated with this theorem that we are going to state. In a way, we have already derived the theorem indirectly in the last few discussions. The central idea in this theorem is, what allows you to do away with the imposters that you have created as a consequence of sampling? So, before I state and prove the theorem formally, let us recapitulate what we have been saying over the last few sessions in the context of sampling. When you sample, you are prone to losing information. I mean, what way do you lose information? It is not that you lose what was originally there, but you cloud what was originally there with a lot of other rubbish in plain language. Unwanted copies of the original waveforms or patterns or information, which you then need to distinguish from the original genuine authentic information. The problem is in sampling, often not that you have disturbed what was originally there, but you have created many unwanted copies and it could be difficult to distinguish what is original and what is a copy. If you can do that, then you are well set to reconstruct from examples. If you cannot, then there is a problem. If the copies are started getting mixed up with the original information, then there is a problem. To state the theorem, we first need to put down the context in which the theorem applies. There again, we have elucidated the concept in the last few sessions. The context is that the signal has a Fourier transform. The Fourier transform is non-zero up to a maximum finite and this is very, very important. I am going to underline this. Finite is important, non-zero only up to a maximum finite frequency, we call that finite frequency fm or of course, the corresponding angular frequency is 2 pi fm omega mm and given this, we now state under what conditions it is possible to reconstruct if we sample at a sampling rate of fs. So, now let us state the Nyquist theorem or the Shannon Nyquist, we take a theorem. The theorem of sampling, Shannon, I am just writing a short form here, Shannon, we take a Nyquist sampling theorem. We have a very simple theorem here. Let me read it out for you and let me show you how simple the theorem is and how simple it is to interpret it given what we have already done. A band limited signal, now remember a band limited signal meaning a signal whose upper frequency is limited. In fact, limited as we said to fm, band limited to fm can be perfectly reconstructed from its samples taken at a rate fs or 1 by Ts. So, here Ts is the sampling interval. So, that fs is greater than twice fm, so simple, all that you need is to ensure that the sampling rate is more than two times the maximum frequency component in the signal. As I said, we have already proved this in all our discussions in the past and the proof is what we are now going to write down and write down essentially taking a Q from what we have learnt in the past, is that right? So, let us write down the proof. Sampling is linear and therefore, each component, each frequency component I mean contributes its own consequence of sampling and the net consequence is the sum. So, let us assume a spectrum, band limited to fm as it were. Now, what we are saying is let us call the signal xt, let us Fourier transform the capital x of f. Now, here f is the cycles per second frequency, let us remember that and it is band limited to fm. The situation is like this, let me sketch it, it is easier to understand this if we sketch it. So, remember when you say band limited to fm, you must include both the positive and the negative frequencies. So, you have some spectrum here and just sketching some spectrum, let us not attribute any great importance to the shape as such. Now, what I am saying is take any tiny part of the spectrum, let us take this part for example, located around frequency f1 and of course, the corresponding negative frequencies minus f1. Now, based on all the discussion that we had and assuming that fs is greater than 2 fm, remember we are obeying the Nyquist principle, this is called the Nyquist principle. We do not have to keep saying Shannon would take a Nyquist, you know people are content calling it the Nyquist rate or Nyquist principle. If we obeyed the Nyquist principle, then if you focus your attention on just this part of the spectrum at f1 and minus f1, what will be the consequence of sampling? It is going to create its imposters at every multiple of fs. So, you would have an imposter at fs minus f1, of course fs minus f1, fs plus f1 and then the corresponding negative frequencies there. So, let me draw those imposters in red. Now, remember f1 is less than fm and therefore, fs minus f1 is going to be greater than fm. So, it is going to occur somewhere here. In fact, fs minus f1 would come somewhere here, let us say and fs plus f1 would similarly come somewhere here and the same thing about the other two. Now, you can keep taking all such f1s, f1 can run from 0 to fm and when you run fs, when you run f1 from 0 to fm, this imposter as we might call it, these are all imposters here. The imposters go from fs all the way to fs minus fm and of course, the net effect of sampling is the sum of all these imposters that are created. So, each little part of the spectrum creates its own imposters that these imposters are all added and remember, let us go back to the diagram here. This point is fs minus fm, remarking it here. Now, fs is greater than 2 fm and therefore, fs minus fm is greater than fm. That is what you see here. fs minus fm is greater than fm and you can in fact see that whatever is the spectrum here is recreated at every multiple of fs. So, therefore, the consequence of sampling on the spectrum is as follows or the spectrum of the sample signal. The spectrum of the sample signal would look like this. You have something going from minus fm to plus fm initially, copy the same thing around every multiple of fs and fortunately, since fs minus fm is greater than fm, let us see what happens. So, I am now going to draw the imposters. The red ones are the imposters or unwanted copies of the original spectrum and because fs is greater than 2 fm, none of the copies pollute the original spectrum. This is the original, none of them pollute the original. So, very simple. We can prove Nyquist's theorem or the principle by simply saying I can retain the original and chop off the copies. Very simple. Let us show that here in green. Reconstruction means retain this. I am showing what I am going to do in green. Retain this chop off all else. So, I need to have a system which retains frequencies just a little bit beyond fm. I am saying just a little bit beyond because we do not want to spoil what is at fm precisely and we have that margin because fs minus fm is beyond fm. So, you have a margin there. You know, you can see it. You have a margin here, this, this margin. So, with that margin, you retain all the frequencies from a little beyond fm to a little before minus fm and chop off all the rest and this system should allow you to reconstruct the original signal from its samples. Now, what is this system? What exactly does it do? How does it work? It will be the subject of our next discussion.