 A warm welcome to the 24th session of the third module in signals and systems. We will now continue the discussion that we had started the last time about what happens when we sample and hold. Let us put all that we did all together first. So, let us look at the signal x t that we gave the ideal sampler, which produced an output x s t and the hold which produced x h t, so called held wave form. Now, let us consider x t to have obeyed the Nyquist principle. So, if it obeys the Nyquist principle, it is band limited. There can be no frequency component more than half of the sampling frequency, which is 1 by T s. And we also know the frequency response of this. We have calculated it. Let us see graphically what is going on here. So, you know let us take a spectrum, a typical prototype spectrum as you might have. So, you know suppose it is band limited, let us first make life simple by making the band falls smoothly from 0 onwards, you know a spectrum something like this perhaps. So, I am showing its Fourier transform in the cycles per second domain. What would happen after sampling? So, let us call the spectrum of the sample signal as x s f. You have the original spectrum and its copies. Remember we gave a name to these copies. We call them aliases. Now, let us sketch the Fourier transform of the hold and then let us put the hold on this. So, sketch of the frequency response of the hold. The first thing is we need to write it in the cycles per second domain. So, we need to write capital H of f, so to speak. Let us sketch its magnitude. Now, notice that we have a main lobe and side lobes here and now let me do one thing. Let me lift that main lobe and put it where it should be in the spectrum that we brought out before. Let me put the side lobes in their correct place. Look at the drawing carefully. You see the main lobe runs all the way from minus 1 by T s to plus 1 by T s. So, it is all the way up to the sampling frequency. However, your signal spectrum should not go beyond half. So, this is where your signal spectrum would lie in this little box here or this little region here that I have marked out in red. So, let us do that superposition. So, if I put the main lobe, now let us not worry too much about the magnitudes, the scale of the magnitudes. So, you are multiplying it by a factor of T s. All that can be controlled by putting a gain or I mean a gain more than one or a gain less than one. So, let us not worry too much about that. It is the form that is important, not so much the amplitudes. Let me put on top of this spectrum, the spectrum of the hold in red. So, the main lobe comes here and then you have the side lobes going away from there. So, therefore, what is the consequence of sampling and holding? You know there are two things that are happening as a consequence of sampling and holding. One thing is the distortion that is going to come in the original spectrum. So, you know this red part, the red main lobe which now sits on top of the original spectrum and some part of the first two aliases, you know this is the first aliase on the positive frequency side and the first aliase here on the negative frequency side. The main lobe sits on the original and the first two aliases. So, the original gets distorted and so do the aliases. So, in the process of sampling and holding, what is happening? One thing is that the aliases are passing through partially. You know you are not point able to stop those aliases. Secondly, the original is getting distorted. That is the pity. Even if you had let the aliases pass, perhaps one could have followed it up with some kind of a filter that removed all frequencies beyond half the sampling frequency. But because of the distortion of the original spectrum, you also need a correction. You need what is called equalization to out to the effect of that main lobe as it lies over the original spectrum. So, let us make a note of that. Let us go back to this drawing. So, you see basically we need equalization here in this part. Let me mark it in green. I will say correction needed or equalization needed and what is the nature of that equalization? We can write that down very easily. It is essentially the reciprocal of the hold system function, the reciprocal of the hold frequency response. So, essentially one by that frequency response. Now, of course, you cannot let this expression apply for all f because there are nulls. So, you know this would only apply in a certain zone. We can even sketch it. We can get an idea of how it will look. Let us sketch that. So, the green spectrum is essentially the sink main lobe corresponding to the hold and we need the reciprocal only in the signal band namely minus 1 by 2 TS to 1 by 2 TS and that reciprocal would look something like this. You know, it would take its minimum at 0 and at 0, of course, it will have the value 1. Let me show it to you in red. You go to maximum at the edges. Now, of course, you can see that if you carry this all the way up to 1 by TS and minus 1 by TS, it will go to infinity. So, that means nothing. So, you see that equalization has a meaning only in the signal band between minus 1 by 2 TS and plus 1 by 2 TS. Beyond that, anyway you have aliases. So, you want to throw them away. So, the net result is that if I wish to reconstruct a signal after being sampled and held, I notionally need to do two things. I, of course, need to cut out all these carbon copies, all these aliases which have been created. But together with that, I also need to equalize. So, now, the ideal reconstructor, what is the ideal reconstructor for a sample and hold situation? We should write that down. So, the ideal reconstructor needs to do equalization in the signal band, stopband in all outside the signal band. And you know the nature of equalization that is required. In other words, when you have sampling and holding in the operation, the ideal reconstructor is not going to be flat in the signal band. That is what we need to understand. Now, a natural question arises, is this the only kind of interpolation? No, we have seen there can also be linear interpolation. How do we capture the effect of linear interpolation? We will talk about it in the next session. Thank you.