 A warm welcome to the 16th session in the third module on signals and systems. We are now at a juncture, where we need to be more realistic about this whole idea of sampling and reconstruction. So, in the previous few sessions, we have identified certain issues with the reconstructor. If we take the ideal reconstructor, it has a fundamental problem, it is unstable. Now if we take a table reconstructor, a practical reconstructor, the first thing we need to do is to allow for what is called a margin, a margin between the original spectrum and the carbon copies, the unwanted copies that are created as a consequence of sampling. And one can make use of that margin to allow a transition band. So, let us look at the situation where we have a signal band limited to FM. So, you have the split-show prototype spectrum here, the original signal spectrum. And let us sample this ideally with a sampling rate of Fs, much greater than FM. So, of course, we assume it is much greater than 2 times FM as well. Let us see what happens to the spectrum. The resultant spectrum looks like this. And of course, you know, the same original spectrum is carbon copied at every multiple of Fs. Now, what we were emphasizing was that this margin is very important because we do not want a brick wall kind of response. So, what we meant by a brick wall response was as follows. We do not want a filter or we do not want a linear shift invariant system to have a response that looks something like this, you know, you do not want it like this. Instead, what we want is something that has a smooth transition. We would want something like this. And why would we want something like this? Because that ensures the resultant system is stable. But in fact, now I am going to put an even greater condition. Look at this blue response here, which I have said is what we might want or accept better than the brick wall one. You know, even this one has a bit of a problem. This flat region here is a bit of a problem. Realizable filters do not really have a flat region in the frequency response. Just like that brick wall is a problem, the flat region also is a problem. So, what is the best we can do for flatness? We can assume that the response varies very slowly around the middle frequency, zero frequency. And if the response varies fast somewhere towards the edges, we would have made sure that all your spectrum, original spectrum is contained in the region where there is slow variation. Not quite clear, right? Let us draw it graphically, then you will understand what I am saying. If you look at the so-called acceptable frequency response, it must have two features. It must not have what I am now going to show in green, must not have a flat bar and must not have a brick wall like this. In fact, both of these make the system unrealizable. Now, why is that so? It is a little beyond the scope of the current discussion. If some of you are knowledgeable, you might want to post on the discussion forum what you think is the answer. Why must you not have a flat region and why must you not have a brick wall? I have already given you some indication of what the problem is. It brings the brick wall brings instability into the system and what does the flat region bring? Think about it. It brings irrationality into the system. So, you know the system would require infinitely sources to realize. So, when I want to realize a system with a given frequency response with finite resources and I also keep it stable, then I need to avoid this flat region at the brick wall. And then practically what are we saying? We are saying something like this. You would like essentially a response of this kind. You know, the response can be shown all the way up to FS or FS by 2 more precisely because FS by 2 is the upper limit on where you can have your signal frequency. But of course, the upper most signal frequency must be much less than FS by 2. Remember that. So, you must have a response that looks something like this perhaps, you know, varies very slowly around 0 and then it can vary fast and go down to FS by 2. And hopefully what you should do is to put the FM in this slowly varying region. So, you know, you can put FM somewhere here maybe put the signal in the slowly varying region. Now, in fact, what I am going to do is to take a very simple example of a low pass filter that we know. Let us take a simple RC circuit and let us look at how it behaves. We shall do that in the next session. And then we shall also ask, can we do something about the whole process of sampling? Here we are making the reconstruction more practical, but can we also do something about the sampling? Can we make the sampling more practical? And does it really affect this reconstruction? We shall answer these questions in the next session. Thank you.