 A warm welcome to the 17th session of the third module on signals and systems. We had initiated a discussion last time on making the whole problem of sampling and reconstruction more practical and more realizable. And we were going to embark on a discussion of what a simple RC circuit could do for us. So, for example, to what extent could a simple RC combination work as a reconstruct? Let us see. So, I have a simple RC circuit like this, resistive capacitive circuit. And here I describe this in terms of its frequency response. So, this is the input and this is the output. Of course, the input and the output are voltages here. The capacitances of value C, the resistance of value R and let us write down the frequency response of this system, of this circuit. Here omega denotes the frequency or the angular frequency. And let RC be denoted as tau, the time constant. This is the frequency response. Now, let us find out, let us call it H omega. This is a function of capital omega and let us find out the magnitude of H omega. It is of course, the magnitude of 1 by 1 plus j tau omega, which is 1 by 1 plus tau omega squared. And then we take the positive square root and we can sketch this. So, this is the so called magnitude response. So, you know indeed, this magnitude response varies somewhat slowly around 0 frequency. And we can show that by looking at the derivative of the magnitude response. So, what we are saying essentially here is that there is a slower variation around 0. And the faster variation as omega increases. And then of course, asymptotically again it goes towards 0. So, let us see the situation. Let us find the derivative of the magnitude. So, let us write down the derivative explicitly. So, of course, we have the expression mod H omega can be written as 1 plus tau omega squared, the whole to the power minus half and you can take d d omega mod H omega, which can be simplified. So, now you can see what is happening. You have tau squared omega divided by 1 plus tau omega squared to the power 3 by 2 in the denominator. Now, let us look at the situation as omega goes towards 0. Because omega goes to 0, this derivative goes towards 0 as well. That is very easy to see. Because you know after all as omega goes to 0, this quantity is negligible. This essentially goes to 1 and this goes to 0. So, around 0 frequency, you observe that the derivative is very small. Of course, it is interesting. You have a numerator which is also increasing in omega and a denominator which is also increasing in omega. In fact, let us look at the way the numerator and the denominator increase in omega. So, let us go back to this and now let me show the same thing in green what is happening as omega tends to infinity. Now, as omega tends to infinity, this dominates. The tau omega squared term dominates in the denominator. So, therefore, we can even neglect the 1 in comparison to that and of course, the numerator remains as it is. So, asymptotically what happens? The derivative goes towards, it goes like this. So, essentially, we are talking about a variation as 1 by omega squared, you know, because you have an omega in the numerator and omega squared to the power 3 by 2 in the denominator. So, omega cubed in the denominator. So, which means as omega tends to infinity, the derivative falls off very fast too. So, it is interesting. As omega tends to 0, the derivative is very small. As omega tends to infinity, the derivative is very small and somewhere in between the derivative is large. That is what, of course, that is what you observe even in the plot. There is a flat region around 0, there is a flat region as you go towards infinity and somewhere in between, there is a steep region. In fact, for those of you who are interested, you can also establish the point where the derivative becomes a maximum and so on, it would be interesting to do. But the point I am trying to make is that there is a flat region around 0 and essentially this is a very simple reconstruct, the kind that we wanted, realizable because you can realize it with finite resources and it also has a steep fall off for higher frequencies and reasonably flat region around 0 frequency. Such a simple circuit allows you to reconstruct, of course, with some inaccuracy provided you are working within the region of flatness and of course, the reconstruction would have a bit of error because after all that region around the freak the magnitude of the response around 0 is not exactly flat. But to any degree that you define, you can define the extent of flatness that you want that you can say that well as long as my variation of the magnitude is within say 5 percent or within 2 percent, I think of it as flat and I put my signal in that zone. So, even if there is a bit of distortion, even if the signal original signal spectrum is slightly distorted on account of it is not being quite flat, you accept it. What I am trying to show you is that it is not too difficult to get a simple reconstruct, a simple rough and ready reconstruct right here. And now the next question is you see in the reconstructor you would have wanted it to have a strictly 0 response after some frequency definitely after fs by 2. Unfortunately, we do not have that in the RC circuit. In fact requiring that that is the next thing we are now going to come to. A realizable stable reconstructor must not have 3 things, we have seen 2 of them, a flat region and a brick wall. It must not have one more thing which I am now going to put before you. So, a stable realizable reconstructor must not have 3 things, let me draw them. It must not have this flat region, it must not have this brick wall and it also must not have this 0 completely 0 frequency response region over an infinite region of frequency. This is the third thing that is forbidden. In fact in a way that completely 0 response after certain frequency is also flat region that is why it is forbidden. The RC circuit of course you know you can see it does not have that completely 0 region. So what is going to happen if you do not have a completely 0 region after a certain frequency where is the trouble going to be? The trouble is going to be that all those carbon copies of the spectrum that you created are going to start and contribute something to the output. You no longer going to be left with only the original spectrum. In fact now you have the following trouble. The original spectrum itself is going to be slightly distorted because the region is not the frequency response the reconstructor is not quite flat and then those carbon copies are also not going to be subjected to an exactly 0 response. So they are also going to play some role in constructing the output. What do we do? What we will do is in fact to go for non-ideal sampling. So all this while we have talked about ideal sampling where you had these ideal impulses which created samples for you. In the next session we shall see what happens if the sampling is not ideal. That means you do not quite have these ideal impulses. You have realistic pulses, can we generalize the sampling theorem there and in fact will it in some sense help us in this whole process of dealing with that non-zero region after half the sampling. We will talk about it in the next session. Thank you.