 So, welcome to this 19th session of discussion. And here we shall now take further that R C circuit go the other way. So, we found the unit impulse response of that R C circuit by using its unit step response. This I wish now to put down as a general principle convenient in many situations. You can clearly appreciate that very often it is very difficult to understand what it means to apply a unit impulse to a system. Of course, one can give it interpretations. Let us give it that interpretation. Now, I had hinted at the interpretation in some qualitative discussion towards the end of the previous session that we had, but now let us complete it by giving it a quantitative interpretation. Quantitative interpretation of unit impulse response of the R C circuit. Now, you know what is the unit impulse there? So, you have this R C circuit back again. You applied a very narrow pulse going to an impulse here and it is a voltage pulse remember. Now, the voltage pulse cannot appear across the capacity cannot appear here. Why a capacitor refuses sudden voltage jumps? You know one way to understand this unit impulse is to think of it as a jerk. A sudden jerk that pumps in a certain area that is whatever quantity when you say there is an impulse of some quantity you are pumping in that quantity lasting for a certain amount of time. So, when you are pumping in a voltage pulse it is as if that voltage lasted for a very short time on some elements of the circuit and created whatever consequences it could. And of course, in the resistive capacitive circuit that sudden jump of voltage could not appear across the capacitor. So, it has to go to the resistance to meet Kirchhoff's voltage law. So, you see let me draw the situation that will make it more clear. So, the unit impulse started from a very narrow pulse as usual. There we go again a delta with a 1 by delta height and this was applied. Now, this sudden change these jumps cannot go to the capacitor. So, they go to the resistance. That means what appears across the resistance is that same pulse going towards an impulse. So, resistor voltage, resistor sustains this whole thing remember there is a 1 sitting there 1 by delta sitting here and with a width of delta this is the voltage sustained across the resistance. And therefore, we need now to find the current in the resistance. Resistor current is the same voltage but divide by r. So, 1 by delta multiplied by 1 by r sustained over a width of delta. And the resistor current is equal to the capacitor current. This is also equal to the capacitor current. Therefore, and now we can integrate the capacitor current. Integrating it gives you the capacitor charge. So, the conclusion is that the capacitor charge is essentially the area captured here 1 by delta into delta divided by r. And therefore, the capacitor voltage jumps to 1 by delta times delta into 1 by r divided by the capacitance at the point of the impulse. And as you can see this expression is essentially 1 by tau. So, now we know what happened. You see there was a charge suddenly pumped into the capacitance. So, it caused the voltage to jump. That charge was equal to 1 by r go back here that you can see that the charge is 1 by r irrespective of delta even as delta tends to 0. The voltage is therefore, 1 by r c that is 1 by tau. And that voltage you know now when you have a voltage put across the capacitance. And when the input has now gone to 0 voltage, obviously that capacitor has to discharge through the resistance. And how will it discharge the resistance exponentially? That is why you get the e raised to the power minus p by tau. So, what we are saying essentially is, essentially capacitor discharges now after charge pump. There was a charge pump at the point where the impulse occurred and the charge pump caused the voltage of 1 by tau and then there is a discharge, an exponential discharge with a time constant of 1 by time constant of tau. So, essentially 1 by tau e raised to the power minus t by tau ut that gives us a physical interpretation of this unit impulse response that we saw. So, what I am trying to illustrate here is that this is the practical way in which we can often calculate unit impulse responses. This also helps us interpret unit impulse responses physically by taking an example of this R C circuit. Now, I am going to leave one exercise for all of you to do. That is remember we had taken two analogous systems, a mechanical system and an electrical system which essentially had the same abstraction. In the mechanical system, you had a frictional element which was like a resistance and a mass element which was like a capacitance and it had the same describing equation. Now, the exercise for you is to put down instead of the R C circuit the abstraction. That means, use the abstract system description relating y 3 and x t and then put down this impulse response in the light of that abstraction and transfer it to the mechanical system and make the same kind of interpretation that I have done for the electrical system. In other words, like I have analyzed the electrical system given a full interpretation to its unit impulse response, unit step response, do the same thing for the mechanical system and then build an abstraction for an impulse response of a system of this kind. If you do this exercise faithfully, it will give you a lot of insights into the whole idea of a step response and impulse response, the relation between a step response and impulse response, the interpretation of an impulse response and the meaning of an impulse response in specific contexts. I encourage you to do this. I would like to spend a few seconds in emphasizing this because it is very often a stumbling block for students on this subject to interpret impulse response. What does the impulse response really mean? What physical interpretation does it have? It is all very well to write things down algebraically and formally. It is a different thing to interpret them in a manner that we can relate to the real world and that is something that we want to do all the time as much as we can in this course. We should not forget that signals and systems is all about abstraction for the purpose of understanding reality and the impulse response is for the linear shift invariant system is a very central idea in signals and systems and I strongly encourage you to understand this idea by taking other such examples of systems interpreting the impulse response and coming to conclusions. Now, what is also true is that the converse would be applicable. Namely, if I took a running integral of the impulse response, I should get the step response. That is the second exercise that I would leave for you to work out here. Calculate the running integral of this unit impulse response and verify that it is indeed the step response that we had earlier. We stop this discussion here and in the next discussion that we are going to have, we are going to use the simple response in many other creative ways to come to some conclusions about other qualities of a linear shift invariant system. But before that, we are going to take a little detail. Namely, we are going to build also another class of systems namely discrete systems and more in the next session. Thank you.