 So, warm welcome to the second session of this course. In the first and introductory session, we had tried to put down the boundaries in which we would like to structure this course namely to understand signals and systems. We had given a number of examples from real life. What I want to do in this session is to talk about the whole concept of abstraction of extracting from several different situations at once, what is common and what is essential to a description of the situation. We had given an example in the previous discussion, the mass and the capacitor. So, if we use the mass as analogous to a capacitor, if we use the velocity of the mass as analogous to the voltage and the force on the mass is analogous to the current in the capacitor. Then we have here two systems, two situations in real life which can be captured by one description. What is that description? Let us write it down. So, in fact, here I bring in the notion of a system description. Let us denote the input as a function of time by x of t. Of course, here I am talking about time being continuous. For the moment, we will only concentrate on a continuous independent variable. This is an independent variable and the input signal is essentially a function of the independent variable, a dependent variable dependent on the independent variable. Of course, we attempted to ask what is the difference between a signal then in any function as we understand it in mathematics. Actually, there is very little to distinguish the two. Most functions that we understand in mathematics are also signals, valid signals. In fact, if we really wish to, we could even say that all functions of an independent variable are signals in that independent variable. However, we would like to exercise a little bit of caution here. We would prefer not to use certain quote unquote pathological functions as examples of signals. So, just for those of you who have a very mathematical mind, let me give you such an example of a pathological signal. You know, in principle, you could define a function of the independent variable time, which is really all the set of real numbers. Time could take any real value. And remember, when you say time can take any real value, you have assumed that you have fixed 0 somewhere in time. So, there is time before that and there is time after that. That is also something that we need to agree on. If time is the independent variable, we have to agree on a 0 in time. And after we have agreed on the 0, there is time after and there is time before. In space, there is no such problem. You could have a function of space, a signal in space. To take an example, suppose I were to look at a scenery in front of me, a three-dimensional object in front of me. And if I were to take a slice of the object along a plane, then I could construct a two-dimensional slice, two-dimensional function there. And you see, there again. Now, let us for example, take a mountainous scene in front of me. Suppose you were to take a slice of that mountainous scene and you were to plot the contours of the mountains in front of me. That would be a one-dimensional signal as a function of space. Sometimes you take photographs. When you take a photograph, again you have a function of two independent variables, the two independent variables in space. So, an independent variable need not be one-dimensional. It could be one-dimensional, it could be two-dimensional, it could be multi-dimensional, three, four in principle. Whatever it is, this is the first abstraction that we have made. We could have a very similar pattern being shown if we record an electrophysiological signal, for example, the electrocardiogram. A very similar looking signal might also come from a pressure wave. So, when we talk about a signal, we are just talking about an independent variable and the dependent variable, which is a function of that independent variable. We have abstracted at the first level. Similarly, we abstract the system. So, for example, for both the mass and the capacitance, let us now write down an abstraction of the system description. So, what is the system description? Let us come back to that now. So, we had an input signal. Similarly, we have an output signal y of t, explicit relation between y of t and x of t in what we call a system description. What is a system description for the mass and the capacitor together? The mass and the capacitor, in this case the input is the force and the output is the velocity. Force we will call x of t, velocity of y of t. Here the input is the current. Here again we will call the current x of t. We will call the mass m, we will call the capacitor c, the value of time n. And the output is the voltage, here again we will denote the voltage at that context by y of t. And in both cases, let us call the constant of proportionality gamma in both cases. So, here gamma becomes equal to the value of m and gamma becomes equal to the capacitance c. In both cases, the relationship that governs them is x of t is gamma times dy of t dt. This is a common system description for both of. So, it is an explicit relation between the input and the output here. So, let me spend a couple of minutes in explaining to you why we need to abstract and have we abstracted before. So, now we are going to talk, you know, after this, once we have agreed that we are going to use this notion of abstraction, then it is all x t and y t for us. We have forgotten what x t was and what y t was in the actual real life context. Why do we do it? Have we done it before? Actually, we have done it all our lives in our studies in mathematics. When we were very young, we had to distinguish between putting two apples and three apples together to get five apples and two oranges and three oranges together to get five oranges. It seemed like there were two different situations, very, very, very long ago. We do not even remember that time now, but there must have been such a time. It was different to add two apples and three apples and two oranges and three oranges, but then we abstract from that. We just say, in both cases, there is the notion of a number of each of those objects. And if you have the same kind of object, then when you add this number to that number, you get a third number. So, the notion of an integer, the notion of a positive integer or a natural number is an abstraction. You do not really have natural numbers in real life, but you abstract situations in real life by associating natural numbers with them. So, from a very basic point in our education and in our understanding of the world around us, we abstract. We have done exactly that in signals and systems. We want to look at different situations, bring out what is common and create an abstraction. And in creating that abstraction, we make our life easier because now we can deal directly with the abstraction and come to conclusions about all the systems that are described by that abstraction. For example, let us take these two systems. Let us draw an input-output relationship for a particular kind of input signal here. Suppose xt is a constant in both cases. So, it xt is equal to 1 for t equal to 0 onwards, t greater than equal to 0 and 0 before. What would yt exhibit in both cases? In both cases, yt would start growing. You see, you have 1 is equal to gamma times dy t dt. So, of course, dy t dt is 1 by gamma and therefore, yt is essentially 1 by gamma times t for t greater than equal to 0. And we will assume that before t equal to 0, there was no xt at all. Therefore, yt need to be 0 too. What it meant is that in the case of the mass, no force acted on the mass at any point in time before r reference point of 0 and therefore, the mass lay at rest in terms of its in inertial conditions. And the capacitor again, the voltage was 0 until we started giving it a current or supplying charge to it. So, let us sketch this input-output relationship. The input looks like this. In fact, this input is a very famous input in signal and systems and is often called a unit step. It looks like that. A step with unit height at time t equal to 0. And the output looks like this. So, it is 0 all the way up to t equal to 0 and then grows linearly. So, 1 by gamma times t is also called a ramp, not a unit ramp of course, but a ramp. So, very simple example of an input-output relationship or what is called a unit step response of a system, which is now common to both situations. Analyze the abstraction and you automatically analyze both of these systems for their input-out behavior. We look at slightly more complex examples of this in the next discussion. Thank you.